1 PRAMANA cfl Indian Academy of Sciences Vol. 55, No. 3 journal of September 2000 physics pp Electron impact single ionization of copper LKJHA &Lambda...

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c Indian Academy of Sciences

physics

Vol. 55, No. 3 September 2000 pp. 447–453

Electron impact single ionization of copper L K JHA , O P ROYy and B N ROYz

Department of Physics, L.N.T. College, Muzaffarpur 842 002, India y Department of Physics, M.S.K.B. College, Muzaffarpur 842 002, India z Professor’s Colony, Aghoria Bazar, Muzaffarpur 842 002, India MS received 15 June 1999; revised 23 August 2000 Abstract. Electron impact single ionization cross sections of copper have been calculated in the binary encounter approximation using accurate expression for E as given by Vriens and Hartree– Fock momentum distribution for the target electron. The BEA calculation based on the usual procedure does not show satisfactory agreement with experiment in this case but a striking modification is found to be successful in explaining the experimental observations. The discrepancy is linked with the ionization of the 3d10 electrons and probably effective single ionization does not take place from 3d shell of copper leading to smaller values of experimental cross sections. Keywords. Electron impact ionization; binary encounter approximation; Hartree–Fock momentum distribution; copper. PACS Nos 79.20.Kz; 79.20.Ap

1. Introduction Electron impact ionization of atoms and ions is one of the most fundamental problems in atomic and molecular physics. The study of this process finds important applications in different fields of current interest and is important from both experimental and theoretical points of view [1,2]. Shah et al [3] developed a crossed beam technique incorporating time of flight spectroscopy for measurement of the electron impact ionization cross section of atomic hydrogen with high precision over a wide energy range. Later on it was used successfully for studies of ionization of stable gas atoms [4,5]. By the use of a specially developed high temperature oven source, McCallion [6] and Shah et al [7] applied this method to study single and multiple ionization of metallic species. Many metallic species are important in fusion energy research and astrophysical applications [8]. Bolorizadeh et al [8] of the Belfast group have carried out accurate experimental measurements of single and multiple ionization of copper by electron impact at energies ranging from near threshold (7.8 eV for single ionization) to 2100 eV. The use of thermal energy copper beams in their measurements obviated metastable contamination ensuring that their experiment contained ground state copper atoms (3d 10 4s) only.

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L K Jha, O P Roy and B N Roy Unfortunately, theoretical calculations of even single ionization cross sections for copper have not been reported so far. Bolorizadeh et al [8] have compared their experimental measurements with the predictions based on the well-known Lotz [9] formula. The nonavailability of theoretical calculations in case of copper is probably due to the complex nature of target as evidenced by Lotz’s empirical calculations of single ionization cross sections. Calculations of ionization cross sections by quantal methods are difficult particularly for complex atomic targets. On the other hand, the binary encounter approximation developed by Gryzinski [10] has been successful in predicting ionization cross sections of atoms and ions. The model of Gryzinski using accurate expression of E as given by Vriens [11] and integrating the cross section over Hartree–Fock velocity distribution of target electrons has yielded satisfactory results in case of several atomic and ionic targets [12–14]. Keeping the above mentioned facts in view, we consider it worthwhile to use the BEA for a theoretical investigation of single ionization cross sections of copper by electron impact.

2. Theoretical methods We have used the Vriens expression [11] in a symmetrical model including exchange and interference for calculating electron impact single ionization cross sections. In the unsymmetrical model of Gryzinski [10], if the incident and the target electron are at the same distance from the nucleus, it is assumed that because of the interaction with the nucleus, the atomic electron has finite potential energy whereas the incident electron has potential energy zero. On the other hand, the symmetrical model of Vriens [11] assumes that the incident electron with initial kinetic energy 12 mV12 (m and V1 being the mass and the velocity of the incident electron respectively) gains a kinetic energy U + 21 mV22 (U; m and V2 being the binding energy, the mass and the velocity of the atomic electron respectively) and simultaneously loses the same amount of potential energy before it interacts with the atomic electron which is bound with this potential energy. This ensures that the total energy of incident electron is conserved. By virtue of the above mentioned considerations, the Vriens expression in the symmetrical model includes exchange and interference (see Vriens [15]). Using dimensionless variables based on the work of Catlow and McDowell [16] the expression for cross section can be written in a convenient form as (see [12])

s2 1 2t2 s4 1 Q (s; t) = 2 2 (s + t + 1) s2 U + 3 s4 U 2 4

i

ln s2 (a2 ); 0 U 2 (s2 + 1)

(1) where

(

1 2 ) 1 = cos s2 U + U ln s2 : In the above expression s 2 = V12 =V02 , t2 = V22 =V02 where V1 and V2 are the velocities =

in atomic units of the incident and target electron respectively. V 0 is the root mean square velocity in atomic unit of the target electron corresponding to the binding energy of the shell under consideration. The equality of the magnitudes of the binding energy and the

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Electron impact ionization mean kinetic energy of the target electron predicted by the virial theorem for the Coulomb potential gives the relation 12 mV02 = U . Thus, in atomic units we have 12 V02 = U and hence V02 = U where U is the ionization potential of the shell of the target in Rydbergs [16]. The above expression for Q i (s; t) has been integrated numerically over the Hartree– Fock velocity distribution for the target electron to obtain the ionization cross section. Thus, the expression for electron impact single ionization cross section for a particular shell of the target is given by

Qi (s) = ne

Z1 0

Qi (s; t)f (t)U 1=2 dt:

(2)

Here ne is the number of equivalent electrons of the atomic shell and f (t) is the momentum distribution for the target electron. The momentum distribution function f (t) is given by (see [16])

f (t) = 4t2 Unl (tU 1=2 ):

(3)

Here

nl =

=+ 1 X 2l + 1 = j m

m

where nlm

l

nlm

(x)j2 ;

l

Z (x) = (21)3 2 =

nlm (r)eix:r dr

is the Fourier transform of the one electron orbital

nlm (r) = Nnl Rnl (r)Ylm ( ) in which Rnl (r) is the Hartree–Fock radial function. In the present work, we have used ionization thresholds and Hartree–Fock radial wavefunctions of 4s and 3d shells of copper

as given by Clementi and Roetti [17]. At this stage we would like to mention that the use of Hartree–Fock wavefunction does not properly take into account the correlation between the electrons in an atom. There are methods by which correlation between electrons can be properly taken into account e.g. configuration interaction approach based on Slater–Condon theory, multiconfiguration Hartree–Fock approach. Calculations using these wavefunctions become very complicated, particularly for heavier targets. With the use of correlated wavefunctions in the BEA the aim of adopting a simplified theoretical approach will not be achieved. In this connection, we would like to mention that Hartree–Fock wavefunctions consider the electron– electron correlation to some extent through antisymmetrization [18]. Moreover, for fast projectiles the effects of electron–electron correlation may not be significant [19]. Thus, the use of Hartree–Fock wavefunction in studies on ionization process using the BEA may be considered to be reasonable.

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L K Jha, O P Roy and B N Roy 3. Results and discussion In order to obtain electron impact single ionization cross sections for copper, we have considered ionization from 4s and 3d shells only. Ionization from deeper inner shells (3p; 3s; 2p; 2s) have not been included in the present calculations as a single vacancy in the shells leads to Auger emission as discussed by Bolorizadehet al [8]. The present results along with experimental data [8] and empirical calculations of Lotz [9] have been shown in figure 1 and table 1. In the figure we have plotted the single ionization cross sections considering ionization from 4s shell including contribution due to only one electron of 3d shell. The reason of adopting this type of approach would be discussed below. The contribution from 4s and 3d shells have been shown separately in the table. Firstly, we would like to discuss our results by considering ionization from 4s shell only. At low incident energies the present results overestimate the cross sections and at 10 eV impact energy the calculated result is two times larger than the experimental value. With increase in energy the calculated results become closer to experimental data up to 20 eV. Beyond this energy value the calculated results underestimate the cross sections but are within a factor of two up to 70 eV impact energy. At still higher energies the discrepancy goes on increasing and at 1000 eV the experimental result is about 4.5 times larger than the calculated value. The peaks obtained in our calculation and experiment appear at 20 and 30 eV respectively. The magnitudes of the peaks are 2:55 10 16 and 3:21 10 16 cm2 respectively, the theoretical cross section being about 20% smaller than the experimental value. The calculations of Lotz show a peak at 50 eV which is much shifted in position as compared to the present calculations and the experiment. If the contribution of 10 electrons of 3d shell is included in the calculations, the cross sections becomes 5 to 6 times larger than the experimental data at all incident energies above 25 eV. In this connection it may be

Figure 1. Electron impact single ionization cross sections of copper in units of 10 16 cm2 . Present results ; experimental results [8] - - - - ; empirical calculation of Lotz [9] xxxx .

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Electron impact ionization Table 1. Electron impact single ionization cross section for copper in unit of 10 cm2 . Energy (eV)

20 30 40 50 60 70 80 100 200 300 400 500 600 700 900 1000

Present result

4s contribution

One 3d electron contribution

Total

2.55 2.33 2.05 1.81 1.62 1.46 1.31 1.12 0.64 0.46 0.33 0.28 0.24 0.21 0.18 0.15

1.06 1.47 1.56 1.56 1.54 1.50 1.46 1.38 1.07 0.89 0.72 0.65 0.58 0.52 0.46 0.41

3.61 3.80 3.61 3.37 3.16 2.96 2.77 2.50 1.71 1.35 1.05 0.93 0.82 0.73 0.64 0.56

Experimental result [8]

Empirical calculation of Lotz [9]

2.80 3.21 3.08 2.90 2.77 2.89 2.76 2.55 2.05 1.63 1.43 1.20 1.12 0.97 0.73 0.66

3.2 3.5 3.6 3.7 3.6 3.54 3.6 3.5 2.8 2.3 1.9 1.7 1.5 1.4 1.07 1.0

16

noted that calculations of single ionization cross sections in the binary encounter approach show good agreement with experimental data in high energy region, being always within a factor of two. At this stage it is worth mentioning the observation made by Lotz who calculated electron impact ionization cross sections of atoms with the help of his empirical formula and found satisfactory agreement with experimental data in most of the cases. In absence of theoretical calculation, experimental data are often compared with the results obtained by Lotz formula, as done by Bolorizadeh et al in case of copper. Lotz has mentioned that he had to reduce the cross section of 3d 10 electrons drastically for copper target in order to get reasonable agreement with experiment. Almost similar difficulties have been observed by Lotz in case of silver (4d 10 5s) [9] which has electronic configuration of similar nature as that of copper. Keeping in view the observation of Lotz, we have made an ad hoc assumption to include contribution of one 3d-electron in order to examine the results. It can be seen from the figure and table that the results so obtained are in good agreement with the experimental data throughout the energy range investigated. The peaks appear in the calculation and the experiment at 26 and 30 eV respectively being close to each other. The magnitudes of the peaks are 3:81 10 16 and 3:21 10 16 cm2 respectively, the theoretical cross section being about 19% larger than experimental value. From the facts given above it is apparent that one faces difficulty in calculation of 3d shell single ionization cross section of copper if contributions from all the ten electrons are taken into account. Similar difficulties have been experienced by earlier workers in case of other atoms and ions involving ionization from fully occupied d-shells. Bell et al [20] have observed difficulties in calculation of electron impact ionization of In + . In

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L K Jha, O P Roy and B N Roy order to obtain satisfactory agreement with experiment the contribution to the ionization cross sections from electrons in the 4d shells was added in at only one half of its calculated value in configuration averaged distorted wave (CADW) approximation. Use of only half of the d-subshell contribution was proposed by Rogers et al [21] earlier and was found to fit the experimental data better in case of other experiments [22,23]. In case of electron impact ionization of gallium, Patton et al [24] have discussed the observation made by Vainshtein et al [25] that autoionization is the dominant Ga 2+ production mechanism since removal of a 3d electron leads mainly to double ionization. Later on Thomason et al [26] have mentioned about the calculation of Younger [27] for electron impact ionization of 4d electron in Cs+ which leads to autoionization and hence double ionization. Now we would like to discuss the possibilities of different physical processes consequent upon ionization of 3d electron in case of copper. After removal of one electron from 3d shell, the target is left in 3d9 4s state. This state is not an autoionizing state and hence autoionization is not possible. At the same time removal of a 3d electron will not lead to an Auger decay since there is only one electron in state higher than 3d. However, it is quite likely that there may be a high probability of immediate de-excitation of the target from 3d9 4s configuration to 3d 10 configuration. In other words, the ionization of a 3d electron in copper may lead to the immediate capture of the 4s electron. This might be due to hybridization of d and s electrons during the scattering. Thus, the consideration of pure s electrons and pure d electrons during the ionization has no physical justification. At this stage we would like to mention that the probability of the above mentioned process may not be unity. However, it is not possible to estimate this probability in the BEA. From the discussion given above it is probable that large number of 3d 10 copper ions and only a small number of 3d 9 4s copper ions might have been detected in the experiment. The former case corresponds to effective single ionization of 4s shell only whereas the latter case corresponds to a small probability of single ionization of 3d shell. The entire discussion suggests that the actual probability of single ionization of d shell in case of copper may be much smaller as compared to the probability obtained by usual method leading to a small contribution from ionization of d shell. We have also calculated electron impact single ionization cross section of scandium (with a single d electron) and found that depending on the binding energies of d and s shells, ionization cross sections are consistent with those obtained in case of copper. Due to non-availability of experimental data in the literature it is not possible to draw any conclusion regarding single ionization of d shell of scandium and hence the results are not presented here. The discussion given above explains why the inclusion of contribution of one 3d electron brings our calculated result in good agreement with the experiment. More elaborate theoretical investigation is required for quantitative understanding of the process of single ionization from 3d shell of copper. It is expected that this work will stimulate other theoretical workers to take up further study of this problem.

Acknowledgements The authors are thankful to Prof. D K Rai for helpful discussion. One of us (LKJ) is thankful to IUCAA, Pune for providing associateship.

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