Tuesday, January 3, 2012

Mikan stacking fault

Never leave a crystallographer with a pile of fruits.

HCP(left)-FCC(right) stacking fault
Even if I do not consider myself a crystallographer, I spent most of my PhD identifying local symmetries in materials. And the pioneers of the land of symmetries in materials are the crystallographers, so I owe them most of my analysis tools ... and the tetris-o-philia.

The FCC side, note how the 3rd layer's fruits sit on top of voids in the first layer
During new year's vacation in my family in law, my daughter (in the background of the upper picture) started emptying the reserve of mandarin to bring them on the living room's table. This got me started at piling the fruits.
The HCP side. 1st layer and 3rd layers sit on top of each other
The pile begins by a layer where the fruits form hexagons. This is the most compact way of packing disks of the same size in 2D, and thus spheres of the same size on the same plane. Real fruits have different sizes, but anyway.

For the second plane, you have two possibilities that are mirror image of each other, a translation, a rotation of 30 degree, etc. In short, this is not a real choice because you have no reference point.

The same alternative has richer consequences in the third plane. Depending on your choice, you end whether with
  • 3rd layer's fruits sitting on top of 1st layer's fruits
  • 3rd layer's fruits sitting on top of 1st layer's voids
Because you have the first layer as a reference, the choice is no more silent. You end with two different crystals: Hexagonal compact (HCP) and Face-centered cubic (FCC).

FCC is left, HCP right, and the grain boundary in the middle (hole in the 3rd layer)
Now let's be messy. I made two different choices of 3rd layer in two different places. The line where the two stacking meet is a stacking fault. It is not possible to pack same-size spheres efficiently on this line. Evidence is the hole you can see on the pictures.
FCC is right, HCP left, and the grain boundary in the middle (hole in the 3rd layer)
Finally, I added a fourth layer, with fruits sitting on top of the ones of the 2nd layer. No fault in this fourth layer, I can continue my stack if I want. However the fruits immediately over the fault line have a little more space to rattle. The same holds for the fruits immediately below (2nd layer).

If you think about the stacking of a crystal of hard spheres, the price paid to have such a stacking fault is the spheres you could not fit in because of the line. In my very small crystal, I could have fit 2 more spheres without the fault. This is a global penalty.

On the other hand, the space gained to rattle by the spheres neighbouring the fault line increases their (vibrational) entropy and thus decreases their free energy. This is a local gain.

When you balance the global penalty with the local gain, you end up with quite a lot of stacking lines.

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