compton scattering in strong magnetic fields
DESCRIPTION
Compton Scattering in Strong Magnetic Fields. Department of Physics National Tsing Hua University G.T. Chen 2006/5/4. Outline. Motivation Relativistic Landau Level Compton Scattering Results Discussion and Future Work. Motivation. The isolated neutron star 1E1207.4-5209. - PowerPoint PPT PresentationTRANSCRIPT
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
Relativistic Landau LevelRelativistic Landau Level
AssumeAssume
iEte
2 2 2 2 2 2( ) (2 )x zE m p e y B eB yp …
……
Relativistic Landau LevelRelativistic Landau Level
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 :
Compton ScatteringCompton Scattering
,p j
,p j
,k
,k ,p j ,p j
,k ,k
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
* *( ) ( )
,
* *( ) ( )
,
( ) ( )
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
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
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
Compton ScatteringCompton Scattering
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
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
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
Discussion & Future WorkDiscussion & Future Work
Cold plasma Cold plasma hot plasma hot plasma
Vacuum polarization should be includedVacuum polarization should be included
Compton ScatteringCompton Scattering
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