K-shell Photorecombinations in Strongly Coupled Plasmas

PHYSICS OF PLASMAS VOLUME 7, NUMBER 10 OCTOBER 2000

K-shell photorecombinations in strongly coupled plasmasYoung-Dae Junga)

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

~Received 2 May 2000; accepted 5 July 2000!

Plasma screening effects on the K-shell photorecombination of the free electron with the ion instrongly coupled plasmas are investigated. The recombination cross section is obtained by theprinciple of detailed balance with the photoionization cross section of the hydrogenic ion. It is foundthat the plasma screening effects on the recombination cross section are found to be less than 20%when the ion-sphere radius (RZ) is greater than two times of the Bohr radius (aZ) of the hydrogenicion with nuclear chargeZ. © 2000 American Institute of Physics.@S1070-664X~00!03710-1#

BRIEF COMMUNICATIONSThe purpose of this Brief Communications section is to present important research results of more limited scope than regulararticles appearing in Physics of Plasmas. Submission of material of a peripheral or cursory nature is strongly discouraged. BriefCommunications cannot exceed four printed pages in length, including space allowed for title, figures, tables, references, and anabstract limited to about 100 words.

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The photorecombination process1–5 is a subject of spe-cial attention in many areas of physics, such as atomicplasma physics, because it is the inverse process to phionization. This photorecombination also has important csequences for x-ray astronomy, since electron captureions is one of the important continuum x-ray emissimechanisms. The photorecombination cross sections byin strongly coupled plasmas must be different from thosefree ions due to the surrounding plasma screening effectthe initial and final states of the electron. A most typicexample of strongly coupled classical plasma may be seethe system of ions inside a highly evolved star.6 The ion-sphere model7,8 has played an important part in elucidatinthe properties of the strongly coupled plasma. Thus, inBrief Communication we investigate the plasma screeneffects on the K-shell photorecombination processes byin strongly coupled plasmas using the ion-sphere mosince the theoretical atomic spectroscopy is essential instudy of plasma parameters. The inverse process to phocombination is the photoionization process, consistingphoton absorption and emission of a bound electron intcontinuum. Since we are dealing here with binary collisio

A1z111e2↔A1z1\v, ~1!

the cross sections~spr : photorecombination,spi : photoion-ization! of both processes have the same dimensions andmutually related by the principle of detailed balance,

spr

spi5

j \v

j e2

gpr

gpi, ~2!

where j \v(5c/L3) and j e2(5v/L3) are the fluxes of pho-tons and electrons for the corresponding channels of thecesses, normalized to one photon per given cubic volu

a!Electronic mail: [email protected]

4341070-664X/2000/7(10)/4347/3/$17.00

Downloaded 24 Oct 2008 to 202.56.207.52. Redistribution subject to AIP

dto--by

nsf

onlin

isgsl,

here-fa,

re

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(L3), wherec is the speed of the light andv is the speed ofthe electron. In Eq.~2!, gpr@52gA1z4pk2dk/(2p)3# andgpi@5gA1z11ge24pq2dq/(2p)3# are, respectively, the statistical weights of the final states for photorecombination aphotoionization processes, wheregA1z, gA1z11, andge2 arethe statistical weights of the internal state of the particlesk andq are the wave numbers of the photon and the electrrespectively. After some algebra, the cross section forphotorecombination of the free electron to the bound statthe ion with chargeZ is found to be

spr5Z2a2

2

~E1Eb!2

Espi , ~3!

where a(5e2/\c' 1137) is the fine structure constant,E

([mv2/2Z2 Ry) is the scaled electron energy,Eb

([Eb /Z2 Ry) is the scaled binding energy, Ry(5me4/2\2

'13.6 eV) is the Rydberg constant, andm is the electron restmass. The recombination into the ground state has the gest probability, due to then23 dependence of the spectruon the principal quantum numbern. For the recombination ofthe free electron to the 1s bound state of the ion with chargZ in strongly coupled plasmas with the ion-sphere radiusRZ ,the binding energyEb(5uE1su) can be given by the receninvestigation9 for the 1s ground state energy and wave funtion using the Ritz variation method including the plasmscreening effect:

Eb5Z82 Ry5Z2~12 d1s!2 Ry, ~4!

whereZ8@5Z(12 d1s)# is the 1s effective charge,d1s is the1s screening constant:

d1s>~121/Z!~aZ /RZ!3

123~121/Z!~aZ /RZ!3 , ~5!

7 © 2000 American Institute of Physics

license or copyright; see http://pop.aip.org/pop/copyright.jsp

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4348 Phys. Plasmas, Vol. 7, No. 10, October 2000 Young-Dae Jung

FIG. 1. The 1s photorecombinationcross sectionss1s

pr in units of pa02 for

Z52 as functions of the scaled electron energy E(5mv2/2Z2 Ry). Thedotted line represents the photorecombination cross section forRZ52, i.e.,including the plasma-screening effects. The solid line represents thphotorecombination cross section foRZ5`, i.e., neglecting the plasmascreening effects.

-

ly , re-

and aZ([a0 /Z5\2/Zme2) is the Bohr radius of a hydrogenic ion with nuclear chargeZ. Then the recombinationcross section to the 1s bound state is given by

s1spr 5

Z2a2

2E@E1~12 d1s!

2#2s1spi , ~6!

wheres1spi is the 1s photoionization cross section in strong

coupled plasmas described by the ion-sphere potential:


V~r !52Ze2

r F12r

2RZS 32

r 2

RZ2D Gu~RZ2r !, ~7!

and u(x) ~51 for x>0; 50 for x,0! is the step function.The 1s photoionization cross section10 s1s

pi including thescreening correction on the bound and continuum statestardation correction, and Coulomb correction:

e ordertron

s1spi 5s1s

pi ~ e,Z,RZ!529p2

3h1s

23~12 d1s!aaZ2e24$12Z2a2e2/@ e2~12 d1s!

2#%22

3e24@ e2~12 d1s!2#21/2 cot21@ e2~12 d1s!2#21/2

12e22p@ e2~12 d1s!2#21/2 A e21

e2~12 d1s!2 ~ tan21 Ae21!22u I 100

~A!~a1s ,q,RZ!u2, ~8!

where the parameterh1s21@512(121/Z)(aZ /RZ)3# represents the plasma-screening effect on the 1s Bohr radius and

e([\v/Z2Ry) is the scaled photon energy. Here,I 100(A)(a1s ,q,RZ) is the dipole acceleration matrix element:

I 100~A!~a1s ,q,RZ!52

6a1s4 q22a1s

6 q3

~RZ1a1s2 q2RZ!3 1tan21~qa1s!1

i

2@Ei~2RZ /a1s2 iqRZ!2Ei~2RZ /a1s1 iqRZ!#

1a1s

2 qe2RZ /a1s

~RZ1a1s2 q2RZ!3 $@4a1sRZ14a1s

3 q2RZ1RZ21a1s

4 q2~221q2RZ2!12a1s

2 ~31q2RZ2!#cos~qRZ!

1@2a1sRZ22a1s5 q4RZ1RZ

21a1s4 q2~261q2RZ

2!12a1s2 ~11q2RZ

2!#sin~qRZ!%, ~9!

where Ei(x)(5*2`x dt et/t) is the exponential integral function11 anda1s(5h1saZ) is the effect Bohr radius of the 1s electron

in strong coupled plasmas. In obtaining Eq.~8!, we restricted ourselves to hydrogenic wave functions withZa!1, so that therelativistic effect for the bound state wave function was neglected, since the relativistic corrections are only of relativ(Za)2. Using the energy conservation,E1(12 d1s)

25 e, the 1s photorecombination cross as a function of the scaled elecenergy (E) is then found to be


4349Phys. Plasmas, Vol. 7, No. 10, October 2000 K-shell photorecombinations in strongly coupled . . .

s1spr ~E,Z,RZ!/pa0

2528p

3a3h1s

23~12 d1s!AE211~12 d1s!

2

E3/2@E1~12 d1s!2#2 S 12

Z2a2@E1~12 d1s!2#2

ED 22

3e24E21/2

cot21E21/2

12e22pE21/2 @ tan21 AE211~12 d1s!2#22U2

6h1s4 E1/222h1s

6 E3/2

~RZ1h1s2 ERZ!3

1tan21~Eh1s!1i

2@Ei~2RZ /h1s2 iE1/2RZ!2Ei~2RZ /h1s1 iE1/2RZ!#

1h1s

2 E1/2e2RZ /h1s

~RZ1h1s2 ERZ!3 $@4h1sRZ14h1s

3 ERZ1RZ21h1s

4 E~221ERZ2!12h1s

2 ~31ERZ2!#cos~E1/2RZ!

1@2h1sRZ22h1s5 E2RZ1RZ

21h1s4 E~261ERZ

2!12h1s2 ~11ERZ

2!#sin~E1/2RZ!%U2

, ~10!

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where RZ([RZ /aZ) is the scaled ion-sphere radius. Hethe plasma screening effects on the continuum state

@2p~12d1s /Z!1/2/ e1/2#

3e24@ e2~12 d1s!2#21/2 cot21@ e2~12 d1s!2#21/2/

~12e22p@ e2~12 d1s!2#21/2!

is reliable forRZ.aZ , since the exponential terme2r /aZ issmall for r 5RZ . Hence, the screening effects on the cotinuum and bound states obtained using the ion-spheretential Eq.~7! are included in Eq.~10!.

In order to explicitly investigate the total plasma screeing effects on the 1s photorecombination cross section, spcifically, we consider five cases of the ion-sphere radiRZ52, 4, 6, 12, and , and we assume thatZ52, since ournonrelativistic result@Eq. ~10!# is valid for Za!1. Figure 1shows the photorecombination cross section for the eleccapture into the 1s bound state as a function of the scalelectron energyE([mv2/2Z2 Ry). The unscreened photorecombination cross section is also illustrated. As we canin this figure, the plasma screening effects are almost inpendent of the incident electron energy for a given iosphere radius. The numerical values of the photorecombtion cross sections in units ofpa0

2 are also listed in Table IIt is noted that the plasma screening effects reduce the

TABLE I. The numerical values of the 1s photorecombination cross sections s1s

pr in units of pa02 for Z52.

RZ s1spr (E550)a (pa0

2) s1spr (E5250)b (pa0

2)

2 5.0963 1029 1.1433 10210

4 6.4693 1029 1.4553 10210

6 6.6123 1029 1.4883 10210

12 6.6653 1029 1.5003 10210

` 6.6723 1029 1.5013 10210

aThe photorecombination cross sections forE(5mv2/2Z2 Ry)550.bThe photorecombination cross sections forE(5mv2/2Z2 Ry)5250.


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torecombination cross sections~e.g.,'23% forRZ52, '3%for RZ54!.

Summarizing, we have obtained the plasma screeneffects on the K-shell photorecombination of the free eltron to the 1s bound state of the hydrogenic ion in strongcoupled plasmas. The charged particle interaction potenin plasmas is given by the ion-sphere potential. The plasscreening effect on the photorecombination cross sectioobtained as a function of the ion-sphere radius and elecenergy. It is found that the plasma screening effects onphotorecombination cross section for interesting domainthe ion-sphere radius,RZ>2aZ , are less than 20%. Thesresults provide useful information for photorecombinatiprocesses in strongly coupled plasmas.

ACKNOWLEDGMENTS

The author is grateful to Professor R. M. More and Pfessor K. Fujima for useful discussions. The author wolike to thank the anonymous referee for suggesting improments to this text.

This work was supported by the Korean MinistryEducation through the Brain Korea~BK21! Project, by theKorea Basic Science Institute through the HANBIT User Dvelopment Program~FY2000!, and by Hanyang UniversitySouth Korea, made in the program year of 2000.

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5V. P. Krainov, H. R. Reiss, and B. H. Smirnov,Radiative Processes inAtomic Physics~Wiley, New York, 1997!, Chap. 3.

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7E. E. Salpeter, Aust. J. Phys.7, 373 ~1954!.8D. Salzmann,Atomic Physics in Hot Plasmas~Oxford University Press,Oxford, 1998!, Chap. 2.

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York, 1996!, Chap. 19.


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K-shell Photorecombinations in Strongly Coupled Plasmas