Reciprocal-space term

Fig. 12 Absolute truncation error of the reduced potential energy from the Ewald summation as a function of a at Rent = 13.0i M and cut = 14 using an equilibrium (filled spheres) and random (open spheres) configiuation of System IV. For a > 0.38J j, the symbols for the equilibrium and random configurations are indistinguishable. Nm = 80 and L = 32.224J m- The hard-sphere radii were retained when generating the random configuration. The corresponding estimated truncation errors of the real-space and reciprocal-space terms of the reduced potential energy according to Eqs. 21 and 22, respectively, are also shown (solid curves)...

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

It is a known property of Fourier transforms that given a convolution product in the reciprocal space, it becomes a simple product of the Fourier transforms of each term in the real space. Then, as the peak broadening is due to the convolution of size and strains (and instrumental) effects, the Fourier transform A 1) of the peak profile I s) is [36] ... 

The theory of band structures belongs to the world of solid state physicists, who like to think in terms of collective properties, band dispersions, Brillouin zones and reciprocal space [9,10]. This is not the favorite language of a chemist, who prefers to think in terms of molecular orbitals and bonds. Hoffmann gives an excellent and highly instructive comparison of the physical and chemical pictures of bonding [6], In this appendix we try to use as much as possible the chemical language of molecular orbitals. Before talking about metals we recall a few concepts from molecular orbital theory. 

The possibility even exists of including dynamical effects with time-dependent friction terms (plus random forces at finite temperatures).77-80 Flowever, it may not be advisable to take advantage of this possibility, as the simulation would become increasingly slow with increasing number of time steps. Moreover, the simulation will slow down considerably in higher dimensions because of the nonorthogonality of the dynamical coupling in reciprocal space. 

Traditionally the performance of HRTEM is judged in terms of its ability to resolve two adjacent atom columns. Resolution is ruled by a few basic principles A position dependent image intensity g(r) is described as a convolution of the specimen function f(r) with a point spread function h(r). It is convenient to express this convolution in real space as a product in reciprocal space ... 

The second term in this expression is evaluated in reciprocal space using Eq. (9.15) convergence function, given by (Nijboer and De Wette 1957)... 

In Eq. (9.21), the second summation is over lattice vectors H. The last two terms of this equation represent the (000) term in the Fourier summation and the self-energy correction. The latter describes the interaction of the point charge with itself, which, as noted above, is included in the reciprocal space summation and must therefore be subtracted. 

Finally, we ignore expansion effects in this chapter thus, we take s = 0 throughout. This means that we may use the terms mean residence time, reactor holding time, space time, and reciprocal space velocity interchangeably. 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

The cracking reaction in all catalytic cracking processes is affected by the following factors (2) catalyst type and inherent activity charge stock characteristics and midboiling point space rate, usually measured in terms of liquid oil volume per volume of catalyst per hour ratio of catalyst to oil, the amount of catalyst in the reaction zone per unit of oil reacted, which in the fixed-bed process becomes the ratio of reciprocal space rate to time on stream, and in the moving-bed process is the ratio of catalyst rate to oil rate temperature and oil partial pressure. 

That T is a series of Bessel rather than trigonometric functions is merely a consequence of using cylindrical polar coordinates (r, j, cz ) for atoms in real space and (R, iji, i/a for points in reciprocal space. Not only is this a convenient framework for describing a helical molecule, but it can lead to economies in computing T. For helices, only Bessel terms with... 

Following Fraser et al. (4), we choose to represent the scattered intensity in terms of a cylindrically symmetric "specimen intensity transform" I (D), where D is a position vector in reciprocal space. Figure 10 shows the Ewald sphere construction, the wavelength of the radiation being represented by X. The angles p and X define the direction of the diffracted beam and are related to the reciprocal-space coordinates (R, Z) and the pattern coordinates (u,v) as follows ... 

Mathematical approximations to the periodic minimal surfaces can be constructed from terms which are each the result of adding symmetry-related sinusoidal density waves for the appropriate symmetry group, and then taking the nodal surface the boundary between regions of positive and of negative density. The waves that correspond to a face-centred figure in real space are the body-centred terms in reciprocal space, namely ... 

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