Primitive vectors

According to the above method, we rewrite the integration, in the irreducible BZ s ment, in terms of the abscissas, a, b and c, along either the reciprocal lattice primitive vectors or any oAer convenient set as follows... 

A space-filling top view and a side view of the zeolite L framework is illustrated in Figure 1.3. The primitive vector c corresponds to the channel axis while the primitive vectors a and b are perpendicular to it, enclosing an angle of 60°. 

The dye molecules are positioned at sites along the linear channels. The length of a site is equal to a number ns times the length of c, so that one dye molecule fits into one site. Thus ns is the number of unit cells that form a site we name the ns-site. The parameter ns depends on the size of the dye molecules and on the length of the primitive unit cell. As an example, a dye with a length of 1.5 nm in zeolite L requires two primitive unit cells, therefore ns = 2 and the sites are called 2-site. The sites form a new (pseudo) Bravais lattice with the primitive vectors a, b, and ns c in favorable cases. 

We consider the case where the sites form a Bravais lattice, which means that the position Rt of an / s-site i can be expressed by the primitive vectors a, b, and ns-c of the hexagonal lattice and the integers na[Pg.21]

In general, for an oblique lattice in two dimensions with primitive vectors ai and ao, the total conductance G can be expanded into a two-dimensional Fourier series. 

For a crystalline surface which has one atom at each lattice point with primitive vectors ai and a , the total force at point r, F(r), is... 

The two-dimensional periodicity can always be described by two primitive vectors ai and a2, as shown in Fig. D.l. A periodic function F has the property... 

Fig. D.l. A surface with two-dimensional periodicity, (a) in real space, the function has two-dimensional periodicity, which is indicated by two primitive vectors, ai and 02. (b) In reciprocal space, two primitive vectors bi and ba are introduced.

Let us come back to our task to find all possible diffraction peaks for a given crystal lattice. What are the possible scattering vectors that lead to constructive interference This question can be answered in an elegant way by defining the so-called reciprocal lattice If a, a2, and <23 are primitive vectors of the crystal lattice, we choose a new set of vectors according to... 

Just as a reminder The dots between the vectors denote the scalar (inner) product and the crosses denote the cross (outer) product of the vectors. These vectors 6 are in units of nr, which is proportional to the inverse of the lattice constants of the real space crystal lattice. This is why one calls the three-dimensional space spanned by these vectors the reciprocal space and the lattice defined by these primitive vectors is called the reciprocal lattice. These primitive reciprocal vectors have the following properties ... 

Why is it useful to introduce such a complicated set of vectors This becomes obvious when we look at the scalar product between a real space lattice vector R and a reciprocal lattice vector q. Expressing these vectors by the corresponding primitive vectors we can write ... 

Taking the primitive translation vectors for one of the real-space cubic lattices from Table 4.1, Eqs. 4.25-4.27 can be used to obtain the primitive translation vectors for the corresponding reciprocal lattice, which are given in Table 4.2. By comparing Tables 4.1 and 4.2, it is seen that the primitive vectors of the reciprocal lattice for the real-space FCC lattice, for example, are the primitive vectors for a BCC lattice. In other words, the ECC real-space lattice has a BCC reciprocal lattice. 

For relatively small CSF expansion lengths and orbital basis sets, the blocks of the Hessian matrix and gradient vector may be explicitly computed and the appropriate equation may be solved directly to determine the wavefunction corrections for the subsequent MCSCF iteration. For example, the solution of the linear equations required for the WNR and PNR methods may use the stable matrix factorization methods found in the LINPACK library . The routines from this library are often available in efficient machine-dependent assembly code for various computers. Even if this is not the case, the FORTRAN versions of these routines use the BLAS library, resulting in the efficient execution of the primitive vector operations. Similar routines are also available for the direct solution of the eigenvalue equations required for the WSCl and PSCI methods. 

D space can be filled without voids or overlapping by identical prismatic cells with well-defined symmetries, and their types are limited to seven. These units cells can be defined by the lengths of three nonplanar primitive vectors ai, a2 and a3 and by the angles a, (3 and 7 between these vectors. They generate the seven simple crystal systems or classes, defined by the sets of all points taken from a given origin of these cells, that are defined by vectors... 

A primitive cell of a BL is a cell of minimum volume that contains only one lattice point, so that the whole lattice can be generated by all the translations of this cell. This definition allows for different primitive cells for the same BL, but their volumes must be the same. The parallelepiped defined by the three primitive vectors ai, a2, and a3 of a simple BL is a primitive cell of this lattice. 

Table B.l. The seven 3D simple crystal systems. The conditions on the primitive vectors of the unit cells and on their orientations are indicated. Angle 7 is taken as the one between ai and a2...

The reciprocal lattice of a BL whose primitive unit cell is defined by three vectors ai, a2 and a3 is generated by three primitive vectors... 

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