Lattice vectors Bravais

LDA methods have been employed to investigate solids in two types of approaches. If the solid has translational symmetry, as in a pure crystal, Bloch s theorem applies, which states that the one-electron wave function n at point (r + ft), where ft is a Bravais lattice vector, is equal to the wave function at point r times a phase factor ... 

The first sum is over cells, the of which is specified by the Bravais lattice vector Rj, while the second sum is over the set of orbitals a which are assumed to be centered on each site and participating in the formation of the solid. If there were more than one atom per unit cell, we would be required to introduce an additional index in eqn (4.68) and an attendant sum over the atoms within the unit cell. We leave these elaborations to the reader. Because of the translational periodicity, we know much more about the solution than we did in the case represented by eqn (4.58). This knowledge reveals itself when we imitate the procedure described earlier in that instead of having nN algebraic equations in the nN unknown... 

Note that we have presumed something further about the solution than we did in our earlier case which ignored the translational periodicity. Here we impose the idea that the solutions in different cells differ by a phase factor tied to the Bravais lattice vector R . The result is that the eigenvector Ua has no dependence upon the site i we need only find the eigenvectors associated with the contents of a single unit cell, and the motions of atoms within all remaining cells are unequivocally determined. Substitution of the expression given above into the equations of motion yields... 

As one of our central missions is to uncover the relation between microscopic and continuum perspectives, it is of interest to further examine the correspondence between kinematic notions such as the deformation gradient and conventional ideas from crystallography. One useful point of contact between these two sets of ideas is provided by the Cauchy-Bom rule. The idea here is that the rearrangement of a crystalline material by virtue of some deformation mapping may be interpreted via its effect on the Bravais lattice vectors themselves. In particular, the Cauchy-Bom mle asserts that if the Bravais lattice vectors before deformation are denoted by Ej, then the deformed Bravais lattice vectors are determined by e = FEj. As will become evident below, this mle can be used as the basis for determining the stored energy function W (F) associated with nonlinear deformations F. 

If we consider a set of platinum atoms (for example) along a line, so that the 1-dimensional Bravais lattice vector is R na, where a is the platinum interatomic distance and n an integer, the harmonic potential energy has the form [2]... 

Let us consider a set of metal atoms of 1-dimensional configuration so that the Bravais lattice vector is a set of R = na, a is the metal interatomic distance, and n an integer as stated above, the harmonic potential energy, t/harm> is... 

In the strict sense, (4.162) is not an idempotency relation, because summation is carried out only over the vectors R, lying within the LUC, whereas the vector difference (i ° — i ) can lie outside the LUC. If we perform summation over aU Bravais lattice vectors, the right-hand side of (4.162) will diverge, because /J°(i2°) does not vanish at infinity. 

We can construct vectors which connect all equivalent points in reciprocal space, which we call G, by analogy to the Bravais lattice vectors defined in Eq. (3.1) ... 

As a final step, we replace the summations over the positions of ions in the entire crystal by summations over Bravais lattice vectors R, R and positions of ions t/, tj within the primitive unit cell (PUC) ... 

Since for all Bravais lattice vectors R and all reciprocal lattice vectors G we have exp( iG R) = 1, the expression for the Madelung energy takes the form... 

Here, as in the discussion of the Smoluchowski effect, we have made expHdt reference to the discrete, periodic stracture of the lattice. The symmetry ofa periodic lattice is substantially lower than the continuous translational and rotational symmetry of the uniform jeUium model. Only translations involving a Bravais lattice vector and a discrete number of point symmetry operations remain and the constant potential has to be replaced by a crystal potential, which reflects the lowered symmetry. This has profound consequences for the electronic structure. 

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