Radiation Physics and Chemistry 85 (2013) 89–94

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M x-ray production cross sections of heavy elements for low and high proton energy B. Deghfel a,b,n, A. Kahoul b,c, S. Heraiz a, N. Belouadah a, M. Nekkab a,b a

Physics Department, Faculty of Sciences, M’sila University, 28000 M’Sila, Algeria LESIMS laboratory, Physics Department, Faculty of Sciences, Ferhat Abbas University of Setif, 19000 Setif, Algeria c Department of Materials Science, Faculty of Sciences and Technology, Mohamed Bachir El Ibrahimi University, Bordj-Bou-Arreridj 34000, Algeria b

H I G H L I G H T S c c c c

M x-ray production cross sections have been calculated from ECPSSR theory. The experimental data have been updated (from 1980 till 2009) for heavy elements by proton impact. Low and high energy procedures are followed to deduce the semi-empirical cross sections. Our results deduced from low-high energy procedure are in good agreement with experiment.

a r t i c l e i n f o

abstract

Article history: Received 9 August 2012 Accepted 20 December 2012 Available online 31 December 2012

The semi-empirical cross sections have been deduced by individual fittings of the updated experimental data (from 1980 till 2009) and normalized to their corresponding theoretical values (ECPSSR model) for elements with 72r Z r 90 by protons energies varying from 0.1 to 4.0 MeV. Also, based on the individual fittings of the elements and the remarkable deviation of the experimental data from the ECPSSR values for low proton energy, we attempt to deduce another semi-empirical cross sections by introducing the low–high proton energy procedure which separates the fitting of the semi-empirical cross sections for low proton energy from those for high proton energy. Our results are presented for selected heavy elements. Finally, a comparison is made between our results and the experiment. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Semi-empirical cross sections M-shell x-ray production cross sections ECPSSR theory

1. Introduction The excitation of x-rays by charged particles is dominated by direct Coulomb ionization. For light-ion impact this process leads to creation of single-vacancy, which decays radiatively or nonradiatively. In the last three decades the inner-shell ionization and x-ray production by charged particles have been extensively studied, mainly because of their importance for the particle induced x-ray emission (PIXE) (Pajek et al., 1990a, 1990b, 1990c, 1999, 2006). Over the years, more accurate data became available for the M-shell ionization by light ions (Busch et al., 1973; Thornton et al., 1974; Ishii et al., 1975; Sarkar et al., 1981; Mehta et al., 1982, 1983; De Castro Faria et al., 1983; Gressett et al.,1989; Jesus and Ribeiro, 1989; Bien´kowski et al., 1990; Pajek et al., 1990a, 1990b, 1990c, 1999, 2006; Cipolla, 1995; Shokouhi et al., 1996; Braich et al., 1996, 1997; Jasko´"a et al., 2000).

n Corresponding author at: Physics Department, Faculty of Sciences, M’sila University, 28000 M’Sila, Algeria. Tel./fax: þ 213 035556453. E-mail address: [email protected] (B. Deghfel).

0969-806X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radphyschem.2012.12.020

Based on the most familiar theory, plane wave Born approximation (PWBA) (Merzbacher and Lewis, 1958), the inner-shell ionization process was further developed by Brandt and Lapicki (1979) by including some effects to give rise to what is known as the ECPSSR model (Brandt and Lapicki, 1979; Basbas et al., 1973a, 1973b). The binding effect (zs); caused by the change in the electron binding energy due to a presence of the charged particle in the vicinity of the nucleus, was included in the PWBA theory by using the united atom approximation (the binding energy in the atom with atomic number Z2 þ 1 for proton impact) (Cohen, 1989; Chen et al., 1983; Chen and Crasemann, 1989). Also, the Coulomb deflection correction combined with the binding and the energy loss effects   of the projectile were introduced as a multiplicative factor C EBs in terms of the exponential integral (Lapicki et al.,1980). Nevertheless, the theoretical predictions deviate significantly, especially for low energies, from the experimental data exceeding sometimes a level of quoted experimental uncertainties. In addition, the ratios of the experimental M-shell x-ray production cross sections normalized to their corresponding values deduced from ECPSSR model tend to unity for high energy of protons, whereas these ratios increase strongly for energies lower, approximately, than 1 MeV. This situation was the

B. Deghfel et al. / Radiation Physics and Chemistry 85 (2013) 89–94

main motivation of the present study to obtain more accurate semiempirical cross sections by dividing the interval of the incident proton energy into low and high subdivisions. Several authors tried to perform the fittings of the available experimental data with analytical functions for both K and L shell (Paul, 1984; Miyagawa et al., 1988; Sow et al., 1993; Orlic et al., 1994; Orlic, 1994; Reis and Jesus, 1996; Strivay and Weber, 2002; Kahoul and Nekab, 2005; Kahoul et al., 2008; Kahoul et al., 2011; Deghfel et al., 2010). More recently, additional results have been presented for M-shell by protons (Pajek et al.,1999; Deghfel et al., 2009) in a universal way by plotting the measured M-shell x-ray production cross as a function of the scaled velocity xM. Based on the ECPSSR calculations and the updated experimental data from 1980 till 2009 (Sera et al., 1980; Mehta et al., 1982,1983; Pajek et al.,1990a, 2006; Cipolla, 1995; RodriguezFerna´ndez et al., 2002; Goudarzi et al., 2006; Phinney et al., 2009), we attempt, in the present paper, to calculate the semi-empirical M-shell x-ray production cross sections for low (0.1–1.0 MeV) and high (1.0–4.0 MeV) energies of proton for elements with 72rZ r90 (about 555 experimental values).

Then, the semi-empirical M-shell x-ray production cross sections have been calculated by defining the normalized cross sections in the usual manner as S ¼ sexp =sECPSSR where sECPSSR refers to our theoretical M-shell x-ray production cross sections 2.5

W

74

2.0 σexp /σECPSSR

90

1.5 1.0 0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

ξM 3.5

Au

79

3.0

The total PWBA ionization cross section from s-shell can be written in a well-known manner by introducing the quantities ys ¼ 2n2 U s =Z 2s (reduced atomic electron binding energy) and Zs ¼ 2mE=M1 Z 2s (reduced ion energy), as (Liu and Cipolla, 1996) Z1 Z 2s

1

Zs

Z

W max

dW W min

Z

Q max Q min

2

9F W,s ðQ Þ9 , 2

ð1Þ

where Z1 is the projectile-charge number, n is the principal quantum number, m is the mass of electron, Us is the observed binding energy of the atomic electron, Zs is an effective nuclear charge for the s-shell seen by an electron in an inner-shell, ao is the Bohr radius of hydrogen, M1 and E are the mass and incident energy of the projectile, respectively and FW,s(Q)is the form factor defined by (Choi, 1973). The PWBA theory was further developed by (Brandt and Lapicki, 1981) by including the energy loss(E), the Coulomb deflection effects of the projectile (C), the perturbed stationary state (PSS) and the relativistic nature of the target’s inner-shell (R). These effects give rise to what is known the ECPSSR model (Cohen, 1989).

2.0 1.5

2.0

2.5

3.0

Bi

3.0

83

2.5 2.0 1.5 1.0 0.5

0.5

1.0

1.5

2.0

2.5

3.0

Th

90

2.5 σexp /σECPSSR

2.5

1.5

1.0

ξM

Hf Ta W Re Os Ir Pt Au Pb Bi Th

3.0

0.5

ξM

0.0

3.5

σexp /σECPSSR

1.5

0.0

4.0

1.0

2.0 1.5 1.0 0.5

0.5 0.0 0.0

2.0

0.5

dQ Q

2.5

1.0

σexp /σECPSSR

sPWBA ¼ 8pa20 s

!2

σexp /σECPSSR

2. ECPSSR theory and the semi-empirical cross sections

0.0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

E(MeV) Fig. 1. Ratios of the experimental Mx-ray production cross sections to the theoretical ones as a function of the energy proton for elements with atomic number 72 r Zr 90.

0.5

1.0

1.5

2.0

2.5

ξM Fig. 2. Ratios of experimental M x-ray production cross sections to the theoretical ones for selected elements as a function of the scaled velocity xM for whole, low (left small figures) and high (right small figures) energy range. The fits are also represented by full lines.

B. Deghfel et al. / Radiation Physics and Chemistry 85 (2013) 89–94

where, xs ¼ 2v1 =ys vs (s ¼M1,y, M5) is an argument appears through the function describing the energy-loss effect (Basbas et al., 1973a, 1973b) and used to distinguish the slow collision from the fast one, and v1 and vs are the projectile and the targetatom electron velocities, respectively. It is important to note that when the scaling based on the reduced velocity parameter xM is used, the data for elements with 72 rZr90 are generally mixed. So, instead of the energy of proton E we used its corresponding reduced velocity parameter xM. For each element separately (individual), the set of these ratios is fitted by a third degree polynomial for the whole energy range as

calculated within the ECPSSR model, which can be written in terms of the PWBA cross section in the well-known manner using the exact limits of integration (Liu and Cipolla, 1996) and the unperturbed nonrelativistic screened hydrogenic wave functions defined by Choi (1973) to evaluate the form factor (Johnson et al., 1979), and sexp refers to the updated experimental data, at our disposal(555 data points), of elements with atomic number 72rZ2 r90 collected from different sources (Sera et al., 1980; Mehta et al., 1982,1983; Pajek et al.,1990a,2006; Cipolla, 1995; Rodriguez-Ferna´ndez et al., 2002; Goudarzi et al., 2006; Phinney et al., 2009). The scatter of these experimental data (see Fig. 1) is partly due to the fact that data are taken from various references and consequently measured in different experimental conditions. Also, it is worth noting that the values of Coster–Kronig factors and fluorescence yields used to convert the theoretical ionization cross sections to the production ones have been reported from ¨ ut ¨ et al. (2002) to allow us this conversion for all elements in Sog this range (72rZ r90). As seen in Fig. 1, the ECPSSR theory predicts the experimental M-shell x-ray production cross sections very well above approximately E¼1 MeV, but for lower energies of proton a large discrepancy is observed (up to a factor of 4). So, it is necessary to adapt a semi-empirical method based on the available experimental data and the ECPSSR calculations for each interval separately; low (0.1–1.0 MeV) and high (1.0–4.0 MeV) energies of proton. Then, we plot S as a function of the scaled velocity parameter xM (Fig. 2) defined as the mean reduced velocity for the M-shell, which is given by Goudarzi et al. (2006) as 



xM ¼ xM1 þ xM2 þ2xM3 þ 2xM4 þ 3xM5 =9:



72Hf

26 47 56 18 31 32 50 91 51 85 68

73Ta 74W 75Re 76Os 77Ir 78Pt 79Au 82Pb 83Bi 90Th

i

ai ðxM Þ :

ð3Þ

For this purpose, we have chosen 74W, 79Au 83Bi and 90Th, which are being the most extensively studied (see Table 1), as a sample of calculation. The fitting results are shown in the same figure (Fig. 2) with a full line. Then, the semi-empirical M-shell x-ray production cross sections are deduced as

ssemp ¼ sECPSSR  S

ð4Þ

On the other hand, we attempt to calculate the semi-empirical M-shell x-ray production cross sections using the normalized experimental data for each element separately and the set of their ratios is fitted by a third degree polynomial for lower energies and by a linear polynomial for higher energies. The fitting results for 74W, 79Au, 83Bi and 90Th are shown in Fig. 2 for low (left small figures) and high (right small figures) energies.  Also,  the total deviation of the experimental cross sections sexp from their corresponding fitted values (ss  emp) is expressed in terms of the root-mean-square error (erms) calculated using the expression " #1=2 X 1 sexp ssemp 2 erms ¼ ð5Þ N ssemp

Table 1 Number of experimental M x-ray production cross sections by proton bombardment of elements with 72r Z r 90. No. of data

3 X i¼0

ð2Þ

Element

91

where N is the number of data.

3. Results and discussion The aim of this contribution is to investigate the reliable semiempirical method for calculating the M x-ray production cross sections by proton impact. Moreover, it is interesting to note that we have reported only the experimental data in the energy range

Table 2 Fitting coefficients for the calculation of semi-empirical M-shell x-ray production cross sections by using the individual fittings with high and low energy procedure (H/L). The associated root-mean-square errors (erms) are also included. With H/L procedure Low energy (0.1–1.0 MeV)

High energy(1.0–4.0 MeV)

Element

a3

a2

a1

a0

erms (%)

a1

a0

erms (%)

72Hf

 0.8535  6 .4990  4.9234 3.7001  7.5523  7.4661  8.5182  9.8146  5.1283  8.9229  6.4990

4.8227 17.9401 16.1825  14.6363 24.5640 23.9889 27.0862 30.9193 15.5881 27.1698 17.9401

 8.8247  16.5927 17.3185 18.8578  25.9552  25.1035  28.0687  32.0580  16.0183  27.5283  16.5927

5.9003 6.1507 6.8724  7.1499 9.7280 9.3641 10.2784 11.8901 6.4836 10.2149 6.1507

04.80 13.81 16.37 00.40 15.08 13.27 14.06 13.32 06.53 07.12 09.94

0.0812  0.0079 0.0466 0.0348 0.0375 0.1290 0.1308 0.1335 0.0186 0.0600 0.0166

0.4832 0.7307 0.5823 0.6557 0.6670 0.4780 0.4431 0.5151 0.6883 0.6682 0.9321

10.76 09.40 04.15 02.70 02.01 02.33 07.25 07.36 07.00 08.36 06.69

73Ta 74W 75Re 76Os 77Ir 78Pt 79Au 82Pb 83Bi 90Th

92

B. Deghfel et al. / Radiation Physics and Chemistry 85 (2013) 89–94

0.1–4 MeV of proton. Therefore, the deduced equations and their corresponding coefficients are only valid within this range. Tables 2 and 3 list the fitting coefficients for the calculation of semi-empirical M-shell x-ray production cross sections and their associated root-mean-square errors (erms) by using the individual fittings with and without high and low energy procedure (H/L), respectively. Also, our results of M-shell x-ray production cross sections have been presented for some selected elements (74W, 79Au 83Bi and 90Th) as an illustration by using the H/L procedure, the whole energy range and the ECPSSR model (Table 4). By considering that the accuracy of the M x-ray production cross sections is expressed by the root-mean-square (erms), we can point out that the cross sections described by the high energy procedure give, generally, a better representation of the experimental data (up to 10.76% for 72Hf) whereas a large discrepancy is observed for low energy procedure ( up to 16.37% for 74W). This is partly due to the fact that the experimental data are significantly scattered for low energy of proton (see Figs. 1 and 2). Also, we attempt to compare our results of the semi-empirical Mshell x-ray production cross sections with other works for selected elements, namely 74W, 79Au, 83Bi and 90Th. So, ratios to the ECPSSR calculations of the experimental M-shell x-ray production cross sections (Pajek et al., 1990a,2006) and those deduced from the whole energy range, low and high energy procedures, are presented as a

Table 3 Fitting coefficients for the calculation of semi-empirical M-shell x-ray production cross sections by using the individual fittings without high and low energy procedure (H/L). The associated root-mean-square errors (erms) are also included. Without H/L procedure Element

a3

a2

a1

a0

erms (%)

72Hf

 0.3247  0.3417  0.1692  0.0760  0.3945  0.4161  0.3852  0.5991  0.0068  0.8000  0.9879

2.2292 1.9699 1.0893 0.5499 2.3677 2.5001 2.3527 3.5364 0.4072 4.2628 4.3949

 4.8796  3.5510 2.2721  1.2503  4.4382  4.6486 4.4970  6.5719  1.3592  7.2251  6.1864

4.0495 2.6707 2.2242 1.6187 3.2971 3.3684 3.3767 4.5985 1.8734 4.6572 3.6943

10.40 13.21 14.54 02.70 15.15 14.01 13.64 14.74 07.16 12.28 09.74

73Ta 74W 75Re 76Os 77Ir 78Pt 79Au 82Pb 83Bi Th 90

Table 4 M-shell x-ray production cross sections for 7.370  101.

74W, 79Au 83Bi

E (MeV)

Semi-emp from H/L procedure

0.2 0.4 0.6 0.8 1.0 2.0 3.0 4.0

7.370E þ01 2.699E þ 02 5.674E þ 02 8.029E þ02 6.680E þ02 1.784E þ 03 2.314E þ 03 2.583E þ 03

0.2 0.4 0.6 0.8 1.0 2.0 3.0 4.0

3.127E þ 01 1.311E þ 02 2.621E þ 02 4.109E þ02 4.853E þ 02 1.160E þ03 1.689E þ 03 2.035E þ03

and

90Th

function of proton energy in Fig. 3. The examination of the obtained results needs some comments: – Our results of the semi-empirical cross sections obtained from the high energy procedure are generally close to the experimental data (deviation is up to 7.6% for 74W; up to 10.3% for 79Au; up to 13.3% for 83Bi; up to 14.5% for 90Th) whereas a large discrepancy is observed between our results deduced from the whole energy range procedure and the experimental data (up to 32.8% for 74W; up to 32.4% for 79Au; up to 32.2% for 83Bi; up to 15.5% for 90Th). – In addition, the derived semi-empirical cross sections deviate significantly from each other for low energy of proton (up to 25.3% for 74W; up to 29.7% for 79Au; up to 23.1% for 83Bi; up to 12.0% for 90Th) whereas a small deviations have been obtained between them for high energy of proton (up to 6.7% for 74W; up to 16.2% for 79Au; up to 21.2% for 83Bi; up to 12.7% for 90Th in the worst case). – Also, the obtained semi-empirical cross sections underestimate the ECPSSR theory and tend generally towards it when going from 74W to 90Th for high energy of proton(ratio to ECPSSR values varies from 0.95 to 0.97for 90Th). – The deviation from the experimental data of the obtained results from the whole energy range procedure (up to 49.1% for 74W; up to 57.1% for 79Au; up to 23.1% for 83Bi; up to 27.7% for 90Th) is significant when compared to that obtained for the low energy procedure (up to 32.6% for 74W; up to 38.9% for 79Au; up to 19.3% for 83Bi; up to 21.9% for 90Th). – For low energy, the semi-empirical cross sections overestimate the ECPSSR theory; their corresponding ratios increase strongly by increasing the energy of proton (up to a factor of 3). 4. Conclusion The semi-empirical cross sections deduced by individual fitting of the updated experimental data and normalized to their corresponding theoretical values of the ECPSSR model have been calculated for elements with 72 r Z r 90 by proton in the energy range 0.1–4.0 MeV. Also, based on the remarkable deviation of the experimental data from the values deduced from the ECPSSR model for low proton energy, we attempt to deduce another semi-empirical cross sections by introducing the procedure low-high proton energy. The

deduced from H/L procedure, whole energy range and ECPSSR model. The value 7.370Eþ 01 denotes

Semi-emp from whole energy range

Theoritical (ECPSSR)

Semi-emp from H/L procedure

Semi-emp from whole energy range

Theoritical (ECPSSR)

7.489E þ 01 3.326E þ 02 6.381E þ 02 9.688E þ 02 1.301E þ03 2.602E þ03 3.264E þ 03 3.546E þ 03

5.359Eþ 01 1.841Eþ 02 3.881Eþ 02 5.932Eþ 02 5.648Eþ 02 1.480Eþ 03 2.130Eþ 03 2.556Eþ 03

79Au 6.063E þ01 2.233E þ02 3.441E þ02 4.572E þ02 5.856E þ02 1.522E þ03 2.319E þ03 2.203E þ03

3.537E þ 01 1.978E þ 02 3.994E þ 02 6.221E þ 02 8.560E þ 02 1.897E þ 03 2.536E þ 03 2.874E þ 03

1.862E þ 01 1.311E þ 02 2.809E þ02 4.466E þ 02 6.242E þ 02 1.494E þ 03 2.108E þ03 2.476E þ 03

8.498Eþ 00 6.670Eþ 01 1.528Eþ 02 2.519Eþ 02 3.441Eþ 02 9.112Eþ 02 1.412Eþ 03 1.777Eþ 03

90Th 8.721E þ00 7.174E þ01 1.507E þ02 2.316E þ02 3.209E þ02 9.382E þ02 1.509E þ03 1.550E þ03

5.460E þ 00 6.052E þ 01 1.502E þ 02 2.513E þ 02 3.596E þ 02 9.513E þ 02 1.466E þ 03 1.834E þ 03

74W

8.285E þ01 2.912E þ02 4.895E þ02 6.906Eþ 02 8.942E þ02 1.815E þ03 2.280Eþ 03 2.105Eþ 03 83Bi

3.284E þ01 1.540Eþ 02 2.534E þ02 3.444E þ02 4.463E þ02 1.198E þ03 1.842E þ03 1.680Eþ 03

B. Deghfel et al. / Radiation Physics and Chemistry 85 (2013) 89–94 3.0 2.5

2.0

σexp /σECPSSR : (Pajek et al..1990a.2006) σref /σECPSSR : Low energy procedure

(a): 74W

1.6

σ/σECPSSR

σ/σECPSSR

1.5 1.0

σexp /σECPSSR : (Pajek et al..1990a.2006) (b): W 74 σref /σECPSSR :High energy procedure σref /σECPSSR : Whole energy range procedure

1.8

σref /σECPSSR : Whole energy range procedure 2.0

1.4 1.2 1.0 0.8 0.6 0.4

0.5 0.0 0.0

0.2 0.2

0.4

0.6

0.8

0.0 1.0

1.0

1.5

2.0

E(MeV)

1.5

σ/σECPSSR

σ/σECPSSR

1.6

σref /σECPSSR : Whole energy range procedure

2.0

1.0

σexp /σECPSSR : (Pajek et al..1990a.2006) σref /σECPSSR :High energy procedure σref /σECPSSR : Whole energy range procedure

1.4 1.2 1.0 0.8 0.6 0.4

0.5

0.2 0.2

0.4

0.6

0.8

0.0 1.0

1.0

1.5

2.0

E(MeV)

σ/σECPSSR

3.5

4.0

σref /σECPSSR : Whole energy range procedure

2.0 1.5

(b): Bi

1.8

83

1.6

σ/σECPSSR

σexp /σECPSSR : (Pajek et al..1990a.2006) σref /σECPSSR : Low energy procedure

2.5

σexp /σECPSSR : (Pajek et al..1990a.2006) σref /σECPSSR :High energy procedure σref /σECPSSR : Whole energy range procedure

1.4 1.2 1.0 0.8 0.6

1.0

0.4

0.5

0.2 0.2

0.4

0.6

0.8

0.0

1.0

1.0

1.5

2.0

3.0

3.5

4.0

σref /σECPSSR : Whole energy range procedure

1.5 1.0

1.6

σ/σECPSSR

σexp /σECPSSR : (Pajek et al..1990a.2006) σref /σECPSSR : Low energy procedure

(b):90Th

1.8

90

2.0

3.0

2.0

(a): Th

2.5

2.5

E(MeV)

E(MeV)

σ/σECPSSR

3.0

2.0

(a): 83 Bi

3.0

1.4 1.2

σexp /σECPSSR : (Pajek et al..1990a.2006) σref /σECPSSR :High energy procedure σref /σECPSSR : Whole energy range procedure

1.0 0.8 0.6 0.4

0.5 0.0 0.0

2.5

E(MeV)

3.5

0.0 0.0

3.5

(b):79Au

1.8

σexp /σECPSSR : (Pajek et al..1990a.2006) σref /σECPSSR : Low energy procedure

2.5

3.0

2.0

(a):79Au

3.0

2.5

E(MeV)

3.5

0.0 0.0

93

0.2 0.2

0.4

0.6

0.8

1.0

E(MeV)

0.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

E(MeV)

Fig. 3. Experimental (Exp.) M x-ray production cross sections and those deduced from the whole energy range and the low (a) and high (b) energy procedures by using the individual fitting for selected elements as a function of proton energy. All these cross sections are normalized to their corresponding ECPSSR calculations from this work.

deduced semi-empirical cross sections have been compared with the experimental data for selected elements. Generally, our results deduced either from low or high energy procedures

are in good agreement with experiment over the whole energy range but in less agreement with ECPSSR model especially for lower proton energy.

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