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 27 204/04/2014

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 27 604/04/2014

SRC – “Aladdin” --MadisonWisconsin

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.