Retarded Dispersion Energy

The retarded dispersion energy between the particles under investigation is readily obtained by applying the integration technique described in Section 3.4. This method is now rigorous. The contradiction of fluctuating electrostatic fields is removed. For the dispersion function G(co) we have to use the ratio of secular determinants obtained in Section 5.5 for finite and infinite separation of the particles considered. 

Let us first consider the attraction between spheres. According to Eq. (5.10) we find the argument Kr of the spherical Bessel functions (5.15) to be imaginary for imaginary frequencies. Both j (Kr) and y (Kr) increase exponentially with increasing r, i.e. the boundary condition (5.42) caused by the cavity with radius Vg turns into 

We are left with the spherical Bessel functions hl Kr) and h Kr) of the third kind, which decrease exponentially with increasing r in the upper and lower half-plane, respectively. This suggests the use of spherical Bessel functions of the third kind from the start. However, in that case it is difficult to interpret the physical relevance of the electromagnetic modes under consideration. They are normalized at imaginary rather than real frequencies. 

The exponential decrease of the potential considered similarly affects the coefficients E, J (Kr2i) Wt , Kr2i) arising in the addition theorem (5.36). The relevant expressions (5.45) entering boundary 

The coefficients % necessary for treating interactions up to dipole-octupole and quadrupole-quadrupole are given in Table 2. 

---

Casimir and Polder also showed that retardation effects weaken the dispersion force at separations of the order of the wavelength of the electronic absorption bands of the interacting molecules, which is typically 10 m. The retarded dispersion energy varies as R at large R and is determined by the static polarizabilities of the interacting molecules. At very large separations the forces between molecules are weak but for colloidal particles and macroscopic objects they may add and their effects are measurable. Fluctuations in particle position occur more slowly for nuclei than for electrons, so the intermolecular forces that are due to nuclear motion are effectively unretarded. A general theory of the interaction of macroscopic bodies in terms of the bulk static and dynamic dielectric properties... 

D. Langbein, Non-retarded dispersion energy between macroscopic spheres. J. Phys. Chem. Solids 32(7), 1657-1667 (1971). doi 10.1016/S0022-3697(71)80059-8... 

Langbein, D. 1970. Retarded dispersion energy between macroscopic bodies. Physical Review B 2, no. 8 3371. doi 10.1103/PhysRevB.2.3371. 

The retarded dispersion energy between two atoms i and j is found to obey a relationship with regard to the separation rather than the nonretarded relationship. 

The retarded dispersion energy between macroscopic particles was treated by Liftshitz [28]. He considered half-spaces. Going half the way from the microscopic to the macroscopic approach, Lifshitz expanded the local fluctuations within the half-spaces in terms of plane waves and coupled them to the outgoing (reflected) radiation field. Then, satisfying the boundary conditions for the radiation field across the surfaces of the half-spaces under consideration, he found their force of attraction from Maxwell s stress tensor in the interspace. 

Finally, let us consider the retarded dispersion energy between halfspaces. Since the cavity now cuts off the internal rather than the external modes, it is the multipole susceptibihties which are affected by... 

The retarded dispersion energy between half-spaces is proportional to the inverse cube of the separation X21. As in the case of spheres or cylinders, we find an additional power xj/ compared with the nonretarded result (4.76). 

The retarded dispersion energy between half-spaces was first considered by Lifshitz in 1955 [28]. We outlined his procedure in Section 5.1. 

The main difference between the quantum theoretical susceptibilities dss( i,j) according to Eqs. (7.68X (7.77) and the respective classical quantities (5.24), (5.25) is the dependence of the former on temperature via the occupation numbers / . /t The retarded dispersion energy between the particles considered, in accordance with the underlying electron-photon-exchange interaction, depends both on Bose statistics and on Fermi statistics. 

---