The Compton profile of neon : comparison between experiment and the impulse approximation

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The Compton profile of neon : comparison between experiment and the impulse approximation

A. Lahmam-Bennani, A. Duguet

To cite this version:

A. Lahmam-Bennani, A. Duguet. The Compton profile of neon : comparison between ex- periment and the impulse approximation. Journal de Physique, 1982, 43 (9), pp.1333-1337.

�10.1051/jphys:019820043090133300�. �jpa-00209512�

The Compton profile ^{of neon :} comparison ^between experiment ^{and the} impulse approximation

A. Lahmam-Bennani and A. Duguet

Laboratoire des Collisions Atomiques ^et Moléculaires, ^Bât. 351, Université ^de Paris-Sud, ^{F 91405} Orsay, ^France (Rep ^{le 16} decembre 1981, ^{révisé le 30 avril} 1982, accepté ^{le 14 mai} 1982)

Résumé.

Les profils Compton ^{total et} de valence du néon sont mesurés par impact électronique ^{à haute} énergie (25 keV). ^Leur partie symétrique ^est comparée à ^des expériences antérieures et à des calculs dans le cadre de l’appro-

ximation de l’impulsion.

Abstract.

The valence and total Compton profiles ^of

^measured using high-energy (25 keV) ^electron scattering. ^Their symmetrical parts

compared ^to previously published experiments ^{and to} impulse approxi-

mation calculations.

Classification

Physics ^Abstracts

34.50H - 34.80D - 34.80G

1. Introduction.

Compton profile (CP) ^determi-

nation is an important ^{tool for} studying ^electronic properties ^{of atoms} and solids. These profiles ^are simply ^{related, within the} framework of the impulse approximation (IA1 ^{to the electron momentum den-}

sity p(p) ^{of the} system ^under study, ^{and hence} they

constitute a very sensitive test of the wavefunction used

to describe this system.

The neon atom was chosen for the present investiga-

tion because of its relative simplicity ^for theoreticians

as well as for experimentalists. ^{Indeed, a} great many theoretical calculations ^are available for the total and valence CP’s of this atom, mostly within the framework of the IA, using ^different wavefunctions. However

they ^{differ at the centre of the} profile by ^{more than} 3 %

from each other, and the best ^{ones are} usually ² %

lower than Eisenberger’s [1] experimental ^{data. These}

are valence CP’s, ^{measured in the} early ^seventies using photon impact ^{at two different} energies, ^but they ^{were shown} by Mendelsohn ^et al. [2] ^to present

some internal inconsistency near q

0. Later elec- tron impact ^data by Wong ^{et al.} [3] ^{did not shed more} light ^{due to the lack of} precision ^{in the} scattering angle measurement, ^{and a} doubt was left by ^the

authors as to the definition of an

effective

valence CP. Such conflicting ^{results warranted a new investi-} gation ^of the Ne valence and total CP’s (the ^later being ^here directly ^measured for the first time), taking advantage ^{of the} high accuracy and precision

achieved with high energy electron impact spectro- scopy.

2. Experiment.

^{2.1 METHOD. - The} apparatus and data acquisition method have been previously

described [4]. Briefly, ^{a 25} keV, ^{0.1 to 200} pA ^{0.25 mm}

FWHM electron beam crosses a gaseous target beam of 1 mm FWHM. The scattering intensity ^is

observed in the angular range 0.70 ^to 200, ^{With an} acceptance angle ^{of 0.010 and an overall} uncertainty

in the scattering angle ^{of ±} 0.0030. The scattered electrons are energy analysed ^{with a} resolution ^of

2-15 eV over an energy-loss range up ^to 7.5 keV, depending ^{on the} scattering angle ^under investigation.

At least 20 000 counts are accumulated at the maxi-

mum of the inelastic profile ^{for each} scattering angle, except ^{for the} measurements taken at - 180 and - 200 where only ^{10 000 counts are reached} because of the weakness of the scattered intensity. ^{Extensive tests}

have been carried out in order to eliminate the effects of energy resolution, multiple scattering ^{and other}

extraneous scattering.

The relative ^cross sections measured at fixed

scattering angle ^{0 are} corrected for detector noise and dead time, ^{and for} analyser transmi’ssion, ^then

transformed (see ^Ref. [4]) ^to generalized ^oscillator strengths (GOS), df(K, E)/dE, ^{where K is the momen-}

tum transfer and E the energy loss. These GOS’s corrected for the missing portion ^{of the} high ^energy-

loss tail are made absolute by ^use of the Bethe ^sum rule.

Following Bonham and Tavard [5], ^{an electron} CP, J(q, K), ^{is then defined from the} spectra using

whose limit for K -+

is simply ^the impulse CP, J,A(q). ^At large ^{but non infinite K} values, ^Gasser

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043090133300

1334

and Tavard [6] have shown through

^series develop-

ment that J(q, K) consists of a leading ^term symme- trical with respect ^to the q variable and corresponding

to the IA, plus ^a first order corrective term which is

antisymmetrical ^with respect to q ^{and whose} magni-

tude decreases with increasing ^K. Hence, ^{if an} asymp- totic limit, J(q), ^is approached ^{for the} experimental J(q, K) profiles ^{as a} function of K, ^its symmetrical part, Js(q)

(J(q) ⁺ J ( - q))12, ^{should be} compared

to the calculated JIA(q).

The profile dependence upon ^{K was} thus investi-

gated using ^the following characteristic quantities :

the maximum intensity ^{of the} profile, Jmax(q, K ), ^{its full}

width at half maximum, Q, ^{and the} Compton ^defect parameters, Aq ^and 6q, respectively characterizing ^the

deviation of the maximum of the profile ^{from the}

q

0 position ^{and the} asymmetry ^{of the} profile. ^All

these quantities ^{were found} [7] ^{to reach constant} values, ^within experimental uncertainties, ^{in the K-}

ranges 4.5 ^{K 7 a.u.} and 13 K 15 _a.u.,

respectively for the valence and total profiles. (The

upper limit, ^K

^{15 a.u. is the} largest ^K value consi- dered in this work.) ^{Moreover the area} under the CP which is in the IA equal ^to the number of electrons

involved, N, ^{is not more than} 1.5 % lower than N. One may thus consider the asymptotic ^{limit to be suffi-} ciently approached ^{in these} K-ranges ^so that Gasser and Tavard’s development may be truncated to the first corrective term (antisymmetrical). ^That is, ^the symmetrical part ^{of the} experimental ^{CP can be} directly compared ^to the IA CP. Note that our experi- mentally ^observed plateau ^for Jmax (q, K) ^{leads to the}

conclusion that the K-dependence ^{of the sum of the} higher order corrective terms is not strong enough ^to

be observed within the statistical accuracy of the

experiment.

«

Asymptotic » valence and total CP’s, J (q), ^were

then determined in the above defined K-ranges ^from

the average of respectively ^{8 and 14} experimental ^runs.

The symmetrical profiles, JS(q), ^were calculated from these average ^ones and ^are shown in tables I and II

for q ranging ^{from 0 to 3 a.u.,} together with the sym- metrical CP of the 1 s inner shell orbital obtained from the difference of the total and valence ones. The numbers shown in parenthesis ^{are the maximum}

deviations from the average values, ^{obtained from a} graphical ^{fit to the} superposition ^{of all the} J(q, K)

CP’s. The standard deviations are about 2-3 times lower. Although the uncertainty ^{of the Is} profile ^is

rather large (about 20 % maximum deviation near

q

0) because this profile ^{is obtained from the diffe-}

rence of two much larger quantities, ^the present

measurements constitute the first experimental ^deter-

mination of ^{a core CP.}

2.2 COMPARISON TO THE IA AND DISCUSSION.

A

large ^{number of} theoretical calculations have been

published ^{for the neon} CP, using wavefunctions of different quality. ^{In order to avoid a} lengthy presen- tation of the following discussion, ^we have chosen to

Table I.

The measured symmetrical ^{total and}

profiles compared ^{to IA} calculations : the VHS-CI calculations by ^{Smith Jr. and} Brown, Ref. [14] ^the full . correlated calculations by Tong ^{and Lam} (TL), Ref [20] ;

the HF-C calculations by ^{Weiss et} al., Ref [9].

Table II.

Same

table I, but for the valence profile, : Eisenberger’s x-ray experiments, Ref. [1]; Wong ^{et al.’s}

electron experiment, Rçf. [3] ; L-Corr. IA calculations

by ^{Naon and} Cornille, Ref. [21].

first compare all the IA CP’s ^to the actual results at the maximum of the profile, i.e. in the key region ^{of low}

momentum density contributions. Comparison ^at all q

values will then be made for only the best calculations.

Table III.

Comparison ^{between the} experimental ^and theoretical values of J(0) for the total and valence pro-

files: GTO : Naon et al., Ref [11]; ^{C-CI : Naon and} Cornille, Ref [12] ; MCHF : Smith Jr. and Brown, Ref [14]

for ^{the total} profile and Mendelsohn et al., Ref. [2] for the valence _{one ;} LDM : Sabin and Trickey, Ref. [ 17], except

for TL : Tong ^and Lam, Ref. [20]. ^Other symbols

^{in tables I and II.}

Table III gives for the total and valence profiles ^the experimentally determined _average values of J (0) ^with

their maximum deviation, ^{as well as the other} existing experiments ^{and the most} representative calculations.

The different calculations have been roughly ^classified

into 3 categories : using Hartree-Fock (HF) ^wave- functions, correlated wavefunctions or local density

models (LDM).

2. 2.1 Total profile.

^The only other direct experi-

mental investigation of the total CP of Ne is due to

Wellenstein’s _group at Brandeis University [8], using

35 keV electron scattering. ^{But their} measurements were limited to a single value of the momentum

transfer, ^K

^10.7 a.u., where the asymptotic ^{IA limit}

is not reached according ^to the above discussion. Also their profile ^{was not Bethe sum} rule normalized but matched to the IA HF one at q

^{0. Hence no} compa- rison can be made here with our results. On the other

hand, Eisenberger’s [1] ^values quoted in table III are

obtained from his experimental ^valence J(O) ^values

shown in table II, ^{to which we have added the Is HF} contribution calculated by Weiss et al. [9]. Hence, ^these results will be discussed below, with the valence profile

ones.

The calculations by ^{Weiss et at} [9], ^who used the HF wavefunctions tabulated by ^Clementi [10] (HF-C),

lead to almost 2 % underestimate of the J(O) ^value.

Naon et al. [11] ^{have also} performed calculations using approximate ^HF gaussian type ^orbitals (GTO) ^wave-

functions. The effect of increasing the size of the

gaussian ^{basis set was found to} improve J (0). However, the agreement ^{with our} experiment ^is poorer than the HF-C calculation even for the largest ^{basis set used} (shown ^{in table} III).

A 1.5 % underestimate of the J (0) value is also observed (5th column of table III) with the value of Naon and Cornille [12] who used the three term confi-

guration interaction wavefunction of Clementi et a1.

[13] (C-CI). This wavefunction takes into account only

radial correlation and gives ^{about 20} % ^{of the correla-}

tion _energy. Two other calculations by ^{Smith and}

Brown [14, 15], ^{who used more} correlated ^wave-

functions, ^{do not} bring ^better agreement ^with experi-

ment : the multiconfiguration Hartree-Fock (MCHF)

wavefunction of Ahlrichs and Hinze where only ^the

L-shell correlations are included, and the second order CI wavefunctions of Viers et al. [16] (VHS-CI). ^The discrepancy ^{seems to become even} larger ^{with the}

inclusion of ^more correlations.

In the sixth column of table III are the values obtain- ed by ^{Sabin and} Trickey [17] using ^{numerical wave-}

functions generated by ^{five LDM for} exchange ^corre-

lation (called ^Xa models). The value of the local

exchange parameter

(usually 2/3

1) ^has

been chosen to be

2/3 ^{in the} X(X2/3 ^{model, and}

a ⁼

(XvT

0.729 97 (the ^so-called

virial theorem »

value) ^{in the} XXyT ^{model, while the} Xa.

^{one has a} stepfunction ^radial dependence ^for

^{The other two} models, Ng ^and HL, ^{due to} Ng [18] ^{and to Hedin and} Lundqvist [19], ^{contain terms} beyond ^{the X a} approxi-

mation which ^are intended to represent correlation corrections.

Except for the first model which leads to about 1 %

overestimate of the experiment, ^the agreement ^is very

satisfactory ^{for the others,} specially ^the XaVT, Ng,

and HL ones where the deviation from the experiment

is lower than 0.4 %. Finally, Tong ^{and Lam} [20] ^have

used wavefunctions with full correlation contributions included, ⁱⁿ the local density approximation. ^The J(q) ^values (labelled ^{TL in table} III) ^are then found as

sum of a first term, J ’(q), ^{and a} correction term,

J2(q). ^{However, as stated} by the authors themselves,

the second term overcorrects at small q values, making J(0) ^too low. It is obvious, comparing ^{the different} slopes dJ (q)ldq from their table II, ^{that this over-} correction is quite large, ^and likely ^{affects not} only ^the J (0) value but also J(0.1).

Our total profile ^is compared ^{in table} I, ^for all q values, ^{to the most} sophisticated ^IA calculations [14, 20] VHS-CI and TL. Figure ¹ gives the relative devia- tion (Js(q) -JIA(q))/Js(q). As discussed above, ^{all the}

IA CP’s are below the measured one near q

0,

1336

Fig. 1.

Total CP : relative deviations of the IA calculations from

symmetrical CP, Js(q). ^The experiment is repre- sented by the horizontal axis, and the solid lines

the

standard deviation

The IA calculations

(- -) by Smith Jr. and Brown, ^Ref. [14], ^and (-) by Tong

and Lam, ^Ref. [20].

except in reference [ 17] (not ^{shown in table I as} only ^a

few points ^are given by ^the authors). Otherwise, they

more or less satisfactorily reproduce Js (q), within,our

experimental uncertainties, up to q - ² a.u., but are

again significantly ^{lower at} larger q ^values.

2.2.2 Valence profi°le.

^{Our valence} J(O) ^{value is} compared in table III to the other existing experi-

mental or theoretical ones. Eisenberger [1] ^{has observ-} ed, ^{in two} separate experiments, ^the scattering ^of

MoKa (17 ³⁷⁴ eV) ^and AgKa (22 163 eV) x-rays

through 1700 from neon. He deduced an

experi-

mental » valence CP from the measured total profile by subtracting theoretical core profiles ^{corrected for}

the

violation » of the IA. The MoKa result agrees with our’s, within the experimental uncertainties. The

AgKa ^{case will be} discussed below. Wong ^{et al.} [3]

measured the valence CP using ^{25 keV electron} scattering, ^{with an} uncertainty ^of (2-3 %) ^{near the}

maximum. Their measurements at three different

scattering angles ^{indicate a definite} dependence ^of J(0) upon the ^momentum transfer K, the 9.5° value

being ^almost 4 % lower than the 7° value. Our corres-

ponding K-range is characterized by ^a plateau, ^where

the measurements taken at 11 different scattering angles ^are randomly ^scattered around the average value with a maximum spread ^{of 1} %. ^{The deviation}

of Wong ^et al.’s results from our’s is respectively ^about 5, ^{3 and} I% ^at 7°, ^{8° and 9.5°.} Only ^{the last two values}

fall within the combined experimental ^{error bars.}

Some inconsistencies are to be noted in reference [3] :

the statement (page 1857) ^that up ^to 80 the profile changes ^to

higher profile

^the scattering angle

increased » is in contradiction with the data shown in their table V. Also, ^{in the same} table, ^the experimental profiles ^{do not} peak at q

0, J(o.1) being larger ^than J(0). ^{We thus} suspect ^{that some errors} might ^be present in this table.

Columns 4 and 5 of our table III give ^the theoretical IA determinations ofJ(O). ^{A 2} % underestimate of the

J(O) ^{value is} again ^{observed with the above cited}

Weiss et al. [9] ^HF-C calculations, ^{as well as with the}

calculations by both Naon and Cornille [21] (L-Corr.)

who used a variational formalism which reproduces

about 98 % of the L-shell correlation energy, and by

Mendelsohn et al. [2] ^who used, ⁱⁿ their MCHF calcu-

lation, ^{the same} correlated Ahlrichs-Hinze wave-

function previously utilized in reference [14]. ^This

wavefunction is said to account for 67 % ^of the valence electrons correlation energy.

Column 7 of table III shows the only calculation which is not within the framework of the IA. Mendel- sohn et al. [2, 22] have calculated L-shell CP’s, ^within

the Bom approximation, ^{with an} effective nuclear

charge determined from the matching ^{of the} impulse hydrogenic CP ^{to the} impulse ^HF profile at q

^{0. The} method is referred to

the exact hydrogenic (EH)

method. While _any IA will be inherently independent ^of

incident particle energy, the ^EH method exhibits the

profile dependence ^on energy. Hence, ^Mendelsohn

et al. [2] ^have performed ^EH calculations for the

Eisenberger [1] x-ray experiments. The MoKa results

agreed ^{well with the} experiment, while the AgKa

results were in slightly poorer agreement. ^Consi-

dering energy dependence arguments, they ^have

shown that there is,

^{at the} very least,... ^{some incon-}

sistency ^{between the two sets of} experimental ^{data ».}

On the other hand, Eisenberger ^has only reported ^the

1 % statistical uncertainty ^as determined by ^{his count}

rates. We believe that the systematic ^{error in} experi-

mental CP studies is usually ^not very small compared

to the statistical uncertainty, ^{either in} photon ^{or elec-}

tron impact work. Hence the 1.3 % difference between the Ag and the Mo experimental J(O) ^values likely

falls within the error bars. If ^one considers also i) ^the

very good agreement of the EH method with our

J(0) determination, ^{the two EH} calculations being

within the experimental uncertainty, ^and ii) ^{the fact}

that ^our result is experimentally ^{observed to be K-} independent, ^one may suspect the silver x-ray value to be slightly ^{too low.}

Our valence profile ^is compared ^{in table} II, ^for all q values, ^{to the} previously existing experiments ^{and to}

the most sophisticated IA calculations [21] ^L-corr.

Figure ² gives ^the corresponding percentage ^deviations from our J.(q) ^values. Eisenberger’s ^MoKa experiment

agrees reasonably ^{well with our valence one within}

the combined experimental uncertainties. Wong ^et

al.’s one differs from the actual values, because these

Fig. ^2.

Valence CP : relative deviations of the previously existing experiments and the IA calculations from

symmetrical CP, J5(q). The actual results

represented by

the horizontal axis, ^and the solid lines

the

standard deviation

(o o o o o) : Eisenberger’s ^MoKa experi- ment, ^Ref. [1]; (-A-) : Wong et al.’s electron experiment,

Ref. [3]; (-.-) : ^{the IA} calculation by ^{Naon and} Comille,

Ref. [21].

authors have only ^{used the}

high energy-loss ^{side »}

of their measured CP. Indeed, their data are found to be in _very good agreement ^{with our} J (q > 0) ^{ones. As}

observed for the total profile, ^{the IA} calculations are

below the experiment ^{near q}

^{0, while} they ^satis- factorily reproduce Js(q) ^{up to q - 1.5 a.u. within our} experimental uncertainties. Above this value, they ^are again ^several percents ^below JS(q).

3. Conclusion

We have shown evidence for the existence of a deviation between JIA(q) ^{and the observ-}

ed symmetrical ^CP, Js(q). ^More confidence in the present experiment ^is given by ^the very good agreement between ^our J(? > 0) values and the « high energy- loss side » electron impact ^data of Wong et ^{al. obtained} using ^{the same incident} energy. On the other hand,

the maximum height, Jmax(q, K ), and the FWHM of the CP have reached constant values for the K-ranges investigated ^{in this work, i.e. the} asymptotic ^limit predicted by Gasser and Tavard’s development ^is reached, ^at least for the top ⁵⁰ % part ^{of the} profile.

Therefore one may conclude that the symmetrical profile, J,(q), ^{is not} fully represented by the IA model and that the inclusion of the next symmetrical term(s)

of the Born’s propagator development is necessary ^to account for the difference. This clearly ^{shows to what}

extent the IA can be used as a comparison ^{model for}

the experiment, ^and points ^{out the need for} extending

Gasser and Tavard’s quantitative calculations on H and He to the Ne atom.

The authors would like to acknowledge ^{Pr. M.}

Rouault for his constant interest in this work.

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