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ECE 407: Homework 7 Solutions (By Farhan Rana)
Problem 7.1
a) The answer follows from elementary vector calculus result that that the gradient of any function is
perpendicular to the surface of constant value of the function. In the present case, the velocity is related to
the gradient of the energy function,
()
()
k
E
k
v
k
c
r
h
r
r
r
∇
=
1
And therefore the velocity must be perpendicular to the surfaces of constant energy.
Problem 7.2
a) First note that:
B
k
B
k
r
r
r
r
.

=
Now take the dot product on both sides of the crystal momentum equation with the magnetic field to get:
()
[]
0
0
.
0
.
.
.

=
⇒
=
⇒
=
⇒
×
−
=
dt
k
d
dt
k
B
d
dt
k
d
B
B
k
v
B
e
dt
k
d
B
c
r
r
r
r
h
r
r
r
r
r
r
h
r
b)
( )
( )
()
()
( )
()
()
()
()
[ ]
[ ]
0
.
.
1
=
×
−
=
∇
=
B
t
k
v
e
t
k
v
dt
t
k
d
t
k
E
dt
t
k
dE
c
c
c
k
c
r
r
r
r
r
r
h
r
h
r
r
c) The complete argument follows in two steps:
i)
Since the energy of the electron remains unchanged and equal to its initial value
o
E
, the
motion in kspace of the electron must be confined to a constant energy surface
corresponding to the initial energy
o
E
of the electron.
ii)
Since the component of its crystal momentum parallel to the magnetic field also remains
unchanged and equal to its initial value

k
r
, the motion in kspace of the electron must also be
confined to the plane in kspace on which the components of all crystal momenta in the
direction of the magnetic field is

k
r
(convince yourself that this later condition defines a
plane in kspace that is perpendicular to the magnetic field).
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 Spring '08
 RANA

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