Question about directions in unit cells?

Question

What is the practical purpose of vectors within crystal unit cells? What information do they provide and how is the formula writtien for them? Do vectors alo relate to moving from one unit cell to another? I am new to this stuff so basic responses would be appreciated.

Answer 1

A vector quantifies the direction and magnitude of a property. The velocity vector, for example, tells you how fast you're going and in what direction. In the case of unit cells, The vector describing the distance and relative direction of the center of one cell vs its neighbors defines the structure of the crystal.

Answer 2

A vector is simply a mathematical idea that means, essentially, "a magnitude and a direction". Vectors have these two properties.

In a crystal lattice, say a cubic structure like a salt crystal, you would have your three perpendicular unit vectors ("up", "left", and "forward") defining the three directions aligned with the atoms in the lattice. However, you could also have a vector pointed in some diagonal direction.

The use of this would be to describe how frequently one would encounter an atom (the lattice spacing) while traveling in a certain direction. If the size of the unit cell was 1 nanometer (making up this number), then traveling in the x-direction, you'd encounter an atom every 1 nanometers. However, if you were traveling in a direction 45 degrees away from this, you might encounter an atom only every 1.414 nanometers. So, vectors can be used to describe the spatial frequency at which atoms are found in the lattice.

The notation for a vector in 3D space is something like this:

-->
V

That is, a variable with a little arrow on top of it pointing sideways. You could also use bold text to indicate a vector. It depends on the choice of the author of the book.

Now, what it means:

-->
V = A*xhat + B*yhat + C*zhat

A vector in 3D space pointing in some random direction is really just the sum of three perpendicular components of different sizes. A, B, and C are the magnitudes of each component, and xhat, yhat, and zhat are the directions of each component. xhat is written as an x with a ^ on top of it (a "hat"). So, A*xhat means "go a distance A in the direction of x". The vector is just the sum of its components.

The total length of the vector is
L = sqrt(a^2 + b^2 + c^2)
which is just the Pythagorean theorem for 3D space.

If you had two legs of a right triangle represented by vectors:

---------------->
^
|
|
|

you the sum of the two vectors would simply be the hypotenuse of the triangle. This illustrates how a vector in a diagonal direction is really just the sum of two vectors (or three in 3D space) at right angles to each other.