# Crystallographic planes and square pyramids

• comicnabster
In summary, the conversation discusses the topic of miller indices and crystallographic directions in materials science. The question asks for the aspect ratio of an AFM silicon tip in the shape of a square pyramid, with each face being a (111) silicon plane, etched out of a [100] silicon wafer. The discussion also mentions the diamond crystal structure of silicon and its relationship to the lattice constant and atomic radius. The solution involves using vector notation and the Pythagorean theorem to relate the [100] vector to the (111) planes.
comicnabster

## Homework Statement

In materials science right now we are learning about miller indices and crystallographic directions, including planes.

What is the aspect ratio (height/width) of an AFM (atomic force microscope) silicon tip in the shape of a square pyramid where each face of the pyramid is a (111) silicon plane? The silicon tip is etched out of a [100] silicon wafer. The questions says that for a [100] wafer the [100] direction points normal to the surface.

## Homework Equations

The book doesn't give any actual equations, only the miller indices definitions, which match what is given on http://en.wikipedia.org/wiki/Miller_index Basically each digit within the brackets represents one of three directions the plane takes, but not the actual value.

Silicon has a diamond crystal structure that is actually two FCC crystal structures offset along the vector (a/4, a/4, a/4) where a represents the lattice constant.

## The Attempt at a Solution

I don't have a good grasp of the crystallographic direction material, unfortunately.

I think the tip of the pyramid relative to the center of the base can be expressed by the miller indices [100] in terms of vector notation. After that it would be a matter of relating the [100] vector to the (111) planes with the Pythagorean theorem.

Would I interpret the (111) plane as having a height of 1 in the z direction and the base having length 1 for both the x and y direction? Then the height would be 1/(sqrt2) (read it as 1 over root 2). But then it doesn't seem to connect with the [100] normal vector, which confuses me.

P.S. I have also derived that in silicon's crystal structure the atomic radius and lattice constant can be related by the equation r = (a(3)^0.5)/8 (read it as "a root 3 over 8"), but I don't get how this is relevant to the question, however the information was provided so I think it relates somehow.

Never mind, solved.

## 1. What are crystallographic planes?

Crystallographic planes are imaginary surfaces within a crystal lattice that are defined by the arrangement of atoms in the crystal structure. These planes can be thought of as layers that make up the crystal.

## 2. How are crystallographic planes represented?

Crystallographic planes are represented by Miller indices, which are a set of three numbers that denote the relative position of the plane within the crystal lattice. The Miller indices are written in square brackets, with no commas between the numbers.

## 3. What is the significance of crystallographic planes?

Crystallographic planes are important in understanding the properties and behavior of crystals. They can affect the crystal's mechanical, electrical, and optical properties, and can also determine how the crystal will fracture or cleave.

## 4. What is a square pyramid in crystallography?

In crystallography, a square pyramid is a geometric shape formed by four triangular faces and a square base. It is often used to describe the arrangement of atoms or molecules in a crystal structure.

## 5. How are square pyramids related to crystallographic planes?

Square pyramids can be used to represent certain crystallographic planes in a crystal structure. The number of sides on the square base corresponds to the Miller index of the plane, and the height of the pyramid represents the distance between adjacent planes in the crystal lattice.

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