Reciprocal lattice diffraction

[big man]

Question 1 — Construct the crystal lattice from the diffraction pattern drawn on page 5 of this exam paper making sure you include the (110) and (220) planes. Explain the procedure used in reconstructing the crystal lattice. What Bravais lattice is represented by the diffraction pattern

Image: http://img329.imageshack.us/img329/4036/reciprocal1bm8.jpg

Problem — I don’t know where to start on this one at all. I don’t know how the reciprocal lattice vectors will be defined and even if I did, I don’t know how to translate this to the crystal lattice (for this case). Yes I do know that the same equations apply as with the case below, but it's not clear since I had some conflicting information from this site here:

http://www.chembio.uoguelph.ca/educmat/CHM729/recip/6reci.htm

Alos, is this pattern of a rhombohedral structure?

Question 2 — Construct the crystal lattice from the diffraction pattern drawn on page 5 of this exam paper making sure you include the (440) and (220) planes. Explain the procedure used in reconstructing the crystal lattice.

Image: http://img141.imageshack.us/img141/4531/reciprocal2lk5.jpg

Problem — With this question I know how to define the reciprocal lattice vectors and from the equations I can see that the crystal lattice vectors are in the same direction. The only thing I’m not sure about is the meaning behind the fact that there are no in-between planes (the set of planes with indices of 1 and 3). This construction of the crystal lattice was a stab-in-the-dark attempt and I don't know the why behind it (I really want to know how and why I'm doing something).

If someone could please take a look at these questions I'd really appreciate since I can't find any information on these types of problems on the internet. Our notes also don't cover it because the lecturer believes that we should be able to research things properly...

Thanks

 

[OlderDan]

See if this helps. Go to this site, and just to be sure it starts in the right mode click on the <001> view button.

http://Newton.umsl.edu/~run/nano/reltutor2.html

I think that your first diagram corresponds to the right side view, with intensities represented by the shading. From the lables in your diagram, you can find the in-plane reciprocal lattice vectors, but there is no direct information about the third reciprocal lattice vector. What your diagram is showing is that unlike the pattern at the web page, there is no intensity at reciprocal-space coordinates 100 or 020 etc. The question I think becomes what third reciprocal lattice space vector will cause the intensity at those points to vanish? And I think the answer is available at the website in the form of a button labeled "add body centering atoms...".

At the top of the page are some links to models of other structures. You can change the views to rotate the crystal, watch the intensity pattern change as you spin the cystal, etc. All pretty neat if you ask me.

 

[big man]

Thanks for replying.

Unfortunately that site keeps timing out so I can't look at it.

I used this site:

http://www.cem.msu.edu/~cem924sg/Reciprocal.html

The FCC of (111) has the same pattern as what you said you would get from a body-centred lattice.

Anyway using that definition of the actual lattice vectors I came up with this for the crystal lattice:

http://img152.imageshack.us/img152/7461/reciprocal3ii2.png

But I don't know if you can do what I did. I actually think that what I should have done is drawn the (100) plane in the place of the (200) plane since the (100) plane will still be in the crystal and it just doesn't diffract. Then the (200) plane will be drawn vertically through the middle of the crystal lattice (intersecting the (220) plane at its half point). So essentially if I did that you'd probably have more of slanted window shape??

Can you tell if this is even close to right though? Sorry for asking this, but it's the only thing that I really can't get and I'd like to know it for my exam on Monday.

 

[OlderDan]

Thanks for replying.

Unfortunately that site keeps timing out so I can't look at it.

I used this site:

http://www.cem.msu.edu/~cem924sg/Reciprocal.html

The FCC of (111) has the same pattern as what you said you would get from a body-centred lattice.

Anyway using that definition of the actual lattice vectors I came up with this for the crystal lattice:

http://img152.imageshack.us/img152/7461/reciprocal3ii2.png

But I don't know if you can do what I did. I actually think that what I should have done is drawn the (100) plane in the place of the (200) plane since the (100) plane will still be in the crystal and it just doesn't diffract. Then the (200) plane will be drawn vertically through the middle of the crystal lattice (intersecting the (220) plane at its half point). So essentially if I did that you'd probably have more of slanted window shape??

Can you tell if this is even close to right though? Sorry for asking this, but it's the only thing that I really can't get and I'd like to know it for my exam on Monday.

I am uncertain how to interpret the numbers labeling the spots in the diagram, and my recollection of this stuff is not what it once was, but I'll throw it out there for what it is worth. If any of this makes sense to you, maybe it will steer you in the right direction.

If you construct a primitive cell in BCC (one atom per primitive cell) you are faced with the non-orthogonal set of primitive vectors from one corner of a cube into three nearest neighbor cube centers. The reciprocal lattice vectors to this set is the FCC set with primitive reciprocal vectors from the corner of a cube into the centers of three faces. The reciprocal lattice to BCC is FCC and vice versa.

An alternative to this view for BCC is to keep the simple cubic structure with two atoms per unit cell, with one atom at the corner of the cube and a second atom at the cube center. Similarly, for FCC you can have the SC structure with one atom at the corner and three additional basis vectors from the corner to the atoms in three faces for a total of 4 atoms per unit cell. This is referred to as a simple cubic latice with a basis. The analysis of the diffraction intensities is simplified by introducing a Structure Factor to account for the effects of the additional atoms in the unit cell.

My interpretation of the diagram is that the numbers represent the coordinates in reciprocal space of a SC lattice, so the reciprocal space vectors are orthogonal. The "missing" dots are a manifestation of the structure factor of the unit cell. For the BCC structure, all the right dots are missing.

http://en.wikipedia.org/wiki/Structure_factor

For an FCC structure, if the diagram indeed represents the coordinates of the reciprocal lattice vectors, then only the dots that have all three coordinates even should be showing (since the last coordinate, 0, is alwasy even).