Last Modified 21 May 2001

The purpose of this page is to develop and annotate the procedures we need to use to calculate the phonon spectrum for hexagonal close-packed (A3) 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 a 76 point mesh. This is the mesh known as the Regular Order (6,3) mesh described in the pre-defined hexagonal 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 six phonon eigenstates, as there are two atoms in the primitive hcp unit cell. Each eigenstate is characterized by a phonon frequency and a six-dimensional polarization vector (one 3-dimensional vector for each atom). The frequencies may be degenerate, but the polarization vectors will be orthogonal in the six-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 fcc 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.
  • For Titanium, we'll sum over the electronic states in the Brillouin zone using the Regular Order (6,6) k-point mesh, or its equivalent for the current structure. This mesh may not be converged for a some problems. 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/c), 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 hexagonal close-packed cell, with primitive vectors a1 = ( ½ a , - ½ 3½ a , 0 ) a2 = (  ½ a ,  ½ 3½ a , 0 ) a3 = ( 0 , 0 , c ) reciprocal lattice vectors b1 = (2 π/a) ( 1 , -3-½ , 0 ) b2 = (2 π/a) ( 1 ,  3-½ , 0 ) b3 = (2 π/c) ( 0 , 0 , 1 ) and basis vectors B1 = 2/3 a1 + 1/3 a2 + 1/4 a3   = ½ a X -½ 3-½ a Y + ¼ c Z B2 = 1/3 a1 + 2/3 a2 + 3/4 a3   = ½ a X +½ 3-½ a Y + ¾ c Z

The First Brillouin zone of a hexagonal lattice, with high symmetry k points marked. The irreducible part of the hcp Brillouin zone is highlighted. All of the k-points in the table below are within or on the boundaries of this zone.

With all of that out of the way, here is the listing of the 76 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.

In the table below,

• Index is the k-point number,
• Lattice Coordinates are the coordinates (l₁,l₂,l₃) of the k-point, i.e.,

  k = l₁ b₁ + l₂ b₂ + l₃ b₃    ,

• 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. (Many points have no designation.) The values of ``x'' listed for some points will be explained on the appropriate pages.
• 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 the GGA version of our tight-binding parametrization of Titanium. Our calculations are compared to the experimental data collected in C. Stassis et al., Phys. Rev. B 19, 181 (1979).
• 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 Titanium in its experimental room temperature hcp structure, with lattice constants, a = 2.95Å = 5.575 Bohr, c = 4.685Å = 8.853 Bohr.

To start, we'll need the Titanium (GGA) Tight-Binding Parameters.

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.