- Email: [email protected]
On the normalization of the positron-impact direct ionization cross-section in the noble gases
Nuclear Instruments and Methods in Physics Research B 192 (2002) 220–224 www.elsevier.com/locate/nimb
On the normalization of the positron-impact direct ionization cross-section in the noble gases ska, G. Laricchia P. Van Reeth, M. Szłuin
*
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
Abstract Difficulties in the normalization of the positron-impact direct ionization cross-sections due to differences among the various data for electron impact ionization are discussed. Consequently, the cross-sections of Kara et al. (J. Phys. B 30 (1997) 3933) and of Moxom et al. (Can. J. Phys. 74 (1996) 367) have been renormalized using more recent and accurate data for electron impact ionization from Sorokin et al. (Phys. Rev. A. 58 (1998) 2900; 61 (2000) 022723). The new normalization for Ne, Kr and Xe lowers the cross-sections by 19%, 8% and 14% respectively, while that for Ar increases the cross-section by 2%. Ó 2002 Published by Elsevier Science B.V.
The accurate normalization of the total ionization cross-sections by positron impact, Qti (eþ ), and of the positron direct single ionization crossþ sections, Qþ i (e ), is important for comparisons with theoretical calculations and also because both these cross-sections can be used to obtain a measure of the positronium formation cross-section by þ t þ the subtraction of Qþ i (e ) from Qi (e ) (see [1] and the contribution of Szłui nska et al. [2] in the present proceedings). Recent investigations of Qti (eþ ) for the noble gases [1,2] have highlighted several problems inherent with the choice of electron impact ionization data for the normalization of the positron data. It has long been established that in the first Born approximation, the cross-sections for ionization depend on the square of the charge of the
*
Corresponding author. Tel.: +44-207-679-7809; fax: +44207-679-2564. E-mail address: [email protected] (G. Laricchia).
projectile and become reasonably accurate at sufficiently high velocities. Therefore, the crosssections for particle and antiparticles (i.e. e and eþ ; p and p ) will merge at sufficiently high energies and relative cross-sections for a given projectile can be normalized to available accurate crosssections for its antiparticle. For instance, a standard procedure for normalizing positron data is to compare the single ionization cross-sections, for both positron and electron impact, over an energy range within which they display a similar energy dependence and then to rescale the positron data so that they match the electron cross-sections in this energy region [3–5]. One of the main difficulties in the normalization procedure is to establish the energy region over which the merging occurs. For the total cross-section, i.e. the sum of all elastic and inelastic contributions, it has been shown that the merging in magnitude and energy dependence occurs from as low as approximately 50 eV for the alkali atom [6] and 200 eV for He [7], while for other noble gases it is expected to occur
0168-583X/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 7 2 - 8
P. Van Reeth et al. / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 220–224
at energies greater than 1000 eV. In the case of direct single ionization, the merging in the energy dependence is seen to occur from approximately 700 eV onwards for all noble gases e.g. [8]. Positron impact ionization data for the noble gases have often been normalized to the measurements of Krishnakumar and Srivastava [9]. The reasons for choosing these experimental results as reference data were that all the noble gases had been studied together over a relatively wide energy range, individual cross-sections, Qnþ (e ), for ioni ization processes up to order n ¼ 5 had been obtained and the quoted uncertainties were low. However, there are significant disagreements between the data of Krishnakumar and Srivastava [9] and measurements of several other groups; for instance approximately a 20% difference with the data of Nagy et al. [10] for Ne. A survey of the literature on experimental measurements and theoretical calculations of Qnþ (e ) for the noble i gases clearly indicates that there are also several discrepancies, both in the magnitude and energy dependence of the high energy data, between theory and experiment, among theories and among experiments. This is clearly shown in the more recent papers of Sorokin et al. [11,12] where a comparison is made between their results and previous available data. Sorokin et al. have developed a method to obtain precise measurements of Qti (e ) from 140 to 4000 eV for Ne, Ar, Kr and Xe. In essence, the method consists of measuring the ratios of Qti (e ) at 1000 eV to the photoionization cross-sections measured from 16 to 1000 eV. From these ratios and the accurate values of the photo-ionization cross-sections, a normalized value of Qti (e ) at 1000 eV is obtained, to which the cross-section for electron impact over the whole energy range is rescaled. The relative standard uncertainties are quoted as between 2% and 3% and are much lower than those of any previous experiment [11,12]. During the course of the work described in [1,2], the disagreements between the results of various groups and also the work of Sorokin et al. [11,12] have made us reconsider the normalization of the direct ionization data of Kara et al. [13] for Ne, Kr and Xe and those of Moxom et al. [14] for Ar which had been normalized to the data of Krish-
221
nakumar and Srivastava [9]. These sets of data were chosen because they are the most recent and accurate available of our group. They have been previously compared with other experimental determinations and the present energy independent renormalization will not affect the level agreement in shape among all the measurements as will be discussed below. We have chosen the total ionization cross-sections of Sorokin et al. [11,12] as reference electron data and have renormalized the data of Kara et al. [13] and Moxom et al. [14] to the cross-section for single ionization from experiments whose Qti (e ) agree well with those of Sorokin et al. [11,12]. When no such agreement was available, Qti (e ) from Krishnakumar and Srivastava [9] was normalized to those of [11,12] and a rescaled Qþ i (e ) was extracted. The normalization constant, Cnþ , for the positron ionizaþ tion data was evaluated by fitting Qþ i (e ) with a þ þ function, f ðEÞ ¼ Cn f ðEÞ, where f ðEÞ is a polynomial fit to the chosen reference electron data, the error in the normalization, DCnþ , being then the error in the fitting parameter Cnþ . The main source of uncertainty in Cnþ arises from the scatter in the positron data at high energies. Furþ thermore, our measured Qþ i (e ) displays a slight oscillatory behaviour at high energies which has been attributed to variation in the positron transport efficiency [13,14] making the fitting procedure more difficult. We have ensured that an integer number of oscillations are included in the energy þ range over which we fit Qþ i (e ) so that their effect averages out. A further source of uncertainty could be the contributions from inner shell ionization which might be different for positrons and electrons at these energies. However, at present, these contributions for positron impact are not precisely known but they are expected, from comparisons with electron data [9–12], to be small. It should be noted that in most cases there are few or no data points at energies above 1000 eV þ t þ for Qþ i (e ) and Qi (e ). This means that the normalization procedure is often performed over a rather limited energy range. For instance, in the present analysis we have used the energy range from 700 up to 1000 eV. In view of the effect that þ an error in Qþ i (e ) would have on other partial cross-sections (e.g. QPs ), it would seem appropriate
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in future work to extend the energy range of the þ measurement for Qþ i (e ) to higher energies in order to increase the confidence level in the þ merging of the energy dependence of Qþ i (e ) and þ Qi (e ) and reduce the uncertainties in the normalization procedure. To normalize the neon data, we have used Qþ i (e ) of Nagy et al. [10] between 800 and 1000 eV as their results for Qti (e ) agree very well with those of Sorokin et al. [11]. This new choice of normalization data had a significant effect on the magniþ tude of Qþ i (e Þ from Kara et al. [13] reducing it by 19%. In the case of argon, the electron data of [9] are found to be 2% smaller than those of [12]. We have therefore rescaled the values Qti (e ) of [9] to those of [12] and extracted new values for argon Qþ i (e ) between 800 and 1300 eV using the same scaling factor as for Qti (e ), since the relative contributions of the higher order ionization processes are relatively constant in this energy region þ (see [4,5]). The Qþ i (e ) of [14] for argon were then renormalized to the rescaled values of Qþ i (e ) and are 2% larger than those quoted in [14]. For krypton, there is a significant difference between the Qti (e ) obtained by Krishnakumar and Srivastava [9] and that of Sorokin et al. [12]. The þ normalized Qþ i (e ) presented in the present work were obtained in a similar manner as that describe earlier in the case of Ar and they are 7% lower then those quoted in Kara et al. [13]. Furthermore, to verify our normalization procedure, we have used the ratios of single to double and double to triple ionization cross-sections from [9,10] to extract a value for the single ionization contribution to the data of Sorokin et al. [12] for Kr and found that these estimates agree very well with the rescaled value Qþ i (e ) obtained for Kr. The xenon data of [13] are normalized to the single ionization data of Nagy et al. [9] whose Qti (e ) agree well with those þ of [12] and the new values of Qþ i (e ) are approximately 14% lower. The differences between the present data and those of Kara et al. [13] for Ne, Kr and Xe and Moxom et al. [14] for Ar are summarized in Table 1 together with the errors in the new normalization, DCnt : The renormalized experimental data of [13] and [14] are presented in Fig. 1 together with the experimental data of Mori and Sueoka [15] and
Table 1 Fractional differences between present and previous data of [13] and [14] and associated errors in the normalization
Ne Ar Kr Xe
Difference between new and old normalization (%)
DCnþ (%)
19 þ2 7 14
0.8 1.2 3.5 2.0
Jacobsen et al. [3] and the distorted wave calculations of Moores [16] and Campeanu et al. [17]. The cross-sections of Moores [16] peak at an energy close to the experimental data but they are larger by up to 60%, the disagreement being greatest for the heavier targets. At higher energies, the theoretical results are larger than the measurements and display a similar energy dependence only from approximately 800 eV onwards. The cross-sections of Campeanu et al. [17] are in fair agreement with the experimental data in the region of the maxima and up to the highest energy they have calculated, 500 eV. It is interesting to note that there are also discrepancies at high energies on the magnitude of the theoretical cross-sections and to a lesser degree on their energy dependence. The differences are more pronounced for the heavier targets and are likely to be the results of the different target wave functions employed. The experimental data of Jacobsen et al. [3] agree well in shape for both Ne and Ar with the present data. In the case of Ar data, they agree well also in magnitude but they do not in the case of Ne. This difference may be accounted for, almost entirely, by the different choice of electron data used for the normalization, the data of Jacobsen et al. [3] being normalized to [9] while the present data being normalized to [10]. The Mori and Sueoka data [15] have somewhat large experimental uncertainties but within these they agree well with the present data. It should be noted that the normalization procedure used by Mori and Sueoka is different to that discussed in the present paper as they have derived their partial inelastic cross-sections using the values of the total scattering cross-sections of [18–20]. We have investigated in detail the effects of the normalization procedure on the uncertainties in the absolute values of the positron-impact direct
P. Van Reeth et al. / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 220–224
223
þ Fig. 1. Comparisons of the renormalized Qþ i (e ), [13] (}); [14] (), with the experimental data of [3] (); [15] (M), and theoretical data of [16] (– – – –), [17] (– – –). For the electron data (——) please refer to the text.
ionization cross-sections when normalized to electron results. Apart from the limiting uncertainty arising from that of the electron data used for the normalization, the main uncertainties in the procedure arise from the limited energy range over þ þ which the matching of Qþ i (e ) and Qi (e ) can be made. We have chosen to renormalize the data of [13,14] using the recent electron ionization crosssections of [11,12]. Clearly, this choice of electron reference data cannot be considered as definitive. However, the data of [11,12] appears to be consistent with several of earlier measurements and to have significantly lower uncertainties. The renormalization of the direct ionization cross-section of [13,14] changes the values for Ne, Kr and Xe significantly and has a noticeable effect on the agreement between the data of [13,14] and the most recent theoretical data of [17] and on the values of QPs [1,2].
Acknowledgements We wish to thank Dr. R. Campeanu for supplying us with his latest cross-sections prior to publication. Thanks are also due to the Workshop Organizers, the Royal Society, the Institute of Physics and UCL for financial support. This work is supported by the Engineering and Physical Science Research Council on grant no. GR/L96837. References [1] G. Laricchia, P. Van Reeth, M. Szłu nska, J. Phys. B, submitted for publication. [2] M. Szłui nska, P. Van Reeth, G. Laricchia, Nucl. Instr. and Meth. B 192 (2002) 215. [3] F.M. Jacobsen, H. Knudsen, U. Mikkelson, D.M. Schrader, J. Phys. B 28 (1995) 4691. [4] H. Knudsen, L. Brun-Nielson, M. Charlton, M.R. Poulsen, J. Phys. B 23 (1990) 3955.
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[5] K. Paludan, G. Laricchia, P. Ashley, V. Kara, J. Moxom, H. Bluhme, U. Mikkelson, S.P. M€ oller, E. Uggerh€ oj, E. Morenzoni, J. Phys. B 30 (1997) L581. [6] T.S. Stein, W.E. Kaupilla, C.K. Kwan, R.A. Lukaszen, S.P. Parikh, Y.J. Wan, S. Zou, M.S. Dabaneh, in: R.J. Drachman (Ed.), Annihilation in Gases and Galaxies, NASA Conferences Publications 3058, 1990, p. 13. [7] T.S. Stein, W.E. Kaupilla, Adv. At. Mol. Phys. 18 (1982) 53. [8] D. Moores, Nucl. Instr. and Meth. B 179 (2001) 316. [9] E. Krishnakumar, S.K. Srivastava, J. Phys. B 21 (1988) 1055. [10] P. Nagy, A. Skutlartz, V. Schmidt, J. Phys. B 13 (1980) 1249. [11] A.A. Sorokin, L.A. Shmaenok, S.V. Bobashev, B. Mobus, G. Ulm, Phys. Rev. A. 58 (1998) 2900.
[12] A.A. Sorokin, L.A. Shmaenok, S.V. Bobashev, B. Mobus, M. Richter, G. Ulm, Phys. Rev. A 61 (2000) 022723. [13] V. Kara, K. Paludan, J. Moxom, P. Ashley, G. Laricchia, J. Phys. B 30 (1997) 3933. [14] J. Moxom, P. Ashley, G. Laricchia, Can. J. Phys. 74 (1996) 367. [15] S. Mori, O. Sueoka, J. Phys. B 27 (1994) 4349. [16] D. Moores, Nucl. Instr. and Meth. 143 (1998) 105. [17] R.I. Campeanu, R.P. McEachran, A.D. Stauffer, Can. J. Phys. 79 (2001) 1231. [18] W.E. Kaupilla, T.S. Stein, G. Jesion, Phys. Rev Lett. 36 (1976) 580. [19] T.S. Stein, W.E. Kaupilla, V. Pol, J.H. Smart, G. Jesion, Phys. Rev A 17 (1978) 1600. [20] W.E. Kaupilla, T.S. Stein, J.H. Smart, M.S. Dababneh, Y.K. Ho, J.P. Downing, V. Pol, Phys. Rev A 24 (1981) 725.
On the normalization of the positron-impact direct ionization cross-section in the noble gases ska, G. Laricchia P. Van Reeth, M. Szłuin
*
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
Abstract Difficulties in the normalization of the positron-impact direct ionization cross-sections due to differences among the various data for electron impact ionization are discussed. Consequently, the cross-sections of Kara et al. (J. Phys. B 30 (1997) 3933) and of Moxom et al. (Can. J. Phys. 74 (1996) 367) have been renormalized using more recent and accurate data for electron impact ionization from Sorokin et al. (Phys. Rev. A. 58 (1998) 2900; 61 (2000) 022723). The new normalization for Ne, Kr and Xe lowers the cross-sections by 19%, 8% and 14% respectively, while that for Ar increases the cross-section by 2%. Ó 2002 Published by Elsevier Science B.V.
The accurate normalization of the total ionization cross-sections by positron impact, Qti (eþ ), and of the positron direct single ionization crossþ sections, Qþ i (e ), is important for comparisons with theoretical calculations and also because both these cross-sections can be used to obtain a measure of the positronium formation cross-section by þ t þ the subtraction of Qþ i (e ) from Qi (e ) (see [1] and the contribution of Szłui nska et al. [2] in the present proceedings). Recent investigations of Qti (eþ ) for the noble gases [1,2] have highlighted several problems inherent with the choice of electron impact ionization data for the normalization of the positron data. It has long been established that in the first Born approximation, the cross-sections for ionization depend on the square of the charge of the
*
Corresponding author. Tel.: +44-207-679-7809; fax: +44207-679-2564. E-mail address: [email protected] (G. Laricchia).
projectile and become reasonably accurate at sufficiently high velocities. Therefore, the crosssections for particle and antiparticles (i.e. e and eþ ; p and p ) will merge at sufficiently high energies and relative cross-sections for a given projectile can be normalized to available accurate crosssections for its antiparticle. For instance, a standard procedure for normalizing positron data is to compare the single ionization cross-sections, for both positron and electron impact, over an energy range within which they display a similar energy dependence and then to rescale the positron data so that they match the electron cross-sections in this energy region [3–5]. One of the main difficulties in the normalization procedure is to establish the energy region over which the merging occurs. For the total cross-section, i.e. the sum of all elastic and inelastic contributions, it has been shown that the merging in magnitude and energy dependence occurs from as low as approximately 50 eV for the alkali atom [6] and 200 eV for He [7], while for other noble gases it is expected to occur
0168-583X/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 7 2 - 8
P. Van Reeth et al. / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 220–224
at energies greater than 1000 eV. In the case of direct single ionization, the merging in the energy dependence is seen to occur from approximately 700 eV onwards for all noble gases e.g. [8]. Positron impact ionization data for the noble gases have often been normalized to the measurements of Krishnakumar and Srivastava [9]. The reasons for choosing these experimental results as reference data were that all the noble gases had been studied together over a relatively wide energy range, individual cross-sections, Qnþ (e ), for ioni ization processes up to order n ¼ 5 had been obtained and the quoted uncertainties were low. However, there are significant disagreements between the data of Krishnakumar and Srivastava [9] and measurements of several other groups; for instance approximately a 20% difference with the data of Nagy et al. [10] for Ne. A survey of the literature on experimental measurements and theoretical calculations of Qnþ (e ) for the noble i gases clearly indicates that there are also several discrepancies, both in the magnitude and energy dependence of the high energy data, between theory and experiment, among theories and among experiments. This is clearly shown in the more recent papers of Sorokin et al. [11,12] where a comparison is made between their results and previous available data. Sorokin et al. have developed a method to obtain precise measurements of Qti (e ) from 140 to 4000 eV for Ne, Ar, Kr and Xe. In essence, the method consists of measuring the ratios of Qti (e ) at 1000 eV to the photoionization cross-sections measured from 16 to 1000 eV. From these ratios and the accurate values of the photo-ionization cross-sections, a normalized value of Qti (e ) at 1000 eV is obtained, to which the cross-section for electron impact over the whole energy range is rescaled. The relative standard uncertainties are quoted as between 2% and 3% and are much lower than those of any previous experiment [11,12]. During the course of the work described in [1,2], the disagreements between the results of various groups and also the work of Sorokin et al. [11,12] have made us reconsider the normalization of the direct ionization data of Kara et al. [13] for Ne, Kr and Xe and those of Moxom et al. [14] for Ar which had been normalized to the data of Krish-
221
nakumar and Srivastava [9]. These sets of data were chosen because they are the most recent and accurate available of our group. They have been previously compared with other experimental determinations and the present energy independent renormalization will not affect the level agreement in shape among all the measurements as will be discussed below. We have chosen the total ionization cross-sections of Sorokin et al. [11,12] as reference electron data and have renormalized the data of Kara et al. [13] and Moxom et al. [14] to the cross-section for single ionization from experiments whose Qti (e ) agree well with those of Sorokin et al. [11,12]. When no such agreement was available, Qti (e ) from Krishnakumar and Srivastava [9] was normalized to those of [11,12] and a rescaled Qþ i (e ) was extracted. The normalization constant, Cnþ , for the positron ionizaþ tion data was evaluated by fitting Qþ i (e ) with a þ þ function, f ðEÞ ¼ Cn f ðEÞ, where f ðEÞ is a polynomial fit to the chosen reference electron data, the error in the normalization, DCnþ , being then the error in the fitting parameter Cnþ . The main source of uncertainty in Cnþ arises from the scatter in the positron data at high energies. Furþ thermore, our measured Qþ i (e ) displays a slight oscillatory behaviour at high energies which has been attributed to variation in the positron transport efficiency [13,14] making the fitting procedure more difficult. We have ensured that an integer number of oscillations are included in the energy þ range over which we fit Qþ i (e ) so that their effect averages out. A further source of uncertainty could be the contributions from inner shell ionization which might be different for positrons and electrons at these energies. However, at present, these contributions for positron impact are not precisely known but they are expected, from comparisons with electron data [9–12], to be small. It should be noted that in most cases there are few or no data points at energies above 1000 eV þ t þ for Qþ i (e ) and Qi (e ). This means that the normalization procedure is often performed over a rather limited energy range. For instance, in the present analysis we have used the energy range from 700 up to 1000 eV. In view of the effect that þ an error in Qþ i (e ) would have on other partial cross-sections (e.g. QPs ), it would seem appropriate
222
P. Van Reeth et al. / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 220–224
in future work to extend the energy range of the þ measurement for Qþ i (e ) to higher energies in order to increase the confidence level in the þ merging of the energy dependence of Qþ i (e ) and þ Qi (e ) and reduce the uncertainties in the normalization procedure. To normalize the neon data, we have used Qþ i (e ) of Nagy et al. [10] between 800 and 1000 eV as their results for Qti (e ) agree very well with those of Sorokin et al. [11]. This new choice of normalization data had a significant effect on the magniþ tude of Qþ i (e Þ from Kara et al. [13] reducing it by 19%. In the case of argon, the electron data of [9] are found to be 2% smaller than those of [12]. We have therefore rescaled the values Qti (e ) of [9] to those of [12] and extracted new values for argon Qþ i (e ) between 800 and 1300 eV using the same scaling factor as for Qti (e ), since the relative contributions of the higher order ionization processes are relatively constant in this energy region þ (see [4,5]). The Qþ i (e ) of [14] for argon were then renormalized to the rescaled values of Qþ i (e ) and are 2% larger than those quoted in [14]. For krypton, there is a significant difference between the Qti (e ) obtained by Krishnakumar and Srivastava [9] and that of Sorokin et al. [12]. The þ normalized Qþ i (e ) presented in the present work were obtained in a similar manner as that describe earlier in the case of Ar and they are 7% lower then those quoted in Kara et al. [13]. Furthermore, to verify our normalization procedure, we have used the ratios of single to double and double to triple ionization cross-sections from [9,10] to extract a value for the single ionization contribution to the data of Sorokin et al. [12] for Kr and found that these estimates agree very well with the rescaled value Qþ i (e ) obtained for Kr. The xenon data of [13] are normalized to the single ionization data of Nagy et al. [9] whose Qti (e ) agree well with those þ of [12] and the new values of Qþ i (e ) are approximately 14% lower. The differences between the present data and those of Kara et al. [13] for Ne, Kr and Xe and Moxom et al. [14] for Ar are summarized in Table 1 together with the errors in the new normalization, DCnt : The renormalized experimental data of [13] and [14] are presented in Fig. 1 together with the experimental data of Mori and Sueoka [15] and
Table 1 Fractional differences between present and previous data of [13] and [14] and associated errors in the normalization
Ne Ar Kr Xe
Difference between new and old normalization (%)
DCnþ (%)
19 þ2 7 14
0.8 1.2 3.5 2.0
Jacobsen et al. [3] and the distorted wave calculations of Moores [16] and Campeanu et al. [17]. The cross-sections of Moores [16] peak at an energy close to the experimental data but they are larger by up to 60%, the disagreement being greatest for the heavier targets. At higher energies, the theoretical results are larger than the measurements and display a similar energy dependence only from approximately 800 eV onwards. The cross-sections of Campeanu et al. [17] are in fair agreement with the experimental data in the region of the maxima and up to the highest energy they have calculated, 500 eV. It is interesting to note that there are also discrepancies at high energies on the magnitude of the theoretical cross-sections and to a lesser degree on their energy dependence. The differences are more pronounced for the heavier targets and are likely to be the results of the different target wave functions employed. The experimental data of Jacobsen et al. [3] agree well in shape for both Ne and Ar with the present data. In the case of Ar data, they agree well also in magnitude but they do not in the case of Ne. This difference may be accounted for, almost entirely, by the different choice of electron data used for the normalization, the data of Jacobsen et al. [3] being normalized to [9] while the present data being normalized to [10]. The Mori and Sueoka data [15] have somewhat large experimental uncertainties but within these they agree well with the present data. It should be noted that the normalization procedure used by Mori and Sueoka is different to that discussed in the present paper as they have derived their partial inelastic cross-sections using the values of the total scattering cross-sections of [18–20]. We have investigated in detail the effects of the normalization procedure on the uncertainties in the absolute values of the positron-impact direct
P. Van Reeth et al. / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 220–224
223
þ Fig. 1. Comparisons of the renormalized Qþ i (e ), [13] (}); [14] (), with the experimental data of [3] (); [15] (M), and theoretical data of [16] (– – – –), [17] (– – –). For the electron data (——) please refer to the text.
ionization cross-sections when normalized to electron results. Apart from the limiting uncertainty arising from that of the electron data used for the normalization, the main uncertainties in the procedure arise from the limited energy range over þ þ which the matching of Qþ i (e ) and Qi (e ) can be made. We have chosen to renormalize the data of [13,14] using the recent electron ionization crosssections of [11,12]. Clearly, this choice of electron reference data cannot be considered as definitive. However, the data of [11,12] appears to be consistent with several of earlier measurements and to have significantly lower uncertainties. The renormalization of the direct ionization cross-section of [13,14] changes the values for Ne, Kr and Xe significantly and has a noticeable effect on the agreement between the data of [13,14] and the most recent theoretical data of [17] and on the values of QPs [1,2].
Acknowledgements We wish to thank Dr. R. Campeanu for supplying us with his latest cross-sections prior to publication. Thanks are also due to the Workshop Organizers, the Royal Society, the Institute of Physics and UCL for financial support. This work is supported by the Engineering and Physical Science Research Council on grant no. GR/L96837. References [1] G. Laricchia, P. Van Reeth, M. Szłu nska, J. Phys. B, submitted for publication. [2] M. Szłui nska, P. Van Reeth, G. Laricchia, Nucl. Instr. and Meth. B 192 (2002) 215. [3] F.M. Jacobsen, H. Knudsen, U. Mikkelson, D.M. Schrader, J. Phys. B 28 (1995) 4691. [4] H. Knudsen, L. Brun-Nielson, M. Charlton, M.R. Poulsen, J. Phys. B 23 (1990) 3955.
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