Vectors lattice plane

The first step in studying the orientation ordering of two-dimensional dipole systems consists in the analysis of the ground state. If the orientation of rigid dipoles is described by two-dimensional unit vectors er lying in the lattice plane, then the ground state corresponds to the minimum of the system Hamiltonian... 

A perfect surface is obtained by cutting the infinite lattice in a plane that contains certain lattice points, a lattice plane. The resulting surface forms a two-dimensional sublattice, and we want to classify the possible surface structures. Parallel lattice planes are equivalent in the sense that they contain identical two-dimensional sublattices, and give the same surface structure. Hence we need only specify the direction of the normal to the surface plane. Since the length of this normal is not important, one commonly specifies a normal vector with simple, integral components, and this uniquely specifies the surface structure. 

The vector / j specifies the position of the atom in a two-dimensional lattice in a plane parallel to the surface and the value of identifies the lattice plane with respect to the surface (1 = 1 for the surface l er. u is the displacement of the vibrating atom from its equilibrium position Rq. The total kinetic energy of the vibrating lattice is... 

By definition, a zone axis is normal to both g and h and other reciprocal lattice vectors in the plane defined by these two vectors. The reciprocal lattice plane passing through the reciprocal lattice origin is called the zero-order zone axis. A G-vector with z - G=n with n O is said to belong to a high order Laue zones, which separate to upper Laue zones (n>0) and lower Laue zones (n<0). 

A structure model was proposed for the compounds with Dj-type patterns by Bursill and Hyde. Figure 2.112 shows a reciprocal lattice plane of rutile with a [111] zone axis. The arrays of superspots for Dj and Dj patterns are parallel to the vectors g(121) and g(l 32), respectively. Those for D2 patterns... 

A 3D lattice can be built up by stacking 2D lattices. If a 2D lattice is defined by two translation vectors, t, and t2, we need to introduce a third translation vector, t3, that defines the stacking pattern. For example, if we stack a set of (identical) oblique lattices (defined by t, and t2) employing a vector t3 that is not orthogonal to the 2D lattice planes, we generate the triclinic lattice, while if we require t3 to be orthogonal to the 2D lattice planes and connect each plane with a point in the nearest neighboring plane we get the primitive monoclinic lattice. 

Fig. 438 Mode] for a structure giving rise to the diffraction pattern in Fig. 4.39 (Phoon et al. 1993). (a) Hexagonal ABC structure (not close-packed) (b) top view (d) front view, showing lattice vectors and some of the lattice planes (c) orientation of unit cell vectors (ajb,c) with respect to the flow direction, rotation axis and the direction of the neutron beam (e) the Bragg diffraction pattern from a twinned structure.

How are the Laue condition and the Bragg condition connected In Fig. A.3 the wave vectors of the incident and outgoing radiation and the scattering vector are drawn for the Bragg reflection of Fig. A.l. We can conclude that for specular reflection, the scattering vector lattice plane. Its length is given by... 

To get constructive interference, we have to fulfill the Bragg condition. Inserting Eq. (A.l) into Eq. (A.6) leads to q = 2it/d. In other words we observe a diffraction peak, if the scattering vector is perpendicular to any of the lattice planes and its norm is equal to... 

The length of each vector bj is 2it/dj, where dj is the distance between the the lattice planes perpendicular to bj. (This is ensured by the numerators a2 x a3 in Eq. A.9.) Therefore, each of these primitive reciprocal lattice vectors fulfills the Laue condition Eq. (A.7) and is a possible scattering vector for constructive interference. 

Another property is that for every set of parallel lattice planes, there are reciprocal lattice vectors that are normal to these planes. The shortest one of these reciprocal lattice vectors is used to characterize the plane orientation. The components (h, k, l) of this vector are called Miller indices and the direction of the plane is denoted by (hkl) for the single plane or hkl for a set of planes (see Section 8.2.1). 

Symmetry plane or symmetry line Graphic symbol Glide vector in units of lattice translation vectors parallel and normal to the projection plane Printed symbol... 

K refers to a particular incidence angle, such that the projection of the incident light wave vector in the lattice plane is K). From (4.16) we get... 

Figure 4.5. Wave vectors around the center of the excitonic Brillouin zone for which coherent emission [solution of equations 4.10 and 4.25] is possible according to the disorder critical value Ac. We notice that r0 is the imaginary eigenvalue for K = 0 (emission normal to the lattice plane) and that K" and K1 indicate, respectively, components of K parallel and perpendicular to the transition dipole moment, assumed here to lie in the 2D lattice. The various curves for constant disorder parameter Ac determine areas around the Brillouin-zone center with (1) subradiant states (left of the curve) and (2) superradiant states (right of the curve). We indicate with hatching, for a large disorder (A,. r ), a region of grazing emission angles and superradiant states for a particular value of A.

Zones bounded by planes defined by equation 38 are consistent with the reduced-vector zones. These reciprocal-lattice planes are simply the planes bisecting the vectors K and normal to them. It is noteworthy that lattices with the same type of translational symmetry have equivalent zone patterns since zone structure is determined by the K vectors, and these are determined by the primitive translation vectors. 

The sums may be carried out with respect to the atomic positions in direct (real) space or to lattice planes in reciprocal space, an approach introduced in 1913 by Paul Peter Ewald (1888-1985), a doctoral student under Arnold Sommerfeld (Ewald, 1913). In reciprocal space, the structures of crystals are described using vectors that are defined as the reciprocals of the interplanar perpendicular distances between sets of lattice planes with Miller indices (hkl). In 1918, Erwin Rudolf Madelung (1881-1972) invoked both types of summations for calculating the electrostatic energy of NaCl (Madelung, 1918). 

It may be recalled that an alternative description for a crystal stmcture can be made in terms of sets of lattice planes, which intersect the unit cell axes at ua, VU2, and was-The reciprocals of the coefficients are transformed to the smallest three integers having the same ratios, h, k, and I, which are used to denote the plane (hkl). Of course, the lattice planes may or may not coincide with the layers of atoms. Any such set of planes is completely specihed by the interplanar spacing, dhU7 and the unit vector normal to the set, since the former is given by the projection of, for example, u ui onto n kh that is dhki = u ui- n ki- The reciprocal lattice vector is defined as ... 

In the Bragg formulation of diffraction we thus refer to reflections from lattice planes and can ignore the positions of the atoms. The Laue formulation of diffraction, on the other hand, considers only diffraction from atoms but can be shown to be equivalent to the Bragg formulation. The two formulations are compared in Fig. 2B for planes with Miller indices (110). What is important in diffraction is the difference in path length between x-rays scattered from two atoms. The distance si + s2 in the Laue formulation is the same as the distance 2s shown for the Bragg formulation. The Laue approach is by far the more useful one for complicated problems and leads to the concept of the reciprocal lattice (Blaurock, 1982 Warren, 1969) and the reciprocal lattice vector S = Q 14n that makes it possible to create a representation of the crystal lattice in reciprocal space. 

We prove this result by showing that it is consistent with the Bragg law. The vector OG, which is designated g, is normal to a set of lattice planes (hkl) and is of magnitude /d(hkl). From Figure 3.6, it is clear that... 

Diffraction vector A vector perpendicular to the lattice planes hkl causing a Bragg reflection. The diffraction vector bisects the directions of the incident and diffracted beams and lies in their plane. 

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