Reciprocal lattice defined

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

In Eq. [30], erfc(a ) denotes the complementary error function, N is number of atoms in the simulation box of volume V, and and are charges of atoms i and j, respectively. K = 27tH, K = IKI, and H stands for a vector of the reciprocal lattice defined for the simulation box and are parameters controlling convergence of the direct and reciprocal sums. 

The inverse relationship between the crystal s real space lattice and its reciprocal lattice defines the distances between adjacent reflections along reciprocal lattice rows and columns in the diffraction pattern. Conversely, measurement of the reciprocal lattice spacings yields the unit cell parameters. Angles between the axes of the reciprocal lattice can similarly be used to determine unit cell axial angles. 

Therefore, we obtain all distinct solutions to our problem if we restrict the allowed values of q to lie in one unit cell of the reciprocal lattice defined by the three vectors 3 means that the integers n in... 

The Structure Factor It is important to always have in mind that the reciprocal lattice defines only the directions in which diffraction spots can appear. Also, the lattice factor G = G(ghk) takes the same value independent of energy and spot hk. The intensity is determined solely by the structure factor I = F = [f(ko,kj) which, as defined in Eq. (3.2.1.4), describes the intensity contribution of each unit... 

Expressing (k) is complicated by the fact that k is not unique. In the Kronig-Penney model, if one replaced k by k + lTil a + b), the energy remained unchanged. In tluee dimensions k is known only to within a reciprocal lattice vector, G. One can define a set of reciprocal vectors, given by... 

Reciprocal lattice vectors are usefiil in defining periodic fimctions. For example, the valence charge density, p (r), can be expressed as... 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

These reciprocal lattice vectors, which have units of and are also parallel to the surface, define the LEED pattern in k-space. Each diffraction spot corresponds to the sum of integer multiples of at and at-... 

The reciprocal lattice is useful in defining some of the electronic properties of solids. That is, when we have a semi-conductor (or even a conductor like a metal), we find that the electrons are confined in a band, defined by the reciprocal lattice. This has important effects upon the conductivity of any solid and is known as the "band theory" of solids. It turns out that the reciprocal lattice is also the site of the Brillouin zones, i.e.- the "allowed" electron energy bands in the solid. How this originates is explciined as follows. 

The vectors which define the so-called reciprocal lattice are given by... 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

In this expression SA is summed over the direct lattice, R( SB is summed over the reciprocal lattice, b,. The parameter r is chosen so as to obtain equally rapid convergence in the sums over the direct and reciprocal lattices. The 6 functions are defined by... 

Electrons diffract from a crystal under the Laue condition k — kg=G, with G = ha +kb +lc. Each diffracted beam is defined by a reciprocal lattice vector. Diffracted beams seen in an electron diffraction pattern are these close to the intersection of the Ewald sphere and the reciprocal lattice. A quantitative understanding of electron diffraction geometry can be obtained based on these two principles. 

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). 

The deviation of the electron beam from the Bragg condition is measured by the distance from the reciprocal lattice vector to the Ewald sphere along the zone axis direction, which approximately is defined by... 

In a distorted crystal, where the atomic displacement from the perfect crystal is given by the vector u, we can define a local reciprocal lattice vector g by g =g-gxad(g.u) (8.19)... 

If we consider a primitive Bravais lattice with cell edges defined by vectors a, 82, and aj, the corresponding reciprocal lattice is defined by reciprocal vectors bj, b2, and bj so that... 

A simple example of this calculation is the simple cubic lattice we discussed in Chapter 2. In that case, the natural choice for the real space lattice vectors has a, = a for all i. You should verify that this means that the reciprocal lattice vectors satisfy b, = 2rr/a for all i. For this system, the lattice vectors and the reciprocal lattice vectors both define cubes, the former with a side length of a and the latter with a side length of 2tt/g. 

The three-dimensional shape defined by the reciprocal lattice vectors is not always the same as the shape of the supercell in real space. For the fee primitive cell, we showed in Chapter 2 that... 

Thus, the scattering of a periodic lattice occurs in discrete directions. The larger the translation vectors defining the lattice, the smaller a i=1 3, and the more closely spaced the diffracted beams. This inverse relationship is a characteristic property of the Fourier transform operation. The scattering vectors terminate at the points of the reciprocal lattice with basis vectors a i=1>3, defined by Eq. (1.21). 

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