Reciprocal space

The reciprocal space is a fundamental concept in solid-state physics. The basis of this space is formed by the basis vectors of the reciprocal lattice. Therefore, the terms of reciprocal space and reciprocal lattice are partly used synonymously.

Its importance for solid-state physics has this space due to the Blochtheorems, which means that the physical effects can be described in a crystal in this space effectively. A central concept of these descriptions is the first Brillouin zone. Using the reciprocal space quasi-particles and their dispersion are described. Depending on the theme while differing perspectives on and approaches have been developed for this term.

The reciprocal space is also called " space of wave vectors " ( " k-space " ) or " Fourier space " or the spatial frequency domain. The standard 3 -dimensional space, in contrast, also referred to as a " real space " or "x - space ".

Definition

The reciprocal of the space is 3-dimensional space of the plane waves. Each vector k of the reciprocal space corresponds to a plane wave with wave vector k:

Related to the reciprocal lattice

A crystal lattice is a regular arrangement of points in space. It is described by its translational symmetry. The set of all translation vectors R that transform a crystal in itself, form the crystal lattice. For the wave vectors G of all waves that do not destroy this translational symmetry, must apply accordingly:

This is exactly the equation for determining the points of the reciprocal lattice. These points represent all the waves in which all unit cells vibrate in the same phase. In reciprocal space, therefore, the same coordinate system as is used in the reciprocal lattice. The relationship between the basis vectors of the real lattice ai and bi of the reciprocal space is:

If the wave vector is a vector of the reciprocal lattice, it is in the literature often referred to as G or K. For a function f (r) where: f (r) is then exactly grating periodically when the Fourier transform F (k) for the reciprocal lattice vectors is non-zero.

The term reciprocal is because the unity of the basis vectors is a reciprocal length. Also, the length of these vectors is inversely proportional ( inversely ) behaves to the length of the vectors of the real lattice.

The first Brillouin zone

For the physical description are waves in which the magnitude of the wave vector is less than, relevant. If a wave has a larger wave vector, so can he so often a vector of the reciprocal lattice will be deducted until the wave vector is in the range without any change in the physical meaning of this wave something. This so- defined volume range is called first Brillouin zone.

Bloch functions

The Blochtheorem describes, among other things, the wave function of an electron in a crystalline solid. It states that the stationary Schrödinger equation with a translation invariant potential can have a solution only plane waves whose amplitude function has the translational symmetry of the lattice. The only difference between cells can therefore be made only in that the waves in the different unit cell have different phases. The wave vector k here has the meaning of an index for the classification of solutions of the Schrödinger equation.

In addition, k is in a conservation laws: Knocks an electron with wave vector k with a phonon with wave vector q, the following selection rule:

Where G is a vector of the reciprocal lattice.

Therefore, is called quasi-momentum or crystal momentum of the electron.

Phonons

Phonons are elastic waves. They are described by a displacement pattern and a wave vector. Phonon each having a certain frequency. In a dispersion curve, these frequencies with respect to the wave vector, which is varied in a certain direction of the grating, is applied. The graphs occurring for each mode is called dispersion branches or boughs. If the crystal 's atoms in the primitive cell, there exists for every wave vector of the Brillouin zone 3s branches, of which 3 acoustic and 3 - 3s are optically. In Figure 1, the dispersion branches of a crystal with two atoms are represented in the primitive cell. The wave vector goes there from the center to the edge of the Brillouin zone. There are three acoustic and three optical branches. Acoustic are those whose frequency vanishes in the center of the Brillouin zone.

Example

Figure 3: crystal in a ferroic phase

Figure 4: crystal in a phase antiferroischen

Figure 5: crystal in an incommensurate phase

At a phase transition, the low-temperature phase can arise from the fact that atoms are displaced in the unit cell with respect to its high temperature location. The structure of the low-temperature phase is then the associated with a phonon displacement pattern. The following three cases are considered hypothetical example in which the phase transition takes place with the same displacement, but at different points in the Brillouin zone. As an example structure of a primitive cubic cell is considered in which an atomic species at the corner points (0, 0, 0) (red circles), and another at the center (0.5, 0.5, 0.5 ) (black circles) the unit cell is. In this case, the directions of real and reciprocal basis vectors are the same. The atom at the center to be deflected in the z- direction. The images show the projections on the xz plane. Here arrows represent the displacement and the blue dot the location in the high temperature phase dar. Figure 2 shows the high-temperature phase - also called Paraphase - dar.

This means that all cells are in the same phase. In all cells, the same operation takes place. The lattice constant is unchanged. This leads to a phase transition ferroic. ( see Figure 3)

This means that there is a phase shift of 180 ° between adjacent cells in the x direction. In one cell, the average atomic moves exactly opposes to its counterpart in the next cell. In the low temperature phase, the atoms adjacent middle cells are deflected in opposite directions. Therefore, the lattice constant doubled in that direction. These phase transitions are called antiferroisch. ( see Figure 4)

This phase transition is particularly interesting if the b1- coordinate of this point is an irrational number. In this case there are 2 incommensurable periodicities in this direction: one is the grating, the other generated by the displacements of the atom. Typically, the phase transition point moves on the b1 - axis with temperature and there is a sequence of phase transitions take place. This leads to a " Devil's staircase ". Such phases are incommensurate (IC - phases) called. ( see Figure 5)

Correspondence in mathematics

In mathematics (specifically in differential geometry of the so-called Riemann spaces ) still corresponds to the notion of x-and k-space the so-called " contravariant " and " covariant " vector components ( upper or lower indices ), where the curvature of space is added (see also the General Theory of Relativity ). We thus used as basis vectors in the first case, the vectors a1, a2 and a3 and writes vectors in the form v = Σ vj aj, in the second case one uses the basis vectors b1, b2 and b3 and write v = Σ vj bj. The b- vectors are defined as above, without distinction of upper and lower indices, which shall be added only by the curvature of space. By using the Einstein summation convention is allowed for the sake of simplicity, the summation sign off by always summed with double indices, and deny at the same time the explicit indication of the basis vectors by implicitly only exploits that ai • bk = 1 for the same values ​​, so for i = k, and = 0 for different values ​​.

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