Electron-Impact Excitations of Hydrogenic Ions in Strongly Coupled Plasmas

8/10/2019 Electron-Impact Excitations of Hydrogenic Ions in Strongly Coupled Plasmas

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THE ASTROPHYSICAL JOURNAL, 530:10851090, 2000 February 202000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

ELECTRON-IMPACT EXCITATIONS OF HYDROGENIC IONS IN STRONGLY COUPLED PLASMAS

YOUNG-DAEJUNGDepartment of Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea; yjung=bohr.hanyang.ac.kr

AND

JUNG-SIKYOONMax-Planck-Institute for Physics of Complex Systems, Strasse 38, D-01187 Dresden, Germany;No thnitzer ysyoon=mpipks-dresden.mpg.de

Received 1999 June 24; accepted 1999 October 6

ABSTRACTPlasma screening eects are investigated on electron-ion collisional excitations in strongly coupled

plasmas. Scaled cross sections for excitation is obtained using the ion-sphere model1s ] 2p0(m\0)

potential with the close-encounter eects. Ground and excited bound wave functions in strongly coupledplasmas are obtained using the Ritz variation and perturbation methods. The semiclassical straight-linetrajectory approximation is applied to the motion of the projectile electron in order to investigate thevariation of the transition probabilities as a function of the impact parameter and ion-sphere radius. Thetransition probability neglecting the screening eects on the target atomic wave functions is found to begreater than that including the screening eects on the target atomic wave functions. It is shown that thetarget screening eects are slightly decreased with an increase of the projectile energy.Subject headings:atomic processes plasmas

1. INTRODUCTION

The detailed study of the motion of electrons in the eldof a nucleus has been made possible by quite recent devel-opments in experimental and calculational techniques.Moreover, electron-ion collisional excitation in denseplasmas (Shevelko & Vainshtein 1993; Jung 1993a; Jung1995a; Yoon & Jung 1996) has received much attentionsince it has applications in many areas of physics, such asatomic physics, astrophysics, and plasma physics. Denseplasmas in laboratory and astrophysical environments canbe classied as weakly and strongly coupled plasmasaccording to the strength of coupling due to Coulomb inter-action in plasmas. The plasmas with values of the plasmacoupling parameter much smaller than unity, ![\ (e2/b)/kT], may be called weakly coupled plasmas; those with theplasma coupling parameter around or greater than unity,where b and T are the interparticle distance and tem-perature, may be called strongly coupled plasmas. Adescription of the strongly coupled plasmas is provided bythe ion-sphere (or Wigner-Seitz) model (Salpeter 1954; Ich-imaru 1986; Kobzev & Iakubov 1995; Jung & Jeong 1996).Astrophysical dense plasmas are those we nd in the inte-riors, surfaces, and outer envelops of astrophysical objects,such as neutron stars, white dwarfs, the Sun, etc. The ion-sphere model has played an important part in elucidatingthe properties of the strongly coupled plasmas (Salpeter1954; Kobzev & Iakubov 1995). The opacity of stellarmatter is a quantity of fundamental importance in the equa-tions of stellar structure. Quantitative analyses of thespectra of astronomical sources of photons and of theatomic processes that populate the atomic energy levels andgive rise to the observed absorption and emission requireaccurate data on transition frequencies, wave functions,transition probabilities, and cross sections. The electron-ioncollision processes in the stellar interior are stronglyaected by the screening of the dense plasma electrons andthe bound-electron wave functions are no longer Coulom-bic. Thus, the emission due to the electron-impact excita-tion of ions in the stellar interior may dier considerably

from that in the stellar atmosphere. We shall carefully inves-tigate the character in strongly coupled plasmas, such asstellar interiors, using a screened ion-sphere model inter-action potential. In particular, we shall investigate theplasma screening eects on electron-impact excitation ofhydrogenic ions in strongly coupled plasmas since there arean enormous number of free electrons produced by ioniza-tion of atoms in hot and dense plasmas. Spectral linesemitted from highly charged ions play an important role inthe diagnostic of hot and dense plasmas. Thus, we obtainedthe screened ground and excited wave functions and itseigenenergies of hydrogenic ions with the ion-sphere poten-tial using the Ritz variation and perturbation methods. Wealso use the straight-line (SL) trajectory method in the semi-classical approximation (SCA) to visualize the motion of theprojectile electron as a function of the impact parameterand ion-sphere radius. There was a similar work using thedipole approximation for inelastic electron-impact excita-tions in strongly coupled plasmas (Jung 1995a). However, inthat paper the maximum position of the transition prob-ability and the target screening eects could not be investi-gated since the impact parameter (b) range was restrictedsuch as where is the rstb 4a

Z, a

Z(4 a

0/Z\ +2/Zme2)

Bohr radius of a hydrogenic ion with nuclear charge Z.Thus, in this paper we investigate the target screening eectson electron-ion collisional excitations in strongly coupledplasma including the close-encounter eects. The resultsshow that the target screening eects are signicant andcannot be neglected in excitation processes in stronglycoupled plasmas. The target screening eects are slightlydecreased with an increase of the energy of the projectileelectron. It is also found that the maximum position of thetransition probability is slightly shifted to the target nucleuswhen the target screening eects are included. The screenedwave functions and energy levels will be quite useful toobtain the absorption and emission coefficients in denseastrophysical plasmas such as in the stellar interior. Theseresults provide a general description of the atomic tran-sition probabilities in strongly coupled plasmas.

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In 2, the target wave functions (1s, 2p) of a hydrogenicion in strongly coupled plasmas are obtained by the Ritzvariation and perturbation methods. Also, its eigenenergiesare obtained. In 3, we obtain a closed form of the 1s ] 2p

0dipole transition probabilities including the close-encountereects and investigate the variation of the transition prob-abilities with a charge of the plasma screening eectsthrough the ion-sphere radius. Finally, in 4, conclusionsare given.

2. SCREENED TARGET WAVE FUNCTIONS AND ENERGIES

Equations2.1. SchrodingerWhen an atom is embedded in plasmas, its wave func-

tions and energies are dierent from those of a free atombecause the nucleus is shielded by the surrounding plasma.In the strongly coupled region (! 1), the concept ofDebye screening as a cooperative phenomenon is no longerapplicable. In this case, the ion-sphere model is known to bequite reliable to describe the interaction potential in strong-ly coupled plasmas (Jung 1995a; Jung & Jeong 1996; Jung& Gould 1991). For simplicity, we consider a collisionsystem consisting of the electron incident on a hydrogenic

ion with nuclear chargeZ

. Then, the ion-sphere radius (orthe Wigner-Seitz radius) is given by the neutralizingRZplasma electron,

RZ\

C3(Z[ 1)4nn

e

D1@3, (1)

where is the density of the plasma electron. Since thenetarget ion is surrounded by the plasma electron, the eective

nuclear chargeZ@can be given by

Z@\Z[ d, (2)

where d is the screening constant due to the plasma elec-trons. Thus, the radial equation for a hydrog-Schrodingerenic ions with nuclear chargeZin strongly coupled plasmas

is given byG[

+2

2m

C d2dr2

[l(l]1)

r2

D[

Z@e2r

HPnl

(r)\Enl

Pnl

(r) , (3)

where n and lare, respectively, the principal and orbitalquantum numbers, and is the radial wave function of theP

nlnlth shell. Here, we consider a simple analytic method toobtain the solutions of equation (3), because the simpleanalytic solutions are more convenient to use for calcu-lating the atomic properties and the transition probabilities.The solutions for equation (3) are assumed to be thehydrogenic forms, then the trial 1sand 2p wave functionsare given, respectively,

P1s(r)4 rR1s(r)\2a~2@3re~r@a, (4)

P2p

(r)4 rR2p

(r)\ 1

2J6a~5@2r2e~r@2a, (5)

where a is the variation parameter, and this variationparameter becomes for vanishing plasma eectsa

Z(\a

0/Z)

in strongly coupled plasma. This parameter will(RZ]O)

be determined for the ground and 2p excited states in fol-lowing subsections.

2.2. Ground State(1s)

In the above section, we mentioned the ion-sphere radiusthis means the radius of a sphere with the characteristicR

Z;

volume Thus, the screening correction on thene~1. d

1sground state due to the plasma electron is obtained by theperturbation method, sinced

1s\Z,

d1s\4n

P0

aZ{3(Z[ 1)4nR

Z3

r2dr

+ (Z[ 1)(a

Z/R

Z)3

1[ 3[1[ (1/Z)](aZ

/RZ

)3

] 6

Z2

C (Z[1)(aZ/R

Z)3

M1[ 3[1[ (1/Z)](aZ/R

Z)3N

D3, (6)

where the upper bound is theaZ{

[4 a0/Z@\Za

Z/(Z [d

1s)]

peak position of the 1swave function, i.e., the screened rstBohr radius, since the screening eect is determined by thecharge interior to A recent investigation (Jung & Goulda

Z{.

1991) shows that the wave functions and energies for manyelectron atoms obtained by the screening eects using thecharge interior to the position of the screened electron arequite reliable and in close agreement with the results ofHartree-Fock calculations and with experimental values.Then the appropriate form of the 1s screening constant is

given by

d1s+ (Z[ 1)

AaZ

RZ

B3]

3

Z(Z[1)2

AaZ

RZ

B6. (7)

The expectation value of the ground-state energy of ahydrogenic ion is given by equations (3) and (4),

SE1s

(a)T\+2

2m

1

a2[

(Z[d1s

)e2a

. (8)

Here, the parameterais obtained from the minimization ofi.e.,SE

1s(a)T, LSE

1s(a)T/La\ 0,

a1s+ aZNC1[A1[1

ZBAaZ

RZB3[

3A1[1

ZB2

AaZ

RZB6

D. (9)Thus, using equations (9) and (10), we can obtain the energyof the ground state as

SE1s

(a1s

)T+[Z(Z[ d1s

)Ry

]C

1[1

Z(Z[1)

AaZ

RZ

B3[

3

Z2(Z[ 1)2

AaZ

RZ

B6D, (10)

whereRy(\me4/2+2+ 13.6 eV) is the Rydberg constant.

2.3. Excited State(2p)

The 2pscreening constant can be obtained byd2p

d2p\4n

P0

4aZ{ 3(Z[1)4nR

Z3

r2dr

+ (Z[ 1)(4a

Z/R

Z)3

M1[3[1[ (1/Z](4aZ/R

Z)3N

] 6Z2

A (Z[ 1)(4aZ/R

Z)3

M1[ 3[1[ (1/Z)](4aZ/R

Z)3N

B3, (11)

where the upper bound is given by 4aZ{

[4 4a0/Z@\

since the screening eect on the 2pstate is4ZaZ/(Z[d

2p)],

determined by the charge interior to the peak position ofthe screened 2pwave function.


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No. 2, 2000 PLASMA SCREENING EFFECTS IN COUPLED PLASMAS 1087

As we obtained in the previous subsection, the appropri-ate form of the 2pscreening constant is found to be

d2p+ (Z[ 1)

A4aZ

RZ

B3]

3

Z(Z[1)2

A4aZ

RZ

B6, (12)

and the expectation value of the 2pexcited state energy canbe obtained by equations (3) and (5),

SE2p(a)T\

+2

2m

1

4a2[

(Z[ d2p

)e2

4a . (13)The parameter a can also be obtained from the mini-mization of thenSE

2p(a)T,

a2p+ a

Z

NC1[

1

Z(Z[1)

A4aZ

RZ

B3[

3

Z2(Z[ 1)2

A4aZ

RZ

B6D,

(14)

thus, the energy of the 2pstate becomes

SE2p

(a2p

)T+ [Z

4 (Z[ d

2p)Ry

]C1[1

Z(Z[ 1)A

4aZ

RZB3[

3

Z2(Z[ 1)2A

4aZ

RZB6

D.(15)

3. ELECTRON-IMPACT EXCITATIONSIn the semiclassical approximation, the cross section for

excitation from the unperturbed atomic state ton[tn,l,m

(r)]a state becomes (Jung 1993b)n@[t

n{,l{,m{(r)]

pn{,n

\2nP

b db oTn{,n

(b) o2, (16)

where is the transition amplitude andbis the impactTn{,n

(b)parameter. From the rst-order time-dependent pertur-

bation theory (Jung 1995b), the transition amplitude Tn{,n(b)is given by the interaction potentialV(r, R),

Tn{,n

\[i

+

P~=

=dteiun{,nSn@ oV(r, R) o nT, (17)

where and are the energies of un{,n

\ (En{[E

n)/+, E

n E

n{atomic states n and n@, respectively and where rand Rarethe position vectors of the target electron and the projectileelectron, respectively. In the limit of high densities and lowtemperatures, the magnitude of the electrostatic interactionenergy is much greater than that of the kinetic energy.Under these conditions, the ion-sphere model (Salpeter1954; Kobzev & Iakubov 1995; Jung & Jeong 1996) isknown to be quite reliable to describe the interaction poten-

tial. In a recent paper (Jung & Jeong 1996), the electron-ionCoulomb bremsstrahlung in strongly coupled plasmas wasinvestigated using the ion-sphere model interaction poten-tial. In the ion-sphere model, the interaction potential isgiven by

V(r, R)\A[

Ze2

R ]

e2

o r[R o

B

]C

1[ R

2RZ

A3[

R2

RZ2

BDh(R

Z[R) , (18)

whereh(x) (\ 1 for x 0 ; \0 for x\ 0) is the step func-tion. As we see in equation (16), the plasma screening eect

would be reduced with an increase of the ion-sphere radiusTo investigate the plasma screening eect on the excita-R

Z.

tion probability in strongly coupled plasmas as a functionof the impact parameter and ion-sphere radius, we use theclassical trajectory method (Jung 1995b) to describe theprojectile motion R(t). When the projectile is moving in thez-direction and a coordinate system is chosen with theorigin at the target atom, in the semiclassical (SL) trajectorymethod, the position of the projectile electron is written as a

function of time,t, and the impact parameter b,R(t)\by] vtz, (19)

where v is the velocity of the projectile electron. Thisstraight-line trajectory method is known to be quite reliableto describe the motion of the high-energy projectiles(McGuire 1997). In the ion-sphere model, the time intervalof the interaction between the projectile electron and targetion is given by

[1

vJR

Z2[ b2\ t\

1

vJR

Z2[b2 ; (20)

then, the transition amplitude becomes

Tn{,n

\[i

+

P~=

=dteiun{,n

]C

1[ R

2RZ

A3[

R2

RZ2

BDFn{,n

(R, r) , (21)

where is the atomic form factor,Fn{,n

(R, r)

Fn{,n

(R, r)\T

n@K 1oR[ ro

KnU

. (22)

For 1s ] 2pdipole transition, there are three possible chan-nels depending on the magnetic substate (m\ 0,^ 1) of the2pstate (Rose 1998). Here, we shall consider the excitation

to m \ 0 state, i.e., state, since we assume that the pro-2p0jectile is moving in the quantizationz-axis.Using the addition theorem (Jeereys & Jeereys 1956)

with the spherical harmonicsYlm

1

oR[ ro\ ;

l/0

=;

m/~l

l 4n2l] 1

r:l

r;l`1

Ylm

(r)Ylm* (R) , (23)

where is the larger (smaller) ofRandr. Then, we canr;

(r:

)easily obtain the atomic form factor for the 1s ] 2p

0(m\ 0) excitation case,

F2p0,1s

\4J2

aZ

(g2p5@2 g

1s3@2)

qR13b65

]C1[A1]R1b6] 12 R12b62] 18 R13b63Be~R1b6[

1

24R13b63(1]R1

Zb6)e~R1Z b6

D, (24)

where the terms proportional to are the close-e~R1b6encounter eects. In the previous research (Jung 1995a),these eects were neglected in the dipole approximation sothat the investigation on the behavior of the transitionprobability for small impact parameters was impossible andalso the plasma screening eects on the atomic states oftarget ion were completely ignored. However, we shall keepthese terms to investigate the behavior of the transition


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0 2 4 6 8

Scaled Impact Parameter

0

0.0001

0.0002

0.0003

0.0004

Scaled

Transition

Probabilities

HaL

0 2 4 6 8


0

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

Scaled

Tran

sition

Probabilities

HbL

1088 JUNG & YOON Vol. 530

probabilities for small impact parameters. Here, R1(4andR/a

Z)\ (v62t2]b62)1@2, R1

Z4R

Z/a

Z, v64 v/a

Z, b64 b/a

Z,

withb6\g1s] (g

2p/2)

g1s\

C1[

1Z

(Z[ 1)Aa

ZRZ

B3 3Z2

(Z[ 1)2Aa

ZRZ

B6D, (25)

g2p\C

1[1

Z(Z[ 1)

A4aZ

RZ

B3 3Z2

(Z[1)2A4a

ZRZ

B6D. (26)

Thus, the close form of the transition amplitude,1s ] 2p0including the plasma screening eects, becomes

T2p0,1s

\8J2ZJv

(g2p5@2g

1s3@2)

P0

(Z2~2)1@2dqq sin [*/(ZJv)q]

R13b65

]C

1[3R12R1

Z

] R132R1

Z3

D

]C

1[A

1]R1b6]1

2R12b62]

1

8R13b63

Be~R1b6

[1

24R13b63(1]R1

Zb6)e~R1Z b6

D, (27)

where v is the scaled projectile energy,[4 12

mv2/(Z2Ry)]andq\ v6t, *4[[(Z [ d

2p)/8]g

2p] [(Z[ d

1s)/2]g

1s.

From equation (16), the scaled semiclassical exci-1s ] 2p0tation cross section can be written as

Z4p2p0,1sna

02 \2

Pdb6 b6P1

2p0,1s(b6) , (28)

where is the scaled transition probability,P12p0,1s

P12p0,1s

(b6)\Z~2 oT2p0,1s

(b6) o 2. (29)

Thus, the scaled transition probability including1s ] 2p0the plasma screening eects on the target wave functions

becomes

b6P12p0,1s

\27

v (g

2p5 g

1s3)b6

K P0

(Z2~2)1@2dq

]qsin [*/(ZJv)q]

R13b65

A1[

3R12R1

Z

] R132R1

Z3

B

]C

1[A

1]R1b6]1

2R12b62]

1

8R13b63

Be~R1b6

[1

24R13b63(1]R1

Zb6)e~R1Z b6

D K2. (30)

If we do not include the plasma screening eects on thetarget wave functions, i.e., the transition

b6\ 3/2, 1s ] 2p

0amplitude is found to be

T2p0,1s@ \

1

ZJv28J2

35

P0

(Z2~2)1@2dq

]qsin [3/(8Jv)q]

R13

A1[

3R12R1

Z

] R132R1

Z3

B

]C

1[A

1]32

R1]98

R12]2764

R13B

e~3@(2R1 )

[964

R13A

1]32

R1Z

Be~3@(2R1Z)

D, (31)

where the prime stands for neglecting the plasma screeningeects on the atomic wave functions. Then, the 1s ] 2p

0transition probability without the plasma screening eectson the atomic wave functions is found to be

b6P12p0,1s@ \

1

v217310

b6K P0

(Z2~2)1@2dq

]qsin [3/(8Jv)q]

R13 C1[

3R1

2R1Z]

R13

2R1Z3D]

C1[

A1]

3

2R1]

9

8R12]

27

64R13

Be~3@(2R1 )

[9

64R13

A1]

3

2R1

Z

Be~3@2(R1Z)

D K2. (32)

In order to specically investigate the target screeningeects on the transition probability, we consider Z \ 2 andtwo cases of the projectile energy: v\ 9 and 25, since thesemiclassical straight-line trajectory method is known to bevalid for high-energy projectiles (Bethe & Jackiw 1986). Wealso consider the three cases of the ion-sphere radius, R1

Z\

8, 12, and 16. Figures 1, 2, and 3 show the scaled 1 s ] 2p0transition probabilities as functions of the scaled impact

FIG. 1.Scaled transition probabilities for1s ] 2p0

b6P12p0,1s

(b6) R1Z\8.

Solid lines represent the transition probability given by eq. (30) (includingthe plasma screening eects on the target wave functions); dashed linesrepresent the transition probability given by eq. (32) (neglecting the plasmascreening eects on the target wave functions). (a)v\ 9 ; (b)v\ 25.


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0 2 4 6 8 10 12


0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Scaled

Transition

Probabilities

HaL

0 2 4 6 8 10 12


0

0.000025

0.00005

0.000075

0.0001

0.000125

0.00015

0.000175

Scaled

Transition

Probabilities

HbL

0 2.5 5 7.5 10 12.5 15


0

0.0005

0.001

0.0015

0.002

0.0025

Scaled

Transition

Probabilities

HaL

0 2.5 5 7.5 1 0 12.5 15


0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

Scaled

Transition

Probabilities

HbL

No. 2, 2000 PLASMA SCREENING EFFECTS IN COUPLED PLASMAS 1089


b6P12p0,1s

(b6) R1Z\

The solid lines represent the transition probability given by eq.12.(30) (including the plasma screening eects on the target wave functions);

the dashed lines represent the transition probability given by eq. (32)(neglecting the plasma screening eects on the target wave functions). ( a)v\ 9 ; (b)v\ 25.

parameter for various ion-sphere radii. The scaled tran-sition probabilities neglecting the target screening eectsare also illustrated. As we see in these gures, the transitionprobabilities neglecting the plasma screening eects on thetarget atomic wave functions are found to be always greaterthan those including the target screening eects since thetarget screening eects reduce the strength of the Coulombinteraction provided by the target ion core. The targetscreening eects are slightly decreased with an increase ofthe projectile energy (i.e., B10.9% for v\9 and R1

Z

\8,B10.5% for v\ 25 and The maximum values of R1Z\8).the scaled transition probabilities are given in1s ] 2p

0Table 1. The maximum positions are also indicated in par-


b6P12p0,1s

(b6) R1Z\

The solid lines represent the transition probability given by eq.16.(30) (including the plasma screening eects on the target wave functions);

the dashed lines represent the transition probability given by eq. (32)(neglecting the plasma screening eects on the target wave functions). (a)v\9 ; (b)v\25.

entheses. The maximum position of the transition probabil-ity corresponds to the position that takes place in the

dipole excitation process. The maximum position1s ] 2p0of the transition probability is slightly shifted to the target

nucleus as the target screening eect is included.

4. SUMMARY AND DISCUSSION

We investigate the target screening eects on electron-ioncollisional excitations in strongly coupled plasmas. Thescaled cross sections for is obtained using the ion-1s ] 2p

0sphere model potential. The screened ground and excitedbound wave functions are obtained using the Ritz variationand perturbation methods including the target screening

TABLE 1

MAXIMUM VALUES OF THESCALED TRANSITIONPROBABILITIES FORZ \21s ] 2p0

Probability R1Z\ 8 R1

Z\12 R1

Z\16

b6P12p0,1s

(v\ 9) . . . . . . . 0.00042558 (b6\1.17) 0.00135136 (b6\ 1.43) 0.00249664 (b6\ 1.62)b6P1

2p0,1s@ (v\ 9) . . . . . . . 0.00047196 (b6\1.16) 0.00137948 (b6\ 1.42) 0.00251603 (b6\ 1.61)

b6P12p0,1s

(v\ 25) . . . . . . 0.00005728 (b6\1.18) 0.00018662 (b6\ 1.44) 0.00035561 (b6\ 1.61)b6P1

2p0,1s@ (v\ 25) . . . . . . 0.00006329 (b6\1.16) 0.00019018 (b6\ 1.43) 0.00035800 (b6\ 1.63)


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1090 JUNG & YOON

eects. The semiclassical straight-line trajectory approx-imation is applied to the motion of the projectile electron inorder to investigate the variation of the transition probabil-ities as a function of the impact parameter and ion-sphereradius. The results show that the target screening eects aresignicant and cannot be neglected. The target screeningeects are slightly decreased as the projectile energyincreases. The maximum position of the transition prob-ability is slightly shifted to the target nucleus with including

the target screening eects. A recent paper (Hong & Jung1996) shows that the line intensities are directly related tothe 1s ] 2p excitation rates. Thus, the 1s ] 2p excitationprobability in dense plasmas would be expected to provideto information of the plasma temperature in the astro-physical dense plasmas. These results provide a general

description of the atomic transition probabilities in denseastrophysical plasmas.

One of the authors (Y.-D. J.) gratefully acknowledgesR. J. Gould for his useful comments and warm hospitalitywhile visiting the University of California, San Diego. Theauthors would like to thank E. Herbst for suggestingimprovements to this text. This work was supported by theKorean Ministry of Education through the Brain Korea

(BK21) Project, by the Research Fund of Hanyang Uni-versity (Project no. HYU-99-040), by the interdisciplinaryresearch program of the Korea Science and EngineeringFoundation through Grant no. 1999-1-111-001-5, and bythe Korea Basic Science Institute through the HANBITUser Development Program (FY2000).

REFERENCESBethe, H. A., & Jackiw, R. 1986, Intermediate Quantum Mechanics (Menlo

Park: Benjamin-Cummings)Hong, W., & Jung, Y.-D. 1996, Phys. Plasmas, 3, 2457Ichimaru, S. 1986, Plasma Physics: An Introduction to Statistical Physics

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