Thank you Roberto,
that's interesting. Here is some bunch of formulae:
http://sepwww.stanford.edu/data/media/public/docs/sep125/jim1/paper_html/node10.html
Still, I don't understand the formula
P = B \delta V / V
in two aspects: 1) what is \delta? A difference? Between what and what?
2) which pressure (uniaxial? hydrostatic? some mixture?) is to monitor
as one varies the volume, keeping c/a.
About inner degrees of freedom in hcp 2-atom cell I still don't understand
either, nor about "distortion of the basal plane" as one uniformly scales
all the cell dimensions.
Whatever; this is far from the Akshu's original question.
Best regards,
Andrei
>
> Hi Andrei and Akshu
>
>> My excuses,
>> I strongly disagree with Roberto's opinion:
>>
>
> Well, not that much ;-)
>
>>> The closest thing to bulk modulus in hcp is obtained by varying
>>> all the cell dimensions simultaneously and homogeneously and
>>> monitoring the pressure while doing that. Then P = B \delta V / V.
>>
>> The definition of bulk modulus (B) known to me is
>> B = -V dP/dV.
>> As the structure is not cubic, varying cell dimensions simultaneously
>> will introduce stress (the relaxed cell won't like to have the same
>> c/a for different volumes). I think the right thing to do is
>> to vary TARGET PRESSURE, let cell parameters free and monitor the
>> VOLUME,
>> then use the formula above.
>> Or - probably better - you extract elastic constants,
>> checking back their definition in hcp, as indepent parameters.
>> Because they may behave differently; the bulk modulus is just
>> some averaged combination of them, and won't tell you much,
>> for a serious test.
>>
>
> The formula I quoted, P = B \delta V / V, will give you just that
> average of elastic modulii Andrei's mentioning
> B = 1/9 {2C11+2C12+4C13+C33}
> This, I seem to recall, is called Voigt's average. Namely one where
> an average deformation is imposed.
>
> Andrei's point of view is an alternative possibility that corresponds
> to impose an average stress (currently a pressure, Reuss average).
>
> Clearly, the former method requires fixed cell while the latter
> variable cell. I believe, however, that my method is a bit better
> for numerical calculations, because it is easier (or more exact) to
> control lattice dimensions than pressure.
>
>
>>>> should i fix the atomic positions?
>>>
>>> No. The hcp 2-atoms cell possesses inner degrees of freedom,
>>
>> How's that??? I always thought there is something like
>> (0 0 0) and (1/3 2/3 1/2), no internal coordinates.
>> These relative coordinates should not change
>> (they will in fact, slightly, due to lack of symmetry constraint
>> in Siesta).
>>
>
> A distortion of the basal plane will generally produce a relative
> displacement between the two atoms of the cell.
> This is because the atomic positions are not centrosymmetric.
> I agree though, that for a uniform expansion/contraction this will
> not happen.
>
> Best,
>
> Roberto
>
>
