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Relative intensity ratios for L-shell X-rays by low-energy proton impact
Nuclear Instruments and Methods 197 (1982) 585-590 North-Holland Publishing Company
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R E L A T I V E I N T E N S I T Y R A T I O S F O R L - S H E L L X-RAYS BY L O W - E N E R G Y P R O T O N I M P A C T Takeshi M U K O Y A M A Institute for Chemical Research, Kyoto University, Kyoto, Japan and Lfiszl6 S A R K A D I Institute of Nuclear Research of the Hungarian Academy of Sciences (.4TOMKI), Debrecen, Hungary Received 17 September 1981
Comparison between theoretical and experimental values for relative intensity ratios of L-shell X-rays by low-energy proton
impact Ep=0.3-0.8 MeV has been made graphically. The calculations for the La, LO and Lv X-ray production cross sections have been performed by the relativistic plane-wave Born approximation theory corrected for the binding-energy and Coulomb-deflection effects (RPWBA-BC). The theoretical predictions are compared with the experimental values for the L,~/La and L,,/L~ X-ray intensity ratios as well as other theoretical models. It is found that the RPWBA-BC gives the best overall agreement with the experimental results. Using an average reduced velocity parameter, we have plotted the experimental L,~/L~ and L,~/L v ratios, normalized to the RPWBA-BC theory, for different target elements. This universal plot shows that the measured L,~/L/~ and L,,/L v values are in agreement with the RPWBA-BC theory within a deviation of 25%.
1. Introduction In recent years extensive experimental works on Kand L-sheU ionization by heavy charged-particle impact have been reported. These experimental results ate usually compared with the theoretical predictions calculated by the plane-wave Born approximation (PWBA) [1]. For low-energy projectiles, the corrections for the binding-energy increase of the target electrons during collisions and for the Coulomb-deflection effect of the projectile should be taken into consideration. Recently, comparison of a large amount of the measured K-shell ionization cross sections for protons and a particles on a variety of target elements with the various theoretical models has been made by Paul [2]. His results indicate that the PWBA theory modified for the binding-energy and Coulomb-deflection effects (PWBA-BC) [3] gives the best overall agreement with the experimental data. More systematic graphical comparison between theory and experiment for K-shell ionization cross sections by various projectiles has also been performed by Paul [4,5]. The experimental data are expressed as the ratio to the theoretical values calculated by the PWBA model corrected for the binding-energy, Coulomb-deflection, polarization, and electronic relativistic effects (PWBABCPR) [6] and plotted against the reduced velocity parameter. Almost all the experimental data are found 0167-5087/82/0000-0000/$02.75 © 1982 North-Holland
to be in fairly good agreement with the theoretical predictions. Similar comparisons have been made by us for the K-shell ionization cross sections by protons and a particles [7] and by heavy ions [8]. The theoretical values have been calculated by the relativistic PWBA model including the corrections for the binding-energy and Coulomb-deflection effects (RPWBA-BC) [9]. The exact values for the limits of integration over momentum transfer are used [10]. It is found that the RPWBA-BC gives better agreement with the experimental values than the PWBA-BCPR. This is due to the fact that the correction derived by Brandt and Lapicki [6] overestimates the electronic relativistic effects for low-energy projectiles [11]. This kind of comparison between theory and experiment is very useful to identify and reject the erroneous experimental data and also to test the limit of validity of theoretical models. We have recently extended the RPWBAzBC model to the case of L-shell ionization cross sections [12]. The calculated values for L-subsheU ionization cross sections by low-energy protons and a particles have been used to compare with the experimental data [13]. The agreement between theory and experiment for L subshells is not so good as that for the K shell. Especially for the L 2 shell the measured values are systematically larger than the theoretical predictions. However, the available experi-
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T. Mvkovama L. Sarkadi / Relative intensi(v ratios for L-shell X-rays
mental data for L-subshell ionization cross sections are scarce because of difficulties in X-ray measurements and data analysis. It is needed to make a more systematic comparison between theory and experiment for L-subshell ionization. As has been discussed in our previous work [13], comparison of theory with experiment should be made not for the total L-shell ionization cross sections, but for the L-subshell ionization cross sections. However, these cross sections are not the directly measurable quantity. From the observed peak intensities of L X-rays the partial X-ray production cross sections corresponding to L X-ray components are estimated and the L-subshell ionization cross sections are converted from these cross sections by the use of atomic parameters, such as Lsubshell fluorescence yields, Coster-Kronig coefficients and partial decay widths. This fact means that the L-subshell ionization cross sections are correlated with each other and the small change in the L X-ray intensities may lead to a large variation in the subshell ionization cross sections. Several experimental results have been reported on relative intensity ratios for L-shell X-rays by low-energy proton impact. These experimental values cannot be converted into the L-subshell ionization cross sections, but it is possible to compare them with the theoretical predictions for the relative intensity ratios. There have also been the experimental data for the partial L-shell X-ray production cross sections, from which the relative X-ray intensity ratios are easily obtained. Comparison between theory and experiment for the relative intensity ratios has the following three advantages: First, these ratios are directly determined from the experimental measurements for X-rays. Second, in these ratios several factors affecting the X-ray intensities, such as geometrical efficiency and correction for absorption of X-rays between the target and the detector, are cancelled out and the experimental errors are small. Third, the X-ray intensity ratios are more sensitive to the difference between theory and experiment than the subshell ionization cross sections. If each subshell ionization cross section agrees with the theoretical predictions fairly well, the measured intensity ratios are sometimes in poor agreement with the theory. It is the purpose of the present work to compare the measured L-shell X-ray intensity ratios with the theoretical predictions. Theoretical values for the L-subshell ionization cross sections by low-energy proton impact are calculated in the RPWBA-BC. These cross sections are converted into the partial X-ray production cross sections and the relative intensities of the L,/L/~ and L , / L r ratios are estimated. Comparison between theory and experiment is made graphically.
2. Theoretical model The theoretical model in the present work is the same as that used in the previous work [13] and is described in detail elsewhere [14]. In the PWBA, the L-shell ionization cross section by charged-particle impact is given by [1] do , 2 M I fqm~dq dE/='*~ra - - - - - , F ~ f ( q ) [2 ,
El Jq,,,in q3
(1)
where q is the momentum transfer, El is the kinetic energy of the ejected electron, a is the fine structure constant, M~ and E t are the mass and energy of the projectile, F,f(q) is the form factor, and qmax and qmin are the upper and lower limits of the momentum transfer. Throughout the present work, the relativistic units (h=m=c=l) areused. The exact values for the limits of momentum transfer are determined from the conservation law of energy and momentum, and used as qm~ and qmin"The calculation for the form factor is made by the use of the relativistic hydrogenic (Dirac) wave functions following the method of Jamnik and Zupanri6 [15]. The analytical expressions for F,/(q) are equivalent to those of Choi [16]. The screening effects are taken into consideration through the inner- and outer-screening method as usual in the PWBA. The binding-energy effect is estimated in the perturbed-stationary-state (PSS) approach developed by Brandt and Lapicki [17] and incorporated into the PWBA model. This can be done by introducing the binding-energy factor c. The effect of Coulomb deflection is also taken into account through the method of Brandt and Lapicki [17]. According to this model, the L,-shell ionization cross section in the RPWBA-BC is expressed as OR,pw"a-"c = CL,(~rdqo, L,) ×
oRPWBA-B /
L,
(2)
where CL,(X) is the correction factor for the CoulombRPWBA-B`tTL,, ~L,, (Li0L,) is the deflection effect, O L, RPWBA cross section corrected for the binding-energy effect, d is one-half of the distance of closest approach in a head-on collision, q0 is the minimum momentum transfer, %, is the binding-energy factor, YL, is the parameter of the target atomic number, ~TL,is the scaled projectile velocity parameter and 0L, is the scaled binding energy of the target electron. The binding energy factor q., is estimated by the use of the relativistic hydrogenic wave functions following the method of Brandt and Lapicki [17] and the analytical expressions are given in ref. 14. The explicit forms of
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T. Mykoyama, L. Sarkadi / Relative intensity ratios for L-shell X-rays
the function CL,(X ) are obtained by Brandt and Lapicki and expressed in ref. 17. In order to estimate the L-shell X-ray intensity ratios, the L-subshell ionization cross sections should be converted into the partial X-ray production cross sections. The L,, L B and Lv X-ray production cross sections are related to the three subshell ionization cross sections by the following relations: ° x = [°1(f,3 +f12f23) + 02f23 + 03] ~ , r ~ o / r , , °~x : [ ° , ( f , 3 +f,2f23) + 02f23 + 03]
(3)
,.,3r,,/r,
q- (01/12 + 02) . 2r2./r2 + ox = ( 0 , / , 2 + % ) ~,2r2,/r2 + o , , < r , ~ / r , ,
(4) (5)
where o, is the L,-subshell ionization cross section, f,j is the Coster-Kronig coefficient for transfer of a vacancy from the ith to the j t h subshell, ~0i is the Li-subshell fluorescence yield, F~ is the total radiative width of the L, subshell and F,~ is the partial radiative width for the transition filling a vacancy in the L~ shell which contributes to the L~ line. The intensity ratios L ~ / L a and L , ~ / L r can be calculated as the ratios of the partial X-ray production cross sections.
3. Comparison of theoretical and experimental L-shell X-ray intensity ratios The theoretical L-subshe11 ionization cross sections have been calculated according to eq. (2). Using eqs. (3)-(5), these cross sections have been converted into the partial X-ray production cross sections. For this purpose, we used the subshell fluorescence yield, ~i, and the Coster-Kronig coefficient, ~j, compiled by Krause [18]. The total and partial radiative widths were taken from the theoretical values calculated by Scofield [19]. References for all the experimental data used in the present work are listed in table 1. In the case of data by AB74, the L , ~ / L r ratios were estimated from the experimental values of L , / L v , and L~/Lv23 ratios assuming
L r = Ly, + Lr2; The experimental data by BH80 were calculated as die ratio of the partial X-ray production cross sections. In this case, estimation of the experimental errors was not made because the errors obtained from those of the X-ray production cross sections are considered to be much larger than those for direct measurements of the X-ray intensity ratios. In fig. 1, the L~/L~ ratios for low-energy protons on gold are shown as a function of incident energy. The different symbols refer to the different measurements identified by the codes and these codes correspond to those in table 1. The experimental data agree well with each other and are almost constant with energy. For comparison, other theoretical values based on the PWBA, the PWBA-BC, the PWBA-BC with the relativistic correction (PWBA-BCR) of Merzbacher and Lewis [1] and the PWBA-BCR by the relativistic correction of Brandt and Lapicki [6] are plotted in the figure. Brandt and Lapicki [6] called their model the Coulomb-deflection-corrected PSS theory with relativistic correction (CPSSR). In order to discriminate between two PWBA-BCR theories, we call the model of Brandt and Lapicki the CPSSR and that of Merzbacher and Lewis the PWBA-BCR, though the only difference between these two models consists in the correction method for the relativistic effect. The PWBA, the PWBA-BC and the PWBA-BCR have been calculated by the computer code DEKY [20], while the CPSSR calculations have been made by the code DEKY2 [21]. In all these calculations, the exact values of the limits of momentum transfer [22] were used instead of the usual approximate ones. It is clear from the figure that all the theoretical models, except for the PWBA-BCR, slightly overpredict the experimental data. On the other hand, the
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AB74 BH80 BU73
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F. Abrath and T.J. Gray, Phys. Rev. A9 (1979) 682. D. Bhattacharya, S.K. Bhattacherjee and S.K. Mitra, J. Phys. B13 (1980) 967. C.E. Busch, A.B. Baskin, P.H. Nettles, S.M. Shafroth and A.W. Waltner, Phys. Rev. A7 (1973) 1601. S.M. Shafroth, G.A. Bissinger and A.W. Waltner, Phys. Rev. A7 (1973) 566.
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Table 1 References for experimental data.
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Fig. 1. The L,/L# ratios for protons on Au as a function of incident energy. Theoretical predictions: . . . . . . PWBA; . . . . . PWBA-BC; . . . . . . . . PWBA-BCR; . . . . . . CPSSR; - RPWBA-BC.
T. Mvkoyarna L. Sarkadi / Relative intensi(v ratios for L-shell X-rays
588
PWBA-BCR values are systematically smaller than the measured values. For low energies, the RPWBA-BC gives better agreement with the experimental values than other theories, but for high energies the PWBA and the PWBA-BC seem to be better. Fig. 2 shows a similar comparison of various theories with the experimental data of the L o / L y ratios for low-energy protons on gold. In this case, the PWBABCR values are too small and out of the range of the figure. The experimental data scatter somewhat, but show the general trend that they increase with energy and then slowly decrease. All the theoretical predictions continuously increase in the energy region concerned, and do not reproduce the behaviour of the experimental values. At low energies the RPWBA-BC is in good agreement with the experimental data, but for higher energies it overpredicts the experimental values. As has been described in the previous work [13], the number of experimental works on the L-shell ionization by low-energy protons is not so large. Especially the tabulated values of X-ray intensity ratios and partial X-ray production cross sections are scarce. For the L-subshell ionization cross sections, we have shown [l 3] that an approximately universal behaviour is obtained if we plot the experimental cross sections, divided by the RPWBA-BC theory, for different target elements as a function of a reduced velocity parameter. It is advantageous to use this property of the cross sections and to show the experimental data for different elements in a u n i v e r s a l plot.
For this purpose, we define an average reduced velocity parameter of the L-shell electrons as
4~fL/2/OL,
~L =
(6)
where 0L is the average scaled binding energy of the
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L-shell electrons. This energy is obtained from 0 L =40L/(Z~LRo¢),
(7)
where OL is the average binding energy of the L-shell electrons, ZZL = Z 2 -- 4.15, Z 2 is the target atomic number and R= is the Rydberg unit. The average binding energy is defined as 0 L = (U, + U2
+ 2U3)/4,
(8)
where U, is the measured value of the Li-shell binding energy. Fig. 3 shows the experimental L , / L # intensity ratios, normalized to the RPWBA-BC theory, for low-energy protons on target elements between gold and uranium as a function of ~ L. The different symbols correspond to the different target elements and the different measurements listed in table 1. It can be seen from the figure
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incident energy. Theoretical predictions: . . . . . . PWBA; . . . . . PWBA-BC; ...... CPSSR; RPWBA-BC.
Fig. 4. The L , , / L y ratios for protons, normalized to t h e R P W B A - B C theory, against the average reduced velocity ~L"
T. Mykoyama, L. Sarkadi / Relative intensity ratios for L-shell X-rays
that the RPWBA-BC slightly overpredicts the experimental data, but is in agreement with all the experimental values within the deviation of 20%. In fig. 4, a similar universal plot for the L,~/Lv ratios is shown, for 1OW-~Lvalues, the prediction of the RPWBA-BC tends to be smaller than the experimental data, but the theory overpredicts the experimental values for the high-~ L region. Almost all the experimental data are in agreement with the RPWBA-BC within a 25% deviation.
4. Discussion
It can be seen from figs. 1 and 2 that the measured values for L-shell X-ray intensity ratios are closest to the RPWBA-BC curve for low energies and to the PWBA and the PWBA-BC for the high-energy region. However, comparison between theory and experiment for L-subshell ionization cross sections shows [12,14] that these nonrelativistic theories are in poor agreement with the experimental data. This fact idicates that the agreement of the PWBA and the PWBA-BC with the measured values in figs. 1 and 2 is fortuitous. On the other hand, the PWBA-BCR is systematically smaller than the measured values, while the CPSSR overpredicts the experimental data. The best overall agreement with experiment is obtained for the RPWBA-BC. It is interesting to note that for the L-subshell ionization cross sections the CPSSR gives almost the same values as the RPWBA-BC [12,14]. However, the relative X-ray intensity ratios calculated with the theories are different from each other for low-energy protons, as shown in figs. 1 and 2. The situation is quite similar to the case for the relative ratios of L-subshell ionization cross sections [14]. This is due to the fact that the relativistic correction proposed by Brandt and Lapicki [6] is not so good for L-shell electrons in the case of low-energy projectiles [23]. As has been described earlier, the relative cross section ratio is more sensitive to the choice of theoretical models than the absolute cross section, and better to test the validity of various theories. From figs. 3 and 4 it is clear that the relative L-shell X-ray intensity ratios, normalized to the RPWBA-BC theory, have a universal behaviour when they are plotted against the average reduced velocity parameter. Most experimental values are in agreement with the calculated ones within the deviation of 25%. Agreement for X-ray intensity ratios is better than for subshell ionization cross sectoins, where the deviation is about 60% for L t and L 3 shells and much larger for the L 2 shell. The main reason for this better agreement can be attributed to the smallness of scatter of the experimental data due to smaller experimental errors. We have discussed the sources of the large uncer-
589
tainty in the L-subshell ionization cross sections in the previous work [13]. For measurements of L-shell X-ray intensity ratios, the situations is more favourable. There is no need to separate each component of the complex L X-ray spectrum. Sometimes it is enough only to separate L,,, La and Lv lines. The relative intensity ratios are determined from these X-ray peaks and directly measurable quantities. Furthermore, some factors common to each X-ray intensity, such as geometrical efficiency and attenuation of X-rays between the target and the detector, are cancelled out in the relative intensity ratios. Thus the X-ray intensity ratios can be determined more accurately than the cross sections for L-subshell ionization or X-ray production. In the present work, we have estimated the theoretical partial x-ray production cross sections from the calculated values of the L-subshell ionization cross sections by the use of eqs. (3)-(5). These equations contain the atomic parameters, such as subshell fluorescence yields, Coster-Kronig coefficients, and total and partial radiative decay widths. The different choice of these parameters leads to different theoretical values of the X-ray intensity ratios. The change in the theoretical intensity ratios due to the choice of the atomic parameters is estimated to be less than 10% and does not influence the general trend observed in the present work. It is worthwhile to make a more systematic comparison between theory and experiment for L-shell X-ray intensity ratios as well as for L-subshell ionization cross sections. However, the experimental data by low-energy proton impact are scarce. There have been reported several measurements for X-ray intensity ratios by lowenergy a particles and heavy ions. In the case of heavy ions the behaviour of the measured intensity ratios deviates strongly from the theoretical predictions [24]. We have pointed out that this discrepancy can be substantially improved by taking into account the contribution of the collision-induced intra-shell transition of a vacancy between L subshells [25]. In the previous work, we have also shown an indication of a similar effect for low-energy a particles [13]. Our estimation of this process is based on a simple two-step approximation. In order to compare these experimental data with the theoretical predictions, more rigorous treatment of the multi-step processes should be done. In conclusion, we have obtained a fairly good agreement between the RPWBA-BC and experiment for relative L-shell X-ray intensity ratios by low-energy protons, but the discrepancy is still large in comparison with the small experimental errors. However, the number of experimental data used for comparison is not so large. Much more experimental data with high precision are urgently needed to test theoretical models in more detail. On the other hand, as has been shown in the previous work [13], there is a large discrepancy between
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T. Mykovama, L. Sarkadi / Relative intensity ratios for L-shell X-ravs
theory and experiment for L2-shell ionization cross sections a n d a possible reason is considered to be the choice of atomic wave functions. It is very i m p o r t a n t to estimate the L-shell X-ray intensity ratios b y the use of more realistic atomic wave functions, such as H a r t r e e Fock wave functions.
References [1] E. Merzbacher and H.W. Lewis, in Handbuch der Physik, ed. S. F!iagge (Springer Verlag, Berlin, 1958) vol. 34, pp. 166-192. [2] H. Paul, Nucl. Instr. and Meth. 169 (1980) 249. [3] G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A7 (1973) 983. [4] H. Paul, Acta Phys. Hung., to be published. [5] H. Paul, At. Data Nucl. Data Tables 24 (1979) 243. [6] W. Brandt and G. Lapicki, Phys. Rev. A20 (1979) 465. [7] T. Mukoyama and L. Sarkadi, Nucl. Instr. and Meth. 179 (1981) 573. [8] T. Mukoyama and L. Sarkadi, Nucl. Instr. and Meth. 186 (1981) 641. [9] T. Mukoyama and L. Sarkadi, Bull. Inst. Chem. Res., Kyoto Univ. 57 (1979) 33. [10] T. Mukoyama and L. Sarkadi, Phys. Rev. A23 (1981) 375.
[11] T. Mukoyama and L. Sarkadi, ATOMKI Krzlemrnyek 23 (1981) 8. [12] T. Mukoyama and L. Sarkadi, Abst. XI ICPEAC, 1979, Kyoto, eds. K. Takayanagi and N. Oda (The Society for Atomic Collision Research, Tokyo, 1979) pp. 671-672. [13] T. Mukoyama and L. Sarkadi, Nucl. Instr. and Meth. 190 (1981) 619. [14] T. Mukoyame and L. Sarkadi, Phys. Rev. A25 (1982) 1411. [15] D. Jamnik and (~. Zupan~i~, Kgl. Dan. Vid. Selsk., Mat.Fys. Medd. 31 (1957) no. 2. [16] B.-H. Choi, Phys. Rev. A4 (1971) 1002. [17] W. Brandt and G. Lapicki, Phys. Rev. AI0 (1974) 474. [18] M.O. Krause, J. Phys. Chem. Ref. Data 8 (1979) 307. [19] J.H. Scofield, At. Data Nucl. Data Tables 14 (1974) 121. [20] T. Mukoyama and L. Sarkadi, Bull. Inst. Chem. Res., Kyoto Univ. 58 (1980) 60. [21] T. Mukoyama and L. Sarkadi, Bull. Inst. Chem. Res., Kyoto Univ., to be published. [22] O. Benka and A. Kopf, At. Data Nucl. Data Tables 22 (1978) 219. [23] T. Mukoyama and L. Sarkadi, Atomic collision research in Japan, no. 7, eds. Y. Hatano et al. (The Society for Atomic Collision Research, Tokyo, 1981) pp. 50-52. [24] L. Sarkadi and T. Mukoyama, J. Phys. B: Atom. Molec. Phys. 13 (1980) 2255. [25] L. Sarkadi and T. Mukoyama, J. Phys. B: Atom. Molec. Phys. 14 (1981) L225.
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R E L A T I V E I N T E N S I T Y R A T I O S F O R L - S H E L L X-RAYS BY L O W - E N E R G Y P R O T O N I M P A C T Takeshi M U K O Y A M A Institute for Chemical Research, Kyoto University, Kyoto, Japan and Lfiszl6 S A R K A D I Institute of Nuclear Research of the Hungarian Academy of Sciences (.4TOMKI), Debrecen, Hungary Received 17 September 1981
Comparison between theoretical and experimental values for relative intensity ratios of L-shell X-rays by low-energy proton
impact Ep=0.3-0.8 MeV has been made graphically. The calculations for the La, LO and Lv X-ray production cross sections have been performed by the relativistic plane-wave Born approximation theory corrected for the binding-energy and Coulomb-deflection effects (RPWBA-BC). The theoretical predictions are compared with the experimental values for the L,~/La and L,,/L~ X-ray intensity ratios as well as other theoretical models. It is found that the RPWBA-BC gives the best overall agreement with the experimental results. Using an average reduced velocity parameter, we have plotted the experimental L,~/L~ and L,~/L v ratios, normalized to the RPWBA-BC theory, for different target elements. This universal plot shows that the measured L,~/L/~ and L,,/L v values are in agreement with the RPWBA-BC theory within a deviation of 25%.
1. Introduction In recent years extensive experimental works on Kand L-sheU ionization by heavy charged-particle impact have been reported. These experimental results ate usually compared with the theoretical predictions calculated by the plane-wave Born approximation (PWBA) [1]. For low-energy projectiles, the corrections for the binding-energy increase of the target electrons during collisions and for the Coulomb-deflection effect of the projectile should be taken into consideration. Recently, comparison of a large amount of the measured K-shell ionization cross sections for protons and a particles on a variety of target elements with the various theoretical models has been made by Paul [2]. His results indicate that the PWBA theory modified for the binding-energy and Coulomb-deflection effects (PWBA-BC) [3] gives the best overall agreement with the experimental data. More systematic graphical comparison between theory and experiment for K-shell ionization cross sections by various projectiles has also been performed by Paul [4,5]. The experimental data are expressed as the ratio to the theoretical values calculated by the PWBA model corrected for the binding-energy, Coulomb-deflection, polarization, and electronic relativistic effects (PWBABCPR) [6] and plotted against the reduced velocity parameter. Almost all the experimental data are found 0167-5087/82/0000-0000/$02.75 © 1982 North-Holland
to be in fairly good agreement with the theoretical predictions. Similar comparisons have been made by us for the K-shell ionization cross sections by protons and a particles [7] and by heavy ions [8]. The theoretical values have been calculated by the relativistic PWBA model including the corrections for the binding-energy and Coulomb-deflection effects (RPWBA-BC) [9]. The exact values for the limits of integration over momentum transfer are used [10]. It is found that the RPWBA-BC gives better agreement with the experimental values than the PWBA-BCPR. This is due to the fact that the correction derived by Brandt and Lapicki [6] overestimates the electronic relativistic effects for low-energy projectiles [11]. This kind of comparison between theory and experiment is very useful to identify and reject the erroneous experimental data and also to test the limit of validity of theoretical models. We have recently extended the RPWBAzBC model to the case of L-shell ionization cross sections [12]. The calculated values for L-subsheU ionization cross sections by low-energy protons and a particles have been used to compare with the experimental data [13]. The agreement between theory and experiment for L subshells is not so good as that for the K shell. Especially for the L 2 shell the measured values are systematically larger than the theoretical predictions. However, the available experi-
586
T. Mvkovama L. Sarkadi / Relative intensi(v ratios for L-shell X-rays
mental data for L-subshell ionization cross sections are scarce because of difficulties in X-ray measurements and data analysis. It is needed to make a more systematic comparison between theory and experiment for L-subshell ionization. As has been discussed in our previous work [13], comparison of theory with experiment should be made not for the total L-shell ionization cross sections, but for the L-subshell ionization cross sections. However, these cross sections are not the directly measurable quantity. From the observed peak intensities of L X-rays the partial X-ray production cross sections corresponding to L X-ray components are estimated and the L-subshell ionization cross sections are converted from these cross sections by the use of atomic parameters, such as Lsubshell fluorescence yields, Coster-Kronig coefficients and partial decay widths. This fact means that the L-subshell ionization cross sections are correlated with each other and the small change in the L X-ray intensities may lead to a large variation in the subshell ionization cross sections. Several experimental results have been reported on relative intensity ratios for L-shell X-rays by low-energy proton impact. These experimental values cannot be converted into the L-subshell ionization cross sections, but it is possible to compare them with the theoretical predictions for the relative intensity ratios. There have also been the experimental data for the partial L-shell X-ray production cross sections, from which the relative X-ray intensity ratios are easily obtained. Comparison between theory and experiment for the relative intensity ratios has the following three advantages: First, these ratios are directly determined from the experimental measurements for X-rays. Second, in these ratios several factors affecting the X-ray intensities, such as geometrical efficiency and correction for absorption of X-rays between the target and the detector, are cancelled out and the experimental errors are small. Third, the X-ray intensity ratios are more sensitive to the difference between theory and experiment than the subshell ionization cross sections. If each subshell ionization cross section agrees with the theoretical predictions fairly well, the measured intensity ratios are sometimes in poor agreement with the theory. It is the purpose of the present work to compare the measured L-shell X-ray intensity ratios with the theoretical predictions. Theoretical values for the L-subshell ionization cross sections by low-energy proton impact are calculated in the RPWBA-BC. These cross sections are converted into the partial X-ray production cross sections and the relative intensities of the L,/L/~ and L , / L r ratios are estimated. Comparison between theory and experiment is made graphically.
2. Theoretical model The theoretical model in the present work is the same as that used in the previous work [13] and is described in detail elsewhere [14]. In the PWBA, the L-shell ionization cross section by charged-particle impact is given by [1] do , 2 M I fqm~dq dE/='*~ra - - - - - , F ~ f ( q ) [2 ,
El Jq,,,in q3
(1)
where q is the momentum transfer, El is the kinetic energy of the ejected electron, a is the fine structure constant, M~ and E t are the mass and energy of the projectile, F,f(q) is the form factor, and qmax and qmin are the upper and lower limits of the momentum transfer. Throughout the present work, the relativistic units (h=m=c=l) areused. The exact values for the limits of momentum transfer are determined from the conservation law of energy and momentum, and used as qm~ and qmin"The calculation for the form factor is made by the use of the relativistic hydrogenic (Dirac) wave functions following the method of Jamnik and Zupanri6 [15]. The analytical expressions for F,/(q) are equivalent to those of Choi [16]. The screening effects are taken into consideration through the inner- and outer-screening method as usual in the PWBA. The binding-energy effect is estimated in the perturbed-stationary-state (PSS) approach developed by Brandt and Lapicki [17] and incorporated into the PWBA model. This can be done by introducing the binding-energy factor c. The effect of Coulomb deflection is also taken into account through the method of Brandt and Lapicki [17]. According to this model, the L,-shell ionization cross section in the RPWBA-BC is expressed as OR,pw"a-"c = CL,(~rdqo, L,) ×
oRPWBA-B /
L,
(2)
where CL,(X) is the correction factor for the CoulombRPWBA-B`tTL,, ~L,, (Li0L,) is the deflection effect, O L, RPWBA cross section corrected for the binding-energy effect, d is one-half of the distance of closest approach in a head-on collision, q0 is the minimum momentum transfer, %, is the binding-energy factor, YL, is the parameter of the target atomic number, ~TL,is the scaled projectile velocity parameter and 0L, is the scaled binding energy of the target electron. The binding energy factor q., is estimated by the use of the relativistic hydrogenic wave functions following the method of Brandt and Lapicki [17] and the analytical expressions are given in ref. 14. The explicit forms of
587
T. Mykoyama, L. Sarkadi / Relative intensity ratios for L-shell X-rays
the function CL,(X ) are obtained by Brandt and Lapicki and expressed in ref. 17. In order to estimate the L-shell X-ray intensity ratios, the L-subshell ionization cross sections should be converted into the partial X-ray production cross sections. The L,, L B and Lv X-ray production cross sections are related to the three subshell ionization cross sections by the following relations: ° x = [°1(f,3 +f12f23) + 02f23 + 03] ~ , r ~ o / r , , °~x : [ ° , ( f , 3 +f,2f23) + 02f23 + 03]
(3)
,.,3r,,/r,
q- (01/12 + 02) . 2r2./r2 + ox = ( 0 , / , 2 + % ) ~,2r2,/r2 + o , , < r , ~ / r , ,
(4) (5)
where o, is the L,-subshell ionization cross section, f,j is the Coster-Kronig coefficient for transfer of a vacancy from the ith to the j t h subshell, ~0i is the Li-subshell fluorescence yield, F~ is the total radiative width of the L, subshell and F,~ is the partial radiative width for the transition filling a vacancy in the L~ shell which contributes to the L~ line. The intensity ratios L ~ / L a and L , ~ / L r can be calculated as the ratios of the partial X-ray production cross sections.
3. Comparison of theoretical and experimental L-shell X-ray intensity ratios The theoretical L-subshe11 ionization cross sections have been calculated according to eq. (2). Using eqs. (3)-(5), these cross sections have been converted into the partial X-ray production cross sections. For this purpose, we used the subshell fluorescence yield, ~i, and the Coster-Kronig coefficient, ~j, compiled by Krause [18]. The total and partial radiative widths were taken from the theoretical values calculated by Scofield [19]. References for all the experimental data used in the present work are listed in table 1. In the case of data by AB74, the L , ~ / L r ratios were estimated from the experimental values of L , / L v , and L~/Lv23 ratios assuming
L r = Ly, + Lr2; The experimental data by BH80 were calculated as die ratio of the partial X-ray production cross sections. In this case, estimation of the experimental errors was not made because the errors obtained from those of the X-ray production cross sections are considered to be much larger than those for direct measurements of the X-ray intensity ratios. In fig. 1, the L~/L~ ratios for low-energy protons on gold are shown as a function of incident energy. The different symbols refer to the different measurements identified by the codes and these codes correspond to those in table 1. The experimental data agree well with each other and are almost constant with energy. For comparison, other theoretical values based on the PWBA, the PWBA-BC, the PWBA-BC with the relativistic correction (PWBA-BCR) of Merzbacher and Lewis [1] and the PWBA-BCR by the relativistic correction of Brandt and Lapicki [6] are plotted in the figure. Brandt and Lapicki [6] called their model the Coulomb-deflection-corrected PSS theory with relativistic correction (CPSSR). In order to discriminate between two PWBA-BCR theories, we call the model of Brandt and Lapicki the CPSSR and that of Merzbacher and Lewis the PWBA-BCR, though the only difference between these two models consists in the correction method for the relativistic effect. The PWBA, the PWBA-BC and the PWBA-BCR have been calculated by the computer code DEKY [20], while the CPSSR calculations have been made by the code DEKY2 [21]. In all these calculations, the exact values of the limits of momentum transfer [22] were used instead of the usual approximate ones. It is clear from the figure that all the theoretical models, except for the PWBA-BCR, slightly overpredict the experimental data. On the other hand, the
I
AB74 BH80 BU73
SH73
F. Abrath and T.J. Gray, Phys. Rev. A9 (1979) 682. D. Bhattacharya, S.K. Bhattacherjee and S.K. Mitra, J. Phys. B13 (1980) 967. C.E. Busch, A.B. Baskin, P.H. Nettles, S.M. Shafroth and A.W. Waltner, Phys. Rev. A7 (1973) 1601. S.M. Shafroth, G.A. Bissinger and A.W. Waltner, Phys. Rev. A7 (1973) 566.
I
I
I
2.0
/
Table 1 References for experimental data.
I
(IZ
-.I
o
o
~
1.5
~, S H 7 3
p o n AU 1,0
0.5
o BHSO
I
I
I
I
I
0.4
0.5
0.6
0.7
0.8
E N E R G Y
(MeV)
Fig. 1. The L,/L# ratios for protons on Au as a function of incident energy. Theoretical predictions: . . . . . . PWBA; . . . . . PWBA-BC; . . . . . . . . PWBA-BCR; . . . . . . CPSSR; - RPWBA-BC.
T. Mvkoyarna L. Sarkadi / Relative intensi(v ratios for L-shell X-rays
588
PWBA-BCR values are systematically smaller than the measured values. For low energies, the RPWBA-BC gives better agreement with the experimental values than other theories, but for high energies the PWBA and the PWBA-BC seem to be better. Fig. 2 shows a similar comparison of various theories with the experimental data of the L o / L y ratios for low-energy protons on gold. In this case, the PWBABCR values are too small and out of the range of the figure. The experimental data scatter somewhat, but show the general trend that they increase with energy and then slowly decrease. All the theoretical predictions continuously increase in the energy region concerned, and do not reproduce the behaviour of the experimental values. At low energies the RPWBA-BC is in good agreement with the experimental data, but for higher energies it overpredicts the experimental values. As has been described in the previous work [13], the number of experimental works on the L-shell ionization by low-energy protons is not so large. Especially the tabulated values of X-ray intensity ratios and partial X-ray production cross sections are scarce. For the L-subshell ionization cross sections, we have shown [l 3] that an approximately universal behaviour is obtained if we plot the experimental cross sections, divided by the RPWBA-BC theory, for different target elements as a function of a reduced velocity parameter. It is advantageous to use this property of the cross sections and to show the experimental data for different elements in a u n i v e r s a l plot.
For this purpose, we define an average reduced velocity parameter of the L-shell electrons as
4~fL/2/OL,
~L =
(6)
where 0L is the average scaled binding energy of the
17-
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16--
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Au
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o
D
-.1
o o
._1
0.8 --
0.25
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[
0.30
0.'55
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I
0.40
0.45
L Fig. 3. The L,~/Lt~ ratios for protons, normalized to the R P W B A - B C theory, against the average reduced velocity ~L"
L-shell electrons. This energy is obtained from 0 L =40L/(Z~LRo¢),
(7)
where OL is the average binding energy of the L-shell electrons, ZZL = Z 2 -- 4.15, Z 2 is the target atomic number and R= is the Rydberg unit. The average binding energy is defined as 0 L = (U, + U2
+ 2U3)/4,
(8)
where U, is the measured value of the Li-shell binding energy. Fig. 3 shows the experimental L , / L # intensity ratios, normalized to the RPWBA-BC theory, for low-energy protons on target elements between gold and uranium as a function of ~ L. The different symbols correspond to the different target elements and the different measurements listed in table 1. It can be seen from the figure
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--
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o
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-
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o
o
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n
o
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o ~ .
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o
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a
_1
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--
/
(MeV)
C3
1 0.25
0.30
D Q
O
--
I
"1
0.55
0,40
0.45
Fig. 2. The L , ~ / L v ratios for protons on A u as a function o f
incident energy. Theoretical predictions: . . . . . . PWBA; . . . . . PWBA-BC; ...... CPSSR; RPWBA-BC.
Fig. 4. The L , , / L y ratios for protons, normalized to t h e R P W B A - B C theory, against the average reduced velocity ~L"
T. Mykoyama, L. Sarkadi / Relative intensity ratios for L-shell X-rays
that the RPWBA-BC slightly overpredicts the experimental data, but is in agreement with all the experimental values within the deviation of 20%. In fig. 4, a similar universal plot for the L,~/Lv ratios is shown, for 1OW-~Lvalues, the prediction of the RPWBA-BC tends to be smaller than the experimental data, but the theory overpredicts the experimental values for the high-~ L region. Almost all the experimental data are in agreement with the RPWBA-BC within a 25% deviation.
4. Discussion
It can be seen from figs. 1 and 2 that the measured values for L-shell X-ray intensity ratios are closest to the RPWBA-BC curve for low energies and to the PWBA and the PWBA-BC for the high-energy region. However, comparison between theory and experiment for L-subshell ionization cross sections shows [12,14] that these nonrelativistic theories are in poor agreement with the experimental data. This fact idicates that the agreement of the PWBA and the PWBA-BC with the measured values in figs. 1 and 2 is fortuitous. On the other hand, the PWBA-BCR is systematically smaller than the measured values, while the CPSSR overpredicts the experimental data. The best overall agreement with experiment is obtained for the RPWBA-BC. It is interesting to note that for the L-subshell ionization cross sections the CPSSR gives almost the same values as the RPWBA-BC [12,14]. However, the relative X-ray intensity ratios calculated with the theories are different from each other for low-energy protons, as shown in figs. 1 and 2. The situation is quite similar to the case for the relative ratios of L-subshell ionization cross sections [14]. This is due to the fact that the relativistic correction proposed by Brandt and Lapicki [6] is not so good for L-shell electrons in the case of low-energy projectiles [23]. As has been described earlier, the relative cross section ratio is more sensitive to the choice of theoretical models than the absolute cross section, and better to test the validity of various theories. From figs. 3 and 4 it is clear that the relative L-shell X-ray intensity ratios, normalized to the RPWBA-BC theory, have a universal behaviour when they are plotted against the average reduced velocity parameter. Most experimental values are in agreement with the calculated ones within the deviation of 25%. Agreement for X-ray intensity ratios is better than for subshell ionization cross sectoins, where the deviation is about 60% for L t and L 3 shells and much larger for the L 2 shell. The main reason for this better agreement can be attributed to the smallness of scatter of the experimental data due to smaller experimental errors. We have discussed the sources of the large uncer-
589
tainty in the L-subshell ionization cross sections in the previous work [13]. For measurements of L-shell X-ray intensity ratios, the situations is more favourable. There is no need to separate each component of the complex L X-ray spectrum. Sometimes it is enough only to separate L,,, La and Lv lines. The relative intensity ratios are determined from these X-ray peaks and directly measurable quantities. Furthermore, some factors common to each X-ray intensity, such as geometrical efficiency and attenuation of X-rays between the target and the detector, are cancelled out in the relative intensity ratios. Thus the X-ray intensity ratios can be determined more accurately than the cross sections for L-subshell ionization or X-ray production. In the present work, we have estimated the theoretical partial x-ray production cross sections from the calculated values of the L-subshell ionization cross sections by the use of eqs. (3)-(5). These equations contain the atomic parameters, such as subshell fluorescence yields, Coster-Kronig coefficients, and total and partial radiative decay widths. The different choice of these parameters leads to different theoretical values of the X-ray intensity ratios. The change in the theoretical intensity ratios due to the choice of the atomic parameters is estimated to be less than 10% and does not influence the general trend observed in the present work. It is worthwhile to make a more systematic comparison between theory and experiment for L-shell X-ray intensity ratios as well as for L-subshell ionization cross sections. However, the experimental data by low-energy proton impact are scarce. There have been reported several measurements for X-ray intensity ratios by lowenergy a particles and heavy ions. In the case of heavy ions the behaviour of the measured intensity ratios deviates strongly from the theoretical predictions [24]. We have pointed out that this discrepancy can be substantially improved by taking into account the contribution of the collision-induced intra-shell transition of a vacancy between L subshells [25]. In the previous work, we have also shown an indication of a similar effect for low-energy a particles [13]. Our estimation of this process is based on a simple two-step approximation. In order to compare these experimental data with the theoretical predictions, more rigorous treatment of the multi-step processes should be done. In conclusion, we have obtained a fairly good agreement between the RPWBA-BC and experiment for relative L-shell X-ray intensity ratios by low-energy protons, but the discrepancy is still large in comparison with the small experimental errors. However, the number of experimental data used for comparison is not so large. Much more experimental data with high precision are urgently needed to test theoretical models in more detail. On the other hand, as has been shown in the previous work [13], there is a large discrepancy between
590
T. Mykovama, L. Sarkadi / Relative intensity ratios for L-shell X-ravs
theory and experiment for L2-shell ionization cross sections a n d a possible reason is considered to be the choice of atomic wave functions. It is very i m p o r t a n t to estimate the L-shell X-ray intensity ratios b y the use of more realistic atomic wave functions, such as H a r t r e e Fock wave functions.
References [1] E. Merzbacher and H.W. Lewis, in Handbuch der Physik, ed. S. F!iagge (Springer Verlag, Berlin, 1958) vol. 34, pp. 166-192. [2] H. Paul, Nucl. Instr. and Meth. 169 (1980) 249. [3] G. Basbas, W. Brandt and R. Laubert, Phys. Rev. A7 (1973) 983. [4] H. Paul, Acta Phys. Hung., to be published. [5] H. Paul, At. Data Nucl. Data Tables 24 (1979) 243. [6] W. Brandt and G. Lapicki, Phys. Rev. A20 (1979) 465. [7] T. Mukoyama and L. Sarkadi, Nucl. Instr. and Meth. 179 (1981) 573. [8] T. Mukoyama and L. Sarkadi, Nucl. Instr. and Meth. 186 (1981) 641. [9] T. Mukoyama and L. Sarkadi, Bull. Inst. Chem. Res., Kyoto Univ. 57 (1979) 33. [10] T. Mukoyama and L. Sarkadi, Phys. Rev. A23 (1981) 375.
[11] T. Mukoyama and L. Sarkadi, ATOMKI Krzlemrnyek 23 (1981) 8. [12] T. Mukoyama and L. Sarkadi, Abst. XI ICPEAC, 1979, Kyoto, eds. K. Takayanagi and N. Oda (The Society for Atomic Collision Research, Tokyo, 1979) pp. 671-672. [13] T. Mukoyama and L. Sarkadi, Nucl. Instr. and Meth. 190 (1981) 619. [14] T. Mukoyame and L. Sarkadi, Phys. Rev. A25 (1982) 1411. [15] D. Jamnik and (~. Zupan~i~, Kgl. Dan. Vid. Selsk., Mat.Fys. Medd. 31 (1957) no. 2. [16] B.-H. Choi, Phys. Rev. A4 (1971) 1002. [17] W. Brandt and G. Lapicki, Phys. Rev. AI0 (1974) 474. [18] M.O. Krause, J. Phys. Chem. Ref. Data 8 (1979) 307. [19] J.H. Scofield, At. Data Nucl. Data Tables 14 (1974) 121. [20] T. Mukoyama and L. Sarkadi, Bull. Inst. Chem. Res., Kyoto Univ. 58 (1980) 60. [21] T. Mukoyama and L. Sarkadi, Bull. Inst. Chem. Res., Kyoto Univ., to be published. [22] O. Benka and A. Kopf, At. Data Nucl. Data Tables 22 (1978) 219. [23] T. Mukoyama and L. Sarkadi, Atomic collision research in Japan, no. 7, eds. Y. Hatano et al. (The Society for Atomic Collision Research, Tokyo, 1981) pp. 50-52. [24] L. Sarkadi and T. Mukoyama, J. Phys. B: Atom. Molec. Phys. 13 (1980) 2255. [25] L. Sarkadi and T. Mukoyama, J. Phys. B: Atom. Molec. Phys. 14 (1981) L225.