Unit-cell symmetry

If the unit-cell contents are symmetric, then the reciprocal lattice is also symmetric and certain sets of reflections are equivalent. In theory, only one member of each set of equivalent reflections need be measured, so awareness of unit-cell symmetry can greatly reduce the magnitude of data collection. In practice, modest redundancy of measurements improves accuracy, so when more than one equivalent reflection is observed (measured), or when the same reflection is observed more than once, the average of these multiple observations is considered more accurate than any single observation. 

In this section, I will discuss some of the simplest aspects of unit-cell symmetry. Crystallography in practice requires detailed understanding of these 

The symmetry of a unit cell is described by its space group, which is represented by a cryptic symbol (like P212121), in which a capital letter indicates the lattice type and the other symbols represent symmetry operations that can be carried out on the unit cell without changing its appearance. Mathematicians in the late 1800s showed that there are exactly 230 possible space groups. 

An example of a symmetry operation is rotation of an object about an axis. To illustrate with a familiar object, if a rectangular table is rotated 180 about an axis perpendicular to and centered on the tabletop (Fig. 4.14), the table looks just the same as it did before rotation (ignoring imperfections such as coffee stains). We say that the table possesses a twofold rotation axis because, in rotating the table one full circle about this axis, we find two positions that are equivalent 0 and 180 . The axis itself is an example of a symmetry element. 

Protein molecules are inherently asymmetric, being composed of chiral amino-acid residues coiled into larger chiral structures such as right-handed helices or twisted beta sheets. If only one protein molecule occupies a unit cell, then the cell itself is chiral, and there are no symmetry elements. This situation is rare in most cases, the unit cell contains several identical molecules 

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Physical Properties. According to Lagowski (Ref 32), X-ray analysis of a single crystal of nitric acid shows a monoclinic unit cell (symmetry P21/a-Cfh) the following dimensions a=16.23, b=8.57, and c=6.3lA, and 0=90°. The unit cell contains 16 molecules, and the calc d is 1.895g/cc at —41.6°... 

In practice, values of 6 can be measured and, since the wavelength of the X-rays is known in any given experiment, values of d can be calculated. These d values are related to the unit cell symmetry and dimensions. If the... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

The striking features of the structure (when compared with that of a normal spinel, Fig. 29) are (a) that the Li(2)04 tetrahedra are very tilted, so that the anion array is grossly distorted from the approximate cubic eutaxy observed in the normal spinel structure and (b) that, by contrast, the cation array is almost exactly that of the normal structure, and very regular indeed. [There is a small tetragonal distortion of the unit cell -symmetry P4i22-which has c/( /2 a) = 0.9697, compared with unity for the equivalent ratio of the cubic cell of the normal spinel structure.]... 

For the crystallographer, one of the most useful ways to describe unit-cell symmetry is by equivalent positions, positions in the unit cell that are superimposed on each other by the symmetry operations. In a P2, cell with an atom located at (x,y,z), an identical atom can be found at (-x, -y, /2 + z), because the operation of a 2j screw axis interchanges these positions. So a P2t cell has the equivalent positions (x, y, z) and (-x, -y, V2 + z). (The V2 means one-half of a unit translation along c, or a distance c/2 along the z-axis.)... 

Unit-cell symmetry can also simplify the search for peaks in a three-dimensional Patterson map. For instance, in a unit cell with a 2X axis (twofold screw) on edge c, recall (equivalent positions, Chapter 4, Section II.H) that each atom at (x,y,z) has an identical counterpart atom at (-x,-y,V2 + z). The vectors connecting such symmetry-related atoms will all lie at (u,v,w) = (2x,2y,V2) in the Patterson map (just subtract one set of coordinates from the other), which means they all lie in the plane that cuts the Patterson unit cell at w = l/2. Such planes, which contain the Patterson vectors for symmetry-related atoms, are called Harker sections or Harker planes. If heavy atoms bind to the protein at... 

The free energy minima can be found by a relaxational OP dynamics [8] dO/dt = -dF/dD with initial conditions t,(f,t = 0) random, with zero mean. In this (OP) strain representation, the (f) variables are effective scalars at each site, with anisotropy of the fourfold unit-cell symmetry... 

In structure determination from X-ray diffraction data, it sometimes happens that, on the Fourier maps, parts of the coming out structure are unclear. Fuzzy electron density maps may present problems in determining even the approximate positions of the respective fragments of the structure being analyzed. For example, the layered structure of the inclusion (intercalation) compound formed by Ni(NCS)2 (4-methylpyridine)4 (host) and methylcellosolve (guest) [1], The guest molecules are (Fig. 11.1) located on twofold crystal axes of unit cell symmetry and are orientationally disordered as shown in the picture. 

The 32 crystallographic point groups, first mentioned in Table 7.1, are now described in Table 7.8 (ordered by principal symmetry axes and also by the crystal system to which they belong). The 230 space groups of Schonflies and Fedorov were generated systematically by combining the 14 Bravais lattices with the intra-unit cell symmetry operations for the 32 crystallographic point... 

A more exact procedure is to solve the Bom-von Karman equations of motions 38) to obtain frequencies as a function of the wave vector, q, for each branch or polarization. These will depend upon unit-cell symmetry and periodicity, force constants, and masses. Thus, for a simple Bravais lattice with identical atoms per unit cell, one obtains three phase-frequency relations for the three polarizations. For crystals having two atoms per unit cell, six frequencies are obtained for each value of the phase or wave vector. When these equations have been solved for a sufficient number of wave-vectors, g hco) can, in principle, be obtained by direct count . Thus, a recent calculation (13) of g to) based upon a normal-mode calculation that included intermolecular forces gave an improved fit to the specific heat data of Wunderlich, and showed additional peaks of 140, 90 and 60 cm in the frequency distribution. Even with this procedure, care must be exercised, since it has been shown that significant features of g k(o) may be rormded out. Topological considerations have shown that significant structure in g hco) vs. ho may arise from extreme or saddle points in the phase-frequency curves (38). 

Crystal family Unit cell symmetry Unit cell shape/parameters... 

NIST - Crystal Data Unit cell, symmetry and references. >200,000... 

Regardless of the nature of the diffraction experiment, finding the unit cell in a conforming lattice is a matter of selecting the smallest parallelepiped in reciprocal space, which completely describes the array of the experimentally registered Bragg peaks. Obviously, the selection of both the lattice and the unit cell should be consistent with crystallographic conventions (see section 1.12, Chapter 1), which impose certain constraints on the relationships between unit cell symmetry and dimensions. 

All three programs TREOR, DICVOL and ITO allow optional input of the information about the measured gravimetric density and formula weight in order to estimate the number of formula units expected in the found unit cell (see section 6.3 in Chapter 6). The latter should be an integer number compatible with the unit cell symmetry, e.g. in a primitive monoclinic lattice it normally should be a multiple of 2 or 4. The agreement between the number of formula units in the unit cell and lattice symmetry may be used as an additional stipulation when selecting the most probable solution. 

This defines a set of equations for the mean field Hamiltonians HPF. These equations have to be solved self-consistently since the thermodynamic values within the angle brackets in (109) involve the mean field Hamiltonians // F. In principle, all // F can be different in practice, we impose symmetry relations. Therefore, we choose a unit cell, compatible with the symmetry of the lattice introduced in Section II,D, and we put Hpf equal to // F whenever P and P belong to the same sublattice. Moreover, we apply unit cell symmetry that relates the mean field Hamiltonians on different sublattices. By using the symmetry-adapted functions introduced in Section II,B, the latter symmetry can be imposed as follows. We select a set of molecules constituting the asymmetric part of the unit cell. Then we assign to all other molecules P Euler angles tip-through which the mean field. Hamiltonian of some molecule P in the asymmetric part has to be rotated in order to obtain HrF. As a result, we... 

Whereas within the family of the cubic Prussian blue analogs a large number of lattice constants have been determined, little attention has been devoted so far to polymeric cyanides not belonging to the cubic system. It must be emphasized, however, that polynuclear cyanides having unit cell symmetries other than cubic are by no means rare exceptions. Hexacyanometalates(III) of Zn2+ and Cd2+ are obtained not only in a cubic modification but also as samples with complicated and not yet resolved X-ray patterns of definitely lower symmetry than cubic (55). The exact conditions for obtaining either modification are not yet known in detail. The hexacyanoferrates(II), -ruthenates(II), and -osmates(II) of Mn2+ and several modifications of the corresponding Co 2+ salts show very complicated X-ray powder patterns which cannot be indexed in the cubic system (55). Preliminary spectroscopic studies show the presence of nearly octahedral M C6-units in these compounds, too. 

Syntheses of crystalline, XRD-single phase, Cr,Si-MFI materials have been achieved from low pH, Na free, reaction mixtures. Cr(III) framework substitution seems unlikely. No effects of Cr content on unit cell symmetry or on IR spectra compared with the silicalite are evident. Accessible Cr (III) cationic species have been detected by IR adsorption experiments. Cr... 

The relation between structure and triboluminescence has been investigated in two polymorphic systems, viz, hexaphenylcarbodisphosphorane and anthranilic acid. ° Only one phase is triboluminescent. A correlation between triboluminescence and unit cell symmetry groups is given. 

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