Neutron scattering matrix elements

Neutron scattering matrix elements We form matrix elements of taken between many-electron atomic states, drawing on the extensive algebra of atomic and nuclear spectroscopy calculations based on work by Racah. The application to f electrons is reviewed by Judd (1963). 

We close this section by noting that the destructive interference has its origin in the spatial wavefunction, which gives cancellations in the scattering matrix element if the amplitudes for particle a (or j3) has opposite signs when the neutron scatters from sites 1 and 2. Fig. 5 is an attempt to illustrate the scattering from a particle that is delocalized over two different sites. 

The coupling of the freeon waves with the appropriate spin function produces a wave function which has nonzero matrix elements with the wave function of a neutron scatterer so that permutation dispersion relation can be and has been experimentally verified. 

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. 

Figure 9. BA was the first system in which the tunneling contribution to the reaction rate at low temperatures was determined [52,147,148] and is the only system for which the tunneling matrix element for intermolecular PT has been measured [53,149,150]. The following sections summarize the optical studies on the material that led to these results and discuss recent work, using NMR and neutron scattering, aimed at a more complete characterization of the proton dynamics.

Ii3/2 multiplets. Frequencies and polarization vectors of phonons in the LiYp4 crystal were obtained at 8000 points in the irreducible part of the Brillouin zone using the rigid ion model of lattice dynamics derived on the basis of neutron scattering data. Matrix elements of electronic operators Vds) were calculated with the wave functions obtained from the crystal-field calculation. The inverse lifetimes of the crystal-field sublevels determine the widths of corresponding absorption lines. 

When introducing the neutron scattering operator, 0 = baCxp (iq-Ra)+ bpexp(iq-Rp), the spatial parts of the matrix elements < I f 0 I i > turn out to have the two forms [12]... 

We can take the algebra for scattering of unpolarised neutrons from an isolated magnetic ion one stage further. First, we note that in the general case the matrix element of the magnetic interaction operator, built up from (29) and (31), is... 

In this and the subsequent section we provide explicit expressions for the orbital and spin matrix elements, respectively, which describe the scattering of neutrons by equivalent, non-relativistic electrons in a single atomic shell. Derivations of the expressions are reported in several references. 

The major effort entering a calculation of matrix elements for magnetic neutron scattering is concentrated in the quantities denoted A K, K ) and B(K, K ). From the discussion in sections 2.2.4 and 6.4 we have found the factor A K, K ) to be associated with both the radial averages OV -i) Uk +i)- emphasise this fact in the tables, the corresponding numbers K — K + are given as subscripts in the form A K, Th values of B K, K ) associated with... 

Thus, we can conclude that elastic scattering, as the most common interaction, is dominated by hydrogen (H). Hydrogen content controls the strength of elastic scattering this is the background for correlation with water content or neutron porosity . But the other elements also interact with neutron radiation this results in a matrix effect for neutron porosity determination. 

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