Marder M.P. Condensed Matter Physics

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Inelastic Scattering from Phonons

373

First

Order.

Now proceed to look at the next order arising from expansion of the

exponential in (13.127). One has

&") = Σ^

(

*"*'

)

!^>e-™((q.û

') (q-û

(t))). (13.130)

II'

To evaluate the average, one must once again return to the explicit expression for

û in terms of creation and annihilation operators. The only difference between the

evaluation of

Eq.

(13.122) and the matrix element appearing in Eq. (13.130) is that

now there is dependence upon both R

and t, requiring use of Eq. (13.53). The

matrix element in Eq. (13.130) is

m'={{q-û

') (q-û

{t)))

1 ^{(q-[Ûr

ilä

+ÛÎ

iMl

'])

—

Υ^

-. - (13 131)

έ-^ M

ΊΑΛΤ,,.^

kv kv kv k

N TMhuty

kv kv kv

(er-q)

y 1"

\.*h-q)

+ e

-i^

A /Ha*

-R>)

(13

133)

where η^ is the occupation number in thermal equilibrium, and M is the mass of a

lattice ion. Placing (13.133) into Eq. (13.130) gives finally

S^(q,

co)

= e-

]T ί|^£[(ΐ+^)

δ{ω

+ ω

ξι/

) + η

ξυ

δ(ω-ω

ξι/

)}.

(13.134)

The first term in Eq. (13.134) corresponds to a neutron creating a phonon and

giving up energy and momentum to it, while the second corresponds to a neutron

destroying a phonon. In this way, the formalism recovers the energy and momen-

tum conservation laws discussed at the beginning of this section, while also giving

explicit formulas describing the extent to which thermal and quantum fluctuations

destroy the effect.

The probability of creating phonons is proportional to

+n^

, because they

can be created at T = 0 when no phonons are initially present. The probability

of destroying phonons, however, is proportional to n^

, because it is impossible

to destroy what does not exist. These two probability factors were deduced by

Einstein (1916) from simple statistical arguments, and they will be discussed again

in Section 21.5.2.

13.4.5 Kohn Anomalies

Figure 13.19 provides a final illustration of the power of inelastic neutron scatter-

ing. This experiment searches for small discontinuities in the slope of the phonon

374

Chapter 13. Phonons

dispersion relation predicted by Kohn (1959). Just as the Peierls distortion (Section

11.5.1) tries to create a static lattice distortion with wave number 2kp, Kohn argued

that phonon dispersion relations should exhibit a singularity whenever q = 2k f.

The traces of the singularity are faint but visible in Figure 13.19.

Γ X K

Wave number

Figure 13.19. Kohn anomalies appear as logarithmic divergences in the

first

derivative of

the phonon dispersion relation. Here neutron scattering data are compared with the theory

Brovman

and Kagan (1974)

141,

showing several Kohn anomalies, marked by arrows,

as well as other singularities due to electron-ion coupling. The combined information

from theory and experiment allows accurate determination of several points on the Fermi

surface.

13.5 The Mössbauer Effect

It is rather remarkable that neutrons are able to scatter elastically from lattices,

even at nonzero temperatures. Because the momentum of the neutron changes dur-

ing the collision, the lattice as a whole must pick up the momentum transferred to

the neutron, and it does so as a whole in the zero-phonon case. Although this fact

had become commonplace by the 1950's, it was still a surprise when Mössbauer

showed that photons impinging upon nuclei of atoms in a lattice could excite nu-

clear transitions in ways that would be forbidden if the lattice were not present.

Creating nuclear excited states by incoming radiation is impossible for nuclei

sitting out in free space. The reason is that when a photon is absorbed by a nucleus,

it imparts both energy and momentum. For energy to be conserved, the recoil

The Mössbauer Effect

375

energy plus the energy of the excited state at the end of the process must be equal

to the initial nuclear energy plus the incoming photon energy. This means that the

incoming photon needs to have more energy than just the energy needed for the

nuclear excitation. However, if this excess energy is greater than the width of the

nuclear excitation (H divided by lifetime), there will be no resonant excitation of

the nucleus. In fact, for nuclear transitions, that is the way it works out, and light

cannot excite nuclear states of free atoms. However, Mössbauer (1958) discovered

that light can excite nuclear states if the nuclei sit in a lattice. The reason is that the

lattice binds the nuclei, and the lattice as a whole takes up the momentum of the

photon, leaving the energy free to go right to the nuclear excitation.

3/2 -

1/2 -

-1/2-

-3/2-

-1/2-

1/2-

(A)

Figure 13.20. (A) Splittings of energy levels of lowest two nuclear states of Fe

in the

spontaneous internal magnetic field seen by the nucleus at room temperature. Note that

the Zeeman splittings are 10 orders of magnitude smaller than separation between the

7=1/2 and / = 3/2 states. (B) Absorption of gamma radiation from Co

as a function

of temperature. The vertical axis gives the speed υ of the Co

source relative to the Fe

sample. The motion produces a Doppler shift

+v/c in the photon energies, comparable to

the splittings produced by internal magnetic fields. [Source: Preston et al. (1962), p. 2212.]

The calculation describing this process is very similar to the one just done for

neutron scattering. If q is the wave vector of an incoming photon, the nuclear

excited state involves an energy Δδ, and has lifetime Γ, then the probability of

exciting the state is

S(q) = - V «#(#-#') H —[

ΚΔε/Η+ίΓ)ΐ

+ e

-i(Ae/H-ir)t^

-iq-Û''

iq-Û<(t)^

(

JO Ζ7Γ

(13.135)

The calculation of the matrix element proceeds as before. The main difference

between this calculation and the one for neutron scattering is the inclusion of a

decay rate Γ. At low temperatures the strength of the resonance absorption is given

by a Breit-Wigner form factor

2Γ

- This factor appears instead of the <5(ω) that

(Δ£//ζ) +Γ appeared for neutron scattering in Eq. (13.129).

(13.136)

376

Chapter 13. Phonons

multiplied by the analog of the Debye-Waller factor, now called the Lamb-Möss-

bauer factor,

/ = exp

3 q

2*2

4 Mhcko

In the Debye approximation. (13.137)

One application of the Mössbauer effect is to take accurate measurements of

the internal magnetic fields in iron. Spontaneous magnetization of ferromagnets

at zero externally applied field is difficult to measure in bulk samples, because

spins arrange themselves into interweaving collections of domains (Section

24.4.1 )

with spins pointing in differing directions. The internal magnetic fields, however,

split the energy levels of nuclei in a way that is independent of the local field

directions, shown in Figure 13.20. The strength of the splitting is

2μ/Β·Ι/Ι,

with

]/2

= 4.6 · 10~

erg G"

and μ

3/2

= -8.0

IO"

erg G"

. Radioactive Co

emits 14.4 keV photons, whose energy can be raised and lowered by parts in 10

through moving the cobalt at speeds on the order of millimeters per second toward

and away from the iron. The Mössbauer line is so sharp that energy shifts of

10~

eV can be detected, and resulting measurements of the spontaneous field in

iron appear in Figure 24.3. The magnitude of the field at room temperature is

330 kOe, which is comparable to the largest magnetic fields that can be generated

in the laboratory.

Problems

Phonon dispersion with alternating springs: Consider a one-dimensional

array of atoms, in which every mass has mass M, but the springs alternate in

strength on every other site between values of

and

%2-

Find the vibrational

frequencies of this array as a function of wave number k.

Optic and acoustic

modes:

The bulk modulus of (three-dimensional) sodium

chloride is 2.4

' ergs/cm

, and the lattice parameter a is 5.6 Â. From these

facts (at the level of dimensional analysis) estimate the spring constant % that

might describe sodium chloride, modeling it as a one-dimensional chain of

atoms of alternating mass. Use the results following Eq. (13.7) to estimate the

frequency of the optical mode and the speed of sound.

Wave Speeds for Two-Dimensional Lattices: Consider a two-dimensional

triangular lattice of particles of mass M and lattice constant a. Let r

be a unit

vector pointing from the equilibrium location

/?,

of particle

to the equilibrium

location Rj of particle j. Let u,· give the two-dimensional displacement of

particle / from its equilibrium location. Suppose that the force on particle / is

Fj = Μω

2_^fij{fij-{Uj — Ui)). where ; indexes nearest neighbors of / (13.138)

(a) Find a set of two equations in two unknowns whose solution would give the

dispersion relation

^ for vibrations of the lattice.

References 377

(b) Take the limit k

—>

0 and find the transverse and longitudinal wave speeds of

the lattice.

Two-phonon scattering: Recall that the basic laws governing neutron scat-

tering from phonons could be deduced from conservation of energy and crys-

tal momentum, without detailed quantum-mechanical calculation. Generalize

this argument to consider processes in which neutrons incident upon a crystal

participate in the creation or destruction of two phonons.

(a) Enumerate the number of distinct ways in which such a process may occur.

(b) Show that two-phonon processes do not produce sharp peaks in neutron ab-

sorption as a function of incident energy, for fixed observation angle, and may

therefore be distinguished from one-phonon processes.

Quantizing vibrations:

(a) Verify if creation and annihilation operators are defined by Eq. (13.42) that

Eq. (13.43) is then satisfied.

(b) Verify that the commutation relation Eq. (13.44) is satisfied.

6. Debye temperature: Use the data of Figure 13.8 to estimate the Debye tem-

perature of diamond, and compare this estimate with the value tabulated in

Table 13.1.

Debye-Waller factor in three dimensions: Adopt the assumptions of the

Debye model, so that polarization vectors e^ consist of a longitudinal mode

parallel to k and two transverse modes perpendicular to it, with the density of

states given by Eq. (13.72).

(a) Show that in three dimensions the Debye-Waller factor

given by Eq.

(

13.126).

(b) Show that in one or two dimensions Eq. (13.124) diverges if T ψ 0.

References

M. Born (1965), Reminiscences of my work on the dynamics of crystal lattices, in Lattice Dynamics,

R. F. Wallis, ed., pp. 1-8, Pergamon, Oxford.

N. Brockhouse (1995), Slow neutron spectroscopy and the grand atlas of the physical world,

Reviews of Modern Physics, 67,

735-751.

E. G. Brovman and Y. M. Kagan (1974), Phonons in nontransition metals, Soviet Physics Uspekhi,

17,

125-152.

Debye (1965), The early days of lattice dynamics, in Lattice Dynamics, R. F. Wallis, ed., pp. 9-13,

Pergamon, Oxford.

A. M. Dirac (1958), The Principles of Quantum Mechanics, 4th ed., Clarendon Press, Oxford.

G. Dolling and R. A. Cowley (1966), The thermodynamic and optical properties of germanium,

silicon, diamond, and gallium arsenide, Proceedings of the Physical Society (London), 88, 463-

504.

378

Chapter 13. Phonons

S. Doniach and E. H. Sondheimer (1974), Green's Functions for Solid State Physicists, Ben-

jamin/Cummings, Reading, MA

A. Einstein (1907), Planck's theory of radiation and the theory of specific heat, Annalen der Physik

(Leipzig),

22, 180-190. In German.

A. Einstein (1916), Quantum theory of radiation, Mitteilungen der Physikalischen Gesellschaft,

Zürich, 16, 47-62. In German. Translated in van der Waerden (1967), pp. 63-77.

J. Goldstone (1963), Field theories with "superconductor" solutions, Nuovo Cimento, 19, 155-164.

R. Golub (1996), Ultracold neutrons: Their role in studies of condensed matter, Reviews of Modern

Physics, 68, 329-347.

S. K. Joshi and A. K. Rajagopal (1968), Lattice dynamics of metals, Solid State Physics: Advances

in Research and Applications, 22, 159-312.

W. T. Kelvin (1904), 19th century clouds over the dynamical theory of heat and light, Appendix

Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light, C. J. Clay and

Sons,

London. Readex Microprint, 1969 (Landmarks of Science)

W. Kohn (1959), Image of the Fermi surface in the vibration spectrum of a metal, Physical Review

Letters, 2, 393-394.

R. S. Krishnan, R. Srinivasan, and S. Devanarayanan (1979), Thermal Expansion of Crystals, Perga-

mon, Oxford.

A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova (1971), Theory of Lattice Dynmaics

in the Harmonic Approximation, 2nd ed., Academic Press, New York.

D. Mermin (1966), A short simple evaluation of expressions of the Debye-Waller form, Journal

of Mathematical Physics, 7, 1038.

R. Mössbauer

(

1958),

Nuclear resonance fluoresence of gamma rays in Ir , Zeitschrift für

Physik,

151,

124-143. In German.

Mozer, K. Otnes, and H. Palevsky (1965), Measured vibrational frequency distributions of Ni, V,

Ti,

and ΤίογΖ^, in Lattice Dynamics, R. F. Wallis, ed., pp. 63-70, Pergamon, Oxford.

G. Nilsson and G. Nelin (1972), Study of the homology between silicon and germanium by thermal-

neutron spectrometry, Physical Review B, 6, 3777-3786.

R. S. Preston, S. S. Hanna, and J. Heberle (1962), Mössbauer effect in metallic iron, Physical Review,

128,2207-2218.

C. G. Shull (1995), Early development of neutron scattering, Reviews of Modern Physics, 67, 753-

757.

G. A. Slack (1979), The thermal conductivity of nonmetallic crystals, Solid State Physics: Advances

in Research and Applications, 34, 1-71.

G. R. Stewart (1983), Measurement of low-temperature specific heat, Review of Scientific Instru-

ments, 54, 1-11.

Y. S. Touloukian and E. H. Buyco (1970a), Specific Heat: Metallic Elements and Alloys, vol. 4 of

Thermophysical Properties of Matter, IFI/Plenum, New York.

Y S. Touloukian and E. H. Buyco

(

1970b),

Specific Heat: Nonmetallic Solids, vol. 5 of Thermophys-

ical Properties of Matter, IFI/Plenum, New York.

Y S. Touloukian, R. K. Kirby, R. E. Taylor, and P. D. Desai (1975), Thermal Expansion: Metallic

Elements and Alloys, vol. 12 of Thermophysical Properties of Matter, IFI/Plenum, New York.

L. van der Waerden (1967), Sources of Quantum Mechanics, North-Holland, Amsterdam.

S. Wei and M. Y Chou (1994), Phonon dispersions of silicon and germanium from first-principles

calculations, Physical Review B, 50, 2221-2226.

J. L. Yarnell, J. L. Warren, and S. H. Koenig (1965), Experimental dispersion curves for phonons in

aluminum, in Lattice Dynamics, R. F. Wallis, ed., pp.

57-61,

Pergamon, Oxford.

14.

Dislocations and Cracks

14.1 Introduction

The perfect crystal presents the perfect starting point for analyzing many features

of condensed matter, but some qualitative phenomena it cannot explain at all. Even

electrical conductivity technically falls in the latter category; Chapter 16 will show

that noninteracting electrons in a perfect crystal oscillate and transport no current.

In the case of electrical resistance, the failings of the perfect crystal are somewhat

academic. But to understand many mechanical properties of solids, the perfect

crystal is tremendously misleading. Wigner said that solid-state physics "deals in

a scientific way with those subjects with which we must deal in our everyday ex-

perience. For example, we are never afraid when dropping a key that it will fly to

pieces, as a glass would." [Wigner (1936)] The key is ductile. It deforms slightly,

or bends, but does not break. The glass is brittle and shatters. Familiar though

these experiences may be, they are not at all easy to explain in a scientific way, for

they result from the dynamics of defects. To motivate the study of these defects, it

is necessary to point out how far the everyday world is from mechanical equilib-

rium, how difficult equilibrium is to attain, and how fortunate that the barriers to

equilibrium are large.

Figure 14.1. Solids of length L are stretched by force F, causing a change of length 5L. In

(A) the force is a shear, while in (B) it is pure tension.

Consider a piece of rock, of area A and height h. According to equilibrium

principles the rock should not be able to sustain its own weight under the force of

gravity if it becomes tall enough. To estimate what that critical height should be,

note that the gravitational potential energy of the rock is pAh

g/2, where p is the

379

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

380 Chapter 14. Dislocations and Cracks

Table 14.1. Shear modulus G/5 versus onset of plastic flow for three

materials

Material Shear modulus G/5

(10

ergs cm

-3

)

Iron

1.0-1.6

Copper 1.0

Titanium 1.0

Yield strength

(10

ergs cm

-3

)

0.02-1

0.005

0.08

Source: Baumeister (1978), pp. 5.2-5.5.

mass density. By cutting the rock into two equal blocks of height A/2 and setting

them side by side, this energy can be reduced to pAh

g/4, for an energy gain

of pAh

g/4. The cost of the cut is the cost of creating new rock surface, which

characteristically equals per unit area Γ =

J m

-2

. Estimating p

—

2000 kg m~

the critical height at which it pays to divide the rock in two is

1.4 cm. (14.1)

So every block of stone more than a few centimeters tall is unstable under its own

weight; changing the material to steel, concrete, or bone does not dramatically

change the estimate.

Obviously rocks can survive to heights of more than 2 cm, so the barriers pre-

venting them from reaching equilibrium must be large. Yet it is not obvious how to

compute the barriers correctly. An easy way to obtain an estimate is by imagining

what happens to the atoms of a solid as one pulls it uniformly at two ends. At first,

the forces between the atoms increase, but eventually they reach a maximum value,

and the solid falls into pieces. Interatomic forces vary greatly between different el-

ements and compounds, but the forces typically reach their maximum value when

the distance between atoms increases by around 20% of their original separation.

Referring to Figure 14.1(A), the shear force needed to deform a solid by 20%

is given by

SL F

f} = —

is the shear

modulus.

(14 2)

LA'

while as sketched in Figure 14.1(B) the force needed to deform a solid by 20% in

tension is given by

6L F

T-Ä-

(l43)

The force per area, §, needed to cause deformations of

20%

is therefore estimated

to be

8=ί

shear

(14.4)

tension.

Dislocations

381

Table 14.2. Failure of solids in tension

Material

Iron

Titanium

Silicon

Glass

Young's

Modulus F/5

(10

ergs cm"

4.0

2.2

3.2

1.4

Theoretical

Strength

) (10

ergs cm"

3.1

1.5

Practical

Strength

) (10

ergs cm"

0.03

0.07

0.04

Ratio

)

0.008

0.009

0.05

0.01

The stress needed in practice to snap solids in tension is compared with

Young's modulus, and with results of detailed calculations that determine the

force per area that would be needed to pull a plane's worth of bonds apart at

once.

The calculations involving realistic accounts of interatomic interactions

produce an answer fairly close to the estimate of Y/5, but quite far from ex-

perimental values. The final column gives the ratio of practical to theoretical

strength. Source: Averbach (1968) and Grigoriev and Meilkhov (1997).

As shown in Tables 14.1 and 14.2, these estimates are in error by many orders

of magnitude. The problem does not lie in the hasty estimates used to obtain the

forces at which bonds separate, but in the whole conception of the calculation. The

argument from equilibrium greatly underestimated the practical resistance of solid

bodies to separation, and now it has been replaced by an equally incorrect argument

that greatly overestimates it. The only way to uncover correct orders of magnitude

is to account for the actual dynamical modes by which solids fail. Ductile crystals

are those in which line defects called dislocations are mobile, allowing the solid

to flow. Brittle materials are those where instead a more severe defect, the crack, is

mobile instead.

Theoretical Strength of Silicon. The theoretical tensile strength of silicon can be

estimated by using the universal cohesive energy of

Eq.

(11.50) and the data of Ta-

ble

11.11.

Simply compute the inward pressure that silicon exerts when uniformly

expanded,

>ψ1.

(14

„

\-nr%/

Equation (14.5) is plotted in Figure 14.2, which gives an estimate for the maximum

tensile stress of 1.5 x 10

erg cm

-3

14.2 Dislocations

Taylor (1934), Orowan (1934), and Polanyi (1934) developed independently the

explanation of how crystals flow. The idea of the dislocation is that one can ob-

tain a macroscopic rearrangement of a crystal by breaking a very small number of

atomic bonds at a time. Figure 14.3 shows how a crystal can be rearranged by the

movement of a configuration in which one line of bonds at a time snaps. This type

382

Chapter 14. Dislocations and Cracks

PL,

1.8 2.0 2.2 2.4

Wigner-Seitz radius, r\y (Â)

2.6

Figure 14.2. Plot of Eq. (14.5), showing that under uniform expansion the maximum

cohesive stress of silicon is around 1.5 erg cm~

when silicon has been expanded by about

15%.

Figure 14.3. A crystal is able to flow in response to external stresses because of disloca-

tions.

As this defect moves through the crystal, only one atomic bond is broken at a time,

but the net effect is to obtain a macroscopic rearrangement of the crystal. (A) Undeformed

crystal. (B) A defect slides backwards, like a bump on a rug being pushed to the back of a

room. (C) Top layer of crystal has been displaced by Burgers vector b.

Figure 14.4. The two basic types of dislocations in three dimensions, (A) edge and (B)

screw, are pictured from a continuum viewpoint, together with their Burgers vectors b.