Crystals reciprocal primitive lattice vectors

The foundation for describing the behavior of electrons in a crystal is the reciprocal lattice, which is the inverse space of the real lattice. The reciprocal primitive lattice vectors are defined by... 

Thus, the reciprocal lattice of a simple cubic lattice is also simple cubic. It is shown in Fig. 5.7 in the xy plane, where it is clear that the bisectors of the first nearest-neighbour (100) reciprocal lattice vectors from a closed volume about the origin which is not cut by the second or any further near-neighbour bisectors. Hence, the Brillouin zone is a cube of volume (2n/a)2 that from eqn (2.38) contains as many allowed points as there are primitive unit cells in the crystal. The second, third, and fourth zones can... 

It has just been stated that a band stracture diagram is a plot of the energies of the various bands in a periodic solid versus the value of the reciprocal-space wave vector k. It is now necessary to discuss the concept of the reciprocal-space lattice and its relation to the real-space lattice. The crystal structure of a solid is ordinarily presented in terms of the real-space lattice comprised of lattice points, which have an associated atom or group of atoms whose positions can be referred to them. Two real-space lattice points are connected by a primitive translation vector, R ... 

The 1st Brillouin zone follows from the reciprocal lattice by construction of the planes which are perpendicular to the lines connecting neighbouring points in the reciprocal lattice at their midpoints. The smallest closed volume which is bounded by these planes is the 1st BZ. For the naphthalene crystal, we find from the lattice parameters at T = 300 K (Table 2.3) the following magnitudes for the reciprocal lattice vectors a = lit 0.145 A ) = 2jt 0.167 A c = 2jr 0.138 A and for the volume V of the primitive... 

We turn our attention next to specific examples of real crystal surfaces. An ideal crystal surface is characterized by two lattice vectors on the surface plane. Hi = aix + aiyS, and H2 = 02xX -I- a2yy. These vectors are multiples of lattice vectors of the three-dimensional crystal. The corresponding reciprocal space is also two dimensional, with vectors bi, b2 such that b aj = IrrStj. Surfaces are identified by the bulk plane to which they correspond. The standard notation for this is the Miller indices of the conventional lattice. For example, the (001) surface of a simple cubic crystal corresponds to a plane perpendicular to the z axis of the cube. Since FCC and BCC crystals are part of the cubic system, surfaces of these lattices are denoted with respect to the conventional cubic cell, rather than the primitive unit cell which has shorter vectors but not along cubic directions (see chapter 3). Surfaces of lattices with more complex structure (such as the diamond or zincblende lattices which are FCC lattices with a two-atom basis), are also described by the Miller indices of the cubic lattice. For example, the (001) surface of the diamond lattice corresponds to a plane perpendicular to the z axis of the cube, which is a multiple of the PUC. The cube actually contains four PUCs of the diamond lattice and eight atoms. Similarly, the (111) surface of the diamond lattice corresponds to a plane perpendicular to the x -I- y -I- z direction, that is, one of the main diagonals of the cube. 

Let us come back to our task to find all possible diffraction peaks for a given crystal lattice. What are the possible scattering vectors that lead to constructive interference This question can be answered in an elegant way by defining the so-called reciprocal lattice If a, a2, and <23 are primitive vectors of the crystal lattice, we choose a new set of vectors according to... 

Just as a reminder The dots between the vectors denote the scalar (inner) product and the crosses denote the cross (outer) product of the vectors. These vectors 6 are in units of nr, which is proportional to the inverse of the lattice constants of the real space crystal lattice. This is why one calls the three-dimensional space spanned by these vectors the reciprocal space and the lattice defined by these primitive vectors is called the reciprocal lattice. These primitive reciprocal vectors have the following properties ... 

The Bom-von Karman contour condition demonstrates that the Bloch wave vector of free electrons in a cubic lattice is, according to Sommerfeld, constituted only by real components. The number of k values (k = p/h) admitted in a primitive cell of a reciprocal lattice is equal to the number of sites in the crystal. The linear momenta operator, p, is... 

First Brillouin zone The primitive cell in the reciprocal lattice containing the point k=0. All physically different electronic states in a crystal can be characterized by a wave vector reduced to the first Brillouin zone. 

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