semiempirical formulas for calculation of l subshell

International Journal of PIXE, Vol. 4, No. 4 (1994) 217-230 @World Scientific Publishing Company

SEMIEMPIRICAL FORMULAS FOR CALCULATION OF L SUBSHELL IONIZATION CROSS SECTIONS

I. ORLIC , C.H. SOW AND S.M. TANG Nat i onal Univ ersity of Singap ore, Phys ics Departm ent 10 Ke n t R i dge Crescent, Singapore 0511

Received 18 March 1994 Revised 11 October 1994 ABSTRACT Presented are n ew parameters for the calculation of L subshell ionization cross sections for proton impact using a semiempirical expression. A similar paper was published by our group in early 1993 but the fitting parameters were obtained by using only 2295 experimental L shell cross section data. Since then a large number of experimental data have become available and therefore a new fitting has been performed employing more than 5000 data points. All available data were fitted separately for L1, L2 and L3 subshells . For targets with low atomic numbers (14 :::; Z :::; 42), only coefficients for Ltot were obtained. Because of the slight Z dependence of the universal function, data were also devised into five sub-groups according to their atomic numbers and fitted separately within each group for L1, L2 and L3 subshells. To extend the energy range of validity of the new fitting function theoretical values were used in the high energy region where experimental data were lacking. Results are compared with ECPSSR predictions and discussed. [( eywords : L shell, ionization cross sections, parametrization, semiempirical fit

1. Introduction

In ion beam applications su ch as Particle Induced X-ray Emission (PIXE) , accurate knowledge of ionization cross sections is required for reliable quantitative analysis. In various application programs cross sections are usually calculated either by using semiempirical expressions 1 •2 •3 or theoretical formulas. 4 •5 •6 At present K shell ionization cross sections can be calculated with quite high confidence (13%) for a wide range of proton energies and almost all target atomic numbers using results of Paul's compilat ions. 7 •8 Unfortunately, the amount of experimental data for reliable L shell cross section calculations is not sufficient. As shown in our recent tabulation 9 and review 10 of L-shell cross sections, experimental data are generally scattered for more than a factor ·of two , and at low energies for as much as factor ten . Also, experimental data deviate significantly from the theoretical predictions. None of the present theories gives satisfactory predictions that agree with experi217

218

I. Orlic, C. H. Sow f3 S. M. Tang

mental data at all energies and for all atomic numbers. These facts motivated us to perform fittings of all available L-subshell ionization cross sections with analytical functions in order to determine a single set of parameters for the calculation of more reliable L shell cross sections. In a similar work carried out in 1988, Miyagawa et ai.2 used only 669 experimental data points for fitting. Later, in our work which was reported on the 6th International PIXE conference (1992) the total number of data used for fitting was 2295. Since than a large number of experimental data have become available calling for a new fitting. The results presented in this work are based on the new set of 5013 data points. Present semiempirical expression is therefore valid for a wider proton energy range and for a wider range of atomic numbers . 2. Data Base Experimental L subshell ionization cross sections for proton impact have recently been tabulated by our group 9 . The tabulation comprises more than 6000 newly compiled experimental data for proton impact published in the period 1982-1992. Together with the data from the former tabulations 11 •12 they make more than 10 000 data points for proton impact only. Experimental data reported as production cross sections were converted to ionization cross sections using standard procedures described in detail elsewhere. 1 •9 The data base used for conversion comprises of L x-ray emission rates of Scofield 13 and fluorescence and Coster-Kronig yields from Krause. 14 This data base was adopted firstly because of the "historical" reason (most 'often used for comparison of experimental data with theoretical prediction), and secondly because of its completeness. Authors are aware of the fact that recent work of Werner and Jitchin 22 •23 and Xu 24 , offers more accurate values of fluorescence yields, but in both cases data are reported for only a limited number of elements. Binding energies for L1, L2 and L3 subshells used in this work are from J .A. Bearden et aJ.l 5 "'-.._./ Beside using experimental data tabulated in compilations (9, 10, 11) more than thousand additional data points were obtained from several recent publications. 16 •17 •18 3. Universal functions for Ll, L2 and L3 subshells 3.1 A single function for all targets

Analytical function used for fitting is the polynomial of the following form: (1)

where x = ln[E j(>.U;)] is the natural logarithm of the normalized energy E j(>.U;) , rr; is the L;( i = 1, 2, 3) subshell ionization cross section in barns, U; is the L; subshell electron binding energy in keV, E is the proton energy in keV, and >. = 1836.109 is the ratio of proton mass to electron mass. Coefficients A 0 to A 9 are obtained by fitting the expression to the experimental data points. To reduce the total number

Semiempirical Formulas for Calculation of . . .

219

of coefficients experimental data were first fitted with a four degree polynomial, and if the fitting was not good the degree of polynomial was increased until a satisfactory fitting was obtained. In the first attempt, the ionization cross section data for all targets were grouped together and fitted separately for 11, 12 and 13 subshells. The coefficients obtained are given in Table 1. All experimental data used for fitting are shown separately for each subshells on Figs. 1a, 1b and 1c. The cross sections calculated using the new scheme are represented with the full line on the same figures. The total deviation of empirical cross sections (uexp) from the corresponding fitted values (uJit) is usually expressed in terms of the root-mean-square error ( rms). It is calculated for each ~ subshell using the following expression:

rms=

[L[(uexp-O"Jit)/uJit]

2

/N]

1

2

(2)

where N is the total number of data within a group . Data for 11 subshell are most scattered (with the average deviation of 35%) while 13 subshell data have an average deviation of only 15.7%. This is expected since the 13 subshell cross sections are usually derived from the most prominent L X-ray lines (La) and 11 from the least ones (1-y lines). Besides, it can be noticed that the fit for 11 data set is not satisfactory. The 'knee' which is expected at around 0.2 < Ej(>.U;) < 0.5 is smoothen out by the mixing of data from targets with different atomic numbers. This problem will be further discussed in the following section. It should also be noted that the coefficients are valid only for the range of Ej(>.U;) specified in Table 1. Since there are no newly published Ltotal ionization cross sections for elements with 14 ~ Z ~ 42, the fitting coefficients given in Table 1 are the same as those reported in our earlier work. 1 3.2 Atomic number dependence It is well known that Equation 1 is not universal for all elements i.e. the coefficients A 0 ···An are functions of the target atomic number. To illustrate this and to get a rough estimate of the deviation from universality, the ECPSSR 19 •20 theoretical prediction was used for the initial study. Theoretical L subshell cross sections were calculated using the algorithm proposed by Smit 5 for three elements- one with low atomic number (Molybdenum, Z = 42), the other with Z = 66 (Dysprosium) and third with high Z (Uranium, Z = 92) in a wide energy range (0.1 ~ E ~ 10 MeV). Fig. 2a, 2b and 2c shows the results obtained for 11, 12 and 13 subshells in the 'universal' coordinate system. The departure of Equation 1 from 'universality' is evident from the fact that the curves for different numbers do not overlap. It can also be seen that the deviation depends strongly on the normalized energy. In order to present the deviation graphically the reduced cross sections for all elements above Z 43 were normalized to the corresponding dysprosium values (Z 66) at three different normalized energies (Ej>.U; = 0.02, 0.06 and 0.2, respectively) .

=

=

220

I. Orlic, C. H. Sow f3 S. M. Tang

Table 1 Fitting coefficients for the calculation of Ll, L2 and L3 subshell ionization cross sections (in barns) for elements with atomic numbers 43sZ:s;92 and for the calculation of Ltot ionization cross section for elements with 14sZ:s;42.

LI

~

L3

I.-tot- only for 14sZs42

Ao

11.31598

11.338279

AI

3.64049

-0.127835

1.9007601

0.2177

A2

5.15819

-0.967713

0.61853352

-0.3758

AJ

3.71198

-0.286649

A.

0.99363

-0.044735

As

0.09089

nns Range ofx No of data

12.728035

0.17539102 -0.0010

-

12.5081

0.0096 0.0073 0.0022

0.35

0.18

0.157

0.3

0.008-0.76

0.008-0.81

0.009-0.87

0.02-9.7

1658

1677

1678

236

O.l

E/>-U

E/AU

E/ >-U

Fig. 1 Plots of normalized ionization cross sections (ap,Z ) for Ll, L2 and L3 subshells and for proton impact vs. the normalized energy, E!AU,. Experimental data are presented with dots and the fitting results with full lines. Each graph contains more than 1650 experimental data points.

Semiempirical Formulas for Calculation of . . .

221

Results are shown in inserts of Figs. 2a, 2b and 2c. Deviations of 50% or more are predicted by the ECPSSR theory at low normalized energies. This certainly gives an indication of 'non-universality ' of the function used for fitting , even though it does not necessarily mean that the same deviation should be expected from the experimental data. To check this, it would be ideal to determine a separate set of coefficients for each element. However, there is not enough data to perform a reliable fitting for each element separately. Having almost 1700 experimental data per subshell, we decided to split the data into five subgroups according to their target atomic numbers (43 ::::; Z ::::; 50, 51 ::::; Z ::::; 60 , 61 ::::; Z ::::; 70, 71 ::::; Z ::::; 80 and 81 ::::; Z ::::; 92) . In such a way, each group will have between 130 and 500 experimental \..__..,- data points which should be sufficient to produce quite reliable fitting results within the group. This will give us opportunity to throw some light on Z-dependence of L subshell ionization cross sections. 3.3 Range of validity

It must be emphasized again that the proposed semiempirical expression and the associated parameters are valid only within the region of experimental data. Outside this region the function might take unpredictable course due to the large number of degrees of freedom of the fitting function.

On the other hand, in some Z-groups experimental data are available for a very limited energy range (0.3 to 4-5 MeV). These two facts would of course imply relatively short range of validity of our fitting procedure. To extend this region we included the ECPSSR predictions in our set of fitting data. This is justified by the results of our recent study where all available experimental data were statistically evaluated and compared with the ECPSSR predictions. 9 , 21 It was shown that the ECPSSR theory gave a very good agreement with the experimental data at all higher "----" energies but failed at lower energies. The final results of evaluation are shown in Figs . 3a, 3b and 3c, for L1, L2 and L3 subshells, respectively. The procedure used for statistical evaluations is described in details elsewhere (Ref. 9, 21) and only briefly outlined here: all experimental data are normalized to the corresponding ECPSSR theoretical values and plotted as a function of a reduced velocity (where: ~i 2nTJ; 12 with TJs mEd(M Z2,R) and 6, n 2E,/(Z2,R) where m is the electron mass, M the reduced mass of the collision system, E 1 the kinetic energy of the projectile, Z 2 , is the effective nuclear charge of the target atom, R the Rydberg constant, n the principal quantum number of the s-shell electron and E, is the experimentally measured value of the hydrogenic ionization energy). Data of each subshell are further grouped into a number of small equidistant intervals of the logarithm of the reduced velocity and averaged within each interval. Average values for various intervals are shown with full circles in Fig. 3. It can be noticed that for reduced velocities higher than 0.8 (corresponding to the energies above 1-2 MeV) the agreement of experimental data with ECPSSR predictions is generally better than 3-5% except at higher energies (~i > 1.3) where the experimental data seems to be 20 to 40% higher than ECPSSR predictions - which can partly be attributed

=

;e.

=

=

222

J. Orlic, C. H. Sow f3 S. M. Tang

== . ..... : ·- ~~ ~•. : ! •1l').U~002 / 1:: ~}, : :+.~~r,~ = • ,;,

··~

i

L2

~ -.

. -~

30

;:;-to

- zo

- ,"

1 ,~ 0.01

-J:

l :'•

_J,_

~

0.1

E/AU

1

I

· --

~

I'

X

:

i I:=±:::±=±:::::±=:S f

-30

r .~ ~-+---~~,~:~\i~,···. ir ';·:~:i'\.: ,·:, to'

to'

to'

to'

41-51 51-60 61-70 71-80 81-92

to'

10-z

10- 1

E/ AU

41-50 51-60 61-70 71-80 81-92 to'

10-z

10- 1

E/A.U

41-50 51-60 61-70 71-80 81-92 to'

10 - z

10- 1

E/A. U

to'

to'

Fig. 4 Results of fitting for Ll, L2 and L3 subs hells for all Z-groups. Experimental data are shown as circles, ECPSSR theoretical predictions as doted lines and fitted function with a full line. Data and fit for different Z-groups are multiplied with different factors, in order to show them on the same graphs.

226

I. Orlic, C. H. Sow f1 S . M. Tang

Table 2 Fitting coefficients for calculation of Ll, L2 and L3 subs hell ionization cross sections (in barns)

4t-SO

Ao

Range of target atomic numbers 7t-80 St-60 6t-70 Lt subsheU

82-92

12.776794 6.562907

28.243087 50.199585

1.035774

6.476722 -25.804787 -54.061629

10.158703

58.281684

3.970908

-56.684589

7.432592

34.130538

3.968233 1.655714

-33.223367 -11.034979 -2.042851 -0.194075 -0.007252

2.332036 0.317946

10.268531 1.525302

0.014479

0.08835

11.274881 -0.187401 -0.943341

11.242637 -0.162515

A4

-1.47817 -1.282343

A6

-0.386544 -0.037932

At

A2 A3

As A7

As A9

nns Range ofx No. of data

-

0.058885 -0.155743

-

-0.042228 -0.003371

-

0.26 0.013-0.72 (1.0)

0.27 0.01-0.45 (0.95)

128

496

11.194798 0.178807 -0.449865

11.241409 0.149635

0.22 0.01-0.42 (0.6)

-

-

-

-

0.18 0.01-0.28 (0.45)

0.32 0.008-0.2 (0.3)

404

115

11.247424 0.203051 -0.219083

11.229924

11.586671 0.730838

0.164514

-0.181546

0.058692

-0.030406

532

L2 subshell

Ao At

A2 A3

A4

As nns Rangeofx No. of data

-0.063528 -0.015364

-0.633269 -0.17834 -0.034743

-0.087241 -0.753908

-0.056713 0.053262 -0.003672

-

0.006474

0.007866

-

-

0.07

0.197

0.168

0.12

0.20

0.015-1.2 (1.5) 195

0.012-0.7 (1.0) 485

0.01-0.44 (0.65) 509

0.01-0.3 (0.47) 395

0.01-0.14 (0.35) 105

L3 subshell

Ao

11.91837

At

A2

0.03064 -0.657644

A3

-0.14532

A4

-0.026059 -0.044735

As nns Rangeofx No. of data

0.124 0.015-0.79 (1.5) 128

11.909485 0.15918 -0.588004

11.878472

11.802538

-0.137007 -0.959475

-0.371796 -1.052238

-0.159466 -0.033184

-0.316505 -0.054154

-0.28766 -0.042608

11.423712 -1.428823 -1.946979 -0.585198 -0.076467

-

-

-

-

0.17

0.22

0.112

0.145

0.013-0.65 (1.1) 494

0.01-0.44 (0.67) 475

0.013-0.33 (0.5) 392

0.01-0.16 (0.35) 114

S emiempirical Formulas for Calculation of . . .

227

to' 1--.l......

. +-;

: ::j : to'

·: T

[...);

1- -- -

•

1---- /

II

. _,....t+++H---i-1-+-i-H-!+1-1

i •: o_o· f

.i

. ... . ........

fHTI

1----". ·i·····

H~

t- -

+·.

to'

i=F·.-:.

~1=

to'

~~~~j'1--

''=

0.1

E/ XU

I

0.01

-·· L-~-~-::':--'-_,_J 50 eo 70 eo ao Af.omtcr l'homber

0.1

0.1

E/!IU

E/ AU

Fig. 5 The Z dependence of the 'universal function' for all Z groups and for all subs hells as calculated from our new fitting coefficients. It is clearly seen that experimental data for L2 and L3 subshells comply with the 'universal law' much better than the ECPSSR predictions (compare with Fig.2).

228

J. Orlic, C. H. Sow e3 S. M. Tang

5 one can easily see that the experimental data obey 'universal law' much better than ECPSSR predictions- at least for 12 and 13 subshells. To examine closely the trend of the deviations (as we have done for the ECPSSR theory, see Fig. 2) reduced cross sections of all Z-groups are normalized to the corresponding dysprosium (Z = 66) values at the three different normalized energies (Ej)..U; = 0.02, 0.06 and 0.2) . Results are shown on inserts of Figs 5a, 5b and 5c. The deviations between fits for different Z-groups are rather erratic and not systematic (compare Fig. 2 and Fig. 5). At higher reduced energies (Ej>..U; = 0.06 and 0.2) and for all subshells, deviations are generally smaller and have the opposite trend than expected from the ECPSSR predictions - having positive values for high Z elements and negative for low Z elements. Differences are higher at the lower reduced energies ( E / >..U; ·.. . . . . .= 0.02). For 12 and 13 subshells the differences increase for both low and high Z elements while for 11 subshell they increase towards high Z elements. In general, we can say that the 'universal law' holds for most of the experimental data for 12 and 13 subshells and definitely better than expected from the ECPSSR theory. When experimental data are inconsistent and scattered as at low energies and for 11 subshell it is difficult to draw any conclusion. More experimental data are needed especially at both low and high energy regions to draw a more definite answer. 4.2 Comparison with the ECPSSR predictions

To throw some more light on Z dependence of 1 subshell cross sections we further compare our fitted values with the ECPSSR predictions. Calculated are cross sections for one representative element of each Z-group at various energies (from 100 keV up to 10 MeV). These cross sections are normalized to their corresponding ECPSSR values and shown on Fig. 6 separately for 13, 12 and 11 subshells. It is clear that the ECPSSR theory gives very good agreement for 13 subshell for all energies above 1 MeV and for all Z-groups (deviations are less then 10%). \..._...,; Below 1 MeV, theoretical predictions are lower than the empirical values by 20-30%. For energies below 500 keV and for higher Z-groups, the ECPSSR values significantly exceed the experimental ones probably due to the underestimated Coulomb corrections. Agreement for 12 subshell is not that good. The ECPSSR theory gives the best results for low Z-groups and for energies above 1 MeV (10-20% deviation) , reaching 50% for energies below 500 keV. For higher Z-groups , theoretical predictions are systematically lower than experimental data by 10-30% except for 61 ::; Z ::; 70 group where the situation is opposite for energies below 500 ke V. The situation with 11 subshell is much more complex. Experimental data are generally 10-50% higher than theoretical predictions, with the maximum deviations more than 50% for energies between 1 and 3 MeV. 5. Conclusion A new set of empirical parameters for calculation of 1 subshell cross sections is presented . These parameters were obtained by fitting a set of more than 5000

Semiempirical Formulas for Calculation of . . .

z.oo

r · · · ·-· ··· · · · · ·+· . ········· ---······. ··+------·····--------....... -i·=T---"l~------------···--·

1 - - Z=46 (Pd) Z=56 (Ba)

.L.aJ - ........................ j ____

1.75 ~---------·-························-····-·-·············l.. ................................ i.................................. j - Z=76 (Os)

······· Z=66 (Rn)

1.Z5 1.00 0.75 0.50

···-··--· •r-•••-•••-•••••

"'"' "'"'u

b "'

~;;: b

z.oo 1.75 1.50 1.Z5 1.00

................................................,,, _______ ____ __

0 .75

i

·········!·······················--·········~

0.50

z. 00

0 . 50

~

............. ). ·······························-~·-·· ...

···-·---··---·-·····+----··-·--·-------········+·····--·-···-····--··· ... ;.." .... ,......................... ; ................................. : ··-·

Ll

-············-··· .....

...l .............. ··---···-·· ····-······ ·

1000

zooo

3000

4000

5000

Energy (keV) Fig. 6 Empirical L subshell ionization cross sections normalized to the corresponding ECPSSR theoretical predictions for all Z groups and for Ll, L2 and L3 subsbells.

230

I. Orlic, C. H. Sow C1 S. M. Tang

experimental data points recently compiled by our group. The data were devised into five Z groups and each set was fitted with the polynomial function in the so called 'universal' coordinate system. Experimental results show that the 'universal law' holds for 12 and 13 subshells better than expected from the ECPSSR theory. For 11 subshell it was found that wave function node is strongly influencing Z dependence of the reduced cross sections. To further discuss Z dependence of the 1 subshell ionization cross sections, the fitted functions were compared with the ECPSSR theoretical predictions. As expected, ECPSSR theory gives relatively good agreement with experimental data for 12 and 13 subshells and for all energies above approximately 1 to 2 MeV, but underestimates experiment for lower energies by more than 20-30% . At even lower \..._.. energies the theory generally underpredicts 12 and over predicts 13 subshell cross sections. Situation with the 11 subshell is still quite uncertain and more reliable experimental data is needed . References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23. 24.

C.H. Sow, I. Odic, K.K . Loh and S.M. Tang, Nucl. Instr. Meth. B75 (1993) 58 Y. Miyagawa, S. Nakamura and S. Miyagawa, Nucl. Instr. Meth. B30 (1988) 115 S.A.E. Johansson and T .B. Johansson, Nucl. Instr. Meth. 137 (1976) 473 D.D . Cohen, Atom. Data Nucl. Data Tables 41 (1989) 288 Z. Smit, Nucl. Instr . Meth. B36 (1989) 254 M.·S. Chen, Atom. Data Nucl. Data Tables 41 (1989) 257 H. Paul and W. Obermann, Nucl. Instr. Meth. 214 (1983) 15 H. Paul and J. Sacher, Atom. Data Nucl. Data Tables 42 (1989) 105 I. Odic, C.H. Sow and S.M. Tang , Atom. Data Nucl. Data Tables 56 (1994) 159-210 I. Odic, Nucl. Instr. Meth. B87 (1994) 285 R.S. Sokhi and D. Crumpton, Atom . Data and Nucl. Data Tables 30 (1984) 49 T.L. Hardt and R.L. Watson, Atom. Data Nucl. Data Tables 17 (1976) 107 J.H. Scofield, Atom . Data. Nucl. Data Tables 14 (1974) 121 M.O. Krause, J. Phys. Chern. Ref. Data 8 (1979) 307 J.A. Bearden, A.F.B ., Rev. Mod. Phys. 39 (1967) 125 C .H. Sow, I. Odic , T . Osipowicz and S.M. Tang, to be published in the Nucl. Instr. Meth . (1994) E. Perillo, Private communication S. Fazinic, I. Bogdanovic, M. Jacsic, I Odic and V. Valkovic, J. Physics B27 (1994) 4229 - 4241 W. Brandt and G. Lapicki, Phys. Rev. A20 (1979) 465 W. Brandt and G. Lapicki, Phys. Rev. A23 (1981) 1717 S.H. Sow, "A study of proton induced L-shell cross sections". Master Thesis, National University of Singapore (1994) W. Jitschin, in AlP Conference Proceedings 215, X-Ray and Inner Shell Processes, Knoxville, Tenn. 1990, "Progress in Measurements of L-Subshell Fluorescence, CosterKronig and Auger Yields" . T.A. Carlson, M.O. Krause and S.T. Manson, eds .. (1990) U. Werner and W. Jitschin, Phys. Rev. A38 (1988) 4009 J.Q.Xu, Phys. Rev. A44 (1991) 373