Radiation Physics and Chemistry 81 (2012) 499–505

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Analytical formulas for calculation of K X-ray production cross sections by alpha ions A. Abdellatif a, A. Kahoul b,c,d,n, B. Deghfel c,d,e, M. Nekkab c,d,e, D.E. Medjadi a a

Physics Department, L’e´cole normale superieure Vieux-Kouba, 16000 Algiers, Alge´rie Department of Materials Science, Faculty of Sciences and Technology, Bordj-Bou-Arreridj University, Bordj-Bou-Arreridj 34000, Algeria c LESIMS laboratory, Physics Department, Faculty of Sciences, Ferhat Abbas University of Setif, 19000 Setif, Algeria d LPCM laboratory, Physics Department, Faculty of Sciences, University of M’Sila, 28000 M’Sila, Algeria e Physics Department, Faculty of Sciences, University of M’Sila, 28000 M’Sila, Algeria b

a r t i c l e i n f o

abstract

Article history: Received 24 February 2011 Accepted 22 December 2011 Available online 11 January 2012

In the present study, different procedures are followed to deduce the semi-empirical and the empirical K X-rayX-ray production cross sections induced by alpha ions from the available experimental data and the theoretical results of the ECPSSR model for elements with 20 r Z r 30. The deduced K X-ray production cross sections are compared with predictions from ECPSSR model and with other earlier works. Generally, the deduced K X-ray production cross sections obtained by fitting the available experimental data for each element separately give the most reliable values than those obtained by a global fit. & 2012 Elsevier Ltd. All rights reserved.

Keywords: K X-ray production cross section ECPSSR model Semi-empirical and empirical cross sections

1. Introduction In the last few decades, a major effort has been made to understand the complex mechanism of the process of the atomic inner-shell ionization by charged particle. A large number of experimental data of K-shell ionization and production cross sections for ion–atom collision have been published namely for proton and alpha impact. Moreover, different theoretical models have been developed to describe the inner shell ionization process by incident charged particles, namely the plane wave Born approximation (PWBA) (Merzbacher and Lewis, 1958), the semi classical approximation (SCA) (Garcia, 1970a) and the classical approach also known as the binary-encounter approximation (BEA) (Choi, 1973). The PWBA theory was further developed by Brandt and Lapicki (1981) to account for energy (E) loss and Coulomb (C) deflection of the projectile and the perturbed stationary state (PSS) and relativistic (R) nature of the target’s inner-shell to give rise to the ECPSSR model, which is being the most advanced approach. Generally, large deviation is observed between the theoretical predictions and the experimental values and between the different experimental data themselves. This situation motivated our research group to find simple analytical formulas for the calculation of the cross sections (Kahoul and Nekab, 2005; Nekab and Kahoul, 2006; Kahoul et al., 2008a, 2008b; Deghfel et al., 2009, 2010). In their

study, Paul and Sacher (1989) and Paul and Bolik (1993) developed a semi-empirical approximation of the K-shell ionization cross sections and regrouped them in tabular form. The first table (Paul and Sacher,1989) contains the semi-empirical K-shell ionization cross sections for protons impact deduced from theoretical predictions and from published experimental data at that time; in the second one, using the same method, Paul and Bolik (1993) performed a calculation of ‘reference’ K-shell ionization cross sections by alpha impact. Since that time, a large number of experimental data have become available calling for new fitting. In the present work, we have calculated the semi-empirical and the empirical K X-ray production cross sections for element from 20Ca to 30Zn using different existing procedures (Paul and Bolik, 1993; Garcia, 1971; Carlos Romo-Kroger, 2000; Lapicki, 2002) for alpha impact. In addition, we have followed the previous procedures for each element separately and presented our results for 23V and 29Cu, as an illustration. The new fitting parameters for the calculation of semi empirical and empirical K X-ray production cross sections for elements in the range 20rZr30 according to the different methods followed here are then tabulated. Finally, the deduced values are compared with the ECPSSR calculations and with the earlier calculations obtained by Paul and Bolik (1993).

2. Analytical formulas n Corresponding author at: Physics Department, Faculty of Sciences, University of M’Sila, 28000 M’Sila, Algeria. Tel.: þ776100072. E-mail address: [email protected] (A. Kahoul).

0969-806X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2011.12.036

The present database of the K X-ray production cross sections rely on the different compilations available in the literature (Heitz et al., 1982; Lapicki, 1989; Paul and Bolik, 1993).

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By keeping only the database for alpha particle whose energies vary from 0.1 MeV to 5.0 MeV, a total of 631 data point have been collected for elements with 20 rZr30. It should be noted that the published K-shell ionization cross sections were converted to the K X-ray production cross sections using the relation sX ¼ oKsI, where oK is the fluorescence yields taken from the compilation of Krause (1979), sX is the K X-ray production cross section and sI is the K-shell ionization one. 2.1. Reference ionization cross sections (semi-empirical formula)

2.5 Z= 20-30 2.0 s =σExp /σECPSSR

ssemp ¼ sECPSSR S where S ¼

P4

ð1Þ i

i¼0

Ai X .

2.2. Scaling of K X-ray production cross sections (empirical formula) In the framework of the plane wave Born theory (PWBA) (Merzbacher and Lewis, 1958) and the binary encounter approximation (BEA) (Garcia, 1970a, 1970b), it is possible to predict a ‘‘universal curve’’ to describe the K X-ray production cross sections for all target-projectile combinations at any energy by introducing an empirical formula. This latter includes E, UK, sexp and z1 variables; where sexp is the experimental K X-ray production cross section, E is the alpha energy in keV, UK is the K-shell binding energy of the target atom and z1 is the atomic number of the projectile (in this case z1 ¼2 for alpha projectile). This is done by plotting lnðsU 2K =z21 Þ vs. ln(E/lUK), where l is the ratio of proton mass to electron mass, in Fig. 2 for all experimental data used in the previous section. It can be seen that the values of lnðsU 2K =z21 Þ are found indeed to be universal, when plotted as a function of the reduced alpha energy ln(E/lUK). Consequently, the experimental data were fitted using the following polynomial: lnðsexp U 2K Þ ¼

3 X

Ai X i

ð2Þ

i¼0

where X¼ ln(E/lUK). It is worth noting that the spread of experimental data in some cases is due to various sources from which the experimental data have been collected and then measured under various experimental conditions. In order to bring the data of different target elements near to a single curve, Carlos Romo-Kroger (1998), corrected the ordinate sU 2K in the previous form by a factor (1 UK/180). This motivated us to apply a new scaling law on the same experimental values used in the two previous sections (formula (1) and (2)) and to fit

24

Z=20-30

20

ln(σExpUK2)

First, we presented a graph of the normalized cross section S¼ sexp/sECPSSR against the reduced velocity parameter X ¼ R log10 ðxK Þ, where : sECPSSR refers to our theoretical production cross sections calculated using a computer program based on the R ECPSSR model of Brandt and Lapicki (1981), xK ¼ ½mRK ðxK =zK Þ1=2 xK is defined as a product of xK ¼2V1/V2KyK (is a measure used to distinguish the slow collision from the fast one) and the function ðmRK Þ1=2 , which introduces the electronic relativistic effect. Here, V1 and V2K are the projectile and K-shell electron velocities, respectively. The parameter yK ¼ n2 EK =Z 22K R is the reduced binding energy, EK is the hydrogenic ionization energy, R is the Rydberg constant, n¼1 is the main quantum number for K-shell, Z2K ¼Z2  0.3 is the effective nuclear charge, Z2 is the target atomic number and zK is the factor that accounts for the perturbed stationary state (For more description of the parameter zK and the function ðmRK Þ1=2 , see the Appendix). It is worth noting that the experimental data of all elements are generally mixed when the R scaling based on the reduced velocity parameter X ¼ log10 ðxK Þ is R 1=2 used. Furthermore, the relativistic factor ðmK Þ is introduced to point out that the electronic relativistic effect has been incorporated in the ECPSSR theory by replacing xK ¼2V1/V2KyK with xRK ¼ ½mRK ðxK =zK Þ1=2 xK . Fig. 1 shows all points ðS,log10 ðxRK ÞÞ corresponding to the elements in the range of 20 rZr30 by alpha impact. The majority of the data are well described by ECPSSR R theory except some cases at low log10 ðxK Þ!0:7, where the theoretical values become larger than the experimental ones. The fitting of the parameter S in this case presents a large error for the calculation of semi-empirical cross sections. To remedy this situation, a criterion is applied by several authors by rejecting the experimental data which fit far from the ECPSSR calculations (Paul, 1982; Paul and Muhr, 1986; Rodriguez-Fernandez et al., 1993; Orlic, 1994). In this work, we use only the experimental data for which the ratio sexp/sECPSSR varies within the range of 0.5–1.5. The rejection of 34 data from the total database (about 5.38%; see Fig. 1) has no much influence on the calculation of the

cross section. Fig. 1 shows the fitting S with a full line. We now define the semi-empirical K X-ray production cross sections as

16

1.5 12 1.0 0.5

8

0.0 -0.8

-0.6

-0.4

-0.2

0.0

R log ξK

Fig. 1. A plot of S¼ sExp/sECPSSR as a function of the reduced velocity parameter R logðxK Þ for K X-ray production cross sections in alpha impact for 20r Z2 r 30. The dots are S and the curve is the fitting S.

-12

-8 -10 ln (E/(λUK))

-6

Fig. 2. A plot of universal K X-ray production cross sections in alpha impact for 20r Z2 r 30 (lnðsExp U 2K Þ vs. ln(E/lUK)). The dots are the experimental values and the curve is a third degree polynomial defined in Table 1.

A. Abdellatif et al. / Radiation Physics and Chemistry 81 (2012) 499–505

501

20

10

Z=20-30

Z (20-30)

16

Ln (σExpZ2K4/z21)

ln(σExpUK2(1-UK/180))

5

0

12

8

4

-5

-12

-2 ln (E/UK)

-4

0

2

Aj X j

ð3Þ

j¼0

4 2 exp Z 2K =z1 Þ ¼

lnðs

Aj X

j

Fig. 4. A plot of universal K X-ray production cross sections in alpha impact for 20r Z2 r 30 (lnðsExp Z 42K =z21 Þ vs. ln(E/UK)). The dots are the experimental values and the curve is a third degree polynomial defined in Table 1.

sV,Cu semp ¼ sECPSSR S

ð5Þ

Pn

where X¼ln(E/UK). All points ðlnðsexp U 2K ð1U K =180ÞÞ, lnðE=U K ÞÞ corresponding to the elements with 20 rZ r30 are shown in Fig. 3. The fit is also presented by a full line in the same figure. Next, we introduce an empirical law for the sK, Z2K, and z1 R dependence of the relativistic scaled velocity xK , defined above, as follows (Merzbacher and Lewis, 1958; Lapicki, 2002; Garcia et al., 1973; Pajek et al., 1990a, 1990b): 3 X

-6

analytical function used for the fitting is

them by a simple third degree polynomial as lnðsexp U 2K ð1U K =180ÞÞ ¼

-8

ln (E/(λUK))

Fig. 3. A plot of universal K X-ray production cross sections in alpha impact for 20r Z2 r30 (lnðsExp U 2K ð1U K =180ÞÞ vs. ln(E/UK)). The dots are the experimental values and the curve is a third degree polynomial defined in Table 1.

3 X

-10

R

where S ¼ i ¼ 0 Ai X i and X ¼ logðxK Þ. The fitting results are also shown in Fig. 5 for 23V and 29Cu with a full lines. In addition, we report the empirical cross sections based on formula (2) by fitting the same experimental data used in the previous section for the elements 23V and 29Cu separately, according to the following expression: 2 lnðsV,Cu exp U K Þ ¼

n X

Aj X j

ð6Þ

j¼0

ð4Þ

j¼0 R

where X ¼ lnðxK Þ and z1 ¼2 for alpha particle. The universal K X-ray production cross sections ðlnðsexp Z 42K =z21 Þ, R lnðxK ÞÞ are shown in Fig. 4 for the same experimental data used here. The resulting distribution is then fitted using a third degree polynomial that is presented by a full line in the same figure. 2.3. Individual fit 2.3.1. Semi-empirical and empirical formula Taking into account the dependence of the collision on the atomic number of the target, we propose that the data may be treated separately to study the difference between the global fit of the elements (20rZ r30) and those obtained when each element is fitted separately. Our results are presented for some selected elements namely 23V and 29Cu, as an illustration. As previously, the rejection criterion of the experimental data, for which the ratio S ¼ sexp/sECPSSR varies within the range of 0.5–1.5, is still going on for the sake of working in the same conditions. The distribution of the normalized cross sections is shown separately in Fig. 5 for both elements 23V and 29Cu. Then, the

where X¼ ln(E/lUK). The fitted data are shown in Fig. 6 for each element separately (23V and 29Cu), where the solid line is the polynomial fit.

2.3.2. Standard values In Table 2 we present the number of the available experimental K X-ray production cross sections by alpha bombardment of elements from 20Ca to 30Zn. With regard to the sufficient experimental data for 29Cu (about 134 data), it is possible to deduce empirical K X-ray production cross-sections for 29Cu by fitting, under the same conditions of rejected data, only the data of 29Cu and their corresponding values can be considered as ‘‘standard ones’’ (Kahoul et al., 2008a) to calculate the cross sections of other elements by assuming that the ratio sEmp/ sECPSSRis roughly the same for all elements. Then, the empirical K X-ray production cross-section for an element X can be evaluated using the following formula:

sXemp ¼ sCu emp

sXECPSSR sCu ECPSSR

ð7Þ

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A. Abdellatif et al. / Radiation Physics and Chemistry 81 (2012) 499–505

2.5 Z= 23 s =σExp /σECPSSR

Z=23

22

2.0 1.5

20

1.0 ln(σExpUK2)

0.5 0.0 -0.6

-0.2

-0.4

0.0

2.5

18

16

Z= 29 s =σExp /σECPSSR

2.0 1.5

14

1.0 -11

0.5

-10

-9

-7

-8

0.0 -0.8

-0.6

-0.4

-0.2

Z= 29

R

log ξK

21

Fig. 5. A plot of S¼ sExp/sECPSSR as a function of the reduced velocity parameter R logðxK Þ for K X-ray production cross sections in alpha impact: (a) for 23V, (b) for 29Cu. The dots are S and the curves are the fittings S; their corresponding coefficients are listed in Table 1.

where

sCu emp ¼ and

1 U 2K-Cu

h

expð159:8712þ 44:78522X þ5:10399X 2 þ 0:20737X 3 Þ

i

ln(σExpUK2)

18

15

12

X ¼ lnðE=lU K-Cu Þ:

Table 2 shows all coefficients Ai, the range of the x-axis (X) and the number of rejected data according to the different procedures followed here.

9

-12 3. Results and discussion The aim of this contribution is to investigate the reliable empirical or semi-empirical method for calculating the K X-ray cross sections by alpha impact. Moreover, it is interesting to note that we have reported only the experimental data in the energy range 0.1–5 MeV of alpha impact. Therefore, the fitting Eqs. (1)–(7) are only valid within this range. Table 3 presents the deduced K X-ray production cross sections (in barn) by alpha bombardment of 23V and 29Cu using the seven procedures described here. Our theoretical production cross sections, calculated using a computer program based on the ECPSSR model of Brandt and Lapicki (1981), and the earlier works of Paul and Bolik (1993), are also listed. On the other hand, these semi-empirical and empirical results are compared to the theoretical values, Fig. 7, by plotting the parameter s¼ s/sECPSSR vs. the alpha energy in MeV; the values of Paul and Bolik (1993), are also introduced. From the examination of the above-mentioned figures and by comparison of the cross section (semi-empirical and empirical) deduced from the seven procedures described here with other

-11

-10 -9 ln (E/(λUK))

-8

-7

Fig. 6. A plot of universal K X-ray production cross sections in alpha impact (lnðsExp U 2K Þ vs. ln(E/lUK)): (a) for 23V and (b) for 29Cu. The dots are the experimental values and the curve is a third degree polynomial defined in Table 1.

Table 1 Number of experimental K X-ray production cross sections by alpha bombardment of elements from 20Ca to 30Zn. Element

No. of data

20Ca

25 10 92 58 51 41 75 39 67 134 39

21Sc 22Ti 23V 24Cr 25Mn 26Fe 27Co 28Ni 29Cu 30Zn

A. Abdellatif et al. / Radiation Physics and Chemistry 81 (2012) 499–505

503

Table 2 Fitting coefficients for the calculation of the semi-empirical and the empirical K X-ray production cross sections for alpha impact according to the different procedures followed here. Z-range

Formulas

A0

A1

A2

A3

A4

N0 of data

Rejected data

Range of X

20–30 20–30 20–30 20–30 23 29 23 29

(1) (2) (3) (4) (5) (5) (6) (6)

1.62677 143.537 9.09471 141.681 0.79101 1.9894  0.65659 159.871

7.99793 40.0312 2.90414 40.9042 0.12089 14.2486  9.63168 44.7852

28.0703 4.63174 0.31718 4.71953 4.00257 55.2968  1.03622 5.10399

33.9042 0.19165 0.1925 0.19461 4.43896 78.8998  0.02221 0.20737

11.4536 – –  – 36.4783 – –

631 631 631 631 58 134 58 134

34 34 34 34 2 8 2 8

 0.8011;  0.0297  11.023;  7.3025  3.4328;0.1296  10.948;  7.3599  0.5881;  0.1116  0.7769;  0.2072  9.9405;  7.7075  8.0457;  10.8541

Table 3 Present K X-ray production cross sections (in barn) by alpha bombardment of

23V

and

29Cu,

were compared with the earlier works of Paul[12] .

Formulas (This work) E (MeV)

ECPSSR

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Paul and Bolik (1993)

6.35E  02 1.18E þ00 5.47E þ00 1.51E þ01 3.17E þ01 5.62E þ01 8.87E þ01 1.29E þ02 1.76E þ02 2.29E þ02

7.03E 02 1.27Eþ 00 5.39Eþ 00 1.39Eþ 01 2.83Eþ 01 5.03Eþ 01 8.17Eþ 01 1.25Eþ 02 1.81Eþ 02 2.52Eþ 02

1.32E  02 2.18E þ00 8.15E þ00 1.90Eþ 01 3.54E þ01 5.82E þ01 8.85E þ02 1.28E þ02 1.76E þ02 2.29E þ02

1.31E  02 2.16E þ 00 8.07E þ00 1.88E þ 01 3.51E þ 01 5.80E þ01 8.83E þ 01 1.27E þ 02 1.77E þ 02 2.38E þ 02

1.37E  02 2.26Eþ 00 8.49Eþ 00 1.99Eþ 01 3.72Eþ 01 6.16Eþ 01 9.43Eþ 01 1.37Eþ 02 1.90Eþ 02 2.57Eþ 02

7.62E  02 1.34E þ00 5.74E þ00 1.48E þ01 2.96E þ01 5.02Eþ 01 7.66E þ01 1.08Eþ 02 1.45E þ02 1.86E þ02

7.77E  02 1.32E þ 00 5.77E þ 00 1.50E þ01 2.98E þ 01 5.05E þ01 7.69E þ 01 1.08E þ02 1.45E þ 02 1.85E þ 02

5.16E  02 1.28Eþ 00 5.87Eþ 00 1.51Eþ 01 2.93Eþ 01 4.86Eþ 01 7.32Eþ 01 1.03Eþ 02 1.38Eþ 02 1.80Eþ 02

7.26E  02 1.28E þ00 – 1.46E þ01 – 5.30E þ01 8.36E þ01 1.22E þ02 1.65E þ02 2.15E þ02

5.91E  03 1.48E  01 7.55E  01 2.22E þ00 4.91E þ00 9.16E þ00 1.52E þ01 2.32E þ01 3.32E þ01 4.53E þ01

5.55E  03 1.67E  01 8.29E  01 2.30Eþ 00 4.83Eþ 00 8.63Eþ 00 1.39Eþ 01 2.08Eþ 01 2.96Eþ 01 4.06Eþ 01

3.58E  03 1.18E  01 5.85E  01 1.57E þ00 3.15E þ00 5.41E þ00 8.40Eþ 01 1.22E þ01 1.68E þ01 2.24E þ01

3.60E  03 1.19E  01 5.90E  01 1.58E þ 00 3.19E þ 00 5.47E þ 00 8.50E þ01 1.24E þ 01 1.71E þ 01 2.28E þ 01

3.80E  03 1.27E  01 6.31E  01 1.69Eþ 00 3.42Eþ 00 5.89Eþ 00 9.17Eþ 01 1.33Eþ 01 1.85Eþ 01 2.47Eþ 01

4.97E  03 1.63E  01 8.21E  01 2.26E þ00 4.62E þ00 8.02Eþ 00 1.25E þ01 1.84E þ01 2.58E þ01 3.52E þ01

4.81E  03 1.62E  01 8.11E  01 2.21E þ 00 4.53E þ 00 7.93E þ 00 1.25E þ 01 1.86E þ 01 2.62E þ 01 3.56E þ 01

4.81E  03 1.62E  01 8.11E  01 2.21Eþ 00 4.53Eþ 00 7.93Eþ 00 1.25Eþ 01 1.86Eþ 01 2.62Eþ 01 3.56Eþ 01

5.30E  03 1.61E  01 – 2.27E þ00 – 8.93E þ00 1.46E þ01 2.21E þ01 3.16E þ01 4.281E þ01

23V

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 29Cu

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

theoretical and experimental works, we notice that:

 By investigation of the results obtained by the global fit (i.e.



formulas (1)–(4)), we observe that the values deduced from the formula (1) coincide very well with the theoretical values over the whole energy range used for both elements 23V and 29Cu. Also, the agreement between our semi-empirical cross sections and those of Paul and Bolik (1993) is excellent for the whole range of the alpha energy for both elements 23V and 29Cu whereas the cross sections deduced from the formulas (2)–(4) are less satisfactory at low energies (E o1.5 MeV). We believe that this disagreement is due to different effective weighting in the four approaches; the semi-empirical cross sections are deduced from the experimental data as well as from the ECPSSR calculations whereas the empirical cross sections depend only on the available experimental data. Furthermore, the production cross sections corresponding to the formulas (5)–(7) agree generally with the ECPSSR calculations. The values deduced from the individual fit lie better close to the ECPSSR predictions and consequently give the ratio s/sECPSSRthe closest to unity. In this case the deviation between the ECPSSR theory and the semi-empirical and the empirical calculations at 1 MeV is about: 4.05% from formula (1), 11.6% from formula (5), 10.8% from formula (6) and 8.17% from formula (7) for 23V; 11.3% from formula (1), 8.7% from

formula (5), 8.2% from formula (6) and 8.2% from formula (7) for 29Cu. This clarifies the influence of the different forms of fitting on the deviation of the results obtained (semi-empirical and empirical cross sections) from the ECPSSR theory. Also, a comparison between the production cross sections deduced from formula (1), on the one hand, and those deduced from formulas (5)–(7), on the other hand, for both elements 23V and 29Cu, enables us to indicate that the ideal situation to determine a reliable cross section is to perform the fitting of the experimental data for each element separately.

4. Conclusion The ECPSSR theory combined with the experimental data has been used to derive the semi-empirical cross sections, whereas the available experimental data are directly fitted to deduce the empirical cross section. The fitted equations and their corresponding coefficients are presented to calculate the semi-empirical and empirical K X-ray production cross sections for elements with atomic numbers 20 rZ r30 by alpha impact and they are only valid in this region. Generally, the KX-ray production cross sections obtained by fitting the experimental data for each element separately give the most reliable values than those obtained by a global fit.

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A. Abdellatif et al. / Radiation Physics and Chemistry 81 (2012) 499–505

given by

3.0 Z= 23

Formula (1) Formula (2) Formula (3) Formula (4) Formula (5) Formula (6) Formula (7) Paul and Bolik (1993)

2.5

σ/σECPSSR

2.0 1.5

W min ¼ yK ,W max ¼ M ZK ,Q min ¼ M ZK Q max ¼ M 2 ZK 1 þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 W 1 1 ZK M

and

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2 W 1 ZK M

1.0

where M is the reduced mass.

0.5

ECPSSR THEORY

0.0

The ECPSSR theory of Brandt and Lapicki (1981), which incorporates energy loss (E) and Coulomb deflection (C) of the projectile as well as perturbed stationary state (PSS) and relativistic (R) effects into the plane-wave Born approximation (PWBA), describes the ionization cross section by the following expression: ! R ECPSSR PWBA mK ðxK =zK Þ ðA3Þ sK ¼ C K ðdq0K zK ÞsK , zK yK ðzK yK Þ2

Z= 29

2.0

1.5 σ/σECPSSR

2

1.0

0.5

0.0 2

1

3 E (MeV)

4

5

Fig. 7. A comparison between the different K X-ray production cross sections deduced in this work (using formulas (1)–(7)) as a function of the alpha energy for 23V and 29Cu. All these cross sections are normalized to their corresponding ECPSSR calculations.

CK represents the Coulomb deflection, d is the half distance of closest approach in a head on collision, q0K is the minimum momentum transfer in the collision, xK ¼2V1/V2KyK is the reduced velocity parameter; where V1 and V2K are the projectile and K-shell electron velocities, respectively, zK ¼ 1þ(2z1/yKZ2K)(gK hK) is a factor that accounts for the perturbed stationary state; more detail for the function CK(dq0KzK), gK and hK can be found in the paper of Liu and Cipolla (1996), mRK is the relativistic correction given as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !   u u ðZ 2K =137Þ2 ðZ 2K =137Þ2 R xK þ 0:4 mK ðA4Þ ¼ t1 þ 1:1 0:4 zK xK =zK xK =zK and 

Appendix

xRK ¼ mRK

Plane-wave born approximation (PWBA)

s0K yK

FK

ZK y2K

, yK

ðA1Þ

with

s0K ¼ 8a20 p

z21 Z 42K

! , yK ¼ 2n2

UK Z 22K

and

ZK ¼ 2

xK zK

1=2

xK

ðA5Þ

xRK is the relativistic reduced ion velocity.

In the PWBA development (Merzbacher and Lewis, 1958), the first-order Born approximation is used in scattering theory to describe the interaction between an incident charged particle and an atomic target. For a system composed of the projectile and the target atom, the PWBA K-shell ionization cross section in the center of mass system is given by the formula: !

sPWBA ¼ K



E1 M 1 Z 22K

:

In the previous expression, s0K denotes a constant cross section for a given combination projectile-target, ZK and yK are dimensionless variables representing the reduced ion energy and the reduced electron binding energy, respectively, M1, E1 and z1 are the mass, the energy and the atomic number of the projectile, Z2K ¼Z2  0.3 and Z2 are the effective atomic number and the atomic number of the target, respectively, a0 is the Bohr radius, UK is the target K-shell binding energy and n ¼1 for the K-shell. 2 F K ðZK =yK , yK Þ follows from the double integration of the squared electron transition form factor and is given by ! Z Z Q max ZK yK W max 2 dQ , y dW 9F WK ðQ Þ9 ðA2Þ ¼ FK K 2 ZK W min Q2 Q min yK The analytical expression of this latter is given by Benka and Kropf (1978). The exact limits of the above integration are

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