Scattering of Thermal Neutrons: A Bibliography (1932–1974)
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The points represent k' values permitted by box normalisation. The derivation shows that, for fixed k, A, and A', the probability of a transition k, A to k', A' is negligible except for a narrow range of fc' values. This enables us to calculate p k ; and it also fixes the normalisation constant of the neutron wavef unctions. The wavevectors of such states form a lattice in k space. A' is the cross-section for neutrons scattered into dft in the direction of k'.
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However, as already mentioned, since k, A, and A' are fixed, the scattered neutrons all have the same energy, deter- mined by conservation of energy. So the potential for the whole scattering system is V-EWr-ffy. The logic in going from 2. We see from 2.
Consider 2. Inserting this result in 2. Inserting this value in 2. The positive sign in 2. We recall that our entire derivation of the cross-section is based on Fermi's golden rule, which, for scattering processes, is equivalent to the Born approximation; both are based on first-order perturbation theory. Now the conditions for this theory to apply do not hold for the nuclear scattering of thermal neutrons. The justification for the use of the golden rule in these circumstances is that, when combined with the pseudopotential, it gives the required result of isotropic scattering for a single fixed nucleus.
Equation 2. However, a positive scattering length does not imply that the actual potential is repulsive. If we simulate the actual potential by a hypothetical 'square-well' or square-barrier of range r and depth or height V, we can solve the Schrodinger equation without approximation.
Solid state physicists have long appreciated the usefulness of thermal neutron scattering in the inves tigation of condensed matter. This technique was first made. Scattering of Thermal Neutrons a Bibliography () (Handbook of Electronic Materials) [Andre Larose] on giuliettasprint.konfer.eu *FREE* shipping on qualifying.
For a repulsive potential b is positive for all values of x. For an attractive potential it may be negative or positive. The actual potential is basically attractive. The details of its shape, depth, and range determine the magnitude and sign of the scattering length. The scattering length defined in 1. If the nucleus is free, the scattering must be treated in the centre-of-mass system. The result is the same as if the nucleus were fixed, but the mass m of the neutron must be replaced by the reduced mass n of the nucleus- t For further discussion of this point see Fermi and Breit The scattering length for this process is called the free scattering length.
Denote it by b f. Since the potential is the same whether the nucleus is fixed or free, the expres- sion for the pseudopotential 2. Repulsive 18 Nuclear scattering - basic theory Inserting this in 2. The reason for doing this will become apparent later in the section. We also need the following results. Let H be the Hamiltonian of the scattering system. The states A and A' are eigenfunctions of H with eigenvalues E k and fv, i. The matrix element in 2. So the square of the matrix element is the sum of N 2 terms, of which a typical member is bfbM'l exp iK. From 2.
Sum over A', average over A In an actual experiment we do not measure the cross-section for a process in which the scattering system goes from a specific state A to another state A'. A ' over all final states A', keeping the initial state A fixed, and then average over all A.
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To carry out the first step we use the closure relation proved in Appendix C. It is a compact expression and may appear simple, but its evaluation, except for the most elementary scattering systems, is not a simple matter.
The properties of the scattering system are contained in the Hamiltonian H. They are therefore contained in the Heisenberg operators, and the eigenstates A. In the next three chapters we shall be concerned with evaluating the cross-section for specific physical systems. We can see at this stage the purpose of expressing the S -function for conservation of energy as an integral with respect to time in 2. By means of 2. There is now no term in A ' outside the two matrix elements, and the sum over A ' can be carried out immediately.
Imagine that we have a large number of scattering systems. They are identical in every way as regards the positions and motions of the nuclei. But each system has a different distribution of the is among the nuclei, every possible distribution being represented once. Now, provided the system contains a large number of nuclei - a condition usually well satisfied - the cross-section we measure is very close to the cross-section averaged over all the systems.
So dV dildE 2. The first term in 2. It therefore gives interference effects. The inco- herent scattering depends only on the correlation between the posi- tions of the same nucleus at different times. It does not give inter- ference effects. The physical interpretation of 2. The actual scattering system has different scattering lengths associated with different nuclei. The coherent scattering is the scattering the same system same nuclei with the same positions and motions would give if all the scattering lengths were equal to b.
The incoherent scattering is the term we must add to this to obtain the scattering due to the actual system.
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Physically the incoherent scattering arises from the random distribution of the deviations of the scattering lengths from their mean value. The simplest case is when the scattering system consists of a single isotope with zero nuclear spin. The quantity b is known as the coherent scattering length of the element or nuclide. It is con- ventional to quote the values of b and b 1 in terms of the two quantities cr coh and a mc defined in 2.
A list of the values of o- coh and o- inc for the elements, together with a description of the methods of measuring these quantities, has been given by Koester A few of the values are given in Table 2. Table 2. The values are taken from Koester The extension of the theory to scattering systems containing more than one element is readily made. The incoherent scattering is the sum of the incoherent scattering from the sodium nuclei and the incoherent scattering from the chlorine nuclei.
Nuclear scattering by crystals 3. We start by considering a Bravais crystal, i. Vo 2tt. The volume of the unit cell in the reciprocal lattice is 2tt 3 Vo Ti. Origin 25 26 Nuclear scattering by crystals From 3. So in 2. Similarly in 2. Origin 3.
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M is the mass of an atom - assumed to be the same for all the atoms. These operators are discussed in Appendix E. Normal modes are discussed in Appendix G, where 3. It is shown in Appendix E.
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- Scattering of Thermal Neutrons.
We first define the probability function for a single bound particle - not necessarily a harmonic oscillator - moving in one dimension. Denote the displacement or position variable by Q. Suppose the particle is not in a single state n, but in an incoherent mixture of states. C is a normalising constant obtained from 3.
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It follows from 3. We now develop the expression exp U exp V. We first prove that UV-VU is a c-number. From 3.
We next use 3. Each displacement has a Gaussian prob- ability function. The probability function for a linear combination of Gaussians is itself a Gaussian. We can therefore apply 3.