Transition probability functions for applications of inelastic electron scattering.

In this work, the transition matrix elements for inelastic electron scattering are investigated which are the central quantity for interpreting experiments. The angular part is given by spherical harmonics. For the weighted radial wave function overlap, analytic expressions are derived in the Slater-type and the hydrogen-like orbital models. These expressions are shown to be composed of a finite sum of polynomials and elementary trigonometric functions. Hence, they are easy to use, require little computation time, and are significantly more accurate than commonly used approximations.

Introduction

Ever since (Geiger and Marsden, 1909) shot α particles on a gold foil (Geiger and Marsden, 1909) and (Rutherford, 1911) subsequently used their results to confirm that atoms consist of a very small nucleus surrounded by an electron cloud (Rutherford, 1911), scientists have used scattering effects to determine the properties of otherwise invisible or inaccessible objects.

In particular, quantum systems are usually investigated by means of scattering experiments. The central quantity in these quantum mechanical scattering systems is the matrix element of the transition operator ,which describes the amplitude for the transition from a product state |I, i〉 = |I〉|i〉 to another product state |F, f〉 = |F〉|f〉 in first order Born approximation. Here, we distinguish between the states of the probe (lower-case letters) and the target (upper-case letters).1

In this work, we limit ourselves to the treatment of the electronic transition of a single, isolated atom from an occupied, tightly bound initial state to a unoccupied, loosely bound final state and the effect that transition has on the probe triggering it. Assuming a Coulomb-like transition potential,with configuration space vectors r, R, r′, R′, momentum vectors k, k′, and inserting appropriate identity operators, the transition matrix element can be written as

Here, we assume that the probe electron was prepared as a plane wave, i.e., |i〉 = |k〉, and that the detector also measures plane waves, i.e., |f〉 = |k′〉.2 In that case, the transition matrix element assumes its commonly used form,or, by virtue of the partial wave or Rayleigh expansion,where the denote the spherical harmonics and the j are the spherical Bessel functions of first kind.

This absolute value squared of the matrix element occurs in scattering cross sections, generalized oscillator strengths (GOS), as well as the dynamic form factor (DFF), and is therefore a central quantity for the interpretation of inelastic electron scattering. Moreover, the complex matrix element enters into the generalization of the DFF: the mixed dynamic form factor (MDFF), which is very important for the quantitative interpretation of, e.g., electron energy loss spectrometry (EELS) data (Kohl and Rose, 1985; Nelhiebel, 1999; Schattschneider et al., 2000; Löffler and Schattschneider, 2010).

Up to this point, the treatment is exact within first-order Born approximation. In order to evaluate the matrix element further, one needs to explicitly specify the functions 〈R|I〉 and 〈R|F〉, however. For the electronic transitions treated here, two common approximations are Slater-type orbitals (STO) (Slater, 1930; Clementi and Raimondi, 1963) and hydrogen-like orbitals (HLO) (Egerton, 1996). In both models, the initial and final states are modeled as a spherical harmonic angular dependence and a (analytic) radial wavefunction ψ:With this, Eq. 5 becomeswhere the angular integral evaluates to Wigner 3j symbols (Nelhiebel, 1999; Nelhiebel et al., 1999; Schattschneider et al., 2006). Hence, we will only deal with the radial integralin this work, for which we will derive an analytical form in the STO and HLO models.

The rationale behind this choice of orbitals is the fact that the tightly bound (and hence strongly localized) initial state assumed here is described well by an atomic model. Since the matrix element can only be non-zero if both the initial and final states are non-zero, this effectively selects a portion of the final state that is close to the nucleus as well.

Crystal and many-body effects, on the other hand, are often relatively small perturbations—especially in the commonly experimentally accessible regions. In addition, for elucidating the underlying fundamentals, an analytical treatment in an isolated atom model is usually advantageous.

In many articles (see, e.g., Schattschneider et al., 1999; Egerton, 1996; Ma et al., 1990, and references therein), eq. 8 is simplified further by taking the small angle limit (also known as dipole-approximation),in which case the integral over R in eq. 8 becomes only a weighting factor (the factor Q can be moved out of the integral).

For increasing momentum transfer, a leading term Q would increase boundlessly, giving rise to a similar effect as the ultraviolet catastrophe. To avoid that, an artificial cutoff Q is sometimes introduced, which is only fighting the symptoms instead of the cause, however, and is not very elegant. In addition, is has recently been shown experimentally using electron energy loss spectrometry (EELS) that a Q dependence is an oversimplification even for Q < Q and can lead to errors of the order of 25% (Löffler et al., 2011; Essex et al., 1999).

Slater-type orbitals

The radial part of STOs is given by (Slater, 1930; Clementi and Raimondi, 1963)where n is an effective quantum number (which is not necessarily an integer). is an effective nuclear charge, with the physical nuclear charge Z and a screening factor s. The normalization constant is given by

Then, eq. 8 becomeswhere we defined and ζ = (ζ + ζ)/a. As derived in appendix A, this integral can be evaluated analytically, yieldingwith the constants as defined in appendix A. This is the first main result.

In table 1, eq. 13 is evaluated for the most important transitions with λ = 0, 1, 2, corresponding to monopole, dipole, and quadrupole transitions, respectively.

Table 1

Weighted radial wave function overlap in the STO model for monopole (λ = 0), dipole (λ = 1), and quadrupole (λ = 2) transitions. For the definitions of the constants N, N, n, ζ refer to the text.

λ = 0	N_IN_FΓ(n)Q(ζ^2+Q^2)^n2sinnarctanQζ
λ = 1	N_IN_FΓ(n−1)Q^2(ζ^2+Q^2)^n2ζsinnarctanQζ−QncosnarctanQζ
λ = 2	N_IN_FΓ(n−2)Q^3(ζ^2+Q^2)^n+12ζ(Q^2−3nQ^2−n^2Q^2+3ζ^2)sinnarctanQζ−Q(Q^2+3nζ^2−n^2Q^2+3ζ^2)cosnarctanQζ

It should be noted that for Q→ ∞, withit is straight forward to derive that eq. 13 behaves asNote that the inclusion of the cos((n − λ)π/2) term is necessary in case n is an integer and n − λ is even. In that case, sin((n − λ)π/2) = 0 and the asymptotic behavior is described by the second term only. Eq. 15 is very useful as it gives a simple approximation to the Q-dependence of 〈j(Q)〉 for large arguments. This is important as it allows to easily determine the maximal Q to be used in numerical simulations (or equivalently to estimate the systematic error introduced by considering only momentum transfers up to a certain maximum Q).

For Q → 0 the behavior of eq. 13 is more generally more complicated3, though. On the one hand, Q is nested deep inside several trigonometric functions, on the other hand many low-order terms cancel due to the unique form of the . In addition, evaluating eq. 13 numerically is also dangerous (because of the division by small numbers). Hence, for this case, evaluating eq. 12 for small Q directly is favorable. Provided that (where Rmax is the supremum of radii for which the product ψ(R)ψ(R) is non-negligible), this becomes

This shows that the usually used approximation eq. 9 is perfectly recovered for small Q. Furthermore, eq. 16 is important for the actual implementation in simulation software packages. As was noted before, inserting Q = 0 into eq. 13 directly would produce a division-by-zero error. Hence, for Q ≈ 0, eq. 16 should be used instead.

Hydrogen-like orbitals

Hydrogen-like orbitals are very similar to STO in so far as the same terms that appear for STOs appear in HLOs as well. There are four key differences, however: (a) in HLOs, the principal quantum number n is an integer, (b) the radial part of the wave function depends on the angular momentum quantum number l, (c) HLOs have an additional factor represented by (generalized) Laguerre polynomials, and it can thus be ensured that (d) HLOs with same l, but different n are orthogonal.

In general, the radial part of the HLOs can be written aswhere the are the generalized Laguerre polynomials, and a = a0m/μ with the electron mass m, the reduced mass μ, and the Bohr radius a04. Typically, ζ is set to be equal to the (unscreened) nuclear charge Z. This does not influence the further calculation here, however, and so we use ζ to indicate that screening may be included, and to preserve the analogy to the STO results. The normalization constant is given by

With this model, the weighted radial wave function overlap eq. 8 becomeswhich can be rewritten aswith and the coefficients p as defined in appendix B.

The integral has exactly the same form as for STOs, with the solution (see appendix A)Since, contrary to the situation for STOs, b is an integer here, this can be simplified further. Withone getsThis is the second main result. It should be emphasized that eq. 23 has the form of a the quotient of two polynomials,As such, it is only marginally more complicated than the dipole approximation (which is simply a linear polynomial in Q), gives but an exact expression of the weighted radial wave function overlap in the framework of the HLO model.

In table 2, eq. 23 is evaluated for the most important transitions with λ = 0, 1, 2, corresponding to monopole, dipole, and quadrupole transitions, respectively.

Table 2

Weighted radial wave function overlap in the HLO model for monopole (λ = 0), dipole (λ = 1), and quadrupole (λ = 2) transitions. For the definitions of the constants N, N, n, ζ refer to the text. The first column shows the spectroscopic notation, the second shows the orbitals involved.

K	1s → 2s	λ = 0	N_IN_F(Q^2+ζ^2)^24ζ+2(Q^2−3ζ^2)ζ_Fa_μ(Q^2+ζ^2)
	1s → 2p	λ = 1	N_IN_FQ(Q^2+ζ^2)^28ζζ_Fa_μ(Q^2+ζ^2)
L	2s → 3s	λ = 0	N_IN_F(Q^2+ζ^2)^212ζ+2(Q^2−3ζ^2)(3ζ_I+4ζ_F)a_μ(Q^2+ζ^2)+16ζζ_F(ζ^2−Q^2)(2ζ_F+9ζ_I)3aμ2(Q^2+ζ^2)^2−16ζ_IζF2(Q^4−10Q^2ζ^2+5ζ^4)3aμ3(Q^2+ζ^2)^3
	2s → 3p	λ = 1	N_IN_FQ(Q^2+ζ^2)^2128ζζ_F3a_μ(Q^2+ζ^2)+64ζ_F(Q^2−5ζ^2)(ζ_F+3ζ_I)9aμ2(Q^2+ζ^2)^2−64ζζ_IζF2(3Q^2−5ζ^2)3aμ3(Q^2+ζ^2)^3
	2s → 3d	λ = 2	N_IN_FQ^2(Q^2+ζ^2)^2128ζζF23aμ2(Q^2+ζ^2)^2+64ζ_IζF2(Q^2−7ζ^2)3aμ3(Q^2+ζ^2)^3
	2p → 3s	λ = 1	N_IN_FQ(Q^2+ζ^2)^224ζζ_Ia_μ(Q^2+ζ^2)+16ζ_Iζ_F(Q^2−5ζ^2)aμ2(Q^2+ζ^2)^2−32ζζ_IζF2(3Q^2−5ζ^2)3aμ3(Q^2+ζ^2)^3
	2p → 3p	λ = 0	N_IN_F(Q^2+ζ^2)^2192ζζ_Iζ_F(ζ^2−Q^2)3aμ2(Q^2+ζ^2)^2−32Qζ_IζF2(Q^4−10Q^2ζ^2+5ζ^4)3aμ3(Q^2+ζ^2)^3
		λ = 2	N_IN_FQ^2(Q^2+ζ^2)^2128ζζ_Iζ_Faμ2(Q^2+ζ^2)^2+64ζ_IζF2(Q^2−7ζ^2)3aμ3(Q^2+ζ^2)^3
	2p → 3d	λ = 1	N_IN_FQ(Q^2+ζ^2)^264ζζ_IζF2(5ζ^2−3Q^2)3aμ3(Q^2+ζ^2)^3

Table 3.

Table 3

Table of the screening constants obtained for several transitions by fitting the model curves to WIEN2k calculations. The fit was performed between 0 and 1.6 /a u. The resulting curves are shown in fig. 4

transition	states	screening constant s
		STO initial state	STO final state	HLO initial state	HLO final state
monopole	2p → 3p	0.00	9.61	7.96	7.96
dipole	2p → 3s	13.64	9.03	8.89	8.37
dipole	2p → 3d	9.81	11.91	9.81	11.91
quadrupole	2p → 3p	10.32	12.48	9.32	9.75

As before, one can study the behavior of eq. 23 for the limit of Q→ ∞ and Q → 0. For Q→ ∞, eq. 21 reduces toThis exhibits exactly the same behavior as eq. 15.

For Q → 0, it is best to start from eq. 20. Using eq. 9 (provided that as before), one obtainsin accordance with predictions.5

Discussion

Eqs. 13 and 23 are the main results of this work. Compared to the dipole approximation for small Q (eqs. 16 and 26), they offer a significant improvement. Unphysical cut-offs are no longer necessary to ensure they tend to zero properly for large Q. In addition, they are simple finite sums of polynomials and—in the case of STOs—elementary trigonometric functions. As such, they are easily implemented into existing simulation programs (Löffler and Schattschneider, 2010; Koch, 2002; Kirkland, 1998; Verbeeck et al., 2009) with very little effort and the increase in computation time is small.

It must be emphasized that STOs are designed to have the same asymptotic behavior as HLOs, but no nodes. In the case of l = n − 1 (in which case the HLOs are also nodeless), they are identical. In all other cases, the missing nodes are problematic. This is particularly evident in the case of l = l which can be allowed in monopole or quadrupole transitions. In those cases, STOs give completely wrong results because the node-less radial wave functions are not orthogonal as they should be, as can be seen from fig. 1.

Fig. 1

Wavefunctions in the STO (top) and HLO (middle) models for 2p and 3p orbitals, as well as their overlap, together with the radial transition amplitude (bottom) for the quadrupole-allowed transition between those two states. Due to the missing node in the 3p wavefunction, the STO overlap is is shifted towards larger r and underestimated in strength. Consequently, 〈j2(Q)〉 is shifted towards smaller Q and severely underestimated in the STO model.

In fig. 2, examples of weighted radial wave function overlaps using both the STO and HLO models are shown in more detail.6 It is clearly evident that using the complete eqs. 13 and 23 gives highly superior results than the approximations for small or large Q alone.

Fig. 2

Weighted radial wave function overlaps 〈j〉 (solid lines) for the example of an isolated Si atom. The top panel shows a 2p → 3d dipole transition calculated using the STO model (for Q ≥ 6, the curve is magnified by a factor of 50), the bottom panel shows a 2p → 3p quadrupole transition calculated using the HLO model. The screening constants for both cases were taken from Slater (1930). The dashed lines show the asymptotic behavior.

The influence of the screening constants is shown in more detail in fig. 3. It is noteworthy that increasing the screening constant s of the initial state tends to have the opposite effect as compared to increasing the screening constant s for the final state. Looking at the 2p → 3s transition, increasing s results in a shift of the first zero crossing to the left and hence also in a reduction of the height of the first peak. Increasing the s has exactly the opposite effect. For different transitions, the tendencies are independent, however. For example, while increasing s for the 2p → 3s transition results in a reduction in height of the first peak, while increasing s for the 2p → 3d transition results in an increase of the height of the first peak.

Fig. 3

Comparison of HLOs using different screening constants (lines) with WIEN2k calculations (dots). a) and b) show dipole-allowed 2p → 3s transitions, c) and d) show dipole-allowed 2p → 3d transitions. In a) and c), the screening constant of the initial state was varied while the screening constant for the final state was set to the optimal value given in tab. 3. In b) and d), the screening constant of the final state was varied while the screening constant for the initial state was set to the optimal value given in tab. 3. To the right of the vertical line, the curves are magnified by a factor of 10.

The formulas even give comparable results to much more sophisticated calculations using a full crystalline environment of the atom. In fig. 4, calculations using both the STO and HLO models are compared to WIEN2k (Blaha et al., 2001) calculations using the TELNES.3 program. It shows that the STO model fails completely for monopole transitions and can typically reproduce the WIEN2k data up only to the first zero crossing. This is caused by the missing nodes in the STO model (which implies non-orthogonal states 2p, 3p). The HLO model, on the other hand, reproduces the general shape very well, only exaggerating the height of the peeks somewhat. The excellent agreement can primarily be attributed to the fact that the initial state—which has a high probability density in close proximity to the nucleus—can be viewed as a filter on the final state. Consequently, the weighted radial wave function overlap is dominated by the shapes of the orbitals close to the nucleus, and crystal effects like bonding play only a secondary role.

Fig. 4

Comparison of several 2p → 3 transitions calculated for crystalline Si using the HLO model (solid line), the STO model (dashed line) and WIEN2k (Blaha et al., 2001) (dots). The screening constants for the STO and HLO models were determined by fitting to the WIEN2k data in the range from 0 to 1.6 /a u (see tab. 3). To the right of the vertical lines, the data is magnified by a factor of 30.

Conclusion

In this work, we have reviewed the fact that the transition probability in Coulomb scattering can be separated in an angular part and a radial part. The former is composed of well-known matrix elements of spherical harmonics. The latter can be expressed in simple independent-atom models such as the STO or the HLO model. Both yield simple algebraic expressions involving only finite sums of polynomials and (in the case of STOs) elementary trigonometric functions. Therefore, they are easy to implement into existing simulations programs without a large increase of the computation time, but with significant improvements in terms of accuracy, especially in the regime of medium momentum transfer.

Moreover, some of the weighted radial wave function overlaps have one or more zeros. Hence, they are suppressed at the corresponding momentum transfers, even though they are not forbidden by other selection rules. This could be exploited, e.g., for measuring faint signals from transitions that are normally hidden under a huge background from another transition with much higher transition probability.7

Finally, the formulas presented here can be exploited in the future to experimentally determine properties of wave functions in atoms, like the screening effects of other electrons.