# Hexagonal System 194 - crystallography

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1. Oct 9, 2014

### atomtm

Hello everyone !
I am currently reading a paper about crystal properties of R2CuIn3 where R=Tb,H0, Er.
It states :
1.that based on bibliography they are crystallized in the hexagonal system P6/mmc. With the use of POWLS software , it was found that there was a second phase Cu2In , so its intensities where subtracted.
2.There are two intensities reversed in (114) and (212) meaning that I114 >>I212.
3. The reversing of these intensities was accomplished by describing an equivalent hexagonal system (194) with (a=b,2c) , where Tb,Ho,Er is at (0,0,0.25) and CU,IN is at (1/3,2/3,z)

My question is about 2 and 3 ? What does it mean that the intensities where reversed and that to eliminate this an equivalent system was proposed with Cu,In at (1/3,2/3,z) . (Who's z? Z is dead :) ) But really what is z for and how can a hexagonal system be equivalent by changing z?

2. Oct 9, 2014

### M Quack

Hi and welcome to PF

2) They used the published crystal structure and calculated the intensities. Apparently that give I(114) < I(212), whereas the measurements give the opposite

2a) They concluded that the published crystal structure is incorrect.

2b) They propose a different but related crystal structure that is obtained by stacking two hexagonal unit cells on top of each other,
hence (a=b, 2c). The space group remains the same, #194. The word "equivalent" is out of place here, imho.

3) They then optimized the match between measured and calculated intensities by fitting the atomic positions.
(0,0,25) is a "fixed" position that is fully determined by symmetry.

(1/3,2/3,z) on the other hand is not fully determined by symmetry. The atoms can shift along the c-axis without modifying the symmetry of the unit cell or of the atomic position. Therefore "z" remains a free parameter that needs to be found by optimizing the agreement between calculation and measurement.

3. Oct 10, 2014

### atomtm

Isn't that the crystal is described by one unit cell repeated ? What is the meaning of putting one unit cell on top of each other ? Do you create a new unit cell?

4. Oct 10, 2014

### M Quack

Yes and no. You start out by stacking two identical unit cells on top of each other to create a new, larger unit cell. Within this new unit cell you can then shift around, add, and remove atoms in a way that was not possible before (because of symmetry) in the old, smaller unit cell.

5. Oct 10, 2014

### atomtm

Ok now I get it!

Can you tell me more about the z positions?
I know that the unit cell can be described by a,b,c and α,β,γ but never thought of the fact that you can change the x,y,z of the atoms positions - these should be placed like a motif in every lattice point right? so how can you decide how the motif is structured?

I have lots of questions , give me the green light and I keep posting :)
Once again I really appreciate your help

6. Oct 10, 2014

### M Quack

Try taking a look at the space group listings in the International Tables for Crystallography.

If you have a position with a free parameter like "z" in this case, that means that you have to have several atoms at related positions in order to keep the space group symmetry. These are called Wyckoff positions, in this case the would be named "4f". The value of z for your crystal should be listed in the paper, probably in a table with all the other relevant parameters.

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list

7. Oct 10, 2014

### atomtm

I currently reading about symmetry operations so that maybe I can understand Wyckoff positions.
Are there any good resources for further studying?

8. Oct 10, 2014

### atomtm

By the way I am following this tutorial: http://www.doitpoms.ac.uk/tlplib/crystallography3/structure.php .
The last animation is about drawing a crystal structure ( ZnS) .
There it says that to draw a crystal you need to know : The crystal system , the Lattice type and the Motif.
and also the lattice parameters .
My question is very basic and that is , how do you find these parameters using crystallography in first place . I mean how can you extract such data?

9. Oct 10, 2014

### M Quack

The crystal system and lattice parameters you find from the observed scattering angles, i.e. d-spacings. Systematically absent reflections tell you the space group. The "motif" or atomic positions within the unit cell you can only find by quantitatively evaluating the peak intensities, e.g. as described above.

Sometimes this leaves ambiguities. In that case you make models for the different possible structures and compare how well each one fits the observed data.

10. Oct 10, 2014

### atomtm

Thank you very much ! Thing are getting more clear now

11. Oct 12, 2014

### atomtm

Another quick question , that I can't find an answer to .
The paper states inter-metallic compounds of rare earth and noble metals with elements of 14,15 group of the periodic table crystallize in the hexagonal crystal system with space group P63/mmc . Their structures belong to the binary structure type of CuIn2. (What does this mean ?)
On these crystal structures the 2 atoms of rare earth are always on 2b(0,0,1/4) and are on the same plane. The other elements are on 4f with the analogy of 1:3 and are on the c axis .

Please forgive my ignorance , on this basic knowledge but I am trying to speed things up , cause of lack of time . And please never mind my bad english .

Last edited: Oct 12, 2014
12. Oct 12, 2014

### M Quack

There are certain "prototype" crystal structures. The most famous ones are the Diamond and Rock Salt structures.

So rather than saying that Silicon crystallizes in the FCC space group Fd-3m with atoms on the 2a site, people just say Silicon crystallizes in the Diamond structure. For experts that know the whole telephone book of possible prototypes by heart (I am not one of them :( ) that provides an instant picture of the crystal structure. For us normal humans that means more literature research.

Without having looked up the details, CuIn2 appears to be one of these prototype structures.

13. Oct 12, 2014

### atomtm

Great thanks!

14. Oct 12, 2014

### M Quack

Not really. I'd google it, maybe on scholar.
But you already have the space group and the atomic positions. The lattice parameters will change with the composition anyways, so there is not all that much you can learn from the prototype structure. Imho it would be more interesting to plug the values into some graphing program and make some 3D views of the thing.

15. Oct 12, 2014

### atomtm

Ok ! Any software you know of?

16. Oct 12, 2014

### M Quack

I draw structures "by hand" using POV-ray. Mostly because I know POV-ray well, and because I usually add planes, vectors, etc into the image that crystallography software cannot do (to the best of my knowledge).

Maybe someone with more hands-on experience in crystallography can point you to a package that is more straight-forward to use.

17. Oct 12, 2014

### atomtm

Ok thank you M Quack !
I really appreciate your help !
So if anyone knows a software for crystal structure modeling please let me know

18. Oct 13, 2014

### atomtm

I found Vesta as a software to visualize crystals .
The problem is that I am not still confident with the free parameter z and the coordinates of equivalent positions that would give me a visual result (Any simple example please ??)
The parameters that I have are : a = b= 4.7028 , c = 7.3790 .
Positions for Tb : 1(a) , x: 0 , y:0 , z: 0.250
Cu : 2(d) , x:0.333 , y : 0.667 , z:0.02960
In : 2(d) , x:0,333 , y: 0.667, z :0.4560

I followed a tutorial for NaCl on youtube where there have to be 4 atoms for Na and 4 for Cl declared in structure parameters . My guess is that there has to be some sort of periodic translations for my case ( using equivalent positions ? ) for constructing Tb2CuIn3 .
Anyone with experience in Vesta that could give me some advice ??