Lattice vector, calculation

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

The most straightforward fype of lattice minimisation is performed at constant volume, where the dimensions of the basic imit cell do not change. A more advanced type of calculation is one performed at constant pressure, in which case there are forces on both the atoms and the unit cell as a whole. The lattice vectors are considered as additional variables along with the atomic coordinates. The laws of elasticify describe the behaviour of a material when... 

Band structure calculations have been done for very complicated systems however, most of software is not yet automated enough or sufficiently fast that anyone performs band structures casually. Setting up the input for a band structure calculation can be more complex than for most molecular programs. The molecular geometry is usually input in fractional coordinates. The unit cell lattice vectors and crystallographic angles must also be provided. It may be nee-... 

However, the calculation has taken account only of the change of the unit cell volume of the compound, AKhl = Fhs Fls which occurs due to the spin-state transition, the individual HS and LS complexes being treated as incompressible spheres. On the basis of detailed X-ray structure investigations [6], the crystal experiences not only a change in size but also a change in shape in the course of the HS LS transition. Every lattice vector x(T) may be expressed according to [70] ... 

The lattice vector and thus (n, m) can be also used to calculate the tube diameter following... 

A simple example of this calculation is the simple cubic lattice we discussed in Chapter 2. In that case, the natural choice for the real space lattice vectors has a, = a for all i. You should verify that this means that the reciprocal lattice vectors satisfy b, = 2rr/a for all i. For this system, the lattice vectors and the reciprocal lattice vectors both define cubes, the former with a side length of a and the latter with a side length of 2tt/g. 

There are many examples where it is useful to use supercells that do not have the same length along each lattice vector. As a somewhat artificial example, imagine we wanted to perform our calculations for bulk Cu using a supercell that had lattice vectors... 

Here, gi is the length of the primitive reciprocal lattice vector. The constants a, p, and are determined either by considering leading Bloch waves or by fitting with first-principles calculations. The method of Harris and Liebsch is used extensively in the treatment of atom scattering data. With some modifications, the method of Harris and Liebsch is also applicable to calculate STM images. We will discuss it in detail in Chapter 5. 

Using now this weak condition, we have to calculate the new resulting ansatz. To this end we have to make closer look at the Kirkwood approximation. The latter has a high symmetry with respect to the lattice vectors, but in the kinetic equations the three-point probabilities appear only in the form of equation (9.1.27) where the lattice sites play different roles l and n are then nearest neighbours and in nearly all terms i — n = 1, Z — m and n — m > 1 holds. Therefore we can suggest that the sites l and n are in all cases more important than the site m. (The case l — m = 1 is an exception which we do not study in detail.) Therefore we can introduce another ansatz instead of (9.1.20) which is non-symmetric ... 

When calculating FE energy states along particular directions in the BZ it is often convenient to work in Cartesian coordinates, that is to use the (e basis rather than the (b basis. The matrix representation of a reciprocal lattice vector bm is... 

The procedure of obtaining the crystal structure from a diffraction experiment should be straightforward. From the diffraction pattern we get the reciprocal lattice vectors and can construct the reciprocal lattice. The atomic arrangement should then be easily calculated by an... 

In calculating the interplanar spacing, or perpendicular distance between adjacent planes of given indices, dku, in the direct lattice (whether or not these planes coincide with lattice points), it is helpful to consider the reciprocal lattice, which defines a crystal in terms of the vectors that are the normals to sets of planes in the direct lattice and whose lengths are the inverse of dku- The relationship between the interplanar spacing and the magnitude of the reciprocal lattice vectors, a, b, c, is given by ... 

The electrical resistivity in directions other than the molecular stacking axis has been measured in a few cases. In (TMTSF)2PF6 one finds pa pb. pc, 1 200(3000) 3 x 104(106), where numbers in parentheses refer to T = 20 K, i.e., just above the transition (88). Here b is a vector in the ab plane perpendicular to a, and c is the reciprocal lattice vector orthogonal to the same plane. The situation in (TMTSF)2PF6 is probably typical for the series, whereas in (ET)2ClO4(TCE)0 5 one finds PilP — 1-2 (86), which suggests a very two-dimensional electronic structure in agreement with other experimental results, as well as band structure calculations (89). 

While this is the essence of what is required to calculate the cross section, it is noteworthy to indicate that Warren deduced that in the neighborhood of the two-dimensional reciprocal lattice vector tj for a finite-sized 2D crystallite of coherence length, L, the structure factor can be approximated by ... 

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