Answer Set 4
1.
K
M
J
M
un
K
vn
M
J
u n+1
M
K
vn+1
The energy of this system is
U=
1
2
n
[J(vn un )2 + K(un+1 vn )2 ].
The equations of motion come from dierentiating U:
M un = force on atom at un =
U
1
= [2J(vn un ) + 2K(un vn1 )]
un
2
M vn = force on a
Answer Set 9
Physics 140B
1. Cylinder b) is aligned with the magnetic eld and hence distorts the eld less. This means that
for a given applied eld, the maximum total eld (which occurs near the sides of the sample)
is smaller for sample b), so that cylinde
Answer Set 3, Physics 140A
1. Since the bi s are primitive vectors of the reciprocal lattice, the unit cell volume is |b1 (b2 b3 )| =
2
( |a1 (a2 a3 )| )3 |(a2 a3 ) [(a1 a3 ) (a1 a2 )]|. From the BAC-CAB rule, with a1 a3 as A, a1 as
B, and a2 as C, the ex
Answer Set 8
Physics 140B
1. a) I didnt specify the saturation magnetization, but Ill assume that the hysteresis loop is almost closed
at 3 Tesla, which means the remanent magnetization is about 2.1 Tesla.
b) Loop starts at zero, is fairly at at rst but g
Answer Set 7
Physics 140B
G&S 9.9: From (9.23), Msat = ngB J = nm = n(4B ). The numbers in the problem show
n = 8/(0.839 109 )3 /m3 , or 1.4 1028 /m3 . This gives Msat = 5.0 105 A/m, close to the
observed value.
1. Im going to switch signs from the proble
Problem Set 9
Physics 140B
Due Wednesday June 9, 2004: no late homework accepted
1. Two identical superconducting cylinders (i.e., same shape, same size, same material) are placed
in a magnetic eld as shown. As the magnetic eld increases, the superconduct
Problem Set 8
Physics 140B
Due Friday May 28, 2004
1. a) Sketch the hysteresis loop from ramping a eld up to 3 Tesla and down to -3 Tesla, if the
loop has squareness 0.7 and coercive eld 1 Tesla. You can assume that the sample has
been prepared by going a
Answer Set 6
Physics 140B
G&S 9.5: As x 0, with the expansion on p. 281 coth
J+1
3J x.
From this, = (ngB J) J+1 gB Jo =
3J
kB T
3n
G&S 9.11: From equation (9.59), P = o 2 ( 2EF ) =
B
P
(9.25), the ratio is Curie =
the ratio is about 103 .
1
1
+ , BJ (x)
Answer Set 4
Physics 140B
1. a) The electric eld runs perpendicular to the interface; call it the x direction. Its magnitude
x
is zero except for dp x dn and is given by E(x) = dp dx 4 (x ). Hence E(x) =
4 Na (x + dp ) for dp x 0 and E(x) = 4 (Na dp Nd x
Problem Set 5
Physics 140B
Due Friday May 7, 2004: no late homework accepted
1. a) Sketch the output voltage as a function of input voltage, for -1 V Vin 1 V.
R
n
p
V
out
V in
p
n
R
b) Repeat a) if the upper resistor has resistance 3R instead of R.
2. a)
Problem Set 7
Physics 140B
Due Friday May 21, 2004
Do G&S 9.9, plus the following:
1. (after Kittel problem 15.5)
Consider a small spherical single-domain particle of a uniaxial ferromagnet, with magnetic
eld applied parallel to the easy axis. Take UK = K
Problem Set 6
Physics 140B
Due Friday May 14, 2004
Do G&S 9.5 and 9.11, plus the following:
1. a) Following Hunds rules, what are the total S, L, and J for a holmium atom, a Ho3+ ion, a
yttrium atom, and a Y3+ ion? You may work in units where = 1.
h
b) Fo
Answer Set 5
Physics 140B
1. a) This time Vout = |Vin |, since the voltage measurement has positive in opposite directions
for the two resistors. (Im dening the upper output terminal as negative for the output
voltage, so that the output voltage is positi
Answer Set 3
Physics 140B
m
G&S 11.2: The change in is = 3 kB T ln mhds . Using the conversion that kB T at 300 K is 25 meV
4
eds
gives = 11 meV. (The chemical potential goes down because the conduction band is broader.)
11.8
G&S 11.3: Donor orbit radius
Answer Set 1
Physics 140B
1. a) Constant-energy curves are concentric spheres. Note that dE = 3k 2 dk. The k-space volume
between spheres at energy E and E + dE is 4k 2 dk = 4( dE . The k-space volume for
3
3
each allowed electron state is (2) (from the s
Answer Set 2, Physics 140B
1. a) Dual lattice atoms at the centers of each triangle in the original lattice gives a honeycomb lattice.
b) There is a path across a random triangle lattice if and only if there is no path from top to
bottom on the correspond
Problem Set 2
Physics 140B
Due Friday, April 16, 2004
1. a) What lattice is dual to the triangle lattice for bond percolation? You can just sketch the
lattice if you dont know its name.
b) What does the duality say about the bond-percolation thresholds of
Problem Set 1
Physics 140B
Due Friday April 9, 2004
1. The concept of density of states, from last term, is crucial for connecting theoretical calculations
with actual measurements. This question gives you a chance to review it.
a) To nd the density of st
Problem Set 4
Physics 140B
Due Friday April 30, 2004
1. a) Find the electric eld for the charge distribution near a pn junction shown in Figure 11.28
and discussed in class. Assume a charge density eNa which extends a distance dp on
the left (p) side and