Basic translation vectors

The geometric stmcture is used conventionally for identification of a particular metal/semiconductor surface phase. The surface structures are usually labeled in accordance with their periodicity with respect to the underlying semiconductor crystal plane. Two methods for the description of the two-dimensional lattices are used conventionally. The first one was proposed by Park and Madden [68P] and it consists in the determination of the matrix which establishes a hnk between the basic translation vectors of the surface under consideration and those of the ideal (unreconstructed) substrate sirrface. That is, if a and b are the basic translation vectors of the substrate lattice, while and As are the basic translation vectors of the surface phase, than they can be linked by the equations... 

The second method for the description of the two-dimensional lattices was proposed by Wood [64W]. This method is the most-used one now though it is less versatile than the above matrix notation. In the Wood description, the ratio of the siuface periods and those of the unreconstructed substrate are given as well as an angle of rotation which makes the unit mesh of the surface to be aligned with the basic translation vectors of the substrate. That is, if a certain adsorbate A induces on the B(hkl) surface the reconstruction with the basic translation vectors of... 

It should be remarked that the Wood description is appropriate only in the cases when the rotation angle is the same for the both basic translation vectors (i.e. for a, and ft with relation to a and b, respectively). Otherwise, the matrix notation only can provide an accurate description However, in the literature sometimes less rigorous notation (of the Wood s-like type) is used with the addition of the necessary conunents or even without them when it concerns the well-known structures. An example is the clean Si(llO) surface. The reconstraction of this... 

TVanslation symmetry of a perfect crystal can be defined with the aid of three non-coplanar vectors 01,02,03 basic translation vectors. Translation tg. through the lattice vector... 

The lattice types are labeled by P (simple or primitive), F (face-centered), I (body-centered) and A B,C) (base-centered). Cartesian coordinates of basic translation vectors written in units of Bravais lattice parameters are given in the third column of Table 2.1. It is seen that the lattice parameters (column 4 in Table 2.1) are defined only by syngony, i. e. are the same for all types of Bravais lattices with the point symmetry F and all the crystal classes F of a given syngony. 

Therefore this lattice is defined by 6 parameters - lengths a, 6, c of basic translation vectors and angles a,/ ,7 between their pairs 02 —03, oi —03 and oi —a2, respectively. 

Syngony F° -crystal classes F Direct lattice types Basic translation vectors Bravais lattice par am. 

All the three translation vectors of a simple orthorhombic lattice Fo are orthogonal to each other, so that the conventional cell coincides with the primitive cell and is defined by three parameters - lengths of the basic translation vectors. For base-centered r, face-centered F/ and body-centered F lattices the conventional unit cell contains two, four and two primitive cells, respectively (see Fig. 2.2). 

The hexagonal lattice is defined by two parameters a - length of two equal basic translation vectors (with the angle 120 degree between them) and c — length of the third basic translation vector orthogonal two the plane of first two vectors. 

Let a<(T i) (i=l,2,3) be the basic translation vectors of the initial direct lattice of type Fi and aj F2) j = 1,2,3) be the basic translation vectors of a new lattice of type F2 with the same point symmetry (symmetrical transformation) but composed... 

Let us demonstrate the procedure of finding the matrix of a symmetrical transformation (4.77) by the example of the rhombohedral crystal system where there is only one lattice type (R). The basic translation vectors of the initial lattice are the following ... 

In the hexagonal crystal system there is only one lattice type (P), but the basic translation vectors may be oriented in two different ways relative to the basic translation vectors of the initial lattice either parallel to them or rotated through an angle of 7t/6 about the -axis. Therefore, two types of symmetrical transformation are possible in this case (with two parameters for each). 

The functions < (k) = (5r,r (k)) are periodic in the reciprocal space with periods determined by the basic translation vectors b of the reciprocal lattice and having the fuU point symmetry of the crystal F is a point group of order np of the crystal). 

Let Bi Fi) (i = 1,2,3) and bj(F2) (j = 1,2,3) be basic translation vectors of the reciprocal lattices corresponding to direct ones determined by basic translation vectors o<(Fi) and Oj(F2), respectively. The transformation (4.77) of the direct lattices is accompanied by the following transformation of reciprocal lattices ... 

Let m be a subset of m corresponding to the stars of vectors gam 9 F) As is proved in [86], for symmetrical transformation (4.77) L points in the initial Brillouin zone (related to the initial basic translation vectors Cj)... 

To generate the set of points for any of the 14 Bravais lattices it is sufhcient to find the inverse of the corresponding matrix from Appendix A, to pick out according to (4.84) L points in the Brillouin zone related to basic translation vectors a and to distribute them over stars. The distribution of these points over stars depends on the symmetry gronp F of the function f K) and can not be made in general form. 

Let us consider some examples. In MgO crystal the superceUs, containing 8, 16 and 32 atoms (S8, S16, S32), are obtained by a linear symmetric transformation of the fee lattice basic translation vectors with matrices... 

Consider, for example, crystals with face-centered cubic Bravais lattices. For the (001), (110) and (111) sections the plane lattices are square, rectangular and hexagonal, respectively. The basic translation vectors of the direct and reciprocal lattices for these three cases are given in Table 11.2 (ai and 02 are given in units a/2, Bi in units 27r/a, where a is the cubic lattice parameter). Note that for a cubic lattice the planes (100), (010) and (001) are equivalent. The equivalence takes place also for (110), (101), and (oil) planes. We see that the vectors Bi(i = 1,2) are now not the translation vectors of the three-dimensional reciprocal lattice. Therefore, the boundaries of BZ-2 do not coincide with those of BZ-3. 

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