Bravais Lattices and Symmetry Reduction
Okay, so youre probably wondering, can we categorize 3D Bravais lattices
like we did in 2D with those five familiar types The short answer is its
complicated. While there are infinite possible Bravais lattices in 3D, we can
classify them based on their symmetry, which is best understood by
examining the unit cell.
Think back to how we create a 3D Bravais lattice from a 2D one. The video
illustrates this nicely: We can start with a two-dimensional Bravais and
translate the whole lattice in the third dimension by a distance say c.
Imagine your favorite 2” lattice maybe hexagonal and then just stacking
copies of it straight up. Perfectly aligned, that’s one option. But, as the video
points out, instead of putting the second layer exactly above the first layer, I
could have placed it at an angle along the x axis or maybe I could have
instead done it along the y axis. See Tiny shifts in how you stack those layers
lead to entirely new lattices. The key takeaway By tweaking the lengths a, b,
and c of the unit cell and the angles between them, I can create infinite
different Bravais lattices in three dimension.
So while were not going to list out all those infinite possibilities that would be
a very long list, the fundamental principle remains: symmetry is key to
classification. The video emphasizes that the best possible way to figure out
the overall symmetry of a lattice is by thinking about its unit cell. Well tackle
that classification process in future videos, but for now, just remember that
3D lattices offer a massive landscape of configurations, all built from the
basic act of translation and variations in unit cell parameters.
Okay, so youre probably wondering, can we categorize 3D Bravais lattices
like we did in 2D with those five familiar types The short answer is its
complicated. While there are infinite possible Bravais lattices in 3D, we can
classify them based on their symmetry, which is best understood by
examining the unit cell.
Think back to how we create a 3D Bravais lattice from a 2D one. The video
illustrates this nicely: We can start with a two-dimensional Bravais and
translate the whole lattice in the third dimension by a distance say c.
Imagine your favorite 2” lattice maybe hexagonal and then just stacking
copies of it straight up. Perfectly aligned, that’s one option. But, as the video
points out, instead of putting the second layer exactly above the first layer, I
could have placed it at an angle along the x axis or maybe I could have
instead done it along the y axis. See Tiny shifts in how you stack those layers
lead to entirely new lattices. The key takeaway By tweaking the lengths a, b,
and c of the unit cell and the angles between them, I can create infinite
different Bravais lattices in three dimension.
So while were not going to list out all those infinite possibilities that would be
a very long list, the fundamental principle remains: symmetry is key to
classification. The video emphasizes that the best possible way to figure out
the overall symmetry of a lattice is by thinking about its unit cell. Well tackle
that classification process in future videos, but for now, just remember that
3D lattices offer a massive landscape of configurations, all built from the
basic act of translation and variations in unit cell parameters.
