Ytterbium to thorium semi-empirical average M-shell fluorescence yields

Radiation Physics and Chemistry 112 (2015) 71–75

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Ytterbium to thorium semi-empirical average M-shell fluorescence yields F. Khalfallah a,b, B. Deghfel c,d, A. Kahoul a,b,n, V. Aylikci e, N. Küp Aylikci f, M. Nekkab c,d a

Department of Materials Science, Faculty of Sciences and Technology, Mohamed El Bachir El Ibrahimi University, Bordj-Bou-Arreridj 34030, Algeria LPMRN laboratory, Department of Materials Science, Faculty of Sciences and Technology, Mohamed El Bachir El Ibrahimi University, Bordj-Bou-Arreridj 34030, Algeria c Laboratory of materials physics and their applications, Physics Department, Faculty of Sciences, University of Mohamed Boudiaf, M’sila 28000, Algeria d Physics Department, Faculty of Sciences, University of Mohamed Boudiaf, M’sila 28000, Algeria e Department of Metallurgical and Materials Engineering, Faculty of Technology, Mustafa Kemal University, Hatay 31040, Turkey f Department of Energy Systems Engineering, Faculty of Technology, Mustafa Kemal University, Hatay 31040, Turkey b

H I G H L I G H T S







The average M-shell fluorescence yields covering the period from 1955 to 2005 are used. We used the weighted and unweighted mean values to deduce the semi-empirical average M-shell fluorescence yield. The obtained results are compared with others works. Reasonable agreement was typically obtained between our result and other works.

art ic l e i nf o

a b s t r a c t

Article history: Received 13 December 2014 Accepted 10 March 2015 Available online 11 March 2015

By using the weighted and unweighted mean values of the experimental average M-shell fluorescence yield reported in the literature covering the period from 1955 to 2005 and the theoretical values based on the non-relativistic Hartree–Fock–Slater (HFS) calculations of McGuire (1972), we produce good estimations for the semi-empirical values of elements in the range of 70r Zr90. These values are obtained by the interpolation of the normalized weighted and unweighted mean values to their corresponding theoretical ones. The results have been compared to other theoretical, empirical and experimental data reported in the literature. Reasonable agreement has been obtained between our result and other works. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Average M-shell fluorescence yield Semi-empirical interpolation Weighted- and unweighted-mean values

1. Introduction The analytical methods based on X-ray fluorescence, measured and calculated X-ray production cross sections, fluorescence yields, vacancy transfer probabilities and intensity ratios of different elements are advantageous for practical applications in a variety of fields including atomic physics, chemistry (X-ray fluorescence surface chemical analysis) and medical research. It is important, for practical purpose, to have accurate average M-shell fluorescence yield (ω¯ M ). Several attempts were made, both experimental and theoretical, to estimate the average M-shell n Corresponding author at: Department of Materials Science, Faculty of Sciences and Technology, Mohamed El Bachir El Ibrahimi University, Bordj-Bou-Arreridj 34030, Algeria. Fax þ 213 35666599. E-mail address: [email protected] (A. Kahoul).

http://dx.doi.org/10.1016/j.radphyschem.2015.03.021 0969-806X/& 2015 Elsevier Ltd. All rights reserved.

fluorescence yields for a wide range of elements. McGuire (1972) has calculated the Auger, Coster–Kronig, super Coster–Kronig and radiative transition rates using the non-relativistic Hartree–Fock– Slater (HFS) wave function with Herman–Skillman potential for elements in the range 20rZ r90. Sampaio et al. (2013) produced a relativistic calculation of M-shell transition rates and fluorescence yields of three elements with similar electronic configurations: Zn, Cd and Hg using the multi-configuration Dirac–Fock (MCDF) code of Desclaux and Indelicato (Desclaux, 1975; Indelicato and Desclaux, 1990; Indelicato et al. 1987). Recently Kaur and Mittal (2014) used the MFCKYLD code to calculate the M subshell fluorescence and Coster–Kronig yields for elements with Z in the range 57 rZr90 and 67rZr 90 on the basis of the non-relativistic Hartree–Fock–Slater (HFS) values of McGuire (1972) and the relativistic Dirac–Hartree–Slater (DHS) values of Chen et al. (1980), (1983) respectively. For the measured values, the authors

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adopted different methods; these methods vary depending on the experimental conditions such as ionization process, the target material and the detectors (Jaffe,1955; Konstantinov, and Sazonova 1968; Hribar et al., 1982; Shatendra et al., 1984; Garg et al. 1991; Mann et al., 1990; Puri et al., 1993; Rao et al., 1995, 1996; Ertuğrul et al., 1996; Durak and Özdemir, 2001; Şahin et al., 2004; Apaydin et al., 2005). Concerning the empirical formulae, in 1955 Burhop (1955) interpolated the available experimental data using the equation ω¯ M = 1.7 × 10−9(Z − σ)4 , where σ = 13 is the screening constant. Hubbell (1989) have proposed the relation ω¯ M = 1.29 × 10−9(Z − 13)4 to determine the average M-shell fluorescence yields in the range 19 rZr 100. Hubbell et al. (1994) fitted the selected measured average M-shell fluorescence yields published in the period 1978–1993 by least-squares using polynomials in Z of the form ∑n anZ n to determine the ω¯ M for elements in the atomic range of 19 rZ r100. Öz et al. (1999) used the M-subshell fluorescence yields ωMi and Coster–Kronig yield fMij and Super Coster–Kronig transition probabilities Sij given by McGuire (1972) to calculate the effective subshell fluorescence yields νi , then the average M-shell fluorescence yields were calculated using least-squares polynomial in Z. Söǧ üt et al. (2002) determined a fitted values of M- subshell fluorescence yields and Coster–Kronig transitions for elements with 20r Zr90 using the McGuire's values (McGuire, 1972). More recently, The Mi (i = 1 − 5)subshell fluorescence and Coster–Kronig yields has been generated by interpolation for elements with 67rZ r92 from the Dirac–Hartree–Slater (DHS) model by Chauhan and Puri (2008). In 2014 our research group Kahoul et al. (2014) have interpolated the weighted and unweighted mean values of the experimental data by using the analytical function (ω¯ M/(1 − ω¯ M))1/4 as function of the atomic number (Z) to deduce the empirical average M-shell fluorescence yield in the atomic range of 70rZr 92. In the same paper we have also employed the famous formula ω¯ M = A × (Z − 13)4 to generalize the average M-shell 1.50

70 ≤ Z ≤ 90 1.25

SW

1.00

0.75

0.50

1.50

70 ≤ Z ≤ 90

fluorescence yield for elements with 19 rZr100. At the present time, no semi-empirical formulae to calculate the average M-shell fluorescence yield are available. We noted that the semi-empirical calculations are deduced by fitting the available experimental data normalized to their corresponding theoretical values, but the experimental data are directly fitted by an analytical formula to deduce the empirical ones. For these reasons, we present in this paper a semi-empirical formulae based on the full experimental data published in the period 1955–2005 and on the theoretical values of McGuire (1972) using the non-relativistic Hartree–Fock– Slater (HFS) approach to determine the semi-empirical average M-shell fluorescence yield for elements in the range 70rZr 90.

2. Calculation procedure of semi-empirical ω¯ M for elements in the range 70rZ r90 As mentioned above, to calculate the semi-empirical average M-shell fluorescence yields we used the weighted and unweighted mean values. This is motivated by the fact that, for the same element, there are various published experimental dataω¯ M−exp obtained from different experimental methods and conditions. In order to get more reliable results we compared these values to the theoretical ones. The fittings for the derivation of the semi-empirical average M-shell fluorescence yields for elements with 70rZr90 was performed using analytical functions based on the available experimental data (Jaffe,1955; Konstantinov and Sazonova 1968; Hribar et al., 1982; Shatendra et al., 1984; Garg et al. 1991; Mann et al., 1990; Puri et al., 1993; Rao et al., 1995, 1996; Ertuğrul et al., 1996; Durak and Özdemir, 2001; Şahin et al., 2004; Apaydin et al., 2005), see also Kahoul et al. (2014). The unweighted N mean values ((ω¯ M)UNW = (1/N) ∑i = 1 (ω¯ M)i , where (ω¯ M)i is the experimental average M-shell fluorescence yields and N is the number of experimental data) and the weighted mean values ( −1 N N ⎡ (ω¯ M)i ⎤ where Δ(ω¯ M)i represents (ω¯ M) W = ∑i = 1 (Δ(ω¯ M)i)−2 ∑i = 1 ⎢ ⎣ (Δ(ω¯ M)i)2 ⎥⎦ the uncertainty of the ith experimental value) for the experimental data have been compared to the theoretical values of McGuire (1972) for elements in the range 70rZr 90 separately by plotting the ratios SW = (ω¯ M)exp − W /(ω¯ M)theo − McGuire and SUNW = (ω¯ M)exp − UNW /(ω¯ M)theo − McGuire ; where (ω¯ M)theo − McGuire refers to the theoretical M-shell fluorescence yields calculated by McGuire (1972) using the non-relativistic Hartree–Fock–Slater wave function with Herman–Skillman potential and (ω¯ M)exp −W , (ω¯ M)exp − UNW are the weighted and unweighted experimental mean values. Then, SW and SUNW has been plotted as a function of the atomic number Z for elements in the range of 70rZr 90 separately. Fig. 1 (a) and (b) shows all points of SW and SUNW vs. Z. Each set of these data has been fitted by a third degree polynomial (noted S¯ W and 3 3 ), as: S¯ = ∑ S¯ =∑ b Z n . The fitting result is a Z n and S¯

(

UNW

W

n=0

n

UNW

n=0

n

also shown in Fig. 1 with a full line. Finally, the semi-empirical average M-shell fluorescence yields have been deduced as follows:

1.25

SUNW

)

(ω¯ M)s−emp −W = S¯ W × (ω¯ M)theo − McGuire

1.00

(1)

and 0.75

Table 1 The fitting coefficients S¯W and S¯UNW to deduce the semi-empirical average M-shell fluorescence yields for elements with 70 rZ r90 using the Eqs. (1) and (2).

0.50 68

72

76

80

84

88

92

S¯W

S¯UNW

Z Fig. 1. The normalized average Weighted and Unweighted M-shell fluorescence yields: (a) for S W and (b) for SUNW as a function of atomic number Z. Each set of these data is fitted (the curve) by a third degree polynomial (noted S¯W and S¯UNW ). The error bars are also included.

a0 a1 a2 a3

70.191347 63.0644  2.66405 7 2.38313 0.033997 0.02991  1.4381  10-4 7 1.24699  10  4

b0 b1 b2 b3

67.7169 771.21663  2.569337 2.69195 0.032797 0.0338  1.38686  10  4 7 1.40938  10  4

F. Khalfallah et al. / Radiation Physics and Chemistry 112 (2015) 71–75

(ω¯ M)s−emp − UNW = S¯UNW × (ω¯ M)theo − McGuire

(2)

The fitting coefficients of S¯ W and S¯UNW are listed in Table 1. It must be emphasized that the fitting formulae (1) and (2) and the associated coefficients are only valid in the region of atomic number 70rZ r90 and the extension out of this region might take an unpredictable course. It is noted that the scatter of the data in Fig. 1 (a) and (b) is partly due to the fact that the data has been taken from various sources and are measured in different experimental conditions. The total deviation of the experimental Weighted and Unweighted means values ((ω¯ M)exp −W and (ω¯ M)exp − UNW ) from the corresponding calculated values ((ω¯ M)s−emp −W , (ω¯ M)s−emp − UNW ) is expressed in terms of the root-mean-square error (εrms) calculated for each formulae (1) and (2) using the expression 1

ε RMS

2⎤ 2 ⎡ N 1 ⎛ (ω¯ M) j (exp) − (ω¯ M) j (cal) ⎞ ⎥ ⎢ ⎜ ⎟ = ∑ ⎟⎥ ⎢ j=1 N ⎜ (ω¯ M) j (cal) ⎝ ⎠⎦ ⎣

(3)

where N is the number of the experimental data.

3. Results and discussion Table 2 shows the values of εrms (%) for the calculation of semiempirical average M-shell fluorescence yields according to the two sets of methods (according to formulae (1) and (2)). The calculated semi-empirical average M-shell fluorescence yields in the atomic Table 2 Root-mean-square error (εrms (%)) for the calculation of semi-empirical average M-shell fluorescence yields. Formulae

εrms (%)

(ω¯ M)s−emp −W

7.68

(ω¯ M)s−emp − UNW

8.28

73

range 70rZr90 according to the two approaches mentioned above have been tabulated in Table 3. In the same table we present the theoretical calculation of McGuire (1972) and Chen et al. (1980, 1983), the fitted values of Kahoul et al. (2014), Öz et al. (1999) and Hubbell et al. (1994) and the experimental results of Apaydin et al. ( 2005) and Puri et al. (1993). We also added in Table 2 the fitted values from Kaur and Mittal (2014) using the nonrelativistic HFS values of McGuire (1972) and the relativistic DHS values of Chen et al. (1980, 1983) respectively. For the fitted average M-shell fluorescence yields of Kaur and Mittal (2014) we used the M-subshell fluorescence yields ωMi(i = 1 − 5) and Coster– Kronig yield fMij (i = 1 − 4, j = 2 − 5) deduced from the MFCKYLD code and the M-shell photoionization cross section σMi(i = 1 − 5) at 8 keV from Scofield 1973. In order to present the deviation of our semi-empirical calculation, Fig. 2 (a) and (b) shows the ratio of the theoretical calculation (McGuire,1972; Chen et al., 1980, 1983), the fitted values (Kahoul et al., 2014; Öz et al.,1999; Hubbell et al.,1994; Kaur and Mittal, 2014) and the experimental results (Apaydin et al., 2005; Puri et al., 1993) to the semi-empirical calculation deduced from formulae (1) and (2) respectively against the atomic number Z. One can observe that the present semiempirical average M-shell fluorescence yields (using the weighted and the unweighted values) agree generally with the theoretical, fitted and experimental values for elements in the full range 70rZr90. Also, the semi-empirical average M-shell fluorescence yields (using the weighted values) agree well with the theoretical values within 0.37–6.96% for the values of McGuire (1972) and 0.37–6.21% for Chen et al. (1980, 1983). These values agree within 0.34–9.65% for the fitted values of Öz et al. (1999). It can be stated that these values are 4.44–14.36% higher than the fitted values given by Hubbell et al. (1994) and 0–6.21% under our fitted values published recently (Kahoul et al., 2014). These values exhibit a good agreement with Kaur and Mittal (2014) using the two sets of calculation (HFS and DHS) in the considered range of elements and the agreement is within 0–8.86% from the values of McGuire and within 0.34–9.65% from those of Chen, respectively. Our calculations agree in 0.87–5.85% with the measured values of Apaydin et al. (2005) (except that for the 71Lu, we observed a deviation of

Table 3 Present semi-empirical average M shell fluorescence yields ω¯ M derived from Eqs. (1) and (2) compared to theoretical, fitted and experimental values of other authors. Z

Z ¼70,Yb Z ¼71,Lu Z ¼72,Hf Z ¼73,Ta Z ¼74,W Z ¼75,Re Z ¼76,Os Z ¼77,Ir Z ¼78,Pt Z ¼79,Au Z ¼80,Hg Z ¼81,Tl Z ¼82,Pb Z ¼83,Bi Z ¼84,Po Z ¼85,At Z ¼86,Rn Z ¼87,Fr Z ¼88,Ra Z ¼89,Ac Z ¼90,Th

This work

Theoretical

(ω¯ M)s−emp −W (ω¯ M)s−emp − UNW

McGuire (1972)

Chen et al. (1980,1983)

Kahoul et al. (2014) (ω¯ M)emp −W (ω¯ M)emp − UNW

Öz et al. Hubbell (1999) et al. (1994)

Kaur and Mittal (2014) McGuire's Chen's values values

Apaydin et al. (2005)

Puri et al. (1993)

0.0145 0.0158 0.0174 0.0195 0.0199 0.0210 0.0225 0.0229 0.0243 0.0269 0.0290 0.0310 0.0328 0.0344 0.0355 0.0368 0.0383 0.0409 0.0439 0.0473 0.0509

0.0154 0.0169 0.0186 0.0208 0.0211 0.0221 0.0234 0.0236 0.0247 0.0270 0.0288 0.0305 0.0320 0.0334 0.0344 0.0357 0.0374 0.0404 0.0442 0.0488 0.0543

– 0.0172 0.0183 0.0193 – – – 0.0240 0.0254 0.0268 – 0.0298 0.0313 0.0329 – – – – – – 0.0451

0.0154 0.0165 0.0177 0.0189 0.0201 0.0214 0.0228 0.0243 0.0258 0.0273 0.0290 0.0307 0.0324 0.0343 0.0362 0.0382 0.0402 0.0424 0.0446 0.0469 0.0493

0.0159 0.0173 0.0186 0.0201 0.0213 0.0226 0.0239 0.0251 0.0264 0.0276 0.0289 0.0303 0.0317 0.0332 0.0348 0.0366 0.0385 0.0407 0.0431 0.0458 0.0486

0.0156 0.0172 0.0188 0.0204 0.0218 0.0229 0.0235 0.0232 0.0249 0.0270 0.0290 0.0307 0.0322 0.0333 0.0344 0.0357 0.0372 0.0394 0.0426 0.0470 0.0541

0.0140 0.0192 – – 0.0188 0.0200 – – – 0.0266 0.0269 0.0305 0.0312 0.0341 – – – – – – –

– 0.0154 0.0176 0.0190 – – – 0.0276 0.0285 0.0300 – – 0.0334 0.0356 – – – – – – 0.0512

0.0142 0.0156 0.0172 0.0193 0.0197 0.0208 0.0223 0.0227 0.0241 0.0266 0.0286 0.0305 0.0322 0.0337 0.0348 0.0360 0.0374 0.0400 0.0430 0.0464 0.0501

Fitted

Experimental

0.0154 0.0165 0.0176 0.0188 0.0201 0.0214 0.0228 0.0243 0.0258 0.0274 0.0290 0.0307 0.0325 0.0344 0.0363 0.0383 0.0404 0.0426 0.0448 0.0471 0.0496

0.0136 0.0146 0.0156 0.0167 0.0179 0.0191 0.0203 0.0216 0.0230 0.0245 0.0260 0.0275 0.0292 0.0310 0.0328 0.0347 0.0366 0.0387 0.0408 0.0430 0.0453

0.0158 0.0162 0.0175 0.0189 0.0203 0.0217 0.0230 0.0242 0.0255 0.0268 0.0283 0.0299 0.0315 0.0331 0.0347 0.0364 0.0381 0.0399 0.0416 0.0432 0.0447

74

F. Khalfallah et al. / Radiation Physics and Chemistry 112 (2015) 71–75

of 20.08%), 1.28–12.78% for Puri et al. (1993) (except that for the two elements 77Ir and 78Pt, we observed a deviation of 21.58% and 18.26% respectively). In Fig. 3, the semi-empirical values using the weighted values are compared with the semi-empirical values using the unweighted ones by the plot of (ω¯M)s−emp − UNW /(ω¯ M)s−emp − UNW against the atomic number Z. As we can see from Fig. 3, the semi-empirical calculation using the unweighted values are close to the calculation using the weighted values for the elements in the range 70rZr90, we observe also that the data deduced from the weighted values are higher with a few percents than those deduced using the unweighted values (about 0.87–2.34%). Finally, by considering that the accuracy of the calculated average M-shell fluorescence yields is expressed by the root-mean-square error (εrms ), it can be pointed out that the first formula (using the weighted values) with εrms = 7.68% gives a better representation of the experimental data (Table 2) than the other ones (using the unweighted values).

1.50

ωM /ωM-s-emp-W

1.25 1.00 0.75 (McGuire, 1972) (Chen et al., 1980, 1983) (Öz et al., 1999) (Apaydin et al., 2005) (Hubbell et al., 1994) (Puri et al., 1993) (Kahoul et al., 2014) (emp-weighted) (Kahoul et al., 2014) (emp-unweighted) (Kaur and Mittal, 2014) (using McGuire's values) (Kaur and Mittal, 2014) (using Chen's values)

0.50 0.25 1.50

ωM /ωM-s-emp-UNW

1.25 1.00 0.75

(McGuire, 1972) (Chen et al., 1980, 1983) (Öz et al., 1999) (Apaydin et al., 2005) (Hubbell et al., 1994) (Puri et al., 1993) (Kahoul et al., 2014) (emp-weighted) (Kahoul et al., 2014) (emp-unweighted) (Kaur and Mittal, 2014) (using McGuire's values) (Kaur and Mittal, 2014) (using Chen's values)

0.50 0.25 68

72

76

80

84

88

92

Z Fig. 2. The normalized theoretical calculation of McGuire (1972) and Chen et al. (1980, 1983), the fitted values of Kahoul et al. (2014), Öz et al. (1999), Hubbell et al. (1994) and Kaur and Mittal (2014) and the experimental results of Apaydin et al. (2005) and Puri et al. (1993) to the present semi-empirical average M-shell fluorescence yields from this work as a function of atomic number Z.

ωs-emp-UNW /ωs-emp-W

1.10

A new set of average M-shell fluorescence yields were produced using simple methods for elements in the atomic region 70rZr90. The deduced semi-empirical fluorescence yields according to the different procedures were generally much closer to each other for the whole considered range of atomic number. In addition to the available experimental and theoretical average M-shell fluorescence yields, the present values can be proposed to workers in the field of atomic inner-shell ionization processes and in other fields as well.

References

1.05

1.00

0.95

0.90 68

4. Conclusion

72

76

80

84

88

92

Z Fig. 3. The normalized semi-empirical average M-shell fluorescence yields (using the unweighted values) to the semi-empirical average M-shell fluorescence yields (using the weighted values) from this work as a function of atomic number Z.

21.51%) and about 0.59–11.52% for Puri et al. (1993) value's (except that for the two elements 77Ir and 78Pt, we observed a deviation of 20.52% and 17.28% respectively). The same remarks can be stated in the case of the semi-empirical average M-shell fluorescence yields (using the unweighted values), the deviation between the present empirical results and the other values is about: 0–8.45% for the values of McGuire (1972), 0–10.26% for Chen et al. (1980, 1983), 0–11.97% for the fitted values of Öz et al. (1999), 2.13–9.83% for the fitted values of Hubbell et al. (1994), 062–8.45% and 0.65– 8.45% for the empirical values of Kahoul et al. (2014) (using the weighted values and unweighted values respectively), for Kaur and Mittal (2014) is within 0–10.66% from the values of McGuire and within 0–11.97% from those of Chen, respectively, 0–5.94% for Apaydin et al. (2005) (except for the 71Lu, we observed a deviation

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