Molecular Imaging. Markus Rudin

are not independent because E^∗ is the negative gradient of a potential dependent on the charges but also on the polarization P_u, as we have seen in Eq. (2.11b). In [2] it is demonstrated that Eq. (2.29) can be expressed as:

      Putting δF^∗ = 0 we have:

      Note that if there were no constraints on δP_u(r), i.e. if its arbitrary variations would not affect E^∗, Eq. (2.24) would be satisfied for arbitrary variations δP_u(r) only if

      that is, if

      which is the usual electrostatic relation between P_u and E^∗ for systems with equilibrium U-type polarization. In our nonequilibrium system, the variations δP_u must satisfy the equation of constraint (2.24) where from we see that:

      The equation for the variation δF is (see Eq. (2.30a)):

      Subtracting Eqs. (2.30b) from (2.30a) one has:

      Equations (2.30a) and (2.31b) are to be satisfied simultaneously. Using the method of Lagrangian multipliers (see, e.g., Refs. [20–25]), the second condition multiplied by a Lagrangian multiplier m is added to the first one to have:

      This is an identity valid for every arbitrary variation of P_u in each volume element. It can be equal to zero only if the term in braces is equal to zero and this condition gives:

      So this is the P_u we were looking for. The first term is the usual electrostatic equilibrium relation between P_u and E^∗. A physical interpretation of Eq. (2.26) was given in the preceding section.

      M. expresses now E^∗ and E in terms of and E_c which can be easily calculated from the known charge distribution. The formulas for E^∗ and E are [1]:

      With the aid of these equations, Eq. (2.26) for P_u becomes:

      Introducing the equations for E^∗, E, and P_u into Eq. (2.13) for the electrostatic free energy of state X^∗, we obtain:

      We also have:

      where from m can be calculated.

      We designate now the charges of the reactants 1 and 2 in state X^∗ by and The corresponding charges in X will be e₁ and e₂. The radii a₁ and a₂ are assumed not to change essentially during the course of the reaction.

      The vector is the negative gradient of the potential which the reacting ions in state X^∗ would exert if they were in a vacuum rather than in a polarized medium. The potential exerted at point r by two ions of charges e₁ and e₂ is:

      where r₁ and r₂ are the distances of the field point r from the centers of the ions. The vector E_c is −∇ψ, ψ being given by the earlier equations. We thus have:

      The expressions for E_c and are introduced into Eq. (2.32) and into Eq. (2.33) and the integrations are performed. The following integrals are used for this purpose:

      where i can be 1 or 2, R is the distance between the centers of the ions and the integration volume excludes the volume physically occupied by the two ionic spheres, that is, r₁ ≥ a₁ and r₂ ≥ a₂ simultaneously.

      With the aid of Eqs. (2.24), (2.32), (2.34), (2.35), (2.25),