Model with external magnetic field

As always, the connected Green s functions are the cumulants of the Green s functions, and consequently their generating function bG B] is given by 

In particular, there exists two two-leg Green s functions, the longitudinal Green s function (in the direction of the magnetic field) 

Incidentally, we note that these functions are important in polymer theory in fact, the field-polymer correspondence and Fig. 11.3 show that the transverse Green s function is related only to intrachain correlations, and the longitudinal Green s function both to interchain and intrachain correlations. 

Let us now consider the vertex functions, and their generating function which we shall determine. First, we can introduce the Legendre transform T(M0) of the function SG(B0) defined by 

This transformation is a special case of (11.5.14). It corresponds to the choice 

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To determine the critical exponents, it is sufficient to study the properties of an isolated polymer. For this purpose, we can content ourselves with a one field theory this is the Landau-Ginzburg model described in Chapter 11. Incidentally, we may recall that the Landau-Ginzburg model with external magnetic field corresponds to the equilibrium ensembles defined in Chapters 9 and 11 consequently, it can be used to calculate approximately the properties of polymer solutions.18... 

Fig. 27 Magnetic heat capacity for PhBABI for 7 < 100 K showing variation with external magnetic field (left) zero-field magnetic heat capacity showing fits (right) to ID AFM chain, 2D AFM square planar, 2D AFM square planar bilayer, singlet-triplet spin pairing (ST), and spin ladder models.

The basic problem of statistical mechanics is to evaluate the sum-over-states in equation 7.2 and obtain Z and F as functions of T and any other variables (such as external magnetic fields) that might appear in %. Any thermodynamic observable of interest can then be obtained in a straightforward manner from equation 7.5. In practice, however, the sum-over-states often turns out to be prohibitively difficult to evaluate. Instead, the physical system is usually replaced with a simpler model system and/or some simplifying approximations are made so that the sum-over-states can be evaluated directly. 

The PEDM is able to explain the anomalous relaxation of solutions of ferritin and akaganeite particles, especially its linear dependence with Bq, the external magnetic field. The model is compatible with the observed dependence of the rate on pH. The relaxation rate predicted by the PEDM is proportional to the number of adsorption sites per particle (q) the values deduced for q from the adjustment of the model to experimental results (from NMR and magnetometry in solutions) are reasonable for hydrated iron oxide nanoparticles (63). 

As an alternative model, imagine a cannister that is filled completely with a gelatinous medium. Imagine also that there is a small magnetized ball suspended in the jelly with adhesion to a reasonable degree at all contact surfaces. By means of an external magnetic field the ball can be inverted by... 

In the triplet model the spin polarization is with respect to the internal molecular states, TjJ>, Ty>, and T > of the triplet and evolves with time according to the time-dependent Schrodinger equation into a spin polarization with respect to the electron spin Zeeman levels Ti>, Tq>, and T i> in an external magnetic field Bq. Consider a simple case of axially symmetric zero-field splitting (i.e., D y 0 and E = 0 D and E are the usual zero-field parameters). Tx>, [Ty>, and TZ> are the eigenstates of the zero-field interaction Hzfs, where Z is the major principal axis. The initial polarization arising from the population differences among these states can be expressed as... 

Fig. 22 Simulated spectra of the five-proton model system (shown in Fig. 19a and with coordinates as in Table 4) obtained with the decoupling schemes as indicated in the figure and at an external magnetic field of 14.1 T. The parameters of the sequences used in the simulations are summarised in Section 7...

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