Marder M.P. Condensed Matter Physics

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Inelastic Scattering from Phonons

363

For an isotropie solid, the coefficient of

linear

expansion ατ is just one-third

of the coefficient of volume expansion ßj. For a crystal, there will generally be

different degrees of contraction along different crystalline axes.

Definition of the Grüneisen parameter does not dramatically simplify the task

of presenting data on thermal expansion data, but does correctly emphasize the fact

that trends in the specific heat, particularly at low temperatures, are likely to be re-

flected in thermal expansion data. For some materials, the Grüneisen parameter is

constant over a wide temperature range, leading the specific heat and thermal ex-

pansion coefficient to become indistinguishable when properly scaled, as shown in

Figure

13.13.

However, even the sign of

the

Grüneisen parameter can change. Fig-

ure 13.14 shows thermal expansion data for Invar, an alloy useful for its extremely

small expansion coefficients.

X, X

<43

<Ji

1CT

10"

-♦ Volume expansion

-o Specific heat

100 200

Temperature T (K)

300

Figure 13.13. Comparison of specific heat Cv and volume expansion coefficient ßj for

aluminum using Eq. (13.90). The comparison employs a bulk modulus B of

GPa from

Eq. (12.29) and Table

12.1,

uses a Grüneisen parameter 77 of 2.2, obtained by matching

specific heat and expansion at 300 K, and obtains the coefficient of bulk expansion from

linear expansion data through ßj « 3αγ- [Source: Touloukian and Buyco (1970a) and

Touloukian et

al.

(1975).]

Thermal

Conductivity.

Phonons also participate in thermal conductivity of solids.

A review focusing upon insulators is provided by Slack (1979).

13.4 Inelastic Scattering from Phonons

The scattering experiments described in Chapter 3 relied upon elastic scattering;

the energy of outgoing particles equals the energy of incoming particles. There is

another class of scattering experiments—inelastic scattering—where the incoming

364

Chapter 13. Phonons

CL.

100 200

Temperature T (K)

300

Figure 13.14. Coefficient of

volume

expansion (β

« 3a

) for

Invar,

Fe-64%

Ni-36% al-

loy in which reduction of

magnetic

order with increasing temperature offsets the otherwise

natural tendency toward thermal expansion and produces an alloy with minimal tendency

to expand. [Source: Touloukian et

al.

(1975), p. 10a.]

beam gains or loses energy at the expense of the sample. The first type of experi-

ment determines static structures, while the second focuses upon excitations.

The theory needed to explain inelastic scattering experiments is available at two

levels.

The first level of explanation derives all its conclusions from conservation

laws.

It is simple and powerful, providing everything needed to deduce phonon

dispersion relations from neutron scattering experiments, and is easily generalized.

The second level of explanation attacks the problem with a full formal apparatus.

It is worth going through the more elaborate analysis because it provides various

constants that the simpler theory leaves undetermined, determines the effects of

thermal fluctuations, and settles questions about the workings of quantum mechan-

ics.

13.4.1 Neutron Scattering

A particularly important type of inelastic scattering experiment is performed with

neutrons, as sketched in Figure 13.15. A beam of neutrons impinges on a sam-

ple.

Some of the neutrons gain or lose energy by destroying or creating lattice

vibrations—phonons—and emerge from the sample scattered in all directions. Al-

most everything that needs to be known in order to interpret such an experiment

can be obtained from two conservation laws.

Energy. Let the wave vector of an incoming neutron of mass m

be k, let its wave

vector after scattering be id, and suppose that it creates a phonon with wave

vector q and energy Ηω^. Then conservation of energy requires

Λ2

_ tr{k

2m„

hiüs„.

(13.92a)

Inelastic Scattering from Phonons 365

Figure 13.15. Schematic drawing of neutron diffraction experiment. The neutrons pass

through a

first

crystal, undergoing Bragg scattering. By choosing only those traveling in a

certain direction, neutrons of a definite energy are selected. These then pass through the

sample crystal, and finally through an analyzer crystal, whose purpose, again, is to sort

the neutrons of various energies according to their Bragg scattering angles. [After Yarnell

al.

(1965)

58.]

If, on the other hand, passage of

the

neutron destroys a phonon and steals

its energy, then

™<ψ1-^.

„3.92b,

Crystal momentum. The study of elastic scattering in Chapter 3 shows that mo-

mentum is not conserved in scattering from a crystal. A neutron can enter with

wave vector k and exit with a different one, κ. However, κ is only permitted

if k

—

= K, where K is a reciprocal lattice vector of the crystal. Crystal mo-

mentum

is conserved, where k is always retracted into the first Brillouin

zone by whatever reciprocal lattice vectors K are necessary. Assuming that

the same conservation law continues to hold for inelastic scattering events,

when a phonon is emitted,

k' + q = k + K (13.93a)

for some reciprocal lattice vector K, and when a phonon is absorbed,

k'-q = k + K. (13.93b)

366

Chapter 13. Phonons

Combining Eqs. (13.92) and (13.93) gives

{k')

_ _

n/n

— 3zHüJ,7_7,\ — — , The reciprocal lattice vectors K disappear from (13.94)

2/iî

*■ '

^ΙΐΙη ωτ because ωτ, s =ΙΛ . Also note

L*J

=ΙΛ .

—κ,ν ku

where the + sign holds for phonon absorption, and the

—

sign holds for emission.

What Eq. (13.94) implies is that for any observation direction, k', there should

be some neutron scattering. The reason is that although the experiment will gaze

at the sample in a fixed direction, the magnitude of k' is arbitrary, and therefore by

scanning through k!, there is every reason to hope that one will obtain a discrete

number of different solutions to Eq. (13.94). If a solution is obtained, the known

difference between k and κ gives the wave vector of the phonon, while the differ-

ence in energies between incoming and outgoing neutrons gives the energy of the

phonon.

Figure 13.16. Graphical construction illustrating the conditions under which inelastic neu-

tron scattering occurs for k' = 0.

Consider first the simplest case, where k and k' lie along the same line. Then

Eq. (13.94) has the graphical interpretation shown in Figure 13.16. Eq. (13.94) is

satisfied whenever the neutron parabola crosses the phonon dispersion curve. By

measuring the outgoing neutrons as a function of energy, several peaks are located,

as shown in Figure 13.17. Each of these peaks constitutes a point on the phonon

dispersion curve. By varying the angles both of k and k!, the phonon dispersion

relations can be mapped out completely. Figure 13.18 shows the end result of

collecting such data for silicon, and it compares the results with density functional

calculations.

13.4.2 Formal Theory of Neutron Scattering

The goal of a formal approach to neutron scattering is to calculate the interaction

between phonons and neutrons to first order in perturbation theory. The nonin-

teracting states about which one perturbs are product states, of a free neutron in

a plane wave times a lattice state with some number of phonons. The interaction

between the ions and the incoming neutrons occurs when the neutrons are within a

distance on the order of 10~

cm from the

ions.

Because this distance is so terribly

short compared to a lattice spacing, one may safely represent the interaction as a

Inelastic Scattering from Phonons 367

·· ··

• · · ·

0.00

0.05 0.10

Energy (eV)

0.15

Figure 13.17. Number of neutron counts as a function of energy at 300 K in Ni. Although

at nonzero temperatures the neutron peaks are not perfectly sharp, they are certainly visible.

[Source: Mozer et al. (1965), p. 68.]

Γ XU Γ L

Wave number k

Figure 13.18. The end result of collecting peaks for absorption and emission of single

phonons is a phonon dispersion curve. These curves display the phonon dispersion relation

of Si, with experimental data from Dolling and Cowley (1966) and. Nilsson and Nelin

(1972) (boxes) and a detailed theoretical calculation involving pseudopotentials due to

Wei and Chou (1994) (solid lines). Compare with the simple estimate in Figure 13.7.

delta function: it is conventionally normalized as

2πΗ a

l d\

Y^5{R

-R

-û

(13.95)

The constant a is the scattering length and is chosen so that the total scattering

cross section of neutrons off a single ion to first order in perturbation theory is

4πα

where m

is the mass of a neutron, R

is the location of an ion, R

is a

position operator for the neutron, and û

is the displacement operator for the ion at

368 Chapter 13. Phonoris

Differential Scattering Cross Section. What an experiment measures is the flux

of particles per energy and per solid angle coming out of the sample. Let 7{k

—>

k')

give the probability of a transition from state k to state k' per unit time. Then in

small volume of V space there are

Well·

———rr" Because there are Vdk'/(2π)

states

the (13.96)

(27Tj reciprocal space volume: See Section 6.3.

transitions per unit time. Rewriting the small k! space volume in terms of neutron

energy £

and solid angle

dQ,,

one has

yvm

hk'd£

ii^hy

(13

97)

However, the number of transitions per time, per energy, per solid angle, and di-

vided by the incident flux,

/ =

hk/Vm

, is just the definition

the differential

scattering cross section

j y (\>

\2 _^

For elastic scattering, as in Eq. (3.2), there

<Τ>(Ϊ ^

Χ'\

no need to bin the radiation according

its

π-i no\

pnprov hp.ransR inaninp and niitpnino

narti-

^ ' ^

dildE,

(2ΤΓΗ)^

energy, because ingoing and outgoing parti-

\ '

cles all are the same. For inelastic scattering,

it is important to keep track of how many par-

ticles have which energy.

Transition Probability

The next goal

to find 7{k

—>

kf).

can be evalu-

ated with the help of Fermi's Golden Rule [originally derived by Dirac; see Dirac

(1958),

p. 180]. The formula reads

y(k^k')=

Σ -j-o{Ì

-£})\{¥\Ù\¥)\

(13.99)

final states

Although the use of this formula should be relatively automatic, it is important

to keep in mind the assumptions under which it was derived. The main assumption

is that scattering occurs

off a

structure that

static on time scales needed for

scattering interaction to take place. Because the interaction of a neutron or an X-

ray with an ion occurs on times that are very short compared to the time scale on

which ions move, the assumption is justified. If an experiment could be conducted

on time scales short compared to phonon motion, one could deduce the precise

locations of the ions. However, normal experiments just see the time average of

this instantaneous scattering, and therefore the appropriate calculation is one that

finds the scattering that would result from any given lattice configuration, and then

takes the time average over it. Equivalently, one can perform

thermal average

instead, and this is in fact what will be done.

The unknown quantity

Eq. (13.99)

the matrix element (Φ

|ί/|Φ

The

wave function (Φ| is a product of

(k\

and (Φ|, where (k\ describes the neutron, and

(Φ j

describes all of the phonons. Moving to a representation where r is the location

Inelastic Scattering from Phonons 369

of the neutron,

one

obtains

2πΗ

dr{k'\r){r\{&\

^±^5(R-R

-u

)\k)\&) (13.100)

^-*Η(φί|^ £$(?_# _ß<)|<|>i)

(13

101)

Ι^

^y^'M«)^).

(13102)

Using Eq. (13.102), with

denoting

the

energy

a phonon state

and

with

Κω

-^-

-^- (13.103)

denoting

the

change

energy

a neutron state,

one has

that

Ê%

δ^-ε^+πω^ί

Σ,&\^-*

Ηύ+

*Ψ)\

(13.104)

Inserting the transition probability Eq. (13.104) into the differential scattering cross

section (13.98)

one

finds that

tin

■

-, -,

S\k-kf,u

(13.105)

dfld£

k H

where

the

inelastic structure factor S

' is

S\q^)

^ô([E

-i

]/h +

u)\

J2(^\e

iHÛ

+âl)

- (13-106)

This structure factor still depends upon

the

initial quantum state "i", which

is why

it carries

the

superscript.

serves

the

same purpose

as the

structure factor defined

in Eq. (3.50),

but

generalizes

it in

three ways.

S ' is

fully quantum mechanical.

describes

dynamical, evolving object,

and

therefore permits studying

in-

elastic events.

After appropriate averaging,

will include

the

effects

temperature.

To proceed,

let Ä

h be the

phonon Hamiltonian,

and

write

]_ r dij'^-E^/n+u) y- ^(tf-^K^k-'^V)

(13 107)

^ J 2π jf

χ(Φ

\β

^'\Φ

)}

370

Chapter 13. Phonons

^ r

—e

ituJ

9-^-*

')(Φ

|β-'9·

''|Φ

)(Φ

|β

)|φΐ) (13.109)

= Jj

I ^ Σ S^^'^\e-^S^W)- co™ettTf

(13.110)

Ζ7Γ ^,

phonon states.

The structure factor still depends upon the initial state of the phonons. The exper-

imental observable is a thermal average over all possible initial states, where the

thermal average of an operator A is

(Α)

' ί^-Α

(13.111)

Σΐ(φΐ|β-^|φΐ)

After averaging, the structure factor becomes

S(q, ω) =

-Υ

"H"') [ ÉL

-^'

^('))

(13.112)

J 2vr

j^j^dt

I drdr'—

>H?-T')

iut

($(?_# -ΰ')δ{Ϋ -R

' -/(/))).(13.113)

From

(

13.113) it is evident that the structure factor is the Fourier transform in space

and time of the density-density correlation function.

13.4.3 Averaging Exponentials

Proceeding further with evaluation of Eq. (13.112) requires finding thermal aver-

ages of exponentials of operators such as

§Ξ(/)

(13.114)

where A is any operator that is linear in harmonic oscillator creation and annihila-

tion operators. For an operator of this form one can write

§ = (l

.). (13.115)

The second term on the right hand side of (13.115) vanishes because it contains

only a single power of a creation or annihilation operator, and so one has

8=1 + ί(Α4) + 1(ΛΛΛΛ) +

. ..

(13.116)

According to Wick's theorem (Doniach and Sondheimer (1974), pp. 52-62), an

expectation value such as (Â1Â2Â3Â4) can be replaced by a sum over products of

the form (AxÂjj^Aé), where the sum is over all unique ways of grouping the op-

erators into pairs. (Â1Â2) can be grouped into one unique pair; (Â1Â2Â3Â4) can be

grouped into the three unique pairs

{A\A2}

(Ατ,Α^),

{Α\Ατ,)

{Α2Α4),

(AyÂ^^Â?,);

Inelastic Scattering from Phonons 371

and a general term with 2/ operators appearing in it can be grouped into (2/)!/(/! 2

)

unique pairs. Using this result,

-(ÂÂ)

~(ÂÂ}

■

^(ÂÂ)

....

(13.117)

/î2\i

For those who do not want to rely upon Wick's

=exp[—

)\. theorem,

straightforward proof

this re- (13.11ο)

■^ suit

provided by Maradudin

al. (1971),

Section VII.2, and

swift and sneaky one by

Mermin(1966).

Identical arguments show for operators A and B linear in creation and annihilation

operators that

„

„ „

„

Because Wick's theorem produces sums over

)

= e

2AB

ß2

> _

all possible paired groupings of

and B, this (13.119)

expectation value equals the expectation value

(exp[A+ß]>.

Equation (13.119) has direct application to the scattering crosssection.

One

needs to find

m={e^-

ül

'^).

(13.120)

Because the mean square displacement of an ion cannot depend upon its location

or upon the time, one has

m = exp[-((q-û

)

)} exp[({q-û

')(q-Û

{t)))}. (13.121)

The first term on the right in Eq. (13.121) is known as the Debye-Waller

factor.

provides

quantitative description of how quantum and thermal fluctuations limit

the size of Bragg scattering peaks. The second term will lead to a physical picture

in which incoming neutrons create or destroy zero, one, two, many phonons as they

pass through the lattice.

Debye-Waller

Factor.

For the Debye-Waller factor, one needs to calculate the

mean-square ionic displacement

2W = ((q-û

)

All the terms

the average (13.111) have the same quantum state on the right

and the left; the operators

such an average must therefore leave the quantum

state unchanged, which requires

â-

and al always to multiply each other in pairs.

Therefore, in Eq. (13.122) one has

= k',

= u', and

1 H le? -q\

1 kv

fi /gt

â +a

(13.123)

M ΊΜΐ,,.ν,

kv kv kv i

N IMhüJr

kv *"

*"'

n e?

q\ / \

ι_κν

'_ i 2

_|_

J j n^

is the occupation number in (13.124)

2MHu^

\ kv j

thermal equilibrium, and M is the

mass of a lattice ion.

372

Chapter 13. Phonons

At T = 0 one can take all the occupation numbers n equal to zero, and thus have

^ 1 H

(er -a)

2W = Y

. (13.125)

N 2Mhu>r

Proceeding further requires some particular form for the phonon dispersion rela-

tion. Problem 7 evaluates Eq. (13.125) within the Debye model, and it shows that

3 q

2W = —- . (13.126)

4 Mhck

Thus the mean-square displacement of ions at zero temperature is given by the ratio

of the energy of a free ion with momentum q to a typical phonon energy. Because

of the factor I/o;, the sum in (13.124) converges for three dimensions, but not for

one or two dimensions. The divergence comes from the Bose occupation factor,

which goes as

jk for low enough k at all temperatures. Fluctuations destroy Bragg

peaks for one- and two-dimensional crystals. This point will be discussed further

in Section 14.3.

13.4.4 Evaluation of Structure Factor

The technique that made it possible to find the Debye-Waller factor makes it pos-

sible to complete the evaluation of the structure factor, (13.112), which is

S(q,u) = y* —^<

') [ —

{<i-"

'<i-û'(t))_ Use Eqs. (13.121) and (13.122).

~^ N J 2π

(13.127)

Equation (13.127) cannot be evaluated immediately. One can, however, expand

the last exponential and evaluate term by term. This expansion is appropriate so

long as ionic displacements u

are small.

Zeroth Order. The zeroth-order term is

The delta function in ω indicates that the energy transfer from the neutrons

is zero; the collisions at this order are completely elastic. The remarkable fact

is that despite the possibility of excitations at arbitrarily low energies provided

by phonons, neutrons have a nonzero chance to pass through the lattice without

exciting a single phonon. The Debye-Waller factor shows that phonons reduce

the amplitude of Bragg peaks even at zero temperature; as temperature rises, the

amplitude declines further, but does not go to zero so long as the lattice retains its

long-range order, as first claimed in Section

3.5.1.