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dubious validity for real liquid helium). We then only make errors in terms
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,

½ k ; y 0 Š 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 Þ y À °sinh k ÞaÀk Š½°cosh k Þak À °sinh k Þay Š
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 Š1=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 Š 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

>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 1ÞeÀ H
Tr 2
À H
" Tr He q
E¼ ¼
Tr eÀ H Tr eÀ H
X
0!q Tr°nq þ 1ÞeÀ H
X
2
q
"
¼ ¼ 0!q °nq þ 1Þ;
À H 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 1Š2
°
1 1 °0!Þ2 exp °0!=kTÞ
¼ °3:8:2Þ
D°!Þ d!:
kT 2 0 ½exp °0!=kTÞ À 1Š2

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

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

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=@HÞeÀ H Š
"
k¼À
Tr½eÀ H Š
 
@F
¼À ; °3:10:7Þ
@H

where the Helmholtz energy, F , is given by

F ¼ À À1 ln ½Tr°eÀ H ފ:

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; pŠ ¼ i0

it follows that

½Lx ; Ly Š ¼ i0Lz ; ½Ly ; Lz Š ¼ i0Lx ; ½Lz ; Lx Š 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 Š ¼ 0sþ ; ½sz ; sÀ Š 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 Š 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 ފ 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 Š
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

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