Cell and supercell of the reciprocal lattice of crystals
The aim of the research is to substantiate the concept of supercell of the reciprocal lattice for crystals with the non-primitive Bravais cell in direct space. The object of research is the types of Bravais supercells for the reciprocal lattice of crystals with different Bravais cells. It is considered Bravais cell <i>P</i>-, <i>I</i>-, <i>F</i>-, <i>S</i>(<i>AB</i>)-types. It is analyzed the analytical relationships between the bases of the crystal lattice and cell parameters ofthe reciprocal lattice. It is shown, when the relationship between parameters of direct <i>a</i><sub>i</sub><i>α</i><sub>i</sub> (<i>i</i> = 1, 2, 3) and reciprocal lattice <i>a</i><sup>*</sup><sub>j</sub><i>α</i><sup>*</sup><sub>j</sub> (<i>j</i> = 1, 2, 3) of the cell <i> ⃗a</i><sup>*</sup><sub>j</sub>⋅<i> ⃗a</i><sub>i</sub> = <i> ⃗a</i><sub>i</sub>⋅<i> ⃗a</i><sup>*</sup><sub>j</sub> = <i>δ</i><sub>ij</sub> the reciprocal lattice can only be <i>P</i>-type. When constructing a reciprocal lattice cell, in addition to the specified requirement (<i>a</i><sup>*</sup><i>α</i><sup>*</sup>), an obligatory condition is necessary: the vertices of the cell should be nodes with non-zero “weight” which is determined by the value of the structural amplitude. In this case, the linear parameters of cells will be <i>n</i> times more than parameters <i> ⃗a</i><sup>*</sup><sub>j</sub>. If the cell volume of the reciprocal lattice is equal to <i>V</i><sup>*</sup>, then the performance of these requirements the choice of parameters of the supercell (<i>a</i><sup>*</sup><sub>ся</sub>)<sub>j=1,2,3</sub>, the volume of the supercell is equal (<i>V</i><sup>*</sup><sub>ся</sub>) = <i>nV</i><sup>*</sup>. The “weight” of the node of the reciprocal lattice can be determined experimentally by X-ray diffraction. From the reflex intensity (<i>I</i>) it is possible to determine the structure factor <i>F</i><sup>2</sup> where <i>F</i><sup>2</sup> = <i>I</i>/<i>PLg</i> (<i>PLg</i>) is a multiplier that depends on the angle of diffraction. Depending on the cell of the Bravais lattice of a crystal “weight” of the individual nodes of the reciprocal lattice can be equal to zero. In this case, it is necessary to go to the supercell, the parameters of which exceed the estimated to an integer number of times. The volume of the supercell is also larger than the volume of the reciprocal lattice cell constructed on the basis <i> ⃗a</i><sup>*</sup>, <i> ⃗b</i><sup>*</sup>, <i> ⃗c</i><sup>*</sup> of <i>n</i> times, where <i>n</i> is an integer. For Bravais cells, space of the crystal is <i>P</i>-type, <i>n</i> = 1, <i>I</i> and <i>F</i> types, <i>n</i> = 8. For diamond, <i>n</i> = 64. The article is intended for specialists in theoretical and applied crystal physics.