NOM B Nuclear Instruments and Methods in Physics Research B74 (1993) 352-361

North-Holland Beam Interactions with Materials 8 Atoms

Parametrization of the total photon mass attenuation coefficients in the energy range 0.1-1000 keV

I. Orlic, K.K. Loh, C.H. Sow, S.M. Tang and P. Thong Department of Physics, National University of Singapore, Singapore 0511

Received 2 November 1992 and in revised form 4 January 1993

It is convenient to generate mass attenuation coefficients using semi-empirical schemes. The validity of most of the existing

schemes is limited to a relatively narrow energy interval (l-40 keV) and their accuracies are poor in some energy regions. In this

work, a semi-empirical scheme flexible enough to give a good fit to data in a very wide photon energy range (0.1-1000 keV) was

employed. Fitting coefficients for the entire range were obtained by utilizing mass attenuation data from two sources: (1)

semi-empirical data of Henke et al. in the low photon energy region, and (2) theoretical values generated with the XCOM code for

fitting in the high energy region. The root mean square of the fit is generally less than 0.2% except for energies below 1 keV where

the available data are scattered. A computer code for generating mass attenuation coefficients based on the proposed scheme has

been developed.

1. Introduction

In almost a century-long history of X-ray spec- troscopy a large number of experimental and theoreti- cal publications dealing with photon-atom interactions have been published. This indicates their importance for scientific, engineering and medical applications and at the same time dissatisfaction with the existing data.

In many applications of X-ray fluorescence (xRF), particle induced X-ray emission (PIXE), electron pho- ton microscopy (EPM) and y-spectroscopy, accurate mass attenuation coefficients (MACS) are required for any reliable quantitative analysis. Many different sources of MACs are available in the literature, rang- ing from exhaustive compilations of experimental data such as, e.g. refs. [l-3] to extensive tabulations of theoretical values, e.g. [4,.5]. The most recent and accu- rate tabulation is the new Lawrence Livermore Na- tional Laboratory (LLNL) data base from Cullen et al.

161. Since it is neither efficient nor practical to store all

the tabulated data in the computer memory for data analysis, many semi-empirical schemes for generating MACs have been introduced. Unfortunately, many of them lack accuracy in some energy regions, especially around absorption edges [7,8]. Another drawback of

Correspondence to: Dr. I. Orlic, Department of Physics, Na-

tional University of Singapore, Lower Kent Ridge Rd., Singa- pore 0511.

existing schemes is that they are valid within a very limited energy span, usually between 1 and 40 keV.

In recent years, there is a growing demand for a unified and more accurate parametrization of MACs for energies below as well as above this energy span. To meet this demand a new scheme which is basically a modification of the simple power law by replacing the constant exponent with a polynomial function of en- ergy has been used. The Klein-Nishina scattering term is also included to improve fitting at higher energies where Compton scattering dominates.

2. Choice of data base

Presently, none of the available theoretical, semi- empirical or experimental tabulations give perfect re- sults for all elements or in all energy ranges. However, it has been pointed out [7-101 that modern theories give predictions which are in agreement with the best experimental values to within l-3%, at least for ener- gies above 1 keV, where reliable experimental values exist.

Recently, Berger and Hubbell approximated differ- ent components of the total mass attenuation cross sections with cubic spline fits to the theoretical values. This work resulted in a computer program named XCOM [lo]. This program provides a convenient means for generating theoretical values of total and partial cross sections in a wide energy range - 1 keV to 100 GeV. Therefore, in our work XCOM total attenuation

0168-583X/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

I. U&c et al. / Parametrization of total photon mass attenuation coeffiicients 353

cross sections were used for fitting at all energies (as well as experimental data) sometimes deviate from above 1 keV (for low 2 elements) or, in order to avoid theoretical predictions by 100% or more 131, but this discontinui~ (see fig. I), above the first absorption deviation is still within the estimated precision of data edge in the energy range l-10 keV for higher 2 which is still very poor at energies below 1 keV. Ac- elements. Since XCOM does not provide MACs for cording to Cullen [6,12], rough estimations of the maxi- energies below 1 keV, semi-empirical tabulated data by mum uncertainty of the photoelectric cross sections in Henke et al. [ll] were found to be suitable and accu- solids are: 1000% for photon energies 10 < E < 100 rate enough. At lower energies, data by Henke et al. eV, lOO-200% for 100 c: E < 500 eV, lo-20% for 0.5

10-l

0.1 1 10 100 11 I I I I 1 ,

0.3 0.5 0.7 1 3 5 7 10 30 50 70 100

Energy (keV)

300 500 1000

Fig. 1. Fitting results for hydrogen, aluminum, yttrium and thorium with the algorithm proposed. To avoid overlapping data, those for yttrium and thorium are multiplied by factors of 10 and 100, respectively. Data by Henke et al. are represented by hollow circles and those from XCOM by filled circles. Residuals, relative deviation between fit and data points (in %I,) are given on the

corresponding upper graphs for aluminum and thorium.

354

Table 1

I. Or& et al. / Parametniation of totaIphoto~ mass attenuation coeficimts

Coefficients for calculating total mass attenuation coefficients using eq. (1) for photon energies above the K absorption edge (k,, k,, k,, k,) and for energies between L, and E, absorption edge (I,, Z,, I,, [J. Z = atomic number, Sy = symbol of corresponding element

z SY EL, E,,<E<E, EK E>E,

4 12 1 3 4 k, 4 kn k,

1 H 2 He 3 Li 4 Be 5 B 6 C 7 N 8 0 9 F

10 Ne 11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar 19 K 20 Ca 21 SC 22 Ti 23 v 24 Cr 25Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr 37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 MO 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 55 Cs

- 7.00266 5.19086 - 0.34413 - 2.38938 3.24818 -0.11121 - 2.50341 3.43696 - 0.12806 - 2.03564 3.43789 - 0.13520 - 2.91417 4.13834 - 0.22931

0.15632 2.00149 0.32080 2.10634 0.28244 0.92539 3.34768 - 0.83777 1.34549 3.81364 - 1.04042 1.42378 3.05305 - 0.24003 1.25060

0.2292 2.76125 0.29160 1.09998 0.2702 3.00797 0.19834 1.11670 0.3200 2.62786 0.68939 1.01331 0.3771 0.68453 3.86200 - 0.43637 0.4378 2.44853 1.46227 0.69759 0.5004 2.03061 2.27680 0.31534 0.5637 2.33976 1.88017 0.55084 0.6282 2.38673 1.99445 0.50415 0.6946 2.53782 1.85115 0.62868 0.7690 2.65431 1.83552 0.65972 0.8461 2.41689 2.80062 - 0.~96 0.9256 2.60591 2.55718 0.18669 1.0081 2.69756 2.72893 0.04244 1.0961 2.75840 2.72647 0.04391 1.1936 2.87225 2.72029 0.04441 1.2977 2.93846 2.73406 0.03264 1.4143 3.02673 2.74409 0.01747 1.5265 3.12353 2.74414 0.01165 1.6539 3.19106 2.76121 - 0.01683 1.7820 3.30117 2.73599 0.01398 1.9210 3.36926 2.74007 0.00224 2.0651 3.46256 2.74320 - 0.01061 2.2163 3.54696 2.73914 - 0.00741 2.3725 3.63965 2.73646 - 0.01290 2.5316 3.71677 2.73020 - 0.01273 2.6977 3.79880 2.72885 - 0.01869

2.8655 3.86387 2.72657 - 0.02278 3.8425 3.93865 2.72262 - 0.02487 3.2240 3.99914 2.71967 - 0.02706 3.4119 4.07189 2.71531 - 0.02854 3.6043 4.12607 2.71194 - 0.03442 3.8058 4.19974 2.70350 - 0.03661 3.0180 4.24309 2.70139 - 0.04152 4.2375 4.30447 2.69926 -0.04319 4.4647 4.35252 2.69160 - 0.04781 4.6983 4.40598 2.68835 - 0.04963 4.9392 4.43690 2.68323 - 0.05424 5.1881 4.51763 2.67963 - 0.05663 5.4528 4.55737 2.66911 - 0.05780 5.7143 4.61780 2.66299 - 0.06277

- 0.04820 -0.11707 -0.16880 - 0.17980 - 0.17486 - 0.16451 - 0.16546 -0.16625

0.04226 -0.12878 - 0.07434 -0.11829 - 0.11481 - 0.14231 - 0.15525 - 0.01904 - 0.06696 - 0.03220 - 0.03423 - 0.03527 - 0.03438 -0.03117 - 0.03096 - 0.02249 - 0.03428 - 0.03125 - 0.02723 - 0.03104 - 0.03036 - 0.03038 - 0.02896 - 0.02881 - 0.03004 -0.03174 - 0.03173 - 0.02941 - 0.02694 - 0.02776 - 0.03320 - 0.02689 - 0.03001 -0.03161 - 0.032% - 0.02658 - 0.02884

0.2838 0.4016 0.5320 0.6854 0.8669 1.0721 1.3050 1.5596 1.8389 2.1455 2.4720 2.8224 3.2029 3.6074 4.0381 4.4928 4.9664 5.4651 5.9892 6.5390 7.1120 7.7089 8.3328 8.9789 9.6586

10.3671 11.1031 11.8667 12.6578 13.4737 14.3256 15.1997 16.1046 17.0384 17.9976 18.9856 19.9995 21.0440 22.1172 23.2199 24.3503 25.5140 26.7112 27.9399 29.2001 30.4912 31.8138 33.1694 34.5614 35.9846

- 6.48071 3.27538 0.04385 - 3.51566 2.60589 0.26361 - 2.36523 2.88866 0.15389 - 1.38914 3.05349 0.07919 - 0.58244 3.07947 0.05871

0.11476 3.09245 0.03213 0.63927 3.08731 0.01587 1.08730 3.07389 0.80035 1.42132 3.06182 - 0.00946 1.81194 3.04034 - 0.02320 2.07992 3.04059 - 0.02978 2.39196 3.00737 - 0.04492 2.62043 2.99387 - 0.05681 2.88077 2.96887 - 0.06174 3.06672 2.95078 - 0.07943 3.28801 2.92821 - 0.08514 3.42667 2.90289 - 0.09061 3.53219 2.88277 - 0.10191 3.75863 2.85955 - 0.09992 3.93329 2.83775 -0.11226 4.00179 2.81618 -0.11678 4.11127 2.79692 -0.11885 4.21463 2.77190 -0.13140 4.35405 2.75878 - 0.13535 4.44148 2.72589 -0.14641 4.56600 2.70395 -0.15515 4.64390 2.67587 -0.16255 4.77308 2.65503 - 0.16936 4.81074 2.62267 - 0.18383 4.89850 2.60534 -0.18393 4.93990 2.57045 - 1.19961 5.00743 2.55814 - 0.19972 5.07635 2.53376 - 0.20666 5.11759 2.50569 - 0.21581 5.19474 2.48145 - 0.22156 5.23284 2.44844 - 0.23505 5.29354 2.41378 - 0.24899 5.34695 2.38309 - 0.25992 5.41166 2.37364 - 0.25521 5.45290 2.33638 - 0.26879 5.50180 2.30251 - 0.28165 6.47503 3.03067 - 0.04965 5.57355 2.24202 - 0.29909 5.60095 2.21314 - 0.30745 5.63478 2.17349 - 0.32262 5.65532 2.14514 - 0.32915 5.68392 2.09840 - 0.34706 5.68137 2.05 134 - 0.36341 5.72030 2.04363 - 0.36014 5.76598 2.05890 - 0.34949 5.74957 1.98639 - 0.37446 5.74700 1.96282 - 0.37979 5.79401 1.93222 - 0.38924 5.80690 1.91476 - 0.39242 5.82191 1.87715 - 0.40213

- 0.00866 - 0.03787 - 0.02935 - 0.02484

- 0.02494 - 0.02538 - 0.02438 - 0.02708 - 0.02568 - 0.03575 - 0.02734 - 0.03063 - 0.02724 - 0.03227 - 0.03094 - 0.02881 - 0.03142 - 0.02689 - 0.02972 - 0.02898 - 0.02784 - 0.03049 - 0.03142 - 0.03205 - 0.03352 - 0.03403 - 0.03476 - 0.03729 - 0.03594 - 0.03856 - 0.03827 - 0.03872 - 0.03969 - 0.04024 - 0.04208 - 0.04399 - 0.04534 - 0.04393 - 0.04558 - 0.04737 - 0.03050 - 0.04910 - 0.04983 - 0.05195 - 0.05230 - 0.05473 - 0.05665 - 0.05573 - 0.05406 - 0.05693 - 0.05738 - 0.05845 - 0.05865 - 0.05948

I. Orlic et al. / Parametrization of total photon mass attenuation coeffiiients 355

Table 1 (continued)

Z SY EL, E,,<E<E, EK E>E,

4 12 13 4 k, kz k3 k,

56 Ba 5.9888 4.65770 2.65946 57 La 6.2663 4.71490 2.65505

58 Ce 6.5488 4.77277 2.64710 59 Pr 6.8348 4.83442 2.64088 60 Nd 7.1260 4.87359 2.63355 61 Pm 7.4279 4.93318 2.62474

62 Sm 7.7368 4.95712 2.61647 63 Eu 8.0520 5.00931 2.61226 64 Gd 8.3756 5.03443 2.60481

65 Tb 8.7080 5.08055 2.60115

66 Dy 9.0458 5.11495 2.59238 67 Ho 9.3942 5.15537 2.58071 68 Er 9.7513 5.19648 2.57148 69 Tm 10.1157 5.23938 2.56466 70 Yb 10.4864 5.26855 2.55541 71 IU 10.8704 5.31163 2.55490 72 Hf 11.2707 5.34305 2.54268 73 Ta 11.6815 5.37806 2.53595 74 w 12.0998 5.41320 2.53128 75 Re 12.5267 5.44913 2.52042 76 OS 12.9680 5.47715 2.52666 77 Ir 13.4185 5.50865 2.50428

78 Pt 13.8799 5.54158 2.50551 79 Au 14.3528 5.57564 2.48980 80 Hg 14.8393 5.60041 2.47759 81 Tl 15.3467 5.62415 2.46173 82 Pb 15.8608 5.65349 2.45575 83 Bi 16.3875 5.69400 2.47086 84 PO 16.9393 5.73466 2.46840 85 At 17.4930 5.76863 2.45600 86 Rn 18.0490 5.74682 2.43686 87 Fr 18.6390 5.78319 2.43220 88 Ra 19.2367 5.80588 2.41932 89 AC 19.8400 5.83228 2.39643 90 Th 20.4721 5.85156 2.40038 91 Pa 21.1046 5.87741 2.35466 92 U 21.7574 5.88224 2.34307

- 0.07172 - 0.07121 - 0.07113 - 0.07717 - 0.07273 - 0.08002 - 0.07625 - 0.08680 - 0.08912 - 0.07597 - 0.08230 - 0.08885 - 0.09459 - 0.09047 - 0.10174 - 0.09208 -0.10077 - 0.09548 - 0.10036 - 0.10577 - 0.09007 -0.10492 - 0.09847 - 0.10793 -0.11241 - 0.12251 - 0.12044 - 0.10349 - 0.09608 - 0.10153 -0.11031 - 0.10993 -0.11330 -0.12322 -0.11549 - 0.14418 - 0.15054

- 0.03577 - 0.03442 - 0.03077 - 0.03325 - 0.02815 - 0.02998 - 0.02487 - 0.03205 - 0.03162 - 0.02286 - 0.02541 - 0.02644 - 0.02802 - 0.02454 - 0.03012 - 0.02484 - 0.02734 - 0.02359 - 0.02631 - 0.02724 - 0.02154 - 0.02495 - 0.02303 - 0.02561 - 0.02570 - 0.02850 - 0.02698 - 0.02277 - 0.01973 - 0.02099 - 0.02272 - 0.02259 - 0.02255 - 0.02407 - 0.02204 - 0.02817 - 0.03027

37.4406 5.84551 1.87466 - 0.39966 38.9246 5.85321 1.82364 - 0.41682 40.4430 5.87476 1.79700 - 0.42193 41.9906 5.90293 1.77311 - 0.42727 43.5689 5.90905 1.74652 - 0.43341 45.1840 5.92214 1.70715 - 0.44407 46.8342 5.92934 1.69904 - 0.44407 48.5190 5.94975 1.67827 - 0.44795 50.2391 5.92827 1.64215 - 0.45629 51.9957 5.95498 1.63471 - 0.45467 53.7885 5.95991 1.61100 - 0.46029 55.6177 5.94804 1.55826 - 0.47717 57.4855 5.96408 1.54243 - 0.47949 59.3896 5.88779 1.43854 - 0.50808 61.3323 6.04546 1.57974 - 0.45878 63.3138 6.07395 1.57891 - 0.45631 65.3508 6.00771 1.48367 - 0.48598 67.4164 6.05159 1.50856 - 0.47273 69.5250 6.00250 1.43225 - 0.49552 71.6764 6.03030 1.43738 - 0.48952 73.8708 6.11400 1.50446 - 0.46496 76.1110 6.09889 1.45990 - 0.47717 78.3948 6.15378 1.53187 - 0.44176 80.7249 6.15303 1.47494 - 0.46441 83.1023 6.19862 1.50676 - 0.45079 85.5304 6.17641 1.46912 - 0.46009 88.0045 6.17648 1.45115 - 0.46277 90.5259 6.30427 1.55736 - 0.42520 93.1050 6.21512 1.43087 - 0.46426 95.7299 6.18945 1.38290 - 0.47598 98.4040 6.24492 1.47699 - 0.43939

101.1370 6.31471 1.51865 - 0.42408 103.9219 6.26760 1.45136 - 0.44373 106.7553 6.32123 1.48583 - 0.42883 109.6509 6.39823 1.55418 - 0.40447 112.6014 6.28773 1.41526 - 0.44597 115.6061 6.30701 1.44776 - 0.43028

- 0.05904 - 0.06098 - 0.06126 -0.06168 - 0.06218 - 0.06326 -0.06312 - 0.06335 - 0.06402 - 0.06359 - 0.06408 - 0.06595 - 0.06606 - 0.06870 - 0.06334 - 0.06301 - 0.06608 - 0.06439 - 0.06672 - 0.06580 - 0.06309 - 0.06429 - 0.05974 - 0.06253 - 0.06097 - 0.06183 - 0.06196 - 0.05792 - 0.06195 - 0.06300 - 0.05879 - 0.05721 - 0.05917 - 0.05744 - 0.05479 - 0.05907 - 0.05716

<E<lkeV,l-S%forl<E<5keV,2%for5<E< 100 keV and l-2% for 0.1 <E < 10 MeV.

The recently published theoretical data base by Cullen et al. [6] was also considered for use in our work. However, those data were not available in a form suitable for fitting and therefore were not used.

3. Semi-empirical scheme

As mentioned before, neither theory nor tabula- tions are suitable for quantitative analysis in various applications of X-ray and y-spectroscopy. Therefore, many semi-empirical schemes have been suggested.

One of the first and widely used schemes was proposed by Victoreen in 1943 [13] followed by Heinrich [14], Theisen [15] and Leroux 1161 in the 1960s. In 1979, Thinh and Leroux [171 proposed a relation based on a simple power law which gave a significant improvement of fits. Because of its simplicity and relatively high accuracy ( f 10% in the energy range l-40 keV), this scheme has been very popular among PIXE and XRF users [l&21]. In 1986, Heinrich [22] proposed another fitting procedure which gave significantly better fits near absorption edges, but it was also valid in a very limited energy range, 0.2-20 keV.

More detailed comparisons between different semi-empirical, theoretical and experimental total mass

356 I. Orlic et al. / Parametrization of total photon mass attenuation coefficients

attenuation coefficients can be found in compilation works by Saloman [3], Campbell et al. [7] and Orlic et al. [8,25].

In an attempt to find a suitable semi-empirical scheme, Campbell [7,23] fitted various empirical ex- pressions to the XCOM data. Since our intention was to fit MACs in a much wider energy range than usual (from 0.1 to 1000 keV) several schemes were tested. The outcome was the same as Campbell’s: the best results were obtained with an expression consisting of

the cubic-corrected power law and the Klein-Nishina term (eq. (1))

P/P = cxp(p, +p,(ln A) +p,(ln A)‘+p,(ln hj3)

+ umZNA/M,

where uKN is the incoherent scattering cross section as given by the Klein-Nishina equation [24], Z and M are

the atomic number and atomic weight, respectively, and NA is the Avogadro’s number. Fitting parameters

Table 2 Coefficients for calculating total mass attenuation coefficients for photon energy between L,, L, and L, absorption edges and for energies between L, and lower energy limit E, or M absorption edge

Z Sy E,/ E, < E < EL) E L3 EL3<E<E,_ E, 2 2 ELM < E < EL, ELI

E Edge m, m2 m3 m4 1 31 1 32 1 21 I 22

16 S 0.1 3.98381 - 1.96207 1.29140 -0.13371 0.2292 17 Cl 0.1 - 2.71337 2.78307 0.23822 - 0.05767 18 Ar 0.1 2.07772 -0.84611 1.21097 - 0.14523 19 K 0.1 8.53825 - 4.62277 1.97295 -0.19719 20 Ca 0.1 5.10006 - 1.85097 1.29907 - 0.14560 21 SC 0.1 7.28382 - 3.26298 1.60507 - 0.16673 22 Ti 0.1 6.39592 - 2.45282 1.38399 - 0.14698 23 V 0.1 2.83998 0.44991 0.61932 - 0.08033 24 Cr 0.1 - 7.38556 8.17416 - - 1.29292 0.07794 25 Mn 0.1 2.13747 1.11467 0.46032 - 0.06687 26 Fe 0.1 2.21508 1.10224 0.47160 - 0.06697 27 Co 0.1 4.76476 - 1.05687 1.09423 - 0.12565 28 Ni 0.1 4.28916 - 0.66344 1.02071 - 0.12207 29 Cu 0.1 2.32551 1.22642 0.45413 - 0.06650 30 Zn 0.1 4.23694 - 0.64439 1.08840 -0.13570 1.0197 31 Ga 0.1 2.60598 0.84383 0.68376 -0.10262 1.1154 32 Ge 0.1 4.27624 - 0.65284 1.15806 -0.15244 1.2167 33 As 0.1 5.14580 - 1.51272 1.47391 -0.19017 1.3231

34 Se 0.1 6.74588 - 3.03410 1.98313 - 0.24674 1.4358 35 Br 0.1 7.26941 - 3.69837 2.28437 - 0.28824 1.5499 36 Kr 0.1 11.95287 - 8.69443 4.04104 - 0.48832 1.6749 37 Rb 0.1 15.89070 - 12.8788 5.51561 - 0.65735 1.8044 38 Sr 0.1085 11.09104 - 8.15432 4.08776 - 0.52451 1.9396 39 Y 0.1487 5.18978 - 1.99566 2.09278 - 0.32184 2.0800 40 Zr 0.2122 6.15438 - 2.90381 2.42622 -0.36813 2.2223 41 Nb 0.2118 5.83085 - 2.56434 2.36756 - 0.37176 2.3705 42 MO 0.2370 6.35098 - 3.21302 2.67445 - 0.41914 2.5202 43 Tc 0.2631 5.57003 - 2.03397 2.13536 - 0.33831 2.6769 44 Ru 0.2901 9.99349 - 8.27969 5.01647 - 0.76636 2.8379 45 Rh 0.3180 6.55219 - 3.56318 2.98931 - 0.48554 3.0038 46 Pd 0.3425 6.32821 - 3.29827 2.95809 -0.49645 3.1733 47 Ag 0.3766 5.65378 - 2.26740 2.49736 - 0.43066 3.3511 48 Cd 0.4129 3.12801 1.54731 0.70511 - 0.16236 3.5375 49 In 0.4530 2.78289 2.29146 0.30064 - 0.09544 3.7301 50 Sn 0.4946 2.57173 2.70219 0.11208 - 0.07092 3.9288 51 Sb 0.5379 2.52338 2.80299 0.07373 -0.06562 4.1322 52 Te 0.5827 4.98562 - 1.29951 2.27938 - 0.44589 4.3414 53 I 0.6293 3.00387 2.36374 0.26067 - 0.09483 4.5571 54 Xe 0.6775 3.00203 2.56894 0.10775 - 0.06179 4.7822 55 Cs 1.2171 3.16785 2.42799 0.12755 -0.05218 5.0119

77.206 - 27.639 1.0428 9.4520 47.827 - 16.433 1.1423 6.2330 20.461 -5.2040 1.2478 5.4560

6.9330 0.6640 1.3586 5.1580 0.0065 3.8833 1.4762 3.9491

- 4.6062 6.2403 1.5960 3.8764 - 5.6370 6.9490 1.7272 3.4100

3.2959 2.4609 1.8639 4.0979 2.0366 3.1926 2.0068 3.9355 1.9840 3.2996 2.1555 3.9621 2.2290 3.2290 2.3067 3.9270 2.724 2.999 2.4647 3.981 2.813 2.999 2.6251 3.996 3.044 2.907 2.7932 4.051 3.320 2.770 2.9669 4.303 3.653 2.586 3.1469 4.091 4.146 2.250 3.3303 4.344 4.111 2.321 3.5237 4.217 3.971 2.454 3.7270 4.328 4.293 2.217 3.9380 4.620 4.834 1.753 4.1561 4.592 4.637 1.940 4.3804 4.584 4.618 1.955 4.6120 4.554 4.577 2.048 4.8521 4.625 4.572 2.073 5.1037 4.599 4.215 2.528 5.3594 4.563

0.2702 0.3200 0.3771 0.4378 0.5004 0.5637 0.6282 0.6946 0.7690 0.8461 0.9256 1.0081 1.0961

- 0.2320 1.1936 1.1180 1.2977 1.4580 1.4143 1.6080 1.5265 2.1968 1.6539 2.2734 1.7820 2.5440 1.9210 2.2097 2.0651 2.3328 2.2163 2.3596 2.3725 2.4170 2.5316 2.425 2.6977 2.447 2.8655 2.452 3.0425 2.302 3.2240 2.502 3.4119 2.337 3.6043 2.487 3.8058 2.420 3.0180 2.196 4.2375 2.238 4.4647 2.279 4.6983 2.323 4.9392 2.318 5.1881 2.375 5.4528 2.482 5.7143

I. Orlic et al. / Parametrization of total photon mass attenuation coefficients 357

Table 2 (continued)

Z SY E,/ E,<E<E,, E L3 EL) < E < EL* ELM EL,< E<E,, EL,

E Edge m, m2 m3 m4 1 31 1 32 1 21 1 22

56 Ba 1.2928 3.19080 2.48392 0.08776 - 0.04429 5.2470 4.363 2.384 5.6236 4.601 2.470 5.9888

57 IA 1.3613 3.23265 2.53710 0.05145 - 0.03744 5.4827 4.401 2.397 5.8906 4.698 2.404 6.2663

58 Ce 1.4346 3.29411 2.53524 0.05465 - 0.03926 5.7234 4.449 2.402 6.1642 4.702 2.470 6.5488

59 Pr 1.5110 3.35891 2.54117 0.04839 - 0.03842 5.9642 4.469 2.438 6.4404 4.757 2.467 6.8348

60 Nd 1.5753 3.41765 2.51947 0.05777 - 0.04029 6.2079 4.463 2.500 6.7215 4.778 2.490 7.1260

61 Pm 1.6540 3.45084 2.59000 0.00563 0.02894 6.4593 4.500 2.528 7.0128 4.854 2.449 7.4279

62 Sm 1.7228 3.49847 2.55440 0.02671 - 0.03367 6.7162 4.536 2.502 7.3118 4.861 2.469 7.7368

63 Eu 1.8000 3.54762 2.56798 0.01585 -0.03181 6.9769 4.553 2.556 7.6171 4.924 2.428 8.0520

64 Gd 1.8808 3.58135 2.57591 0.00022 - 0.02718 7.2428 4.583 2.543 7.9303 4.927 2.467 8.3756

65 Tb 1.9675 3.61719 2.61430 - 0.03278 -0.01918 7.5140 4.633 2.537 5.2516 4.985 2.423 8.7080

66 Dy 2.0468 3.67219 2.57359 - 0.00002 - 0.02888 7.7901 4.655 2.572 8.5806 5.014 2.420 9.0458

67 Ho 2.1283 3.72181 2.55757 0.01760 - 0.03537 8.0711 4.695 2.558 8.9178 5.047 2.426 9.3942 68 Er 2.2065 3.77845 2.52004 0.05588 - 0.04817 8.3579 4.727 2.581 9.2643 5.086 2.414 9.7513

69 Tm 2.3068 3.82800 2.53261 0.03019 - 0.03886 8.6480 4.767 2.592 9.6169 5.128 2.398 10.1157

70 Yh 2.3981 3.86458 2.50726 0.06374 - 0.05192 8.9436 4.802 2.569 9.9782 5.082 2.815 10.4864

71 Lu 2.4912 3.91793 2.48402 0.08777 - 0.06146 9.2441 4.847 2.560 10.3386 5.181 2.431 10.8704 72 hF 2.6009 3.94527 2.52598 0.03404 - 0.04239 9.5607 4.866 2.629 10.7394 5.214 2.377 11.2707 73 Ta 2.7080 3.99165 2.50855 0.05734 - 0.05289 9.8811 4.911 2.584 11.1361 5.237 2.468 11.6815

74 w 2.8196 4.02988 2.51689 0.04574 - 0.05071 10.2068 4.940 2.624 11.5440 5.276 2.318 12.0998 75 Re 2.9317 4.07043 2.52499 0.03041 - 0.04512 10.5353 4.979 2.629 11.9687 5.302 2.373 12.5267

76 OS 3.0485 4.10244 2.53312 0.01448 - 0.03948 10.8709 5.006 2.661 12.3850 5.323 2.314 12.9680

77 Ir 3.1737 4.14636 2.52893 0.01458 - 0.04009 11.2152 5.050 2.673 12.8241 5.355 2.361 13.4185

78 Pt 3.2960 4.18356 2.53390 0.00520 - 0.03783 11.5637 5.084 2.666 13.2726 5.375 2.278 13.8799 79 Au 3.4249 4.22770 2.52409 0.01426 - 0.04262 11.9187 5.125 2.690 13.7336 5.406 2.303 14.3528

80 Hg 3.5616 4.26093 2.51954 0.02092 - 0.04665 12.2839 5.156 2.643 14.2087 5.430 2.337 14.8393 81 Tl 3.7041 4.29302 2.51974 0.01369 - 0.04376 12.6575 5.184 2.630 14.6979 5.451 2.323 15.3467 82 Pb 3.8507 4.32934 2.51778 0.00460 - 0.03614 13.0352 5.220 2.644 15.2000 5.512 2.502 15.8608 83 Bi 3.9991 4.36991 2.52184 - 0.01213 - 0.02654 13.4186 5.257 2.642 15.7111 5.536 2.453 16.3875 84 PO 4.1494 4.41866 2.52158 - 0.00971 - 0.03298 13.8138 5.301 2.625 16.2443 5.555 2.387 16.9393 85 At 4.3170 4.46126 2.51394 0.00640 - 0.04593 14.2135 5.347 2.642 16.7847 5.574 2.335 17.4930 86 rN 4.4820 4.45433 2.51411 - 0.00666 - 0.03871 14.6194 5.334 2.637 17.3371 5.535 2.281 18.0490 87 Fr 4.6520 4.49562 2.51108 - 0.01358 - 0.02823 15.0312 5.376 2.647 17.9065 5.650 2.494 18.6390 88 Ra 4.8220 4.52666 2.50733 0.00692 - 0.05016 15.4444 5.410 2.653 18.4843 5.644 2.411 19.2367 89 AC 5.0020 4.56932 2.50607 - 0.01644 - 0.03124 15.8710 5.454 2.662 19.0832 5.761 2.594 19.8400 90 Th 5.1823 4.59153 2.50190 - 0.00815 - 0.03948 16.3003 5.490 2.696 19.6932 5.883 2.800 20.4721 91 Pa 5.3669 4.63772 2.49776 - 0.00936 - 0.03490 16.7331 5.525 2.666 20.3137 5.798 2.534 21.1046 92 U 5.5480 4.65216 2.49170 - 0.02703 -0.01144 17.1663 5.541 2.672 20.9476 5.829 2.571 21.7574

pi, pz, p3 and p4 are constants for one element and within energy regions enclosed by two adjacent absorp- tion edges or for energies beyond the K absorption edge.

3.1. Fitting procedure and results

Data files containing tabulated data of Henke et al. and XCOM calculated values of MACs for the entire energy range (0.1-1000 keV) were created for all ele- ments (1 I 2 5 92). Fitting of expression (1) to the data was performed with a Marquardt non-linear least square fitting routine available within the SIGMA- PLOT i scientific graphic package. Klein-Nishina

’ SIGMAPLOT is a trade mark of Jandel Corporation, USA.

scattering cross sections were calculated from the fol- lowing expression,

ln( 1 + 2k)

k 1 ln(1 + 2k) 1 + 3k

+ 2k - (1+2k)2 ’ 1 (2)

where k = E/mc2, r = 2.817939 X lo-i3 cm mc2 = 511.0034 keV and E is photon energy in keV.’

Fitting results for hydrogen, aluminum, yttrium and thorium are presented in fig. 1. To avoid overlapping, data for yttrium and thorium are multiplied by factors

Tab

le 3

2

Coe

ffic

ien

ts

for

calc

ula

tin

g to

tal

mas

s at

ten

uat

ion

coe

ffic

ien

ts

for

phot

on

ener

gies

bet

wee

n

M,,

M,,

. . . ,

M,

abso

rpti

on

edge

s an

d fo

r en

ergi

es b

etw

een

M

, an

d lo

wer

en

ergy

2

lim

it E

, $ 6’

Z

E

o E

o <

E <

&is

, %

E

t+

< E

< &

iVI,

&

a E

M,

< E

< &

E

MU

E

M,

<

E

<E

M&

, E

MU

<E

<E

M&

I,

3

9

41

0.10

85

42

0.10

85

43

0.10

85

44

0.10

85

45

0.10

85

46

0.13

28

47

0.13

28

48

0.14

87

49

0.17

17

50

0.17

17

51

0.15

11

52

0.15

11

53

0.15

11

54

0.18

33

55

0.21

22

56

0.27

70

57

0.19

26

58

0.21

22

59

0.31

17

60

0.31

17

61

0.27

70

62

0.27

70

63

0.27

70

n1

n2

113

n4

m42

m41

m32

m31

m22

m21

ml2

m

l1

s a 11

5.01

3 10

6.66

2 3.

5107

89

.575

0 80

.589

8 -

6.38

73

29.0

389

11.0

289

174.

946

42.9

485

39.8

331

37.0

196

14.2

756

23.1

086

32.5

527

31.6

390

28.6

047

3.91

81

4.69

93

20.5

246

16.9

684

11.8

798

5.64

37

- 69

.420

8 -

67.3

126

2.38

89

- 57

.974

0 -

57.4

238

1.04

08

- 18

.528

2 -

6.35

56

- 13

8.96

3 -

31.1

679

- 30

.666

3 -

28.3

109

- 9.

7352

-

18.4

405

- 27

.556

1 -

28.1

233

- 24

.129

0 -

0.98

35

0.01

30

- 16

.599

7 -

13.1

825

- 7.

7898

-

1.48

28

14.9

364

15.2

602

- 0.

4178

13

.842

4 15

.081

3 1.

9749

5.

5664

2.

6658

38

.388

0 9.

1280

9.

5340

8.

8955

3.

9179

6.

7995

9.

7373

10

.435

1 8.

8027

1.

6398

0.

6969

6.

5042

5.

4255

3.

5371

1.

4474

- 1.

0543

-

1.14

08

0.03

24

- 1.

1025

-

1.30

52

- 0.

3214

-

0.54

70

- 0.

3046

-

3.50

71

- 0.

8669

-

0.95

02

- 0.

8939

-

0.45

54

- 0.

7732

-

1.08

74

- 1.

2218

-

1.00

79

- 0.

2735

-

0.09

75

-0.7

721

- 0.

6587

-

0.43

81

- 0.

2096

0.73

30

0.79

05

0.84

73

0.88

61

0.93

49

0.98

43

1.02

69

1.08

02

1.13

09

0.79

61

3.76

8 2.

088

1.06

22

0.84

85

3.55

4 2.

190

1.12

34

0.90

13

3.19

4 2.

376

1.18

54

0.95

11

3.31

7 2.

355

1.24

22

0.99

99

5.48

4 1.

419

1.29

74

96.3

73

- 35

.542

1.

0515

6.

609

0.92

1 1.

3569

89

.021

-

33.2

67

1.10

60

6.35

2 1.

023

1.41

98

83.5

64

-31.

660

1.16

06

6.31

6 1.

039

1.48

06

2.36

4 2.

621

1.06

50

3.98

3 2.

072

1.21

71

3.89

7 2.

098

1.13

67

3.95

7 2.

097

1.29

28

3.87

0 2.

132

1.20

44

3.97

0 2.

114

1.36

13

3.85

2 2.

162

1.27

28

4.00

7 2.

120

1.43

46

5.31

4 1.

551

1.37

74

2.48

9 2.

863

1.51

10

3.87

9 2.

193

1.40

28

3.99

3 2.

167

1.57

53

3.84

5 2.

234

1.47

14

4.03

9 2.

170

1.65

40

3.82

5 2.

255

1.54

07

3.98

6 2.

206

1.72

28

3.82

5 2.

278

1.61

39

4.00

5 2.

220

1.80

00

64

0.2770

6.6183

-2.4260

1.7586 -0.2443

1.1852

76.154 -29.235

1.2172 8,597

-0.048

1.5440 3.792

2.308

1.6883 4.032

2.217

1.8808

65

0.1717 -5.6112

10.4330

-2.6765

0.2602 1.2412

73.753 -28.683

1.2750 6.289

1.062

1.6113 3.863

2.296

1.7677 4.027

2.242

1.9675

66 0.2122

1.4535

3.1211 -0.1907

-0.0138

1.2949

69.682 -27.456

1.3325 6.096

1.157

1.6756 3.809

2.342

1.8418 4.060

2.243

2.0468

67

0.2770

4.0777

0.1743

0.9232 -0.1528

1.3514

65.046 -25.898

1.3915 5.446

1.471

1.7412 3.811

2.363

1.9228 4.092

2.245

2.1283

68 0.2122

1.6315

2.7196

0.0767 -0.0607

1.4093

60.848 -24.477

1.4533 5.353

1.523

1.8118 3.847

2.368

2.0058 4.087

2.269

2.2065

69

0.1926 -5.0355

9.4382 -2.1146

0.1726 1.4677

63.572 -26.240

1.5146 5.841

1.257

1.8845 3.889

2.370

2.0998 4.114

2.278

2.3068

70 0.1085 -1.8380

6.2849 -1.0736

0.0593 1.5278

54.494 -22.411

1.5763 5.900

1.222

1.9498 3.916

2.373

2.1730 4.125

2.287

2.3981

71

0.1085 -0.3276

4.8522 -0.6175

0.0113 1.5885

57.769 -24.531

1.6394 6.893

0.681

2.0236 3.876

2.422

2.2635 4.129

2.309

2.4912

72

0.1085 -0.1868

4.8402 -0.6304

0.0127 1.6517

55.928 -24.196

1.7164 7.170

0.527

2.1076 3.916

2.420

2.3654 4.158

2.311. 2.6009

73

0.1085 -1.9405

6.5977 -1.1795

0.0672 1.7351

53.589 -23.567

1.7932 6.974

0.627

2.1940 3.951

2.420

2.4687 4.146

2.343

2.7080

74

1.1085 -1.1407

5.9151 -0.9671

0.0439 1.8092

50.662 -22.588

1.8716 6.894

0.671

2.2810 4.019

2.408

2.5749 4.178

2.344

2.8196

75

0.1085 -0.3842

5.1929 -0.7225

0.0155 1.8829

30.913 -12.446

1.9489 6.854

0.666

2.3673 4.063

2.405

2.6816 4.214

2.346

2.9317

76 0.1085 -1.8806

6.8129 -1.2684

0.0733 1.9601

54.605 -25.802

2.0308 6.900

0.581

2.4572 4.032

2.445

2.7922 4.228

2.356

3.0485

77 0.1085 -2.9368

7.9106 -1.6078

0.1054 2.0404

48.393 -22.958

2.1161 7.249

0.343

2.5507 4.126

2.410

2.9087 4.307

2.327

3.1737

78

0.1328

1.5113

3.4177 -0.1106

-0.0581

2.1216

45.233 -21.695

2.2019 7.209

0.344

2.6454 4.166

2.408

3.0265 4.323

2.338

3.2960

79 0.1511

3.5698

1.1260

0.7499 -0.1652

2.2057

43.502 -21.199

2.2911 7.102

0.393

2.7430 4.199

2.414

3.1478 4.317

2.370

3.4249

80

0.2122

3.6427

1.0962

0.7581 -0.1655

2.2949

40.458 -19.895

2.3849 6.847

0.540

2.8471 4.187

2.445

3.2785 4.382

2.344

3.5616

81

0.1926

3.6296

1.1014

0.7912 -0.1770

2.3893

31.751 -15.013

2.4851 6.055

1.064

2.9566 4.191

2.467

3.4157 4.314

2.419

3.7041

82 0.1717

4.6393

-0.1055

1.2868 -0.2451

2.4840

20.254

-8.092

2.5856 5.339

1.574

3.0664 4.259

2.443

3.5542 4.453

2.332

3.8507

83

0.1926

5.7697

-1.3933

1.7810 -0.3078

2.5796

9.171

-1.075

2.6876 4.475

2.212

3.1769 4.277

2.459

3.6963 4.383

2.420

3.9991

84 0.2122

5.6131

-1.3279

1.8314 -0.3245

2.6830 -0.200

5.161

2.7980 3.586

2.895

3.3019 4.327

2.459

3.8541 4.460

2.395

4.1494

85 0.3117

5.1757

-0.6452

1.5552 -0.2925

2.7867 -4.893

8.501

2.9087 3.141

3.276

3.4260 4.372

2.457

4.0080 4.494

2.402

4.3170

86

0.3924

4.2011

0.6074

1.0490 -0.2281

2.8924 -7.059

10.172 3.0215 3.222

3.251

3.5280 4.382

2.442

4.1590 4.500

2.386

4.4820

87 0.3924

4.5198

0.1763

1.2693 -0.2654

2.9999 -2.615

7.198

3.1362 3.649

2.961

3.6638 4.405

2.458

4.3270 4.523

2.398

4.6520

88

0.3924

5.5170

-1.1906

1.9034 -0.3623

3.1049

0.627

4.924

3.2484 3.982

2.716

3.7918 4.391

2.501

4.4895 4.511

2.442

4.8220

89

0.3924

6.3317

-2.2638

2.3875 -0.4354

3.2190

2.581

3.529

3.3702 4.255

2.525

3.9098 4.460

2.475

4.6560 4.599

2.387

5.0020

90 0.2770

4.1761

0.8164

1.0031 -0.2383

3.3320

3.793

2.621

3.4908 4.355

2.455

4.0461 4.495

2.462

4.8304 4.576

2.437

5.1823

91

0.2770

4.3146

0.8409

0.9488 -0.2273

3.4418

3.639

2.790

3.6112 4.369

2.484

4.1738 4.526

2.477

5.0009 4.603

2.457

5.3669

92 0.2770

3.9954

1.2801

0.7662 -0.2038

3.5517

3.662

2.788

3.7276 4.409

2.460

4.3034 4.527

2.492

5.1822 4.614

2.458

5.5480

360 I. &kc et ai. / P~ra~eiri~ation of total photon tnass atten~t~~ coefficieizts

of 10 and 100, respectively. Also shown in the same figure are the residuals of aluminum and thorium fits. For all energies above 1 keV, root mean square devia- tions of fits were generally below 0.2%. At lower energies, deviations were sometimes higher (5-20%) due to scattered semi-empirical data.

Fitting parameters obtained for all elements are given in tables l-3. Those for energies between the L, and K absorption edges (I,, I,, I, and E,f and for all energies above K absorption edges (k,, k,, k, and kJ are shown in table 1. Parameters for all energies be- tween the M and L, edges Cm,, m2, m3 and m,) and L,-L, and L,-L, edges are shown in table 2. Fitting coefficients for energies below the M absorption edges are tabulated together with the corresponding absorp- tion edge energies in table 3. Generally, four fitting parameters were used in all energy regions except in regions between L,, L,, L, and M,. . . M, absorption edges where two parameters were found sufficient. Energy I$, in tables 2 and 3 represents the lower cut-off energy limit.

4, Conclusion

Semi-empirical schemes are convenient means for generating mass attenuation coefficients, but they are often inaccurate and only valid in a relatively narrow energy interval (l-40 keV). The scheme presented in this paper is the first one capable of fitting total mass attenuation coefficients for the entire energy span of 0.1-1000 keV. Fitting coefficients for all elements were obtained. The mass attenuation coefficients utilized in our work are: (1) the semi-empirical data of Henke et al. [ll] at lower photon energies, and (2) the theoreti- cal values generated with the computer code XCOM [lo] at higher energies. The root mean square devia- tions of the first are generally below 0.2% for ail energies between 1 and 1000 keV. At lower energy RMS deviations of fits are 5-20% due to great uncer- tainties in the data used. A computer code for calculat- ing mass attenuation coefficients based on the de- scribed scheme has been developed and is available from the authors on request.

The authors are grateful to Dr. J. Hubbell for his valuable suggestions and encouragement as well as for providing us with the XCGM code and the numerous bibliographic data. Thanks are due to Prof. J.L. Camp- bell for giving us his unpublished data and also to Dr.

D.E. Cullen for the LLNL’s recent theoreticai data base.

References

Ill J.H. Hubbell, At. Data, 3 (1971) 241. [2] J.H. Hubbell, H.M. Gerstenberg and E.B. Saloman, Bib-

liography of Photon Total Cross Section (Attenuation Coefficient) Measurements 10 eV to 13.5 GeV, National Bureau of Standards Internal Report NBSIR no. 86-3461 (1986).

I31

141

151

161

[71

Bl

191 1101

Ill1

f121

[131

1141

[151

1161

1171 [181

[191

1201

E.B. Saloman and J.H. Hubbell, X-Ray Attenuation Co- efficients (Total Cross Sections): Comparison of the Ex- perimental Data Base with the Recommended Values of Henke and the Theoretical Values of Scofield for Ener- gies between 0.1-100 keV, National Bureau of Standards Internal Report NBSIR no. 86-3431 (1986). W.H. McMaster, N.K. Del Grande, J.H. Mallett and J.H. Hubbell, Compilation of X-Ray Cross Sections, Report UCRL-50174, 1979, Lawrence Livermore National Labo- ratory, Livermore, California. J.H. Scofield, Report UCRL-51326, 1973, Lawrence Liv- ermore National Laboratory, Livermore, California. DE. Cullen, M.H. Chen, J.H. Hubbell, S.T. Perkins, E.F. Plechaty, J. Rathkopf and J.H. Scofield, Tables and graphs of photon-interaction cross sections from 10 eV to 100 GeV derived from the LLNL evaluated photon data library, Lawrence Livermore National Laboratory, Liver- more, CA, UCRL-504~, vol. 6, Part A, Rev. 4 (1989). J.L. Campbell and D. DeForge, X-Ray Spectrom. 18 (1989) 235. D.C. Creagh and J.H. Hubbell, Acta Crystallogr. A4 (1990) 402. D.C. Creagh, Nucl. Instr. and Meth. A255 (198’7) 1. M.J. Berger and J.H. Hubbell, XCOM: Photon Cross sections on a Personal Computer, NBSIR 87-3597 (1987). B.L. Henke, P. Lee, R.L. Shimabukuro and B.K. Fu- jikawa, At. Data Nucl. Data Tables 27 (1982) 1. D.E. Cullen and J.A. Ratkopf, Summary of the coordina- tion meeting for contributors to the new Livermore pho- ton cross section data base, Lawrence Livermore Na- tional Laboratory, Livermore, CA, UCID-21448 SUM (1988). J.A. Victoreen, J. Appl. Phys. 14 (1943) 95; 19 (1948) 855 and 20 (1949) 1141. K.F.J. Heinrich, Electron Microprobe, eds. T.D. McKin- ley et al. (John Wiley, New York, 1966) p. 296. R. Theisen and D. Vollath, Tables of X-Ray Mass Atten- uation Coefficients (Stahleisen M.B.H., Dusseldorf, 1967). J. Leroux, Advances in X-Ray Analysis and W.M. Mueller, vol. 5 (Plenum Press, New York, 1962) p. 153. J.P. Thinh and J. Leroux, X-Ray Spectrom. 8 (1979) 85. J.L. Campell, J.A. Cookson and H. Paul, Nucl. Instr. and Meth. 212 (1983) 427. I. Orlic, J. Makjanic, D. Raos and V. Valkovic, X-Ray Spectrom. 17 (19881 139. I. Grlic, J. Makjanic, G.H.J. Tros and RD. Vis, Nucl. Instr. and Meth. B49 (1990) 166.

I. Orlic et al. / Parametrization of total photon mass attenuation coefficients 361

[21] D.D. Cohen and E. Clayton, Nucl. Instr. and Meth. B22 (1987) 59.

[22] K.F.J. Heinrich, Proc. 11th Int. Congr. on X-Ray Optics and Microanalysis, London, Ontario, 1986, eds. J.D. Brown and R.H. Packwood.

[23] J.L. Campbell, Tables of Coefficients for a Semi-Em- pirical Representation of the XCOM X-Ray Mass Atten- uation Coefficients, unpublished, Private communication.

[24] 0. Klein and Y. Nishina, Z. Phys. 52 (1929) 853. [25] I. Orlic and SM. Tang, Singapore J. Phys. 9 (1992) 57.