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Apr 25, 2002
Notes for Lecture
Notes for Lecture #35
Synchrotron Radiation
For this analysis we will use the geometry shown in Fig. 14.9 of
Jackson. A particle with charge q is moving in a circular trajectory
with radius r and speed v. Its trajectory as a function of time t
is given by
|
rq(t) = rsin(vt/r) |
^
x
|
+ r(1-cos(vt/r)) |
^
y
|
. |
| (1) |
Its velocity as a function time is given by
|
vq(t) = v cos(vt/r) |
^
x
|
+ v sin(vt/r) |
^
y
|
. |
| (2) |
The spectral intensity that we must evaluate is given by the expression
(Eq. 14.67 in Jackson)
|
|
d2I
dwdW
|
= |
q2 w2
4 p2 c
|
|
ê
ê
|
|
ó
õ
|
¥
-¥
|
|
^
r
|
×( |
^
r
|
×b)ei w(t - [^(r)]·rq(t)) dt |
ê
ê
|
2
|
. |
| (3) |
After some algebra, this expression can be put into the form
|
|
d2I
dwdW
|
= |
q2 w2 b2
4 p2 c
|
{ |C||(w)|2 + |C^(w)|2 }, |
| (4) |
where the amplitude for the light polarized along the y-axis is given by
|
C||(w) = |
ó
õ
|
¥
-¥
|
dt sin(vt/r) ei w(t -[(r)/ c] cosqsin(vt/r)) |
| (5) |
and the amplitude for the light polarized perpendicular to [^(y)] and [^(r)]
is given by
|
C^(w) |
ó
õ
|
¥
-¥
|
dt sinqcos(vt/r) ei w(t -[(r)/ c] cosqsin(vt/r)). |
| (6) |
We will analyze this expression for two different cases. The first case, is
appropriate for man-made synchrotrons used as light sources. In this case,
the light is produced by short bursts of electrons moving close to the speed
of light (v » c(1 - 1/(2g2)) passing a beam line
port. In addition q » 0 and the relevant integration times t
are close to t » 0. This results in the form shown in Eq. 14.79
of your text. It is convenient to rewrite this form in terms of a critical
frequency
The resultant intensity is then given by
|
|
d2I
dwdW
|
= |
3 q2 g2
4 p2 c
|
|
æ
ç
è
|
w
wc
|
ö
÷
ø
|
2
|
(1 + g2 q2)2 |
ì
í
î
|
|
é
ê
ë
|
K2/3 |
æ
ç
è
|
w
2wc
|
(1+g2 q2) |
ö
÷
ø
|
ù
ú
û
|
2
|
+ |
g2 q2
1 + g2 q2
|
|
é
ê
ë
|
K1/3 |
æ
ç
è
|
w
2wc
|
(1+g2 q2) |
ö
÷
ø
|
ù
ú
û
|
2
|
|
ü
ý
þ
|
. |
| (8) |
By plotting this expression as a function of w, we see that
the intensity is largest near
w » wc.
The second example of synchroton radiation comes from a distant charged
particle moving in a circular trajectory such that the spectrum represents
a superposition of light generated over many complete circles. In this
case, there is an interference effect which results in the spectrum consisting
of discrete multiples of v/r. For this case we need to reconsider
Eqs. 5 and 6. There is a very convenient Bessel function
identity of the form:
|
e-iasina = |
¥
å
m = -¥
|
Jm(a)e-ima. |
| (9) |
Here Jm(a) is a Bessel function of integer order m.
In our case a = [(wr)/ c] cosq and
a = [vt/( r)].
Analyzing the ``parallel'' component we have
|
C|| = |
c
-iwr
|
|
¶
¶cosq
|
|
ó
õ
|
¥
-¥
|
dt ei w(t -[(r)/ c] cosqsin(vt/r)) = |
c
-iwr
|
|
¶
¶cosq
|
|
¥
å
-¥
|
Jm |
æ
ç
è
|
wr
c
|
cosq |
ö
÷
ø
|
2 pd(w-m |
v
r
|
). |
| (10) |
In determining this result, we have used the identity
|
|
ó
õ
|
¥
-¥
|
dt ei(w- m [v/( r)])t = 2 pd(w-m |
v
r
|
). |
| (11) |
Eq. 10 can be simplified to show that
|
C|| = 2 pi |
¥
å
-¥
|
Jm¢ |
æ
ç
è
|
wr
c
|
cosq |
ö
÷
ø
|
d(w-m |
v
r
|
), |
| (12) |
where Jm¢(a) º [(d Jm(a))/ da]. The ``perpendicular''
component can be analyzed in a similar way, using integration by parts
to eliminate the extra cos(vt/r) term in the argument. The result
is
|
C^ = 2 p |
tanq
v/c
|
|
¥
å
-¥
|
Jm |
æ
ç
è
|
wr
c
|
cosq |
ö
÷
ø
|
d(w-m |
v
r
|
). |
| (13) |
In both of these expressions, the sum over m includes both negative and
positive values of m. However, only the positive values of w
and therefore positive values of m are of interest, and if we needed
to use the negative m values, we could use the identity
Combining these results, we find that the intensity spectrum for this case
consists of a series of discrete frequencies which are multiples of v/r.
|
|
d2I
dwdW
|
= |
q2 w2 b2
c
|
|
¥
å
m = 0
|
d(w-m |
v
r
|
) |
ì
í
î
|
|
é
ê
ë
|
Jm¢ |
æ
ç
è
|
wr
c
|
cosq |
ö
÷
ø
|
ù
ú
û
|
2
|
+ |
tan2q
v2/c2
|
|
é
ê
ë
|
Jm |
æ
ç
è
|
wr
c
|
cosq |
ö
÷
ø
|
ù
ú
û
|
2
|
|
ü
ý
þ
|
. |
| (15) |
These results were derived by Julian Schwinger (Phys. Rev. 75, 1912-1925
(1949)). The discrete case is similar to the result quoted in Problem 14.15 in
Jackson's text. It should have some implications for Astronomical
observations, but I have not yet found any references for that.
File translated from TEX by TTH, version 2.20.
On 25 Apr 2002, 20:31.