Directed lattice

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.

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

The subscript labels a, b,... (i, j,...) correspond to unoccupied (occupied) bands. The Mulliken notation has been chosen to define the two-electron integrals between crystalline orbitals. Two recent studies demonstrate the nice converging behaviour of the different direct lattice sums involved in the evaluation of these two-electron integrals between crystalline orbitals [30]. According to Blount s procedure [31], the z-dipole matrix elements are defined by the following integration which is only non zero for k=k ... 

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

A is the volume of the unit cell in the direct lattice of the crystal The range of integration is restricted to the first Brillouin zone of the crystal, and the volume of the zone is (27t)3/A. 

In this expression SA is summed over the direct lattice, R( SB is summed over the reciprocal lattice, b,. The parameter r is chosen so as to obtain equally rapid convergence in the sums over the direct and reciprocal lattices. The 6 functions are defined by... 

The geometrical aspect concerns the position of the diffracted beams on a pattern it only depends on the direct lattice of the crystal through the Bragg law =2dhkisin9B - dhu being the interplanar distance of the diffracted (hkl) lattice planes and 0b the Bragg angle. In other words, it only depends on the lattice parameters of the crystal a, b, c, a, P and y. 

Dynamic smdies of the alloy system in CO and H2 demonstrate that the morphology and chemical surfaces differ in the different gases and they influence chemisorption properties. Subnanometre layers of Pd observed in CO and in the synthesis gas have been confirmed by EDX analyses. The surfaces are primarily Pd-rich (100) surfaces generated during the syngas reaction and may be active structures in the methanol synthesis. Diffuse scattering is observed in perfect B2 catalyst particles. This is attributed to directional lattice vibrations, with the diffuse streaks resulting primarily from the intersections of 111 reciprocal lattice (rel) walls and (110) rel rods with the Ewald sphere. 

We note that in one dimension na is a direct lattice vector, whereas m(2n/a) is a reciprocal lattice vector. Their product is an integral multiple of 2n. 

Example 16.3-1 Find the reciprocal lattice of the fee direct lattice. From Figure 16.2(a),... 

The direct lattice and reciprocal lattice unit cells are marked on the crystal pattern of a planar hexagonal net in Figure 16.10, using eqs. (13), (14), and (18). The scales chosen for... 

We now describe a general method for the construction of the BZ. It is a consequence of the SP relation eqs. (7)—(9) that every reciprocal lattice vector b , is normal to a set of planes in the direct lattice. In Figure 16.11(a), bm is a reciprocal lattice vector that connects lattice point O to some other lattice point P. Let 1 be the plane through Pi that is normal to b , and let 0 be the plane parallel to 1 through O. Let a be the lattice vector from O to some... 

Similar to the direct lattice, all the possible points that lie at the reciprocal lattice can be represented as follows ... 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

The method described above has been used by Avgul et al. (I) to calculate the dispersion interactions between a number of adsorbates and the cleavage surface of graphite. A prime disadvantage of the method of direct lattice sums, apart... 

The concept, incorporated in this approximation, that matter is distributed continuously in each layer plane, while it might not be permissible for a localized adsorbed film, is actually close to the truth for a mobile adsorbed film, in which the rapid translation of the molecules along the surface prevents their responding to its fine structure. This approximatioi i produces just the sort of average that is desired and is so difficult to obtain from the direct lattice sum. 

A repeat of Fig. 2.4.) The primitive direct-lattice unit cell in a triclinic (lowest-symmetry) crystal is an oblique parallelopiped with sidesa, b, c, interfacial angles ot, ft, and y and unit vectors ea, eb, and ec. 

Note also that in a triclinic crystal a and a are not collinear in a monoclinic crystal (b unique setting) b is parallel to b, but a and c form the obtuse angle [>, while a and c form a smaller acute angle /T given by fJ = 180 — fi. The reciprocal lattice vectors and the direct lattice vectors are a ying-yang duo of concepts, as are position space and momentum space, or space domain and time domain. Fourier transformation helps us walk across from one space to other, as convenience dictates Some problems are easy in one space, others in the space dual to it this amphoterism is frequent in physics. The directions of the direct and reciprocal lattice vectors are shown as face normals in Fig. 7.22. 

Equation (8.4.2) suggests that a wavefunction uk(r) needs to be found by standard quantum-chemical means for only the atoms or molecules in the one direct-lattice primitive unit cell. For each of the Avogadro s number s worth of fermions in a solid, the factor exp(ik R) in Eq. (8.4.2) provides a new quantum "number," the wavevector k, that guarantees the fermion requirement of a unique set of quantum numbers. The Bloch waves were conceived to explain the behavior of conduction electrons in a metal. 

In calculating the interplanar spacing, or perpendicular distance between adjacent planes of given indices, dku, in the direct lattice (whether or not these planes coincide with lattice points), it is helpful to consider the reciprocal lattice, which defines a crystal in terms of the vectors that are the normals to sets of planes in the direct lattice and whose lengths are the inverse of dku- The relationship between the interplanar spacing and the magnitude of the reciprocal lattice vectors, a, b, c, is given by ... 

Derive the expression for in terms of the direct lattice, for each of the crystal systems with orthogonal axes. 

The dot product is also useful for calculating the angle between a plane normal and any direction in a direct lattice. In general, the angle between any two directions, specified... 

Show that the angle between the sets of planes given by (hi k l ) and (/i2 212) in the orthorhombic direct lattice is given by ... 

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