Problem(1((A&M1.4) - University of Floridamaslov/phz6426/hw/HW2_solution.pdf · HW2_solution...
Transcript of Problem(1((A&M1.4) - University of Floridamaslov/phz6426/hw/HW2_solution.pdf · HW2_solution...
Problem 1 (A&M 1.4)
, , .
a) Seek steady-‐state solution of the form: ,
.
,
,
.
,
,
The solution is: where
The current density is: where
b) From Maxwell equations,
Look for a solution of this form
Plugging in where
c) for polarization , we have . For the plot, let
Below is plot of as function of , assuming that .
Below is plot of as function of , assuming that , and .
(two plots are just for different y-‐axis range)
Now assuming , and ,
For large , one can rewrite as
It is positive for , real solutions for exist.
For small but positive , one can rewrite it as
If is larger and term is ignored, then it is positive for , and real solutions for exist.
d) For (but still >0)
=1cm, =10 kilogauss. cm/s, esu. Taking a typical metallic electron density of
, the helicon frequency is
Problem 2 (A&M 1.5)
a)
For we have or
Similarly for
For we have
Similarly for
Finally the continuity gives for and for
The first two equation give
where we used the continuity equations in the last step.
Subtracting the third and fourth equation yields
And using the relation between and we get
This gives and
In order for a solution to exist we need and hence we need to be below the bulk Plasmon frequency.
This gives and
Since we need and
Finally, which is positive.
b) If we have and hence , or
Below is plot of , assuming and
c) If the second factor needs to be large and we need for a solution, with small.
This gives or
Also becomes very large, and also becomes very large. Hence the solution
is localized at the surface.
Using the solution for we get and hence . The wave is circularly polarized in vacuum and
has elliptic polarization in the metal.
Problem 3
(a)
and
(b)
When near top of the band,
Let then
and
Similarly:
Problem 4
Suppose :
Case of 1D:
Case of 2D:
Case of 3D:
If we take and , then we have
For case of ,
Problem 5
The Sommerfeld expansion suggests that:
The correction is:
For 3D free electron gas,
For 2D free electron gas,