Section 6.1.1 covers X-ray scattering from atoms and ions. Scattering is described by the Thomson formula, including coherent (Rayleigh) and incoherent (Compton) X-ray scattering. Atomic scattering factors, calculated using relativistic Hartree–Fock or Dirac–Slater wavefunctions, give the X-ray scattering from an atom (in fact, its ensemble of electrons) in terms of that from a single electron. Free-atom scattering factors are tabulated for neutral atoms from atomic number 1 (hydrogen) to 98 (californium) over a scattering range of sin θ/λ from 0 to 6 Å⁻¹, and for ions from H¹⁻ to Pu⁶⁺ over 0 to 2 Å⁻¹. Analytical fits to the scattering factors are given and methods for interpolation of the tabulated factors are described. Perturbations from free-atom electron density for bound atoms are handled with generalized scattering factors expressed as spherical harmonics. Probability density functions for atom displacement due to temperature are described in terms of generalized temperature factors related to atom vibration symmetries. The final parts of Section 6.1.1 describe the role of atomic scattering factors in the computation of crystal structure factors by summation over unit-cell atoms, and the reflecting power of small crystals. Section 6.1.2 presents the basic equations governing magnetic scattering of neutrons. They are used to define the useful intermediate quantities of the magnetic interaction vector, the magnetic structure factor and the magnetic form factor, which are used in calculations of magnetic cross sections. A brief account of the way in which the magnetic scattering depends upon the neutron spin direction (neutron polarization) is included. Formulae for the scattering of neutrons by the nuclei of an atom are given in Section 6.1.3. The scattering cross sections for a single nucleus, for an element containing a mixture of isotopes, and for a single crystal are considered.