Spin-line models equations

Figure 1. The relation between central density and the mass of various degenerate star models. Chandrasekhar s curve is for white dwarfs with a mean molecular weight 2 of atomic mass units. Rudkjobing s curve is the same except for inclusion of the relativistic spin-orbit effects Rudkjobing (1952). The curve labeled Oppenheimer and Volkoff is for a set of neutron star models. The solid line marked Wheeler is a set of models computed with a generalized equation of state, from Cameron (1959).

Recently, the stochastic models for the Mossbauer line shape problem have been discussed by several investigators.20 Such models can be treated in a systematic way as we have described in the above. For example, in a 57Fe nucleus, the spin in the excited state is / = and that in the ground state is / = i, so that the Hamiltonian is a 6 x 6 matrix. If a two-state-jump model is adopted, the dimension of the matrix equation, Eq. (63), is 6 x 2 = 12. If the stochastic operator is of the type (26), then the equation is a set of six differential equations. These equations can be solved, if necessary, by computers to yield the line shape functions for various values of parameters. 

Quantitative calculations of the CIDNP effect can be performed as described in Section II.B.5. One has to set up the nuclear spin system of the intermediate radical pair or biradical, choose a diffusional model, compute reaction probabilities for every nuclear spin state by solving the stochastic Liouville equation numerically or approximately, establish a correlation between the nuclear spin states in the paramagnetic intermediates and the nuclear spin states in the products to obtain the populations of the latter, and finally apply Eq. 61 or the formalism of the preceding section to get line intensities. This approach, which for all but the simplest systems is impracticable except on a computer, is often necessary with the usual uncertainty of the parameters entering the calculations of the radical pair mechanism, a reasonable accuracy can be expected. However, qualitative relationships between signal intensities, especially signal phases, and parameters of the reaction mechanism as well as magnetic properties of the intermediates are... 

Equation (43) can be substituted into Eq. (36) to give a prediction of the pressure dependence of the lifetime of Cr YAG based on a model that considers spin-orbit coupling of the zero phonon E and T2 states. Figure 14 includes a representative lifetime prediction based on this simple spin-orbit coupling model (dot-dash line). As in the pure electronic model, the prediction assumes that fE(P = 0) = 114 s and A (cm ) = 828 cm -i-9.8P (kbar). Since fj fE, we further assume that pressure induced changes in spin-orbit coupling do not significantly affect fx and use f = 6135 s. The final model parameter Vgo was set equal to a typical ambient pressure value, 202 cm [254], and was assumed to be constant with pressure. 

Fig. 2. Larmor frequency and angular dependences of the longitudinal proton spin relaxation time Tj for the two /i-alkyl-cyano-biphenyls 5CB and 8CB, respectively, at temperatures in the middle of the nematic mesophases. The upper diagram shows the dispersion profiles Ti(v) with the director H a igned parallel (A = 0) and perpendicular (A = 90 ) to the external 2 eman field, Bq, and suggests to distinguish at least four relaxation regimes. The lower diagram shows ri(A) in different frequency ranges of the dispersion profile. The solid lines are model fits of equation (3a) to the experimental 5CB data points as discussed in the text.

Favre (1993) found that his data for constant volume perfusion of two different mouse-mouse hybridoma lines could not fit the classical constant volume filtration equation. He suggested that this could be due to a monolayer type of cell buildup instead of a cake in the conventional filtration. According to Favre, a model principally based on stochastic and steady plugging of the pores in the screen by the cells leading to an equilibrium state can possibly give a rational design procedure for spin filters. 

---