3b Lattice Vibrations PDF
3b Lattice Vibrations PDF
3b Lattice Vibrations PDF
Lattice Dynamics
Phonons
Thermal energy
Heat capacity
Einstein model
Density of states
Debye model
Anharmonic effects
Thermal expansion
Thermal conduction by phonons
Neutron scattering
Phonons
So far we discussed a classical approach to the lattice vibrations.
Solving the problems of vibrations of atoms in lattice we have found that
coupled harmonic oscillations of atoms give rise to non coupled normal modes
of oscillation
Hamiltonian of a harmonic oscillator: =
+
Similarly any normal mode of the lattice is quantized and the quantum of
vibration is called a phonon in analogy with the photon, which is the quantum
of the electromagnetic wave travelling wave of lattice vibrations
Since the phonon moves in the crystal its wave vector is k = but, although it
interacts with other particles (such as electrons) as it had a momentum ,
however a phonon does not carry physical momentum because the phonon
coordinates are relative coordinates of atoms
Phonons
If Phonons are to be considered as particles (quasi-particle to study collisions
with other phonons or other particles), a mode with a well defined k is not
localized,
travelling waves of a single frequency and wavelength imply a vibration of
the whole lattice.
space
=0
As observed, in these
points we have stationary
waves superposition of
two travelling waves with
opposite directions (+k and
k) due to Bragg reflection
Phonons
The lattice with s atoms in a unit cell is described by 3s independent
oscillators (3 acoustic modes and 3s-3 optical modes)
The frequencies of normal modes of these oscillators will be given by the
solution of 3s linear equations as we have discussed before.
Each mode has a frequency p(k), where p denotes a particular mode, i.e.
p=1,3s. The energy of this mode is given by
1
( =
+ ) (
)
2
the integer np(k) is the occupation number of the normal mode p,k.
+ ( + / )
1 1
= 2 2
( + 2 / )
1
( / )
= partition function of the oscillator
((/ ))
+
E=
This is the mean energy of phonons. The first term in the above equation is the
zero-point energy, the minimum energy of the oscillator even at T=0 K.
Rearranging the result
= = <n()> + =
+
<n()>=
mean excitation number of phonon at frequency
1
h
=
+
2
T
1 = 1 + + ! +
high temperature limit
the energy steps are now small compared with the energy of the
harmonic oscillatorclassical limit corrected by the zero point
energy (negligible at high temperatures).
G. Bracco-Material Science SERP CHEM 12
Heat Capacity C
Heat capacity C can be found by differentiating the average energy of
phonons
= +
()
=
( )
= =
( )
E
E
=
E
1
Where E= has the dimensions of T and is the Einstein temperature.
so that the whole solid had a heat capacity given by the previous results for 1D
oscillator multiplied by 3N (3 degrees of freedom for the N oscillators)
E
A. Einstein, Ann. Physik, vol.
E
= 3 = 3 22, p. 186 (1907)
E
1
These energy differences can be so small depending on the system that the
energy can be considered as continuous This is the case of classical mechanics.
But on atomic scale the energy can only jump by a discrete amount from one
value to another.
This discussion on DOS is valid also for electrons ( , boson E(k), fermion).
G. Bracco-Material Science SERP CHEM 16
Density of States
In some cases, each particular energy level can be associated with more than one
different state (mode for phonons or wavefunction for electrons)
This energy level is said to be degenerate.
The density of states is the number of discrete states per unit energy interval, and
so that the number of states between and + is g() and
=N number of stated between and .
k k k k
G. Bracco-Material Science SERP CHEM 17
any k occupy a volume and in this volume there is a single value dn=1
density in k between k and k+dk (k)= = = (constant density)
Number of states between (k)(k)+d g()d = 2 (k)dk= 2 dk=
dk g()d = dk
G. Bracco-Material Science SERP CHEM 18
Density of States (1D)
It is worth noting that the density of states is independent from the specific boundary
condition, in fact choosing the vanishing of the vibration at the boundaries to get
standing waves:
any k occupy a volume and in this volume there is a single value dn=1
density in k between k and k+dk (k)= = = (constant density)
Number of states between (k)(k)+d g()d = (k)dk= dk
g()d = dk density equal to the case of Born-von Karman BC.
= 1 ( )
(k)= sin ( )
cos ( )
2
( )
Multiplying by 2 = 2 2
g()
= =
2N
( 2 )
2 1/ 2
for 1D g()=
max
any k occupy a volume = and in this volume there is a single
x y
point dn=1
as in the isotropic case, g() is the area between the two lines
divided by the density of k points in the kx , ky plane
=
g()= d 1st BZ
Simple square lattice
Curves =
but d=|| d=|vg | ( vg group velocity)
d
g()=
| v | dk
+d
g
d
This calculation has to be done for any dispersion curve and the
results summed up to get the total g().
G. Bracco-Material Science SERP CHEM 23
For 3D, the calculation of !() is a very heavy task and is performed
numerically.
In this model only the acoustic branches are considered and the true
dispersion curves are approximated by linear dispersion curves.
This model gives better results than the Einstein model.
for a cubic
sample with periodicity
# $
$
radius of a sphere
in k space,
for the real crystal k within the 1st BZ
= ! + =
" 2
v3s
+
# = 3
V V %&
g() = 3 ( + ) ( + )=
$
%&
! =
a parameter that provides a way to say if the temperature is high (T>D ) or low
(T<D).
Neglecting the zero point energy which does not contribute to the heat capacity Cv
x3
= 9 ' x
D ex 1
and the dependence on T is outside
the integral and in # = D/T
For T>>D (high temperature limit)
x x = x3= (D/T)3 ,
2
ex1+x
3
Cv= 3 (Dulong-Petit law)
G. Bracco-Material Science SERP CHEM 30
The Debye model
For T0 # (low temperature limit)
x3
e
x x= $
' ' D
The energy depends on the fourth power of T, therefore the heat capacity
Cv= =
' D
The heat capacity Cv= =
' D
is in agreement with experiments in both
high and low T limit.
Even for intermediate temperature the
heat capacity calculated by means of the
Debye model shows a good agreement
with experiments.
Debye frequency and Debye temperature scale with the velocity of sound in
the solid. So solids with low densities and large elastic moduli have high *# .
Debye energy # can be used to estimate the maximum phonon energy in
a solid.
Solid Ar Cs Pb
93 38 105
At high temperature the measured heat capacity can be also greater than the
Dulong-Petit value due to anharmonic effects.
G. Bracco-Material Science SERP CHEM 34
Einstein & Debye models vs. real crystals
The two models can be applied to approximate
acoustic and optical dispersion curves as schematically
shown in the figure.
The different DOS are shown in the figure below.
For real crystals the DOS is given by
d)
gs()= | v | for any mode of polarization s
g
g()= * gs()
Anharmonic Effects
Up to now we have considered only vibrations with small amplitude and with
this assumption the interaction potential between atoms has been
approximated by an harmonic potential retaining only term up to second order.
The harmonic potential allows us to calculate crystal properties for T not too
high (below the melting point) to fulfill the requirements of small vibration
amplitude.
One could argue that anharmonic terms can added only to increase the
precision of the calculation.
exp .( ! - )
4
In a similar way the denominator
exp
]exp [
]exp [
exp
][1 +
] [1 +
exp
][1 +
+
exp
.
!
So <x> =
2
The thermal expansion does not involve the symmetric term but only the
asymmetric cubic term.
G. Bracco-Material Science SERP CHEM 38
Anharmonic Effects: Thermal expansion
The linear expansion coefficient
= at constant pressure
generally depends on the crystal directions, for cubic crystal only a single value
Related to thermal expansion, there is the change in the phonon frequency
and this effect is described by the Grneisen parameter
(5)
sk=
for each mode and a total value where each sk is summed with a
weight given by the contribution of the mode to the specific heat
sk cvsk
= cvsk
In this way
thermodynamic
quantities depend on
the volume of the
crystal.
,
, = , , +G crystal momentum conservation
G. Bracco-Material Science SERP CHEM 40
Anharmonic Effects: Thermal conductivity
Consider a rod with the two ends maintained at different temperatures. Thermal
conductivity is defined as the energy j transmitted per unit time across unit
area per unit temperature gradient
/ = 0 in 1D 1 = 0
The thermal energy transfer is a random process involving scattering introduces
mean free path of phonons in the problem.
Kinetic theory: Lets assume that the energy contributed by a phonon at a point
depends on the position of its last collision with an average collision time
phonons coming from the high temperature end bring more energy than those
coming from the low temperature end, thus, although there is no net number
flux, there can be energy flux travelling from the high T end to the low T end.
Number of phonons arriving at x per unit time per unit area of cross section is
1/2 n vx, where vx is the phonon speed in x direction.
Net energy flux: 1 = v {E(T[x- vx])-E(T[x+vx])} v 2 ( )
Averaging and assuming isotropy < v 2 >= < v: 2 >= < v; 2 >= < v2 >
=
1 = v2 = v
In the Debye model the phonon velocity is constant (sound velocity) therefore
the temperature dependence of = 2 is that of and
decrease as .
0
Pure NaCl
=321 K
Melting
temp
Neutron scattering
|:8
| = |:8
| and the Laue condition :8
= :8
+ 7.
On the other hand the interaction with the solid can be inelastic and phonons
can be created or annihilated. In case of single phonon exchange
Energy cons. 8
=
(
) (+ for annihilation for creation)
Crystal momentum cons. :8
= :8
+
+ 7.
In this way the dispersion curve (
) can be measured.
G. Bracco-Material Science SERP CHEM 48
Neutron scattering
The neutron beam produced by a
reactor has a very broad energy
distribution and the elastic
scattering with a crystal
(monochromator) can be employed
to select an energy in a small range
(quasi-monochromatic beam).
This energy selected beam can be
employed for inelastic scattering.
Neutrons have an energy similar to
phonon energies (10-100 meV)
therefore during an inelastic
scattering there is a relatively huge
energy exchange, that can easily be
measured.
The same technique can be used for
X-rays to get monochromatic beams
but the energy of phonons is in the
keV range and, for X-rays, phonon
energy exchange is negligible. G. Bracco-Material Science SERP CHEM 49
Neutron scattering
E=