Reciprocal-lattice vectors

Now let us generalize this to three dimensions. We must expand n r) in Fourier series. 

This requires exp(zG-T) = 1, which means that G-T = 2ti x integer. 

To facilitate the construction of the vector G, we define G as a translation vector in reciprocal space  

and C are unit cell vectors in reciprocal space h, k, and are integers. 

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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 ... 

The reciprocal lattices shown in figure B 1.21.3 and figure B 1.21.4 correspond directly to the diffraction patterns observed in FEED experiments each reciprocal-lattice vector produces one and only one diffraction spot on the FEED display. It is very convenient that the hemispherical geometry of the typical FEED screen images the reciprocal lattice without distortion for instance, for the square lattice one observes a simple square array of spots on the FEED display. 

One of the spots in such a diffraction pattern represents the specularly reflected beam, usually labelled (00). Each other spot corresponds to another reciprocal-lattice vector = ha + kb and is thus labelled (hk), witli integer h and k. 

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 ... 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

Pounds per square inch psi Reciprocal lattice vector (cir- G... 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at t, a reciprocal lattice vector. 

Figure 2 View looking down on the real-space mesh (a) and the corresponding view of the reciprocal-space mesh (b) for a crystal plane with a nonrectangular lattice. The reciprocal-space mesh resembles the real-space mesh, but rotated 90°. Note that the magnitude of the reciprocal lattice vectors is inversely related to the spacing of atomic rows.

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

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-... 

Thus the point group part of the operation works on the momentum coordinates and the translation part gives rise to a phase factor. We notice that this phase factor reduces to 1 in the diagonal elements, or in general when the difference between the the two arguments of N(p,p ) is a reciprocal lattice vector. 

This sum over a// reciprocal space vectors of the form (IV.2) should be carefully distinguished from the expansion (III.4) of the density of a periodic crystal. If the density has the "little period", the e>mansion (IV.3) reduces to a sum over all reciprocal lattice vectors. The general case (IV.3) and the periodic case (III.4) actually represent two extreme cases. The presence of "more and more symmetry" in the density can be gauged... 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

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.

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 ... 

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