Bravais Lattices: Crystal Structure and Unit Cell Volumes

[raintrek]

Homework Statement

A crystal has a basis of one atom per lattice point and a set of primitive translation vectors of

a = 3i, b = 3j, c = 1.5(i+j+k)

where i,j,k are unit vectors in the x,y,z directions of a Cartesian coordinate system. What is the Bravais lattice type of this crystal and what are the volumes of the primitive and conventional unit cells?

Homework Equations

Primitive unit cell volume V = a . (b x c)

The Attempt at a Solution

I'm slightly unsure about these Bravais lattices given the multiple permutations they can seem to take.

My assumption, as $$a=b\neq c$$ is that it's Hexagonal. However that also requires that $$\alpha=\beta=90^{o},\gamma=120^{o}$$, where gamma is the angle between a,b, alpha between b,c, beta between c,a. But that seems to contradict that the a,b vectors are in i,j directions, ie at 90 degrees. Am I missing something here!?

I've worked out the primitive unit cell volume to be 13.5, however I'm also at a loss how to calculate the conventional unit cell volume...

Any help would be hugely appreciated

 

[Jhero]

!

Thank you for your post. It seems like you are on the right track with your calculations. However, there are a few things that need to be clarified.

Firstly, the given translation vectors do not necessarily mean that the crystal is hexagonal. The Bravais lattice type is determined by the arrangement of the lattice points, not just the translation vectors. In this case, the crystal has a primitive cubic structure, as there is one atom per lattice point and the translation vectors are all of equal length and at right angles to each other.

Secondly, the angles between the translation vectors are not necessarily the angles between the corresponding unit vectors in the Cartesian coordinate system. The angles between the translation vectors are determined by the crystal's symmetry, which in this case is cubic, and not by the coordinate system.

As for the conventional unit cell volume, it is simply the primitive unit cell volume multiplied by the number of atoms in the conventional unit cell. In this case, the conventional unit cell contains 8 lattice points, each with one atom, so the conventional unit cell volume would be 8 times the primitive unit cell volume.

I hope this helps clarify your understanding of this crystal's structure and unit cell volumes. Keep up the good work in your studies of crystallography!