General theory of scattering

An electromagnetic wave consists of an electrical and a magnetic field which vary periodically with position and time, and which travel through space at a velocity c = 3-0 X 10 cm/s (in vacuum). The electrical field strength may be given by  

Consequently, m varies with time and changes essentially in phase with if the frequency of the light is sufficiently below the resonant frequency for 

For a system consisting of a collection of particles, each of which is small as compared with the wavelength of light, the total scattered amplitude is the sum of the amplitudes from the particles and is  

The scattered intensity is often described in terms of the Rayleigh ratio given by  

The summation for F is over a unit volume of scattering material. 

We turn now to a first consideration of expressions for the orbital and spin matrix elements which arise when there is no restriction placed on the magnitude of the scattering vector. The matrix elements can be expressed in terms of Racah tensors, as mentioned before. 

Let us begin with the orbital, or current, matrix element. Write p = -iftV and work with spherical vector components labelled by = 0, 1. Atomic states are labelled by 6, J, M, where 6 contains all the quantum numbers required to define a state other than J, M. The magnetic neutron operator contains the quantity K X V, and the many-electron matrix element is found to be 

Our definition of spherical harmonics Yq k) and Clebsch-Gordan coefficients are in accord with the standard references. The quantity A(K, K ) is the product of radial integrals, and various factors which arise from the vector coupling coefficients of n equivalent electrons. The latter component of A(K, K ) is essentially embodied in the Racah tensor. Let 

The spin matrix element has a similar structure, with replaced by 

We can take the algebra for scattering of unpolarised neutrons from an isolated magnetic ion one stage further. First, we note that in the general case the matrix element of the magnetic interaction operator, built up from (29) and (31), is 

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