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This course deals with crystalline solids and is intended to provide students with basic physical concepts and mathematical tools used to describe solids. Key words in this lecture are: The Monatomic Chain, Hooke's Law, Equation of Motion, Amplitude, Nth Atom, Phonon Dispersion Relaion, Normal Mode Frequencies, First Brillouin Zone, Standing Wave, Group Velocity, Kittel,s Notaions, Acoustic Branch, Wavelenght
Typology: Slides
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One dimensional model # 1:
The Monatomic Chain
-^ Consider a
with nearest-neighbor,
type forces
( F = - kx
)^ between the atoms.
-^ This is equivalent to a force-springmodel, with masses
m^
& spring
constants
K.
-^ To illustrate the procedure for treating the interatomic potential in theharmonic approximation, consider just two neighboring atoms.
Assume
that they interact with a known potential
V(r)
. Expand
V(r)
in a
Taylor’s series in displacements about the equilibrium separation,keeping only up through quadratic terms in the displacements:
This potential energy is the same asthat associated with a spring with
spring constant:
ar Vd dr K
^
2 2
) (^
a rK
Force
r^
R
V(R)^0
a
Repulsive
Attractive
nd^
a^
a
U^ n-^
U^ n^
U^ n+
l
..
1
1
(^
2
)
n^
n
n^
n
mu
K u
u^
u
^
^
^
^
^
^
^
^
^
^
0
0
0
0
1
1
2
e^
e^
2
e^
e
n^
n^
n^
n
i kx
t^
i kx
t^
i kx
t^
i kx
t
m^
K^
A^
A^
A
A
^
^
^
^
^
^
^
kna^
(^
k^ n^
a
(^
k^ n^
a
kna
^
^
^
^
^
^
2
e^
e^
e^
2 e
e^
e
ika^
ika
i kna
t
i kna
t
i kna
t
i kna
t
m^
K^
A^
A^
A
A
^
^
^
^
^
^
^
Cancel Common Terms & Get:
2
e^
2
e ika^
ika
m^
K
^
^
^
^
^
^
^
^
^
^
2
e^
e^
2
e^
e
i kna
t^
i kna
ka^
t^
i kna
t^
i kna
ka^
t
m^
K^
A^
A^
A
A
^
^
^
^
^
^
^
^
^
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-^ Mathematical Manipulation
Solution to the NormalMode Eigenvalue Problem for the monatomic chain.
-^ The
physical significance
of these results is that, for the monatomic
chain, the only allowed vibrational frequencies
ω^
must be related to
the wavenumber
k = (
π/λ
)^ or the wavelength
λ^ in this way.
-^ This result is often called the
“Phonon Dispersion Relation
”^ for
the chain, even though
these are
classical
lattice vibrations
&
there are
no
(quantum mechanical
)^ phonons
in the classical theory.
-^ After more manipulation,
this simplifies to
-^ The maximum allowed frequency is:
-^ We started from the Newton’s 2
nd^ Law equations of motion for
N^
coupled
harmonic oscillators
.^ If one atom starts vibrating, it does not continue with
constant amplitude, but transfers energy to the others in a complicated way.That is, the vibrations of individual atoms are not simple harmonic because ofthis
exchange of energy among them.
-^ On the other hand, our solutions represent the oscillations of
-^ As we already said, these are called the
Normal Modes
of the system. They
are a
collective property of the system
as a whole & not a property of any of
the individual atoms. Each mode represented by
ω(k)
oscillates independently
of the other modes.
Also, it can be shown that the
number of modes is the
same as the original number of equations N.
Proof of this follows.
4
sin
2 K^
ka m
p a Nk k Nap
p Na
^
2
2
To establish
which wavenumbers are possible
for the one-dimensional
chain, reason as follows: Not all values are allowed because of periodicity.In particular, the
th n atom is equivalent to the (N+n)
th^ atom
. This means
that the assumed solution for the displacements:must satisfy the periodic boundary condition:This, in turn requires that
there are an integer number of wavelengths
in
the chain. So, in the first BZ,
there are only N allowed values of k.
^
0
exp n^
n
u^
A^
i kx
^
^
^
Na^
n N n^
u u^
Points
A and C
have the same
frequency
& the
same atomic displacements.
It can
be shown that the
group velocity
v^ = (dg^
ω/dk)
there is negative, so that a
wave at that
ω^ & that
k^ moves to the left.
The
green curve
(below) corresponds to
point B
in the
ω(k)
diagram. It has the
same frequency & displacement as points A and C
, but
v^ = (dg^
ω/dk)
there is
positive, so that a wave at that
ω^ & that
k
moves to the right.
Points
A & C
are symmetrically
equivalent; adding a multiple of
^2 π/a
to^ k
does not change either
ω^ or
v^ ,g^
so point
A^ contains no physical information that is different frompoint
B. T^ The points
k = ±
π/a
have special
significance
x
2
2 n n^
n^
n k^
a ^
^
^
^
Bragg reflection occurs at k= ± n
π/a
2
2 4 sin
ka 2
m^
K ^
u n
x
a u n
k
K V
K m (^2) /^ s k
C
A B 0 ω(k)
π (dispersion relation) /a 2
π/a 0
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k = (
/a) = (
/
);^
= 2a
k^
0;^
^
Group Velocity, v
in the 1g
st^ BZ
-^ This means that a wavewith
λ^ corresponding to a zone edgewavenumber
-^ That is, it must be a^ Standing Wave!
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