Bragg Reflections: Calculating Conditions in hcp Crystal Lattice

[marie2010]

hi,
when calculating conditions for Bragg reflections one usually works with a (1x1) cell. For example, for the hcp crystal lattice forbidden reflections are given by the condition h+2k= 3n and l=odd. Does it mean that if we double the cell, let's say to (2x2) the condition is h+2k=(3/2)n. Can someone please help...
Thanks.

 

[Jacques_L]

hi,
when calculating conditions for Bragg reflections one usually works with a (1x1) cell. For example, for the hcp crystal lattice forbidden reflections are given by the condition h+2k= 3n and l=odd. Does it mean that if we double the cell, let's say to (2x2) the condition is h+2k=(3/2)n. Can someone please help...
Thanks.

False.
The Miller indexes of planes are covariant to the units of the crystalline cell.
If you double the cell, the same line is denoted by half the previous indexes of line (contravariant index), but the same plane is denoted by the double of the previous Miller indexes (covariant).
The lecture for these metric relations is at http://jacques.lavau.perso.sfr.fr/syntaxe3.htm or http://jacques.lavau.perso.sfr.fr/syntaxe3.pdf, but in french.

 

[marie2010]

hi,
thank you for the answer. So ( I just want to make sure I get it right), if we double or triple etc the unit cell, the condition for the forbidden reflection will be the same, i.e. h+2k=3n and l=odd and the conditions for the allowed reflections will not change as well.
So, only the intensity of the bragg peaks change since we have more atoms now, correct?
Thanks in advance.

 

[Jacques_L]

I did not carry you.

You whish to buy some fuel for your car. The price of the gas-oil is 1.25 €/l.
This figure "1.25" is contravariant to the currency unit, and covariant to the volume unit.
Now convert this price in pound per gallon.

Let's take a centered cubic lattice.
The atoms are at coordinates [0, 0, 0], [0, 0, 1], [0, 1, 0], [1, 0, 0], [1, 0, 1], [1, 1, 0], [0, 1, 1], [1, 1, 1], and [1/2, 1/2, 1/2].
If we double the mathematical cell, leaving unchanged the physical lattice, the coordinates of the atoms become [0, 0, 0], [0, 0, 1/2], [0, 1/2, 0], [1/2, 0, 0], [1/2, 0, 1/2], [1/2, 1/2, 0], [0, 1/2, 1/2], [1/2, 1/2, 1/2], [1/4, 1/4, 1/4] and so on... up to [1, 1, 1]
The coordinates of points or vectors are contravariant to the mathematical parameter.

The Miller indexes of planes behave just the reverse : covariant.
So the plane (1, 2, 0) is re-indexed as (2, 4, 0) in the new system of coordinates, with double the primitive units.

 

[Dickfore]

The Bragg reflection conditions are actually valid for a periodic, thus infinite crystal lattice. The choice of the unit cell should not change the position of the peaks. Of course, some indices may change due to a different choice of a basis. If you double the size of the cell, then you will double the number of atoms per unit cell. This would make the structure factor different and retrieve the same peak structure.

Also, I thought English was the only allowed language on these forums?! What's with the French references?

 

[Jacques_L]

Also, I thought English was the only allowed language on these forums?! What's with the French references?

I cannot have already translated into english years of previous works.
But an english translation of the russian authors Sirotine and Shaskolaskaïa is surely available in your University.

 

[marie2010]

Thank you all or merci beaucoup...

I get it now.