of order (N À N0 Þ=N0 if we write

X X

0k ay ak þ 1 k °ak aÀk þ ay ay Þ:

H ¼ 1 N 2 V0 þ °3:4:1Þ

k Àk

k

2 2

k k

While the ¬rst term is a constant, and the second is an old friend, the third

term is an awkward one. In perturbation theory it leads to divergences, the

pictorial representations of which are aptly known as ˜˜dangerous diagrams.™™

The major advance we have made, however, is to reduce our original

Hamiltonian, which contained interactions represented by a product of

four operators, to a quadratic form, in which only products of two operators

are present. It is then in principle always possible to diagonalize the

Hamiltonian.

The trick that Bogoliubov used to get rid of the o¬-diagonal terms ak aÀk

and ay ay was to de¬ne a new set of operators. He wrote

k Àk

k ¼ °cosh k Þak À °sinh k Þay ;

Àk

where the k are left arbitrary for the time being. One can show that the

obey the same commutation relations as the a,

½k ; y 0 ¼ kk 0 :

k

91

3.4 Bogoliubov™s theory of helium

Now suppose we had a Hamiltonian

X

0!k y k :

H¼ k

k

P

This would pose no di¬culties; the energies would just be k nk 0!k . Our

approach now is to write out y k in terms of the a™s and see if we can choose

k

!k and k in such a way as to make it equal to the kth component of our

approximate Hamiltonian, (3.4.1). Substituting, we have

y k ¼ ½°cosh k Þy À °sinh k ÞaÀk ½°cosh k Þak À °sinh k Þay

Àk

k k

¼ °cosh2 k Þay ak þ °sinh2 k ÞaÀk ay À °cosh k sinh k Þ

Àk

k

‚ °ay ay þ aÀk ak Þ:

k Àk

Then, if !k ¼ !Àk and k ¼ Àk ;

X X X

y y

0!k k k ¼ 0!k °cosh 2k Þa k ak þ 0!k sinh2 k

k k k

X

0!k °sinh 2k Þ°ak aÀk þ ay aÀk Þ:

À1 k

2

k

This is identical to (3.4.1) except for a constant if we choose ! and such that

!k cosh 2k ¼ k ; 0!k sinh 2k ¼ Àk :

Then

02 !2 ¼ 02 2 À 2

k

k k

and

0!k ¼ ½°E k þ N0 Vk Þ2 À °N0 Vk Þ2 1=2 :

Thus Bogoliubov™s transformation from the a™s to the ™s has diagonalized

the Hamiltonian. Within the approximation that N0 is large compared with

everything else in sight we can say that the excitations of the system above its

ground state are equivalent to Bose particles of energy 0!k .

The interesting thing about these excitations is the way the energy varies

with k for small k. We can write

q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

¬

0!k ¼ E 2 þ 2E k N0 Vk ;

k

92 Boson systems

and since for small enough k we shall have that E 2 , which varies as k4 , will be

k

small compared with E k N0 Vk , we shall ¬nd

r¬¬¬¬¬¬¬¬¬¬

NVk

!k ™ k:

M

That is, the excitations will look more like phonons than like free particles,

and will have the dispersion law predicted from the elementary arguments

used at the beginning of this section. When k becomes large, so that

E k ) N0 Vk , then the excitations will once again be like particles. The detailed

shape of the graph of ! against k will depend upon the form of Vk . If we

choose a form of Vk like Fig. 3.4.1 then we should ¬nd that ! behaves as in

Fig. 3.4.2, starting o¬ with a ¬nite gradient, but then dipping down again to a

minimum at some value of k.

This is the form of the dispersion relation for liquid 4 He that is found

experimentally, and is in accord with the super¬‚uid properties of this sub-

stance at low temperatures. We consider a heavy particle of mass M0 pro-

jected into a container of liquid helium at zero temperature, and investigate

the mechanism by which the particle is slowed down. Since its energy E M is

p2 =2M0 , a heavy particle has a lot of momentum but not much energy, as

shown in Fig. 3.4.3. If the particle is slowed down by the helium it will only

give up a small amount of energy even though it loses a considerable amount of

momentum. Now if the helium is in its ground state, then all the excitations

available in Fig. 3.4.2 require a lot of energy for each bit of momentum they

provide. The massive particle is not capable of providing this energy, and hence

cannot cause an excitation and will experience no viscous force. It is only when

Figure 3.4.1. One possible form that the e¬ective interaction between helium atoms

might take.

93

3.5 Phonons in one dimension

Figure 3.4.2. An interaction of the form shown in Fig. 3.4.1 would lead to a disper-

sion curve with a minimum as shown here.

Figure 3.4.3. For a given momentum p a heavy particle has very little energy.

the particle has such a large momentum, pc , that its velocity is equal to the

gradient of the dotted line in Fig. 3.4.2 that excitations will be caused.

In fact liquid 4 He at low temperatures is found to have super¬‚uid proper-

ties for motions below a certain critical velocity, but the magnitude of this

velocity is only about 1 cm sÀ1 , rather than the 104 cm sÀ1 predicted by this

theory. The discrepancy is accounted for by low-energy excitations in the

form of vortex rings not included in the Bogoliubov theory.

3.5 Phonons in one dimension

In the case of the Bogoliubov theory of helium we started with a system

containing a ¬xed number of Bose particles. It was the fact that the total

number of particles had to be conserved that obliged the k ¼ 0 state to

contain a macroscopic number of particles, and which, in turn, gave the

94 Boson systems

system its remarkable properties. We now turn back to the situation that we

encountered with the harmonic oscillator, where we start with a Hamiltonian

and transform it in such a way that the excitations appear as the creation of

an integral number of bosons. We return to a linear chain of interacting

atoms as the ¬rst such system to consider.

Once again we let the displacements of the atoms from their equilibrium

positions, l, be yl , and abbreviate the notation yl1 ; yl2 ; . . . by writing

y1 ; y2 ; . . . . Then the Hamiltonian will be

X p2

l

H¼ þ V°y1 ; y2 ; . . .Þ:

2m

l

We expand V in a Maclaurin series to get

!

X @

V°y1 ; y2 ; . . .Þ ¼ V°0; 0; . . .Þ þ V°y1 ; y2 ; ; . . .Þ

yl

@yl y1 ¼y2 ¼ÁÁÁ¼0

l

!

1X @2

þ V°y1 ; y2 ; ; . . .Þ

yy0

2! l;l 0 l l @yl @yl 0 y1 ¼y2 ¼ÁÁÁ¼0

!

1X @3

þ V°y1 ; y2 ; ; . . .Þ

y y 0 y 00

3! l;l 0 ;l 00 l l l @yl @yl 0 @yl 00 y1 ¼y2 ÁÁÁ¼0

þ higher terms: °3:5:1Þ

The ¬rst term on the right-hand side may be eliminated by suitable choice

of the zero of energy, and all the terms in the summation forming the second

term must be zero by virtue of the de¬nition of y ¼ 0 as the equilibrium

positions of the atoms. Thus the ¬rst set of terms we need to consider are

the set

X @2 V

:

yl yl 0

@yl @yl 0

l;l 0

We could write this double summation in matrix notation. If we abbreviate

@2 V=@yl @yl 0 by Vll 0 then we can represent the double sum as

0 10 1

ÁÁÁ

V11 V12 y1

B CB C

°y1 ; y2 ; . . .ÞB V21 CB y 2 C:

@ A@ A

. .

. .

. .

95

3.5 Phonons in one dimension

Now we can always diagonalize a ¬nite matrix like Vll 0 . That is, we can ¬nd

some matrix T such that TVT À1 is diagonal. If T has elements Tql this means

X

TVT À1 ¼ Tql Vll 0 °T À1 Þl 0 q 0 ¼ Vq qq 0 ; °3:5:2Þ

l;l 0

where the Vq are a set of numbers de¬ned by T and V. Then

X X

yl °T À1 Þlq Tql 00 Vl 00 l 000 °T À1 Þl 000 q 0 Tq 0 l 0 yl 0

yl Vll 0 yl 0 ¼

l;l 0 ;l 00 l 000

l;l 0

0

q;q

X

yl °T À1 Þlq Vq qq 0 Tq 0 l 0 yl 0

¼ °3:5:3Þ

l;l 0 q;q 0

X

˜

¼ y q y q Vq ;

q

where

X

yl °T À1 Þlq

yq ¼

l

and

X

˜

yq ¼ Tql yl :

l

y

Because yl is a physical observable it must be its own conjugate, and yl ¼ y l .

If we can choose T such that (T À1 Þlq ¼ T * we should have that yq ¼ yy , and

˜

P

ql q

y

1

we could write the potential energy as 2 q yq yq Vq .

What we have done here is really no more complicated than the elementary

approach of Section 1.2 “ we have changed from the particle coordinates yl to

the collective coordinates yq . We can similarly de¬ne collective momenta, pq ,

using the inverse transformation:

X

pq ¼ Tql pl :

l

This follows from the fact that

X @yq @ X À1

@ @

pl ¼ Ài0 ¼ Ài0 ¼ Ài0 °T Þlq :

@yl @yl @yq @yq

q q

96 Boson systems

Thus on multiplication by T we have

X

@

Ài0 ¼ Tql pl :

@yq l

The kinetic energy remains diagonal in the new coordinates, since

1X2 1X

pq °T À1 Þlq py 0 °T À1 Þ* 0

pl ¼ q lq

2M l 2M l;q;q 0

1X

pq T * °T À1 Þ* 0 py 0

¼ ql lq q

2M l;q;q 0

1X

pq py :

¼ q

2M q

Thus if we ignore all terms in the Hamiltonian that are of order y3 or higher

(this is known as the harmonic approximation) we can write

X 1

1

pq py þ M!2 yq yy ;

H¼ q

q q

2M 2

q

where M!2 ¼ Vq . Because pq ¼ Ài0@=@yq the commutation relations for the

q

collective coordinates are similar to those for particles, and we have

½yq ; pq 0 ¼ i0qq 0 :

We then see that by de¬ning operators

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

1

°M!q yq þ ipy Þ

aq ¼ °3:5:4Þ

q

2M0!q

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

1

ay ¼ °M!q yy À ipq Þ; °3:5:5Þ

q q

2M0!q

which are a simple generalization of (3.2.1), we can write

X

0!q ay aq

H¼ þ: °3:5:6Þ

1

q 2

q

97

3.5 Phonons in one dimension

Thus to know all about the excitation spectrum of the linear chain we simply

need to know the transformation matrix Tql .

The matrix we need is, of course, the one that will make the yq the collective

coordinates for phonons. We thus need to have Tql proportional to eiql and so

we write

X X

À1=2 Àiql À1=2

yq ¼ N yl ; pq ¼ N eiql pl ; °3:5:7Þ

e

l l

where N is the total number of atoms in the chain. In order to avoid di¬-

culties with the ends of the chain we adopt the device of introducing periodic

boundary conditions, as was done in Section 2.1 for the electron wavefunc-

tions. That is, we specify that

ylþNa yl ;

with a once again the distance between atoms, so that the ends of the chain

are e¬ectively joined. This restricts the possible values of q, since from expres-

sion (3.5.7) we must have

eiql ¼ eiq°lþNaÞ

if the yq are to be uniquely de¬ned. We then have

2n

q¼ ;

Na

where n is an integer. It then follows that

yq yqþg ; pq pqþg

where g ¼ 2=a, which shows that there are only N distinct collective co-

ordinates. The inverse transformations are found to be

X X

À1=2 À1=2

eÀiql pq ;

yl ¼ N e yq ; pl ¼ N °3:5:8Þ

iql

q q

where the summations proceed over all N distinct values of q. We note that

yy ¼ yÀq and py ¼ pÀq , so that since !q ¼ !Àq we can, by making use of

q q

98 Boson systems

expressions (3.5.4) and (3.5.5), write

s¬¬¬¬¬¬¬¬¬¬¬¬¬

0

°ay þ aq Þ

yq ¼ °3:5:9Þ

2M!q Àq

r¬¬¬¬¬¬¬¬¬¬¬¬¬

M0!q y

pq ¼ i °a q À aÀq Þ: °3:5:10Þ

2

These relations allow us to write any operator in terms of phonon annihila-

tion and creation operators.

The frequencies !q that appear in the Hamiltonian (3.5.6) are given by

r¬¬¬¬¬¬

Vq

!q ¼

M

and

X

Tql Vll 0 °T À1 Þl 0 q

Vq ¼

l;l 0

X 0

¼ N À1 eiql Vll 0 eÀiql :

l;l 0

As Vll 0 is, by the translational invariance of the system, a function only of

(l À l 0 ), we have

X iqL

Vq ¼ e VL ;

L

where we have written L for l À l 0 .

This result for the frequencies is identical to that which we obtained by

classical methods in Section 1.2. For the particular case where there were

interactions only between nearest neighbors we had

X X

V¼ 2 K°yl À ylþa Þ ¼

2

K°y2 À yl ylþa Þ;

1

l

l l

so that

l ¼ l0

Vll 0 ¼ 2K if

l ¼ l0 Æ a

¼ ÀK if

¼ 0 otherwise:

99

3.6 Phonons in three dimensions

Then

Vq ¼ 2K À K°eiqa þ eÀiqa Þ

qa

¼ 4K sin2 ;

2

and

r¬¬¬¬¬

K qa

!p ¼ 2 sin

2

M

as before.

3.6 Phonons in three dimensions

The theory of phonons in three-dimensional crystals is not very much more

di¬cult in principle than the one-dimensional theory. The basic results that

we found merely become decorated with a wealth of subscripts and super-

scripts. We ¬rst consider the simplest type of crystal, known as a Bravais

lattice, in which the vector distance l between any two atoms can always be

written in the form

l ¼ n1 l1 þ n2 l2 þ n3 l3 :

Here the n are integers and the li are the basis vectors of the lattice. It is con-

venient to de¬ne a set of vectors g such that eig Á l ¼ 1 for all l. These form the

reciprocal lattice. We can calculate the useful property that sums of the form

P iq Á l

vanish unless q is equal to some g, in which case the sum is equal to N,

le

the total number of atoms. Thus we can de¬ne a function Á°q) by the equation

X iq Á l X

¼N qg NÁ°qÞ:

e

l g

The Hamiltonian of a lattice of atoms interacting via simple potentials can

be written in analogy with Eq. (3.5.1) as

X1

H¼ °pil Þ2

2M

l;i

1 X i j ij X

1

yil ylj0 yk00 Vll 0 l 00 þ Á Á Á ;

ijk

þ yl yl 0 Vll 0 þ °3:6:1Þ

l

2! l;l 0 ;i; j 3! l;l 0 ;l 00 ;i; j;k

where pil and yil represent the ith Cartesian component of the momentum and

displacement, respectively, of the atom whose equilibrium position is l. The

ij ijk

tensor quantities Vll 0 , Vll 0 l 00 , etc., are the derivatives of the potential energy

100 Boson systems

with respect to the displacements as before. In the harmonic approximation

only the ¬rst two terms are retained. The Hamiltonian can then be written in

matrix notation as

0 1 0 10 1

xy

xx xz

px yx0

Vll 0 Vll 0 Vll 0

l l

X1 X x y z B yx

x y z B yC 1 yz C BC

yy

°yl ; yl ; yl Þ@ Vll 0 Vll 0 Vll 0 A yy0 A:

H¼ °pl ; pl ; pl Þ@ pl A þ @l

2M 2 l;l 0

l zy

zx zz

pz yz0

Vll 0 Vll 0 Vll 0

l l

°3:6:2Þ

Collective coordinates may be de¬ned as in the one-dimensional problem.

We put

X X

yiq ¼ N À1=2 eÀiq Á l yil ; piq ¼ N À1=2 eiq Á l pil :

l l

From these de¬nitions one can see that

yqþg yq ; pqþg pq ;

for any reciprocal lattice vector g, and so we only need to consider N non-

equivalent values of q. It is usually most convenient to consider those for

which jqj is smallest, in which case we say that we take q as being in the ¬rst

Brillouin zone. (We note also that since there are only 3N degrees of freedom

in the problem it would be an embarrassment to have de¬ned more than N

coordinates yq .) With this restriction on q the inverse transformations are

X X

yil ¼ N À1=2 eiq Á l yiq ; pil ¼ N À1=2 eÀiq Á l piq ;

q q

and may be substituted into (3.6.2) to give

8 0 x1

> pq

X> 1< BC

°pxy ; pyy ; pzy ÞB py C

H¼

>2M q q q @ q A

q> :

pz

q

0 x 19

0 xx xz 1

xy

yq >

Vq Vq Vq >

CB y C=

B yx

1 xy yy zy B

Vq CB yq C ;

yy yz

þ °yq ; yq ; yq Þ@ Vq Vq A@ A>

2 >

;

zx zy zz

yz

Vq Vq Vq q

101

3.6 Phonons in three dimensions

where

X 0

eiq Á °lÀl Þ Vll 0 :

ij

¼

ij

Vq

l0

The Hamiltonian has thus been separated into a sum of N independent terms

governing the motions having di¬erent wavenumbers q. To complete the

ij

solution we now just have to diagonalize the matrix Vq . This can be achieved

ij

merely by rotating the coordinate system. The matrix Vq will have three

mutually perpendicular eigenvectors which we can write as the unit vectors

1 2 3

s1 , s2 , and s3 , with eigenvalues Vq , Vq , and Vq . Then in the coordinate system

de¬ned by the s

X & 1 sy s 1 s sy s '

H¼ ppþ Vyy : °3:6:3Þ

2M q q 2 q q q

q;s

ij

The three directions s that describe the eigenvectors of Vq are the directions

of polarization of the phonons, and are functions of q. If it happens that one

of the s is parallel to q we say that there can be longitudinally polarized

phonons in the crystal. Since the s are mutually perpendicular it follows

that there can also be transversely polarized phonons of the same wavenum-

ber; for these s Á q ¼ 0. It is usually only when q is directed along some

symmetry direction of the lattice that this will occur. However, if q and s

are approximately parallel it is still useful to retain the terminology of long-

itudinal and transverse polarizations.

The frequencies of the phonons described by expression (3.6.3) are given by

r¬¬¬¬¬¬¬¬

s

Vq

!qs ¼ :

M

We can write the Hamiltonian in the concise language of second quantization

by de¬ning annihilation and creation operators

1

aqs ¼ p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ °M!qs yq þ ipy Þ Á sq

q

2M0!qs

°3:6:4Þ

1

ay ¼ p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ °M!qs yy À ipq Þ Á sq :

qs q

2M0!qs

Then

X

0!qs °ay aqs þ 1Þ:

H¼ qs 2

q;s

102 Boson systems

3.7 Acoustic and optical modes

In solving the dynamics of the Bravais lattice we diagonalized the

Hamiltonian in two stages. First we transformed from the yl to the yq and

thereby reduced the double summation over l and l 0 to a single summation

over q. We then rotated the coordinate system for each q so as to eliminate

ij

terms o¬ the diagonal of the matrix Vq . This completed the separation of the

Hamiltonian into terms governing the motion in the 3N independent modes

of vibration.

Not all lattices, however, are of the simple Bravais type, and this leads to a

further stage that must be included in the task of diagonalization of the

Hamiltonian. In a lattice with a basis the vectors l no longer de¬ne the

equilibrium positions of atoms, but rather the positions of identical groups

of atoms. The equilibrium position of an atom is then given by the vector

l þ b, where l is a vector of the Bravais lattice, and b is a vector describing the

position of the atom within the group (Fig. 3.7.1). There may be several

di¬erent types of atom within the group, each having a di¬erent mass Mb .

The harmonic Hamiltonian then takes on the rather complicated form

X X 1ij

1 ij

H¼ °pl;b Þ þ

i2

ylb yl 0 b 0 Vlb l 0 b 0 :

2Mb 2

l;b;l 0 ;b 0 ;i; j

l;b;i

One can look upon a lattice with a basis as a set of interlocked Bravais

lattices, and this suggests that we de¬ne collective coordinates for each

Figure 3.7.1. In a lattice with a basis the vectors l now de¬ne the position of some

reference point of a group of atoms, while the vectors b de¬ne the positions of

individual atoms of this group relative to the reference point.

103

3.7 Acoustic and optical modes

sublattice separately. We write for each of the nb possible values of b

X

À1=2

ylb eÀiq Á l

yqb ¼ N

l

and

X 0

eiq Á °lÀl Þ Vlbl 0 b 0 ;

ij ij

¼

Vqbb 0

l0

which reduces the Hamiltonian to

X X 1 i ij

1 iy i jy

H¼ pqb pqb þ yqb Vqbb 0 yqb 0 :

2Mb 2

qbb 0 ij

q;b;i

It is not enough now just to rotate the coordinate system to complete the

diagonalization of H; we also need to form some linear combination of the

yiqb that will remove terms of the form Vqbb 0 when b 6¼ b 0 . We then ¬nd that

for each q there are 3nb distinct modes of vibration. The polarization direc-

tions of these modes are in general ill-de¬ned since the nb atoms that form the

basis group may be moving in quite di¬erent directions. It is only the collec-

tive coordinate formed by the linear combination of the yiqb that has a speci¬c

direction in which it vibrates.

The 3nb di¬erent modes that one ¬nds in this way form the various

branches of the phonon spectrum of the crystal. The lowest frequencies of

vibration will be found in the three modes in which all the atoms within the

basis move more or less in phase. For vanishingly small values of q these can

be identi¬ed as the three modes of ordinary sound, for which ! is propor-

tional to jqj. For this reason these three are said to form the acoustic branch

of the phonon spectrum. In the other modes the atoms within the basis move

to some extent out of phase, and ! tends to a nonzero value as jqj tends to

zero. (There is some parallel here with plasma oscillations, in which the ions

and electrons also move out of phase.) Because the frequencies of these

phonons may be high enough to be excited by infrared radiation, they are

said to lie in the optical branch of the phonon spectrum (Fig. 3.7.2).

An understanding of the way in which the phonon spectrum splits into

acoustic and optical branches is helped by considering the problem of the

linear chain when alternate atoms have di¬erent masses. This is solved clas-

sically in many texts on solid state physics. An instructive variation of this

system, to be solved quantum mechanically, is given in Problem 3.4.

104 Boson systems

Figure 3.7.2. There are two atoms in the basis of the diamond lattice, and so this

structure has a phonon dispersion curve with acoustic and optical branches.

3.8 Densities of states and the Debye model

We have found that in the harmonic approximation the lattice may be con-

sidered as a gas of independent phonons of energies 0!q , where now the

subscript q is intended to specify the wavenumber and polarization of a

phonon as well as the branch of the spectrum in which it lies in the case of

a lattice with a basis. It is useful to de¬ne a function D°!Þ to be the density of

phonon states “ that is, the number of states per unit frequency range near a

given frequency. We write

X

D°!Þ ¼ °! À !q Þ; °3:8:1Þ

q

Ð!

from which it is seen that !12 D°!Þ d! is the number of phonon states with

frequencies between !1 and !2 .

This function is important in the interpretation of many experiments.

There are, for instance, many processes that could occur in crystals but are

forbidden because they do not conserve energy. Some of these nevertheless

take place if it is possible to correct the energy imbalance by absorbing or

emitting a phonon in the process. The probability of these phonon-assisted

processes occurring will be proportional to D°!Þ among other things. As

another example we might consider the speci¬c heat of the phonon gas,

which we could calculate by ¬nding the variation with temperature of the

105

3.8 Densities of states and the Debye model

average expectation value of the Hamiltonian. According to Section 3.3 we

should have

X

0!q °nq þ 1ÞeÀH

Tr 2

ÀH

" Tr He q

E¼ ¼

Tr eÀH Tr eÀH

X

0!q Tr°nq þ 1ÞeÀH

X

2

q

"

¼ ¼ 0!q °nq þ 1Þ;

ÀH 2

Tr e q

where

1

"

nq ¼ :

exp °0!q =kTÞ À 1

Note that , the chemical potential, is zero in this case because the number of

phonons is not conserved. Then

" 1 X °0!q Þ exp °0!q =kTÞ

2

dE

Cv ¼ ¼

dT kT 2 q ½exp °0!q =kTÞ À 12

°

1 1 °0!Þ2 exp °0!=kTÞ

¼ °3:8:2Þ

D°!Þ d!:

kT 2 0 ½exp °0!=kTÞ À 12

Thus the function D°!Þ is all that we require to calculate the speci¬c heat of a

harmonic crystal.

Unfortunately, it is a tedious job to calculate D°!Þ for even the simplest

crystal structure and set of force constants. One would like, however, to have

some model for D°!Þ in order to interpret experiments. A popular and con-

venient model is the one ¬rst proposed by Debye in 1912, in which D°!Þ is

proportional to !2 below a certain cuto¬ frequency, !D , above which it is

zero (Fig. 3.8.1). The foundation for this model comes from consideration of

the form of !q when qÀ1 is much greater than the lattice spacing. Then ! is

proportional to jqj in the acoustic branch of the spectrum, so that the density

of states in frequency is proportional to the density of states as a function of

jqj. By arguments similar to those we used in considering electron states

(Section 2.1), one can show that the density of states is uniform in q-space.

Thus one knows that the exact D°!Þ certainly varies as !2 in the limit of

small !. The Debye model is an extrapolation of this behavior to all ! up

to !D .

106 Boson systems

Figure 3.8.1. In the Debye model the phonon density of states D°!Þ, which may be a

very intricate shape, is approximated by part of a parabola.

It is convenient to express the cuto¬ parameter in temperature units rather

than frequency units. This is achieved by de¬ning

0!D ¼ k‚;

where ‚ is known as the Debye temperature. The cuto¬ frequency is

expected to correspond to a wavelength of the order of the lattice spacing,

a, and so one has the useful approximate relation for the Debye model

0!q qa‚

™ :

kT T

The constant of proportionality of D°!Þ to !2 is ¬xed by stipulating that the

total number of modes must be equal to 3N, where N is the number of atoms

in the crystal. Thus if D°!Þ ¼ A!2 one has

° !D

!2 d! ¼ 3N;

A

0

so that

3

0

D°!Þ ¼ 9N !2 °! !D Þ: °3:8:3Þ

k‚

107

3.9 Phonon interactions

Substitution of Eq. (3.8.3) into the speci¬c heat formula (3.8.2) gives the well

known Debye result

3 ° ‚=t

x4 ex

T

Cv ¼ 9Nk dx;

‚ °ex À 1Þ2

0

from which Cv is found to vary as T 3 at very low temperatures.

In some physical problems in which the phonon spectrum only enters in a

minor way, it is occasionally desirable to have an even simpler approximation

for D°!Þ. In these cases one may use the Einstein model, in which it is

assumed that a displaced atom experiences a restoring force caused equally

by every other atom in the crystal, rather than by the near neighbors alone.

Then all vibrations have the same frequency, and

D°!Þ ¼ 3N °! À !E Þ: °3:8:4Þ

Because this model neglects all the vibrational modes of low frequency, its

use is appropriate only for describing the optical modes of vibration.

3.9 Phonon interactions

While the picture of a lattice as a gas of independent phonons may be an

excellent approximation with which to calculate the speci¬c heat, there are

many physical properties that it completely fails to explain. We know, for

instance, that sound waves are attenuated in passing through a crystal, which

shows that phonons have a ¬nite lifetime. We also know that if we heat a

substance then its elastic constants will change, or it may even undergo a

martensitic transformation and change its crystal structure. The fact that the

elastic constants change implies that the frequencies of the long-wavelength

phonons also change. This means that !q must be a function not only of q,

but also of all the occupation numbers of the other phonon states. To explain

these phenomena we must return to the lattice Hamiltonian (3.6.1), and

rescue the higher-order terms that we previously neglected.

The term of third order in the displacements was

X

1

yil ylj0 yk00 Vll 0 l 00 ;

ijk

H3 ¼ l

3! l;l 0 ;l 00 ;i; j;k

108 Boson systems

where for simplicity we consider a Bravais lattice, so that there is no summa-

tion over b. We can substitute for the yil with the yiq and write

X 0

Á °l 0 ÀlÞ iq 00 Á °l 00 ÀlÞ ijk ijk

Vll 0 l 00 ¼ Vq 0 q 00

eiq e

l 0 ;l 00

to obtain

X

1 0 00

ei°qþq þq ÞÁl i j k ijk

H3 ¼ yq yq 0 yq 00 Vq 0 q 00

3!N 3=2 l;q;q 0 ;q 00 ;i; j;k

X i i k ijk

1

yq yq 0 yq 00 Vq 0 q 00 Á°q þ q 0 þ q 00 Þ:

¼ 1=2

3!N q;q 0 q 00

i; j;k

From (3.6.4)

s¬¬¬¬¬¬¬¬¬¬¬¬¬¬

X 0

°ay þ aqs Þsi ;

yiq ¼

2M!qs Àqs

s

where si is the ith Cartesian component of the unit polarization vector s, and

so

3=2 X

0

1

°!qs !q 0 s 0 !q 00 s 00 ÞÀ1=2 si s 0j s 00k

H3 ¼

3!N 1=2 2M q;q 0 ;q 00 ;i; j;k

s;s 0 ;s 00

‚ Vq 0 q 00 Á°q þ q 0 þ q 00 Þ°ay þ aqs Þ°ay 0 s 0 þ aq 0 s 0 Þ°ay 00 s 00 þ aq 00 s 00 Þ: °3:9:1Þ

ijk

Àqs Àq Àq

The third-order term in the Hamiltonian thus appears as a sum of products

of three annihilation or creation operators, and can be interpreted as repre-

senting interactions between phonons. As in the case of electron“electron

interactions we can draw diagrams to represent the various components of

(3.9.1), although the form of these will be di¬erent in that the number of

phonons is not conserved. In the case of electron interactions the diagrams

always depicted the mutual scattering of two electrons, as there were always

an equal number of annihilation and creation operators in each term in the

Hamiltonian. The interactions represented by expression (3.9.1), however,

are of the four types shown in Fig. 3.9.1. Some terms will be products of

three creation operators, and will be represented by Fig. 3.9.1(a). It is, of

course, impossible to conserve energy in processes such as these, and so

109

3.9 Phonon interactions

Figure 3.9.1. The anharmonic term in the Hamiltonian that is of third order in the

atomic displacements gives rise to processes involving three phonons. These are the

four possible types of three-phonon interactions.

the three phonons created in this way would have to be very short-lived. They

might be quickly annihilated by a process such as that shown in Fig. 3.9.1(d),

which represents a product of three annihilation operators. The processes of

Figs. 3.9.1(b) and 3.9.1(c) are more like scattering events, except that one of

the phonons is created or destroyed in the process. Such interactions may

conserve energy if the wavenumbers and polarizations are appropriate, and

would then represent real transitions.

The fact that the term Á°q þ q 0 þ q 00 Þ appears in the expression for H3

implies a condition that is equivalent to the conservation of momentum

in particle interactions. Because this function vanishes unless the vector

q þ q 0 þ q 00 is zero or a reciprocal lattice vector, g, the total wavenumber

must be conserved, modulo g. Thus in Fig. 3.9.1(a) the sum of the wave-

numbers of the three created phonons must either vanish, in which case we

call the interaction a normal process, or N-process, or else the total wave-

number is equal to a nonzero reciprocal lattice vector, in which case we call

the interaction an Umklapp process, or U-process.

The distinction between N-processes and U-processes is to some extent

arti¬cial, in that whether a scattering is designated as N or U depends on

the de¬nition of the range of allowed values of q. It remains a useful concept,

however, in discussing phonon interactions by virtue of the fact that there is a

well de¬ned distinction between N- and U-processes within the framework of

the Debye model. This is of importance in the theory of thermal conductivity

as a consequence of a theorem ¬rst proved by Peierls. He pointed out that

the heat current density is calculated from the group velocity @!=@q of the

phonons as

X @!qs

J ¼ À1 0!qs nqs :

@q

q;s

In the Debye model !qs ¼ vjqj, where the velocity of sound, v, is independent

110 Boson systems

of q or s, so that

X

À1

J¼

0v2 qnqs :

q;s

This quantity is conserved when H3 contains only terms describing N-

processes, and so the energy current should remain constant in time. This

indicates that thermal resistivity “ the ability of a solid to support a steady

temperature gradient “ must be due to U-processes or impurities in this

model.

Now that we have expressed the third-order anharmonic part, H3 , of the

Hamiltonian in terms of the aq and ay , it is straightforward in principle to use

q

perturbation theory to ¬nd the change in energy of the system caused by

phonon interactions. If the unperturbed lattice is in the eigenstate jfni gi, then

¬rst-order perturbation theory gives an energy shift of

hfni gjH3 jfni gi;

which clearly vanishes because of the fact that each term in H3 is a product of

an odd number of annihilation or creation operators. Just as in the anhar-

monic oscillator of Section 3.2, the perturbation cannot recreate the same

state jfni gi that it operates upon. We must then go to second order in per-

turbation theory, allowing the possibility of H3 causing transitions into vir-

tual intermediate states jfnj gi. The qualitative result of including phonon

interactions in the Hamiltonian is to give the energy a set of terms that

will not be linear in the occupation numbers, nq . As in the case of the inter-

acting electron system, it is meaningless to talk about the energy of one

particular phonon in an interacting system. But we can ask how the energy

of the whole system changes when we remove one phonon from the unper-

turbed state, and to evaluate this we need to form @E=@nqs . The result we ¬nd

will contain a term 0!qs arising from the di¬erentiation of the unperturbed

energy, and also a set of terms arising from di¬erentiation of products like

nqs nq 0 s 0 . The energy required to introduce an extra phonon into the qth mode

is thus a function of the occupation numbers of the other modes. For a

crystal in equilibrium these occupation numbers are functions of the tem-

perature, as dictated by the Bose“Einstein distribution formula for their

average value nqs . In particular the energy required to introduce phonons

of long wavelength, as in a measurement of the elastic constants of the

material, will depend on the temperature. The inclusion of phonon interac-

tions is thus necessary for the calculation of all properties at temperatures

near the Debye temperature, and in particular for the thermal expansion and

thermal conductivity.

111

3.10 Magnetic moments and spin

3.10 Magnetic moments and spin

The classical idea of a magnetic substance is that of an assembly of atoms

containing circulating electrons. By using the laws of electromagnetism

one may show that the average magnetic ¬eld h due to a single circulating

electron of mass m and charge e is of the form associated with a magnetic

dipole, i.e.,

3°k Á rÞr À r2 k

h¼

r5

at large distances r from the atom. Here the magnetic dipole moment, k, is

given by

e

k¼ r ‚ v; °3:10:1Þ

2c

averaged over a period of the particle™s orbital motion. The magnetization M

of a macroscopic sample of unit volume is then given by

X

M¼ ki ;

i

where the sum proceeds over all contributing electrons. While the de¬nition

(3.10.1) is quite adequate for the calculation of magnetic moments of classical

systems, it is not su¬ciently general to be useful in the framework of quan-

tum mechanics. We can, however, derive an expression for k in terms of the

Hamiltonian of the electron which may then be interpreted as de¬ning the

magnetic moment operator of a quantum-mechanical system.

To achieve this we consider the motion of the electron from the point of

view of formal classical mechanics. In the presence of an externally applied

magnetic ¬eld H an electron experiences the Lorentz force,

e

F¼ v ‚ H;

c

so that in a potential V°r) the equation of motion is

e

m_ ¼ ÀrV þ v ‚ H: °3:10:2Þ

v

c

(Note that we are considering e¬ects on a microscopic scale here, and do not

make any distinction between the magnetic induction B and the magnetic

¬eld H. If the atom we are considering is located within a sample of magnetic

material we should say that H is the sum of an applied ¬eld H0 and the dipole

¬elds hi of the other atoms. It is only when one is considering the average

112 Boson systems

¬eld in a macroscopic body that it is useful to make the distinction between B

and H.) Now Lagrange™s equation states that

d @L @L

¼ ;

dt @v @r

and in order for this to be equivalent to Eq. (3.10.2) it is su¬cient to write the

Lagrangian

1 e

L¼ mv2 À V þ v Á A;

2 c

where A is a vector potential de¬ned by H ¼ r ‚ A. The momentum p is then

de¬ned by

@L e

p¼ ¼ mv þ A; °3:10:3Þ

@v c

and the classical Hamiltonian is

@L 1

H ¼ vÁ À L ¼ mv2 þ V: °3:10:4Þ

@v 2

If one then di¬erentiates the Hamiltonian with respect to the applied mag-

netic ¬eld, keeping p and r constant, one ¬nds

!

1 X2

@H @

¼ vi þ V°rÞ

m

@H @H 2 i

X @vi

¼m vi

@H p;r

i

eX @Ai

¼À :

v

c i i @H r

For a uniform ¬eld, H, it is convenient to write

A ¼ 1 H ‚ r; °3:10:5Þ

2

which is consistent with the de¬nition of A. Then

e X @°H ‚ rÞi

@H e

¼À ¼ À r ‚ v:

vi

@H @H

2c i 2c

113

3.10 Magnetic moments and spin

By comparison with (3.10.1) we then have

@H

k¼À : °3:10:6Þ

@H

It is this expression that is taken as a de¬nition of the quantum-mechanical

operator that represents the magnetic moment of a system. If in particular a

system is in an eigenstate of energy E i then its magnetic moment is À@E i =@H.

If it is a member of an ensemble of systems at temperature T then by

Eq. (3.3.1) its average magnetic moment is

Tr½°@H=@HÞeÀH

"

k¼À

Tr½eÀH

@F

¼À ; °3:10:7Þ

@H

where the Helmholtz energy, F , is given by

F ¼ ÀÀ1 ln ½Tr°eÀH Þ:

To illustrate this we might consider the magnetic moment due to a single

spinless electron. In the absence of a magnetic ¬eld the Hamiltonian is

12

H0 ¼ p þ V°rÞ:

2m

As is seen from substituting for v from Eq. (3.10.3) in (3.10.4), the presence of

a magnetic ¬eld modi¬es the Hamiltonian to

2

1 e

H¼ p À A þ V°rÞ; °3:10:8Þ

2m c

which is equivalent to adding to H0 a perturbation

e e2

H1 ¼ A À pÁA À AÁp :

2mc c

Use of the relation (3.10.5) then gives

e2

e

H1 ¼ À r ‚pÁH þ °H ‚ rÞ2 :

2

2mc 8mc

114 Boson systems

We then ¬nd that when H is taken in the z-direction

@H e2 H 2

e

z ¼ À ¼ °r ‚ pÞz À °x þ y2 Þ: °3:10:9Þ

@H 2mc 2

4mc

This shows that in the limit of small applied ¬elds the magnetic moment

due to a spinless nonrelativistic electron is proportional to its orbital

angular momentum, L ¼ r ‚ p, with a constant of proportionality equal to

e=2mc.

In solids the form of the potential V°r) that acts on an electron bound to a

particular atom is frequently so lacking in symmetry that the eigenstates have

no net orbital angular momentum. That is to say, the states of the electron

are formed out of mixtures of equal amounts of the two degenerate wave-

functions corresponding to orbital angular momenta L and ÀL. The only

magnetic moment observed is then due to second-order e¬ects such as the

second term in expression (3.10.9), which, being intrinsically negative, leads

to diamagnetic e¬ects in which the induced moment is in the opposite direc-

tion to H. One says that the strong magnetic moment one would expect from

the ¬rst term in expression (3.10.9) is quenched.

Having thus considered and disposed of the orbital angular momentum as

an important source of magnetic e¬ects in solids we now restore to the

electron its spin, and ask how this property modi¬es our picture. As spin

has been shown to be a consequence of the relativistic nature of the electron

we might take as our starting point the Dirac equation, which describes the

relativistic motion of an electron or positron by means of a wavefunction

having four components. In the nonrelativistic limit the electron and positron

parts of this equation may be separated by means of the Foldy“Wouthuysen

transformation to give an equation for the two-component wavefunction

describing the electron alone. The fact that the electron wavefunction does

have two components is consistent with the electron possessing a degree of

freedom corresponding to a spin angular momentum s of 1 0 that can point

2

either up or down.

The most important terms from our point of view that are contained in this

reduction of the Dirac equation are given by the Hamiltonian

2

1 1

e e

H¼ p À A À 3 2 p4 þ V°rÞ À sÁr ‚ A

2m 8m c

c mc

!

02

1 e

þ 2 2 s Á rV°rÞ ‚ p À A þ 2 2 r2 V: °3:10:10Þ

2m c 8m c

c

115

3.10 Magnetic moments and spin

This di¬ers from expression (3.10.8) in having a term in p4 , which is a rela-

tivistic correction to the kinetic energy, and in having two terms involving

the spin angular momentum s. In this section we consider only the ¬rst of the

terms containing s. Since r ‚ A ¼ H the presence of this term in the

Hamiltonian shows the electron to have a magnetic moment of es=mc due

to its spin. The ratio of the magnetic moment of a substance to its angular

momentum is a quantity that can be determined by experiment, and is

found to be close to a value of e=mc in many ferromagnetic substances.

This indicates that it is the spin of the electron rather than its orbital motion,

which from (3.10.9) would have led to a value of e=2mc for this ratio, that is

principally responsible for magnetic properties in these materials.

The operator s, being an angular momentum, has the same com-

mutation properties as the orbital angular momentum L. Because of the

de¬nition

L¼r‚p

and the relation

½r; p ¼ i0

it follows that

½Lx ; Ly ¼ i0Lz ; ½Ly ; Lz ¼ i0Lx ; ½Lz ; Lx ¼ i0Ly ;

or, more concisely,

L ‚ L ¼ i0L:

We then also have the relation

s ‚ s ¼ i0s: °3:10:11Þ

It is useful to de¬ne two new operators, sþ and sÀ , known as spin raising and

lowering operators, by writing

sþ ¼ sx þ isy ; sÀ ¼ sx À isy : °3:10:12Þ

We then ¬nd that

½sz ; sþ ¼ 0sþ ; ½sz ; sÀ ¼ À0sÀ : °3:10:13Þ

116 Boson systems

It then follows that if the state j"i is an eigenfunction of sz having eigenvalue

À

2 0, then s j"i is the eigenfunction j#i of sz having eigenvalue À 2 0.

1 1

sz °sÀ j"iÞ ¼ °sÀ sz À 0sÀ Þj"i

¼ °1 0sÀ À 0sÀ Þj"i

2

¼ À 1 0°sÀ j"iÞ:

2

The naming of sþ may be similarly justi¬ed by showing that it transforms j#i

into j"i. As these are the only two possible states for a spin- 1 particle we have

2

sþ sþ ¼ sÀ sÀ ¼ 0:

The spin raising and lowering operators remind us rather strongly of boson

creation and annihilation operators. We recall that the number operator for a

boson state can have an eigenvalue equal to any one of the in¬nite spectrum

of natural numbers (Fig. 3.10.1(b)). The operator sz , which has eigenvalues

Æ 1 0 (Fig. 3.10.1(a)) could be considered as operating within the ˜˜ladder™™ of

2

the boson system if we could somehow disconnect the bottom two rungs

from the rest of the spectrum. The procedure that enables one to make the

correspondence between the spin system and the boson system is known as

the Holstein“Primako¬ transformation. We de¬ne boson operators ay and a

as usual so that

½a; ay ¼ 1; ay a ¼ n;

Figure 3.10.1. In the Holstein“Primako¬ transformation a direct correspondence is

achieved between the two possible states of a particle of spin 1 0 and the n ¼ 0 and

2

n ¼ 1 states of a harmonic oscillator.

117

3.11 Magnons

and make the identi¬cation

sþ ¼ 0°1 À nÞ1=2 a; sÀ ¼ 0ay °1 À nÞ1=2 : °3:10:14Þ

Substitution of these expressions in (3.10.11), (3.10.12), and (3.10.13) shows

that these relations satisfy the commutation relations for spin operators, and

that

À Á

sz ¼ 0 1 À n : °3:10:15Þ

2

This form for sz seems to suggest that it can have all the eigenvalues 1 0, À 1 0,

2 2

À 3 0, etc. While this is so one can also see that the operator sÀ , which trans-

2

forms the state with sz ¼ þ 1 0 to that with sz ¼ À 1 0, is not capable of further

2 2

lowering the spin, as the factor (1 À nÞ then gives zero. There is in e¬ect a

1=2

barrier separating the lowest two levels of the boson system from the other

states. This correspondence paves the way for the description of a ferromag-

net in terms of a gas of interacting bosons. We shall in particular consider a

model of a ferromagnetic insulator. This is distinguished from the ferromag-

netic conductor considered in Section 2.8 by the fact that the spins are

considered as bound to a particular lattice site in the manner of the classical

model of Section 1.4.

3.11 Magnons

In a ferromagnet an atom carrying a magnetic moment is not free to orient

itself at random, but is in¬‚uenced by the moments carried by other atoms in

the crystal. The simplest model of such a situation is due to Weiss, who

assumed that there was an e¬ective magnetic ¬eld Hm acting on each atom

proportional to the macroscopic magnetization of the whole crystal. This

model is very similar in concept to the Einstein model of lattice dynamics

introduced at the end of Section 3.8, where it was assumed that the restoring

force on a displaced atom was due equally to every other atom in the crystal.

The term mean ¬eld model is now used generically to refer to theories such as

the Weiss or Einstein models in which a sum of di¬erent forces is approxi-

mated by an overall average. Because a mean ¬eld theory ignores the dom-

inance of interactions between neighboring atoms, there are no low-

frequency phonons in the Einstein model of lattice dynamics. As a consequence

the lattice speci¬c heat is incorrectly predicted to vary exponentially at low

temperatures. In a similar way the Weiss model of ferromagnetism does not

support the existence of magnons of low frequency, and the magnetization of

118 Boson systems

a ferromagnet of spinning electrons is also incorrectly predicted to vary

exponentially at low temperatures. What is needed to rectify this situation

is a theory based on a model in which the interaction between near neighbors

is emphasized.

The simplest such model of a ferromagnet is one in which neighboring

spins interact only through the z-component of their magnetic moments. A

lattice of N ¬xed electrons (¬xed so that we can neglect kinetic and potential

energies) would then have the Hamiltonian

!

X X 0

H¼À sz °lÞ ju0 j þ Jll 0 sz °l Þ ;

l0

l

where u0 ¼ eHz =mc, the l and l 0 are lattice sites, and the Jll 0 are functions

only of l À l 0 . This is known as the Ising model, and is of great interest to

those who study the statistical mechanics of phase transitions. It is of less

interest to us, however, as it is no more able to support magnons than was the

Weiss model. We could say in classical terms that because the x- and y-

components of magnetic moment are ignored, the tilting of one moment

does not induce its neighbor to change its orientation. It is thus necessary

to introduce interactions between the x- and y-components of spin, leading us

to the Heisenberg model, for which

!

X X 0

H¼À s°lÞ Á u0 þ Jll 0 s°l Þ : °3:11:1Þ

0

l l

We rewrite this in terms of boson operators by making use of the Holstein“

Primako¬ transformation. From Eqs. (3.10.12), (3.10.14), and (3.10.15) we

¬nd

s°lÞ Á s°l 0 Þ ¼ sx °lÞsx °l 0 Þ þ sy °lÞsy °l 0 Þ þ sz °lÞsz °l 0 Þ

¼ 1 ½sþ °lÞsÀ °l 0 Þ þ sÀ °lÞsþ °l 0 Þ þ sz °lÞsz °l 0 Þ

2

y y

¼ 1 02 ½°1 À nl Þ1=2 al a l 0 °1 À nl 0 Þ1=2 þ a l °1 À nl Þ1=2 °1 À nl 0 Þ1=2 al 0

2

þ 02 °1 À nl Þ°1 À nl 0 Þ: °3:11:2Þ

2 2

At very low temperatures the magnetization of the specimen, which will be

directed in the z-direction, will be close to its saturation value of (Ne=mcÞ 1 0.

2

119

3.11 Magnons

That is, the total z-component of spin will be close to 1 N0. In terms of the

2

boson number operators, nl , we have from (3.10.15) that

X X

sz °lÞ ¼ 2 N0 À nl 0;

1

l l

showing that the expectation value of the nl will all be small compared with

unity at low temperatures. We take this as justi¬cation for neglecting terms of

order n2 in expression (3.11.2). Replacing (1 À nl Þ1=2 by unity in this way we

¬nd that

y y

0

s°lÞ Á s°l Þ ™ 2 0 al a l 0 þ al al 0 þ 2 À nl À nl 0 :

12 1

As we are looking for spin waves, now is clearly the time to transform from

local to collective coordinates. The magnon creation and annihilation opera-

tors are de¬ned by

X iq Á l y X Àiq Á l

ay ¼ N À1=2 e al ; aq ¼ N À1=2 al ;

e

q

l l

from which