PHYSICAL REVIE% A VOLUME 26, NUMBER 3

Brief Reports

SEPTEMBER 1982

Brief Reports are short papers which report on completed research which, while meeting the usual Physical Review standards of

are included in Brief Reports. ) A Brief Report may be no longer than 3/~ printed pages and must be accompanied by an abstract. The

same publication schedule as for regular articles is followed, and page proofs are sent to authors.

Energy eigenvalues for Yukawa potentials

A. E. S. Green

(Received 2 April 1982)

A simple formula is given for the energy levels of a particle bound by a Yukawa potential

which is accurate at the 1'/0 level over the entire range of Z for a large range of n and l.

The bound states of the Yukawa, Debye-Huckel,or screened Coulomb potential have been the subjectof many studies' ' in atomic, nuclear, particle,solid-state, and plasma physics. Recently motivatedby an electron-transport problem, Green et al. " (tobe referred to as GRSSG) developed an approximateanalytic representation for the phase shifts for elasticscattering by a Yukawa potential. This potential maybe cast in the form

Y= ——-- e'2Z—r

by using the screening distance d as the unit of lengthand t'/2p, d' as the corresponding unit of energy.

The GRSSG study provided an accurate formula

ergy (E„t=0). These are given to within 0.4%(Zt, —=0.839908 is exact2) by

Z„I= (JZ~ + [(n —l —1)/Sl] j' . (4)

Thus Eq. (4) can serve as a precise boundary condi-tion in formulas relating energy eigenvalues to Z.

Our approach to the representation of nonvanish-ing eigenvalues largely rests upon an observation thatthe arrays of E„I given in Tables I and II of GRHtake on a simpler behavior when represented by thecombination variable

Y„,= Z(1+n'E„,) .

This is illustrated in Fig. 2 where the dots represent

v = (JZ —JZi)SI+ I (2) 10

for the number of bound states of a given angularmomentum, I, sustainable by a given Z, where

ZI=Zp(1+ul+PI ), S(=Sp(1+pl +SI ), (3)

and Zp=0. 839908, a=2.7359, P=1.6242, Sp=1.1335, y =-0.019102, and 5 = —0.001 684.

Figure 1 illustrates this relationship. The pointsrepresent 45 data points given in Table III of Rogers,Graboske, and Harwood" (to be referred to asRGH). Equation (2) is represented by solid straightlines. The dashed lines represent the loci of pointswith the same total quantum numbers n = v+ 1.

Equations (2) and (3) may also be applied to othershort-range potentials by the use of appropriate con-stants.

In GRSSG, Eq. (2) was incorporated into analyticformulas for the phase shift to insure satisfaction ofLevinson's theorem. For our present study of boundstates we may invert Eq. (2) to solve for the criticalvalues Z„& which just lead to bound states at zero en-

~ 8

~ 6Z

p 4

2

'0 4 6 8

CRITICAL X = ~ VALUES

10

FIG. 1. Number of bound states vs critical X=JZvalues for various angular momentum states. The solidcurves represent iso-l lines. The dashed curves representiso-n lines (adapted from Ref. 15).

26 1759 1982 The American Physical Society

&RIEF REPQRTS 26

100

CLU

+

w10&I

%hf

~C

10

FK;G. 2. Contours of

1001 l I I

Io1id

1000

poor ns states and the

Ref. 12), The oior n, n —1 states.

Y for n=l to 9 and the x 's

oints for intermediatel

uggest that as Z ll Yt umber n meruantum n

n onY Z-21 b h li

Z,„(Z —Z„I)Z„(Bn2+Bn'+ (Z —Z„()

(6)

where B=1.2464 g ater precisionand for re

,„=W (n + ~)'with A =1.9875 = . 95

re rei . 2 t E (6' ~ and the dashed lines

i ines in

I 1 o ldf all in betwer intermediate

o di RGHwell.

Schrodinger eiit the cor-

from th}1 percentage de a

ua y

ing 488 RGH "data"h o odi

th 1

Co bi i E. (5( ' )

'0

y, we arrive at thes

e inal formula

for ener 1gy evels in a Yukawu awa potential

E„,(z, d, ,)=- ","' "2p1 71

—A (n + o.)'+Bn'Z-Z —Z„(+Bn

tial

&(r) =-r H(e' I)+1-

which reduces t E .has been u

o q. (1 when H =1.used to provide accuralf 1

h er straightforward to'

qar to generalize Eq.

(9)

This isis is good at the 1% levelZf

Th' 1' f 1ula appears simpler and mo

o er approximate rean more ac-

p

Z''p y

region above criticalit i

g t of the bindu e useful in at ', ar

o1 d- t t d 1an plasma physics.e ukawa function m

G -Slli -Z h' - ac or 6 poten

26 BRIEF REPORTS 1761

(8) to apply to Eq. (9) by determining the depen-dence of the parameters in Z„i, Z,„, and Bupon H.Generalizations for neutrals and positive ions shouldalso be straightforward.

When 2Z/r in Eq. (9) is replaced by 2Z (i.e., r isreplaced by I or the scale length) Eq. (9) encom-passes the Einstein-Bose function (H ~oo), theMaxwell-Boltzmann function (H = I) and theFermi-Dirac function (H « I). In the latter casethe potential has the flat bottom and diffuse boun-dary characteristics of nuclear potentials. The resem-blance of Fig. 2 to Figs. 3—7 of Green and Lee"suggests that Eq. (4) can serve a critical role in thedevelopment of eigenvalue formulas for a variety of

nuclear potentials including the Woods-Saxon' po-tential. Such formulas and Eq. (8) should greatly fa-cilitate the process of inferring approximate potentialsfrom experimental binding energies.

ACKNOWLEDGMENTS

The writer greatly appreciates the computationalsupport of Mr. J. M. Schwartz which was essential tothe completion of this effort. The assistance of PaulSchippnick, Brian Walters, and Linda Green is alsoacknowledged. This work was supported in part bythe U. S. Department of Energy under Contract No.DE-AS-05-76 EV03798.

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