Basic vectors

This result can be generalized into the statement that any arbitrary vector in n dimensions can always be expressed as a linear combination of re basic vectors, provided these are linearly independent. It will be shown that the latent solutions of a singular matrix provide an acceptable set of basis vectors, just like the eigen-solutions of certain differential equations provide an acceptable set of basis functions. 

If all the A s are different, as assumed, it follows that c = 0. In the same way all of the Q can be shown to be zero. The linear relationship can therefore not exist and it follows that if A has n distinct eigenvalues the n corresponding eigenfunctions provide a set of basic vectors as discussed before. 

The most interesting corollary of the results of the previous chapter is, that using basic vector operations and features of vectors, inequalities relating the elements of density matrix can be formulated. Vectors D are completely determined by the configurational coefficients of the underlying full-CI type wave function, but we do not need the knowledge of these coefficients when deriving the inequalities. 

For our calculations we choose a spherical coordinate framework with the polar axis directed along the vector //eff- Then the basic vectors of the problem are expressed as... 

But our concern here is with the two-dimensional cases layered misfit structures, in which the lack of commensurability is between the intralayer periodicities of layers of two types, which alternate regularly through the structure. The layers may be simple or complex (i.e. composite groups of several, physically distinct layers). In most cases the two layer types compensate each other s valency and consequently alternate with strict regularity, forming double-layer or two-component layered structures. Both intralayer identity vectors of one layer set A) may differ fi-om those of the other layer set (B), so that each layer set has its own periodicities, and the vectors defining the net common to both (if it exists) are more or less complicated resultants (e.g. lowest common multiples) of these basic, intralayer vectors. In some cases the basic vectors are identical in one... 

In the SC and SS structures, the nodes of the component lattices of A and B coincide at multiples of the basic vector(s) and there is a coincidence ( super -) lattice, and its coincidence unit cell. The former represents a set of coinciding nodes of the true component lattices, and in the ideal case - with structurally-independent layer sets - it is not connected with common structural changes (modulation) in them. For growth considerations it is important that the SC and SS (as well as CC) structures have a coincidence mesh (coincidence net) parallel to the layers, whereas the IS structures have only a coincidence row in the layer plane. 

In the case of semi-commensurability in one or two directions it is likely that, structurally, the two layer sets are not quite independent. Semi-coherent structural and/or compositional modulation is then present i.e. a cooperative periodic variation in the size and/or content of the component subcells. Each modulation vector (one only in Fig. 2a, two in Fig. 2 b) will be equal to (or a multiple or sub-multiple of) that of the coincidence net. For each modulated layer set, A or B, a true-structure (component) lattice and unit cell can then be defined, based on its modulation period or periods plus the basic vectors or vector in the direction(s) in which there is no visible modulation of the basic structure. The longer-range modulation pattern of the two layer sets is imposed on the short range approximate periodicities which, in turn, describe sub-motifs manifested as a subnet (or subcell) of each layer set. If, as may be the case, the separate component unit cells of each of the two sets are identical, then they are also the coincidence cell of the two sets (Fig. 2 b). In the more general case, when this is not so, the vectors of the coincidence net will be multiples of the identity vectors of the unit nets of the two layer sets (or some simple summations of them). 

The intensity is thus the product of the structure factor, which only depends on the atomic positions within the unit cell, by the form factor, related to the shape of the crystal. In the limit of large N, the 5 function tends to a periodic array of Dirac delta funetions with Q spacing of 2jt/a, i.e. the intensity is nonzero only if Q.a, = 27di, Q.a2 = 27dc and Q.a,=27d, with h, k, I integers. This expresses that Q is a vector of the reciprocal lattice of basic vectors b, and... 

Directions in a hexagonal lattice are best expressed in terms of the three basic vectors a, a2, and c. Figure 2-11(b) shows several examples of both plane and direction indices. Another system, involving four indices, is sometimes used to designate directions. The required direction is broken up into four component vectors, parallel to a, a2, as, and c and so chosen that the third index is the... 

By extension, similar relations are found for all the planes of the crystal lattice. The whole reciprocal lattice is built up by repeated translations of the unit cell by the vectors bj, b2, bs. This produces an array of points each of which is labeled with its coordinates in terms of the basic vectors. Thus, the point at the end of the... 

A,B and C defined above then become multiples of the ciystal cell s basic vectors and the vectors R can be expressed as a Unear combination of these basic vectors. If we define R = ua + vb + wc, then we can write ... 

We will use the same notations as in the last two sectiorrs and assume that the vector 5 which characterizes the displacement of arty cell located in a position defined by the vector R can be expressed as a linear combirration of the cell s basic vectors. We have 8(R) = Xa+Yb + Zc.By using relation [5.65], we obtain ... 

Here k0) is the state corresponding to the wave vector Jc0. The state a) is the orbital state of the scattering nucleus, with energy E (the spin-orbit coupling is neglected). Finally, the state i is the spin state of the neutron and of the nucleus. This state belongs to the subspace, the basic vectors of which are... 

The reciprocal lattice is defined by the following relations between its basic vectors (kj.kg.kg) and those of the original lattice (Ri,R R,) ... 

Any position in real space is given by the real-space vector R which is a linear combination of the three basic vectors i, 2/ arid 3, namely... 

Likewise, any given reciprocal lattice vector K is constructed from the reciprocal basic vectors g, g, and 3 according to... 

Let B be the basis matrix associated with J. Since each set of basic vectors J is uniquely associated with a column index set, we shall, without confusion, let J be this set of indices and J the set of non-basic columns. The simplest tableau associated with J is... 

The body-centered cubic lattice is generated by the basic vectors of length t. 

The solution of the homogeneous set (45) of linear equations gives different sets of coefficients Aj. For a given set, the number of invariants is well defined, and it can indeed be higher than the number of elements, which gives to this method its optimal character (with respect to a balance of the elements only). On the contrary, the exact nature of the invariants is arbitrary, because, as for the stoichiometric equations, each linear combination of invariants is itself an invariant. This indetermination vanishes by fixing the basic vectors a priori. 

Every crystal has a lattice as its geometrical basis. A lattice may be described as a regular, infinite arrangement of points (or motifs) in which every point has the same environment as any other point the motif is repeated periodically at intervals a, b, and c along three noncoplanar basic vectors a, b, and c. 

Therefore, scale separated by distances 2%ja These peaks form a one-dimensional reciprocal lattice with basic vector 2tt/a, shown in Fig. 5.8b. 

In the three-dimensional-lattice, there are three basic vectors a, b, and c, Fig. 5.9a, and we can introduce a concept of the reciprocal three-dimensional lattice. It is a lattice in the wavevector space having the dimension of inverse length for each coordinate in the inverse space. Such a lattice may be built by translations of the elementary cell shown in Fig. 5.9b. The basic vectors of the reciprocal lattice are a, b, c and the vector of the reciprocal lattice is given by... 

This appendix describes a few basic vector and tensor operations that may be useful in understanding the material presented in the book. The vectors and tensors are presented only in the Cartesian coordinate system [1,2],... 

---