Download The Mathematical Theory of Symmetry in Solids: by Christopher Bradley, Arthur Cracknell PDF

By Christopher Bradley, Arthur Cracknell

This ebook offers the whole idea of the irreducible representations of the crystallographic element teams and area teams. this is often vital within the quantum-mechanical research of a particle or quasi-particle in a molecule or crystalline stable as the eigenvalues and eigenfunctions of a method belong to the irreducible representations of the crowd of symmetry operations of that procedure. the idea is utilized to offer whole tables of those representations for all of the 32 aspect teams and 230 area teams, together with the double-valued representations. For the distance teams, the gang of the symmetry operations of the okay vector and its irreducible representations are given for the entire distinct issues of symmetry, strains of symmetry and planes of symmetry within the Brillouin quarter. purposes ensue within the digital band constitution, phonon dispersion family members and choice ideas for particle-quasiparticle interactions in solids. the speculation is prolonged to the corepresentations of the Shubnikov (black and white) element teams and house teams.

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Extra info for The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups

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From what has so far been written the convention may already be clear, but, in case this is not so, it is worth emphasizing that we interpret unitary operators as acting on the points of space and that the coordinates of these points are always referred to a fixed set of axes. Operators that move the points of a space S Y M M E T R Y A N D T H E SOLID STATE 25 and leave the axes fixed are said to be active. Operators that move the axes and leave the points fixed are said to be passive. In this book we shall always use the active convention.

T r is the vector that is produced from r by the application of the active pointgroup operator R{. 3) 46 S Y M M E T R Y AND THE S O L I D STATE as can easily be verified by applying {R2 I V 2J to both sides of eqn. 2). From eqn. 3) it follows immediately that the inverse of [R^ \ V j } is {R^1 \ — -Rf 1 ^}. The Seitz operators for all the 230 space groups are identified by various authors (Faddeyev 1964, Henry and Lonsdale 1965, Koptsik 1966, Kovalev 1965, Lyubarskii 1960, Miller and Love 1967) and they are listed in Chapter 3.

3 , C'3 , C'2 1 , Cj 2 , C23 £, CJ . C7 , adi, "a2, Vd3 DM E, CJ, C3 , C 2 1 , C 22 , C 23 , /, 57, 5^ , a,,,, od2, ad} c. „,, (T^j-CTi,3 C 3h C6(, #6 £>„,, ^5 6(3 , 6(3 , 6 3 , 6 3 , 6 2 , 6 2 j i 6 2 2 , 6. 3 (i) The arbitrary numbers in column 1 are those of Koster, Dimmock, Wheeler, and Statz (1963). (ii) The labels of the symmetry operations can be identified from Figs. / = 1, 2, 3, a n d 4 ; m = x,y,andz; p = a, b, c, d, e, and/. , C2a, and 6'2I); sometimes alternatives of this kind are important when one considers the space groups (see Chapter 3).

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