Cromer–Mann coefficients - Revision history

BrianMcMahon at 15:19, 8 June 2021

2021-06-08T15:19:13Z

BrianMcMahon: Style edits to align with printed edition

2017-05-13T10:57:00Z

Style edits to align with printed edition

BrianMcMahon at 15:10, 10 April 2008

2008-04-10T15:10:09Z

BrianMcMahon at 15:05, 10 April 2008

2008-04-10T15:05:50Z

BrianMcMahon at 14:25, 10 April 2008

2008-04-10T14:25:48Z

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== Definition ==

The set of nine coefficients <math>a_i, b_i, c (i=1,\dots, 4)</math> in a parameterization of the [[atomic scattering factor|scattering factors]] for neutral atoms as a function of <math>(\sin \theta)/\lambda</math>:

<math>f(\sin\theta/\lambda) = \sum_{i=1}^4 a_i \exp[-b_i(\sin\theta/\lambda)^2] + c</math>

for <math>0\lt(\sin\theta)/\lambda\lt2.0\,\mathrm{\AA}^{-1}</math>.

== History ==

Atomic scattering factors for non-hydrogen atoms were calculated from relativistic Hartree–Fock wavefunctions by Doyle, P. A. & Turner, P. S. [(1968). [http://dx.doi.org/10.1107/S0567739468000756 ''Acta Cryst.'' A'''24''', 390–397]. ''Relativistic Hartree–Fock and electron scattering factors''] using the wavefunctions of Coulthard, M. A. [(1967). [http://dx.doi.org/10.1088/0370-1328%2F91%2F1%2F309 ''Proc. Phys. Soc.'' '''91''', 44–49. ''A relativistic Hartree–Fock atomic field calculation''], and in 1968 by Cromer, D. T. & Waber, J. T. using the unpublished wavefunctions of J. B. Mann (1968) [''International Tables for X-ray Crystallography'' (1974), Vol. IV, p. 71. Birmingham: Kynoch Press]. The latter are based on a more exact treatment of potential that allows for the finite size of the nucleus, but the effect on the scattering factors is small. The calculations of Cromer & Waber (1968[link]) were originally made for [0\lt(\sin\theta)/\lambda\lt2.0\,{\rm \AA}^{-1}], but these have been extended to 6 Å−1 by Fox, O'Keefe & Tabbernor (1989[link]); this has been done because there are increasing numbers of applications for high-angle scattering factors.

== See also ==

Intensity of diffracted intensities.
P. J. Brown, A. G. Fox, E. N. Maslen, M. A. O'Keefe and B. T. M. Willis. ''International Tables for Crystallography'' (2006). Vol. C, ch. 6.1, pp. 554-595 [http://dx.doi.org/10.1107/97809553602060000600 doi:10.1107/97809553602060000600], especially Table 6.1.1.4.

@@ Line 5: / Line 5: @@
 <math>f^0(\sin\theta/\lambda) = \sum_{i=1}^4 a_i \exp[-b_i(\sin\theta/\lambda)^2] + c</math>
-for <math>0 < (\sin\theta)/\lambda < 2.0\,\mathrm{\AA}^{-1}</math>.
+for <math>0 < (\sin\theta)/\lambda < 2.0\,\mathrm{Å}^{-1}</math>.
 This expression is convenient for calculation in crystal structure software suites.

@@ Line 11: / Line 11: @@
 == History ==
-Atomic scattering factors for non-hydrogen atoms were calculated from relativistic Hartree&ndash;Fock wavefunctions by Doyle, P. A. &amp; Turner, P. S. [(1968). [http://dx.doi.org/10.1107/S0567739468000756 ''Acta Cryst.'' A'''24''', 390–397]. ''Relativistic Hartree–Fock and electron scattering factors''] using the wavefunctions of Coulthard, M. A. [(1967). [http://dx.doi.org/10.1088/0370-1328%2F91%2F1%2F309 ''Proc. Phys. Soc.'' '''91''', 44–49]. ''A relativistic Hartree–Fock atomic field calculation''], and in 1968 by Cromer, D. T. &amp; Waber, J. T. using the unpublished wavefunctions of J. B. Mann [''International Tables for X-ray Crystallography'' (1974), Vol. IV, p. 71. Birmingham: Kynoch Press]. The latter are based on a more exact treatment of potential that allows for the finite size of the nucleus. Subsequent calculations [Fox, A. G., O'Keefe, M. A. &amp; Tabbernor, M. A. (1989). [http://dx.doi.org/10.1107/S0108767389007567 ''Acta Cryst.'' A'''45''', 786–793]. ''Relativistic Hartree–Fock X-ray and electron atomic scattering factors at high angles''] extended the useful range to 6 Å<sup>−1</sup> to accommodate the increasing numbers of applications for high-angle scattering factors.
+Atomic scattering factors for non-hydrogen atoms were calculated from relativistic Hartree&ndash;Fock wavefunctions by Doyle, P. A. and Turner, P. S. [(1968). [https://doi.org/10.1107/S0567739468000756 ''Acta Cryst.'' A'''24''', 390–397]. ''Relativistic Hartree–Fock and electron scattering factors''] using the wavefunctions of Coulthard, M. A. [(1967). [https://doi.org/10.1088/0370-1328%2F91%2F1%2F309 ''Proc. Phys. Soc.'' '''91''', 44–49]. ''A relativistic Hartree–Fock atomic field calculation''], and in 1968 by Cromer, D. T. and Waber, J. T. using the unpublished wavefunctions of J. B. Mann [''International Tables for X-ray Crystallography'' (1974), Vol. IV, p. 71. Birmingham: Kynoch Press]. The latter are based on a more exact treatment of potential that allows for the finite size of the nucleus. Subsequent calculations [Fox, A. G., O'Keefe, M. A. and Tabbernor, M. A. (1989). [https://doi.org/10.1107/S0108767389007567 ''Acta Cryst.'' A'''45''', 786–793]. ''Relativistic Hartree–Fock X-ray and electron atomic scattering factors at high angles''] extended the useful range to 6 Å<sup>−1</sup> to accommodate the increasing numbers of applications for high-angle scattering factors.
 == See also ==
-Intensity of diffracted intensities.
+*Intensity of diffracted intensities. P. J. Brown, A. G. Fox, E. N. Maslen, M. A. O'Keefe and B. T. M. Willis. ''International Tables for Crystallography'' (2006). [https://doi.org/10.1107/97809553602060000600 Vol. C, ch. 6.1, pp. 554-595], especially Table 6.1.1.4.
 [[Category:Structure determination]]
 [[Category:X-rays]]

← Older revision		Revision as of 15:10, 10 April 2008
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	Intensity of diffracted intensities.		Intensity of diffracted intensities.
	P. J. Brown, A. G. Fox, E. N. Maslen, M. A. O'Keefe and B. T. M. Willis. ''International Tables for Crystallography'' (2006). Vol. C, ch. 6.1, pp. 554-595 [http://dx.doi.org/10.1107/97809553602060000600 doi:10.1107/97809553602060000600], especially Table 6.1.1.4.		P. J. Brown, A. G. Fox, E. N. Maslen, M. A. O'Keefe and B. T. M. Willis. ''International Tables for Crystallography'' (2006). Vol. C, ch. 6.1, pp. 554-595 [http://dx.doi.org/10.1107/97809553602060000600 doi:10.1107/97809553602060000600], especially Table 6.1.1.4.
		+
		+
		+	[[Category:Structure determination]]
		+	[[Category:X-rays]]

@@ Line 1: / Line 1: @@
 == Definition ==
-The set of nine coefficients <math>a_i, b_i, c (i=1,\dots, 4)</math> in a parameterization of the [[atomic scattering factor|scattering factors]] for neutral atoms as a function of <math>(\sin \theta)/\lambda</math>:
+The set of nine coefficients <math>a_i, b_i, c\, (i=1,\dots, 4)</math> in a parameterization of the non-dispersive part of the [[atomic scattering factor]] for neutral atoms as a function of <math>(\sin \theta)/\lambda</math>:
-<math>f(\sin\theta/\lambda) = \sum_{i=1}^4 a_i \exp[-b_i(\sin\theta/\lambda)^2] + c</math>
+<math>f^0(\sin\theta/\lambda) = \sum_{i=1}^4 a_i \exp[-b_i(\sin\theta/\lambda)^2] + c</math>
-for <math>0\lt(\sin\theta)/\lambda\lt2.0\,\mathrm{\AA}^{-1}</math>.
+for <math>0 < (\sin\theta)/\lambda < 2.0\,\mathrm{\AA}^{-1}</math>.
 == History ==
-Atomic scattering factors for non-hydrogen atoms were calculated from relativistic Hartree&ndash;Fock wavefunctions by Doyle, P. A. &amp; Turner, P. S. [(1968). [http://dx.doi.org/10.1107/S0567739468000756 ''Acta Cryst.'' A'''24''', 390–397]. ''Relativistic Hartree–Fock and electron scattering factors''] using the wavefunctions of Coulthard, M. A. [(1967). [http://dx.doi.org/10.1088/0370-1328%2F91%2F1%2F309 ''Proc. Phys. Soc.'' '''91''', 44–49. ''A relativistic Hartree–Fock atomic field calculation''], and in 1968 by Cromer, D. T. &amp; Waber, J. T. using the unpublished wavefunctions of J. B. Mann (1968) [''International Tables for X-ray Crystallography'' (1974), Vol. IV, p. 71. Birmingham: Kynoch Press]. The latter are based on a more exact treatment of potential that allows for the finite size of the nucleus, but the effect on the scattering factors is small. The calculations of Cromer & Waber (1968[link]) were originally made for [0\lt(\sin\theta)/\lambda\lt2.0\,{\rm \AA}^{-1}], but these have been extended to 6 Å−1 by Fox, O'Keefe & Tabbernor (1989[link]); this has been done because there are increasing numbers of applications for high-angle scattering factors.
+Atomic scattering factors for non-hydrogen atoms were calculated from relativistic Hartree&ndash;Fock wavefunctions by Doyle, P. A. &amp; Turner, P. S. [(1968). [http://dx.doi.org/10.1107/S0567739468000756 ''Acta Cryst.'' A'''24''', 390–397]. ''Relativistic Hartree–Fock and electron scattering factors''] using the wavefunctions of Coulthard, M. A. [(1967). [http://dx.doi.org/10.1088/0370-1328%2F91%2F1%2F309 ''Proc. Phys. Soc.'' '''91''', 44–49]. ''A relativistic Hartree–Fock atomic field calculation''], and in 1968 by Cromer, D. T. &amp; Waber, J. T. using the unpublished wavefunctions of J. B. Mann [''International Tables for X-ray Crystallography'' (1974), Vol. IV, p. 71. Birmingham: Kynoch Press]. The latter are based on a more exact treatment of potential that allows for the finite size of the nucleus. Subsequent calculations [Fox, A. G., O'Keefe, M. A. &amp; Tabbernor, M. A. (1989). [http://dx.doi.org/10.1107/S0108767389007567 ''Acta Cryst.'' A'''45''', 786–793]. ''Relativistic Hartree–Fock X-ray and electron atomic scattering factors at high angles''] extended the useful range to 6 Å<sup>−1</sup> to accommodate the increasing numbers of applications for high-angle scattering factors.
 == See also ==