Positron annihilation in a strong magentic field.

<strong>Positron</strong> <strong>annihilation</strong> <strong>in</strong>
a strong magnetic field
Jerzy Dryzek 1,2 , Sylwia Lewicka 2
1)
Opole University, Poland
2)
Institute of Nuclear Physics PAN, Poland
<strong>Positron</strong>s <strong>in</strong> Astrophysics – March 2012
1/37

Strong magnetice field, it means close to the value
equal to:
B
c
3 2
0
13
0
≡ = 4.414×
10 G
e
m
In that field several <strong>in</strong>terest<strong>in</strong>g phenomena occur:
- the electron mas is affected by the strong magnetic field,
as B <strong>in</strong>creases first it is gett<strong>in</strong>g slightly lower than m 0
,
then slightly higher.
-magneic field distorts atoms <strong>in</strong>to long, th<strong>in</strong> cyl<strong>in</strong>ders and
molecules <strong>in</strong>to strong, polymer-like cha<strong>in</strong>s.
-photons can rapidly split and merge with each other.
- vacuume itself is unstable, at extremely large B~10 51 -10 53 G.
Duncan, arXiv:astro-ph/0002442v1 (2000)
2/37

Outl<strong>in</strong>e
1. Introduction,
2. Cross section for s<strong>in</strong>gle and two-photon <strong>annihilation</strong> proceses
3. <strong>Positron</strong>ium atom <strong>in</strong> strong magnetic field.
4. Summary
3/37

Classical electron gyrat<strong>in</strong>g <strong>in</strong> a magnetic field, satify<strong>in</strong>g
the equation:
dp e
= v × B
dt c
where
p = m v
0
Cyklotron frequency:
ϖ =
C
e B
cm
a
0
gyr
agyr
=
3 2
2 c m0
13
ϖ
C
= mc
0
⇒ B0 = = 4.414×
10 G
e
cp
eB
p
critical magnetic field straight
radius of gyration:
B
1
0.386[pm]
mc B b
0
⇒ agyr
= =
0
b
=
B
B
0
4/37

Pulsars and magnetars:
Magnetic field
SGR 1806-20 8x10 14 G
PSR J1846-0258 4.9x10 13 G
SGR 0418+5729 < 7.5x10 12 G
SGR 1900+14 7x10 14 G
Swift J1834.9-0846 1.4x10 14 G
SGR 0501+4516 1.9x10 14 G
CXO J164710.2-455216 [Wd 1]

<strong>Positron</strong> <strong>annihilation</strong> or <strong>Positron</strong>ium <strong>in</strong> a strong magnetic
field have been considered by many authors:
-Kroupa and Robl (1954)
-Ternov et aL, 1968
-Wunner (1979)
-Daugherty and Bussard (1980)
-Kam<strong>in</strong>ker et al,1987, 1990
-Herold, Ruder and Wunner (1985)
-Le<strong>in</strong>son and A. Perez (2000)
6/37

The Dirac equation <strong>in</strong> a magnetic field
µ ( ) ν
⎡
⎣−γ ( i∂ µ
+ qAµ
) − m ⎤
0 ⎦ψ ± ( x ) = 0,
u
↑
n
ψ
E + m
( x) exp( iE t) u ( x)
( + )
( + ) n 0
( + ) ↑↓
= −
( + )
n n
2En
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎞
n−1
⎟
⎟
⎟
0 ⎟
⎟
p
⎟
( + ) ⎟
I
( + ) n−1
⎟
n
+ m ⎟
0 ⎟
⎟
2nb m
⎟
0 ( )
( ) I
+ ⎟
+ n ⎟
En
+ m
0
⎟
⎠
⎜
( x)
= ⎜
,
⎜ E
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−
I
= c = 1
µ
A ( x) = (0, A) = (0,0, xB,0)
( We adopted <strong>in</strong> many equations that follow)
Solution for electrons
u
0
⎛
⎞
⎜
⎟
⎜
⎟
⎜ ( + )
I
⎟
⎜ n ⎟
⎜
⎟
⎜
⎟
↓ ⎜ 2nb m0
( )
n
( x)
( ) I
+ ⎟
= ⎜−
+ n−1
⎟
⎜ En
+ m ⎟
⎜
0 ⎟
⎜
⎟
⎜ p
⎟
⎜−
I
( + ) ⎟
⎜ ( + ) n ⎟
⎜ En
+ m
0
⎟
⎝
⎠
n
2
i ⎡ b( x−a)
⎤ ⎛ x−a⎞
⎛ iayb⎞
In
= exp ⎢− H exp exp( )
1/4
2 ⎥ n
ipz
n
⎜ ⎟ ⎜−
2 ⎟
L( πλ / b) 2 n!
⎣ 2λC
⎦ ⎝ λ ⎠ ⎝ λC
⎠
C
Johonsn and Lippmann, PR, 76 (1949) 828,
Bhattacharya, arXiv:0705.4275v2 [hep-th]
7/37

Solution for positrons
ψ
E + m
( x) exp( iE t) v ( x)
( −)
( −) n 0
( −)
↑↓
=
( −)
n n
2En
v
⎛
⎜
E
⎜
⎜ −
( −)
n
− p
+ m
0
n−1
2nb m I
↑
0
n
( x) = ⎜ ,
( −)
n ⎟
En
+ m0
⎜
⎜
⎜
⎝
I
n−1
0
I
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
v
⎛
⎞
⎜ 2nb m ⎟
⎜
0
−
( ) I ⎟
⎜ −
n−1
⎟
⎜ En
+ m0
⎟
⎜
⎟
⎜
⎟
↓ ⎜ p
⎟
n
( x)
⎜
I
( −)
n ⎟
⎜ En
+ m ⎟
⎜
0 ⎟
⎜
⎟
⎜ 0 ⎟
⎜
⎟
( −)
⎜ I
⎝ n
⎟⎠
= ,
n
2
i ⎡ b( x−a)
⎤ ⎛ x−a⎞
⎛ iayb⎞
In
= exp ⎢− H exp exp( )
1/4
2 ⎥ n
ipz
n
⎜ ⎟ ⎜−
2 ⎟
L( πλ / b) 2 n!
⎣ 2λC
⎦ ⎝ λ ⎠ ⎝ λC
⎠
C
8/37

Landau levels
The transverse motion of the positron (e+) and
electron (e‐) is quantized, they energy obeys
the relation :
m0 1+
4b
m0 1+
2b
∆ E > m 0
E = m + ( p ) + 2 n bm
( ± ) 2 ( ± ) 2 2
n
0 ± 0
n
±
=
0,1,2,...
is the number of the Landau level,
m 0
p - is the longitud<strong>in</strong>al momentum, along
the magnetic field l<strong>in</strong>e.
In calculations it is sufficient to assume: n=0
9/37

Two-photon <strong>annihilation</strong>
<strong>Positron</strong> annhilation is related to two photons emission
.
γ( , ε )
k 1 1
γ( k 2 , ε 2 )
γ( k 1 , ε 1 ) γ( k 2 , ε 2 )
e - ( p )
( q ) e +
e - ( p)
( q)
e +
.
cross section
2
≈ α
10/37

S<strong>in</strong>gle photon <strong>annihilation</strong>
In a field B close to B 0
the s<strong>in</strong>gle photon <strong>annihilation</strong> is possible:
.
γ( k, ε)
e - () p () q e +
.
cross section
≈ α
(In free space this process is forbidden.)
11/37

Differential cross section for s<strong>in</strong>gle photon <strong>annihilation</strong>
(In the center of mass (CM) system)
2 2 2
2
2
d σ1 γ αλC
m0
ϖ ⎡
2 (1 − cos θ)
⎛ ϖ ⎞ ⎤
= (1 −cos θ )exp ⎢− ⎜ ⎟ ⎥δϖ ( cos θ) δ(2 E0
−ϖ)
dΩdω
2 2 2
E b 2b m
0
− m0E
⎢
0
⎣
⎝ 0 ⎠ ⎥
⎦
For <strong>annihilation</strong> at rest,
due to the delta function, it <strong>in</strong>dicates
the l<strong>in</strong>e of emission of the photon
perpendicular to the magnetic field
l<strong>in</strong>e.
E 0
is the total energy of e + −12
λ C
= 2.426×
10 m
is the Compton wavelenght
B
<strong>in</strong> CM θ = π /2
θ
ϖ
a fan beam
Daugherty and Bussard, The Astrophysical Journal, 238 (1980) 296
12/37

Differential cross section for s<strong>in</strong>gle photon <strong>annihilation</strong>
direction of magnetic l<strong>in</strong>e
d σ αλ m ⎡ θ E ⎤ m
dΩ b ⎢ b m
⎣
⎥
⎦ ⎣ ⎦
2 2
2
2
1γ
0
2s<strong>in</strong> ⎛ ⎞
C
0 ⎡ 2
0 ⎤ 2
= exp ⎢− ⎜ ⎟ ⎥Θ 2E0
s<strong>in</strong>
2 2 2
⎢ −
E0 m0E0
0
s<strong>in</strong>θ
⎥
− ⎝ ⎠
θ
b=1
E = 1.1m
0 0
E = 2m
0 0
E = 1.4m
0 0
0
Ω= π −2arcs<strong>in</strong> m E 0
13/37

Differential cross section for s<strong>in</strong>gle photon <strong>annihilation</strong>
direction of magnetic l<strong>in</strong>e
d σ αλ m ⎡ θ E ⎤ m
dΩ b ⎢ b m
⎣
⎥
⎦ ⎣ ⎦
2 2
2
2
1γ
0
2s<strong>in</strong> ⎛ ⎞
C
0 ⎡ 2
0 ⎤ 2
= exp ⎢− ⎜ ⎟ ⎥Θ 2E0
s<strong>in</strong>
2 2 2
⎢ −
E0 m0E0
0
s<strong>in</strong>θ
⎥
− ⎝ ⎠
θ
b=10
E = 2m
0 0
E = 1.4m
0 0
E = 1.1m
0 0
0
Ω= π −2arcs<strong>in</strong> m E 0
14/37

Total cross section for
s<strong>in</strong>gle photon <strong>annihilation</strong>.
1e+5
1e+4
1e+3
σ
1γ
2 2
αλ m 0 1 ⎡ 2⎛
0
exp
E ⎞
= C ⎢− ⎜ ⎟
2 2 2
E b b m
0
− m0E
⎢
0 ⎣ ⎝ 0 ⎠
2
⎤
⎥
⎥
⎦
(n=0, lowest Landau level)
E 0
is the total energy of e +
σ 1γ
(b) (barn)
1e+2
1e+1
1e+0
1e-1
1e-2
1e-3
⎛ E
2⎜
⎝mc
0
2
0
⎞
⎟
⎠
2
λ C
= ×
−12
2.426 10 m
is the Compton wavelenght
1e-4
1e-5
1e-6
E
= 3m c
0 0
2
1e-7
0.1 1 10 100 1000
b magnetic field (<strong>in</strong> B 0
)
Wunner PRL 42 (1979) 82.
15/37

The s<strong>in</strong>gle photon <strong>annihilation</strong> spectra for different
angles related to the direction of a magnetic field.
Annihilation rate
Annihilation rate (cm 3 s -1 MeV -1 srad -1 )
p
p
λ ∼ nn dpη p dpη p dσ
1e-12
1e-13
1e-14
1e-15
1e-16
1e-17
1e-18
+ −
+ − + + − −
0 0∫
( ) ∫ ( )
+
1γ
+ 2 2 − 2 2
( p ) + m0 ( p ) + m0
θ=π/2
mc
0
2 s<strong>in</strong>
2
θ
θ=π/4
θ=π/6
b=1
B
e +
e-
θ
1e-19
1.0 1.5 2.0 2.5 3.0
photon energy (MeV)
16/37

The s<strong>in</strong>gle photon <strong>annihilation</strong> spectra for different
magnetic field strength.
1e-11
1e-12
Annihilation rate (cm 3 s -1 MeV -1 srad -1 )
1e-13
1e-14
1e-15
1e-16
1e-17
1e-18
1e-19
1e-20
1e-21
100
b=0.1
10
1
1e-22
1e-23
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
photon energy (MeV)
17/37

Total cross section for
two-photon <strong>annihilation</strong>.
σ
σ
2γ
limit for small E 0
2 2 3 2
2
C mE
0 0 1 ⎡ 1⎛ E ⎞ ⎤ ⎡
0 1 ⎛ E ⎞⎤
0
exp ⎢ ⎜ ⎟ ⎥Erfi
⎜ ⎟
2 2
E b b m
0
− m ⎢
0
⎝ 0 ⎠ ⎥ ⎢⎣
b ⎝m0
⎠⎥⎦
πα λ
= − ⎢ ⎥
2
⎣ ⎦
Total cross section for two-photon <strong>annihilation</strong> for
free particles(Dirac formula) :
2 2
2 2
1 αλm
0 0 0 2 2
2γ
= E0 − m0 E0 + m0 − E0 + m0 E0 −m0
4π
σ
⎡
⎛ E + E −m
C
0
⎢
( )( ) ( )( 5 ) ln⎜
⎟ ( 3 )
2 2
E + m E −m
⎢
⎜ m ⎟
0
0 0 0 0
2λ
⎣
≅
αλ
2 2
m C 0
E
− m
2 2
0 0
⎝
(n=0, lowest Landau level)
problem for small b or b=0
⎞
⎠
⎤
⎥
⎥
⎦
limit for small E 0
:
σ
2λ
≅
1
4π
αλ
2 2
m C 0
E
− m
2 2
0 0
18/37

Total cross section for
two-photon <strong>annihilation</strong>.
1e+2
E 0
=1.1m 0
c 2
1e+1
σ 2γ
(barn)
1e+1
1e+0
1e-1
1e-2
1e-3
Dirac dependency
0.1
1
5
2 m 0
c 2
3 m 0
c 2
1e+0
1e-1
1e-2
1e-3
σ 2γ
(barn)
1e-4
1e-5
b=10
1e-4
1e-6
1 2 3 4 5 6
total positron energy <strong>in</strong> (m 0
c 2 )
1e-5
0.01 0.1 1 10
magnetic field (<strong>in</strong> B 0 )
19/37

Differential cross section for two-photon <strong>annihilation</strong> <strong>in</strong> CM
(n=0, lowest Landau level)
2 2 3 2 2
dσ
αλC Em
0 0
1 ⎛ E0
2
⎞
=
2 exp ⎜−
2 s<strong>in</strong> θ ⎟
dΩ π p E + b ⎝ bm0
⎠
2 2
( 0
2bm0)
CM
E = p + m
B
e +
e-
2 2
0 0
p
p
θ
Dirac (Heitler) formula for free <strong>annihilation</strong>:
.
dσ
αλC m ⎡ E + p + p s<strong>in</strong> θ 2p
s<strong>in</strong> θ
dΩ E p ⎣ E − p E − p
2 2 2 2 2 2 2 4 4
0 0
= ⎢
−
2 2 2 2 2 2 2
D,CM
16π 0 0
cos θ (
0
cos θ)
⎤
⎥
⎦
20/37

Differential cross section for two-photon <strong>annihilation</strong>
<strong>in</strong> CM system.
Free particles <strong>in</strong> CM (Dirac formula)
direction of motion of positron
and electron.
b=0.1
direction of magnetic l<strong>in</strong>e
E=1.5m 0
2m 0
3m 0
E=1.5m 0
2m 0
3m 0
21/37

Differential cross section for two-photon <strong>annihilation</strong>
<strong>in</strong> CM system.
E=1.5m 0
Dirac formula <strong>in</strong> CM
b=10
3m 0
2m 0
3m 0
2m 0
direction of magnetic l<strong>in</strong>e
E=1.5m 0
22/37

Differential cross section for two-photon <strong>annihilation</strong>
<strong>in</strong> CM system.
E=1.5m 0
Dirac formula <strong>in</strong> CM
direction of motion of positron
and electron <strong>in</strong> CM
b=100
3m 0
2m 0
3m 0
direction of magnetic l<strong>in</strong>e
2m 0
E=1.5m 0
23/37

Differential cross section for two-photon <strong>annihilation</strong>
<strong>in</strong> Laboratory system.
Dirac formula <strong>in</strong> Lab
direction of motion of positron
and electron <strong>in</strong> Lab
b=0.1
direction of magnetic l<strong>in</strong>e
3m 0 3m 0
2m 0
E=1.5m 0
2m 0
E=1.5m 0
24/37

Differential cross section for two-photon <strong>annihilation</strong>
<strong>in</strong> laboratory system
Free particles <strong>in</strong> Lab (Dirac formula)
direction of motion of positron
and electron <strong>in</strong> Lab
b=10
direction of magnetic l<strong>in</strong>e
3m 0
3m 0
2m 0
2m 0
E=1.5m 0
E=1.5m 0
25/37

The spectrum of the <strong>annihilation</strong> l<strong>in</strong>e (511 keV) for
<strong>annihilation</strong> at rest as seen perpendicular to
the magnetic field direction.
B
π/2
e + e-
ω
26/37

Comparison
1e+5
1e+4
E 0
=3m 0
c 2
σ 1γ
σ 2γ
1e+3
total cross section (barn)
1e+2
1e+1
1e+0
1e-1
1e-2
1e-3
1e-4
1e-5
Dirac dependency
1e-6
1e-7
0.1 1 10
magnetic field (<strong>in</strong> B 0 )
27/37

Daugherty and Bussard, The Astrophysical Journal, 238 (1980) 296
28/37

In „<strong>annihilation</strong> <strong>in</strong> flight” process positron or electron
contributs to the energy of emitted photons
hν 1
p
hν m<strong>in</strong>
p
hν max
hν 2
Free particles <strong>in</strong> Lab (From Dirac formula)
2
dσ 2γ
2π rm ⎡
o 0
1 1 E+ 3m0 1 E+
m ⎤
0
=− ⎢ − +
2 2⎥
dEγ E − m0 ⎢⎣
E + m0 2 Eγ( E + m0 − Eγ) 2 Eγ ( E+ m0
−Eγ)
⎥⎦
,
E
=
(m<strong>in</strong>) 0 0
γ
2 2
E+ m0 + E −m0
dσ/dEγ (barn/MeV)
( E+
m ) m 1
m
2
0
( E+
m ) m 1
(max) 0 0
Eγ
= E+
m0
2 2
E+ m 2
0
− E −m0
0.4
E +
=0.2 MeV
0.3
0.2
0.5 MeV
0.1
1 MeV
1.5 MeV
2 MeV
2.5 MeV
Kontrovich et al., The 10th International Symposium
on Orig<strong>in</strong> of Matter and Evolution of Galaxis (OMEG-2010)
0.0
0 1 2 3 4
E γ
(MeV)
255.5 keV
Spectra of <strong>annihilation</strong> <strong>in</strong> flight photon
29/37

<strong>Positron</strong>ium Ps <strong>in</strong> a strong magnetic field. The
problem can be attacked by solution of the Bethe-
Salpeter equation.
(Le<strong>in</strong>son and Perez: JHEP11(2000)039 )
The <strong>Positron</strong>ium wave function:
⎛0⎞
2 2
⎜ ⎟
mb ⎧ mb
⎫ 1
φ00l
=
↓↑ 0 ⎨− + ⎜ ⎟
0
+ ⎬
2π
⎩ 4
⎭ ⎜ 0 ⎟
⎜ 0⎟
⎝ ⎠
2
1 ∂ (0, l) ↓↑
(0, l) l (0, l)
− f (, zx
2
0) + V (, zx0) f (, zx0) = ε00( x0) f (, zx0)
↓↑ ↓↑ ↓↑ ↓↑
2µ
∂z
( )
0
ˆ P2
(0, l ) 0 0
2 2
( r) f ( z, x ) exp [( x x ) y ] 0 0 0 1
00
where V 00
(z, x 0
) is an effective potential
expressed by the elliptic <strong>in</strong>tegral of the first k<strong>in</strong>d
0 2
and x0 = P2 m0b
(P 20
- is the transverse momentum
characteriz<strong>in</strong>g the motion of the mass center across
the magnetic field)
30/37

The case of small x 0
means the small distances
between the centers of Landau orbits <strong>in</strong> the plane
orthogonal to the magnetic field, then the effective
2
potential reads:
e
V00
( z)
≅
2b
z +
2
π m
0
( δ )
(0,0) 2
f( z) ≡ f ( z,0) = δ exp − z
↓↑
ρ( P, P , P ) =Φ ˆ ( P, P , P ) =
x y u x y u
2 2 2
( − Px
+ Py
m b)
4δα
exp 2( ) /
π b (( P mα) + δ )
2
6 2
0
2 2 4 2
u
m ⎛
0
22 b ⎞
δ ≡ α ln
2 ⎜
π α ⎟
⎝ ⎠
The momentum spectrum of <strong>annihilation</strong> radiation
∫
∫
N( P) = dP dP ρ( P, P , P ); i, j, k∈{ x, y, z}
i j≠i, k k≠i,
j i j k
2 6
2 ⎧⎪
2P
⎫
xy , ⎪
2δ
N( Pxy
,
) = exp , ( ) .
2 ⎨− 2 ⎬ N Pz
=
4 2 2
πmb 0 ⎪⎩
mb
0 ⎪⎭
π( δ + Pz
)
31/37

Intensity of the <strong>annihilation</strong> l<strong>in</strong>e versus the photon energy
c
1e+1
1e+0
1e-1
(a)
α 1
ξ
0
= =
b 137.04b
small x 0
, ξ 0
= 0.01
large x 0
, ξ 0
= 0.01
small x 0
, ξ 0
= 0.1
large x 0
, ξ 0
= 0.1
small x 0
, ξ 0
= 1.0
large x 0
, ξ 0
= 1.0
10
(b)
0.13
FWHM[keV] = 14.9×
b
Intensity [arb.u.]
1e-2
1e-3
1e-4
FWHM (keV)
1
FWHM[keV] = 14.1×
b
0.39
1e-5
x 0
large
1e-6
0.1
x 0
small
1e-7
515 520 525 530 535 0.001 0.01 0.1 1
Energy [keV]
b = B/B 0
Lewicka and Dryzek, Materials Sciene Forum, 666 (2011) 31
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The case of large x 0
. The created <strong>Positron</strong>ium has a quasi
momentum of the order of 2m 0
, moreover, a typical pulsar
magnetic field b ≤ 0.1 giv<strong>in</strong>g the limit of large x 0
. In such a
case the effective potential V 00
(z) takes on the analytical form:
1
V00
( u = mα
z)
=−
2 2
( x mα)
+ u
0 0
Ground states of <strong>Positron</strong>ium <strong>in</strong> strong magnetic field as a
function of b, calculated from the Schröd<strong>in</strong>ger-like equation
<strong>in</strong> the case of large x 0
.
Magnetic field
<strong>in</strong> b unit
Bound<strong>in</strong>g energy E b
(eV)
0 (vacuum) 6.80
0.0073 9.13
0.073 51.9
0.73 197.2
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Annihilation rate of positronium dependent on the
magnetic field b (compared to the values obta<strong>in</strong>ed by
Wunner and Herold [3] <strong>in</strong> the non-relativistic regime).
B [G]
λ[s -1 ]
λ (for small x 0
)
λ[s -1 ]
(Wunner and
Herold results)
1∙10 12 7.40∙10 11 7.2∙10 12 0.729927
(ξ 0
=0.01)
3∙10 12 2.53∙10 12 2.5∙10 13 0.226552
(B=10 13 G)
5∙10 12 4.46∙10 12 4.2∙10 13 0.243309
(ξ 0
=0.03)
1∙10 13 9.56∙10 12 7.9∙10 13 0.145985
(ξ 0
=0.05)
b = B/B 0
λ[s -1 ]
λ (for large x 0
)
3.8457∙10 11
τ = 0.003 ns
1.81645∙10 8
τ = 5.5 ns
3.57567∙10 8
τ = 2.8ns
7.22707∙10 5
τ = 1.4 µ s
<strong>in</strong> vaccum
Wunner and Herold: Astrophys. Space Sci. 63 (1979) p. 503
( → γ ) = ( ± )
λ 1 S 2 7.9909 0.0017 10
1 9
exp 0
τ = 0.125 ns
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Daugherty and Bussard suggested to us<strong>in</strong>g the narrow widths of
the <strong>annihilation</strong> l<strong>in</strong>e (two photons <strong>annihilation</strong>) to place an upper
limit on the magnetic field strength.
∆E
E
γ
γ
FWHM
~0.35b
In the case of <strong>Positron</strong>ium formation it could be as follows:
0.13
FWHM[keV] = 14.9×
b
FWHM[keV] = 14.1×
b
0.39
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Summary
Ω hν= −m c 2 0
1
2
B
p-Ps
Possible <strong>annihilation</strong> channels
<strong>in</strong> a strong magnetic field:
-s<strong>in</strong>gle photon (most important
signature the l<strong>in</strong>e of 1022 keV)
-two-photon (not important for b>1)
-<strong>annihilation</strong> <strong>in</strong> flight
(signature the l<strong>in</strong>e of 255,5 keV)
-<strong>Positron</strong>ium lifetime is much longer
than free particle.
Cross sections depends
on the magnetic field strength.
ω
pulsar polar cap surface
Possible use for determ<strong>in</strong>ation
of the upper limit of the magnetic
field strenght.
36/37

Thank you for your attention
37/37