A simple procedure for evaluating effective scattering cross-sections in STEM

Uhramicroscopy 16 (1985) 265-268 North-Holland, Amsterdam

265

SHORT N O T E

A SIMPLE PROCEDURE FOR EVALUATING EFFECTIVE SCATrERING C R O S S - S E C T I O N S IN S T E M H. K O H L lnstitut ffw Angewandte Physik, Technische Hochsehule Darmstadt, Hochschulstrasse 2, D -6100 Darmstadt, Fed. Rep. of Germany Received 30 November 1984

The quantitative interpretation of STEM images relies on a precise knowledge of the effective scattering cross-sections. To determine these cross-sections, it is necessary to take into account the finite angles of both the illuminating and the recorded beams. In the following a procedure is outlined by which at least two of the four integrations can be performed analytically. Single atom differential cross-sections require only a single numerical integration. Although the proposed method is rather simple, it does not seem to be commonly known.

1. Introduction

the energy window

The quantitative interpretation of electron micrographs necessitates an accurate knowledge of the cross-sections involved. For a Fixed-Beam Electron Microscope (FBEM) the angular width of the illuminating beam can often be neglected. In this case the effective scattering cross-section is defined as [1]

W ( E ) = { 1' 0,

oerf(,o)= f A(o)

d20.

(1)

Here d o / d O denotes the differential scattering cross-section, and A(O) describes the transmission properties of the objective aperture. For a circular aperture subtending an angle a 0, the aperture function

fl, A(O)=/0,

if0 <101<~0,

otherwise,

(3) '

we find [2-4] Oeff(O~0; E l , E 2 )

=fW(E)

~(o)

dZo(E, O) dO d E (4)

/ d20 dE.

In a Scanning Transmission Electron Microscope (STEM) the situation is more complicated, because the objective aperture angle a 0 and the acceptance angle of the bright-field detector 130 are of the same order of magnitude (fig. 1). Then the effective scattering cross-section takes the form 1 : oe. ( - 0 , / 3 o )

(2)

:

da(/3-a)

=

(5)

× d2a d2/3,

otherwise,

has a "top-hat" distribution. If we consider image formation by inelastically scattered electrons, we must use the double-differential cross-section d2o/dg2dE. It is advantageous to characterize the action of the energyselecting filter or spectrometer by an energy window W(E), where E is the energy loss. Using

ifE'<~E<~E2'

where DBF(/3) =

o,

for 0 ~<{/31~
(6)

is the detector function. For large illumination angles the neglect of the finite beam divergence can lead to serious errors, as has been shown by

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H. Kohl / Effective scattering croxs-section.s" in S T E M

266

object,ve aDerture

denote their transmission functions by A(a) and D ( ~ ) respectively. Substituting the scattering angle 0 =/3 - a in eq. (5), we obtain ,do(0)

object plane

Here

~A = f A(~) d2~ aperture i ~ Ictl

Fig. 1. Schematic ray diagram in a STEM. The scattering angle 0 is the difference between the angular directions ~ and a of the scattered and incident electron respectively.

several authors [1,5,6]. The integral (5) can be evaluated analytically if the differential scattering cross-section is approximated by appropriate model functions [7,8,19] (e.g. the genz-Wentzel model for elastic scattering [9-121 or the dipole approximation for inelastic processes [13]). Currently the obtainable accuracy of the quantitative interpretation of images is limited by the inaccuracies of the available effective crosssections. The differential cross-sections, however, are known to a higher degree of accuracy. (For elastic cross-sections see, e.g., the tables of Bonham and Sch~fer [14]; for inelastic cross-sections, the calculations by Leapman et al. [15] and the hydrogenic model by Egerton [2,16],) The direct numerical evaluation of the multiple integral (5) is very time consuming, as has been shown by Joy and Maher [6]. However, a simple procedure exists which allows an analytical evaluation of two of the four integrations in (5) and hence reduces the numerical effort considerably. Our procedure: hardly reduces the numerical expenditure of phase: contrast calculations for STEM. In this case efficient approximation techniques should be applied

[z01. 2. Calculation

The proposed method is quite general and valid for arbitrarily shaped apertures and detectors. We

(8)

is the solid angle subtended by the objective aperture. The integration over the angle ~ does not affect the differential scattering cross-section. The normalized cross-correlation, F ( 0 ) -= ~l 4 fA ( a ) D(0 + ~) d~a,

(9)

of the aperture and detector functions can be interpreted as probability to register an electron which has been scattered by an angle 0. This probability is given by the integral over the angular detection efficiency D(0 + ~) multiplied by the angular probability distribution function A (a)/[2~ for the incident electrons. The cross-correlation function can be visualized as the common area of the aperature and detector, displaced by a distance 0. Inserting eq. (9) into relation (7), we obtain

m.= f F(O)

d20.

(10)

Comparing this equation with the FBEM formula (1), we find that the only difference is the replacement of the aperture transmission function A(O) by the normalized cross-correlation function F(0). This change of the integrand does not severely complicate the remaining numerical integration. In many cases both F(O) and d c ( O ) / d ~ depend only on the absolute value 0 of 0. Then the effective cross-section is given by

m,r=f F ( O ) ~ 2 , , O dO.

(11)

The cross-correlation function is closely connected with the contrast transfer function for incoherent imaging (e.g. ref. [17]). For convenience

H. Kohl / Effectioe scattering cross-sections in S T E M

we shall list two of the most commonly encountered types.

267

for the effective scattering cross-section %r,(ao, rio) = f o

,~,,+fl,,

do(0)

FBv(ao, rio; 0 ) - T ~ 2 ~ r O

(15)

3. The cross-correlation function 3.1. Bright field In the STEM bright-field mode the aperture and detector functions are generally given by eqs. (2) and (6) respectively. (We assume that the objective aperture is uniformly illuminated.) Then the cross-correlation function can be interpreted as the common area of two circles of radii a o and flo displaced by the scattering angle 0 (fig. 2). From geometrical considerations we find

v
forO<~O<~[ao-flo[,

+ Bo

0,

o~.(,~,,. B,);&. £ 2 ) = f w ( £ ) ×

Fm.(a o,

rio" 0)

d2o(E, 0) 2rr0 dO d E . d12dE (16)

3.2. Dark field

~ - 1[arccos(x) + ( .0~/a0~) arccos(y)

=

Similarily we can use the relations (12), (13) and (14) to calculate an effective inelastic scattering cross-section. Here we must also integrate over the energy window (3) yielding

For d2o/da2 d E one can use, e.g., the hydrogenic cross-sections given by Egerton [2,16].

&F(,x0, flo; 0) (*q2 ~

dO.

02) 2 l'

for [a 0 - fl0[< 0 < a o + rio, otherwise. (12)

Dark-field images in STEM are generally obtained by employing an annular detector, whose inner and outer angles are denoted by fl~ and f12 respectively. In this case the detector function DDv(fl) is given by

Here 0 < = m i n ( a o, fl0},

DDv(fl)=

~o + 05 -rid x

2a00

(13)

y

2floO

(14)

Inserting these expressions into eq. (11) we obtain

1,

f°rfll ~<)fl]~< f12,

O,

otherwise

(17)

The corresponding cross-correlation function can be expressed as the difference of two bright-field cross-correlation functions

gDv(ao, fl,, flz; 0) = Fuv(a o, f12; O) - F u v ( a o, fi,: 0).

(18)

This function is non-vanishing only if fll - a0 ~< 0 ~< a o + fl2.Eusemann [18] has shown that this expression can be further simplified.

4. Conclusion

Fig. 2. G e o m e t r i c a l i n t e r p r e t a t i o n of the cross-correlation function as the c o m m o n area of two circles.

We have shown that the standard evaluation of effective scattering cross-sections in STEM can be facilitated considerably. The remaining numerical integration must be taken over the differential

268

tl. Kohl / EfJectit,e scattering cross-sections in S T E M

c r o s s - s e c t i o n m u l t i p l i e d b y a f u n c t i o n F, w h i c h describes the probability that an electron deflected b y a n a n g l e 0 f r o m its o r i g i n a l d i r e c t i o n will b e r e c o r d e d b y t h e d e t e c t o r . T h i s p r o b a b i l i t y is g i v e n by the normalized cross-correlation between the aperture function A(~) and the detector function D(,8). For most recording geometries the crossc o r r e l a t i o n f u n c t i o n c a n b e v i s u a l i z e d as t h e c o m m o n a r e a o f t w o c i r c l e s o r o f a s i n g l e circle a n d a ring. In t h e s e c a s e s t h e c o r r e l a t i o n f u n c t i o n c a n b e evaluated by simple geometric considerations.

Acknowledgements I w o u l d like to t h a n k P r o f e s s o r H. R o s e a n d D r . R. E u s e m a n n for v a l u a b l e d i s c u s s i o n s .

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[5] M.S. lsaacson, in: Scanning Electron Microscopy/1978, Vol. 1. Ed. O. Johari (SEM, AMF O'Hare, IL, 1978} p. 763. [6] D.C. Joy and D.M. Maher, J. Microscopy 124 (1981) 37. [7] R. Eusemann, H. Rose and J. Dubochet, J. Microscopy 128 (1982) 239. [8] H. Kohl, PhD Dissertation, TH Darmstadt (1983), unpublished. [91 G. Wentzel, Z. Physik 40 (1927) 590. [10] G. Moliere, Z. Naturforsch. 2a (1947) 133. {11] S. Fltigge and H. Marschall, Rechenmethoden der Quantentheorie (Springer, Berlin, 1952) p. 262. [12] F. Lenz, Z. Naturforsch. 9a (1954) 185. [13] H. Bethe, Ann. Physik (Ser. 5) 5 (1930) 325. [14] R.A. Bonham and L. SchMer, Complex Scattering Factors for the Diffraction of Electrons by Gases, in: International Tables for X-Ray Crystallography, Vol. IV (Kynoch, Birmingham, 1974) p. 176. 115] R.D. Leapman, P. Rez and D.F. Mayers. J. ('hem. Phys. 72 (1980) 1239. [16] R.F. Egerton, in: Proc. 39th Annual EMSA Meeting, Atlanta, 1981 (Claitor's, Baton Rouge, LA, I981) p. 198. [171 M. Born and E. Wolf, Principles of Optics (Pergamon, 1965) oh. 9.5.2. [181 R. Eusemann, PhD Dissertation. TH Darmstadt (19841, unpublished. [19] A.J. Craven, T.W. Buggy and R.P. Ferrier, in: Quantitative Microanalysis with High Spatial Resolution, Eds. E.W. Lorimer, M.H. Jacobs and P, Doig (Metals Society, London, 1981) p. 141. [20] E. Zeitler and M. lsaacson, in: Proc. 32nd Annual EMSA Meeting, St. Louis, 1974 (CIaitor's Baton Rouge, LA, 1974) p. 370.