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

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. Wangk and V. H. Smith Jrk

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[link]; Kelley & Fink, 1982b[link]; Mawhorter, Fink & Archer, 1983[link]; Mawhorter & Fink, 1983[link]; Miller & Fink, 1985[link]; Hilderbrandt & Kohl, 1981[link]; Kohl & Hilderbrandt, 1981[link]). 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[link]; Seip, 1965[link]), 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)[link] becomes (Yates & Bonham, 1969[link]) [\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[link]; Tavard & Roux, 1965[link]; Tavard, Rouault & Roux, 1965[link]; Iijima, Bonham & Ando, 1963[link]; Bonham, 1967[link]) [\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[link]; Bonham & Iijima, 1965[link]) 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[link]; Kohl & Bartell, 1969[link]; Liu & Smith, 1977[link]; Epstein & Stewart, 1977[link]; Sasaki, Konaka, Iijima & Kimura, 1982[link]; Shibata, Hirota, Kakuta & Muramatsu, 1980[link]; Horota, Kakuta & Shibata, 1981[link]; Xie, Fink & Kohl, 1984[link]). 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[link]; Breitenstein, Endesfelder, Meyer & Schweig, 1984[link]; Breitenstein, Mawhorter, Meyer & Schweig, 1984[link]; Wang, Tripathi & Smith, 1994[link]). 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[link]). For CO2, the calculations have been confirmed in essence by an experimental set of data (McClelland & Fink, 1985[link]). 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[link]; Hoerni, 1956[link]; Bunyan, 1963[link]; Gjønnes, 1964[link]; Bonham, 1965a[link], 1966[link]) 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[link]) 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[link]; Bartell & Wong, 1972[link]; Wong & Bartell, 1973[link]; Bartell, 1975[link]); 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[link]; Kelley & Fink, 1982a[link]; Ketkar & Fink, 1985[link]).

References

Bartell, L. S. (1975). Modification of Glauber theory for dynamic scattering of electrons by polyatomic molecules. J. Chem. Phys. 63, 3750–3755.
Bartell, L. S. & Miller, B. (1980). Extension of Glauber theory to account for intratarget diffraction in multicenter scattering. J. Chem. Phys. 72, 800–807.
Bartell, L. S. & Wong, T. C. (1972). Three-atom scattering in gas-phase electron diffraction: a tractable limiting case. J. Chem. Phys. 56, 2364–2367.
Bonham. R. A. (1965a). Multiple elastic intramolecular scattering in gas electron diffraction. J. Chem. Phys. 43, 1103–1109.
Bonham, R. A. (1966). Dynamic effects in gas electron diffraction. Trans. Am. Crystallogr. Assoc. 2, 165–172.
Bonham, R. A. (1967). Some new relations connecting molecular properties and electron and X-ray diffraction intensities. J. Phys. Chem. 71, 856–862.
Bonham, R. A. & Fink, M. (1974). High energy electron scattering, Chaps. 5 and 6. New York: Van Nostrand Reinhold.
Bonham. R. A. & Iijima, T. (1965). Preliminary electron-diffraction study of H2 at small scattering angles. J. Chem. Phys. 42, 2612–2614.
Bonham, R. A. & Su, L. S. (1966). Use of Hellman–Feynman and hyperviral theorems to obtain anharmonic vibration–rotation expectation values and their application to gas diffraction. J. Chem. Phys. 45, 2827–2831.
Breitenstein, M., Endesfelder, A., Meyer, H. & Schweig, A. (1984). CI calculations of electron-scattering cross sections for some linear molecules. Chem. Phys. Lett. 108, 430–434.
Breitenstein, M., Endesfelder, A., Meyer, H., Schweig, A. & Zittlau, W. (1983). Electron correlation effects in electron scattering cross-section calculations of N2. Chem. Phys. Lett. 97, 403–409.
Breitenstein, M., Mawhorter, R. J., Meyer, H. & Schweig, A. (1984). Theoretical study of potential-energy differences from high-energy electron scattering cross sections of CO2. Phys. Rev. Lett. 53, 2398–2401.
Breitenstein, M., Mawhorter, R. J., Meyer, H. & Schweig, A. (1986). Vibrational effects on electron–molecule scattering for polyatom in the first Born approximation: H2O. Mol. Phys. 57, 81–88.
Bunyan, P. J. (1963). The effect of multiple elastic scattering in gas electron diffraction. Proc. Phys. Soc. 82, 1051–1057.
Epstein, J. & Stewart, R. F. (1977). X-ray and electron scattering from diatomic molecules in the first Born approximation. J. Chem. Phys. 66, 4057–4064.
Gjønnes, J. (1964). A dynamic effect in electron diffraction by molecules. Acta Cryst. 17, 1075–1076.
Hilderbrandt, R. L. & Kohl, D. A. (1981). A variational treatment of the effects of vibrational anharmonicity on gas-phase electron diffraction intensities. Part I. Molecular scattering function. J. Mol. Struct. Theochem. 85, 25–36.
Hoerni, J. A. (1956). Multiple elastic scattering in electron diffraction by molecules. Phys. Rev. 102, 1530–1533.
Horota, F., Kakuta, N. & Shibata, S. (1981). High energy electron scattering by diborane. J. Phys. B, 14, 3299–3304.
Iijima, T., Bonham, R. A. & Ando, T. (1963). The theory of electron scattering from molecules. I. Theoretical development. J. Phys. Chem. 67, 1472–1474.
Karle, I. L. & Karle, J. (1950). Internal motion and molecular structure studies by electron diffraction. III. Structure of CH2CF2 and CF2CF2. J. Chem. Phys. 18, 963–971.
Kelley, M. H. & Fink, M. (1982a). The molecular structure of dimolybdenum tetraacetate. J. Chem. Phys. 76, 1407–1416.
Kelley, M. H. & Fink, M. (1982b). The temperature dependence of the molecular structure parameters in SF6. J. Chem. Phys. 77, 1813–1817.
Ketkar, S. N. & Fink, M. (1985). Structure of dichromium tetraacetate by gas-phase electron diffraction. J. Am. Chem. Soc. 107, 338–340.
Kohl, D. A. & Arvedson, M. (1980). Elastic electron scattering from molecular potentials. J. Chem. Phys. 73, 3818–3822.
Kohl, D. A. & Bartell, L. S. (1969). Electron densities from gas-phase electron diffraction intensities. II. Molecular Hartree–Fock cross sections. J. Chem. Phys. 51, 2896–2904.
Kohl, D. A. & Bonham, R. A. (1967). Effect of bond formation on electron scattering cross sections for molecules. J. Chem. Phys. 47, 1634–1646.
Kohl, D. A. & Hilderbrandt, R. L. (1981). A variational treatment of the effects of vibrational anharmonicity on gas-phase electron diffraction intensities. Part II. Temperature dependence. J. Mol. Struct. Theochem. 85, 325–335.
Liu, J. W. & Smith, V. H. (1977). A critical study of high energy electron scattering from H2. Chem. Phys. Lett. 45, 59–63.
McClelland, J. J. & Fink, M. (1985). Electron correlation and binding effects in measured electron-scattering cross sections of CO2. Phys. Rev. Lett. 54, 2218–2221.
Mawhorter, R. J. & Fink, M. (1983). The vibrationally averaged, temperature-dependent structure of polyatomic molecules. II. SO2. J. Chem. Phys. 79, 3292–3296.
Mawhorter, R. J., Fink, M. & Archer, B. T. (1983). The vibrationally averaged, temperature-dependent structure of polyatomic molecules. I. CO2. J. Chem. Phys. 79, 170–174.
Miller, B. R. & Fink, M. (1981). Mean amplitudes of vibration of SF6 and intramolecular multiple scattering. J. Chem. Phys. 75, 5326–5328.
Miller, B. R. & Fink, M. (1985). The vibrationally averaged, temperature-dependent structure of polyatomic molecules. III. NO2. J. Chem. Phys. 83, 939–944.
Pulay, P., Mawhorter, R. J., Kohl, D. A. & Fink, M. (1983). Ab initio Hartree–Fock calculation of the elastic electron scattering cross section of sulphur hexafluoride. J. Chem. Phys. 79, 185–191.
Sasaki, H., Konaka, S., Iijima, T. & Kimura, M. (1982). Small-angle electron scattering and electron density in carbon dioxide. Int. J. Quantum Chem. 21, 475–485.
Schomaker, V. & Glauber, R. (1952). The Born approximation in electron diffraction. Nature (London), 170, 290–291.
Seip, H. M. (1965). Studies on the failure of the first Born approximation in electron diffraction. I. Uranium hexafluoride. Acta Chem. Scand. 19, 1955–1968.
Shibata, S., Hirota, F., Kakuta, N. & Muramatsu, T. (1980). Electron distribution in water by high-energy electron scattering. Int. J. Quantum Chem. 18, 281–285.
Tavard, C., Rouault, M. & Roux, M. (1965). Diffraction des rayons X et des électrons par les molécules. III. Une méthode de detémination des densités électroniques moléculaires. J. Chim. Phys. 62, 1410–1417.
Tavard, C. & Roux, M. (1965). Calcul des intensités de diffraction de rayons X et de électrons par les molécules. C. R. Acad. Sci. 260, 4933–4936.
Wang, J., Tripathi, A. N. & Smith, V. H. Jr (1994). Chemical binding and electron correlation effects in X-ray and high energy electron scattering. J. Chem. Phys. 101, 4842–4854.
Wong, T. C. & Bartell, L. S. (1973). Three atom scattering in gas-phase electron diffraction. II. A general treatment. J. Chem. Phys. 58, 5654–5660.
Xie, S.-D., Fink, M. & Kohl, D. A. (1984). Basis set dependence of ab initio SCF elastic, Born, electron scattering cross sections for C2H4. J. Chem. Phys. 81, 1940–1942.
Yates, A. C. & Bonham, R. A. (1969). Use of relativistic electron scattering factors in electron diffraction analysis. J. Chem. Phys. 50, 1056–1058.








































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