Molecular scattering factors for electrons

Ross, A. W.; Fink, M.; Hilderbrandt, R.; Wang, J.; Smith, V. H. Jr

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 390-391

Section 4.3.3.3. Molecular scattering factors for electrons

A. W. Ross,^d M. Fink,^d R. Hilderbrandt,^f J. Wang^k and V. H. Smith Jr^k

4.3.3.3. Molecular scattering factors for electrons

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The simplest theory of molecular scattering assumes that a molecule consists of spherical atoms and that each electron is scattered by only one atom in the molecule. If only single scattering is allowed within each atom, the molecular intensity can be written as $[\eqalignno{ I(s) &=I_a(s)+I_m(s) \cr &=\left[{{4I_0}\over {a^2s^4R^2}}\right]\Biggl[\sum^M_{i=1}\{[Z_i-F_i(s)]^2+S_i(s)\} \cr & +\sum^M_i\,\sum^M_{j\neq i}[Z_i-F_i(s)][Z_j-F_j(s)] \cr & \times \int\limits^\infty_0 {\rm d} r\,P_{ij}(r,T)(\sin sr)/sr\Biggr], & (4.3.3.1)}]$ where M is the number of constituent atoms in the molecule, [F_i(s)] and [S_i(s)] are the coherent and incoherent X-ray scattering factors, and $[P_{ij}(r, T)]$ is the probability of finding atom i at a distance r from atom j at the temperature T (Bonham & Su, 1966; Kelley & Fink, 1982b; Mawhorter, Fink & Archer, 1983; Mawhorter & Fink, 1983; Miller & Fink, 1985; Hilderbrandt & Kohl, 1981; Kohl & Hilderbrandt, 1981). The constant [I_0] is proportional to the product of the intensities of the electron and molecular beams and R is the distance from the point of scattering to the detector. The single sum is the atomic intensity [I_a(s)] and the double sum is the molecular intensity [I_m(s)] . This expression, referred to here as the independent atom model (IAM), may be improved by replacing the atomic elastic electron scattering factors by their partial wave counterparts. This modification is necessary to explain the failure of the Born approximation observed in molecules containing light and heavy atoms in proximity (Schomaker & Glauber, 1952; Seip, 1965), and may be written as $[\eqalignno{ I(s) &=I_a(s)+I_m(s) \cr&={I_0\over R^2}\Biggl\{\sum^M_{i=1} [|\,f_i|^2+4S_i(s)/(a^2s^4)] \cr &+\sum^M_i \sum^M_{j\neq i}\,|\,f_i|\,|\,f_j|\cos (\eta_i-\eta_j) \cr & \times \int\limits^\infty_0\,{\rm d} r\,P_{ij}(r,T)(\sin sr)/sr\Biggr\}. & (4.3.3.2)}]$ This is the most commonly used expression for the interpretation of molecular gas electron-diffraction patterns in the keV energy range. If it is necessary to consider relativistic effects in the scattering intensity, equation (4.3.3.2) becomes (Yates & Bonham, 1969) $[\eqalignno{ I(s) &=I_a(s)+I_m(s) \cr &={I_0\over R^2}\Biggl\{\sum^M_{i=1} [|\,f_i|^2+|g_i|^2+4S_i(s)/(a^2s^4)] \cr & +\sum^M_i \sum^M_{j\neq i} [|\,f_i|\,|\,f_j|\cos (\eta ^f_i - \eta ^f_j)+|g_i| |g_j|\cos (\eta^g_i-\eta^g_j)] \cr & \times \int\limits^\infty_0\, {\rm d} r\,P_{ij}(r,T)(\sin sr)/sr\Biggr\}, & (4.3.3.3)}]$ where [|g_i|] and $[\eta^g_i]$ refer to the scattering-factor magnitude and phase for electrons that have changed their electron spin state during the scattering process and $[|\,f_i|]$ and $[\eta^f_i]$ refer to retention of spin orientation. The incident electron beam is assumed to be unpolarized and no attempt has been made to consider relativistic effects on the inelastic scattering cross section, which is usually negligible in the structural s range.

If it is necessary to consider binding effects, the first Born approximation may usually be used in describing molecular scattering, since binding effects are largest for molecules containing small atoms where the Born approximation is most valid.

The exact expression for I(s) in the first Born approximation can be written as (Bonham & Fink, 1974; Tavard & Roux, 1965; Tavard, Rouault & Roux, 1965; Iijima, Bonham & Ando, 1963; Bonham, 1967) $[\eqalignno{ I(s)&={{4I_0}\over {a^2s^4R^2}}\,\Biggr\{\sum^M_{i=1} (Z^2_i+Z_i) \cr& +\sum^M_i \sum^M_{j\neq i}\,Z_iZ_j \int\limits^\infty_0\,{\rm d} r\,P_{ij}(r,T)(\sin sr)/sr \cr& -2\sum^M_{i=1} Z_i \Biggl\langle\int\limits^{}_{} {\rm d} r\,\rho(r+r_i)(\sin sr)/sr\Biggr\rangle_{\rm vib} \cr &+ \Biggl\langle\int{\rm d} r\, \rho_c(r)(\sin sr)/sr\Biggr\rangle_{\rm vib}\Biggr\}, }]$ where $[\rho(r)=\textstyle\sum\limits^N_{i=1}\int{\rm d} r_1\ldots\int{\rm d} r_N\,|\psi(r_1,\ldots,r_N)|{}^2\delta(r-r_i)]$ and $[\rho_c(r)=\textstyle\sum\limits^N_i \sum\limits^N_{j\neq i} \int {\rm d} r_1\ldots\int{\rm d} r_N\,|\psi(r_1,\ldots,r_N)|{}^2\delta(r-r_i+r_j).]$ The brackets $[\langle\,\rangle_{\rm vib}]$ denote averaging over the vibrational motion, $[\delta(r)]$ is the Dirac delta function, and $[\psi(r_i,\ldots,r_n)]$ is the molecular wavefunction. Binding effects appear to be proportional to the ratio of the number of electrons involved in binding to the total number of electrons in the system (Kohl & Bonham, 1967; Bonham & Iijima, 1965) so that binding effects in molecules containing mainly heavy atoms should be quite small.

The intensities, I(s), for many small molecules have been calculated based on molecular Hartree–Fock wavefunctions. In most cases, a distinctive minimum has been found at about s = 3–4 Å⁻¹ and a much small maximum at s = 8–10 Å⁻¹ in the cross-sectional difference curve between the IAM and the molecular HF results (Pulay, Mawhorter, Kohl & Fink, 1983; Kohl & Bartell, 1969; Liu & Smith, 1977; Epstein & Stewart, 1977; Sasaki, Konaka, Iijima & Kimura, 1982; Shibata, Hirota, Kakuta & Muramatsu, 1980; Horota, Kakuta & Shibata, 1981; Xie, Fink & Kohl, 1984). Further studies using correlated wavefunctions (accounting for up to 60% of the correlation energy) showed that in the elastic channel the binding effects are only weakly modified; only the maximum at s = 8–10 Å⁻¹ is further reduced. However, strong effects are seen in the inelastic channel, deepening the minimum at s = 3–4 Å⁻¹ significantly (Breitenstein, Endesfelder, Meyer, Schweig & Zittlau, 1983; Breitenstein, Endesfelder, Meyer & Schweig, 1984; Breitenstein, Mawhorter, Meyer & Schweig, 1984; Wang, Tripathi & Smith, 1994). Detailed calculations on CO₂ and H₂O averaging over many internuclear distances and applying the pair distribution functions $[P_{ij}(r)]$ showed that vibrational effects do not alter the binding effects (Breitenstein, Mawhorter, Meyer & Schweig, 1986). For CO₂, the calculations have been confirmed in essence by an experimental set of data (McClelland & Fink, 1985). However, more molecules and more detailed analysis will be available in the future. The binding effects make it desirable to avoid the small-angle-scattering range when structural information is the main goal of a diffraction analysis.

The problem of intramolecular multiple scattering may necessitate corrections to the molecular intensity when three or more closely spaced heavy atoms are present. This correction (Karle & Karle, 1950; Hoerni, 1956; Bunyan, 1963; Gjønnes, 1964; Bonham, 1965a, 1966) appears to be more serious for three atoms in a right triangular configuration than for a collinear arrangement of three atoms. A case study by Kohl & Arvedson (1980) on SF₆ showed the importance of multiple scattering. However, their approach is too cumbersome to be used in routine structure work. A very good approximate technique is available utilizing the Glauber approximation (Bartell & Miller, 1980; Bartell & Wong, 1972; Wong & Bartell, 1973; Bartell, 1975); Kohl's results are reproduced quite well using the atomic scattering factors only. Several applications of the multiple scattering routines showed that the internuclear distances are rather insensitive to this perturbation, but the mean amplitudes of vibration can easily change by 10% (Miller & Fink, 1981; Kelley & Fink, 1982a; Ketkar & Fink, 1985).

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 390-391