Simple Cubic Direct Lattice

3 Lattice Types in Reciprocal Space 6.1.3.1 Simple Cubic Direct Lattice 

The reciprocal lattice vectors form a simple cubic lattice in reciprocal space with a lattice 

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Note in Figure 16.3, that increasing k beyond tt/a simply repeats the curve between —ti/a and 0, so that aU relevant information is contained within the region from—it/a and tt/a. Recall that the reciprocal lattice vector for a simple cubic direct lattice has magnitude 2tt/a and that the first Brillouin zone is formed by planes that are perpendicular bisectors... 

Fig. 36. Schematic temperature variation of intcrfacial stiffness kn I K and interfacial free energy, for an interface oriented perpendicularly to a lattice direction of a square a) or simple cubic (b) lattice, respectively. While for tl — 2 the interface is rough for all non zero temperatures, in d — 3 il is rough only for temperatures T exceeding the roughening transition temperature 7r (see sect. 3.3). For T < 7U there exists a non-zero free energy tigT.v of surface steps, which vanishes at T = 7 r with an essential singularity. While k is infinite throughout the noil-rough phase, k Tic reaches a universal value as T - T . Note that k and fml to leading order in their critical behavior become identical as T - T. ...

FIQ. 1 Sketch of the BFM of polymer chains on the three-dimensional simple cubic lattice. Each repeat unit or effective monomer occupies eight lattice points. Elementary motions consist of random moves of the repeat unit by one lattice spacing in one lattice direction. These moves are accepted only if they satisfy the constraints that no lattice site is occupied more than once (excluded volume interaction) and that the bonds belong to a prescribed set of bonds. This set is chosen such that the model cannot lead to any moves where bonds should intersect, and thus it automatically satisfies entanglement constraints [51],... 

The easiest ciystal lattice to visualize is the simple cubic stracture. In a simple cubic crystal, layers of atoms stack one directly above another, so that all atoms lie along straight lines at right angles, as Figure 11-26 shows. Each atom in this structure touches six other atoms four within the same plane, one above the plane, and one below the plane. Within one layer of the crystal, any set of four atoms forms a square. Adding four atoms directly above or below the first four forms a cube, for which the lattice is named. The unit cell of the simple cubic lattice, shown in... 

The simple cubic lattice is built from layers of spheres stacked one directly above another. The cutaway... 

Thus, the planes of the lattice are found to be important and can be defined by moving along one or more of the lattice directions of the unitcell to define them. Also important are the symmetry operations that can be performed within the unit-cell, as we have illustrated in the preceding diagram. These give rise to a total of 14 different lattices as we will show below. But first, let us confine our discussion to just the simple cubic lattice. 

The strict generalization of the one-dimensional model treated in Sec. III,A leads to a crystal with its surface completely covered by adsorbed atoms. For this system, the tight-binding approximation gives a difference equation and boundary conditions which can be solved directly. The results show an important new feature. For a simple cubic lattice, the energy levels are given by the usual equation... 

Kittel and Abrahams 12S) have predicted an approximately Lorentzian magnetic resonance line shape for a system of spins which are randomly distributed over a small fraction of a large number of possible sites. This effect has been observed in electron spin resonance (124)- Kittel and Abrahams estimate that appreciable deviations from Gaussian shape will occur when the fraction of sites occupied, f, is less than 0.1, in the case of spins of / = H iu a simple cubic lattice with the magnetic field directed... 

The shape of the curve in Fig. 2.1 is simple it has a single minimum at a value of a we will call g0. If the simple cubic metal exists with any value of a larger or smaller than a0, the total energy of the material could be reduced by changing the lattice parameter to a0. Since nature always seeks to minimize energy, we have made a direct physical prediction with our calculations DFT predicts that the lattice parameter of our simple cubic material is a0. 

Figure 3.7. Imaginary component of the dynamical susceptibiUty versus temperature (the real component is shown in the inset) for a spherical sample and spins placed in a simple cubic lattice. The anisotropy axes are all parallel, and the response is probed along their common direction. The dipolar interaction strength hi = 5j/2a is hi = 0 (thick lines), 0.004, 0.008, 0.012, and 0.016 from (a) right to left and (b) top to bottom. The frequency is coxo/tr = 2k x 0.003.

In particular, let us consider the band structure along where kr = (0,0,0) and kx = (2n/a)(l, 0,0) with a the edge length of the face-central cubic unit celL (Note that the X point for fee is In/a not nfa like for simple cubic.) In this direction the two lowest free-electron bands correspond to Ek = (H2/2m)k2 and k+I = (H2/2m)(k + g)2 respectively. The term g is the reciprocal lattice vector (2n/a)(2,0,0) that folds-back5 the free-electron states into the Brillouin zone along so that Ek and k+l... 

Fig. 7.1 Left-hand panel The s band structure for a simple cubic lanice in the 100> and 111> directions. Right-hand panel The s band density of states for a simple cubic lattice.

Fig. 7.2 The p band structure for a simple cubic lattice in the j100> direction.

Comparison between asymptotic formulas based on direct enumeration and Monte Carlo estimates. Simple cubic lattices. (Mazur and McCrackin15)... 

As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. 

Coincidence grain boundary in a simple cubic lattice that corresponds to a 36.9° rotation about a <100> direction. One fifth of the atoms (dark circles) in the boundary have common sites. 

Fig. XIV-1.—Number of vibrations of one direction of polarization, per unit frequency range, in a simple cubic lattice with lattice spacing d, and constant velocity of propagation v.

The angle between two sets of planes in any type of direct-space lattice is equal to the angle between the corresponding reciprocal-space lattice vectors, which are the plane normals. In the cubic system, the [h k /] direction is always perpendicular to the (h k 1) plane with numerically identical indices. For a cubic direct-space lattice, therefore, one merely substimtes the [h k 1] values for [u v w] in Eq. 10.57 to determine the angle between crystal planes with Miller indices h k l ) and (h2 h)- With all other lattice types, this simple... 

A schematic sketch of the selenium. structure showing how the zig-zag chain can be coiled up along a [111] direction in a simple cubic lattice. Every third atom along the coil is translationally equivalent one such set is shaded. Other chains may be added such that the isolated shaded atoms shown are translationally equivalent to the shaded atoms in the chain and there is an atom site at every cube corner. The structure can then be distorted to reduce intrachain atom distances, increase interchain distances, and increase the bond angles to 105°. 

NaCl, Figure 7-7(a), is made up of face-centered cubes of sodium ions and face-centered cubes of chloride ions combined, but olfset by half a unit cell length in one direction so that the sodium ions are centered in the edges of the chloride lattice (and vice versa). If all the ions were identical, the NaCl unit cell would be made up of eight simple cubic unit cells. Many alkali halides and other simple compounds share this same structure. For these crystals, the ions tend to have quite different sizes, usually with the anions larger than the cations. Each sodium ion is surrounded by six nearest-neighbor chloride ions, and each chloride ion is surrounded by six nearest-neighbor sodium ions. 

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