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Calculation of differential cross section for dielectronic recombination with one-electron
uranium
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2015 J. Phys.: Conf. Ser. 583 012005
(http://iopscience.iop.org/1742-6596/583/1/012005)
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17th International Conference on the Physics of Highly Charged Ions
Journal of Physics: Conference Series 583 (2015) 012005
IOP Publishing
doi:10.1088/1742-6596/583/1/012005
Calculation of differential cross section for
dielectronic recombination with one-electron uranium
K N Lyashchenko and O Yu Andreev
Department of Physics, St. Petersburg State University, Ulyanovskaya 3, Petergof,
St. Petersburg, 198504, Russia
E-mail: [email protected]
Abstract. Calculation of the differential cross section for the dielectronic recombination with
one-electron uranium within the framework of QED is presented. The contribution of the QED
corrections and the interfernce of the photon multipoles is investigated.
1. Introduction
The electron recombination, in particular, the dielectronic recombination with one-electron
uranium is under intensive investigation during the last decade [1, 2, 3, 4, 5]. The present
paper is devoted to calculation of the differential cross section of the dielectronic recombination
with one-electron uranium within the framework of QED.
We study a process, where an ion captures an electron and goes into a single excited state
with emission of one photon
e− + U 91+ (1s) → U 90+ (r) + γ → U 90+ (1s1s) + γ + γ 0 ,
(1)
where r denotes one of the single excited states (1s2s), (1s2p). The first emitted photon (γ)
can be registered in the experiment [5]. The interference between the first (γ) and the second
emitted (γ 0 ) photons can be neglected [2].
If the energy of the initial state is close to the energy of a double excited state, the cross
section shows resonances. The resonances are explained by the dielectronic recombination
e− + U 91+ (1s) → U 90+ (d) → U 90+ (r) + γ → . . . ,
(2)
where d is one of the double excited states (2s2s), (2s2p), (2p2p).
2. Electron recombination with a bare nucleus
We consider first the electron recombination with a bare nucleus. In the zeroth order of the
QED perturbation theory the amplitude of electron recombination is written as [6]
Z
U
= e
d3 r ψ b (r)γ ν A(k,λ)∗
(r)ψpµ (r) ,
ν
(3)
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
1
17th International Conference on the Physics of Highly Charged Ions
Journal of Physics: Conference Series 583 (2015) 012005
IOP Publishing
doi:10.1088/1742-6596/583/1/012005
where e is the electron charge; ψp,µ is a wave function of the incident electron with certain
momentum p and polarization µ; ψb = ψnjlm is the wave function of the bound electron; 4(k,λ)∗
vector Aν
describes the emitted photon with wave vector k = (ω, k) and polarization λ
(k,λ)
(A0
= 0, the transverse gauge is used); γ ν – Dirac gamma matrices.
In the transverse gauge the photon wave function reads (the relativistic units are used)
A
(k,λ)
r
(r) =
2π ikr (λ)
e e .
ω
(4)
This function can be presented as [7]
A
(k,λ)
r
=
2π X l
∗
i gl (ωr)(e(λ) , Yjlm
(k))Yjlm (r) ,
ω jlm
(5)
where gl (x) = 4πjl (x) and jl (x) is the spherical Bessel function.
The
pwave function ψp,µ (r) describes the incident electron with certain momentum (p), energy
( = 1 + p2 ) and polarization (µ). It is convenient to expand the wave function ψp,µ (r) in
series over the wave functions of electron with certain energy, total angular momentum (j), its
projection (m), and parity (l) [6]
Z
ψpµ (r) =
d
X
apµ,εjlm ψεjlm (r) ,
(6)
jlm
apµ,εjlm =
(2π)3/2 l iφjl +
i e [Ωjlm (p)υµ (p)]δ( − ε) ,
√
p
(7)
where Ωjlm (p) is the spherical spinor and υµ (p) is the spinor with certain projection on the
electron momentum (p), the phase φjl is the Coulomb phase shift. The wave function ψεjlm are
normalized as
Z
+
d3 r ψεjlm
(r) ψε0 j 0 l0 m0 (r) = δ(ε − ε0 )δjj 0 δll0 δmm0 .
(8)
Accordingly, the electron recombination amplitude can be written as
Uif
= −e
Z
3
d r
X X
ψn+b jb lb mb (r)~
α
r
jp lp mp jlm
(2π)3/2
× √
p
2π
(−i)lp glp (ωr)(e(λ)∗ Yjp lp mp (k))Yj∗p lp mp (r)
ω
il eiφjl [Ω+
jlm (p)υµ (p)]ψεjlm (r) ,
(9)
where α
~ are Dirac alpha matrices.
3. Electron recombination with a one-electron ion
The initial state (the bound (1s) electron and the incident electron) is described by
Ψi =
1
√ det{ψp,µ , ψnb jb lb mb } .
2
(10)
This function can be expanded over two-electron functions with certain total momentum (in the
j-j coupling scheme)
Ψi =
X
(0)
(0)
hΨJM j1 j2 l1 l2 | Ψi i ΨJM j1 j2 l1 l2 ,
JM j1 j2 l1 l2
2
(11)
17th International Conference on the Physics of Highly Charged Ions
Journal of Physics: Conference Series 583 (2015) 012005
IOP Publishing
doi:10.1088/1742-6596/583/1/012005
where
(0)
ΨJM j1 j2 l1 l2
X
= N
j1 j2
(m1 m2 ) det{ψn1 j1 l1 m1 , ψn2 j2 l2 m2 } .
CJM
(12)
m1 m2
Here, n – the principle quantum number, j – total momentum of electron, l – parity, m – total
j1 j2
(m1 m2 ) – Clebsch-Gordan coefficients.
momentum projection, N – normalization constant, CJM
The interaction with the quantized electromagnetic and electron-positron fields leads to
various correction to the amplitude: interelectron interaction correction, electron self-energy and
vacuum polarization corrections. For consideration of these corrections the line-profile approach
(LPA) was employed [3, 4]. Within the LPA we introduce the set of two-electron configurations
(g) which includes all the two-electron configurations composed by 1s, 2s, 2p, 3s, 3p, 3d electrons
and the incident electron e.
Within the LPA there is also introduced the matrix V which is defined by the one and
two-photon exchange, electron self-energy and vacuum polarization matrix elements. Matrix
V = V (0) + ∆V is considered as a block matrix
V
=
V11 V12
V21 V22
"
=
(0)
∆V12
V11 + ∆V11
(0)
∆V21
V22 + ∆V22
#
.
(13)
Matrix V11 is defined on set g, which contains configurations mixing with the reference state ng
∈ g. Matrix V (0) is a diagonal matrix: sum of Dirac energies. Matrix ∆V11 is not a diagonal
matrix, but it contains a small parameter of the QED perturbation theory. Matrix V11 is a finite
matrix and can be diagonalized numerically:
diag
V11
= B + V11 B ,
B+B = I .
(14)
Taking into account the interaction with the quantized fields leads to the substitution of
Φng
=
X
(0)
Bkg ng Ψkg +
kg ∈g
X
[∆V21 ]klg
k∈g,l
/ g ∈g
Blg ng
(0)
Eng
−
(0)
Ek
(0)
Ψk + · · · .
(15)
(0)
for Ψng functions, where ng = (JM j1 j2 l1 l2 ) is a reference state. Here, E (0) is the energy of the
corresponding two-electron configuration.
The amplitude of the electron recombination is given as a matrix element of the photon
emission operator (Ξ(0) ) with the bra and ket-vectors given by the corresponding Φng functions:
Uif
= hΦf |Ξ(0) |Φi i ,
(0)
Ξab = −
Z
d3 r ψa+ (r)(~
αA(k,λ)∗ (r))ψi (r) .
(16)
4. Results
We preset calculation of the diferential cross section for the electron recombination with oneelectron ion of uranium being in the ground state (1s) as a function of the incident electron energy
(e ) and the angle of the emitted photon in the rest frame of the ion. The calculation is performed
for the kinetic energy of the incident electron (e − me c2 ) within the interval [62.5, 64.2] keV.
It supposed that the process of electron recombination is registered by registration of emitted
photons with energy near the value ω ≈ e +1s−r, where r denotes a single excited configuration:
(1s2s), (1s2p). The higher orders of the multipole expansion of the photon (0 < l0 < 10) were
taken into account, what led to changes of 2 − 6% in the peaks area and up to 30% in the rest
area of the differential cross section. The total cross section of the electron recombination was
also calculated and found good agreement with the previous works [4, 5].
3
17th International Conference on the Physics of Highly Charged Ions
Journal of Physics: Conference Series 583 (2015) 012005
IOP Publishing
doi:10.1088/1742-6596/583/1/012005
180
58
160
55
52
140
49
46
120
43
degree
40
37
100
34
31
80
28
25
60
22
19
16
40
13
10
20
7,0
4,0
63,85
63,90
63,95
64,00
64,05
64,10
64,15
energy of the incident electron (keV)
Figure 1. The differential cross section for recombination of non-polarized electrons
with non-polarized ions. The left graph corresponds to the full QED calculation and the
right graph represents the calculation with disregard of the Breit interaction.
Results for the differential cross section (in barn/str) for recombination of non-polarized
electron with non-polarized ions are presented in Fig.(1).
Results for the differential cross section (in barn/str) for recombination of polarized (µ = 1/2)
electron with polarized (m = 1/2) ions are presented in Fig.(2).
180
180
58
160
64
160
61
55
58
52
140
140
49
55
52
46
43
120
49
120
46
43
37
100
34
31
80
28
degree
degree
40
40
100
37
34
80
31
25
22
60
28
25
60
22
19
16
40
13
19
40
16
13
10
20
7,0
10
20
7,0
4,0
63,85
63,90
63,95
64,00
64,05
energy of the incident electron (keV)
64,10
64,15
4,0
63,85
63,90
63,95
64,00
64,05
64,10
64,15
energy of the incident electron (keV)
Figure 2. The differential cross section for recombination of polarized electron with
polarized ions. The left graph corresponds to the full QED calculation and the right
graph represents the calculation with disregard of the Breit interaction.
The presented calculation shows that the differential cross section of the electron and
dielectronic recombination is very sensitive to the QED corrections and to the interference of
the photon multipoles.
Acknowledgments
The authors acknowledge the support by RFBR grants 14-02-20188 and St. Petersburg State
University for a research Grant No. 11.38.227.2014.
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