# Download Handbook of numerical analysis by P. G. Ciarlet, Jacques Louis Lions, Philippe G. Ciarlet PDF

By P. G. Ciarlet, Jacques Louis Lions, Philippe G. Ciarlet

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Example text

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).