Unit cell dimensions of crystal structure

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Discussion Overview

The discussion revolves around the selection of specific crystallographic planes, particularly the (111) plane, for determining the unit cell dimensions of a cubic crystal structure, specifically NiAl. Participants explore the implications of using different planes in the context of Bragg's law and d-spacing calculations, as well as considerations for non-cubic structures.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why the (111) plane is used for calculating the unit cell dimensions instead of other planes.
  • Another participant asserts that any plane (e.g., (100), (110), (111)) can be used to find the lattice parameter in a cubic structure, as they yield the same result.
  • A participant inquires whether a different plane would need to be selected if the crystal structure were not cubic.
  • There is a discussion about whether the formula for d-spacing changes for non-cubic structures and what aspects of the formula would be affected.
  • One participant explains that for non-cubic orthogonal axes, the formula for d-spacing includes the lattice parameters a, b, and c, while in cubic structures, these parameters are equal, simplifying the formula.
  • There is speculation about why certain planes might be omitted in discussions of Bragg peaks, particularly in BCC structures, and whether these omitted planes yield the same d-spacing results.
  • A later reply challenges the assumption that (100) planes in BCC have the same d-spacing as (110) planes, suggesting that the structure factor may play a role in determining which planes contribute to Bragg peaks.

Areas of Agreement / Disagreement

Participants express differing views on the selection of planes for calculating unit cell dimensions and the implications of using different planes in non-cubic structures. The discussion remains unresolved regarding the specific reasons for omitting certain planes in the context of BCC lattices.

Contextual Notes

Participants note that the formula for d-spacing changes depending on the crystal structure and whether the axes are orthogonal. There is uncertainty about the structure factor and its influence on the presence of Bragg peaks for different planes.

MightyQuinn11
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Hello all,

I already have the answer to my question, but what I would like to know is why. My question is: why in the question below is the plane 111 used to find ao and not another plane?

An ordered compound of NiAl has a cubic structure with one formula unit/cell. One Al atom at the origin and one Ni at 1/2 1/2 1/2. The nine lines of lowest bragg angle on a diffraction pattern have d-spacing of: 4.05, 2.86, 2.34, 2.03, 1.81, 1.65, 1.43, 1.35, 1.28.

QUESTION: Evaluate the unit cell dimensions of NiAl.

the formula to use is: dhkl=ao/sqrt(h^2+k^2+l^2)

The answer is: ao= sqrt(3)*2.34= 4.05A

Thank you for any clarification on this topic.
 
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The formula you have comes form the unit cell being cubic. Using that formula you can find the lattice parameter using any of the planes (100, 110, 111, 200, 210, etc). They will all give you the same lattice parameter for the unit cell.

The easiest way would be to use the (100) plane seeing as a0 = sqrt(1)*4.05 = 4.05 A.
 
Thank you, this is very useful. Now, if it was not cubic would I have to select another plane? Maybe a plane that contains a peak?
 
If it were not cubic, would you still have the same formula for the d-spacing? If not, what part of it would change?
 
It comes from the same formula, for orthogonal axes the formula is 1/d^2=h^2/a^2+k^2/b^2+l^2/c^2. Since cubic has a=b=c, b and c are not needed in the formula reducing it to the formula below in my previous post. Now, if it is not orthogonal, the formula changes. I cannot find all of the formulas for the other structures.

I am now thinking the reason they omit certain planes is because they would give the same result? For example, in BCC the planes 100, 010, 001 and 110 give the same d-spacing result? If not, why do they omit the first 3 planes only listing 110?

Correct me if I am wrong and help with any underlying physics that i am not clear enough on. Thank you for any help.
 
MightyQuinn11 said:
It comes from the same formula, for orthogonal axes the formula is 1/d^2=h^2/a^2+k^2/b^2+l^2/c^2. Since cubic has a=b=c, b and c are not needed in the formula reducing it to the formula below in my previous post. Now, if it is not orthogonal, the formula changes.
Correct.

I am now thinking the reason they omit certain planes is because they would give the same result? For example, in BCC the planes 100, 010, 001 and 110 give the same d-spacing result? If not, why do they omit the first 3 planes only listing 110?
You can check very easily that in a bcc lattice (or most any lattice, for that matter), the (100) planes have a different d-spacing than the (110) planes. There is a different reason that the (100) family of planes do not contribute Bragg peaks in the bcc lattice. Do you know what the structure factor is, and have you calculated it for different lattices? If not, you should look that up.
 

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