The Lattice and the sublattice and their importance to construct the magnetic structure
Introduction
The crystal structure is presented always by a lattice which be composed of one sublattice or more. There 14 Bravais lattices.
Check this link to know more about this subject Crystal lattice
Lattice and sublattice
The sublattice may be primitive P , centered B or face centred FC.
The primitive cell of elements is always composed of one sublattice. The centred one is composed of 2 sublattices B and the face centred one is composed of 4 sublattices P.
For the compounds, the crystal structure is always composed of more than one sublattice. For binary compounds with primitive cell, there are 2 sublattices P. For binary compounds with centered cell, there are 2 sublattices B or 4 sublattices P. For binary compounds with face centered cell, there are 2 sublattices FC, 8 sublattices P.
We can apply that for ternary and quaternary compounds.
A sublattice may be decomposed into 2 or more sublattices.
Examples:
1_ The NaCl structure is the superposition of two identical fcc Bravais sublattices shifted by 1/2 of the edges of their unit cell; one NaC .Cl / ion has 6 Cl .NaC/ nns along h100i directions.
2_ The CsCl structure is the superposition of two identical sc sublattices translated by 1/2 of the diagonal of their unit cell; one CsC .Cl / ion has 8 Cl .CsC/ nns of the other sublattice along h111i directions. The symmorphic space group of CsCl is O1 h .Pm3m/.
3_ The fluorite .CaF2/ lattice is the superposition of a fcc sublattice of Ca++ ions with a sc sublattice of F ions. The lengths of the edges of the unit cells of the Ca++ and F- sublattices are in the ratio of 2 to 1, respectively, and the F- unit by 1/4 along the diagonal of the Ca++ cubic cell. The CaF2 lattice is thus made of unit cells containing each four Ca++ ions and eight F ions.
Reference: Crystal structures
Sublattice and the construction of the magnetic structure
Example 01: We take the example of the element of Chromium Cr
The lattice of Chromium Cr is of type B. We can decompose this lattice into 2 sublattices P shifted by the translation (1/2,1/2,1/2). We will get a new structure with 2 inequivalent atoms: Cr1 (0,0,0) and Cr1 (1/2,1/2,1/2).
This new structure with a new space group which is a subgroup of the parent space group of the first structure. We can use this new structure for the magnetic calculation : FM or AFM. The configuration of the magnetic calculation will be done in the case.inst file .
Example 02: We take the example of the binary compound Nickel Oxide NiO
The lattice of NiO is of type NaCl structure which is composed of 2 sublattices FCC. The compound consists of 2 atoms Ni and O , and we will be interested only to the magnetic atom Ni. We can either decompose every sublattice FCC into 4 sublattices P or only the sublattice of the magnetic atom.
The problem is the sublattice of type FCC lies in the fact that an phenomenon called geometrical frustration (to learn about this phenomenon click here ) occured. This phemonenon states that there is no AFM configuration possible, and to solve the problem, the nature forces the atomic arrangement to get a structural distortion. There are 3 types of structural distortion: tetragonal distortion along the direction [001], orthorhombic distortion along the direction [011] and the rhombohedral distortion olong the direction [111]
In the case of NiO, the distortion is rhombohedral, and we have to make deviation of the 3 angles from the ideal value of 90 to 89.99 and proceed to get the new structure and after do a relaxation.
To know how to get the new magnetic structure of NiO check this link magnetic structure of NiO
Example 03: We take the example of the heusler compound
In the example of the NiO, the geometrical frustration (manifested by the degeneracy of the magnetic configurations) is removed by the structural distortion. In this case with the compound of GdPtBi, the geometrical frustration is removed by the strong fourth-nearest neighbor interaction along the cube diagonal. Read this article ( here )to learn about that.
We will add other examples later
Sublattice and the construction of the magnetic structure
Example 01: We take the example of the element of Chromium Cr
The lattice of Chromium Cr is of type B. We can decompose this lattice into 2 sublattices P shifted by the translation (1/2,1/2,1/2). We will get a new structure with 2 inequivalent atoms: Cr1 (0,0,0) and Cr1 (1/2,1/2,1/2).
This new structure with a new space group which is a subgroup of the parent space group of the first structure. We can use this new structure for the magnetic calculation : FM or AFM. The configuration of the magnetic calculation will be done in the case.inst file .
Example 02: We take the example of the binary compound Nickel Oxide NiO
The lattice of NiO is of type NaCl structure which is composed of 2 sublattices FCC. The compound consists of 2 atoms Ni and O , and we will be interested only to the magnetic atom Ni. We can either decompose every sublattice FCC into 4 sublattices P or only the sublattice of the magnetic atom.
The problem is the sublattice of type FCC lies in the fact that an phenomenon called geometrical frustration (to learn about this phenomenon click here ) occured. This phemonenon states that there is no AFM configuration possible, and to solve the problem, the nature forces the atomic arrangement to get a structural distortion. There are 3 types of structural distortion: tetragonal distortion along the direction [001], orthorhombic distortion along the direction [011] and the rhombohedral distortion olong the direction [111]
In the case of NiO, the distortion is rhombohedral, and we have to make deviation of the 3 angles from the ideal value of 90 to 89.99 and proceed to get the new structure and after do a relaxation.
To know how to get the new magnetic structure of NiO check this link magnetic structure of NiO
Example 03: We take the example of the heusler compound
In the example of the NiO, the geometrical frustration (manifested by the degeneracy of the magnetic configurations) is removed by the structural distortion. In this case with the compound of GdPtBi, the geometrical frustration is removed by the strong fourth-nearest neighbor interaction along the cube diagonal. Read this article ( here )to learn about that.
We will add other examples later
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