Group theory space groups

Abeles, F. Optical properties of solids. Amsterdam North Holland Publish. Co. 1972. Bradley, C. J., Cracknell, A. P. Mathematical theory of symmetry in solids Representation theory for point groups and space groups. Oxford Qarendon Press 1972. Becher, H. J. Angew. Chem. Intern. Ed. Engl. 77 26 (1972). 

In all of these stmctures the atomic positions are fixed by the space group syimnetry and it is only necessary to detennine which of a small set of choices of positions best fits the data. According to the theory of space groups, all stmctures composed of identical unit cells repeated hi three dimensions must confomi to one of 230 groups tliat are fomied by coinbinmg one of 14 distinct Bmvais lattices with other syimnetry operations. 

H. Weyl, Space, Time, Matter, Dover Books, New York, 1950 The Theory of Groups and Quantum Mechanics, Dover Books, New York, 1950. 

Actually this statement is not quite correct because a state of the physical system is represented by each observer not by a vector in his (private) Hilbert space, but by a ray, since the normalization and phase of the state is not pertinent to the description of the state. What one has, therefore, is a correspondence between rays rather them between vectors. We shall, however, ignore this fact in the interests of simplicity. Its relevance to the present discussion is fully treated by E. P. Wigner, Nuovo Cimento, 3, 517 (1956) Rev. Mod. Phys., 29, 255 (1957) and Group Theory, Academic Press, New York, 1959. 

Magnetic space groups, 744,758 in describing nonlocalized states, 753 representation of, 742 Magnetic structure application of Landau-Lifshjtz theory, 762... 

Sink (in graph theory), 258 "Slack variables, 294 Slightly-ionized gases, 46 "Slow time, 362 Small parameter methods, 350 S-matrix, 599,649,692 Smirnova, T. S., 726 Smoluchowski, R., 745 Sokolov, A. V., 768 Sommerfeld, C. M., 722 Sonine polynomials, 25 Source (in graph theory), 258 Space group... 

The crystal structures of hematite and corundum have been determined through the use of Taue and spectral photographs, interpreted with the aid of the theory of space groups. The unit of structure is a rhombohedron with a = 55° 17 and a = 5.420 = = 0.010 A. for hematite, and with a = 55° 17 and a = 5.120 = = 0.010 A. for corundum. The space group underlying the atomic arrangement is D. ... 

R. W. G. Wyckoff, The Analytical Representation of the Results of the Theory of Space Groups Publications Carnegie Institution No. 318 (1922). 

Crystals of the intermetallic compound magnesium stannide, MgjSn, have been prepared and investigated by means of Laue and spectral photographs with the aid of the theory of space-groups. The intermetallic compound has been found to have the calcium fluoride structure, with dwo = 6.78 0.02 A. U. The closest approach of tin and magnesium atoms is 2.94 0.01 A. U. 

These are the only cubic arrangements provided by the theory of space groups in which every atom occupies an invariant position, and all of the atomic positions are those corresponding to the A1 arrangement. Very recently we have found that the cubic phase Pt3Cu is highly probably also based upon the cubic structure ABCa (Tang, 1951). 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

The concept of a coset space is discussed in detail in books on group theory (Gilmore, 1974) and is reviewed in Chapter 3 of Iachello and Arima (1987). The coset spaces of interest for algebraic models with structure U(n) are the spaces U(n)/U(n - 1) U(l). These spaces are complex spaces with (n - 1) complex variables (coordinates and momenta). 

---