Last Modified 12 July 2002

The purpose of this page is to develop and annotate the procedures we need to use to calculate the phonon spectrum for body-centered cubic (A2) structure solids using the frozen phonon approximation and an appropriate total energy calculation program. In the examples we will use the NRL static program.

Note that this is more a collection of notes than a tutorial. So you can expect that as we go on you'll find that write-up at each k-point in each structure will change. In addition, the descriptions will probably become more and more terse with time. For these reasons, I'd recommend that you go through these pages in the order in which they are written. At the moment, that means start with the bcc lattice, going through the k-points in order.

In the supercell approximation phonon frequencies can only be determined at those wave vectors which lead to reasonably sized supercells of the primitive lattice. Thus we'll derive procedures to calculate the phonons on an 55 point mesh. This is the mesh known as the Regular Order-8 mesh described in the pre-defined bcc k-point mesh page. However, there are some differences that must be noted:

• The current calculation is in the reciprocal space of lattice vibrations, not the space of electron momentum. In fact, we will see that there are two type of k-point meshes involved.
• The first, described on this page, is the k-point mesh of the phonons. At each phonon k-point there will be three phonon eigenstates, as there is only one atom in the primitive bcc unit cell. Each eigenstate is characterized by a phonon frequency and polarization vector. The frequencies may be degenerate, but the polarization vectors will be orthogonal in a three-dimensional space.
• The second k-point mesh is that needed for the calculation of the total energy of a given structure by summing over the eigenvalues of the occupied states in the Brillouin zone. This is a reciprocal space of electron crystal momentum variables. In a typical bcc material, there is one s orbital, three p orbitals, and five d orbitals for each atom, and so nine electronic eigenvalues at each k-point in the primitive unit cell.
  • In all calculations shown on this we will use only the k-point mesh equivalent to the Regular Order-8 k-point mesh to sum over the Brillouin zone. We'll provide alternate k-point sets so that you can check convergence yourself.
• The two reciprocal lattices represent different things, the space of phonon vibrations and the space of allowed electron eigenstates. So, even though both meshes might include the point (2π/a, 2π/a, 2π/a), we need to remember that the quantities coming out of the calculations are different.

The current calculations would not have been possible without the use of Prof. Harold Stokes' frozsl program, which determines the displacements needed to calculate frozen-phonon frequencies, and his findsym web page, which determines the space group of an arbitrary lattice+basis.

Of course,

This software and any accompanying documentation are released as is. The U.S. Government makes no warranty of any kind, expressed or implied, concerning this software and any accompanying documentation, including without limitation, any warranties of merchantability or fitness for a particular purpose. In no event will the U.S. Government be liable for any damages, including any lost profits, lost savings, or other incidental or consequential damages arising out of the use, or inability of use, of this software or any accompanying documentation, even if informed in advance of the possibility of such damages.

Remember that the primitive cell for all of this is the bcc unit cell, with primitive vectors a1 = ( -½ a , ½ a , ½ a ) a2 = ( ½ a , -½ a , ½ a ) a3 = ( ½ a , ½ a , -½ a ) and reciprocal lattice vectors b1 = (2 π/a) ( 0 , 1 , 1 ) b2 = (2 π/a) ( 1 , 0 , 1 ) b3 = (2 π/a) ( 1 , 1 , 0 )

The First Brillouin zone of a body-centered cubic lattice, with high symmetry k points marked. Note that this is also the Wigner-Seitz cell of a face-centered cubic lattice in real space.

With all of that out of the way, here is the listing of the 55 k-point mesh for the phonons. Clicking on a given k-point link will take you to a page showing how we do the calculations. You may note that the ordering is a bit unfamiliar. We've done this so that you can construct k-point meshes equivalent to the Order-0, Order-1, Order-2, Order-4 and Order-8 meshes of the bcc lattice. The ``Order'' of a mesh refers to the number of divisions made between the origin (Gamma) and the high-symmetry point H, at (0,2π/a,0).

In the table below,

• Index is the k-point number,
• Order is the smallest order k-point mesh which includes this point,
• Lattice Coordinates are the coordinates (l₁,l₂,l₃) of the k-point, i.e.,

  k = l₁ b₁ + l₂ b₂ + l₃ b₃    ,

• Cartesian Coordinates are the coordinates of the same point in the real reciprocal space,
• A Relative Weight is assigned to each point based on the number of equivalent points in the unit cell. When doing summations over the Brillouin Zone these weights should be normalized so that they sum to unity.
• Atoms indicates the number of atoms contained in the supercell.
• Notation is the designation of this point according to the scheme in the figure above. (Some points have no designation.)
• Clicking on the Index for a given k-point will take you to a page describing the calculations needed to determine the phonon frequencies at that point. The examples here use our tight-binding parametrization of Niobium. Our calculations are compared to the experimental data of
  • Y. Nakagawa and A.D.B. Woods, Phys. Rev. Lett. 11, 271 (1963) and
  • Y. Nakagawa and A.D.B. Woods, in Lattice Dynamics, Proceedings of the International Conference Held at Copenhagen, edited by R. F. Wallis (Pergamon Press, Inc., New York, 1963), pp. 39-48.
• This page is under construction, so not all k-points are hooked up at this time.

In the following pages, we will determine phonon frequencies of niobium at the experimental lattice constant, 3.30Å.

To start, we'll need the Niobium Tight-Binding Parameters.

We'll also need to note the location of the space group files directory.

Return to

• The Phonon Frequency Calculation Home Page;
• The static Home Page;
• The DoD Tight-Binding Molecular Dyamics Home Page;

  or to the
  
  

• NRL Code 6390 Home Page.