[Pw_forum] Space Group Issue

[Paolo Giannozzi]

On Tue, Sep 19, 2017 at 9:37 PM, Amar Singh <amarsingh122014 at rediffmail.com>
wrote:


> I tried your suggestion, with approximate values of lattice parameters for
> rhombohedral setting (as I have lattice parameters for only hexagonal
> settings) [...] if the unit cell given by me has a volume ~ 64 A3 which
> fits one BaTiO3 unit, how can space group generate two units of BaTiO3 for
> same volume ??
>

maybe your approximate values of the lattice parameters are not well
approximated

​My other problem is that the CIF file which I have (true data reported
> from X-ray diffraction) has lattice parameters for hexagonal setting
> ​ a = b = 5.5 A, c = 13 A and alpha = beta = 90, gama = 120.
> ​Now for this unit cell, R3c should be able to generate 6 units of BaTiO3.
> ​How can I write my input script to give me this unit cell optimized with
> 30 atoms (6 + 6 + 18)
>

There are 1001 different ways to provide structural data to QE. For
instance, you can do what I did years ago for trigonal and hexagonal Al2O3
(see below): write down atomic positions in  crystal axis in the two cases,
write down the relation between the two set of axis.

Paolo
---
alpha-Al2O3:
R-3c (space group: 167, hexagonal axis) Z=6
positions Al: 12c  (0,0,z) (0,0,-z+1/2) (0,0,-z) (0,0,z+1/2)
positions  O: 18c  (x,0,1/4) (0,x,1/4) (-x,-x,1/4)
                   (-x,0,3/4) (0,-x,3/4) (x,x,3/4)
+ (0,0,0); (2/3,1/3,1/3); (1/3,2/3,2/3) (obverse setting) or
+ (0,0,0); (1/3,2/3,1/3); (2/3,1/3,2/3) (reverse setting)
z(Al)=0.35228, x(O)=0.3064
a=4.758, c=12.99

R-3c (space group: 167, rhombohedral axis) Z=2
positions Al: 4c  (z,z,z) (-z+1/2,-z+1/2,-z+1/2) (-z,-z,-z)
(z+1/2,z+1/2,z+1/2)
positions  O: 6e  (x,-x+1/2,1/4) (1/4,x,-x+1/2) (-x+1/2,1/4,x)
                  (-x,x+1/2,3/4) (3/4,-x,x+1/2) (x+1/2,3/4,-x)
z(Al)=0.35228, x(O)=1/4-0.3064=-0.0564
a=5.13, alpha=55.6 (cos=0.565)

Relation between rhombohedral A_r, B_r, C_r and hexagonal A_h, B_h, C_h:

A_r = 2/3 A_h + 1/3 B_h + 1/3 C_h       A_h = A_r - B_r
B_r =-1/3 A_h + 1/3 B_h + 1/3 C_h       B_h = B_r - C_r
C_r =-1/3 A_h - 2/3 B_h + 1/3 C_h       C_h = A_r + B_r + C_r

a_r = sqrt(3a_h^2+c_h^2)/3 ; sin(alpha/2)= 3/(2*sqrt(3+(c_h/a_h)^2)

Relation between rhombohedral axis with threefold axis around 001:

a=( s*sqrt(3)/2, -s/2, r)   b=( 0, s, r)   c = (-s*sqrt(3)/2, -s/2, r)

s = 2a_0/sqrt(3) * sin(alpha/2),  r = sqrt(a_0^2-s^2)

and around 111: a=(p,q,q), b=(q,p,q), c=(q,q,p)   (p<q)

r = (p+2q)/sqrt(3) ;  s = (q-p)*sqrt(3/2)
q = (r+s/sqrt(2))/sqrt(3) ;  p = (r-s*sqrt(2))/sqrt(3)

-- 
Paolo Giannozzi, Dip. Scienze Matematiche Informatiche e Fisiche,
Univ. Udine, via delle Scienze 208, 33100 Udine, Italy
Phone +39-0432-558216, fax +39-0432-558222
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