Transcript Document

Lecture 9
•The field of sound waves.
•Thermodynamics of crystal lattice.
•Phonons and second sound.
•The Debye model.
•The Debye temperature.
•Specific heat of the solid in the Debye model.
1
In the same way that the energy of the electromagnetic
radiation is quantized in the form of photons so the energy
of the elastic waves, or sound waves, inside a solid medium
can be considered to be quantized in the form of phonons.
The energy of a phonon of frequency  is again   and,
because the phonons have integral angular momentum,
the assembly of phonons in the solid may again treated as
a boson gas.
To illustrate this point, we may
consider the Hamiltonian of a classical
solid composed of N atoms whose
positions in space specified by the
coordinates (x1, x2,.…., x3N).
2
In the state of the lowest energy, the values of these
coordinates may be denoted by ( x1 , x2 ,....., x3 N ) .
Denoting the displacements ( xi  xi )
of the atoms from
their equilibrium positions by the variables i (i=1,2,…3N),
the kinetic energy of the system in the configuration (xi) is
given by
3N
3N
K  21 m xi2  21 m i2
i 1
i 1
(9.1)
and the potential energy by
3
  
   ( xi )   ( xi )   
( xi  xi ) 
 x 

i 
i  ( xi ) ( xi )
(9.2)
  2 


( xi  xi )( x j  x j ) .....



i , j  xix j 
( xi ) ( xi )
The main term in this expansion represents the (minimum)
energy of the solid when all the N atoms are at rest at their
mean positions ; this energy may be denoted by the
symbol 0.
The next set of terms in the expansion is identically equal
to zero, because the function (xi) has its minimum value
at (xi)=(x i) and hence all its derivatives must vanish there.
The second-order terms of the expansion represent the
harmonic component of the atomic vibrations.
4
If we assume that the overall amplitudes of the atomic vibrations are
not very large 
xj
j
 1
we may retain only the harmonic terms of the expansion and neglect
all the successive ones; we are then working in the so-called
harmonic approximation. Note that the inharmonic components are
important at phase transition (from one crystal symmetry to another
and solid-liquid phases)
 1 2

H   0   2 m i   ij i j 
i, j
 i

(9.3)
where
i , j
2


 
1

 2 
 xix j  ( x
(9.4)
i
)  ( xi )
5
We now introduce a linear transformation, from the coordinates
i
to
the so-called normal coordinates qi, and choose the transformation
matrix in such a way that the new expression for the Hamiltonian does
not contain the cross terms, i.e.
H     21 m(qi2  i2qi2 )
(9.5)
i
where i (i=1,2,..3N) are the characteristic frequencies of the socalled normal modes of the system and are determined essentially by
the quantities
function
ij
or, in turn, by the nature of the potential energy
(xi).
The expression (9.5) suggests that the energy of the solid, over and
above the (minimum) value
0, may be considered as arising from a
set of 3N one-dimensional, non interacting, harmonic oscillators,
whose characteristic frequencies i are determined by the nature of
the interatomic interactions in the system.
6
Classically, each of the 3N normal modes of vibration corresponds to a
wave of distortion of the lattice points, i.e a sound wave.
Quantum-mechanically, these modes give rise to quanta, called
phonons, in very much the same way as the vibrational modes of the
electromagnetic field give rise to photons.
However, there is one important difference, i.e. while the number of
normal modes in the case of electromagnetic field is indefinite, the
number of normal modes (or the number of phonon energy levels) in
the case of a solid is fixed by the number of lattice sites in it.
This introduces certain differences in the thermodynamic behavior of
the sound filed in contrast to thermodynamic behavior of the radiation
field; however, at low temperatures, when the high-frequency modes
of the solid are not very likely to be excited, these differences become
rather insignificant and we obtain a striking similarity between the
two sets of results.
7
The thermodynamics of the solid can now be studied along the lines of
a system of harmonic oscillators. First of all, we note that the
quantum-mechanical eigenvalues of the Hamiltonian (9.5) would be
E{ni }   0   (ni  21 )i
(9.6)
i
where the numbers ni denote the “states of excitation” of the various
oscillators (or, equally well the occupation numbers of the various
phonon levels). The internal energy of the system is then given by
i


1
E (T )   0   2 i    i / kT
1
i

 i e
(9.7)
The expression within the curly brackets gives the energy of the solid
at absolute zero. The term 0 is necessarily negative and larger in
1
magnitude than the total zero-point energy,  2 i of the oscillators
i
8
together they determine the binding energy of the lattice. The last
term in the formula represents the temperature dependent part of
energy, which determines the specific heat of the solid:
2  / kT

E
(


/
kT
)
e


i
CV (T )  
  k
 / kT
2

T
(
e

1
)

V
i
i
(9.8)
i
To proceed further, we must have knowledge of the frequency spectrum
of the solid. To acquire this knowledge from first principles is not an
easy task.
Accordingly, one either obtains this spectrum through experiment or else
makes certain plausible assumptions about it. Einstein, who was the first
to apply quantum concept to the theory of solids (1907), assumed, for
simplicity, that the frequencies
(common) value by
i are all equal in value!
Denoting this
E, the specific heat of the solid is given by
CV (T )  3NkE ( x)
(9.9)
9
where E(x) is so-called Einstein function:
2
with
x
x e
E ( x)  x
(e  1) 2
(9.10)
 E  E
x

kT
T
(9.11)
At sufficiently high temperatures, when T>>E and hence x<<1, the
Einstein result tends towards the classical one, viz. CV=3Nk.
At sufficiently low temperatures, when T<<E and hence x>>1, the
specific heat falls at an exponentially fast rate and tends to zero as
T0.
The dashed curve in Fig. 9.1 depicts the variation of the specific heat
with temperature as, given by the Einstein formula (9.9)
10
CV/3Nk
T3-law
1.0
0.5
0
0
0.5
1.0
T/E
Fig.9.1 The specific heat of a solid, according to the Einstein model
(dashed line), and according to the Debye model (solid line). The
circles denote the experimental results for copper.
11
The theoretical rate of fall, however, turns out to be rather too fast in
comparison with the observed rate. Nevertheless, Einstein’s approach
to the problem did at least provide a theoretical basis for
understanding the observed departure of the specific heat of solids
from the classical law of Dulong and Petit, whereby CV=3R5.96
calories per oK of the substance.
Debye (1912) on the other hand, allowed a continuous spectrum of
frequencies, cut off at an upper limit
D such that the total number of
normal modes of vibration is equal to 3N, that is
D
 g ( )d  3 N
(9.12)
0
where g()d denotes the number of normal modes of vibration
whose frequency lies in the range (,+d).
12
For g(), Debye adopted the Rayleugh expression (8.49),
2
 1   1   d
2  4   d    2 3
     c
2
modified so as to suit the problem under study. Writing cL for the
velocity of propagation of the longitudinal modes and cT for the
velocity of propagation of the transverse modes eqn.(9.12) becomes
D
  2 d   2 d 
0 V  2 2 cL3  2 2 cT3   3 N
(9.13)
whence we obtain for the cut-off frequency
N 1
2
 3  3
  18
V  c L cT 
3
D
2
1
(9.14)
13
Accordingly, the Debye spectrum may be written as
9 N 2
 3
g( )    D
0

for
for
  D
(9.15)
  D
Before we proceed further to calculate the specific heat of solids on
the basis of the Debye spectrum, two remarks appear in order.
•First, the Debye spectrum is only an idealization of the actual situation
obtaining in a solid; it may be compared with a typical spectrum.
While for low-frequency modes ( the so called acoustical modes) the
Debye approximation is reasonably valid, there are serious
discrepancies in the case of high-frequency modes ( the so-called
optical modes).
At any rate, for “averaged” quantities, such as the specific heat, the
finer details of the spectrum are not very important. In fact, Debye
approximation serves the purpose reasonably well; things indeed
improve if we take account of the various peaks in the spectrum by
including in our result a number of “suitably weighted” Einstein
terms.
14
•Second, the longitudinal and the transverse modes of the solid
should have their own cut-off frequencies, D,L and
than having a common cut-off at
D,T say, rather
D, for the simple reason that, of the
3N normal modes of the lattice, N are longitudinal and 2N transverse.
Accordingly, we should have, instead of (9.13),
 D ,L

0
 d
V
2 3  N
2 cL
 D ,T
2
and

0
We note that the two cuts-offs D,L and
 2 d
2 3  2N
2 cT
(9.16)
D,T correspond to a common
wavelength min  (4V / 3N )1/ 3  which is comparable to the
mean interatomic distance in the solid. This is quite reasonable
because, for wavelengths shorter than
min, it would be rather
meaningless to speak of a wave of atomic displacements.
In the Debye approximation, formula (9.8) for the specific heat of the
solid becomes
15
CV (T )  3 NkD( x0 )
(9.17)
where D(x0) is the so called Debye function:
3
D( x 0 )  3
x0
x0
x 4 e x dx
 e
0
x
 1
(9.18)
2
with
 D  D
x0 

kT
T
(9.19)
where D being the so-called Debye temperature of the solid.
Integrating by parts, the expression for the Debye function becomes
x0
3x 0
12 x 3 dx
D ( x 0 )   x0
 3 x
e  1 x0 0 e  1
(9.20)
16
For T>>D, which means x0 <<1, the function
D(x0)
may be
expressed as a power series in x0:
x02
(9.21)
D( x0 )  1 
.......
20
Thus, as T , CV  3Nk; moreover, according to this theory, the
classical result should be applicable to within ½ percent so long as
T>3D. For T<<D, which means x0 >>1, the function D(x0) may
be written as
x
12 0 x 3dx
D( x0 )  3  x
 0(e  x0 ),
x0 0 e  1
(9.22)
whence
4
4  T 
D( x0 )  3 


5x0
5  D 
4
4
3
(9.23)
17
Thus, at low temperatures the specific heat of the solid follows the
Debye T3-law:
3
3
 T 
12  T 

  464.4
 cal / mole /
CV  Nk
5  D 
 D 
4
o
K
(9.24)
Thus, while
•in the limit T we recover the well-known classical behavior
(CV=const),
• in the limit T0 we obtain the typical phonon behavior
(CVT3).
It is clear from eqn. (9.24) that a measurement of the low-
temperature specific heat of a solid should enable us not only to
check the validity of the T3-law but also to obtain an empirical value
of the Debye temperature D.
18
The value of
D can also be obtained by computing the cut-off
frequency D from a knowledge of the parameters N/V, cL and cT ;
see formulae (9.14) and (9.19).
 1
2
3
2 N
  
 D  18
V  c L3 cT3 
 D  D
x0 

kT
T
1
(9.14)
(9.19)
The closeness of these estimates is another evidence in favor of
Debye’s theory. Once D is known, the whole of temperature range
can be covered theoretically by making use of the tabulated values of
the function D(x0). A typical case is shown in Fig. 9.1. We note that
not only was T3-law obeyed at low temperatures, the argument
between theory and experiment was good throughout the range of
observations.
19
CV/3Nk
T3-law
1.0
0.5
0
0
0.5
1.0
T/E
Fig.9.1 The specific heat of a solid, according to the Einstein model
(dashed line), and according to the Debye model (solid line). The
circles denote the experimental results for copper.
20
21
Back to Dispersion Curves
We know we can measure the phonon dispersion
curves - the dependence of the phonon frequencies
upon the wavevector q.
To calculate the heat capacity, we begin by summing
over all the energies of all the possible phonon
modes, multiplied by the Planck Distribution.
U
U q, p   n  q, p
q
sum over all
wavevectors
p
q
p
Planck
Distribution
sum over all
polarizations
22
Density of States
U    n  q , p  
q
p
   g ( )
p
q
q , p
e
 / k B T
1
p
q , p
e
 / k BT
1
d
number of modes
g()
unit frequency
23
Density of States: One
Dimension
us  u0 sin(sqa  q, p t )  u0 sin(sqa) e
iq, pt
determined by the dispersion relation
If the ends are fixed, what modes, or wavelengths, are allowed?
24
Density of States: One
Dimension
To calculate the density of states, use
number of modes
g()
unit frequency
There is one mode per interval  q =  / L with allowed values...
( N  1) 
 2 3
q   ,
, ,...,

L
L L L

So, the number of modes per unit range of q is L / .
25
Density of States: One
Dimension
There is one mode for each
mobile atom.
To generalize this, go back to the
definition...the number of modes
is the product of the density of
states and the frequency unit.
dN
g ( )d 
d
d
dN dq

d
dq d
L dq

d
 d
L d

 d / dq
L d

 vg
26
Density of States: One
Dimension
monatomic lattice
diatomic lattice
• Knowing the dispersion curve we can calculate the group
velocity, d/dq.
• Near the zone boundaries, the group velocity goes to
zero and the density of states goes to infinity. This is
called a singularity.
27
Periodic Boundary Conditions
• No fixed atoms – just require that u(sa) = u(sa + L).
• This is the periodic condition.
• The solution for the displacements is
i ( sqat )
us  u0 sin(sqa) e
• The allowed q values are then,
2 
 2 4
q  0,
,
,..., N

L
L
L

28
Density of States: 3 Dimensions
• Let’s say we have a cube with
sides of length L.
• Apply the periodic boundary
condition for N3 primitive cells:
i ( qx xq y y qz z )
e
i ( qx ( x L)q y ( y  L)qz ( z  L))
e
2 
 2 4
qx, y,z  0,
,
,..., N

L
L
L

29
Density of States: 3 Dimensions
qz
There is one allowed value of q per
volume (2/L3) in q space or
3
V
 L 



 2 
8 3
allowed values of q per unit volume of
q space, for each polarization, and for
each branch.
The total number of modes for each
polarization with wavevector less than
q is
 V 4
N   3  q 3
 8  3
qy
qx
dN
g ( ) 
d
Vq2 dq

2 2 d30