The Theory of Inelastic Neutron Scattering Spectroscopy

There have been many exeellent publications that have addressed the theoretical basis for our understanding of neutron scattering experiments and it is not our aim to make an original contribution to this well-established and non-controversial field. Unfortunately, whilst some publications are approachable [1,2], most are written for a readership comfortable with mathematical manipulation, and aim to treat all the possible physical cases. Our book, however, is focused on a materials chemistry theme using the technique of molecular vibrational spectroscopy and, moreover, almost exclusively from hydrogenous systems. It is, therefore, appropriate to restrict the scope and style of this theoretical presentation to reflect the theme of the book and the interests of its readership. 

We begin immediately by deseribing a simple neutron scattering measurement, shown diagrammatically in Fig. 2.1. 

A beam of monoenergetic neutrons falls on a sample and is scattered onto a detector of small area, dA, a distance df from the sample. This is positioned at the scattering angle, 0, which is defined with respect to the unscattered, straight through, beam. It is found empirically that neutrons are scattered differently by different elements. Some substances scatter neutrons evenly in all directions whilst others scatter most neutrons in specific directions. The latter is of course a consequence of diffraction, more familiar through the scattering of X-rays, and is a coherent event. 

The number of neutrons scattered out of the beam is constant for a fixed mass of a given element. It follows a law similar to the Beer-Lambert law for optical absorption and linear approximations to this law can be exploited [3] (for simplicity we shall assume that the element does not absorb neutrons). The initial neutron intensity, or flux on the sample, is Jj and the unscattered flux after the sample is Jf. Then, since the mass of the sample is the product of its thickness, dg, and concentration, C, we can write  

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