# Figuring out Bravais lattice from primitive basis vectors

• spaghetti3451
In summary, the primitive basis vectors of a lattice are ##\mathbf{a} = \frac{a}{2}(\mathbf{i}+\mathbf{j})##, ##\mathbf{b} = \frac{a}{2}(\mathbf{j}+\mathbf{k})##, ##\mathbf{c} = \frac{a}{2}(\mathbf{k}+\mathbf{i})##, where ##\mathbf{i}##, ##\mathbf{j}##, and ##\mathbf{k}## are the usual three unit vectors along cartesian coordinates, and the Bravais lattice is
spaghetti3451

## Homework Statement

Given that the primitive basis vectors of a lattice are ##\mathbf{a} = \frac{a}{2}(\mathbf{i}+\mathbf{j})##, ##\mathbf{b} = \frac{a}{2}(\mathbf{j}+\mathbf{k})##, ##\mathbf{c} = \frac{a}{2}(\mathbf{k}+\mathbf{i})##, where ##\mathbf{i}##, ##\mathbf{j}##, and ##\mathbf{k}## are the usual three unit vectors along cartesian coordinates, what is the Bravais lattice?

## The Attempt at a Solution

There are seven different crystal systems and fourteen different Bravais lattices.

The common length of the primitive unit cell is ##a##, so the crystal system is either cubic or trigonal (rhombohedral).

Furthermore, the basis vectors are oriented at ##90°## to each other, so the crystal system must be cubic.

Finally, the Bravais lattice is face-centred cubic, because, if the basis vectors originate from one corner of the primitive unit cell, then they point to lattice sites at the centre of three adjacent (to the corner) faces of the primitive unit cell.

It's an FCC. Why? (Hint: Draw out the vectors)

1. How many lattice points does FCC have, and what are their locations?
2. Using the info above, write out the lattice vectors.

In my mind, I translated from the origin (at the corner of one primitive unit cell) by the the lattice vector ##\mathbf{R} = n_{1} \mathbf{a} + n_{2} \mathbf{b} + n_{3} \mathbf{c}##, where I used ##(n_{1}, n_{2}, n_{3}) = (0,0,1),## ##(n_{1}, n_{2}, n_{3}) = (0,1,0),## and ##(n_{1}, n_{2}, n_{3}) = (1,0,0)##. Each time, I found that the system is invariant under translation.

You are over-complicating things.

1. All three lattice vectors have the same length - What does this tell you about its structure? (Cubic, orthorhombic, tetragonal ...)

3. Now draw 5 more neighbouring unit cells.

4. What is the bravais lattice type?

1. FCC has four lattice points. They are located one at the corner of a chosen unit cell, and the other three at the centres of each of the three faces which intersect at the corner.

2. The lattice vectors are ##\mathbf{R} = n_{1} \mathbf{a} + n_{2} \mathbf{b} + n_{3} \mathbf{c}##, where
##(n_{1}, n_{2}, n_{3}) = (0,0,0)##, ##(1,0,0)##, ##(0,1,0)##, and ##(0,0,1)##.

failexam said:
1. FCC has four lattice points. They are located one at the corner of a chosen unit cell, and the other three at the centres of each of the three faces which intersect at the corner.

Yes. Good. Now using the corner as (0,0,0) how do you get to the other three lattice points? Hence write down the lattice vectors.
failexam said:
2. The lattice vectors are ##\mathbf{R} = n_{1} \mathbf{a} + n_{2} \mathbf{b} + n_{3} \mathbf{c}##, where
##(n_{1}, n_{2}, n_{3}) = (0,0,0)##, ##(1,0,0)##, ##(0,1,0)##, and ##(0,0,1)##.
Wrong. You just told me the lattice points. What are the lattice vectors?

The lattice vectors are ##\mathbf{R} = \mathbf{0}##, ##\mathbf{R} = \mathbf{a}##, ##\mathbf{R} = \mathbf{b}##, and ##\mathbf{R} = \mathbf{c}##.

failexam said:
The lattice vectors are ##\mathbf{R} = \mathbf{0}##, ##\mathbf{R} = \mathbf{a}##, ##\mathbf{R} = \mathbf{b}##, and ##\mathbf{R} = \mathbf{c}##.

No. Using the corner as (0,0,0) how do you get to the other three lattice points? Hence write down the lattice vectors.

In that case, the three lattice vectors are ##\mathbf{R} = \mathbf{a}##, ##\mathbf{R} = \mathbf{b}##, and ##\mathbf{R} = \mathbf{c}##.

failexam said:
In that case, the three lattice vectors are ##\mathbf{R} = \mathbf{a}##, ##\mathbf{R} = \mathbf{b}##, and ##\mathbf{R} = \mathbf{c}##.

Oops. That's right. To ensure you get all the marks, draw out the cubes to show that the repeating motif is indeed an FCC.

## 1. What is a Bravais lattice?

A Bravais lattice is a mathematical concept used to describe the periodic arrangement of atoms in a crystal. It is characterized by a set of primitive basis vectors that define the unit cell of the crystal.

## 2. How do you determine the Bravais lattice from primitive basis vectors?

To determine the Bravais lattice from primitive basis vectors, you need to first identify the three basis vectors that define the unit cell. Then, you can use these vectors to determine the type of lattice by comparing them to the seven types of Bravais lattices (cubic, tetragonal, orthorhombic, monoclinic, triclinic, rhombohedral, and hexagonal).

## 3. What are primitive basis vectors?

Primitive basis vectors are the minimum set of vectors that can be used to define the unit cell of a crystal. They are typically chosen to be the shortest and most symmetric vectors, and they must have the same origin.

## 4. How many primitive basis vectors are needed to define a unit cell?

The number of primitive basis vectors needed to define a unit cell depends on the type of lattice. For example, a cubic lattice requires three primitive basis vectors, while a hexagonal lattice requires six.

## 5. Can you have more than one type of Bravais lattice for a crystal?

No, a crystal can only have one type of Bravais lattice. However, it is possible for a crystal to exhibit different symmetries, which can be described using different sets of primitive basis vectors.

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