Problem Set 9
Physics 140A
Due Tuesday March 16, 2004
1. a) Draw the band structure of a one-dimensional Fermi gas in the rst Brillouin zone. Include
h
2
all curves lying completely below E = 30( )2 2m .
a
2
h
b) Let o = ( )2 2m . Describe how the band st
Problem Set 8
Physics 140A
Due Tuesday March 9, 2004
Do Gersten & Smith problems 7.5 and 7.8, plus the following:
1. A thermocouple, used to measure temperatures, consists of two strips of two dierent metals,
connected together as shown. The voltage is me
Answer Set 9, Physics 140A
1. a) Five portions of the free electron parabola are needed, with the last two having highest energy
h
2
E = 25 2m ( )2 at the Brillouin zone edge.
a
b) There are four degeneracies in the portion of the free electron band struc
Answer Set 8
Physics 140A
3
1
1 k
G&S 7.5 For the density of states per unit volume, (EF ) = 22 ( 2m )3/2 EF = 22 EF =
h
2
F
1
3n
2
2 E (3 n) = 2E . Here n is the number of electrons per volume. With z the number
2 F
F
3n z
3z
of electrons per atom, the d
Answer Set 7
Physics 140A
p2
p2
f
i
1. a) Energy conservation is Ei = Ef + h(k), or 2M = 2M + h(k), where ps represent neutron
momenta and M is the neutron mass. Using momentum conservation, we can rewrite
p2
2
h
i
this as 2M (pi2Mk) = h(k). To solve gra
Problem Set 7
Physics 140A
Due March 3, 2004
1. a) Assume that a one-dimensional crystal with lattice constant a has exactly the monatomic
chain dispersion relation. For a neutron incident on the crystal (with wave vector parallel
to the crystal), show ho
Problem Set 6
Physics 140A
Due Tuesday February 24, 2004
Do Gersten & Smith, problems 5.2, 5.4 (start by calculating the equilibrium lattice constant, then expand
the potential around that atomic spacing), and 5.6, plus the following:
1. Verify that in th
Answer Set 6
Physics 140A
G&S 5.2 First get the density of states. The constant- curves are circles of radius vs , and d = vs dk.
Then the area with frequency between and + d is 2 vs d . Dividing by the area per allowed k,
vs
(2)2
A
A
(where A is the area
Problem Set 4
Physics 140A
Due Tuesday February 10, 2004
I recommend a computer or graphing calculator for problems 2 and 5, but it isnt strictly necessary.
1. In class we looked at normal modes of a linear chain with a basis, assuming two alternating
mas
Problem Set 5
Physics 140A
Due Tuesday February 17, 2004
This will count as a half-problem set only. Part of its purpose is to get the 130A and 140A due dates onto
dierent days.
2
1. The bulk modulus of a material, V VU , measures how compressible it is.
Answer Set 5
Physics 140A
1. My denition of bulk modulus was missing a factor of V ; it should be V
2U
.
V 2
a) Since springs are 1D creatures, we rst need to relate a 3D compression to what happens to the
d
d
d
springs. Note that dL = dV dV = 3V dV , whe
Problem Set 2
Physics 140A
Due Thursday January 22, 2004
Do Gersten & Smith, problem 2.2, and the following:
1. We wrote the cohesive energy of a crystal by summing the interactions between all pairs of
atoms in a crystal. In reality, of course, an atom i
Problem Set 3
Physics 140A
Due Tuesday February 3, 2004
1. From the bi s (gi s in G&S), show that the unit cell volume in the reciprocal lattice is
where V is the unit cell volume of the real lattice. (Use the BAC-CAB rule.)
(2)3
V ,
2. Gersten & Smith, p
Answer Set 2
Physics 140A
G&S 2.2: These lattice constants give info on the dierences between pairs of ionic radii. Subtracting a(NaCl) from a(KCl) gives 0.066 nm, which should be the dierence between the
K and Na radii. Similarly subtracting a(NaF) from
Answer Set 1
Physics 140A
Gersten 1.8: Consider a triangle of spheres, with a fourth sphere resting above the center of the triangle.
c
The center of the fourth sphere is 2 above the plane of the other three, and together the four form a
regular tetrahedr
Problem Set 1
Physics 140A
Due January 15, 2004
Do Gersten 1.8, plus the following:
1. Do Gersten 1.5, calculating out to tenth-nearest neighbors. On the course web page there is a C+ template
which includes a routine you may nd useful. (You are not requi