. . . . YS:: A .&=V:=W,.ATOMIC, MOLECULAR, AND OPTICAL PHYSICS

THIRD SERIES, VOLUME 45, NUMBER 7 PART A 1 APRIL 1992

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Time-dependent wave-packet theory for electron scattering

Burke RitchieLawrence Livermore National Laboratory, University of CaliforniaLive, rmore, California 94550

(Received 2 December 1991)

Time-dependent wave-packet theory is formulated for fairly-low-energy elastic electron scattering.The Schrodinger equation is solved numerically in the time and two spatial dimensions for e-Hescattering in the static-local-exchange approximation. The angular distribution is calculated from asingle pass through the region of the target and shows good agreement with the angular distributioncalculated with knowledge of the phase shifts from the solution of the set of time-independent radialequations.

PACS number(s): 34.80.Bm

There has been much recent work in the use of numeri-cal methods to solve the Schrodinger equation in the timeand two spatial dimensions, with applications primarily toion-atom charge exchange [1-7] and strong electromag-netic-field atomic ionization [8]. These methods have alsobeen applied to wave-packet scattering of muon and hy-drogen [9]. In this paper I apply these methods to elasticelectron scattering.

The general theory of wave-packet elastic scattering hasbeen available for some time [10]. The scattering offairly-low-energy electrons (above about 1 Ry), however,presents some new problems. The width of the wave pack-et must be large compared to the wavelength of the pro-jectile [10]. For 1-Ry electrons this means that the widthmust also be large compared to the size of the potential.In most applications [9,11] the width is small compared tothe size of the potential, but, for heavy-particle scattering,the spread in the momentum due to the uncertainty prin-ciple is still small compared to the momentum itself. Forelectrons the uncertainty in the momentum must be re-duced to acceptable values by use of a very large Gaussianwave packet requiring much machine storage, somethingwhich is increasingly becoming available. However, thelarge size of the wave packet also has an advantage,namely, that the spreading of the wave packet beyond thegrid boundaries during the collision is not a problem. This

+ V(p, z) v'(t, p, z) .

I make the substitution,

(t pz)~e i (hk /2m)i tlt(t pz) (2)

is because the size of the potential, a, is small compared tothe initial width w and the collision is over before the wavepacket can spread [10] appreciably beyond the initialwidth, i.e., ht/2tnw «1 for w»a. An equivalent state-ment is ha/2mvw, which holds if the electron velocity v

is not too small.In what follows I test the theory on e-He scattering in

the static-local-exchange approximation. The use of cy-lindrical coordinates, i.e., t, p, and z, means that the angu-lar distribution can be obtained in a single calculation.Another advantage of this coordinate system is found inelectron-linear-molecule scattering, in which, for fixed-nuclei theory [12], the t,p, z equation is solved once foreach molecular orientation, and the angular distribution isobtained by averaging over molecular orientations.

In an azimuthally symmetric problem the Schrodingerequation is

rl l'i 8 1 t) tlih +(t,p, z) = +— p8t 2m tiz p r)p r)p

R4207 @1992The American Physical Society

(~ ~

BURKE RITCHIE

and solve the resulting equation for the slowly varyingwave y. V is the static-local-exchange potential studiedby Riley and Truhlar [13],

V V, +V,„,V ~ —,

' (EII—V, ) ——,' [(EII—V, ) +a ]'

a' -4np(e ii'2 ) 2/m,

(3a)

(3b)

(3c)

where V, is the static potential, Eo is the scattering-electron energy, and p is the total electron density of thetarget. Initially,

1.2 =

1.00.9

OIOCO~ 0.80CO p7c~ 0.6C—0.55Ico p.4tA

o 0.3

Il

I I I I I I I I Il

I I I I I I I I I1

I I I I I I I I Il

I I I I I I I I I1

I I I I I I I I Il

I I-

l —[(z —z ) +p j/2~I/0 3/4

e''X W

(4) 0.2.IstIV~~~~.„~&$

where w 10 a.u. and zo —30 a.u. The grid boundariesare —50 and +50 a.u. for z and 0.05 and 50 a.u. for p.(The Coulomb singularity necessitates the use of 0.05rather than 0.0 for the minimum value of p, and this hasbeen found to give acceptably accurate results in previousapplications [14].) A spatial mesh size of 0.1 a.u. and atemporal mesh size of 0.2 a.u. were found to be adequate.1 use the Peaceman-Rachman alternating-direction-implicit method [15] to integrate the time-dependentSchrodinger equation, with initial conditions specified by

Eq. (4).1 propagate (Fig. 1) llr and lire (undistorted by the po-

tential) through the region of scattering and stop atzo +30 a.u. (The results are insensitive to zo as long asit is outside the range of the potential. ) The scatteringwave function is then

(5)

The wave function at some later time t can be expressed in

terms of the wave function at the temporal, spatial point

0.5 1.0 1.58 (rad)

2.0 2.5 3.0

t', zo by means of the Huygens-Fresnel integral,' 3/2

e (r,p, z) = m2niii'2 (r —r')

drlei[m/2h(r —I'))lr —r'( @ (& p z ) (6)

At a distant detector (t » t'),r ' 3/2

IIV„(r,p, z) = e'"' dr'e '"'Iir„(r',p', z') (7)m

2niht'where k is in the direction of the outgoing electron. Thecurrent of electrons in cm s ' at the detector is [16]

FIG. 2. Angular distribution for k 2 a.u. Dash-dotted line,

static-exchange, wave-packet calculation; solid line, static-exchange, phase-shift calculation; dashed line, static, wave-

packet calculation; dotted line, static, phase-shift calculation.

ft t) Ysc s, t) Ysc

a ~" e

5

co 4cI'OC7CO

CL

I I I I I

-50 -40 -30 -20 -10 0 10 20 30 40 50

z (cm)

~01

0.9

0.8

@alp 7ag 0.6~~c~ 0.5C~ 0.4CP

I I I I I I I I I1

I I I I I I I I Ii

I I I I I I I I I1

I I I I I I I I I1

I I I I I I I I I1

I I

FIG. 1. Squared modulus of the wave function vs z for theminimum value of p for k 2 a.u. with (solid line) and without

(dashed line) the potential in the static-exchange approxima-

tion, showing the distortion of the wave packet due to the poten-

tial. The time is chosen so that the wave packet is centered atthe origin of the coordinate system, where the potential is

strongest.

0.1

0.5 1.0 1.58 (rad)

2.0 2.5 3.0

FIG. 3. Angular distribution for k 3 a.u. See Fig. 2 for la-

bels.

TIME-DEPENDENT WAVE-PACKET THEORY FOR ELECTRON. . . R4209

Substituting Eq. (7) into Eq. (8) I obtain3

lh

2x p fdr'e '" 'y„(t', p', z') (9)

where in Eqs. (7)-(9) I have used r vt. The scatteringpower in s ' is P=r j. By the same reasoning thecurrent of electrons along z, incident uniformly on a planeperpendicular to z, is

dz'e ' ' tile(t', O, z')2m't 4

and the cross section is

der PJinn

(10)

Figures 2 and 3 show results for the angular distribu-tion at k 2 and 3 a.u. Results from Eq. (9) are com-pared against standard radial phase-shift results [13] forpartial waves to order 9. The agreement improves with in-

creasing k for a Gaussian width fixed at 10 a.u. , as expect-ed, and is poorest for k 2 a.u. for small angles. Thesmall, irregular ripples in the k =2 a.u. results are typicalfor cases for which the method is beginning to fail because

the width is not large enough. For example, for k I a.u.the results show irregular, large-amplitude oscillationsabout a mean which qualitatively agrees with the correctangular distribution. I am confident that, for k 1 a.u. ,use of a larger width and larger grid boundaries wouldlead to the kind of agreement observed at larger k, ac-cording to the validity criterion a/2kw ((I; however, Ihave not performed this calculation because, for the kvalues chosen, the problem is already fairly large and ex-pensive (1000&500 mesh points in the z and p dimensions,respectively).

In conclusion I have formulated wave-packet scatteringtheory for fairly-low-energy electron scattering andworked out an example for e-He scattering. The methodlooks very promising, especially as larger, faster comput-ers become available. The usefulness of the method forscattering from noncentral potentials or laser-assistedscattering, in which the potential is explicitly time depen-dent, is obvious.

I wish to thank Michael D. Feit for helpful discussion.This work was performed under the auspices of the U.S.Department of Energy by Lawrence Livermore NationalLaboratory under Contract No. W-7405-ENG-48.

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