I know that atomic form factors for x-ray scattering are obtained by Fourier transform of the electronic charge density.

I am looking for a citable elementary source (book, review, article) which covers the systematic and explicit derivation of the relationship between electronic charge density and the atomic form factor, to which people unfamiliar with the connection can be referred to for better context.

  • $\begingroup$ Something like this: web.physics.ucsb.edu/~fratus/phys103/LN/Scattering.pdf $\endgroup$
    – Karsten
    Feb 7, 2021 at 13:19
  • $\begingroup$ Or 7.3.33 in ocw.mit.edu/courses/physics/… ? $\endgroup$
    – Karsten
    Feb 7, 2021 at 13:24
  • $\begingroup$ try IUCR Texts on Crystallography -2, ed C. Giacovazzo publ. IUCR/OUP, chapter 3. for a brief but clear examination of scattering off electrons and atoms, and R. James, 'The Optical Principles of the Diffraction of X-rays vol II' , publ. Bell 1965, chapter 3 which is an older but still excellent book on all the basics. $\endgroup$
    – porphyrin
    May 28 at 16:45

2 Answers 2


The International Union of Crystallography (IUCr) compiled a freely accessible Online Dictionary.* To cite a section from the entry about the atomic scattering factor (often anonymously used for atomic form factor):

The scattering from a crystal of an X-ray beam results from the interaction between the electric component of the incident electromagnetic radiation and the electrons in the crystal. Tightly bound electrons scatter coherently (Rayleigh scattering); free electrons scatter incoherently (Compton scattering). The scattering process from atomic electrons in a crystal lattice has both coherent and incoherent components, and is described as Thomson scattering.

The scattering amplitude from a neutral atom depends on the number of electrons ($Z$ = the atomic number) and also on the Bragg angle $\theta{}$ – destructive interference among waves scattered from the individual electrons reduces the intensity at other than zero scattering angle. For $\theta{} = 0$ the scattering amplitude is normally equal to $Z$. However, the scattering factor is modified by anomalous scattering if the incident wavelength is near an absorption edge of the scattering element.

The X-ray scattering factor is evaluated as the Fourier transform of the electron density distribution of an atom or ion, which is calculated from theoretical wavefunctions for free atoms.

enter image description here

Scattering factor of stationary C and Fe atoms plotted as a function of Bragg angle for incident X-ray wavelength of $\pu{0.70930 Å}$. Ticks on the horizontal axis correspond to Bragg angle increments of 10 degrees; ticks on the vertical axis are increments of 5 electrons.

The lemma equally picks three references from the International Tables (chiefly IT) for additional information about the theory and listing the values. You may find it in print in groups active in e.g., (chemical) crystallography, earth sciences, solid state physics, structure biology; or in research libraries with the corresponding subscription:

  • C. Colliex, J. M. Cowley, S. L. Dudarev, M. Fink, J. Gjønnes, R. Hilderbrandt, A. Howie, D. F. Lynch, L. M. Peng, G. Ren, A. W. Ross, V. H. Smith Jr, J. C. H. Spence, J. W. Steeds, J. Wang, M. J. Whelan and B. B. Zvyagin. International Tables for Crystallography (2006). Vol. C, ch. 4.3, pp. 259-429 doi:10.1107/97809553602060000593. (preview of the content)
  • Intensity of diffracted intensities: P. J. Brown et al., 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 doi:10.1107/97809553602060000600 (preview of the content)
  • Neutron techniques: I. S. Anderson, P. J. Brown, J. M. Carpenter, G. Lander, R. Pynn, J. M. Rowe, O. Schärpf, V. F. Sears and B. T. M. Willis. International Tables for Crystallography (2006). Vol. C, ch. 4.4, pp. 430-487 doi:10.1107/97809553602060000594 (preview of the content)

* This reference already is among the site's resources for learning chemistry.


Here is a possible source: Modern X-ray analysis on single crystals: a practical guide by Peter Luger, De Gruyter, Inc. 2014.

The relevant excerpt describing how to estimate the density of an atom j, and with that, the scattering factor (really scattering function) is below.

enter image description here

So atomic scattering factors are derived by taking the Fourier transform of the computed electron density of a stationary atom (the density has spherical symmetry, so the scattering factor is a function that only depends on $s = \sin \theta / \lambda$).

  • 1
    $\begingroup$ Thanks for your time. But this is not what I am looking for. The atomic scattering factors are calculated by taking a Fourier transform of the charge density. But this relation itself has an analytical derivation which starts with differential scattering cross section. That is what I am interested in. $\endgroup$
    – R.U.
    Apr 19, 2019 at 18:45

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