Basis vector choice

There are many such basis sets possible. Any basis vectors choice is acceptable from the point of view of mathematics. For economic reasons we choose one of the possible vector sets that give the least volume parallelepiped with sides a, 02 and 03. This parallelepiped (arbitrarily shifted in space, Fig. 9.1) represents our choice of the unit cell, which together with its content (motif) is to be translationally repeated. ... 

Any other modes we can think of are a linear combination of these. For example the double antisite, which is formed by exchanging a pair of A and B atoms, is equivalent to n + ni - 2 T. This would be an equally good choice as a basis vector instead of one of the three above. 

The choice of representation is arbitrary and one basis can be mapped into another by unitary transformation. Thus, let ip(n, q) and ip l, q) be two countable sets of basis vectors, such that... 

The biorthogonality and completeness relations presented above do not uniquely define the reciprocal basis vectors and mi a list of (3N) scalar components is required to specify the 3N components of these 3N reciprocal basis vectors, but only (3N) —fK equations involving the reciprocal vectors are provided by Eqs. (2.186-2.188), leaving/K more unknowns than equations. The source of the resulting arbitrariness may be understood by decomposing the reciprocal vectors into soft and hard components. The/ soft components of the / b vectors are completely determined by the equations of Eq. (2.186). Similarly, the hard components of the m vectors are determined by Eq. (2.187). These two restrictions leave undetermined both the fK hard components of the / b vectors and the Kf soft components of the K m vectors. Equation (2.188) provides another fK equations, but still leaves fK more equations than unknowns. Equation (2.189) does not involve the reciprocal vectors, and so is irrelevant for this purpose. We show below that a choice of reciprocal basis vectors may be uniquely specified by specifying arbitrary expressions for either the hard components of the b vectors or the soft components of m vectors (but not both). 

Thus, the Stokes parameters are equivalent to the ellipsometric parameters although less easily visualized, they are operationally defined in terms of measurable quantities (irradiances). Additional advantages of the Stokes parameters will become evident as we proceed. Note that Q and U depend on the choice of horizontal and vertical directions. If the basis vectors 0 ( and , ... 

The algorithm for the FFT (the reverse butterfly in our case) is well known (ref. 1,2) and will not be discussed here in detail. On the other hand, the FHT has been often neglected in spite of some advantages it offers. Due to the fact that both transformations rotate the time domain into the frequency space and vice versa, the only conceptual difference between both transformations is the choice of basis vectors (sine and cosine functions vs. Walsh or box functions). In general, the rotation or transformation without a translation can be written in the following form (ref. 3) ... 

This new representation is mathematically identical to the equations (4.1) and (4.2) regardless of the choice of the matrix of new basis vectors Af. 

In the triclinic crystal system, the reduction becomes more complicated due to possible multiple choices of the basis vectors in the lattice. 

If one looks up the term component in practically any text on physical chemistry or thermodynamics, one finds it is defined as the minimum number of chemical formula units needed to describe the composition of all parts of the system. We say formulas rather than substances because the chemical formulas need not correspond to any actual compounds. For example, a solution of salt in water has two components, NaCl and H2O, even if there is a vapor phase and/or a solid phase (ice or halite), because some combination of those two formulas can describe the composition of every phase. Similarly, a mixture of nitrogen and hydrogen needs only two components, such as N2 and H2, despite the fact that much of the gas may exist as species NH3. Note that although there is always a wide choice of components for a given system (we could equally well choose N and H as our components, or N10 and H10), the number of components for a given system is fixed. The components are simply building blocks , or mathematical entities, with which we are able to describe the bulk composition of any phase in the system. The list of components chosen to represent a system is, in mathematical terms, a basis vector, or simply the basis . 

Using this choice, the direction cosines between the basis vectors of the two coordinate systems and the instantaneous angular velocities about the molecular axes — Uo, oib, o)c — are related to the Eulerian angles and their time derivatives (f>, 6 and X through... 

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. 

As in the earlier three-dimensional example, where it was convenient to use hexagonal axes involving linear combinations of the rhombohedral basis vectors, it may also be useful here to use an alternative coordinate system to bring out certain symmetry properties. For molecules with a Tj frame, one choice is to take ... 

The next three lines contain the translational vectors spanning the unit cell in real space. Since computer time increases as NQ to some power, and since the primitive cell contains the smallest number of atoms per cell, the natural choice of translational vectors is the primitive vectors. In special case one may, however, make other choices as long as LAT is selected accordingly. The next NQ lines contain the basis vectors giving the positions of the individual atoms in the unit cell. 

The recognition of such relations is also of practical importance. For instance, if during the refinement of a mica structure the homo-octahedral model fails, only the choice between the related meso- or hetero-octahedral models has to be made. All such polytypes have the same framework of all atoms except those octahedrally coordinated. Therefore, they have identical or very similar basis vectors, and the space-group type of the homo-octahedral polytype is their common supergroup. Also their diffraction patterns are closer to one another than to those of other polytypes the geometry in reciprocal space is virtually the same and also the distribution of intensities is very similar owing to the fact that the framework of non-octahedral atoms in an average mica represents about 70 % of the total diffraction power. 

Varying the bulk composition can change the numerical values of the isopleths but not their relative orientations. These are controlled by stoichiometry, and by the choice, and weighting, of the basis vectors. Adding silica to the bulk composition would simply increase the labels on the quartz isopleths. Decreasing silica, however, would mean that the zero isopleth for quartz would truncate the triangle on a line just north of vertex C which lies upon the zero isopleth for quartz in the initial bulk composition. Truncation of the physically accessible portions of a modal space must obviously occur at any zero isopleths that are encountered. 

Of interest in the theory of chemical reactivity is also the intra-reactant decoupled representation [35,36] (see also Sect. 3.1). It corresponds to the choice of the (m -1- n) basis vectors consisting of eigenvectors of qA.A and Iib.b, (I a>5... 

Thus the functions cVv 2iri i and c / /2ir 4 i form an orthonormal basis in the space 2l- The functions xy, yz, xz, x — j/, y — z, z — X are linearly dependent (see Example 2.15), and the vector space of these six functions is only five-dimensional. Only five functions are required to span the space xy, yz, xz, x — y, 3z — r is one such choice of basis vectors as was shown in Example 2.15. As they stand, these functions are not square integrable, but they appear in quantum theory (d orbitals) with each multiplied by the factor e" . 

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