IV.%20Vibrational%20Properties%20of%20the%20Lattice

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IV. Vibrational Properties of the Lattice

1. Heat CapacityEinstein Model
2. The Debye Model Introduction
3. A Continuous Elastic Solid
4. 1-D Monatomic Lattice
5. Counting Modes and Finding N(?)
6. The Debye Model Calculation
7. 1-D Lattice With Diatomic Basis
8. Phonons and Conservation Laws
9. Dispersion Relations and Brillouin Zones
10. Anharmonic Properties of the Lattice

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A. Heat CapacityEinstein Model (1907)
Having studied the structural arrangements of
atoms in solids, we now turn to properties of
solids that arise from collective vibrations of
the atoms about their equilibrium positions.
For a vibrating atom
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Classical Heat Capacity
For a solid composed of N such atomic oscillators
Giving a total energy per mole of sample
So the heat capacity at constant volume per mole
is
This law of Dulong and Petit (1819) is
approximately obeyed by most solids at high T ( gt
300 K). But by the middle of the 19th century it
was clear that CV ? 0 as T ? 0 for solids.
Sowhat was happening?
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Einstein Uses Plancks Work
Einstein (1907) model a solid as a collection
of 3N independent 1-D oscillators, all with
constant ?, and use Plancks equation for energy
levels
occupation of energy level n (probability of
oscillator being in level n)
classical physics (Boltzmann factor)
Average total energy of solid
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Some Nifty Summing
Using Plancks equation
Which can be rewritten
So we obtain
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At lastthe Heat Capacity!
Using our previous definition
Differentiating
Now it is traditional to define an Einstein
temperature
So we obtain the prediction
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Limiting Behavior of CV(T)
High T limit
Low T limit
These predictions are qualitatively correct CV ?
3R for large T and CV ? 0 as T ? 0
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But Lets Take a Closer Look
High T behavior Reasonable agreement with
experiment
Low T behavior CV ? 0 too quickly as T ? 0 !
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B. The Debye Model (1912)
Despite its success in reproducing the approach
of CV ? 0 as T ? 0, the Einstein model is clearly
deficient at very low T. What might be wrong
with the assumptions it makes?
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Details of the Debye Model
Pieter Debye succeeded Einstein as professor of
physics in Zürich, and soon developed a more
sophisticated (but still approximate) treatment
of atomic vibrations in solids. Debyes model of
a solid 3N normal modes (patterns) of
oscillations Spectrum of frequencies from ?
0 to ?max Treat solid as continuous
elastic medium (ignore details of atomic
structure)
This changes the expression for CV because each
mode of oscillation contributes a
frequency-dependent heat capacity and we now have
to integrate over all ?
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C. The Continuous Elastic Solid
We can describe a propagating vibration of
amplitude u along a rod of material with Youngs
modulus E and density ? with the wave equation
By comparison to the general form of the 1-D wave
equation
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D. 1-D Monatomic Lattice
By contrast to a continuous solid, a real solid
is not uniform on an atomic scale, and thus it
will exhibit dispersion. Consider a 1-D chain of
atoms
p atom label p ? 1 nearest neighbors p ?
2 next nearest neighbors cp force constant
for atom p
For atom s,
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1-D Monatomic Lattice Equation of Motion
Now since c-p cp by symmetry,
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1-D Monatomic Lattice Solution!
The dispersion relation of the monatomic 1-D
lattice!
Often it is reasonable to make the
nearest-neighbor approximation (p 1)
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Dispersion Relations Theory vs. Experiment
In a 3-D atomic lattice we expect to observe 3
different branches of the dispersion relation,
since there are two mutually perpendicular
transverse wave patterns in addition to the
longitudinal pattern we have considered.
Along different directions in the reciprocal
lattice the shape of the dispersion relation is
different. But note the resemblance to the
simple 1-D result we found.
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E. Counting Modes and Finding N(?)
We will first find N(k) by examining allowed
values of k. Then we will be able to calculate
N(?) and evaluate CV in the Debye model.
First step simplify problem by using periodic
boundary conditions for the linear chain of atoms
We assume atoms s and sN have the same
displacementthe lattice has periodic behavior,
where N is very large.
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First finding N(k)
Since atoms s and sN have the same displacement,
we can write
This sets a condition on allowed k values
independent of k, so the density of modes in
k-space is uniform
So the separation between allowed solutions (k
values) is
Thus, in 1-D
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Next finding N(?)
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N(?) at last!
A very similar result holds for N(E) using
constant energy surfaces for the density of
electron states in a periodic lattice!
This equation gives the prescription for
calculating the density of modes N(?) if we know
the dispersion relation ?(k). We can now set up
the Debyes calculation of the heat capacity of a
solid.
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F. The Debye Model Calculation
We know that we need to evaluate an upper limit
for the heat capacity integral
If the dispersion relation is known, the upper
limit will be the maximum ? value. But Debye
made several simple assumptions, consistent with
a uniform, isotropic, elastic solid
3 independent polarizations (L, T1, T2) with
equal propagation speeds vg continuous,
elastic solid ? vgk ?max given by the
value that gives the correct number of modes per
polarization (N)
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N(?) in the Debye Model
First we can evaluate the density of modes
Next we need to find the upper limit for the
integral over the allowed range of frequencies.
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?max in the Debye Model
Since there are N atoms in the solid, there are N
unique modes of vibration for each polarization.
This requires
The Debye cutoff frequency
Now the pieces are in place to evaluate the heat
capacity using the Debye model! This is the
subject of problem 5.2 in Myers book. Remember
that there are three polarizations, so you should
add a factor of 3 in the expression for CV. If
you follow the instructions in the problem, you
should obtain
And you should evaluate this expression in the
limits of low T (T ltlt ?D) and high T (T gtgt ?D).
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Debye Model Theory vs. Expt.
Better agreement than Einstein model at low T
Universal behavior for all solids!
Debye temperature is related to stiffness of
solid, as expected
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Debye Model at low T Theory vs. Expt.
Quite impressive agreement with predicted CV ? T3
dependence for Ar! (noble gas solid)
(See SSS program debye to make a similar
comparison for Al, Cu and Pb)
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G. 1-D Lattice with Diatomic Basis
Consider a linear diatomic chain of atoms (1-D
model for a crystal like NaCl)
In equilibrium
Applying Newtons second law and the
nearest-neighbor approximation to this system
gives a dispersion relation with two branches
?-(k) ? ? 0 as k ? 0 acoustic modes
(M1 and M2 move in phase) ?(k) ? ? ?max as
k ? 0 optical modes (M1 and M2 move out of
phase)
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1-D Lattice with Diatomic Basis Results
These two branches may be sketched schematically
as follows
gap in allowed frequencies
In a real 3-D solid the dispersion relation will
differ along different directions in k-space. In
general, for a p atom basis, there are 3 acoustic
modes and p-1 groups of 3 optical modes, although
for many propagation directions the two
transverse modes (T) are degenerate.
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Diatomic Basis Experimental Results
The optical modes generally have frequencies near
? 1013 1/s, which is in the infrared part of
the electromagnetic spectrum. Thus, when IR
radiation is incident upon such a lattice it
should be strongly absorbed in this band of
frequencies.
At right is a transmission spectrum for IR
radiation incident upon a very thin NaCl film.
Note the sharp minimum in transmission (maximum
in absorption) at a wavelength of about 61 x 10-4
cm, or 61 x 10-6 m. This corresponds to a
frequency ? 4.9 x 1012 1/s.
If instead we measured this spectrum for LiCl, we
would expect the peak to shift to higher
frequency (lower wavelength) because MLi lt
MNaexactly what happens!
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H. Phonons and Conservation Laws
Crystal momentum is analogous to but not
equivalent to linear momentum. No net mass
transport occurs in a propagating lattice
vibration, so the linear momentum is actually
zero. But phonons interacting with each other or
with electrons or photons obey a conservation law
similar to the conservation of linear momentum
for interacting particles.
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Phonons and Conservation Laws
Lattice vibrations (phonons) of many different
frequencies can interact in a solid. In all
interactions involving phonons, energy must be
conserved and crystal momentum must be conserved
to within a reciprocal lattice vector
Compare this to the special case of elastic
scattering of x-rays with a crystal lattice
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I. Brillouin Zones of the Reciprocal Lattice
Each BZ contains identical information about the
lattice
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Wigner-Seitz Cell--Construction
For any lattice of points, one way to define a
unit cell is to connect each lattice point to all
its neighboring points with a line segment and
then bisect each line segment with a
perpendicular plane. The region bounded by all
such planes is called the Wigner-Seitz cell and
is a primitive unit cell for the lattice.
1-D lattice Wigner-Seitz cell is the line
segment bounded by the two dashed planes
2-D lattice Wigner-Seitz cell is the shaded
rectangle bounded by the dashed planes
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1st Brillouin Zone--Definition
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1st Brillouin Zone for 3-D Lattices
For 3-D lattices, the construction of the 1st
Brillouin Zone leads to a polyhedron whose planes
bisect the lines connecting a reciprocal lattice
point to its neighboring points. We will see
these again!
bcc direct lattice ? fcc reciprocal lattice
fcc direct lattice ? bcc reciprocal lattice
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IJ. Anharmonic Properties of Solids
Two important physical properties that ONLY occur
because of anharmonicity in the potential energy
function

1. Thermal expansion
2. Thermal resistivity (or finite thermal
   conductivity)

Thermal expansion In a 1-D lattice where each
atom experiences the same potential energy
function U(x), we can calculate the average
displacement of an atom from its T0 equilibrium
position
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IThermal Expansion in 1-D
Evaluating this for the harmonic potential energy
function U(x) cx2 gives
Now examine the numerator carefullywhat can you
conclude?
Thus any nonzero ltxgt must come from terms in U(x)
that go beyond x2. For HW you will evaluate the
approximate value of ltxgt for the model function
Why this form? On the next slide you can see
that this function is a reasonable model for the
kind of U(r) we have discussed for molecules and
solids.
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Do you know what form to expect for ltxgt based on
experiment?
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Lattice Constant of Ar Crystal vs. Temperature
Above about 40 K, we see
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Thermal Resistivity
When thermal energy propagates through a solid,
it is carried by lattice waves or phonons. If
the atomic potential energy function is harmonic,
lattice waves obey the superposition principle
that is, they can pass through each other without
affecting each other. In such a case,
propagating lattice waves would never decay, and
thermal energy would be carried with no
resistance (infinite conductivity!). Sothermal
resistance has its origins in an anharmonic
potential energy.
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Phonon Scattering
There are three basic mechanisms to consider
1. Impurities or grain boundaries in
polycrystalline sample
2. Sample boundaries (surfaces)
3. Other phonons (deviation from harmonic
behavior)
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Temperature-Dependence of ?
The low and high T limits are summarized in this
table
CV ? ?
low T ? T3 nph ? 0, so ? ? ?, but then ? ? D (size) ? T3
high T 3R ? 1/T ? 1/T
How well does this match experimental results?
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Experimental ?(T)
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Phonon Collisions (N and U Processes)
How exactly do phonon collisions limit the flow
of heat?
2-D lattice ? 1st BZ in k-space
No resistance to heat flow (N process phonon
momentum conserved)
? Predominates at low T ltlt ?D since ? and q will
be small
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Phonon Collisions (N and U Processes)
What if the phonon wavevectors are a bit larger?
2-D lattice ? 1st BZ in k-space
Two phonons combine to give a net phonon with an
opposite momentum! This causes resistance to
heat flow. (U process phonon momentum lost in
units of hG.)

• More likely at high T gtgt ?D since ? and q will be
  larger