Symmetry, crystal

Many of the geometric shapes that appear in the crystalline state are readily recognized as being to some degree symmetrical, and this fact can be used as a means of crystal classification. The three simple elements of symmetry which can be considered are  

It must be remembered, however, that while some crystals may possess a centre and several different axes and planes of symmetry, others may have no element of symmetry at all. 

A crystal possesses a centre of symmetry when every point on the surface of the crystal has an identical point on the opposite side of the centre, equidistant from it. A perfect cube is a good example of a body having a centre of symmetry (at its mass centre). 

If a crystal is rotated through 360 about any given axis, it obviously returns to its original position. If, however, the crystal appears to have reached its original 

This relationship states that the number of edges is two less than the sum of the number of faces and corners. 

---

Periodic boundary conditions can also be used to simulate solid state con dition s although TlyperChem has few specific tools to assist in setting up specific crystal symmetry space groups. The group operation s In vert, Reflect, and Rotate can, however, be used to set up a unit cell manually, provided it is rectangular. 

In crystals, the response of the crystal to a longitudinal loading may produce deformation controlled by the crystal symmetry that is not uniaxial... 

We have performed full-potential calculations on TisSia in its proposed stable crystal structure. The enthalpy of formation obtained from these calculations agrees well with the value deduced from experiment. Due to the low crystal symmetry, the possibility of a more complex bonding character arises. The charge density in this phase differs considerably from that in the hypothetical unstable structure, so two-electron bonds can be excluded in this phase. We have also showed that the opening of a quasigap in the Si DOS has its origin in the Ti-Si interaction. 

Typically we fit up to the / = 3 components of the one center expansion. This gives a maximum of 16 components (some may be zero from the crystal symmetry). For the lowest symmetry structures we thus have 48 basis functions per atom. For silicon this number reduces to 6 per atom. The number of random points required depends upon the volume of the interstitial region. On average we require a few tens of points for each missing empty sphere. In order to get well localised SSW s we use a negative energy. 

Creation operator, 505 representation of, 507 Critical value, 338 Crystallographic point groups irreducible representations, 726 Crystallographic symmetry groups construction of mixed groups, 728 Crystal, eigenstates of, 725 Crystal symmetry, changes in, 758 Crystals... 

Coincidence of crystal symmetry between monomer Space... 

The most characteristic feature of any crystal is its symmetry. It not only serves to describe important aspects of a structure, but is also related to essential properties of a solid. For example, quartz crystals could not exhibit the piezoelectric effect if quartz did not have the appropriate symmetry this effect is the basis for the application of quartz in watches and electronic devices. Knowledge of the crystal symmetry is also of fundamental importance in crystal stmcture analysis. 

The 230 space-group types are listed in full in International Tables for Crystallography, Volume A [48], Whenever crystal symmetry is to be considered, this fundamental tabular work should be consulted. It includes figures that show the relative positions of the symmetry elements as well as details concerning all possible sites in the unit cell (cf. next section). 

No ferroelectricity is possible when the dipoles in the crystal compensate each other due to the crystal symmetry. All centrosymmetric, all cubic and a few other crystal classes are... 

B. K. Vainshtein, Modem Crystallography I Fundamentals of Crystals Symmetry and Methods of Structural Crystallography. 2nd ed. Springer, 1994. 

In crystalline solids, the Raman effect deals with phonons instead of molecular vibration, and it depends upon the crystal symmetry whether a phonon is Raman active or not. For each class of crystal symmetry it is possible to calculate which phonons are Raman active for a given direction of the incident and scattered light with respect to the crystallographic axes of the specimen. A table has been derived (Loudon, 1964, 1965) which presents the form of the scattering tensor for each of the 32 crystal classes, which is particularly useful in the interpretation of the Raman spectra of crystalline samples. 

C5HuN05 L-Arabinose st/n-oxime (SARBOX)243 P21212I Z = 4 D = 1.61 R = 0.04 for 1,387 intensities. The acyclic molecule has the planar, zigzag conformation. The terminal OH and N-OH groups are oriented—syn and ap, respectively. Despite the difference in crystal symmetry, the hydrogen bonding is remarkably similar to that of the anti compound (see preceding abstract), with the same spiral of O-H N bonds. 

The chemical bonding and the possible existence of non-nuclear maxima (NNM) in the EDDs of simple metals has recently been much debated [13,27-31]. The question of NNM in simple metals is a diverse topic, and the research on the topic has basically addressed three issues. First, what are the topological features of simple metals This question is interesting from a purely mathematical point of view because the number and types of critical points in the EDD have to satisfy the constraints of the crystal symmetry [32], In the case of the hexagonal-close-packed (hep) structure, a critical point network has not yet been theoretically established [28]. The second topic of interest is that if NNM exist in metals what do they mean, and are they important for the physical properties of the material The third and most heavily debated issue is about numerical methods used in the experimental determination of EDDs from Bragg X-ray diffraction data. It is in this respect that the presence of NNM in metals has been intimately tied to the reliability of MEM densities. 

---