Identifying BBravais Lattice with vectors Given

[PsychonautQQ]

Homework Statement

Given that the primitive basis vectors of a lattice area (a/2)(I+J),(a/2)(j+k), (a/2)(k+i), where I j and k are the usual three unit vectors along Cartesian coordinates, what is the bravais lattice?

Homework Equations

The Attempt at a Solution

So just drawing the three vectors given gives the obvious. The first is on the z-origin and makes a 45 degree angel between the x and y axis's. The second vector is on the x-origin and makes a 45 degree angel between the y and z azis. The third vector is on the y-origin and makes a 45 degree angel between the z and x axis. How do I use these three lines to identify the type of bravais lattice?

 

[BruceW]

It seems like you have a good idea of how to visualize the lattice vectors. That's good, although I'm not sure about your terminology. I don't know what you mean by "vector is on the x-origin"... It seems like you mean "vector is perpendicular to the x-axis". OK, anyway, you have visualized them. So, do you have a table of the properties of the different Bravais lattices? you need this so you can compare the properties of your lattice vectors to the lattice vectors of the various Bravais lattices.

 

[PsychonautQQ]

It seems like you have a good idea of how to visualize the lattice vectors. That's good, although I'm not sure about your terminology. I don't know what you mean by "vector is on the x-origin"... It seems like you mean "vector is perpendicular to the x-axis". OK, anyway, you have visualized them. So, do you have a table of the properties of the different Bravais lattices? you need this so you can compare the properties of your lattice vectors to the lattice vectors of the various Bravais lattices.

Cool, yeah I have a table of different properties. It looks like A = B = C but I'm having trouble figuring out the angles between each vector... any tips?

 

[BruceW]

yep, I agree, the length of each side is the same. And you are told that it is primitive lattice. So now there are only a few possible options. And yes, the next thing to do is work out the angles. If you can visualize it very well in your head, then you could work out the relationship between the angles just by thinking of it. But maybe this is a bit tricky. So, otherwise, how would you work out the angle between two vectors? And then use this method to find the three angles.

hint: think of a mathematical operation between two vectors.

 

[paranormal]

The given vectors form a primitive basis for a lattice, meaning that they are the smallest set of vectors that can be used to generate the entire lattice. This implies that the lattice is a primitive lattice, meaning that the basis vectors are not repeated or duplicated to form the lattice points.

To identify the type of Bravais lattice, we need to consider the symmetry of the lattice. In this case, we can see that the lattice has a 45 degree rotational symmetry about its origin, as each of the basis vectors are rotated by 45 degrees from the previous one. This suggests that the lattice is a tetragonal lattice, specifically a primitive tetragonal lattice, as the basis vectors are not repeated or duplicated.

We can also consider the lattice parameters, which are the lengths of the basis vectors. In this case, all three basis vectors have the same length of a/2, indicating that the lattice is cubic. However, as the basis vectors are not parallel to the Cartesian axes, it is not a conventional cubic lattice. Instead, it is a body-centered tetragonal lattice, where the basis vectors are rotated by 45 degrees from the Cartesian axes.

In summary, the given vectors form a primitive body-centered tetragonal lattice, with a lattice parameter of a/2.