1.
Chapter 2
Phonons
Disregarding point symmetry, we can simplify the crystal structure by the space
group [1, 2], representing the thermodynamic state in equilibrium with the sur-
roundings at given values of p and T. In this approach, the restoring forces secure
stability of the lattice, where the masses at lattice points are in harmonic motion.
In this case, we realize that their directional correlations in the lattice are ignored
so that a possible disarrangement in the lattice can cause structural instability.
In this chapter, we discuss a harmonic lattice to deal with basic excitations in
equilibrium structure. Lattice vibrations in periodic structure are in propagation,
speciﬁed by frequencies and wavevectors in virtually continuous spectra. Quantum
mechanically, on the other hand, the corresponding phonons signify the dynamical
state in crystals. In strained crystals, as modulated by correlated constituents, low-
frequency excitations dominate over the distorted structure, which is however
thermally unstable as discussed in this chapter.
2.1 Normal Modes in a Simple Crystal
A crystal of chemically identical constituent ions has a rigid periodic structure
in equilibrium with the surroundings, which is characterized by translational
symmetry. Referring to symmetry axes, physical properties can be attributed to the
translational invariance, in consequence of energy and momentum conservations
among constituents.
Constituents are assumed to be bound together by restoring forces in the lattice
structure. Considering a cubic lattice of N3
identical mass particles in a cubic crystal
in sufﬁciently large size, we can solve the classical equation of motion with nearest-
neighbor interactions. Although such a problem should be solved quantum mechan-
ically, classical solutions provide also a useful approximation. It is noted that the
lattice symmetry is unchanged with the nearest-neighbor interactions, assuring
M. Fujimoto, Thermodynamics of Crystalline States,
DOI 10.1007/978-1-4614-5085-6_2, # Springer Science+Business Media New York 2013
11
2.
structural stability in this approach. In the harmonic approximation, we have linear
differential equations, which can be separated into 3N independent equations; this
one-dimensional equation describes normal modes of N constituents in collective
motion along the symmetry axis x, y, or z [3]. Denoting the displacement by a vector
qn from a site n, we write equations of motion for the components qx;n; qy;n and qz;n
independently, that is,
€qx;n ¼ o2
qx;nþ1 þ qx;nÀ1 À 2qx;n
À Á
; €qy;n ¼ o2
qy;nþ1 þ qy;nÀ1 À 2qy;n
À Á
and
€qz;n ¼ o2
qz;nþ1 þ qz;nÀ1 À 2qz;n
À Á
;
where o2
¼ k m= and k and m are the mass of a constituent particle and the force
constant, respectively. As these equations are identical, we write the following
equation for a representative component to name qn for brevity:
€qn ¼ o2
qnþ1 þ qnÀ1 À 2qn
À Á
; (2.1)
which assures lattice stability along any symmetry direction.
Deﬁning the conjugate momentum by pn ¼ m _qn, the Hamiltonian of a harmonic
lattice can be expressed as
H ¼
XN
n¼0
p2
n
2m
þ
mo2
2
qnþ1 À qn
À Á2
þ
mo2
2
qn À qnÀ1ð Þ2
& '
: (2.2)
Each term in the summation represents one-dimensional inﬁnite chain of identi-
cal masses m, as illustrated in Fig. 2.1a.
Normal coordinates and conjugate momenta, Qk and Pk , are deﬁned with the
Fourier expansions
qn ¼
1
ﬃﬃﬃﬃ
N
p
XkN
k¼0
Qk exp iknað Þ and pk ¼
XkN
k¼0
Pk exp iknað Þ; (2.3)
Fig. 2.1 (a) One-
dimensional monatomic
chain of the lattice constant a.
(b) A dispersion curve o vs: k
of the chain lattice.
12 2 Phonons
3.
where a is the lattice constant. For each mode of qn and pn, the amplitudes Qk and
Pk are related as
QÀk ¼ Qk
Ã
; PÀk ¼ Pk
Ã
and
XN
n¼0
exp i k À k0
ð Þna ¼ Ndkk0 ; (2.4)
where dkk0 is Kronecker’s delta, that is, dkk0 ¼ 1 for k ¼ k0
, otherwise zero for k ¼ k0
.
Using normal coordinates Qk and Pk, the Hamiltonian can be expressed as
H ¼
1
2m
X2p a=
k¼0
PkPk
Ã
þ QkQk
Ã
m2
o2
sin2 ka
2
'
; (2.5)
from which the equation of motion for Qk is written as
€Qk ¼ Àm2
o2
Qk; (2.6)
where
ok ¼ 2o sin
ka
2
¼ 2
ﬃﬃﬃﬃ
k
m
r
sin
ka
2
: (2.7)
As indicated by (2.7), the k-mode of coupled oscillators is dispersive, which are
linearly independent from the other modes of k0
6¼ k . H is composed of N
independent harmonic oscillators, each of which is determined by the normal
coordinates Qk and conjugate momenta Pk. Applying Born–von Ka´rman’s boundary
conditions to the periodic structure, k can take discrete values as given by k ¼ 2pn
Na
and n ¼ 0; 1; 2; . . . ; N. Figure 2.1b shows the dispersion relation (2.7) determined
by the characteristic frequency ok:
With initial values of Qkð0Þ and _Qkð0Þ speciﬁed at t ¼ 0, the solution of (2.7)
can be given by
QkðtÞ ¼ Qkð0Þ cos okt þ
_Qkð0Þ
ok
sin okt:
Accordingly,
qnðtÞ ¼
1
ﬃﬃﬃﬃ
N
p
XkN
k¼0
X
n0¼n;nÆ1
qn0ð0Þ cos ka n À n0
ð Þ À oktf g þ
_qn0 ð0Þ
ok
sin ka n À n0
ð Þ À oktf g
!
;
(2.8)
where a n À n0
ð Þ represents distances between sites n and n0
so that we write it as
x ¼ a n À n0
ð Þ in the following. The crystal is assumed as consisting of a
large number of the cubic volume L3
where L ¼ Na, if disregarding surfaces.
2.1 Normal Modes in a Simple Crystal 13
4.
The periodic boundary conditions can then be set as qn¼0ðtÞ ¼ qn¼NðtÞ at an
arbitrary time t. At a lattice point x ¼ na between n ¼ 0 and N , (2.8) can be
expressed as
q x; tð Þ ¼
X
k
Ak cos Ækx À oktð Þ þ Bk sin Ækx À oktð Þ½ ;
where Ak ¼ qkð0Þ
ﬃﬃﬃ
N
p and Bk ¼
_qkð0Þ
ok
ﬃﬃﬃ
N
p , and x is virtually continuous in the range 0 x
L, if L is taken as sufﬁciently long. Consisting of waves propagating in Æ x
directions, we can write q x; tð Þ conveniently in complex exponential form, that is,
q x; tð Þ ¼
X
k
Ck exp i Ækx À okt þ jkð Þ; (2.9)
where C2
k ¼ A2
k þ B2
k and tan jk ¼ Bk
Ak
. For a three-dimensional crystal, these one-
dimensional k-modes along the x-axis can be copied to other symmetry axes y and z;
accordingly, there are 3N normal modes in total in a cubic crystal.
2.2 Quantized Normal Modes
The classical equation of motion of a harmonic crystal is separable to 3N indepen-
dent normal propagation modes speciﬁed by kn ¼ 2pn
aN along the symmetry axes. In
quantum theory, the normal coordinate Qk and conjugate momentum Pk ¼ Àih @
@Qk
are operators, where h ¼ h
2p and h is the Planck constant. For these normal and
conjugate variables, there are commutation relations:
Qk; Qk0½ ¼ 0; Pk; Pk0½ ¼ 0 and Pk; Qk0½ ¼ ihdkk0 ; (2.10)
and the Hamiltonian operator is
Hk ¼
1
2m
PkP
y
k þ m2
o2
kQkQ
y
k
: (2.11a)
Here, P
y
k and Q
y
k express transposed matrix operators of the complex conjugates Pk
Ã
and Qk
Ã
, respectively.
Denoting the eigenvalues of Hk by ek, we have the equation
HkCk ¼ ekCk: (2.11b)
For real eigenvalues ek , Pk and Qk should be Hermitian operators, which are
characterized by the relations P
y
k ¼ PÀk and Q
y
k ¼ QÀk , respectively. Deﬁning
operators
14 2 Phonons
5.
bk ¼
mokQk þ iP
y
k
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2mek
p and b
y
k ¼
mokQ
y
k À iPk
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2mek
p ; (2.12)
we can write the relation
bkb
y
k ¼
1
2mek
m2
o2
kQ
y
k Qk þ P
y
k Pk
þ
iok
2ek
Q
y
k P
y
k À PkQk
¼
Hk
ek
þ
iok
2ek
QÀkPÀk À PkQkð Þ:
From this relation, we can be derive
Hk ¼ hok b
y
k bk þ
1
2
; if ek ¼
1
2
hok: (2.13)
Therefore, Hk are commutable with the operator b
y
k bk, that is,
Hk; b
y
k bk
h i
¼ 0;
and from (2.12)
bk0 ; b
y
k
h i
¼ dk0k; bk0 ; bk½ ¼ 0 and b
y
k0 ; b
y
k
h i
¼ 0:
Accordingly, we obtain
Hk; b
y
k
h i
¼ hokb
y
k and bk; Hk½ ¼ hokbk:
Combining with (2.11b), we can derive the relations
Hk b
y
k Ck
¼ ek þ hokð Þ b
y
k Ck
and Hk bkCkð Þ ¼ ek À hokð Þ bkCkð Þ;
indicating that b
y
k Ck and bkCk are eigenfunctions for the energies ek þ hok and
ek À hok , respectively. In this context, b
y
k and bk are referred to as creation and
annihilation operators for the energy quantum hok to add and subtract in the energy
ek; hence, we can write
b
y
k bk ¼ 1: (2.14)
Applying the creation operator b
y
k on the ground state function Cknk-times, the
eigenvalue of the wavefunction b
y
k
nk
Ck can be given by nk þ 1
2
À Á
hok, generating a
2.2 Quantized Normal Modes 15
6.
state of nkquanta plus 1
2 hok. Considering a quantum hok like a particle, called a
phonon, such an exited state with nk identical phonons is multiply degenerate by
permutation nk! Hence, the normalized wavefunction of nk phonons can be
expressed by 1ﬃﬃﬃﬃ
nk!
p b
y
k
nk
Ck. The total lattice energy in an excited state of n1; n2;
::::: phonons in the normal modes 1, 2,. . ... can be expressed by
U n1; n2; :::::ð Þ ¼ Uo þ
X
k
nkhok; (2.15a)
where Uo ¼
P
k
hok
2 is the total zero-point energy. The corresponding wavefunction
can be written as
C n1; n2; :::::ð Þ ¼
b
y
1
n1
b
y
2
n2
:::::
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n1! n2! :::::
p C1C2:::::ð Þ; (2.15b)
which describes a state of n1; n2; :::: phonons of energies n1hok1
; n2hok2
; :::::. The
total number N ¼n1 þ n2 þ ::::: cannot be evaluated by the dynamical theory;
however, we can determine the value in thermodynamics, as related to the level
of thermal excitation at a given temperature.
2.3 Phonon Field and Momentum
In a one-dimensional chain of identical mass particles, the displacement mode qk is
independent from each other’s modes, and hence representing normal modes in a
three-dimensional crystal. However, this model is only approximate, in that these
normal modes arise from the one-dimension harmonic chain model, where mutual
interactions between different normal modes are prohibited. For propagation in
arbitrary direction, the vibrating ﬁeld offers more appropriate approach than the
normal modes, where quantized phonons move in any direction like free particles in
the ﬁeld space.
Setting rectangular coordinates x; y; z along the symmetry axes of an orthorhom-
bic crystal in classical theory, the lattice vibrations are described by a set of
equations
px;n1
2
2m
þ
k
2
qx;n1
À qx;n1þ1
À Á2
þ qx;n1
À qx;n1À1
À Á2
n o
¼ ex;n1
;
py;n2
2
2m
þ
k
2
qy;n2
À qy;n2þ1
À Á2
þ qy;n2
À qy;n2À1
À Á2
n o
¼ ey;n2
;
and
pz;n3
2
2m
þ
k
2
qz;n3
À qz;n3þ1
À Á2
þ qz;n3
À qz;n3À1
À Á2
n o
¼ ez;n3
;
(2.16)
16 2 Phonons
7.
whereex;n1
þ ey;n2
þ ez;n3
¼ en1n2n3
is the total propagation energy along the direction
speciﬁed by the vector q n1; n2; n3ð Þ and k is the force constant.
The variables qx;n1
; qy;n2
; qz;n3
in (2.16) are components of a classical vector
q n1; n2; n3ð Þ , which can be interpreted quantum theoretically as probability
amplitudes of components of the vector q in the vibration ﬁeld. We can therefore
write the wavefunction of the displacement ﬁeld as C n1; n2; n3ð Þ ¼ qx;n1
qy;n2
qz;n3
,
for which these classical components are written as
q x; tð Þ ¼
X
kx
Ck;x exp i Ækxx À okx
t þ jkx
À Á
;
q y; tð Þ ¼
X
ky
Cky
exp i Ækyy À oky
t þ jky
;
q z; tð Þ ¼
X
kz
Cky
exp i Ækzz À okz
t þ jkz
À Á
;
and hence, we have
C n1; n2; n3ð Þ ¼
X
k
Ak exp i Æk:r À
n1ex;n1
þ n2ey;n2
þ n3ez;n3
h
t þ jk
:
Here, Ak ¼ Ckx
Cky
Ckz
; jk ¼ jkx
þ jky
þ jkz
, and k ¼ kx; ky; kz
À Á
are the ampli-
tude, phase constant, and wavevector of C n1; n2; n3ð Þ , respectively. Further
writing
n1ex;n1
þ n2ey;n2
þ n3ez;n3
h
¼ ok n1; n2; n3ð Þ ¼ ok; (2.17a)
the ﬁeld propagating along the direction of a vector k can be expressed as
C k; okð Þ ¼ Ak exp i Æk:r À okt þ jkð Þ; (2.17b)
representing a phonon of energy hok and momentum Æ hk. For a small kj j, the
propagation in a cubic lattice can be characterized by a constant speed v of propaga-
tion determined by ok ¼ v kj j, indicating no dispersion in this approximation.
The phonon propagation can be described by the vector k, composing a recipro-
cal lattice space, as illustrated in two dimensions in Fig. 2.2 by
kx ¼
2pn1
L
; ky ¼
2pn2
L
and kz ¼
2pn3
L
;
2.3 Phonon Field and Momentum 17
8.
in a cubic crystal, where aN ¼ L. A set of integers n1; n2; n3ð Þ determines an
energy and momentum of a phonon propagating in the direction of k, where ok
¼ ok n1; n2; n3ð Þ and kj j ¼ 2p
L
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
n2
x þ n2
y þ n2
z
q
. In the reciprocal space, all points on a
spherical surface of radius kj jcorrespond to the same energy hok , representing a
sphere of a constant radius kj j and energy ek. Quantum mechanically, we can write
the phonon momentum as p ¼ hk to supplement the energy hok, characterizing a
phonon particle.
2.4 Thermal Equilibrium
In thermodynamics, a crystal must always be in thermal contact with the surround-
ings. At constant external pressure p, the quantized vibration ﬁeld can be in
equilibrium with the surroundings at a given temperature T. A large number of
phonons are in collision-free motion, traveling in all directions through the lattice,
colliding with surfaces to exchange their energy and momentum with the surround-
ings. Assuming the crystal volume as unchanged, the average of phonon energies
can be calculated with the Boltzmann probability at T.
In equilibrium, the total energy of a crystal can be expressed asU þ Us, whereUs
is the contribution from the heat reservoir and Urepresents the energy of a stable
crystal. In this case, the total energy U þ Us should be stationary with any
thermodynamic variation around equilibrium. Using probabilities w and ws for
keeping the crystal in equilibrium with the surroundings, the product wws should
be calculated as maximizing U þ Us to determine the most probable value. Setting
this variation problem as
2πny
ky =
a
2πnx
kx =
a
dk
k
4
3
2
1
0
1 2 3 4
Fig. 2.2 Two-dimensional
reciprocal lattice. A lattice
point is indicated by kx; ky
À Á
.
Two quarter-circles of radii k
and k + dk show surfaces of
constant ek and ekþdk for small
kj j in kxky plane.
18 2 Phonons
9.
d wwsð Þ ¼ 0 and d U þ Usð Þ ¼ 0
for arbitrary variations dw and dws, these variations can be calculated as
wsdw þ wdws ¼ 0 and dU þ dUs ¼ 0;
respectively. We can therefore write
d ln wð Þ
dU
¼
d ln wsð Þ
dUs
;
which is a common quantity between U and Us. Writing it as equal to b ¼ 1
kBT , we
can relate b to the conventional absolute temperature T. Therefore,
d ln wð Þ
dU
¼ b ¼
1
kBT
or w ¼ wo exp À
U
kBT
: (2.18)
Here, w is called the Boltzmann probability; wo is the integration constant that can
be determined by assuming U ¼ 0 at T ¼ 0 K, where kB is the Boltzmann constant.
Quantum theoretically, however, T ¼ 0 is fundamentally unreachable, as stated in
the third law of thermodynamics. Accordingly, we write U ¼ 0 þ Uo at 0 K, where
Uo ¼ 1
2 Nhoo is the zero-point energy.
Although we considered only vibrations so far, physical properties of a crystal are
also contributed by other variables located at lattice points or at interstitial sites.
Though primarily independent of lattice vibrations, these variables can interact with
the lattice via phonon scatterings. If accessed by random collisions of phonons,
energiesei of these variables are statistically available with Boltzmann’s probabilities
wi, so we can write equations
U ¼
X
i
ei and w ¼ Piwi;
where
wi ¼ wo exp À
ei
kBT
and
X
i
wi ¼ 1; (2.19)
as these wi are for exclusive events. In this case, the function Z ¼
P
i
exp À ei
kBT
,
called the partition function, is useful for statistical calculation. In such a system
as called microcanonical ensemble, thermal properties can be calculated directly
with Z.
The Boltzmann statistics is a valid assumption for a dynamical system under the
ergodic hypothesis. Despite of the absence of rigorous proof, the Boltzmann statis-
tics can usually be applied to phonon gas in a ﬁxed volume. Thermodynamically,
2.4 Thermal Equilibrium 19
10.
however, it is only valid for isothermal processes, because the volume is not always
constant in adiabatic processes of internal origin. The anharmonic lattice cannot be
ergodic in strict sense, whereas the harmonicity is essentially required for stable
crystals at constant volume and pressure.
2.5 Speciﬁc Heat of a Monatomic Crystal
The speciﬁc heat at a constant volume CV ¼ @U
@T
À Á
V
is a quantity measurable with
varying temperature under a constant external pressure p. The phonon theory is
adequate for simple monatomic crystals, if characterized with no structural
changes.
For such a crystal, the speciﬁc heat and internal energy are given by quantized
phonon energies ek ¼ nk þ 1
2
À Á
hok , for which the wavevector k is distributed
virtually in all directions in the reciprocal space. Assuming 3N phonons in total,
the energies ek are degenerate with the density of k-states written as gðkÞ, which is a
large number as estimated from a spherical volume of radius kj j in the reciprocal
space. In this case, the partition function can be expressed as
Zk ¼ gðkÞ exp À
ek
kBT
¼ gðkÞ exp À
hok
2kBT
X1
nk¼0
exp À
nkhok
kBT
;
where the inﬁnite series on the right converges, if hok
kBT 1. In fact, this condition is
satisﬁed at any practical temperature T lower than the melting point so that Zk is
expressed as
Zk ¼ gðkÞ
exp À
hok
2kBT
1 À exp À
hok
kBT
:
The total partition function is given by the product Z ¼ PkZk , so that ln Z ¼P
k ln Zk; the free energy can therefore be calculated as the sum of ln Zk, namely,
F ¼ kBT
P
k
ln Zk, where
ln Zk ¼ À
hok
2
þ kBT ln gðkÞ À kBT ln 1 À exp À
hok
kBT
'
:
By deﬁnition, we have the relation F ¼ U À TS ¼ U þ T @F
@T
À Á
V
from which we
can derive the formula U ¼ ÀT2 @
@T
F
T
À Á
. Using the above ln Zk, we can show that the
internal energy is given by
U ¼ Uo þ
X
k
hok
exp hok
kBT À 1
and Uo ¼
1
2
X
k
hok: (2.20)
20 2 Phonons
11.
The speciﬁc heat at constant volume can then be expressed as
CV ¼
@U
@T
V
¼ kB
X
k
hok
kBT
2
exp
hok
kBT
exp
hok
kBT
À 1
2
: (2.21)
To calculate CV with (2.21), we need to evaluate the summation with the number
of phonon states on energy surface ek ¼ h ok þ 1
2
À Á
in the reciprocal space. In
anisotropic crystals, such a surface is not spherical, but a closed surface, as
shown in Fig. 2.3a. In this case, the summation in (2.21) can be replaced by a
volume integral over the closed surface, whose volume element is written as d3
k
¼ dk:dS ¼ dk? dSj j. Here, dk?is the component of k perpendicular to the surface
element dSj j ¼ dS, as illustrated two-dimensionally in Fig. 2.3b. We can write
dok ¼
dek
h
¼
1
h
gradke kð Þj jdk?;
where 1
h gradke kð Þj j ¼ vg represents the group velocity for propagation, and hence
dk? ¼ dok
vg
. Using these notations, we can reexpress (2.21) as
CV ¼ kB
ð
ok
hok
kBT
2
exp
hok
kBT
exp
hok
kBT
À 1
2
D okð Þdok; (2.22a)
where
a b
n
kx
dS
dS
0
kz
kx
ky
ky
Fig. 2.3 (a) A typical constant-energy surface in three-dimensional reciprocal space, wheredSis a
differential area on the surface. (b) The two-dimensional view in the kxky-plane.
2.5 Speciﬁc Heat of a Monatomic Crystal 21
12.
D okð Þ ¼
L
2p
3 þ
S
dS
vg
(2.22b)
is the density of phonon states on the surface S.
Tedious numerical calculations performed in early studies on representative
crystals resulted in such curves as shown in Fig. 2.4a, for example, of a diamond
crystal. However, in relation with dispersive longitudinal and transversal modes,
the analysis was extremely difﬁcult to obtain satisfactory comparison with experi-
mental results. On the other hand, Einstein and Debye simpliﬁed the functionD okð Þ
independently, although somewhat oversimpliﬁed for practice crystals. Neverthe-
less, their models are proven to be adequate in many applications to obtain useful
formula for Uand CV for simple crystals [4].
2.6 Approximate Models
2.6.1 Einstein’s Model
At elevated temperatures T, we can assume that thermal properties of a crystal are
dominated by n phonons of energy hoo. Einstein proposed that the dominant mode
at a high temperature is of a single frequency oo, disregarding all other modes in the
vibration spectrum. In this model, using the expression (2.22b) simpliﬁed as D ooð Þ
¼ 1, we can express the speciﬁc heat (2.22a) and the internal energy as
6
ω
a b
4
2
0 01 2 2 4 6 8
ωD
ω
2π
D (ω)
trans.
trans.
long.
long.
3 2π
k
Fig. 2.4 (a) Examples of practical dispersion curves. Longitudinal and transverse dispersions are
shown by solid and broken curves, respectively. (b) The solid curve shows an example of an
observed density function, being compared with the broken curve of Debye model.
22 2 Phonons
13.
CV ¼ 3NkB
x2
exp x
exp x À 1ð Þ2
and U ¼ 3NkB
1
2
x þ
x
exp x À 1
; (2.23)
respectively, where x ¼ YE
T ; the parameter YE ¼ hoo
kB
is known as the Einstein
temperature. It is noted that in the limit x ! 0, we obtain CV ! 3NkB.
At high temperatures, U can be attributed to constituent masses, vibrating
independently in degrees of freedom 2; hence, the corresponding thermal energy
is 2 Â 1
2 kBT, and
U ¼ 3NkBT and CV ¼ 3NkB: (2.24)
This is known as the Dulong–Petit law, which is consistent with Einstein’s
model in the limit of T ! 1.
2.6.2 Debye’s Model
At lower temperatures, longitudinal vibrations at low frequencies are dominant
modes, which are characterized approximately by a nondispersive relation o ¼ vgk.
The speed vg is assumed as constant on a nearly spherical surface for constant
energy in the k-space. Letting vg ¼ v, for brevity, (2.22b) can be expressed as
D oð Þ ¼
L
2p
3
4po2
v3
: (2.25)
Debye assumed that with increasing frequency, the density D oð Þ should be
terminated at a frequency o ¼ oD, called Debye cutoff frequency, as shown by the
broken curve in Fig. 2.4b. In this case, the density function D oð Þ / o2
can be
normalized as
RoD
0
D oð Þdo ¼ 3N, so that (2.25) can be replaced by
D oð Þ ¼
9N
o3
D
o2
: (2.26)
Therefore, in the Debye model, we have
U ¼ 3NkBT
ðoD
0
ho
2
þ
ho
exp ho
kBT À 1
0
@
1
A 3o2
do
o3
D
and
2.6 Approximate Models 23
14.
CV ¼ 3NkB
ðoD
0
exp
ho
kBT
exp
ho
kBT
À 1
2
ho
kBT
2
3o2
do
o3
D
:
Deﬁning Debye temperature hoD
kBT ¼ YD and ho
kBT ¼ x, similar to Einstein’s model,
these expressions can be simpliﬁed as
U ¼
9
8
NkBYD þ 9NkBT
T
YD
3 ð
YD
T
0
x3
exp x À 1
dx
and
CV ¼ 9NkB
T
YD
3 ð
YD
T
0
x4
exp x
exp x À 1ð Þ2
dx:
Introducing the function deﬁned by
Z
YD
T
¼ 3
T
YD
3 ð
YD
T
0
x3
dx
exp x À 1
; (2.27)
known as the Debye function, the expression CV ¼ 3NkBZ YD
T
À Á
describes
temperature-dependent CV for TYD. In the limit of YD
T ! 1, however, these U
and CV are dominated by the integral
ð1
0
x3
dx
exp x À 1
¼
p4
15
;
and hence the formula
U ¼
9
8
NkBYD þ 9NkBT
T
YD
3
p4
15
and
CV ¼ 9NkB
T
YD
3
p4
15
: (2.28)
24 2 Phonons
15.
can be used at lower temperatures than YD. In the Debye model, we have thus the
approximate relation CV / T3
for TYD, which is known as Debye T3
-law.
Figure 2.5 shows a comparison of observed values of CV from representative
monatomic crystals with the Debye and Dulong–Petit laws, valid at low and high
temperatures, respectively, showing reasonable agreements.
2.7 Phonon Statistics Part 1
Quantizing the lattice vibration ﬁeld, we consider a gas of phonons hok; hkð Þ. A
large number of phonons exist in excited lattice states, behaving like classical
particles. On the other hand, phonons are correlated at high densities, owing to
their quantum nature of unidentiﬁable particles. Although dynamically unspeciﬁed,
the total number of phonons is thermodynamically determined by the surface
boundaries at T, where phonon energies are exchanged with heat from the sur-
roundings. In equilibrium, the number of photons on each k-state can be either one
of n ¼ 1; 2; . . . ; 3N. Therefore, the Gibbs function can be expressed by G p; T; nð Þ,
but the entropy ﬂuctuates with varying n in the crystal. Such ﬂuctuations can be
described in terms of a thermodynamic probability g p; T; nð Þ, so that we consider
that two phonon states, 1 and 2, can be characterized by probabilities g p; T; n1ð Þ and
g p; T; n2ð Þ in an exclusive event, in contrast to the Boltzmann statistics for indepen-
dent particles.
At constant p, the equilibrium between the crystal and reservoir can therefore be
speciﬁed by minimizing the total probability g p; T; nð Þ ¼ g p; T; n1ð Þ þ g p; T; n2ð Þ,
considering such binary correlations dominant under n ¼ n1 þ n2 ¼ constant,
leaving all other niði 6¼ 1; 2Þ as unchanged. Applying the variation principle for
small arbitrary variations dn1 ¼ Àdn2, we can minimize g p; T; nð Þ to obtain
Cv
T
T3
-law
3R
0.6
0.4
0.2
0 .2 .4 .6 .8 1.0 1.2
Al
Cu
Ag
Pb 95K
215K
309K
396K
Dulong-Petit
ΘD
Fig. 2.5 Observed speciﬁc
heat CV 3R= against T YD= for
representative metals. R is the
molar gas constant. In the
bottom-right corner, values of
D for these metals are shown.
The T3
-law and Dulong–Petit
limits are indicated to
compare with experimental
results.
2.7 Phonon Statistics Part 1 25
16.
dgð Þp;T ¼
@g1
@n1
p;T
dn1 þ
@g2
@n2
p;T
dn2 ¼ 0;
from which we derive the relation
@g1
@n1
p;T
¼
@g2
@n2
p;T
:
This is a common quantity between g1 and g2, which is known as the chemical
potential. Therefore, we have equal chemical potentials m1 ¼ m2 in equilibrium
against phonon exchange. Writing the common potential as m, a variation of the
Gibbs potential G for an open system at equilibrium can be expressed for an
arbitrary variation d n as
dG ¼ dU À TdS þ pdV À mdn; (2.29)
where dn represents a macroscopic variation in the number of phonons n.
Consider a simple crystal, whose two thermodynamic states are speciﬁed by the
internal energy and phonon number, Uo; Noð Þ and Uo À e; No À nð Þ , which
are signiﬁed by probabilities go and g, as related to their entropies S Uo; Noð Þ and
S Uo À e; No À nð Þ, respectively. Writing the corresponding Boltzmann relations,
we have
go ¼ exp
S Uo; Noð Þ
kB
and g ¼ exp
S Uo À e; No À nð Þ
kB
:
Hence,
g
go
¼
exp S Uo À e; No À nð Þ=kBf g
exp S Uo; Noð Þ=kBf g
¼ exp
DS
kB
;
where
DS ¼ S Uo À e; No À nð Þ À S Uo; Noð Þ ¼ À
@S
@Uo
No
e À
@S
@No
Uo
n:
Using (2.29), we obtain the relations
@S
@Uo
No
¼
1
T
and
@S
@No
Uo
¼ À
m
T
so that
26 2 Phonons
17.
g ¼ go exp
mn À e
kBT
: (2.30)
For phonons, the energy e is determined by any wavevector k, where kj j ¼ 1; 2;
::::; 3N, and N can take any integral number. The expression (2.30) is the Gibbs
factor, whereas for classical particles, we use the Boltzmann factor instead. These
factors are essential in statistics for open and closed systems, respectively. For
phonons, it is convenient to use the notation l ¼ exp m
kBT , with which (2.30) can be
written as g ¼ goln
exp À e
kBT
. The factor l here implies a probability for the
energy level e to accommodate one phonon adiabatically [5], whereas the conven-
tional Boltzmann factor exp À e
kBT
is an isothermal probability of e at T. Origi-
nally, the chemical potential m was deﬁned for an adiabatic equilibrium with an
external chemical agent; however, for phonons l is temperature dependent as
deﬁned by l ¼ exp m
kBT . Here, the chemical potential is determined as m ¼ À @G
@n
À Á
p;T
from (2.29), which is clearly related with the internal energy due to phonon
correlations in a crystal.
For phonon statistics, the energy levels are en ¼ nho, and the Gibbs factor is
determined by e ¼ ho and n. The partition function can therefore be expressed as
ZN ¼
XN
n¼0
ln
exp À
ne
kBT
¼
XN
n¼0
l exp À
e
kBT
'n
:
Consideringl exp À e
kBT
1, the sum ofthe inﬁnite series evaluated for N ! 1is
Z ¼
1
1 À l exp À
e
kBT
:
With this so-called grand partition function, the average number of phonons can
be expressed as
nh i ¼ l
@ ln Z
@l
¼
1
1
l
exp À
e
kBT
À 1
¼
1
exp
e À m
kBT
À 1
: (2.31)
This is known as the Bose–Einstein distribution. It is noted that the energy e is
basically dependent on temperature, whereas the chemical potential is small
and temperature independent. Further, at elevated temperatures, we consider that
for e m; (2.31) is approximated as nh i % exp mÀe
kBT % exp Àe
kBT , which is the
Boltzmann factor. However, there should be a critical temperature Tc for nh i ¼ 1
to be determined by e Tcð Þ ¼ m, which may be considered for phonon condensation.
2.7 Phonon Statistics Part 1 27
18.
So far, phonon gas was speciﬁcally discussed, but the Bose–Einstein statistics
(2.31) can be applied to all other identical particles characterized by even parity;
particles obeying the Bose–Einstein statistics are generally characterized by even
parity and called Bosons. Particles with odd parity will be discussed in Chap. 11 for
electrons.
2.8 Compressibility of a Crystal
In the foregoing, we discussed a crystal under a constant volume condition. On the
other hand, under constant temperature, the Helmholtz free energy can vary with a
volume change DV, if the crystal is compressed by
DF ¼
@F
@V
T
DV;
where p ¼ À @F
@V
À Á
T
is the pressure on the phonon gas in a crystal. At a given
temperature, such a change DFmust be offset by the external work À pDV by
applying a pressure p, which is adiabatic to the crystal.
It is realized that volume-dependent energies need to be included in the free
energy of a crystal in order to deal with the pressure from outside. Considering an
additional energy Uo ¼ UoðVÞ, the free energy can be expressed by
F ¼ Uo þ 9NkBT
T
YD
3 ð
YD
T
0
x
2
þ ln 1 À exp Àxð Þf g
!
x2
dx
¼ Uo þ 3NkBTZ
YD
T
; (2.32)
where Z YD
T
À Á
is the Debye function deﬁned in (2.27), for which we have the relation
@Z
@ ln YD
T
¼ À
@Z
@ ln T
V
¼ À
T
YD
@Z
@T
: (2.33)
Writing z ¼ z T; Vð Þ ¼ TZ YD
T
À Á
for convenience, we obtain
@z
@V
T
¼ À
g
V
@z
@ ln YD
T
¼
gT
V
@Z
@ ln T
V
;
where the factor
28 2 Phonons
19.
g ¼ À
d ln Y
d ln V
is known as Gru¨neisen’s constant. Using (2.30), the above relation can be
reexpressed as
@z
@V
T
¼
g
V
T
@z
@T
V
À z
'
:
From (2.29), we have NkBz T; Vð Þ ¼ F À Uo; therefore, this can be written as
@ F À Uoð Þ
@V
'
T
¼
g
V
T
@ F À Uoð Þ
@T
V
À F þ Uo
'
: (2.34)
Noting Uo ¼ UoðVÞ, the derivative in the ﬁrst term of the right side is equal to
T @F
@T
À Á
V
¼ ÀTS; hence, the quantity in the curly brackets is À U þ Uo ¼ Uvib that
represents the energy of lattice vibration. From (2.31), we can derive the expression
for pressure in a crystal, that is,
p ¼ À
dUo
dV
þ
gUvib
V
; (2.35)
which is known as Mie–Gru¨neisen’s equation of state.
The compressibility is deﬁned as
k ¼ À
1
V
@V
@p
T
; (2.36)
which can be obtained for a crystal by using (2.32). Writing (2.32) as pV ¼ ÀV dUo
dV
þgUvib and differentiating it, we can derive
p þ V
@p
@V
T
¼ À
dUo
dV
À V
d2
Uo
dV2
þ g
@Uvib
@V
T
:
Since the atmospheric pressure is negligible compared with those in a crystal, we
may omit p, and also from (2.31)
@Uvib
@V
T
¼
g
V
T
@Uvib
@T
V
À Uvib
'
in the above expression. Thus, the compressibility can be obtained from
2.8 Compressibility of a Crystal 29
20.
1
k
¼ ÀV
@p
@V
T
¼
dUo
dV
þ V
d2
Uo
dV2
À
g2
V
TCV À Uoð Þ; (2.37)
where CV ¼ @Uvib
@T
À Á
V
is the speciﬁc heat of lattice vibrations.
If p ¼ 0, the volume of a crystal is constant, that is, V ¼ Vo and dUo
dV ¼ 0, besides
Uvib ¼ const: of V. Therefore, we can write 1
ko
¼ Vo
d2
Uo
dV2
V¼Vo
, meaning a
hypothetical compressibility ko in equilibrium at p ¼ 0. Then with (2.34) the
volume expansion can be deﬁned as
b ¼
V À Vo
Vo
¼
kogUvib
V
: (2.38)
Further, using (2.32)
@p
@T
V
¼
g
V
@Uvib
@T
V
¼
gCV
V
;
which can also be written as
@p
@T
V
¼
1
V
@V
@T
p
À
1
V
@V
@p
T
;
and hence we have the relation among g; k, and b, that is, g ¼ À V
CV
b
k .
Such constants as Y; k; b, and g are related with each other and are signiﬁcant
parameters to characterize the nature of crystals. Table 2.1 shows measured values of
YDby thermal and elastic experiments on some representative monatomic crystals.
Exercise 2
1. It is important that the number of phonons in crystals can be left as arbitrary,
which is thermodynamically signiﬁcant for Boson particles. Sound wave propa-
gation at low values of k and o can be interpreted for transporting phonons, which
is a typical example of low-level excitations, regardless of temperature. Discuss
why undetermined number of particles is signiﬁcant in Boson statistics. Can there
be any other Boson systems where the number of particles if a ﬁxed constant?
Table 2.1 Debye
temperatures YD determined
by thermal and elastic
experimentsa
Fe Al Cu Pb Ag
Thermal 453 398 315 88 215
Elasticb
461 402 332 73 214
a
Data: from Ref. [3]
b
Calculated with elastic data at room temperature
30 2 Phonons
21.
2. Einstein’s model for the speciﬁc heat is consistent with assuming crystals as a
uniform medium. Is it a valid assumption that elastic properties can be attributed
to each unit cell? What about a case of nonuniform crystal? At sufﬁciently high
temperatures, a crystal can be considered as uniform. Why? Discuss the validity
of Einstein’s model at high temperatures.
3. Compare the average number of phonons nh i calculated from (2.26) with that
expressed by (2.31). Notice the difference between them depends on the chemi-
cal potential: either m ¼ 0 or m 6¼ 0: Discuss the role of a chemical potential in
making these two cases different.
4. The wavefunction of a phonon is expressed by (2.17b). Therefore in a system of
many phonons, phonon wavefunctions should be substantially overlapped in the
crystal space. This is the fundamental reason why phonons are unidentiﬁable
particles; hence, the phonon system in crystals can be regarded as condensed
liquid. For Boson particles 4
He, discuss if helium-4 gas can be condensed to a
liquid phase at 4.2K.
5. Are the hydrostatic pressure p and compressibility discussed in Sect. 2.8 ade-
quate for anisotropic crystals? Comment on these thermodynamic theories
applied to anisotropic crystals.
References
1. M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964)
2. R.S. Knox, A. Gold, Symmetry in the Solid State (Benjamin, New York, 1964)
3. C. Kittel, Quantum Theory of Solids, (John Wiley, New York, 1963)
4. C. Kittel, Introduction to Solid State Physics, 6th edn. (Wiley, New York, 1986)
5. C. Kittel, H. Kroemer, Thermal Physics (Freeman, San Francisco, 1980)
References 31
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