Exam 3 Solid State Physics/Physics of Materials
May 9, 2014
This is a take home exam. You have three hours to complete the exam. You may use
your notes and your book. Please return the completed exam to my office (S2-258) by
5pm Friday May 16.
1. The ther
Homework #3 Solutions:
1. a The average energy of an atom in an ideal gas is simply given by:
Eatom =
3
k BT
2
For N atoms, the total energy is just N<Eatom> = E = N E =
3
NkBT
2
2
E
3
Since the ideal gas law gives: PV = NkBT
2E
we can write P =
3V
b. We
Homework #4 Solutions:
1.a For the Hexagonal Space Lattice with primitive translation vectors:
a1 =
3a
a
3a
a
x + y; a 2 =
x + y; a 3 = cz
2
2
2
2
The volume of the primitive cell is:
Vcell = a1 (a 2 a 3 )
3a
a
From the definitions above, a 2 a 3 =
x +
Solid State Physics Homework #7 Solutions
Spring 2014
1. The equations of motion for the two atoms in the nath primitive cell are:
U harm
= 2Ku1 (na) + Ku2 (na) + Ku2 (n 1)a)
u1 (na)
U harm
!
M 2u2 (na) =
= 2Ku2 (na) + Ku1 (na) + Ku1 (n + 1)a)
u2 (na)
!
Homework #9 Solutions:
1. The built-n voltage in a p-n junction is given by:
np
e = Eg + kBT ln c v
N c Pv
Eg
but N c Pv = n e
2 kBT
i
so that:
np
n p
e = Eg + kBT ln c v = kBT ln c 2 v .
N c Pv
ni
nc is determined by the donor concentration, ND
Homework #6 Solutions Spring 2014
1.a. For free electrons in 2 dimensions, the density of states in k-space is (accounting for
2
A
spin): g(k)=
2 =
2 2
2
L
(
)
If we place m electrons on each lattice site of a square 2D lattice with side, a, we will
have
Homework #2 Solutions:
1. a If this fictitious Lithium solid has a density of 0.5 g/cm2 (note this an incredibly
large mass density!), the number density of atoms is:
n!
atoms
g
atoms 1 mole
atoms
= 0.5 2 x 6.022x10 23
x
= 4.33x10 22
2
cm
cm
mole
6.941g
c
Homework #1 Solutions:
1. a. From statistical thermodynamics, we know that (assuming no degeneracy) the
probability of finding an atom with energy Ei is given by:
Ni
e
= P(Ei ) =
N
N
Ei
kB T
!
"e
!
(1)
Ej
kB T
j =1
where the sum in the denominator is the
1. a. The density of states in k space for phonons in d dimensions is just:
L
g(k) =
2
d
d
L
g(k)dk = Bg(k)k dk = Bk d1dk
2
d1
(1)
We want to find g( ) and we know that for low frequencies, we have an approximately linear dispersion
relationship:
Homework #5 Solutions
Spring 2014
1.a fcc lattice:
The distance between nearest neighbors in the fcc lattice is just
spheres that just touch will be half this value:
a 2
. Thus the radius of
2
a 2
. The volume of such a sphere is
4
3
4 3 4 a 2
. There are