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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 27: Continue reading Chap. 14 – Synchrotron radiation Radiation from electron synchrotron devices Radiation from astronomical objects in circular orbits. Radiation from charged particle in circular path. z. r. - PowerPoint PPT Presentation
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PHY 712 Spring 2014 -- Lecture 27 104/04/2014
PHY 712 Electrodynamics10-10:50 AM MWF Olin 107
Plan for Lecture 27:
Continue reading Chap. 14 – Synchrotron radiation
1. Radiation from electron synchrotron devices
2. Radiation from astronomical objects in circular orbits
PHY 712 Spring 2014 -- Lecture 27 304/04/2014
Power distribution for circular acceleration
y
z
x
b
b
.r
q
42
3
2
/
5
22222
/
5
22
3
2
ˆ1
1ˆˆ1
4
ˆ1
ˆˆ
4
v
Rβ
βRRββ
Rβ
ββRR
c
q
d
tdPdtP
c
q
c
q
d
tdP
rrrr
cRtt
cRtt
rr
r
r
Radiation from charged particle in circular path
PHY 712 Spring 2014 -- Lecture 27 404/04/2014
Spectral composition of electromagnetic radiation -- continued
2
/ˆ
2
222
ˆˆ4
:integral following on the
dependsintensity spectral theclears,dust When the
cttirr
rqretdtc
qI Rrβrr
PHY 712 Spring 2014 -- Lecture 27 504/04/2014
Synchrotron radiation light source installations
Synchrotron radiation center in Madison, Wisconsin
Ec=0.5 GeV and 1 GeV; lc = 20 Å and 10 Å
http://www.src.wisc.edu/
PHY 712 Spring 2014 -- Lecture 27 704/04/2014
http://www.bnl.gov/ps/
Brookhaven National Laboratory – National Light Source
Ec=0.6 GeV; lc = 20 Å
PHY 712 Spring 2014 -- Lecture 27 804/04/2014
y
z
x
b
b
.r
q
2
/ˆ
2
222
ˆˆ4
:iprelationshintensity Spectral
cttirr
rqretdtc
qI Rrβrr
y
x
r
sinˆcosˆˆ
:choose e,conveniencFor
/sinˆ/cosˆ
/cos1ˆ
/sinˆ
zxr
yxβ
y
xR
rrr
r
rrq
vtvtt
vt
vtt
Top view:
PHY 712 Spring 2014 -- Lecture 27 904/04/2014
y
z
x
b
b
.r
q
sinˆcosˆˆ
:choose e,conveniencFor
/sinˆ/cosˆ
/cos1ˆ
/sinˆ
zxr
yxβ
y
xR
rrr
r
rrq
vtvtt
vt
vtt
/cossin/sinˆˆ
cosˆsinˆ ˆ
:directionson polarizati orthogonal twoanalyze tochoose
can then wedirection; ˆ in the is vector Poynting he t
zone,radiation in the shown that previous have that weNote
||
||
rr vtvt
εεβrr
zxεyε
r
PHY 712 Spring 2014 -- Lecture 27 1004/04/2014
||
||
ˆ ˆ ˆ sin cos
ˆ ˆ
sin / sin cos /r rvt vt
ε y ε x z
r r β
ε εy
z
x
b
b
.r
q ||ε
ε
2 2 2 2( ( )/ )
2
2
ˆ
2 2 22 2
2
( cos sin( / ))
( cos sin( / ))
( )e4
| ( ) | | ( ) |4
( ) sin( / )
ˆ
e
( ) sin cos( / )e
ˆ qi t t c
i t vtc
i t vtc
d I qdt
d d c
d I qC C
d d c
C dt vt
C dt vt
r Rr r
PHY 712 Spring 2014 -- Lecture 27 1104/04/2014
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 electro2
ns moving close to the speed
of light passing a beam line port. In addition, because
o
( (1 1/ (2 ))
radiation ports, 0, and the relevant integration
ti
f the de
mes a
sign of the
re
Tclose to 0.
v c
t t
3
his results in the form shown in Eq. 14.79
of your text. It is convenient to rewrite this form in terms of a critical
frequ3
en y . 2
c c
c
22 32 2 22 2 2 2 2 2
2/32
232 2
2 2 21/32 2
3(1 ) (1 )
4 2
(1 )1 2
c c
c
d I qK
d d c
K
PHY 712 Spring 2014 -- Lecture 27 1204/04/2014
Modified Bessel functions
0
331
23
3/2
0
331
23
3/1 sin 3 cos 3 xxxdxKxxdxK
2/122
2/3223
32
3222
2
2
1 and 1
3 where
3
1
2
3
61
2ˆ
2
11 ,0 ,0 oflimit In the
/sincosˆ
r
rrrqr
r
rrrqr
tcx
c
xxtct
tt
cvt
vtc
ttt
Rr
Rr
Exponential factor
Some details:
PHY 712 Spring 2014 -- Lecture 27 1304/04/2014
c
c
By plotting the intensity as a function of , we see that the
intensity is largest near ω ω . The plot below shows the
intensity as a function of ω/ω for γθ=0 , 0.5 an : d 1
2d I
d d
c ω/ω
22 32 2 22 2 2 2 2 2
2/32
232 2
2 2 21/32 2
3(1 ) (1 )
4 2
(1 )1 2
c c
c
d I qK
d d c
K
γθ=0
γθ=0.5
γθ=1
PHY 712 Spring 2014 -- Lecture 27 1404/04/2014
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 consistingof discrete multiples of v/r. For this case we need to reconsider the analysis. There is a very convenient Bessel function identity of the form:
sin Here is a Bessel function of integer e ( )e order .
In our case
( )
co s . and
mia im
mm
J mJ a a
vta
c
( cos sin( / )) ( cos sin( / ))( ) sin( / )e e
c
=
os
cos 2 ( ).cos
i t vt i t vtc c
mm
cC dt vt dt
i
c vJ m
i c
PHY 712 Spring 2014 -- Lecture 27 1504/04/2014
Astronomical synchrotron radiation -- continued:
( cos sin( / ))( ) sin cos( / )e
tan
Similarly:
=2 cos ( )./
i t vtc
mm
C dt vt
vJ m
v c c
( )
e 2 ( ).
) 2 cos ( ),
( )where (
Note t :
)
hat
(
vi m t
mm
mm
vdt m
vC i J m
c
dJ aJ a
da
PHY 712 Spring 2014 -- Lecture 27 1604/04/2014
Astronomical synchrotron radiation -- continued:In both of the expressions, the sum over m includes both negative and positive values. However, only the positive values of w and therefore positive values of m are of interest. Using the identity: theresult becomes:
( ) ( 1) ( ),mm mJ a J a
2
2 22 2 2 2
2 20
tan( ) cos cos .
/m mm
d I
d d
q vm J J
c c v c c
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. For information on man-made synchrotron sources, the following web page is useful: http://www.als.lbl.gov/als/synchrotron_sources.html.