Lifshitz

The continuum treatment of dispersion forces due to Lifshitz [19,20] provides the appropriate analysis of retardation through quantum field theory. More recent analyses are more tractable and are described in some detail in several references [1,3,12,21,22],... 

A related approach carries out lattice sums using a suitable interatomic potential, much as has been done for rare gas crystals [82]. One may also obtain the dispersion component to E by estimating the Hamaker constant A by means of the Lifshitz theory (Eq. VI-30), but again using lattice sums [83]. Thus for a FCC crystal the dispersion contributions are... 

Good, van Oss, and Caudhury [208-210] generalized this approach to include three different surface tension components from Lifshitz-van der Waals (dispersion) and electron-donor/electron-acceptor polar interactions. They have tested this model on several materials to find these surface tension components [29, 138, 211, 212]. These approaches have recently been disputed on thermodynamic grounds [213] and based on experimental measurements [214, 215]. 

Landau L D and Lifshitz E M 1980 Statistical Physics part 1,3rd edn (Oxford Pergamon)... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

For a conserved order parameter, the interface dynamics and late-stage domain growth involve the evapomtion-diffusion-condensation mechanism whereby large droplets (small curvature) grow at tlie expense of small droplets (large curvature). This is also the basis for the Lifshitz-Slyozov analysis which is discussed in section A3.3.4. 

Landau L D and Lifshitz E M 1959 Fluid Mechanics (Reading, MA Addison-Wesley) eh 2, 7, 16, 17. (More reeent editions do not have ehapter 17.)... 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

Sigaud G, Hardouin F and Aohard M F 1977 An experimentai system for a nematio-smeotio A-smeotio C Lifshitz s point Solid Saste Commum 23 35-6... 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

Flough D B and White L R 1980 The calculation of Flamaker constants from Lifshitz theory with applications to wetting phenomena Adv. Colloid Interface Sc/. 14 3-41... 

L, D, Landau and E, M, Lifshitz, Quantum Mechanics, Pergamon Press, Oxford, U.K., 1965,... 

Several texts are available for further reading on turbulent flow, including Tennekus and Lumley (ibid.), Hinze (Turbulence, McGraw-HiU, New York, 1975), Landau and Lifshitz (Fluid Mechanics, 2d ed.. Chap. 3, Pergamon, Oxford, 1987) and Panton (Jncompressible Flow, Wiley, New York, 1984). 

Equation (2.2) defines the statistically averaged flux of particles with energy E = P /2m -f V Q) and P > 0 across the dividing surface with Q =0. The step function 6 E — Vq) is introduced because the classical passage is possible only at > Vq. In classically forbidden regions, E < Vq, the barrier transparency is exponentially small and given by the well known WKB expression (see, e.g., Landau and Lifshitz [1981])... 

Of special interest is the case of parabolic barrier (1.5) for which the cross-over between the classical and quantum regimes can be studied in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the parabolic barrier transparency [Landau and Lifshitz 1981],... 

The transition described by (2.62) is classical and it is characterized by an activation energy equal to the potential at the crossing point. The prefactor is the attempt frequency co/27c times the Landau-Zener transmission coefficient B for nonadiabatic transition [Landau and Lifshitz 1981]... 

Since the Hamiltonian is unbounded, the energy of each state is complex [Landau and Lifshitz 1981]... 

Landau, L.D. and E.M. Lifshitz, 1981, Quantum Mechanics (Pergamon, Oxford). 

Band gap engineetring confined hetetrostruciutres. When the thickness of a crystalline film is comparable with the de Broglie wavelength, the conduction and valence bands will break into subbands and as the thickness increases, the Fermi energy of the electrons oscillates. This leads to the so-called quantum size effects, which had been precociously predicted in Russia by Lifshitz and Kosevich (1953). A piece of semiconductor which is very small in one, two or three dimensions - a confined structure - is called a quantum well, quantum wire or quantum dot, respectively, and much fundamental physics research has been devoted to these in the last two decades. However, the world of MSE only became involved when several quantum wells were combined into what is now termed a heterostructure. 

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