Marder M.P. Condensed Matter Physics

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Theory of Scattering from Crystals

but not lying within it. Placing the origin at this lattice site, find the three points u,

v, and w where the plane intersects the coordinate axes, measured in units of the

lattice spacing a. The inverses / = 1/w, j = l/v and k = \/w of these intercepts

are integers equal to Miller indices. This point of view is closely allied with the

second law of crystal habit, and therefore rooted in the origins of crystallography

(Problem 9 in Chapter

2).

If the plane is parallel to one of the coordinate axes, one

might for example get u = oo, leading to i = 0.

Miller Indices for Hexagonal

Lattices.

In hexagonal lattices, Miller indices make

use of four numbers (ijkl). The last integer, /, points along the z axis (the axis of

three- or sixfold symmetry), while /, j, and k refer to three axes at 120° to one

another, and conventionally

k=-(i + j). (3.26)

The precise meaning of i, j, and k is most clear if one uses the crystallographer's

definition of

the

Miller indices as inverse distances along crystal axes, as illustrated

in Figure 3.5.

Figure 3.5. The (10

1 1 )

lattice plane illustrated for an hexagonal lattice. The distances

to the plane along the coordinate axes are a, oo, —a, and —c, and the inverses of these

distances, in units of

the

lattice parameters a and c, lead to the Miller indices.

3.2.7 Scattering from a Lattice with a Basis

The precise conditions for sharp scattering peaks are modified if the crystal is not

a Bravais lattice, but is instead a lattice with a basis. In this case, every vector in

the crystal can be written in the form

R =

U[+vi>

(3.27)

for some / and /', where w/ is a vector in a Bravais lattice and v\i is an element in

the basis. Assuming all atoms to be identical, as in the diamond or hep lattices,

scattering from such a lattice can be calculated by regrouping the basic sum.

Σ S* = Σ

+ii

)

(3.28)

R "'

Chapter 3. Scattering and Structures

= (E «**) (E e*'") R-al, £„,

QD,,

= (

Σ/

Q) (Σ, Ο,)

■

(3-29)

/oc

( Σ e-^-W

)

( Σ

·^'"^

) ·

(3.30)

The Bravais lattice vectors and basis vectors appear in a completely symmetrical way.

However, the sum over Bravais lattice vectors might have 10

elements while the

sum on the basis might have 8, so the sums have qualitatively different character.

The first term in Eq. (3.30) is precisely the sum over Bravais lattice vectors that

appeared previously in Eq. (3.10), and it therefore is nonzero only for vectors a

that lie in the reciprocal lattice K given by Eq. (3.18). However, the strength of the

peaks is modulated by the function

p-. = I \

e I . Sum / over all vectors in the basis. (3.31)

In some cases, the result may be that F$ vanishes, and certain peaks disappear

altogether. These cases are called extinctions.

Example: Diamond Lattice. The diamond lattice, described in Section 2.2.6, is

built from the fee lattice with the basis

£, = (0 0 0),

= ^(l 1 1)· (3-32)

Because the reciprocal lattice of an fee lattice is bec with lattice spacing 4π/α, the

reciprocal lattice vectors are

—

4-TT

47Γ

4"7T

K = h—

1 -\)+h — (-1 1 \) + h — {\ -1 1). SeeEq.(2.4). (3.33)

2a 2a 2a

Therefore,

-K=^{h+h + h), (3.34)

and the modulation factor F^ is

F£ = |l+£>('»

+/2+/3

(3.35)

'4if/

= 4, 8, 12, ...

2if/,+/2 + /

isodd (3.36)

0if/i+/

= 2, 6, 10,

So some of

the

peaks are four times as bright as they were before the basis, some are

twice as bright, and some have disappeared. The latter are examples of extinctions.

3.3 Experimental Methods

It remains to determine the implications of these calculations for scattering experi-

ments, and it is probably best to begin by summarizing the mathematical results so

Experimental Methods

far. Radiation with wave vector ko impacting a crystal produces a scattered wave

along direction k, where the magnitudes of k and ko are the same, but only if the

wave vector q = ko

—

k equals one of the reciprocal lattice vectors K of the crystal.

The reciprocal lattice vectors are completely determined by the underlying Bravais

lattice of the crystal, and any decoration of the Bravais lattice with a basis serves

only to modify the strengths of the scattering peaks, not their positions.

The prescription for a successful experiment at first seems clear. Shine a

monochromatic X-ray source at a crystalline sample, put a camera behind it, and

start clicking the shutter. But a moment's reflection shows that this brilliant idea

will not work. Each point on the photographic paper catching a scattered X-ray

corresponds to a single scattering direction k, so the experiment scans through a

two-dimensional space of scattering vectors. However, the reciprocal lattice is a

discrete set of points in three dimensions, and it is therefore exceptionally unlikely

that any given two-dimensional surface will cut through any of them. In order to

visualize this point, it is convenient to look at the Ewald construction, shown in

Figure 3.6.

The Ewald construction has the first advantage of showing the necessary scale

for the wavelength of incident radiation. In order to resolve atomic structure, its

k vector should be comparable to the spacing of reciprocal lattice points, although

somewhat larger. One therefore needs wavelengths on the order of an angstrom,

which for electromagnetic waves requires the X-ray portion of the spectrum. It also

permits one to imagine strategies by which to carry out scattering experiments;

Figure 3.6. For incoming radiation ko there will be outgoing radiation along direction k

only if q = k

— ko

kor — ko

lies in the reciprocal lattice. For a fixed orientation of

crystal

and wave number

ko,

all an experiment can do is scan over observation directions r. This

produces candidate <f s in a spherical shell shown at the left (called the Ewald sphere; a

two-dimensional cross-section appears on the right), and all the reciprocal lattice vectors

K are shown as dots. For the direction and magnitude of incoming radiation displayed here,

an attempted scattering experiment would end with an unexposed piece of film, because

there are no intersections of the spherical shell with the reciprocal lattice vectors. This

graphical representation of Eq. (3.18) is called the Ewald construction.

Chapter 3. Scattering and Structures

there are three traditional solutions to the problem that monochromatic radiation

generally does not produce sharp scattering peaks after contact with a fixed crystal:

the Laue method, the rotating crystal method, and the powder method.

3.3.1 Laue Method

The first solution is the Laue

method,

found as an accidental offshoot of the manner

in which the first X-rays were generated. X-rays were produced by bombarding

a tungsten anode with electrons accelerated through about 510

eV. A slightly

higher voltage would excite sharp lines in the tungsten spectrum, corresponding to

ionization energies of inner core electrons, but at this voltage the X-ray spectrum

is continuous, resulting from multiple collisions, and looks as in Figure 3.7(A).

X-rays generated in this fashion are called Bremsstrahlung ("braking radiation").

1.0

(B)

Figure 3.7. (A) Schematic indication of the continuous spectrum resulting from electron

collisions upon tungsten at around 50 keV. (B) Ewald construction. The gray points are the

scattering peaks that would be visible when incoming radiation has a wave vector between

The continuous component of the radiation was essential to the success of the

experiment. As shown in Figure 3.7, if the incident radiation spans a range of wave

vectors, the chance that some intermediate wave vector will actually satisfy condi-

tion (3.18) becomes considerable. The particular frequency of incoming radiation

needed to cause scattering can be obtained from Eq. (3.6) by setting q = K and

writing

kl = kl-2k

-K + K

(3.37)

=/■

= —^ z7, Because K = q when scattering occurs. (3.38)

-K

while the relation between the Bragg angle

and the reciprocal lattice vector K is

K lc

—-^ = Sin Θ Using

Eq.

(3.6) (3.39)

Kko

The Laue method is not a good method for precise structure identification,

because the intensities of spots depend upon both the intensity of incoming X-

■R

■33

(A)·

0.2 0.4 0.6 0.!

Wavelength λ (Â)

Experimental Methods 57

rays at varying frequency and properties of the scattering sample. However, for

a substance whose lattice parameters are known, it provides an excellent means

of orienting the sample. For this purpose, one needs to be able to identify the

reciprocal lattice vectors corresponding to spots observed on a photographic image.

One way to simplify the task is with the Mauguin abacus, which is a ruler whose

marks are graduated to indicate D tan

to the left of the origin and D(tan 20

—

tan

Θ)

to the right of the origin, where D is the distance between sample and film.

The reason for this choice is illustrated in Figure 3.8.

D tan

2Θ

D tan0

Figure 3.8. The parallel lines within the crystalline sample indicate a set of Bragg planes.

they

were extended, they would contact

backing piece of

film

at height

tan

Θ,

but the

radiation bouncing off

the

planes hits at height

tan

2Θ

instead. This observation provides

a simple geometrical way to deduce Bragg angles

from Laue photographs.

3.3.2 Rotating Crystal Method

A second method is the rotating crystal method. In contrast to the continuous ra-

diation used for a Laue pattern, this method relies upon an intense monochromatic

beam. A crystal is rotated about one or more axes, and in the course of rotation a

variety of scattering peaks is recorded on a cylindrical film, as shown in Figure 3.9.

Because all the scattering spots on the film result from one frequency of incoming

radiation, the intensity of the scattering is meaningful, and the size of the spots

recorded on the film can be used for quantitative analysis.

The scattering geometry shown in Figure 3.9 is not employed in practice, in

part because reciprocal lattice vectors parallel to the rotation axes cannot be im-

aged, and in part because scattering spots all cluster tightly on parallel lines on the

film, and the results are difficult to decipher accurately. Instead, there is a vari-

ety of experimental arrangements where the sample and flat plates of film rotate

simultaneously. The operation of one of these, the precession camera, is indicated

schematically in Figure 3.10(A). Spots appear on the film whenever film and sam-

ple are rotated through a Bragg angle

Θ,

and the image of the spots on the film

provides an undistorted projection of the reciprocal lattice.

k ^^

58 Chapter 3. Scattering and Structures

Figure 3.9. When a crystal is rotated about a fixed axis, numerous reciprocal lattice vectors

generate scattering peaks when they intersect the Ewald sphere. (A) The geometry of the

experiment. (B) The rotation axis comes out of the page, and scattering occurs whenever a

reciprocal lattice point intersects the Ewald sphere.

(A)

Figure 3.10. (A) Schematic description of a precession camera. The lines in the sample

indicate Bragg planes, from which incoming radiation reflects to hit the film in a specular

fashion. Whenever both film and sample are rotated through the Bragg angle

Θ,

a spot

hits the film at distance 2D sin

from the origin, a distance that according to Eq. (3.8) is

proportional to the magnitude of the relevant reciprocal lattice vector. By rotating sample

and film about multiple axes one obtains an undistorted projection of the reciprocal lattice.

(B) Precession image of K

Cr0

. (Courtesy of

Steinfink, University of Texas.)

Experimental Methods 59

3.3.3 Powder Method

Figure 3.11. The powder method generates every scattering peak from a monochromatic

beam that could be produced by a crystal at any imaginable angle. Whenever a reciprocal

lattice point contacts the Ewald sphere, rotating the crystal about the direction of the in-

coming wave continues to produce a scattering peak. Therefore scattering from powdered

samples is in the form of rings.

The third general technique for structure determination is the powder

method,

also called the Debye-Scherrer

method;

this method is suitable for materials when

a perfect single crystal is not available. The incoming beam needs again to be

monochromatic, but now the scattering sample is either polycrystalline, with in-

dividual crystallites on the order of a micron, or else a powder made of particles

roughly that size. Because small crystals are present in all orientations in the sam-

ple,

the net effect is the same as a rotating crystal experiment that turns the crystal

through all possible directions, and there is a ring of scattering peaks, as shown in

Figure 3.11, corresponding to every reciprocal lattice vector of magnitude less than

twice that of the incoming beam; the Bragg angles corresponding to the scattering

are

Θ = Sin" ' (K/2k

)

Eq- (3-8)· K, is the magnitude of a re- (3.40)

ciprocal lattice vector.

and the radius r on film of the scattering ring due to reciprocal lattice vector K is

f = D

tan(2$).

D is the distance from sample to film. (3.41)

From the series of rings recorded on film, complicated crystals can be worked

out. The process proceeds in stages. The first step is to compare the locations of

the rings with those produced by various known crystal structures and to pick out

reasonable candidates, trying to identify the Bravais lattice first, and then working

out decorations and extinctions next. The second step settles on a set of candidates

and carries out a refinement process, varying precise locations of the atoms, varying

scattering intensities of the species, and allowing for thermal motion, in order to

produce a best fit to the data. In practice, this procedure is usually computerized,

Chapter 3. Scattering and Structures

J L

Simple cubic

_l_

Basis at (.25 .25 .25)

0.0

Body-centered cubic

-U

0.5 1.0

Bragg angle

(radians)

1.5

Figure 3.12. Scattering intensity as a function of Bragg angle

for three monatomic

lattices. At the top is a simple cubic lattice with lattice constant a and incoming radiation

of wavenumber

= 10/α. Each lattice site contains two atoms, to ease comparison with

the remaining cases. In the middle is the same lattice, but decorated with basis atoms at

the origin and at (.25, .25, .25). Finally, at the bottom the basis moves to (.5, .5, .5), so

that the lattice is now body-centered cubic. Notice the increase in scale of the scattering

pattern as the physical lattice develops structure on smaller scales.

with locations and intensities of rings being compared automatically with databases

of known structures.

A simple example of

the

results to be expected from powder diffraction appears

in Figure 3.12. The intensities of lines in this figure are calculated according to the

results obtained in Problem 7.

3.4 Further Features of Scattering Experiments

The discussion so far has presumed that X-rays lie behind all scattering exper-

iments, and that the scattering form factors f(r) contain no useful information.

Neither of these points is correct, and the corrections deserve some discussion al-

though there is no possibility of describing all the refinements and elaborations of

structure determination since 1912.

There are three types of particles widely used in scattering experiments: pho-

tons,

neutrons, and electrons. They differ in the relations between energy and

momentum, in the strength and nature of their interaction with condensed matter,

and in cost and ease of generation. Some characteristics are displayed in Table 3.2.

3.4.1 Interaction of X-Rays with Matter

The interaction of X-rays with condensed matter is actually quite complicated. The

easiest way to get an idea of what it should be is to recall the expressions in classical

electromagnetism for the interaction of an electromagnetic wave with an isolated

charged particle. The charged particle vibrates at the frequency of the incoming

Further

Features

of Scattering Experiments 61

Table 3.2. Characteristic values associated with different types of radi-

ation

X-rays Neutrons Electrons

Ö ~e

1.67· 10^

kg 9.11·

60keV

Charge

Mass

Typical energy

Typical wavelength

Typical attenuation length

0 0

0 1.67·

12keV 0.02

1Â 2À

100 μηι 5 cm

0.05Â

1 μπι

Typical atomic form factor, / 10~

Â 10~

Â 10 Â

Source: Eberhart (1991).

radiation, and it reradiates a spherical wave. The scattering cross-section for this

process is

( 1

+ cos

260 e

/atom(r) = —

~ '- = -T-

P{r) See Jackson (,999). (3.42)

2 m

Eq. (14.124); Pis a

polarization factor.

f(f) =

JP(r) = 2X2- \QT

JP{r)

TO.

Putting in values for the (3.43)

ItlC * * electron and expressing the

answer m meters.

Because the .nuclei of atoms are so much heavier than the electrons, only the elec-

trons contribute to X-ray scattering. However, the electrons are not tightly localized

in space, but instead are characterized by a number density n(7), which peaks up in

the vicinity of ion cores, but does not completely vanish between them, particularly

in metals. An expression for X-ray scattering which takes this fact properly into

account is

f(r) = —

φψ) ( dr

n{f)S-

= ~

>Jp{f)n{q)

(3.44)

J mc

L v

=> /atom(g) = n . P(r) |w(^) | . n(q) is the Fourier transform of the (3.45)

m C electron density.

Hahn and Cochran (1992) have tabulated f(f) for all the elements.

3.4.2 Production of X-Rays

The oldest way to generate the monochromatic X-rays needed for rotating crystal

and powder methods is to collide electrons upon a metal anode at energies great

enough to ionize core electrons, and thereby create a bright resonance at a precise

frequency, such as the Ka line of copper at 8.98 keV. Although a good fraction of

the emitted X-ray power is contained in the desired line, a continuous component

to the radiation is unavoidable. One way to omit the continuous component is to

obtain a crystal arranged to have a scattering peak at precisely 8.98 keV, and then

use the scattered beam from the reference crystal as the incoming radiation for

Chapter 3. Scattering and Structures

further experiments. The highly monochromatic beam produced in this way has

unfortunately low amplitude, and a brighter but less narrow beam can be chosen

by filtering the X-rays through an element that absorbs most of the continuous

radiation while leaving the sharp line alone; a good candidate for such a filter is

the element one atomic number down in the periodic table, which in the case of

copper is nickel. Electron collisions with a metal do not provide an efficient way

to generate X-rays; 99% of incoming energy is converted to heat. Even employing

tricks such as rapidly rotating the target anode to keep it cool, the maximum X-ray

power that can practically be extracted from conventional tubes is around 100 W,

and by the time the X-ray has been filtered, fewer than 10 W may be left. The

situation is very different at a synchrotron, where radiation is generated by rapidly

accelerating electrons in huge rings. The total power emitted as X-rays in a broad

band up to 200 keV is on the order of 10

W, so even after filtering to

0.1%

of the

200 keV bandwidth to get a monochromatic beam, 1000 W of power is left. The

relative intensities of radiation at various frequencies available from the different

types of X-ray sources are compared in Figure 3.13.

Figure 3.13. Overview of various means of producing radiation, relative to the intensity

available from a synchrotron. Intensity for sources spanning a range of wavelengths is

measured by finding the number of photons in a frequency interval that is 0.1% of the

total wavelength range. Synchrotron radiation is orders of magnitude more intense than

any source apart from lasers, and spans a broad range of wavelengths. [After Brefeld and

Guertler(1990), p. 285.]