Double space groups

Elliot, R. J. (1954) Spin-orbit coupling in band theory - character tables for some double space groups. Phys. Rev. 96, 280-7. 

Hurley, A. C. (1966) Ray representations of point groups and the irreducible representations of space groups and double space groups. Phil. Trans. Roy. Soc. (London) A260, 1-36. 

The four-component DHF LCAO equations for ID-, 2D- and 3D-periodic systems were at first presented by Ladik [547]. The resulting somewhat compUcated generaUzed matrix eigenvalue equation for solids is described (for details we refer the reader to [547]). It was also shown for ID and 2D systems how MP2 methods could be applied iu their relativistic form. With the help of these, on the one hand, the total energy per unit cell (including correlation effects) can be computed. On the other hand, the relativistic band structure can also be corrected for correlation. Note that the symmetry of crystaUine orbitals changes, compared with the nonrelativistic case, as the symmetry of the DHF Hamiltonian is described by double space groups. Finally,... 

In a relativistic APW approach, symmetrization with double space groups is necessary for the following reasons ... 

LaNis has the CaCu, structure, space group P6/mmm [26] the hexagonal metal lattice is shown in Fig. 4. The crystal structure of LaNi5D7 has been determined 127, 28] and is illustrated in Fig. 5. There are three types of interstitial D sites La2Ni4 tetrahedra, and NiA tetrahedra. The unit cell is doubled along the oaxis because of the formation of a superlattice which is a consequence of long-range correlations between occupied and unoccupied... 

For this scheme, m, n, and p have been defined for minerals as being even numbers. Notice also that the formula for bismuthite and aikinite have been doubled in keeping with the reported model where only (l) = Cu and (ll) = Pb are known [78]. Also defined are the terms A (A=m/4) and B (B=n/2). It has been observed that for most mineralogical systems, when A-i-B is even, the corresponding space group was the non-centrosymmetric, Pmc2i, and when A-vB is... 

Figure 16, Relation between symmetry in (a) the single space and (b) the double space of a system whose molecular symmetry group has a a plane in the single space, and is isomorphic with C2v in the double space.

To see why this is so, let us attempt to apply the procedure of Section II.B to a bound-state wave function. This is illustrated schematically in Fig. 19. It is clear immediately that we cannot construct an unsymmetric in the double space, because each bound-state eigenfunction must be an irreducible representation of the double-space symmetry group. Thus a bound-state function in the double space is necessarily symmetric or antisymmetric under R2k, and is thus either a Fq or a Fn function. For a Fq function, we have Fn = 0 (since and Fn cannot form a degenerate pair), which implies [from Eq. (6)] that... 

A further report dealt with the synthesis, variable temperature magnetic susceptibility measurements, and crystal structure determination at various temperatures (115, 136, 140, 150 and 231 K space group P-1) of [Fe(isoxa-zole)6][Fe(isoxazole)4(H20)2](BF4)4 [73]. The molecular structure of this well-defined double salt consists of two mononuclear Fe(II) dications,... 

In the case of P Cu-phthalocyanine (space group P2i/c) only 8 different zones were found at 100 kV via SAED using a rotation-tilt holder (a=14.50 A, b= 4.70 A, c= 19.32 A, b=123°). With a rotation-double-tilt holder and in nano diffraction mode it was possible to detect 17 different zones at 300 kV (initial zone [001] and tilt about a and b ) from one crystal at room temperature (a=14.50A, b=4.84A, c=19.4lA, p=120.3°). In Fig. 5 the initial zone (middle) and two other zones obtained through different tilt series (a left hand side b right hand side) are shown exemplarily. 

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