Neutron Scattering Cross Sections

of neutrons per second scattered into dX2 with energy between E and E -t dE 0dX2d  

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Neutron scattering depends upon nuclear properties, which are related to fluctuations in the neutron scattering cross section a between the scatterer and the surroundings. The scattered amplitude from a collection of scatterers can thus be written as (similar to (B 1.9.29)) ... 

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

Table 4.1-1 Comparison of the neutron scattering cross-sections for Li and Li with all the other atoms present in a binary mixture of LiSCN with AICI3.

FIGURE 2.13 (a) X-ray scattering factors for hydrogen, carbon, chloride and ferrous ions (b) the neutron scattering cross sections for several elements, as a function of sinOA. 

The density and thermal neutron cross-section values in Table 6 pertain to the thermal neutron attenuation gauging process. In this method, advantage is taken of the large thermal neutron scattering cross-section of hydrogen as compared to most other elements. In its simplest form, when a beam of thermal neutrons of intensity IQ traverses a sample of thickness x, the intensity 1 of neutrons measured by a thermal neutron detector will be... 

Fig. 8. Difference in the inelastic neutron scattering data between LaFe4Sb 2 and CeFe4Sb 2 vs. energy loss (Keppens et al., 1998). CeFe4Sbi2 was used as a reference compound since the neutron scattering cross section of Ce is much smaller than that of La. The difference spectra therefore reflect the vibrational density of states (DOS) associated with the La atoms. The peak at 7 meV (78 K) corresponds to the quasi-localized La mode. The second broader peak at about 15 meV corresponds to the hybridization of La and Sb vibrational modes. Both peaks can be accounted for using lattice dynamic models based on first-principles calculations (Feldman et al., 2000).

A number of studies have treated the effects of impurities on phase transitions from a theoretical perspective (Halperin and Varma 1976, Hock et al. 1979, Levanyuk et al. 1979, Weyrich and Siems 1981, Lebedev et al. 1983, Bulenda et al. 1996, ScWabl and Tauber 1996). By and large, however, theoreticians have focused on the way in which local interactions between defect fields and the order parameter produce an anomalous central peak in neutron scattering cross-sections of impure ferroelectrics up to 65°C above the critical temperature (Shirane and Axe 1971, Shapiro et al. 1972, Muller 1979). 

X-rays are scattered by electron density - the more electrons an atom has, the more intensely it scatters. Neutrons, however, are scattered by a parameter of the atomic nucleus, which is different for different isotopes (but shows no general trend with nuclear mass). Hydrogen and deuterium have very different (but large) neutron scattering cross-sections and therefore neutron diffraction, which requires access to an atomic reactor, is used where location of hydrogen atoms is critical. 

However, because of the nuclear spin incoherence of " Cl, the total incoherent cross section of chlorine is oinc (Cl) = 5.30 bam. Spin incoherence is particularly important for an understanding of the neutron scattering cross section of hydrogen nuclei. In = 1/2. 

In principle all the rotational transitions of dihydrogen are observable by INS spectroscopy but some transitions are rather weak since the incoherent neutron scattering cross section used generally in this book is inapplicable ( 2.1). This derives from the strict nuclear spin correlation found in dihydrogen, Transitions within the separate para- or orthohydrogen manifolds are controlled by the coherent cross section of hydrogen, 1.76 bam, and these transitions are too weak to be observed in its INS spectmm. 

To obtain further insight into the meaning of the inelastic neutron spectra, it is necessary to have specific theoretical models with which to compare the experimental results. In the harmonic approximation it is possible to calculate the incoherent inelastic neutron spectrum i.e., the neutron scattering cross section for the absorption or emission of a specific number of phonons can be obtained with the exact formulation of Zemach and Glauber.481 A full multiphonon inelastic spectrum can be evaluated by use of Fourier transform techniques.482 The availability of the normal-mode analysis for the BPTI136 has made possible detailed one-phonon calculations483 for this system the one-phonon spectrum arises from transitions between adjacent vibrational levels and is the dominant contribution to the scattering at low frequencies for typical experimental conditions.483 The calculated one-phonon neutron en-... 

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