[Pw_forum] rotational ASR in nanotube

[Stefano Baroni]

On Aug 25, 2004, at 12:22 PM, WU Gang wrote:

> Hello All.
>
> Recently I calculated the phonon-dispersion curves for nanotubes, but
> I found that at Gamma point, there is always some imaginary phonon
> frequencies, even when I applied translational acoustic summing rule
> (ASR) on the final force constant matrix. After look up some
> reference, a so-called rotational ASR seems to be important in reduce
> these unstable frequencies.

Nice problem of physics!

>  But What does this rule mean in nanotube?

It simply means that you cannot pay any energy to twisting a tube 
around its axis.
In free space, this is trivially rotational invariance. With periodic 
boundary conditions,
of course, such a twist would correspond to a non-trivial periodic mode 
which has got
zero frequencies. The same, by the way would hold or any molecule.

> And how can I apply this rule to the result force constant matrix?

Once you understand what physically this "rotational ASR" means,
it should not be difficult to figure out how to impose it onto the 
dynamical matrix that
you calculate. We would leave this to you as an exercice (Hint: figure 
out which atomic
displavement pattern would correspond to a rigid twist of the tube and 
see how the ASR is
imposed in the translational case).

> Thank you very much!

You are most welcome!

Stefano Baroni

---
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