Compton Scattering in Strong Magnetic Fields

Compton Scattering in Compton Scattering in Strong Magnetic FieldsStrong Magnetic Fields

Department of PhysicsDepartment of PhysicsNational Tsing Hua UniversityNational Tsing Hua University

G.T. ChenG.T. Chen2006/5/42006/5/4

OutlineOutline

MotivationMotivation Relativistic Landau Level Relativistic Landau Level Compton ScatteringCompton Scattering ResultsResults Discussion and Future WorkDiscussion and Future Work

MotivationMotivation The isolated neutron star 1E1207.4-5209The isolated neutron star 1E1207.4-5209

Four absorption features are seen in the pn spectrum at the harmonically spaced energies of ~0.7kev , ~1.4kev , ~2.1kev , ~2.8kev

A. De Luca et al. ,A&A ,2004

MotivationMotivation

Using three independent statistical analyseUsing three independent statistical analyses indicated that there is no third or four line.s indicated that there is no third or four line. (Kaya Mori et al.,ApJ,2005)(Kaya Mori et al.,ApJ,2005)

Spectral lines:Spectral lines:

observed timing : B ~ 2.6 x10observed timing : B ~ 2.6 x1012 12 GG

cyclotron lines: B~ 8 x10cyclotron lines: B~ 8 x101010 G (electron) G (electron)

B~ 1.6 x10B~ 1.6 x1014 14 G (proton)G (proton)

We attempt to construct a model finding We attempt to construct a model finding

the origin of these lines the origin of these lines

MotivationMotivation

Compton ScatteringCompton Scattering

,p j ,p j

,k ,k

BremsstrahlungBremsstrahlung

,p j

,p j

,k Z

Relativistic Landau LevelRelativistic Landau Level


Dirac equation with magnetic field: ( )Dirac equation with magnetic field: ( )

Choose Landau gauge Choose Landau gauge

( ( ) )i p eA mt

0

0

0 0 1

1 0

0

0

ii

i

0 0y z

x

A A A

A yB

1c

B Bz

( )i p mt

( ) 0p m


AssumeAssume

iEte

2 2 2 2 2 2( ) (2 )x zE m p e y B eB yp …

……


Energy eigenvalueEnergy eigenvalue

oror

2 2 2 2z cE m p m j 1

2

c

j n s

eB

m

2 2 2 (1 2 )zcr

BE p m j

B

2 3134.414 10cr

m cB G

e


Feynman diagrams of Compton scattering :Feynman diagrams of Compton scattering :


,p j

,p j

,k

,k ,p j ,p j

,k ,k


S-matrix:S-matrix:

andand

wherewhere

2 12 4 4

1 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

f F i

fi

f F i

x A x G x x A x xS ie d x d x

x A x G x x A x x

2

fi

Vd S d

T

332 2 (2 )

x zx z

L L Vd dp dp d k


The differentiate cross sectionThe differentiate cross section2 2

0

2

( ) 2 ( ) 3

1( ) ( ) ( )( )

4

( )

z z

ij i ijj j z

j

r md E E p k p k E m E m

EE

a e b e d k dp

( , , , , , , )z zd d k p j k p j


Integrate final electron momentum Integrate final electron momentum 2 2

0

2

( ) 2 ( ) 3

1( ) ( )( )

4

( )ij i ijj j

j

r md E E E m E m

EE

a e b e d k

( , , , , , )zd d k p j k j


* *( ) ( )

,

* *( ) ( )

,

( ) ( )

22

( ) ( )

22

fq iq

js

j

p p k

q i q f

js

j

p p

G k G kE m

aiE E E

G k G kE m

biE E E

k

2

1( )( ) sin( )

2

2 c

j j

km


Integrate over the final photon azimuth and Integrate over the final photon azimuth and average over initial yieldsaverage over initial yields

3 3

3

2 20

2 2 *

1( ) ( )( )

4

2 (2 ) Re( )j j j j j j j jj j

r md E E E m E m

EE

a b J a b dk d


We assume the distribution of initial electrons is We assume the distribution of initial electrons is 1D relativistic thermal distribution 1D relativistic thermal distribution

KK11=modified Bessel fn. of the 2nd kind =modified Bessel fn. of the 2nd kind

T = the electron temperature parallel to the fieldT = the electron temperature parallel to the field

1

( )2 ( )

E

Tef p

mmK

T


From the energy conservation and parallel From the energy conservation and parallel momentum conservation, we havemomentum conservation, we have

wherewhere

2

2

[ 2 ( )][ 2 ( )]

2 2 4 ( 2 )

cc

c

q m j jk kp q m j j

q q q m m j

2 2( ) ( )q k k


Redistribution function:Redistribution function:

Apply into our case, because of the delta Apply into our case, because of the delta function of energy, the integration over function of energy, the integration over momentum reduces to a summation.momentum reduces to a summation.

( ) ( ) ( , , )j j jjj

dpn p dp w p p


Therefore, the Compton scattering differentiate Therefore, the Compton scattering differentiate cross sectioncross section

3 3

3

2 20

2 2

2 2 *

1 ( )( )( )

2 [ 2 ( )] 4

2 (2 ) Re( )

jj p c

j j j j j j j jj j

r m E m E md f p

q m j j qm

a b J a b dk d

( , , , , , , )d d j j


11

3 3

3

2 20

2 2

2 2 *

1 ( )( )( )

2 [ 2 ( )] 4

2 (2 ) Re( )

jj p c

j j j j j j j jj j

r m E m E md f p

q m j j qm

a b J a b dk d

2d

d d

ResultsResults

ResultsResults

11

22

12

3

38 3.28 10

0.0167 /

10

c kev B G

g cm

T kev

12

3

6

10 11.58

1 /

10 86.2

cB G kev

g cm

T K ev

0, 0 ~ 3j j

12

3

38 3.28 10

0.0167 /

10

c kev B G

g cm

T kev

12

3

38 3.28 10

0.0167 /

10

c kev B G

g cm

T kev

0, 0 ~ 3j j

12

3

38 3.28 10

0.0167 /

10

c kev B G

g cm

T kev

0, 0 ~ 3j j

0, 0 ~ 3j j

12

3

38 3.28 10

0.0167 /

10

c kev B G

g cm

T kev

0, 0j j

12

3

6

10 11.58

1 /

10 86.2

cB G kev

g cm

T K ev

Discussion & Future WorkDiscussion & Future Work

The relativistic calculation is given major The relativistic calculation is given major correctioncorrection

The appearance of higher harmonicsThe appearance of higher harmonics

The energy shift of the resonance

The decrease below the Thomson value at w>wc


Cold plasma Cold plasma hot plasma hot plasma

Vacuum polarization should be includedVacuum polarization should be included


Two photon processTwo photon process

>>Thank You<<>>Thank You<<


is the inverse of the half-life of the is the inverse of the half-life of the electron state j’’electron state j’’

j

,p j

,p j

,k

H. Herold et al. ,A&A, 1982


Compton ScatteringCompton Scattering12

3

6

10 11.58

1 /

10 86.2

cB G kev

g cm

T K ev

12

3

38 3.28 10

0.0167 /

10

c kev B G

g cm

T kev

?

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Compton Scattering in Strong Magnetic Fields