The transition rate—Fermis Golden Rule

the sample, of N atoms, is represented by the wavefunction, and the N volume elements Avn contain the individual atoms, whilst the single volume dv contains the neutron. The integral is taken over all the N+l volume elements. 

We can now proceed to determine the specific scattering cross section for, say a (1 —0) transition, i f. Substituting into Eq. (A2.8)  

The normalised density of final momentum states becomes  

This cross section is for neutrons that are scattered into a solid angle of d/2 that lies in the direction given by the final wavevector, k. In the experiment the incident neutron energy is fixed and the initial and final states of the sample are also fixed, even if not known. The total energy of the system is conserved and that which is lost by the neutron is gained by the sample, so  

Since the energy is conserved, the final energy of the neutrons is also fixed. All of the scattered neutrons appear with precisely the same energy, which is conveniently represented by a delta-function in energy. The delta-function has several useful mathematical properties, including  

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