To compute scattering, we need to tabulate the probability for scattering of our probe by atoms. This probability function is described with several names and formulations.
For x-rays, the probability for scattering is usually called the form factor and is written as f (\(2\theta \)) or f (Q), with units of length. X-ray photons are predominantly scattered by the electrons in an atom, and since the electrons are smeared out across a large region of the atom, the probability for interaction is greatest when the electron has the greatest average path through the atom – which means that the greatest scattering probability is for forward scattering, where the momentum (path) of the photon is not changed. The form factor for forward scattering, f (0), is proportional to the number of electrons in the atom or ion and by conventionf (0)=Z where Z is the number of electrons for the species. The larger the angle of scattering, the lower the probability of scattering. The probability of scattering thus is a function of the scattering angle but if tabulated as a function of Q (or equivalently \(\sin \theta /\lambda \)) is largely independent on the energy of the photon. Form factor values are computed for isolated atoms using high-level quantum theory (typically Hartree-Fock) and are tabulated in the International Tables for Crystallography Volume C, but in GSAS-II form factors are computed for the specified x-ray wavelength from a set of coefficients tabulated by D. T. Cromer and J. T. Waber (1971) in the International Tables Vol. IV.
These form factors are only an approximation, since bonding considerations in the solid state do affect the distribution of electrons significantly. Thankfully, bonding most greatly affects the valence electrons, which are already quite diffuse, so even though these form factors are from idealized computations, made for isolated atoms in vacuum, rather for the actual environment of an atom in a real material, they do work pretty well. There are more complex treatments available for fitting diffraction data, but they can only be used with extremely accurate data collected over a very wide Q range. At the time that I write this, the capability to treat non-spherical atom scattering is being added to GSAS-II, but is not yet complete. However, it is not clear if powder diffraction studies will ever be made at sufficient precision to detect the differences between spherical and non-spherical scattering contributions.
A more controversial question is how to describe atoms with formal non-zero oxidation states: should one use scattering factors computed for ions or for neutral atoms? I side with the neutral atom choice. To illustrate why, consider a mixed iron oxide where Fe is nominally Fe+3. The picture taught in first-year chemistry is that Fe has transferred three electrons to neighboring O atoms, However, is not an accurate picture of the chemical bonding, as the electrons are shared in molecular orbitals generated between the Fe and O atoms and on average the valence electrons still have a high probability of being found close to the Fe nucleus. This means that the scattering from this solid-state “ion” is not much changed from that of a neutral Fe atom, as would be by the complete removal of three electrons. My understanding is that even in a highly ionic material such as NaCl, in the solid state, the actual electron density distribution is quite far from what would be the case if there were complete transfer of an electron from each sodium to chlorine.
Another reason that one may safely use neutral atom scattering factors, even for ions, is that this choice has a very limited impact on the computation. We can understand how the choice of a neutral atom versus a charged-atom species affects the scattering calculation using a utility that GSAS-II offers, where scattering factor curves can be viewed, called PlotXNFF (found in the Calculate menu). If this is opened, a window labeled “Plot Xray, Neutron and Magnetic Form Factors” is displayed. As an example, click on this window and in the associated menu select the element Fe. This will open a plot where the neutron, x-ray, electron and magnetic scattering factors can be visualized, as seen in the screen image below in Fig. 2.1. For Q=0, the value will be the number of electrons in the atom, so at 0 degrees (where one cannot measure scattering) f (0) for Fe+3 is 23 electrons while it is 26 for neutral Fe. The differences between the Fe and Fe+3 scattering factor curves become increasingly close as Q increases from zero. Note that by \(\sin \theta /\lambda =0.1\) (Q=0.6) the curves for the ionic forms of Fe and the neutral element have largely converged and above double that value the curves are largely indistinguishable. Since few materials have more than a few diffraction peaks at these low values of Q and many small-cell materials have none, computed diffraction patterns will show very slight – if any – differences between neutral and ionic treatment of atomic scattering. Thus, there is very little reason in my opinion to use charged atom form factors. Not only are they a better chemical description of the electron distribution, they are simpler. Based on what we have now seen, they will not change the fitting appreciably.
Despite my comments, if one persists in using charged-atom scattering factors, a common question will be “what should I use for a species that is not provided, such as Fe+4?” Certainly, form factors for Fe+3 and Fe+4 will be just about the same except at very low Q. Feel free to substitute a similar valence scattering factor curve. Achieving charge balance for scattering factor selection is not important, since again this only affects low-angle scattering and again only in a second-order fashion.
In contrast to x-rays, neutrons are usually scattered from the nucleus of the atom, and this scattering can usually be considered as a point-point interaction that does not vary with the angle of scattering. The probability for this type of scattering is thus a constant independent of angle, but does depend on the neutron energy and, in particular, on the interactions at the quantum level between the nucleus of the atom and the neutron, which depends on the isotope of the atom. In fact, the scattering probability may be very different for isotopes of the same element. The scattering probability for neutrons is commonly referenced as a cross-section, \(\sigma \), with units of area (length2). There is also a quantity related to the square-root of the cross-section, which is called the scattering length, b, and has units of length. Finally, while all atoms scatter x-rays with no phase inversion, most isotopes do scatter neutrons with phase inversion, but a few do not. For those unusual isotope types that do not invert the scattering phase, we indicate this using a negative value for b.
There are separate probabilities for coherent and incoherent neutron scattering, which can vary greatly. As a fairly extreme example, the coherent and incoherent scattering lengths for the 1H isotope of hydrogen are -3.7406 fm and 25.274 fm, respectively. Since scattering probabilities go as the square of the scattering lengths, a neutron is \(\approx \)45 times more likely to be scattered incoherently than coherently by a 1H atom. As was discussed previously, this incoherent scattering occurs at all diffraction angles and adds greatly to the background, making it much harder to measure diffraction with significant precision in the presence of a significant atomic percentage of 1H. For this reason, samples containing deuterium (2H), for which the coherent and incoherent scattering lengths are 6.671 fm and 4.04 fm, respectively, are greatly preferred for neutron diffraction.
Noting that almost all elements exist as a mixture of isotopes, the net coherent scattering from any element is dependent on the weighted sums of the scattering lengths, which can work towards canceling each other if the signs of b are opposite. While normally 2H is only 0.015% of the natural isotopic makeup of hydrogen, if we enrich the level of 2H to 36% at a site, we have a net coherent scattering of 0.0075 fm [ = (-3.7406*0.64) + (6.671*0.36)], meaning that there is nearly no coherent scattering from this site. The overall probabilities for scattering from the 1H and 2H atoms are unchanged by their mixture, but all of their scattering now becomes incoherent. A similar effect occurs with vanadium, which at natural isotopic composition is 0.25% 51V with b= 7.6 fm and the remainder 50V with b = -0.402 fm. Note that (0.25*7.6 + 99.75*-0.402)/100 = -0.382 fm, which is a relatively weak diffracting material. Vanadium is a preferred as a container material for neutron diffraction, due to the very small levels of diffraction seen from that metal.
Note that there are potentially two ways that incoherent scattering can arise. It can be an intrinsic property of the neutron-nucleus interaction, typified by 1H, but can also arise from having a mixture of isotopes. GSAS-II provides coherent neutron scattering lengths for elements at natural abundance and for isotopes where the scattering lengths have been reported in the literature, so it is not necessary to look up any of these values.
There is one common exception to the angular-independent scattering of neutrons by atoms, and that occurs in materials with ordered unpaired electrons. Depending on the type of ordering, materials with unpaired electrons may be ferromagnetic, paramagnetic or exhibit related types of magnetic phenomena such as ferrimagnetism. Unpaired electrons have a net spin, as do neutrons, and these unpaired electrons can scatter neutrons from this spin-spin interaction. These unpaired electrons are always valence electrons and thus are more dispersed from the nucleus than the average for all the electrons. This means that scattering of neutrons by unpaired electrons falls with increasing angle similarly to what is seen in x-ray scattering, but the decrease with Q is even faster for magnetic neutron scattering than what is seen in the form factor for x-rays. Different scattering length curves are tabulated for different valence states for atoms that exhibit magnetic effects. These are automatically assigned in GSAS-II when a material with magnetic scattering is defined. In GSAS-II to change the valence state for an atom, click on the type entry in the Atoms table. This brings up a periodic table and under each element there are choices for valence state.
