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7.E. Heat Capacity Of A Debye Solid
Consider a simple crystalline solid with one atom at each
lattice site.
Since the size and shape of the solid are fixed macroscopically,
the motion of the
atoms must be restricted to oscillations about their equilibrium
positions.
The degrees of freedom of such vibrations is 3 N , where N is
the number of atoms.
f the amplitudes of the vibrations are small enough, the
oscillations become
harmonic, i.e., the potential energy is quadratic in atomic
displacements. Thus, the
atomic vibrations of a solid can be approximated as a set of 3 N
coupled harmonic
oscillators.
!y means of a transformation into the so"called normal
coordinates , the vibrations
can be de"coupled into independent normal modes .
Typical measured values of the heat capacity of monatomic solids
are shown in #ig.
$.%. &t high temperatures, V C approaches a constant value
of ' cal() mole, in
agreement with the classical theory. #or low temperatures, V C
drops as 3T , which
can only be explained in terms of quantum theory.
The *ebye theory is a quantum theory of harmonic oscillations in
a continuum.
&ccording to classical elastic theory, there are % types of
waves governed by the wave
equations
( )%
%% %
+,T
T
t c t =
u rT =u transverse, doubly degenerate
( )
%%
% %
+, L
Lt c t
= u r L =u longitudinal
where c is the phase velocity. The propagating modes thus obey
linear dispersion
i ic k = ,i L T =and are called sound waves .
-et the solid be a rectangular lattice of sides , , x y z L L L
. The normal modes are
standing waves which vanishes at the surfaces. This means the
allowable wave
vectors are
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i ii
k n L = where , ,i x y z = and +,%, ,i in N = L a/
where iii
L N
a= , with ia being the lattice spacing, is the number of sites
in the ith
direction. Thus, the total number of sites is x y z N N N N = .
0ith each site having 3
degrees of freedom, the total degrees of freedom is 3N .
n the quantized version of the theory, the hamiltonian is
( ) + %
,, ,
+%i ii L T T
H n =
= + k k k h $.'%/where the sum over k involves N modes for each
branch i. The partition function is
( ) ( ) + %
,, ,
+exp
% N N i ii L T T Z T Tr n
=
= + k
k
k h
( ),
,
+exp
%i
i ii n
n
=
= +
k
k
k
k h
( )
( )( )
+ +exp
% + expii i
=
k
k k
hh
( )
++
%sinh%
ii
=
k k h $.'3/
1ence,
( ) ( )+ln ln %sinh% N ii
Z T = k k h
ln N E Z
=
( )
( )( )
+cosh
+%+ %sinh%
i
ii
i
=
k
k
k
k
hh
h
( ) ( )
( ) ( )( )
+ +exp exp
+% %+ + %exp exp% %
i i
ii
i i
+ =
k
k k
k
k k
h hh
h h
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( )( )
( )exp + +
%exp +i
ii i
+ =
k
k k
k
hh
h
( ) ( )+ +
% exp + ii i
= +
k
k k
hh
$.'2/
( ),+% i ii
n = + k k k h
where the average occupation of the mode ( ), ik
( ),+
exp +i in
=
k
k h $.' /
is called the Planck's formula .
The sum over k can be approximated by an integral
( )d = k
k k
where the density of states in k- space, ( ) k , can be
calculated from eq a/ as
( ) 3 3 x y z L L L V
= =k
so that3
3
ik
V d k
>=
k
where the condition ik > from eq a/ restricts the integration
to the + st quadrant.
4sing k c
= , we have
( ) ( )33ik
V f d k f
>=
k ( )%%%
V dk k f
=
( )%% 3%V d f
c
= b/
where we5ve used the fact that integration of the angular part
over the + st quadrant
gives
+
+6 % st Q
d d = =
Summing over the branches gives
( ) ( )%% 3%i i i i i ii i i
V f d f
c
=
k
#or the special case that i f f = , we have
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( ) ( )%3 %+
%ii i i
V f d f
c
= k
( )%% 33% V d f c = where
3 3 3 3
3 + % +
i i T Lc c c c= = +
To restrict the total number of modes to 3N , we must introduce
a cutoff *ebye/
frequency D so that
%% 3
33
%
DV N d
c
=
3
% 3
%
DV
c
= $.'6/
+( 3%' D D
N c c k
V
= =
$.'7/
*efining the density of states in "space by
( )i
d g = k
we have
( ) %% 33
%V g
c
= %37
D
N
= $.$ /
8q $.'2/ thus becomes
( ) ( )+%
D
E d g n
= + h $.$+/
where
( )( )
+
exp +
n
=h
0ith the help of $.$ /, we get
( )337 +
%
D
D
N E d n
= + h
( )
2 3
3
76 exp +
D
D
D
N d
= +
hh
$.$%/
The heat capacity is therefore
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( )( )
2
%3 %
exp7
exp +
D
N D B
N C d
k T
=
h hh
h
( )% 2
%3 %7
+
D x
x B
x D B
N k T x edxk T e
= h
h where B
xk T
= h
( )
3 2
%7+
D x x B
B x D
k T x e Nk dx
e = h
( )
3 2
%7+
D x x
B x D
T x e Nk dx
T e
= #or low T , we have D x . 4sing
( )
2 2
%
2++
x
x
x edx
e
=
we get
32+% N B
D
T C Nk
T
; $.$2/
which is the famous *ebye 3T rule.
