Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

2.6. Properties of C(W~K)
a) Signs, limiting values
According to (2.43) the response function C is defined as
where f satisfies
f" (2) =' I K 1- ~ iwu0u (z) 15 (z)
C has the dimension of a lenqtl~.. Let
-i+
C = g - ih or C = [~[e
Then g - > 0, h - > 0 or 0 - c - < s/2
(2. GO)
Proof: Talce the comp1.e~ conjugate of (2.59) , mu1 tip1.y by f and in-
-
tegrate over z. Integration by parts yields
-f(o) frX(0) = I C [fl
m
0
[2-i-(~2--iw~00)
[fl21dz.
Division by 1 f' (0) 1 leads to the result.
The limiting values of C for w -* 0 and w + CQ are
1
f ; tanh (KH) for W+O
I I for LO+ m -
Ji.wuocr (0)
- In (2.62a) H is the depth of a possib1.e perfect conduc-tor. If
absent, then 1-1 i. 'm, C I jk.
Proof: For w=O the solution of (2,59) vanislling at z=X is
f - sinlz~c(H-z), - whence (2.G2a). - For high frequencj.es f tends t(
.---
'the solutio~l for a uniform ha1.f-space, i.e. f -. exp{fidll,~)~ yield:
(2.62b). This limit is attainecl, if the penetration depth for a
uniform halfspace with u - ~'(oJ,
V
is small co~npared with the scale leng-Lh I/K of the external fie12
and the scale l.engt11 lu(o)/ol (0) 1 of corducti-vity variatiol?.