Moment density

The magnetic field seen by the probe neutron is solely due to the magnetic dipole moment density of the unpaired electrons. In other words, the magnetisation density is simply related to the electron spin density by a multiplicative factor, and there is no ambiguity in its definition. 

P polarisation, nett dipole moment densities per unit volume... 

Examples of p(r) are energy density, charge density, current density (see Section 4.6), difference density (difference between a final density and an initial density), electric moment density, magnetic moment density, local reactivity functions (see Section 4.5.2), force density, etc. Note that, for ensuring the stability of matter, the net force density must vanish everywhere in space. The concept of a PDF has generated many significant developments in interpretative quantum chemistry. 

Then the momental densities n (p) = (4ti) u t p) p for various values of the angular momentum quantum number have the following expansions [187] ... 

The unique piezoelectric and pyroelectric properties of semicrystalline films of PVDF arise from changes in the polarization imparted to the overall film by the crystalline P-phase. The polar nature of the P-phase is, in turn, a direct result of the parallel alignment of the dipole moment of the repeat units in the unit cell (Figure 11.1). The crystal polarization is defined as the dipole moment density of the crystal ... 

These results express the fact that any linear combination of conserved densities (a generalized moment density) is itself a conserved density in thermodynamics. We have shown, therefore, that if the free energy of the system depends only on K moment densities p,... pK, we can view these as the densities of K quasi-species of particles and can construct the phase diagram via the usual construction of tangencies and the lever rule. Formally this has reduced the problem to finite dimensionality, although this is trivial... 

In previous work, the authors originally arrived independently at two definitions of a reduced free energy in terms of moment densities (moment free energy for short) [31, 32]. Though based on distinctly different principles, the two approaches led to very similar results. We explain this somewhat surprising fact in the present work at the same time, we describe the two methods in more detail and explore the relationship between them. We also discuss issues related to practical applications and give a number of example results for simple model systems. 

The second approach rederives the entropy of mixing in the ideal part of the free energy in a form that depends explicitly only on the chosen moment densities. As described in Section n.B, the expression that results is intractable in general because it still contains the full complexity of the problem. However, in situations where there are only infinitesimal amounts of all... 

As explained in the previous section, truncatable here means that the excess free energy / depends only on K moment densities p,. Note that, in the first (ideal) term of (6), we have included a dimensional factor R(a) inside the logarithm. This is equivalent to subtracting T dap a) In R(a) from the free energy. Since this term is linear in densities, it has no effect on the exact thermodynamics it contributes harmless additive constants to the chemical potentials p a). However, in the projection route to a moment free energy, it will play a central role. 

At this point, the factor R(a) in (6), which is immaterial if all conservation laws are strictly obeyed, becomes central. Indeed, maximizing the entropy over all distributions p a) at fixed moment densities p,- yields... 

The corresponding minimum value of / then defines our projected free energy as a function of the moment densities pp. 

Here spr is the projected entropy of an ideal mixture. The first term appearing in it, p0 = J dop a), is the zeroth moment, which is identical to the overall particle density p defined previously. If this is among the moment densities used for the projection (or more generally, if it is a linear combination of them), then the term — Tp0 is simply a linear contribution to the projected free energy/pr(p,) and can be dropped because it does not affect phase equilibrium calculations. Otherwise, p0 needs to be expressed—via the A —as a function of the pit and its contribution cannot be ignored. We will see an example of this in Section V. 

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