Date: Nov 13, 1998 (Friday)
Time allowed: 50 minutes.
Answer all questions.
1. Surface plasmons. Consider a semi-infinite plasma on
the positive side of the plane z = 0. The negative side (z<0) is vacuum.
Given solution of Laplace's equation 2f=0
in the plasma is f i(x,z)
= A cos kx e-kz and that of vacuum is f0(x,z)
= A cos kx ekz. (a) Write down all boundary conditions of electric
fields at the boundary z=0. (b) Show that the frequency w
s of a surface plasma oscillation is 
, where w p is the plasma frequency.
(c) The electron concentration in a copper sample is 8x1022
cm-3, mean free path is ~400, and the Fermi velocity is 1.6x108
cm s-1. Mass of an electron is 9.1110-28 g. Estimate
the plasma frequency at the surface of this sample.
Solution: HTML Format PDF Format
2. Conductivity for free electrons at high frequency.
(a) Conductivity is defined as = j/E. Since j and E are not necessary in
phase, can be complex. Let the conductivity of a metal be s
(w) =s '(w)
+ is"(w), where s'(w)
and s"(w) are the
real part and imaginary part of the function respectively. Use Kramers-Kronig
relation, show that at high frequency , 
.
(b) At very high frequency, the electrons in the metal are essentially
oscillated by the electric field without any drifting. Write down the equation
of motion of an electron and then show that 
.
(c) Further prove that 
.
(d) The electron concentration in a copper sample is 8x1022
cm-3, mean free path is ~400, and the Fermi velocity is 1.6108
cm s-1. Give a measure on the meaning of "high frequency" for
the above results to be valid.
Hint. In case if you forget Kramer-Kronig relation, you can derive
it by calculating the integral 
,
and compare real and imaginary parts. Assume s(w)
has no pole in the upper half of the complex plane, and s'(w)
is even and s"(w)
is odd.
Solution: HTML Format PDF Format