Nuclear Instruments and Methods in Physics Research B 164±165 (2000) 241±249
www.elsevier.nl/locate/nimb
Energy-angle distribution of low-energy hydrogen ions in thin
aluminum and gold foils
M. Fama *, G.H. Lantschner, J.C. Eckardt, C.D. Denton, N.R. Arista
Instituto Balseiro, Centro At
omico Bariloche, Comisi
on Nacional de Energõa At
omica, RA-8400 Bariloche, Argentina
Abstract
Energy and angular distributions of hydrogen ions in thin solid foils of elements with signi®cantly di€erent electronic
structures and atomic masses, such as Al and Au, have been measured at 9 keV using the transmission technique. The
results are compared with Monte Carlo simulations using a classical scattering approach and an impact-parameter
independent electronic energy loss, as well as with a simpli®ed model containing the main physical features. In both
cases, the e€ect of foil roughness on angular dependence of the energy loss has been included and it has been found to
be responsible for the variations observed at small angles. The measured angular distributions have also been compared
with standard multiple scattering functions corresponding to Thomas±Fermi and Lenz±Jensen potentials and with
calculations based on 1=rn power potentials. Ó 2000 Elsevier Science B.V. All rights reserved.
PACS: 34.50.Bw; 34.50.-s; 34.20.Cf; 02.70.Lq
Keywords: Stopping power; Energy loss; Angular distributions; Multiple scattering
1. Introduction
The energy distributions of protons as a function of the exit angle after traversing thin solid
foils are determined by physical factors such as the
elastic and inelastic energy loss cross-sections (as a
function of the single scattering angle), and topological factors like the foil thickness and density
inhomogeneities. The elastic scattering of the
protons with the atomic nuclei determine the re-
*
Corresponding author. Tel.: +54-2944-445237; fax: +542944-445299.
E-mail address: [email protected] (M. FamaÂ).
sulting angular distributions. This is described by
theories based on the transport equation and collision theory [1±4]. In most cases, the electronic
energy loss is described as an angle-independent
magnitude formulated either as the stopping
power dE=dx or the stopping cross-section per
atom S [5,6]. Only in the last two decades
some measurements performed at higher energies
showed the need for considering impact parameter-dependent electronic energy losses Qinelastic per
single collision to describe this process [7±14], because neither the geometrical path enlargement
nor the elastic energy loss could explain the experimental results. These e€ects have been also
found to be relevant in similar studies with heavy
ions [15±19].
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 1 0 8 6 - 1
242
M. Fam
a et al. / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 241±249
Foil thickness inhomogeneities also generate an
observation angle dependence of the energy loss
because the thinner sectors transmit the particles
into a narrower cone than the thicker ones, so that
when observing projectiles in the forward direction
one collects preferentially those which came out
from thinner sectors and therefore lost less energy
[20,21].
The standard multiple scattering calculations
are valid as long as the small angle approximation
holds, the energy loss is negligible compared to the
projectile energy, and the path length is equivalent
to the penetrated depth. At the present energies
not all of these conditions hold. Essentially, there
is a di€erence between the travelled path length
and the penetrated depth. This a€ects the tails of
the distribution. The energy loss e€ect in the
multiple scattering distribution is adequately taken
into account, for low-energy ions, by using an
average energy corresponding to the mean velocity
of the projectiles in the target [22]. But to our
knowledge there is not a simple way to take into
account the breakdown of the other approximation in calculating angular distributions.
Angular dispersions of protons in the low-energy range have been previously measured for
other materials such as carbon, for energies between 1 and 20 keV [23], and between 3 and 54 keV
[24]. These papers show angular halfwidth as a
function of energy and foil thickness. In [25] angular distributions of hydrogen, helium and argon
ions in VYNS ®lms are compared at energies
ranging from 4 to 30 keV. However, a detailed
study of the applicability of the multiple scattering
theory in the range of very low energies has not yet
been accomplished.
Regarding the angular dependence of proton
energy loss at low energies, a simple approximation proposed by previous authors [23,26] considers a geometrical increase of the energy loss,
proportional to the secant of the observation angle. H
ogberg [27] proposed a somewhat di€erent
path length enlargement for the energy loss of
protons in C, considering a single de¯ection in the
middle of the foil. All these authors claim an independence of the energy loss with the observation
angle and they ascribe the observed di€erences to
path length e€ects. On the other hand, Blume et al.
[28] studying protons in gold observed an increase
of the energy loss with observation angle beyond
the simple length enlargement proposed by
H
ogberg [27].
The purpose of this work is to obtain a better
knowledge of the energy and angular distributions
of low-energy hydrogen ions, after traversing thin
solid foils and to test the applicability at these low
energies of theoretical models used at higher energies.
In this work, we present measurements of energy spectra and angular distributions of 9 keV
protons after traversing polycrystalline aluminum
and gold targets.
The angular distributions are compared with
tabulated values using the multiple scattering
theory [4] for the Thomas±Fermi and Lenz±Jensen
potentials, and with distributions corresponding to
rÿn power potentials calculated following the
multiple scattering formalism of the same reference.
Monte Carlo simulations of the angular dependence of the energy loss have been made assuming a random atom distribution in the solid,
rÿn scattering potentials, and a Gaussian-like foil
thickness distribution. As will be shown below, a
very good reproduction of the experimental data
can be achieved with these simulations.
We also present a simpli®ed model containing
the main physical features of the process, which
reproduces the experimental data with a surprisingly good degree of accuracy. These calculations
take into account the variations with observation
angle of the energy loss, due to: variations of the
electronic energy loss through changes in the path
lengths, the e€ect of foil roughnesses, and the
variations of the nuclear energy loss with the
scattering angle.
2. Experimental procedure
The proton beam was generated by a small
accelerator with a hot discharge ion source followed by focusing and accelerating stages, a Wien
®lter for mass selection, electrostatic beam steerers, and an 18° de¯ector to get rid of neutral
particles. The beam-energy spread is as small as
M. Fam
a et al. / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 241±249
some eVs, and the intensity is variable between
10ÿ13 and 10ÿ9 A/mm2 and it is very stable. The
target holder was preceded by two diaphragms of
4 and 1 mm diameter respectively separated by 58
cm, de®ning the angular divergence to less than
0:4° and the beam spot size on the target to 1 mm
diameter. The multiple target holder allowing foil
removal during the experiment, and a rotatable
127° cylindrical electrostatic energy analyzer with
the detection system, were located in the same
chamber. The energy and angular resolutions of
the analyzer were respectively 1% and 0:58°. A
combination of a sorption pump for the preliminary vacuum and a di€usion pump with a UHVsuitable LN2 trap maintained an oil-free high
vacuum.
The self-supported targets were made by evaporation under clean vacuum conditions on a very
smooth plastic substrate [29] which is subsequently
dissolved. The foil thicknesses were determined by
matching the proton energy-loss measurements at
9 keV to previous stopping cross-section S determinations [30], which were based on comparisons
with absolute empirical values at higher energies
[31] 1. The values obtained were 20.0 nm for Al
and 14.3 nm for Au. The roughness has been determined through a simple in situ procedure based
on measurements of the energy straggling X of
protons transmitted through the foil. For Gaussian foil thickness distributions this straggling can
be expressed as [32]
s
2
q
dE
r2x  X20 ‡ q2 DE2 ;
X ˆ X20 ‡
dx
…1†
where X0 is the energy straggling of an ideal foil of
thickness h xi, rx the standard thickness deviation,
DE the energy loss and q is a parameter characterizing the foil roughness, de®ned as q ˆ rx =h xi.
Taking the experimental energy loss values DE…0†
at h ˆ 0, and using in Eq. (1) the theoretical X0 values, X0 …Al† ˆ 140 eV and X0 …Au† ˆ 164 eV,
resulting from [6], we extracted the following ap1
The stopping cross-sections were calculated using the
STOP program from TRIM-90 with the coecients from the
®le SCOEF-88.
243
proximate values of the roughness parameter:
q…Al† ˆ 15% and q…Au† ˆ 11%. These values have
been checked by analyzing a foil of the same
production batch with an atomic force microscope.
A TEM analysis previously made on similar
targets [33] revealed a polycrystalline structure of
random orientation with grain size of 20 nm.
The foil thickening due to ion beam bombardment was held within insigni®cant limits by using a
low current density of 10ÿ10 A/mm2 , and short
irradiation times. This was checked by comparison
of the energy loss at the beginning and the end of
the measurement series.
The energy loss at large observation angles h is
very sensitive to departures of the foil orientation
from orthogonality to the incoming beam direction. This was carefully adjusted and checked by
comparison of a spectrum corresponding to a large
h with its similar corresponding to ÿh.
The energy spectra for the di€erent observation
angles h have been measured maintaining a constant incident number of projectiles for each value
of the angle. So these measurements allowed the
determination of the angular distributions by integration of the individual energy spectra.
The energies of the distributions have been
determined by ®tting them with Gaussians and
taking its central value. Given the nearly Gaussian
shape of body of the measured distributions, the
resulting values are very close (within 10 eV) to
the most probable value. The di€erences between
these values and the mean values are also small
(10±25 eV depending on the spectra, over a total
energy loss of about 1400 eV), which is less than
the experimental uncertainties. The energy losses
DE…h† are obtained by subtracting the energies of
the corresponding distributions determined in the
preceding way, from the incident beam energy.
3. Models and computer simulations
3.1. Angular distributions
Using the Sigmund and Winterbon formalism
[4] we calculated the angular distributions for rÿn
power potentials varying the n values and search-
244
M. Fam
a et al. / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 241±249
ing the best ®t to our experimental data. In this
®tting procedure the large angle tails of the distributions were not considered because of the
breakdown of an assumption required for the validity of the formalism, namely the equivalence of
travelled path length and penetrated depth.
The scaled angular distribution of a particle
beam after traversing a thin layer is given by
Z 1
z dz J0 …~
az† exp‰ÿsD…z†Š
…2†
f1 …s; a~† ˆ
0
in terms of the reduced thickness and angle units
s ˆ pa2 Nx;
a~ ˆ
Ea
a;
2Z1 Z2 e2
2=3
…3†
2=3 1=2
where a ˆ 0:8853a0 =…Z1 ‡ Z2 † is the screening radius, N the atomic density, x the penetrated
depth, E the ion energy, Z1 and Z2 the atomic
numbers of the ion and target, respectively, and e
the elementary charge. In Eq. (2) J0 is the zeroorder Bessel function of the ®rst kind, z is an integration variable and
Z 1
~
f …/†
~
d/~
‰1 ÿ J0 …z/†Š:
…4†
D…z† ˆ
/~2
0
For a power potential V …r† / rÿn the scattering
~ ˆ k/~1ÿ2=n
function is analytically given by f …/†
[34], where k is a numerical constant evaluated
following [34,35]. For this kind of potential one
obtains
D…z† ˆ cz2=n ;
cˆÿ
kC…ÿ1=n†
:
22=n‡1 C…1 ‡ 1=n†
…5†
This formalism was also used to calculate the derivatives of the multiple scattering function with
respect to the foil thickness, necessary for the
evaluation of the foil roughness e€ect as it will be
described below.
3.2. Energy loss as a function of observation angle
3.2.1. Monte Carlo simulations
Monte Carlo simulations of the energy loss as a
function of the observation angle have been performed following the scheme of [36,37] using the
binary collision model. A random distribution of
atoms in the solid is assumed. In this case, one has
a random probability ka of occurrence of a collision, related to the total cross-section rtot and to
the travelled distance between collisions D by the
expression
ka ˆ 1 ÿ exp…ÿN rtot D†;
rtot being the total scattering cross-section and N
the atomic density. In reduced units [34] of path
length, energy and mass, this expression is
ka ˆ 1 ÿ exp‰ÿ…M1 ‡ M2 †Jtot n=4lŠ;
…6†
where n is the reduced travelled distance between
2
collisions and Jtot ˆ …r0 =a† is the reduced total
scattering cross-section, with r0 ˆ 0:562N ÿ1=3 being half the distance between neighboring atoms in
an fcc lattice. With Eq. (6), n can be written in
terms of the random number ka as
nˆÿ
4l
ln…1 ÿ ka †
:
M1 ‡ M2
Jtot
…7†
An electronic energy loss proportional to this
travelled path length was included, taking the
DE=Dx values corresponding to a zero observation
angle previously measured in this laboratory [30].
Using a second random number kb , the reduced
scattering angle /~ in each collision is obtained
through the expression [36,37]
/~ ˆ J ÿ1 ‰J …e† ‡ …kb ÿ 1†Jtot Š;
…8†
where J is the reduced scattering cross-section, e
the reduced energy, and J ÿ1 the inverse function of
J.
The reduced cross-section was derived following the Lindhard formalism [34] using an rÿn potential extracted from a ®t of the measured angular
distribution.
In terms of the center of mass scattering angle
/; the elastic energy loss Qnucl …/† is
Qnucl …/† ˆ 4
M1 M2
2
…M1 ‡ M2 †
E sin2 …/=2†:
The e€ect of foil roughness was included in this
simulation by adding a random thickness generator, reproducing the Gaussian-like target thickness
distribution with a roughness coecient q similar
M. Fam
a et al. / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 241±249
to the one determined experimentally, as previously described.
3.2.2. Simple model calculation
Guided by the results of the Monte Carlo
simulations we present here a simple model which
includes the main physical features relevant for the
angular dependence of the energy loss DE. Although a rigorous treatment requires a more
elaborated theory [10,11], this model leads to a
remarkably good reproduction of the experimental
DE…h† ÿ DE…0† data. These calculations include
the following three contributions to the DE-variation with the observation angle h:
1. Changes in the electronic energy loss due to
path length increase.
2. The angular dependence of the nuclear scattering.
3. The e€ect of foil roughness.
The path length increase with the observation
angle (originated in the multiple scattering process)
is estimated
by the simple expression
1
1
ÿ
1
which,
although
it is not derived from
2 cos…h†
multiple scattering theory, behaves closely like the
multiple scattering path length enlargement, as can
be observed from our own Monte Carlo simulations, as well as the simulations of [38].
Under these assumptions the variation of the
electronic energy loss can then be written as
1
1
ÿ 1 DEelec …0†;
DEelec …h† ÿ DEelec …0† ˆ
2 cos…h†
245
jectiles collected at larger angles. Following [21]
the foil roughness e€ect is given by
DE…h†rough ÿ DE…0†rough
2
ˆ DEq
!
~ j o ln j f1 …s; 0† j
o ln j f1 …s; h†
ÿ
;
o ln s
o ln s
…11†
which requires a previous knowledge of the variation of the multiple scattering function with the
reduced foil thickness s. DE is the angular average
of the energy loss, which we evaluated with our
angular distributions and DE…h† data.
The total variation of the energy loss
DE…h† ÿ DE…0† with observation angle h is given
by the sum of Eqs. (9)±(11).
4. Results and discussion
Figs. 1 and 2 show the measured energy spectra
for di€erent observation angles for Al and Au
targets of 20 and 14.3 nm thickness respectively
together with the ®tted Gaussians. In both cases
…9†
where DEelec …0† is the electronic energy loss measured in the forward direction.
The nuclear (elastic) contribution for M1 M2 ,
is approximated by
DEnucl …h† ÿ DEnucl …0†
' DEnucl …h† ' 4
M1 M2
…M1 ‡ M2 †
2
E sin2 …h=2†:
…10†
As described in the introduction, the roughness
e€ect arises from the preferential scattering onto
small angles of the projectiles traversing thinner
sectors, and therefore losing less energy than pro-
Fig. 1. Energy spectra of 9 keV protons after travesing 20 nm
thick Al foils at di€erent angles, with the corresponding ®tted
Gaussians.
246
M. Fam
a et al. / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 241±249
Fig. 4. Same as Fig. 3, for a gold target. In this case the power
potential for calculating the angular distribution is 1=r2:8 : The
line types have the same meaning as in Fig 3.
Fig. 2. Same as Fig. 1, for 14.3 nm thick Au targets.
Fig. 3. Experimental angular distribution for 9 keV protons in
Al foils resulting from the integration of the spectra of Fig. 1,
together with the calculated distributions for the Thomas±
Fermi and Lenz±Jensen potentials [4] (dot-dash and dotted lines
respectively), as well as the present calculations with a 1=r2
potential (full line).
one can clearly observe a shift of the distributions
towards lower energy with increasing observation
angle. The angular distributions resulting from the
integration of these spectra are depicted in Figs. 3
and 4. In these ®gures we also show the wellknown tabulated values of multiple scattering
distributions from [4] corresponding to Lenz Jensen and Thomas±Fermi potentials. We observe
that the experimental distributions are narrower
than those calculated with the Thomas±Fermi and
Lenz±Jensen potentials especially for Au. A similar discrepancy was observed by other authors at
higher energy experiments [39,40].
The solid lines in Figs. 3 and 4 show the calculations of the angular distributions resulting
from the best ®tting power potential rÿn . The most
suitable powers n were determined to be 2 for Al
and 2.8 for Au. This adjusted potential was used in
the previously discussed numerical simulations
and to evaluate the foil roughness e€ect through
Eq. (11).
Figs. 5 and 6 show the energy losses as a
function of the angle of emergence DE…h† ÿ DE…0†
referred to the energy loss at h ˆ 0; corresponding
to the same data of the previous ®gures. A common feature of the depicted angular dependences
is the important increase for small angles followed
by a change in slope and a further increase at large
angles.
In these ®gures, one can also observe the results
of the previously mentioned model calculation,
showing separately the contributions of Eqs. (9)±
(11). One can observe a remarkably good agreement with the experimental data using the previously speci®ed q values. In spite of its simplicity,
this model calculation clearly shows that the ®rst
M. Fam
a et al. / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 241±249
247
Fig. 5. DE…h† ÿ DE…0† data for 9 keV protons in Al resulting
from the measurements of Fig. 1, together with the model
calculations with its di€erent contributions and the Monte
Carlo simulation, as described in the text.
Fig. 7. Protons in Al: Path length enlargements (a), and elastic
energy loss (b), resulting from the simple model (dashed line)
(see text) and the Monte Carlo simulations (step line).
Fig. 6. DE…h† ÿ DE…0† data for 9 keV protons in Au resulting
from the measurements of Fig. 2, together with the model
calculations with its di€erent contributions and the Monte
Carlo simulation, as described in the text.
increase at small angles is mainly due to foil
roughness, whereas the stronger increase at large
angles results as the composed e€ect of path length
enlargement and growing elastic energy loss.
The Monte Carlo simulations depicted in these
®gures show also a remarkably good agreement
with the experimental data. We have performed a
set of simulations with di€erent roughness coecients in order to test its in¯uence, and have con®rmed that the initial increase of DE…h† ÿ DE…0† at
angles K 10° is due to the e€ect of foil roughness.
Figs. 7 and 8 show path length enlargements and
elastic energy losses, resulting from the Monte
Carlo simulations compared to our simpli®ed
model. A very good agreement can be observed.
We point out that for hydrogen ions, neither
these simulations nor the simple model include an
impact parameter dependent single-scattering inelastic energy loss as required for higher energies
[7±14]. So the good agreement with the experimental data indicates that if there were an impact
parameter dependence of such a single-scattering
inelastic energy loss in the case of protons, its e€ect
would be too small to be observable in the present
low-energy experiment.
248
M. Fam
a et al. / Nucl. Instr. and Meth. in Phys. Res. B 164±165 (2000) 241±249
dence of the electronic energy loss at the low energy of these experiments.
3. The angular dependence of DE at small angles is
dominated by foil roughness e€ects.
4. The angular dependence of DE at larger angles
is essentially due to path length enlargements
and to the increase of the elastic contribution.
5. Our Monte Carlo simulations reproduce very
closely the angular dependence of the energy
loss.
Acknowledgements
This work was supported in part by the argentine Consejo Nacional de Investigaciones
Cientõ®cas y Tecnicas (project PIP 4267) and the
Agencia Nacional de Promoci
on Cienti®ca y Tecnol
ogica (project PICT0355). We thank Dr. Esteban Sanchez for the AFM foil analysis. One of us,
M.F., thanks the FOMEC program for a doctoral
fellowship.
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