CHAPTER 2. CRYSTAL STRUCTURE
2.3
28
Determination of Bulk Structure
If we seek to obtain useful information about the crystal structure we have to use
a tool with appropriate properties. Direct-space information is ruled out by the
long wavelength of radiation used, because we cannot resolve details finer than
the wavelength. Hence, we work with diffraction techniques, using radiation of a
wavelength comparable with atomic dimensions. Diffraction gives us information in
Fourier form, which can be analyzed in terms of the average spacings of lines and
planes of atoms in a solid, the angles between lines and planes, the symmetries of
point groups, and the location of particular species of atoms. There are three types
of excitations one uses for crystallographic investigations which have quite different
energies for wavelengths useful for diffraction.
For diffraction of neutrons at a crystal, we have to consider the wavelength
λ=
h
h
=√
Mv
2M E
(2.7)
Thus neutrons with a wavelength of 1 Å move at a speed of 4000 m/s, and have a
kinetic energy of 0.08 eV. This energy is comparable with lattice vibrational energy,
and neutrons do not interact with the solid intensively.
Electrons have a kinetic energy
1
1
E = mv 2 = p2 /m
2
2
(2.8)
Using the de Broglie wavelength λ = h/mv for electrons, one obtains a relationship
between the wavelength and energy in appropriate units:
λ = h(2mE)−1/2 and λ(Å) ≈
q
150/E (eV).
(2.9)
Thus, electrons with a wavelength of 1 Å have an energy of about 150 eV and
move at a speed about 7 × 106 m/s. These electrons interact with those of the
solid so strongly that they cannot penetrate the solid at all without losing energy.
Electrons in this energy range are used in low-energy electron diffraction (LEED).
For bulk structural work, one uses high-energy electrons in a transmission electron
microscope (TEM).
The photon energy is E = hν and the wavelength λ is related to the frequency
ν over speed of light c = νλ. Using appropriate units, we find for photons E(eV ) =
12400/λ(Å). For a wavelength of 1 Å the energy is about 12.4 keV easily accessible
in the laboratory. The suggestion to use x-rays for structural studies first came
from von Laue in 1912. He regarded the crystal as a 3DIM diffraction grating, and
a diffraction pattern provides information about the regular arrangements of atoms.
X-rays are up to date the principal source of information about the crystallography
of solids.
CHAPTER 2. CRYSTAL STRUCTURE
2.3.1
29
X-Ray Diffraction
W.L. Bragg in 1913 proposed an expression for the geometric condition which must
be satisfied if waves are to be diffracted by a parallel set of planes. The Bragg
diffraction condition specifies that the angles of incidence and reflection are equal,
i.e., that reflection is specular. Bragg’s Law is not concerned with the arrangement
of atoms within a single reflecting plane.
We can see from Fig. 2.9, that the x-rays are reflected from planes, and constructive interference for specular reflection occurs for monochromatic radiation
whenever the Bragg condition is satisfied:
nλ = 2d sin θ.
(2.10)
Here, d is the spacing of the planes of atoms and n is an integer. All specularly
reflected components will be able to combine constructively in phase to form the
intensity of reflected radiation. This formulation is independent of the atomic arrangement within each plane.
Figure 2.9: A Bragg (specular) reflection from a particular family of
lattice planes, separated by a distance d. Incident and reflected rays are
shown for the two neighboring planes. The path difference is 2d sin θ.
We note that Bragg’s results, presented in Eq. 2.10, are identical to those of von
Laue. If we analyze diffraction in terms of initial and final wave vectors ko and k0
in reciprocal space, we end up with Laue equations:
a · ∆k = 2πh
b · ∆k = 2πk
c · ∆k = 2πl
(2.11)
Eq. 2.11 ties together real- and reciprocal-space entities. It can be expressed more
elegantly as
∆k = G,
(2.12)
CHAPTER 2. CRYSTAL STRUCTURE
30
which led Ewald in 1921 to suggest a geometric interpretation of the diffraction
requirement. This is illustrated in Fig. 2.10. Given the incident wave vector ko , a
sphere of radius |k| is drawn to intersect the origin. The condition for diffraction,
given in Eq. 2.12, is fulfilled whenever the sphere goes through a reciprocal lattice
point.
Figure 2.10: The Ewald construction.
In considering the Bragg diffraction of an x-ray beam by a single crystal, we must
note that Eq. 2.10 requires a suitable combination of d, θ, and λ for constructive
interference. Thus monochromatic x-rays directed on a crystal at an arbitrary angle
will usually not be reflected. Either x-rays of many wavelengths (white light) must
be used at a single angle of incidence (the Laue method ) , or monochromatic rays
must be allowed to encounter the crystal at a variety of angles. This is accomplished
mechanically in the rotating crystal , oscillating crystal , and Weissenberg methods,
while the Debye-Scherrer method uses a powdered sample so that every angle of
incidence is encountered for some of the crystallites.
The Laue method is very simple in concept and operation. A narrow beam of
unmonochromatized x-rays falls on a single crystal, as illustrated in Fig. 2.11. For
any wavelength satisfying the Bragg condition, a diffracted beam will emerge. The
figure shows two positions in which a photographic plate can be placed to record
a set of diffraction spots which is characteristic of the structure and orientation of
CHAPTER 2. CRYSTAL STRUCTURE
31
Figure 2.11: A Laue flat plane camera. The Laue spots can be recorded
in either forward or backward direction in order to orient the crystal.
The crystal can be rotated about three orthogonal axes. From JSB.
the crystal. The Laue method is rarely used for investigation of new structures.
Its main use lies in the rapid and convenient orientation of known crystals which
is achieved by tuning the angles necessary for perfect alignment of a major crystal
plane with the axis of the Laue camera.
The rotating-crystal method is a technique in which the crystalline sample is
rotated through the angle θ about a fixed axis, while monochromatic x-rays are
presented to the sample. The variation in the angle brings different atomic planes
into position for reflection. The essential aspects of the experimental arrangement
are sketched in Fig. 2.12. The sample is rotated and a cylinder of photographic
film, placed coaxially with the rotation axis, records a spot whenever the Bragg
condition is fulfilled. The inclination of these spots to the direction of the incident
beam has a component of θ in the plane perpendicular to the cylinder axis.
Elaborations of the rotating crystal experiment to give much more information
about the crystal symmetry include the oscillating crystal method, the Weissenberg
method.
Instead of using a single x-ray wavelength and a time-dependent angle of incidence, we may present a crystalline sample with every θ simultaneously. This is
the Debye-Scherrer technique of using a finely powdered crystalline sample in which
the crystalline orientations are random. Rays which for one crystallite or another
satisfy the Bragg condition emerge from the sample as a series of cones concentric
with the incident beam direction. Thus a photographic plate records a series of
concentric circles.
The scattering amplitude F of x-rays is generally given by
F =
Z
dV n(r)e−i∆k·r
(2.13)
CHAPTER 2. CRYSTAL STRUCTURE
32
Figure 2.12: The crystal is rotated through θ, while diffraction intensity
is recorded at 2θ. From JSB.
If the diffraction condition ∆k = G is met,
FG = N
Z
dV n(r)e−iG·r = N · SG .
(2.14)
The relative intensities of the reflections depend on the number, position, and electronic distribution of the atoms in the unit cell. The function SG is called the
structure factor .
X
fj e−iG·rj
(2.15)
SG =
j
is obtained by summation over the cell, which finally results in
S(hkl) =
X
fj ei2π(pj h+qj k+rj l)
(2.16)
j
with rj = pj a + qj b + rj c and G = ha∗ + kb∗ + lc∗
Considering the basis of the bcc structure we obtain for the structure factor of
the bcc lattice
S = 0 when h + k + l = odd integer
S = 2f when h + k + l = even integer
As a consequence, diffraction spectrum does not contain lines such as (100), (300),
(111), or (221).
Similarly, we obtain the structure factor for the fcc lattice
S(hkl) = f [1 + e−iπ(k+l) + e−iπ(h+l) + e−iπ(h+k) ]
If all indices are even or odd integers, S = 4f . In an fcc lattice, no reflections can
occur for which the indices are partly even and partly odd. We thus have a tool to
differentiate between bcc and fcc if we just know that the sample is cubic.
CHAPTER 2. CRYSTAL STRUCTURE
2.3.2
33
X-Ray Absorption Fine Structures
When a specimen is presented to electromagnetic radiation a portion of intensity is
absorbed causing interband transitions in the material. The probability Wif for a
transition of an electron from an initial state i with the energy Ei to a final state f
of the energy Ef as a result of absorption of a photon with the energy hν is given
by the Fermi Golden Rule:
4π 2
Wif =
|hf |V |ii|2 δ(Ei − EF + hν)
h
(2.17)
The interaction with the electromagnetic field is described by the operator:
V =−
e2
e
pA +
A2 ,
mc
2mc2
(2.18)
with the vector potential A and the momentum p. In the dipole approximation
(k · r 1) the absorption cross section is expressed as:
σ = 2παhω
X
|hf |r~|ii|2 δ(Ei − EF + hν)
(2.19)
f
Here, α is the fine structure constant, ~ the polarization vector and r is the position
of the particle. Using the density of the particles nc we obtain the absorption coefficient µ = nc σ which is an experimentally observable quantity. The measurement
of µ and the interpretation of data demonstrate that geometric, electronic, and
chemical properties of the material under investigation are interrelated, and just
one experimental result contains all this information.
The shape of the absorption cross section is characteristic to the absorbing
atom. The sharp absorption edge reveals the binding energy of the particular inner
shell. X-ray absorption spectrum thus carries information about the identity of
atoms constituting the sample. If the atom is bound in a molecule, a solid, or even
placed in a liquid (cf. Fig. 2.13), thus having neighbors, one observes modulations
of the absorption coefficient beyond the absorption edge extending to a wide energy
range.6
Two regions in energy are arbitrarily differentiated (cf. Fig. 2.13):
1. A range as narrow as about few 10 eV near the edge. Here, the structures are
due to the creation of localized excitations, variations of the density of states, as
well as many-body excitations, referred to as x-ray absorption near-edge structure
(XANES).
2. The higher energy range. The absorption coefficient shows a more or less simple
modulation, especially from atoms having a symmetric environment, called extended
x-ray absorption fine structure (EXAFS).
A convincing example is the absorption spectrum from the ferrocene molecule,
Fe(C5 H5 )2 , as shown in Fig. 2.14. In the molecule, the central Fe atom is surrounded
6
A. Di Cicco et al., Phys. Rev. B 54, 9086 (1996).
CHAPTER 2. CRYSTAL STRUCTURE
34
Figure 2.13: K -edge spectra of gaseous (lower curve), liquid (0.1 – 0.75 GPa), and
solid Kr as a function of pressure at room temperature. Ref. [6]. The vertical line
indicates the arbitrary separation of the spectra in XANES and EXAFS regions.
by 10 carbon atoms at exactly the same distance. Owing to identical contributions
from 10 neighbors, absorption measurements from the ferrocene molecule show enhanced sinusoidal oscillations.7
In crystalline materials the shape of the oscillations in the extended region can
look complicated because it consists of superposition of oscillations with different
periodicities (frequencies), as illustrated in Fig. 2.15 for Cu and Ni.
Additional observations include:
1. The energy distance between extrema decreases with growing distance between
the absorbing atom and its neighbors.
2. With increasing temperature thermal amplitude of atoms grow around their zerotemperature position, and the EXAFS oscillations are effectively damped. This
situation is shown for a Cu crystal in Fig. 2.15(left) and clearly demonstrate the
link between the absorption of photons, an electronic process, and the position of
the absorbing atoms, a geometric effect.
7
T.K. Sham and R.A. Holroyd, J. Chem. Phys. 80, 1026 (1983).
CHAPTER 2. CRYSTAL STRUCTURE
35
Figure 2.14: A ball-and-stick model of the ferrocene molecule and the absorption
coefficient µ. Ref. [7].
Historically, there are two different explanations for the observed oscillations:
a. Near-order theory claims that the observed structures originate from the matrix
element for absorption |Mif |2 , where |Mif | = hf |r · |ii. This case describes the situation where excited electrons do not travel long distances in the crystal. Through
multiple scattering they are confined near the absorbing atom.
b. Far-order theory places the origin of the structures in the density of states
P
N (E) = f δ(Ei − Ef + hν). If electrons are allowed to travel long distances after
excitation in the crystal, they sample information in a larger volume around the
absorbing atom.
Figure 2.15: The absorption coefficient of single-crystalline copper and nickel.
P. Aebi, unpublished
CHAPTER 2. CRYSTAL STRUCTURE
36
Today we know that both theories do correctly apply and the absorption spectra
are influenced by the matrix elements as well as the density of states that are
unoccupied prior to the excitation. The only crucial issue is that for kinetic energies
little over the Fermi energy, excited electrons travel long distances in the crystal.
They have a long mean free path, as displayed in Fig. 1.14. Hence the near-order
theory better describes the oscillations in the absorption coefficient. This is the
XANES regime. For excitations with higher photon energies, kinetic energy of
excited electrons are proportionally higher, the mean free path is shorter and the
absorption coefficient is rather influenced by the proximity of the absorbing atom.
This is the EXAFS regime.
Figure 2.16: Three possible high-symmetry positions of the absorbed
atoms (•) on an fcc (100) surface.
The determination of site-specific information about adsorbed species on otherwise clean surfaces is always a challenge due to the extremely small signal from
the adsorbates. Figure 2.16 shows schematically the ordered adsorption of foreign
atoms on a (100) surface of an fcc crystal. The amount of adsorbates are referred to
as 0.5 ML, that means that there are 1/2 foreign atoms per surface host atom. The
ordering of adsorbates shows that the periodicity in both directions is doubled with
an addition adsorbate atom in the middle of the adsorbate square thus formed. If
we take an adsorbate unit cell aligned with the substrate cell, this adsorbate configuration is written as c(2x2), c designating centered . If we seek the smallest unit
√
√
of adsorbates, than 2 × 2 − 45◦ is appropriate to describe the adsorbate order.
Nevertheless, the adsorbate structure has three different high-symmetry positions
on the surface not accounted for by the simple description of adsorbate geometry.8
The top position is connected with the simplest binding geometry. In the bridge
configuration, an adsorbate atom has two substrate neighbors, while in the hollow
configuration, there are four substrate neighbors, as illustrated in Fig. 2.16.
8
D.D. Vvedensky et al., Phys. Rev. B 35, 7756 (1987).
CHAPTER 2. CRYSTAL STRUCTURE
37
Figure 2.17: Comparison between computations and XANES measurements
indicate that the hollow-site positions for oxygen atoms on Cu(100) are more
probable. Ref. [8].
Figure 2.17 shows on the left-hand side the experimentally determined absorption spectrum of oxygen on Cu(100). Also shown are calculated absorption spectra
for three different adsorption sites, the top, bridge, and hollow site. The similarity
between the computed and measured curves indicates that adsorption prefers the
hollow site for oxygen atoms on Cu(100). Calculated curves for different heights of
O on the Cu surface indicates a value of 0.5 Å as the most probable position.
These are results of multiple scattering calculations that are very cumbersome.
The EXAFS regime, on the other hand, involves the close proximity of the absorbing atom, and a simple scattering procedure delivers successful results. For the
interpretation of EXAFS data, one considers the absorption coefficient
µ = µo [1 + χ(k)]. µo is the atomic contribution without the neighbors and hence
without the oscillations. The contribution from the neighbors is contained in χ(k)
which can be represented as
χ(k) =
m X Ni
2 2
t (2k) e−2Ri /Λ sin[2kRi + 2δi (k)] e−2k σi
2 i
2
4πh k i Ri
(2.20)
CHAPTER 2. CRYSTAL STRUCTURE
38
Figure 2.18: Data processing for EXAFS measurements on Ni. P. Aebi, unpublished.
Here, the following structural and electronic parameters are used. Their proper
values can be extracted from a successful experiment:
Ni
number of atoms in the i-th coordination shell
Ri
average interatomic distance to the i-th coordination shell
ti (2k) electron backscattering factor
Λ
electron mean free path
δi
phase shift for the scattering at the potential of the i-th atom
2
σi
Debye-Waller factor to consider deviations in Ri
Figure 2.18 presents an example of EXAFS data-evaluation process. An absorption spectrum is obtained from a crystalline Ni sample at 77 K, as shown in the
figure. First, a smooth background is subtracted from data to eliminate the atomic
contribution µo of Ni and obtain just the oscillatory part χ(E). The next step is to
convert the abscissa from E to k using
k = [2m/h̄2 (h̄ω − Eo )]1/2
(2.21)
with Eo the inner potential. Subsequently, the resulting spectrum is truncated
at kmin (≈ 2Å−1 ) in order to eliminate the multiple-scattering XANES signal
CHAPTER 2. CRYSTAL STRUCTURE
39
and kmax (≈ 10 Å−1 ) in order to account for more energetic absorption edges.
Fourier transformation converts data to real space where the atom-atom distance
can directly be obtained but a phase shift δ. Phase shifts have to be calculated. One also uses known values from comparable measurements with related
substances. The truncation of kmax and kmin limits the resolution scale in real
space by ∆R ≈ π[2(kmax − kmin )]−1 (Å).
Figure 2.19: EXAFS oscillations for (a) crystalline and (b) amorphous germanium.
Only the oscillatory part χ of the absorption edge is shown. Ref. [9].
An example how one obtains useful structural information without an elaborate
phase-shift calculation is illustrated for germanium in Fig. 2.19. One measures
a crystalline sample for which the interatomic distance is well known from x-ray
diffraction data. One accurately obtains herewith the phase shift. This value is
then used in amorphous Ge. As seen in Fig. 2.19, no data analysis is necessary
to extract usable information from measured data. Visual inspection shows that
there are strong harmonics in the crystalline substance, whereas in the amorphous
material the fundamental frequency dominates. This comparison can be interpreted
that in the amorphous material every atom has its nearest-neighbors in an ordered
coordination sphere and SRO prevails. In the crystalline sample, on the other hand,
the contribution of the coordination shells beyond the first one is equally important
producing higher harmonics in the spectrum.9
9
D.E. Sayers et al., Phys. Rev. Lett. 27, 1204 (1971).
CHAPTER 2. CRYSTAL STRUCTURE
40
Figure 2.20: EXAFS following the K edges in an Fe-based steel
containing 30 % Ni and 20 % Cr. Unknown report.
Figure 2.20 displays the x-ray absorption spectrum obtained from a polycrystalline sample of stainless steel containing predominantly Cr, Fe, and Ni. This
example demonstrates that the EXAFS method contains element-specific spectral
information from a sample which is not necessarily crystalline. Hence, we can obtain
from a multicomponent system crucial information about the identity and geometry of each component. We learn about bond lengths and coordination numbers.
There is no requirement on the crystallinity of the samples. All these properties
show that the method suits in an excellent way to study biomolecules for which
the large number of atoms often hinders the application of conventional diffraction
methods.