Neutron scattering theory

This is very similar to the familiar electron Schrodinger equation, but here the particle is the neutron. It is positioned at some general coordinate, r, and has a mass, /Wn. The solutions are, y/, where  

We invoke the Bom approximation, which views the scattering potential, V(r), as so weak a perturbation that, for incident, i, and final, f, conditions  

In free space the incident, i, neutron experiences no perturbing field and at long distances from the sample  

We shall need this later to apply a well known result of quantum mechanics, involving time dependent perturbation theory, Fermi s Golden Rule [1]. 

Here it is convenient to recall the expression for the de Broglie wavelength. A, of a neutron and the related neutron momentum vector, k. Readers should familiarise themselves with aspects of the manipulation of k ( 2.3). The neutron velocity is v. 

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The parameters of neutron scattering theory of polymer networks are A, the macroscopic stretching of the sample, or linear degree of swelling, f, the network functionality, K. which accounts for restricted junction fluctuations and a, a measure of the degree to which chain extension parallels the macroscopic sample deformation. The functionality is known from knowledge of the chemistry of network formation, and A is measured. Both K and a must be extracted from experiments. 

Even though TEM and SEM played major roles in the study of IPN morphological features, there are various shortcomings, such as staining artifacts, difficulties in sample preparation for very rubbery materials, and the two-dimensional viewing limit for the former. Recently, various scattering techniques have been applied to measure the phase dimensions of IPN s via statistical treatment. The principles of neutron scattering theory as applied to the phase separated materials have been described in a number of papers and review articles [33-36]. 

The MO model is discussed first and then the necessary elements of neutron scattering theory, with a brief description of experimental techniques. Finally there is a discussion of the data so far obtained. 

B. The Neutron Scatteing Method 1. Neutron Scattering Theory... 

It is important to realize that the wave incident on each nucleus is not the same as the incident wave on the entire sample. A good discussion on treating the problem of multiple scattering has been given by Sears [15] but falls beyond the scope of this introduction to neutron scattering theory. It can be shown [16] that the effective potential for a neutron scattered by a collection of nuclei is simply the volume average of the Fermi potenhals... 

A detailed description of the neutron scattering theory can be found in references 20 and 21. 

This last expression, (6.3), has been derived for fixed nuclei but the separation process into incoherent and coherent terms carries over into all kinds of neutron scattering theory. Moreover, it is particularly important for hydrogen. Remembering that the number of quantum states associated with a compound nuclear spin of J will be 27 -I- 1, we find that the average scattering length taken over the two spin states of hydrogen can be written ... 

The principles of neutron scattering theory as applied to the solution of polymer problems have been described in a number of papers and review articles (8-23). The coherent intensity in a SANS experiment is given by the scattering cross-section dl/dG, which is the probability that a neutron will be scattered into a solid angle, for unit volume of the sample. The quantity d /dG expresses the neutron scattering power of a sample and is the counterpart of the Rayleigh ratio, R(0), used in lightscattering. 

Hughes, D. J., and J. A. Harvey Neutron Cross Sections. Brookhaven National Laboratory Report BNL 325- This is a collection of total and reaction cross sections presented graphically as a function of energy. It extends and brings up to date the earlier report by Adair Neutron Cross Sections of the Elements. Rev. Mod. Phys. 22, 249 (1950). This latter report has a very clear introduction in which neutron scattering theory is summarised. 

Fundamental neutron scattering theory shows that da/dfl is given by ... 

Eckert J and Kearley GJ eds (1992) Spectroscopic applications of inelastic neutron scattering theory and practice. Spectrochimica Acta 48A 269-478. 

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