Zero-point energy crystal structure

Inert Gases. The calculation of 7 should be relatively straightforward for crystals of inert gases, in which only one kind of interaction may be expected. These crystals have a face-centered cubic structure. If each atom is treated as a point source of attractive and repulsive forces, only the forces between the nearest pairs of atoms are considered, the zero point energy is neglected, and no re-arrangement of atoms in the surface region is permitted, then the calculated 7 still depends on the equation selected to represent the interatomic potential U. 

Dunitz wrote of these equations Debye s paper, published only a few months after the discovery of X-ray diffraction by crystals, is remarkable for the physical intuition it showed at a time when almost nothing was known about the structure of solids at the atomic level. Ewald described how The temperature displacements of the atoms in a lattice are of the order of magnitude of the atomic distances The result is a factor of exponential form whose exponent contains besides the temperature the order of interference only [h,k,l, hence sin 9/M]. The importance of Debye s work, as stressed by Ewald,was in paving the way for the first immediate experimental proof of the existence of zero-point energy, and therewith of the quantum statistical foundation of Planck s theory of black-body radiation. ... 

Let us consider the vdW interaction in helium in detail. Liquid helium is the only substance that does not solidify down to 0 K in the absence of external pressure. This is explained by the quantum character of the substance, whose zero point energy (ZPE) exceeds the crystal lattice energy [30]. At the same time, the macroscopic properties of helium (its crystal structure and thermochemical characteristics) do not differ fundamentally from those of other rare gases this allows to treat it in classical terms. Table 4.1 lists the structural and thermodynamic properties of rare gas molecules and crystals (see also [31]). 

Having adopted this convention we may ask. Does there exist a physical state of a substance for which the conventional entropy is actually zero Now perfect crystals are known to have a very orderly structure, and at very low temperatures the lattice vibrations will all be in their lowest states which correspond to the zero-point energy. Therefore it may be expected that a crystal will have a very low entropy at temperatures approaching the absolute zero, and in one of the original forms (Planck s version) of the third law it was asserted that the entropy of a pure substance is actually zero under such conditions. On the other hand, from (13 51), based on the Einstein approximation, it is seen that... 

Typically we fit up to the / = 3 components of the one center expansion. This gives a maximum of 16 components (some may be zero from the crystal symmetry). For the lowest symmetry structures we thus have 48 basis functions per atom. For silicon this number reduces to 6 per atom. The number of random points required depends upon the volume of the interstitial region. On average we require a few tens of points for each missing empty sphere. In order to get well localised SSW s we use a negative energy. 

The limit in accmacy of cell parameters could be largely due to the neglect of thermal effects lattice energy minimization yields the static crystal structure, which should approximate the structme at T = 0 K (neglecting zero-point vibrations). However, the crystal structures to which empirical parameterization is performed are unavoidably... 

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