Lattice-unit-vectors

We also need to define how large the unit-cell is in terms of both the length of its sides and its volume. We do so by defining the unit-cell directions in terms of its "lattice unit-vectors". That is, we define it in terms of the x, y, z directions of the unit cell with specific vectors having directions corresponding to ... 

Liquid-phase adsorption characteristics examined by atomic force microscopy (AFM) were compared for two pyridine base molecules, pyridine and d-picoline. on (010) surfaces of two natural zeolites, heulandite and stilbite. These adsorption systems formed well-ordered. two-dimensioncJ (quasi-)hexagonal adlayers. The 2D lattice structures of the ordered adlayers w ere dependent on the adsorbate/substrate combinations. Although there existed certain habit in the orientation of the 2D lattice unit vector of the adsorbed phase with respect to the substrate(OlO) lattice vectors, the molecular arrays w ere incommensurate with the substrate atomic arrangements. 

Graphene lattice and its Brillouin zone. Left lattice structure of graphene, made out of two interpenetrating triangular lattices (ai and aj are the lattice unit vectors). Right corresponding Brillouin zone. The Dirac cones are located at the K and K points... 

Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

We choose as the length unit the length of the amphiphile, which is about 20, and in this unit the lattice constant is a = 1. The unit lattice vectors are denoted by e and Cj denotes the ith component of the unit vector where i= 1,..., i/, and d is the dimension of the system. We concentrate on the symmetric case ow = 0- The generalization to ow 0 is straightforward. 

The common periodic structures displayed by surfaces are described by a two-dimensional lattice. Any point in this lattice is reached by a suitable combination of two basis vectors. Two unit vectors describe the smallest cell in which an identical arrangement of the atoms is found. The lattice is then constructed by moving this unit cell over any linear combination of the unit vectors. These vectors form the Bravais lattices, which is the set of vectors by which all points in the lattice can be reached. 

Fig. 16. (a) The chiral vector OA or Ch = nhi + md2 is defined on the honeycomb lattice of carbon atoms by unit vectors ai and a of a graphene layer and the chiral angle with respect to the zigzag axis (9 = 0°). Also shown are the lattice vector... 

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... 

There is another case when the orientational Hamiltonian for nonpolar molecules (2.3.1) on a symmetric two-dimensional lattice is reducible to a quasidipole form, viz. the case of planar orientations of long molecular axes ( s = 90° in expression (2.3.2)) when one can invoke the transformation for doubled orientation angles qy of the unit vectors ej and r ... 

Consider the approximation of four discrete molecular orientations along the axes of a square lattice ( At/4 kBT, AU4 < 0). To conveniently describe orientations, we introduce, at each lattice site, two spin variables, a m = l and s m = 1, which are related to unit vectors nm as follows (Fig. 2.18) ... 

Here /(R) and pflX) denote the shift and generalized momentum for the molecular vibration of the low frequency a>9 and reduced mass m, at the Rth site of the adsorbate lattice bi+(K) and K) are creation and annihilation operators for the collectivized mode of the adsorbate that is characterized by the squared frequency /2(K) = ml + d>, a,(K)/m , with O / iat(K) representing the Fourier component of the force constant function /jat(R). Shifts i//(R) for all molecules are assumed to be oriented in the same arbitrary direction specified by the unit vector e they are related to the corresponding normal coordinates, ue (K), and secondary quantization operators ... 

We now introduce a Fourier transform procedure analogous to that employed in the solution theory, s 62 For the purposes of the present section a more detailed specification of defect positions than that so far employed must be introduced. Thus, defects i and j are in unit cells l and m respectively, the origins of the unit cells being specified by vectors R and Rm relative to the origin of the space lattice. The vectors from the origin of the unit cell to the defects i and j, which occupy positions number x and y within the cell, will be denoted X 0 and X for example, the sodium chloride lattice is built from a unit cell containing one cation site (0, 0, 0) and one anion site (a/2, 0, 0), and the translation group is that of the face-centred-cubic lattice. However, if we wish to specify the interstitial sites of the lattice, e.g. for a discussion of Frenkel disorder, then we must add two interstitial sites to the basis at (a/4, a/4, a]4) and (3a/4, a/4, a/4). (Note that there are twice as many interstitial sites as anion-cation pairs but that all interstitial sites have an identical environment.) In our present notation the distance between defects i and j is... 

The structure of SWCNTs is characterized by the concept of chirality, which essentially describes the way the graphene layer is wrapped and is represented by a pair of indices (n, m). The integers n and m denote the number of unit vectors (a a2) along the two directions in the hexagonal crystal lattice of graphene that result in the chiral vector C (Fig. 1.1) ... 

The symbols for plane groups, the Hermann-Mauguin symbol, have been the standard in crystallography. The first place indicates the type of lattice, p indicates primitive, and c indicates centered. The second place indicates the axial symmetry, which has only 5 possible vales, 1-, 2-, 3-, 4-, and 6-fold. For the rest, the letter m indicates a symmetry under a mirror reflection, and the letter g indicates a symmetry with respect to a glide line, that is, one-half of the unit vector translation followed by a mirror reflection. For example, the plane group pAmm means that the surface has fourfold symmetry as well as mirror reflection symmetries through both x and y axes. 

Calculate the volume of a unit cell from the lattice translation vectors. 

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