Jackson 7.22 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
UMASS LOWELL
C.S.BAIRD
PROBLEM: Use the Kramers-Kronig relation to calculate the real part of a(w), given the imaginary part of (w) for positive w as
(a) J(e1e)=X[0(w-w)-0(w-w)],W>wi>0
m&y
(b) 3(e1e)= (w2-w2)+yw2
SOLUTION: (a) Inserting the first expression for the imaginary part:
R(e(w)/e)=I+3 TT%
mp.
zm-zim
mp
TT
z3-z13
dw'- TT 3-233 TT
3
mp
[z1n(w2-w1)
T2
R(c(w)/e)=1+[1n(|w3-w1)-In(|}-w|)]
We can plot these to see how they compare to the harmonic model (as shown on the next page)
6.0 - 4.0 - 2.0 - Re(e/e0) 0.0 -2.0 -4.0 -
1
3
5
7
9
11
13
15
3
2.5 2.0 - 1.5 - Im(/0) 1.0 - 0.5 - 0.0 T -1
1
3
5
7
9
11
13
15
3
This looks quite similar to the results from the harmonic model. The difference is that the flat top of the imaginary part of the dielectric constant leads to an extended region of anomalous dispersion in the real part. Also the unrealistically steep walls of the imaginary part leads to infinite peaks in the real part.
Ayw
2 we have:
R(e(w)/eo)=1+2xp mp TT ((w2-w2)2+y2w2)(w2-w3)
Let us use partial fraction decomposition to break up the fraction:
W 12
A
B
((w2-w2)2+y2w2)(wi2-w2)((w2-w2)2+y2w2) (zm-zm)
(w2-w2)A-w2+((w-w2)+y2w2)B=0
This must be true for all w' so we can set w' = o to pick out B:
W2
(w2-w2)2+yw2
Plug this back in and solve for A:
w-w2w2 (w-w)+yw
So that finally we have:
3z13-43
((w2-w2)+yw2)(w2-w2)((w2-w2)+yw2)[(w3-w2)2+yw'
We can further decompose the first fraction in the brackets:
wg-w i2 w2 A B (w-w'2)+yw2 (w-w'2+iyw') (w-w'-iyw')
Multiply out, choose B =w? -- A , and solve for A and B. This leads to the expansion:
wg-w 2 w2 w? (23-13).3 1 3-233 1 (w3-w'2)2+x2w' L 2 2iy I(w-w'+iyw') [ 2iy I(w2-wi_iyw')
Plugging this back in, the whole integrand in completely expanded form becomes:
((w-w)2+y2w2)(w2-w2) ((w2-w2)2+y2w2) (zm-1m),m 1 zm-3).m 2iy m) 2iy (w-w-iw
This may seem like a lot of work that just leads to a less useful form, but we must do this to solve the integral analytically.
3R(e(w)/eo)=I+2Xy TT ((w2-w2)+y2w2) /
(w'2-w-iyw')
W
.mp
mp o w'2-w+iyw
2ix
iV
2
3-3
m+m
[.m p