Doublet Distribution Functions

The doublet distribution function (or pair distribution function) f,0, for a pair of two molecules a and is defined as  

If 0L = p, the terms in the double sum with i = j are to be omitted. The interpretation of the doublet distnbution function is as follows  

The doublet distribution function in configuration space is obtained by integrating the above function over all momenta  

If the centers of masses of the two molecules a and p are quite far apart, so that the internal motions of the two molecules are uncorrelated, then the pair distribution function can be factorized  

Since Pap varies rapidly with the vector distance = r — r between the centers of mass of the two molecules, but weakly with the location of the center of mass r of the two-molecule pair, Eq. (1.22), it is useful to make a further change of variables, for which the Jacobian is unity  

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We consider the pair formulation because it is most suitable for this application. The kinetic equation for the doublet distribution function... 

In order to use the flux expressions developed in the foregoing sections, it is necessary to have the singlet and doublet distribution functions. The partial... 

In order to make use of the flux expressions in Sects. 6, 7, and 8, it is necessary to have the singlet distribution function and - unless the short-range force assumption is used - the doublet distribution function as well. Virtually nothing is known about the doublet distribution function. If we knew how to make a reasonable guess of this function (possibly obtainable from molecular or Brownian dynamics), then we could estimate the contributions to the fluxes in Table 1 that involve the molecule-molecule interactions. 

Two mechanisms have been proposed to explain the appearance of an asymmetric doublet in randomly oriented substances with no magnetic ordering. One mechanism is based on the combination of the directional quantities — the angular distribution function of the magnetic dipole radiation and the Debye-Waller factor which becomes anisotropic in systems of lower than cubic symmetry. This mechanism predicts an asymmetry which should decrease as the temperature is lowered, in contradiction to the experimental observations in hemoglobin. The second mechanism is based on magnetic interactions described by the general Hamiltonian Eq. (234). 

In order to use the expressions for the mass flux vector, the stress tensor, and the energy flux vector in Table 1, it is necessary to know the singlet and doublet configuration-space distribution functions. For example, we need to solve Eq. (10.6) or Eq. (10.7) to get the distribution function for a single polymer molecule. This cannot, however, be done until something is inserted for the double-bracket... 

We confine our attention here to dilute solutions of several polymer speaes m a solvent. According to Sect. 7, the stress tensor is a sum of four contnbutions, the first three of which involve the singlet distribution function (s), whereas the fourth involves the doublet distribution fiinction(4) ... 

NMR in Fig. 1 (top and middle). If the sample is partially ordered, like a partially aligned liquid crystal polymer, the NMR line shape reflects the orientational distribution function of the coupling tensors [28, 29]. Figure 1 (bottom) depicts the special case of a completely aligned sample. Such a sample with macroscopically uniform alignment, for nematic phases usually induced by the magnetic field, gives rise to a simple doublet. 

The birth and death functions now have the same units as the population balance. To attempt a solution, an integral or continuous approach will be used in place of this discrete summation. This suggests that there is a continuous distribution of particle sizes (i.e., the sizes of interest for the population balance are much larger than that of singlets, doublets, etc.). Some key substitutions for this integration are necessary ... 

A simplified method was proposed by Kundig et al. (9) allowing evaluation of the particle size and the size distribution of a solid by analysis of its spectrum as a function of temperature. By variation of the temperature it is possible to follow the variation of the relative spectral areas of the sextet and doublet. Assuming that particles for which Tj. < Tl and > Tl contribute exclusively to one of the two components (paramagnetic and magnetic), the temperature at which — tl and at which the hyperfine split takes place can be determined V is calculated from Eq. (19) at the temperature T at which the spectral areas of the two components are equal). The dependence of the two spectral areas on temperature in the range in which both components are observed yields the particle size distribution. 

The Mossbauer spectra of iron in numerous minerals have been studied, but a few examples will serve to illustrate this technique. In the rockforming silicate minerals, Mdssbauer spectroscopy has been used to study the oxidation state, spin state and coordination of iron, and its distribution between different sites in a structure. Thus, Fig. 2.46 (after Williams et al., 1971) shows the spectrum of an augite [essentially (Ca,Mg,Fe)2Si20J, the structure of which contains two kinds of sixfold-coordinate sites that may be occupied by iron (the Ml and M2 sites). The Mdssbauer spectrum can be fitted to three quadrupole doublets peaks 1 and 1 have parameter characteristic of Fe (in both Ml and M2 sites), peaks A and A have parameters characteristic of Fe in Ml, and peaks C and C of Fe + in the M2 sites. These assignments, based chiefly on comparisons with endmember compositions and related species, also enable estimates of site populations to be made on the basis of the areas under the peaks. Studies of the variation in site populations as a function of composition and thermal treatment have led to important advances in understanding intercrystalline order-disorder equilibria, as pioneered in the work of Virgo and Hafner (1970). 

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