What Is the Packing Fraction in a Simple Cubic Crystal Structure?

[Timiop2008]

Problem:
A)
In a simple cubic arrangement the atoms sit at the corners of imaginary cubes with their curved surfaces just touching. Show that the ratio of filled space to empty space between the atoms is pi/6 or about 0.5. This "packing fraction" is related to the density of the material. The volume of a sphere of radius r is 4/3 pi r³.

B)
If the atoms in the simple cubic arrangement in part A have diameter d, show that largest impurity atom that can just fit in between the host atoms has a diameter of 0.4d.

Please Help! I don't understand how to approach this Question!

 

[Kurdt]

This is really just about calculating volumes. If there are 8 spheres at the corner of a cube what fraction of the volume of each sphere is in the cube. Compare that to the volume of the cube.

 

[thomate1]

A) To calculate the packing fraction, we need to determine the volume of the atoms and the volume of the empty space between them. In a simple cubic arrangement, each atom is located at the corner of a cube and the distance between the centers of adjacent atoms is equal to the diameter of the atoms (d).

The volume of one atom can be calculated using the formula for the volume of a sphere (V = 4/3 πr³), where r is the radius of the atom. In this case, the radius is equal to half of the diameter (r = d/2). Thus, the volume of one atom is V = 4/3 π(d/2)³ = πd³/6.

The volume of the imaginary cube surrounding each atom is (d³), and since there are eight atoms at the corners of the cube, the total volume occupied by the atoms is 8πd³/6.

The remaining volume is the empty space between the atoms, which can be calculated by subtracting the volume occupied by the atoms from the total volume of the cube, which is d³. Thus, the empty space is equal to d³ - 8πd³/6 = d³(1 - 8π/6) = d³(1 - 4π/3).

The packing fraction is then equal to the volume occupied by the atoms (8πd³/6) divided by the total volume (d³). Simplifying this, we get 8πd³/6d³ = 8π/6 = 4π/3. This is equivalent to π/6 or approximately 0.5.

This means that in a simple cubic arrangement, 50% of the space is occupied by the atoms, while the other 50% is empty space between the atoms. This packing fraction is important in determining the density of the material, as a higher packing fraction means a higher density.

B) In order to determine the largest impurity atom that can fit in between the host atoms, we need to consider the space between the atoms. From part A, we know that the empty space between the atoms is equal to d³(1 - 4π/3).

To find the maximum diameter of an impurity atom that can fit in this space, we need to find the radius of the largest sphere that can fit in this volume. We can use the