Space, direct

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

In direct space successive layers are sheared homogeneously along cylindrical surfaces, one relative to the adjacent one, as a consequence of the circumference increase for successive layers. In diffraction space the locus of the corresponding reciprocal lattice node is generated by a point on a straight line which is rolling without sliding on a circle in a plane perpendicular to the tube axis. Such a locus... 

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

Azimuth dependence is clearly present for achiral tubes such as for instance the (10, 10) tube of Fig. 12, where it reflects the 20-fold rotation symmetry of this tube in direct space. 

The integrals in Eqs. (17) and (18) are called convolution integrals. In Fourier space they are products of the Fourier transforms of c r). Thus, Eq. (18) is a geometric series in Fourier space, which can be summed. Performing this summation and returning to direct space, we have the OZ equation... 

Thus, in the case of CuPt, Xcc(q) peaks in correspondence of the vector joining L and X, that is parallel to the (111) directioa exactly commensurate with the lattice and corresponds, in the direct space, to twice the distance between (111) planes. This is precisely the mechanism found by Clark et al.". 

In order to And rjopt, we minimise an expression of the total CPU time, T, required for the Ewald sums. We assume that T is proportional to the number of vectors used in both the reciprocal and direct space sums, G /(2j ) and R /a . and to t, and t<, the CPU times required for the evaluation of a single term in each series. In formulae. 

Space within 5 ft of any edge of such equipment, extending in all directions. Space between 5 ft and 8 ft of any edge of such equipment, extending in all directions. Also, space up to 3 ft above floor or grade level within 5 ft to 25 ft horizontally from any edge of such equipment. ... 

Space between 5 ft and 10 ft from open end of vent, extending in all directions. Space above the roof and within the shell... 

The electrostatic potential [Pg.289]

N.2 Computational speedup for the direct and reciprocal sums Computational speedups can be obtained for both the direct and reciprocal contributions. In the direct space sum, the issue is the efficient evaluation of the erfc function. One method proposed by Sagui et al. [64] relies on the McMurchie-Davidson [57] recursion to calculate the required erfc and higher derivatives for the multipoles. This same approach has been used by the authors for GEM [15]. This approach has been shown to be applicable not only for the Coulomb operator but to other types of operators such as overlap [15, 62],... 

Examination of the EXAFS formulation in wave vector form reveals that it consists of a sum of sinusoids with phase and amplitude. Sayers et al32 were the first to recognize the fact that a Fourier transform of the EXAFS from wave vector space (k or direct space) to frequency space (r) yields a function that is qualitatively similar to a radial distribution function and is given by ... 

Lengyel and Kalman have reported an electron diffraction study of water 20>. The high energy electrons used permit the structure function to be probed at (large) values of s unattainable in either X-ray or neutron diffraction. This feature is valuable in that the wider the range of s for which data are available, the more accurate is the inversion of the observed h(s) to the direct space function h(R). Typical data, for D2O at 25 °C, are shown in Fig. 5. Note the similarities, and differences, between the electron diffraction, X-ray diffraction and neutron diffraction data. 

To discuss the influence of aberrations in the HRTEM image formation process in more detail, it is convenient to work in Fourier space, where the real-space quantities I r), T (r), V r), and T r) are related to their counterparts 7(g), T (g), Vp(g) and T(g) by a Fourier Transformation. Distances, d, in direct space correspond to spatial frequencies, g, in Fourier space. With this approach, the electron wave can be expressed as... 

One of the drawbacks of ellipsometry is that the raw data cannot be directly converted from the reciprocal space into the direct space. Rather, in order to obtain an accurate ellipsometric thickness measurement, one needs to guess a reasonable dielectric constant profile inside the sample, calculate A and and compare them to the experimentally measured A and values (note that the dielectric profile is related to the index of refraction profile, which in turn bears information about the concentration of the present species). This procedure is repeated until satisfactory agreement between the modeled and the experimental values is found. However, this trial-and-error process is complicated by an ambiguity in determining the true dielectric constant profiles that mimic the experimentally measured values. In what follows we will analyze the data qualitatively and point out trends that can be observed from the experimental measurements. We will demonstrate that this... 

Since the scalar product of two functions in direct space is equal to the scalar product of their Fourier transforms in reciprocal space, we have... 

The integrals Ma a can be evaluated in direct space using ellipsoidal coordinates, or alternatively in momentum space using methods discussed in reference [17], The result is ... 

Bricogne, G. (1976). Methods and programs for direct-space exploitation of geometric redundancies. Acta Crys-tallogr. A 32,832-847. 

In contrast to partitioning methods that involve dimension reduction of chemical reference spaces, MP is best understood as a direct space method. However, -dimensional descriptor space is simplified here by transforming property descriptors with continuous or discrete value ranges into a binary classification scheme. Essentially, this binary space transformation assigns less complex -dimensional vectors to test molecules, with each dimension having unity length of either 0 or 1. Thus, although MP analysis proceeds in -dimensional descriptor space, its dimensions are scaled and its complexity is reduced. 

The Fourier transform of the direct space S function is a <5 function in reciprocal space, representing the reciprocal lattice. We thus obtain... 

Expressions (3.42) and (3.43) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same /, in. This is summarized in the statement that the spherical harmonic functions are Fourier-transform invariant. It means, for example, that a dipolar density described by the function dl0, oriented along the c axis of a unit cell, will not contribute to the scattering of the (hkO) reflections, for which H is in the a b plane, which is a nodal plane of the function dU)((l, y). 

The conclusion on the equivalence of direct-space and reciprocal-space minimization is not completely flawless, because weights are assigned to the observations in the least-squares refinement, so a weighted difference density is minimized. 

When the electrostatic moments are to be obtained by integration over direct space, it is advantageous to use the deformation density rather than the total... 

Not surprisingly, formalisms with very diffuse density functions tend to yield large electrostatic moments. This appears, in particular, to be true for the Hirshfeld formalism, in which each cos 1 term in the expansion (3.48) includes diffuse spherical harmonic functions with / = n, n — 2, n — 4,... (0, 1) with the radial factor rn. For instance when the refinement includes cos4 terms, monopoles and quadrupoles with radial functions containing a factor r4 are present. For pyridin-ium dicyanomethylide (Fig. 7.3), the dipole moment obtained with the coefficients from the Hirshfeld-type refinement is 62.7-10" 30 Cm (18.8 D), whereas the dipole moments from the spherical harmonic refinement, from integration in direct space, and the solution value (in dioxane), all cluster around 31 10 30 Cm (9.4 D) (Baert et al. 1982). 

Like the potential, other electrostatic functions can be expressed as Fourier summations over the structure factors (Stewart 1979). The electric field, being the (negative) gradient of the potential, is a Fourier series in which the power of the magnitude of H increases from —2 to —1, as expected from the reciprocal relationship between direct space and Fourier space. Starting with... 

The expression for the electric field gradient in direct space then becomes... 

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