# Proving Greatest Density of Points in {111} & {110} Planes

• Chillguy
In summary, the lattice planes with the greatest densities of points are the {111} planes in a fcc bravis lattice and the {110} planes in a bcc bravis lattice. This can be determined by maximizing the value of d/v, where d is the spacing of planes and v is the unit volume. In the fcc case, the reciprocal space coordinates can be used to find the maximum value, while in the bcc case, the coordinates are slightly different but the same process applies.
Chillguy

## Homework Statement

Prove that the lattice planes with the greatest densities of points are the {111} planes in a fcc bravis lattice and the {110} planes in a bcc bravis lattice.

## Homework Equations

d/v=points per unit area where d is the spacing of planes and v is the unit volume.

## The Attempt at a Solution

In the fcc case
$$d=\frac{2\pi}{hb_1+kb_2+lb_3}\\ b_1=\frac{2\pi}{a}(y-x+z)$$
Which is in terms of the reciprocal space. So we simply need to maximize this value. Does this immediately tell us it is in the {111} miller index because the reciprocal is given in terms of 3 coordinates?

In the bcc case, we have d=\frac{2\pi}{hb_1+kb_2+lb_3}\\b_1=\frac{2\pi}{a}\big(\frac{x+y+z}{2}\big)And we see that the same process applies in finding the maximum.

## 1. What is the significance of proving the greatest density of points in {111} & {110} planes?

The greatest density of points in {111} & {110} planes is significant because it provides insight into the atomic arrangement and symmetry of a crystal structure. This information is crucial for understanding the physical properties and behavior of materials.

## 2. How is the greatest density of points in {111} & {110} planes determined?

The greatest density of points in {111} & {110} planes is determined by calculating the packing fraction, which is the ratio of the total volume occupied by atoms to the total volume of the unit cell. The plane with the highest packing fraction has the greatest density of points.

## 3. What is the relationship between the greatest density of points in {111} & {110} planes and crystallographic directions?

The greatest density of points in {111} & {110} planes is related to the crystallographic directions that are perpendicular to these planes. These directions are known as close-packed directions and are important for understanding the arrangement of atoms in a crystal lattice.

## 4. Why is it important to prove the greatest density of points in {111} & {110} planes?

Proving the greatest density of points in {111} & {110} planes is important for confirming the theoretical predictions of crystallography and for validating the models used to describe crystal structures. It also has practical applications in materials science and engineering.

## 5. What are some methods used to prove the greatest density of points in {111} & {110} planes?

Some methods used to prove the greatest density of points in {111} & {110} planes include X-ray diffraction, electron diffraction, and computer simulations. These techniques allow scientists to analyze the atomic arrangement and symmetry of crystal structures and determine the planes with the highest density of points.

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