Bethe surface and compton profile for CO2 obtained by use of 35 keV incident electrons

Volume 60, number 3

CREMICAL

BETHE SURFACE AND CGMPTON PROFILE OBTAINED

BY USE OF 35 keV INCIDENT

15 January 1979

PHYSICS LETTJXS

FOR CO,

ELECTRONS

Azzedine LAHMAM BENNANI, Alain DUGUET and H.F. WELLENSTEIN Luboraroirede iMh&ns Atomiques et Molhhires (Difficrion Hectronkpe),

*

(inivemite de Paris SC&, 91405 Orsay, Face

Receiwd 10 April1978 Revised manuscript received 21 September 1978

Electron impact spectra for CO* have been obtained at 25 diffexent scattering angks ranging from 1.12O to 14.060. The measured intensities were converted to generalized oscillator strengths and normalized by use of the Bethe sum rule, leading to the mapping of the Bethe surface over large momentum transfa K and energy loss E ranges. Substantial deviations from the binary encounter theory were obsexved for K dues smalier than 3 au A diswussionis give-n on the possiiility of extracting partial Compton profiles from the data The total Compton profile was obtained at large K values and found to be in good agreement with r-t calculations. within rhe experimentai uncertainties.

1. Jn~oduction Theoretical and experimental determinations of profiics (CP’s) are of special interest because of their simple relation to the electron momentum densities in the target. However, in spite of ‘he large revival in the different experimental techniques used for this determination, and in the rapidly increasing number of studies published on this subject (see for instance the excellent review by Williams [l] and references therein), the CO2 molecule, which is one of the most easily experimentally accessible, has not yet been so investigated, to our knowledge, though several CP calculations [24] on this molecule do exist in the literature. Nevertheless it is only at large momentum transfer K that high energy electron impact spectroscopy (HEELS) does lead to the CPJ(Q) (5 ] . It was then decided to measure electron impact spectra at many different scattering angles, with CO2 as a target, in order to study the dependence of J(s) on K and to so define the K region where a plateau is reached. Compton

+ Permanent address: Physics Department,BrandeisUniversity, W&ham,

Massachusetts 02154. USA.

2. Apparatus and method The high energy electron spectrometer has been in part described elsewhere [6,7] so that only a brief description of the experimental arrangement will be presented here: a 35 keV electron beam (0 to 200 M intensity, 0.4 mm fwhm), issued from a Steigerwald [I31type electron gun, intersects at right angles a CO2 gas jet produced by expansion through a multiarray nozzle 191. The scattered electrons are analyzed by a 127O cylindrical electrostatic energy analyzer, and detected by a plastic sciutillator-photomuhipher arrangement, while the intensity of the non-scattered electrons is measured by a beam trap. The rotation of the electron gun about the gasjet axis defines the scattering angle, the analyzer system re maining tied. The overall angular precision in this study was 0.05” ticluding alI error sources, by the extent of the scattering volume, i.e. the hatersection of electron and gas beam, and is about 0.02”. The energy resolution of the electrostatic analyzer was set to 6 eV at the smallest angles. In order to increase the large angle count rate, we enlarged the width of the analyzer slits, resulting in a resolution of 12 eV. However, by assuming gaussian shapes for both the Compton profile and the analyzer resolution function, it can e&y be shown [lo] that this degrading in reso405

CHEMICAL PHYSICS LJzl-rERs

Volume 60,num~r3

does not appreciably affect the Compton profile &ape, as the perturbation is <8 X 10m5 zt the maximum of the CP and <8 X lOA at h&its maximum as long as the instrument resolution function is
3. Results and dkcussion Energy loss spectra have been measured at 25 cWferent angIea rang@ from 1.12O to 14.06O. According to the relativistic binary encounter theory, the maximum of the inelastic profile for each spectrum should be Iocated at an energy loss given by EBE = E. sin28 (1 + $/32cos26 + ...) , where E. is the incident energy, and fl the ratio of the incident electron velocity to the velocity of !ight.

b

However,

15 January1979

due to the relatively large aqplar

ty in the present experiment,

uncertain-

*O.OS”, it was inrpossible

to rigorously check this prediction (an error of 0.05O corresponds to 8.5 eV error in EpE for a typical angle of 8”). The scattering angle was then slightly adjusted so as to match the observed inekstic peak with the peak predicted by the impulse approximation. However this adjustment was applied only if it did not exceed the estimated uncertahrty in 6, namely for 8 > 3O. It is to be noted that the effect of this procedure is to screen the so called Compton defect [1 11, AE=E, - EBE, where Em is the observed energy loss at the maximum of the inelastic profile, but however, an upper limit is thus determined for the relative defect A.&/EBE which decreases from 3.3% at 3” to 0.7% at 14O. For scattering an&s smaller than 3”, the binary encounter theory was found to fail catastrophically in predictiug the most probable energy loss, the relative Compton defect becoming as much as 135% at l-12”. This is not surprising since, in the impulse approximation [12] (IA), the target electron is assumed to be free, or in other words its binding energy is assumed to be negligible compared to the energy loss suffered by the incident electron, but this is precisely not the case at such small artglee (for instance,

GOS (ad CO2 Bethe

Energy

loss

35 keV Surface

E (ad

Fii 1. The Bethe surfacefor CQ obtained with 35 keV incidentelectrons.afiK,E)/dE representsthe generzked ~.sci&tor strer@ at a given valueof energylossE and momentumtsamferK

Volume60, r.umber3

at 1.12”,E,isbere31.5eVandE~Eis13.4eV,w~e the fust ionization potential of CO2 is 14.4 ev). Using the previously described technique 1771,the measured energy loss spectra were corrected for bacicground and termination errors, and then normalized by means of the Bethe sum rule after conversion to generalizedoscillator strengii (GOS), dj&,.E)/dE. The so obtained GOS, when plotted as a function of the energy loss E and of log K2 Iead to the threedimensional plot commonly caped B&be surface [ 131. Such a plot, as shown in fig. 1 for COz, contains ah the information concerning the inelastic scattering of structurelesscharged particles, and is dominated at large K values by the Bethe ridge which corresponds to the scattering of the incident electron by an electron initially free and at rest. As a check for our energy loss calibration, the averagevalue for the ionization thresholds of the carbon and oxygen K shells, as determined from all the spectra, were found to bc Ck = = 532 + 4 eV, in good agreement 286+4eVandOk with the determination by Wight and Brion [ 141, respectively 288 eV and 535 eV. The main features of this Bethe surface which are of interest for the present study will be discussedlater. The GUS were then converted to a Compton profile using the Bonbam delition [lS] of an electron CP: J(4, K) =>2K31”

df(&g%=,

where .T(q,K) is a momentum transfer dependent profile, whicir should approach the limit J(q), for sufficiently large K, usually defined in X-ray scattering as the CP of the target_ The intensity maxima of the J(q, K) profiles were determined, for B > 3”, by least squares fitting the top of the curves to an even polynomial of the form A + Bq2 + Cq4, and by a graphical fit for 8 < 3”. T&se valuesJ(q = 0, K) are of special interest since, within the framework of the IA, the maximum of the CP, J(O), is directly related to Q-l>, where p is the electron momentum in the target before ejection: J(0) = $v@-1,

15 January1979

cHEMx!AJL. PHYSlc!3 LETrERs

,

N being the number of electrons in the target. Fig. 2 gives the dependence of J(O,K) or of
-*____&-=_

e

8 * 0

c

fdeg.) 12

8

*

K ( a.~.)

FQ_ 2. bfsimum valueof theC’P,J(4 = 0, m (crosses),plotte-dzs a functionof mcmentumtrim&rK (Or scattering angle 6). The Fidel aad broken lines are eye inter-

polations to the data.

gy loss side of the inelastic protie (see the Bethe surface in fig. 1). This means that for those angles, only the valence electrons contribute to the J(q = 0). On the other hand, the CP moves to higher energy losses as the scatteringangle increases,and is exactly centered on the carbon K threshold and on the oxygen K threshold respectively at 8 = 5_05Oand 6_98O:in this small angular range, the observed J(O) includes the contribution of the 16 valence electrons and that of the two 1s eectrons of the carbon, leading to a higher value for J(0). Above %7”, all the electrons of the molecule are in- . volved and J(0) is then even higher. The 1s electrons being faster than the other ones, the addition of their contribution must lower the per electron, in good agreement with our observations in fig. 2. The separation in 3 regions on tbis figure could suggestthe definition of partial CP for the CO2 molecule as previously done by Wong et a!. [16] for N2, Ne and Ar: region l(0 < 5”) would correspond to the valence CP; its difference with the intermediate region 2 would 407

iSJanwsyf979

give the CP of *&e 20~ orbital; and again the difference with region 3 (0 > 7-) would lead to the CP of the sum of the orb&As 10~ aud lo,. But several reasons make these determinations very imprecise: (i) The cl35 ferences are to be taken on very ciose quantities and the relative precision on the final result will be poor(ii) The very rapid decrease of 2(0,X) and WWf> for a&es 6 5 3O ctearfy shows a defect of the W, and in spite of the spread in the measurements, it seems that, even at 9, the limit of validity of the IA is not yet reached. The profile one would then deduce will not be a good valence profile, The situation is even worse because, as to obtain a significant part of the CE’, the carbon K shell has to be quite far from its maximum, i_e, B 5 3”. @ii) ‘I&e intermediate region between 5O and 7” is too narrow and strongly perturbed by the K shehs. However, above 7.5” J(O,K) becomes independent of K within the experimental accuracy: the Cl? as determined in the framework of the IA will be the total CP for tire target. Table 1 gives the value of./(O) and
f 1st CO,-35keV

-2

II

2

t

6

8

10

-4fa.H JFig.3.Experimental ComptonprofileforCC& asobtainedby

thesupa?rpodtionofffie~e~tprofilesm~ to 10*z3ngdarsgtteringrange.

inthe?

mined at each scattering angle in region 3. The average values for J(q) and the standard deviations were then obtained by a fitto t&s Sgure,and are summa&ed in table 2. They are compared to the previously mentionTable 2 ExperM?n?.slComptonprofile for C& as compsredto the mostrecent~~tions,Theex__r?erimen&f.ratuesare obtainedbyafittofig.3

Td~lel ExpeBnental vaiuesfor Jfo) and Q3” ) compared to the most

retest theoreticalcd~ions mlis

Theory

0.0

ezqerime-nt ref. j21 r& [3f

ref. I41

total

i(O) 7.46 ~0.14
7.382 7.333 7.480 0.67il 0.6666 0.680

total-

J
7-33 Hx?O 0.814+0.022

6.929 0.7699

vaIe3.I~ J@> 6.85 *OK!0 0.855~0.025

6621 0.8276


408

0.2 0.4 O-6 0.8 1-o 1.2 2-4 16 1.8 2.0 3.0 4-O 6.0

7-46+ 0.14 7.28 6.61 5.71 4.84 3.94*Q-24 3.15 246 1.98 1.63 1.30* 0.10 0_67*0.08 0.41r 0.06 0.18iO.05

7.3821 7.2683 6.8985 6.2422 5.3483 4.3509 326011 26024

7333 7.201 6.798 6.132 5.275 4.349 3.471

1.9913

2110 1.652 1.317 0.592

1.5531 1.2494 0.6157

2714

7.4804 7.3576 6.9550 6.2406 5.2872 4.2616 3.3252 2.!%40 1.9908 1.5720 1.2740 0.6110

CHEMICAL PHYSICS LETrERs

Volume 60, number 3

ed theories: (i) the selkonsistent field (SCF) calculations by Lindner [2] using gaussiantype orbitals and a basis (?s, 3p) contracted to (4s, 2p) for both C and 0, (ii) the caldations bfEpstein 141 using SnyderBasch double zeta gaussianwavefunctions; and (iii) the calculations by Roux and Epstein [3] using wavefimcticns computed from minimum basis sets of Slater type orbitals. The agreement, as seen from table 2, is very good close to Q = 0 and for large Q values, but is less satisfactory in the intermediate Q range where the theoretical values are systematically above the experiment (about +lO% at Q = 1 au). This can also be expressed by the quantity &, the fwhm of the CP, which is related to the bond formation in *be molecule, As the outer shell electrons contribute most to the small momenta, they are responsible for the more significant part of the CP. But those outer regions are distorted by the chemical bonds, and the effect is to increase the mean momentum of the electrons and hence to broaden the profile. Table 3 gives the experimental and theoretical A4 values.Also shown is the value obtained by adding the CP of the constituent atoms, namely C C 2 X 0. The atomic profile for oxygen was taken from the X-ray measurement of Eisenberger [ 171 plus a core contribution calculated by Weisset al. [18] using the HartreeFock wavefunctions tabulated by Clementi [ 191, and for carbon from the same calculations by Weiss et al. As seen in table 3, the experiment as well as the three molecular calculations give higher Aq values than does the sum of atoms. This result is consistent with the hypothesis that the electron densities contract during the bond formation_ However the molecular calculations lead to slightly higher Aq values compared to the experiment. It is to be noted here that all the CF's, either obtained by photon scattering or theoretical ones, are normalized to the number of electrons in the target. Hence the CP of the sum of atoms which is the narrower one, is necesjarily the higher, leading to Table 3 Experimental2nd theoreticA v&es of @, full Mdth at half the maximumof the CP

A&u)

Thk experimerit

Ref.[21 Ref.131 Ref.[4] Sumof

208 i 0.1

228

atoms 230

2.20

192

15 January 1979

J(0) = 8.427 to be compared for instance to the experimental value 7.46. To checkfurther the validity of the IA, we have also considered the following point: It is obvious that the total Compton scattered intensity (i.e. the area under the CP) has to be equal to the X-ray incoherent scatteringfactor

In the Wailer-Hartree [20] theory, S&K)

is given by

N

fij = (*iI

exp(iK 0 r) I*$

,

where ‘ki is the wavefunction for the ii& electron of the target. The presence of the non diagonal terms & is due to the exclusion principle which forbids electronic transitionsto occupied states. But within the framework of the IA, Sti,(K) = N is valid for sufficiently large K values [2 11. Hence we have integrated the high energy loss side of our CP over 4, and the results are summarizedin table 4 for the large scattering angles (region 3 of fig. 2). It cau be seen thatG J(q) dq is systematicallylower than N/2 (N = 22 for CO-& and that it has a slight tendency to increase as 8 increases.The asymptotic limit for Si&K) is not reached at the largestangle used here (~3% deviation at e = 149. This is an a posteriori justification of not having used the relation K, J(4) & = N to normalize our resultsas usually done in X-ray scattering, but of ratherhaving used the Bethe sum rule. However, it TabIe 4 Vadation with the scatteringangleof the zrea under the hi& energy loss side of the CF. The limitN/2 = 11 is not rezckd

7.94 8.06 9.06

10.34 10.40 10.60

9.94 10.06 12-06 14.06

10.44 10.50 10.57 10.65

409

Volume 60, number 3

ctiEMICAL PHYSICS LETTERS

cannot be ruled out tb.zt part of the deviation observed between 6 J(q) dq and N/2 could be dueto the useof thissumrule2t constant 0 ratherthan constant K: fi changesby X% over the whole spect_ralrange investigated.and this change becomes less as 6 ticreasa.

4. Conclusion The Bethe suhce, of which only few measurements exist in the literature [16,22,23], has been mapped over large momentum transfer and energy loss i-es. A critical invest.igation of the determinationof the CO* CP using HEEIS has been carriedout and the range of validity of the IA has been clearly defmed withiu the experimentaluncertainties.At large momentum transfervalues, the CP was detemGned and found to be in good agreementwith recent theoretical calculations_It is hoped that new improvements -k experimental techniques will lead to more accurateresults.

Refmce5 [l] B-G_ ‘SViEmq Pbysica Saipta 15 (1977) 69. (21 P_ Lindner, upp!da university, pxivate a3mlnuni~tion (1977).

133M. Rouxand LR Epstein,Chem, Phys. Letters 18 (1973) 18

410

15 January 1979

[4] LR Epstein,Brand&U niversity,priMte cosnnmrdcation (1976). [51 U Bonham and C. Tamed, J. Chem Phys. 59 (1973)

-

4691. [6J k Tnhmam-Bemani,B_ Nguyw, J. Pebay and M. Lecas, J_Phys. E8 (1975) 651, [7) k LahmamBcmmni, KF. WeIknstein, A_ Dugaet and AJI. Bar&i, Chem. Pbys. Letters41(1976) 470. C81KM- Sty. OMk 5 (1949) 468 and M. Rol?autt,J. 191 B. Nguy&, A_ ~Benaani Pbys Es i1975) 909, I101 k Labmam_Bermaz&XI&e de Doctor&, Orsay, France (1978). ’ IllI kD. Barks, W. Ruecknerand I-I-F.Wellenstein,Phil_ Hag 36 (1977) 201. v21 P. Eisenbergerand PM_ Pl&zrmnn, Phys_Rev_ A2 (2970) 41.5. 1131 56 Inckuti, Rev. Mo& Pbys 43 (1971) 297. (141 G-R Wiit and CF. Brian, J_Ekctron Spe&y_ 3 (1974) 191. 1151 flk Bonhzn 2nd H_F_Welknstein, in: Compton sutteriug, ed. B. Williams (Ascot Press, 1976) cb_ 8. 1161 TX. Wang, J-S_Lee, ftF. Wellensteinand RA. Bonbzm, Phys_Rev_ Al2 (1975) 1846. 1171 P_ Eisenberger,Phys_Rev_AS (1972) 628. P81 RJ. Weiss,A_ Harvey and W-C. Phillips,Phil_Mag_ 17 (1968) 241. 1x91 E, Ucmenti, IBM J. Res_Develop. 9 (1965) 2. [201 L WaRer and D.R Hartreq Rot. Roy. Sac Al24 <1929) 119. 1211 R Cumt, P-D, de Cicco and RJ. Weiss,Phys_Rev. B84 (1971) 4256. 1221 H_F_Weknstcin, fLk Bonham and RC. ulsh, Pbys. Rev. A8 (1973) 304. v31 RC Ukh, ELF. Wellensteinand RA Bonbam, J. C&em_ Phys_60