Shape identity vector

In other words, Np ss is the number of symmetrically independent points in the reciprocal lattice limited by a sphere with the diameter d N (= 1/t/v) as established by Eq. 5.3 after substituting the Bragg angle, 0, of the iV observed Bragg peak for Qhu. Additional restrictions are imposed on Nposs in high symmetry crystal systems when reciprocal lattice points are not related by symmetry but when they have identical reciprocal vector lengths due to specific unit cell shape (e.g. h05 and h34 in the cubic, or 05/ and 34/ in the... 

Since all the cells of the lattice shown in Fig. 2-1 are identical, we may choose any one, for example the heavily outlined one, as a unit cell. The size and shape of the unit cell can in turn be described by the three vectors a, b, and c drawn from one corner of the cell taken as origin (Fig. 2-2). These vectors define the cell and are called the crystallographic axes of the cell. They may also be described in terms of their lengths a, b, c) and the angles between them (a, P, y). These lengths and angles are the lattice constants or lattice parameters of the unit cell. 

Figure 1.5 Optically induced potential energy landscapes for four identical interacting particles as a function of the vector positions of one of them, when the other three are at (0, 0, 0), (0, 0, 3), and (0, 0, 6). Scale in lla k lAits c units. Polarization and wavevector as in Figure 1.3. Black circular shapes represent local divergences in energy shift in the proximity of the fixed particles. Copyright (2008) by the American Physical Society.

The choice of unit cell shape and volume is arbitrary but there are preferred conventions. A unit cell containing one motif and its associated lattice is called primitive. Sometimes it is convenient, in order to realise orthogonal basis vectors, to choose a unit cell containing more than one motif, which is then the non-primitive or centred case. In both cases the motif itself can be built up of several identical component parts, known as asymmetric units, related by crystallographic symmetry internal to the unit cell. The asymmetric unit therefore represents the smallest volume in a crystal upon which the crystal s symmetry elements operate to generate the crystal. 

A crystal consists of a large number of unit cells arranged regularly in three-dimensional space, with each unit cell having the identical atomic content. The shape and size of the unit cell are defined by the three unit cell vectors a, b, c. The origin of each unit cell is on a lattice point, whose position is specified as... 

All these dipoles are conservative ones with respect to the entity number, the basic quantity in this case of capacitive dipoles. The interesting feature is that the separability is linked to the symmetry between energies-per-entity (here efforts) When the dipole is inseparable, both efforts are equal in magnitude but opposed in direction (for vectors) or value (for scalars). The converse is not true such symmetry may be found in peculiar cases for separable dipoles. For instance, the two potential values Vi and V2 of a capacitor may be equal in magnitude and opposite however, this happens only in case of equal pole capacitances. This is a frequent case in electrodynamics when the two capacitor plates are strictly identical, as in the case of planar capacitor, but this is not general as nonidentical shapes or geometries can also be found. Note that, in physical chemistry, this never happens, because it would correspond to identical partners in a chemical reaction of to identical phases in an interface ... 

A graphic visualization of the formation of the sp hybridization is shown in Fig. 2.10. The four hybrid sp orbitals (known as tetragonal hybrids) have identical shape but different spatial orientation. Connecting the end points of these vectors (orientation of maximum probability) forms a regular tetrahedron (i.e., a solid with four plane faces) with equal angles to each other of 109° 28. ... 

The basic concept to connect both scales of simulation is illustrated in Fig, 7, The model system is a periodic box described by a continuum, tessellated to obtain finite elements, and containing an atomistic inclusion, Any overall strain of the atomistic box is accompanied by an identical strain at the boundary of the inclusion. In this way, the atomistic box does not need to be inserted in the continuum, or in any way connected (e,g, with the nodal points describing the mesh at the boundary of the inclusion). This coupling via the strain is the sole mechanism to transmit tension between the continuum and the atomistic system. The shape of the periodic cells is described by a triplet of continuation (column) vectors for each phase (see also [21]), A, B, and C for the continuous body, with associated scaling matrix H = [ABC], and analogously a, b, and c for the atomistic inclusion, with the scaling matrix h = [abc] (see Fig, 8),... 

In another approach [34-36], introducing a reference body with the shape and dimensions identical to those of a real MHM and with the conductivity tensor one may define the vector of polarization as... 

Here S,- is the transfer matrix operating between lines (or planes) i and / -F 1. In the case of regular shapes, such as squares or cubes, all planes being identical, S2 = S3 = = S 1 = S, with u and v being the vectors extracting the states accessible to the first and last lines (or planes), respectively. Superscript T denotes the transpose. Elements of u are either 0 or the number of ways to transit, depending on the possible occurrence of the state i at the starting line (plane). A similar definition holds for v. 

The slope A or A is an adjustable parameter for fitting to measurements, which represents both the intrinsic properties and the extrinsic artifacts such as defects, the pileup of dislocations, shapes of indentation tips, strain rates, load scales, and directions in the test. The and b correspond, respectively, to the shear modulus and the Burger s vector modulus reduced by atomic size, d. The bulk modulus B is related to the shear modulus and the Poisson ratio vhy — B/[2(l -I- v)]. Using the dimensionless form of the normalized, yield strength aims to minimizing the contribution from artifacts due to processing conditions, crystal orientations, and the purity of the specimens if the measurement is conducted under the identical conditions throughout the course of the experiment. For convenience, we use both X = K and K as indicators of the dimensionless form of sizes. 

The end-to-end vectors of the subchains have distributions in their length and orientation. Equation [10] clearly indicates that the deviatoric part (measurable part) of the stress tensor due to the entropy elasticity of the polymer chains, hereafter referred to as the polymeric stress, reflects the orientational anisotropy of the subchains specified by the configuration tensor S(n,t). Consequently, the polymeric stress relaxes, even though the material keeps its distorted (e.g., sheared) shape, when the orientational anisotropy induced by tbe applied strain relaxes tbrougb tbe tbermal motion of tbe cbains. (In tbis relaxed state, S(n,t) is equal to 1/3 and tbe subcbain tension is transmitted isoUopically in aU tUrecrions to balance tbe isotropic pressure.) Thus, tbe relaxation time of the polymeric stress is identical to the orientational relaxation time of the polymer chains. 

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