Impact parameters for ionization by high-energy electrons

Ultramicroscopy 80 (1999) 125}131

Impact parameters for ionization by high-energy electrons M.P. Oxley, L.J. Allen* School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia Received 21 December 1998; received in revised form 11 May 1999

Abstract Root-mean-square impact parameters for K- and L-shell ionization by electrons with energies between 60 and 400 keV, and pertinent to energy dispersive X-ray (EDX) analysis, are calculated for a wide range of atoms. Hartree}Fock wave functions with relativistic corrections are used for the bound states and Hartree}Slater wave functions for continuum states. These impact parameters di!er substantially from those given by widely used approximate formulas. Impact parameters for K- and L-shell ionization fall on a single curve, suggesting that the impact parameters are a single-valued function of the ionization threshold energy for a given incident beam energy. For the range of incident beam energies considered, impact parameters for threshold energies up to 45 keV are given in a parameterized form.  1999 Elsevier Science B.V. All rights reserved. Keywords: X-ray microanalysis; Inelastic electron scattering theory

1. Introduction The degree of delocalization of the atomic innershell ionization interaction plays an important part in the interpretation of ionization cross sections when considering ionization in a cystalline environment, where channelling of fast incident electrons leads to a variation in the observed cross section as a function of orientation [1}4]. This orientation dependence is the basis of atom location by channelling enhanced microanalysis (ALCHEMI). Early implementations of ALCHEMI [5}10] assumed that the inner-shell ionization interaction was completely localized at the atomic site. This is not a good assumption for less tightly bound atomic orbitals which may be ionized by incident

* Corresponding author.

electrons with non-negligible impact parameters. Delocalization may also lead to a signi"cant reduction in atomic resolution when using techniques such as electron energy loss spectroscopy (EELS) [11]. Attempts have been made to obtain information on the delocalization from experiment in particular cases [12}14]. Nevertheless, the use of a priori information on the delocalization is the more usual approach. Theoretical estimates of ionization interaction impact parameters [15}17] provide some measure of the delocalization, and may be used in conjunction with the calculated electron intensity distribution across the unit cell in the calculation of C-factors [17,18] which correct for di!ering delocalizations. Similar approaches have been used by other authors [19]. Previous estimates of impact parameters based on a screened hydrogenic model [20] have provided results that di!er substantially from the earlier estimates of Bourdillon [15]. We have previously

0304-3991/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 9 9 ) 0 0 1 0 2 - 3

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compared [21] the half-width at half-maximum (HWHM) of the ionization potential with estimates of the root-mean-square (rms) impact parameter b provided by Pennycook [17] which are based   on an oscillator strength model and a small angle approximation for the momentum transfer. However, the relationship between these two quantities is in general not transparent. In this paper we compare impact parameter estimates calculated using the approximate formulation of Pennycook [17] with those calculated from "rst principles using realistic atomic models for both K- and L-shell ionization. We calculate impact parameters pertinent to energy dispersive X-ray (EDX) analysis using realistic Hartree}Fock bound state wave functions and Hartree}Slater continuum wave functions. The momentum transfer is handled exactly. It is found that impact parameter values calculated in this way can di!er substantially from those given by approximate formulas and this leads to a correspondingly large di!erence in delocalization corrections which use impact parameters. We provide a simple parameterization for EDX rms impact parameters. 2. Theory 2.1. Ionization cross section The di!raction of incident electrons in a crystalline environment makes the cross section for ionization a function of incident beam orientation, the site of the inelastic event within the unit cell and the depth within the crystal. Unlike the case of an isolated atom, the phase of the inelastic scattering matrix is an essential part of the physics. This is taken into account by a general expression describing the cross section for inelastic scattering from a crystal of thickness t which is based on the oneparticle SchroK dinger equation with the absorptive scattering represented by a nonlocal term [22}24]. The expression, which implicitly assumes integration over all "nal states of the scattered electron, is as follows: p"N< 





1! BGH(t) CGuCHH k0 0 u  u GH



# BGH(t) CGuCHH , h kh E uh GH 

where N< is the total crystal volume and  exp[i(jG!jHH)t]!1 BGH(t)"aGaHH . i(jG!jHH)t

The Bloch wave eigenvalues jG in the BGH(t), the Bloch state amplitudes aG and Fourier coe$cients CGu, which represent the eigenvector of the ith state, come from solution of the Bethe scattering equations [23], where u denotes a reciprocal lattice vector. This result can also be obtained with suitable approximations from the formalism of Dudarev and coworkers [25]. While in this paper the kh u describe inner-shell ionization, they may in  general describe any speci"c inelastic scattering under consideration. It is important to note that while the kh u refer to a speci"c form of absorptive  scattering, the eigenvector components CGu in Eq. (1) and complex eigenvalues jG in Eq. (2) come from solution of the total scattering equations [23] and hence, in principle, take into account all forms of absorptive scattering concurrently occurring. In particular, the inclusion of thermal di!use scattering (TDS) is crucially important to obtain accurate cross sections [23]. The "rst term in Eq. (1) (the factor in square brackets multiplied by k0 0) ac counts for ionization by electrons which have been `dechannelleda or absorbed from the dynamical elastic beams by wide angle (mainly TDS) scattering. The second term represents the dynamical contribution to p (which is attenuated by the absorptive scattering). It has been demonstrated that, in energy dispersive X-ray (EDX) spectroscopy, integration over the dynamical states of the inelastically scattered electron averages in such a way that an e!ective plane wave representation of the scattered electrons is a reasonable approximation [26] and this is used here. The cross section for an amorphous solid can be calculated using a single `beama (u"h"0) and Eq. (1) reduces to p"N< k .  

(1)

(2)

(3)

For inner-shell ionization the inelastic scattering coe$cients kh u in Eq. (1) take the following 

M.P. Oxley, L.J. Allen / Ultramicroscopy 80 (1999) 125}131

form [23]:



4F[site] Ki kh u"  (2p)k < a    F@H(Qh,j)F@(Qu,j) ; n J dX dX di, K G )Y "Qh""Qg " KJ (4)




 

where j is the ejected electron wave vector and F@(Qu,j) the transition matrix element for the atom of species b. The crystallographic site term, F[site], is given by F[site]" exp[!M (u!h)]exp[i(u!h) ) s L], @ @ @L (5) where the vectors s L describe the position of each @ atom of type b within the unit cell and M (q)"q1u2 is the Debye}Waller factor for @  @ atoms of type b. For EDX, the integration over the solid angle dX "sin h dh d extends over all )Y space (K is the wave vector of the scattered electron). The range of integration over the magnitude of the ejected electron wave vector is determined by the threshold energy E for ionization.The number  of electrons in an orbital is taken into account by the factor n J. The atomic transition matrix element K for atom species, b, F@(Qh,j) is given by



F@(Qh,j)" b@H(j,r)exp[iQh, ) r]u@ (r) dr. 

(6)

Here Qh"q#h, with q the momentum transfer. The wave functions for the bound and continuum states are given by u@ (r) and b@(j,r), respectively.  The sum over the azimuthal quantum number m of J the initial bound state in Eq. (4) is required for other than s-orbitals. Analytic evaluations of the integral inside the round brackets in Eq. (4) for K-shell ionization in a linear momentum representation and using a screened hydrogenic model have been presented previously [27]. This integral may be calculated in an angular momentum representation [28] which allows the use of more realistic atomic wave functions and is not limited to K-shell ionization. Calculations in this paper, using both Eqs. (1) and

127

(3), are based on a recently presented implementation of this method [21]. 2.2. Impact parameters To characterise the degree of delocalization of the electron impact ionization interaction, Pennycook [17] proposed the use of root-meansquare impact parameters b . Using our notation,   b is given in terms of the ionization cross section   de"ned in Eq. (3) as





 1 dp 1 b " d*E dX , (7)   )Y p d*E dX q )Y where *E is the energy lost by the incident electron. Pennycook uses a simpli"ed oscillator strength model to describe the di!erential cross section dp/d*E dX which assumes the oscillator )Y strength is unity up to some cuto! scattering angle h "(*E/E ), where E is the energy of the inci   dent electron. Beyond this angle the oscillator strength is assumed to be zero. Using a small angle approximation to calculate q, and taking the limit E ;E , where E is the threshold energy of the    atom being ionized, Pennycook [17] approximates the rms impact parameter appropriate to X-ray emission as






\

v E 16E  b " ln  ln , (8)   E E E    where v is the speed of the incoming electron. In the results presented here we numerically integrate Eq. (7) using the appropriate form of the di!erential cross section derived from Eqs. (3) and (4) for the single beam case and treat the momentum transfer

q exactly.

3. Results In Fig. 1 we compare the results calculated using Eq. (7) with those obtained from Eq. (8) for threshold energies which lie in the approximate range 300 eV}45 keV for incident electron energies of 100 keV (Fig. 1(a)) and 400 keV (Fig. 1(b)). This encompasses the range of threshold and incident energies usually encountered in EDX measurements.

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Fig. 1. Root-mean-square impact parameters b for ionization   as a function of threshold energy E for (a) 100 keV incident  electrons and (b) 400 keV incident electrons. In each case calculated values of b for K-shell ionization are shown by the   open circles and those for L-shell ionization are shown by the "lled triangles. Fits to these calculated values using Eq. (9) are shown by the solid lines. The values of b given by Pen  nycook's approximate formula, Eq. (8), are shown for comparison purposes by the dashed lines.

In our calculations, the bound state wave functions in Eq. (6) and threshold energies E are cal culated using Cowan's program RCN [29], which calculates Hartree}Fock wave functions with relativistic corrections. The continuum wave functions were calculated by solving the SchroK dinger equation using a Hartree}Slater potential. K-shell results (shown by the hollow circles for each element) are calculated for all elements in the periodic table from carbon (Z"6, E "309 eV) to tin  (Z"50, E "29.4 keV) with additional results  given for caesium (Z"55, E "36.2 keV) and neo dymium (Z"60, E "43.8 keV) to cover a range  of threshold energies up to approximately 45 keV. The solid triangles show some L-shell results

Fig. 2. Root-mean-square impact parameters b as a function   of incident energy E for (a) the carbon K-shell, (b) the germa nium L-shell, (c) the germanium K-shell and (d) the neodymium K-shell. The results calculated here are shown by the solid lines and the linearity of the results can be seen by comparison with the dotted straight lines. The results obtained from the estimate of Pennycook (Eq. (8)) are shown by the dashed lines.

ranging from calcium (Z"20, E "390 eV) to  mercury (Z"80, E "13.4 keV). The e!ective  threshold energies for L-shells can be estimated from E "(pE#pE)/(p#p), where p(E)     and p(E) are the cross section (threshold energy)  for the 2s and 2p orbitals, respectively. The crosssection values are de"ned by Eq. (3). In practice we found that E +0.2E#0.8E gave a good empiri   cal "t for all the L-shell cases considered here. We note that the L-shell results closely follow those for K-shell ionization as a function of threshold energy, consistent with results found by Holbrook and Bird [30]. This suggests that in practice the impact parameters for K- and L-shell ionization are a single-valued function of the ionization threshold energy for a given incident beam energy. The dashed line shows the corresponding results calculated using Eq. (8). It is clear for both incident energies that the results calculated using Eq. (7) and the approximate expression given by Eq. (8) di!er

M.P. Oxley, L.J. Allen / Ultramicroscopy 80 (1999) 125}131

Fig. 3. Variation of Ge L-shell C factor, C , as a function of %* incident energy E for [1 0 0] axial chanelling. The results ob tained from experiment by Pennycook are shown by the open circles. Our theoretical results found from "rst principles using Eq. (12) are shown by the solid line. Pennycook's calculation from Ref. [17] which uses Eq. (13) is shown by the dashed line. C for 100 keV incident electrons estimated from Eq. (14) %* using the "rst principles value of b , Eq. (7), and Pennycook's   estimate, Eq. (8), are shown by the solid and open triangles, respectively.

signi"cantly. This is particularly true for small threshold energies where delocalization is expected to be most signi"cant. In general, the estimates of Pennycook for b are less than our results. This is   due to the use of a small angle approximation in calculating q, which overestimates this quantity for angles up to about 603 (c.f. Eq. (7)). The fact that, for 100 keV electrons, Pennycook's estimate exceeds our values for E around 30 keV and above is due to  the fact that his assumption that E ;E is no   longer valid and this becomes the dominant consideration. We note that for E '45 keV the  calculated results using Eq. (8) diverge strongly from those calculated here. Comparison of Fig. 1(a) and 1(b) show that the agreement between the present results and those given by Eq. (8) is signi"cantly worse at low threshold energies for 400 keV incident electrons than for the 100 keV case. The solid lines shown in Fig. 1 represent a parametric form of the impact parameter as a function of threshold energy for a "xed incident energy. This parametric form can be expressed as  a b " L,   EL L 

(9)

129

where the +a , are constants. In the region of low L threshold energies and hence large impact parameters, this parameterization is extremely close to the values obtained from Eq. (7) for both incident energies. For higher vales of E , closer to 45 keV,  the parametric form given by Eq. (9) di!ers only slightly from the exact calculation. The constants +a , must be determined as a function of E for this L  form of parameterization to be generally applicable. In Fig. 2 we plot b as a function of incident   energy E , which is shown plotted on a common  log axis, for given values of threshold energy E .  The impact parameters calculated using Eq. (7) are shown by the solid line and can be seen to be almost linear in each case. Dotted straight lines are given as a visual guide in each case. For low to moderate threshold energies the impact parameter varies almost linearly with log (E ). The results for   the Ge K-shell are slightly non-linear. However, even in the extreme case of the Nd K-shell, where the theshold energy is 43.8 keV, the variation from the linear form is at most about 2.5%. The value of b calculated using Eq. (8) is shown by the dashed   lines. We see that for low-to-moderate threshold energies the dashed lines are also approximately linear. The slope of the results calculated using Eq. (8) is, however, much smaller than that obtained using Eq. (7). This leads to a larger discrepancy at high incident energies when compared to low incident energies. This is consistent with the greater di!erence between Eqs. (7) and (8) observed at low threshold energies for 400 keV incident electrons in Fig. 1(b) when compared to the 100 keV results shown in Fig. 1(a). For the Nd K-shell the variation of Eq. (7) with log (E ) is decidely non-linear.   Here, for low incident energies, E +E and the   failure of the approximation E E used to derive   Eq. (8) is clearly illustrated. The linear nature of our results suggests a parameterization, for a "xed threshold energy E , of the form  b "c #c log (E ), (10)       where c and c are constants. Combining Eqs. (9)   and (10) we get an expression in terms of both E and E which may be written as follows:    c #c L log (E )    . b " L (11)   EL R L

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Table 1 Parameters for use in Eq. (11) which parameterizes the impact parameter, b (in), for K- and L-shell ionization pertinent to   EDX as a function of the energy of the incident electron (60}400 keV) and threshold energy (approximately 300}45 keV)

where b,1/q. Pennycook suggests that to a good approximation C may be calculated from

n

c L 

c L 

0 1 2 3

6.5333;10\ !1.4794;10\ 3.8261;10\ !2.5131;10\

!1.0057;10\ 2.4589;10\ !1.5115;10\ 3.0998;10\

However, as we have shown above, the value of b calculated from Eq. (7) can di!er substantially   from that given by Eq. (8). This can lead to a signi"cant variation in the value of the C-factor calculated from Eq. (14). For example, for 100 keV incident electrons and using the approximation given by Eq. (14), b calculated using Eqs. (7) and   (8) yields C values of 0.51 and 0.74, respectively. %* These values have been estimated using the electron intensity distribution calculated by Pennycook (Fig. 8 of Ref. [17]) and are indicated by the open and "lled triangles in Fig. 3. In Ref. [17] the ratio of C factors given in Eq. (12) was measured as a function of incident energy for [1 0 0] axial channelling with a sample thickness of 645As . Assuming that C "1, the values of %) C derived from this data are shown in Fig. 1 by %* the hollow circles. The dashed line shows the value of C calculated by Pennycook by numerical %* integration of Eq. (13) and using an oscillator strength model of the di!erential cross section occuring inside the integral. As noted by Pennycook the trend of his theoretical results matches that of the experiment but di!erences of approximately 30% can be seen. The solid line represents theoretical calculations of Eq. (12), where the ratios of characteristic X-ray counts have been calculated as ratios of theoretical cross sections. The channelling cross sections are calculated using Eq. (1) and the non-channelling cross sections have been calculated using Eq. (3), with the input atomic wave functions calculated as discussed in Section 3. We assume, as does Pennycook, that C "1 to ob%) tain C . This matches the experimentally derived %* C much more closely than Pennycook's calcu%* lations (dashed line). In our calculations (solid line), we have used 69 beams and included absorption due to TDS calculated from an Einstein model [31}33] to calculate the ratios of X-ray counts that appear on the right-hand side of Eq. (12). The thermal motion of the Ge atoms has been taken into account using a projected mean square thermal displacement of 1k2"0.0072 As  [34]. An

The parameters +c L, c L, required to calculate   b from Eq. (11) are given in Table 1. These are   appropriate for threshold energies ranging between 300 eV and 45 keV, and incident energies between 60 and 400 keV. The threshold energy E and  incident energy E must be entered in keV.  The resulting rms impact parameter is given in units of As . The application of impact parameter calculations to correct for delocalization was proposed by Pennycook [17]. He introduced C-factors [18] in an attempt to correct the assumption commonly used in ALCHEMI that the ionization interaction is perfectly localized. Pennycook examined the case of [1 0 0] axial channelling in Ge and attempted to correct for the di!ering delocalizations of the Ge Kand L-shells. The ratio of C-factors for this case is de"ned by









N! N! C %)!1 , %*" %*!1 (12) C N0 N0 %) %* %) where N! and N0 are the measured characteristic X-ray counts in the channelling and random or non-channelling orientations, respectively. The subscripts GeK and GeL refer to the Ge K- and L-shells. Pennycook has assumed that C "1 %) and C does not appear explicitly in the corre%) sponding Eq. (9) in Ref. [17]. Pennycook's theoretical calculations of C-factors are based on the distribution of electron intensity I across the unit cell renormalised to a maximum value of one. They are de"ned by



1 @  dp C" I(b) d*E db, p  d*E db @

(13)

C+I(b

).  

(14)

M.P. Oxley, L.J. Allen / Ultramicroscopy 80 (1999) 125}131

additional mean absorption with a mean-free path equal to the mean-free path for TDS at each energy has been included. Part of this additional absorption can be considered to be associated with delocalized electronic excitations and its inclusion provides good agreement with the Ge K-shell channelling results presented by Pennycook in Fig. 2 of Ref. [17]. Futhermore, we note that our theoretical calculations for C are insensitive to sensible %* variations about the mean absorption included in the calculations, even though the individual channelling strengths N!/N0!1 are. 4. Summary and conclusions We have calculated root-mean-square impact parameters appropriate for EDX measurements using realistic atomic models and have shown that they di!er signi"cantly from estimates based on oscillator strength models with approximate treatment of momentum transfer. C-factors encompassing similar approximations are also signi"cantly di!erent from those we have calculated. A parameterized form of the root-mean-square impact parameter appropriate for EDX measurements based on K- and L-shell ionization with threshold energies in the range of 300 eV}45 keV has also been provided. This parameterization is valid for incident energies between 60 and 400 keV. Acknowledgements We would like to thank Dr C. J. Rossouw for suggesting these calculations and for useful discussions and advice. L. J. A. acknowledges "nancial support from the Australian Research Council. References [1] P. Duncumb, Philos. Mag. 7 (1962) 2101. [2] C.R. Hall, Proc. Roy. Soc. Lond. Ser. A 295 (1966) 140.

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[3] D. Cherns, A. Howie, M. H. Jacobs, Z. Naturf. 28a (1973) 565. [4] A.J. Bourdillon, P.G. Self, W.M. Stobbs, Philos. Mag. A 44 (1981) 1335. [5] J. Taft+, J. Appl. Crystallogr 15 (1982) 378. [6] J. Taft+, Z. Lilienthal, J. Appl. Crystallogr 15 (1982) 260. [7] J.C.H. Spence, J. Taft, Scanning Electron Microscopy/ 1982/II, in: O. Johari (Ed.), SEM, Chicago, 1982, p. 523. [8] J. Taft+, J.C.H. Spence, Science 218 (1982) 40. [9] J. Taft+, J.C.H. Spence, Ultramicroscopy 9 (1982) 243. [10] J.C.H. Spence, J. Taft+, J. Microscopy 130 (1983) 147. [11] S.J. Pennycook, in: S. Amelinckx, D. van Dyck, J. van Landuyt, G. van Tendeloo (Eds.), Handbook of Microscopy, Methods II, VCH Verlagsgesellschaft mbH, Weinheim, 1997, p. 595 [12] J. Bentley, Proceedings 44th EMSA, San Francisco Press, 1986 p. 704. [13] J.C.H. Spence, T. Kuwabara, Y. Kim, Ultramicroscopy 26 (1988) 77. [14] W. NuK chter, W. Sigle, Philos. Mag. A 71 (1995) 165. [15] A.J. Bourdillon, Philos. Mag. A 44 (1984) 839. [16] S.J. Pennycook, J. Narayan, Phys. Rev. Lett. 54 (1985) 1543. [17] S.J. Pennycook, Ultramicroscopy 26 (1988) 239. [18] S.J. Pennycook, Scanning Microscopy 2 (1988) 21. [19] Z. Horita, M.R. McCartney, H. Kuninaka, Philos. Mag. A 75 (1997) 153. [20] C.J. Rossouw, V.W. Maslen, Ultramicroscopy 21 (1987) 277. [21] M.P. Oxley, L.J. Allen, Phys. Rev. B 57 (1998) 3273. [22] L.J. Allen, C.J. Rossouw, Phys. Rev. B 47 (1993) 2446. [23] L.J. Allen, T.W. Josefsson, Phys. Rev. B 52 (1995) 3184. [24] L.J. Allen, T.W. Josefsson, Phys. Rev. B 53 (1996) 11285. [25] S.L. Dudarev, L.-M. Peng, M.J. Whelan, Phys. Rev. B 48 (1993) 13408. [26] T.W. Josefsson, L.J. Allen, Phys. Rev. B 53 (1996) 2277. [27] V.W. Maslen, J. Phys. B 16 (1983) 2065. [28] D.K. Saldin, P. Rez, Philos. Mag. B 55 (1987) 481. [29] R.D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, Berkeley, 1981. [30] O.F. Holbrook, D.M. Bird, Proceedings in Microscopy and Analysis, Jones and Begall, New York, 1995, p. 278. [31] C.R. Hall, P.B. Hirsch, Proc. Roy. Soc. (London) Ser. A 286 (1965) 158. [32] C.J. Humphreys, P.B. Hirsch, Phil. Mag. 18 (1968) 115. [33] L.J. Allen, C.J. Rossouw, Phys. Rev. B 42 (1990) 11644. [34] N.M. Butt, J. Bashir, B.T.M. Willis, G. Heger, Acta Crystallogr A 44 (1988) 396.