Indian J. Phys. «7B <1). 1-7 (1993)
Electron impact double ionization of Ar^ and Ar^-
L K Jha, S N Chaitcrjcc and B N Roy
Dcpanmcni of Physics, Pihar Universiiy, Mu/jirr}irpur-S42 tX)l, Bihar, India Received 20 August 19^2 , accepted 24 August 1992
A b stract : Electron impact double loni/.ulion cross sections for Ar"*” and Ai^"^ have been calculated in the modified binary encoiinlcr model including the comributions from inner'shell ionizalion-Augcr process. 'I’hc clfecls o( Coulomb attraction between the target ion and the incident electron have been incorporated in the calculations 'I’hc calculated cross sections are in fair agreement with the avall.iblc experimental results.
K eyw ords : Electron, ion, double iom/.aiion, cro.ss section, modified binary-encounter model.
PACS No. : 34 50. E‘a
1 . Introduction
The cro.ss scclionii for single and muliipic ioni/alion of positive ions from the ground and excited slates arc essential in the study of laboratory and astrophysical plasma and that of solar corona. These cross sections form an integral part of the theories of intcrsteller media and models of siellcr interior [1-6]. Apart from this such studies provide information about the ftKussing effect of the Coulombic field of the tiirgcl ion on the ioni/>ation cross section.
Substantial experimental evidences now exist to show that in many double
^ionization processes, indirect mechanism is more important than the direct double {if)nization. In ease of electron impact ionization of ions, single ionization of the next inner
i
shell following an Auger transition has significant contribution to the double ionization cross section 15, 7], These contributions have been found to increase with the increase in the charge of the target ion as well as with the increase in the atomic number of the target il81.Quantum mechanical studies of electron impact direct double ionization of atoms and
^ns arc complicated and arc limited to few two electron systems [9, lOJ. On the other hand, he modified binary encounter (BE) method proposed by Roy and Rai [H] has been found to
|ive reasonably ticcuratc values of clcciron impact double ionization cross sections for atoms id ions [12-151. In the present work, we have calculated ciccu-on impact double ionization ross sections for Ar“^ and Ar"^"^ in the binary encounter approximation for which Kperimcnial results are available in literature. The effects of Coulombic field of the target m on the ionization cross section have been incorporated following the meibod suggested
© 1993 lACS
L K Jha , S N Chatterjee and B N Roy
by Thomas and Garcia |16J. The contributions from Auger process to the double ionization cross section have also been included in the calculations.
2 . Method of calculation
As the incident electron comes close to a positive ion due to the Coulombic attraction, its kinetic energy is increased and in the impact parameter is reduced. Effects of these two factors on the ionization cross section have been incorporated through the experssion [16],
a , (£,) - a, (CO 1^2+ 2 Ci | J [ ' Ex
where E\ is the incident energy, Z'c is the charge on the target ion, {E\) is the ionization cross section at the increased electron energy E'l and ^ is the collision radius whose value depends upon the ionic radius (r) and electron-electron separation
Electron impact double ionization cross section of an ion including the contribution from Auger emission is given by
( n = Qc/“ +
where Qa" is the cross section for direct ejection of two electrons and Q/\*‘is that for Auger proce.sos. The expression for()"d in terms of dimensionless variables; 5, .S' and t is given by
Im -
C'l -
u„
(
1 4Y u ,
J [2.S'^(;V'^ + /^ + 1) 1
(AEf
^ 3{AE)' J U.
______1________ ______4 /V . - A E f + LJ,
AE{S’W ,
AEY
---— j a f ( t ) U, L(AE) c//x8.797x10“’’' (toio^
2 2 2Z'
where S' = 5 -f 7 7 7. Other quantities appearing in the above expression have been defined qUt
by Roy and Rai [ 111. In the derivation of the above expression for Qd^ an accurate Hartree- Fock (HF) velocity distribution and a hydrogcnic velocity distribution have been used for the first and the second ejected electron respectively. The use of HF velocity distribution also for the second ejected electron would have been more appropriate but it would involve complex numerical integration making the expression for Q / complicated. The value of
has been taken as
( ) / = q/ (3/2, 3p) + Qd (3/;, 3.V)
where Q''d (3/?, 3s) sixmds for electron impact direct double ionization cross section with the first electron is ejected from the 3p-sub shell and the other from the 3 5-sub shell.
Electron impact double ionization o f Ar^ and
In order to obtain Qa“ w c have calculated the electron impaci single ionization cross sections for Ip and 25 sub shells using the relation
Qa - ne
6
is + S ' f n P + r + 1)
(S'^ - 1) 2(^
T
5''* - 1 ? P "1<p In 5'^
+ 1)
(see [17]).
Using single ionization cross sections the values of 0a” have been obtained as explained in the next section.
In the present work the values of Harirce-Fock binding energies of the sub shells reported by dementi and Roetti 118] have been used. CrysUil ionic radius for Ar^ reported in Lange’s Handbcx)k of Chemistry [ 19| has been used as radius of 3 / 7 sub shell of Ar"^, The radii of other sub shells have been approximalcly calculated using Bohr model of atom. For these calculations the effective nuclear charge has been obuined using Slater’s rule. In order to construct the momentum distribution function /(/) for the Uirgct electron (sec [2 0]) the HF radial wave functions reported by CIcincmi and Roetti [ 18] have been used.
3 . Results and discussion
Wc have compared our calculated cross sections with the experimental measurements of Muller Cl al 121] in case of Ar" and Ar"^“" and with those of Pindzola et al [22] in case of Ar"^"". Thresholds for electron impact double ionization of Ar^ and Ar^"" arc 63.3 eV and
166.0 eV respectively. Experimental cross section curve for Ar"^ shows some structure close to an impact energy 260 eV which has been aiinbuted to the contribution from ionization of 2/7-sub shell followed by Auger process leading to double ionization of Ar"^. In case of Ar'^'' a sharp increase in the electron impact double ionization cross section has been found m experiment at about 3(X) cV impact energy which is close to the ionization energy (320 eV) of 2/7-sub shell of Ar'^^. This has been attributed to the contribution to double ionization cross section due to single ionization of 2/7-sub shell of Ar'*’*^ followed by an Auger process. The experimental observations do not show any contribution from excitation-autoionization process to the electron impaci double ionization of Ar^ and Ar"*"*^.
In the present work* we have incorporated ihp contributions from L-shell ionization of Ar“^ and Ar^^ followed by Auger transition to the electron impact double ionization cross section using the relation
Q / ; q + 2^^^ = 0.89 Qip^ + 0.047 02^’ + 0.052 Qts\
as suggested by Muller and FrodI 17J. Here 0zr' stand for elecion impact single ionization cross sections for 2p and 2s sub shell of the target ion. In the above equation, 0.89 and 0.052 are the weight factors lor the single Auger process following single ionization of 2p and 25 sub shell respectively. The factor 0.047 is the probability of Auger process connected with a shake up transition 25^ 2p^ 3 5^ 3p^ 2s^ 2p^ 3s^ 3p^ nl ^ e.
L K Jha, S N Chatierjee and B N Roy
While using the above relation for calculating L-shcll contributions to electron im pact double ionization cross section for Ar* and Ar we have assumed that the probability of rearrangement process subsequent to L-shcll ionization for Ar^ ions is the sam e as that o f Ar atom (see (21 ]). It is not physically justified, one should use appropriate experimental or theoretical values of probability for rearrangement for separate targets. To the best o f our knowledge values of these quantities lor Ar and Ar are not available in literature, so we have used the approximate values suggested by Muller and FrodI [7j.
F igu re I. Hlcciroii irnpHCi iloiihic ioni/alion cross section for Ar* : Curve A — Direct double lom/^lion cross scciions.
Curve B — Cross sections including the cllecls ol inner-shell loniziiUon-Auger process and O — I:x[)cnmciilal results o( Muller el al [211
Our calculiiied results for Ar"" arc shown in Figure 1 along with the experimental (lata. The present cross sections arc always within a factor 2 from the experimental results.
Moreover, the shapes ol the expcrimcntai and theoretical cross section curves are is close agreement. The expenmcnuil cross section shows a peak of magnitude 4.46 x 10“^* cm^ at about 180 eV impact energy. The calculated peak has been found at about 160 eV impact energy and is of magnitude 2.8x 1 0 cm^. Thus we observe that although the magnitude ol the calculated peak is smaller than the cxpcrimcnuil results, the theoretical and experimental positions of peak arc close to each other. The calculated cross section curve shows a break, as observed in experiment at about 270 cV impact energy due to onset of ionization of 2p- sub shell followed by an Auger proccs.s. It is seen from calculated results the beyond 300 eV impact energy, the Auger process has a significant contribution to electron impact double ionization cross .section for Ar’'. The agreement between the experimental and calculated results improves with increase in impact energy. The underestimation of the cross
sections in the presscnt calculation may partly be attributed to the use of crude values of sub shell radii for At* to which the present results are sensitive. If accurate values of sub shell radii for Ar* we have used in the present work the results would improve leading to belter agreement with experimental cross sections, further, while evaluating the direct double ionization corss section for Ar* we have used the binding energy of Ar^* for the ejection of the second electron. This approach implies that, after the ejection of the First electron, the target ion gets sufficient lime for rearrangement before the second electron is ejected. This approach may be justified at impact energies close to the threshold but with increase in impact energy the collision becomes fast and consequently the probability of rearrangement decreases. If the allowance of partial rearrangement is made in the present calculation the results are expected to increase leading lo better agreement with ciCperiment [14].
Electron impact double ionization o fA r* andAr^* 5
l^igurc 2. 'nicciton impact double ioni/.aiion cross section Tor Ar** ' Curves A and B — Same as in Tigurc 1.
O — F-xpcrimerilal results ol Muller et al |211 and
• — Kxpcriincnial data of Pirid/x)la el al [22].
Figure 2 shows the results for Ar"'*'. For calculation of direct double ionization cross section we have used binding energy of Ar"'" for both the ejected electrons instead of the binding energy of Ar^'*’ for the second ejected electron. We have performed calculations with the latter choice also but obtained cross sections much smaller than the experimental values. This might be attributed to the fact that the charge on Ar^^ being large, due to Coulomb attraction, the energy of the incident electron at the time of collision will be much higher than the impact energy and hence the target ion, after the ejection of the first electron, may not be able to rearrange itself to acquire the binding energy of Ar^^. The present results arc always within a factor 2 as compared with the experimental data.
Moreover, the shape of calculated ionization curve is in reasonably good agreement with
that obtained experimentally. At impact energies slightly lower than the threshold of L- shcll ionization the agreement between the theory and experiment is relatively poor. This might be due to the fact that because of Coulomb attraction between the incident electron and the target ion the L-shcll ionizatron process starts at an impact energy below the ionization energy o f2p-sub shell. But in the method proposed by Thomas and Garcia [16]
for inclusion of effects of Coulomb attraction between the incident electron and the target ion on the ionization cross section, it could not be possible to calculate the ionization cross section for 2/>sub shell below the threshold. As mentioned in case of Ar"^ results, the undcrestimaiion of the prescni calculaiions may partly be attributed to the use of inaccurate values of sub shell radii. Apart from this, the use of probability of L-shcll ionization-Auger transition process the same in the case t)f Ar"^^ and Ai might be partly responsible for the low values or the present cross sections, lixperimental evidences show tht)t the contribution from indircci process to electron impact double loni/ations of ions increasl^ with increase in ionic charge ol the uirget. If accurate values ol these probabilities are used in the present calculations, the results would be higher leading to a better agreement with experiments.
Apart from these discrepancies the overall agreement between the present calculations and die experimental results is reasonably good.
In the prc.sent work it is concluded that m ease of Ar^, the L-shell ionization-Auger process has significant contribution to electron impact double ionization cross section but in ease of Ar'^", this process is mainly responsible for double ionization above the ionization threshold of 2/?-sub shell. This might be due to small number of valence electrons |)rcsen( m Ar*^’ as compared to ihat jircscnt in Av*.
Aekii o wi cdgni ent
One of us (SNC) is grateful to the Council of Scientific and Industrial Research, (CSIR), New Delhi, for the award of a Research Associatcship. BNR is thankful to the CSIR, New Delhi for sanction of the rescLirch scheme No. 03(0720)/92 EMR II.
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Electron impact double ionization ofAr^ and Ar^* 7
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