Lattice, inverse

Let us now construct the so called biorthogonal basis bi,b2,b3 with respect to the basis vectors a, a2, of the primitive lattice, i.e. the vectors that satisfy the 

The vectors bi, bi and hi form the basis of a lattice in a 3D space. This lattice will be called the inverse lattice. 

We have only a single biorthogonality relation bi a = 2tt, i.e. after skipping the index ba = lTr. Because of the single dimension, we have to have = ( ), where a = a. Therefore, 

This time we have to satisiy bioi = 2iT,t2a2 = t.TT, Pia2 = V, =it- i tus means 

Electronic Motion in the Mean Field Periodic Systems 

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What this means is that the primitive lattice is composed of points at the corners of the lattice, whereas the inversion lattice hcis an additional point at the center of the lattice, i.e.- "body-centered". Face-centered has points in the middle of each face of the lattice in addition to those at the corners of the lattice. 

A single hexaaluminate phase was obtained at 0.4diffraction line at lower angle, Mn-rich hexaalruninate phase, decreased and Mn-poor hexaaluminate phase increased inversely. Lattice constants dependence on Pr concentration in Mn-poor hexaalmninate were shown in Figure 5. With increasing Pr content, lattice constant rose steeply at x<0.2 and then approached... 

Fig. 6 Correlation between the formal potentials of the hexacyanometalate units of the solid hexacyanometalates and the inverse lattice constant L of the solid compounds [53, 55].

With (9.4-6) and using X of the order of ten times the inverse lattice constant together with N specified by (9.7), the structure constants may be evaluated with an accuracy of a few per cent by summation over 3 shells in the direct lattice plus less than ten points of the reciprocal lattice. For an accuracy exceeding one part per thousand the summations must include 6-8 shells of both lattices. 

Fig. 9.3. Construction of the inverse lattice in 2-D. In order to satisfy the biorthogonality relatirais Eq. (9.15), the vector b has to be orthogonal to 02. while f>2 must be perpendicular to i. The lengths of the vectors b and b2 also follow from the biorthogonalify relations b a = b2 02 = 2n.

Fig. 9.5. Construction of the FBZ as a Wigner-Seitz unit cell of the inverse lattice in 2-D. The circles represent the nodes of the inverse lattice. We cut the lattice in the middle between the origin node W and all the other nodes (here, it turns out to be sufficient to take only the nearest and the next nearest neighbors) and remove all the sawed-off parts that do not contain W. Finally we obtain the FBZ in the form of a hexagon. The Wigner-Seitz unit cells (after performing all allowed translations in the inverse lattice) reproduce the complete inverse space.

Let us imagine two inverse space vectors k and k" related by the equality k" = k - -Kg, where Kg stands for an inverse lattice vector. Taking into account the way the FBZ has been... 

If we decide to draw the function in space, we would obtain Fig. 9.6b. When asked this time, we would answer that the wavelength is equal to A, = 2a, which, by the way, is equal to There is a problem. Does the wave correspond to k = or A = — It corresponds to both of them. Well, does it contradict the theorem that the contains all different states No. it does not. Both functions are from the border of the FBZ their k values differ by (one of the inverse lattice vectors), and therefore both functions represent the same state. 

This time, we will consider the crystal as 2-D rectangular lattice therefore, the corresponding inverse lattice is also 2-D, as well as the wave vectors k = (kx, ky). 

The crystal lattice basis vectors allow the formation of the basis vectors of the inverse lattice. Linear combinations of them (with integer coefficients) determine the inverse lattice subject to translational symmetry. 

A special [Wigner-Seitz) unit cell of the inverse lattice is called the First Brillouin Zone (FBZ). 

The Wigner-Seitz unit cells (after performing all allowed translations in the inverse lattice) reproduce the complete inverse space. 

The crystal lattice basis vectors allow the formation of the basis vectors of the inverse lattice. 

Linear combinations of them (with integer coefficients) determine the inverse lattice. 

Bloch function (p. 435) symmetry orbital (p. 435) biorthogonal basis (p. 436) inverse lattice (p. 436)... 

Here uflk) designates the y-component of displacement of the molecule at the / -site in the /-cell, whose position vector is R(/A ). The wave vector q is a vector in the inverse lattice. 

The basis vectors of the inverse lattice bi, bj, bg are defined in terms of the basis vectors of the lattice aj, a, a, as follows ... 

A vector composed of integral multiples of the inverse lattice basis vectors is called an inverse lattice vector. ... 

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