The helium (e, 2e) satellite spectrum

Chemical

Physics X1 (1983) 329-333

North-Holland

Publishing

329

Company

THE HELIUM

(e, 24

SATELLITE

SPECTRUM

F.P. LARKINS

and J.A. RICHARDS

Rrceived

2 May

1983

Two configuration-interaction-typr wawfuncIions each xcounting for = 98% of Ihe ground-zkne c~rrrelaiion snsr_gy in ratio leading IO cirher Ihc ,I = 2 or rhe II = 1 fmal ion helium have been used to predict the differential (c. _7~) cross-xclion states of helium. The findings are compared with known experimental data ;Ind IWO previous Ihenrcric:d c~lculrltions. The predicted ratios using the various wavefuncrions are in reasonable agsemcnt with one anoIhrr and xvith experiment up to q = 2.0 au. Thereafter.

the dispersion

for various

wavefunctions

increases

significantly.

Furthermore.

the difiermrial

cros+sec-

for populating the He’(Zp) IO He+(%) final ion SIZWS differ by an order of msgniwde for corrrlsIed \vsvcfunciions of equal quality in ground-state-energy terms. i\ reformulation of the rlccwon-impxt cross-xxlion thcary beyond Ihe plane-wave impulse approximation with the use of non-configurakw-imerxtion-I>pr correlawd funcrions is rsquircd IO assist the resolution of this problem. Lion ratio

1. Introduction The differential cross-section ratio for electronimpact ionisation of helium leading to either the )I = 2 or II= 1 He+ final states was measured by McCarthy et al. [l] and compared with a theoretical prediction [1.2] obtained by using a sophisticated correlated ground-state wavefunction developed by Joachain and Vanderpoorten [3] (hereafter denoted JVl). The JVl wavefunction (ref. [3]. eq. (2.8)) used a (r, . r,) basis and accounted for some 98.0% of the correlation energy. Larkins [4] used a simple four-term configuration-interactiontype wavefunction proposed by Taylor and Parr [5] (hereafter denoted TP) to predict the a(n = ‘)/a( tr = 1) cross-section ratio. The wavefunction was dominated by a single unrestricted (1s. 1s’) configuration (coefficient 0.99811). The TP function accounted for some 85% of the correlation energy. Despite its simplicity the predictions obtained with this wavefunction were also in good agreement with experiment and with the theoreti0301-0104/83/0000-0000/903.00

0 1953 North-Holland

cal findings using the sophisticated JVl correlated function in the ion-recoil momentum range q = O-3 au. At higher 4 values however there \vere significant differences in the calculated cross-section ratio using the two correlated functions. Furthermore. the calculated o(2p)/a(Zs) cross-section ratio differed by an order of magnitude for the two functions used. The present study. using t\vo other sophisticated ground-state CI \vsvefunctions for helium_ was undertaken to endeavour to further elucidate the problem highlighted in the previous \vork [cl] and to provide additional information for evaluating

the

sensitix*ity

of

the

(e.

k)

criminate betlveen the quality lated wavefunctions.

technique

of different

to

dis-

corre-

2. Theoretical considerations Detailed accounts of (5. 2s) cross-section theory may be found in the literature as it pertains to the

Hurtrre-Fock model [6] and to correlated wavefunctions of the configuration-interaction type 17.S.4). Only the essential equations required for the present work are presented here using the notation given in ref. (41. In the plane-wave impulse approsimation for a symmelric geometry the dilkrcntial cross section for atoms may be written in atomic units as

where (x&( N - 1)1x:( N - 1)) is the overlap matrix between the residual (N - I)-electron CSF. xr;;;‘
1) exp(iy.r))lz=I,‘(q).

LI\

where k,,. k, and k,, are the momenta of the incident and scattered electrons respectively. T( y ) is the Coulomb T matrix. 4 is ths ion recoil a summation over nwn~rntuni and I,, represents final states and an averaging over initial states. \k,;( X ) is the wavefunction for the h’-electron initial state of the target and T,r(N - 1) is the wavefunction for the jth final-ton state. It will be convenient to write eq. (I) for a symmetric geometry with k, = k,, as a(y)=

X(k,,.k,.

rl)f’(q).

(7)

where the f,‘( (1) term represents the E:,, term in c’y. ( I ) and S( k,,. k,. q ) the remainder. For the helium problem when the initial state JI,:( A’ ) may be represented by a correlated wavefunction of the configuration-interaction type (3) N here the x_hA( A’) are the configuration-state functions (CSFs) espressed as antisymmetrised products Or one-electron functions 0r appropriate symmetry and the h,,, are the coefficients of the CSF in the initial-state wavefunction. The jth final ion Irt:tte a,‘( A’ - I) for the He’ ion may be represented b> (f$ .\’ - 1) = x;( N - 1).

(4)

ivhcrs x’,( N - I) is the hydrogen-like electron functionThen (9i(A’)14:(hr-

= 5

~h,(xKV’,,I

exact

one-

I) exp(iy-r))

1)1x;@‘-

W&‘Zq).

(5)

x (x,‘$( h’ - l))x:_( N - I))

x+::(q)+$gq)-

(6)

and 11h .,,,- are the number of equivalent target electrons with the same trl and exponent values in the state being ionised. ?X is a constant_ In this paper we are interest in the behaviour of the cross-section ratio for the II = 2 helium satellite to the tI - I main line with EA = El,. From eqs. (I) and (2)

!I h ,,,

It should be noted that even when the satellite line intensity is measured at the same 4 values as the main line intensity (i.e. q, = q,,) the Coulomb T matrices do not cancel since T(y) is a function of k”, a variable which depends on the excitation energy to the final ion state. The differential (e. 2e) cross-section ratio. a( II = 2)/a( )I = 1) has been calculated using eq. (6) with qb = q,,_ The factor k~“‘*‘l7;,,(q)l’/ ki;“‘(T,,,i,( 9)1’ depends on the incident-electron energy E. Since the cross-section ratio depends on this factor, if experimental data taken at more than one incident energy are to be compared in a single figure the ratio I,t,(q)/I&i,(q). which is independent of E, should be considered. For the present investigation, the two Cl wavefunctions used for \k;f(N) are a 20-term function

proposed hy Nesbct and Watson [9] (hereafter denoted NW) and the alternative function proposed by Joachain and Vanderpoorten (ref. [3]. cq. (2.4)). using an (r,. r2) basis (hereafter denoted JV2) rather than the (I-, . r,) basis. The latter had been used by McCarthy et al. [ 1.21 in previous theoretical work. Full details of the NW ~~~vefu~~-tion characteristics may he found in the original paper. Briefly. the wavefunction contains configurations based upon s. p_ d and f type orbitals and is dominated by the Hartree-Fock type (Is’) COIIfiguration (coefficient 0.99596). Some 97.7%. of the correlation energy is accounted for with this lvavefunction. The JV2 function is a 60-term function and accounts for some 96.6%1 of the correlation energy.

3. Results

CCE _. --

2.

L’5’5

G ;

-r-.

30..

, --.---_-_____

_-_-___-._____

/ I

and discussion

The incident-energy-independent ratio I’( II = 2)/l’(n = 1) 3s a function of the ion recoil momentum q is shown in fig. 1 for the NW and the JVZ functions_ The experimental results of Dixon et al. [2] and the theoretical results with the the the JVl function and TP function. Hartree-Fock (HF) function as presented in fig. 1 of our earlier paper [4] are also given for comparison. The HF value is independent of y in the plane-wave impulse approximation model. The cross-section ratio calculated with the NW and JV2 functions are in close agreement for q 5 0.75 au, but slightly higher than the values u3th the TP and JVl functions. The limiting value for the ratio at y = 0 au is 0.00s with the present calculations. For q values greater than 1.0 the ratio calculated with the JVl and JV2 functions are found to be in close agreement up to a masinwnl value of 0.076 at q = 4.0 au. The rapid fluctuation in the ratio beyond q = 3 au for the JVI function reported earlier by other workers [2] is undoubtedly due to a numerical instahilit_v in the calculations_ The maximum in the NW curve is near q = 3.0 au where the rtltio has a value of 0.61. \vhile the TP function yielded a maximum value of 0.099 at q = 7.0 au. Clearly. the results in fig. 1 demonstrate that the ratio is most sensitive to variations

in the nature of the CI Lx-avefunctian at q \alue~ greater than 2 au. The value for the ratio at q = 10.0 XI v.w O.O-l5 with the JV2 function and 0.010 \vith the XL\‘ function. The value at high q may be compared with the II = 2 to II = 1 photninnisstion cross-setenergy of 1457 cV dstion ratio at a photon termined experimentally by Carlson et al. [IO] to be 0.05 & 0.00s. The cross-section ratio derived \vith the ;\;I\ functiou is dominated by the ~(%)/a( 1s) component. in common with the prediction using the TP function. but it is in contrast to the use of rhe J\ functions lvhere the a(7p)/a( Is) component is more significant. In particular. the data for the a(Zp)/u(Zs) ratio determined \vith the three correlated \vavefunctions corresponding to a symnlerric co-planar geometry \vith E = SO0 eV at 8 = 15. -lY and 53” are presented in rahls 1. Tine TP and _lVl data have been presented prrviousl? (ref. [4]. rahlc:

Tahlr I Differential (e, 2r) cross-section ratio for populating the He’ (2~) IO He’(k) final ion SIRIUS (B). Kinematic conditions: symmc~ric coplanar geometry. + = 0. Or,= 8,,. E = SO0 eV. TP: Taylor and Parr [S] wavefunction JB: Joachain and Vandsrpoorten 131 wavrfuncIions. JVl. cq. (2.6): JV2. eq. (2.4). NW: Nesbei and Watson 191 wavefunction. @(deg)

NW

TP”’

JVI ”

JV2

4s 49 53

0.3h 2.04 3.07

0.26 1.40 2.74

I.8 14.1 23.2

3.0 13.3 22.4

a’ Ref. 141: h’ Ref. [I].

3). While in fig. 1 the total a( II = 2)/a( II= 1) ratio with the NW function more closely parallels the JV ratios than the TP ratio. the contribution to the total t1 = 2 cross section from the process populating the 2p final ion state is an order of magnitude greater with the JV functions than with the other IWO correlated wavefunctions. The present investisation with the JVZ function confirms the results of previous workers with the JVl function. but it does nothing to resolve the ambiguity which exists in this area. It is interesting to note that a similar problem has existed with the calculation of the cl-oss section in o(2p)/a(ls) photoionisation helium. but the problem has now been substantially resolved [ 1 1- 141. The theoretical predictions and the esperimentat ratios shown in fig. 1 are in agreement to within experimental error up to q = 2 au. However. for q < 2 au it is not possible to confidently discriminate between the quality to the various correlated wavefunctions. For q > 3 au there are large uncertainties associated with the experimental vah~es, hence more precise measurements especially for the t1 = 2 state are required. Furthermore. the plane-wave impulse approximation is also clearly inadequate at the higher q values leading to uncertainties in the accuracy of theoretical predictions. Further refinements to both the experiment and to the theory are required before the results can be profitably used to assess the quality of various helium correlated wavefunctions.

4. Conclusion Four different correlated wavefunctions of the configuration-interaction type have now been used to calculate the a(n = 2)/0(11 = 1) and the a(2p)/u(2s) cross-section ratios for the electronimpact ionisation of helium within a plane-wave impulse approximation. Three of them are of very different constructions from one another. Significant variations in the ion-recoil-momentum dependence of the cross-section ratios have been found for high-quality functions which account for up to 98% of the correlation energy. The most serious discrepancy however is in the relative importance of the ~(2s) and u(2p) contributions to the total a( )I = 2) cross section. The findings underline once again the conclusion that correlated wavefunctions of high quality as judged by an energy criterion may not necessarily be suitable for the calculation of other atomic properties. Further experiments are required as well as calculations using correlated helium wavefunc[ions. such as Hylleraas-type functions, which are not of the configuration-interaction type. This approach would require a reformulation of the (e. 2e) reaction cross-section theory beyond the planewave impulse approximation. Furthermore. in view of recent work on the helium photoionisation u(2p)/u(2s) ratio, which has demonstrated that the photon dependence of this ratio can be predominantly accounted for using a relaxed Hartree-Fock theory [12,13], the assumptions in the (e. 2e) plane-wave impulse approximation theory which led to the conclusion that the Hartree-Fock U(PZ= 2)/u(11= 1) ratio is independent of q and that u(2p)/u(2s) is zero warrant re-examination. References [I] LE. hlccarthy. [2] 131 [4] (51

-4. Ugbabr. E.Weigold and P.J.O. Teubner. Phys. Rev. Letters 33 (1974) 459. A.J. Dixon, I.E. McCarthy and E. Weigold. J. Phys. B9 (1916) Ll95. C.J. Joachain and R. Vanderpoorten. Physica 46 (1970) 333. F.P. Larkins. J. Phys. B14 (1981) 1477. G.R. Taylor and R.G. Parr. Proc. Nail. Acad. Sci. US 38 (1952) 154.

161 I.E. XlcCnr0ly and E. Weigotd. Phys. Rspt. 27C (1976) 2775. 171 V.G. Lsvin. Phys. Leirers 39A (1972) 125. [S] V.G. Levin. V.G. Neudulchin. A.V. Pavlitchenkov and Yu.F. Smirnov. 3. Chrm. Phys. 63 (1975) IS4t. [9] R.K. Neshct and R.E. Warson. Phys. Rev. 110 (19%) 1073. [ 101 TA. Carlson. iM.0. Krause and W.E. Xloddeman. J. Phys. (Paris) IOC4 (1971) 76.

[ 111 D.A. Shirk?. P.H. Kohrin. D.W. Lindtc. C.&l. Truesdate. S.H. Soulhworth. U. Beckr and H.G. Kcrkhoff. AIP Conf. Proc. 94 (19S2) 569. [ 121F.P. Larkins. AIP Conf. Proc. 94 (19~) 530. [ 131J.A. Richards. and F.P. Larkins. J. Electron Spccln_.. to he published. 1141 K.X. Berringon. P.G. Burke. W.C. Fan and K.T. Taylor. J. Phys. B t 5 ( t 9SO) L603.