Friday, February 27, 2009
Friday, February 20, 2009
Symmetry of crystals
Symmetry of crystals
Here we are going to summarize the symmetry of crystals, including the translational symmetry, rotational symmetry, the combination of them, and a lot of terminologies.
Translational symmetry:
Translational symmetry is the most fundamental symmetry of a crystal, or the definition of a crystal. In principle, a crystal may not have rotational symmetry, but will always have translational symmetry.
Translational symmetry is described by the primitive lattice, or the basis a1,a2,a3. However, they are not unique. One has may ways to choose a1,a2,a3, because by definition, as long as all the points in the lattice can be reproduced by n1a1+n2a2+n3a3, you have the basis. However, there are some rules for chosen primitive axis to unify the usage of them. (e.g. a1=i, a2=j, and a3=k).
Note that we want to avoid the term sc, bcc, fcc and etc. here because they contain information more than just translational symmetry. Let us emphasize that translational symmetry means only primitive axis. To exagerate, for every point on the lattice, we can put a potato. It will still be a lattice as long as the potato have the same orientations.
Primitive cell and unit cell
Primitive cells are the cells that has the smallest repeating
volume. The axes are defined by a1,a2 and a3. As mentioned above, the choice of primitive cells is infinite. Most of them do not reflect the symmetry of the lattice. For example, for fcc a1=(i+j)/2, a2=(k+j)/2, a1=(i+k)/2. This will generate a primitive cell that has no obvious symmetry.
It is always nice to choose a primitive cell to reflect the symmetry of the lattice. So called W-S cell is to do that. W-S cell has all the symmetry of the lattice.
Sometimes, for simplicity, people also choose unit cell that's larger than the primitive cell, just to retain the symmetry. For example, choose the cube for bcc.
Rotational symmetry:
If we talk about rotational symmetry alone, there are infinite number of rotational symmetry, for example, Cn, where n is an integer. However, since we are talking about crystal, we should focus on the rotational symmetries that are only allowed in crystals. (Note that, here, introduction of rotational symmetry alreay implies the coupling between translation and rotational symmetry.) It is well know only n=1,2,3,4,6 is valid for Cn in a crystal.
We have to keep in mind that this is only the rotational symmetry of the lattice (not the crystal). Only when the crystal have simple lattice, we have have the same rotational symmetry for the crystal too.
Lattice system and Bravais lattice
Continuing the discussion of rotational symmetry, we said n can only be 1,2,3,4,6. This combined with the translational symmetry, will only result 7 lattice systems and 14 Bravais lattices. Just to list them, they are
Triclinic
Monoclinic (s, bc 2)
Orthorhombic (s,bc,bc1,fc 4)
Tetragonal (s, bc)
Cubic (sc, bcc, fcc 3)
Trigonal
Hexagonal
Symmetry group (macroscopic symmetry)
So far, we just talked about the microscopic view of a crystal. Next, we need to talk about macroscopic symmetry, e.g. the symmetry of the physical properties such as dielectric constant.This has to do with the complexity of the lattice point.
Let's define:
simple lattice: lattice points are spheres (isotropic)
complex lattice: lattice points have structures (e.g. a potato)
To emphasize again, when we talk about the rotational lattice, we did not care about the structure of every lattice point. Here we have to because we care about the rotational symmetry now.
Obviously, for the same lattice, the macroscopic symmetry is very different, depending on if you put a sphere on lattice points or if you put a potato. For example, for sc lattice, simple lattice gives cubic macroscopic symmetry (Oh), complex lattice (potato) gives C1 macroscopic symmetry.
A more academic example is the comparison between diamond and ZnS. They both have fcc lattice. But diamond has Oh macroscopic symmetry and ZnS has Td.
----------------------------------------------------
Space group
Another effect of coupling the translational and rotational symmetry is the space group. When the unit cell has several atom of the same element, there can be screw axis, glide plane which are combination of point group operator and translations.
For example, for BiFeO3. If we don not consider the glide plane, the point group symmetry is C3. With glide plane, the point group will be C3v. the space group is R3c ={{C3|R}{C3v|τ+R}}.
Reciprocal space and Brillouvin zone.
The reciprocal lattice can be found by the definition:
bi=2πajXak/V, where V is the unit cell volume.
The Brillouvin zone is chosen to be the W-S primitive cell of the reciprocal space.
The symmetry of wave functions
Monday, February 16, 2009
Symmetry properties of wave functions in magnetic crystals
Symmetry properties of wave functions in magnetic crystals
| space operator | time reversal | Full space group | ||
| Paramagnetic | H | θ | G=H+Hθ | |
| magnetic (below TN) | ℋ | G'=ℋ+ℋa0 (a0=v0θ) | ||
Extra timer-reversal degeneracy
u belongs to ℋ
u={σ|τ}
σk=k+Kq
v0={ρ0|τ0}
ρ0k=-k+Kq'
Relation between representation of ℋ and G'
| case 1 | case 2 | case 3 | ||
| ℋk | Δ(i)(u) | Δ(i)(u) | Δ(i)(u), Δ(i)(v0-1uv0)*=Δ(j)(u) | |
| G'k | D(i)(u) | D(i)(u) | D(i)(u) | |
| Relation | D(i)(u) =Δ(i)(u) | D(i)(u) =2Δ(i)(u) | D(i)(u) =Δ(i)(u)+Δ(j)(u) | |
| degeneracy | no new degeneracy | degeneracy doubled | Δ(i)(u) and Δ(j)(u) are now degenerate | |
| Criteria: ∑χΔ(i){u'u} where u=u={σ|τ} u'=v0-1uv0 The sum is over σ, and there are M of them. The ni is the dimension of Δ(i)(u) ω=1 for single group, =-1 for double group | ωχ{Δ(i)(v02)}M/ni | -ωχ{Δ(i)(v02)}M/ni | 0 | |
| If v02={E|Rn} or {E'|Rn} where E=ωE | ωM | -ωM | 0 |
Sunday, February 8, 2009
Subscribe to:
Posts (Atom)