Unit cells of symmetry

Figure 2. Stereoscopic view of the four Mo2( 5-CsH5)2(CO)4-(m2-H)(m2-P(CH3)2) molecules in the triclinic unit cell of symmetry Cl...

Figure 9. Stereographic drawing of the molecular packing of four [Ct2(CO)io(m2- )]-anions and four [(P/igP)2N]+ cations in the monoclinic unit cell of symmetry Cz/c. The anions lie on a crystallographic center of symmetry whereas the [(PhsP N] cations contain a crystallographic twofold rotation axis which bisects the P-N-P bond angle...

The Raman spectrum of a single crystal of PDA-TS at 295 K is shown in Fig. 7 (17). Althou the unit cell of symmetry contains 92 atoms and 66 totally symmetric Ag modes, only four lines and overtones and combinations of those modes are present in the spectrum. This is in agreement %rith the predictions of a planar model for the vibrational modes of the backbone for which the eigenvectors are shown in Fig. 8 (19). Mode 1 (2086 cm ) involves considerable stretching of the triple bond and mode 2 (1485 cm ) stretching of the double bond. Modes 3 (1203 cm ) and 4 (952 cm ) approximately describe rotation about those bonds. 

Ewald summation was invented in 1921 [7] to permit the efl5.cient computation of lattice sums arising in solid state physics. PBCs applied to the unit cell of a crystal yield an infinite crystal of the appropriate. symmetry performing... 

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

Regarding the electronic structure, the number of energy bands for ( ,0) zigzag carbon nanotubes is In, the number of carbon atoms per unit cell, with symmetries... 

Magnetic symmetry associated with the group P62m, 757 Magnetic unit cell of MnF2,754 Malkin, I. G, 349... 

The unit cell of cubic diamond corresponds to a face-centered packing of carbon atoms. Aside from the four C atoms in the vertices and face centers, four more atoms are present in the centers of four of the eight octants of the unit cell. Since every octant is a cube having four of its eight vertices occupied by C atoms, an exact tetrahedral coordination results for the atom in the center of the octant. The same also applies to all other atoms — they are all symmetry-equivalent. In the center of every C-C bond there is an inversion center. As in alkanes the C-C bonds have a length of 154 pm and the bond angles are 109.47°. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

Finally, we note that, to the best of our knowledge, only one report exists about EPR spectra of non-Kramers lanthanide ions in molecular magnets. In 2012, Hill and coworkers [51] performed a multifrequency study on powder and single crystal samples of NagHofWgOj H20, in both the pure form and when doped into the isostructural Y3+ derivative. While crystallizing in a triclinic unit cell, the symmetry of the lanthanide ion in this family is very close to Did. For this reason, susceptibility data had been previously fitted by a purely axial Hamiltonian [50]. 

The symmetry of the structure is imposed on the field (j)(r) by building the field inside a unit cell of smaller polyhedron, replicating it by reflections, translations, and rotations [21-23]. This procedure reduces the number of independent variables one order of magnitude for G structure and two orders of magnitude for D structure. 

In crystals of any material, the atoms present are always arranged in exactly the same way, over the whole extent of the solid, and exhibit long-range translational order. A crystal is conventionally described by its crystal structure, which comprises the unit cell, the symmetry of the unit cell, and a list of the positions of the atoms that lie in the unit cell. 

Polymers invariably form helical structures, and the helix symmetry is denoted by u, indicating that there are u repeat units in V turns of the helix. The helix pitch is denoted by P and the molecular repeat distance is c = vP. X-ray diffraction patterns from non-crystalline specimens contain diffracted intensities restricted to layer lines that are spaced by 1/c. On a diffraction pattern from a polycrystalline specimen, diffraction signals, or Bragg reflections, occur only at discrete positions on the layer lines, the positions being related to the lateral dimensions of the unit cell of the crystal. The meridian (vertical axis) of the diffraction pattern is devoid of diffracted intensity unless the layer line number J, is a multiple of u, so that u can be determined straightforwardly. The diffracted intensities can be calculated using standard expressions (2), for model structures (i.e. given the atomic coordinates). 

In some circumstances the magnitudes of the translation vectors must be taken into account. Let us demonstrate this with the example of the trirutile structure. If we triplicate the unit cell of rutile in the c direction, we can occupy the metal atom positions with two kinds of metals in a ratio of 1 2, such as is shown in Fig. 3.10. This structure type is known for several oxides and fluorides, e.g. ZnSb20g. Both the rutile and tlie trirutile structure belong to the same space-group type PAjmnm. Due to the triplicated translation vector in the c direction, the density of the symmetry elements in trirutile is less than in rutile. The total number of symmetry operations (including the translations) is reduced to... 

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