Simple Translation

An infinite three-dimensional crystal lattice is described by a primitive unit cell which generates the lattice by simple translations. The primitive cell can be represented by three basic lattice vectors such as and h defined above. They may or may not be mutually perpendicular, depending on the crystal... 

In the above treatment of the problem of the particle in a box, no consideration was given to its natural symmetry. As the potential function is symmetric with respect to the center of the box, it is intuitively obvious that this position should be chosen as the origin of the abscissa. In Fig. 4b, x =s 0 at the center of the box and the walls are symmetrically placed at x = 1/2. Clearly, the analysis must in this case lead to the same result as above, because the particle does not know what coordinate system has been chosen It is sufficient to replace x by x +1/2 in the solution given by Eq. (68). This operation is a simple translation of the abscissa, as explained in Section 1.2. The result is shown in Fig. 4b, where the wave function is now given by... 

For fractionation processes that are chiefly dependent on molecular velocity, this relationship can be rearranged into a simple translational isotopic fractionahon factor. 

Under the classification of Amelinckx, stacking faults give rise to fringes. The stmcture in the region below the fault is derived from that above by a simple translation r parallel to the fault. In contrast, across a twin boundaiy the region below the fault is related by a similar displacement vector r which increases linearly from the boundaiy and the fringes thus produced are known as fringes. 

If [TO4] groups are linked to form chains, multiplicity is the number of linked chains the observed multiplicities in this case are m = 2,3, 4, and 5. Analogously, multiplicity is defined as the number of mutually linked sheets. If several [TO4] groups are linked in a chain, periodicity (p) is the number of groups that defines the structural motive (i.e., after p groups, the chain is obtained by simple translation). Dimensionality of [TO4] groups is the degree of condensation of a crystal structure. The condensation of an infinite number of tetrahedra may lead to infinite unidimensional chains d = 1), bidimensional sheets d = 2), and tridimensional lattices d = 3). The dimensionality of an isolated [TO4] tetrahedron is zero d = 0). 

Johnson, N. L. 1949. Bivariate distribntions based on simple translation systems. 

This is a sub-group of the space group, from which the effects of simple translations have been factored off . Isomorphous means here isomorphous to the factor group . 

A fundamental characteristic of isotactic polymers is the presence of translational symmetry with periodicity equal to a single monomer unit In the representation 4 and 5 successive monomer units can be superimposed by simple translation (30-32). In a syndiotactic structure this superimposition is not possible for two successive groups. The corresponding symmetry operator, if one... 

Conductance behavior is dependent on the material and what is conducted. For instance, polymeric materials are considered poor conductors of sound, heat, electricity, and applied forces in comparison with metals. Typical polymers have the ability to transfer and mute these factors. For instance, as a force is applied, a polymer network transfers the forces between neighboring parts of the polymer chain and between neighboring chains. Because the polymer matrix is seldom as closely packed as a metal, the various polymer units are able to absorb (mute absorption through simple translation or movement of polymer atoms, vibrational, and rotational changes) as well as transfer (share) this energy. Similar explanations can be given for the relatively poor conductance of other physical forces. 

Unlike crystalline subsystems, one cannot understand the structure of an amorphous particle from a knowledge of the spatial organization of a small group of its constituent atoms. The reason is that the atoms are not arranged in a regular, periodic array which would enable one to define the whole space occupied by the particle by simple translational repetitions of a basic structural motif of atoms. The spatial organization of the ions comprising the amorphous material in bone mineral is, at present, completely unknown. 

Restricting ourselves to curves whose maximum is at = 0 (all the other curves may be obtained from them by simple translation along the -axis), we obtain the picture presented in Fig. 1. The higher the maximum of the curve, the more sharply it is bent in accord with the greater value of e 6 and the larger is the temperature gradient, since the amount of evacuated heat is larger. 

Fig. 22. The effect of the form of the scattering object (W) on the interference function. In (A) the two arrays labeled W are related by simple translational symmetry. The array on the right is obtained by shifting the array on the left by L toward the right. This means that the full transform of the array in (A), shown as (B), is a product of the transform from one W array and a set of equidistant Cos-squared fringes as in (C). The fringe spacing is related to 1/L. In (D) the diffracting object is one array W on the right and a similar array -W on the left, except that they are mirror images they are not related by a simple translation. Interference effects still occur (E), but the interference function is far from simple (F) it consists of unevenly-spaced peaks whose positions are not easily predicted. (From Knupp and Squire, 2005.)...

The structure of cubic zinc sulfide (zinc blende, sphalerite) may be described as a ccp of S atoms, in which half of the tetrahedral sites are filled with Zn atoms the arrangement of the filled sites is such that the coordination numbers of S and Zn are both four, as shown in Fig. 10.1.7. The crystal belongs to space group 7 2 — / 43m. Note that the roles of the Zn and S atoms can be interchanged by a simple translation of the origin. 

If four nuclei are not coplanar, then they define a tetrahedron. If two tetrahedral nuclear arrangements are congruent, then simple translation and rotation are sufficient to superimpose them exactly. Furthermore, even if the two tetrahedra are not... 

Simple translation is the most obvious symmetry element of the space groups. It brings the pattern into congruence with itself over and over again. The shortest displacement through which this translation brings the pattern into coincidence with itself is the elementary translation or elementary period. Sometimes it is also called the identity period. The presence of translation is seen well in the pattern in Figure 8-1. The symmetry analysis of the whole pattern was called by Budden the analytical approach. The reverse procedure is the... 

Because simple translation of the entire solid is not of interest, this class of motion is eliminated to give a parameter related only to local deformations of the solid this parameter is the displacement gradient, V . The gradient of a vector field Vu is a second-rank tensor, specified by a 3 by 3 matrix. The elements of this displacement gradient matrix are given by (Vu),y = dujdxj, also denoted Uij in which i denotes the i" displacement element and j denotes a derivative with respect to the y spatial coordinate, i.e. [1],... 

Just as the effect of simple translation was eliminated by taking the gradient of the displacement vector, the contributions due to rotations can be eliminated, resulting in a parameter that describes only the local stretching of the solid. This... 

It was easy to show that we can formulate the method also in the case of a combined symmetry operation (for instance helix operation = translation + rotation) instead of simple translation ( ). In this case k is defined on the combined symmetry operation and from going from one cell to the next one, one has (1) to put the nuclei in the positions required by the symmetry operation and (2) one has to rotate accordingly also the basis set. 

The fourth equation of set (4.1) expresses the time derivative of Aq (the relative orientation between real and virtual body) in terms of v, a, and Aq themselves. For simplicity, the explicit analytical expression is omitted. The corresponding (very simple) translational term is given by the last equation of the set (4.1). 

Analogous to equation (8) the vibrational densities of states D " u) are calculated for the cases of CO and CS2 at two different pressures, respectively. They are shown in Figure 4 while Table 4 contains information about their positions and widths. Since under the similarity transformation the trace of V is preserved and because the relative band widths are small, the spectra are centred around the natural oscillator frequency uiq. The widths of the distributions depend on the strength of coupling between two oscillators which, apart from factors of positional and orientational correlation, scale linearly with transition dipole (dfijd(,Y and density p (equation (14)). CO is regarded as a reference system because of its simple translational and minor orientational structure. The last column in Table 4 expresses the influence of liquid structure on band width. All densities of states... 

Figure 3.9. Schematic diagram of simulation with periodic boundary conditions in which adjacent cells are generated by simple translations of coordinates.

Another classification is based on the presence or absence of translation in a symmetry element or operation. Symmetry elements containing a translational component, such as a simple translation, screw axis or glide plane, produce infinite numbers of symmetrically equivalent objects, and therefore, these are called infinite symmetry elements. For example, the lattice is infinite because of the presence of translations. All other symmetry elements that do not contain translations always produce a finite number of objects and they are called finite symmetry elements. Center of inversion, mirror plane, rotation and roto-inversion axes are all finite symmetry elements. Finite symmetry elements and operations are used to describe the symmetry of finite objects, e.g. molecules, clusters, polyhedra, crystal forms, unit cell shape, and any non-crystallographic finite objects, for example, the human body. Both finite and infinite symmetry elements are necessary to describe the symmetry of infinite or continuous structures, such as a crystal structure, two-dimensional wall patterns, and others. We will begin the detailed analysis of crystallographic symmetry from simpler finite symmetry elements, followed by the consideration of more complex infinite symmetry elements. 

It is practically obvious that simultaneously or separately acting rotations (either proper or improper) and translations, which portray all finite and infinite symmetry elements, i.e. rotation, roto-inversion and screw axes, glide planes or simple translations can be described using the combined transformations of vectors as defined by Eqs. 1.38 and 1.39. When finite symmetry elements intersect with the origin of coordinates the respective translational part in Eqs. 1.38 and 1.39 is 0, 0, 0 and when the symmetry operation is a simple translation, the corresponding rotational part becomes unity, E, where... 

For purposes of comparison, it is possible to classify the various types of potential functions which may be represented by the functional form used in Eq. (3.32) with a few simple considerations. The restrictions we shall make are always to locate the origin in the minimum, or if more than one, in the deepest minimum second minima or inflection points are restricted to negative values of the coordinate Z and the positive values of Z always represent the most rapidly rising portion of the function. These restrictions do not eliminate any unique shape of potential function. Any other functions described by Eq. (3.32) are related to those already included by a simple translation of the origin or by rotation about the vertical axis. These operations, at most, change the eigenvalues by an additive constant. The different types of potential functions are summarized in Table 3.1. 

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