# Vector Algebra for Diamond Unit Cell: Neighbouring Atoms and Bond Angles

• spaghetti3451
In summary, the unit cell of diamond consists of a cube with carbon atoms at each corner and at the centre of each face. The atoms at positions displaced by 0.25A(i + j + k) are determined by adding 0.25A to each of the original points. To find the vectors joining the atom at 0.25A(i + j + k) to its four nearest neighbours, one can roughly draw out the points and determine the nearest neighbours visually. The angle between the carbon bonds in diamond can then be calculated using basic vector operations.
spaghetti3451

## Homework Statement

A unit cell of diamond is a cube of side A, with carbon atoms at each corner, at
the centre of each face and, in addition, at positions displaced by 0.25A(i + j + k) from each of those already mentioned; i, j, k are unit vectors along the cube axes. One corner of the cube is taken as the origin of the coordinates. What are the vectors joining the atom at 0.25A(i + j + k) to its four nearest neighbours? Determine the angle between the carbon bonds in diamond.

## The Attempt at a Solution

The atoms at the corners are (0,0,0), (A,0,0), (0,A,0), (0,0,A), (A,A,0), (A,0,A), (0,A,A), (A,A,A).

The atoms at the centre of each face are (0.5A,0.5A,0), (0.5A,0,0.5A), (0,0.5A,0.5A), (0.5A,0.5A,A), (0.5A,A,0.5A), (A,0.5A,0.5A).

I am having trouble figuring the atoms at positions displaced by 0.25A(i + j + k) from each of those already mentioned.

Any help would be greatly appreciated.

"positions displaced by 0.25A(i+j+k) from each of those already mentioned" this tells you the displacement vector from the points already mentioned to some new points. You have written down displacement vectors of the original points from the origin, so how would you work out the displacement vectors of the new points from the origin? Hint- what is one of the simplest operations you can do with two vectors?

Well, the trouble is figuring out if it's addition or subtraction, because both give displacements of 0.25A(i+j+k) from each of the original points. But, now I'm beginning to think 'displaced by' instructs you to add (by convention). So, the displaced positions are (0.25A,0.25A,0.25A), (1.25A,0.25A,0.25A), (0.25A,1.25A,0.25A), (0.25A,0.25A,1.25A), (1.25A,1.25A,0.25A), (1.25A,0.25A,1.25A), (0.25A,1.25A,1.25A), (1.25A,1.25A,1.25A), (0.75A,0.75A,0.25A), (0.75A,0.25A,0.75A), (0.25A,0.75A,0.75A), (0.75A,0.75A,1.25A), (0.75A,1.25A,0.75A), (1.25A,0.75A,0.75A).

So, do I have to use trial and error to find out the the vectors joining the atom at 0.25A(i + j + k) to its four nearest neighbours?

yes, 'displaced by' usually means addition, and you've got all the new points correct. You could use trial and error, but I think it is best to roughly draw out the points, then hopefully you will see which are the nearest neighbours.

## 1. What is a vector in the context of diamond unit cells?

A vector in the context of diamond unit cells is a quantity that has both magnitude and direction and is used to represent the position of atoms within the unit cell. It is typically represented by an arrow pointing from the origin of the unit cell to the position of the atom.

## 2. How are neighbouring atoms determined in a diamond unit cell?

In a diamond unit cell, neighbouring atoms are determined by the vectors connecting each atom to its nearest neighbours. These vectors are known as bond vectors and are used to calculate bond lengths and angles between neighbouring atoms.

## 3. What is the significance of bond angles in a diamond unit cell?

Bond angles in a diamond unit cell are significant because they determine the shape and stability of the crystal structure. The angles between neighbouring atoms can affect the strength of the bonds between them and can also impact the overall properties of the material.

## 4. How is vector algebra used in diamond unit cells?

Vector algebra is used in diamond unit cells to calculate the position of atoms, bond lengths, and bond angles. It involves the manipulation of vectors using mathematical operations such as addition, subtraction, and multiplication to determine the relationships between neighbouring atoms.

## 5. Can vector algebra be applied to other crystal structures?

Yes, vector algebra can be applied to other crystal structures as well. It is a fundamental concept in crystallography and is used to describe the positions of atoms in various crystal lattices. However, the specific calculations and equations may differ depending on the type of crystal structure being studied.

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