Reciprocal Nearest Neighbors

Begin at an unused point and trace a path of unused nearest neighbors until a reciprocal nearest neighbor pair is found. 

The absorption spectrum in the form of Equation (9) is expected to have maxima for values of 1/X which correspond to reciprocal distances from the center of the atom under study to nearest neighbor atoms. As... 

If the initial state corresponds to the Ising model with nearest-neighbor interactions, Eqs. (4)-(9), we obtain the following expressions for the average reciprocal relaxation times of n 1, n 0, and 01 ... 

Figure 6.3a presents a unit cell of graphene sheet comprising two kinds of carbon atoms a and (1, where a = 2.46 A and x2 are the lattice constant and the vector connecting two carbon atoms, respectively. The corresponding Brillouin zone (BZ) in the reciprocal lattice is shown in Figure 6.3b. Assuming that only the nearest neighbors overlap and resonance integrals work in the system, the Jt-band energies are calculated from the secular equation as expressed by...

Nearest-neighbor effects refer to the reciprocal influence of adjacent amino acids on protein folding. In some cases, this term also refers to amino acids that are close in the 3-D structure of the protein (<8-10 A away), but distant in the sequence. Early studies found a nonrandom assorting of amino acids in secondary structures by pair-wise analysis of protein sequences (43). More recently, it was reported that the preference of pairs of amino acids for secondary structure was determined, in part. 

Analogous results were obtained for the uncomplexed CD-St heptachromophoric cyclodextrin discussed in Section 7.2.4.2. In this case however, and owing to the larger Forster radius for homotransfer (22 A), the leveling of the anisotropy occurs in ca. 100 ps, and the average reciprocal rate constant for transfer is only 2.4 ps [14]. Nevertheless, a similar nearest neighbor distance (7 A) is recovered. 

We deduce for an expression which coincides (in the limit case of negligible interaction between nearest neighbors) with the reciprocal of the characteristic Debye-Hiickel length. Following current methods we establish the relation between x and the given experimental results. We study the... 

What this means is that a standing wave will occur for nearest neighbor planes of the reciprocal lattice. Since all atoms in this example are equivalent, we can replace 2 FI by a constant, C., as follows ... 

The sum over the reciprocal distance values can be expressed as a convergent row, based on Tq, the distance between central ion and nearest neighbors. 

Fig. 98. Left The Gaussian-broadened Gaussian relaxation fimction (bottom) (explanation see text). ZF pSR asymmetry spectrum in polycrystalline CeCuo2Nio.jSn at 0.08K (top). The dashed line is a fit of the static Gaussian Kubo-Toyabe relaxation function. The solid line is a fit of the static Gaussian-broadened Gaussian function. From Noakes and Kalvius (1997). Right The minimum polarization achieved by the Monte Carlo RCMMV static ZF muon spin relaxation functions, as a function of the reciprocal of the moment-magnitude correlation length, in units of the magnetic-ion nearest-neighbor separation in the model lattice. The horizontal line represents the 1/3 asymptote, above which the minimum polarization cannot rise. The dashed line is a...

Among all possible equivalent choices of a unit cell in the reciprocal lattice, one is particularly useful. It can be obtained by connecting one reciprocal lattice point to all its nearest neighbors and letting orthogonal planes pass through their midpoints. The volume within these planes is known as the first Brillouin zone. It includes all points that are closer to that reciprocal lattice point than to any other lattice point. 

Period of the chain is equal to a. Let us suppose the linear relationship between the interaction force between the nearest neighbors and atomic displacement. Every internal motion of the lattice could be represented by the superposition of the mutually orthogonal waves as follows from the lattice dynamic theoiy (see e.g. Bom and Huang 1954 Leibfried 1955). Aiy lattice wave could be described by the wave vector K from the first Brillouin zone in the reciprocal space. Dispersion curve co K) has two separated branches (for 2 atoms in the primitive unit), which could be characterized as acoustic and optic phonons. If we suppose also the transversal waves (along with longimdinal ones), we can get three acoustic and three optical phonon branches. There is always one longitudinal (LA or LO) and two mutually perpendicular transversal (TA or TO) phonons. 

The range of k-values between — Ti/a < k < n/siisknownsiSthQ rstBrillouinzone (BZ). The first BZ is also defined as the Wigner-Seitz primitive cell of the reciprocal lattice, whose construction is illustrated in Figure 2.75. First, an arbitrary point in the reciprocal lattice is chosen and vectors are drawn to all nearest-neighbor points. Perpendicular bisector lines are then drawn to each of these vectors the enclosed area corresponds to the primitive unit cell, which is also referred to as the first Brillouin zone. 

A system s interaction can be identified with any type of reciprocation of balances. The number of parameters is often greater than the number of balance laws, so it is necessary to have other axioms and interconnections. Among the most important are causality, determinism, equipresence, objectivity, material invariance, time irreversibility, and the decisive effect of nearest neighbors, and nearby history. 

---