Reciprocal lattice vector, definition

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 ... 

The two-dimensional Bragg condition leads to the definition of reciprocal lattice vectors at and aj which fulfil the set of equations ... 

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

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). 

You can check this last statement by using the definition of the reciprocal lattice vectors.)... 

Because of the translational symmetry of the reciprocal lattice (Section 16.3) and the definition of the Brillouin zone (BZ), the BZ faces occur in pairs separated by a reciprocal lattice vector. For example, the cubic faces of the first BZ of the simple cubic (sc) lattice occur in pairs separated by the reciprocal lattice vectors b (2rc/a)[[l 0 0]] (see eq. (16.3.27)). In general, for every k vector that terminates on a BZ face there exists an equivalent vector k (Figure 17.1) such that... 

The factor 2-77 is usually omitted from the definition of a reciprocal-lattice vector in crystallography. This is because Bragg s law defines the deviation of a diffracted ray from the direct ray in terms of the half-wavelength of the radiation and the quantity 1 jd, which, in crystallography, is taken as the reciprocal-lattice vector ... 

Use of the reciprocal lattice unites and simplifies crystallographic calcnlations. The motivation for the reciprocal lattice is that the x-ray pattern can be interpreted as the reciprocal lattice with the x-ray diffraction intensities superimposed on it. See Section 14.2 for the definition of the reciprocal lattice vectors a b and c in terms of the direct basis vectors a, b, and c. Table 14.2 shows the parallel between the properties of the direct lattice and the reciprocal lattice, and Table 14.3 relates the direct and reciprocal lattices. 

A more precise definition of h aj, k ua and l ta can be given in terms of the maximum reciprocal lattice vector length, which is d, max 2/X. 

Let us start with a few definitions. A lattice plane of a given 3D BL contains at least three noncollinear lattice points and this plane forms a 2D BL. A family of lattice planes of a 3D BL is a set of parallel equally-spaced lattice planes separated by the minimum distance d between planes and this set contains all the points of the BL. The resolution of a given 3D BL into a family of lattice planes is not unique, but for any family of lattice planes of a direct BL, there are vectors of the reciprocal lattice that are perpendicular to the direct lattice planes. Inversely, for any reciprocal lattice vector G, there is a family of planes of the direct lattice normal to G and separated by a distance d, where 2jt/d is the length of the shortest reciprocal lattice vector parallel to G. A proof of these two assertions can be found in Ashcroft and Mermin [1]. 

Here, the angles (R,) denote the orientations of the N2 molecules on the positions R of the triangular lattice with lattice constant a as in the definition of the Hamiltonian (2.5) itself. The corresponding reciprocal lattice vectors Q and phases distinguish the three components a = 1, 2, 3 of the order parameter. 

We suppose that jR is a symmetry operator that corresponds to some proper or improper rotation, and that r is a vector in the real lattice. The vector Rr is also a vector in the real lattice since K is a symmetry operator. There are as many points in reciprocal lattice space as in the direct lattice, and each direct lattice vector corresponds to a definite vector in the reciprocal lattice. It follows that Rr corresponds to a reciprocal lattice point if r is a reciprocal lattice vector. Thus the operators R, S,. . . , that form the rotational parts of a space group are also the rotational parts of the reciprocal lattice space group. It now follows that the direct and reciprocal lattices must belong to the same crystal class, although not necessarily to the same type of translational lattice (see Eqs. 10.28-10.31). 

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 ... 

Figure 6.8 Definition and properties of the two-dimensional reciprocal lattice a, and a2 are base vectors of the surface lattice, and a and a2 are the base vectors of the reciprocal lattice. The latter is equivalent to the LEED pattern.

In choosing beam optics to measure xrd-rsm, one must consider resolution function in the reciprocal space. The resolution function is defined by the incident beam divergence and the acceptance window of the diffracted beam side optic. Figure 6.3 schematically shows the definition of the resolution function in the reciprocal space. The X-ray detector is located at the tip of the scattering vector H in the reciprocal space. The incident beam divergence 5u> and the acceptance window of the diffracted beam optic 520 define the resolution function (gray area in Figure 6.3) in the reciprocal space. The form of the obtained diffracted intensity distribution of the crystal by xrd-rsm is defined by the convolution of the resolution function and the reciprocal lattice point of the crystal. Therefore, a resolution function smaller than... 

This construction places a reciprocal lattice point at one end of h. By definition, the other end of h lies on the surface of the sphere. Thus, Bragg s law is only satisfied, when another reciprocal lattice point coincides with the surface of the sphere. Diffraction is emanating from the sample in these directions. To detect the diffracted intensity, one simply moves the detector to the right position. Any vector between two reciprocal lattice points has the potential to produce a Bragg peak. The Ewald sphere construction additionally indicates which of these possible reflections satisfy experimental constraints and are therefore experimentally accessible. 

According to the definitions given by Eqs. (1.1) to (1.3), the product G R = 2jrx integer. Therefore each vector of the reciprocal lattice is normal to a set of planes in the direct lattice, and the volume of a unit cell of the reciprocal lattice is related to the volume of the direct lattice I4 by... 

Consider a polycrystalline sample placed at a point O. A crystal within this sample diffracts for a family of planes (hkl) if and only if the reciprocal lattice point associated with this family is on the Ewald sphere with radius 1/X. By definition, the tip of the vector kg =(l/A,)So is the origin of the reciprocal lattice and the scattering vector S, which hnks point C with lattice point hkl, has a norm equal to 1/dhki. For all the crystals contained in the sample which diffract for a given family of planes (hkl), we observe a set of lattice points hkl located on a sphere with radius 1/dhn, which is centered in C. By definition, the directions [hkl] are normal to the (hkl) planes and therefore the distribution map of the hkl lattice points on the surface of this sphere is the hkl pole figure. This configuration is shown in Figure... 

Takeuchi et al (1972) defined the vector r as r = MA1+VA2, i.e. with respect to a basis with interaxial angle 60° correspondingly in the multiplicity of the mesh (Eq. 8) and in the length of the vector (Eq. 9) the term uv has opposite sign. Their definitions of (m, v) and n values correspond to reciprocal lattice values in om treatment. 

In the case of plane-wave basis sets the scaling proceeds on the reciprocal-space vectors as G- (1+ ) G, which is seen by the definition aj b< = 6 4, where and b are real- and reciprocal-lattice primitive translation vectors, respectively. Thus one finds the derivative of reciprocal-space vectors given by... 

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