Procedure

This experiment is about finding the primitive vectors for a given lattice. In theory, a lattice is infinite, but here we have a finite lattice. For a finite lattice with $N$ sites, the Bravais lattice definition is changed slightly as $$R = n_{1}a_{1} + n_{2}a_{2} + n_{3}a_{3}$$ where $0 \leq n_{1} < N_{1}$, $0 \leq n_{2} < N_{2}$, $0 \leq n_{3} < N_{3}$ and $N = N_{1}N_{2}N_{3}$
The experiment contains 6 lattices which are square, rectangular, simple cubic, body centered cubic, face centered cubic and honeycomb lattice. The first two lattices are planar lattices, meaning they only extend in two dimensions. It is to be noted that the number of primitive vectors $d$ is equal to the number of dimensions of the lattice and hence the first two lattices have only two primitve vectors.

Finding primitive vectors

Primtive vectors can usually be found with ease for simple lattices, usually the vectors being the coordinate axes themselves. For the more complex lattices, viewing the vectors as translations will give a more clear picture. The vectors picked should encorporate lattice points in all directions from the origin. Satisying this condition alone helps identify the primtive vectors for the most part. Once the directions have been set, find the unit distance between lattice points in that direction by finding the closest lattice point in that direction. The vector now so obtained is the primitive vector that is required.

The steps to the experiment are given below

• The interface will contain a canvas in which the different finite lattices in question are shown.
• The interface contains a dropdown to select lattice, a button to select atoms (which is a toggle switch), a button to clear your choices and a button to check the choice made.
• In the select mode, left click can be used to select an atom and right click can be used to de-select an atom.
• It is expected that atoms are selected sequentially as head and tail of primitive vectors of the lattice. It is okay to have a common atom as the tail for many vectors. It is expected that the right head atoms are chosen given a common tail, to create the primitive vectors.
• If the vectors obtained do not have a common origin, they are brought to a common origin during evaluation of the selection, retaining the validity of the selection if the magnitudes and directions of the vectors obtained are correct.
• Once selections are made in the above said order, the selections can be evaluated.
  The last lattice, the honeycomb lattice is a counter case to the Bravais lattice definition. It has been proven that the honeycomb lattice is **not** a Bravais lattice. Since there is no right combination of atoms that will give the primitve vectors, it can never be a Bravais lattice. It is left as an exercise for the user to prove why it is impossible to adhere to the Bravais lattice definition for the case of a honeycomb lattice.

Note

Some lattices have been showed with an atom radius that is smaller than the mathematically proven one to help with the visibility of the lattice points to the user.