States ladder model for electrons backscattered in X-ray microanalysis. Application of the invariant embedding method

Nuclear Instruments and Methods in Physics Research B 160 (2000) 235±242

www.elsevier.nl/locate/nimb

States ladder model for electrons backscattered in X-ray microanalysis. Application of the invariant embedding method S.P. Heluani a

a,*

, C. Ho€mann

a,b

Lab. De Fõsica del S olido, Inst. de Fõsica, Facultad de Ciencias Exactas y Tecnologõa, Universidad Nacional de Tucum an. Av. Independencia 1800, 4000 Tucum an, Argentina b Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Catamarca, Argentina Received 7 December 1998; received in revised form 11 August 1999

Abstract The advantage of the Invariant Embedding method for the study of transport process in electron probe microanalysis (EPMA), is illustrated considering a model where the energy of the impinging electrons slows down on a ladder of states. Di€erential equations systems for the probability of the electrons being backscattered and for the contribution of these electrons to the detected X-ray intensity are obtained. Theoretical predictions of the model are compared with experimental results. It is shown that the proposed model describes the general trend of experimental data and leads to a good agreement for a wide range of atomic numbers. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Microanalysis; X-rays; Invariant embedding; Backscattered

1. Introduction This work is concerned with the application of the Invariant Embedding Principle [1] to the study of random walk of electrons injected into solids. In particular, the method is applied to the calculation of the functionals relevant in electron probe microanalysis (EPMA), scanning electron microscopy (SEM), Auger electron microscopy and similar techniques. When an electron beam penetrates a solid target, elastic and inelastic scattering processes occur producing backscattered electrons, secondary *

Corresponding author. E-mail address: [email protected] (S.P. Heluani).

electrons, absorbed electrons and characteristic and continuum X-rays spectra. Quantitative microanalysis in EPMA is performed by comparing the X-ray intensity detected from one element in the sample with that detected from a reference standard containing a known amount of the same element [2,3]. Castaing [3] showed that the generated X-ray intensity ratio may be related to the weight (mass) concentration of the element of interest. To transform the ratio of the detected X-ray intensity into mass concentrations it is necessary to take into account correction factors, for example for ¯uorescence production and for absorption of the generated radiation. To estimate the absorption correction, the most important of all correction factors, it is necessary

0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 5 9 6 - 0

236

S.P. Heluani, C. Ho€mann / Nucl. Instr. and Meth. in Phys. Res. B 160 (2000) 235±242

to know the X-ray production as a function of the sample depth, which is usually called the /(qz) function, with qz the mass depth. Experimental determination of this function has been performed. Although, knowledge of the analytical form of this function is particularly important in strati®ed specimens, thin ®lms, and in quantitative analysis of light elements as in new superconducting materials research (where experimental determination of /(qz) for light elements, is usually a dicult matter [4]). In the last decade the major e€orts to set up a theory capable of explaining the results in EPMA were aimed to obtain analytical approximations for the /(qz) function [4±7]. Although this function is important, the ultimate interest in EPMA is to obtain relationships between the detected X-ray characteristic intensity and quantities as the electrons incident energy, concentration of elements present, the size and atomic number of the sample for example. Another quantity of interest is the fraction of backscattered electrons which is used to estimate the absorption corrections factor in EPMA. This fraction is also of great importance for its applications in SEM and in ®lm thickness determinations [8]. In spite of the fact that most theoretical development in EPMA in the last ten years is the accuracy with which the ionisation distribution function and the di€erent aspects of electron backscattering are calculated, the analytical approximations mostly used by the microanalyst in quantitative analysis are of a semi-empirical nature (see [4] for a review). Most of them do not take into account the degradation of the electron energy and the straggling process. They assume an average value for the energy and for the step length of the electron into the solid sample. This is a coarse approximation for the process of electron scattering in the energy range of electrons in EPMA [9]. Several models described with the Boltzmann equation attempted to take account of the energy degradation process in EPMA [10±12]. Generally they use the integrated path length as the variable, instead of the electron energy. They do not take into account that the degradation process is a stochastic one. Instead, they suppose that the de-

gradation is a continuous process (continuousslowing-down approximation) and they consider that the path length is given by the reciprocal of the stopping power. The continuous-slowing-down model is a good approximation when the particles dissipate their energy in small quantities so that the slowing down may be regarded for many purposes as a continuous process. In the scattering process involved in EPMA, an electron occasionally loses a large fraction of its energy in a single collision (specially when Bremstrahlung is important); this rare event has a great in¯uence on the range-energy straggling (the distribution in path length of radiation particles of various energies). Recently Werner [13] considered the energy variable as a discrete variable in the Boltzmann equations and used a distribution to the stochastic process governing multiple collisions. His results agree well with experimental data. Independently of the important contribution to the knowledge of di€erent physical processes connected to the scattering of electrons travelling within the sample, several of the models mentioned above are not suitable to be used in iterative procedures, necessary to quantitative analysis with EPMA. Therefore, in order to create simple rules or formulas for the user of the EPMA, it is important to develop simple theoretical models that, considering the physical processes involved, can be treated also analytically. 1.1. Invariant embedding principle Recently a simple theoretical procedure that considers a di€erent approach to describe the scattering of electrons in EPMA was presented by us [14,15]. The fundamental ideas of the mathematical formulation employed were developed by the astrophysicist Ambarzumian [16] and by Chandrasekhar [1] for describing physical processes occurring in radiative transfer. The method, now known as the invariant embedding principle, was further developed and generalised by Bellman, Kalaba and Wing [17], who applied the Ambarzumian±Chandrasekhar method to problems related to neutron-transport theory, scattering and di€usion theory.

S.P. Heluani, C. Ho€mann / Nucl. Instr. and Meth. in Phys. Res. B 160 (2000) 235±242

In most experimental situations in microanalysis and SEM the microanalyst can only detect particles (or intensity of radiation) which emerge from the sample when particles are injected into the surface. The traditional transport formulation, the Boltzmann equation, puts the emphasis on the internal ¯uxes and the application of boundary conditions gives the backscattered and transmitted particles. This method is uneconomical, since much information about internal ¯uxes is obtained, which is quite useless to the experimentalist. The invariant embedding method reverses the roles and concentrates on the boundary ¯uxes as the primary objects of investigation and yields equations for the observable quantities directly as function of the size of the sample. These equations are obtained by regarding the size of the sample as a variable rather than a constant. Thus the sample is ®guratively embedded in a class of samples of di€erent sizes [18]. With this method Dashen [19] derived an integro-di€erential equation for the backscattered ¯ux applicable only to a bulk target. In our previous work, using invariant embedding method on a simple model of scattering, we obtained an analytical expression for the backscattered fraction, detected intensities and /(qz) function. This simple model considers all physical parameters involved (as the cross sections) as time independent. With the aim of removing this coarse approximation, in the following section we describe a state ladder model for the energy ``states'' of the electrons in EPMA. The details of the invariant embedding method of calculation will be illustrated by the estimation of the probability that electrons which impinge on the surface of a sample with some initial energy, are backscattered with di€erent energies. In Section 4 the contribution of these electrons to the detected X-ray intensity is calculated. In Section 5 the results of calculations are presented and compared with relevant experimental results. 2. States ladder model Consider a beam of electrons with energy E0 impinging perpendicularly on a solid target of thickness s; some of the electrons will be absorbed

237

into or transmitted through the target and some will be backscattered out. The energy range of electrons, from 0 to E0 , is divided into intervals or steps in a ladder. The transition (in the ladder) from one step to the following is described by equations corresponding to a memoryless process. De®ning: · ``sj '' as the probability per unit time that an electron is slowed down from Ej to Ej‡1 , where Ej ; Ej‡1 ; Ej‡2 ; . . . ; indicate di€erent energy ``states'', Ej P Ej‡1 . · ``rj '' is the collision probability per path length unit of the electrons in the energy state Ej . · ``gj '' is the X-rays production eciency per path length unit of the electrons in the energy state Ej . · ``Rj;k (s)'' is the probability that electrons are backscattered with energy Ej when impinging on a sample of thickness s with energy Ek . The details of the invariant embedding method of calculation can be illustrated by the estimation of this probability, Rj;k (s), as shown below. Consider now a new layer of thickness ds added to the sample. Thus, an electron, with energy E0 , impinges on a sample of length …s ‡ ds†. Following the usual procedure in the invariant embedding method, we may consider the interval (0,s) as a subsample lying in the new …0; s ‡ ds† sample. The diagrams in Fig. 1 (typical in this theory), show the di€erent possible paths of the electrons across the new layer with its respective probabilities. Taking into account all paths that yield terms of order lower than (ds)2 , the probability Rj;0 …s ‡ ds† can be written in terms of Rj;k (s). To the aim of illustrating how the probabilities of the di€erent paths are calculated, consider for example the expression for the probability associated to the ®rst diagram in Fig. 1 [expression (1)]: It is the probability that the impinging electron passes across the interval …s ‡ ds; s† without suffering any collision, will be backscattered from a ``subsample'' of size s with energy Ej , and escapes from the sample at s ‡ ds with the same energy Ej . For the second diagram, expression (2) is the probability that one electron su€ers a collision in the new layer without changing its energy state, then is backscattered from the subsample (0,s) with energy Ej and escapes at s ‡ ds. Expression

238

S.P. Heluani, C. Ho€mann / Nucl. Instr. and Meth. in Phys. Res. B 160 (2000) 235±242

The corresponding initial condition is Rj;0 …0† ˆ 0. The latter expression represents a system of coupled non-linear di€erential equations. Its solution gives information about the fraction of incident electrons that are backscattered as well as about their energy distribution. 3. Contribution of backscattered electrons to the detected X-ray intensity

Fig. 1. All possible trajectories, and their respective probabilities, of an electron that after impinging on sample surface with : collision energy E0 was backscattered with energy Ej . without change of energy. A





: interaction with loss of energy.

(4) shows the probability that the electron, when crossing the interval …s ‡ ds; s† for the ®rst time, experiences a degradation of the energy, from E0 to Ej , then is backscattered from the subsample (0,s) without changing its energy state and escapes from the new layer at s ‡ ds. The interpretation of the other diagrams is made in the same way. In Eq. (7) d0;j is the Kronecker delta. The probabilities of any other possible paths are of the order of (ds)2 and their contribution will vanish. It is important to emphasise the fact that the details of the particle behaviour in (0,s) are of no interest. To obtain Rj;0 …s ‡ ds† all probabilities illustrated in Fig. 1 must be summed. Then, letting ds ! 0, we ®nd the following di€erential equation:   dRj;0 …s† 1 s0 sj r0 ‡ rj ‡ ‡ Rj;0 …s† ˆ r0 d0;j ÿ v0 vj 2 ds 2 s0 sjÿ1 ‡ Rj;1 …s† ‡ Rjÿ1;0 …s† v0 vjÿ1 ‡

1X Rk;0 …s†rk Rj;k …s†: 2 k6j

…8†

We call Jj;0 (s) the mean value of the contribution to the detected X-ray intensity of an electron that penetrates into a solid target of thickness s, with energy E0 , with the condition that it is backscattered with energy Ej , E0 P Ej . These intensities can be written in the form: Ij;0 …s† ˆ Rj;0 …s†Jj;0 …s†. The contributions to Ij;0 …s ‡ ds† of the electron paths shown in Fig. 1 are respectively ‰1 ÿ s0; =v0 dsŠ…1 ÿ r0 ds†‰1 ÿ sj; =vj dsŠ  …1 ÿ rj ds†‰…g0 ‡ g† ds ‡ Jj;0 …s† eÿlds ŠRj;0 …s†; …9†

1=2 r0 ds‰1 ÿ s0 =v0 dsŠ‰1 ÿ rj dsŠ  ‰1 ÿ sj; =vj dsŠ  ‰…g0 ‡ gj † ds ‡ Jj;0 …s† eÿlds ŠRj;0 …s†;

…10†

‰1 ÿ s0; =v0 dsŠ  …1 ÿ r0 ds†1=2 rj ds‰1 ÿ sj; =vj dsŠ  ‰…g0 ‡ gj † ds ‡ Jj;0 …s† eÿlds ŠRj;0 …s†Š;

…11†

s0 =v0 ds‰1 ÿ r0 dsŠRj;1 …s†‰1 ÿ rj dsŠ  ‰1 ÿ sj =vj dsŠ  ‰…g0 ‡ gj † ds ‡ Jj;0 …s† eÿlds Š;

…12†

‰1 ÿ s0 =v0 dsŠ  ‰1 ÿ r0 dsŠRjÿ1;0 …s†sjÿ1 =vjÿ1 ds‰1 ÿ rj dsŠ  ‰…g0 ‡ gj † ds ‡ Jjÿ1;0 …s† eÿlds Š;

…13†

‰1 ÿ s0 =v0 dsŠ‰1 ÿ r0 dsŠ‰1 ÿ rj dsŠ  ‰1 ÿ sj; =vj dsŠRk Rk;0 …s† 1=2 rk ds  ‰1 ÿ sk =vk dsŠRj;k …s†  ‰…g0 ‡ gj ‡ gk † ds ‡ …Jk;0 …s† ‡ Jj;k …s†† eÿlds Š; …14†

S.P. Heluani, C. Ho€mann / Nucl. Instr. and Meth. in Phys. Res. B 160 (2000) 235±242

1=2 r0 d0;j dsg0 ds;

…15†

where l ˆ l0 cosec w was used, and l0 is the X-ray absorption coecient. For the interpretation of these equations, consider, for example, Eq. (10); it is built up by multiplying the probability of the ®rst trajectory in Fig. 1 by its contribution to the detected intensity as follows: the term …g0 ‡ g† ds indicates the contributions of the trajectory within the new layer …s ‡ ds; s† and the term Jj;0 (s)eÿl ds represents the contribution to the detected intensity of the sample (0, s) taking into account the absorption of the new layer. After summing all terms (9)±(15), the di€erential equation for Ij;0 (s) can then be written:   dIj;0 …s† s0 sj r0 ‡ rj ‡ ‡ ‡ l Ij;0 …s† ˆÿ v0 vj 2 ds ‡ …g0 ‡ gj † Rj;0 ‡ ‡

s0 sjÿ1 Ij;1 …s† ‡ Ijÿ1;0 …s† v0 vjÿ1

 1X  rk Ij;k Rk;0 ‡ Rj;k Ik;0 : 2 k6j

…16†

The procedure to obtain the mean value of the contribution to the detected X-ray intensity of absorbed electrons, with the condition that they lose the X-rays production eciency inside the sample, is similar and not presented in this work. Analytical solutions for bulk samples (s®1) are easy to obtain considering only two states of energy, E0 and E1 . The solutions for the two state model are s0 …R1;1 ‡ R0;0 † v0 R1;0 ˆ r   r  ; 4s0 s0 4s1 s1 ‡ r0 ‡ v1 v1 ‡ r1 v0 v0

…17†



I1;0 ˆ

s0 1 …I0;0 ‡ I1;1 † ‡ R10 …r1 I1;1 ‡ r0 I0;0 † v0 2  s0 s1 ‡ ‡ …g0 ‡ g1 †R10 v0 v1   r0 ‡ r1 ‡ l ÿ ‰r0 R0;0 ‡ r1 R1;1 Š ; ‡ 2

where we denote Ii;j …1† ˆ Ii;j ; Ri;j …1† ˆ Ri;j .

…18†

239

Ii;i (1) and Ri;i (1) can be written explicitly: s   2si 2si 2si ; ‡1ÿ Ri;i …1† ˆ vi ri vi ri vi ri ‡ 1 r   …19† 2si 2si 2si ‡ 1 ÿ ‡ 1 vi ri vi ri vi ri r Ii;i …1† ˆ 2gi   : l0 ‡ r v2si rii v2si rii ‡ 1 In [15] we considered the e€ect of anisotropy on the velocity of the incoming electrons. It is possible to distinguish between the collision probabilities until the ``®rst e€ective collision'' takes place, and the collision probabilities of the electrons moving in random walk. The ®rst ``e€ective'' collision is the one in which the electrons start to move in a random walk. If it is taken into account, Ri;i is written as follows: Ri;i …1† ˆ

2si ri ‡ r0i ‡ vi r i 2ri s  2 2si ri ‡ r0i r0 ÿ ‡ ÿ i; vi ri 2ri ri

where r0i is the collision probability, per path length unit, until ``the ®rst e€ective collision''.

4. Results and discussion Our aim in this paper is to illustrate how the invariant embedding method can be generalised to take into account di€erent energies of the electrons. Assuming simple expressions for the parameters involved, experimental results can be successfully reproduced by the calculations. The screened Rutherford cross section was used to compute the ri . The si parameter was estimated evaluating the reciprocal of the time required for a particle to travel an electron-range associated with each energy Ei . The physical basis for the electronrange energy equation is the BetheÕs stopping power [20]. The energy intervals may be selected considering experimental observation. The experimental relationship between backscattering coecients and atomic number shows an initial rapid rise after which it slowly

240

S.P. Heluani, C. Ho€mann / Nucl. Instr. and Meth. in Phys. Res. B 160 (2000) 235±242

approaches a limiting value of 0.55 for a wide range of energies. The origin of this behaviour is the increase of the angular scattering in the elastic collisions with atomic number Z. The variation of the backscattering coecients g with the atomic number can be expressed in a closed analytical form for the case of a bulk sample and a two state model, g ˆ R1;0 …1†‡ R0;0 …1†. Fig. 2 shows these results for di€erent values of E1 /E0 , for an assumed target density q ˆ 0:23 Z gm/cm3 and energies of the incident electrons of 20 and 10 KeV. The theoretical results seem to be a general trend of experimental data measured by various authors [21±27], the discrepancy may be due to the assumed variations of q. For E1 =E0 ˆ 0:5 the agreement between our results and representative experimental points is very good for low atomic numbers, Z < 20. For high Z, the comparison with experimental data and other theoretical results shows good agreement for E1 =E0  1. The latter agrees with the fact that for heavy elements the energy distribution of backscattered electrons is strongly peaked at high energies (near the incident energy). With a target of high atomic number, elastic scattering is strong and many of the primary electrons will be backscattered before they have travelled far. Consequently, they escape with little energy loss. With

Fig. 2. Backscattering probability g versus atomic number Z. Present work: A: E1 /E0 ˆ 0.5; hhhh: E1 /E0 ˆ 0.9; +++: E1 / E0 ˆ 0, E0 ˆ 20 keV. Present work: dddd: E1 /E0 ˆ 0.9, E0 ˆ 10 keV. Experimental results: DD: Heinrich 10/49 keV; : Bishop 10/30 keV; 5: Reimer 20 keV; : Kanter 10 keV; 1±3 keV; °H, Sorensen and Schou [25]; d: N2 , Sorensen and Schou [26] 1±3 keV; j: Werner analytical results [27] 1/10 keV.

low atomic number materials the electrons traverse, on average, a signi®cant distance before reemerging from the target, hence the energy of the backscattered electrons will be considerably diminished from the incident energy. The behaviour of the theoretical results shown in Fig. 2 is a consequence of the atomic number dependence of the ratio of elastic to inelastic scattering probabilities, i.e., the ratio of ri to si . Consideration of oblique incidence in backscattering coecients can be made in an easy way, by replacing in Eqs. (1)±(7) (only when we consider the probability at beginning of the trajectories) ds for ds/cos h0 , where h0 is the angle of incidence measured to the surface normal. For example, in Eq. (1), ‰1 ÿ s0; =v0 dsŠ…1 ÿ r0 ds† has to be replaced In by ‰1 ÿ s0; =v0 ds= cosh0 Š…1 ÿ r0 ds= cosh0 †. Figs. 3±5 g is plotted versus h0 for aluminium, copper and silver, respectively. The increase of backscattering coecients, when the electron beam is not perpendicular to the target surface, is because the angular scattering required for primary electrons to escape from the surface is smaller. Although in [15] we obtained analytical results for thin ®lms in good agreement with experimental data considering only one state of energy, we believe that the contribution of Eq. (8) is important for other possible applications (such as inhomogeneous materials, thickness calculations or estimations of probabilities associated with the electrons scattering).

Fig. 3. Backscattering probability g versus h0 for Cu (see also [28]).

S.P. Heluani, C. Ho€mann / Nucl. Instr. and Meth. in Phys. Res. B 160 (2000) 235±242

Fig. 4. Backscattering probability g versus h0 for Al.

241

or angular distributions of the electrons in X-ray microanalysis. In spite of the fact that the expression used for the cross sections and the density are roughly estimated, the theoretical results shown in the last section are in agreement with experimental ones. The model presented here is a better approximation compared with recent models used in microanalysis, because it takes into account the di€erent values of the parameters (e.g., the cross section, or path length) in successive energy intervals. The application of proper boundary condition (taking into account experimental condition in EPMA) is not an easy task using the Boltzmann equation. The method described here converts a two-point boundary value problem into an initialvalue problem and this is its most important advantage. It simpli®es the computational solutions. As Bellman and Kalaba [29] pointed out, the invariant embedding approach, in the course of obtaining the solution for a given thickness, automatically grinds out the solution for any intermediate thickness. This may be useful in inhomogeneous materials where the traditional approach must be carried out for each stage. Acknowledgements

Fig. 5. Backscattering probability g versus h0 for Ag.

On the other hand, the procedure described in Section 3 can be applied to obtain the angular distribution of the backscattered electrons. We can calculate the probabilities that an electron impinging the sample surface with an angle h0 from the normal can be backscattered with an angle hj . This result is important for scanning microscopy. This work is now in progress. 5. Conclusion This work intends to prove that the invariant embedding method is a very powerful tool for obtaining functionals related to electron scattering in EPMA. The method can also be used to clarify some theoretical concepts of energy degradation,

This research was partially ®nanced by Foncyt and Consejo de Investigaciones de la Universidad Nacional de Tucuman. We would like to thank Lic. Gabriela Simonelli for her valuable help. References [1] S. Chandrasekhar, Radiative Transfer, Dover, New York, 1960. [2] J.B. Reed, Electron Microprobe Analysis, Cambridge University Press, Cambridge, 1993. [3] R. Castaing, Thesis University of Paris, France, 1951. [4] A.P. Mackenzie, Rep. Prog. Phys. 56 (1993) 557. [5] G. Love, M.G.C. Cox, V.D. Scott, J. Phys. D.: Appl. Phys. 11 (1978) 7. [6] R.H.Y Packwood, J.D. Brown, X-Ray Spectrometry 10 (1981) 138. [7] M. Villafuerte, S.P. Heluani, Brizuela H. Ho€mman, XRay Spectrometry 26 (1997) 3. [8] A. Chan, J.D. Brown, X-ray Spectrometry 26 (1997) 279.

242 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

S.P. Heluani, C. Ho€mann / Nucl. Instr. and Meth. in Phys. Res. B 160 (2000) 235±242 U. Fano, Phys. Rev. 93 (1954) 117. D.B. Brown, R.E. Ogilvie, J. Appl. Phys. 37 (1966) 4429. J.P. Spencer, C.J. Humphreys, Phil. Mag. A 42, 433. D.B. Brown, D.B. Wittry, D.F. Kyser, J. Appl. Phys. 40 1627. W.S.M. Werner, Phy. Rev. B 55 (22) (1997) 14925. S.P. Heluani, H. Ho€mman, Avances en An alisis por Tecnicas de Rayos X (1997) 107. S.P. Heluani, H. Ho€mman, X-ray Spectrometry 27 (1998) 390. V.A. Ambarzumian, Theoretical Astrophysics, Pergamon Press, New York, 1958. R. Bellman, R.Y. Kalaba, M.G. Wing, J. Math. Mech. 8 (1959) 249. M.G. Wing, An Introduction to Transport Theory, Wiley, New York, London, 1962. R.F. Dashen, Phys. Rev. A 134 (4a) (1964) 1025.

[20] H.A. Bethe, Ann. Phys. Leipz 5 (1930) 325. [21] H. Kanter, Ann. Phys. 20 (6) (1957) 144. [22] H.E. Bishop, Proc. IV Congr. Inter. Optique des rayonsX et Microanalyse, Paris, 1965, pp. 153±158. [23] K.F.J. Heinrich, Proc. IV Congr. Inter. Optique des rayonsX et Microanalyse, Paris, 1965, pp. 159±167. [24] L. Reimer, C. Tollkamp, Scanning 3 (1980) 35. [25] H. Sorensen, J. Schou, J. Appl. Phys. 53 (1982) 5230. [26] H. Sorensen, J. Schou, J. Appl. Phys. 49 (1978) 5311. [27] S. Wolfgang, M. Werner, Phys. Rev. B 55 (22) (1997) 14925. [28] H. Drescher, L. Reimer, H. Seidel, Z. Angew. Phys. 29 (1970) 331. [29] Bellman, Kalaba, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness, Elsevier, New York, 1963.