Centering translations

With the chemical structure of PbTX-1 finally known and coordinates for the molecule available from the dimethyl acetal structure, we wanted to return to the natural product crystal structure. From the similarities in unit cells, we assumed that the structures were nearly isomorphous. Structures that are isomorphous are crystallographically similar in all respects, except where they differ chemically. The difference between the derivative structure in space group C2 and the natural product structure in P2. (a subgroup of C2) was that the C-centering translational symmetry was obeyed by most, but not all atoms in the natural product crystal. We proceeded from the beginning with direct methods, using the known orientation of the PbTX-1 dimethyl acetal skeleton (assuming isomorphism) to estimate phase... 

If we now invoke the A centering (i.e., point 2 is translated by b/2 + c/2), point 2 becomes point 3. But clearly, point 1 will go directly to 3 if a 2X operation is carried out about a 2, axis cutting the b edge of the cell at bl4. In a similar way, every twofold rotation can be coupled with the -centering translation to generate one of the 2, axes. 

There are two important points to note about the list of special positions and their symmetries. The point i, 5, z is identical to 0, 0, z, because they are related by the centering translation. The point at i, 0, z may not appear equivalent to the listed one, 0, z since the a and b directions in the lattice are not equivalent. However, it is because the two twofold axes at the centers of the a and b edges are interchanged by the twofold rotations about the axes at a = b = i, and so on. 

Crystal symmetries that entail centering translations and/or those symmetry operations that have translational components (screw rotations and glides) cause certain sets of X-ray reflections to be absent from the diffraction pattern. Such absences are called systematic absences. A general explanation of why this happens would take more space and require use of more diffraction theory than is possible here. Thus, after giving only one heuristic demonstration of how a systematic absence can arise, we shall go directly to a discussion of how such absences enable us to take a giant step toward specifying the space group. 

Fauquignon Y. deLongueville, Naval Intelligence Support Center Translation No 3768, ADA 021183 (1976) 49) B.C. Taylor ... 

Double glide plane (pair of planes in centered cells only) Two coexisting glides of 1/2 (related by a centering translation) parallel to and perpendicular to the projection plane e... 

Identity, axial and centering translations, twofold rotation, and inversion. 

The data set A shown in Table 3.2 and Figure 3.3 will be used to discuss some characteristics of principal components from the user s point of view. The data in Figure 3.3 are mean-centered translation does not affect the principal components because only variances are considered. The variability of the data set is partly represented by variance vxi of feature Xi, partly by variance vX2 of feature X2 ... 

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Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

These centering translations, summarized below, should be memorized ... 

Body-, face-, or base-centering translations, if present, must begin and end on atoms of the same kind. For example, if the structure is based on a body-centered Bravais lattice, then it must be possible to go from an A atom, say, to another A atom by the translation ... 

What is the Bravais lattice of CsCl Figure 2-18(a) shows that the unit cell contains two atoms, ions really, since this compound is completely ionized even in the solid state a caesium ion at 0 0 0 and a chlorine ion at The Bravais lattice is obviously not face-centered, but we note that the body-centering translation i i i connects two atoms. However, these are unlike atoms and the lattice is therefore not body-centered. It is, by elimination, simple cubic. If one wishes, one may think of both ions, the caesium at 0 0 0 and the chlorine at as being... 

The sodium ions are clearly face-centered, and we note that the face-centering translations (0 0 0, 0, 0 0 when applied to the chlorine ion at ... 

The number of atoms per unit cell in any crystal is partially dependent on its Bravais lattice. For example, the number of atoms per unit cell in a crystal based on a body-centered lattice must be a multiple of 2, since there must be, for any atom in the cell, a corresponding atom of the same kind at a translation of from the first. The number of atoms per cell in a base-centered lattice must also be a multiple of 2, as a result of the base-centering translations. Similarly, the number of atoms per cell in a face-centered lattice must be a multiple of 4. 

Here the terms corresponding to the face-centering translations appear in the first factor the second factor contains the terms that describe the basis of the unit cell, namely, the Na atom at 0 0 0 and the Cl atom at The terms in the first bracket, describing the face-centering translations, have already appeared in example (d), and they were found to have a total value of zero for mixed indices and 4 for unmixed indices. This shows at once that NaCl has a face-centered lattice and that... 

Since the structure is face-centered, we know that the structure factor will be zero for planes of mixed indices. We also know, from example (e) of Sec. 4-6, that the terms in the structure-factor equation corresponding to the face-centering translations can be factored out and the equation for unmixed indices written down at once ... 

Here, a, represents the velocity at the particle boundary, af are the velocity components of the particle center (translation), and co is the angular velocity about the particle center. The distance between the particle center and the boundary point is given as r and the components of the counterclockwise normal to the position vector form the center of the particle to the corresponding node is denoted by. ... 

Type lb needs more than one vector to connect all nets, and these are called partial interpenetration vectors (PIV). These are included in the Z-symbol as where Zn is the number of integer translations such as the [1,0,0], [0,1,0] and [0,0,1] vectors, and Z the number of centering translations, that is where x,y and z in the vector fx,y,z] has a values of 0.5 or 0. In this case, the degree of interpenetration will be ... 

Diamond is face-centered cubic with eight atoms per unit cell. Carbon atoms at 0 0 0 and positions and the other positions (six) are given by face-centering translations. Deriving simplified expression for the structure factor, find the rule for systematic absences and also the intensity of 2 2 2 reflection. 

Table 14.5 Combinations of symmetry operations in organic crystals A, twofold rotation M, mirror reflection G, glide reflection S, twofold screw rotation 1, inversion tbrongb a point Ct, centering translation. The labels preceding each space group symbol are as follows C, cluster, R, row, L, layer and 3D, fuU three-dimensional structure. When several possihihties are given for an arrangement, they depend on the relative orientation of the symmetry operations...

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