Lifshitz bounds

The term molecular crystal refers to crystals consisting of neutral atomic particles. Thus they include the rare gases He, Ne, Ar, Kr, Xe, and Rn. However, most of them consist of molecules with up to about 100 atoms bound internally by covalent bonds. The dipole interactions that bond them is discussed briefly in Chapter 3, and at length in books such as Parsegian (2006). This book also discusses the Lifshitz-Casimir effect which causes macroscopic solids to attract one another weakly as a result of fluctuating atomic dipoles. Since dipole-dipole forces are almost always positive (unlike monopole forces) they add up to create measurable attractions between macroscopic bodies. However, they decrease rapidly as any two molecules are separated. A detailed history of intermolecular forces is given by Rowlinson (2002). 

Feng, W. Y., Ling, Y., and Lifshitz, C., Reactivity of mixed and neat proton bound dimers of acetonitrile and methyl acetate, J. Rhys. Chem. 100, 35-39 (1996). 

T < T then pv C) is defined only outside the interval C < C < CK In this case, Lifshitz assumes that C(r) has no values in the interval and is therefore discontinuous (see Fig. 14.12). This conclusion is open to question. In fact, we can admit that pv(C) remains bounded and if a solution C(r) of (14.4.36) exists, it must be continuous since A[C(r)] must remain bounded. Now, if C(r) is continuous, there exists no interval C1 < C < C in which is not defined, since /[Pg.675]

At this point we digress slightly to recall the classic Stokes oscillating boundary layer problem (Lighthill 1978, Landau c Lifshitz 1987). In this problem a large volume of fluid, say in the region z<0, is bounded by a solid... 

Capillary waves do not only broaden the width of the interface but they can also destroy the orientational order in highly swollen lamellar phases (see Fig. 1 for a phase diagram extracted from Monte Carlo Simulations). Those phases occur in mixtures of diblock-copolymers and homopolymers. The addition of homopolymers swells the distance between the lamellae, and the self-consistent field theory predicts that this distance diverges at Lifshitz points. However, general considerations show that mean-field approximations are bound to break down in the vicinity of lifshitz points [61]. (The upper critical dimension is du = 8). This can be quantified by a Ginzburg criterion. Fluctuations are important if... 

To summarize, if p(ej ) is unbounded the energy band is unbounded. If the band is bounded and there is sufficient symmetry remaining, the bound is a normal band edge. Otherwise, it is a Lifshitz limit. 

Dispersion forces are universal because they attract all molecules together, regardless of their specific chemical nature. The potential energy of dispersion attraction between two isolated molecules decays with the sixth power of the separation distance. Based on the so-called Hamaker theory (i.e., the method of pair-wise summation of intermolecular forces) or the more modern Lifshitz macroscopic treatment of strictly additive London forces, it is possible to develop the so-called Lifshitz-Van der Waals expression for the macroscopic interactions between macroscopic-in-size objects (i.e., macrobodies) [19, 21], Such an expression strongly depends on the shapes of the interacting macrobodies as well as on the separation distance (non-retarded or retarded interaction). For two portions of the same phase of infinite extent bounded by parallel flat surfaces, at a distance h apart, the potential energy of macroscopic attraction is ... 

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