Surface theorem

Likewise, the second term on the LHS of (3.55) is transformed using the Leibnitz theorem for the area A[ (see app A). The surface theorem yields ... 

We prove that in some cases one may guarantee the existence of at least two stable periodic solutions on three-dimensional constant-energy surfaces (isoenergy surfaces) of an integrable system by proceeding merely from the data on onedimensional homologies of these surfaces (Theorem 2.1.1) or from the data on the fundamental group. Such stable solutions may be effectively found. 

Since an actual crystal will be polyhedral in shape and may well expose faces of different surface tension, the question is what value of y and of r should be used. As noted in connection with Fig. VII-2, the Wulff theorem states that 7,/r,- is invariant for all faces of an equilibrium crystal. In Fig. VII-2, rio is the... 

We noted in Section VII-2B that, given the set of surface tension values for various crystal planes, the Wulff theorem allowed the construction of fhe equilibrium or minimum firee energy shape. This concept may be applied in reverse small crystals will gradually take on their equilibrium shape upon annealing near their melting point and likewise, small air pockets in a crystal will form equilibrium-shaped voids. The latter phenomenon offers the possible advantage that adventitious contamination of the solid-air interface is less likely. 

The stoi7 begins with studies of the molecular Jahn-Teller effect in the late 1950s [1-3]. The Jahn-Teller theorems themselves [4,5] are 20 years older and static Jahn-Teller distortions of elecbonically degenerate species were well known and understood. Geomebic phase is, however, a dynamic phenomenon, associated with nuclear motions in the vicinity of a so-called conical intersection between potential energy surfaces. 

State basis in the molecule consists of more than one component. This situation also characterizes the conical intersections between potential surfaces, as already mentioned. In Section V, we show how an important theorem, originally due to Baer [72], and subsequently used in several equivalent forms, gives some new insight to the nature and source of these YM fields in a molecular (and perhaps also in a particle field) context. What the above theorem shows is that it is the truncation of the BO set that leads to the YM fields, whereas for a complete BO set the field is inoperative for molecular vector potentials. 

Variational transition state theory (VTST) is formulated around a variational theorem, which allows the optimization of a hypersurface (points on the potential energy surface) that is the elfective point of no return for reactions. This hypersurface is not necessarily through the saddle point. Assuming that molecules react without a reverse reaction once they have passed this surface... 

Theorems of Pappus (for volumes and areas of surfaces of revolution)... 

These theorems are useful for determining volumes V and surface areas S of solids of revolution if the centers of gravity are known. If S and V are known, the centers of gravity may be determined. 

Crystals have spatially preferred directions relative to their internal lattice structure with consequences for orientation-dependent physico-chemical properties i.e., they are anisotropic. This anisotropy is the reason for the typical formation of flat facetted faces. For the configuration of the facets the so-called Wullf theorem [20] was formulated as in a crystal in equihbrium the distances of the facets from the centre of the crystal are proportional to their surface free energies. ... 

Flachen-satz, m. theorem of conservation of areas, -wert, m. surface value, -winkel, m. plane angle, -zahl,/. number of faces square number. 

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

M. Menon and R. E. AUen, New technique for molecular-dynamics computer simulations Hellmann- Feynman theorem and subspace Hamiltonian approach , Phys. Rev. B33 7099 (1986) Simulations of atomic processes at semiconductor surfaces General method and chemisorption on GaAs(llO) , ibid B38 6196 (1988). 

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