Name:
____________________
2.57 Midterm Exam No. 2
Fall, 2004
Take Home Exam
Distributed: Friday, November 19, 7:00 pm
Due: November 24, 2:30 pm
Four questions, each counts for 25 points
Rules
:
(1) You are required to finish these problems independently, without consultation to
other fellow students in the class or anyone else.
Replies to specific questions will be shared with
other students by em
ail.
You can check whatever references you can find. You are on your own honor.
(2) State all the assumptions you made in solving each problem.
1. An fcc crystal has a lattice constant b and one atom at each lattice point.
The phonon
dispersions for transverse and longitudinal phonons are degenerate and are given by the
following relation,
ω
=
2
K
m
sin
ka
2
where a is the equivalent lattice constant of an isotropic crystal that has the same number of
phonon modes as the fcc crystal, and
2
2
4
π
k
2
=
k
x
+
k
y
+
k
2
(k ,k ,k
=
±
2
π
,
±
,...)
z
x
y
z
L
L
The phonon relaxation time
τ
in the crystal is a constant. Answer the following questions:
(a) Relate the equivalent lattice constant “a” to the lattice constant of the fcc crystal “b”.
(b) Derive an integral expression for the phonon specific heat (per unit volume) of the crystal
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 Fall '04
 GangChen
 Heat, Heat Transfer, Snell's Law, Fundamental physics concepts, Thermal conductivity

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