[Pw_forum] Bulk modulus of trigonal structures

[Michael Mehl]

Finding the bulk modulus wth ev.x is essentially calculating

B(V) = V E''(V).

What the heck is E(V)?  That's actually rather simple:  E(V) is the minimum
energy of the the system at fixed volume.  For your system,

E(V) = min eng(V,c/a)     for all values of c/a at fixed volume V,

where eng(V,c/a) is the energy you get from the relax mode for given values
of V and c/a.

For your tetragonal system, you want E(V) for several different volumes.
Here, V = a*a*c, or a*a*c/2 if you've got a body-centered system.

So:

1) Pick a volume V and initial value c/a.

2) Use pw.x's relax mode to find the equilibrium positions of the atoms for
this cell.  Save the energy.

3) Adjust c/a -- keeping the volume fixed -- until you find the minimum.
That is, the lowest energy eng(V,c/a) for all values of c/a at this
particular volume.  Plot eng(V,c/a) versus c/a at fixed V if that helps you
see what's going on.  There will be at least one minimum, somewhere, for
any reasonable choice of volume.

4) Repeat for enough volumes to get a good number from ev.x.  Ideally your
volumes should bracket the equilibrium volume.  Again, plotting helps.

5) Use the pair (V,E(V)) as input to ev.x.  Out will come the equilibrium
volume and bulk modulus.

Note that your procedure only gives you the minimum energy structure at one
volume -- equilibrium.

This is ''easily'' generalized to other types of lattices, though you might
want to look at the cell_dofree='shape" option for that.

Alternatively, you can compute the bulk modulus from the elastic constants,
but for a tetragonal system this is not trivial -- read up on it in the
literature.  I have the formula in several of my papers from the early
1990s, but unfortunately I don't have the references on this computer.

On Sat, Jun 10, 2017 at 2:50 PM, Nadire Nayir <nadire.nayr42 at gmail.com>
wrote:

> Dear all,
> I deal with the cell optimization and the equation of states of the
> trigonal structures (especially, GeO2). To get the optimized structure, I
> started from the experimental structure and performed a series of the relax
> calculations with "relax" keyword to it. During the relax calculations,
> first, I changed celldm(1)=a and celldm(3)=c/a parameters by keeping c
> parameter constant and I found the optimized celldm(1), after that, I kept
> celldm(1) fixed at its optimized value and I started to change only
> celldm(3) parameter until getting the optimized c parameter.
>
> After getting the optimized celldm(1) and celldm(3), I calculated the
> equation of states of the structure by applying SCF calculation and
> finally, I used ev.x to calculate the bulk modulus with the non-cubic
> option, which is ~140 GPa.
>
> However, the bulk modulus value I found is so different from the
> experimental value which is 42 GPa.
>
> When I applied the same optimization process to the cubic structures
> (Si-diamond and Ge-diamond), I achieved the almost same results with their
> experimental values.
>
> I did not understand why this procedure does not work for the trigonal
> structures. I wonder, is there anything that I missed?
>
> By the way, I also applied vc-relax calculation to the trigonal structure,
> again, I got the larger bulk modulus than the experimental one!
>
> I would greatly appreciate if you could kindly help me with this problem.
>
> Best regards,
> Nadire
>
>
> --
> Nadire Nayir
> Research Assistant
> Middle East Technical University
> Physics Department
>
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>



-- 

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Michael J. Mehl, Ph.D.

Kinnear Chair in Physics

The United States Naval Academy

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mmehl at usna.edu

Library of Crystallographic Prototypes <http://aflow.org/CrystalDatabase/>
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