Understanding Crystal Symmetries & Dielectrics

[Seidhee]

Hi,

I am currently reading the Feynman Lectures on Physics, and I have just finished the chapter about the geometry and the symmetries of crystals, and there is something I do not quite understand.

There are 230 different possible symmetries which are grouped into seven classes (triclinic --> cubic).

My question is : why is a cubic crystal (for example) necessarily invariant under rotations, and thus necessarily an isotropic dielectric ?
Couldn't we imagine a cubic lattice but with an internal pattern which is not invariant under rotation (with x-axis oriented arrows for instance) ?

What I do not understand is that, for instance, we know there are cubic crystals which are not centrosymmetric, but Feynman says that every cubic crystal is invariant under rotations and thus isotropic dielectrics.
If I can find cubic pattern which are not centrosymmetric, I do not see why I couldn't find cubic patterns which are not invariant under such rotations...Thanks.

 

[gckaufman]

Hi siedhee!
I believe their are several misunderstandings in your interpretation of crystal symmetries and whatnot. First off, the 230 different symmetries refers to the 230 crystallographic space groups. Space groups themselves are not single symmetry operations but combinations of the 32 point groups (which themselves are composed of rotations and reflections) and the translation inherent inside the thirteen bravais lattices. Now because each point group is compatible with only a single bravais lattice, each of the 230 space groups are also only compatible with a single bravais lattice. This also applies in the reverse. Every bravais lattice has very specific point groups symmetries which they need to conform to.
So to answer your questions, by definition, a cubic crystal system needs to conform to very specific symmetries or else it is not a cubic crystal system. Just like a trigonal system, in addition to possessing specific and fixed definitions for the lengths and angles in the lattice, needs to possesses specific symmetry. Additionally, centrosymmetry does not imply invariance under rotation. Centrosymmetry just means there is an inversion center in the point group which is an entirely different symmetry operation from a rotation. So you can have a crystal system without inversion (centrosymmetry) which does not possesses a rotation.
Let me know if this helps!

 

[Seidhee]

Hi :)

Thanks for your explanations.

I completely know that a centrosymmetry cannot be obtained from rotations (in 3D), and thus these are two independent things, but I just talked about centrosymmetry because I know there are cubic crystals which are not centrosymmetric, but I wanted to know if there are also cubic crystals which are not invariant under rotations.

Indeed, maybe it is Feynman's definition of a cubic lattice which is not very clear (he changes very quickly to another subject), but basically what I understood was that the definition of a cubic crystal is to have three primitive vectors of the same length and at right angles to each other. Nothing more.
What I understood with such a definition for a cubic crystal lattice was that only translation symmetries were needed, not necessarily rotational symmetries ( because, like I said, translation symmetries, even at right angles, does not imply rotational symmetries; we could figure it out with a weird pattern ).

But, if I understood well what you said, you're telling me that in the definition of a cubic crystal we must necessarily add the rotational symmetries to the translations ?
If that is the case, all the rest makes sense : a cubic crystal is necessarily an isotropic dielectric, and the ellipsoid of the polarisability is a sphere.

But then, twhat about the zincblende, for example ? http://en.wikipedia.org/wiki/Cubic_...:Sphalerite-unit-cell-depth-fade-3D-balls.png

It is a cubic pattern, however we can see that this unit cell is not invariant under a rotation of Pi/2 about about the first vertical edge... The arrangement of the atoms are not exactly the same when we look at it from the front or from the left (the positions of the blue atoms are changed).

Thanks again :)

 

[DrDu]

While Feynman's definition is correct for a cubic lattice, it is not correct for a crystal for the very reason you noted, i.e., the basis may break the symmetry of the lattice.
However, this is a subtle point. If the basis were completely unsymmetric, then it would be a highly improbable coincidence that all basis vectors had equal length and 90 degree angles. So in practice you may ask what is the least symmetry a basis must have to force the lattice to be cubic. It turns out that this/ these symmetry(s) is(are) less than the full symmetry of the cubic lattice. Especially, the symmetry group does not contain inversion.