# Is there any difference between Bravais lattice and primitive lattice?

So far I just had the first lecture in solid state physics...I am still confused if there is any difference between Bravais lattice and primitive lattice?

Bravais lattices are those that fill the whole space without any gaps or overlapping, by simply repeating the unit cell periodically. The whole lattice can be generated starting from one point by operating the translation vector $$\vec{R}\ =\ n_1 {\bf{a_1}}\ +\ n_2 {\bf{a_2}}\ +\ n_3{\bf{a_3}}$$ where $(n_1,n_2,n_3)$ are integers and $({\bf{a_1}},{\bf{a_2}},{\bf{a_3}})$ are the basis vectors which span the lattice.

In three dimensions, there are exactly 14 types of Bravais lattices. A primitive lattice is one such example. A primitive lattice is generated by repeating a primitive unit cell, which contains a single lattice point.

An example of a primitve unit cell in three dimension would be a cube with lattice points only at the vertices. Now, each of these lattice points is shared by 8 unit cells (4 above, 4 below). So, in each unit cell, one lattice point is counted as $\frac{1}{8}$. Thus, the total no. of lattice points in each unit cell is $\frac{1}{8}\times 8\ =\ 1$

• You just wrote what is written in wikipedia.. It didnt give me a clear picture of the difference between those two lattices.. How do I differentiate Bravais lattice from Primitive lattice? – Zen Sep 5 '16 at 17:47
• Why is a cube centered on a base is not a bravais lattice? Translation vectors applied repeatedly can cover all the lattice points.. Still a base centered cubic is not a bravais lattice (as what i read in wikipedia).. Why is that so? – Zen Sep 5 '16 at 18:08
• The main point is that all primitive lattices are Bravais lattices, not the other way round. Also, base centered cubic IS a Bravais lattice. In the base centered cubic, apart from the 8 atoms at the corners, you have one centered at two bases. So, there are $\frac{1}{2}\times2\ +\ \frac{1}{8}\times 8\ =\ 2$ atoms in each unit cell, unlike a primitve unit cell where you have only 1. That's why a base centered cubic is not a primitive lattice (also called simple cubic), but another kind of Bravais lattice. Just wrote whatever little I could learn in my undergraduate days. Hope this helps! – Sucheta Sep 6 '16 at 5:14
• However, in the base centered cubic lattice, the atoms placed at the corners may be different from those placed at the center of the bases. That's why I chose not to refer to them as lattice points. – Sucheta Sep 6 '16 at 5:15

This can often get confusing as physicists and mathematicians tend to have different definitions for lattices, Bravais lattices, unit cells and so on. In physics, we often use lattice to refer to any periodic [1] packing, while we use Bravais lattice to refer to mathematical lattices, namely [2]:

A Lattice is an infinite set of points defined by integer sums of a set of linearly independent primitive basis vectors.

Now I guess by primitive lattice, you meant primitive unit cells (let me know if otherwise). Unit cells [2]:

A unit cell is the repeated motif which is the elementary building block of the periodic crystal.

And a primitive unit cell [2]:

A primitive unit cell for a periodic crystal is a unit cell containing only a single lattice point.

[1]: A periodic packing can be thought of as a union of many translates of a lattice.

[2]: Simon, Steven H. The Oxford solid state basics. OUP Oxford, 2013.