Applications of invariant embedding: positron backscattering from surfaces

Nuclear Instruments and Methods in Physics Research B 215 (2004) 509–524 www.elsevier.com/locate/nimb

Applications of invariant embedding: positron backscattering from surfaces Lev G. Glazov a, Imre P
azsit b

b,*

a Institute of High Current Electronics, 4 Akademichesky Ave., 634055 Tomsk, Russia Chalmers University of Technology, Reactor Physics, Department of Reactor Physics, S-412 96 G€oteborg, Sweden

Received 13 July 2003; received in revised form 8 September 2003

Abstract An effective invariant-embedding technique is developed for the detailed description of charged-particle backscattering from solids, with example calculations concentrating on quasielastic positron backscattering. The invariantembedding formalism considered incorporates elastic and inelastic interactions, as well as particle annihilation effects, and allows a very efficient numerical scheme (based upon a recurrence in the number of elastic collisions) for accurate and fast evaluation of various distributions of backscattered particles, including those resolved at once in numerous variables. Not invoking drastic simplifications characteristic for existing analytical-based approaches and easily handling realistic collision cross-sections, the calculation scheme allows to test the accuracy of simulation programs and – within its (albeit more narrow) range of validity – offers significant advantages compared to those as regards accuracy, computing speed, and resolution. Capabilities of the approach are illustrated by calculation results pertaining to elastic and inelastic positron backscattering for various targets and energies in the keV range. Distributions in (dependences on) numerous variables (incidence and/or emission directions, numbers of elastic and/or inelastic collisions, path length, energy, target atomic number) and their combinations are exemplified, and their nontrivial features are briefly discussed.
2003 Elsevier B.V. All rights reserved. Keywords: Positron backscattering; Invariant embedding

1. Motivation and goals A fast positron injected into a material will rapidly slow down and be thermalised by inelastic collisions, similarly as fast neutrons do. After thermalisation, it will diffuse at thermal energies

*

Corresponding author. Tel.: +46-31-772-3081; fax: +46-31772-3079. E-mail address: [email protected] (I. P
azsit).

and will possibly be trapped at a defect before annihilation [1–3]. However, some of the positrons can be scattered back to the surface, either before slowing down or after having been thermalised. The properties of positron annihilation at the surface differ from those in the bulk, for the same reason as in the case of positrons trapped at defects, i.e. different electron density and momentum distribution at the annihilation site. The surface annihilation is therefore interfering in the measurement with the quantification of defects.

0168-583X/$ - see front matter
2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2003.09.009

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azsit / Nucl. Instr. and Meth. in Phys. Res. B 215 (2004) 509–524

Further, pre-thermalisation backscattering affects the spatial distribution of positrons remaining in the target which also affects the evaluation of a measurement. Hence it is important to know characteristics of positrons reflected back to the surface. Quantification of pre-thermalisation backscattering requires transport calculations, based either on deterministic kinetic equations or computer simulation techniques. Deterministic descriptions of positron transport so far have not been aimed to supply quantitative results [4,5] for backscattering. Quantitative results have been obtained by Monte-Carlo simulations so far, cf., for example, [6–8]. On a broader note, adequate description of backscattering is one of the central tasks for the kinetic theory of particle-matter interactions. Various approaches have been developed and may be applied to evaluate relevant quantities (e.g. the total backscattering coefficient, the albedo, etc.) and distributions (e.g. in energy and/or emission direction, collision numbers, etc.) characterizing backscattering in a specific case. Not attempting a comprehensive survey here, we note that methodologically various descriptions of backscattering found in the literature range from (semi-)analytical approaches (e.g. [9–14], to mention only a few prominent treatments geared to electron backscattering and applicable as well for positrons with appropriate minor modifications) to full-scale numerical simulations of the particle transport process; the actual accuracies delivered by different treatments and their regions of validity also vary significantly. The literature of the last decades on backscattering and related problems of particle transport reflects a distinct trend of replacement of the analytical-based approaches with numerical simulations. Partly this is due to wider applicability of the Monte-Carlo-based schemes as well as relative ease of their implementation and ad hoc modifications. To a large degree, however, the trend is stimulated by shortcomings of existing analyticalbased treatments: for many problems, those just do not deliver a priori reliable quantitative results. For complex interaction laws and the free-surface boundary conditions, the relevant kinetic equa-

tions are easy to write down but notoriously hard to solve without drastic simplifications. Therefore, the analytical-based treatments typically involve rough models and approximations that e.g. assume a simple structure of either the solutions to be found or the collision cross-sections, or both. The arising results are no doubt useful for qualitative analysis but hardly can a priori claim good quantitative accuracy; at best, the available accuracy estimates are based upon comparison to results of numerical simulations (cf. e.g. [15]) which then serve as a standard of reference and need to be performed anyway. All in all, at present the computer-simulation techniques dominate the field and have unchallenged authority, at least wherever quantitative results are essential. The absence of competitive alternatives has its drawbacks, however. We mention two important points: i(i) Generally, the simulation techniques discourage a detailed description of e.g. the backscattering distributions. This may sound paradoxical as Monte-Carlo simulations of particle transport may take considerable CPU time and deliver huge sets of data; however, these data concern individual particle trajectories and typically provide with rather modest resolution of the resulting (averaged) distributions. The statistical requirements cause a drastic increase of computation effort with extending the set of variables to be resolved or improving resolution in each of those. Subsequently, in the literature one rarely encounters simulations with more than a single variable resolved. For example, calculations of the total backscattering yield (no variables resolved), the distributions in energy or polar emission angle (1 variable) are readily performed by Monte-Carlo techniques (cf. e.g. [6–8] for positron backscattering); meanwhile, e.g. the collision-number (or path length) distribution of particles emitted into a given solid angle (three variables resolved) is highly relevant for electron spectroscopy techniques, but typically it is not calculated and model simplifications are invoked in order to skip resolving azimuths and/or other variables.

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One would encounter even greater difficulties simulating e.g. the incidence-angle dependence of backscattering into a given direction at a given energy (four variables), etc. (ii) There is also a significant methodological difficulty: reasonably looking but in fact inaccurate simulation results are hardly falsifiable in practice. While there are numerous simulation programs in use, one can actually do little (short of creating oneÕs own code) to verify the correctness and accuracy of specific results: deviations from analytical estimates are justifiably attributed to inaccuracies of the latter; moderate discrepancies with experimental results (if available) may be readily ascribed to the uncertainties in measurements or the interaction laws assumed for simulations; comparisons of results by different programs for exactly the same parameters and assumptions are rarely performed and usually inconclusive; finally, most relevant publications do not at all bother to offer any explicit statements on the accuracy of simulations performed for specific interaction laws. In these circumstances, an error or inaccuracy in an algorithm may in principle stay unidentified forever, and whether one chooses to rely upon specific results becomes basically a matter of belief in qualifications of a particular programÕs author(s) rather than rational verification. In view of the above, it is worthwhile to develop analytical-based alternatives to the simulation techniques that might challenge the latter on their own ground – producing accurate quantitative results. Such an approach is considered in the present paper, with illustrative calculations treating elastic and nearly elastic backscattering of
1 keV positrons from solids. Similar calculations – geared to REELS and EPES applications and limited to the case of normal incidence – have been performed recently for electron backscattering [16]; in particular, the first comparison [16] to available Monte-Carlo results exposed significant inaccuracies of the latter in certain cases, and thus confirmed the need for the schemes capable to test simulation programs. In the present work, the approach and results are extended to positron

511

backscattering and non-normal incidence; the issues well covered in [16] will only be given a cursory discussion. Based upon numerical solution of deterministic kinetic equations, the present approach is not as universally applicable as numerical simulations. However, within its range of validity, it represents a perfect tool for testing simulation programs and offers significant advantages compared to the latter: • it is a priori capable to deliver a much better accuracy and resolution for a given set of cross-sections, while requiring a modest computation effort (CPU times of the order of seconds or minutes on a PC); • the present technique allows to easily resolve several variables at once, in which case the computing speed may exceed that of simulations by many orders of magnitude; • the calculation scheme is simple and transparent, and can be easily reproduced from scratch. The present technique is based upon the prominent invariant-embedding approach [9,17, 18] to transport problems. The invariant-embedding description of backscattering is well-suited for the task outlined above: the equations of this approach involve no spatial variable(s) and it naturally incorporates the free-surface boundary condition, albeit at the cost of the arising equations becoming non-linear and involving integrations over the variables characterizing both the incident and outcoming particles. Though the latter features create serious difficulties for analytical studies, they can be reasonably handled in course of a numerical evaluation after appropriate transformation of the equations. The present calculation scheme utilizes (with appropriate modifications) the version of the invariant-embedding formalism recently developed by Vicanek [18] in his fundamental treatise on electron spectroscopy applications. The original calculation scheme implemented in [18] actually still included MonteCarlo simulations, in contrast to the present work. Note that the Vicanek formalism as well as numerical-evaluation results depicted below are limited to the region of small relative energy loss;

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generalization of the scheme to arbitrary energy losses is possible and pertinent general equations are actually listed in Section 2 below, but their effective numerical evaluation requires extensive analysis and non-trivial transformation to be considered in a forthcoming work.

2. General invariant-embedding equations for backscattering Let a semi-infinite homogeneous medium be bombarded with particles (e.g. positrons) of initial energy E0 and incidence direction X0 . Introduce the distribution Y  YðE0 ; X0 ; E; XÞ of backscattered particles: YðE0 ; X0 ; E; XÞ dE d2 X is the probability for an incident particle to be eventually backscattered with energy (E; dE) into solid angle (X; d2 X). Here X0 ¼ ðh0 ; /0 Þ, X ¼ ðh; /Þ with h0 and h being the polar incidence and emission angles with respect to the inward surface normal; the distribution Y may be nonvanishing only for E 6 E0 and 0 6 h0 < p=2 < h 6 p. Further, we introduce the differential cross-section rðE; X; E0 ; X0 Þ dE0 d2 X0 for a collision E ! ðE0 ; dE0 Þ, X ! ðX0 ; d2 X0 Þ (E0 6 E); the Rassociated total collision R cross-section is rt ðEÞ  dE0 d2 X0 rðE; X; E0 ; X0 Þ. Additionally to (elastic and inelastic) collisions conserving the number of particles, in the case of positron transport one may need to account for the particle-loss processes, e.g. annihilation or trapping at defects; the effective cross-section for particle loss at energy E will be denoted rl ðEÞ. Since the probability of trapping is negligible for fast positrons, rl ðEÞ characterizes annihilation in flight. The basic idea of the invariant-embedding approach is that adding a layer of (the same) material at the surface should not change any characteristics of backscattering. Considering collisions that may occur in the added layer and equating their net effect to zero [9,18], one obtains a nonlinear integral equation for the distribution Y: 



rt ðE0 Þ þ rl ðE0 Þ rt ðEÞ þ rl ðEÞ YðE0 ; X0 ; E; XÞ þ cos h0 j cos hj ¼JþJ YþY JþY J Y

ð1Þ

for cos h < 0 < cos h0 , where J ¼ JðE0 ; X0 ; E; XÞ 

1 rðE0 ; X0 ; E; XÞ j cos h0 j

and, for brevity and transparency, the operator notation is used in the right-hand side with symbol denoting the integrations of the type Z E0 dE0 J Y E Z

d2 X0 JðE0 ; X0 ; E0 ; X0 ÞYðE0 ; X0 ; E; XÞ: ð2Þ In this and similar integrals, effective limits for integrations over angles are determined by the condition of YðE0 ; X0 ; E; XÞ being nonvanishing only for cos h < 0 < cos h0 . Thus, despite the common notation, the operators J in various terms may actually differ from each other due to different integration limits and applicable ranges of the arguments. From Eq. (1), one easily obtains also an equation for the azimuth-integrated distribution; it may be written in the same form as (1) but with second-order integration in place of the third-order one in Eq. (2). In principle, for arbitrary cross-sections one may directly tabulate solutions of Eq. (1) e.g. by an appropriate recurrence procedure. An alternative (though related) scheme may be based upon the expansion of the distribution Y into contributions Yn of particles that emerge from the surface after having suffered a certain number nð P 1Þ of collisions; the functions Yn are sequentially determined by the relations:   rt ðE0 Þ þ rl ðE0 Þ rt ðEÞ þ rl ðEÞ Yn ðE0 ; X0 ; E; XÞ þ cos h0 j cos hj X ¼ J Yn1 þ Yn1 J þ Ym J Yn1m m

ð3Þ for n P 2, with the single-collision initialization Y1 ðE0 ; X0 ; E; XÞ 1  rt ðE0 Þ þ rl ðE0 Þ rt ðEÞ þ rl ðEÞ ¼ þ cos h0 j cos hj

JðE0 ; X0 ; E; XÞ:

ð4Þ

L.G. Glazov, I. P
azsit / Nucl. Instr. and Meth. in Phys. Res. B 215 (2004) 509–524

In Eq. (3), the summation in the last term is formally limited by the condition Yn ¼ 0 for n < 1, so this formula is easily checked to represent an explicit recurrence: the right-hand side contains only functions with indices n  1 or smaller. Though a straightforward numerical evaluation of the distribution from Eqs. (1) or (3) has become feasible with modern computer powers, the arising scheme would require a considerable computation effort and could use some optimization. By far more effective schemes can be obtained either by means of a suitable transformation of the invariant-embedding equations in the general case, or by limiting consideration to the range of small relative energy losses. The former option will be explored in a study presently in preparation, while the latter is applied below to backscattering of positrons from solid surfaces.

3. Nearly elastic backscattering: Vicanek’s formalism Considering only the distribution of backscattered particles that suffered small relative energy loss, one may introduce a number of convenient simplifications. First, one neglects the variation of relevant cross-sections with instantaneous energy loss, though the cross-sections (as well as all resulting distributions) still depend on initial energy E0 of the incident particles; this parametric dependence will be accounted for in calculations below but omitted in the formulae for brevity. Second, disregarding also the angular deflections associated with inelastic collisions, one arrives at the customary model with deflections and energy loss decoupled and exclusively governed by elastic and inelastic collisions, respectively. A detailed invariant-embedding treatment of (electron) backscattering within that model was published recently by Vicanek [18]; here, modifying notation as appropriate [16] and incorporating the particleloss cross-section rl , we will list the basic relations for the distributions integrated over azimuths. It is convenient to express the distributions as functions of direction cosines g0 ¼ cos h0 , g ¼ j cos hj; 0 < g0 ; g 6 1. Further, in view of angular deflections and energy losses being de-

513

coupled, it is convenient to consider separate distributions in the numbers of elastic and inelastic collisions: for a particle incident with the direction cosine g0 , Wn ðg0 ; gÞ dg [or Vm ðg0 ; gÞ dg] will signify the probability of backscattering within the direction-cosine interval (g; dg) after having suffered nð P 1Þ elastic [or mð P 0Þ inelastic] collisions. The advantage of this separation is that Vm and all other distributions of interest are readily expressed in terms of Wn [18]; meanwhile, the latter is insensitive to inelastic interactions and is governed exclusively by the differential cross-section rel ðX0 ; XÞ d2 X for elastic collisions. R Now, we denote kel  1=N rel ðX; X0 Þ dX0 , kin and kl  1=N rl the elastic, inelastic and particleloss mean free paths, N being the target atomic density. For axially symmetric scattering, the normalized elastic-collision cross-section may be expressed as a function of the scattering angleÕs cosine: KðX0 XÞ  N kel rel ðX0 ; XÞ. Defining the kernels J þ and J  as 1 K ðg0 ; gÞ; g0  Z 2p  qffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffi K g0 g þ 1  g20 1  g2 cos / d/; K ðg0 ;gÞ  J ¼ J ðg0 ; gÞ 

0

ð5Þ one obtains a simplified version of the recurrence (3) and (4) for the distribution Wn ,   1 1 kel Wn ðg0 ; gÞ þ g0 g k~el ¼ dn1 J  þ J þ Wn1 þ Wn1 J þ X þ Wm J  Wn1m ; ð6Þ m

where k~el  kel kl =ðkel þ kl Þ and the symbol denotes now the single-order integration Z 1 J W  J ðg0 ; g0 ÞW ðg0 ; gÞ dg0 : ð7Þ 0

Eq. (6) represents an explicit recurrence formula for the functions Wn that may be sequentially evaluated starting from k~el  gg J ðg0 ; gÞ 0 g0 þ g kel k~el g : ¼ K  ðg0 ; gÞ g0 þ g kel

W1 ðg0 ; gÞ ¼

ð8Þ

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In general Eqs. (6) and (8) differ from similar formulae derived earlier for electron backscattering [16,18] by the additional factor kel =k~el that accounts for particle-loss effects which are seen to deliver only the extra factor ðk~el =kel Þn in Wn . Note that differing integration limits in various terms of Eq. (1) have been translated into differing kernels J þ and J  in Eq. (7). The physical meaning of this distinction is clear from Eq. (5): the kernels K þ ðg0 ; gÞ and K  ðg0 ; gÞ determine single-collision probability distributions for transformations of the direction cosine conserving and inverting its sign, respectively. For an arbitrary elastic-collision cross-section, the recurrence (7) and (8) allows straightforward numerical tabulation of the functions Wn . The reader is referred to [16] for details of a very simple numerical-evaluation scheme utilized in our calculations depicted below. In particular, tabulation of the functions Wn ðg0 ; gÞ up to n ¼ nm for certain energy and target material was estimated [16] to require
Ng3 n2m =2 floating-point arithmetic operations, Ng being the number of grid intervals in each of the variables g0 , g. For example, for Ng
nm
100 the calculation amounts to
5 · 109 arithmetic operations, equivalent to CPU time
1 minute on a PC. It is worth stressing that at this modest cost we tabulate Wn as a function of the three variables n, g0 , g (with
100-points resolution in each), i.e. we at once obtain the complete dependence on the incidence angle of the distribution of backscattered particles in both emission angle and the number of elastic collisions; obviously, getting comparably detailed data by means of Monte-Carlo simulations would require by far larger computation efforts. Also, involving no statistical errors, the present technique typically delivers a better accuracy of the calculated distributions for specific cross-sections, though is subject to the applicability limitations listed above. Once Wn have been tabulated, a relatively minor calculation delivers various other distributions of interest. A detailed derivation of relevant expressions in terms of Wn has been given by Vicanek [18, Sections 2.2–2.4] for electron backscattering; the particle-loss effects are readily incorporated into his derivation by means of the simple substitution kel ! k~el . Thus, we plainly list the resulting for-

mulae for the path length (Q), inelastic collision number (Vm ) and energy (Y ) distributions for particles incident in the direction g0 : Qðg0 ; g; RÞ ¼

1 1 R=k~el X ðR=k~el Þn1 Wn ðg0 ; gÞ; e ðn  1Þ! k~el n¼1

ð9Þ Vm ðg0 ; gÞ  ¼

1 X n¼1 1 X n¼1

Vmn ðg0 ; gÞ k~mel knin ðn þ m  1Þ! Wn ðg0 ; gÞ; m!ðn  1Þ! ðk~el þ kin Þmþn ð10Þ

Y ðE0 ; g0 ; E; gÞ ¼ Y0 ðE0 ; g0 ; E; gÞ þ

1 X

Vm ðg0 ; gÞjm ðE0  EÞ:

ð11Þ

m¼1

Physically, Qðg0 ; g; RÞ dg dR, Vm ðg0 ; gÞ dg and Y ðE0 ; g0 ; E; gÞ dE dg represent the probabilities for ultimate escaping the surface within the directioncosine interval (g; dg) having, respectively, travelled the path length (R; dR), experienced m inelastic collisions, and lost the energy (E0  E; dE) within the target. The separate terms Vmn (n P 1, m P 0) in Eq. (10) determine the more detailed number distribution in both elastic (n) and inelastic (m) collisions. The Y0 term in Eq. (11) signifies the contribution of elastically backscattered particles (the Ôelastic peakÕ), and formally reduces to Y0 ðE0 ; g0 ; E; gÞ ¼ V0 ðg0 ; gÞdðE0  EÞ if energy losses in elastic collisions and surfaceplasmon excitations are strictly disregarded; jm ðT Þ denotes the m-fold convolution of the normalized (bulk) excitation cross-section jðT Þ [18]. In a numerical procedure, Eqs. (9)–(11) readily deliver respective distributions of backscattered particles but only up to certain values of the path length R, the number of inelastic collisions m, and the energy loss E0  E. The reasons for the latter limitations are twofold. One – rather technical – limitation concerns convergence of the series (9)– (11) which generally becomes slower with increasing R, m, E0  E. Therefore, tabulation of Wn up to n ¼ nm implies certain limitations on the ranges of R, m, E0  E that allow truncating the

L.G. Glazov, I. P
azsit / Nucl. Instr. and Meth. in Phys. Res. B 215 (2004) 509–524

series at n 6 nm and neglect of the n > nm terms. This condition is easily monitored in course of numerical calculations; for all results depicted below, the relative magnitude of the truncated terms is below 103 . Another set of limitations originates in the assumptions (cf. the beginning of this section) entering the present treatment that favour sufficiently small energy losses and corresponding R and m. Note, however, that similar limitations (i.e. to sufficiently small n) do not apply to Wn if intended for use as auxiliary quantities for calculation of Q, Vm , Y within the appropriate ranges of R, m, E0  E.

4. Illustrative results and discussion In this section, we present some example distributions of backscattered positrons illustrating capabilities of the calculation technique. It is worth stressing that selected dependences plotted for each energy and target represent only a minor fraction of the data actually obtained in the calculations: e.g., with Ng ¼ nm ¼ 100, one obtains
Ng2 nm ¼ 106 values only for Wn ðg0 ; gÞ in a single run for specific energy and target. Following a few examples illustrating the behavior of Wn and Q, we concentrate on the angle-resolved inelastic-collision number distributions (Vm ) rather than the associated energy spectra Y . While Vm and Y are trivially related by Eq. (11), for our purpose the characterization of backscattering by Vm has important advantages: distributions of elastically (V0 ) and inelastically (V1 ; V2 ; . . .) backscattered positrons are presented in a common way and may be directly compared; various features are seen more clearly, not being obscured by the interplay of angular-deflection and energyloss statistics; finally, not relying on the accuracy of specific elementary excitation spectra, the results for Vm are more generally valid and accurate than those for Y . 4.1. Elastic-collision number and path length distributions These distributions are governed exclusively by the differential cross-sections for elastic collisions.

515

For the latter, here and below we used the tables of cross-sections for elastic positron-atom collisions 1 by Dapor and Miotello [19]; corresponding dependences of rel ðX0 ; XÞ on X0 X are exemplified by Fig. 1. In specific cases treated below, annihilation corrections are negligible, so further we do not distinguish between kel and k~el in figures and discussion. The adopted positron-atom cross-sections will suffice for illustrative purposes, but the results might be somewhat affected by solid-state effects that are known to significantly modify the scattering law at sufficiently small deflection angles [19]. Yet, except for the distribution Wn , the solidstate corrections to the results would be by far smaller than those to the cross-sections or the elastic mean free path. This situation is best illustrated by considering the simple transformation of the cross-section from rel ðX0 ; XÞ to rel ðX0 ; XÞ þ rf dðX0  XÞ where rf is a constant and the last term models a change of cross-section at negligibly small scattering angles. Though having no influence on actual particle trajectories, formal introduction of the fictitious ÔcollisionsÕ with zero deflection angle changes the mean free path kel and the distribution Wn (so one can consider kel and Wn as auxiliary quantities defined apart from a transformation of this type; cf. explicit formulae in [16]). As expected, however, this change has no effect on Q, Vm , Y and other quantities that do not refer explicitly to the number of elastic collisions; substituting the new kel , Wn into Eqs. (9)–(11), one easily demonstrates that analytically. For a variety of incidence and emission angles, Figs. 2 and 3 exemplify distributions of backscattered positrons in the number of elastic collisions (Wn ) and the path length (Q) calculated from Eqs. (6) and (9); the test case is backscattering of 2 keV positrons from aluminum target. In disaccord with

1 More precisely, the cross-sections used were obtained by a cubic spline over the logarithms of table values in the cosine of the scattering angle. At large scattering angles, the resolution (5) of the cross-section tables [19] is significantly lower than that of our calculations; deviations of results due to different (reasonable) interpolations of the tables are distinguishable within the present techniqueÕs accuracy, though typically are not well visible in the figures.

516

L.G. Glazov, I. P
azsit / Nucl. Instr. and Meth. in Phys. Res. B 215 (2004) 509–524 σ el (Ω0,Ω) 2

[Å / srad]

E0 = 2keV

0.0010

Al

0

10

0.0005

C

-1

10

0.0000 -1.0

-0.5

Ag

0.0

Au

-2

10

Cu Al

-3

10

C

-4

10

-1.0

-0.5

0.0

0.5

1.0

Cosine of scattering angle: Ω0 · Ω Fig. 1. Examples of the differential cross-sections rel ðX0 ; XÞ for elastic positron-atom collisions used in the present calculations. For positron energy E0 ¼ 2 keV and several elements (C, Al, Cu, Ag, Au), the cross-sections from [19] are plotted as functions of the cosine X0 X of the scattering R angle; apart from normalization, the plotted dependences represent the respective functions KðX0 XÞ  rel ðX0 ; XÞ= rel ðX0 ; X0 Þ d2 X0 entering Eq. (5). In a linear scale, the insert shows the same dependences for lighter elements (C, Al) and scattering angles above 90 (i.e. 1 6 X0 X 6 0).

the results by the P1 -approximation [10] and the exponential-decrease model [20], the distributions shown are nonmonotonic, each having a single maximum at n > 1 or R > 0. This, however, is by no means a universal feature: the shape of the distribution depends on g0 , g and specifics of the cross-section; e.g. in recent electron-backscattering calculations, a variety of shapes has been found, ranging from monotonic distributions to those with two maxima [16,18,21]. Except for the case of oblique incidence and emission, the distributions in Fig. 2 decrease rather slowly with n. This ensures that various functions of interest (e.g. V0 ; V1 ; . . ., cf. Eq. (10)) receive a significant (if not dominating) contribution from positrons that have experienced numerous elastic collisions prior to re-emission; therefore, caution is indicated with respect to various models (e.g. [9,12–14]) based upon an accurate description of one or a few first collisions and rough approximations for the rest. Comparison of Figs. 2 and 3 shows that the continuous distributions QðRÞ closely follow respective discrete ones (Wn versus n). This effect – due to statistical correlation between the path length and the number of elastic collisions – has

been earlier discussed and quantified by various methods for electron backscattering [16,18,21]. In a different way, the same Wn ðg0 ; gÞ results are presented in Fig. 4 which – for a representative set of g0 and n – shows Wn versus g, i.e. the distributions of positrons emitted after a certain number of elastic collisions in the cosine of emission angle. Even for a common underlying scattering law, the shapes of the distributions plotted are observed to significantly vary with n and the incidence angle, again affirming the need for accurate calculation techniques and discouraging the models assuming simple angular distributions. Generally, the distributions get smoother and ultimately become monotonic with increasing n; this is only a qualitative trend, however: even at the largest n shown, the normalized distributions for different incidence angles significantly deviate from each other, as well as from e.g. the
g dependence corresponding to the isotropic particle flux. 4.2. Elastic backscattering Additionally to the cross-section for elastic scattering, evaluation of Vm requires the mean free path for inelastic collision kin , cf. Eq. (10). For our

L.G. Glazov, I. P
azsit / Nucl. Instr. and Meth. in Phys. Res. B 215 (2004) 509–524 Wn

η0 = 1

0.006

λel Q

η=1

η = 0.5 Ση

Ση

0.002

η= 1

0.004

η = 0.5

0.004

η0 = 1

517

0.002

η = 0.2

η = 0.2 0.000

0.000

η = 0.5 0 η = 0.5

0.010

η0 = 0.5

η = 0.5

0.010

η= 1

η= 1 0.005

Ση

0.005

Ση η = 0.2

η = 0.2 0.000 0.06

0.000 0.06

η = 0.2 0

η = 0.2 0

η = 0.2

η = 0.2

0.04

0.04

Ση

Ση 0.02

0.02

η= 0.5

η= 1

η= 1 0.00

0.00

0

20

40

60

80

100

Number of elastic collisions: n

Fig. 2. Distributions of backscattered positrons in the number of elastic collisions: Wn ðg0 ; gÞ versus n for various g0 , g, Al target and initial energy E0 ¼ 2 keV. Top, middle and bottom subfigures show the distributions for g0 ¼ 1 (normal incidence), 0.5 and 0.2, respectively. The applicable values of g are indicated at the plots: g ¼ 1 (normal emission; circles), 0.5 (up triangles), 0.2 (down triangles); the plots marked Rg (squares) show the R 1 distributions integrated over all emission directions, i.e. Wn ðg0 ; gÞ dg versus n. 0

calculations depicted below, we adopted the kin results of [22]. Figs. 5–8 aim to give an impression of all relevant dependences (on incidence and emission directions, energy, target) of the distribution 1 X Wn ðg0 ; gÞ V0 ðg0 ; gÞ ¼ ð12Þ n ~ n¼1 ð1 þ kel =kin Þ characterizing elastically backscattered positrons. Figs. 5 and 6 demonstrate distributions in the co-

0

η = 0.5

25

50

75

Relative path length: R / λel

Fig. 3. Dimensionless path length distributions for positrons backscattered from Al target at the initial energy E0 ¼ 2 keV: kel Qðg0 ; g; RÞ versus R=kel for various g0 , g ( ¼ 1, 0.5, 0.2) indicated at the graphs. The curves marked Rg show the respectiveR distributions integrated over directions of emission, i.e. 1 kel 0 Qðg0 ; g; RÞ dg versus R=kel .

sine g of emission angle, while Figs. 7 and 8 – dependences on the incidence direction; Figs. 5 and 7 illustrate variation of the respective functions with positron energy, Figs. 6 and 8 – with the target atomic number. The dependences plotted quantify some predictable general trends: decrease of V0 with increasing positron energy or with approaching normal incidence, the latter being especially pronounced for smaller g or for V0 integrated over all emission directions; the peak values of g and g0 shift to the left with decreasing, respectively, g0 and g, etc. These are readily explained in qualitative

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Wn 0.006

V0

η0 = 1

500 eV

η0 = 1

n = 50 1000 eV

n = 20 0.004

0.004

1500 eV

n = 10

2000 eV

n= 5

3000 eV

0.002

0.002

n= 1

4000 eV 0.000

0.000

n = 10

η0 = 0.5

0.010

η0 = 0.5

500 eV 0.02

1000 eV

n = 20

1500 eV

n= 5

2000 eV

n= 1

3000 eV

0.01

0.005

n = 50

4000 eV

0.00

0.000 0.06

η0 = 0.2 n= 1

0.10

500 eV

η0 = 0.2

1000 eV

n= 5

1500 eV

0.04

2000 eV 3000 eV 0.05

n = 10 0.02

4000 eV

n = 20 n = 50

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Cosine of emission angle: η

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Cosine of emission angle: η

Fig. 4. Distributions in emission direction of positrons backscattered after certain numbers n ( ¼ 1, 5, 10, 20, 50, as indicated at the curves) of elastic collisions: Wn ðg0 ; gÞ versus g for Al target, E0 ¼ 2 keV. The distributions are shown for a few incidence directions: g0 ¼ 1 (normal incidence; top subfigure), g0 ¼ 0:5 (middle), and g0 ¼ 0:2 (bottom).

Fig. 5. Distributions of elastically backscattered positrons in the cosine g of emission angle: V0 ðg0 ; gÞ versus g for various directions of incidence (g0 ¼ 1, top; g0 ¼ 0:5, middle; g0 ¼ 0:2, bottom). The distributions are shown for a common target (Al) and various positron energies (E0 ¼ 0:5, 1, 1.5, 2, 3, 4 keV) indicated at the curves.

terms; here we will only trace the origins of a few less obvious features observed in the graphs. Consider the minor oscillations present e.g. in the distributions for Al target of Fig. 5 for g0 ¼ 1 (normal incidence, top) but not observed for other values of g0 . At first sight, these may look as just numerical inaccuracies but in fact are not. It is worth mentioning that a similar though more pronounced effect has been occassionally found for electron backscattering, see e.g. [15,16]; how-

ever, those cases are primarily attributed to oscillations of the relevant cross-sections rel , while the positron cross-sections [19] used for Fig. 5 decrease monotonically with scattering angle, cf. examples in Fig. 1. Yet, the present effect is also traced down to be due to specific properties of the elastic collision cross-sections. First, it is easily checked that oscillations in V0 come primarily via the first (
W1 ) term in Eq. (12); for a specific energy, this is con-

L.G. Glazov, I. P
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V0

0

V0

Au

=1

519

All emission angles -1

10

Cu

0.010

500 eV

Ag

1000 eV 2000 eV -2

10

0.005

4000 eV

Al C

0.000

-3

10

Au

500 eV

0.04

0

Cu

= 0.5

η =1

0.02 1000 eV

Ag

2000 eV

0.01

0.02

Al

4000 eV

C 0.00

0.00 0.15 0

Au

η = 0.5

500 eV

= 0.2 0.06

1000 eV

Ag Cu

0.10

2000 eV

Al

0.03

0.05

4000 eV

C 0.00

0.00 0.0

0.2

0.4

0.6

0.8

η = 0.2

1.0

Cosine of emission angle: η

-1

10

Fig. 6. Same as Fig. 5 but for a common positron energy E0 ¼ 2 keV and various targets – C (graphite), Al, Cu, Ag, Au – indicated at the curves.

500 eV 1000 eV 2000 eV

-2

10

firmed by the g0 ¼ 1 curves in Fig. 4. Meanwhile, in accordance with Eqs. (5) and (9), the function W1 at g0 ¼ 1 is directly related to the elastic collision cross-section at scattering angles >90: g g
KðgÞ : W1 ð1; gÞ
K  ð1; gÞ 1þg 1þg Here KðgÞ is just the normalized cross-section expressed as a function of the cosine of the scattering angle (>90). Further, with the relevant data from [19], the functions KðgÞ for Al decrease monotonically with g, but their slopes demonstrate pronounced oscillations, cf. e.g. the insert in Fig. 1.

4000 eV -3

10

0.0

0.2

0.4

0.6

0.8

1.0

Cosine of incidence angle: ηo Fig. 7. Dependences of the probability for elastic backscattering on the direction cosine g0 of incident positrons. The top subfigure shows the total (integrated overR all emission direc1 tions) probability of elastic backscattering: 0 V0 ðg0 ; gÞ dg versus g0 ; the three lower ones demonstrate the dependences resolved in emission direction: V0 ðg0 ; gÞ versus g0 for g ¼ 1, 0.5, 0.2 (top to bottom). The results depicted are for a common target (Al) and various positron energies (E0 ¼ 0:5, 1, 2, 4 keV) indicated at the curves.

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L.G. Glazov, I. P
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All emission angles -1

10

Au

Al -2

C

10

Cu,Ag

-3

10

η= 1

0.04

Au Cu Ag 0.02

Al C 0.00

η= 0.5

Au 0.10

0.05

Cu

Ag

Al C

0.00

η = 0.2 -1

10

Al

Au

C

-2

10

Ag,Cu

10

-3

0.0

0.2

0.4

0.6

0.8

1.0

Cosine of incidence angle: ηo

Fig. 8. Same as Fig. 7 but for a common positron energy E0 ¼ 2 keV and various targets indicated at the curves.

After multiplying with the monotonically increasing and smooth function g=ð1 þ gÞ, one obtains the products
W1 ð1; gÞ exhibiting the moderate oscillations (exemplified by the n ¼ 1, g0 ¼ 1 curve in Fig. 4). Note that such sensitivity to the minor

details of cross-sections is characteristic for W1 and (close to) normal incidence: the effect smears out in higher-order Wn and in W1 for grazing incidence due to relevant integrations (over polar angles and azimuths, respectively). Finally, as W2 ; W3 ; . . . are relatively smooth functions, the oscillations in W1 ð1; gÞ are directly projected onto Vm ð1; gÞ, and – due to the weight factors in Eq. (10) – are best seen in V0 ð1; gÞ. Though the oscillation effect for specific Al cross-sections is relatively minor, its reliable reproduction represents a challenging test for the accuracy of statistical sampling in Monte-Carlo simulation programs. Such tests are not always passed satisfactorily, as demonstrated by a recent comparison [16] of analytical-based and simulations results for elastic electron backscattering. Another interesting feature is observed in Figs. 6 and 8 and concerns variation of the elastic backscattering probabilities with the target atomic number at a fixed energy. In the examples shown, the basic tendency is increase of V0 with the target atomic number Z, from C to Au. There is a notable exception however: the curves for Cu (Z ¼ 29) are consistently above those for Ag (Z ¼ 47). As will be evident from the figures of the next section, this effect of ÔCu–Ag inversionÕ is not limited to V0 , but takes place also for Vm , at least for not too large m 6¼ 0. At first sight, this finding may seem surprising as various parameters pertaining to the adopted cross-sections change monotonically with Z; this concerns e.g. the elastic (kel ) and inelastic (kin ) mean free paths, the ratio kel =kin entering Eqs. (10) and (12), as well as the total and (first and higher-order) transport cross-sections of elastic collisions. However, the origin of the inversion is easily traced, again leading to an explanation in terms of specific properties of the elastic-collision cross-section. Ultimately, the effect originates in the likewise inversion in the normalized elastic-collision crosssection KðgÞ that governs the distribution of scattering angles in a single elastic collision. In accordance with the data from [19], KðgÞ for Cu consistently exceeds that for Ag at scattering angles P 10; this may be quantified by the respective ratios of the transport and total crosssections of elastic collisions, cf. Table 1. Here the

L.G. Glazov, I. P
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521

Table 1 Ratios of the first (r1 ) and second (r2 ) transport cross-sections to the total cross-section r0 of elastic collisions Target

Z¼6C

Z ¼ 13 Al

Z ¼ 29 Cu

Z ¼ 47 Ag

Z ¼ 79 Au

r1 =r0 r2 =r0

0.035 0.080

0.039 0.085

0.078 0.158

0.074 0.148

0.087 0.164

Evaluated from data of [19].

Vm Vm

η0 = 1

0.006

η= 1

η0 = 1

m = 20 m = 10

0.006

η= 0.5

m= 5 m= 1

0.003

Ση

0.003

m= 0

η = 0.2 0.000 0.015

0.000

η= 0.5

m = 10

m= 5

η0 = 0.5

η0 = 0.5

0.015

Ση 0.010

m= 1

0.010

m= 0

η =1

0.005

m = 20

η = 0.2

0.005

0.000

0.000

η0 = 0.2

m= 0

η0 = 0.2

0.08

0.06

η = 0.2

m= 1 Ση

m= 5

0.03

η = 0.5

η= 1 0.00

0

0.04

10

20

m = 20

30

Number of inelastic collisions: m

Fig. 9. Distributions of backscattered positrons in the number of inelastic collisions: Vm ðg0 ; gÞ versus m for various g0 , g, Al target and initial energy E0 ¼ 2 keV. Top, middle and bottom subfigures show the distributions for g0 ¼ 1 (normal incidence), 0.5 and 0.2, respectively. The applicable values of g are indicated at the plots: g ¼ 1 (normal emission; circles), 0.5 (up triangles), 0.2 (down triangles); the plots marked Rg (squares) show R 1 the distributions integrated over all emission directions, i.e. 0 Vm ðg0 ; gÞ dg versus m.

total (r0 ) and transportR (r1 , r2 ) cross-sections are defined through r0 ¼ rel ðX; X0 Þ dX0 and r1;2 ¼

m = 10

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Cosine of emission angle: η Fig. 10. Distributions in emission direction of positrons backscattered after certain numbers m ( ¼ 0, 1, 5, 10, 20, as indicated at the curves) of inelastic collisions: Vm ðg0 ; gÞ versus g for Al target, E0 ¼ 2 keV. The distributions are shown for a few incidence directions: g0 ¼ 1 (normal incidence; top subfigure), g0 ¼ 0:5 (middle), and g0 ¼ 0:2 (bottom).

R

½1  P1;2 ðX X0 Þrel ðX; X0 Þ dX0 , Pi being the Legendre polynomials. As is readily checked numerically, this property of the cross-sections is

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L.G. Glazov, I. P
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Vm

Vm

All emission angles

m= 0

Au

Cu

0.1

Ag

0.010

m= 1

η0 =1, all emission angles

m= 5 m = 10 0.005

0.01

Al

m = 20

C m= 0

η=1

m = 20

0.010

0.000

η0 = 1, η = 1

Au

m= 1 0.02

m = 10

Cu Ag

m= 5 0.005

0.01

Al C

0.000

m= 0 0.04

0.00 0.015

η = 0.5

Cu m= 1

Ag 0.010

m= 5 0.02

η0= 1, η= 0.5

m = 10 0.005

m = 20 0.00

0.1

Au

Al

m= 0

C 0.000

η = 0.2

m= 1

η0 = 1, η= 0.2

m= 5

Au

0.006

Cu

m = 10

Ag

0.01

m = 20

0.003

Al 0.0

0.2

0.4

0.6

0.8

1.0

Cosine of incidence angle: η0

C 0.000

0

5

10

15

Number of inelastic collisions: m Fig. 11. Probabilities for positron backscattering after certain numbers m ( ¼ 0, 1, 5, 10, 20, as indicated at the curves) of inelastic collisions: dependences on the direction cosine of incident particles for Al target and E0 ¼ 2 keV. The top subfigure shows the total (integrated over emission directions) probabilities of elastic (m ¼ 0) and inelastic (m P 1) backscatR1 tering: 0 Vm ðg0 ; gÞ dg versus g0 ; the three lower ones demonstrate the dependences resolved in emission direction: Vm ðg0 ; gÞ versus g0 for g ¼ 1, 0.5, 0.2 (top to bottom).

Fig. 12. Distributions of backscattered positrons in the number m of inelastic collisions; for normal incidence (g0 ¼ 1), E0 ¼ 2 keV, and various targets – C (graphite; squares), Al (circles), Cu (up triangles), Ag (down triangles), Au (diamonds) – also indicated at the graphs. The top and the lower three subfigures show, respectively, the distributions integrated over and reR1 solved in emission direction: 0 Vm ðg0 ; gÞ dg and Vm ðg0 ; gÞ for g ¼ 1; 0:5; 0:2 versus m.

L.G. Glazov, I. P
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Vm

effect is somewhat compensated by the increase of kin =kel with Z; but compensation is only partial as kin =kel values for Cu and Ag (1.76 and 1.79, respectively) happen to be close in the specific case.

η0 = 0.2, all emission angles

Au Cu

0.06

523

Ag Al

4.3. Inelastic backscattering

C

This section exemplifies the inelastic collision number distribution Vm ðg0 ; gÞ. The set of relevant variables – m, g0 , g, not to mention variation with the positron energy and the target atomic number – is too large to aim at a comprehensive overview. Therefore, we just present a few graphical examples, not attempting an extensive discussion of the results. Figs. 9–11 demonstrate dependences of Vm ðg0 ; gÞ on each of the three variables (m, g, g0 , respectively). These results are for the same test case (2 keV eþ fi Al) as in Figs. 2–4 above, that depicted corresponding elastic collision number (Wn ) and path length (Q) distributions. For various sets of incidence and emission angles, Figs. 12 and 13 illustrate variation with the target type of the distributions of backscattered positrons in the number of inelastic collisions.

0.03

0.00

η0 = 0.2, η = 1

Au 0.03

Cu Ag

0.02

Al

0.01

C 0.00

Au Cu

0.08

η0 = 0.2, η = 0.5

Ag

0.04

Al C

5. Conclusion and outlook

0.00 0.15

η0 = 0.2 η = 0.2 Ag Cu

0.10

Al C

0.05

Au 0.00

0

5

10

15

The invariant embedding formalism has been proved to be very effective in the treatment of quasielastic electron and positron backscattering, as was demonstrated for positrons in this paper. As regards positron transport, extensions of this work will include treatment of energy loss below the elastic peak. As was indicated in Section 2, this can be most effectively achieved by means of a suitable transformation of the invariant-embedding equations. The method and the results will be reported in a subsequent work.

Number of inelastic collisions: m Fig. 13. Same as Fig. 12 but for oblique incidence: g0 ¼ 0:2.

directly translated into the values of Wn for Cu exceeding those for Ag (at least for not too large n), and – according to Eqs. (12) and (10) – is further relayed to a likewise effect in V0 ; V1 ; . . .. The

Acknowledgements The authors are grateful to S. Tougaard, P. Sigmund and M. Vicanek for illuminating discussions. This work has been supported by the Swedish Royal Academy of Sciences (KVA).

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