Casimir limit

Phonon velocity is constant and is the speed of sound for acoustic phonons. The only temperature dependence comes from the heat capacity. Since at low temperature, photons and phonons behave very similarly, the energy density of phonons follows the Stefan-Boltzmann relation oT lvs, where o is the Stefan-Boltzmann constant for phonons. Hence, the heat capacity follows as C T3 since it is the temperature derivative of the energy density. However, this T3 behavior prevails only below the Debye temperature which is defined as 0B = h( DlkB. The Debye temperature is a fictitious temperature which is characteristic of the material since it involves the upper cutoff frequency ooD which is related to the chemical bond strength and the mass of the atoms. The temperature range below the Debye temperature can be thought as the quantum requirement for phonons, whereas above the Debye temperature the heat capacity follows the classical Dulong-Petit law, C = 3t)/cb [2,4] where T is the number density of atoms. The thermal conductivity well below the Debye temperature shows the T3 behavior and is often called the Casimir limit. 

The equation of radiative transfer will not be solved here since solutions to some approximations of the equation are well known. In photon radiation, it has served as the framework for photon radiative transfer. It is well known that in the optically thin or ballistic photon limit, one gets the heat flux as q = g T[ - T ) from this equation for radiation between two black surfaces [13]. For the case of phonons, this is known as the Casimir limit. In the optically thick or diffusive limit, the equation reduces to q = -kpVT where kp is the photon thermal conductivity. The same results can be derived for phonon radiative transfer [14,15]. 

Fig. 99. The temperature dependence of Kl/T for a single crystal of TmV04 (Daudin and Salce 1982). Dashed lines show calculated values of Kl for boundary scattering (the so-called Casimir limit).

Again, taking the limit L —> oo, we regain the Stefan-Boltzmann contribution alone, while making (3 —> oo the Casimir term at zero temperature is recovered. The third term, which stands for the correction... 

Again, as (3 —> 0, the Stefan-Boltzmann term dominates, while in the limit (5 —> oo, we find the Casimir energy at zero temperature, the second term in the right hand side of Eq. (38). 

Although not much used in actual calculations, we also quote the case of the strict normal-mode limit. In this case, H can be written in terms of Casimir invariants of Eq. (4.48)... 

Ya.B. applied formal perturbation theory to the interaction of an atom with the electrons of a metal, where the latter are assumed to be free. Meanwhile, Casimir and Polder and Lifshitz neglected the spatial dispersion of the dielectric permittivity of the metal. Therefore, in the region of small distances, frequencies of order ui0 are important at small distances in the sense indicated above, as are arbitrarily small frequencies at large distances. In both limits the dielectric permittivity of the metal is not at all close to one. Meanwhile, the perturbation theory used by Ya.B. corresponds formally to an expansion in powers of e - 1. and is therefore not applicable in this case. Neglecting the spatial dispersion is valid, however, only at distances r > a (a is the Debye radius in the metal) of the atom from the surface. At the opposite extreme, r a, the wave vectors kj 1/r > a vF/u>0 Me of importance (vF is the electron speed at the Fermi boundary). In this region of strong spatial dispersion perturbation theory can be applied, and the (--dependence satisfies Zeldovich s law. 

A rule of thumb for the validity of linear relationships is that processes should be slow and the thermodynamic states near equilibrium. But even then, not all flows can be coupled. Coupling is limited to certain cases. Casimir shows that coupling is only possible between driving forces of the same tensorial character, such as scalar, vectorial, and so forth. For more details, see Ref. [2],... 

It is not heretical to consider the electromagnetic vacuum as a physical system. In fact, it manifests some physical properties and is responsible for a number of important effects. For example, the field amplitudes continue to oscillate in the vacuum state. These zero-point oscillations cause the spontaneous emission [1], the natural linebreadth [5], the Lamb shift [6], the Casimir force between conductors [7], and the quantum beats [8]. It is also possible to generate quantum states of electromagnetic field in which the amplitude fluctuations are reduced below the symmetric quantum limit of zero-point oscillations in one quadrature component [9]. 

If the Hamiltonian now contains the Casimir operators of both G, and G[, which do not commute, then the labels of neither provide good quantum numbers. Of course, in general such a Hamiltonian has to be diagonalized numerically. In this way one can proceed to break the dynamical symmetries in a progressive fashion. In (61) all the quantum numbers of G, up to G remain good. If we add another subalgebra beside Gz only those quantum numbers provided by G, on will be conserved, etc. In applications, the different chains are found to correspond to different limiting cases such as the normal versus the local mode limits for coupled stretch vibrations (99). 

There are some indications that the pseudo-Casimir force in liquid crystals can be studied systematically with existing experimental techniques [69]. The lower limit for the separation of the substrate between which the force could be studied is roughly given by a molecular size where the system starts to behave as a continuum. The upper limit depends on the sensitivity of the force apparatus which is in the case of standard apparatuses about 10 pN [70]. This means that these apparatuses are precise enough to detect the pseudo-Casimir force at distances up to ss 40 nm [69]. 

In the application to strong electrolytes serious doubt exists whether the application of the complete cq. (40) is permissible because it implies certain internal inconsistencies which have been analys ed most extensively by Kirkwood But Casimir extending Kirkwood s analysis has shown that these inconsistencies do not arise (remain very small) when the complete eq. (40) is applied to the double layer on a large plane interface or on a large particle if the electrolytic concentrations in the whole system remain so small that in the bulk of the solution the limiting laws of Debye and Huckel form a reasonable approximation. 

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