Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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BookID 160928 ChapID 02 Proof# 1 - 29/07/09

58 2 Lattice Vibrations

, q

,andq

can be converted to integrals as follows: 459



⇒

2π/N

⇒

2π



. (2.109)

Therefore, the three-dimensional sum



becomes 460



(2π)



q =

(2π)



q. (2.110)

In these equations L

, L

,andL

are equal to the length of the crystal in 461

the x, y,andz directions, and V = L

is the crystal volume. For any 462

function f (q), we can write 463



f (q)=

(2π)



q f (q). (2.111)

Now, it is convenient to introduce the density of states g(ω) deﬁned by

464

g(ω)dω =

The number of normal modes per unit volume

Whose frequency ω

qλ

satisﬁes ω<ω

qλ

<ω+dω.

(2.112)

From this deﬁnition, it follows that

465

g(ω)dω =



qλ

ω<ω

qλ

<ω+dω

(2π)





ω<ω

qλ

<ω+dω

q. (2.113)

Let S

(ω) be the surface in three-dimensional wave vector space on which 466

qλ

has the value ω.ThendS

(ω) is an inﬁnitesimal element of this surface 467

of constant frequency (see Fig. 2.10). The frequency change dω in going from 468

the surface S

(ω)tothesurfaceS

(ω +dω) can be expressed in terms of dq, 469

an inﬁnitesimal displacement in q space as 470

dω =dq · [∇

qλ

]

qλ

=ω

or dω =dq

⊥

|∇

qλ

=ω

. (2.114)

Here, dq

⊥

is the component of dq normal to the surface of constant frequency 471

(ω). The volume element d

q in wave vector space can be written d

q = 472

⊥

(ω), and using (2.114) allows us to write 473

()S

()Sd

ω+ ω

Fig. 2.10. Constant frequency surfaces in three-dimensional wave vector space

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2.6 Heat Capacity of Solids 59

q =

dω

|∇

qλ

=ω

(ω). (2.115)

With this result, we can express the density of states as

474

g(ω)=

(2π)





(ω)

|∇

qλ

. (2.116)

475

In (2.116) the integration is performed over the surface of constant frequency 476

(ω). The denominator contains the magnitude of the gradient of ω

qλ

(with 477

respect to q) evaluated at ω

qλ

= ω. 478

2.6.3 Debye Model 479

To evaluate (2.107) and obtain the speciﬁc heat, Debye

introduced a simple 480

assumption about the phonon spectrum. He took ω

qλ

= s

|q| for all values of 481

q in the ﬁrst Brillouin zone. Then, the surfaces of constant energy are spheres 482

(i.e., S

(ω) is a sphere in q space of radius q =

). In addition, Debye replaced 483

the Brillouin zone by a sphere of the same volume. Since



q∈1

1=N,we 484

can write 485

N =



2π





|q|<q

q =

(2π)

πq

. (2.117)

In (2.117) we have introduced q

,theDebyewavevector. A sphere of radius 486

contains the N independent values of q associated with a crystal contain- 487

ing N atoms. From (2.117), q

=6π

N/V ,whereV is the volume of the 488

crystal. 489

The density of states for the Debye model is very simple since |∇

490

qλ

| = s

. Substituting this result into (2.116) gives 491

g(ω)=

(2π)





4πq



≤q

. (2.118)

If we introduce the unit step function θ(x)=1forx>0andθ(x)=0for

492

x<0, g(ω) can be expressed 493

g(ω)=

2π



θ(s

− ω)

2 θ(s

− ω)



. (2.119)

Here, of course, s

and s

are the speed of a longitudinal and of a transverse 494

sound wave. Figure 2.11 shows the frequency dependence of the three- 495

dimensional density of states in the Debye model. Any summation over allowed 496

values of wave vector can be converted into an integral over frequency by using 497

the relation 498

P. Debye, Annalen der Physik 39, 789 (1912).

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BookID 160928 ChapID 02 Proof# 1 - 29/07/09

60 2 Lattice Vibrations



()

Fig. 2.11. Three-dimensional density of states in the Debye model



qλ

f (ω

qλ

)=V



dωg(ω)f (ω). (2.120)

Here, f (ω

qλ

) is an arbitrary function of the normal mode frequencies ω

qλ

. 499

Making use of (2.120), the expression for the speciﬁc heat [(2.107)] can be 500

written 501

= k



dω



¯hω





¯hω/Θ

− 1



−1



1 − e

−¯hω/Θ



−1

g(ω). (2.121)

Here,wehaveintroducedΘ=k

T . We deﬁne the Debye temperature T

by 502

= k

=¯hs

. Remembering that V =6π

−3

and that the integral 503

dω goes from ω =0toω = ω

= s

for longitudinal waves and from ω =0 504

to ω = s

for transverse waves, it is not diﬃcult to demonstrate 505

that 506

=3Nk











, (2.122)

507

where the Debye function F

(x) is deﬁned by 508

(x)=



− 1)(1 − e

−z

)

. (2.123)

Behavior at Θ  Θ

509

In this limit, x which equals

is much smaller than unity. Therefore, 510

we can expand the exponentials for small argument to obtain 511

(x) 



≈ 1. (2.124)

In this limit, C

=3Nk

, in agreement with the classical Dulong–Petit law. 512

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2.6 Heat Capacity of Solids 61

Behavior at Θ  Θ

513

In this limit, x is much larger than unity, and because of the exponential in 514

the denominator of the integral little error arises from replacing the upper 515

limit by inﬁnity. This gives 516

(x) 



∞

− 1)(1 − e

−z

)

. (2.125)

The integral is simply a constant. Its value can be obtained analytically

517



∞

− 1)(1 − e

−z

)

. (2.126)

The result for C

at very low temperature is 518



1+2











. (2.127)

519

This agrees with the observed behavior of the speciﬁc heat at very low 520

temperature, viz. C

= AT

,whereA is a constant. 521

2.6.4 Evaluation of Summations over Normal Modes 522

for the Debye Model 523

In our calculation of the recoil free fraction in the M¨ossbauer eﬀect (See 524

(2.56)), and in the evaluation of (2.89), the mean square displacement u

· u

 525

of an atom from its equilibrium position, we encountered sums of the form 526

I = N

−1



qλ

¯n

qλ

¯hω

qλ

. (2.128)

These sums can be performed by converting the sums to integrals through the

527

standard prescription 528



f (ω

qλ

) →

(2π)



q f (ω

qλ

), (2.129)

or by making use of the density of states g(ω) and the result that

529



qλ

f (ω

qλ

)=V



dωg(ω)f (ω). (2.130)

For simplicity, we will use a Debye model with the velocity of transverse and

530

longitudinal waves both equal to s.Then 531

g(ω)=

3ω

2π

θ(sk

− ω). (2.131)

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62 2 Lattice Vibrations

The summation in (2.128) can then be written as 532

I =



dω

3ω

2π

¯hω



¯hω/Θ

− 1



. (2.132)

Let z =

¯hω

, and make use of k

=6π

. Then (2.132) can be rewritten 533

I =







/Θ

dzz



− 1



. (2.133)

First, let us look at the high temperature limit of (2.133). If Θ  Θ

, 534

then for values of z appearing in the integrand

−1



. This corresponds to 535

the classical equipartition of energy since the energy of a mode of frequency 536

qλ

is given by 537

¯hω

qλ



¯hω

qλ

/Θ

− 1



 ¯hω

qλ



¯hω

qλ



and this is equal to Θ for every mode (the

is negligible if Θ  ¯hω

qλ

)as 538

required by classical statistical mechanics. With this approximation 539

I 







/Θ

dz =

9Θ

. (2.134)

At very low temperature, Θ  Θ

, we can approximate the upper limit 540

by ∞ in the term proportional to (e

− 1)

−1

, since the contribution from very 541

large values of z is very small. This gives 542

I =









∞

dzz

− 1



/Θ



. (2.135)

The ﬁrst integral in the square bracket is a constant, while the second is

543





. The second term is much larger than the ﬁrst for Θ  Θ

,soitis 544

a reasonable approximation to take 545

I =

4Θ

. (2.136)

(see, for example, Fig. 2.12).

546

2.6.5 Estimate of Recoil Free Fraction in M¨ossbauer Eﬀect 547

Equation (2.56) gave the probability of starting in a lattice state |n

>= 548

,...,n

> and ending, after the γ-ray emission, in the same state. If 549

we assume that the crystal is in thermal equilibrium at a temperature Θ, then 550

(2.56) is simply 551

P (¯n

, ¯n

)=e

−2E(K)I

, (2.137)

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BookID 160928 ChapID 02 Proof# 1 - 29/07/09

2.6 Heat Capacity of Solids 63

Fig. 2.12. Behavior of an integral I for Θ ≤ Θ

where ¯n

is the Bose–Einstein distribution function, E(K) is the recoil energy, 552

and I is given by (2.132). We have just evaluated I using a simpliﬁed Debye 553

model at both high (Θ  Θ

)andlow(Θ Θ

) temperatures. If we use 554

(2.134) and (2.137), we ﬁnd that at (Θ

 Θ), P  e

−

9E(K)

2Θ

. Remember that 555

E(K)  2 × 10

−3

eV. For a typical crystal Θ

 300K · k

≈ 2.5 × 10

−2

eV, 556

giving for P , P  e

−

≈ 0.7. This means that at very low temperature, 70% 557

of the γ rays are emitted without any change in the number of phonons in the 558

crystal. 559

At high temperature (let us take Θ = 400 K, larger than but not much 560

larger than Θ

 300 K) I 

9Θ

giving P (¯n

, ¯n

)  e

−

9E(K)

2Θ

4Θ

.Thisgives 561

P (¯n

, ¯n

) at Θ = 400 K of roughly 0.14, so that, even at room temperature 562

the M¨ossbauer recoil free fraction is reasonably large. 563

2.6.6 Lindemann Melting Formula 564

The Lindemann melting formula is based on the idea that melting will occur 565

when the amplitude of the atomic vibrations



i.e.,

(δR)

1/2



becomes

566

equal to some fraction γ of the interatomic spacing. Recall that u

· u

 = 567

¯h

I where I is given by (2.128) (see (2.89)). We can use the Θ  Θ

limit 568

for I to write 569

(δR)



9¯h

MΘ

. (2.138)

570

The melting temperature is assumed to be given by Θ

Melting

MΘ

9¯h

, 571

where r

is the atomic spacing and γ is a constant in the range (0.2 ≤ γ ≤ 572

0.25). This result is only very qualitative since it is based on a very much 573

oversimpliﬁed model. 574

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64 2 Lattice Vibrations

Some Remarks on the Debye Model 575

One can obtain an intuitive picture of the temperature dependence of the 576

speciﬁc heat by applying the idea of classical equipartition of energy, but only 577

to modes for which ¯hω < Θ. By this we mean that only modes whose energy 578

¯hω is smaller than Θ = k

T can be thermally excited at a temperature Θ and 579

make a contribution to the internal energy U, and such modes contribute an 580

energy Θ. Thus, we can write for U 581

U =



qλ



¯n

qλ



¯hω

qλ

 3

(2π)



Θ/¯hs

Θ4πq

dq. (2.139)

In writing (2.139) we have omitted the zero point energy since it does not

582

depend on temperature and put ¯hω[¯n(ω)]  Θ for all modes of energy less 583

than Θ. This gives (using V =

6π

and ¯hsk

=Θ

) 584

U =3N





Θ. (2.140)

Diﬀerentiating with respect to T gives

585

=12Nk





. (2.141)

This rough approximation gives the correct T

temperature dependence, 586

but the coeﬃcient is not correct as might be expected from such a simple 587

picture. 588

Experimental Data 589

Experimentalists measure the speciﬁc heat as a function of temperature over 590

a wide range of temperatures. They often use the Debye model to ﬁt their 591

data, taking the Debye temperature as an adjustable parameter to be deter- 592

mined by ﬁtting the data to (2.122) or some generalization of it. Thus, if 593

you see a plot of Θ

as a function of temperature, it only means that at 594

that particular temperature T one needs to take Θ

=Θ

(T ) for that value 595

of T to ﬁt the data to a Debye model. It is always found that at very low 596

T and at very high T the correct Debye temperature Θ

=¯hs



6π



1/3

597

agrees with the experiment. At intermediate temperatures these might be 598

ﬂuctuations in Θ

of the order of 10% from the correct value. The rea- 599

son for this is that g(ω), the density of states, for the Debye model is 600

a considerable simpliﬁcation of the actual of g(ω) for real crystals. This 601

can be illustrated by considering brieﬂy the critical points in the phonon 602

spectrum. 603

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2.6 Heat Capacity of Solids 65

2.6.7 Critical Points in the Phonon Spectrum 604

Remember that the general expression for the density of states was given by 605

(2.116). Points at which ∇

qλ

= 0 are called critical points; the integrand 606

in (2.116) becomes inﬁnite at such points. 607

Suppose that q

is a critical point in the phonon spectrum. Let ξ = q−q

; 608

then for points in the neighborhood of q

we can write 609

= ω

+ α

, (2.142)

where ξ

are the components of ξ,andω

= ω(q

). If α

, α

,andα

are all 610

negative, by substituting into the expression for g(ω) and evaluating in the 611

neighborhood of q

,oneobtains 612

g(ω)=

0ifω>ω

Constant (ω

− ω)

1/2

if ω<ω

(2.143)

Thus, although g(ω) is continuous at a critical point, its ﬁrst derivative is

613

discontinuous. 614

In three dimensions there are four kinds of critical points: 615

1. Maxima: Points at which all three α

are negative. 616

2. Minima: Points at which all three α

are positive. 617

3. Saddle Points of the First Kind :Pointsatwhichtwoα

’s are positive and 618

one is negative. 619

4. Saddle Points of the Second Kind: Points at which one α

is positive and 620

the other two are negative. 621

The critical points all show up as points at which

dg(ω)

dω

is discontinuous. A 622

rough sketch of g(ω)vs.ω showing several critical points is shown in Fig. 2.13. 623

It is not too diﬃcult to demonstrate that in three dimensions the phonon spec- 624

trum must have at least one maximum, one minimum, three saddle points of 625

each kind. As an example, we look at the simpler case of two dimensions. Then 626

Fig. 2.13. Behavior of the density of states at various critical points

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66 2 Lattice Vibrations

Fig. 2.14. Behavior of critical points in two dimensions

the phonon spectrum must have at least one maximum, one minimum, and 627

two saddle points (there is only one kind of saddle point in two dimensions) 628

(see Fig. 2.14). This can be demonstrated as follows: 629

1. We know ω

is a periodic function of q;valuesofq which diﬀer by a 630

reciprocal lattice vector K give the same ω

. 631

2. For a Brillouin zone of a two-dimensional square, we can consider ω(q

) 632

as a function of q

for a sequence of diﬀerent ﬁxed values of q

. Because 633

ω(q

) is a periodic function of q

there must be at least one maximum 634

and one minimum on each line q

= constant. 635

3. Consider the locus of all maxima (represented by X’s in Fig. 2.14). Along 636

this locus ω(q) must have at least one maximum and one minimum as a 637

function of q

. These points will be an absolute maximum and a saddle 638

point. 639

4. Doing the same for the locus of all minima (represented by O’s in Fig. 2.14) 640

gives one absolute minimum and another saddle point. 641

Because of the critical points, the phonon spectrum of a real solid looks 642

quite diﬀerent from that of the Debye model. However, the Debye model is 643

constructed so that 644

1. The low frequency behavior of g(ω) is correct because for very small ω, 645

qλ

= s

|q| is a very accurate approximation. 646

2. The total area under the curve g(ω) is correct since k

,theDebyewave 647

vector is chosen so that there are exactly the correct total number of modes 648

3N. 649

Because of this, the Debye model is good at very low temperature (where only 650

very low frequency modes are important) and at very high temperature (where 651

only the total number of modes and equipartition of energy are important). 652

In Fig. 2.15 we compare g(ω) for a Debye model with that of a real crystal. 653

We note that

Debye

(ω)dω ≈

Actual

(ω)dω. 654

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2.7 Qualitative Description of Thermal Expansion 67

Fig. 2.15. Comparison of the density of states g(ω) for a Debye model and that of

arealcrystal

Fig. 2.16. Comparison of the potential felt by an atom and the harmonic approxi-

mation to it

2.7 Qualitative Description of Thermal Expansion 655

We have approximated the interatomic potential in a crystal by 656

V (R)=V (R



+higherterms. (2.144)

In Fig. 2.16 we show a sketch of the potential felt by one atom and the

657

harmonic approximation to it. There are two main diﬀerences in the two 658

potentials: 659

1. The true interatomic potential has a very strong repulsion at u = R − R

660

negative (i.e., close approach of the pair of atoms). 661

2. The true potential levels oﬀ as R becomes very large (i.e., for large 662

positive u). 663

For a simple one-dimensional model we can write x = x

+ u,wherex

is the 664

equilibrium separation between a pair of atoms and u = x−x

is the deviation 665

from equilibrium. Then, we can model the behavior shown in Fig. 2.16 by 666

assuming that 667

V (x)=V

+ cu

− gu

− fu

. (2.145)