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ElectroncaptureintheLorentziandistributionplasma
Young-Dae Jung
a)
and Jung-Sik Yoon
Department of Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea
Received 5 April 1999; accepted 21 May 1999
Electron captures from hydrogenic ions by protons in a generalized Lorentzian
kappa
velocitydistribution plasma are investigated using the semiclassical Bohr–Lindhard model. The potentialenergy between the projectile ion and the released electron is obtained by introduction of the plasmadielectric function. The straight-line trajectory approximation is applied to the motion of theprojectile ion in order to obtain the electron capture probability as a function of impact parameter,collision energy, and spectral index
. It is found that the semiclassical capture probability includingthe dynamic plasma screening effect is increased with increasing the spectral index. The plasmascreening effects on the electron capture probabilities are found to be more effective for lowprojectile velocities. It is also found that the dynamic plasma screening effect on the electron captureprobability in the Lorentzian distribution plasma is stronger than that in the Maxwellian distributionplasma since the effective Debye length in the Lorentzian plasma is smaller than the Debye lengthin the Maxwellian plasma. ©
1999 American Institute of Physics.
S1070-664X
99
01009-5
I.INTRODUCTION
Electron capture
1–6
has received much attention, sincethis process is one of the basic atomic processes in atomiccollision physics.
1
The charge capture processes have beenwidely investigated using various methods
2–7
depending onthe range of the collision energy between the projectile ionand target system. For intermediate and high energy projec-tiles, the classical Bohr–Lindhard
2
method has been knownto be quite reliable
5,8
for calculating the electron captureprobability and capture cross section, since the de Brogliewavelength of the projectile is smaller than the collision di-ameter for the capture interaction. Recently, dynamic plasmascreening effects on the electron capture processes in Max-wellian velocity distribution plasmas were investigated usingthe semiclassical version of the Bohr–Lindhard model,
9
since when the projectile velocity is smaller than the electronthermal velocity, the charged projectile ion polarizes the sur-rounding plasma electrons so that the static description of thescreening effects may not be reliable.In general, most plasmas exist in a state very far re-moved from thermal equilibrium. These non-Maxwelliandistributions are common in laboratory and space plasmas.Non-Maxwellian plasmas are usually generated when highenergy electrons are injected from outside the plasma andwhenever they are produced by strong external interactions.In laboratory plasmas, the coupling of the injected externalenergy with the target system most often generates superther-mal electrons escaping from the Maxwellian distribution.
10,11
Thus, in this work we investigate the semiclassical electroncapture process in a non-Maxwellian velocity distributionplasma represented by the Lorentzian
kappa
distributionmodel.
12,13
The dynamic interaction energy is obtained bythe introduction of the plasma dielectric function in theLorentzian distribution plasma. The straight-line trajectorymethod is applied to describe the motion of the projectile ionas a function of the impact parameter, collision velocity, andspectral index
.In Sec. II, we derive the semiclassical expression of theelectron capture cross section by the projectile ion from ahydrogenic target ion in a generalized Lorentzian
kappa
distribution plasma using the semiclassical Bohr–Lindhardmodel including the dynamic interaction potential. We alsoderive the electron capture radius including the dynamicplasma screening effects. In Sec. III, we obtain a closed formof the scaled semiclassical electron capture probability as afunction of the impact parameter, Debye length, projectilevelocity, and electron thermal velocity for various spectralindices. We also investigate the variation of the captureprobabilities with changing the energy of the proton projec-tile and the ratio of the thermal electron velocity to the pro- jectile velocity. The results show that the semiclassical cap-ture probability including the dynamic plasma screeningeffect is found to be increased as an increase of the spectralindex. The plasma screening effect on the electron capture inthe Lorentzian distribution plasma is stronger than that in theMaxwellian plasma, since the Debye length (
) in theLorentzian distribution plasma is smaller than the Debyelength
in the Maxwellian distribution plasma. It is alsofound that the plasma screening effects on the electron cap-ture probabilities are more effective for low projectile veloci-ties. Finally, in Sec. IV, the conclusions of this work aresummarized.
II.CAPTURECROSSSECTION
The electron capture cross section using the Bohr–Lindhard method
6
is given by
c
2
db bP
b
,
1
a
Electronic mail: yjung@bohr.hanyang.ac.krPHYSICS OF PLASMAS VOLUME 6, NUMBER 9 SEPTEMBER 1999
36741070-664X/99/6(9)/3674/4/$15.00 © 1999 American Institute of Physics
Downloaded 24 Oct 2008 to 202.56.207.52. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
where
b
is the impact parameter and
P
(
b
) is the electroncapture probability. In the Bohr–Lindhard model, the elec-tron capture phenomena happens when the distance betweenthe projectile ion and a released electron from the target sys-tem is smaller than the classical capture radius
R
c
.
2,5,8
Thiscapture distance can be determined by equating the kineticenergy of the released electron in the frame of the projectileion and the binding energy provided by the charged projec-tile. In the semiclassical Bohr–Lindhard model,
14
the elec-tron capture probability is deﬁned by the ratio of the colli-sion time to the electron orbital time using the quantummechanical description of the target ion with nuclear charge
Z
T
:
P
b
t
c
r
2
d
3
r
,
2
where
(
/
E
2
r
/
Z
T
e
2
) is the electron release time,
t
c
is the electron capture time, and
(
r
) is the bound statewave function of the target ion.If the charged projectile ion is injected into dense plas-mas, the plasma electrons can be polarized by the projectile.Thus, we have to consider the dynamic motion of the sur-rounding plasma electrons during the collision time interval.The electron capture radius (
R
c
) including the dynamicplasma screening effects can be obtained by the Fouriertransformation of the dynamic interaction potential in themomentum space and the kinetic energy of the released elec-tron in the frame of the projectile ion:
Z
P
e
2
2
d
3
q
e
i
q
ã
R
c
q
2
q
,
12
m
v
p
2
,
3
where
Z
P
is the charge of the projectile ion,
q
is the momen-tum transfer,
m
is the electron rest mass,
v
p
is the projectilevelocity, and
(
q
,
) is the plasma dielectric function. Inorder to obtain the electron capture radius, this integral has tobe evaluated by mean of contour integration in the plane of the complex variable
q
.Generally most plasmas are known to be far from theMaxwellian distribution. This is the case for plasmas of fu-sion interest, which are conﬁned by magnetic or laser ﬁelds,and for plasmas generated from solid or gaseous targetthrough electric discharge or electron beam. These so-calledsuperthermal electrons escape the Maxwellian distributioncorresponding to the bulk of electrons. Thus, we assume thatthe plasma has a high energy tail for the particle distributionwith the spectral index
of the form
f
v
n
m
2
E
3/2
1
1/2
1
12
m
v
2
/
E
1
,
4
where
n
is the number density,
v
is the velocity of theplasma electron,
E
is the characteristic energy,
is thespectral index, and
represents the gamma function. Thisdistribution is the so-called Lorentzian
or kappa velocity
distribution
12,13
and has some interesting features, ﬁrst, athigh velocities the distribution obeys an inverse power law,and second, for all velocities, in the limit as
˜
the dis-tribution becomes a simple Maxwellian distribution. Usingthis Lorentzian distribution function, the plasma dielectricfunction can be given in the form
12
q
,
1
1
q
2
2
1
1/2
z
Z
z
,
5
where
z
/
q
,
(2
3)/
1/2
(
T
e
/
m
)
1/2
is the ther-mal velocity in the Lorentzian distribution plasma,
q
ã
v
p
,
v
p
is the projectile velocity, and the spectral index
should be greater than 3/2. Here
3/2
1/2
1/2
is the Debye length in the Lorentzian distribution plasma
13
and
kT
4
n
e
e
2
is the Debye length in the Mawellian plasma. The function
Z
(
z
) in Eq.
5
is called the modiﬁed plasma dispersionfunction:
12
Z
z
1
1/2
1
3/2
1/2
dx
x
z
1
x
2
/
1
.
6
The approximation
q
v
p
is quite reliable since it has beenknown that the longitudinal component of the plasma dielec-tric function determines the dynamic polarization potential.
15
Thus, Eq.
5
can be rewritten as
q
,
1
1
q
2
2
1/2
3/2
3/2
z
¯
Z
z
¯
,
7
where
z
¯
/(2
3)
x
1
, the velocity ratio
x
(
v
T
/
v
P
) isthe key parameter for investigating the dynamic screeningeffect on the electron capture process, and
v
T
(
T
e
/
m
)
1/2
.Since the Lorentzian distribution becomes the Maxwelliandistribution in the limit as
˜
, the plasma dielectric func-tion, Eq.
7
, reduces to the form for the Maxwellian forlarge spectral indices. By means of contour integration withperturbational calculations in Eqs.
3
and
7
, the electroncapture distance in the Lorentzian distribution plasma can beobtained by
R
R
0
,
8
where
R
0
(
2
Z
P
e
2
/
m
v
p
2
) is the capture distance withoutplasma screening effects and
1
12
R
0
,Re
z
¯
18
R
0
2
,Re
z
¯
,
9
with
(
z
¯
)
/(
3/2)
(
1/2)/
z
¯
Z
(
z
¯
)
and
,Re
is the real part of
. The parameter
represents the dy-namic plasma screening effects on the electron capture radiusin the Lorentzian distribution plasma.
III.SEMICLASSICALCAPTUREPROBABILITY
In this section, we consider the semiclassical method toevaluate electron capture probability by the proton projectile
3675Phys. Plasmas, Vol. 6, No. 9, September 1999 Electron capture in the Lorentzian distribution plasma
Downloaded 24 Oct 2008 to 202.56.207.52. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
from hydrogenic target ions in the Lorentzian distributionplasma. For heavy ion projectiles, the projectile path can bedescribed by the straight-line trajectory, i.e.,
r
(
t
)
byˆ
v
P
tzˆ
. This impact parameter method has a strong appeal inaiding the physical intuition. Usually, calculations based onthe impact parameter method are mathematically more trac-table than fully quantum mechanical treatments.
16
Thus, thecollision time
t
c
in Eq.
2
is given by 2(
R
2
2
)
1/2
/
v
P
where
is the distance between the projectile ionand the released electron. Then, the electron capture prob-ability from the 1
s
ground state of the hydrogenic target ionis found to be
P
2
Z
T
e
2
v
P
R
R
d
3
r
1
s
r
2
1
/
R
2
r
.
10
Since
1
(
/
R
)
2
is a slowly varying function, it can beapproximated by its average value over the integral region,
12
1
/
R
2
1
R
2
0
R
d
1
/
R
2
13.
11
Since the main purpose of this paper is the investigationof the dynamic plasma screening effects on the electron cap-ture probability using the plasma dielectric function, the un-screened atomic bound wave function is retained throughoutthis paper. For hydrogenic ions, the unscreened bound statewave function
17
is given by
1
s
r
1
a
Z
T
3/2
e
r
/
a
Z
T
,
12
where
a
Z
T
(
a
0
/
Z
T
) is the ﬁrst Bohr radius of target ionwith nuclear charge
Z
T
. Thus the electron capture probabil-ity from the 1
s
target electron is given by
P
2
Z
T
e
2
3
v
P
R
a
Z
T
2
I
a
Z
T
,
b
,
R
,
13
where the integral
I
(
a
Z
T
,
b
,
R
) is given by using
r
z
b
:
I
a
Z
T
,
b
,
R
R
d
3
r
e
2
r
/
a
Z
T
r
0
R
d
02
d
dze
2
r
(
,
,
z
)/
a
Z
T
r
,
,
z
.
14
After some algebra using the inverse Fourier transforma-tion, the integral
I
(
a
Z
T
,
b
,
R
) is found to be
I
a
Z
T
,
b
,
R
4
R
0
dq
J
0
q
b
J
1
q
R
q
2
2/
a
Z
T
2
,
15
where
b
is the impact parameter,
q
is the perpendicularcomponent of the momentum transfer
q
, and
J
n
is the
n
thorder Bessel function. Finally, we obtain the scaled semiclas-sical electron capture probability including the dynamicplasma screening effects in the Lorentzian distributionplasma as the following form:
b
¯
ã
P
b
¯
83
R
¯
02
v
¯
,Re2
b
¯
0
dQ J
0
Qb
¯
J
1
QR
¯
0
,Re
Q
2
4,
16
where
Q
q
a
Z
,
R
¯
0
(
R
0
/
a
Z
T
)is the scaled electron cap-ture radius without including the plasma screening effects,and
b
¯
(
b
/
a
Z
T
) is the scaled impact parameter.In order to explicitly investigate the dynamic plasmascreening effects on the electron capture probability by theproton projectile (
Z
P
1) from the hydrogenic target ionwith nuclear charge
Z
T
2, we consider two cases of thecollision velocity
v
P
:
v
P
/
v
Z
T
1
intermediate energy
and3
high energy
, where
v
Z
T
(
Z
T
e
2
/
) is the electron orbitalvelocity in the target ion, since the Bohr–Lindhard model isknown to be valid for
v
p
and
v
Z
T
Refs. 2 and 5
and weconsider two cases of the velocity ratio:
x
4/5 and 5/4.Tables I and II show numerical values of the maximum elec-tron capture probabilities by the proton projectile from thehydrogenic ion in Lorentzian distribution plasma includingthe dynamic plasma screening effects for
a
0.1. Thescaled electron capture probabilities in the Maxwellian dis-tribution plasma are also given in Tables I and II. The posi-tions of the maximum capture probabilities are also indicatedin parentheses. As we see in Tables I and II the electroncapture probability in the Maxwellian distribution plasma isfound to be always greater than that in the Lorentzian distri-bution plasma. It is found that the semiclassical captureprobability including the dynamic plasma screening effect isincreased with increasing the spectral index
. It is alsofound that the plasma screening effects on the electron cap-ture probabilities are more effective for low projectile veloci-ties. The dynamic plasma screening effect on the electroncapture probability in the Lorentzian distribution plasma isstronger than that on the Maxwellian distribution plasma,
TABLE I. The maximum values of the scaled semiclassical electron captureprobabilities for
a
0.1 and
v
P
/
v
Z
T
1. Here
a
(
a
Z
/
) is the scaledreciprocal Debye length,
v
P
is the projectile velocity, and
v
Z
T
is the electronorbital velocity in the target ion, and
v
T
(
T
e
/
m
)
1/2
.
v
T
/
v
P
5/4
v
T
/
v
P
4/5
b
¯
ã
P
(
2) 2.193 08
10
1
(
b
¯
0.75
2.194 77
10
1
(
b
¯
0.75
b
¯
ã
P
(
3) 2.315 12
10
1
(
b
¯
0.77
2.320 67
10
1
(
b
¯
0.77
b
¯
ã
P
(
5) 2.359 65
10
1
(
b
¯
0.77
2.366 45
10
1
(
b
¯
0.77
b
¯
ã
P
(
) 2.487 62
10
1
(
b
¯
0.79
2.648 97
10
1
(
b
¯
0.80
TABLE II. The maximum values of the scaled semiclassical electron cap-ture probabilities for
a
0.1 and
v
P
/
v
Z
T
3. Here
a
(
a
Z
/
) is thescaled reciprocal Debye length,
v
P
is the projectile velocity, and
v
Z
T
is theelectron orbital velocity in the target ion, and
v
T
(
T
e
/
m
)
1/2
.
v
T
/
v
P
5/4
v
T
/
v
P
4/5
b
¯
ã
P
(
2) 1.391 86
10
4
(
b
¯
0.30
1.392 00
10
4
(
b
¯
0.30
b
¯
ã
P
(
3) 1.401 49
10
4
(
b
¯
0.30
1.401 92
10
4
(
b
¯
0.30
b
¯
ã
P
(
5) 1.404 90
10
4
(
b
¯
0.30
1.405 41
10
4
(
b
¯
0.30
b
¯
ã
P
(
) 1.414 40
10
4
(
b
¯
0.30
1.425 81
10
4
(
b
¯
0.30
3676 Phys. Plasmas, Vol. 6, No. 9, September 1999 Y.-D. Jung and J.-S. Yoon
Downloaded 24 Oct 2008 to 202.56.207.52. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
since the Debye length (
) in the Lorentzian plasma issmaller than the Debye length
in the Maxwellian plasma.The difference between that dynamic plasma screening ef-fects on the Lorentzian and Maxwellian distribution plasmasis found to be more signiﬁcant when the velocity ratio
x
issmaller than unity, i.e., when the projectile velocity issmaller than the velocity of the plasma electron. For highenergy projectiles, the difference between the electron cap-ture processes in the Lorentzian and Maxwellian distributionplasmas is found to be quite small.
IV.CONCLUSIONS
We investigate the electron capture process from hydro-genic ions by protons in generalized Lorentzian
kappa
ve-locity distribution plasmas. The semiclassical Bohr–Lindhard model is applied to obtain the electron captureradius and capture probability. The dynamic electron–ion in-teraction potential is obtained by introduction of the plasmadielectric function in the Lorentzian distribution plasma. Thestraight-line trajectory approximation is applied to the mo-tion of the projectile ion to obtain the electron capture prob-ability as a function of impact parameter, collision energy,and spectral index
. The results show that the semiclassicalcapture probability including the dynamic plasma screeningeffect is increased as an increase of the spectral index
. It isalso found that the plasma screening effects on the electroncapture probabilities are more effective for low projectilevelocities. The dynamic plasma screening effect on the elec-tron capture probability in the Lorentzian distribution plasmais found to be stronger than that on the Maxwellian distribu-tion plasma, since the effective Debye length in the Lorent-zian plasma is smaller than the Debye length in the Maxwell-ian plasma. The difference between the dynamic plasmascreening effects on the Lorentzian and Maxwellian distribu-tion plasmas is found to be more signiﬁcant when the pro- jectile velocity is smaller than the velocity of the plasmaelectron. These results provide useful information for elec-tron capture processes in non-Maxwellian distribution plas-mas.
ACKNOWLEDGMENTS
This work was supported by the Korea Science and En-gineering Foundation through Grant No. 981-0205-016-2and by the Korea Research Foundation through the BasicScience Research Institute Program
1998-015-D00128
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3677Phys. Plasmas, Vol. 6, No. 9, September 1999 Electron capture in the Lorentzian distribution plasma
Downloaded 24 Oct 2008 to 202.56.207.52. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp

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