Optical Constants and Inelastic Electron-Scattering Data for 17 Elemental
Metals
Wolfgang S. M. Wernera…
Institut für Allgemeine Physik, Vienna University of Technology, Wiedner Hauptstraße 8–10, A 1040 Vienna, Austria
Kathrin Glantschnig
Chair of Atomistic Modelling and Design of Materials, University of Leoben, Franz-Josefstraße 18, A 8700 Austria and Institut für
Physik, Fachbereich Theoretische Physik, University of Graz, Universitätsplatz 5, A 8010 Graz, Austria
Claudia Ambrosch-Draxl
Chair of Atomistic Modelling and Design of Materials, University of Leoben, Franz-Josefstraße 18, A 8700 Austria
共Received 27 January 2009; accepted 16 September 2009; published online 10 December 2009兲
Two new sets of optical data, i.e., values for the real 共␧1兲 and imaginary 共␧2兲 parts of
the complex dielectric constant as well as the energy loss function 共ELF兲 共Im兵−1 / ␧其兲, are
presented for 16 elemental metals 共Ti, V, Fe, Co, Ni, Cu, Zn, Mo, Pd, Ag, Ta, W, Pt, Au,
Pb, and Bi兲 and 1 semimetal 共Te兲 and are compared to available data in the literature. One
data set is obtained from density functional theory 共DFT兲 calculations and gives ␧ from
the infrared to the soft x-ray range of wavelengths. The other set of optical constants,
derived from experimental reflection electron energy-loss spectroscopy 共REELS兲 spectra,
provides reliable optical data from the near-ultraviolet to the soft x-ray regime. The two
data sets exhibit very good mutual consistency and also, overall, compare well with
optical data found in the literature, most of which were determined several decades ago.
However, exceptions to this rule are also found in some instances, some of them systematic, where the DFT and REELS mutually agree significantly better than with literature
data. The accuracy of the experimental data is estimated to be better than 10% for the
ELF and ␧2 as well as for ␧1 for energies above 10 eV. For energies below 10 eV, the
uncertainty in ␧1 in the experimental data may exceed 100%, which is a consequence of
the fact that energy-loss measurements mainly sample the absorptive part of the dielectric
constant. Electron inelastic-scattering data, i.e., the differential inverse inelastic mean free
path 共IMFP兲 as well the differential and total surface excitation probabilities are derived
from the experimental data. Furthermore, the total electron IMFP is calculated from the
determined optical constants by employing linear response theory for energies between
200 and 3000 eV. In the latter case, the consistency between the DFT and the REELS
data is excellent 共better than 5% for all considered elements over the entire energy range
considered兲 and a very good agreement with earlier results is also obtained, except for a
few cases for which the earlier optical data deviate significantly from those obtained
here. © 2009 American Institute of Physics. 关doi:10.1063/1.3243762兴
Key words: dielectric function; density functional theory; electron scattering.
CONTENTS
1.
2.
3.
4.
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density Functional Theory Calculations. . . . . . .
Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Extracting Optical Constants From REELS. . . . .
4.1. Electron-solid interaction parameters. . . . . .
4.2. Multiple scattering. . . . . . . . . . . . . . . . . . . .
1016
1017
1020
1021
1021
1022
a兲
Electronic mail: [email protected].
© 2009 American Institute of Physics.
0047-2689/2009/38„4…/1013/80/$45.00
1013
5.
6.
7.
8.
4.3. Deconvolution of multiple scattering from
loss spectra. . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Retrieval of optical data from the
single-scattering distributions. . . . . . . . . . . .
4.5. Consistency test and error analysis. . . . . . .
Results and Discussion. . . . . . . . . . . . . . . . . . . . .
5.1. Optical constants. . . . . . . . . . . . . . . . . . . . . .
5.2. Inelastic electron-scattering data. . . . . . . . .
Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Deconvolution Algorithm For
1023
1024
1024
1029
1029
1035
1038
1041
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1014
9.
10.
11.
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
REELS Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix B: Values of the Drude-Lorentz
Parameters for the Dielectric Function. . . . . . . .
Appendix C: Values of Optical Constants
Derived from DFT and REELS. . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1041
1042
13.
1092
1092
List of Tables
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Structural and convergence parameters: For
each element, crystal structure and lattice
parameters are given. The angles between the
basis vectors different from 90° are ␥ = 120°
for the trigonal Te lattice and ␥ = 110.33° for
the monoclinic Bi case. In addition, the k
meshes used in the calculation of the
electronic structure and the optical spectra, the
radius of the muffin-tin spheres RMT, and the
RKmax value, i.e., the product of the smallest
muffin-tin radius and the plane wave cutoff,
are listed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results of sum-rule checks on the three sets of
optical data discussed in the present work. . . . .
Fit parameters in Eq. 共33兲 describing the IMFP
values derived from the REELS measurements..
Values for the surface excitation parameter as
in units of aNFE = 2.896 eV−1/2 derived from the
REELS measurements, compared with the
theoretical values of Chen 共Ref. 79兲. . . . . . . . . .
Drude-Lorentz parameters for Ti
共E3p1/2 = 32.6 eV兲, V 共E3p3/2 = 37.3 eV兲, and Fe
共E3p1/2 = 52.7 eV兲. . . . . . . . . . . . . . . . . . . . . . . . . .
Drude-Lorentz parameters for Co
共E3p1/2 = 58.9 eV兲, Ni 共E3p3/2 = 66.2 eV兲, and
Cu 共E3p3/2 = 75.1 eV兲. . . . . . . . . . . . . . . . . . . . . . .
Drude-Lorentz parameters for Zn
共E3d5/2 = 10.1 eV兲, Mo 共E4p3/2 = 35.5 eV兲, and
Pd 共E4p3/2 = 50.9 eV兲. . . . . . . . . . . . . . . . . . . . . . .
Drude-Lorentz parameters for Ag
共E4p3/2 = 58.3 eV兲, Te 共E4d5/2 = 40.4 eV兲, and Ta
共E4f7/2 = 21.6 eV兲. . . . . . . . . . . . . . . . . . . . . . . . . .
Drude-Lorentz parameters for W
共E4f7/2 = 31.4 eV兲, Pt 共E5p3/2 = 51.7 eV兲, and Au
共E5p3/2 = 57.2 eV兲. . . . . . . . . . . . . . . . . . . . . . . . . .
Drude-Lorentz parameters for Pb
共E5d5/2 = 18.1 eV兲 and Bi 共E5d5/2 = 23.8 eV兲. . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Ti calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of V calculated with
DFT and derived from REELS data. The last
14.
15.
1019
1039
1040
16.
1041
1043
17.
1043
1043
18.
1043
1043
1043
19.
1044
20.
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Fe calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Co calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Ni calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Cu calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Zn calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Mo calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Pd calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Ag calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
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1047
1050
1053
1056
1059
1062
1065
1068
OPTICAL CONSTANTS FOR 17 METALS
21.
22.
23.
24.
25.
26.
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Ta calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of W calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Pt calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Au calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Pb calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
Values of the real 共␧1兲 and imaginary 共␧2兲
parts of the complex dielectric constant and
the ELF 共Im兵−1 / ␧共␻兲其兲 of Bi calculated with
DFT and derived from REELS data. The last
two columns contain values for the volume
and surface single-scattering loss distributions
the DIIMFP 共wb兲 and the DSEP 共ws兲. . . . . . . . . .
1071
3.
1074
1077
4.
1080
5.
1083
6.
1086
1089
List of Figures
1.
2.
Imaginary part of the complex dielectric
function and reflectivity of silver in the energy
range of 0 – 25 eV for two different
broadenings ⌫dr = 0.10 eV 共black line兲 and ⌫dr
= 0.02 eV 共red line兲 of the intraband
contributions. For the interband part of the
spectrum a Lorentzian broadening of 0.10 eV
was adopted.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019
Oscillator strengths for optical transitions of
silver visualized for three planes in the BZ.
7.
From top to bottom: Position of the plane in
the BZ; oscillator strength for transitions in the
energy ranges of 3.5–5 and 19.5– 21.5 eV,
respectively. From left to right:
z = 共1 / 34兲共␲ / a兲, z = 0.5共␲ / a兲, and z = 共33/ 34兲
⫻共␲ / a兲 planes. Dark areas correspond to
regions with high oscillator strengths.. . . . . . . . .
共a兲 Comparison of raw REELS data for
1500 eV electrons backscattered from a Au
sample 共red curve兲 with the corresponding data
after a Richardson-Lucy deconvolution 共blue
curve兲. 共b兲 Illustration of the elastic peak
removal procedure for the data shown in 共a兲.
Blue curves: spectrum before elimination of
the elastic peak; red curves: elastic peak; black
curves: loss spectrum after removal of the
elastic peak.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DIIMFP and DSEP for medium energy
electrons in Al and Pt. The DIIMFP is
presented for 1000 eV 共solid curves兲 and
3000 eV 共dashed curves兲 while the DSEP is
shown for 1000 eV for two different angles of
surface crossing 共0°, solid curves; 70°, dashed
curves兲. 共a兲 Al; 共b兲 Pt.. . . . . . . . . . . . . . . . . . . . . .
共a兲 Elastic scattering cross section of Au for
several energies in the range of scattering
angles between 100° and 140°. 共b兲 PLD Q共s兲
calculated by means of a MC simulation for
Au for 700 and 3400 eV for the geometrical
configuration used in the present work and for
5 and 40 keV for the experimental setup of
Ref. 58. 共c兲 Reduced partial intensities for
volume scattering calculated by means of Eq.
共17兲 for the PLDs shown in 共b兲.. . . . . . . . . . . . . .
共a兲 Results of the deconvolution procedure
equation 共23兲 applied to all possible energy
combinations between 300 and 3400 eV. Note
that the scattering geometry employed in the
present work corresponds to a deep minimum
in the elastic-scattering cross section at 300 eV
共see Fig. 5兲. 共b兲 Same as 共a兲 for energies larger
than 300 eV, thus avoiding the deep minimum
in the scattering cross section at 300 eV. 共c兲
Same as 共b兲 but using Oswald’s formula for
the energy and angular dependence of the
surface excitation parameter equation 共30兲
instead of Eq. 共20兲.. . . . . . . . . . . . . . . . . . . . . . . .
Comparison of deconvolution of REELS
spectra for Au measured in significantly
different energy ranges, corresponding to the
PLDs and partial intensities shown in Figs.
5共b兲 and 5共c兲. 共a兲 Loss spectra for 700 and
3400 eV electrons, measured in the present
work. 共b兲 Measurements of Ref. 58 for 5 and
40 keV. 共c兲 Comparison of the
single-scattering loss distributions obtained
with Eq. 共23兲 using the partial intensities
1015
1019
1020
1022
1025
1026
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1016
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
shown in Fig. 5共c兲.. . . . . . . . . . . . . . . . . . . . . . . .
Influence of the chosen dispersion constant ␣
in the retrieval of the dielectric function from
the single-scattering loss distributions for the
Co 3p semicore transition. Dashed red curve
with open circles: ELF values of ␧ retrieved
from REELS using ␣ = 0.0 in Eq. 共26兲; solid
red curve with filled circles: ELF values of ␧
retrieved from REELS using ␣ = 0.5; black
curve: Palik’s data 共Ref. 1兲 for the ELF 关that
were taken from Henke’s work 共Ref. 2兲兴. Blue
curves: results of DFT calculations. Note that
the DFT calculations accurately reproduce the
position of the semicore edge binding energy
E3p3/2 = 58.9 eV derived from photoemission
data 共Ref. 59兲, as do the REELS data when a
value of ␣ = 0.5 is used for the dispersion
constant.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the dispersive 共␧1兲 and
absorptive 共␧2兲 parts of ␧ for Au derived from
the present REELS measurements 共red curves
with error bars兲, DFT calculations 共blue
curves兲, and Palik’s data 共black curves兲.. . . . . . .
REELS data and optical constants derived
from them for Ti. Upper panel: deconvoluted
REELS data at the employed energies, as
indicated in the legend; second panel from
above: DIIMFP 关wb共T兲兴 and DSEP 关ws共T兲兴
共data points兲 as well as the best fit of these
data to the Drude-Lorentz model of Eq. 共25兲
共solid curves兲; lower three panels: comparison
of the loss function Im兵−1 / ␧其 as well as the
dispersive 共␧1兲 and absorptive 共␧2兲 parts of the
dielectric function extracted from REELS 共red
curves兲 with results from DFT calculations
共blue curves兲 and Palik’s data set 共black
curves兲.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for V.. . . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Fe.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Co.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Ni.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Cu.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Zn.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Mo.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Pd.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Ag.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Te.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Ta.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for W.. . . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Pt.. . . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Au.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Pb.. . . . . . . . . . . . . . . . . . . .
Same as Fig. 10 for Bi.. . . . . . . . . . . . . . . . . . . .
Comparison of values for the refractive index
n and extinction coefficient k derived from
REELS 共black full curves兲 and DFT 共red
dashed curves兲 with the values given in Palik’s
1027
28.
29.
30.
31.
32.
33.
34.
35.
1028
36.
1028
book 共blue dotted curves兲 as a function of the
photon wavelength for Cu, Ag, and Au.. . . . . . .
Same as Fig. 27 for Fe, Co, and Ni.. . . . . . . . . .
Same as Fig. 27 for Ti, V, and Zn.. . . . . . . . . . .
Same as Fig. 27 for Mo, Ta, and W.. . . . . . . . . .
Same as Fig. 27 for Pd, Pt, and Pb.. . . . . . . . . . .
Same as Fig. 27 for Te and Bi.. . . . . . . . . . . . . .
Comparison of IMFP values for Ti, Fe, Co,
and Pd taken from different sources.. . . . . . . . . .
Comparison of REELS spectra with fitted
spectra for determination of the surface
excitation parameter.. . . . . . . . . . . . . . . . . . . . . . .
First-order reduced surface partial intensities
共data points兲 derived from the REELS spectra.
Solid line: fit of these data to Eq. 共20兲; dashed
line: results of Ref. 43 based on Oswald’s
energy and angular dependence of the average
number of surface excitations, Eq. 共30兲;
dashed-dotted line: theoretical results by Chen
共Ref. 79兲 based on the optical constants of
Palik 共Ref. 73兲.. . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the Padé approximation 关Eq.
共23兲, dots兴 for the DIIMFP and DSEP for Pt
with the second-order truncated 关Eq. 共A6兲, red
curves兴 and the full second-order 关Eq. 共A2兲,
blue curves兴 expansion.. . . . . . . . . . . . . . . . . . . . .
1037
1037
1038
1038
1039
1039
1040
1040
1041
1042
1. Introduction
1028
1029
1029
1030
1030
1031
1031
1032
1032
1033
1033
1034
1034
1035
1035
1036
1036
The response of a solid to an external electromagnetic perturbation determines many important fundamental and technological properties of a material and its surface. The susceptibility of the solid to become polarized by incoming light
or charged particles is described by the frequency- 共␻兲 and
momentum- 共q兲 dependent dielectric function ␧共␻ , q兲 which
can be measured by probing a surface with elementary particles such as photons1–4 or electrons.5,6 Energy-loss measurements with electrons mainly sample the energy loss
function 共ELF兲,
再 冎
Im −
1
␧2
= 2
,
␧
␧1 + ␧22
共1兲
where ␧1 and ␧2 are the real 共dispersive兲 and imaginary 共absorptive兲 parts of the dielectric function, respectively. These
quantities are related to the index of refraction n and extinction coefficient k via
␧1 = n2 − k2 ,
␧2 = 2nk.
共2兲
Since the rest mass of the photon is zero, conservation of
energy and momentum requires that the particle be absorbed
during an interaction with the solid-state electrons in the
overwhelming majority of cases. This implies that optical
spectra can be divided into regions of high opacity and transparency for which accurate simultaneous determination of
the absorptive 共imaginary兲 and dispersive 共real兲 parts of the
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OPTICAL CONSTANTS FOR 17 METALS
dielectric function is not straightforward. Furthermore, to experimentally cover a wide wavelength range, different
sources and monochromators as well as other optical components are needed. In the case of transmission experiments,
accessing a wide wavelength range over which ␧ varies significantly generally also requires measurements on a series of
samples with different thicknesses. Therefore, optical data
found in the literature usually consist of a synthesis of various data sets employing a Kramers-Kronig analysis.7 Investigating the dielectric response of a nanostructure down to
small dimensions is also problematic since photons are rather
difficult to focus. Measurements of the optical constants by
means of light scattering have in the past not always been
carried out under ultrahigh vacuum 共UHV兲 conditions which
implies that the investigated surfaces were not always well
defined.
The drawbacks mentioned above for measurements of the
optical response using photons can be avoided by using electrons as probing particles instead: First of all, they are easy
to focus in the sub-nanometer range. Furthermore, electron
scattering experiments are routinely carried out in UHV
equipment where sample cleaning and monitoring facilities
are available and it is easy to access the spectral range between the 共near兲 infrared and soft x-ray regions in a single
experiment. Both electron energy loss spectroscopy in
transmission5 and reflection8,9 can be used for this purpose,
the latter technique being the simplest from the experimental
point of view, in particular, when applied to nanostructured
surfaces.
An inherent drawback of using probing particles with a
finite rest mass is that they experience multiple scattering
since in this case energy and momentum conservation makes
it highly unlikely for them to be absorbed during an interaction. Therefore, features due to multiple scattering need to be
eliminated from the raw experimental data by means of an
appropriate deconvolution procedure properly accounting for
the physics governing the transport of the electrons from the
source to the detector. The main processes that need to be
considered in this connection are deflections arising in the
course of an elastic interaction10 as well as energy losses
deep inside the solid 共“volume” or “bulk” excitations11兲 and
additional losses that may occur during the crossing of the
vacuum-solid boundary 共“surface” excitations12兲.
For electron energy-loss spectra measured in the transmission electron microscope 共TEM兲, procedures for spectrum
deconvolution have been developed several decades ago1,5,6
and the measurement of optical data in the TEM is nowadays
a matter of routine. Such measurements need to be carried
out at high enough energies of several hundreds of a keV
where the mean free path for inelastic scattering is of the
order of the thickness of films that can be prepared in practice. In this energy regime the loss spectrum is dominated by
volume energy losses, since the time it takes for an electron
to cross the solid-vacuum boundary is too small to give rise
to surface excitations to a significant extent.
Reflection electron energy-loss spectroscopy 共REELS兲
represents an attractive alternative to energy-loss measure-
1017
ments in transmission since the experiment is extremely
simple,13,14 In many an UHV apparatus the required components are available, and REELS can therefore be conveniently combined with other surface characterization techniques. Owing to the energy dependence of the electron
reflection coefficient, which decreases rapidly above several
keV, energy-loss measurements in reflection are most easily
carried out in the medium energy range 共100 eV– 10 keV兲.
Then, the low-loss part of the spectrum is dominated by surface excitations, a fact which severely complicates data
analysis, in particular, the elimination of plural scattering
effects from the raw data.
Existing sources of optical constants1–4 are often a synthesis of experimental and theoretical data and the reliability of
these data sets below the soft x-ray regime is not well
known. Self-consistent field calculations have been used for
some decades to determine ␧ in the hard x-ray range for free
atoms.2–4 On the other hand, solid-state calculations based on
density functional theory 共DFT兲 beyond the ground state that
allow one to access the soft x-ray range and below have only
become available recently.15–18 While DFT has become a
standard tool for describing material properties in a
parameter-free manner, as far as ground state properties are
concerned, the treatment of excited states with the same reliability and hence predictive power as the ground state is
still a challenge; nevertheless significant progress has been
achieved during the past years such that theoretical spectroscopy has become an emerging field in solid-state theory.
In the present paper, two new sets of optical data for 17
elemental materials are presented. One is derived from DFT
calculations made for energies between the infrared and soft
x-ray regions. The other set is extracted from experimental
energy-loss measurements in reflection between the visible
and soft x-ray ranges. The method to extract the dielectric
function from REELS is outlined and its consistency and
accuracy are analyzed in detail. A comparison with earlier
results compiled in Palik’s books1 is performed. Generally,
very good consistency is found between the DFT calculations and the present measurements; while the overall agreement with Palik’s data is satisfactory, significant differences
are observed in several instances, some of these deviations
being systematic. This comparison proves the reliability of
DFT results which therefore represent a novel way to theoretically obtain information of the response of a solid to an
electromagnetic perturbation. Furthermore the results demonstrate that REELS, owing to its experimental simplicity, is
a very attractive alternative to the other experimental techniques to obtain information on optical constants of solids
addressed above.
2. Density Functional Theory Calculations
DFT can be regarded as the standard model in computational solid-state science. Especially ground state properties
such as equilibrium volumes, atomic positions, phonon frequencies, and elastic constants are obtained with high accuracy. Since the Hohenberg-Kohn theorem19 is valid for the
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1018
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
ground state only, the interpretation of the Kohn-Sham 共KS兲
eigenstates in terms of quasiparticle energies and wave functions should be handled with care. This is particularly true
for semiconductors, which suffer from the band-gap problem. In contrast, for metals the KS band structure can be
taken as a good approximation to the quasiparticle energies,
where this approximation has proven successful for a variety
of materials.
The electronic structure is obtained by solving the KS
equations with the linearized augmented planewave method
using the WIEN2K code.20 It is based on a decomposition of
the unit cell 共with volume ⍀兲 into two different volume
types, the nonoverlapping muffin-tin spheres centered at the
atomic positions and the interstitial region. In these areas
appropriate basis sets are used.21–23 In the interstitial region a
plane wave basis is adopted to account for the moderate
change of the wave functions in this part of the unit cell:
␾kជ +Gជ 共rជ兲 =
1
冑⍀ e
ជ 兲·rជ
i共kជ +G
,
rជ 苸 I.
共3兲
The rapid oscillations of the wave functions close to the
atom cores are described by atomiclike basis functions inside
the muffin-tin spheres,
␣ ជ
ជ 兲u␣l 共r,El兲
␾kជ +Gជ 共Sជ ␣ + rជ兲 = 兺 关Alm
共k + G
lm
␣ ជ
ជ 兲u̇l␣共r,El兲兴Y lm共r̂兲,
+ Blm
共k + G
兩rជ兩 艋 R␣ ,
共4兲
where Sជ ␣ represents the position of the atomic nucleus ␣ and
R␣ = RMT is the radius of the muffin-tin sphere. ul␣共r , El兲 are
the solutions of the radial Schrödinger equation for given
linearization energies El, u̇␣l 共r , El兲 denote the corresponding
energy derivatives, and Y lm共r̂兲 represent spherical harmonics.
␣ ជ ជ
␣ ជ ជ
The expansion coefficients Alm
共k + G兲 and Blm
共k + G兲 are obtained by the condition that the two basis sets match in value
and slope at the sphere boundaries.
To improve the description of semicore states, local orbitals 共LOs兲 can be introduced as additional basis functions.
Their form is similar to that of the basis function inside the
muffin-tin sphere,
␣ ␣
␣ ␣
␾LO共Sជ ␣ + rជ兲 = 兺 关Ãlm
ul 共r,El兲 + B̃lm
u̇l 共r,El兲
lm
␣ ␣
+ C̃lm
ul 共r,Elo兲兴Y lm共r̂兲,
共5兲
with the additional linearization energy Elo corresponding to
␣
␣
the semicore states. The variational parameters Ãlm
, B̃lm
, and
␣
C̃lm are determined by the conditions that the basis set must
be normalized and that the LO should be zero in value and
slope at the sphere boundary. Thus, LOs are totally confined
inside the atomic spheres.
For all elemental solids presented in this work, exchange
and correlation effects were treated by the generalized gradient approximation 共GGA兲 parametrized by Perdew et al.24
Spin-polarized calculations were performed for the ferromagnetic elements Fe, Ni, and Co. For Au, Pb, Pd, Pt, and
W, spin-orbit 共SO兲 interaction was taken into account via a
second variational scheme on top of the GGA calculation. In
this procedure, the KS equations are solved neglecting SO
coupling at first; the wave functions obtained in this way
form the basis set for the eigenvalue problem including the
SO part of the Hamiltonian.25 Structural and convergence
parameters used in the electronic structure and optical calculations are listed in Table 1.
We calculate the dielectric response within the random
phase approximation, where the excited electron and the created hole are regarded to be independent 共independent particle approximation兲. Due to the effective screening of the
Coulomb interaction this approach is well suited for metals.
The imaginary part of the interband dielectric function is
given by
ប 2e 2
兺
␲m2␻2 n,n
⬘
冕
k
pn⬘,n,k pn⬘,n,k
⫻关f共␧n,k兲 − f共␧n⬘,k兲兴␦共␧n⬘,k − ␧n,k − ␻兲dk,
共6兲
where pn⬘,n,k is the momentum matrix element between
bands n and n⬘ for a k point in the irreducible wedge of the
Brillouin zone 共BZ兲 and f共␧n,k兲 is the occupation number of
the corresponding single particle state with energy ␧n,k. The
intraband contributions 共n = n⬘兲 are described by a Drude-like
lineshape with a lifetime broadening of 0.1 eV. The same
value is also used for the interband spectra. To obtain the real
part of the dielectric function a Kramers-Kronig transformation is performed. The details regarding this framework are
described in Ref. 17. The BZ integration is carried out by the
improved tetrahedron method.26
In the following, a procedure for analyzing features in the
optical spectra is presented, with silver serving as an example. Figure 1 shows the complex dielectric function for
two different values of the intraband broadening ⌫dr. ⌫dr
= 0.02 eV corresponds to the experimentally determined lifetime ␶ of 31⫻ 10−15 s 共Ref. 27兲 and ⌫dr = 0.1 eV is a universal value chosen for all elemental metals in this work. The
interband broadening ⌫inter was set to 0.1 eV. As can be seen
from Fig. 1, the intraband broadening affects the low-energy
region up to 2.6 eV. Because of a higher overlap between the
interband and intraband contributions for ⌫dr = 0.1 eV, the
first minimum is narrowed and decreases in depth. The spectrum shows two interesting features in the energy range up to
25 eV, ie., a strong peak at 4 eV and a broad structure
around 20 eV. To investigate the origin of these two structures, the oscillator strength of transitions contributing to a
given energy range is calculated for each k point in the BZ
by
OE共k兲 =
兺冕
n,n⬘
Emax
␻=Emin
dEpn⬘,n,k pn⬘,n,k␦共␧n⬘,k − ␧n,k − ␻兲,
共7兲
where the sum runs over all band combinations n , n⬘ for
which the related transition energies lie within the energy
range of interest, i.e., band energies fulfill
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OPTICAL CONSTANTS FOR 17 METALS
1019
TABLE 1. Structural and convergence parameters: For each element, crystal structure and lattice parameters are
given. The angles between the basis vectors different from 90° are ␥ = 120° for the trigonal Te lattice and ␥
= 110.33° for the monoclinic Bi case. In addition, the k meshes used in the calculation of the electronic structure
and the optical spectra, the radius of the muffin-tin spheres RMT, and the RKmax value, i.e., the product of the
smallest muffin-tin radius and the plane wave cutoff, are listed
Lattice parameters
k mesh
Material
Crystal
structure
a 共a.u.兲
b 共a.u.兲
c 共a.u.兲
scf
Optic
RMT
共a.u.兲
RKmax
Ag
Au
Cu
Ni
Pb
Pt
Pd
Fe
Mo
Ta
V
W
Co
Ti
Zn
Bi
Te
fcc
fcc
fcc
fcc
fcc
fcc
fcc
bcc
bcc
bcc
bcc
bcc
hcp
hcp
hcp
Monoclinic
Trigonal
7.7290
7.7101
6.8219
6.6518
9.3541
7.4077
7.3510
5.4235
5.9526
6.2361
5.7259
5.9715
4.7432
5.5747
5.0359
6.2437
8.4229
7.7290
7.7101
6.8219
6.6518
9.3541
7.4077
7.3510
5.4235
5.9526
6.2361
5.7259
5.9715
4.7432
5.5747
5.0359
12.6120
8.4229
7.7290
7.7101
6.8219
6.6518
9.3541
7.4077
7.3510
5.4235
5.9526
6.2361
5.7259
5.9715
7.6912
8.8439
9.3481
11.5595
11.2042
34⫻ 34⫻ 34
31⫻ 31⫻ 31
31⫻ 31⫻ 31
34⫻ 34⫻ 34
31⫻ 31⫻ 31
31⫻ 31⫻ 31
31⫻ 31⫻ 31
34⫻ 34⫻ 34
34⫻ 34⫻ 34
31⫻ 31⫻ 31
34⫻ 34⫻ 34
34⫻ 34⫻ 34
29⫻ 29⫻ 15
23⫻ 23⫻ 11
34⫻ 34⫻ 16
20⫻ 9 ⫻ 20
19⫻ 19⫻ 12
58⫻ 58⫻ 58
58⫻ 58⫻ 58
58⫻ 58⫻ 58
58⫻ 58⫻ 58
58⫻ 58⫻ 58
58⫻ 58⫻ 58
58⫻ 58⫻ 58
58⫻ 58⫻ 58
58⫻ 58⫻ 58
53⫻ 53⫻ 53
58⫻ 58⫻ 58
58⫻ 58⫻ 58
62⫻ 62⫻ 33
59⫻ 59⫻ 32
70⫻ 70⫻ 32
42⫻ 18⫻ 42
40⫻ 40⫻ 26
2.5
2.5
2.1
2.25
2.5
2.5
2.5
2.33
2.5
2.5
2.4
2.5
2.3
2.6
2.4
2.5
2.5
11
11
10
11
11
11
11
10
10
11
10.5
11
11
10
10.5
11
11
Emin 艋 ⑀n⬘,k − ⑀n,k 艋 Emax .
共8兲
The resulting oscillator strengths OE共k兲 are visualized in Fig.
2 for selected planes in the BZ, depicted in the first row. In
the second row, the oscillator strengths for transitions contributing to the feature around 4 eV are displayed, where
dark gray areas correspond to high oscillator strengths. K
points with kz values smaller than 0.5␲ / a in a ring close to
the BZ boundaries are dominant. High contributions are
found in the region around the X points 共see the first and
third panels兲. They can be ascribed to transitions from valence bands of d character to p-like conduction bands. Moreover, k points in an area around the L point are involved in
7.0
Γdr = 0.10 eV
Γdr = 0.02 eV
6.0
5.0
X
ε2(ω)
4.0
X
L
3.0
2.0
1.0
0.0
1.0
Reflectivity
0.8
3.5-5.0 eV
0.6
0.4
0.2
0
0
5
10
15
20
25
19.5-21.0 eV
Energy (eV)
FIG. 1. 共Color online兲 Imaginary part of the complex dielectric function and
reflectivity of silver in the energy range of 0 – 25 eV for two different broadenings ⌫dr = 0.10 eV 共black line兲 and ⌫dr = 0.02 eV 共red line兲 of the intraband
contributions. For the interband part of the spectrum a Lorentzian broadening of 0.10 eV was adopted.
FIG. 2. Oscillator strengths for optical transitions of silver visualized for
three planes in the BZ. From top to bottom: Position of the plane in the BZ;
oscillator strength for transitions in the energy ranges of 3.5–5 and
19.5– 21.5 eV, respectively. From left to right: z = 共1 / 34兲共␲ / a兲, z
= 0.5共␲ / a兲, and z = 共33/ 34兲共␲ / a兲 planes. Dark areas correspond to regions
with high oscillator strengths.
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WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
p → sd and d → pd transitions 共middle panel兲.
A different situation is found for the features in the
19.5– 21.5 eV range 共bottom panels兲. They stem from a region of k points around the center of the BZ, with high
oscillator strengths close to the kz axis for small kz values. As
kz increases, these areas move towards the BZ boundary and
become strongest along K − L. The optical transitions creating this feature involve valence bands of d character and
conduction bands of pf and f character.
(a)
Intensity (arb. units)
1020
x50
raw
deconvoluted
1480
1490
1500
energy E (eV)
3. Experiment
(b)
0.02
-1
Intensity(eV )
Loss spectra were acquired in UHV on a VG Microlab 310
F equipped with a hemispherical analyzer that was operated
in the constant ⌬E mode at 20 eV pass energy. The primary
electrons were normally incident on the target. The detection
angle was 60° with respect to the surface normal. According
to the manufacturer, the full polar opening angle of our spectrometer amounts to 24°. The base pressure of the system
during the measurements was ⬃10−9 mbar. A single sample
holder containing over 30 solids was used in order to ensure
identical experimental conditions for the measurements of all
samples. Of these samples, the following 17 elemental solids
are considered in the present work: Ag, Au, Bi, Co, Cu, Fe,
Mo, Ni, Ta, Te, Ti, Pb, Pd, Pt, V, W, and Zn.
REELS data were measured for energies of 50, 100, 150,
200, 300, 500, 700, 1000, 1500, 2000, 2500, 3000, and
3400 eV. The REELS spectra for primary energies below
300 eV contain significant intensity from low-energy secondary electrons whose tail can exceed kinetic energies of
100 eV and therefore interferes significantly with the loss
spectrum for primary energies below ⬃300 eV. For this reason, only spectra measured with primary energies of 300 eV
or more are considered in the present analysis. Count rates in
the elastic peak were kept well below the saturation count
rate of the channeltrons and the spectra were corrected for
detector dead time. The samples were cleaned by means of
3 keV Ar+ ion bombardment. This also served to amorphize
some specimens such as silicon. Sample cleanliness was
checked before and after the measurement of each sample by
means of Auger electron spectroscopy. The primary beam
current was stable throughout the measurements to within
2%. To minimize diffraction effects, the primary beam was
scanned over a large area of the sample 共0.5⫻ 0.5 ␮m2兲.
Three complete sets of data were acquired in the course of
half a year to check the reproducibility of the experimental
procedure. The results were found to agree within the experimental error imposed by counting statistics.
The raw REELS data were converted to loss spectra 关corresponding to Eq. 共22兲 below兴. The spectra were first subjected to a Richardson-Lucy deconvolution to remove the
experimental energy broadening caused by the thermal fluctuations of the electron gun and the finite resolution of the
analyzer.28 This is necessary to make a precise elimination of
the elastic peak possible, as illustrated in Fig. 3.28 It is assumed that the experimental broadening is symmetric and
can be described by a Gaussian, with a width and position
Elastic
REELS
Loss
0
Deconvoluted
0.02
Raw
0
-5
0
5
Energy loss T (eV)
10
FIG. 3. 共Color online兲 共a兲 Comparison of raw REELS data for 1500 eV
electrons backscattered from a Au sample 共red curve兲 with the corresponding data after a Richardson-Lucy deconvolution 共blue curve兲. 共b兲 Illustration
of the elastic peak removal procedure for the data shown in 共a兲. Blue curves:
spectrum before elimination of the elastic peak; red curves: elastic peak;
black curves: loss spectrum after removal of the elastic peak.
determined by fitting an appropriate portion of the elastic
peak of the raw data. The spectrum is then subjected to a
Richardson-Lucy deconvolution using a 共normalized兲 Gaussian as 共de兲convolution kernel. In the resulting spectrum the
elastic peak is always distinctly separated from the loss features and is then fitted by an asymmetric lineshape accounting for final state effects in the recoil process,29
f 0共T兲 = a1G共T兲 + a2
⳵3
⳵3
G共T兲
+
a
G共T兲 + a4共T兲, 共9兲
3
⳵T3
⳵T4
where G共T兲 is a Gaussian centered around the origin and
共T兲 represents the error function. The elastic peak is then
subtracted from the spectrum which is subsequently divided
by the elastic peak area giving the loss spectrum in absolute
units 关see Eq. 共22兲 in the next section兴. Figure 3共a兲 compares
the REELS data for 1500 eV electrons backscattered from a
Au sample before 共blue curve兲 and after 共red curve兲 deconvolution. The resulting loss spectrum is represented in Fig.
3共b兲 by the black solid curves. It should be noted that the
fourth derivative in Eq. 共9兲 needs to be included in the fitted
peak shape to properly account for final state effects in the
elastic recoil process at the low energies considered here and
furthermore, that any attempt to fit the elastic peak with a
symmetric function leads to much larger misfits than the
ones shown in the lower panel of Fig. 3共b兲. The latter also
demonstrates the necessity for a deconvolution prior to the
elastic peak elimination: Owing to the large dynamic range
of REELS data which is caused by the fact that the elastic
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OPTICAL CONSTANTS FOR 17 METALS
peak is much sharper than any other features in the spectrum,
it overlaps with the loss features unless the broadening can
be reduced significantly during the experiment. Then only a
comparably small energy range of the elastic peak can be
used to fit its shape and the resulting fit is generally worse
than after deconvolution, leading to spurious oscillations in
the loss spectrum around zero energy loss. It can be seen in
the lower panel of Fig. 3共b兲 that these oscillations may be
comparable in intensity to the real loss features, which complicates quantitative interpretation of the loss spectra.
4. Extracting Optical Constants From
REELS
The procedure to extract the dielectric function from 共reflection兲 electron energy-loss spectra consists of two steps:
eliminating multiple scattering from the spectra to reveal the
single-scattering loss distribution in an inelastic interaction
and subsequent determination of the optical constants from
the latter. Since in a reflection experiment the momentum
transfer is not fixed by the geometry, as in transmission electron energy-loss measurements,5 the single-scattering loss
distribution represents an average over the BZ and therefore
the second step cannot be achieved with a Kramers-Kronig
analysis, as is usually done in a transmission experiment.
Furthermore, since the reflection coefficient is very small for
the typical energies used in the TEM, a reflection experiment
is usually performed at much lower energies where it is no
longer admissible to neglect surface excitations. Therefore,
the procedure to eliminate multiple scattering also needs to
be modified to take into account the effects of surface excitations.
In the following, the fundamental parameters for the
electron-solid interaction in the relevant energy range are
presented and the procedure to eliminate multiple scattering
to retrieve the single-scattering loss distributions as well as
the method to extract the dielectric function from them are
outlined. Finally, a detailed consistency check and an error
analysis are performed.
4.1. Electron-solid interaction parameters
When an electron hits the surface of a solid, it interacts
strongly with the ionic and electronic subsystems of the
solid. Therefore its degrees of freedom are subject to fluctuations. Changes in the direction of motion 共referred to as
elastic collision in the following兲 are mainly brought about
by interaction with the screened Coulomb field of the
nucleus that is assumed to be static during the interaction.
Energy losses of the probing particle 共referred to as inelastic
collisions in the following兲 mainly occur via polarization of
the solid-state electrons as a response of the loosely bound
solid-state electrons to the incoming probing particle representing a strong perturbation. The incoming particle is decelerated by the polarization field it induces in the solid. In a
semi-infinite solid with a planar surface, two types of inelastic excitations exist: bulk or volume excitations 共denoted by
the subscript “b” in the following兲 taking place deep inside
1021
the solid where the influence of the vacuum-solid boundary
can be neglected and surface excitations 共subscript “s”兲 that
take place in a narrow depth region zs near the surface 共zs
= v / ␻s, where v is the electron speed and ␻s is the surface
plasmon frequency兲 both in vacuum and inside the solid.
These surface modes are governed by the boundary conditions of Maxwell’s equations at the interface of two media
with different dielectric susceptibilities and have a somewhat
lower characteristic frequency than volume excitations 共␻s
= ␻b / 冑2 for a free-electron material with a planar surface兲.
Since energy and momentum conservation governs the kinematics of the interaction, any change in the direction of
motion is always accompanied by a change in the speed of
the electron and vice versa. However, the recoil energy loss
suffered during an elastic collision is typically of the order of
some 10 meV which is much smaller than the energy loss
during an inelastic collision, which is typically of the order
of 10 eV. Furthermore, the scattering angles during an inelastic collision are much smaller than deflections caused by
elastic collisions: milliradians for inelastic scattering compared to scattering angles of the order of ␲ for elastic scattering. Therefore, to a good approximation, deflections and
energy losses can be treated as independent processes. Note
that this follows immediately from energy and momentum
conservation and by considering the fact that the masses of
the interaction partners in an inelastic collision are very similar, while they are very different for an elastic interaction.
When coherent scattering 共diffraction兲 is neglected,
elastic-scattering processes are completely specified by the
differential cross section for elastic scattering d␴e共␪s兲 / d⍀,
i.e., the distribution of polar scattering angles ␪s in an individual interaction. This quantity can be calculated by means
of the partial-wave expansion method.10,30 Usually a static
atomic potential is used for such calculations, which is justified by the fact that solid-state effects mainly affect the
small-angle scattering part of the cross section which is immaterial for the momentum relaxation process 共see Ref. 31
for a detailed discussion兲. Inelastic scattering is described by
the differential inelastic mean free path 共DIIMFP兲 for bulk
scattering Wb共T兲, i.e., the distribution of energy losses per
unit path length. Surface excitations are most conveniently
expressed via the so-called differential surface excitation
probability 共DSEP兲 Ws共T兲, i.e., the distribution of energy
losses in a single surface crossing.
While calculations of the elastic-scattering cross section
have been performed over several decades, only in the very
recent past has the advent of density functional calculations
made ab initio calculations for the inelastic-interaction characteristics possible. To date, empirical data for the susceptibility of any given solid to become polarized by an incoming
charged particle have been used instead. The relationship between the inelastic-interaction characteristics and the macroscopic response function of a solid is provided by linear response theory. The DIIMFP is expressed in terms of the
dielectric function ␧共␻ , q兲 of the solid via the well known
formula11
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WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
1
␲E
冕
再 冎
q+
Im
q−
−1
dq
,
␧共␻,q兲 q
共10兲
(a)
differential loss probability (arb. units)
W b共 ␻ 兲 =
where a quadratic dispersion is usually assumed such that the
momentum transfer q is limited by q− and q+ given by
q⫾ = 冑2E ⫾ 冑2共E − ␻兲.
共11兲
Note that atomic units are used in this section and T = ␻ is
used interchangeably throughout this work to denote the energy loss or the corresponding characteristic frequency. The
differential inverse mean free path normalized to unity area
is related to the unnormalized DIIMFP via wb共T兲 = ␭iWb共T兲,
where ␭i is the 共total兲 inelastic mean free path 共IMFP兲, i.e.,
the average distance the particle travels in between successive inelastic collisions, measured along its trajectory.
For the DSEP, the expression by Tung et al. that includes
the recoil term32 is employed in the present work,
Ws共␻, ␪,E兲 =
Ps+共␻, ␪,E兲
+
Ps−共␻, ␪,E兲,
共12兲
where the quantity Ps⫾共␻ , ␪ , E兲 is defined as
Ps⫾共␻, ␪,E兲
1
=
␲E cos ␪
冋
⫻Im
and
冕
q+
q−
共␧共␻,q兲 − 1兲2
␧共␻,q兲共␧共␻,q兲 + 1兲
冋 冉 冑 冊册
qs⫾ = q2 −
␻ + q2/2
2E
兩qs⫾兩dq
q3
2 1/2
cos ␪ ⫾
冉
册
␻ + q2/2
冑2E
共13兲
冊
sin ␪ .
共14兲
The normalized DSEP is calculated via ws共T , ␪兲
= Ws共T , ␪兲 / 具ns共␪ , E兲典, where 具ns共␪ , E兲典 is obtained by integrating Eq. 共13兲 over the energy loss and represents the average number of surface excitations experienced during a
single surface crossing.
To avoid confusion, it is noted that although the same
symbol 共“W” in Wb and Ws兲 is used to denote the distribution
of energy losses in a surface and bulk excitation, the physical
meaning of these quantities is quite different. While the DIIMFP is the volume-scattering probability per unit path
length and energy, the DSEP represents the surface excitation
probability 共SEP兲 per unit energy, since this quantity is obtained by integrating over the path the electron takes through
the surface-scattering zone.32 The normalized distribution of
energy losses wb共T兲 and ws共T兲 are physically equivalent
quantities and both have the dimension of reciprocal energy,
which is the reason why the same symbol is chosen for these
quantities.
Examples of the DIIMFP and DSEP for Si and Pt are
given in Fig. 4. The DIIMFP is shown for two energies 共1000
and 3000 eV兲, while the DSEP is given for 1000 eV for two
different surface-crossing angles of 0° and 70°. It is seen that
for the considered energy-loss range, being significantly
smaller than the probing energy, the shape of the DIIMFP
hardly depends on the energy, or, in other words, that the
(b)
differential loss probability (arb. units)
1022
Si
DIIMFP
DSEP
70°
1000eV
0°
3000eV
0
0
10
20
30
energy loss (eV)
40
50
Pt
1000eV
DSEP
DIIMFP
3000eV
70°
0°
0
0
5
10
15
20
25
energy loss (eV)
30
35
40
FIG. 4. DIIMFP and DSEP for medium energy electrons in Al and Pt. The
DIIMFP is presented for 1000 eV 共solid curves兲 and 3000 eV 共dashed
curves兲 while the DSEP is shown for 1000 eV for two different angles of
surface crossing 共0°, solid curves; 70°, dashed curves兲. 共a兲 Al; 共b兲 Pt.
normalized distribution of energy losses is independent of
the incoming energy wb共T , E兲 ⯝ wb共T兲 to a good approximation. The same holds for the DSEP 共not shown兲. On the other
hand, the angular dependence of the normalized DSEP is
seen to be very weak as well, ws共T , E , ␪兲 ⯝ ws共T兲. Of course
the total SEP 具ns共E , ␪兲典 exhibits a pronounced angular and
energy dependence, while the energy dependence of the
IMFP, ␭i共E兲, which governs the total bulk scattering probability, is appreciable as well.
4.2. Multiple scattering
Owing to their finite rest mass, electrons suffer intense
multiple scatering when traveling through a solid. This is in
contrast to the case of photon transport in matter where the
vanishing rest mass makes it highly unlikely for the photon
not to be absorbed during an interaction. The energy and
direction of motion of a charged particle are changed repeatedly by multiple-scattering processes. For noncrystalline materials, where coherent scattering can be neglected 共see Ref.
33 for the limitations of the standard model for particle transport where diffraction effects are neglected兲, the fluctuations
after multiple scattering can be expressed in terms of the
fluctuation distributions for a single collision by solving a
linearized Boltzmann-type kinetic equation.34 This was done
in the so-called quasielastic approximation, valid for energy
losses significantly smaller than the incoming energy 共T / E
Ⰶ 1兲, where the interaction characteristics are taken to be
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OPTICAL CONSTANTS FOR 17 METALS
1023
具ns典ns −具n 典
e s.
n s!
共19兲
independent of the energy. Green’s function of the problem
can then be expressed in terms of the 共nb − 1兲-fold selfb兲
convolution of the DIIMFP, w共n
b 共T兲, and the 共ns − 1兲-fold
self-convolution of the DSEP, ws共ns兲共T兲, weighted with the
collision statistics Anb,ns, i.e., the number of times a given
scattering process occurs for the considered boundary conditions 关Eq. 共12兲 in Ref. 34兴. The resulting REEL spectrum
Y共E兲 is then found by superposition,8
⬁
⬁
Y共E兲 =
兺兺
n =0 n =0
b兲
Anb,nsw共n
b 共T⬘兲 丢
A ns =
The quantity 具ns共E , ␪兲典 represents the average number of surface excitations in a single surface crossing, which is given
in terms of a material parameter as, the so-called surface
excitation parameter, by the formula32
具ns共E, ␪兲典 =
ws共ns兲共T⬙兲
as
共20兲
冑E cos ␪ ,
s
b
丢 f 0共E + T⬘ + T⬙兲,
共15兲
where the energy distribution at the source is given by f 0共E兲
and the symbol “ 丢 ” denotes a convolution over the energy
loss T. The partial intensities Anb,ns represent the number of
electrons that arrive in the detector after being 共nb , ns兲-fold
inelastically scattered in the solid. Since the width of the
surface-scattering zone v / ␻s is smaller than, or of the order
of, the elastic mean free path ␭e, the electron path in the
surface-scattering zone is rectilinear to a good approximation. A most beneficial consequence of this fact is that the
partial intensities for volume and surface scatterings are uncorrelated to a good approximation,35
Anb,ns = Anb ⫻ Ans .
共16兲
Although 共small兲 effects of deflections in the surfacescattering zone have been experimentally observed for a
rather pathological case,36 implying that Eq. 共16兲 is not
strictly true, this approach nonetheless has proven to constitute an effective approximation.35–38
The volume partial intensities Anb are given by an integral
over all possible lengths of the paths, s, taken by the particle
in the solid,
and ␪ is the off-normal polar angle of surface crossing. Plural
surface scattering is usually assumed to be described by the
Poisson stochastic process. The fact that the normalized
DSEP does not significantly depend on the energy and
surface-crossing angle 共see Fig. 4兲 was utilized in Eq. 共19兲,
allowing one to combine the effect of surface excitations
along the in- and outgoing parts of the trajectory by writing8
具ns典 = 具ns共␪i兲典 + 具ns共␪o兲典,
共21兲
where the ␪i and ␪o indicate the incident and outgoing polar
directions of surface crossing.
Finally, for quantitative analysis of REELS, the elastic
peak needs to be removed and the energy scale converted
into an energy-loss scale. Furthermore, the spectrum is divided by the area of the elastic peak. These transformations
then give the reduced loss spectrum y共T兲 which is used in the
following analysis in the compact form:
⬁
y共T兲 =
⬁
共n 兲
兲
␣n ,n w共n
兺
兺
b 共T⬘兲 丢 ws 共T⬙兲,
n =0 n =0
b
s
b s
b
共22兲
s
共17兲
where the reduced partial intensities ␣nb,ns = Anb,ns / Anb=0,ns=0
have been introduced and the zero-order partial intensity is
taken to be zero, ␣nb=0,ns=0 = 0, to account for the fact that the
elastic peak has been removed from the spectrum.
Here Q共s兲 is the distribution of path lengths and Wn共s兲 is the
stochastic process for multiple scattering, which, in the
quasielastic regime, is given by39
4.3. Deconvolution of multiple scattering from loss
spectra
A nb =
冕
⬁
0
Wnb共s兲Q共s兲ds.
冉冊
s
Wn共s兲 =
␭i
n −s/␭i
e
n!
.
共18兲
The path length distribution 共PLD兲 Q共s兲 can be calculated
analytically34 or using a MC technique.31,34
Since there exists a unique straight line path connecting
any two points in space, the part of the PLD relevant for
surface excitations resembles a ␦ function, and Qs共s兲 ⬇ ␦共s
− v / ␻scos␪i兲 when the passage through the surface-scattering
zone is approximately rectilinear. In this case an equation
similar to Eq. 共17兲 for the surface partial intensities can
readily be integrated giving
To extract the dielectric function from an energy-loss
spectrum, the single-scattering loss distributions wb共T兲 and
ws共T兲 need to be retrieved, i.e., a method to eliminate multiple scattering from Eq. 共22兲 is needed. Recalling the convolution theorem, it is immediately obvious that in Fourier
space, the spectrum is given by a bivariate power series in
the variables wb共T兲 and ws共T兲. This result implies that a
unique solution of these quantities cannot be found by reverting the series equation 共22兲 since a single equation with two
unknowns has no unique solution. However, when two loss
spectra y 1共T兲 and y 2共T兲, with a different sequence of partial
intensities ␣nbns and ␤nb,ns, are measured, reversion of the
bivariate power series becomes possible using the formula8,40
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1024
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
N
N
␧2共␻,q兲 = 兺
w共T兲 = 兺 兺 ak,lY k,l共T兲
i
k=0 l=0
−
冕
T
N
N
兺 兺 bk,lY k,l共T − T⬘兲w共T⬘兲dT⬘ ,
T⬘=0 k=0 l=0
共23兲
where the quantity Y k,l共T兲 is the 共k , l兲-th order cross convolution of the two REELS spectra,
Y k,l共T兲 =
y 共k兲
1 共T
− T ⬘兲
共l兲
丢 y 2 共T⬘兲.
共24兲
Since w共T = 0兲 ⬅ 0, the integration on the right hand side of
Eq. 共23兲 can always be carried out over the energy-loss range
for which the loss distribution is already known. For more
details on the 共trivial兲 numerical treatment of the Volterra
integral equation of the second kind see, e.g., Ref. 41.
The expansion coefficients ak,l and bk,l are entirely determined by the reduced partial intensities of the two
spectra,8,40 which, together with the two loss spectra, are
therefore the only input parameters of the procedure. The
reduced volume partial intensities can always be calculated
with sufficient accuracy when a reasonable estimate for the
IMFP is used, such as the TPP-2M predictive formula.42
While a similar guideline has been published for the surface
excitation parameter,43 it is not needed to know the absolute
value of this parameter at all, since the expansion coefficients
turn out to scale with the value of as in Eq. 共20兲.40 Thus,
apart from the functional form of the dependence of the SEP
on the energy and surface-crossing direction, such as Eq.
共20兲, no information on surface excitations whatsoever is
needed as input to the procedure. Note, finally, that the equations to retrieve the surface and volume single-scattering loss
distributions are identical and given by Eq. 共23兲; merely the
expansion coefficients are different.40
Calculation of the expansion coefficients in Eq. 共23兲 is
quite tedious, as explained in Refs. 8 and 44. Therefore, in
Appendix A, the synopsis of a recently developed simplified
deconvolution procedure45 is given which is equivalent to the
procedure employed here and can be used for the same purpose with considerably less numerical effort.
4.4. Retrieval of optical data from the singlescattering distributions
The final step in the analysis is the retrieval of the dielectric function from the single-scattering loss distributions.
This is achieved by fitting the theoretical expressions for the
DIIMFP and DSEP to the corresponding experimental results
using a suitable model for the dielectric function.
The electromagnetic response of the solid is described in
terms of the fit parameters f i, ␻i, and ␥i using a model for the
dielectric function in terms of a set of Drude-Lindhard oscillators for the evaluation of the theoretical expressions for the
DIIMFP and DSEP,46
␧1共␻,q兲 = 1 − 兺
i
f i共␻2 − ␻i共q兲2兲
共␻2 − ␻i共q兲2兲2 + ␻2␥2i
,
f i␥ i␻
,
共␻2 − ␻i共q兲2兲2 + ␻2␥2i
共25兲
where ␻ = T is the energy loss and q is the momentum transfer. In the above expressions, f i represents the oscillator
strength, ␥i is the damping coefficient, and ␻i is the energy
of the i-th oscillator. A quadratic dispersion,
␻i共q兲 = ␻i + ␣q2/2,
共26兲
was used for the transition described by the i-th oscillator.
The value of the dispersion coefficient ␣ was chosen to be
unity, except for those oscillators with an energy higher than
the most loosely bound core electrons 共semicore transitions兲,
for which ␣ = 0.5 was used. Those values of the DrudeLorentz parameters in the above equation that minimize the
deviation between the experimental data and the theoretical
expressions for the DIIMFP and DSEP 共as described below兲
are taken to parametrize the dielectric function. Note that a
dielectric function of the form of Eq. 共25兲 implicitly satisfies
the Kramers-Kronig dispersion relationships as well as the
perfect screening ps-sum rule.
The 共normalized兲 distributions wb共T兲 and ws共T兲 obtained
from the experimental data are simultaneously fitted to the
above theoretical expressions by minimizing the function
2
2
Xb␹DIIMFP
共pb, f i, ␻i, ␥i兲 + Xs␹DSEP
共ps, f i, ␻i, ␥i兲.
共27兲
Here the function ␹ is the least-squares difference between
theory and experiment, defined in the usual way, and Xb and
Xs are weight factors chosen as Xb = 1.0 and Xs = 0.01, emphasizing the DIIMFP in the fitting procedure. The reason
for this choice of Xs is that, although the experimental singlescattering loss distributions for surface scattering are reasonably described by Eq. 共12兲, they differ in detail for some
materials 共see Sec. 5.1 below兲, while the theoretical expression for the DIIMFP does match the experimental results in
much better detail. In consequence, the overall agreement of
the simultaneous fit of experiment to theory is significantly
improved by reducing the weight of the surface term in the
fitting procedure. As discussed further below, the physical
reason for this behavior is related to the approximation of
depth-independent surface-scattering characteristics, made in
the model of Tung et al. The fit parameters pb and ps are
multiplicative scaling factors for the DIIMFP and the DSEP
that compensate for an eventual mismatch of the surface excitation parameter and IMFP used as input. Using the
TPP-2M formula42 to estimate the IMFP, the value of pb
deviated from unity by less than 5% for all cases studied,
while the returned value of ps scales with the value of the
surface excitation parameter as, as expected.
2
4.5. Consistency test and error analysis
In the above procedure to measure the dielectric function,
the following of assumptions have been introduced that
should be investigated in some detail in order to judge the
reliability of the method and to arrive at a meaningful error
estimate for the resulting optical constants.
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OPTICAL CONSTANTS FOR 17 METALS
共iv兲
共v兲
The first four assumptions above pertain to the deconvolution of multiple scattering providing the single-scattering
loss distributions that will be discussed first, while the last
assumption concerns the extraction of optical constants from
the single-scattering loss distributions and will be dealt with
at the end of this section.
In Fig. 5, the calculation of the reduced volume partial
intensities is illustrated. This is achieved most easily by
means of a MC simulation for the PLD Q共s兲, such as the
algorithm described in Ref. 31. The only input parameter for
such calculations is the differential elastic-scattering cross
section for the considered energy and the geometrical configuration employed in the experiment. The relevant portion
of the differential elastic-scattering cross section for Au is
shown in Fig. 5共a兲 for several energies. It is seen that the
cross section exhibits deep minima at scattering angles varying with the energy. These minima are more pronounced for
high-atomic-number elements and low energies. For energies
in excess of some tens of a keV, these oscillations disappear
and the differential cross section approaches a Rutherford
shape. It has been known for some time47 that if the experimental configuration corresponds to scattering angles coinciding with a deep minimum in the cross section, the shape
of the loss spectrum changes significantly. The explanation is
that when the value of the cross section is small for the
considered scattering geometry, the probability for the electron to be backscattered into the detector after a single elastic
collision is also small and plural elastic scattering predominates for the electron to experience the required net direc-
300 eV
500
700
1000
2000
3000
3400
0.1
e
2
d ()/d (Å /sr)
Au
0.001
100
(b)
110 120 130
Scattering angle 140
s
700 eV
0.001
-1
PLD Q(s) (Å )
共iii兲
(a)
3400 eV
-5
5 keV
10
40 keV
-7
10
(c)
0
nb
共ii兲
An essential assumption that makes the outlined procedure possible at all is that the partial intensities for
surface and volume scatterings are uncorrelated 关Eq.
共16兲兴.
The model to calculate the reduced volume partial
intensities via Eq. 共17兲 on the basis of state-of-the-art
calculations for the elastic-scattering cross section is
assumed to be accurate enough for the present purpose.
The functional form of the energy and angular dependence of the surface-scattering probability is supposed
to be known. In the present work, it is assumed to be
given by Eq. 共20兲. The scaling properties of the deconvolution coefficients in Eq. 共23兲 with respect to the
parameter as make any knowledge concerning the
value of this parameter immaterial for the determination of the optical constants.8,40,44
The shapes of the normalized DIIMFP and DSEP are
supposed not to depend on the primary energy of the
reflected electrons and the surface-crossing direction.
In the theoretical relationships between the DIIMFP
and the DSEP and the dielectric function 关Eqs. 共10兲
and 共13兲兴, it is assumed that the dispersion of the oscillators in the Drude-Lorentz expansion used as
model dielectric function 关Eq. 共25兲兴 is of the form
␻i共q兲 = ␻i + ␣q2 / 2 and that an appropriate value of ␣
can be chosen.
Reduced bulk partial intensity 共i兲
1025
1000
Pathlength s(Å)
1.6
3400 eV
40 keV
1.2
5 keV
0.8
0.4
700 eV
0
0
2
4
6
8
10
Number of inelastic collisions n
b
FIG. 5. 共Color online兲 共a兲 Elastic scattering cross section of Au for several
energies in the range of scattering angles between 100° and 140°. 共b兲 PLD
Q共s兲 calculated by means of a MC simulation for Au for 700 and 3400 eV
for the geometrical configuration used in the present work and for 5 and
40 keV for the experimental setup of Ref. 58. 共c兲 Reduced partial intensities
for volume scattering calculated by means of Eq. 共17兲 for the PLDs shown
in 共b兲.
tional change. In consequence, the path length traveled in the
solid is larger than for a case where the differential elastic
cross section is larger. This can be clearly observed in Fig.
5共b兲 where the PLDs for 700 and 3400 eV, corresponding to
a maximum and a minimum in the differential cross section,
respectively, demonstrates the expected differences: For
700 eV, the PLD decreases monotonically with the traveled
path length, while for 3400 eV it exhibits a maximum. As a
consequence, the sequence of partial intensities for different
energies also becomes qualitatively different, as can be seen
in Figure 5共c兲 where the partial intensities calculated from
the PLDs of Fig. 5共b兲 with Eq. 共17兲 are shown for four energies.
Let us now address the uncertainty of the reduced volume
partial intensities obtained in this way. It is easy to see from
Eq. 共17兲 that the absolute value of the partial intensities
scales with the value used for the IMFP. This is in fact the
basis of the so-called elastic peak 共ns = nb ⬅ 0兲 electron spectroscopy 共EPES兲 technique to measure the IMFP.34,48–51
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1026
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
0.02
However, the reduced partial intensities are virtually unaffected by the choice of the IMFP value. This can be seen by
employing the common rules of error propagation to Eq.
共17兲. For the reduced partial intensities one finds that an
uncertainty ⌬␭ in the IMFP leads to an error in the first-order
reduced partial intensity given by
0.01
0
-0.005
-0.01
共28兲
-0.015
300 eVE3400 eV
(b)
500 eVE3400 eV
0.015
-1
0.01
b
In Fig. 5共b兲 it is seen that the width of the PLD is large
compared to the value of the IMFP. The IMFP, on the other
hand, governs the width of the Poisson distribution 关Eq. 共18兲兴
for any value of n. Taking into account Eq. 共17兲, it follows
that the difference between the reduced partial intensities of
subsequent order is always considerably smaller than unity,
implying that ␣1 is always quite close to unity 共the zeroorder partial intensity is unity per definition, ␣0 ⬅ 1兲. In consequence, the error in the partial intensities due to an uncertainty in the IMFP is always negligible if a reasonable
estimate of the IMFP is available. For example, for 0.9
⬍ ␣1 ⬍ 1, the error in the partial intensities will be at least ten
times smaller than the error in the IMFP, as follows from Eq.
共28兲.
The second input parameter for calculation of the partial
intensities is the elastic-scattering cross section based on a
static atomic potential. The accuracy of this quantity has
been studied in detail by Jablonski and co-workers.10,48,52
One of the main conclusions of these works was that the
cross sections calculated in this way are appropriate and reliable for the description of elastic scattering in solids, except
for scattering angles close to a deep minimum in the cross
section. Also, the effects of absorption can reach up to 30%
in the vicinity of a deep minimum53,54 adding to the uncertainty in this range of scattering angles. Moreover, for experimental geometries where the net scattering angle coincides with a deep minimum, most electrons experience plural
elastic deflections before reaching the detector leading to a
violation of assumption 共i兲 above, since in these cases the
part of the trajectory in the surface-scattering zone is likely
to be nonrectilinear. This expectation has indeed been proven
experimentally.36 Finally, near a deep minimum the derivative of the cross section is relatively high implying that in
MC calculations, the solid angle of acceptance of the analyzer needs to be accurately known.
The above discussion may be summarized by stating that
quantitative interpretation of REELS may be problematic
when, for a particular element and energy and for a particular
experimental scattering geometry, there is a deep minimum
in the differential elastic-scattering cross section. On the
other hand, it is clear that for the deconvolution procedure to
yield numerically stable results, the spectra need to be sufficiently different. In other words, the employed combination
of energies and/or surface-crossing directions should be chosen in such a way that the sequence of partial intensities for
the two considered spectra is sufficiently different. The extent to which the spectra differ is to first order determined by
the determinant40
(a)
0.005
w (T) (eV )
冏 冏
⌬␣1
⌬␭
=
兩1 − ␣1兩.
␣1
␭
0.015
0.005
0.02
(c)
0.015
0.01
0.005
0
-0.005
-0.01
0
500 eVE3400 eV,
Oswald formula
50
100
Energy loss T (eV)
150
FIG. 6. 共a兲 Results of the deconvolution procedure equation 共23兲 applied to
all possible energy combinations between 300 and 3400 eV. Note that the
scattering geometry employed in the present work corresponds to a deep
minimum in the elastic-scattering cross section at 300 eV 共see Fig. 5兲. 共b兲
Same as 共a兲 for energies larger than 300 eV, thus avoiding the deep minimum in the scattering cross section at 300 eV. 共c兲 Same as 共b兲 but using
Oswald’s formula for the energy and angular dependence of the surface
excitation parameter equation 共30兲 instead of Eq. 共20兲.
⌬ = ␣1,0␤0,1 − ␣0,1␤1,0 ,
共29兲
where ␣nb,ns and ␤nb,ns represent the partial intensities of the
two spectra. The above considerations are fully supported by
the outcome of the consistency test shown in Fig. 6. The
result of the deconvolution procedure equation 共23兲 for the
DIIMFP of Au is shown for all possible combinations of
incident energies between 300 and 3400 eV for which ⌬
艌 0.1 in Fig. 6共a兲. Note from Fig. 5 that the differential cross
section at 300 eV exhibits a deep minimum at the scattering
angle of 120° employed in our experiment. Comparison with
the results in Fig. 6共b兲 where this deep minimum is avoided
by disregarding the spectrum for an incident energy of
300 eV shows that the consistency is dramatically improved.
A statistical analysis of the rms deviation from the average of
these results leads to an error estimate for the DIIMFP of
ⱗ10%. In Fig. 6共c兲 a similar result is shown, from an analysis in which the energy and angular dependence given by Eq.
共20兲 is replaced with the formula proposed by Oswald,49
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OPTICAL CONSTANTS FOR 17 METALS
1
.
共30兲
0.03
700 eV
3400 eV
-1
asⴱ冑E cos ␪ + 1
REELS (eV )
具ns共E, ␪兲典 =
1027
0.02
0.01
Low Energy
(a)
0
-1
REELS (eV )
5 keV
40 keV
0.02
0.01
High Energy
(b)
0
-1
w (T), w (T) (eV )
b
0.01
s
It is seen that in this case the consistency of results is slightly
worse, in particular, for small energy losses, where negative
excursions are seen in the DIIMFP in Fig. 6共c兲, which are not
physically realistic. The analysis so far demonstrates that
when deep minima in the cross section are avoided, results
consistent to within ⱗ10% are obtained for the considered
experimental spectra.
The assumption that the shapes of the DIIMFP and DSEP
are approximately independent of the energy and surfacecrossing direction was studied in detail earlier8 by analyzing
model spectra with Eq. 共23兲 and comparing the results with
experimental data. The results verified the model for the dynamic interaction of an energetic electron with a surface,
summarized by Eqs. 共10兲 and 共13兲 in great detail. It was
found that this approximation leads to small spurious replicas of the DIIMFP and DSEP when the loss features are very
sharp, such as for free-electron materials, such as Al and Si.
For solids with a broad loss distribution, such as those studied in the present work, the approximation 共iv兲 mentioned
above consitutes an appropriate approach.
While the consistency for different incident energies is
apparent from the above, the question remains whether the
same level of consistency is also obtained when analyzing
different geometrical configurations. Furthermore, Yubero
and co-workers suggested that there may be an influence of
interference effects between the in- and outgoing parts of the
trajectory on the energy-loss spectrum when the field set up
during the penetration of the electron into the solid affects
the surface crossing on its way out.55–57 The magnitude of
the interference effect is predicted to scale reciprocally with
the speed of the incoming electron and should be negligible
at energies above several keV. Figure 7 compares the result
of the deconvolution for a set of spectra taken at rather low
energies of 700 and 3400 eV 关Fig. 7共a兲兴, measured in the
present work, with the spectra of Ref. 58 measured for incident energies of 5 and 40 keV 关Fig. 7共b兲兴. The shape of the
loss spectra is seen to significantly depend on the energy: For
700 eV the spectrum is decreasing more or less monotonically with energy loss, while for energies above ⬃3 keV an
increase is seen at small energy losses and a flat energy-loss
dependence is observed at higher energies. Note that the
shape of these spectra can be explained perfectly by considering the collision statistics imposed by elastic scattering, as
follows from the partial intensities shown in Fig. 5共a兲. Considering the fact that these sets of spectra were measured at
opposite sides of the world on different samples using a different spectrometer in a different energy range and with different geometrical setups, the consistency between the
single-scattering loss distributions shown in the lower panel
of Fig. 7 is remarkable and gives confidence in the experimental procedure as well as the method of analysis employed. It also suggests that the difference in shape of loss
spectra taken at different energies is indeed caused by the
(c)
Surface (x0.25)
Low Energy
High Energy
0.0
0.01
Volume
0.0
0 10 20 30 40 50 60 70 80
Energy loss T (eV)
FIG. 7. 共Color online兲 Comparison of deconvolution of REELS spectra for
Au measured in significantly different energy ranges, corresponding to the
PLDs and partial intensities shown in Figs. 5共b兲 and 5共c兲. 共a兲 Loss spectra
for 700 and 3400 eV electrons, measured in the present work. 共b兲 Measurements of Ref. 58 for 5 and 40 keV. 共c兲 Comparison of the single-scattering
loss distributions obtained with Eq. 共23兲 using the partial intensities shown
in Fig. 5共c兲.
influence of elastic scattering rather than by interference effects, for which no experimental evidence whatsoever can be
found in the literature.
In concluding this section, the model for the dielectric
function and, in particular, the choice of the dispersion constant ␣ deserve to be discussed. First of all, it should be
noted that the physical significance of the fit parameters describing the oscillators in the Drude-Lorentz expansion of
the dielectric function, Eq. 共25兲, is limited in that the characteristic frequencies ␻i and damping constants ␥i in the
low-energy region 共i.e., for energies clearly below the ionization edges of the semicore states兲 do not necessarily reflect the correct energies describing the inter- and intraband
transitions, etc. The values of the fit parameters, together
with the Drude-Lorentz expansion, should rather be regarded
as a convenient numerical representation of the data. Also, in
this energy-loss range, the choice of the dispersion constant
is immaterial, as can be seen from Eq. 共11兲 which shows that
for low-energy losses ␻ Ⰶ E the range of momentum transfers over which the averaging is performed is small. Indeed,
changing the dispersion constant ␣ between unity and 0.5
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WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
Co 3p
2
1.5
REELS, α=0.0
REELS, α=0.5
DFT
Palik (Henke)
-1
60
ω (eV)
65
70
FIG. 8. 共Color online兲 Influence of the chosen dispersion constant ␣ in the
retrieval of the dielectric function from the single-scattering loss distributions for the Co 3p semicore transition. Dashed red curve with open circles:
ELF values of ␧ retrieved from REELS using ␣ = 0.0 in Eq. 共26兲; solid red
curve with filled circles: ELF values of ␧ retrieved from REELS using ␣
= 0.5; black curve: Palik’s data 共Ref. 1兲 for the ELF 关that were taken from
Henke’s work 共Ref. 2兲兴. Blue curves: results of DFT calculations. Note that
the DFT calculations accurately reproduce the position of the semicore edge
binding energy E3p3/2 = 58.9 eV derived from photoemission data 共Ref. 59兲,
as do the REELS data when a value of ␣ = 0.5 is used for the dispersion
constant.
during the retrieval of ␧ from the single-scattering loss distributions does not lead to any significant change in the resulting values for ␧ for ␻ ⱗ 50 eV.
For the semicore transitions, i.e., the ionization edges of
the core states with the lowest binding energy, the situation is
different. This is illustrated in Fig. 8 that compares the value
of the ELF function for the Co 3p edge retrieved from
REELS data with the present procedure using different values of the dispersion constant ␣ with DFT calculations and
1
0
1
()
2
-2
0
20
40
60
Energy loss, (eV)
2
()
80
REELS
Palik
DFT
8
4
0
700 eV
3400 eV
Surface (x0.16)
0.02
Volume
0
2
20
40
60
Energy loss, (eV)
80
FIG. 9. 共Color online兲 Comparison of the dispersive 共␧1兲 and absorptive 共␧2兲
parts of ␧ for Au derived from the present REELS measurements 共red curves
with error bars兲, DFT calculations 共blue curves兲, and Palik’s data 共black
curves兲.
Im[-1/ε (ω)]
55
1
0
2
ε1 (ω)
0
50
0
wb,s(T) (eV )
0.5
0
0.03
0
1
0
-2
REELS
DFT
ε2 (ω)
ELF, Im {-1/ε(ω)}
Ti
=58.9 eV
3p1/2
-1
E
2.5
REELS (eV )
1028
5
0
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 10. 共Color online兲 REELS data and optical constants derived from them
for Ti. Upper panel: deconvoluted REELS data at the employed energies, as
indicated in the legend; second panel from above: DIIMFP 关wb共T兲兴 and
DSEP 关ws共T兲兴 共data points兲 as well as the best fit of these data to the DrudeLorentz model of Eq. 共25兲 共solid curves兲; lower three panels: comparison of
the loss function Im兵−1 / ␧其 as well as the dispersive 共␧1兲 and absorptive 共␧2兲
parts of the dielectric function extracted from REELS 共red curves兲 with
results from DFT calculations 共blue curves兲 and Palik’s data set 共black
curves兲.
with the optical constants taken from Palik’s book.1 For this
larger energy-loss range, the entire BZ is sampled by Eq.
共11兲 and the influence of dispersion becomes more important.
At this stage it should be noted that the employed DrudeLorentz expansion model for the dielectric function only
roughly accounts for the effects of dispersion through a
single parameter, the dispersion constant ␣, which is a fundamental weakness of this model. Three effects of dispersion
play a role in electron energy-loss spectra: the dispersion of
共1兲 the initial and 共2兲 the final state and 共3兲 the fact that the
scattered electron itself can transfer momentum to the solid.
It should be clear that the details of these phenomena are
difficult to describe with a single dispersion parameter in ␧,
which complicates the choice of ␣. Therefore, the retrieval
was performed for different values of ␣ and the results were
compared with other results for the dielectric function. It was
found that for ␣ = 0.5 the characteristic energies ␻ for the
semicore transitions compare best to the binding energies
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OPTICAL CONSTANTS FOR 17 METALS
1029
wb,s(T) (eV-1)
REELS (eV )
-1
0
700 eV
3000 eV
Surface (x0.16)
0.02
Volume
2
0
0.02
Volume
1
0
-2
REELS
DFT
Palik
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
ε2 (ω)
ε2 (ω)
-2
0
Surface (x0.16)
0
0
2
ε1 (ω)
ε1 (ω)
0
2
5
700 eV
3400 eV
0
Im[-1/ε (ω)]
0
0.02
0
-1
0.03
0
Im[-1/ε (ω)]
Fe
wb,s(T) (eV )
REELS (eV-1)
V
5
0
REELS
DFT
Palik
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 11. 共Color online兲 Same as Fig. 10 for V.
FIG. 12. 共Color online兲 Same as Fig. 10 for Fe.
derived from photoemission data found in the literature.59
This value of ␣ also gives very good agreement with the
DFT calculations for the semicore transitions, as can be seen
in Fig. 8. For ␣ = 0, the retrieved ELF agrees closely with the
data found in Palik’s books. In this energy region Palik’s data
were taken from Henke’s work,2 which is a synthesis of empirical and theoretical data for free atoms, and therefore is
not expected to be representative for the semicore ionization
edges in a solid. The same is true for the more recent compilation by Chantler,3,4 which gives ␧ values very close to
those obtained by Henke for most of the transitions considered in the present work. Therefore, although our approach
introduces a small systematic uncertainty in the retrieved dielectric function for energies beyond the semicore edges, we
have chosen a value of ␣ = 0.5 throughout this work for all
transitions involving semicore levels, while for lower energies ␣ was taken to be unity.
In the above, the error in the retrieved DIIMFP was derived from a statistical analysis of data derived from different
energy combinations and was estimated to be ⱗ10%. Assuming that the resulting error in the ELF Im兵−1 / ␧共0 , ␻兲其 is
of the same order of magnitude, the common rules of error
propagation can be used to estimate the uncertainty in ␧1 and
␧2. The results of this analysis are shown for Au in Fig. 9
which again also shows Palik’s data and the results of DFT
calculations. The uncertainty in ␧1 is seen to be rather large
below 20 eV. This is a fundamental characteristic of the derivation of optical data from absorption measurements that
mainly sample ␧2. Within the estimated uncertainty, the three
data sets agree satisfactorily both for the real as well as the
imaginary parts of ␧. It should be noted again at this point
that the employed procedure determines in the first place the
ELF, Im兵−1 / ␧共0 , ␻兲其, for which the uncertainty is estimated
to amount to ⱗ10%.
5. Results and Discussion
5.1. Optical constants
The results of the data analysis according to the procedure
outlined above are given in Figs. 10–32. A comparison of the
optical constants Im兵−1 / ␧其, ␧1, and ␧2 as a function of frequency or energy loss ␻ derived from REELS, DFT calculations, and Palik’s data are presented in Figs. 10–26. In Figs.
27–32 the refractive index n and extinction coefficient k derived from the same data are shown as a function of the
photon wavelength. Numerical values corresponding to the
data in the lower three panels of Figs. 10–32 are given in the
Appendixes.
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1030
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
Ni
-1
700 eV
3400 eV
0
Surface (x0.16)
0
0.02
Volume
1
ε1 (ω)
ε1 (ω)
0
REELS
DFT
Palik
ε2 (ω)
ε2 (ω)
0
0.01
Volume
1
0
-2
-2
0
Surface (x0.16)
0
2
0
2
5
700 eV
3400 eV
0
Im[-1/ε (ω)]
0
0.02
-1
wb,s(T) (eV-1)
0
Im[-1/ε (ω)]
REELS (eV )
0.02
wb,s(T) (eV )
REELS (eV-1)
Co
5
0
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
REELS
DFT
Palik
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 13. 共Color online兲 Same as Fig. 10 for Co.
FIG. 14. 共Color online兲 Same as Fig. 10 for Ni.
In Figs. 10–26, the upper panels display the pair of
energy-loss spectra used for the analysis. The raw data were
subjected to a Richardson-Lucy deconvolution, and subsequently the elastic peak was fitted to a function of the form
given by Eq. 共9兲 and removed from the spectra, which were
finally divided by the elastic peak area to give the loss spectrum in absolute units.
The primary energies chosen for the analysis are indicated.
These were selected according to the guidelines given in the
previous section, i.e., those energy combinations were selected for which the contributions of surface excitations are
sufficiently different. This is indicated by the value of the
determinant ⌬ 关Eq. 共29兲兴 using Eq. 共20兲 to estimate the surface excitation parameter. On the other hand, energies close
to a deep minimum in the differential elastic-scattering cross
section were avoided. Values of ⌬ for all energy combinations were in the range ⌬ ⬇ 0.2− 0.4. For most materials the
overall shape of the two spectra is very similar, in particular,
for energy losses in excess of ⬃30 eV where the effects of
multiple inelastic scattering dominate. Significant differences
in the high-energy-loss tail can be observed for some materials, such as Ag, Au, W, Pd, Pt, Mo, Ta, Pb, and Bi, which is
attributable to the influence of elastic scattering on the shape
of energy-loss spectra measured in reflection, as discussed in
detail in the previous section 共see Figs. 5 and 7兲. The fact
that such differences mainly occur for materials with high
atomic number, for which the effects of elastic scattering are
most pronounced, supports these considerations.
The influence of surface excitations on REELS is evident
in the differences of the spectral pair at energies ⱗ20 eV: for
the lower energy of the pair, the corresponding features are
significantly enhanced as compared to the region of higher
energy losses. For some materials, such as, Ti, the spectral
pair is very similar, and at first sight the attempted separation
of the distribution of individual losses inside the solid and
during the surface crossing may seem questionable. However, it should be kept in mind that the sequence of partial
intensities is generally different for the spectral pair allowing
a stable decomposition of the spectra to be made.
The second panel from the top in Figs. 15–26 shows the
result of the partial-intensity analysis 关Eq. 共23兲兴 to retrieve
the energy-loss distributions for single surface- and volumescattering processes, represented by the blue 共surface兲 and
red 共bulk兲 data points. The results for the DIIMFP are in
absolute units; the units for the DSEP are arbitrary since a
value of as ⬅ 1 was chosen throughout for the calculation of
the SEP with Eq. 共20兲, as the normalized DSEP scales with
SEP and accurate values for this quantity are not known for
all materials under study.43
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OPTICAL CONSTANTS FOR 17 METALS
1031
Zn
0.01
-1
Volume
0
ε1 (ω)
ε1 (ω)
0
0.02
Volume
1
0
2
0
2
0
0
-2
-2
REELS
DFT
Palik
REELS
DFT
ε2 (ω)
ε2 (ω)
Surface (x0.16)
0
Im[-1/ε (ω)]
1
0
0.06
0
Surface (x0.16)
0
5
700 eV
3400 eV
-1
wb,s(T) (eV-1)
0
Im[-1/ε (ω)]
REELS (eV )
0.03
wb,s(T) (eV )
REELS (eV-1)
Cu
700 eV
3400 eV
3
0
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 15. 共Color online兲 Same as Fig. 10 for Cu.
FIG. 16. 共Color online兲 Same as Fig. 10 for Zn.
Comparison with the loss spectra in the upper panels
clearly shows that the contribution of plural inelastic scattering becomes significant above ⬃20 eV and that the lowenergy-loss regions in the loss spectra are dominated by surface excitations. These observations emphasize the need for
spectral decomposition before information on the optical
constants can be retrieved from loss spectra.
The solid curves drawn through the data points represent
the results of a simultaneous fit of the volume and surface
single-scattering loss distributions 共DIIMFP and DSEP, respectively兲 to the theoretical expressions for these quantities,
Eqs. 共10兲 and 共13兲, using a Drude-Lorentz model for the
optical constants. For the bulk single-scattering loss distributions this procedure always leads to a very good match of the
experimental data and the fitted loss distribution. The only
exceptions to this statement are the sharp low-loss features in
the DIIMFP at about 5 eV in Cu, Ag, and Au, which almost,
but not quite,60 coincide with the surface plasmon loss feature, representing the dominant structure in the DSEP. It
should be emphasized that the fact that these features are
attributed to volume scattering 共and are therefore observable
in the DIIMFP after deconvolution兲 rather than surface excitations not only follows from a partial-intensity analysis on a
spectral pair taking into account the collision statistics, as in
Figs. 15, 19, and 24, but can also be demonstrated directly
experimentally by covering the surface with a small amount
of another material and observing the changes in the 共double兲
peak structure.60 We also note that ␧1 increases rapidly with
increasing energy loss in this region so that the conditions for
prominent surface and volume energy losses are satisfied for
similar energy-loss values.
The fit of the data for the DSEP is slightly worse than for
the DIIMFP for the same optimum values of the fitted
Drude-Lorentz parameters, in particular, for materials with a
pronounced “begrenzungs” effect that display a slight negative excursion in the DSEP at the energy of the volume plasmon. This effect is caused by the coupling of the surface and
the bulk plasmon,12,35 i.e., the polarization of the surface
dipoles leads to a decrease in the polarization for frequencies
corresponding to volume excitations. Since, per definition,
the DSEP describes the influence of the surface on the electron energy loss it consists of the “pure” surface term and the
begrenzungs term and becomes 共slightly兲 negative whenever
the magnitude of the former exceeds the latter. This effect
can be clearly identified in the present data set for Ti, V, W,
Mo, Ta, Pb, Te, and Bi. Since the absolute value of the intensity of the DSEP for these materials near the energy of the
volume plasmon is quite small, the deviation of the fit from
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1032
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
wb,s(T) (eV-1)
-1
REELS (eV )
300 eV
3400 eV
Surface (x0.16)
0
0.02
Volume
0
Im[-1/ε (ω)]
ε1 (ω)
ε1 (ω)
0
0.01
Volume
1
0
2
0
-2
0
Surface (x0.16)
0
2
0
-2
REELS
DFT
Palik
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
ε2 (ω)
3
0.03
0
0
2
ε2 (ω)
300 eV
3400 eV
-1
0.03
0
Im[-1/ε (ω)]
Pd
wb,s(T) (eV )
REELS (eV-1)
Mo
REELS
DFT
Palik
4
0
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 17. 共Color online兲 Same as Fig. 10 for Mo.
FIG. 18. 共Color online兲 Same as Fig. 10 for Pd.
the data points is nonetheless moderate. It should be noted
that for materials with a pronounced begrenzungs effect,
such as the free-electron materials, the predicted negative
excursion is satisfactorily reproduced by the deconvoluted
DSEP.35 The second reason why the fitted dielectric function
matches the volume data better than the surface data is that
the theoretical expression for the DIIMFP 关Eq. 共10兲兴 is exact,
while the model for the DSEP is based on the approximation
that the actual shape of the trajectory the electron takes has
no influence on the normalized DSEP, allowing one to integrate the effect of surface excitations over depth along a
V-shaped trajectory. This approach then gives rise to the expression in Eq. 共13兲 in the framework of the model of Tung
et al.32 Indeed, in a recent work,61 this model has been studied in detail and it was found that the assumption of simple
rectilinear surface crossing gives rise to changes in the shape
of the DSEP similar to the deviations between the fitted function and the data points observed here. However, the influence of the actual trajectory shape on the collision statistics
was found to be negligible, which implies that the basis of
the deconvolution procedure, i.e., Eq. 共16兲, retains its validity and the procedure should thus return the correct shape of
the volume single-scattering loss distribution. This is the rea-
son why the DSEP was given a relative weight of 0.01 in the
fitting procedure.
The sets of Drude-Lorentz parameters giving the best fits
of the experimental single-scattering loss distributions to the
theoretical expressions for these quantities 关Eqs. 共10兲 and
共13兲兴 is given in Appendix B. The corresponding optical constants, i.e., Im共−1 / ␧兲, ␧1, and ␧2, are displayed in the lower
three panels of Figs. 15–26. For comparison, the results of
the DFT calculations and the corresponding data taken from
Palik are also included in these figures. In general, the three
data sets agree reasonably well for all examined cases in
view of the experimental uncertainties involved in the analysis of the present experimental data and the data sets of Palik.
The important exception to this rule are the strong deviations
observed in all cases for the semicore edges, where Palik’s
data are consistently different from the other two data sets.
The DFT and REELS results show a very good mutual
agreement for Fe, Co, and Ni, while for the other materials
the DFT and REELS data for the semicore edges agree satisfactorily but differ in detail. Palik’s data, on the other hand,
always essentially deviate from the DFT and REELS results.
This can be attributed to the fact that most of Palik’s data in
this energy range have been taken from Henke’s tables2
which are based on atomic calculations.
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OPTICAL CONSTANTS FOR 17 METALS
1033
0
0.01
0
-1
REELS (eV )
0.02
0
0
-1
wb,s(T) (eV-1)
Surface (x0.16)
wb,s(T) (eV )
0.05
0
Im[-1/ε (ω)]
Te
300 eV
3400 eV
Volume
2
0
2
ε1 (ω)
ε1 (ω)
0
2
0
0
-2
REELS
DFT
Palik
ε2 (ω)
ε2 (ω)
-2
0
Volume
0
1
5
Surface (x0.16)
0.04
Im[-1/ε (ω)]
REELS (eV-1)
Ag
300 eV
2500 eV
REELS
DFT
Palik
4
0
0 10 20 30 40 50 60 70 80
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
Energy Loss ω (eV)
FIG. 19. 共Color online兲 Same as Fig. 10 for Ag.
For energy losses below the semicore edges 共 ⱗ 50 eV兲,
the three data sets are consistent within the involved errors,
although for most materials the DFT and REELS data agree
in more detail with each other than with Palik’s data. For
example, for the position of the peaks below 30 eV in the
ELF for Ag, Pd, and Au 共Figs. 19, 18, and 24兲, the DFT and
REELS data are accurately consistent, while the peaks appear a bit shifted in Palik’s data. For Fe, Co, and Ni 共Figs.
12–14兲 the strong deviations of the latter in this energy range
are particularly dramatic.
When comparing ␧1 and ␧2 instead of the ELF, similar
general observations can be made. It should be kept in mind,
however, that for the REELS data, the volume singlescattering loss function that is used in the fit procedure is
essentially governed by the ELF rather than the real and
imaginary parts of the dielectric function. Consequently,
small deviations in the retrieved ELF for small energy losses
can lead to large uncertainties in ␧1 and ␧2 共see Fig. 9兲.
Therefore, for energies below ⬃5 eV, where the loss function assumes small values, the uncertainty in the real and
imaginary parts of the dielectric function derived from
REELS becomes appreciable 共that may exceed 100% in the
case of ␧1兲. Indeed, the deviations of the REELS data from
the other data sets in this energy range generally seem to be
FIG. 20. 共Color online兲 Same as Fig. 10 for Te.
more significant than for the ELF, as expected. This is even
more apparent when the refractive index and the extinction
coefficient derived from the three data sets are plotted
against the photon wavelength, as shown in Figs. 27–32
where the REELS data deviate from the other data sets for
wavelengths above ⬃0.2 ␮m. The DFT results generally
agree quite well with Palik’s data for wavelengths ranging
from the soft x-ray regime up to the near infrared. The origin
of the rather large uncertainty in the REELS data from the
near-UV to the infrared regime 共i.e., energy losses between
0.1 and 5 eV兲 lies in the dynamic range of the REELS spectrum, i.e., the fact that tails of the elastic peak in this energyloss range assumes values comparable to the loss features
which makes their separation problematic. The only way to
improve this situation seems to be the use of a better monochromated primary beam.
An additional test of the consistency of the derived data is
provided by the sum rules for optical constants. The data
presented here were subjected to two sum rule tests: The
f-sum rule or Thomas-Reiche-Kuhn sum rule,
Zeff =
2
␲⍀2p
冕
␻max
␻ Im兵− 1/␧共␻兲其d␻ ,
共31兲
0
where ប⍀ p = 冑4␲nae2 / me is the generalized plasmon energy
and na is the atomic density. The expectation value of the
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1034
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
-1
REELS (eV )
0.03
0
500 eV
3400 eV
0.02
Volume
1
0
REELS
DFT
Palik
␻max
0
1
Im兵− 1/␧共␻兲其d␻ .
␻
REELS
DFT
Palik
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 22. 共Color online兲 Same as Fig. 10 for W.
f-sum rule is the total number of electrons per atom and
should approach Z, the atomic number, for sufficiently large
values of ␻max ⲏ Ec, where Ec is the largest binding energy
for the inner-shell levels. Therefore, the main contribution to
the f-sum rule comes from energies beyond the range of
interest here 共 ⬃ 1 – 80 eV兲. The ps-sum rule for metals, on
the other hand, emphasizes energy losses below 100 eV, and
is therefore more relevant in the present context:
冕
0
0
0 10 20 30 40 50 60 70 80
FIG. 21. 共Color online兲 Same as Fig. 10 for Ta.
2
␲
Volume
2
-2
5
Energy Loss ω (eV)
Peff =
0.02
ε2 (ω)
ε2 (ω)
-2
0
Surface (x0.16)
0
2
ε1 (ω)
ε1 (ω)
0
2
5
0
700 eV
3400 eV
0
Im[-1/ε (ω)]
Im[-1/ε (ω)]
0
0.03
0
Surface (x0.16)
-1
0
wb,s(T) (eV-1)
W
wb,s(T) (eV )
REELS (eV-1)
Ta
共32兲
The expectation value of the ps-sum rule is unity when ␻max
is significantly larger than ⬃100 eV. Since the DFT and
REELS data provide values of the optical constants over a
limited energy range 共 ⱗ 100 eV兲, they were complemented
by Henke’s data above 80 eV in order to perform the sum
rule tests.
The results are summarized in Table 2. It is seen that the
deviations from the expectation value for the f-sum rule is of
the same order of magnitude for the three data sets, of the
order of 3%–5%, but except for Pd and Pt, the REELS and
DFT data yield values of Zeff consistently closer to the
atomic number of the considered element. Since the used
optical data are essentially identical for energies above
80 eV, this suggests that the REELS and DFT data are generally superior to Palik’s data in the UV regime. As far as the
REELS data are concerned this is not surprising since electron energy-loss measurements mainly sample the UV regime. A similar conclusion can be drawn from the results for
the ps-sum rule, which is most sensitive to the UV regime,
where the REELS data exhibit deviations typically of the
order of 2% and never in excess of 4%, while Palik’s data
show much larger deviations from the expected value.
The analysis so far can be summarized by stating that the
DFT calculations for the 17 examined materials agree satisfactorily with earlier optical data available in the literature,
as compiled in Palik’s books, for wavelengths ranging from
the near infrared to the soft x-ray region. In the UV regime,
these data differ in detail, however. The DFT data exhibit
significantly more fine structure in the optical constants, a
result probably caused simply by the finite energy resolution
of the experiments and the fact that detailed optical measurements in this regime are more difficult. Indeed, the number
of earlier measurements 共or calculations which are mostly for
free atoms兲 in the UV range is generally far smaller than
measurements in the near UV to infrared range, indicating
the need for simple experimental methods that produce reliable data in the UV range.
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OPTICAL CONSTANTS FOR 17 METALS
1035
wb,s(T) (eV-1)
-1
REELS (eV )
700 eV
3400 eV
Surface (x0.16)
0
0.02
Volume
0
Im[-1/ε (ω)]
0
1
0
Surface (x0.16)
0
0.01
Volume
0
1
0
-2
REELS
DFT
Palik
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
ε2 (ω)
ε2 (ω)
-2
0
700 eV
3400 eV
0
2
ε1 (ω)
ε1 (ω)
0
2
5
0.02
-1
0.02
0
Im[-1/ε (ω)]
Au
wb,s(T) (eV )
REELS (eV-1)
Pt
5
0
REELS
DFT
Palik
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 23. 共Color online兲 Same as Fig. 10 for Pt.
FIG. 24. 共Color online兲 Same as Fig. 10 for Au.
Important deviations between DFT and Palik’s data are
mainly seen for the semicore ionization edges in the far UV
where the DFT calculations predict much more pronounced
peaks in the optical data than Palik’s data. It is believed that
this result reflects the difference between solid-state calculations, such as the present DFT calculations, and atomic calculations, such as the far UV values for the optical constants
in Palik’s data sets.
The assessment of the uncertainties associated with optical
constants derived from electron energy-loss measurements in
reflection, together with a comparison with the DFT results
and Palik’s data, leads us to conclude that the energy-loss
measurements provide accurate values 共with an uncertainty
better than 10%兲 for the optical constants in the range between the near-UV and the soft x-ray regime. Thus, such
measurements can provide useful information to further improve theoretical techniques to calculate the optical response,
in particular, for energy losses in the UV regime and beyond.
mount importance.42,62–69 In calculations of the stopping
power of charged particles in matter66–69 the relevant transport process is dominated by momentum relaxation rather
than by energy fluctuations,31 such as scattering of ions in
solids, inelastic electron backscattering, x-ray production by
electron bombardment, and emission of secondary electrons.
When energy fluctuations dominate the particle transport,
i.e., when discrete 共multiple兲 energy losses are observable in
the corresponding spectra, the quantities of relevance are the
total IMFP ␭i, the DIIMFP wb共T兲, the DSEP ws共T兲, and the
integral SEP 具ns典.
The DSEP and DIIMFP are graphically represented in the
second panel from the top in Figs. 10–26. These quantities
can also be derived from the corresponding Drude-Lorentz
parameters given in Appendix B using Eqs. 共10兲 and 共13兲.
Accurate knowledge of the DIIMFP and DSEP is essential
for a detailed analysis of the shape of a photoelectron line in
an electron spectrum as well as the inelastic background accompanying it.70,71 The inelastic background in an x-ray photoelectron spectrum is in fact similar to the REELS spectrum
recorded at the same energy, apart from the fact that the
collision statistics, i.e., the partial intensities, are generally
essentially different for an emission geometry compared to
the case of a reflection geometry. Therefore, the present op-
5.2. Inelastic electron-scattering data
Calculation of the inelastic-interaction characteristics of
charged particles with solids within the framework of linear
response theory represents an important example where optical constants, in particular, in the UV regime, are of para-
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1036
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
wb,s(T) (eV-1)
Surface (x0.16)
0
0.03
Volume
1
0.04
0
0
0.03
0
-2
1
0
-2
REELS
DFT
ε2 (ω)
REELS
DFT
ε2 (ω)
Volume
0
2
ε1 (ω)
0
2
2
0
Surface (x0.16)
0
Im[-1/ε (ω)]
0
ε1 (ω)
700 eV
3400 eV
-1
REELS (eV )
500 eV
3000 eV
-1
0.04
0
Im[-1/ε (ω)]
Bi
wb,s(T) (eV )
REELS (eV-1)
Pb
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
3
0
0 10 20 30 40 50 60 70 80
Energy Loss ω (eV)
FIG. 25. 共Color online兲 Same as Fig. 10 for Pb.
FIG. 26. 共Color online兲 Same as Fig. 10 for Bi.
tical constants in Appendixes A and B are believed to be
useful in peak-shape analysis31,70,72 where multiple inelastic
scattering needs to be eliminated from an experimental spectrum in order to reveal the intrinsic photoemission lineshape
which contains important information on the electronic structure of a solid. It should be noted that even for a very common material, such as Au, for which quite a large number of
data have been published for the optical constants 共see Ref.
73 and references therein兲, the present opical data, in particular, those for the ELF, are believed to lead to an improvement
in inelastic background analysis. This can be seen by comparing the peaks in the ELF in Fig. 24 for Au with the corresponding REELS data in the upper panel. The two interband transitions at around 25 and 35 eV in Palik’s optical
constants are clearly located at a slightly higher energy loss
than the corresponding peaks in the REELS. In the ELF and
␧2 derived from REELS, on the other hand, the location of
these peaks is obviously consistent with the REELS itself.
While the chosen example may not lead to dramatical differences in the background analysis, it is nonetheless important
to conclude that in general an analysis of REELS spectra
such as in the present work can provide, in a straightforward
way, data needed in background analysis.
The quantity of foremost importance for quantitative
analysis of electron spectra is the total IMFP. It is not only
needed to convert measured intensities of a certain element
in an electron spectrum into chemical composition,74 but is
also essential for nanoscale calibration in the depth
dimension,75 which can be established accurately by means
of data for electron-beam attenuation provided a reliable
value for the IMFP is available. Figure 33 shows a comparison of IMFP values derived from four sets of optical constants. The four materials chosen for this figure are those for
which the deviations between Palik’s data for the ELF and
the two other data sets are the largest 共Fe, Co兲 as well as Ti
and Pd for which the optical data used by Tanuma and
co-workers,42,65,68,69,76 deviate significantly from the present
results. For all other materials considered here a very good
agreement between the IMFP results from the four sets of
optical constants was found. In Fig. 33, IMFP values calculated with Penn’s algorithm63 are shown as solid lines for
energies between 100 and 10 000 eV using the optical constants calculated with DFT, those derived from REELS, and
those taken from Palik’s and Tanuma’s77 data sets. Above
80 eV the REELS and DFT data have again been complemented by Henke’s data.2 In addition, the data points represent results of EPES 共Ref. 48兲 measurements of the
IMFP.51,78 The first apparent feature of this figure is the fact
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OPTICAL CONSTANTS FOR 17 METALS
Frequency ω (eV)
10
100
Frequency ω (eV)
10
100
Cu
1037
Fe
1
10
1
10
n
n
0.1
1
0.1
1
k
k
1
0.01
0.001
10
n
0.1
1
k
0.01
0.001
0.1
0.01
Au
1
10
0.1
0.01
Co
1
10
n
0.1
1
k
0.01
0.001
0.1
0.01
Ni
1
n
10
n
0.1
1
0.1
1
k
k
0.01
0.1
0.01
0.1
REELS
DFT
Palik
0.001
0.01
Extinction coefficient k
Ag
0.01
Refractive Index n
Refractive Index n
0.001
0.1
Extinction coefficient k
0.01
REELS
DFT
Palik
0.01
0.1
Wavelength (μm)
1
FIG. 27. 共Color online兲 Comparison of values for the refractive index n and
extinction coefficient k derived from REELS 共black full curves兲 and DFT
共red dashed curves兲 with the values given in Palik’s book 共blue dotted
curves兲 as a function of the photon wavelength for Cu, Ag, and Au.
that the DFT and REELS results for the IMFP are indistinguishably similar, leading to the very important conclusion
that DFT nowadays provides accurate enough optical data to
calculate the reference length used in in-depth nanoscale
calibration by means of electron-beam attenuation. The importance of this finding can only be appropriately appreciated after familiarizing oneself with the serious problems related to IMFP measurements in the past three decades, as
outlined in an excellent review by Powell and Jablonski.48
In view of the significant differences for the ELF for Co
according to Palik’s data on one hand and the REELS and
DFT results on the other hand, in particular, in the UV range,
the difference of ⬃20% in the corresponding IMFP values
sets a rather moderate upper limit to the uncertainty of IMFP
values in use nowadays. It should be emphasized again that
for all other materials examined here a much better mutual
agreement in the IMFP values of typically better than 5% is
found.
The IMFP values derived from the REELS data have been
fitted to an equation of the form
0.001
0.01
0.01
0.1
Wavelength (μm)
1
FIG. 28. 共Color online兲 Same as Fig. 27 for Fe, Co, and Ni.
␭i共E兲 = k1E p1 + k2E p2 .
共33兲
The resulting fit parameters are given in Table 3.
As a final result of this work, values of the total surface
excitation parameter have been derived from the REELS
spectra. This was done by fitting the REELS spectra to Eq.
共15兲 using a single quantity, the zero-order partial intensity,
or the average number of surface excitations in a single surface crossing 具ns共E兲典 as a free parameter. The values for the
DIIMFP and DSEP derived from the deconvolution of
REELS spectra were used in the fitting procedure, along with
the appropriate volume partial intensities. The higher-order
surface partial intensities were calculated according to the
Poisson stochastic process. A result of the fitting procedure
for all considered energies is shown for Cu, Fe, Ag, and Au
in Fig. 34. In spite of the fact that the shape of the loss
spectra is significantly different, as a result of the energy
dependence of the shape of the elastic cross section, as discussed above, the fit is satisfactory in all cases, indicating
that the approximations involved in Eq. 共15兲 and, in particular, in Eq. 共16兲 are reasonable and lead to a detailed understanding of the electrodynamical interaction of electrons with
plane surfaces in the considered energy range.
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1038
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
Frequency ω (eV)
10
100
Frequency ω (eV)
10
100
Ti
Mo
1
10
1
10
n
n
0.1
1
0.1
1
k
k
1
0.01
0.001
10
n
0.1
1
k
0.01
0.001
0.1
0.01
Zn
1
10
0.1
1
10
n
0.1
1
k
0.01
0.001
0.1
0.01
W
1
n
10
n
0.1
1
0.1
k
1
k
0.01
0.1
0.01
REELS
DFT
Palik
0.001
0.01
0.01
Ta
Extinction coefficient k
V
0.01
Refractive Index n
Refractive Index n
0.001
0.1
Extinction coefficient k
0.01
0.01
0.1
Wavelength (μm)
0.1
REELS
DFT
Palik
1
0.001
0.01
0.01
0.1
Wavelength (μm)
1
FIG. 29. 共Color online兲 Same as Fig. 27 for Ti, V, and Zn.
FIG. 30. 共Color online兲 Same as Fig. 27 for Mo, Ta, and W.
For each energy, such a fit gives the zero-order surface
partial intensity, being equal to the average number of surface excitations for a single surface crossing, 具ns共E兲典. The
energy dependence of this quantity is compared with results
from other sources in Fig. 35. The filled circles are the
present results, and the solid lines represent a fit of the energy dependence of the zero-order partial-intensity Eq. 共20兲
using the surface excitation parameter as as a free parameter.
For all studied cases, the energy dependence of the zeroorder partial intensity is seen to be well described by Eq.
共20兲. Therefore, the magnitude of surface excitations can be
accurately predicted by using the values of the surface excitation parameter given in Table 4, in combination with Eq.
共20兲. The dashed curves in Fig. 35 were derived in a very
similar way in Ref. 43 but using Palik’s data to calculate the
DIIMFP and DSEP. The resulting fit was also considerably
worse than the ones shown in Fig. 34. Thus the differences in
the resulting values of the zero-order partial intensities are
attributed to the systematic error caused by using Palik’s
data. The present results for the surface excitation parameter
are therefore believed to be significantly more reliable than
the earlier ones. The dashed-dotted lines are theoretical calculations by Chen79 using Palik’s optical data as input.
6. Summary
Two new sets of optical constants have been presented for
16 elemental metals 共Ti, V, Fe, Co, Ni, Cu, Zn, Mo, Pd, Ag,
Ta, W, Pt, Au, Pb, and Bi兲 and one semimetal 共Te兲 and are
compared with existing data in the literature. Values are
given for the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant as well as the ELF 共Im兵−1 / ␧其兲.
These values have been calculated from the Drude-Lorentz
expansion coefficients derived from experimental results for
the DIIMFP 共and DSEP兲 which are also given in graphical
and tabular forms. The method to determine such data using
DFT calculations has been given and a detailed analysis of
the experimental determination of optical constants from
REELS spectra is also described, along with an analysis of
the reliability of the results. The latter analysis showed that
accurate results can be expected for optical constants ranging
from the visible to the soft x-ray regime of wavelengths. In
principle such measurements can also provide optical constants in the infrared regime; however, in this case the use of
a monochromated electron source is mandatory since the
thermal spread in the source energy distribution cannot be
adequately eliminated numerically. The agreement between
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OPTICAL CONSTANTS FOR 17 METALS
Frequency ω (eV)
10
100
Frequency ω (eV)
10
100
Pd
1039
Te
1
10
1
n
10
n
1
0.1
1
k
0.01
Pt
1
10
n
0.1
1
k
0.01
0.1
Extinction coefficient k
Refractive Index n
0.001
k
0.1
Refractive Index n
0.01
0.01
0.001
0.1
0.01
Bi
1
10
n
0.1
1
k
0.01
0.001
0.1
REELS
DFT
Palik
0.01
Pb
0.001
0.01
1
Extinction coefficient k
0.1
0.01
0.1
1
Wavelength (μm)
10
n
0.1
FIG. 32. 共Color online兲 Same as Fig. 27 for Te and Bi.
1
k
0.01
0.1
REELS
DFT
Palik
0.001
0.01
共iii兲
0.01
0.1
Wavelength (μm)
1
FIG. 31. 共Color online兲 Same as Fig. 27 for Pd, Pt, and Pb.
the DFT and REELS results is satisfactory for all studied
materials, while larger deviations are observed with Palik’s
data.
The experimental data for the optical constants were derived from a particular pair of energy-loss spectra for each
material, selected according to outlined criteria that the sequence of partial intensities be sufficiently different for the
two spectra while at the same time avoiding measurements
where the combination of primary energy and scattering
angle corresponding to a deep minimum in the elasticscattering cross section. A detailed consistency check for
many different pairs of spectra with arbitrary energy combinations demonstrates that this is essential to appropriately
separate surface and bulk contributions contained in an
energy-loss spectrum. Further consistency checks also support other key assumptions that need to be made in order to
perform the analysis:
共i兲
共ii兲
The collision statistics 共partial intensities兲 for surface
and bulk scattering are uncorrelated.
Calculation of the volume partial intensities using
elastic-scattering cross sections for free atoms on the
basis of a Boltzmann-type kinetic equation 共i.e., ne-
glecting coherent scattering and diffraction兲 is adequate for the present purpose.
The functional form of the energy and angular dependence of the surface-scattering probability is appropriately described by Eq. 共20兲 rather than by Eq. 共30兲.
Plural surface scattering is described by the Poisson
stochastic process.
TABLE 2. Results of sum-rule checks on the three sets of optical data discussed in the present work
f
sum
Ti
V
Fe
Co
Ni
Cu
Zn
Mo
Pd
Ag
Te
Ta
W
Pt
Au
Pb
Bi
22
23
26
27
28
29
30
42
46
47
52
73
74
78
79
82
83
ps
sum
REELS
DFT
Palik
REELS
DFT
Palik
20.8
22.3
25.9
27.3
29.9
30.6
16.2
40.1
43.0
46.6
55.4
69.9
71.5
74.4
74.5
79.9
80.8
20.7
22.0
25.2
26.7
27.8
29.5
29.4
40.3
43.2
44.5
53.7
70.9
72.7
76.0
75.2
77.9
72.6
–
21.0
24.0
24.2
26.7
28.3
–
39.7
45.6
50.6
69.1
70.5
77.7
75.2
75.0
–
–
1.023
1.029
1.012
1.035
1.000
1.037
1.021
1.022
1.026
1.018
1.027
1.041
1.039
1.028
1.008
1.024
1.018
0.979
0.990
1.007
1.016
1.017
1.016
1.034
0.991
0.997
1.005
0.991
1.007
1.012
1.012
0.992
1.009
1.003
–
1.016
0.944
0.845
1.011
0.995
–
1.013
1.143
1.148
1.378
1.025
0.986
1.105
1.107
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
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1040
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 3. Fit parameters in Eq. 共33兲 describing the IMFP values derived
from the REELS measurements
IMFP (Å)
Ti
REELS
EPES
10
DFT
Tanuma
IMFP (Å)
Fe
REELS
EPES
Palik
DFT
Tanuma
10
Ti
V
Fe
Co
Ni
Cu
Zn
Mo
Pd
Ag
Te
Ta
W
Pt
Au
Pb
Bi
k1
p1
k2
p2
0.046
0.043
0.043
0.042
0.041
0.040
0.042
0.038
0.045
0.048
0.050
0.035
0.034
0.039
0.044
0.048
0.054
0.85
0.85
0.84
0.83
0.83
0.84
0.86
0.85
0.83
0.82
0.87
0.85
0.84
0.83
0.83
0.85
0.84
4.2
6.1
16.9
22.3
24.0
14.1
4.7
3.5
15.7
29.3
12.5
3.9
5.0
10.5
14.0
16.2
13.6
−0.17
−0.23
−0.40
−0.45
−0.45
−0.34
−0.11
−0.13
−0.42
−0.52
−0.36
−0.10
−0.16
−0.31
−0.35
−0.42
−0.35
IMFP (Å)
Co
REELS
EPES
Palik
DFT
Tanuma
10
The uncertainty in the experimental data has been analyzed and is estimated to amount to about 10% for the ELF
3400
3000
0.08
2500
2000
1500
1000
700
500 eV
Pd
IMFP (Å)
0.04
1000
Energy (eV)
10000
FIG. 33. 共Color online兲 Comparison of IMFP values for Ti, Fe, Co, and Pd
taken from different sources.
共iv兲
共v兲
The shapes of the normalized DIIMFP and DSEP are
independent of the energy of the reflected electrons
and the surface crossing direction to a good approximation.
The dispersion of the characteristic frequency of any
electronic transition, as well as the dispersion of the
involved initial and final states, can appropriately be
described by a single dispersion parameter ␣ in a
Drude-Lorentz expansion of the dielectric constant
using oscillator energies of the form ␻i共q兲 = ␻i
+ ␣q2 / 2. This follows from the fact that the results
show no pronounced variation around a reasonable
choice of this parameter of ␣ = 0.5.
3000
2500
2000
1500
1000
700
0.04
500 eV
-1
100
3400
0.08
REELS (eV )
10
Cu
0
0.12
REELS
EPES
Palik
DFT
Tanuma
Ag
0
3400
3000
0.08
2500
2000
1500
0.04
1000
700
500 eV
Au
0
3400
3000
0.08
2500
0.04
0
Fit
Raw
0
2000
1500
1000
700
500 eV
20
40
60
Energy loss (eV)
Fe
80
FIG. 34. 共Color online兲 Comparison of REELS spectra with fitted spectra for
determination of the surface excitation parameter.
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
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OPTICAL CONSTANTS FOR 17 METALS
First order reduced surface scattering partial intensity
1
0.1
1
Cu
This work
Fit
Chen
Werner et al. Surf. Sci
Ag
0.1
1
Au
0.1
1
Fe
0.1
100
Energy (eV)
10000
FIG. 35. First-order reduced surface partial intensities 共data points兲 derived
from the REELS spectra. Solid line: fit of these data to Eq. 共20兲; dashed line:
results of Ref. 43 based on Oswald’s energy and angular dependence of the
average number of surface excitations, Eq. 共30兲; dashed-dotted line: theoretical results by Chen 共Ref. 79兲 based on the optical constants of Palik 共Ref.
73兲.
1041
may exceed 100%, which is a consequence of the fact that
energy-loss measurements mainly sample the absorptive part
of the dielectric constant. Below ⬃3 – 5 eV, the uncertainty
in the present REELS data 共for ␧1 as well as for ␧2兲 is additionally increased as a result of the large dynamic range of
REELS experiments, leading to difficulties in the separation
of the elastic peak from the energy-loss portion of the spectra. The accuracy of the DFT results is difficult to estimate
quantitatively, but the comparison with the experimental results and Palik’s data shows that for energies below 10 eV,
where, for the reasons mentioned above, the experimental
data are known to be less reliable, the agreement between
DFT and Palik’s data is comparable to the agreement of all
three data sets at higher energies.
The above may be summarized by stating that the REELS
data provide optical constants with an experimental accuracy
of about 10% for energies between the near-ultraviolet regime and the 共far兲 soft x-ray regime of energies, while the
DFT results provide similarly reliable data for energies ranging from the infrared to the soft-x-ray regime.
Electron-scattering data derived from the optical constants
have also been compared with literature values, again showing very good agreement. In particular, values for the electron IMFP derived from REELS and DFT show a very good
consistency and agree better than 5% for the majority of all
cases studied for energies between 50 and 5000 eV. An important conclusion in this context is that nanoscale calibration by means of electron-beam attenuation is nowadays possible on the basis of DFT calculations without the need for
an experimental calibration of the electron attenuation
length.
7. Acknowledgments
and ␧2 as well as for ␧1 for energies above 10 eV. For energies below 10 eV, the uncertainty in ␧1 in the present data
TABLE 4. Values for the surface excitation parameter as in units of aNFE
= 2.896 eV−1/2 derived from the REELS measurements, compared with the
theoretical values of Chen 共Ref. 79兲
Metal
as
Chen 共Ref. 79兲
Ti
V
Fe
Co
Ni
Cu
Zn
Mo
Pd
Ag
Te
Ta
W
Pt
Au
Pb
Bi
2.8
3.3
3.8
3.8
4.1
4.3
4.8
3.9
2.9
3.4
3.8
3.3
3.4
2.7
2.6
3.1
3.2
–
–
2.51
–
–
2.45
–
–
–
2.34
–
–
–
–
3.06
–
–
Support by the Austrian Science Foundation FWF 共Project
Nos. P16227 and P20891-N20兲 is gratefully acknowledged.
8. Appendix A: Deconvolution Algorithm
For REELS Spectra
In the present work, multiple-scattering contributions in
REELS spectra were eliminated by the deconvolution procedure equation 共23兲. Calculation of the coefficients needed
using this Padé expansion is very tedious as described in
Refs. 8 and 44. A simplified deconvolution procedure was
recently developed45 which leads to the same results and
may be used for the same purpose while the numerical effort
is dramatically reduced. Therefore, for completeness, a synopsis of this simplified deconvolution procedure is given below.
The simplified algorithm for retrieval of the DIIMFP and
DSEP can be summarized as follows: Let y 1共T兲 and y 2共T兲 be
the measured spectra with the reduced volume partial intensities ␣n and ␤n, respectively. In the first step, the intermediate spectra y ⴱ1共T兲 and y ⴱ2共T兲 are derived by
y ⴱ1共T兲 = y 1共T兲 −
冕
T
y 1共T − T⬘兲y ⴱ1共T⬘兲dT⬘ ,
0
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WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
冕
T
y 2共T − T⬘兲y ⴱ2共T⬘兲dT⬘ .
-1
The DIIMFP and DSEP can then be retrieved by means of
the series expansion
ⴱ
ⴱ
丢 y 1共T兲 + u02y 2 丢 y 2共T兲,
共A2兲
where the expansion coefficients are different for the surface
and bulk 共subscripts “s” and “b,” below兲 single-scattering
loss distributions. To bring the expressions for the expansion
coefficients into a more tractable form, it is useful to introduce the shorthand notation
1
冉
冉
␮=
冑E 1
␯=
1
冑E 2
1
cos
1
␪i1
cos
␪i2
+
+
1
冊
冊
cos ␪o1
1
cos
␪02
,
␮l
,
l!
␯l
␤k,l = ␤k
l!
where the subscript “k” in ␣k and ␤k indicate the fact that
these quantities represent the bulk partial intensities, the expressions for the expansion coefficients read
⌬ = ␣ 1␯ − ␤ 1␮ ,
ub10 = ␯/⌬,
ub01 = − ␮/⌬,
us10 = − ␤1/⌬,
共A3兲
us01 = ␣1/⌬,
冉
冎
冊
␣21
␣2
␮
2
2
+ ␣ 1␤ 1 − ␣ 2 ␮ ␯ − 1 ␯ 2 ,
3 共 ␤ 2 − ␤ 1兲 ␮ +
2
2
⌬
␮␯
ub11 = 3 兵␤21␮ − 2␤2␮ − ␣21␯ + 2␣2␯其,
⌬
␤21
␯
+ ␣ 1␤ 1 − ␤ 2 ␯ ␮
ub20 = − 3 共␣2 − ␣21兲␯2 +
2
⌬
再
−
us02 =
冉
冊
␤21 2
␮ ,
2
再冉
冎
共A4兲
冋
1
␣ 1␤ 1
␮2 ␣2␤1 + ␣1 ␤21 − ␤2 −
2
⌬3
+ ␯2
us11 =
冎
0
0
␣31
,
2
␮␯
兵␤1共␣21 − 2␣2 − ␣1␤1兲 + 2␣1␤2其,
⌬3
册冊
− ␮␯␣21␤1
2nd order, truncated
2nd order, full
Padé
10 20 30 40 50 60 70 80
energy loss T (eV)
FIG. 36. 共Color online兲 Comparison of the Padé approximation 关Eq. 共23兲,
dots兴 for the DIIMFP and DSEP for Pt with the second-order truncated 关Eq.
共A6兲, red curves兴 and the full second-order 关Eq. 共A2兲, blue curves兴
expansion.
s
u20
=−
␣k,l = ␣k
再
Volume
0.02
,
where the indices “1” and “2” refer to the two spectra y 1共T兲
and y 2共T兲 and the superscript i and o indicate the ingoing and
outgoing polar direction of surface crossing. Furthermore, by
writing
ub02 =
Surface (x0.16)
0.04
s
w共T兲 = u10y ⴱ1共T兲 + u01y ⴱ2共T兲 + u11y ⴱ1 丢 y ⴱ2共T兲 + u20y ⴱ1
ⴱ
Pt
共A1兲
0
b
y ⴱ2共T兲 = y 2共T兲 −
w (T), w (T) (eV )
1042
再冉
冎
冋
1
␣ 1␤ 1
2
2
3 ␯ ␣ 1␤ 2 + ␤ 1 ␣ 1 − ␣ 2 −
2
⌬
+ ␮2
册冊
␤31
.
2
− ␮␯␤21␣1
共A5兲
Equation 共A2兲 can also be truncated after the mixed
second-order term, giving for the deconvoluted singlescattering loss distributions
w共T兲 = u10y ⴱ1共T兲 + u01y ⴱ2共T兲 + u11y ⴱ1 丢 y ⴱ2共T兲.
共A6兲
In this case, the expansion coefficients for the mixed secondorder term are given by
␮ ␯ 共 ␣ 1 − ␤ 1兲
b
=
,
u11
共 ␣ 1␯ 兲 2 − 共 ␤ 1␮ 兲 2
␣ 1␤ 1共 ␯ − ␮ 兲
s
=
,
共A7兲
u11
共 ␣ 1␯ 兲 2 − 共 ␤ 1␮ 兲 2
while u01 and u10 are still given by Eq. 共A3兲. The above
deconvolution scheme 关Eqs. 共A1兲, 共A6兲, and 共A7兲兴 is extremely simple and yet accurate enough for most practical
purposes.
In Fig. 36 the red curves are the single-scattering loss
distributions for Pt obtained by means of Eq. 共A6兲 that are
compared with the results for the DIIMFP and DSEP obtained with the second-order Padé approximation equation
共23兲, calculating the coefficients in the usual way.44 Considering the complexity of the calculation of the former Padé
coefficients in comparison to the simplicity of the coefficients ukl, the agreement between the resulting singlescattering loss functions is remarkable. The agreement can
be even further improved by taking into account the quadratic terms in the intermediate spectra, leading to the full
second-order expansion 关blue curves, Eq. 共A2兲兴. The latter is
also shown in these figures and is seen to agree excellently
with the results of the Padé approximation.
9. Appendix B: Values of the Drude-Lorentz
Parameters for the Dielectric Function
This appendix contains values of the Drude-Lorentz parameters for the dielectric function as given in Tables 5–10.
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
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OPTICAL CONSTANTS FOR 17 METALS
TABLE 5. Drude-Lorentz parameters for Ti 共E3p1/2 = 32.6 eV兲, V 共E3p3/2
= 37.3 eV兲, and Fe 共E3p1/2 = 52.7 eV兲
Ti
␻i
共eV兲
␥i
共eV兲
0.000 0.027
1.0
0.5
3.7
1.2
5.5
3.0
9.9
10.5
22.8
10.8
37.2
6.5
41.7
5.1
47.5
4.1
80.6
37.1
V
␻i
共eV兲
Ai
共eV2兲
116.1
7.8
22.3
36.5
197.8
28.7
112.6
116.5
49.8
0.1
␥i
共eV兲
0.000 0.027
1.0
9.8
3.6
3.6
8.2
7.3
11.5
1.5
15.4
11.1
40.1
4.0
46.3
8.9
54.1
3.3
82.1
11.8
TABLE 8. Drude-Lorentz parameters for Ag 共E4p3/2 = 58.3 eV兲, Te 共E4d5/2
= 40.4 eV兲, and Ta 共E4f7/2 = 21.6 eV兲
Fe
␻i
共eV兲
Ai
共eV2兲
143.3
0.2
73.6
108.3
14.2
145.9
106.8
214.8
29.6
95.7
␥i
共eV兲
1043
Ag
Ai
共eV2兲
0.000
8.0
0.049
1.1
0.5 168.5
3.9
2.0
31.5
8.9
10.5 225.2
19.0
8.4
58.1
32.8
148.6
19.1
44.0
231.9 1278.6
53.1
2.1 150.1
63.3
16.8 246.8
TABLE 6. Drude-Lorentz parameters for Co 共E3p1/2 = 58.9 eV兲, Ni 共E3p3/2
= 66.2 eV兲, and Cu 共E3p3/2 = 75.1 eV兲
Te
␻i
共eV兲
␥i
共eV兲
Ai
共eV2兲
0.000
1.0
4.9
10.3
13.2
21.2
30.2
43.0
65.7
0.065
93.8
0.7
22.0
6.3
3.3
3.7
16.7
38.8
98.4
103.9
23.8
318.3
89.6
63.5
43.3
313.3
519.9
␻i
共eV兲
␻i
共eV兲
␥i
共eV兲
0.000 0.027
1.0
0.5
4.0
3.6
10.1
12.8
15.1
32.3
20.8
7.7
26.1
1.2
58.6
1.4
66.1
37.1
Ni
␻i
共eV兲
Ai
共eV2兲
133.4
19.0
71.8
133.3
407.5
38.3
3.2
91.8
645.3
␥i
共eV兲
Cu
Ai
共eV2兲
0.000
0.027
32.0
1.0
0.5
127.3
3.9
1.6
44.7
6.2
7.9
111.2
7.7
2.3
13.3
13.6
10.8
237.3
22.9
8.8
83.7
33.2
0.9
4.6
40.5
32.4
160.1
63.4
1.9
86.2
79.6
109.5
1843.5
␻i
共eV兲
0.000
1.0
1.0
4.0
4.8
7.2
14.2
16.9
24.3
59.4
␥i
共eV兲
Ai
共eV2兲
0.027
86.3
0.5
19.8
0.5
21.9
1.7
30.2
0.5
2.5
11.3
157.7
10.1
130.9
68.2
108.6
3.0
40.0
55.1
1022.4
␻i
共eV兲
␥i
共eV兲
Mo
Ai
共eV2兲
0.000
0.8
85.9
0.9
0.5
0.002
4.2
0.5
3.6
5.9
1.4
9.5
10.4
3.5
14.3
23.6
26.4
16.1
35.7
114.7 1257.7
␻i
共eV兲
␥i
共eV兲
0.000 0.027
1.0
0.5
5.9
4.9
11.8
1.0
14.0
7.3
21.7
9.6
34.7
0.5
36.6
1.9
39.2
3.4
43.0
6.1
46.6
0.5
49.4
6.6
56.4
20.6
Ai
共eV2兲
␻i
共eV兲
␥i
共eV兲
Ai
共eV2兲
139.3
138.2
161.8
8.6
92.5
6.5
12.3
108.4
180.6
411.2
TABLE 9. Drude-Lorentz parameters for W 共E4f7/2 = 31.4 eV兲, Pt 共E5p3/2
= 51.7 eV兲, and Au 共E5p3/2 = 57.2 eV兲
Pt
Au
␻i
共eV兲
␥i
共eV兲
Ai
共eV2兲
␻i
共eV兲
␥i
共eV兲
Ai
共eV2兲
␻i
共eV兲
␥i
共eV兲
Ai
共eV2兲
0.000
1.0
2.7
3.9
5.0
7.0
11.1
16.7
22.9
38.4
41.1
48.5
80.2
0.027
0.5
0.9
0.5
0.8
2.5
6.0
7.7
0.6
3.1
0.9
8.9
65.1
168.0
54.3
47.4
25.5
33.4
37.9
164.2
125.1
2.9
98.4
11.1
221.9
495.5
0.000
3.5
4.5
9.0
9.1
9.7
18.2
18.7
26.4
28.2
38.9
39.2
49.6
53.5
72.4
0.2
0.8
0.9
0.5
11.5
0.5
4.9
36.1
55.5
4.0
6.0
25.0
0.5
5.7
47.6
179.1
16.6
7.5
2.1
367.4
1.0
121.3
18.5
486.5
36.9
12.1
3.3
14.5
87.2
421.4
0.000
4.0
7.3
12.8
18.9
19.9
28.9
38.7
64.3
0.2
1.5
3.3
11.8
71.0
2.9
3.9
13.0
51.9
113.1
44.6
54.8
184.9
728.1
65.7
50.0
74.7
544.0
TABLE 7. Drude-Lorentz parameters for Zn 共E3d5/2 = 10.1 eV兲, Mo 共E4p3/2
= 35.5 eV兲, and Pd 共E4p3/2 = 50.9 eV兲
Zn
␥i
共eV兲
0.000
0.027 102.3 0.000 0.027
1.0
2.3
19.8 1.0
7.9
6.6
3.7
92.5 5.4
7.2
9.0
405.0
0.1 9.5
0.5
13.4
5.6
44.2 13.7
7.5
39.7
0.9
29.4 18.3
1.9
48.9
0.5
3.5 25.5
3.6
59.4
4.2
95.8 36.8
4.9
72.4
8.6
236.1 45.1
7.6
86.0
10.8
177.8 70.4
70.0
W
Co
Ta
Pd
Ai
共eV2兲
177.0
114.1
88.8
15.4
201.5
56.5
6.9
55.4
102.0
94.9
5.7
73.9
99.9
␻i
共eV兲
␥i
共eV兲
0.000 0.027
3.7
2.6
6.1
1.9
12.1
7.4
18.8
5.3
23.7
5.2
29.9
7.5
38.9
5.7
44.0
3.7
52.0
10.6
54.8
0.5
64.6
15.3
Ai
共eV2兲
138.9
59.9
8.3
281.0
121.6
32.8
119.3
68.3
40.7
217.3
3.4
176.4
TABLE 10. Drude-Lorentz parameters for Pb 共E5d5/2 = 18.1 eV兲 and Bi
共E5d5/2 = 23.8 eV兲
Pb
Bi
␻i
共eV兲
␥i
共eV兲
Ai
共eV2兲
␻i
共eV兲
␥i
共eV兲
Ai
共eV2兲
0.000
1.9
8.7
19.5
24.4
35.9
37.9
45.4
59.9
0.4
0.5
4.2
2.1
4.8
10.9
48.6
9.7
21.6
161.7
6.3
64.5
66.1
42.9
126.8
0.6
120.1
400.8
0.000
3.9
6.6
11.3
24.1
27.5
40.2
55.8
84.2
0.087
1.8
4.5
6.2
0.6
1.9
9.6
21.7
48.4
108.4
23.6
50.2
44.8
6.6
13.2
99.3
326.8
373.1
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1044
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 11. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ti calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Ti
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
5.812
6.799
9.039
9.622
3.674
−1.163
−3.570
−4.749
−4.804
−3.433
−2.755
−3.440
−4.892
−5.655
−8.458
−9.576
−8.658
−7.080
−5.583
−3.065
−0.546
−1.970
−3.447
−2.567
−1.733
−0.470
0.601
−0.283
−0.443
−0.661
−0.970
−1.108
−0.958
−0.731
−0.498
−0.368
−0.322
−0.230
−0.165
−0.069
−0.084
0.008
0.103
0.187
0.258
0.322
0.386
0.444
0.509
0.538
0.570
0.617
51.273
49.843
39.345
38.053
39.479
33.160
27.125
23.926
19.609
17.267
16.906
16.717
15.876
14.873
14.037
9.965
6.784
4.875
3.795
2.833
3.494
6.058
3.264
2.263
1.244
1.195
1.972
2.461
2.571
2.409
2.266
1.716
1.317
1.055
0.966
0.892
0.829
0.759
0.683
0.651
0.550
0.447
0.388
0.342
0.321
0.279
0.261
0.238
0.228
0.218
0.201
0.163
0.019
0.020
0.024
0.025
0.025
0.030
0.036
0.040
0.048
0.056
0.058
0.057
0.058
0.059
0.052
0.052
0.056
0.066
0.083
0.163
0.279
0.149
0.145
0.193
0.273
0.724
0.464
0.401
0.378
0.386
0.373
0.411
0.497
0.641
0.818
0.959
1.048
1.206
1.383
1.519
1.777
2.234
2.408
2.250
1.891
1.540
1.204
0.938
0.732
0.648
0.551
0.399
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−447.600
−189.661
−109.799
−74.201
−49.772
−34.555
−24.626
−17.761
−12.773
−9.016
−6.197
−4.395
−4.024
−4.737
−4.994
−4.481
−3.749
−3.120
−2.686
−2.292
−2.099
−1.825
−1.509
−1.235
−1.034
−0.902
−0.821
−0.773
−0.739
−0.709
−0.674
−0.632
−0.581
−0.523
−0.458
−0.390
−0.319
−0.248
−0.178
−0.108
−0.041
0.023
0.084
0.142
0.196
0.247
0.294
0.338
0.377
0.413
0.445
0.474
28.594
16.723
19.374
9.230
4.557
3.018
2.496
2.422
2.638
3.144
4.025
5.352
6.719
6.945
6.007
5.080
4.570
4.367
4.306
4.170
3.796
3.366
3.039
2.822
2.670
2.545
2.422
2.290
2.146
1.993
1.836
1.681
1.531
1.391
1.262
1.145
1.040
0.947
0.864
0.792
0.728
0.673
0.625
0.584
0.548
0.518
0.492
0.470
0.450
0.433
0.417
0.402
0.000
0.000
0.002
0.002
0.002
0.003
0.004
0.008
0.016
0.034
0.074
0.112
0.110
0.098
0.098
0.111
0.131
0.152
0.167
0.184
0.202
0.230
0.264
0.297
0.326
0.349
0.370
0.392
0.417
0.445
0.480
0.521
0.571
0.630
0.700
0.782
0.878
0.988
1.110
1.240
1.368
1.484
1.571
1.617
1.616
1.572
1.497
1.404
1.306
1.210
1.121
1.040
0.0
0.0
0.1
0.1
0.1
−0.2
−0.4
−0.9
−1.2
−0.6
1.1
2.9
4.0
4.3
4.3
4.2
4.4
5.0
6.0
7.7
7.6
7.7
9.7
12.0
12.1
11.5
12.4
14.1
15.0
14.8
15.6
18.5
20.4
20.0
20.1
22.2
24.8
26.9
28.4
29.3
30.9
34.4
38.2
38.8
37.2
37.7
39.0
37.4
34.6
32.8
30.9
29.3
1.2
1.9
2.5
3.1
3.7
4.8
8.3
17.5
33.5
51.6
62.7
63.8
60.2
56.6
55.3
56.2
58.3
59.9
60.4
60.3
63.7
66.4
62.1
56.8
59.3
63.6
63.0
61.3
63.8
70.7
73.8
69.9
65.7
69.0
75.2
72.8
63.1
55.6
54.0
53.5
47.5
36.5
23.9
18.4
22.2
20.8
11.3
8.6
13.8
15.4
13.9
13.6
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1045
TABLE 11. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ti calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ti
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.670
0.721
0.769
0.814
0.859
0.898
0.933
0.969
1.008
1.046
1.090
1.133
1.185
1.241
1.305
1.383
1.479
1.585
1.719
1.915
2.204
2.771
4.014
2.453
0.520
0.340
0.677
0.936
0.640
−1.133
−1.064
−0.581
−0.324
−0.142
−0.021
0.065
0.179
0.306
0.441
0.570
0.415
0.393
0.377
0.385
0.416
0.453
0.497
0.514
0.517
0.529
0.543
0.563
0.147
0.136
0.129
0.124
0.121
0.124
0.123
0.119
0.117
0.112
0.107
0.103
0.098
0.093
0.089
0.085
0.091
0.102
0.108
0.123
0.154
0.254
1.617
3.220
3.144
1.949
1.662
1.835
2.725
2.471
0.861
0.568
0.408
0.327
0.272
0.213
0.139
0.114
0.124
0.252
0.336
0.307
0.262
0.211
0.169
0.143
0.133
0.145
0.141
0.124
0.110
0.102
0.312
0.252
0.212
0.183
0.161
0.151
0.139
0.125
0.114
0.101
0.089
0.079
0.069
0.060
0.052
0.044
0.041
0.040
0.036
0.033
0.032
0.033
0.086
0.197
0.310
0.498
0.516
0.432
0.348
0.334
0.460
0.860
1.502
2.573
3.652
4.298
2.702
1.065
0.589
0.649
1.178
1.237
1.242
1.093
0.841
0.632
0.503
0.510
0.491
0.422
0.359
0.311
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.501
0.526
0.549
0.573
0.596
0.620
0.644
0.669
0.694
0.720
0.747
0.773
0.800
0.827
0.855
0.882
0.910
0.938
0.967
0.995
1.024
1.052
1.079
1.103
1.124
1.138
1.143
1.136
1.113
1.076
1.029
0.982
0.942
0.915
0.898
0.884
0.864
0.825
0.760
0.675
0.588
0.520
0.482
0.473
0.484
0.506
0.534
0.560
0.578
0.579
0.558
0.519
0.387
0.372
0.357
0.341
0.325
0.310
0.296
0.282
0.270
0.259
0.250
0.243
0.238
0.234
0.232
0.233
0.235
0.240
0.249
0.260
0.275
0.295
0.320
0.351
0.389
0.433
0.484
0.538
0.591
0.635
0.665
0.679
0.680
0.676
0.677
0.690
0.715
0.749
0.777
0.780
0.747
0.684
0.607
0.533
0.471
0.426
0.397
0.385
0.388
0.402
0.414
0.409
0.966
0.897
0.831
0.767
0.705
0.646
0.589
0.535
0.487
0.443
0.404
0.370
0.341
0.316
0.296
0.279
0.266
0.256
0.249
0.246
0.245
0.247
0.253
0.262
0.275
0.292
0.314
0.341
0.372
0.407
0.443
0.476
0.504
0.523
0.536
0.549
0.569
0.604
0.658
0.733
0.827
0.927
1.010
1.050
1.034
0.973
0.897
0.834
0.802
0.809
0.858
0.936
28.6
27.7
25.5
23.6
22.5
21.0
18.6
17.5
18.2
17.1
14.9
14.3
14.1
13.1
12.0
11.2
10.3
9.4
9.3
9.7
9.1
8.0
7.7
8.4
9.3
9.2
8.9
9.2
8.7
8.3
9.5
9.7
8.3
8.3
10.3
11.5
10.8
10.4
12.0
13.6
12.7
12.3
13.7
15.8
16.8
16.4
15.6
15.7
17.3
18.4
16.8
15.3
10.1
6.1
8.1
10.0
8.3
8.0
11.0
11.5
6.4
4.8
7.2
6.7
6.0
5.2
3.2
4.1
8.1
9.1
5.5
2.1
2.9
6.5
6.4
2.9
1.2
3.6
4.3
2.8
5.4
6.9
2.7
2.0
7.7
9.4
3.3
−0.5
3.0
5.7
1.8
−0.3
4.8
7.1
2.9
−1.2
−0.6
2.2
4.5
6.2
4.6
0.9
3.6
7.3
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1046
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 11. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ti calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ti
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.574
0.590
0.600
0.608
0.627
0.643
0.662
0.661
0.667
0.680
0.692
0.706
0.706
0.714
0.725
0.731
0.736
0.747
0.757
0.759
0.758
0.763
0.771
0.779
0.784
0.791
0.799
0.807
0.809
0.812
0.809
0.812
0.819
0.824
0.825
0.827
0.829
0.833
0.835
0.832
0.834
0.837
0.841
0.844
0.847
0.850
0.096
0.089
0.085
0.072
0.063
0.060
0.064
0.064
0.057
0.049
0.047
0.048
0.049
0.043
0.040
0.041
0.037
0.034
0.038
0.042
0.039
0.033
0.029
0.028
0.028
0.025
0.025
0.028
0.032
0.035
0.036
0.029
0.028
0.031
0.032
0.031
0.031
0.031
0.033
0.032
0.028
0.025
0.024
0.023
0.022
0.022
0.285
0.251
0.232
0.192
0.160
0.144
0.145
0.146
0.127
0.105
0.097
0.096
0.098
0.083
0.076
0.077
0.069
0.060
0.065
0.073
0.067
0.056
0.049
0.047
0.045
0.041
0.039
0.043
0.049
0.053
0.055
0.045
0.042
0.046
0.047
0.046
0.045
0.044
0.047
0.046
0.040
0.036
0.034
0.032
0.030
0.030
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.480
0.458
0.455
0.466
0.484
0.504
0.525
0.545
0.564
0.582
0.598
0.613
0.627
0.640
0.652
0.663
0.674
0.684
0.693
0.702
0.710
0.718
0.726
0.733
0.739
0.746
0.752
0.758
0.764
0.769
0.774
0.779
0.784
0.789
0.793
0.797
0.801
0.805
0.809
0.813
0.816
0.820
0.823
0.826
0.830
0.833
0.377
0.328
0.277
0.232
0.196
0.167
0.145
0.128
0.114
0.102
0.093
0.084
0.078
0.072
0.067
0.062
0.058
0.055
0.051
0.049
0.046
0.044
0.041
0.039
0.038
0.036
0.034
0.033
0.031
0.030
0.029
0.028
0.027
0.026
0.025
0.024
0.023
0.022
0.022
0.021
0.020
0.020
0.019
0.018
0.018
0.017
1.012
1.035
0.976
0.856
0.718
0.592
0.489
0.407
0.343
0.293
0.253
0.221
0.195
0.173
0.155
0.140
0.127
0.116
0.106
0.098
0.091
0.084
0.078
0.073
0.068
0.064
0.060
0.057
0.054
0.051
0.048
0.046
0.044
0.041
0.039
0.038
0.036
0.034
0.033
0.032
0.030
0.029
0.028
0.027
0.026
0.025
16.3
17.2
17.0
17.1
17.1
16.1
15.0
15.6
16.6
15.3
12.8
11.0
10.5
11.0
10.4
9.2
8.3
7.3
7.1
7.6
8.0
7.7
6.4
5.5
5.2
4.7
5.5
6.4
5.3
4.2
4.2
4.6
3.9
2.0
1.1
2.4
3.2
3.1
3.5
4.0
3.1
0.8
0.5
2.6
3.0
1.0
5.4
3.0
1.2
1.3
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1047
TABLE 12. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of V calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
V
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−209.448
−80.837
−35.526
−13.731
−0.230
5.941
5.912
3.891
5.740
7.038
4.130
−4.903
−11.641
−13.812
−15.732
−14.030
−12.191
−9.981
−8.197
−5.954
−3.185
−3.799
−4.655
−3.820
−3.038
−2.275
−1.480
−0.522
0.437
−0.077
0.014
0.563
0.120
−0.170
−0.340
−0.405
−0.364
−0.290
−0.207
−0.127
−0.069
−0.069
−0.587
−0.493
−0.340
−0.186
−0.097
−0.078
0.047
0.130
0.226
0.217
52.784
21.302
14.875
12.682
14.499
19.382
23.941
22.714
22.225
25.386
30.735
32.995
26.728
21.725
16.148
11.544
8.189
6.249
5.364
3.761
3.171
6.461
3.741
2.521
1.771
1.235
0.860
0.662
1.571
1.867
1.681
1.795
2.396
2.243
2.089
1.840
1.645
1.506
1.416
1.376
1.390
1.559
1.343
0.974
0.778
0.685
0.666
0.566
0.511
0.468
0.474
0.503
0.001
0.003
0.010
0.036
0.069
0.047
0.039
0.043
0.042
0.037
0.032
0.030
0.031
0.033
0.032
0.035
0.038
0.045
0.056
0.076
0.157
0.115
0.105
0.120
0.143
0.184
0.293
0.932
0.591
0.535
0.595
0.507
0.416
0.443
0.466
0.518
0.579
0.640
0.691
0.721
0.718
0.640
0.625
0.817
1.080
1.360
1.470
1.735
1.939
1.984
1.719
1.677
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−562.518
−245.332
−134.086
−82.588
−54.671
−37.947
−27.262
−20.172
−15.399
−12.203
−10.098
−8.720
−7.781
−7.069
−6.452
−5.866
−5.294
−4.738
−4.210
−3.275
−2.534
−1.987
−1.608
−1.352
−1.170
−1.017
−0.865
−0.695
−0.505
−0.312
−0.231
−0.538
−0.765
−0.680
−0.551
−0.450
−0.378
−0.328
−0.292
−0.264
−0.239
−0.213
−0.185
−0.152
−0.116
−0.076
−0.033
0.011
0.056
0.102
0.148
0.193
32.050
10.631
5.797
4.446
4.209
4.441
4.904
5.476
6.061
6.563
6.892
6.987
6.843
6.507
6.056
5.568
5.098
4.680
4.326
3.808
3.490
3.294
3.152
3.015
2.859
2.684
2.506
2.348
2.241
2.235
2.401
2.502
2.139
1.824
1.651
1.544
1.464
1.392
1.321
1.247
1.170
1.091
1.012
0.934
0.859
0.788
0.722
0.662
0.607
0.557
0.512
0.472
0.000
0.000
0.000
0.001
0.001
0.003
0.006
0.013
0.022
0.034
0.046
0.056
0.064
0.070
0.077
0.085
0.094
0.106
0.119
0.151
0.188
0.223
0.252
0.276
0.300
0.326
0.357
0.392
0.425
0.439
0.413
0.382
0.415
0.481
0.545
0.597
0.640
0.681
0.722
0.767
0.820
0.883
0.957
1.044
1.144
1.258
1.382
1.511
1.634
1.737
1.802
1.816
0.1
0.1
0.1
0.2
0.2
−0.1
−0.4
−1.2
−2.5
−3.7
−3.5
−1.7
0.4
1.7
2.2
2.3
2.4
2.8
3.3
4.9
6.1
7.1
8.2
9.4
10.6
11.8
12.2
12.4
13.6
14.9
14.8
13.7
13.9
14.9
15.1
16.2
18.5
19.1
18.4
19.9
21.8
21.5
21.8
24.9
27.7
27.6
27.5
29.7
32.5
35.2
36.7
35.5
0.6
1.0
1.3
1.6
1.9
2.7
5.1
11.9
25.2
42.7
54.5
54.8
49.2
44.4
42.8
44.2
46.9
49.6
51.2
51.5
51.9
54.5
55.5
55.0
55.2
57.5
61.5
63.7
61.8
59.4
62.3
68.4
68.8
65.5
65.7
65.0
58.5
57.4
63.2
58.9
49.9
52.4
56.1
46.3
34.5
35.3
38.3
30.9
21.7
14.1
8.2
10.1
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1048
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 12. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of V calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
V
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.201
0.297
0.349
0.389
0.427
0.424
0.485
0.522
0.542
0.588
0.644
0.702
0.760
0.799
0.840
0.871
0.913
0.959
0.998
1.034
1.065
1.106
1.161
1.212
1.256
1.296
1.377
1.497
1.657
1.901
2.447
1.990
2.003
2.531
2.556
1.250
−0.517
−0.751
−0.934
−0.840
−0.454
−0.230
−0.095
0.006
0.097
0.175
0.243
0.308
0.363
0.446
0.549
0.645
0.425
0.364
0.354
0.330
0.326
0.294
0.244
0.250
0.211
0.170
0.146
0.132
0.133
0.137
0.138
0.139
0.134
0.137
0.145
0.153
0.161
0.153
0.158
0.172
0.191
0.185
0.159
0.152
0.160
0.188
0.358
1.571
1.007
1.331
2.668
3.425
3.353
2.460
1.523
0.786
0.518
0.417
0.359
0.301
0.254
0.218
0.189
0.170
0.145
0.126
0.144
0.359
1.922
1.647
1.432
1.268
1.131
1.104
0.827
0.746
0.623
0.453
0.334
0.259
0.224
0.209
0.191
0.178
0.157
0.146
0.143
0.140
0.139
0.122
0.115
0.115
0.118
0.108
0.083
0.067
0.058
0.051
0.058
0.244
0.200
0.163
0.195
0.258
0.291
0.372
0.477
0.594
1.091
1.840
2.605
3.322
3.433
2.788
1.992
1.374
0.950
0.584
0.446
0.658
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.237
0.279
0.321
0.361
0.400
0.438
0.474
0.510
0.544
0.577
0.609
0.640
0.670
0.700
0.729
0.757
0.786
0.814
0.842
0.870
0.898
0.927
0.956
0.986
1.017
1.050
1.083
1.119
1.156
1.194
1.232
1.268
1.295
1.302
1.270
1.180
1.037
0.889
0.785
0.737
0.729
0.739
0.753
0.764
0.765
0.757
0.737
0.708
0.672
0.634
0.598
0.567
0.436
0.404
0.375
0.349
0.326
0.306
0.288
0.272
0.257
0.245
0.233
0.224
0.215
0.208
0.202
0.197
0.193
0.191
0.190
0.189
0.191
0.193
0.198
0.204
0.213
0.225
0.240
0.260
0.286
0.321
0.366
0.427
0.508
0.610
0.731
0.844
0.907
0.893
0.821
0.737
0.669
0.626
0.604
0.598
0.603
0.614
0.624
0.630
0.627
0.613
0.590
0.557
1.772
1.674
1.538
1.383
1.223
1.072
0.935
0.814
0.711
0.623
0.549
0.487
0.435
0.390
0.353
0.322
0.296
0.273
0.255
0.239
0.226
0.216
0.208
0.201
0.197
0.195
0.195
0.197
0.202
0.210
0.222
0.239
0.262
0.295
0.340
0.401
0.478
0.563
0.637
0.678
0.684
0.668
0.648
0.636
0.635
0.646
0.669
0.702
0.742
0.789
0.836
0.881
34.5
35.5
35.7
33.4
30.5
28.9
27.5
24.2
21.4
20.7
19.6
17.6
16.1
15.4
14.6
12.8
11.0
10.0
9.7
10.3
10.6
9.0
8.1
9.2
9.3
8.2
7.7
7.6
6.4
4.8
4.8
5.7
5.8
6.1
7.4
8.1
7.9
7.7
8.1
9.2
10.4
10.4
9.9
10.6
11.5
10.9
10.5
11.7
12.9
12.8
12.4
12.9
12.8
5.6
−1.4
2.4
10.5
11.4
8.8
14.0
18.4
15.4
14.5
16.7
16.7
15.3
14.9
17.3
19.5
20.0
19.3
14.0
8.0
10.1
13.1
8.4
4.6
7.5
7.8
4.3
7.0
13.5
13.0
7.6
6.4
7.4
5.9
3.1
1.3
3.3
6.5
3.7
−1.5
−1.3
1.4
0.6
−0.7
2.4
3.4
0.4
−0.3
0.7
0.6
0.5
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1049
TABLE 12. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of V calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
V
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.433
0.443
0.446
0.465
0.480
0.492
0.504
0.521
0.542
0.568
0.579
0.597
0.615
0.612
0.624
0.634
0.635
0.645
0.653
0.671
0.687
0.700
0.708
0.717
0.729
0.736
0.737
0.737
0.743
0.751
0.746
0.756
0.757
0.761
0.751
0.741
0.748
0.764
0.768
0.771
0.774
0.777
0.788
0.799
0.793
0.794
0.304
0.269
0.222
0.200
0.182
0.166
0.151
0.129
0.112
0.110
0.106
0.099
0.102
0.110
0.095
0.094
0.089
0.078
0.069
0.059
0.059
0.062
0.062
0.062
0.063
0.071
0.074
0.075
0.070
0.075
0.072
0.071
0.074
0.079
0.083
0.068
0.052
0.049
0.052
0.048
0.049
0.040
0.036
0.046
0.047
0.042
1.087
1.003
0.895
0.780
0.691
0.613
0.545
0.448
0.366
0.330
0.305
0.271
0.262
0.284
0.237
0.228
0.217
0.185
0.161
0.130
0.124
0.125
0.122
0.120
0.117
0.130
0.134
0.137
0.125
0.132
0.128
0.123
0.128
0.135
0.145
0.124
0.092
0.084
0.088
0.081
0.082
0.066
0.058
0.071
0.075
0.067
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.545
0.532
0.527
0.529
0.537
0.550
0.566
0.585
0.603
0.619
0.627
0.616
0.586
0.552
0.534
0.536
0.548
0.565
0.582
0.599
0.616
0.630
0.644
0.657
0.669
0.681
0.692
0.702
0.711
0.720
0.729
0.738
0.745
0.753
0.760
0.768
0.774
0.781
0.787
0.794
0.800
0.806
0.811
0.817
0.822
0.828
0.520
0.480
0.440
0.403
0.370
0.341
0.319
0.303
0.295
0.297
0.308
0.323
0.327
0.304
0.263
0.222
0.188
0.163
0.143
0.128
0.116
0.107
0.099
0.092
0.086
0.081
0.076
0.073
0.069
0.066
0.063
0.061
0.059
0.057
0.055
0.054
0.053
0.052
0.051
0.050
0.050
0.049
0.049
0.049
0.049
0.049
0.916
0.935
0.934
0.911
0.870
0.815
0.755
0.699
0.654
0.629
0.632
0.667
0.726
0.766
0.742
0.660
0.561
0.471
0.399
0.341
0.297
0.261
0.232
0.208
0.189
0.172
0.158
0.146
0.135
0.126
0.118
0.112
0.105
0.100
0.095
0.091
0.088
0.084
0.082
0.079
0.077
0.075
0.074
0.073
0.072
0.072
13.2
12.9
14.0
14.9
13.6
12.9
14.1
14.8
14.3
13.8
13.9
13.5
12.3
12.2
13.2
13.0
12.1
12.2
12.2
11.6
12.2
12.4
9.9
8.1
9.0
9.7
8.3
6.8
6.8
6.8
6.0
5.6
5.2
5.2
6.4
6.1
4.1
3.2
3.9
5.5
6.5
5.3
3.1
2.6
4.4
5.6
2.5
5.6
2.3
−2.4
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1050
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 13. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Fe calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Fe
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−121.135
−44.593
−21.687
−12.071
−6.571
−2.809
−1.020
0.299
−0.271
−3.785
−7.452
−9.978
−12.033
−11.333
−9.111
−7.145
−5.305
−4.149
−3.737
−3.006
−0.875
−2.031
−2.930
−2.009
−1.017
0.015
0.244
0.236
−0.275
−0.401
−0.214
−0.014
−0.064
−0.202
−0.254
−0.236
−0.198
−0.155
−0.089
−0.005
0.083
0.170
0.261
0.361
0.438
0.474
0.371
0.235
0.217
0.162
0.068
0.049
68.667
43.976
35.929
29.833
25.884
23.737
22.815
22.856
24.683
24.655
22.970
19.331
15.646
10.479
7.753
6.103
5.326
5.558
5.352
4.113
3.585
6.103
3.661
2.515
1.957
2.088
2.619
2.981
2.997
2.562
2.275
2.336
2.316
2.216
2.036
1.857
1.707
1.569
1.439
1.334
1.255
1.197
1.159
1.153
1.200
1.305
1.383
1.318
1.264
1.229
1.158
1.015
0.004
0.011
0.020
0.029
0.036
0.042
0.044
0.044
0.041
0.040
0.039
0.041
0.040
0.044
0.054
0.069
0.094
0.116
0.126
0.158
0.263
0.148
0.166
0.243
0.402
0.479
0.378
0.333
0.331
0.381
0.436
0.428
0.431
0.448
0.484
0.530
0.578
0.631
0.692
0.750
0.793
0.819
0.821
0.790
0.736
0.677
0.674
0.735
0.768
0.800
0.860
0.983
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
157.124
186.611
158.602
−85.798
−102.565
−69.923
−47.877
−33.944
−24.733
−18.384
−13.936
−10.913
−9.058
−8.074
−7.491
−6.907
−6.204
−5.443
−4.705
−3.456
−2.551
−1.933
−1.529
−1.275
−1.120
−1.026
−0.962
−0.909
−0.856
−0.798
−0.732
−0.659
−0.581
−0.499
−0.416
−0.334
−0.254
−0.179
−0.108
−0.045
0.009
0.053
0.087
0.109
0.122
0.128
0.132
0.135
0.143
0.155
0.172
0.194
35.710
90.339
242.890
233.642
89.018
40.072
22.358
14.618
10.850
8.988
8.183
7.985
8.014
7.874
7.355
6.583
5.807
5.169
4.692
4.112
3.824
3.670
3.562
3.458
3.336
3.189
3.021
2.837
2.646
2.457
2.275
2.104
1.949
1.809
1.685
1.577
1.484
1.406
1.340
1.287
1.243
1.208
1.177
1.147
1.116
1.079
1.036
0.986
0.932
0.876
0.821
0.768
0.001
0.002
0.003
0.004
0.005
0.006
0.008
0.011
0.015
0.021
0.031
0.044
0.055
0.062
0.067
0.072
0.080
0.092
0.106
0.143
0.181
0.213
0.237
0.255
0.269
0.284
0.301
0.320
0.342
0.368
0.398
0.433
0.471
0.514
0.559
0.607
0.655
0.700
0.741
0.776
0.804
0.826
0.845
0.864
0.886
0.914
0.950
0.995
1.048
1.107
1.167
1.224
−0.1
−0.1
−0.2
−0.2
−0.2
−0.2
−0.4
−0.8
−1.4
−1.8
−1.3
−0.2
0.7
0.9
0.7
0.5
0.5
0.8
1.5
3.1
4.5
5.7
6.1
6.1
7.3
8.5
8.4
8.4
9.0
9.6
10.3
11.0
11.3
12.1
12.7
12.7
14.2
16.5
17.2
17.3
18.0
19.1
19.7
19.4
19.3
19.9
20.4
20.8
20.7
21.3
23.2
24.0
0.9
1.4
1.8
2.3
2.7
2.6
3.9
8.1
16.7
29.0
40.2
46.1
48.9
51.2
54.6
59.0
63.8
68.1
71.3
74.2
74.6
76.4
81.6
85.8
83.8
79.9
78.8
77.4
75.3
73.3
69.9
67.4
67.7
65.2
63.7
67.0
64.2
54.8
50.2
50.5
49.9
45.2
39.6
37.9
38.1
36.8
33.7
30.8
31.5
30.7
24.3
22.1
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1051
TABLE 13. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Fe calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Fe
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.070
0.084
0.127
0.208
0.272
0.308
0.386
0.342
0.333
0.357
0.418
0.459
0.491
0.469
0.523
0.538
0.550
0.569
0.573
0.576
0.587
0.623
0.637
0.664
0.678
0.672
0.680
0.697
0.709
0.717
0.725
0.744
0.749
0.738
0.731
0.743
0.746
0.757
0.775
0.789
0.796
0.811
0.826
0.837
0.859
0.873
0.896
0.920
0.942
0.956
0.976
0.997
0.914
0.818
0.717
0.651
0.635
0.589
0.606
0.624
0.568
0.493
0.479
0.448
0.474
0.439
0.413
0.426
0.404
0.407
0.393
0.389
0.349
0.339
0.329
0.323
0.346
0.328
0.318
0.305
0.308
0.301
0.295
0.291
0.304
0.305
0.283
0.271
0.256
0.238
0.231
0.223
0.218
0.204
0.202
0.190
0.183
0.179
0.172
0.170
0.177
0.178
0.177
0.174
1.087
1.210
1.353
1.395
1.329
1.334
1.173
1.233
1.309
1.330
1.184
1.088
1.017
1.063
0.930
0.904
0.869
0.831
0.814
0.805
0.750
0.674
0.641
0.593
0.598
0.586
0.564
0.527
0.515
0.498
0.481
0.455
0.466
0.479
0.461
0.433
0.412
0.378
0.353
0.332
0.320
0.291
0.279
0.258
0.237
0.225
0.206
0.194
0.193
0.189
0.180
0.170
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.220
0.248
0.278
0.308
0.338
0.368
0.397
0.425
0.452
0.478
0.503
0.527
0.550
0.572
0.593
0.613
0.632
0.650
0.668
0.685
0.702
0.718
0.733
0.748
0.762
0.776
0.790
0.804
0.817
0.830
0.842
0.855
0.867
0.879
0.891
0.903
0.916
0.928
0.940
0.953
0.965
0.979
0.992
1.006
1.020
1.036
1.052
1.069
1.087
1.107
1.129
1.154
0.718
0.673
0.632
0.594
0.561
0.531
0.504
0.480
0.457
0.437
0.419
0.403
0.388
0.374
0.361
0.349
0.338
0.328
0.319
0.310
0.302
0.294
0.287
0.281
0.275
0.269
0.264
0.259
0.254
0.250
0.246
0.242
0.239
0.236
0.233
0.230
0.228
0.226
0.225
0.223
0.222
0.222
0.221
0.221
0.222
0.223
0.225
0.227
0.231
0.235
0.241
0.248
1.272
1.308
1.326
1.326
1.307
1.272
1.225
1.168
1.106
1.042
0.978
0.916
0.857
0.801
0.750
0.702
0.658
0.618
0.582
0.548
0.517
0.489
0.463
0.440
0.418
0.398
0.380
0.363
0.347
0.333
0.319
0.307
0.295
0.284
0.274
0.265
0.256
0.248
0.240
0.233
0.226
0.220
0.214
0.209
0.204
0.199
0.194
0.190
0.187
0.183
0.180
0.178
23.4
23.5
24.3
24.7
24.5
23.6
23.3
23.5
22.8
21.8
21.0
19.6
18.6
18.7
18.4
17.5
16.7
15.8
14.9
14.7
13.7
12.5
12.3
11.9
11.3
11.2
10.8
9.8
9.7
9.7
9.4
9.6
9.9
9.0
7.7
7.1
7.1
7.0
6.8
6.7
6.1
5.1
5.2
6.3
5.9
4.7
4.8
5.1
4.3
3.9
4.4
4.5
25.0
25.4
21.3
16.0
14.6
16.0
13.6
10.1
11.6
12.7
12.5
13.3
13.0
9.8
7.8
8.3
9.4
11.0
10.9
8.4
9.5
12.6
12.2
11.9
12.7
10.7
9.8
11.7
11.5
10.3
10.2
8.4
6.1
8.0
11.6
12.4
11.4
10.9
10.6
9.3
9.8
12.6
11.7
7.5
8.5
12.2
10.9
7.8
9.7
11.9
8.9
6.9
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1052
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 13. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Fe calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Fe
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
1.027
1.060
1.104
1.156
1.225
1.323
1.487
1.728
1.921
1.768
1.146
−0.010
−0.002
0.010
0.203
0.302
0.369
0.420
0.466
0.506
0.543
0.576
0.619
0.677
0.757
0.672
0.712
0.701
0.671
0.675
0.689
0.694
0.693
0.698
0.701
0.700
0.700
0.704
0.711
0.719
0.730
0.736
0.734
0.738
0.735
0.741
0.170
0.169
0.170
0.174
0.179
0.195
0.224
0.453
0.820
1.514
1.848
1.834
0.856
0.572
0.419
0.349
0.298
0.257
0.224
0.198
0.177
0.154
0.135
0.129
0.218
0.199
0.180
0.243
0.214
0.192
0.181
0.182
0.178
0.170
0.168
0.162
0.152
0.143
0.133
0.126
0.122
0.127
0.124
0.122
0.117
0.109
0.157
0.147
0.136
0.127
0.117
0.109
0.099
0.142
0.188
0.280
0.391
0.545
1.168
1.748
1.933
1.641
1.326
1.058
0.839
0.671
0.543
0.434
0.337
0.271
0.351
0.405
0.334
0.442
0.432
0.390
0.356
0.354
0.348
0.329
0.323
0.314
0.297
0.276
0.254
0.237
0.223
0.228
0.225
0.218
0.211
0.194
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
1.182
1.213
1.250
1.293
1.345
1.408
1.483
1.565
1.622
1.534
1.093
0.531
0.314
0.332
0.409
0.485
0.549
0.600
0.641
0.673
0.698
0.718
0.733
0.744
0.751
0.755
0.756
0.756
0.753
0.749
0.744
0.739
0.733
0.728
0.723
0.718
0.714
0.711
0.709
0.708
0.708
0.708
0.710
0.711
0.714
0.716
0.258
0.270
0.288
0.312
0.347
0.399
0.482
0.620
0.858
1.228
1.546
1.399
1.012
0.728
0.563
0.470
0.416
0.384
0.365
0.354
0.348
0.346
0.345
0.347
0.348
0.350
0.352
0.352
0.352
0.351
0.348
0.344
0.338
0.332
0.324
0.315
0.305
0.295
0.285
0.275
0.264
0.254
0.244
0.234
0.225
0.216
0.176
0.175
0.175
0.176
0.180
0.186
0.198
0.219
0.255
0.318
0.431
0.625
0.901
1.137
1.163
1.031
0.878
0.757
0.671
0.612
0.572
0.545
0.526
0.515
0.508
0.506
0.505
0.507
0.509
0.512
0.515
0.518
0.519
0.518
0.516
0.512
0.506
0.498
0.488
0.476
0.463
0.448
0.433
0.418
0.402
0.386
3.5
3.0
3.9
4.5
3.1
2.3
2.9
4.0
4.3
3.4
3.1
4.3
5.6
6.5
7.4
8.2
9.0
9.5
9.3
9.0
9.4
9.6
9.3
8.6
8.7
8.9
8.5
8.3
8.0
8.3
9.3
9.2
8.2
7.4
6.9
7.1
7.5
7.3
7.3
7.6
7.4
7.3
7.9
7.8
7.1
7.0
10.5
11.9
6.6
5.0
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1053
TABLE 14. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Co calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Co
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−107.960
−21.595
7.463
−2.923
−8.233
−8.132
−14.932
−15.453
−14.114
−11.590
−9.627
−8.842
−9.154
−8.455
−7.090
−5.789
−4.495
−3.055
−2.177
−1.739
−0.747
−3.518
−2.770
−1.747
−0.833
−0.445
−0.338
−0.170
0.170
0.180
−0.024
0.177
0.141
0.033
−0.018
0.015
0.111
0.173
0.263
0.320
0.373
0.455
0.457
0.486
0.474
0.441
0.405
0.375
0.353
0.341
0.326
0.316
51.266
33.538
45.855
54.923
43.145
40.200
33.796
26.390
19.689
16.011
13.889
13.049
11.116
8.557
6.923
5.745
4.929
4.560
5.233
4.616
6.263
5.088
3.164
2.384
2.161
2.412
2.381
2.199
2.123
2.434
2.292
2.121
2.176
2.105
1.904
1.778
1.614
1.543
1.476
1.436
1.386
1.404
1.402
1.394
1.425
1.411
1.390
1.357
1.312
1.272
1.232
1.190
0.004
0.021
0.021
0.018
0.022
0.024
0.025
0.028
0.034
0.041
0.049
0.053
0.054
0.059
0.071
0.086
0.111
0.151
0.163
0.190
0.157
0.133
0.179
0.273
0.403
0.401
0.412
0.452
0.468
0.409
0.436
0.468
0.458
0.475
0.525
0.562
0.617
0.640
0.657
0.663
0.673
0.645
0.645
0.640
0.632
0.646
0.663
0.685
0.711
0.734
0.759
0.785
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−501.815
−203.650
−119.311
−91.209
−61.840
−42.714
−30.446
−22.251
−16.616
−12.702
−10.014
−8.222
−7.070
−6.335
−5.833
−5.427
−5.042
−4.648
−4.245
−3.453
−2.747
−2.163
−1.702
−1.348
−1.079
−0.878
−0.726
−0.609
−0.517
−0.442
−0.377
−0.319
−0.266
−0.215
−0.166
−0.118
−0.071
−0.026
0.018
0.061
0.102
0.140
0.175
0.206
0.232
0.251
0.264
0.271
0.271
0.268
0.265
0.265
36.182
27.709
42.469
22.506
10.778
7.132
5.951
5.691
5.842
6.183
6.574
6.902
7.079
7.056
6.841
6.480
6.043
5.589
5.160
4.444
3.931
3.574
3.321
3.132
2.981
2.848
2.725
2.606
2.489
2.373
2.258
2.147
2.039
1.937
1.840
1.749
1.665
1.587
1.517
1.453
1.395
1.345
1.300
1.262
1.228
1.199
1.172
1.144
1.113
1.078
1.036
0.990
0.000
0.001
0.003
0.003
0.003
0.004
0.006
0.011
0.019
0.031
0.046
0.060
0.071
0.078
0.085
0.091
0.098
0.106
0.116
0.140
0.171
0.205
0.238
0.269
0.297
0.321
0.343
0.364
0.385
0.407
0.431
0.456
0.482
0.510
0.539
0.569
0.600
0.630
0.659
0.687
0.713
0.736
0.755
0.772
0.786
0.799
0.812
0.828
0.848
0.874
0.906
0.943
0.0
−0.1
−0.1
−0.1
−0.1
−0.2
−0.3
−0.6
−1.0
−1.1
−0.2
1.4
2.4
2.5
2.2
1.8
1.6
1.5
1.5
2.7
5.0
6.0
5.9
6.5
7.4
8.1
8.8
9.0
8.9
9.4
10.3
10.9
11.1
11.7
12.6
13.7
14.9
14.7
13.7
14.3
16.0
16.6
16.3
17.2
17.8
16.7
17.0
18.7
18.5
17.1
17.6
19.5
0.7
1.1
1.4
1.8
2.1
2.0
3.2
7.1
15.7
29.5
43.2
50.8
53.0
53.3
54.4
57.6
62.3
67.5
71.5
72.8
69.7
74.4
84.7
89.3
89.0
86.8
82.9
80.9
79.8
75.3
69.1
65.8
64.5
62.4
61.1
58.5
52.6
50.5
52.7
50.5
44.3
40.3
38.4
35.0
34.7
38.1
34.2
26.5
27.8
33.9
33.5
26.6
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1054
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 14. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Co calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Co
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.304
0.286
0.253
0.276
0.281
0.253
0.191
0.196
0.228
0.264
0.268
0.263
0.315
0.346
0.365
0.426
0.463
0.472
0.471
0.495
0.509
0.535
0.562
0.572
0.581
0.581
0.583
0.583
0.593
0.613
0.621
0.625
0.633
0.650
0.672
0.685
0.700
0.719
0.726
0.732
0.741
0.749
0.752
0.757
0.768
0.780
0.793
0.800
0.811
0.821
0.817
0.820
1.157
1.121
1.062
1.003
0.994
0.977
0.901
0.808
0.737
0.700
0.673
0.601
0.554
0.528
0.482
0.452
0.462
0.472
0.441
0.424
0.404
0.388
0.385
0.388
0.382
0.380
0.369
0.354
0.328
0.317
0.313
0.301
0.281
0.265
0.256
0.252
0.242
0.245
0.248
0.243
0.239
0.238
0.237
0.226
0.219
0.212
0.213
0.211
0.209
0.215
0.218
0.205
0.808
0.837
0.891
0.927
0.932
0.959
1.062
1.170
1.239
1.251
1.283
1.397
1.365
1.326
1.320
1.170
1.081
1.058
1.059
0.999
0.958
0.889
0.829
0.812
0.790
0.789
0.775
0.761
0.715
0.664
0.647
0.625
0.586
0.537
0.495
0.473
0.441
0.424
0.422
0.409
0.395
0.385
0.381
0.362
0.344
0.324
0.316
0.308
0.298
0.298
0.305
0.286
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.269
0.279
0.294
0.313
0.337
0.364
0.396
0.427
0.420
0.372
0.392
0.424
0.452
0.478
0.501
0.523
0.543
0.562
0.580
0.598
0.615
0.631
0.646
0.661
0.676
0.690
0.704
0.717
0.729
0.742
0.754
0.766
0.777
0.788
0.799
0.809
0.820
0.830
0.840
0.849
0.859
0.868
0.877
0.886
0.894
0.903
0.911
0.919
0.927
0.935
0.943
0.951
0.941
0.892
0.844
0.799
0.759
0.724
0.699
0.692
0.714
0.662
0.594
0.554
0.526
0.503
0.483
0.465
0.449
0.433
0.420
0.407
0.395
0.384
0.373
0.363
0.354
0.346
0.338
0.331
0.324
0.318
0.312
0.307
0.302
0.298
0.294
0.290
0.286
0.283
0.281
0.278
0.277
0.275
0.273
0.272
0.272
0.271
0.271
0.271
0.271
0.272
0.273
0.274
0.982
1.021
1.057
1.084
1.100
1.102
1.084
1.046
1.041
1.148
1.173
1.139
1.093
1.045
0.997
0.950
0.904
0.860
0.818
0.778
0.740
0.704
0.670
0.638
0.609
0.581
0.555
0.531
0.509
0.488
0.469
0.451
0.435
0.419
0.405
0.392
0.380
0.369
0.358
0.349
0.340
0.332
0.324
0.317
0.311
0.305
0.300
0.295
0.291
0.287
0.283
0.280
20.3
19.8
19.6
20.0
20.9
21.1
20.7
20.9
20.9
20.4
20.5
21.3
21.9
21.5
19.8
18.6
18.4
18.0
17.5
17.3
16.9
15.9
15.1
15.1
14.6
13.7
13.3
12.7
11.5
10.6
10.6
11.3
11.5
10.1
8.8
9.0
9.8
9.9
9.2
8.5
8.3
7.7
7.5
8.2
7.8
6.7
6.4
6.5
6.6
6.4
5.6
5.6
21.9
22.5
24.0
22.9
18.3
16.1
17.4
15.3
12.7
13.1
13.7
10.6
4.2
2.4
8.4
11.3
7.8
6.6
6.7
5.2
4.8
6.3
6.2
4.5
5.5
7.1
6.3
6.0
7.7
10.4
10.3
5.2
2.1
6.6
11.7
10.1
4.9
2.9
4.7
5.5
5.6
8.9
10.0
5.6
5.2
8.2
7.8
6.5
5.7
6.6
9.9
8.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1055
TABLE 14. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Co calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Co
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.834
0.839
0.847
0.853
0.862
0.872
0.886
0.898
0.911
0.924
0.935
0.960
0.985
1.021
1.061
1.115
1.191
1.321
1.640
1.582
1.392
0.086
0.148
0.425
0.466
0.506
0.548
0.580
0.602
0.623
0.631
0.652
0.685
0.721
0.741
0.776
0.735
0.759
0.731
0.719
0.719
0.723
0.716
0.715
0.721
0.728
0.203
0.200
0.199
0.194
0.189
0.184
0.181
0.178
0.178
0.176
0.171
0.162
0.159
0.158
0.160
0.163
0.171
0.188
0.287
0.978
1.355
1.798
0.482
0.430
0.377
0.325
0.281
0.258
0.236
0.222
0.201
0.175
0.159
0.156
0.159
0.212
0.203
0.204
0.221
0.205
0.190
0.186
0.178
0.163
0.150
0.139
0.275
0.269
0.263
0.254
0.243
0.232
0.221
0.213
0.207
0.199
0.189
0.171
0.159
0.148
0.139
0.129
0.118
0.106
0.103
0.283
0.359
0.555
1.897
1.175
1.050
0.899
0.742
0.641
0.566
0.509
0.459
0.384
0.321
0.287
0.277
0.327
0.348
0.330
0.379
0.366
0.343
0.333
0.327
0.303
0.276
0.253
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.959
0.967
0.975
0.983
0.992
1.000
1.009
1.019
1.029
1.040
1.053
1.067
1.083
1.103
1.127
1.158
1.198
1.253
1.328
1.421
1.445
0.995
0.403
0.365
0.455
0.533
0.589
0.629
0.658
0.679
0.695
0.707
0.716
0.722
0.727
0.731
0.733
0.735
0.736
0.737
0.737
0.737
0.737
0.737
0.737
0.737
0.276
0.277
0.279
0.282
0.284
0.287
0.290
0.294
0.298
0.302
0.308
0.314
0.321
0.331
0.343
0.360
0.385
0.425
0.500
0.655
0.998
1.417
1.117
0.721
0.537
0.453
0.409
0.385
0.369
0.358
0.351
0.344
0.339
0.335
0.330
0.326
0.322
0.317
0.313
0.308
0.304
0.299
0.294
0.289
0.284
0.278
0.277
0.274
0.271
0.269
0.267
0.265
0.263
0.261
0.260
0.258
0.256
0.254
0.252
0.249
0.247
0.245
0.243
0.243
0.248
0.268
0.323
0.473
0.792
1.104
1.083
0.925
0.795
0.707
0.648
0.607
0.578
0.557
0.541
0.528
0.518
0.509
0.501
0.495
0.489
0.483
0.478
0.472
0.467
0.461
0.455
0.449
6.2
6.1
5.7
5.8
5.9
5.9
5.9
5.4
5.1
5.6
5.8
5.4
5.1
4.7
4.5
5.0
5.5
4.9
3.7
3.2
3.9
5.3
5.3
4.4
5.4
7.4
7.6
7.0
8.5
10.2
8.7
6.8
7.3
8.0
7.6
7.1
7.4
7.9
8.0
7.1
6.0
5.9
6.7
7.4
7.2
7.0
4.4
3.5
5.2
6.6
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1056
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 15. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ni calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Ni
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−158.422
−61.253
−38.025
−26.202
−18.963
−14.840
−14.053
−13.486
−10.674
−8.490
−7.375
−6.702
−5.608
−4.513
−3.434
−2.354
−1.142
0.609
1.483
−1.126
−3.203
−2.102
−1.139
−0.447
−0.417
−0.352
−0.351
−0.173
−0.036
−0.150
0.065
0.291
0.377
0.416
0.505
0.653
0.587
0.491
0.431
0.288
0.216
0.193
0.194
0.211
0.234
0.246
0.257
0.272
0.313
0.371
0.434
0.498
79.503
52.551
43.332
32.429
26.763
23.273
21.248
15.632
12.483
10.728
9.830
8.230
6.892
5.867
5.090
4.507
4.098
4.338
6.323
8.154
5.108
3.420
2.926
2.955
3.033
2.862
2.633
2.354
2.399
2.127
1.856
1.816
1.847
1.803
1.741
1.777
1.926
1.916
1.908
1.862
1.727
1.602
1.491
1.397
1.319
1.257
1.189
1.106
1.025
0.964
0.925
0.905
0.003
0.008
0.013
0.019
0.025
0.031
0.033
0.037
0.046
0.057
0.065
0.073
0.087
0.107
0.135
0.174
0.226
0.226
0.150
0.120
0.141
0.212
0.297
0.331
0.324
0.344
0.373
0.423
0.417
0.468
0.538
0.537
0.520
0.527
0.530
0.496
0.475
0.490
0.499
0.524
0.570
0.615
0.659
0.700
0.735
0.766
0.804
0.853
0.892
0.903
0.886
0.848
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
33.880
119.796
−22.964
−112.616
−79.874
−53.413
−36.711
−25.791
−18.275
−12.903
−9.046
−6.503
−5.365
−5.591
−6.390
−6.746
−6.452
−5.809
−5.081
−3.774
−2.777
−2.042
−1.559
−1.360
−1.280
−1.088
−0.825
−0.575
−0.370
−0.217
−0.114
−0.054
−0.031
−0.034
−0.054
−0.078
−0.101
−0.114
−0.116
−0.105
−0.081
−0.047
−0.005
0.042
0.092
0.142
0.190
0.233
0.270
0.298
0.317
0.327
58.369
146.571
256.452
114.300
46.863
24.412
15.339
11.169
9.221
8.505
8.631
9.397
10.469
11.101
10.605
9.269
7.853
6.728
5.917
4.923
4.383
4.089
3.958
3.845
3.553
3.192
2.921
2.749
2.644
2.581
2.540
2.505
2.466
2.414
2.344
2.257
2.155
2.042
1.924
1.806
1.693
1.587
1.492
1.408
1.337
1.277
1.229
1.191
1.162
1.140
1.121
1.102
0.013
0.004
0.004
0.004
0.005
0.007
0.010
0.014
0.022
0.036
0.055
0.072
0.076
0.072
0.069
0.071
0.076
0.085
0.097
0.128
0.163
0.196
0.219
0.231
0.249
0.281
0.317
0.349
0.371
0.385
0.393
0.399
0.405
0.414
0.426
0.442
0.463
0.488
0.518
0.552
0.589
0.629
0.670
0.709
0.745
0.774
0.795
0.808
0.816
0.821
0.826
0.834
0.1
0.1
0.2
0.2
0.3
0.0
−0.4
−1.1
−2.6
−4.2
−3.9
−0.9
2.1
2.9
2.0
0.4
−0.7
−1.1
−0.7
1.3
3.2
4.5
5.7
6.6
6.6
6.9
8.2
9.2
9.3
10.1
11.2
11.4
10.4
9.8
10.6
11.4
11.8
12.3
12.5
12.4
12.9
13.8
14.3
14.9
15.4
15.4
15.5
16.1
16.6
17.0
17.4
17.0
0.5
0.7
1.0
1.2
1.5
2.4
4.7
11.7
26.9
50.0
68.9
72.3
66.4
61.7
61.9
65.5
69.9
73.2
74.9
76.3
79.6
82.8
81.1
79.0
81.3
81.7
76.5
72.9
73.7
70.3
62.8
60.7
65.4
67.9
63.3
58.5
56.1
54.1
54.0
54.7
52.7
50.5
49.0
46.2
44.7
45.8
44.5
41.1
39.5
37.3
33.0
31.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1057
TABLE 15. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ni calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ni
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.557
0.602
0.634
0.632
0.612
0.564
0.460
0.361
0.273
0.223
0.200
0.210
0.245
0.275
0.323
0.411
0.494
0.548
0.564
0.546
0.542
0.540
0.556
0.582
0.607
0.613
0.607
0.614
0.611
0.606
0.607
0.612
0.619
0.627
0.637
0.645
0.655
0.667
0.681
0.693
0.701
0.710
0.719
0.728
0.736
0.738
0.743
0.751
0.760
0.765
0.770
0.776
0.907
0.928
0.966
1.027
1.048
1.111
1.125
1.092
1.014
0.922
0.810
0.711
0.624
0.551
0.471
0.426
0.413
0.445
0.480
0.500
0.476
0.455
0.426
0.414
0.415
0.423
0.420
0.410
0.406
0.395
0.377
0.360
0.344
0.331
0.317
0.305
0.293
0.281
0.274
0.270
0.264
0.260
0.256
0.253
0.250
0.251
0.242
0.237
0.235
0.234
0.231
0.231
0.801
0.758
0.723
0.706
0.711
0.716
0.761
0.826
0.919
1.024
1.164
1.294
1.389
1.454
1.443
1.215
0.996
0.894
0.875
0.912
0.914
0.912
0.868
0.811
0.767
0.763
0.770
0.752
0.755
0.754
0.738
0.714
0.687
0.658
0.626
0.599
0.569
0.536
0.508
0.488
0.470
0.454
0.439
0.425
0.415
0.413
0.396
0.383
0.372
0.366
0.357
0.352
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.328
0.324
0.318
0.315
0.315
0.322
0.334
0.352
0.373
0.398
0.425
0.453
0.481
0.508
0.536
0.563
0.589
0.616
0.642
0.670
0.701
0.738
0.746
0.629
0.647
0.679
0.703
0.721
0.737
0.750
0.761
0.772
0.781
0.790
0.797
0.805
0.811
0.818
0.824
0.829
0.834
0.839
0.844
0.849
0.853
0.858
0.862
0.866
0.870
0.874
0.878
0.882
1.079
1.051
1.015
0.973
0.926
0.876
0.827
0.780
0.737
0.697
0.662
0.631
0.604
0.580
0.559
0.542
0.526
0.514
0.503
0.497
0.496
0.513
0.589
0.566
0.488
0.461
0.449
0.441
0.436
0.431
0.426
0.422
0.419
0.415
0.412
0.409
0.405
0.402
0.399
0.396
0.392
0.389
0.386
0.383
0.380
0.377
0.374
0.371
0.368
0.365
0.362
0.359
0.848
0.869
0.897
0.931
0.968
1.006
1.039
1.065
1.080
1.081
1.070
1.046
1.014
0.975
0.932
0.888
0.843
0.799
0.756
0.714
0.672
0.635
0.652
0.791
0.743
0.685
0.646
0.617
0.595
0.576
0.560
0.546
0.533
0.522
0.511
0.502
0.493
0.484
0.476
0.469
0.462
0.455
0.448
0.442
0.435
0.429
0.423
0.418
0.412
0.407
0.401
0.396
16.3
16.3
16.8
17.3
17.6
17.6
17.8
18.4
18.7
18.5
18.3
18.2
18.3
18.6
18.2
16.9
16.2
16.7
17.2
16.5
14.9
13.7
13.6
13.8
13.3
13.3
13.5
12.7
12.3
12.8
12.5
11.3
10.9
11.2
11.3
10.6
10.1
9.7
9.8
10.4
10.1
9.0
8.7
9.2
9.4
9.2
8.8
8.4
8.2
8.2
8.2
8.2
33.0
31.9
28.5
24.8
22.3
22.1
22.5
20.8
18.2
18.3
18.7
16.9
15.5
14.4
14.1
14.5
12.8
9.9
6.6
5.5
9.0
13.7
13.4
9.1
8.1
6.8
4.6
7.5
7.8
3.0
2.3
5.5
5.8
3.1
1.8
3.5
4.3
4.3
3.1
0.2
0.5
3.3
3.0
0.9
0.5
0.2
0.5
2.4
2.2
0.6
0.4
0.5
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1058
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 15. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ni calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ni
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.779
0.785
0.783
0.781
0.780
0.780
0.781
0.788
0.803
0.818
0.820
0.820
0.824
0.831
0.839
0.849
0.858
0.860
0.864
0.869
0.877
0.889
0.910
0.933
0.950
0.958
0.971
0.996
1.045
1.148
1.466
0.583
0.596
0.650
0.649
0.660
0.679
0.697
0.715
0.732
0.748
0.768
0.793
0.825
0.798
0.768
0.230
0.231
0.231
0.228
0.222
0.214
0.203
0.191
0.182
0.186
0.190
0.188
0.180
0.176
0.172
0.169
0.172
0.176
0.173
0.167
0.161
0.155
0.149
0.156
0.172
0.180
0.183
0.183
0.182
0.197
0.330
1.258
0.413
0.395
0.331
0.279
0.241
0.214
0.194
0.180
0.167
0.157
0.152
0.185
0.221
0.205
0.348
0.345
0.347
0.345
0.337
0.327
0.312
0.290
0.269
0.264
0.269
0.266
0.253
0.244
0.234
0.225
0.224
0.228
0.223
0.213
0.203
0.190
0.175
0.174
0.185
0.190
0.187
0.179
0.162
0.145
0.146
0.654
0.785
0.684
0.624
0.544
0.464
0.402
0.353
0.316
0.284
0.255
0.234
0.259
0.322
0.324
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.886
0.890
0.894
0.898
0.902
0.906
0.910
0.914
0.918
0.923
0.927
0.932
0.937
0.942
0.948
0.954
0.961
0.968
0.977
0.986
0.997
1.009
1.024
1.041
1.063
1.090
1.124
1.166
1.215
1.246
1.153
0.815
0.551
0.519
0.561
0.608
0.648
0.678
0.703
0.722
0.738
0.751
0.762
0.771
0.778
0.785
0.356
0.354
0.351
0.349
0.346
0.344
0.342
0.340
0.338
0.336
0.335
0.333
0.331
0.330
0.329
0.328
0.328
0.328
0.328
0.328
0.330
0.332
0.337
0.343
0.352
0.368
0.393
0.436
0.515
0.664
0.901
1.017
0.814
0.597
0.473
0.405
0.366
0.341
0.325
0.313
0.305
0.298
0.292
0.287
0.283
0.280
0.391
0.386
0.381
0.376
0.371
0.367
0.362
0.358
0.353
0.349
0.344
0.340
0.336
0.331
0.327
0.322
0.318
0.313
0.309
0.304
0.299
0.295
0.290
0.285
0.281
0.278
0.277
0.281
0.296
0.333
0.421
0.599
0.843
0.954
0.878
0.758
0.661
0.592
0.542
0.506
0.478
0.456
0.439
0.425
0.413
0.403
8.1
7.5
6.8
7.5
8.5
7.5
6.0
6.5
7.9
7.9
7.0
7.1
7.9
7.4
6.4
6.8
7.3
6.8
6.4
6.6
6.2
5.9
6.4
6.5
5.9
5.8
6.4
5.8
4.9
5.5
6.1
5.7
5.5
5.4
5.0
5.4
6.4
6.4
6.0
6.6
7.3
7.2
7.2
7.7
7.7
7.3
−0.3
1.6
4.9
1.5
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1059
TABLE 16. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Cu calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Cu
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−299.196
−130.859
−69.909
−40.759
−23.639
−13.577
−10.040
−7.983
−6.180
−4.739
−3.917
−3.285
−2.609
−1.802
−1.069
−0.881
−0.326
−0.387
−1.537
−2.102
−1.560
−1.023
−0.706
−0.682
−0.582
−0.330
−0.303
−0.203
−0.010
0.227
0.309
0.399
0.532
0.498
0.453
0.433
0.422
0.395
0.240
0.178
0.163
0.171
0.191
0.211
0.218
0.245
0.293
0.354
0.423
0.496
0.566
0.633
61.762
18.836
8.228
4.587
3.677
6.510
8.153
7.972
7.459
7.155
6.950
6.444
5.922
5.503
5.653
5.677
5.632
6.684
6.586
4.852
3.826
3.345
3.257
2.988
2.685
2.375
2.304
2.031
1.808
1.783
1.757
1.699
1.742
1.804
1.774
1.738
1.707
1.732
1.674
1.536
1.410
1.299
1.204
1.132
1.044
0.949
0.864
0.793
0.741
0.707
0.691
0.692
0.001
0.001
0.002
0.003
0.006
0.029
0.049
0.063
0.079
0.097
0.109
0.123
0.141
0.164
0.171
0.172
0.177
0.149
0.144
0.174
0.224
0.273
0.293
0.318
0.356
0.413
0.427
0.487
0.553
0.552
0.552
0.558
0.525
0.515
0.529
0.542
0.552
0.549
0.585
0.642
0.700
0.757
0.810
0.854
0.918
0.987
1.038
1.051
1.018
0.948
0.865
0.787
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−288.397
−91.601
−73.064
−80.642
−55.729
−38.032
−26.696
−19.112
−13.794
−9.934
−7.103
−5.121
−3.996
−3.742
−4.026
−4.207
−3.982
−3.825
−4.101
−3.056
−2.248
−1.687
−1.281
−0.971
−0.726
−0.525
−0.356
−0.214
−0.097
−0.002
0.070
0.120
0.148
0.157
0.151
0.136
0.117
0.101
0.092
0.092
0.102
0.121
0.150
0.186
0.227
0.273
0.322
0.373
0.427
0.482
0.539
0.598
34.594
49.326
86.333
41.391
17.567
9.757
6.718
5.428
4.945
4.932
5.268
5.898
6.705
7.351
7.393
6.838
6.206
6.013
5.185
3.918
3.429
3.128
2.902
2.714
2.553
2.414
2.296
2.198
2.118
2.054
2.002
1.959
1.918
1.876
1.826
1.765
1.693
1.611
1.520
1.426
1.331
1.239
1.152
1.071
0.998
0.934
0.877
0.829
0.790
0.760
0.740
0.733
0.000
0.005
0.007
0.005
0.005
0.006
0.009
0.014
0.023
0.040
0.067
0.097
0.110
0.108
0.104
0.106
0.114
0.118
0.119
0.159
0.204
0.248
0.288
0.327
0.362
0.396
0.425
0.451
0.471
0.487
0.499
0.509
0.518
0.529
0.544
0.563
0.588
0.618
0.655
0.699
0.747
0.800
0.854
0.906
0.952
0.987
1.005
1.003
0.980
0.938
0.883
0.819
−0.1
−0.1
−0.1
−0.2
−0.2
−0.2
−0.4
−0.8
−1.5
−1.0
1.8
4.8
5.3
3.8
2.0
0.8
0.3
0.3
0.7
2.4
4.3
5.8
7.0
7.5
7.6
8.9
10.7
11.7
11.7
12.1
13.1
12.8
12.2
13.4
14.2
13.4
13.5
14.6
15.1
15.3
15.7
16.1
17.0
18.2
18.3
17.7
17.7
18.9
20.4
19.8
17.5
17.0
0.6
0.9
1.2
1.5
1.8
1.9
3.3
8.7
22.3
46.1
71.2
88.4
97.8
101.4
102.2
101.9
101.6
101.3
101.0
101.1
106.4
118.3
128.1
126.2
115.8
103.8
94.0
86.9
80.5
72.7
64.8
62.1
58.8
49.7
45.5
47.8
44.5
38.5
39.4
41.3
37.9
34.9
34.6
33.8
32.9
32.4
30.3
26.9
21.8
20.1
24.5
23.5
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1060
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 16. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Cu calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Cu
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.695
0.757
0.818
0.812
0.650
0.545
0.429
0.342
0.322
0.328
0.356
0.363
0.416
0.511
0.608
0.664
0.635
0.627
0.626
0.640
0.669
0.700
0.709
0.707
0.712
0.704
0.707
0.712
0.723
0.732
0.734
0.741
0.749
0.761
0.770
0.775
0.774
0.773
0.776
0.785
0.794
0.796
0.800
0.805
0.810
0.818
0.815
0.810
0.812
0.798
0.786
0.786
0.706
0.741
0.813
0.983
1.040
1.011
0.980
0.869
0.752
0.659
0.582
0.515
0.410
0.359
0.367
0.412
0.456
0.438
0.420
0.387
0.373
0.379
0.388
0.387
0.394
0.384
0.372
0.358
0.348
0.346
0.340
0.330
0.323
0.317
0.317
0.320
0.320
0.313
0.306
0.297
0.299
0.297
0.295
0.295
0.291
0.297
0.304
0.303
0.306
0.314
0.301
0.291
0.719
0.660
0.611
0.605
0.692
0.766
0.856
0.997
1.123
1.216
1.250
1.297
1.201
0.920
0.727
0.675
0.747
0.748
0.740
0.692
0.636
0.599
0.594
0.596
0.595
0.597
0.584
0.564
0.540
0.527
0.520
0.502
0.485
0.466
0.458
0.455
0.456
0.450
0.440
0.422
0.415
0.412
0.407
0.402
0.393
0.393
0.402
0.405
0.406
0.427
0.424
0.414
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.656
0.708
0.742
0.727
0.632
0.481
0.370
0.340
0.360
0.400
0.444
0.486
0.525
0.560
0.592
0.621
0.647
0.671
0.693
0.713
0.732
0.749
0.765
0.780
0.793
0.806
0.817
0.828
0.838
0.848
0.856
0.864
0.871
0.878
0.884
0.889
0.894
0.899
0.903
0.906
0.909
0.911
0.914
0.915
0.916
0.917
0.918
0.918
0.918
0.917
0.916
0.915
0.743
0.777
0.841
0.933
1.013
1.004
0.897
0.766
0.660
0.583
0.528
0.489
0.460
0.438
0.420
0.406
0.395
0.386
0.378
0.372
0.367
0.362
0.359
0.356
0.354
0.352
0.351
0.350
0.349
0.349
0.349
0.349
0.350
0.351
0.352
0.353
0.354
0.355
0.357
0.358
0.359
0.361
0.362
0.364
0.366
0.367
0.368
0.370
0.371
0.372
0.373
0.374
0.756
0.703
0.669
0.666
0.711
0.810
0.952
1.090
1.168
1.167
1.110
1.029
0.944
0.866
0.797
0.738
0.687
0.644
0.607
0.575
0.547
0.523
0.503
0.485
0.469
0.455
0.443
0.433
0.423
0.415
0.408
0.402
0.397
0.392
0.389
0.385
0.383
0.380
0.379
0.377
0.376
0.376
0.375
0.375
0.376
0.376
0.377
0.378
0.379
0.380
0.381
0.383
17.9
17.1
14.9
14.1
15.0
15.2
15.0
15.2
16.0
17.1
17.7
17.1
16.3
16.9
17.9
16.4
13.3
11.5
12.3
13.9
13.3
11.7
11.4
11.7
11.5
11.2
11.3
10.7
9.2
8.8
9.0
9.0
9.1
9.0
8.6
8.7
9.1
9.1
8.5
7.9
7.9
7.9
7.5
7.8
7.8
7.3
7.7
7.4
6.3
6.0
6.1
6.2
15.8
15.6
22.6
23.4
19.0
18.4
19.5
18.8
16.9
12.8
9.8
11.8
14.8
9.8
2.0
5.7
15.7
19.9
14.7
5.9
5.3
10.8
11.6
8.5
7.8
8.7
6.7
7.1
12.8
14.3
10.6
9.3
10.9
11.5
9.4
6.9
6.8
8.2
8.7
7.7
6.6
8.8
11.3
7.9
6.0
7.8
6.7
7.5
10.8
10.4
10.0
9.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1061
TABLE 16. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Cu calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Cu
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.781
0.778
0.783
0.785
0.792
0.788
0.795
0.792
0.793
0.794
0.808
0.811
0.813
0.812
0.811
0.821
0.838
0.858
0.857
0.842
0.830
0.825
0.827
0.829
0.831
0.824
0.816
0.807
0.805
0.807
0.820
0.834
0.842
0.846
0.850
0.853
0.854
0.862
0.860
0.867
0.877
0.885
0.898
0.924
0.976
0.898
0.288
0.272
0.264
0.257
0.249
0.249
0.240
0.235
0.233
0.220
0.211
0.218
0.213
0.210
0.201
0.186
0.187
0.192
0.220
0.225
0.221
0.212
0.206
0.204
0.205
0.208
0.204
0.193
0.179
0.161
0.149
0.147
0.149
0.151
0.148
0.149
0.148
0.146
0.144
0.138
0.136
0.141
0.133
0.138
0.191
0.247
0.415
0.400
0.386
0.376
0.362
0.365
0.348
0.345
0.341
0.324
0.302
0.309
0.302
0.298
0.288
0.262
0.253
0.248
0.281
0.296
0.299
0.292
0.284
0.280
0.280
0.288
0.288
0.281
0.263
0.238
0.215
0.204
0.204
0.204
0.199
0.198
0.197
0.191
0.190
0.179
0.173
0.175
0.161
0.158
0.193
0.285
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.914
0.912
0.910
0.908
0.906
0.904
0.901
0.898
0.895
0.892
0.889
0.886
0.882
0.879
0.875
0.872
0.868
0.865
0.861
0.858
0.855
0.851
0.848
0.845
0.842
0.839
0.836
0.833
0.830
0.828
0.825
0.823
0.821
0.819
0.817
0.815
0.813
0.811
0.810
0.809
0.807
0.806
0.805
0.804
0.804
0.803
0.375
0.376
0.376
0.377
0.377
0.377
0.377
0.376
0.376
0.375
0.374
0.373
0.372
0.370
0.368
0.367
0.364
0.362
0.360
0.357
0.354
0.351
0.348
0.345
0.341
0.338
0.334
0.330
0.326
0.322
0.318
0.314
0.310
0.306
0.301
0.297
0.292
0.288
0.284
0.279
0.275
0.270
0.266
0.261
0.257
0.253
0.384
0.386
0.388
0.390
0.391
0.393
0.395
0.397
0.399
0.401
0.402
0.404
0.406
0.407
0.409
0.410
0.411
0.412
0.413
0.413
0.414
0.414
0.414
0.414
0.414
0.413
0.412
0.411
0.410
0.409
0.407
0.405
0.403
0.400
0.398
0.395
0.392
0.389
0.385
0.381
0.378
0.374
0.370
0.365
0.361
0.356
6.9
7.3
6.5
5.4
5.6
6.8
7.1
6.3
5.8
6.2
6.4
5.4
4.9
5.6
6.5
6.5
6.0
5.9
6.4
6.8
6.7
6.4
6.1
6.1
6.4
6.5
6.4
6.6
6.3
5.6
6.0
6.9
6.9
6.3
5.8
5.2
5.5
6.4
6.3
5.0
4.9
6.4
6.7
5.5
4.9
5.4
5.6
4.2
8.2
10.9
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1062
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 17. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Zn calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Zn
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−275.430
−101.234
−27.773
8.567
−13.915
−30.897
−40.810
−37.319
−30.798
−25.283
−20.922
−17.484
−14.745
−12.536
−10.741
−9.286
−8.035
−6.967
−6.049
−4.572
−3.458
−2.568
−1.706
−1.177
−0.884
−0.617
−0.484
−0.411
−0.312
−0.289
−0.254
−0.160
−0.068
0.049
0.177
0.251
0.292
0.342
0.394
0.440
0.478
0.511
0.514
0.454
0.408
0.382
0.367
0.413
0.473
0.534
0.591
0.630
67.392
23.545
19.008
49.900
69.615
59.184
42.263
24.856
15.829
10.759
7.669
5.664
4.304
3.353
2.674
2.154
1.734
1.422
1.184
0.867
0.703
0.550
0.538
0.766
0.876
0.960
1.046
1.044
1.016
1.014
0.896
0.798
0.719
0.645
0.629
0.644
0.640
0.619
0.611
0.609
0.614
0.633
0.680
0.687
0.645
0.585
0.496
0.409
0.357
0.326
0.317
0.323
0.001
0.002
0.017
0.019
0.014
0.013
0.012
0.012
0.013
0.014
0.015
0.017
0.018
0.020
0.022
0.024
0.026
0.028
0.031
0.040
0.057
0.080
0.168
0.388
0.565
0.737
0.787
0.830
0.899
0.912
1.033
1.205
1.379
1.542
1.472
1.347
1.293
1.238
1.156
1.079
1.014
0.956
0.935
1.013
1.108
1.198
1.304
1.211
1.018
0.832
0.705
0.644
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−179.123
−129.337
−92.759
−67.690
−50.584
−38.689
−30.201
−23.981
−19.306
−15.706
−12.869
−10.577
−8.670
−7.021
−5.647
−5.612
−5.426
−4.480
−3.669
−2.666
−2.550
−2.205
−1.644
−1.158
−0.771
−0.467
−0.238
−0.085
−0.012
0.010
0.032
0.081
0.150
0.224
0.295
0.359
0.416
0.466
0.509
0.547
0.580
0.609
0.634
0.656
0.676
0.693
0.708
0.721
0.733
0.744
0.753
0.762
276.550
133.940
72.687
42.949
27.174
18.182
12.745
9.294
7.015
5.464
4.386
3.634
3.134
2.899
3.156
3.698
2.645
2.117
1.977
2.164
2.118
1.545
1.206
1.054
0.991
0.980
1.005
1.049
1.083
1.068
1.002
0.918
0.843
0.784
0.740
0.707
0.681
0.660
0.642
0.627
0.614
0.602
0.591
0.580
0.571
0.561
0.552
0.543
0.534
0.526
0.517
0.509
0.003
0.004
0.005
0.007
0.008
0.010
0.012
0.014
0.017
0.020
0.024
0.029
0.037
0.050
0.075
0.082
0.073
0.086
0.114
0.184
0.193
0.213
0.290
0.430
0.629
0.831
0.943
0.947
0.923
0.936
0.997
1.081
1.150
1.179
1.166
1.125
1.070
1.012
0.956
0.906
0.861
0.822
0.787
0.757
0.730
0.706
0.685
0.666
0.649
0.633
0.619
0.606
0.0
0.0
0.1
0.1
0.1
0.0
0.0
−0.1
−0.2
−0.6
−1.0
−1.2
−1.2
−0.8
−0.4
0.0
0.4
0.8
1.2
2.4
3.6
5.4
9.6
16.6
24.9
29.2
28.0
26.7
26.6
27.7
29.6
29.5
27.9
29.0
30.5
30.4
31.3
31.8
30.0
26.1
23.5
24.9
25.8
22.7
20.0
19.4
19.8
20.8
19.9
17.9
17.9
18.5
0.2
0.3
0.4
0.4
0.5
0.5
0.8
1.8
4.2
9.1
15.7
22.0
26.0
27.4
27.7
27.8
28.8
30.9
33.8
41.2
49.6
56.5
65.3
94.4
159.8
229.7
221.7
156.0
112.9
102.1
94.1
86.9
86.9
78.4
70.4
69.8
63.7
55.7
54.4
61.8
64.4
51.7
42.6
47.4
51.0
49.8
43.7
32.8
32.3
38.5
33.8
25.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1063
TABLE 17. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Zn calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Zn
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.663
0.690
0.740
0.775
0.797
0.783
0.749
0.718
0.689
0.676
0.676
0.676
0.692
0.738
0.784
0.804
0.790
0.778
0.784
0.793
0.804
0.811
0.821
0.826
0.830
0.841
0.849
0.856
0.856
0.857
0.859
0.863
0.868
0.870
0.864
0.857
0.855
0.852
0.849
0.849
0.841
0.839
0.833
0.833
0.833
0.839
0.840
0.845
0.847
0.847
0.847
0.844
0.315
0.306
0.310
0.335
0.377
0.417
0.437
0.428
0.409
0.371
0.341
0.310
0.267
0.238
0.241
0.273
0.295
0.276
0.264
0.252
0.248
0.244
0.244
0.239
0.242
0.231
0.238
0.240
0.245
0.244
0.240
0.242
0.243
0.250
0.255
0.256
0.251
0.250
0.246
0.245
0.243
0.237
0.231
0.222
0.213
0.208
0.205
0.199
0.200
0.197
0.198
0.198
0.584
0.537
0.482
0.470
0.485
0.529
0.582
0.613
0.637
0.624
0.594
0.561
0.486
0.395
0.358
0.378
0.415
0.405
0.385
0.364
0.350
0.340
0.333
0.323
0.323
0.304
0.306
0.304
0.309
0.307
0.302
0.302
0.299
0.306
0.314
0.320
0.316
0.317
0.316
0.313
0.317
0.311
0.310
0.298
0.288
0.279
0.275
0.264
0.264
0.261
0.262
0.263
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.770
0.777
0.783
0.788
0.794
0.798
0.803
0.806
0.810
0.814
0.817
0.820
0.822
0.825
0.827
0.830
0.832
0.834
0.836
0.838
0.840
0.841
0.843
0.845
0.846
0.848
0.849
0.851
0.852
0.853
0.854
0.856
0.857
0.858
0.859
0.860
0.861
0.862
0.863
0.864
0.865
0.866
0.867
0.868
0.869
0.870
0.870
0.871
0.872
0.873
0.874
0.874
0.501
0.493
0.485
0.477
0.469
0.462
0.454
0.447
0.440
0.432
0.425
0.418
0.412
0.405
0.398
0.392
0.386
0.379
0.373
0.367
0.362
0.356
0.350
0.345
0.339
0.334
0.329
0.324
0.319
0.314
0.310
0.305
0.301
0.296
0.292
0.288
0.283
0.279
0.275
0.271
0.268
0.264
0.260
0.257
0.253
0.249
0.246
0.243
0.239
0.236
0.233
0.230
0.594
0.583
0.572
0.562
0.552
0.543
0.534
0.526
0.517
0.509
0.502
0.494
0.487
0.479
0.472
0.465
0.459
0.452
0.445
0.439
0.433
0.426
0.420
0.414
0.408
0.403
0.397
0.391
0.386
0.380
0.375
0.370
0.365
0.360
0.355
0.350
0.345
0.340
0.335
0.331
0.326
0.322
0.318
0.313
0.309
0.305
0.301
0.297
0.293
0.289
0.285
0.281
18.1
16.9
15.1
13.5
13.0
12.3
12.0
12.7
11.6
10.0
11.1
12.6
12.3
12.0
11.6
9.5
8.5
10.0
10.8
9.8
9.6
9.4
8.0
8.5
9.0
6.9
6.3
8.5
9.5
8.1
6.1
5.7
7.2
8.3
7.8
6.7
6.3
7.2
8.0
7.0
6.1
6.4
6.8
6.9
6.7
5.9
6.3
7.7
7.6
6.4
6.6
7.9
25.4
29.4
31.0
29.5
27.5
28.1
25.1
21.2
27.4
32.7
24.8
16.5
15.6
14.3
15.1
23.5
25.9
16.6
11.6
15.6
15.7
14.8
19.5
17.9
15.2
21.4
20.6
10.6
7.4
16.0
24.9
22.0
13.1
11.6
16.7
17.5
14.3
11.7
11.3
13.0
11.2
9.3
10.9
11.8
12.2
14.1
11.6
6.7
6.4
9.9
9.4
4.0
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1064
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 17. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Zn calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Zn
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.838
0.831
0.837
0.849
0.856
0.861
0.858
0.858
0.856
0.859
0.860
0.856
0.854
0.854
0.856
0.860
0.863
0.861
0.862
0.863
0.862
0.859
0.853
0.849
0.847
0.847
0.847
0.851
0.863
0.868
0.870
0.871
0.869
0.866
0.867
0.871
0.873
0.873
0.872
0.872
0.875
0.877
0.879
0.880
0.880
0.881
0.195
0.183
0.168
0.163
0.164
0.167
0.169
0.168
0.165
0.162
0.164
0.164
0.159
0.153
0.147
0.145
0.147
0.145
0.144
0.144
0.144
0.146
0.143
0.136
0.131
0.122
0.114
0.103
0.101
0.104
0.104
0.107
0.107
0.103
0.099
0.095
0.096
0.095
0.093
0.089
0.086
0.084
0.083
0.082
0.080
0.079
0.263
0.253
0.230
0.219
0.216
0.217
0.221
0.220
0.216
0.212
0.214
0.215
0.211
0.204
0.196
0.190
0.191
0.190
0.188
0.188
0.188
0.192
0.191
0.184
0.178
0.166
0.156
0.141
0.133
0.136
0.136
0.138
0.139
0.136
0.130
0.124
0.124
0.124
0.121
0.115
0.111
0.108
0.107
0.106
0.103
0.101
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.875
0.876
0.877
0.877
0.878
0.879
0.879
0.880
0.881
0.882
0.882
0.883
0.883
0.884
0.885
0.885
0.886
0.887
0.887
0.888
0.888
0.889
0.890
0.890
0.891
0.891
0.892
0.893
0.893
0.894
0.894
0.895
0.895
0.896
0.896
0.897
0.898
0.898
0.899
0.899
0.900
0.900
0.901
0.901
0.902
0.902
0.227
0.224
0.221
0.218
0.215
0.212
0.210
0.207
0.204
0.202
0.199
0.196
0.194
0.192
0.189
0.187
0.184
0.182
0.180
0.178
0.175
0.173
0.171
0.169
0.167
0.165
0.163
0.161
0.159
0.157
0.155
0.154
0.152
0.150
0.148
0.146
0.145
0.143
0.141
0.140
0.138
0.137
0.135
0.134
0.132
0.131
0.278
0.274
0.270
0.267
0.263
0.260
0.256
0.253
0.250
0.246
0.243
0.240
0.237
0.234
0.231
0.228
0.225
0.222
0.219
0.217
0.214
0.211
0.209
0.206
0.203
0.201
0.198
0.196
0.193
0.191
0.189
0.186
0.184
0.182
0.180
0.177
0.175
0.173
0.171
0.169
0.167
0.165
0.163
0.161
0.159
0.157
7.3
5.9
6.6
8.1
7.6
6.1
6.1
7.1
7.8
6.9
5.0
4.6
5.8
6.4
5.7
5.3
5.1
5.7
7.5
7.5
5.4
4.8
5.4
5.5
5.8
5.8
5.3
5.2
5.2
5.3
5.3
4.8
5.0
5.8
5.2
3.9
3.9
5.0
5.4
4.7
4.1
4.9
5.1
4.6
4.7
5.2
5.2
10.5
5.9
−0.5
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1065
TABLE 18. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Mo calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Mo
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−245.480
−90.107
−38.529
−14.020
−0.141
4.460
4.258
1.759
−2.094
−2.053
−1.119
0.053
−0.124
−0.536
−1.829
−9.162
−12.729
−14.486
−13.564
−12.066
−10.568
−8.084
−5.450
−6.134
−4.952
−3.765
−2.754
−1.894
−1.126
−0.341
0.749
1.165
0.348
−0.051
−0.138
−0.254
−0.156
0.069
0.329
0.729
0.725
−0.156
−0.803
−1.139
−1.369
−1.926
−1.256
−0.869
−0.631
−0.440
−0.333
−0.317
66.346
31.720
25.541
23.209
25.336
30.154
32.189
34.183
29.695
26.661
24.459
23.949
24.665
25.181
27.525
28.691
23.656
18.351
14.703
10.204
6.528
4.427
3.450
3.860
2.241
1.555
1.192
0.995
0.891
0.848
1.118
2.416
3.003
2.765
2.652
2.412
2.151
2.015
1.995
2.357
2.994
3.530
3.031
2.828
2.371
1.572
0.899
0.818
0.748
0.731
0.785
0.758
0.001
0.003
0.012
0.032
0.039
0.032
0.031
0.029
0.034
0.037
0.041
0.042
0.041
0.040
0.036
0.032
0.033
0.034
0.037
0.041
0.042
0.052
0.083
0.073
0.076
0.094
0.132
0.217
0.432
1.014
0.618
0.336
0.329
0.362
0.376
0.410
0.462
0.496
0.488
0.387
0.316
0.283
0.308
0.304
0.316
0.254
0.377
0.574
0.781
1.004
1.080
1.123
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−565.328
−164.779
−171.790
−194.298
−138.958
−98.839
−72.983
−55.691
−43.596
−34.810
−28.233
−23.191
−19.258
−16.151
−13.680
−11.712
−10.150
−8.918
−7.951
−6.587
−5.657
−4.892
−4.173
−3.479
−2.822
−2.215
−1.661
−1.160
−0.702
−0.274
0.135
0.269
−0.730
−0.810
−0.709
−0.710
−0.770
−0.840
−0.890
−0.904
−0.882
−0.832
−0.760
−0.678
−0.590
−0.504
−0.421
−0.346
−0.278
−0.217
−0.163
−0.115
85.961
141.347
223.146
103.268
43.207
22.220
13.377
9.033
6.679
5.327
4.534
4.079
3.839
3.742
3.739
3.793
3.873
3.952
4.003
3.958
3.691
3.282
2.848
2.470
2.179
1.978
1.860
1.817
1.849
1.973
2.272
3.035
3.219
2.652
2.501
2.439
2.352
2.218
2.048
1.859
1.672
1.497
1.344
1.212
1.102
1.011
0.937
0.875
0.821
0.774
0.730
0.686
0.000
0.003
0.003
0.002
0.002
0.002
0.002
0.003
0.003
0.004
0.006
0.007
0.010
0.014
0.019
0.025
0.033
0.042
0.051
0.067
0.081
0.095
0.112
0.136
0.171
0.224
0.299
0.391
0.473
0.497
0.439
0.327
0.295
0.345
0.370
0.378
0.384
0.394
0.411
0.435
0.468
0.510
0.564
0.629
0.705
0.792
0.888
0.989
1.092
1.197
1.305
1.417
−0.1
−0.1
−0.1
−0.1
−0.2
−0.2
−0.3
−0.6
−1.2
−1.9
−2.1
−1.6
−1.0
−0.5
−0.3
−0.1
−0.1
−0.1
−0.2
−0.2
0.4
1.3
1.7
2.1
3.1
4.6
6.4
8.8
11.5
13.0
12.8
11.8
10.4
9.7
10.2
10.8
10.5
10.8
11.6
11.7
11.5
11.7
12.5
13.8
14.8
15.4
16.4
18.6
21.0
22.1
23.2
25.0
0.7
1.0
1.3
1.6
2.0
1.7
2.6
5.6
11.8
21.3
30.1
33.7
32.8
30.2
28.1
27.3
28.0
30.1
33.1
40.0
45.2
48.7
52.4
56.9
62.2
68.4
74.0
77.8
78.1
75.9
73.5
71.3
68.3
64.2
59.9
57.8
57.8
55.1
51.1
50.6
52.1
52.0
50.5
48.7
47.3
47.2
47.5
45.0
40.9
39.5
38.3
34.8
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1066
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 18. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Mo calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Mo
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.226
−0.294
−0.245
−0.059
0.072
0.174
0.262
0.338
0.358
0.399
0.430
0.482
0.550
0.614
0.714
0.817
0.915
0.960
0.971
1.028
1.084
1.201
1.310
1.480
1.709
2.175
2.817
2.377
1.597
0.861
0.356
−0.574
−0.324
−0.692
−0.537
−0.578
−0.434
−0.253
−0.148
−0.058
0.021
0.081
0.130
0.186
0.237
0.286
0.365
0.458
0.477
0.461
0.443
0.387
0.730
0.685
0.441
0.358
0.339
0.337
0.333
0.358
0.371
0.337
0.325
0.282
0.252
0.222
0.189
0.202
0.242
0.340
0.308
0.328
0.289
0.283
0.307
0.296
0.317
0.405
1.185
2.272
2.700
2.795
2.889
2.022
1.972
1.384
1.092
0.916
0.610
0.483
0.429
0.379
0.341
0.319
0.286
0.258
0.241
0.215
0.202
0.228
0.313
0.347
0.376
0.400
1.249
1.234
1.731
2.722
2.825
2.341
1.855
1.476
1.396
1.234
1.118
0.904
0.689
0.522
0.347
0.285
0.271
0.328
0.297
0.282
0.229
0.186
0.170
0.130
0.105
0.083
0.127
0.210
0.274
0.327
0.341
0.458
0.494
0.578
0.737
0.780
1.089
1.625
2.084
2.575
2.924
2.949
2.895
2.552
2.107
1.680
1.162
0.872
0.960
1.041
1.112
1.292
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.069
−0.024
0.021
0.067
0.114
0.162
0.211
0.260
0.309
0.358
0.406
0.455
0.503
0.550
0.598
0.647
0.695
0.745
0.797
0.852
0.911
0.975
1.049
1.139
1.264
1.406
1.124
1.299
1.359
1.168
0.858
0.776
0.801
0.782
0.671
0.517
0.410
0.376
0.386
0.408
0.423
0.425
0.417
0.408
0.405
0.414
0.435
0.470
0.524
0.554
0.373
0.431
0.643
0.600
0.558
0.516
0.477
0.441
0.408
0.378
0.352
0.328
0.308
0.291
0.276
0.263
0.253
0.246
0.240
0.237
0.237
0.240
0.246
0.258
0.277
0.309
0.375
0.652
0.621
0.665
0.908
1.179
1.160
1.012
0.980
1.041
1.099
1.071
0.965
0.848
0.760
0.707
0.677
0.656
0.633
0.600
0.560
0.519
0.483
0.458
0.458
0.596
0.483
0.425
1.537
1.664
1.791
1.905
1.982
1.998
1.936
1.797
1.606
1.393
1.185
0.998
0.839
0.707
0.600
0.514
0.444
0.388
0.343
0.306
0.277
0.253
0.235
0.222
0.216
0.272
0.377
0.312
0.340
0.428
0.557
0.622
0.612
0.614
0.663
0.757
0.878
0.986
1.046
1.061
1.062
1.073
1.102
1.140
1.172
1.178
1.143
1.064
0.946
0.900
1.296
1.159
26.4
27.8
30.0
31.1
31.3
32.6
33.7
32.6
30.5
28.6
26.7
24.5
22.5
20.4
18.3
16.9
16.2
14.7
13.2
12.6
11.4
9.9
9.0
8.5
7.7
7.2
7.5
8.0
8.3
8.1
7.8
8.2
9.0
9.6
9.6
9.9
10.9
11.6
11.7
12.6
13.9
14.4
14.5
14.9
15.4
15.5
15.7
16.4
16.7
16.1
15.5
15.6
31.8
29.5
26.1
22.3
18.6
14.4
10.8
10.4
11.3
10.6
10.6
11.7
12.0
12.2
13.3
13.5
12.3
12.8
12.8
10.7
10.6
12.2
11.6
9.5
9.5
10.0
8.5
6.4
6.8
9.0
9.8
8.6
6.3
5.7
8.0
8.6
5.7
4.5
4.5
3.3
2.3
2.2
2.6
2.1
1.3
0.8
0.5
−0.1
−1.1
−0.8
0.7
0.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1067
TABLE 18. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Mo calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Mo
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.355
0.359
0.367
0.374
0.378
0.397
0.408
0.407
0.416
0.426
0.455
0.485
0.510
0.509
0.504
0.506
0.519
0.538
0.556
0.575
0.592
0.600
0.605
0.616
0.623
0.628
0.639
0.650
0.664
0.652
0.659
0.665
0.677
0.686
0.692
0.689
0.694
0.698
0.702
0.707
0.714
0.723
0.731
0.739
0.746
0.752
0.339
0.303
0.279
0.266
0.238
0.214
0.211
0.204
0.170
0.140
0.125
0.115
0.126
0.137
0.124
0.110
0.086
0.076
0.067
0.065
0.067
0.073
0.068
0.066
0.067
0.062
0.057
0.055
0.064
0.064
0.057
0.050
0.047
0.048
0.053
0.051
0.046
0.043
0.040
0.035
0.030
0.027
0.027
0.027
0.029
0.032
1.406
1.373
1.313
1.262
1.192
1.054
0.998
0.983
0.843
0.699
0.559
0.463
0.458
0.492
0.458
0.408
0.310
0.258
0.215
0.194
0.189
0.200
0.183
0.171
0.170
0.155
0.138
0.129
0.145
0.150
0.130
0.113
0.101
0.101
0.110
0.108
0.095
0.088
0.081
0.070
0.058
0.052
0.050
0.048
0.052
0.056
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.458
0.466
0.462
0.452
0.441
0.434
0.432
0.435
0.443
0.453
0.465
0.478
0.490
0.502
0.514
0.525
0.535
0.545
0.554
0.562
0.571
0.579
0.586
0.594
0.601
0.608
0.615
0.621
0.628
0.635
0.641
0.648
0.654
0.660
0.666
0.672
0.678
0.684
0.690
0.695
0.701
0.706
0.711
0.716
0.721
0.726
0.417
0.417
0.413
0.404
0.386
0.363
0.337
0.310
0.286
0.264
0.245
0.229
0.215
0.203
0.192
0.183
0.174
0.167
0.159
0.153
0.146
0.140
0.134
0.128
0.122
0.117
0.111
0.106
0.102
0.097
0.092
0.088
0.084
0.080
0.077
0.073
0.070
0.067
0.064
0.061
0.058
0.056
0.054
0.051
0.049
0.047
1.086
1.066
1.075
1.099
1.123
1.134
1.122
1.085
1.028
0.959
0.887
0.816
0.750
0.691
0.639
0.592
0.551
0.514
0.480
0.449
0.421
0.395
0.370
0.347
0.325
0.305
0.286
0.268
0.251
0.235
0.220
0.206
0.193
0.181
0.170
0.160
0.150
0.141
0.133
0.125
0.118
0.111
0.105
0.100
0.094
0.089
16.1
16.6
16.8
16.2
15.5
15.8
16.6
16.9
16.6
15.6
14.9
15.4
15.7
14.3
13.3
13.9
14.4
13.1
11.4
11.1
11.3
10.8
10.0
9.6
9.8
9.6
8.6
8.1
8.6
8.7
7.5
6.9
7.2
7.4
7.2
7.1
6.8
5.9
5.3
5.3
6.0
6.1
5.3
4.7
4.8
5.1
−0.5
−1.6
−2.5
−1.6
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1068
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 19. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pd calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Pd
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−142.506
−68.572
−44.612
−33.219
−27.262
−22.162
−18.074
−14.881
−12.544
−10.462
−8.655
−7.296
−6.170
−5.219
−4.601
−3.939
−3.234
−2.537
−2.579
−2.392
−1.845
−1.299
−0.761
−0.061
0.564
0.885
1.402
1.220
1.386
1.259
1.453
1.518
1.424
1.435
1.510
1.460
1.291
1.260
1.175
1.045
0.981
1.012
1.034
1.150
1.270
1.435
1.714
1.541
0.863
0.310
−0.231
−0.145
109.816
73.262
53.255
40.430
31.145
23.870
18.900
15.321
12.957
10.558
9.171
8.093
7.163
6.605
5.948
5.352
4.865
4.849
4.860
3.892
3.100
2.519
2.004
1.633
1.735
1.682
1.866
2.129
2.232
2.119
2.075
2.224
2.283
2.219
2.292
2.390
2.380
2.326
2.384
2.323
2.181
2.077
1.971
1.916
1.948
1.989
2.215
2.760
3.244
3.203
2.737
2.228
0.003
0.007
0.011
0.015
0.018
0.022
0.028
0.034
0.040
0.048
0.058
0.068
0.080
0.093
0.105
0.121
0.143
0.162
0.161
0.187
0.238
0.314
0.436
0.612
0.521
0.465
0.343
0.354
0.323
0.349
0.323
0.307
0.315
0.318
0.304
0.305
0.325
0.332
0.338
0.358
0.381
0.389
0.398
0.384
0.360
0.331
0.282
0.276
0.288
0.309
0.363
0.447
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−545.897
−238.405
−130.490
−80.442
−53.195
−36.727
−26.030
−18.742
−13.660
−10.153
−7.884
−6.616
−6.063
−5.859
−5.670
−5.326
−4.816
−4.203
−3.560
−2.411
−1.676
−1.115
−0.471
0.132
0.625
1.002
1.262
1.403
1.423
1.326
1.127
0.863
0.581
0.327
0.135
0.019
−0.026
−0.012
0.044
0.123
0.209
0.282
0.321
0.306
0.229
0.103
−0.033
−0.137
−0.185
−0.178
−0.134
−0.069
30.623
9.671
4.781
3.262
2.826
2.866
3.183
3.708
4.406
5.229
6.070
6.745
7.042
6.861
6.295
5.555
4.825
4.209
3.741
3.223
2.966
2.596
2.309
2.213
2.256
2.392
2.590
2.827
3.072
3.293
3.450
3.512
3.465
3.322
3.110
2.867
2.626
2.410
2.234
2.105
2.027
1.998
2.012
2.049
2.077
2.057
1.967
1.817
1.640
1.467
1.320
1.205
0.000
0.000
0.000
0.001
0.001
0.002
0.005
0.010
0.021
0.040
0.061
0.076
0.082
0.084
0.088
0.094
0.104
0.119
0.140
0.199
0.256
0.325
0.416
0.450
0.412
0.356
0.312
0.284
0.268
0.261
0.262
0.269
0.281
0.298
0.321
0.349
0.381
0.415
0.448
0.473
0.488
0.491
0.485
0.477
0.476
0.485
0.508
0.547
0.602
0.672
0.750
0.827
0.0
0.0
0.0
0.0
0.0
−0.1
−0.4
−1.0
−2.2
−2.9
−1.9
0.4
2.1
2.6
2.5
2.3
2.1
2.1
2.4
3.9
6.6
9.4
11.9
13.5
13.6
11.9
10.0
9.7
9.9
9.3
8.9
8.7
8.4
8.7
9.2
9.3
9.5
9.7
9.9
10.5
11.2
11.6
11.5
11.3
11.7
11.9
11.4
11.4
12.2
12.7
13.3
14.4
0.6
0.9
1.3
1.6
1.9
2.3
4.6
11.9
27.6
49.4
65.6
68.8
65.1
61.3
60.5
63.2
68.4
75.0
81.6
89.0
88.4
86.2
83.3
76.6
68.4
63.2
59.7
54.2
48.3
45.5
43.5
41.4
39.8
36.8
34.3
33.5
32.5
31.8
32.1
30.7
27.7
25.8
26.2
26.9
25.6
24.8
25.5
24.9
23.6
23.5
23.0
20.6
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1069
TABLE 19. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pd calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Pd
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.245
−0.220
−0.207
−0.260
−0.199
−0.136
−0.047
0.131
0.221
0.312
0.368
0.321
0.388
0.527
0.630
0.541
0.371
0.349
0.199
0.123
−0.099
0.016
0.089
0.185
0.285
0.367
0.457
0.518
0.509
0.540
0.535
0.581
0.607
0.598
0.578
0.539
0.549
0.560
0.582
0.564
0.579
0.619
0.634
0.690
0.672
0.667
0.672
0.687
0.672
0.671
0.725
0.788
2.044
1.831
1.739
1.507
1.338
1.158
1.017
0.948
0.922
0.936
0.992
0.955
0.871
0.870
1.018
1.185
1.157
1.160
1.119
1.129
0.868
0.669
0.565
0.476
0.433
0.414
0.412
0.461
0.478
0.495
0.471
0.472
0.507
0.534
0.574
0.510
0.493
0.446
0.476
0.447
0.414
0.403
0.388
0.434
0.418
0.438
0.424
0.423
0.439
0.407
0.375
0.403
0.482
0.538
0.567
0.644
0.731
0.852
0.982
1.035
1.026
0.961
0.887
0.941
0.958
0.841
0.710
0.698
0.783
0.790
0.866
0.875
1.137
1.494
1.727
1.824
1.612
1.352
1.090
0.958
0.981
0.923
0.927
0.843
0.810
0.830
0.865
0.927
0.905
0.871
0.842
0.864
0.817
0.739
0.703
0.653
0.667
0.689
0.671
0.650
0.681
0.660
0.562
0.515
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.000
0.062
0.112
0.148
0.178
0.213
0.255
0.304
0.357
0.406
0.449
0.481
0.499
0.502
0.490
0.467
0.440
0.416
0.401
0.397
0.406
0.425
0.452
0.485
0.521
0.559
0.597
0.635
0.669
0.698
0.719
0.729
0.725
0.709
0.687
0.668
0.660
0.666
0.683
0.707
0.734
0.756
0.765
0.748
0.709
0.669
0.649
0.652
0.669
0.692
0.714
0.734
1.121
1.062
1.017
0.975
0.929
0.879
0.832
0.794
0.769
0.757
0.758
0.768
0.784
0.801
0.811
0.809
0.792
0.760
0.718
0.669
0.619
0.573
0.533
0.500
0.474
0.456
0.446
0.444
0.450
0.464
0.485
0.509
0.533
0.547
0.548
0.534
0.510
0.485
0.464
0.454
0.456
0.474
0.504
0.537
0.551
0.533
0.496
0.458
0.431
0.416
0.411
0.414
0.892
0.939
0.972
1.003
1.038
1.074
1.098
1.098
1.070
1.025
0.976
0.935
0.907
0.897
0.904
0.927
0.965
1.012
1.062
1.105
1.129
1.126
1.091
1.031
0.956
0.876
0.802
0.740
0.692
0.660
0.644
0.644
0.659
0.683
0.710
0.730
0.733
0.715
0.681
0.643
0.611
0.595
0.601
0.633
0.683
0.729
0.744
0.722
0.681
0.639
0.605
0.583
15.7
16.4
16.7
17.5
18.8
19.3
18.7
19.1
20.5
20.6
19.6
19.0
18.7
18.4
18.5
18.4
17.7
17.4
17.8
18.2
18.4
18.6
18.4
18.0
17.8
17.5
17.0
16.5
15.9
15.1
14.5
14.5
14.2
13.1
12.7
13.0
12.6
12.1
12.3
12.6
12.0
11.3
11.6
11.8
11.1
10.5
10.9
11.5
10.9
10.4
10.9
11.2
18.3
17.7
17.7
16.1
13.2
11.6
12.7
11.9
8.4
6.9
7.4
7.0
6.9
6.7
5.4
4.4
5.2
6.0
4.9
3.4
2.9
2.6
2.1
1.1
0.7
0.8
0.9
1.0
1.4
2.0
1.8
0.8
0.9
2.7
3.1
1.4
1.0
1.8
1.2
0.1
0.8
1.9
1.0
−0.1
0.8
1.8
0.5
−0.9
0.2
0.9
−0.9
−2.1
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1070
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 19. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pd calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Pd
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.912
1.041
0.318
0.463
0.525
0.552
0.537
0.562
0.642
0.771
0.465
0.541
0.569
0.567
0.603
0.653
0.624
0.575
0.588
0.574
0.599
0.595
0.620
0.603
0.613
0.546
0.511
0.525
0.546
0.571
0.585
0.605
0.594
0.601
0.602
0.607
0.616
0.626
0.637
0.650
0.660
0.663
0.667
0.668
0.672
0.676
0.415
1.098
0.603
0.453
0.420
0.417
0.387
0.327
0.303
0.348
0.588
0.359
0.358
0.332
0.302
0.308
0.412
0.353
0.330
0.315
0.305
0.292
0.294
0.327
0.324
0.325
0.277
0.223
0.195
0.183
0.175
0.179
0.176
0.165
0.157
0.142
0.131
0.123
0.114
0.110
0.110
0.110
0.110
0.104
0.098
0.090
0.413
0.480
1.296
1.080
0.929
0.871
0.883
0.774
0.602
0.487
1.046
0.851
0.792
0.768
0.665
0.592
0.737
0.776
0.725
0.735
0.676
0.665
0.625
0.695
0.673
0.804
0.819
0.686
0.581
0.510
0.469
0.450
0.459
0.424
0.407
0.366
0.330
0.302
0.273
0.253
0.246
0.244
0.240
0.227
0.212
0.193
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.750
0.761
0.766
0.765
0.758
0.745
0.727
0.706
0.684
0.663
0.645
0.634
0.632
0.649
0.528
0.544
0.562
0.574
0.584
0.594
0.604
0.613
0.621
0.629
0.635
0.640
0.644
0.647
0.648
0.648
0.646
0.644
0.641
0.638
0.635
0.631
0.629
0.627
0.625
0.625
0.625
0.627
0.629
0.632
0.635
0.639
0.423
0.436
0.452
0.469
0.485
0.499
0.509
0.513
0.511
0.503
0.490
0.474
0.460
0.472
0.491
0.399
0.370
0.350
0.334
0.320
0.309
0.300
0.294
0.288
0.284
0.281
0.279
0.278
0.276
0.274
0.272
0.269
0.265
0.259
0.253
0.246
0.237
0.228
0.219
0.209
0.199
0.188
0.178
0.169
0.159
0.151
0.570
0.567
0.571
0.583
0.599
0.621
0.647
0.674
0.702
0.727
0.747
0.757
0.752
0.732
0.944
0.876
0.817
0.775
0.737
0.703
0.672
0.645
0.622
0.603
0.587
0.575
0.566
0.560
0.556
0.554
0.553
0.552
0.550
0.547
0.542
0.535
0.526
0.513
0.498
0.481
0.461
0.440
0.418
0.395
0.372
0.349
10.7
10.4
10.5
10.2
9.8
10.0
10.6
10.9
10.2
9.7
10.5
11.5
11.5
11.1
11.2
11.4
11.6
11.8
12.3
12.3
11.4
11.2
11.8
11.7
10.9
10.7
11.1
11.1
10.7
10.4
10.4
10.1
9.7
9.3
9.2
9.7
10.0
9.6
9.4
9.2
8.8
8.7
8.9
8.7
8.3
8.1
−1.2
−1.1
−2.0
−1.8
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1071
TABLE 20. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ag calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Ag
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−315.856
−140.655
−77.567
−47.965
−31.729
−21.829
−15.290
−10.668
−7.155
−4.194
−1.376
−0.239
0.183
0.294
−0.472
−0.778
−1.011
−1.089
−1.076
−0.883
−0.578
−0.252
0.173
0.536
0.881
0.890
0.978
1.120
1.161
1.129
1.187
1.299
1.438
1.497
1.360
1.165
0.908
0.803
0.762
0.760
0.780
0.814
0.839
0.864
0.925
1.027
1.199
1.425
1.267
1.063
0.347
0.260
64.244
19.452
8.304
4.299
2.528
1.634
1.147
0.875
0.747
0.794
1.655
3.255
4.126
5.126
5.480
5.124
4.824
4.404
4.008
3.292
2.739
2.287
1.980
1.852
1.850
1.941
1.857
1.796
1.876
1.830
1.745
1.702
1.767
1.917
2.113
2.209
2.107
1.940
1.793
1.668
1.567
1.491
1.440
1.367
1.302
1.245
1.259
1.444
1.951
2.234
2.008
1.612
0.001
0.001
0.001
0.002
0.002
0.003
0.005
0.008
0.014
0.044
0.357
0.306
0.242
0.194
0.181
0.191
0.199
0.214
0.233
0.283
0.349
0.432
0.501
0.498
0.441
0.426
0.422
0.401
0.385
0.396
0.392
0.371
0.340
0.324
0.335
0.354
0.400
0.440
0.472
0.496
0.512
0.517
0.518
0.523
0.511
0.478
0.416
0.351
0.361
0.365
0.484
0.604
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−381.046
−167.668
−92.057
−56.909
−37.787
−26.256
−18.776
−13.650
−9.980
−7.253
−5.158
−3.488
−2.104
−0.893
0.242
1.387
2.507
2.474
−1.599
−2.937
−1.497
−0.666
−0.153
0.193
0.441
0.627
0.769
0.878
0.956
1.002
1.012
0.980
0.905
0.793
0.664
0.542
0.449
0.394
0.378
0.392
0.430
0.483
0.548
0.620
0.697
0.775
0.850
0.906
0.915
0.831
0.637
0.413
52.998
17.070
8.164
4.990
3.618
2.959
2.629
2.467
2.399
2.390
2.422
2.489
2.598
2.773
3.074
3.658
4.957
7.803
9.249
4.175
2.939
2.589
2.435
2.347
2.289
2.250
2.227
2.221
2.232
2.259
2.298
2.339
2.368
2.366
2.317
2.219
2.084
1.932
1.780
1.639
1.516
1.414
1.332
1.272
1.236
1.227
1.252
1.320
1.432
1.564
1.633
1.559
0.000
0.001
0.001
0.002
0.003
0.004
0.007
0.013
0.023
0.041
0.075
0.136
0.232
0.327
0.323
0.239
0.161
0.116
0.105
0.160
0.270
0.362
0.409
0.423
0.421
0.412
0.401
0.389
0.379
0.370
0.365
0.364
0.368
0.380
0.399
0.425
0.459
0.497
0.538
0.577
0.610
0.633
0.642
0.635
0.614
0.582
0.547
0.515
0.496
0.499
0.531
0.599
0.0
0.0
0.0
0.0
0.1
0.0
0.0
−0.1
−0.2
−0.3
0.8
6.1
15.6
22.0
19.3
11.4
4.9
2.0
1.4
3.8
7.4
9.4
11.0
13.1
13.9
12.7
11.5
11.8
12.2
11.7
11.4
11.3
11.4
11.9
11.9
11.3
11.2
11.6
12.2
13.3
14.3
14.5
14.2
14.4
15.1
15.2
14.4
13.3
13.0
13.2
13.4
13.3
0.1
0.2
0.2
0.3
0.4
0.3
0.5
1.2
4.0
13.1
34.9
73.6
120.2
158.5
175.1
165.3
142.2
119.3
103.7
96.5
105.4
110.9
106.4
97.5
87.8
77.9
67.2
55.5
46.3
41.9
38.9
35.6
32.9
30.1
27.8
26.8
26.1
25.4
24.2
21.9
19.9
19.4
19.0
18.0
17.2
16.1
15.5
16.9
18.1
16.6
14.6
14.2
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1072
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 20. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ag calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ag
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.356
0.328
0.251
0.150
0.213
0.319
0.513
0.680
0.800
0.764
0.789
0.807
0.803
0.867
0.805
0.804
0.715
0.556
0.504
0.481
0.404
0.457
0.552
0.642
0.708
0.753
0.738
0.675
0.680
0.697
0.679
0.649
0.629
0.606
0.621
0.620
0.652
0.647
0.661
0.639
0.627
0.633
0.632
0.584
0.614
0.665
0.590
0.554
0.569
0.454
0.485
0.519
1.423
1.410
1.314
1.113
0.904
0.729
0.630
0.656
0.739
0.821
0.807
0.858
0.829
0.890
0.953
0.980
1.087
1.022
0.934
0.868
0.776
0.625
0.559
0.549
0.575
0.616
0.698
0.677
0.646
0.686
0.666
0.679
0.648
0.617
0.580
0.554
0.552
0.565
0.560
0.561
0.524
0.540
0.551
0.523
0.480
0.525
0.539
0.500
0.529
0.459
0.354
0.339
0.661
0.673
0.734
0.883
1.048
1.151
0.955
0.735
0.623
0.653
0.634
0.618
0.623
0.577
0.613
0.610
0.642
0.755
0.829
0.881
1.013
1.042
0.905
0.769
0.691
0.650
0.676
0.741
0.734
0.717
0.736
0.769
0.795
0.825
0.803
0.801
0.756
0.765
0.747
0.776
0.785
0.779
0.783
0.852
0.790
0.731
0.844
0.898
0.876
1.101
0.981
0.883
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.272
0.238
0.270
0.329
0.396
0.463
0.528
0.589
0.647
0.702
0.754
0.803
0.847
0.879
0.889
0.865
0.803
0.727
0.675
0.659
0.670
0.694
0.722
0.751
0.779
0.803
0.826
0.845
0.861
0.874
0.884
0.891
0.895
0.896
0.894
0.888
0.880
0.868
0.855
0.839
0.823
0.805
0.788
0.770
0.754
0.739
0.726
0.714
0.705
0.697
0.692
0.688
1.376
1.179
1.019
0.900
0.813
0.751
0.706
0.674
0.653
0.641
0.640
0.650
0.673
0.711
0.764
0.821
0.854
0.839
0.784
0.718
0.661
0.618
0.588
0.569
0.557
0.552
0.551
0.553
0.558
0.566
0.575
0.585
0.596
0.608
0.620
0.631
0.641
0.649
0.655
0.659
0.660
0.658
0.654
0.646
0.637
0.625
0.611
0.596
0.580
0.563
0.546
0.530
0.699
0.815
0.917
0.981
0.994
0.964
0.909
0.841
0.773
0.709
0.654
0.609
0.575
0.557
0.556
0.577
0.622
0.680
0.732
0.756
0.746
0.716
0.678
0.641
0.608
0.581
0.559
0.542
0.530
0.522
0.517
0.515
0.515
0.519
0.524
0.532
0.541
0.552
0.565
0.579
0.593
0.609
0.624
0.639
0.654
0.667
0.679
0.689
0.696
0.701
0.703
0.703
13.4
13.7
14.4
16.0
17.7
17.9
17.2
17.4
17.8
17.1
15.6
15.0
14.7
13.9
13.3
13.1
13.0
13.4
13.7
13.3
13.3
13.7
13.6
13.2
13.4
13.8
12.8
11.6
11.9
12.3
12.0
11.5
11.1
10.8
11.1
11.3
10.8
10.3
10.2
10.5
10.6
10.6
10.3
10.2
10.5
10.7
10.5
10.4
10.3
10.1
10.0
10.4
14.8
15.1
13.7
10.8
8.6
9.1
9.7
7.2
4.5
5.0
6.6
6.0
5.1
6.0
6.5
5.7
5.5
5.1
4.7
5.2
4.7
3.4
3.6
4.3
3.0
1.3
2.7
4.2
2.8
1.5
1.6
1.6
1.5
1.8
1.2
0.9
1.7
1.9
1.6
1.3
0.9
0.4
0.5
1.0
0.9
0.3
−0.9
−1.7
−1.1
−0.2
−0.4
−2.1
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1073
TABLE 20. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ag calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ag
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.549
0.596
0.623
0.667
0.659
0.667
0.652
0.654
0.672
0.691
0.705
0.728
0.748
0.717
0.703
0.739
0.666
0.542
0.582
0.609
0.635
0.667
0.722
0.696
0.669
0.669
0.684
0.702
0.734
0.718
0.691
0.674
0.672
0.677
0.686
0.694
0.705
0.710
0.713
0.709
0.701
0.688
0.688
0.690
0.696
0.705
0.287
0.280
0.265
0.288
0.299
0.308
0.305
0.283
0.273
0.265
0.283
0.263
0.310
0.334
0.334
0.334
0.428
0.325
0.240
0.219
0.194
0.180
0.185
0.247
0.228
0.204
0.186
0.180
0.185
0.224
0.229
0.209
0.189
0.173
0.162
0.155
0.151
0.153
0.155
0.159
0.159
0.145
0.130
0.114
0.099
0.090
0.748
0.645
0.579
0.545
0.570
0.571
0.589
0.557
0.518
0.484
0.491
0.439
0.473
0.534
0.551
0.509
0.683
0.815
0.606
0.522
0.439
0.377
0.333
0.453
0.457
0.417
0.370
0.343
0.323
0.396
0.432
0.420
0.387
0.354
0.327
0.306
0.290
0.290
0.290
0.302
0.308
0.292
0.266
0.234
0.200
0.179
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.685
0.684
0.685
0.686
0.688
0.691
0.694
0.697
0.701
0.705
0.709
0.713
0.717
0.720
0.724
0.727
0.730
0.733
0.735
0.737
0.739
0.741
0.742
0.743
0.744
0.744
0.744
0.744
0.744
0.744
0.743
0.743
0.742
0.741
0.741
0.740
0.739
0.738
0.737
0.737
0.736
0.736
0.735
0.735
0.735
0.734
0.514
0.498
0.483
0.469
0.456
0.444
0.432
0.422
0.412
0.403
0.395
0.388
0.381
0.375
0.369
0.364
0.359
0.355
0.351
0.347
0.343
0.340
0.336
0.333
0.330
0.327
0.323
0.320
0.317
0.314
0.310
0.307
0.303
0.299
0.295
0.291
0.287
0.283
0.278
0.274
0.269
0.264
0.260
0.255
0.250
0.245
0.700
0.695
0.688
0.679
0.669
0.658
0.647
0.635
0.623
0.611
0.600
0.589
0.578
0.568
0.559
0.551
0.543
0.535
0.529
0.522
0.517
0.512
0.507
0.503
0.498
0.495
0.491
0.488
0.484
0.481
0.478
0.475
0.471
0.468
0.464
0.461
0.457
0.452
0.448
0.443
0.438
0.433
0.427
0.421
0.415
0.409
11.2
11.0
9.9
9.7
10.5
10.7
10.5
10.5
10.5
10.4
10.1
10.2
11.1
11.3
10.3
9.8
10.2
10.0
9.3
9.4
10.2
9.6
8.0
8.1
9.6
9.4
7.7
7.6
9.1
9.3
8.1
7.4
8.3
9.3
8.6
7.7
8.1
8.7
8.6
7.9
7.5
7.6
7.9
8.4
8.7
8.0
−3.8
−3.1
−0.6
−0.7
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1074
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 21. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ta calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Ta
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−288.473
−121.442
−60.953
−33.328
−19.051
−10.073
−3.000
2.022
4.715
5.365
4.746
3.321
1.950
1.509
1.898
2.753
3.887
5.040
4.191
−8.114
−9.550
−7.493
−5.120
−3.453
−2.236
−1.292
−1.679
−2.052
−2.040
−1.814
−1.631
−1.191
−1.126
−1.015
−0.682
−0.442
−0.473
−0.459
−0.830
−1.062
−1.199
−0.926
−0.775
−0.863
−0.548
−0.275
−0.061
−0.058
0.113
0.221
0.264
0.273
64.569
21.888
12.115
9.805
8.694
7.549
7.375
9.025
11.395
13.703
15.239
15.867
15.133
13.908
12.912
12.549
13.000
14.948
18.390
19.484
10.263
6.282
4.382
3.647
3.355
3.977
4.366
3.972
3.294
2.807
2.536
2.317
2.223
1.965
1.834
2.011
2.057
2.147
2.285
1.921
1.441
1.080
1.185
0.741
0.519
0.429
0.564
0.500
0.462
0.497
0.561
0.506
0.001
0.001
0.003
0.008
0.020
0.048
0.116
0.106
0.075
0.063
0.060
0.060
0.065
0.071
0.076
0.076
0.071
0.060
0.052
0.044
0.052
0.066
0.096
0.145
0.206
0.227
0.200
0.199
0.219
0.251
0.279
0.341
0.358
0.402
0.479
0.474
0.462
0.445
0.387
0.399
0.410
0.534
0.591
0.573
0.910
1.652
1.752
1.973
2.040
1.681
1.460
1.529
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−541.808
−238.276
−131.924
−82.720
−56.042
−40.018
−29.690
−22.689
−17.770
−14.223
−11.623
−9.695
−8.259
−7.187
−6.385
−5.782
−5.325
−4.970
−4.686
−4.237
−3.855
−3.482
−3.101
−2.707
−2.292
−1.829
−1.213
−1.837
−2.140
−1.634
−1.325
−1.115
−0.971
−0.879
−0.824
−0.791
−0.764
−0.730
−0.683
−0.621
−0.545
−0.459
−0.367
−0.280
−0.236
−0.252
−0.230
−0.156
−0.073
0.008
0.083
0.154
64.899
33.368
22.804
17.751
14.864
13.041
11.818
10.962
10.343
9.880
9.518
9.217
8.947
8.687
8.422
8.141
7.839
7.517
7.177
6.462
5.745
5.070
4.464
3.943
3.516
3.205
3.182
4.357
2.705
2.328
2.169
2.072
1.999
1.932
1.856
1.763
1.651
1.526
1.395
1.268
1.152
1.052
0.976
0.932
0.920
0.866
0.748
0.647
0.577
0.525
0.485
0.455
0.000
0.001
0.001
0.002
0.004
0.007
0.012
0.017
0.024
0.033
0.042
0.052
0.060
0.068
0.075
0.082
0.087
0.093
0.098
0.108
0.120
0.134
0.151
0.172
0.200
0.235
0.274
0.195
0.227
0.288
0.336
0.374
0.405
0.429
0.450
0.472
0.499
0.533
0.578
0.636
0.709
0.798
0.898
0.984
1.020
1.064
1.221
1.459
1.707
1.904
2.001
1.974
−0.1
−0.1
−0.1
−0.1
−0.2
−0.2
−0.2
−0.2
0.1
1.0
2.0
2.3
2.2
2.0
2.0
2.0
2.1
2.3
2.6
3.2
3.7
4.0
4.3
5.0
6.1
6.4
6.5
7.4
7.8
8.0
9.3
10.0
9.6
9.7
11.2
13.0
13.8
14.1
14.5
14.9
15.8
17.6
19.3
20.5
21.7
23.1
24.4
26.0
29.0
32.1
34.0
35.1
0.7
1.0
1.4
1.7
2.0
2.1
3.0
6.0
12.3
22.7
33.7
40.1
40.6
37.6
34.6
33.1
33.1
34.1
35.6
39.1
41.8
43.2
44.5
45.9
46.8
49.6
52.3
52.0
53.5
56.2
55.5
56.3
61.4
64.5
62.6
60.7
62.6
65.0
65.9
67.0
66.9
64.6
62.2
59.9
57.8
56.6
55.5
50.7
42.4
36.0
32.2
25.8
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1075
TABLE 21. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ta calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ta
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.366
0.416
0.462
0.592
0.098
0.273
0.384
0.468
0.544
0.612
0.668
0.750
0.830
0.887
0.929
0.987
1.054
1.112
1.200
1.324
1.488
1.664
2.073
1.963
1.311
1.550
1.849
1.897
1.766
1.486
1.160
0.969
0.624
−0.122
−0.776
−0.595
−0.309
−0.140
−0.024
0.052
0.103
0.152
0.213
0.273
0.295
0.326
0.327
0.339
0.308
0.328
0.326
0.315
0.510
0.516
0.536
0.688
0.643
0.379
0.311
0.276
0.243
0.231
0.201
0.182
0.188
0.200
0.210
0.195
0.191
0.196
0.168
0.160
0.188
0.245
0.395
1.311
0.874
0.787
0.894
1.376
1.663
2.003
2.114
2.036
2.895
1.957
1.999
1.042
0.791
0.702
0.609
0.566
0.555
0.490
0.468
0.468
0.446
0.468
0.451
0.451
0.442
0.401
0.398
0.365
1.295
1.175
1.070
0.835
1.521
1.739
1.273
0.937
0.685
0.540
0.413
0.305
0.259
0.242
0.231
0.193
0.167
0.153
0.114
0.090
0.084
0.087
0.089
0.235
0.352
0.260
0.212
0.251
0.283
0.322
0.364
0.401
0.330
0.509
0.435
0.724
1.097
1.370
1.639
1.752
1.740
1.862
1.769
1.595
1.559
1.440
1.452
1.417
1.524
1.493
1.503
1.569
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.220
0.282
0.341
0.395
0.443
0.479
0.499
0.505
0.511
0.529
0.558
0.595
0.635
0.675
0.714
0.754
0.792
0.831
0.869
0.907
0.946
0.984
1.021
1.056
1.086
1.107
1.112
1.090
1.033
0.945
0.842
0.754
0.699
0.677
0.680
0.698
0.724
0.751
0.777
0.799
0.813
0.819
0.812
0.792
0.757
0.711
0.657
0.603
0.555
0.517
0.491
0.477
0.431
0.413
0.402
0.398
0.401
0.411
0.421
0.418
0.398
0.368
0.339
0.316
0.300
0.289
0.282
0.279
0.280
0.284
0.291
0.303
0.319
0.341
0.370
0.408
0.457
0.520
0.594
0.677
0.754
0.804
0.810
0.773
0.711
0.645
0.589
0.547
0.521
0.509
0.509
0.520
0.540
0.568
0.599
0.631
0.659
0.675
0.677
0.662
0.632
0.591
0.545
0.497
1.842
1.650
1.446
1.265
1.124
1.032
0.988
0.973
0.949
0.887
0.795
0.696
0.608
0.536
0.478
0.432
0.396
0.368
0.347
0.331
0.320
0.314
0.314
0.318
0.329
0.347
0.374
0.411
0.461
0.523
0.593
0.663
0.715
0.738
0.728
0.695
0.655
0.618
0.590
0.573
0.567
0.572
0.588
0.616
0.654
0.702
0.760
0.825
0.894
0.959
1.013
1.047
35.2
34.1
31.7
28.7
27.2
26.3
24.5
22.6
21.7
20.8
19.5
18.9
18.4
16.9
15.0
13.7
12.9
11.9
10.8
9.9
8.9
8.8
9.4
9.0
7.8
7.8
9.3
9.8
9.1
8.8
9.4
10.4
10.6
10.5
11.3
11.7
11.0
10.9
11.4
11.4
11.6
11.8
11.1
10.1
9.9
11.0
12.1
11.3
10.6
12.1
13.1
12.4
18.6
15.4
17.2
18.7
14.0
9.9
12.2
14.7
13.6
13.0
12.8
10.0
7.7
9.0
11.3
11.3
9.2
8.2
8.3
8.4
8.9
7.0
3.6
5.3
10.1
9.8
5.8
5.8
9.0
9.1
7.2
6.9
8.3
7.9
5.0
4.7
7.7
7.2
2.9
1.0
1.1
0.8
1.7
3.7
4.7
3.4
1.3
2.5
4.1
2.2
2.3
4.2
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1076
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 21. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Ta calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Ta
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.314
0.339
0.368
0.400
0.425
0.422
0.432
0.450
0.466
0.490
0.518
0.553
0.545
0.540
0.541
0.568
0.587
0.593
0.598
0.603
0.615
0.622
0.640
0.661
0.681
0.674
0.659
0.645
0.647
0.659
0.662
0.674
0.692
0.698
0.702
0.707
0.715
0.721
0.727
0.729
0.736
0.740
0.741
0.742
0.746
0.751
0.322
0.279
0.252
0.235
0.235
0.233
0.202
0.192
0.167
0.154
0.143
0.150
0.184
0.159
0.138
0.126
0.126
0.134
0.125
0.120
0.114
0.105
0.098
0.097
0.114
0.132
0.137
0.123
0.101
0.099
0.088
0.076
0.074
0.078
0.077
0.073
0.073
0.072
0.072
0.070
0.069
0.071
0.071
0.068
0.064
0.062
1.591
1.445
1.267
1.094
0.994
1.002
0.887
0.802
0.683
0.583
0.493
0.458
0.556
0.500
0.441
0.372
0.349
0.362
0.336
0.318
0.290
0.263
0.233
0.218
0.239
0.280
0.301
0.285
0.235
0.224
0.196
0.165
0.152
0.159
0.155
0.144
0.141
0.137
0.135
0.130
0.126
0.129
0.128
0.123
0.115
0.110
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.472
0.475
0.483
0.494
0.507
0.521
0.536
0.550
0.564
0.578
0.592
0.604
0.616
0.628
0.638
0.649
0.658
0.667
0.676
0.684
0.692
0.699
0.706
0.712
0.718
0.724
0.730
0.735
0.740
0.745
0.749
0.754
0.758
0.762
0.765
0.769
0.772
0.776
0.779
0.782
0.785
0.787
0.790
0.793
0.795
0.798
0.452
0.411
0.374
0.342
0.315
0.291
0.270
0.253
0.237
0.224
0.213
0.203
0.194
0.187
0.180
0.174
0.169
0.164
0.159
0.156
0.152
0.149
0.146
0.143
0.140
0.138
0.136
0.134
0.132
0.130
0.128
0.126
0.124
0.123
0.121
0.120
0.118
0.117
0.115
0.114
0.113
0.111
0.110
0.108
0.107
0.106
1.058
1.042
1.003
0.948
0.884
0.817
0.751
0.689
0.633
0.583
0.539
0.500
0.466
0.435
0.409
0.386
0.365
0.347
0.331
0.316
0.303
0.291
0.280
0.271
0.262
0.254
0.246
0.239
0.233
0.227
0.221
0.216
0.211
0.206
0.202
0.198
0.194
0.190
0.186
0.183
0.179
0.176
0.173
0.169
0.166
0.164
12.9
14.5
15.0
14.3
13.8
14.8
14.8
12.8
12.2
13.4
13.3
11.7
10.6
10.3
10.5
10.0
9.4
9.5
9.7
8.9
7.9
8.5
9.6
9.0
7.5
6.8
7.5
7.9
7.0
6.2
6.4
6.5
5.5
5.0
5.6
6.0
5.5
4.7
4.9
5.6
5.3
4.3
4.1
4.9
5.0
4.1
1.6
−2.1
−2.3
0.1
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1077
TABLE 22. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of W calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
W
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−156.936
−50.462
−21.027
−8.119
1.527
6.747
6.848
5.737
5.081
5.268
5.303
3.868
2.568
2.255
2.454
2.866
3.041
3.621
3.042
−10.483
−11.305
−9.064
−7.398
−5.512
−4.155
−2.766
−2.084
−0.783
−1.313
−1.402
−1.747
−1.720
−1.805
−1.384
−1.591
−1.567
−1.046
−0.671
−0.316
−0.112
0.086
−0.450
−1.021
−1.133
−1.265
−0.985
−1.429
−1.229
−0.980
−0.721
−0.448
−0.298
54.751
30.936
29.63
23.739
21.326
23.220
24.892
24.423
22.972
21.961
21.915
22.053
20.794
19.453
18.511
18.417
18.631
19.774
23.437
22.284
12.727
8.316
6.286
4.547
3.564
3.571
3.408
3.754
3.995
4.281
3.927
3.580
3.225
2.803
2.708
2.165
1.833
1.761
1.859
2.031
2.365
2.960
2.548
2.290
1.819
1.647
1.607
0.981
0.743
0.584
0.536
0.611
0.002
0.009
0.022
0.038
0.047
0.040
0.037
0.039
0.042
0.043
0.043
0.044
0.047
0.051
0.053
0.053
0.052
0.049
0.042
0.037
0.044
0.055
0.067
0.089
0.119
0.175
0.214
0.255
0.226
0.211
0.213
0.227
0.236
0.287
0.275
0.303
0.412
0.496
0.523
0.491
0.422
0.330
0.338
0.351
0.371
0.447
0.348
0.397
0.491
0.678
1.098
1.321
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−591.677
−213.138
−153.784
−135.890
−90.863
−60.175
−39.721
−25.668
−18.551
−21.449
−23.295
−18.804
−12.725
−8.725
−15.518
−14.260
−10.131
−7.494
−8.485
−10.067
−6.983
−5.094
−4.315
−3.808
−3.130
−2.433
−1.874
−1.506
−1.331
−1.310
−1.374
−1.444
−1.467
−1.429
−1.340
−1.224
−1.104
−0.997
−0.914
−0.861
−0.833
−0.822
−0.814
−0.799
−0.770
−0.725
−0.666
−0.596
−0.518
−0.436
−0.351
−0.262
58.689
72.950
114.349
52.032
22.973
14.008
11.889
13.929
19.871
23.118
16.895
11.390
10.101
14.450
15.600
8.807
7.253
8.646
10.712
6.051
4.077
3.967
3.963
3.547
3.110
2.903
2.896
2.995
3.111
3.167
3.115
2.959
2.735
2.495
2.275
2.095
1.958
1.859
1.785
1.720
1.652
1.568
1.467
1.351
1.226
1.102
0.985
0.877
0.782
0.699
0.628
0.570
0.000
0.001
0.003
0.002
0.003
0.004
0.007
0.016
0.027
0.023
0.020
0.024
0.038
0.051
0.032
0.031
0.047
0.066
0.057
0.044
0.062
0.095
0.115
0.131
0.160
0.202
0.243
0.266
0.272
0.270
0.269
0.273
0.284
0.302
0.326
0.356
0.388
0.418
0.444
0.465
0.483
0.500
0.521
0.548
0.585
0.633
0.697
0.780
0.889
1.030
1.213
1.448
−0.1
−0.1
−0.1
−0.2
−0.2
−0.2
−0.3
−0.4
−0.6
−0.2
0.7
1.4
1.6
1.4
1.2
1.0
0.9
1.0
1.1
1.4
1.5
1.9
2.6
2.9
3.2
4.3
5.7
6.7
7.2
7.5
7.4
7.2
7.5
8.0
8.0
8.1
8.7
9.2
9.7
10.7
11.2
10.8
10.7
11.7
12.9
13.3
14.1
15.4
16.8
18.6
20.2
21.7
0.8
1.1
1.5
1.9
2.3
2.2
2.9
5.5
10.5
17.5
23.7
26.8
27.0
25.6
24.1
23.3
23.4
24.4
26.0
30.2
34.9
38.6
42.4
50.0
58.1
61.3
62.3
64.1
66.0
66.2
66.1
67.3
67.7
67.2
67.3
67.6
67.9
69.2
68.6
64.3
61.2
62.3
63.6
61.6
60.9
63.2
63.0
61.0
60.6
60.2
57.5
52.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1078
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 22. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of W calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
W
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.249
−0.256
−0.203
−0.120
−0.039
0.121
0.249
0.321
0.374
0.438
0.523
0.645
0.654
0.688
0.710
0.772
0.851
0.835
0.763
0.827
0.851
0.950
0.874
1.008
1.161
1.362
1.688
2.033
1.819
1.496
1.216
0.903
0.882
0.267
0.008
−0.420
−0.127
0.033
0.114
0.191
0.333
0.392
0.483
0.582
0.710
0.797
0.770
0.788
0.715
0.512
0.502
0.082
0.624
0.586
0.522
0.394
0.335
0.254
0.280
0.309
0.306
0.288
0.260
0.316
0.359
0.374
0.379
0.366
0.408
0.606
0.431
0.428
0.391
0.402
0.379
0.289
0.266
0.262
0.336
0.810
1.374
1.661
1.728
1.703
1.800
2.138
1.804
1.028
0.747
0.650
0.587
0.465
0.438
0.414
0.389
0.384
0.415
0.602
0.634
0.770
0.872
0.955
0.942
1.209
1.382
1.433
1.664
2.322
2.944
3.204
1.997
1.554
1.310
1.049
0.761
0.612
0.646
0.610
0.585
0.501
0.458
0.569
0.561
0.494
0.446
0.378
0.418
0.262
0.187
0.136
0.113
0.169
0.264
0.332
0.387
0.458
0.448
0.461
0.554
0.833
1.302
1.533
1.643
1.841
1.447
1.275
1.012
0.790
0.614
0.604
0.637
0.635
0.685
0.813
0.826
0.824
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.166
−0.059
−0.165
−0.100
−0.009
0.066
0.131
0.191
0.247
0.299
0.349
0.397
0.442
0.486
0.529
0.570
0.610
0.650
0.690
0.730
0.770
0.811
0.853
0.896
0.942
0.991
1.042
1.097
1.153
1.204
1.233
1.207
1.078
0.857
0.662
0.583
0.603
0.672
0.618
0.486
0.560
0.632
0.689
0.733
0.768
0.795
0.813
0.822
0.821
0.809
0.786
0.753
0.529
0.547
0.582
0.422
0.368
0.335
0.310
0.290
0.272
0.257
0.244
0.234
0.224
0.217
0.211
0.206
0.203
0.201
0.201
0.202
0.205
0.210
0.217
0.229
0.244
0.265
0.295
0.338
0.399
0.488
0.614
0.779
0.941
1.000
0.913
0.764
0.648
0.620
0.720
0.556
0.439
0.401
0.391
0.396
0.409
0.430
0.456
0.486
0.519
0.552
0.581
0.604
1.721
1.807
1.590
2.244
2.715
2.871
2.734
2.406
2.016
1.651
1.345
1.102
0.913
0.766
0.651
0.561
0.490
0.434
0.388
0.352
0.322
0.299
0.281
0.267
0.258
0.252
0.252
0.256
0.268
0.289
0.324
0.378
0.460
0.577
0.718
0.827
0.827
0.741
0.800
1.019
0.867
0.715
0.624
0.570
0.540
0.526
0.524
0.533
0.550
0.575
0.608
0.648
24.4
27.5
28.9
29.9
32.8
37.1
39.7
39.2
36.4
33.0
31.2
29.0
25.7
22.8
20.4
18.4
16.8
14.5
12.5
12.0
12.3
11.6
10.3
9.7
9.2
8.9
8.9
8.4
7.5
7.3
7.7
8.1
8.4
8.7
8.5
9.0
10.7
11.4
11.1
11.1
11.8
12.9
13.0
11.5
10.2
10.3
10.9
10.9
10.3
9.9
9.6
9.6
46.2
39.5
35.3
31.7
23.7
11.1
0.7
−1.7
2.4
5.6
1.6
−0.7
2.9
5.1
5.3
5.3
5.1
5.9
6.9
6.2
5.0
5.6
6.5
6.8
6.9
5.2
3.2
3.9
5.2
4.4
3.6
4.9
5.9
5.9
6.8
5.5
2.2
3.3
6.3
6.1
4.2
2.1
1.7
3.9
5.2
3.6
1.3
1.1
1.3
0.0
−0.4
0.8
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1079
TABLE 22. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of W calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
W
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.011
0.124
0.186
0.225
0.262
0.274
0.320
0.360
0.381
0.405
0.403
0.457
0.490
0.493
0.498
0.481
0.489
0.483
0.476
0.489
0.500
0.509
0.520
0.535
0.546
0.561
0.560
0.571
0.586
0.597
0.604
0.610
0.614
0.632
0.618
0.604
0.617
0.640
0.650
0.656
0.668
0.674
0.676
0.683
0.693
0.703
0.643
0.516
0.459
0.412
0.369
0.348
0.300
0.286
0.282
0.267
0.259
0.235
0.243
0.261
0.270
0.263
0.265
0.248
0.223
0.206
0.191
0.184
0.168
0.158
0.151
0.153
0.150
0.132
0.127
0.129
0.123
0.126
0.118
0.128
0.144
0.115
0.090
0.083
0.087
0.082
0.081
0.083
0.081
0.076
0.072
0.076
1.555
1.831
1.869
1.869
1.802
1.772
1.560
1.354
1.253
1.136
1.130
0.888
0.812
0.837
0.842
0.874
0.856
0.842
0.806
0.730
0.668
0.628
0.561
0.508
0.472
0.452
0.446
0.386
0.354
0.347
0.324
0.326
0.301
0.308
0.357
0.303
0.231
0.199
0.203
0.189
0.179
0.179
0.174
0.161
0.149
0.152
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.712
0.667
0.622
0.582
0.549
0.525
0.510
0.502
0.501
0.505
0.512
0.522
0.533
0.545
0.558
0.571
0.584
0.596
0.608
0.620
0.631
0.642
0.652
0.662
0.671
0.680
0.688
0.696
0.703
0.711
0.717
0.724
0.730
0.736
0.741
0.746
0.751
0.756
0.760
0.764
0.768
0.772
0.776
0.779
0.782
0.785
0.616
0.617
0.605
0.582
0.550
0.514
0.475
0.437
0.401
0.369
0.339
0.313
0.290
0.270
0.252
0.237
0.224
0.212
0.202
0.193
0.185
0.179
0.173
0.167
0.163
0.159
0.155
0.152
0.149
0.146
0.144
0.142
0.140
0.138
0.137
0.135
0.134
0.133
0.132
0.131
0.130
0.129
0.128
0.127
0.126
0.125
0.695
0.748
0.804
0.859
0.911
0.952
0.978
0.986
0.974
0.943
0.899
0.845
0.787
0.729
0.673
0.620
0.573
0.530
0.492
0.458
0.429
0.402
0.379
0.359
0.341
0.325
0.311
0.299
0.288
0.278
0.269
0.260
0.253
0.246
0.240
0.235
0.230
0.225
0.221
0.217
0.214
0.210
0.207
0.204
0.201
0.198
9.5
9.4
9.7
10.9
11.8
11.6
11.4
11.7
12.5
13.0
12.6
12.3
12.4
12.1
11.4
11.0
10.9
11.3
11.5
10.7
9.9
9.5
9.5
9.6
9.4
8.4
7.7
7.7
7.2
7.0
7.9
7.8
6.3
6.0
7.0
7.0
6.4
6.7
6.9
6.5
5.7
4.6
4.7
5.5
4.9
4.2
2.1
2.2
1.1
−1.0
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1080
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 23. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pt calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Pt
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−123.344
−32.952
−44.891
−40.081
−33.179
−26.710
−21.394
−17.271
−13.732
−11.458
−9.901
−8.355
−7.181
−6.385
−5.611
−4.607
−3.868
−3.223
−2.493
−1.352
−0.141
0.686
1.560
1.086
1.316
0.686
0.775
−0.128
−0.506
−0.289
−0.071
0.132
0.126
0.508
0.649
0.830
0.860
0.830
0.830
0.799
0.964
1.196
1.456
1.805
1.928
1.358
1.423
0.385
−0.684
−0.553
−0.500
−0.715
72.533
101.354
81.698
57.069
40.443
29.831
23.502
18.644
15.691
13.892
11.956
10.503
9.459
8.469
7.295
6.473
5.880
5.274
4.740
4.070
3.883
3.817
4.288
4.800
4.863
5.058
4.839
5.073
4.098
3.496
3.088
3.176
2.571
2.513
2.448
2.479
2.539
2.523
2.479
2.313
2.160
2.130
2.187
2.402
3.003
3.406
3.868
4.657
3.630
3.095
2.768
2.752
0.004
0.009
0.009
0.012
0.015
0.019
0.023
0.029
0.036
0.043
0.050
0.058
0.067
0.075
0.086
0.103
0.119
0.138
0.165
0.221
0.257
0.254
0.206
0.198
0.192
0.194
0.201
0.197
0.240
0.284
0.324
0.314
0.388
0.382
0.382
0.363
0.353
0.358
0.363
0.386
0.386
0.357
0.317
0.266
0.236
0.253
0.228
0.213
0.266
0.313
0.350
0.340
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−635.547
−294.704
−165.622
−103.964
−69.907
−49.132
−35.496
−26.009
−19.067
−13.748
−9.570
−6.797
−7.017
−7.938
−6.730
−5.354
−4.825
−4.448
−3.694
−2.255
−1.320
−0.756
−0.433
−0.261
−0.168
−0.077
−0.353
−0.442
−0.583
−0.505
−0.445
−0.382
−0.309
−0.227
−0.136
−0.039
0.062
0.162
0.257
0.337
0.387
0.387
0.313
0.156
−0.061
−0.279
−0.439
−0.520
−0.533
−0.500
−0.442
−0.372
216.023
68.276
29.962
16.097
10.023
7.046
5.511
4.755
4.506
4.718
5.574
7.492
9.412
8.097
6.530
6.182
6.126
5.495
4.904
4.480
4.444
4.487
4.519
4.511
4.462
4.425
4.617
4.178
3.845
3.558
3.330
3.122
2.933
2.766
2.622
2.503
2.412
2.352
2.327
2.339
2.391
2.476
2.573
2.639
2.623
2.503
2.301
2.069
1.846
1.652
1.490
1.358
0.000
0.001
0.001
0.001
0.002
0.003
0.004
0.007
0.012
0.022
0.045
0.073
0.068
0.063
0.074
0.092
0.101
0.110
0.130
0.178
0.207
0.217
0.219
0.221
0.224
0.226
0.215
0.237
0.254
0.276
0.295
0.316
0.337
0.359
0.380
0.399
0.414
0.423
0.425
0.419
0.408
0.394
0.383
0.378
0.381
0.395
0.419
0.455
0.500
0.554
0.617
0.685
0.0
0.0
−0.1
−0.1
−0.1
−0.1
−0.2
−0.3
−0.5
−0.4
0.4
1.6
2.3
2.3
2.3
2.5
3.0
3.7
4.3
4.9
5.7
6.9
6.8
6.0
5.9
6.4
6.3
5.8
5.9
6.6
7.4
8.2
9.2
9.5
8.8
8.5
9.1
9.8
10.1
10.3
10.2
9.8
9.5
9.5
9.6
9.5
9.4
9.3
9.6
10.4
11.4
12.0
0.4
0.6
0.8
1.0
1.3
1.3
1.9
4.2
9.9
20.3
32.0
39.7
42.6
42.8
42.8
43.8
46.3
50.4
56.2
69.7
74.6
67.6
61.5
61.2
61.5
58.7
55.9
56.0
56.5
54.3
50.8
48.1
45.4
44.1
44.7
44.4
42.5
39.4
36.7
35.3
35.2
34.8
33.5
32.3
30.9
29.3
29.5
31.3
32.1
30.6
28.5
28.6
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1081
TABLE 23. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pt calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Pt
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.983
−0.890
−0.733
−0.651
−0.384
−0.139
0.023
−0.039
0.023
−0.051
−0.025
0.128
0.426
0.386
0.131
−0.204
−0.372
−0.341
−0.305
−0.194
−0.084
0.078
0.147
0.184
0.196
0.283
0.324
0.324
0.371
0.401
0.375
0.372
0.405
0.409
0.419
0.442
0.480
0.479
0.480
0.497
0.523
0.567
0.609
0.555
0.579
0.610
0.554
0.558
0.612
0.677
0.701
0.764
2.315
1.867
1.628
1.430
1.163
1.145
1.251
1.301
1.277
1.269
1.088
0.986
1.114
1.383
1.628
1.513
1.211
0.979
0.809
0.673
0.563
0.537
0.552
0.573
0.517
0.511
0.550
0.533
0.511
0.545
0.577
0.515
0.507
0.499
0.484
0.469
0.439
0.471
0.447
0.426
0.399
0.395
0.439
0.441
0.415
0.439
0.444
0.376
0.333
0.348
0.346
0.339
0.366
0.436
0.511
0.579
0.775
0.861
0.799
0.768
0.783
0.787
0.918
0.997
0.783
0.671
0.611
0.649
0.755
0.911
1.082
1.373
1.737
1.825
1.693
1.582
1.691
1.497
1.350
1.370
1.281
1.191
1.219
1.276
1.203
1.199
1.182
1.129
1.037
1.043
1.039
0.994
0.923
0.827
0.778
0.878
0.818
0.777
0.881
0.830
0.686
0.600
0.567
0.485
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.298
−0.223
−0.150
−0.080
−0.013
0.051
0.111
0.167
0.216
0.254
0.271
0.258
0.218
0.170
0.140
0.137
0.155
0.184
0.217
0.251
0.284
0.315
0.345
0.374
0.400
0.425
0.449
0.472
0.492
0.511
0.528
0.542
0.553
0.562
0.569
0.577
0.587
0.598
0.610
0.624
0.639
0.654
0.670
0.685
0.701
0.718
0.734
0.752
0.771
0.791
0.813
0.840
1.250
1.161
1.089
1.030
0.983
0.947
0.921
0.907
0.905
0.917
0.940
0.963
0.965
0.930
0.867
0.794
0.728
0.673
0.629
0.593
0.564
0.539
0.519
0.501
0.486
0.473
0.462
0.453
0.446
0.440
0.436
0.432
0.428
0.423
0.415
0.405
0.394
0.382
0.371
0.361
0.352
0.344
0.338
0.332
0.328
0.324
0.322
0.320
0.320
0.321
0.324
0.329
0.757
0.831
0.901
0.965
1.017
1.053
1.070
1.066
1.045
1.013
0.982
0.969
0.986
1.040
1.125
1.222
1.313
1.382
1.421
1.430
1.415
1.382
1.336
1.283
1.226
1.169
1.112
1.059
1.011
0.968
0.931
0.900
0.876
0.855
0.836
0.815
0.789
0.760
0.728
0.695
0.662
0.630
0.601
0.573
0.547
0.523
0.500
0.479
0.459
0.441
0.423
0.405
12.3
13.1
14.3
15.3
15.9
16.9
17.9
18.4
18.2
18.2
18.4
18.2
18.2
18.8
19.0
18.1
18.1
20.0
21.4
20.6
20.2
21.3
22.3
21.9
20.5
19.3
19.0
19.3
19.7
19.2
17.9
17.0
16.8
16.0
15.0
15.1
15.5
15.0
14.4
13.7
12.9
12.6
12.7
12.6
12.1
11.8
11.4
10.8
10.3
10.1
10.2
10.2
30.2
29.1
25.7
23.9
23.6
21.7
18.3
15.8
15.4
16.1
15.6
14.0
11.5
8.7
8.0
10.5
10.3
4.8
1.2
3.1
3.9
0.6
−3.4
−4.1
−1.1
2.2
1.5
−2.2
−4.7
−4.2
−1.6
0.0
−2.0
−2.4
0.5
0.1
−2.9
−3.2
−1.9
−0.3
1.6
1.9
0.5
−0.4
0.3
1.1
0.9
1.1
2.0
2.3
1.1
0.2
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1082
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 23. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pt calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Pt
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.903
1.276
0.477
0.326
0.420
0.465
0.487
0.491
0.498
0.508
0.526
0.561
0.605
0.571
0.537
0.566
0.555
0.554
0.534
0.553
0.558
0.571
0.580
0.588
0.600
0.631
0.674
0.799
0.572
0.543
0.576
0.583
0.581
0.589
0.603
0.637
0.571
0.592
0.635
0.625
0.631
0.643
0.669
0.618
0.624
0.634
0.329
0.656
1.148
0.631
0.510
0.474
0.457
0.431
0.412
0.376
0.356
0.342
0.350
0.439
0.362
0.355
0.349
0.351
0.319
0.304
0.286
0.277
0.267
0.264
0.237
0.224
0.223
0.265
0.455
0.290
0.265
0.264
0.246
0.227
0.220
0.226
0.228
0.178
0.166
0.207
0.178
0.177
0.199
0.194
0.168
0.155
0.356
0.318
0.743
1.251
1.168
1.076
1.024
1.010
0.987
0.942
0.883
0.793
0.717
0.847
0.863
0.796
0.812
0.816
0.824
0.762
0.728
0.687
0.655
0.635
0.571
0.499
0.442
0.373
0.852
0.765
0.660
0.646
0.618
0.570
0.534
0.494
0.604
0.466
0.385
0.477
0.413
0.398
0.409
0.463
0.403
0.363
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.874
0.923
1.015
1.052
0.569
0.692
0.748
0.772
0.776
0.762
0.733
0.694
0.655
0.623
0.603
0.594
0.593
0.597
0.605
0.615
0.625
0.635
0.645
0.654
0.663
0.671
0.678
0.685
0.691
0.697
0.703
0.707
0.712
0.716
0.720
0.723
0.726
0.729
0.732
0.734
0.737
0.739
0.741
0.742
0.744
0.746
0.339
0.360
0.423
0.828
0.543
0.431
0.430
0.449
0.476
0.502
0.521
0.525
0.513
0.487
0.455
0.421
0.390
0.363
0.341
0.322
0.307
0.295
0.284
0.276
0.269
0.262
0.257
0.252
0.248
0.245
0.241
0.238
0.236
0.233
0.231
0.228
0.226
0.224
0.222
0.220
0.217
0.215
0.213
0.211
0.209
0.206
0.386
0.366
0.349
0.462
0.877
0.648
0.577
0.563
0.575
0.603
0.644
0.693
0.741
0.779
0.797
0.795
0.775
0.743
0.706
0.669
0.634
0.601
0.573
0.547
0.525
0.506
0.489
0.473
0.460
0.448
0.437
0.428
0.419
0.411
0.404
0.397
0.390
0.385
0.379
0.374
0.368
0.364
0.359
0.354
0.349
0.345
9.5
8.8
8.9
8.8
8.0
7.8
8.3
8.6
8.7
8.9
9.3
9.3
8.9
9.0
9.6
10.3
10.5
9.9
9.6
10.4
10.9
10.0
9.0
9.5
10.5
10.2
9.1
8.4
8.3
8.4
8.7
8.6
7.8
7.6
8.0
8.1
7.4
6.7
7.2
7.6
7.2
6.8
6.5
6.5
6.9
6.3
1.3
1.6
−0.4
−0.9
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1083
TABLE 24. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Au calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Au
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−289.567
−126.211
−67.156
−39.078
−23.319
−13.471
−6.685
−3.447
−2.222
−1.594
−0.992
−0.804
−0.873
−0.860
−0.875
−0.788
−0.353
0.150
0.608
0.829
1.382
1.892
0.873
0.862
0.075
0.197
−0.027
−0.045
0.218
0.586
0.710
1.026
0.868
0.976
0.936
0.719
0.710
0.749
0.822
0.918
0.957
0.979
1.123
1.265
1.549
1.333
1.224
0.924
0.410
0.206
−0.246
−0.328
59.936
18.250
7.925
4.353
3.240
3.246
4.448
6.621
7.642
7.866
7.826
7.923
7.627
7.164
6.684
5.930
5.329
4.966
4.916
4.688
4.384
5.251
5.129
5.182
4.448
4.113
3.736
3.166
2.715
2.532
2.411
2.460
2.629
2.418
2.663
2.480
2.302
2.157
2.050
2.002
2.013
1.915
1.856
1.878
1.981
2.627
2.851
3.245
2.595
2.691
2.502
2.084
0.001
0.001
0.002
0.003
0.006
0.017
0.069
0.119
0.121
0.122
0.126
0.125
0.129
0.138
0.147
0.166
0.187
0.201
0.200
0.207
0.207
0.169
0.189
0.188
0.225
0.243
0.268
0.316
0.366
0.375
0.382
0.346
0.343
0.356
0.334
0.372
0.397
0.414
0.420
0.413
0.405
0.414
0.394
0.366
0.313
0.303
0.296
0.285
0.376
0.369
0.396
0.468
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−404.801
−184.411
−101.906
−62.665
−41.020
−27.810
−19.125
−13.069
−8.644
−5.295
−2.745
−0.957
−0.164
−0.693
−2.154
−3.269
−3.440
−2.997
−2.340
−1.087
−0.233
0.118
0.001
−0.297
−0.447
−0.400
−0.257
−0.100
0.033
0.132
0.198
0.238
0.260
0.273
0.284
0.297
0.318
0.347
0.385
0.435
0.494
0.563
0.642
0.726
0.806
0.854
0.808
0.588
0.240
−0.020
−0.098
−0.064
127.357
40.136
17.819
9.848
6.436
4.844
4.117
3.878
3.982
4.395
5.158
6.339
7.883
9.258
9.472
8.378
6.919
5.754
4.989
4.311
4.272
4.495
4.630
4.429
3.991
3.552
3.219
2.988
2.828
2.711
2.615
2.526
2.437
2.344
2.246
2.145
2.045
1.947
1.856
1.774
1.705
1.654
1.626
1.633
1.687
1.807
1.994
2.164
2.144
1.904
1.614
1.381
0.001
0.001
0.002
0.002
0.004
0.006
0.011
0.021
0.044
0.093
0.151
0.154
0.127
0.107
0.100
0.104
0.116
0.137
0.164
0.218
0.233
0.222
0.216
0.225
0.247
0.278
0.309
0.334
0.354
0.368
0.380
0.392
0.406
0.421
0.438
0.457
0.478
0.498
0.517
0.532
0.541
0.542
0.532
0.511
0.483
0.452
0.431
0.430
0.461
0.525
0.617
0.723
0.0
−0.1
−0.1
−0.1
−0.1
−0.2
−0.2
−0.3
0.5
3.6
7.8
8.9
7.2
5.1
3.7
3.3
3.4
4.1
5.0
6.7
7.0
6.2
6.0
6.6
6.8
6.7
7.5
8.7
9.3
9.4
9.8
10.4
10.2
9.7
10.2
11.0
11.2
11.3
11.8
12.2
12.9
12.6
11.7
11.7
12.2
12.2
11.6
11.0
10.9
11.2
11.4
11.9
0.7
1.0
1.3
1.7
2.0
2.2
3.7
9.4
23.3
47.0
70.0
79.1
74.3
64.3
56.8
54.4
56.3
61.6
68.3
78.4
79.5
75.4
69.4
64.1
62.8
63.6
61.0
54.8
49.8
48.5
46.2
41.6
39.5
38.7
36.0
32.5
30.7
29.6
28.3
26.0
22.6
21.8
23.1
21.6
19.0
17.9
18.0
19.1
20.1
20.1
19.7
18.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1084
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 24. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Au calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Au
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
−0.293
−0.164
−0.069
0.052
0.291
0.373
0.396
0.485
0.522
0.590
0.783
0.895
0.590
0.363
0.331
0.189
0.015
−0.022
0.115
0.212
0.279
0.357
0.404
0.329
0.407
0.427
0.454
0.428
0.430
0.396
0.381
0.340
0.384
0.434
0.422
0.444
0.469
0.522
0.502
0.482
0.465
0.460
0.413
0.441
0.440
0.421
0.414
0.408
0.442
0.470
0.475
0.486
1.690
1.513
1.333
1.111
1.093
1.152
1.105
1.084
1.161
1.032
1.113
1.384
1.614
1.543
1.436
1.446
1.265
1.014
0.865
0.819
0.792
0.763
0.828
0.800
0.739
0.759
0.753
0.792
0.762
0.769
0.740
0.708
0.607
0.626
0.597
0.581
0.563
0.557
0.610
0.618
0.585
0.609
0.560
0.523
0.542
0.492
0.470
0.429
0.386
0.379
0.352
0.349
0.574
0.653
0.748
0.898
0.854
0.786
0.802
0.769
0.717
0.730
0.601
0.509
0.547
0.614
0.661
0.680
0.791
0.985
1.136
1.144
1.123
1.075
0.976
1.069
1.038
1.001
0.974
0.977
0.996
1.028
1.068
1.148
1.176
1.078
1.116
1.087
1.049
0.955
0.977
1.006
1.047
1.046
1.158
1.118
1.113
1.174
1.199
1.224
1.121
1.040
1.008
0.976
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.012
0.097
0.179
0.256
0.327
0.394
0.455
0.514
0.567
0.614
0.650
0.665
0.645
0.583
0.497
0.426
0.390
0.389
0.407
0.434
0.463
0.492
0.518
0.542
0.564
0.582
0.597
0.609
0.619
0.627
0.632
0.635
0.637
0.638
0.638
0.638
0.639
0.640
0.643
0.645
0.649
0.653
0.658
0.663
0.668
0.674
0.680
0.686
0.691
0.697
0.702
0.708
1.214
1.097
1.013
0.951
0.906
0.874
0.854
0.844
0.847
0.864
0.897
0.945
1.001
1.039
1.032
0.976
0.896
0.818
0.755
0.706
0.670
0.643
0.622
0.607
0.596
0.588
0.582
0.577
0.573
0.569
0.565
0.560
0.554
0.547
0.539
0.530
0.519
0.508
0.497
0.485
0.474
0.463
0.453
0.443
0.433
0.424
0.416
0.408
0.401
0.395
0.389
0.383
0.823
0.905
0.957
0.980
0.976
0.951
0.912
0.865
0.815
0.769
0.731
0.708
0.706
0.732
0.786
0.861
0.938
0.997
1.026
1.028
1.010
0.981
0.949
0.916
0.886
0.859
0.837
0.819
0.805
0.794
0.787
0.781
0.778
0.775
0.773
0.770
0.766
0.760
0.753
0.744
0.734
0.723
0.710
0.697
0.683
0.669
0.655
0.641
0.628
0.615
0.603
0.592
13.4
14.8
15.0
15.0
15.9
17.2
17.5
17.1
16.6
15.9
15.7
15.8
15.4
14.7
14.5
15.0
15.3
15.2
15.8
16.2
16.0
16.2
16.6
16.5
16.2
15.9
15.3
14.9
15.0
14.9
14.7
14.6
14.2
13.8
13.4
12.9
13.2
13.6
13.3
12.8
12.5
12.5
12.3
12.2
12.5
12.5
11.9
11.4
11.4
11.3
11.2
11.0
16.7
15.9
17.9
19.0
15.0
10.0
9.5
10.5
9.8
9.2
8.8
8.3
7.7
7.7
8.4
7.2
6.0
5.5
4.5
4.6
5.5
4.2
2.1
1.5
1.5
0.7
0.8
1.9
1.4
−0.1
−1.1
−1.0
−1.2
−1.5
−0.5
0.8
−0.8
−3.2
−3.8
−2.6
−0.9
−0.8
−1.6
−2.4
−3.2
−3.2
−1.9
−0.7
−0.4
−1.1
−1.9
−2.0
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1085
TABLE 24. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Au calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Au
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.488
0.547
0.576
0.602
0.608
0.632
0.651
0.662
0.700
0.756
0.700
0.637
0.600
0.597
0.612
0.640
0.511
0.500
0.543
0.557
0.548
0.540
0.556
0.576
0.575
0.576
0.589
0.599
0.613
0.622
0.624
0.636
0.652
0.670
0.674
0.682
0.686
0.695
0.705
0.681
0.674
0.676
0.692
0.717
0.690
0.667
0.305
0.282
0.294
0.298
0.299
0.287
0.313
0.305
0.319
0.347
0.435
0.446
0.422
0.390
0.371
0.441
0.404
0.328
0.290
0.296
0.290
0.262
0.238
0.228
0.229
0.211
0.196
0.187
0.177
0.177
0.165
0.155
0.148
0.151
0.154
0.157
0.158
0.159
0.180
0.180
0.165
0.152
0.136
0.148
0.186
0.158
0.921
0.744
0.703
0.661
0.652
0.595
0.600
0.574
0.539
0.501
0.640
0.738
0.784
0.768
0.725
0.729
0.953
0.919
0.765
0.745
0.753
0.727
0.649
0.595
0.598
0.561
0.508
0.474
0.435
0.423
0.397
0.361
0.331
0.320
0.323
0.320
0.318
0.313
0.341
0.363
0.343
0.317
0.274
0.275
0.365
0.337
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.713
0.718
0.722
0.726
0.730
0.734
0.738
0.741
0.744
0.747
0.749
0.752
0.754
0.756
0.757
0.759
0.760
0.761
0.762
0.763
0.764
0.765
0.765
0.766
0.766
0.766
0.766
0.766
0.766
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.767
0.768
0.768
0.768
0.378
0.373
0.369
0.365
0.361
0.357
0.354
0.351
0.347
0.344
0.342
0.339
0.336
0.333
0.331
0.328
0.325
0.323
0.320
0.318
0.315
0.312
0.310
0.307
0.304
0.301
0.298
0.295
0.292
0.289
0.286
0.282
0.279
0.276
0.273
0.269
0.266
0.262
0.259
0.255
0.252
0.248
0.244
0.241
0.237
0.234
0.581
0.571
0.561
0.552
0.544
0.536
0.528
0.522
0.515
0.509
0.504
0.498
0.493
0.489
0.484
0.480
0.476
0.472
0.468
0.465
0.461
0.458
0.454
0.451
0.448
0.444
0.441
0.437
0.434
0.431
0.427
0.423
0.420
0.416
0.412
0.408
0.404
0.399
0.395
0.391
0.386
0.382
0.377
0.372
0.367
0.362
11.0
11.1
10.7
10.3
10.2
10.0
9.9
9.8
9.4
9.1
9.5
9.8
9.0
8.0
8.2
9.2
9.1
8.3
8.3
8.4
8.0
7.9
8.2
8.5
8.6
8.2
7.7
7.8
8.2
8.3
7.9
7.4
7.6
8.0
7.7
7.2
7.5
7.6
7.1
7.1
7.4
7.2
6.5
6.3
6.8
6.9
−1.8
−2.1
−2.1
−1.9
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1086
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 25. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pb calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Pb
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−296.591
−124.313
−60.575
−32.451
−17.019
−5.973
−6.995
−9.950
−8.887
−13.705
−15.826
−13.482
−10.804
−9.180
−9.354
−8.770
−7.919
−7.022
−6.157
−4.614
−3.316
−2.261
−1.418
−1.028
−0.942
−0.816
−0.645
−0.649
−0.542
−0.428
−0.298
−0.164
−0.031
0.097
0.211
0.323
0.416
0.505
0.617
0.754
0.955
0.907
0.888
0.896
0.946
0.984
0.896
0.851
0.833
0.817
0.796
0.781
63.202
20.144
12.326
11.877
12.106
16.910
23.855
21.094
20.849
21.232
13.909
9.569
7.496
7.491
6.516
5.055
3.889
2.997
2.323
1.439
0.983
0.809
0.909
1.259
1.294
1.188
1.185
1.060
0.909
0.772
0.649
0.549
0.468
0.408
0.364
0.328
0.306
0.271
0.232
0.219
0.314
0.460
0.475
0.470
0.478
0.560
0.616
0.592
0.570
0.547
0.527
0.465
0.001
0.001
0.003
0.010
0.028
0.053
0.039
0.039
0.041
0.033
0.031
0.035
0.043
0.053
0.050
0.049
0.050
0.051
0.054
0.062
0.082
0.140
0.320
0.477
0.505
0.572
0.651
0.686
0.812
0.990
1.272
1.672
2.128
2.317
2.058
1.548
1.148
0.825
0.535
0.355
0.311
0.445
0.468
0.459
0.426
0.437
0.521
0.551
0.560
0.565
0.579
0.563
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−415.274
−227.300
−137.764
−90.129
−61.967
−44.707
−38.383
−31.461
−24.905
−20.026
−16.357
−13.523
−11.284
−9.479
−8.001
−6.774
−5.742
−4.865
−4.113
−2.898
−1.972
−1.273
−0.777
−0.495
−0.435
−0.549
−0.704
−0.779
−0.740
−0.622
−0.463
−0.292
−0.124
0.036
0.185
0.325
0.457
0.583
0.704
0.825
0.947
1.075
1.211
1.357
1.503
1.594
1.448
0.870
0.317
0.190
0.277
0.404
308.901
113.945
52.871
28.889
18.467
15.392
13.143
7.515
4.858
3.493
2.668
2.125
1.749
1.484
1.293
1.158
1.063
1.001
0.966
0.961
1.034
1.179
1.391
1.641
1.856
1.931
1.813
1.562
1.277
1.024
0.824
0.672
0.559
0.476
0.415
0.371
0.341
0.322
0.314
0.317
0.334
0.368
0.428
0.534
0.718
1.039
1.504
1.782
1.489
1.050
0.765
0.616
0.001
0.002
0.002
0.003
0.004
0.007
0.008
0.007
0.008
0.008
0.010
0.011
0.013
0.016
0.020
0.025
0.031
0.041
0.054
0.103
0.208
0.392
0.548
0.559
0.511
0.479
0.479
0.513
0.586
0.713
0.922
1.251
1.704
2.089
2.009
1.524
1.049
0.727
0.528
0.406
0.331
0.285
0.260
0.251
0.259
0.287
0.345
0.453
0.642
0.922
1.155
1.134
0.1
0.2
0.2
0.3
0.4
0.3
0.1
−0.2
−1.0
−2.3
−3.0
−1.9
0.3
2.3
3.4
3.5
3.2
2.8
2.5
3.6
7.4
12.8
18.2
22.6
24.3
25.2
27.2
30.0
32.9
32.9
30.3
33.2
45.7
61.6
66.7
59.0
48.3
38.3
29.9
25.2
22.7
19.7
16.1
14.0
15.7
19.5
21.0
19.7
18.5
18.9
20.8
24.3
0.3
0.5
0.6
0.8
1.0
0.9
2.2
6.4
16.6
34.8
52.6
59.0
54.7
47.4
43.0
43.1
46.8
52.7
59.5
72.8
84.5
96.4
103.8
98.7
92.2
92.1
96.6
98.6
93.9
94.0
99.3
88.0
56.2
18.8
4.6
15.6
24.6
28.8
31.9
28.3
22.2
19.1
20.9
26.2
26.7
20.6
19.1
23.4
23.4
19.7
17.8
11.6
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1087
TABLE 25. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pb calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Pb
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.803
0.827
0.873
0.905
0.954
0.999
0.951
0.906
0.958
1.062
1.083
1.050
1.051
1.069
1.135
1.157
1.160
1.189
1.180
1.165
1.161
1.146
1.133
1.112
1.063
1.067
1.086
0.989
0.937
0.978
0.923
0.996
0.983
0.952
0.966
0.906
0.935
0.952
1.053
0.850
0.756
0.704
0.778
0.816
0.748
0.762
0.772
0.876
0.611
0.432
0.576
0.701
0.422
0.377
0.355
0.346
0.336
0.390
0.412
0.373
0.274
0.284
0.373
0.379
0.363
0.324
0.333
0.389
0.403
0.431
0.484
0.499
0.527
0.546
0.573
0.597
0.613
0.571
0.634
0.675
0.584
0.592
0.549
0.532
0.589
0.575
0.591
0.593
0.559
0.568
0.691
0.669
0.725
0.562
0.522
0.551
0.518
0.558
0.490
0.625
0.695
0.520
0.315
0.338
0.513
0.456
0.400
0.369
0.328
0.339
0.384
0.389
0.275
0.235
0.284
0.304
0.294
0.260
0.238
0.261
0.267
0.269
0.297
0.311
0.324
0.338
0.356
0.375
0.407
0.390
0.401
0.471
0.479
0.453
0.476
0.417
0.449
0.465
0.461
0.506
0.471
0.462
0.436
0.572
0.661
0.693
0.594
0.568
0.626
0.625
0.586
0.540
0.812
1.137
0.731
0.559
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.520
0.609
0.669
0.695
0.690
0.669
0.651
0.648
0.662
0.689
0.724
0.760
0.798
0.833
0.867
0.898
0.926
0.952
0.974
0.992
1.006
1.016
1.021
1.021
1.016
1.008
0.996
0.982
0.968
0.955
0.944
0.936
0.932
0.931
0.933
0.937
0.942
0.949
0.955
0.960
0.963
0.963
0.959
0.951
0.939
0.923
0.905
0.886
0.867
0.851
0.839
0.830
0.548
0.528
0.536
0.555
0.567
0.556
0.521
0.473
0.424
0.381
0.348
0.324
0.309
0.300
0.297
0.298
0.304
0.313
0.325
0.340
0.357
0.376
0.395
0.414
0.431
0.445
0.456
0.462
0.464
0.461
0.455
0.447
0.437
0.428
0.420
0.414
0.411
0.410
0.413
0.418
0.426
0.436
0.448
0.459
0.469
0.476
0.479
0.477
0.470
0.459
0.446
0.431
0.961
0.812
0.730
0.702
0.711
0.735
0.750
0.735
0.685
0.614
0.540
0.475
0.422
0.383
0.354
0.333
0.320
0.312
0.309
0.309
0.313
0.320
0.330
0.341
0.353
0.367
0.380
0.392
0.403
0.410
0.414
0.415
0.413
0.408
0.401
0.395
0.389
0.384
0.381
0.381
0.385
0.391
0.400
0.412
0.426
0.441
0.457
0.471
0.483
0.491
0.494
0.493
26.0
23.4
20.7
21.3
22.6
20.9
17.7
17.7
20.4
19.8
16.1
14.5
15.4
15.9
15.2
13.2
11.5
11.7
12.7
12.2
10.7
10.9
12.2
11.7
10.1
10.0
11.7
12.1
9.7
8.7
10.9
13.1
12.6
10.7
9.5
11.0
13.5
12.7
10.5
10.2
10.5
10.4
10.6
10.9
11.2
10.9
10.5
11.8
12.8
11.6
10.4
11.5
4.7
8.8
14.4
12.1
7.1
6.7
9.6
5.9
−1.8
0.5
10.0
10.8
5.2
3.9
5.2
8.8
12.3
10.4
5.3
3.8
5.4
6.1
6.9
8.8
7.9
4.4
0.7
2.3
8.8
8.6
2.1
−2.1
−2.2
−0.6
2.4
0.7
−6.1
−4.6
0.9
0.3
0.0
1.3
−0.2
−2.0
−2.6
−1.8
−1.1
−4.7
−5.7
−1.6
1.7
−1.5
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1088
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 25. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Pb calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Pb
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.732
0.600
0.486
0.599
0.687
0.715
0.755
0.722
0.740
0.776
0.796
0.820
0.759
0.652
0.682
0.708
0.696
0.704
0.639
0.669
0.699
0.683
0.693
0.718
0.721
0.721
0.708
0.723
0.739
0.749
0.761
0.759
0.764
0.752
0.765
0.788
0.803
0.801
0.777
0.777
0.780
0.790
0.784
0.778
0.780
0.780
0.424
0.462
0.358
0.205
0.217
0.228
0.255
0.280
0.244
0.246
0.271
0.313
0.399
0.326
0.262
0.276
0.257
0.268
0.248
0.193
0.196
0.186
0.165
0.158
0.159
0.168
0.145
0.128
0.120
0.117
0.123
0.122
0.122
0.120
0.093
0.094
0.110
0.123
0.135
0.110
0.105
0.105
0.109
0.104
0.089
0.084
0.592
0.806
0.983
0.510
0.419
0.404
0.402
0.467
0.401
0.372
0.383
0.406
0.543
0.613
0.490
0.478
0.467
0.473
0.527
0.399
0.373
0.373
0.326
0.293
0.292
0.306
0.277
0.237
0.215
0.203
0.207
0.207
0.203
0.206
0.157
0.148
0.168
0.188
0.217
0.179
0.169
0.165
0.174
0.169
0.145
0.137
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.824
0.822
0.822
0.825
0.828
0.833
0.837
0.841
0.845
0.849
0.851
0.853
0.853
0.852
0.850
0.847
0.842
0.837
0.830
0.823
0.814
0.806
0.797
0.787
0.778
0.769
0.760
0.752
0.745
0.738
0.732
0.727
0.722
0.719
0.716
0.714
0.713
0.712
0.713
0.713
0.714
0.716
0.718
0.720
0.723
0.726
0.416
0.401
0.388
0.377
0.368
0.360
0.355
0.351
0.349
0.348
0.348
0.350
0.352
0.354
0.356
0.359
0.362
0.364
0.365
0.366
0.365
0.364
0.362
0.358
0.354
0.348
0.341
0.333
0.325
0.316
0.306
0.296
0.286
0.275
0.264
0.254
0.243
0.233
0.223
0.213
0.204
0.195
0.186
0.178
0.170
0.163
0.488
0.480
0.469
0.458
0.448
0.438
0.429
0.422
0.417
0.414
0.412
0.412
0.413
0.416
0.420
0.425
0.430
0.437
0.444
0.451
0.458
0.466
0.473
0.479
0.484
0.488
0.491
0.493
0.492
0.490
0.487
0.481
0.474
0.464
0.454
0.442
0.429
0.415
0.400
0.385
0.370
0.354
0.339
0.323
0.309
0.294
13.3
12.9
10.5
9.9
11.9
13.6
12.6
10.3
9.7
11.6
13.0
11.3
9.6
10.7
11.3
10.5
10.4
11.2
11.2
10.8
11.4
11.2
10.0
10.0
11.3
12.0
12.2
11.8
10.7
10.2
11.3
12.4
10.9
9.6
10.9
12.9
12.4
10.8
10.0
9.3
9.5
10.2
9.5
9.0
10.1
11.4
−8.8
−7.8
2.5
4.7
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1089
TABLE 26. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Bi calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲
Bi
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−178.373
−78.799
−42.858
−25.888
−16.419
−10.360
−5.745
−1.787
−4.159
−3.687
−3.447
−3.770
−3.191
−2.519
−1.834
−1.117
−0.279
0.784
0.003
−1.268
−0.822
−0.332
0.168
0.690
1.340
0.715
−0.326
0.120
1.349
−0.972
−0.213
0.040
0.099
0.200
0.291
0.346
0.415
0.490
0.536
0.562
0.598
0.631
0.663
0.696
0.736
0.768
0.799
0.833
0.853
0.876
0.911
0.952
36.667
11.133
4.797
2.546
1.589
1.200
1.340
3.737
5.981
4.322
4.687
3.356
2.453
1.870
1.423
1.181
1.120
1.733
3.492
1.975
1.206
0.807
0.654
0.646
1.215
1.983
1.988
0.753
1.637
0.961
0.526
0.499
0.434
0.355
0.310
0.277
0.224
0.198
0.210
0.184
0.157
0.144
0.124
0.101
0.093
0.082
0.075
0.076
0.075
0.066
0.056
0.053
0.001
0.002
0.003
0.004
0.006
0.011
0.039
0.218
0.113
0.134
0.138
0.132
0.151
0.190
0.264
0.447
0.841
0.479
0.286
0.359
0.566
1.060
1.434
0.723
0.371
0.446
0.490
1.296
0.364
0.515
1.634
1.990
2.193
2.139
1.717
1.410
1.008
0.708
0.634
0.525
0.411
0.344
0.274
0.205
0.169
0.138
0.116
0.109
0.102
0.085
0.068
0.059
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
−416.687
−185.820
−103.181
−64.569
−43.467
−30.669
−22.307
−16.528
−12.362
−9.282
−7.013
−5.452
−4.584
−4.306
−4.270
−4.120
−3.772
−3.324
−2.875
−2.165
−1.758
−1.555
−1.414
−1.247
−1.046
−0.834
−0.637
−0.476
−0.359
−0.284
−0.240
−0.210
−0.177
−0.132
−0.076
−0.010
0.060
0.132
0.202
0.270
0.334
0.394
0.450
0.503
0.552
0.599
0.642
0.684
0.724
0.762
0.801
0.840
73.052
22.209
9.673
5.253
3.374
2.507
2.126
2.025
2.125
2.401
2.851
3.442
4.040
4.372
4.250
3.815
3.340
2.971
2.731
2.514
2.421
2.279
2.049
1.785
1.548
1.367
1.244
1.167
1.116
1.070
1.012
0.935
0.842
0.742
0.645
0.558
0.482
0.418
0.365
0.321
0.285
0.255
0.230
0.209
0.192
0.178
0.166
0.156
0.148
0.142
0.137
0.134
0.000
0.001
0.001
0.001
0.002
0.003
0.004
0.007
0.014
0.026
0.050
0.083
0.108
0.116
0.117
0.121
0.132
0.149
0.174
0.228
0.270
0.299
0.331
0.376
0.444
0.533
0.637
0.735
0.812
0.873
0.935
1.018
1.138
1.306
1.528
1.791
2.041
2.173
2.094
1.824
1.478
1.157
0.899
0.705
0.561
0.455
0.376
0.317
0.271
0.235
0.208
0.186
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.2
1.2
2.7
4.1
4.7
4.6
4.5
4.7
5.3
6.4
7.7
9.8
10.4
11.3
13.1
14.9
17.1
20.1
22.7
24.5
26.5
29.6
32.6
34.6
36.6
40.1
44.2
47.4
50.5
53.6
54.0
51.0
46.6
40.8
34.3
29.5
26.1
22.5
18.7
15.7
13.4
11.5
10.6
9.5
0.4
0.7
0.9
1.1
1.3
1.5
2.3
5.2
11.7
22.6
34.3
42.4
46.6
48.5
50.5
53.7
58.1
63.8
70.0
82.2
90.3
91.8
93.7
100.8
106.1
104.9
102.0
101.1
99.0
90.6
80.1
73.2
67.6
57.2
45.7
39.4
32.3
21.4
14.8
14.2
14.2
15.8
18.4
18.2
15.8
15.2
15.8
14.6
14.6
15.9
13.1
9.7
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1090
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
TABLE 26. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Bi calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Bi
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
1.003
1.110
1.123
0.902
0.862
0.922
0.892
0.894
0.913
0.906
0.916
0.921
0.932
0.934
0.954
1.002
1.029
1.043
1.059
1.069
1.068
1.073
1.051
1.055
1.068
1.067
1.076
1.093
1.104
1.104
1.117
1.108
1.109
1.087
1.082
1.082
1.087
1.065
1.053
1.052
1.016
1.005
1.003
1.015
0.992
0.993
0.999
0.970
0.948
0.916
0.876
0.825
0.053
0.071
0.329
0.364
0.193
0.187
0.207
0.166
0.159
0.142
0.129
0.121
0.101
0.095
0.053
0.052
0.071
0.082
0.097
0.116
0.129
0.145
0.147
0.141
0.144
0.145
0.145
0.152
0.166
0.177
0.196
0.216
0.230
0.254
0.242
0.251
0.267
0.279
0.287
0.306
0.306
0.295
0.280
0.295
0.299
0.294
0.315
0.349
0.338
0.351
0.336
0.302
0.052
0.058
0.240
0.385
0.247
0.211
0.247
0.200
0.185
0.169
0.151
0.141
0.114
0.108
0.058
0.052
0.067
0.075
0.086
0.100
0.112
0.124
0.131
0.124
0.124
0.125
0.123
0.125
0.133
0.142
0.153
0.169
0.180
0.203
0.197
0.204
0.213
0.230
0.241
0.255
0.272
0.269
0.258
0.264
0.279
0.274
0.287
0.329
0.334
0.364
0.382
0.391
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
34.5
35.0
35.5
36.0
36.5
37.0
37.5
38.0
38.5
39.0
39.5
40.0
40.5
41.0
41.5
42.0
42.5
43.0
43.5
44.0
44.5
45.0
45.5
46.0
46.5
47.0
47.5
0.881
0.927
0.987
1.075
1.089
0.745
0.850
0.926
0.984
1.029
1.037
0.958
0.867
0.860
0.889
0.922
0.952
0.979
1.002
1.023
1.042
1.060
1.076
1.091
1.104
1.117
1.127
1.136
1.143
1.147
1.147
1.143
1.136
1.124
1.107
1.088
1.067
1.046
1.027
1.011
0.998
0.989
0.984
0.981
0.981
0.983
0.985
0.988
0.992
0.996
0.999
1.001
0.134
0.138
0.153
0.208
0.487
0.316
0.188
0.173
0.188
0.230
0.307
0.369
0.319
0.240
0.193
0.170
0.159
0.155
0.154
0.157
0.161
0.167
0.175
0.184
0.195
0.208
0.222
0.238
0.255
0.274
0.294
0.314
0.334
0.353
0.368
0.379
0.385
0.385
0.381
0.373
0.362
0.351
0.339
0.328
0.319
0.311
0.305
0.300
0.297
0.296
0.296
0.297
0.169
0.157
0.153
0.173
0.342
0.482
0.248
0.194
0.187
0.207
0.262
0.350
0.374
0.301
0.234
0.193
0.171
0.158
0.150
0.146
0.145
0.145
0.147
0.151
0.155
0.161
0.168
0.176
0.186
0.197
0.210
0.224
0.239
0.254
0.270
0.285
0.299
0.310
0.317
0.321
0.321
0.318
0.313
0.306
0.299
0.293
0.286
0.281
0.277
0.274
0.273
0.272
8.0
7.4
7.5
7.5
8.3
9.7
10.4
9.6
8.4
8.2
8.2
8.3
8.6
9.0
8.9
7.8
7.2
7.3
6.8
5.9
5.6
5.7
5.7
4.9
4.4
5.4
6.5
6.0
4.9
5.0
6.3
6.9
5.8
5.2
5.7
5.4
5.1
6.2
7.4
7.3
6.4
6.4
7.1
6.9
6.3
6.9
7.7
7.3
6.1
5.8
6.3
6.2
11.2
10.8
6.3
4.9
7.9
10.2
9.4
9.0
8.7
7.0
7.5
9.7
9.0
4.6
2.3
4.7
5.7
3.5
4.0
6.8
7.3
6.3
5.7
7.5
9.5
6.6
3.1
4.4
6.9
5.8
1.5
0.6
5.0
5.9
3.2
4.4
5.8
2.1
−1.7
−0.9
2.0
1.4
−1.0
−0.3
1.4
0.2
−2.3
−1.6
1.7
3.5
2.8
2.1
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
OPTICAL CONSTANTS FOR 17 METALS
1091
TABLE 26. Values of the real 共␧1兲 and imaginary 共␧2兲 parts of the complex dielectric constant and the ELF
共Im兵−1 / ␧共␻兲其兲 of Bi calculated with DFT and derived from REELS data. The last two columns contain values
for the volume and surface single-scattering loss distributions the DIIMFP 共wb兲 and the DSEP 共ws兲—Continued
Bi
DFT
REELS
E
共eV兲
␧1
␧2
ELF
E
共eV兲
␧1
␧2
ELF
wb
共meV−1兲
ws
共arb. un.兲
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
0.845
0.870
0.859
0.882
0.872
0.888
0.865
0.863
0.815
0.837
0.838
0.840
0.845
0.845
0.843
0.846
0.840
0.862
0.875
0.856
0.838
0.855
0.850
0.849
0.856
0.852
0.856
0.864
0.859
0.853
0.847
0.857
0.862
0.860
0.857
0.862
0.865
0.867
0.875
0.880
0.881
0.883
0.883
0.881
0.883
0.885
0.248
0.243
0.227
0.228
0.223
0.230
0.234
0.256
0.216
0.191
0.186
0.176
0.164
0.171
0.157
0.153
0.136
0.136
0.133
0.170
0.129
0.123
0.129
0.115
0.113
0.115
0.101
0.105
0.105
0.102
0.090
0.084
0.084
0.083
0.073
0.069
0.065
0.059
0.055
0.057
0.058
0.058
0.058
0.055
0.052
0.050
0.319
0.298
0.287
0.275
0.276
0.273
0.292
0.315
0.304
0.259
0.253
0.239
0.221
0.230
0.214
0.207
0.188
0.179
0.170
0.223
0.179
0.165
0.175
0.157
0.151
0.155
0.136
0.138
0.140
0.138
0.124
0.113
0.112
0.112
0.099
0.093
0.087
0.079
0.072
0.073
0.074
0.074
0.074
0.071
0.066
0.063
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
52.5
53.0
53.5
54.0
54.5
55.0
55.5
56.0
56.5
57.0
57.5
58.0
58.5
59.0
59.5
60.0
60.5
61.0
61.5
62.0
62.5
63.0
63.5
64.0
64.5
65.0
65.5
66.0
66.5
67.0
67.5
68.0
68.5
69.0
69.5
70.0
70.5
1.004
1.005
1.005
1.005
1.003
1.001
0.997
0.992
0.987
0.980
0.973
0.965
0.956
0.947
0.938
0.929
0.920
0.911
0.902
0.894
0.887
0.880
0.874
0.869
0.864
0.860
0.857
0.854
0.853
0.851
0.850
0.850
0.850
0.850
0.851
0.851
0.852
0.854
0.855
0.856
0.858
0.859
0.861
0.862
0.864
0.865
0.299
0.302
0.306
0.310
0.314
0.319
0.324
0.328
0.332
0.336
0.339
0.341
0.342
0.342
0.342
0.340
0.337
0.333
0.329
0.324
0.317
0.311
0.304
0.297
0.289
0.281
0.274
0.266
0.258
0.251
0.244
0.237
0.231
0.224
0.218
0.213
0.208
0.203
0.198
0.194
0.189
0.186
0.182
0.179
0.176
0.173
0.273
0.274
0.277
0.280
0.285
0.289
0.295
0.300
0.307
0.313
0.319
0.326
0.332
0.337
0.343
0.347
0.351
0.355
0.357
0.358
0.358
0.357
0.355
0.352
0.348
0.343
0.338
0.332
0.326
0.319
0.312
0.305
0.297
0.290
0.283
0.276
0.270
0.263
0.257
0.251
0.245
0.240
0.235
0.230
0.226
0.222
6.3
6.9
6.7
6.1
6.4
6.9
7.0
7.3
7.3
7.0
7.0
7.0
6.7
6.8
7.3
7.1
6.4
6.3
6.4
6.7
7.1
6.9
6.8
6.7
6.5
7.0
7.6
7.0
6.0
6.2
7.5
7.7
6.8
6.2
6.4
6.8
6.8
6.4
6.3
6.5
6.0
5.7
6.3
6.4
5.6
5.2
0.9
−0.6
1.0
2.9
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
J. Phys. Chem. Ref. Data, Vol. 38, No. 4, 2009
Downloaded 20 Mar 2013 to 129.6.105.191. Redistribution subject to AIP license or copyright; see http://jpcrd.aip.org/about/rights_and_permissions
1092
WERNER, GLANTSCHNIG, AND AMBROSCH-DRAXL
10. Appendix C: Values of Optical
Constants Derived from DFT and REELS
This appendix contains values of optical constants calculated by means of DFT and derived from REELS data as
given in Tables 11–26.
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49
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50
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51
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52
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60
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M. Novak, Ph.D. thesis, University of Debrecen, 2008.
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71
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73
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