Indi an Journal o f Pure & Appli ed Phys ics Vol. 40, Jul y 2002, pp. 50 1-507

Theoretical determination of electron impact ionization f S •+ s. ++ d c 7 + • 0 I , I an r lOllS M Alfaz Uddin & M G o la m M o rtuza

Department o f Phys ics, Uni versity of Rajshahi , Raj shahi 6205 , Bangladesh

Received 20 September 200 I ; revised 18 January 2002; accepted I 0 April 2002

Semi -empirical formul ae BEBA, BEB and Pattard-Rost models are employed to compute the to tal cross-section of single ioni zation of Si +, Si ++ and Cr

7+ ions by electron impact. The parameters o f the semi-empirical formulae are obtained

by fittin g wi th the ex perimental data over a wide range of energy. It is found that, the semi -empirical formul ae provide excellent fit , wh ile the BEBA and BEB models give qualitati ve fit to the ex perimental data. The parameter-free Pattard-Rost model is in fai rl y good agreement with the data.

!Introduction

Electron impact ioni zation of atoms and molecules is important for the basic understanding of kinematics and dynamics of colli sion phenomena. The data for total ioni zati on cross-sections are al so needed in a wide range of applications such as pl asma kinetics, pl asma etching of semiconductors, flu orescent lamps, ion gauges, mass spectrometry , fu sion devices , astrophys ics, radi ati on phys ics and X-ray lasers, etc.

In fu sion researc h, accurate data for ionizati on of impurity ions by e lectrons are indi spensable for mode lling of power balance, charge-state di stribut ions, impurity compos ition and for spectroscopic di agnos tics . Suffic ient information about e lectron impact ioni zation of ions is still lacking despite its importance in contro lled fu sion research. The study of the e lectron impact ioni zati on cross-section of s ilicon and chromium ions is of interes t because, these are not only fo und as impurities in fusion dev ices, but are also ve ry important as astra -phys ica ll y abundant e lements.

The accuracy of standard theoretica l methods fo r ioni zation cross-sections depends both on the quality of the wave functi on as we ll as on the co lli sion theory. Many theories work well in hi gh inc ident ene rgies bur, few can be trusted in the low energies, particularl y, near the thresholds. Al so, the theories that require continuum wave functi on are lengthy and arduous to use. Thi s si tuation underscores the need for ana lytica l fo rmul ae for the ionizati on cross-secti ons.

E lectron-atom co llis ion fa ll s into two broad categories, vi z. hard and soft. The Mott cross-secti on formali sm 1, which is a generali zed Rutherford cross-section\ taking into account the exchange effects, describes the colli s ion of two free electrons . Thi s formula accounts we ll the hard co llision. Conversely, using the Born approximation , Bethe3 derived a formula for the dipo le interaction invol ving the fast inc ident electrons and thus expl aining well for the soft co li is ions.

Attempts have been made by many workers (see, e .g. Mull er et a /. 4 and references there in) to combine the dipo le inte rac ti on with the free e lectron co lli sions in order to obtain a correct parameter free formul a for ionization cross-secti on. All ea rl y attempts had only limited success because, they failed to find the correct mixing rati o between the soft and hard colli sion terms.

The earl y success ful atte mpt to calcul ate the ioni zati on cross-sec tion combining the soft and hard collis ions was made by Khare et a /.5 in 1974 , for rare gases and Ja in & Khare6·7 in 1976, fo r molecules. By extending the approach of them, Saxena et al.x proposed another mode l for the atomic and molecul ar ionization c ross-section, using the phenomenological equation of M ayo! & Sa l va rt~ and introducing a multipl ying fac tor. Recentl y, Khare et a l.10 gave a mode l for the total cross-section of single ioni zati on by modifying the ionizati on mode l of Saxena et al.x and utili zing the multiplying factor of BED mode l propounded by

502 INDIAN J PURE & APPL PHYS, VOL 40, JULY 2002

Kim & Rudd 11 • The authors refer to the simpler version of th is model as , the BEBA model of Khare et af.III. The resulting expression splits into Mott and Bethe cross-sections, sum of which is almost clo e to BEB model, simpler version of BED model of Kim & Rudd 11 • The BED/BEB models combine the binary encounter theory 12 with the dipole interaction of Bethe theory' . The combination of binary encounter theory and the Bethe theory is achieved by requiring that asymptotic form at high energy, E, of the former, matches with that of the latter, both in the ionization and stopping cross-sections.

Deutsch & Mark 13 introduced another model , known as the DM model in the literature, for the calculation of total cross-section of electron impact ionization . The formu la consists of an energy-dependent part using class ica l binary encounter approximation of Gryzinsk i 1 ~ and another part equal to the weighted sum of the squared radii of the max imum charge density of the outer electron shell s. The approach has been applied to single and multiple ionization of atoms, molecules, radicals and clusters. Other approaches represent the ioni zati on cross-section in compact and analytical forms. These requ ire adjustable parameters that must be fitted with experimental/theoretical ioni zat ion cross-section data. There has been a continuous effort to prov ide simple, semi-empirical formulae for the ioni zati on cross-section of atoms and ions. Probab ly, Lotz15 gives the best known fo rmula among them. Over the years, many different parametrizations have been proposed. A common feature of most of thi s is, the inc lusion of correct asymptot ic behaviour as in Bethe' s formula. Pattard & Rozt1r. gave a semi-empirical formula for the ioni zati on cross-section with incorporation of the correct threshold and hi gh energy behaviour of cross-section . Despite being parameter-free, the formul a provides a good fit to ex perimental data over a wide range of incident energies and for several class of projectil e ions.

The BEBA and BEB models have structural elegance, providing insight into the ionization process and also provide straightforward methods for cross-secti ons for si ngle ioni zation from the knowledge of the form factors of the atomic/molecul ar specimens. The models have been app li ed with fairly good success to molecules. But their test cases on atomic ions are not so conspicuous as on the molecular specimens. Kim &

Rudd 11 applied with success, BEB/BED models to the ionization of singly-charged atomic ions. It may be of interest to apply the models to many-electron ions. Although, the BED and Khare et a! . III models have the versati lity in applications over a wide range of energy and to a broad class of targets, they are in, most cases, arduous to implement them owing to the requirement of the knowledge of the oscillator strengths. However, BEBA, BEB, Pattard-Rost and the semi-empirical models require inputs whi ch are not hard to obtain as in the case of the BED and the sophisticated form of the models of Khare et a!. III Despite being simple in structure, the semi-empirical formulae provide normally good fit to the data even though indirect ionization such as exc itat ion-auto-ionization processes are involved . The BEBA, BEB models are simple in sophisticacy, from the implementat ion point of view. These considerations lead the authors to choose BEBA, BEB and the semi-empirical models for the calculations of ionization cross-sections.

The present work by the authors calcul ates the total cross-section of single ion ization of ions Si +, Si ++, Ct-7+ by electron impact. These ions are chosen for their analysis because of their theoretical as well as practical importance, as specified earlier. They have made comparative studi es of the efficacy of the BEBA, BEB, Pattard-Rost and semi-empirical models to predict the total ionizat ion cross-sections of the aforesaid ions. They also propose two semi-empirical formulae that agree well over the entire energy range of the experimental data.

2 Theoretical Background

2.1 Semi-empirical formula

The total cross-section of single ionizat ion by electron impact in compact semi-empirical form, as suggested by Kim & Rudd 11 is:

? [ ( J ] 4rra~ I lnt CJ(£)=-- a ln t+b 1- - +c --t I t + I

.. . ( I )

where a, b and c are the fitting parameters, a., is the Bohr rad ius, t=EII, where E is the incident kinetic energy and I is the ionization threshold . The authors also suggest the following two semi-empirical formulae:

I c CJ (£) =-[a ln(t) +b ]x

IE ... (2)

UDDIN & MORTUZA: ELECTRON IMPACT IONIZATION

--- -- ------·

I • Expt . ,--BEBA of Khare et al I·· ·· BEB

0 2 00 600 BOO 1000 1200

Energy (eV)

Fig. I -Ionisation of Si+

20 ,-------------------1 I

1 €'1 4

:5:: ,, ~ 1 2 f T -~ 1 0 1 ~- ~ i : j ~.

'"" ~. ¥

2 ~~

0 ~--200 400

I a (E) =-[a In(!) +b ](I+ ex)

IE

with x=l-llt

E -----

• Expt . Equ . 1.

- -- Equ . 2

:-~~ttard et at

600 eoo

Energy(eV)

Fig. 2 -Ionisation of Si+

... (3) AN

a pm = 12 (1+u+l )

1---+-+---[

2 1-1 5-1 2

1+1 21 2 2(1+1) 2

1000 1200

I 1+11 (1+1 \l l(t+l ) -7 n - 2-)j

50.3

Each of the above three formulae satisfies the well-known classical scaling law as well as the Bethe behaviour3. Expression in Eq. (3) can be used to account for the non-zero cross-section values below the thresholds.

. .. (4)

2.2 BEBA formulae of Khare et al. 10

, The total ionization cross-section of single ionization by electron impact per orbit in the BEBA model is given by:

and

CY = AN f-1 IJ w}lw PB (t+u+l) I OJ"' '\ Q .. . (5)

where, A= 4 rta} R2 , R=l3.6 eV, N , I and u are the number of e lec trons , ionizati on potential, the

504 INDIAN J PURE & APPL PHYS , VOL 40, JULY 2002

average orbital kinetic energy di vided by / , of the orbit concerned. The quantity Q is the recoil energy of the target and is given by:

Neglecting the very small contributi on to the cross-section due to the transverse interact ion \11, the total cross-section per each orbit in the BEBA model is given by:

Q=-l-2mc2 CJ (£) = (J PM + (J PB . . . (7)

[ {E(E + zmc 2 lF -{(E - w l(E - w + zmc 2 JF] 2.3 BEB model of Kim & Rudd 11

The total cross-section of s ingle ioni zat ion per orb it in BEB model is g iven by:

E

UDDIN & MORTUZA: ELECTRON IMPACT IONIZATION 505

AN a (E) = -:----

!2(t+u+1)

[1- .!__ ~+ _!__ln(t)(l- ~)]

t t+I 2 t-

•••••• •• .. e • • •

•• • • • • • I •

.. ~ .

• Expt.

---·- Ground state -Metastable state

Energy{eV)

Fig. 5- Ionisation of Cr7+

. . . (8)

BEBA and BEB cross-sections for most ions with Burgess 17 denominator (t + u + 1), introduced as ad hoc part in the formulae in Eqs (4) and (8) to represent the effective kinetic energy of the incident electron as seen by the target electron, are too low compared to the reliable experimental data. The attractive force between the positive ion and electron reduces the Burgess denominator from the neutral value. But, the exact form of the denominator is not clearly understood. However, replacing (t + u + I ) in Eqs (4) and (8) by (t + I) sometimes improves the situation for ions. Kim & Rudd 11 recommended this prescription for open shell ions. For closed shell ions (t + u + I) is usually used .

2.4 Formula of Pattered & Rose 6

The model of Pattard & Rost 1r. . may be used to compute the s ingle as well as multiple ionization of atoms/ions by charge particles. The total cross-section for single ionization by electron impact is :

aE=a f(x) ( ) max f (I) . .. (9)

X where f (x) = 2 and x (x + 1)

E- I

£max

Gmax is the maximum cross-section and £"'"' is the corresponding incident kinetic energy .

3 Results and Discussion

3.1 General plan

The authors have computed the total cross-sections for single ionization using the model s described in section 2. The orbital constants, namely, orbital ionization potential , kinetic energy are calculated using a relativistic Hartee-Fock program 1H. The parameters of the semi-empirical formulae given in Eqs ( 1-3) are obtained by using the method of least-square fitting. In the calculations of BEBA and BEB cross-sections, they have taken the Burgess denominator as (t + u + I ) for Si ++ and (t + I) for Si+ and Cr7+ as in section 2.3.

The authors have taken the experimental data of total single ionization cross-section of Si+, Si ++ and that for Cr7+ from references 1 ~· 20 respectively . They compared the calculated cross-section with the experimental values and also among themselves in Figs (I to 6). Table I provides the optimum values of the parameters of the semi-empirical formul ae given in Eqs (1-3), standard errors of estimate and the correlation coefficients .

Table I -Parameters of semi-empirical formulae

Ion Eq. Parameters of the formulae St. No. error

a b c estimate

0.932 0.000 1.272 0.7 1 Si+

2 4539 13188 0.970 0.76

0.192 0.001 -0.127 0. 12 Si ++

3 854 284 5.39 0. 13

0. 13 1 -0.33 1 0.394 0.18 Cr7+

3 83.4 9. 1 3 10 0.1 9

Corr. coeffi-cient

0.98

0.98

0.97

0.97

0.99

0.99

Note: In Eq. ( I) the parameters a and b are dimensionless while in Eqs (2) and (3) the unit of a and b is I o-17 cm2e Y2. The parameter cis dimensionless in all the equations

3.2 Ionization of Si+

In Fig. 1, the authors compared the BEBA cross-sections with the BEB cross-sections and also with

506 INDIAN J PURE & APPL PHYS , VOL 40, JULY 2002

the experimental data. It is observed that, the two cross-sections are almost indistingui shable from each other throughout the entire range of incident energy . The calculated BEBA and BEB cross-sections under=estimate the corresponding experimental values from threshold to around 200 eV but slighti y overestimates the measurements beyond this energy. The experimental curve ri ses sharply from the ground-state threshold indicating a significant contribution from the exc itation auto-ionization, not reproducible by BEBA and BEB model s meant for direct ionization . Fig. 2 shows the prediction of Eqs ( I ) and (2) and compares with the experimental data. Both of the formulae given in Eqs ( I ) and (2) provide excellent fit to the data and, the three sets of cross-sections are also indi stinguishable from one another. The di screpancies with the experimenta l data are within the uncertamttes of the measurement. The parameters of the formul ae given in Eqs ( I ) and (2) are presented in Table I .

Eq. (9) of Pattard & Rost 16• gives a reasonably good fit to the data, almost throughout the entire range and reproduces the data in shape at almost all points (Fig. 2). The max imum is reproduced in magnitude as well as in positi on. For energies beyond 400 eV the agreement provided by Eq . (9) is within the uncertainties of the measurement.

3.3 Ionization of Si++

The comparison of BEBA and BEB among themselves and also wi th experimental data is shown in Fig. 3. Although, both of the models compare well with the experimental data up to around 100 eV, the disagreement is pronounced beyond thi s energy . The BEBA and BEB cross-sections overestimate the ex perimenta l cross-sections, a lmost throughout the entire range, notwithstanding the pos ition of the maxima of cross-secti ons a lmost coinc ide (Fig. 3) . In the above calcul ation, the populati on of 65 % ground-state and 35% metastable-s tate, as is assessed in the experimenta l data, is taken into account. It may be too much to expect a si mple theory, such as the BEBA/BEB model, to be va lid for all ions of any complex ities . However, the success of the BEBA and BEB models in reproducing the shape and pos tll ons of max ima presented here, clearly demonstrates that the main aspects of the ioni zation colli sions are embedded in the square brackets of

Eqs (4) and (8). Fig. 4 presents the result of calculation using the Eqs ( I ), (3) and (9) along with the experimental data. The agreement between Eqs (I) and (3) is excellent below 800 e V . Eq . (9) produces reasonably fai r fit throughout the whole energy range.

I I

] .. N-

E + u - 25 ,. ..

0 ~

1 c 2 .Q u .. ~ 15

1 ., " e I () 1

1 ;5 j f 01

lOO 800 1000 1200 1ol00 1&00

Energy (eV)

Fig. 6- Ioni sation of Cr7+

3.4 Ionization of Cr7+

The BEBA cross-sections from the ground as we ll as metastable states of Cr7+ along with experimenta l cross-section are presented in Fig. 5. Although, the magn itudes di ffe r, the shapes of the cross-sections are s imilar to one another. The threshold energy is lower than that expected from ground-state configurat ion of the ion . Also, the cross-section values calculated from the metastable configuration of the ton are c loser to the experimental values. Thi s fact suggests that metastable component is present in the incident ion beam and major contribution comes from this component. Although, BEBA cross-sections remain well be low the experimenta l va lues throughout the whole range, the models demonstrate the power of predicting the structure of the struck target ion. Fig. 6 shows the predicted cross-sections of the Eqs ( I) and (3) a long with the experimental c ross-section. The Eqs ( I ) and (3) prov ide exce llent fit to the data as is the case with Si+ and Si++.

4 Conclusion

It is found that the three semi-empirical formulae given in Eqs (I)- (3) produce re li able total

UDDIN & MORTUZA: ELECTRON IMPACT IONIZATION 507

cross-section of single ionization by electron impact. The parameter-free formula in Eq. (9) also dictates fairly good agreement with the experimental data. Although, the theoretical curves of BEBA and BEB models are similar in shape with the corresponding experimental data, discrepancies of their prediction are noticeable. For ionization cross-section, the predicted values may differ from the reality by more than one order of magnitude since, indirect ionization processes, such as the excitation-auto-ionization are not included in the BEBA and BEB models. The direct ionization may be predicted but, the situation for the indirect ionization is not clear. The discrepancies of the BEBA and BEB models indicate further modifications of the theory. The modifications may require from multiple contributions beyond the dipole interaction. Another possibility, of course, is to refine the Burgess denominator, representing the correlation between the colliding electrons. From the discussion of the examples presented in this paper, it is clear that one has to try other innovative ideas to see if better understanding of the true meaning of the Burgess denominator would emerge. The authors hope that , this worthy goal is achieved soon. So that, the most reliable ionization theory can be found without going through hours of supercomputer time and long complicated series of equations.

It is concluded that, the semj-empirical formulae given in Eqs ( I )-(3) can be reliably used to interpolate and extrapolate into the void region, when experimental data set is available. The BEBA, BEB and Eq. (9), providing straightforward methods of calcul ations of ionization cross-section, may be used where experimental data is not ava ilable. The authors also observe that, although, BEBA and BEB model s are propounded from the different points of view, they generate almost the same total cross-sections of single ionization. In all the three test cases, the Eqs ( 1-3) are clearly seen to be better in performance than the Eq. (9). Statistical regression analysis as well as the graphical comparison of the ex perimental data with the corresponding theoretical values indicates slightly better success of the Eq. ( I ) in reproducing data than the Eqs (2) and (3) for two Si ions and demonstrates almost equi va lence in performance with the Eq. (3) for the Cr ion . These very small discrepancies may sometimes arise from the uncertainties of the optimized parameter values produced by the fitting program which depends , to

some extent, on guess values for its efficient functioning. Moreover, it may be rllisleading to draw any definite conclusion regarding superiority of one model over another model on the basis of very small discrepancies and from a study of only a few test cases. However, it may be safe to say that all three Eqs ( I )-(3) seem to represent data excellent( y.

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