Electron Captures by Positrons From Hydrogenic Ions in Nonideal Classical Plasmas

8/10/2019 Electron Captures by Positrons From Hydrogenic Ions in Nonideal Classical Plasmas

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Eur. Phys. J. D 12 , 351354 (2000) THE EUROPEANP HYSICAL JOURNAL Dc

EDP SciencesSociet` a Italiana di FisicaSpringer-Verlag 2000

Electron captures by positrons from hydrogenic ionsin nonideal classical plasmasYoung-Dae Jung a

Department of Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea

Received 21 January 2000 and Received in nal form 27 April 2000

Abstract. In nonideal classical plasmas, the electron captures by positrons from hydrogenic ions are inves-tigated. An effective pseudopotential model taking into account the plasma screening effects and collectiveeffects is applied to describe the interaction potential in nonideal plasmas. The classical Bohr-Lindhardmodel has been applied to obtain the electron capture radius and electron capture probability. The modiedhyperbolic trajectory method is applied to the motion of the projectile positron in order to visualize theelectron capture probability as a function of the impact parameter, nonideal plasma parameter, projectilevelocity, and plasma parameters. The results show that the electron capture probability in nonideal plas-mas is always greater than that in ideal plasmas descried by the Debye-H uckel potential, i.e. , the collectiveeffect increases the electron capture probability. It is also found that the collective effect is decreased withincreasing the projectile velocity.

PACS. 52.20.-j Elementary processes in plasma

1 Introduction

Electron capture process in atom-charged particle colli-sions has been of great interest since this process is oneof the basic processes in atomic collision physics [1]. Theelectron capture processes have been investigated widelyusing various methods [26] depend on the state of theprojectile and target system. Collision processes involv-ing the positrons and electrons have received many at-tentions since theses processes have many applicationsin atomic, plasma physics, and astrophysics. Especially,the annihilation of positrons has been widely investigatedin astrophysical plasmas [79]. The theoretical investiga-tions for positron-electron direct annihilation and positronformation also have been widely performed. Recently, in

weakly coupled ideal plasmas, the electron capture bypositron from target atom was investigated using theBohr-Lindhard (BL) model with the Debye-H uckel poten-tial [10]. When the relative interaction velocity vP of theprojectile is greater than the ground state orbital velocityZ (= Ze 2 / ) of the hydrogenic ion with nuclear chargeZ , i.e. , intermediate and high energy projectiles, the clas-sical BL model has been known to be quite reliable sincethe de Broglie wave length of the projectile is smaller thanthe collision diameter for the capture interaction [3,4]. TheDebye-H uckel effective potential describes the propertiesof a low density plasma and corresponds to a pair correla-tion approximation. The plasmas descried by the Debye-

Huckel model can be called the ideal plasmas since thea e-mail: [email protected]

average energy of interaction between particles is smallcompared to the average kinetic energy of a particle [11].

However, multiparticle correlation effects caused by simul-taneous interaction of many particles should be taken intoaccount with increasing the plasma density. It is necessaryto take into account not only short-range collective effectsbut also long range effects in the case of a plasma with amoderate density and temperature. In this case, the inter-action potential cannot be described by the Debye-H uckelmodel because of the strong collective effects of nonidealparticle interaction [12]. Then, the electron capture prob-ability by the positron projectile from the hydrogenic iontarget in nonideal plasmas would be different from that inideal plasmas. Thus, in this paper we investigate electroncapture processes in positron-hydrogenic ion collisions innonideal plasmas. A pseudopotential model including theplasma screening effects and collective effects is appliedto describe the interaction potential between the projec-tile positron and the target ion in nonideal plasmas. Theclassical BL model has been applied to obtain the elec-tron capture radius and electron capture probability. Themodied hyperbolic trajectory method is applied to themotion of the projectile positron in order to visualizethe electron capture probability as a function of the im-pact parameter, nonideal plasma parameter, projectile ve-locity, and plasma parameters.

In Section 2, we derive the electron capture radiusand electron capture cross-section by positrons from hy-drogenic ions in nonideal plasmas using the BL modeland the modied hyperbolic orbit trajectory method withthe pseudopotential model including the plasma screening


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effects and collective effects. In Section 3, we obtain thescaled electron capture probability as a function of theimpact parameter, nonideal plasma parameter, projectilevelocity, and plasma parameters. We also investigate the

variation of the scaled electron capture probability withchanging the scaled modied impact parameter. The re-sults show that the electron capture probability in non-ideal plasmas is found to be always greater than that inideal plasmas, i.e. , the collective effect increases the elec-tron capture probability. It is also found that the collectiveeffect is decreased with increasing the velocity of the projectile positron. Finally, in Section 4, a discussion is given.

2 Electron capture cross-section

Using the classical trajectory method, the electron capturecross-section can be given by [5]

C = 2 db b P c (b), (1)where b is the impact parameter and P c (b) is the electroncapture probability. In the BL model, the electron cap-ture process happens when the distance between the projectile and a released electron is smaller than the electroncapture radius Rc [24]. This electron capture radius isdetermined by equating the kinetic energy of the releasedelectron in the frame of the projectile and the binding en-ergy provided by the projectile. In the classical BL model,

the electron capture probability is dened by the ratio of the collision time to the electron orbital time

P c (b) = t c

t cdt

, (2)

where t = 0 is arbitrary chosen as the instant at whichthe projectile positron makes its closest approach to thetarget ion and tc is the electron capture time within theelectron capture radius. Here, the electron orbital time in the ground state of the hydrogenic ion with nuclearcharge Z is given by

= aZ

/Z

, (3)

where aZ (= a0 /Z = 2 /me 2 Z 2 ) is the rst Bohr radius.For intermediate and high projectile velocities P Z ,this classical expression of the electron capture cross-section (Eq. (1)) is known to be valid [3,4].

In a recent paper [12], an integro-differential equationfor the effective potential of the particle interaction tak-ing into account the simultaneous correlations of N par-ticles was obtained on the basis of a sequential solutionof the chain of Bogolyubov equations for the equilibriumdistribution function of particles of a classical nonidealplasma and an analytic expression for the pseudopoten-tial of the particle interaction in a nonideal plasmas wasalso obtained by application of the spline-approximation.Using the pseudopotential taking into account the plasma

screening effects and collective effects, the interactionpotential V (r ) between the projectile positron and thetarget ion with charge Z in nonideal plasmas can be rep-resented by

V ( r ) = Ze 2

r er/ 1 + f (r )/ 2

1 + c( ) , (4)

where r is the position vector of the projectile positronfrom the target ion, is the Debye length,

f (r ) = (e r/ 1)(1 e2 r/ )/ 5

and ( e2 /kT e) is the nonideal plasma parameter,

c( ) = 0.008617+ 0 .455861 0.108389 2 + 0 .009377 3

is the correlation coefficient for different values of ,and T e is the electron temperature. When 1, i.e. ,weakly nonideal or rare ideal plasmas, the pseudopoten-tial (Eq. (4)) goes over into the Debye-H uckel potentialV ( r ) (Ze 2 /r )er/ . Using this pseudopotential, theelectron capture radius Rc in nonideal plasmas can beobtained by

e2

RceR c / 1 + f (Rc )/ 2

1 + c( ) = 12

2P , (5)

since the kinetic energy of the released electron in theframe of the projectile positron has to be smaller than thebinding energy provided by the projectile positron, where

(= m/ 2) is the reduced mass and m is the electronmass. After some algebra using the perturbation calcula-tion since the electron capture radius Rc is usually smallerthan the Debye length , the electron capture radius in-cluding the plasmas screening effect and collective effectis found to be

Rc =

R0 /1 + R0 / + c( )

+R0

3 (1/ 2 3 / 2 / 5)[1 + R0 / + c( )]3

,

(6)

where R0 4e2 /m 2P .For projectiles such as electrons and positrons rather

than heavy nuclei, the curved trajectory method must beapplied to describe the projectile motion in Coulomb eldsbecause of the small mass ratios ( m/M 1). For positronprojectiles, the useful parametric representation [13] of thehyperbolic orbit trajectory for r (t) in x-y plane is given by

x = d (2 1)1 / 2 sinh w,y = d (cosh w ),

r (t) | r (t)| = d( cosh w + 1) ,

t = dP

( sinh w + w), < w < (7)

where d and are half of the distance of closest approachin a head-on collision and the eccentricity, respectively. In-cluding the plasma screening effects and collective effects,


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Young-Dae Jung: Electron captures by positrons from hydrogenic ions in nonideal classical plasmas 353

the parameters d and are given by

d =

d0 /1 + d0 / + c( )

+d0

3 (1/ 2 3 / 2 / 5)[1 + d0 / + c( )]3

, (8)

= (1 + b2 /d 2 )1 / 2 , (9)where d0 Z eff e2 /m 2P and Z eff is the effective charge of the target ion seen by the projectile positron, for example,Z eff = Z 5/ 16 for one-electron atoms. The hyperbolicorbit trajectory method has been widely used to investi-gate the electron-ion collisional impact excitation [14,15]and bremsstrahlung [16] processes. Using equations (2, 7),the electron capture probability using the hyperbolic orbittrajectory method is given by

P c (b) = 2dP

( sinh wc + wc ), (10)

where the parameter wc is determined by the electron cap-ture time tc :

wc = ln (R c /d 1)(1 + b2 /d 2 )1 / 2

+(Rc /d 1)2

(1 + b2 /d 2 ) 1

1 / 2

.

(11)

From equation (11), the maximum impact parameter bmaxis given by (R 2c 2Rc d)1 / 2 . However, this is physically un-reasonable since the electron capture process is also possi-ble in the region ( R2c 2Rc d)1 / 2 b Rc . Thus, we haveto modify the impact parameter in order to obtain thecorrect electron capture probability and electron capture

cross-section. In the Bohr-Lindhard model, the electronrelease radius Rr is taken to be that the initial electronenergy should be equal to the height of the electrostaticpotential barrier given by the total potential energy actingon the electron. For collisions between the projectile ionwith charge z and hydrogenic ion target with charge Z ,the electron release radius Rr is given by 2(1+2 z/Z )aZ ,which is the independent of the projectile velocity. WhenRc > R r , i.e. , low velocities, the electron capture cross-section is geometrical with the maximum of the cross-section C = R 2r [17]. For high velocities, the electroncapture process only happens when the distance betweenthe projectile and a release electron is smaller than theelectron capture radius Rc [4]. Physically, the maximumimpact parameter should be equal to the electron captureradius R c since the electron capture probability has beendened by the electron capture time. Since the electroncapture process is only proceeded within the electron cap-ture sphere with the radius Rc , after some straightforwardmanipulations, the modied impact parameter b can beobtained by the shortest distance from the tangent [16] atthe position [ x(wc ), y(wc )]:

b = d(2 1)1 / 2 cosh wc + 1 cosh wc 1

1 / 2

, (12a)

= b(1 2d/R c )1 / 2 . (12b)From equations (11, 12b), we can readily show that themodied maximum impact parameter becomes bmax = Rc .

For b > R c , the trajectory of the projectile would bea straight line since the capture probability is zero fort > |t c | . However, for b Rc , i.e. , t | t c | , the trajectoryof the projectile would be then the screened hyperbolic

orbit. Hence, our modication on the impact parameter isquite reliable and can be applied to investigate the elec-tron capture probability.

3 Electron capture probability

From equations (10, 11, 12b), the electron capture prob-ability as a function of the modied impact parameter isgiven by

P c (b ) = 2dP

Rcd

R cd

2 1 b 2

R2c

1 / 2

12

ln 1 + R c

dRcd

2b 2

R2c

+ lnR cd

1 +Rcd

Rcd

2 1 b 2

R2c

1 / 2

.

(13)

Then, the electron capture cross-section (Eq. (1)) can berewritten as

C = 2R 2c

db b P c (b ), (14)

where b ( b /R c ) the scaled modied impact parameter.Thus, the scaled differential electron capture cross-sectionin units of R 2c becomes

dC / dbR 2c

= 2 b P c (b ) (15a)

= 22RcP

b P c (b , R c /d ). (15b)

Here, b P c (b , R c /d ) is the scaled electron capture proba-bility including the plasma screening effects and collective

effects through the radio R c /d (Eqs. (6, 8)).In order to investigate the plasma screening effects andcollective effects on the electron capture probability by thepositron projectile we consider the two cases of the projectile velocity P / Z = 1 and 3 since the classical BLmodel is known to be valid for intermediate and high en-ergy projectiles [3,4] and we choose a = 0 .1 and = 1.Figures 1 and 2 show the scaled electron capture proba-bilities b P c (b , R c /d ) by the positron projectile from thehydrogenic ion target with nuclear charge Z = 2 as func-tions of the scaled impact parameter including the plasmascreening effects and collective effects for P / Z = 1 and 3,respectively. As we can see in these gures, the electroncapture probability in nonideal plasmas ( = 1) describedby the pseudopotential is always greater than that in idealplasmas ( = 0) descried by the Debye-H uckel potential.


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354 The European Physical Journal D

0 0.2 0.4 0.6 0.8 1

Scaled Impact Parameter

0

0.1

0.2

0.3

0.4

0.5

0.6

d e l a c S

e r u t p a C

y t i l i b a b o r

P

Fig. 1. The scaled electron capture probability b P c (b , R c /d )by the positron from the hydrogenic ion with nuclear chargeZ = 2 as a function of the scaled impact parameter b ( b /R c )for a = 0 .1 and P / Z = 1. The solid line represents the elec-tron capture probability in ideal plasmas, i.e. , = 0. The dot-ted line represents the electron capture probability in nonidealplasmas with = 1.

Thus, it is found that the nonideality ( = 0) of plasmas,i.e. , the collective effect, increases the electron captureprobability. It is also found that the collective effect isdecreased with increasing the projectile velocity.

4 Discussion

We investigate the plasma screening effects and collec-tive effects on the electron captures by positrons fromhydrogenic ions in nonideal classical plasmas. An effec-tive pseudopotential model taking into account the plasmascreening effects and collective effects is applied to de-scribe the interaction potential in nonideal plasmas. Theclassical Bohr-Lindhard model has been applied to obtainthe electron capture radius and electron capture proba-bility using the modied hyperbolic trajectory method.The scaled electron capture probability by positrons fromhydrogenic ions is obtained as a function of the impactparameter, nonideal plasma parameter, projectile veloc-ity, and plasma parameters. It is found that the electroncapture probability in nonideal plasmas is always greaterthan that in ideal plasmas descried by the Debye-H uckelpotential, i.e. , the collective effect increases the captureprobability. It is important to note that the collective ef-fect is decreased with increasing the velocity of the projectile positron. These results provide a useful informationof the electron capture processes by positrons from targetions in nonideal classical plasmas.

The author gratefully acknowledges Prof. Richard M. Moreand Dr. Izumi Murakami for useful comments and stimulatingdiscussions. The author would like to thank the anonymousreferee for suggesting improvements to this text. This workwas supported by the Korean Ministry of Education throughthe Brain Korea (BK21) Project, by the Korea Basic Science

0 0.2 0.4 0.6 0.8 1

Scaled Impact Parameter

0

0.2

0.4

0.6

0.8

d e l a c S

e r u t p a C

y t i l i b a b o r

P

Fig. 2. The scaled electron capture probability b P c (b , R c /d )by the positron from the hydrogenic ion with nuclear chargeZ = 2 as a function of the scaled impact parameter b (b /R c ) for a = 0 .1 and P / Z = 3. The solid line representsthe electron capture probability in ideal plasmas, i.e. , = 0.The dotted line represents the electron capture probability innonideal plasmas with = 1.

Institute through the HANBIT User Development Program(FY2000), and by Hanyang University, South Korea, made inthe program year of 2000.

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Electron Captures by Positrons From Hydrogenic Ions in Nonideal Classical Plasmas