Reciprocal space vector

This sum over a// reciprocal space vectors of the form (IV.2) should be carefully distinguished from the expansion (III.4) of the density of a periodic crystal. If the density has the "little period", the e>mansion (IV.3) reduces to a sum over all reciprocal lattice vectors. The general case (IV.3) and the periodic case (III.4) actually represent two extreme cases. The presence of "more and more symmetry" in the density can be gauged... 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

Coordinate system Description Components of direct space vector Components of reciprocal space vector... 

Any function used as the local part of the wavefunction description must be periodic in the cell dimensions. In a plane-wave basis this is ensured by choosing a Hnear combination of plane-waves with particular reciprocal space vectors ... 

The treatment of applying periodic boundary conditions discussed here is markedly different from that traditionally employed in simulations of planar Couette flow. The PBC method that is commonly used is called the Lees-Edwards boundary condition. In its simplified form applied to cubic boxes, it represents a translation of the image boxes in the y direction, at a rate equal to y. Further details on this method can be found elsewhere. In contrast to the method involving the dynamical evolution of h presented here, the Lees-Edwards method is much harder to develop and implement for noncubic simulation cells. Also, in simulations involving charged particles, the Coulom-bic interaction is handled in both real and recipro l spaces. The reciprocal space vectors k of the simulation cell represented by h can be written " " as follows ... 

From Equation (2), we deduce that diffraction is observed only when the indices h, k, l in d take integral values. These reciprocal space vectors form a lattice, the reciprocal lattice, and the mathematical relationship between the real and reciprocal lattices (and between other aspects of the diffraction pattern) is a FT, as we will explain below. The interpretation of the Ewald construction is that diffraction is observed when the scattering vector s-s0 is equal to a reciprocal space vector A bki with integral indices h, k, l. This occurs whenever such a... 

Fourier Transform for the hkt) profile component Average dislocation contrast factor Interplanar distance between hkl) planes Scattering (reciprocal space) vector (<7 = 2 sin 0/2) Scattering vector in Bragg condition for the hkl) planes... 

The basic output from STR is the canonical structure constants used in CANON and LMTO to calculate band structures. In addition, STR produces a file with real and reciprocal space vectors which is used by the combined correction term programme COR. This file may also be read by STR next time the same crystal structure is encountered, thus saving the time used to generate these vectors. 

An edited output corresponding to the bcc data in Table 9.2 is given in Sect.9.2.4. In a production run one would normally suppress the printing of the real and reciprocal space vectors and of the structure constants, i.e. NOVCGN = 0 (if one has a suitable set of vectors in VEC/XXX) and NOWRT = 0. 

RESULT FROM VECGEN FOR RECIPROCAL SPACE VECTORS... 

The basis input for COR is the basis vectors giving the positions of the atoms in the cell, and the reciprocal-space vectors generated by STR. The basic output of COR is the correction-term structure constants used by LMTO to perform the most accurate band calculations allowed by the present collection of computer programmes. 

In the presence of cylindrical symmetry, various functions of r, including the pair distribution function g(r), are independent of and can be written as a function of R and Z alone. The reciprocal space vector q is similarly represented by cylindrical coordinates ( , a, ) (see Figure 4.6), where... 

The problem has now been transformed from treating an infinite number of orbitals (electrons) to only treating those within the unit cell. The price is that the solutions become a function of the reciprocal space vector k within the first Brillouin zone. For a system with Mbasis functions, the variation problem can be formulated as a matrix equation analogous to eq. (3.51). 

As mentioned in Sect. 2.5, in principle the FMM, multigrid methods, or tree codes can handle this situation, but they are much too slow for the normally only small number of charges involved, and error estimates are not easy to obtain. Also, a modified Ewald method in which the summation of the reciprocal-space vectors was modified [70], similar to the one used by Kawata and Mikami [71] exists, but also here the approximations made seem hard to control which render the method rather useless. 

In the case of plane-wave basis sets the scaling proceeds on the reciprocal-space vectors as G- (1+ ) G, which is seen by the definition aj b< = 6 4, where and b are real- and reciprocal-lattice primitive translation vectors, respectively. Thus one finds the derivative of reciprocal-space vectors given by... 

Now we can use the -function relations that result from summing the complex exponential exp( ik R) over real-space lattice vectors or reciprocal-space vectors within the BZ, which are proven in Appendix G, to simplify Eq. (3.84). With these relations, / takes the form... 

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