Marder M.P. Condensed Matter Physics

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Scattering and Structures

3.1 Introduction

The scattering of X-rays from crystals provided information on the locations of

atoms within solids that had been the object of speculation for centuries, and it

rapidly led to a huge scientific enterprise of structure determination, but before the

first careful experiments of 1912 it was far from obvious that anything could come

of it at all.

M. von Laue was struck in 1912 by the intuition that X-rays might scatter

off crystals in the way that ordinary light scatters off a diffraction grating. This

hunch preceded any mathematical attempt to quantify the size and character of the

effect, and if it seems obvious in retrospect, one might keep in mind that neither

the periodic character of crystals nor the wave nature of X-rays was known with

certainty at the time. Laue discussed

his idea with colleagues Sommerfeld, Wien and others with the result of

encountering a strong disbelief in a significant outcome of any diffraction

experiment based upon the regularity of the internal structure of crystals.

It was argued that the inevitable temperature motion of the atoms would

impair the regularity of the grating to such an extent that no pronounced

diffraction maxima could be expected. —Ewald (1962), p. 42

This argument against diffraction effects was in fact quite reasonable. Consider,

for example, the prospect of scattering off

NaCl.

The study of elastic deformations

was well developed in 1912, and salt crystals were known to be anisotropie, but

characterized approximately by Young's modulus Y (defined in Section 12.3.2) of

5-10

ergs cm

-3

. The chlorine atom was known to have a mass of 35 g mole

-1

and sodium 23 g mole

-1

; from salt's known density of 4.29 g cm~

, a charac-

teristic spacing of d = 2.5 · 10

-8

cm between sodium and chlorine could be de-

termined. To account for the observed Young modulus, one would need a spring

constant between atoms on the order of % = Yd =10

dyne cm

-1

. Making use

of the equipartition theorem, the characteristic excursion of atoms due to thermal

fluctuations should therefore be expected to be x =

^/2kßT/%

ss 2

-9

cm. This

distance is fortunately smaller than the interatomic spacing; it has to be, otherwise

the crystals would melt. However, the best estimate for the wavelength of X-rays

based upon diffraction around small slits was also of order 10~

cm. A diffraction

grating whose vibrations are the same size as the incoming waves hardly seemed

to pose a promising experiment, and Sommerfeld was reluctant to devote resources

or personnel. The work was carried out by Friedrich et al. (1912) with equipment

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

Chapter 3. Scattering and Structures

pirated from elsewhere; it successfully produced diffraction spots from copper sul-

fate on only the second attempt, the first having failed when photographic plates

were placed in front of the crystal in the mistaken expectation that reflected X-rays

would produce the largest effect. Laue's mathematical theory for the directions of

X-ray maxima followed only a few hours later. At first the theory did not encom-

pass the possibility of a lattice with a basis, with the result that some predicted

spots were mysteriously absent from the experiment.

There was some question, at first, about whether the spots observed on the

photographic plate were truly due to diffraction from the crystal. The doubts were

largely dispelled by a series of experiments which seemed to rule out any other

possibility. First the crystalline sample was removed, and it was checked that the

diffraction pattern disappeared. Second, the crystal was replaced by a powdered

sample of the same material, and again distinct spots disappeared. Finally, the ori-

entation of the crystal was altered slightly, and the spots were observed slightly

displaced on the photographic plate. Several years were to pass before it could cor-

rectly be explained why thermal agitation did not obliterate the X-ray interference,

but by summer of 1912 the phenomenon was well on its way to wide acceptance

throughout Germany.

The discovery of X-ray diffraction contains many elements that have been ab-

solutely characteristic of important developments in condensed matter physics.

The enterprise began with a theoretical notion to which, however, there were com-

pelling theoretical objections. The experiment rapidly encountered a highly regular

set of phenomena with which theory agreed only uneasily in its first incarnation.

A rapid simultaneous development of theory and experiment in concert then pro-

duced a new set of concepts and a powerful new experimental tool.

3.2 Theory of Scattering from Crystals

3.2.1 Special Conditions for Scattering

X-rays created a world-wide sensation at the end of the nineteenth century because

they could produce images of the interior of the human body. This means that

they can travel through several centimeters of solid matter, attenuating weakly, and

come out the other side. When X-rays travel through crystals this is usually what

happens.

However, there is an important exception. When the wavelength of the X-

rays is chosen precisely right, and simultaneously the orientation of the crystal is

precisely right, there is constructive interference between the waves scattered by

successive atoms. Intense narrow beams of X-rays emerge from the crystal in a

finite number of special directions. Friedrich, Knipping, and von Laue were not

so lucky as to stumble exactly upon the frequency of radiation needed to create

the effect. Their X-rays contained a broad spectrum of radiation, and happened

to contain the special waves (Section 3.3.1). The same considerations apply to

neutrons and electrons, two other types of waves that can reveal atomic positions

in condensed matter. Their interaction with matter is mainly linear, so it is enough

Theory of Scattering from Crystals 45

Figure 3.1. Geometry of a scattering experiment. Radiation of wave vector

arrives at

a sample, inducing a circular ring of radiation to depart from each atom. If

is chosen

just right, the scattered radiation from the atoms adds constructively in certain directions.

It is detected at position r, which points in the same direction as the scattered radiation k.

The Bragg angle

characterizing the scattering is half the scattering angle at which the

radiation is observed. This figure shows the amplitude of the radiation field due to a square

crystal composed of 25 atoms; the angular range of constructive interference becomes

much narrower for larger crystals. Figure 3.2 contains a more precise characterization of

the waves involved in this figure.

Figure 3.2. Illustration of Bragg scattering at angle

Θ —

26.56° from the (21) planes of a

square lattice. The magnitudes of

ko,

k, and K are determined using Eqs. (3.38) and (3.39).

Chapter 3. Scattering and Structures

to pick

single incoming plane wave, study what happens, and build a theory of

more elaborate scattering experiments by superposition.

3.2.2 Elastic Scattering from Single Atom

In a scattering experiment, a plane wave moves into a sample of condensed matter,

travels through it, interacts, comes out, and is measured by detectors far away.

the simplest class of scattering experiments, the frequency of the outgoing radiation

is the same as that of the radiation sent in, and the scattering is called elastic. This

phenomenon is most easily explained by invoking wave-particle duality to view

the incoming radiation as photons, neutrons, or electrons that bounce without any

change of energy off atoms in the sample.

Begin by considering a plane wave that collides with a single atom sitting at the

origin and scatters off it. Whether the incoming wave is a neutron or an electron

described by quantum mechanics, or an X-ray described by classical electromag-

netism, the scattered wave ψ takes a particularly simple form far from the atom:

Jkor

ψ « Ae~

"[e'

°'

+ f(f) 1,

See, for example, Schiff (1968) p. 115 for the

(3.1)

quantum mechanical version, or Jackson (1999)

Eq. (9.8) for the electromagnetic version.

where the particle enters as a plane wave along ko, the scattering is measured at

a distance

that is much larger than the range

interaction with the atom,

angle

2Θ

relative to the

axis, and

is a form factor containing details about the

interaction between the scattering potential and the scattered wave. It is related to

the differential scattering cross-section of the atom by

= =

|f(r)| The intensity of scattered radiation into solid

(3.2)

dÇl atom angle

dii

at distance

relative to the incom-

ing beam is dfi

da/dQ/r

Forms for

will be given in Section 3.4, but for the moment assume

to be known.

Next ask how Eq. (3.1) changes when the same incoming plane wave scatters

off a particle located at R rather than at the origin. In this case, when the radiation

arrives at point R, it has phase exp(i7?

· ko)

relative to what it would have had at the

origin. On the other hand, to reach the observation point, it then has only to travel

a distance

\r —

rather than

as previously. Therefore

-, - _ -

ikj\r—R\

Ae~

iu}>

ik(yR

iko

'^~

"> +/(r) —1. The leading exponential is

(3.3)

ft\

unchanged from Eq. (3.1).

For sufficiently large r, one may use the approximation

ko\r-R\~k

r-ko--R.

(3.4)

Using Eq. (3.4) and defining

Theory of Scattering from Crystals

- r

k =

KQ —,

k points in the observation direction (3.5)

r r and has the same magnitude as

k(>.

and

q =

k.Q

—

k (3.6)

gives

- _ g'kor+il'R In the denominator of

Eq.

(3.3) one needs only the

ψ ~

Ae~

[e °'

+ /(f) ]. first term on the right side of Eq. (3.4). However, (3.7)

f within the exponential function, one has to keep

all terms that are large compared to 2π, and re-

quires both terms on the right side of Eq. (3.4).

Note that

n = 2kn sin Θ. Square both sides of

Eq.

(3.6) and use the fact that (3.8)

ko-k =

£Q

cos

2Θ.

The quantity hq describes the momentum

transfer

between the incoming and out-

going waves, and the angle

is called the Bragg angle.

3.2.3 Wave Scattering from Many Atoms

When one has a large assembly of scatterers, shown in Figures 3.1 and 3.2, the

angular dependence of the scattered radiation is the product of two pieces. The

first results from the fact that each individual scatterer emits radiation with differ-

ent intensities in different directions, described by the form factor f. The second,

which modulates the first, results from interference between the radiation com-

ing from the various objects and therefore contains information about their spatial

correlation.

So consider a large collection of scatterers, placing the origin somewhere in

the middle of them, and observe the scattering from far away. If the scatterers

are dilute, then the total scattered radiation due to the collection will simply be a

sum of the radiation due to each one. Ignoring the contributions that arise when

the light which has scattered off one atom then scatters off of another one before

being observed (multiple

scattering),

as well as ignoring contributions arising from

changes in the state of the scatterer (inelastic

scattering

), one has

ψ-Αβ-^'Ιβ^

ΣΜϊ)

r+iq-R,

Summing many terms of the sort that arise in (3 9)

Eq.(3.7).

The largest contribution to ψ comes from the incoming beam exp

[iko ·

r],

which also

exits the sample as shown in Figure

3.1.

However this bright spot is restricted to an

outgoing angle of

= 0, so with the understanding that one will restrict attention

to nonzero scattering angles it is possible to drop the first term on the right-hand

side of

Eq.

(3.9). The intensity per unit solid angle divided by the intensity \A\

the incoming beam, / =

\ip\

is then

~ (R R )

ReCa

that

' Σι

2 =

Σ//'

P-

The

functions /,

fitfe'W·-"'''. still depend upon the scattering direction r, but the de- (3.10)

. ., pendence is not made explicit.

Chapter 3. Scattering and Structures

This basic equation for weak elastic scattering applies to scattering off matter in all

forms.

The equation is posed in terms of intensity rather than amplitude because

intensity is the quantity measured by experimental instruments.

3.2.4 Lattice Sums

Now ask what happens when all of the scatterers in the sum (3.10) are identical

and arranged in a Bravais lattice. In this case, the scattering intensity is equal to

/ = /

atom

'Q(

i'\ /

alom

is the scattering from a single atom, de- (3.11)

^—' fined in Eq. (3.2).

Inspection of Eq. (3.11) leads to a plausible guess for the wave vectors q that

will produce strong scattering. If one could choose q so that exp(iq-R) =

for all

then all terms in Eq. (3.11) would be one, and summing them would produce a

large final result. Otherwise, one might expect terms in Eq. (3.11) to alternate in

sign and to cancel out on average.

This guess is correct, but to justify it one must understand sums of the type

appearing in Eq. (3.11). Such sums determine the behavior of

any

waves interacting

with a periodic lattice; they are important not only for X-ray scattering, but also for

conduction electrons in lattices (Chapter 7). The mathematics of the sum is best

explained in one dimension, and then generalized.

One-Dimensional Sum. In one dimension, lattice points must be of the form

la, where / is an integer and a is the interparticle spacing. So the relevant sum in

Eq. (3.11) becomes

Y^e

ilaq

(3.12)

/=o

This sum is calculated in Appendix A, which yields the results

JNaq _ i

7^τ

13)

ÙÎNaqjl

sin aq/2

A plot of Eq. (3.14) appears in Figure 3.3. The graph contains a number of very

sharp identical peaks, separated by regions where the scattering intensity is nearly

zero.

The locations of

the

peaks are determined by searching for the points at which

the denominator of Eq. (3.14) vanishes. They occur whenever

aq/2 = Ιπ

=>■

q = 2πΙ/α.

/ is an

integer.

(3.15)

Glancing back at Eq. (3.12), one sees that peaks in the sum Σ

correspond exactly

to the choices for q such that all terms in the sum (3.12) equal 1 and thus add

coherently. For any other choice of q, the terms in the sum add with different

phases and signs, giving a much smaller result.

Theory of Scattering from Crystals

JUL

, il , IL· il

2π 4π 6π 8π

a a a a

Wave number q

Figure 3.3. Plot of |Σ

from Eq. (3.14), for N = 30. Note the sharp peaks centered

around points where q = 2πΙ/α.

Because the peaks in Figure 3.3 are so sharp, it is natural to view Y,

as a sum

of delta functions. However, because the delta function is defined to have unit

area lying beneath it, one must find the area under each peak. It is calculated in

Appendix A and is 2-KN/L. Thus one has the identity

N—Ì

OO n.

aq=

J2 Ν^δ(ς-2πΙ'/α).

(3.16)

/=0 /' = -oo

3.2.5 Reciprocal Lattice

Returning now to Eq. (3.11), one can state that the requirement for the sum to yield

a sharp peak is that q be chosen such that

q-R = 2nl (3.17)

for all R in the Bravais lattice, and with / some integer depending upon R. Once q

has been chosen in this fashion, all the terms in the sum (3.11) equal one and add

together, producing a Bragg peak. Any scattering vector q with this property will

be denoted by the symbol K, and the collection of scattering vectors K satisfying

C\p[iK R] =

\OTKR

= 2πΙ The value of the integer / will depend upon (3.18)

the

choice of R and K. This equation de-

mands

only that the dot product produce 2π

times

some integer.

is called the reciprocal lattice. The importance of the reciprocal lattice is that it

identifies the collection of wave vectors K for which coherent scattering can oc-

cur off a Bravais lattice. The strength of the scattering is given by the analog of

1UUU

800

600

400

200

50 Chapter 3. Scattering and Structures

Eq. (3.16), which according to Eq. (A.25) is

R K

where V is the volume of the system.

Bragg Planes. Why should there be any K that have property (3.18), and why do

they form a lattice? First of all, notice that if one finds any two K that have this

property, then their sum has it. Second of all, suppose that one has found a vector

K so that

KR = 0 (3.20)

for at least three noncollinear Bravais lattice points R. The set of all r that satisfy

■

r = 0 is a plane passing through the origin, so the R that satisfy Eq. (3.20) must

lie in such a plane

1Ό

(Figure 3.4), and K must be the normal to that plane. Thus the

direction of every K corresponds to a plane containing points in the Bravais lattice.

However, it is not enough to guarantee that K

■

R = 0 for R lying on the plane

7Ό;

one needs Eq. (3.18) to hold for all R. To achieve this goal, it is essential to note

that the whole Bravais lattice can always be constructed by taking the points that

lie in the plane

CPo,

all of which are of the form

ΙχΟ,-ι

-\-ljfl2 Having chosen a plane, one can then find two (3.21)

primitive vectors that span it.

and stacking multiple copies of this plane next to one other, separated by a third

primitive vector

a-},

as shown in Figure 3.4. There is an infinite number of ways to

decompose a crystal in this fashion into Bragg planes.

Figure 3.4. Any Bravais lattice can be regarded as composed of parallel planes of points,

the Bragg planes. In this case, a simple cubic lattice is built out of parallel planes normal

to (110) (stereo pair).

Choose the magnitude of K so that

-Κ =

2ττ.

(3.22)

Theory of Scattering from Crystals 51

Once Eq. (3.22) is satisfied, one has for any point in the Bravais lattice

K-R

= K-(l

ai+l

)

=2π/

. (3.23)

The integer

IT,

simply counts by how many lattice planes the one containing R is

distant from the origin.

Because the magnitudes of the vectors K are not arbitrarily small according to

Eq. (3.22) and because the sum of any two is another such vector, the set of K's

must themselves form a lattice. One can find primitive vectors for the reciprocal

lattice, by choosing any three primitive vectors a\, a

, and a

for the Bravais lattice

and then building three primitive vectors b\, b

, and b

for the reciprocal lattice

from

- a? x «3

£ι=2π^-^—— (3.24a)

a\ -a

ατ,

= 2vr^^—— (3.24b)

-«3 xa\

a\ x a

-a\ x a

-* at x a

= 2π ^ ^ \, (3.24c)

and in general K — ^ m\Î>i.

e m are arbitrary integers. (3.24d)

1=1

This construction works because if one takes the dot product of b\ with any prim-

itive vector αιι, the result is either zero or

27Γ.

Therefore, any vector in the direct

lattice dotted into linear combinations of the è/'s gives an integer times 2π as re-

quired.

Table 3.1 records the reciprocal lattices of four common Bravais lattices. The

reciprocal lattice of a fee lattice of spacing a is, a bcc lattice of spacing 4π/α, the

reciprocal lattice of a bcc lattice of spacing a is an fee lattice of spacing 4π/α, and

the reciprocal lattice of an hexagonal lattice is another hexagonal lattice twisted by

30°.

3.2.6 Miller Indices

A traditional notation describing reciprocal lattice vectors, lattice planes, and lat-

tice points is the Miller index. The notation is most commonly used for lattices of

cubic and hexagonal symmetry.

In cubic crystals, one begins by defining three perpendicular coordinate axes x,

and z pointing along the edges of the conventional unit cell.

•

[ijk]

refers to a direction

+ jy + kz (3.25)

in the lattice specified by the three integers i,

and k.

Chapter 3. Scattering and Structures

Table 3.1. Conventional primitive vectors of four common Bravais

lattices and their reciprocal lattices.

Lattice

fee

bec

hex

Lattice

Spacing

a, c

Primitive

Vectors

(10 0)

(0 10)

(0 0 1)

(22°)

( 2 0 2)

(0 \ \)

fi i _i)

I 2 2

f_i i i)

V 2 2

f i _i Ì)

\ 2

2 2^

( 1 f o)

H f

(001)

Reciprocal

Lattice

Spacing

2π

4π

4π 2π

\βα e

Reciprocal

Lattice

Primitive

Vectors

(100)

(0 10)

(0 0 1)

fi i _\\

l 2 2

f_i i ì)

v 2 2 2/

f i _i i)

\ 2 2

(Ho)

( io Ì)

è £)

( f 20)

(f-^o)

(0 0 1)

• (ijk) refers to a lattice plane perpendicular to the normal vector [ijk]. (ijk)

may also refer to the reciprocal lattice vector of smallest magnitude perpen-

dicular to the plane

(ijk).

• {ijk} refers to the complete collection of lattice planes perpendicular to [ijk].

• ijk refers to the X-ray diffraction peak resulting from scattering off the lattice

planes

{ijk}.

• Negative integers are denoted by / rather than — i.

• (ijk) refers to the set of all lattice planes or reciprocal lattice vectors related

to (ijk) by rotational symmetry.

Examples. For a simple cubic lattice, the reciprocal lattice vector pointing along

(2π/α, 0, 0) is called (100), while the one along (2π/α, 2π/α, 2π/α) is called

(111) and the one along (2π/α, —2π/α, 2π/α) is called (111), and so forth. For

an fee lattice, the smallest reciprocal lattice vector is (111), and it has magnitude

2ν

3π/α. The collection of reciprocal lattice points referred to by (100) is (100),

(010),

(001), (TOO), (0Ϊ0), and (00Ϊ).

Alternate Definition. An alternate prescription used by crystallographers to find

the Miller index of a plane is to find a lattice point as near as possible to the plane