Positron annihilation in a strong magnetic field

37 Pages·2012·1.8 MB·English

by  Jerzy Dryzek

Positron annihilation in
a strong magnetic field
Jerzy Dryzek1,2, Sylwia Lewicka2
1)Opole University, Poland
2)Institute of Nuclear Physics PAN, Poland
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Positrons in Astrophysics – March 2012
Strong magnetice field, it means close to the value
equal to:
c3 m2
B ≡ 0 = 4.414×1013 G
0
e (cid:61)
In that field several interesting phenomena occur:
- the electron mas is affected by the strong magnetic field,
as B increases first it is getting slightly lower than m ,
0
then slightly higher.
-magneic field distorts atoms into long, thin cylinders and
molecules into strong, polymer-like chains.
-photons can rapidly split and merge with each other.
- vacuume itself is unstable, at extremely large B~1051-1053 G.
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Duncan, arXiv:astro-ph/0002442v1 (2000)
Outline
1. Introduction,
2. Cross section for single and two-photon annihilation proceses
3. Positronium atom in strong magnetic field.
4. Summary
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Classical electron gyrating in a magnetic field, satifying
(cid:71)
the equation: (cid:71)
dp e (cid:71)
= v × B
(cid:71) (cid:71)
dt c
p = m v
where
0
Cyklotron frequency: critical magnetic field straight
e B c3 m2
ϖ = (cid:61)ϖ = m c2 ⇒ B = 0 = 4.414×1013 G
C C 0 0
c m e (cid:61)
0
radius of gyration:
c p
a =
gyr (cid:61) B 1
e B
⇒ a = 0 = 0.386[pm]
gyr
m c B b
0
a p (cid:17) (cid:61)
gyr B
b =
B
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0
Pulsars and magnetars:
Magnetic field
SGR 1806-20 8x1014 G
PSR J1846-0258 4.9x1013 G
SGR 0418+5729 < 7.5x1012 G
SGR 1900+14 7x1014 G
Swift J1834.9-0846 1.4x1014 G
SGR 0501+4516 1.9x1014 G
CXO J164710.2-455216 [Wd 1] <1.5x1014 G
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From McGillSGR/AXP OnlineCatalog& astro-ph.SR0910.4859
Positron annihilation or Positronium in a strong magnetic
field have been considered by many authors:
-Kroupa and Robl (1954)
-Ternov et aL, 1968
-Wunner (1979)
-Daugherty and Bussard (1980)
-Kaminker et al,1987, 1990
-Herold, Ruder and Wunner (1985)
-Leinson and A. Perez (2000) 
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The Dirac equation in a magnetic field
⎡−γµ(i∂ + qA ) − m ⎤ψ(±) (xν) = 0,
⎣ ⎦
µ µ 0
(cid:61) = c =1
( We adopted  in many equations that follow) 
(cid:71)
Aµ(x) = (0, A) = (0,0, xB,0)
Solution for electrons
E(+) + m
ψ(+)(x) = n 0 exp(−iE(+)t) u↑↓(x)
2E(+) n n
n
⎛ I ⎞ ⎛ 0 ⎞
⎜ n−1 ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ 0 ⎟ ⎜ I(+) ⎟
⎜ ⎟ ⎜ n ⎟
⎜ ⎟ ⎜ ⎟
⎜ p ⎟ ⎜ 2nb m ⎟
u↑(x) = ⎜ I(+) ⎟, u↓(x) = ⎜− 0 I(+)⎟
n ⎜⎜ E(+) +m n−1 ⎟⎟ n ⎜⎜ E(+) +m n−1⎟⎟
⎜ n 0 ⎟ ⎜ n 0 ⎟
⎜ ⎟
⎜⎜⎜⎜−E(2+n) b+mm0 In(+)⎟⎟⎟⎟ ⎜⎜⎜⎜⎜−E(+)p+m In(+)⎟⎟⎟⎟⎟
⎜⎜ ⎟⎟ ⎝ n 0 ⎠
⎝ n 0 ⎠
in ⎡ b(x−a)2 ⎤ ⎛ x−a ⎞ ⎛ iayb⎞
I = exp⎢− ⎥ H ⎜ ⎟exp⎜− ⎟exp(ipz)
n L(πλ /b)1/4 2nn! ⎣ 2λ2 ⎦ n ⎝ λ ⎠ ⎝ λ2 ⎠
C C
C
Johonsn and Lippmann, PR, 76 (1949) 828, 7/37
Bhattacharya, arXiv:0705.4275v2 [hep-th]
Solution for positrons
E(−) + m
ψ(−)(x) = n 0 exp(iE(−)t) v↑↓(x)
2E(−) n n
n
⎛ −p ⎞
I
⎜ E(−) + m n−1⎟ ⎛⎜ 2nb m ⎞⎟
− 0 I
⎜ n 0 ⎟ ⎜ ⎟
⎜ E(−) + m n−1⎟
⎜ ⎟ ⎜ n 0 ⎟
− 2nb m ⎜ ⎟
v↑(x) = ⎜ 0 I ⎟, ⎜ p ⎟
n E(−) + m n v↓(x) = ⎜ I ⎟,
⎜ n 0 ⎟ n ⎜⎜ E(−) + m n ⎟⎟
⎜ ⎟ ⎜ n 0 ⎟
I
⎜ ⎟
0
⎜ n−1 ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
0
⎝ ⎠ ⎜ I(−) ⎟
⎜⎜ ⎟⎟
⎝ n ⎠
in ⎡ b(x −a)2 ⎤ ⎛ x − a ⎞ ⎛ iayb ⎞
I = exp − H exp − exp(ipz)
⎢ ⎥ ⎜ ⎟ ⎜ ⎟
n L(πλ /b)1/4 2nn! ⎣ 2λ2 ⎦ n ⎝ λ ⎠ ⎝ λ2 ⎠
C C
C
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Landau levels
The transverse motion of the positron (e+) and 
electron (e‐) is quantized, they energy obeys 
the relation :
E(±) = m2 + ( p(±))2 + 2n b m2
n 0 ± 0
m 1+ 4b
0
n = 0,1, 2,...
±
m 1+ 2b
0
∆E > m is the number of the Landau level, 
0
m
0 p -
is the longitudinal momentum, along
the magnetic field line.
In calculations it is sufficient to assume: n=0 9/37
Two-photon annihilation
Positron annhilation is related to two photons emission
.
γ(k ,ε ) γ(k ,ε ) γ(k ,ε ) γ(k ,ε )
1 1 2 2 1 1 2 2
ee-- (p) (q) e+ e-(p) (q) e+
.
2
cross section ≈α
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