Non-polarized beam and target

The differential cross-section L (q) calculated above depends on the initial spin state i of the neutron-nuclei system. Let us average over these initial states i. We allow, that the spin states of the nuclei j (j = 1,. . ., AO are distributed at random and that these states and the initial state of the incident neutron are totally uncorrelated. In such a condition, the distribution of spin states for the incident neutron has no effect on the cross-section it will be assumed here that the orientation of these spins is at random (non-polarized incident beam). The total number of spin states is 

The diagonal terms j = / can easily be calculated with the help of the result obtained in Section 3, for the scattering by one nucleus. In this simple case, the 

Let us now evaluate the last term. The neutron-nuclei spin states being uncorrelated, we have 

This shows that the operators s j contained in the right-hand side of (6.4.32) do not contribute to the cross-section. Consequently, this cross-section is 

Z(q) can be written as the sum of a part which is called coherent because it contains all the -dependent terms, and of an incoherent part which does not depend on q 

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