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Extra info for Fundamentals of Crystals: Symmetry, and Methods of Structural Crystallography
Crystalline. The lowest energy state at T = 0, which actually is a lattice, is unique. Indeed, according to the third principle of thermodynamics the entropy S = 0 at T = 0, and the state at T = 0 is unique, because S = In N (N being the number of states). Thus, the thermodynamic principle of the minimum of energy of a system consisting of an infinite number of particles can be realized only within the framework of the principle of symmetry and specifically within the framework of three-dimensionally periodic translational symmetry.
30 2. Fundamentals of the Theory of Symmetry be regarded as a space of rn dimensions with the coordinates of a point in it Xl"'"' Xi," "' Xm o The set of coordinates will be denoted by x. They may have either identical meanings (as in the three Cartesian coordinates) or distinct ones (for instance, some may mean distances, others angles, while still others, some physical parameters ). 1) g[Xb X2, ... , xm] = X'b x~, ... , x:n; g[x] = x'. We call F a symmetric object (function, figure) and g, a symmetry operation or transformation, provided F does not change when 9 acts on the initial variables, F(X1,' ..
Atoms or their groups, however, are of definite size, and therefore such operations are inapplicable to them. , of the atomism of crystalline substance, since not all the points of the substance are identical. Thus, points of different atoms or points of the center and the periphery of the same atom are not equivalent. A symmetry operation of infinite order which ensures discreteness is an operation of discrete, infinitely repeating transfers - translations. , crystalline. The lowest energy state at T = 0, which actually is a lattice, is unique.