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ECE215A/Materials206A
Fundamentals of Solids for Electronics
E.R. Brown/Winter 2008
1
NOTES 7: Energy of Lattice Waves and their Quantization (Phonons)
Classical Analysis
So far we have addressed only the
mechanical behavior
of lattice waves – the description of
the wave phenomenology consistent with Newton’s law and the kinematics of the crystal lattice.
This led to two different types of plane waves, one (the longitudinal) polarized along the direction of
wave propagation, and the other (transverse) polarized in one of the lateral directions in the
perpendicular plane.
Now we take the analysis one step further to understand the wave energy.
In
addition to being essential to the statistical mechanics, the energy also leads to a quantum
mechanical description of lattice waves and the introduction of the particle
dual
of lattice waves –
the
phonon
.
We start with a monatomic linear lattice of atomic mass m, spacing a, and nearest neighbor
interaction (spring constant) C.
We consider the longitudinal wave
()
cos
rA
t
s
k
a
s
ω
∆=
−
.
The
total energy of the wave is the energy of each atom summed over all atoms.
From mechanics we
know that the instantaneous energy of a massspring system has a kinetic energy term associated
with each mass and a potential energy term associated with each spring.
The kinetic energy depends
only on the velocity of the individual masses in motion.
The potential term depends only on the
force constant and the displacement of the atoms from their equilibrium positions.
The kinetic
energy associated with the
sth
mass is
2
11
2
22
dr
s
KE
M
M
ss
dt
υ
∆
⎛⎞
→=
⎜⎟
⎝⎠
The potential energy associated with the
sth
spring is
PE
C
r
r
C
r
r
s
s
s
→
∆
−∆
=
∆ −∆
++
By summing over all atoms we get the total energy:
2
2
1
s
Ut
M
C
r
r
dt
∆
=+
∆
−
∆
+
∑∑
(instantaneous)
(1)
For longitudinal wave
cos
t
s
k
a
s
−
, we have
sin
s
A
ts
k
a
dt
ωω
∆
=−
−
and
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Fundamentals of Solids for Electronics
E.R. Brown/Winter 2008
2
()
(
)
{ }
2
11
22 2
2
sin
cos
cos
22
S
U t
M
A
t ska
CA
t ska
t ska ka
ωω
ω
⎡
⎤
=−
+
−
−
−
−
⎣
⎦
∑
The last term has the form
2
cos
cos
,
with
t
ska
ka
αβ
α
α ω
β
⎡⎤
−−
= −
=
⎣⎦
.
So we can use the
trigonometric identity
( )
cos
cos
cos
sin
sin
βα
−=
+
2
2
sin
cos
cos
1 sin sin
S
Ut
M A
C
A
ωα
=+
−
+
⎢⎥
∑
( )
1
1
2
2
2
2
sin
cos
cos
1
2cos
cos
1 sin sin
sin
sin
2
2
s
MA
C
A
+−
+
−
+
=
∑
To get the average, we integrate over the period of wave,
τ
21
,
o
UU
t
f
dt
τ
π
≡
==
∫
2
sin
;
cos
;
2 sin
cos
sin 2
0
oo
o
dt
dt
dt
dt
ττ
ααα
=
=
∫∫
∫
∫
So,
12
1
2
2
2
cos
1
sin
2
cos
2cos
1 sin
44
2
1c
o
s
42
UM
A
C
A
A
C
A
A
C
A
ωβ
⎡
⎤
−
+
⎢
⎥
⎣
⎦
−
+
+
−
for linear monatomic lattice we have
2
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 Winter '08
 Brown

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