Lattice with a basis

Figure 16.1. (a) Portion of a 2-D (space) lattice, (b) Lattice with a basis, primitive unit cell, k = [00]. 

In a lattice with a basis, each molecule or ion of the primitive cell can have a different spin or a different orientation. This leads to the necessity of considering sublattices. For two sublattices with opposite spins of different moduli, the total spin is the difference between spins. This is a ferrimagnet. If two sublattices have opposite spins of the same moduli, the resulting spin is zero and we have an antiferromagnet. 

Silicon will serve as the paradigmatic example of slip in covalent materials. Recall that Si adopts the diamond cubic crystal structure, and like in the case of fee materials, the relevant slip system in Si is associated with 111 planes and 110> slip directions. However, because of the fact that the diamond cubic structure is an fee lattice with a basis (or it may be thought of as two interpenetrating fee lattices), the geometric character of such slip is more complex just as we found that, in the case of intermetallics, the presence of more than one atom per unit cell enriches the sequence of possible slip mechanisms. 

Fig. 1.3-1 A periodic crystal can be described as a convolution of a mathematical point lattice with a basis (set of atoms). Open circles, mathematical pomts, filled circles, atoms...

Graphene is a one-atom-thick planar layer of graphite, where the atoms are packed in a hexagonal ( honeycomb ) crystal lattice. The carbon atoms are sp -bonded with a bond length of 1.42 A. The crystal lattice has two atoms per unit cell, A and B, and it is rotationally symmetric for rotations of 120° around any lattice point. One can view the honeycomb lattice as a triangular Bravais lattice with a basis of two atoms per unit cell. Fig. 2a. 

Such types of lattices are called lattices with a basis. The diamond lattice can be constructed by placing not one, but two atoms (a diatomic basis) on each site of a face-centred cubic lattice (figure 1.17). Another example of a lattice with a basis, the hexagonal close-packed structure, was already discussed in section 1.2.2. It can also be constructed by placing a diatomic basis on each site of a Bravais lattice, in this case a simple hexagonal lattice. 

The electronic structure is determined using the ab initio all-electron scalar-relativistic tight-binding linecir muffin-tin orbital (TB-LMTO) method in the atomic-sphere approximation (ASA). The nnderlying lattice, zincblende structure, refers to an fee Bravais lattice with a basis which contains a cation site (at a(0,0,0)), an anion site (at o(j,, )), and two interstitial sites occupied by empty spheres (at a(, 5, h) and a(, , )) which in turn are necessary for a correct description of open lattices . ... 

Primitive and nonprimitive vectors in an fee reciprocal lattice in which the primitive vectors and their translation sites are denoted by prime symbols. Trar slatioi s along these primitive vectors produce a nonprimitive fee reciprocal lattice with a basis of (000), (110), (101), (011). Notice that primitive trai slatioi s (100) are equivalent to (110), (200) to (220), etc... 

I have to say a word about the notation. It is the same as that used in other books about this subject. This notation might sometimes look complicated. The basic concepts (linear oscillator. Hook s law, etc.) are simple but the lattice with a basis introduces an unavoidable complex notation. Experience has shown, however, that students become accustomed to the notation very quickly. Therefore, there is no reason to be discouraged by this. Whenever possible I have tried to use a simpler or condensed notation. Appendix Q contains the most important physical constants and units used in this book. 

Fig. 2.13. Two-dimensional Bravais lattice with the basis vectors a)s a2, and the reciprocal lattice vectors bi, b2. The solid and dashed arrows at angles A and 0A give the ferroelectric (k = 0) and antiferroelectric (k = bi/2) configurations of dipoles in the ground state.

Up to this point we did not make any specific assumptions about the real space lattice. It could contain more than one atom per lattice point and more than more than one type of atoms. In such a case the lattice would be described using a Bravais lattice plus a basis (see Section 8.2.2. To obtain the intensity of the diffracted wave for crystals with a basis, we simply have to sum up the contributions from all scattering points within the unit cell. The scattering probability for a crystal of N unit cells with an electron density ne(r) is proportional to ... 

There are, however, other possibilities for an ordered starting structure. One of these is the BC-8 structure of Si, which is a crystalline polymorph created at high pressure[36,37]. The BC-8 structure is body-centered cubic with a basis of 8 atoms per BC-2 lattice site. Thus the BC-8 unit cell consists of 16 atoms. This offers the possibility of supercells containing N = 16n atoms with periodicity and strained tetrahedral bonding built in. Weaire and Taylor describe the BC-8 structure as a kind of half-way house between the simplicity of the FC-2 structure and the complexity of the random network that characterizes the amorphous material [38]. 

Basis Group of atoms associated with each and every lattice point. We can describe crystal structures in terms of a Bravais lattice and a Basis ... 

NaCl, GaAs fee lattice with a two-atom basis... 

The results obtained on the first two subjects were used by me to calculate the optimal geometry and multiplication constant for oxide lattices and to obtain estimates for the same quantity in metal lattices. I arrived at the conclusion - around the middle of November - that the multiplication constant in Fermi s Columbia pile could be increased by about 5% by going over to a lattice with a considerably smaller lattice constant. I expected a further increase of another 5% if one could replace the oxide by metal. This last increase was not really based on the measurements of Creutz and Wilson, but on a theory of the resonance absorption which I developed around this time. On the basis of these calculations and because it became evident that a very considerable improvement in the multiplication constant can be achieved by using materials of a higher purity, I became convinced that a chain reaction is possible in a graphite-uranium mixture and estimated the multiplication constant obtainable with an oxide-graphite lattice as 1.02, with a metal-graphite lattice as 1.07. 

Most of the technologically-important semiconductors, such as Ge, Si, GaAs, GaSb, and (Hg,Cd)Te have a facecubic lattice with a two-atom basis. In the case of Si and Ge, the atoms on the A and B sites are identical forming the sodiamond structure. In the case of binary semiconductors such as GaAs, the A sites are occupied Ga atoms and the B sites As atoms the crystal is said to have the zinc-blende structure. As stated above, in each case, the underlying lat-... 

The isotherm for bromide adsorption (Fig. 14c) shows that increasing the potential in the double-layer region causes a continuous increase in the bromide coverage. The data shown in Fig. 15 and in other experiments, however, are not consistent with the notion that the additionally adsorbed bromide leads to a continuous compression of the incommensurate bromide adlayer, as this would cause the sharp peak at/ = 0.67 to shift gradually to higher wave vector. An alternative explanation of our results is that bromide forms a series of high-order commensurate (HOC) structures on the Pt(lll) surface, that is, at all potentials the structure corresponds to a close-packed monolayer but that the periodicity depends critically on the ratio of the bromide and Pt lattice parameters. At 0.2 V, the unit cell corresponds to a (3 x 3) structure with a basis of four Br atoms. At 0.6 V, the unit cell is a (7 x 7) structure containing 25 Br atoms, that is, with a 7 5 ratio of the Br and Pt lattice parameters. 

Decoration of the Lattice with the Basis At this point we have to recall that in a real crystal structure we have not only the lattice, but also the basis. In [3.8], there are standardized sets of general and special positions (i. e. coordinates x, y, z) within the unit cell (Wyckoff positions). An atom placed in a general position is transformed into more than one atom by the action of all symmetry operators of the... 

Fig. 1.17. Diamond lattice, constructed as a face-centred cubic lattice with a diatomic basis...

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