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 For large
, and , 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, , the helicon frequency is esu. Taking a typical metallic electron density of 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 This gives and we get and
In order for a solution to exist we need This gives Since we need Finally, and and and hence we need to be below the bulk Plasmon frequency. 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
This gives for a solution, with small. or Also becomes very large, and also becomes very large. Hence the solution is localized at the surface. Using the solution for we get
has elliptic polarization in the metal. and hence
. The wave is circularly polarized in vacuum and 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 For case of
, then we have , Problem 5 The Sommerfeld expansion suggests that: The correction is: For 3D free electron gas, For 2D free electron gas,