Calculate the planar packing fraction (fractional area occupied by atoms) of the (110) plane of aluminum at 300K
0.554 atm/cm
Calculate the linear packing density (atoms/m) in the [100] direction of Al at 300K.
2.47⋅10^9 m
The planar packing fraction refers to the fractional area occupied by atoms in a specific crystal plane. It can be calculated using the formula:
Planar packing fraction = (Number of atoms per unit area) x (Atomic radius)^2
To calculate the planar packing fraction for the (110) plane of aluminum at 300K, we need to know the number of atoms per unit area and the atomic radius of aluminum.
The number of atoms per unit area in a crystal plane can be calculated using the formula:
Number of atoms per unit area = (Number of atoms per unit cell) / (Area of the unit cell)
The (110) plane of aluminum has a face-centered cubic (FCC) crystal structure. In an FCC structure, there are 4 atoms per unit cell.
The area of the unit cell for the (110) plane can be calculated based on the lattice constant of aluminum (a), which is 0.4049 nm.
Area of the unit cell = (a^2) / 2
Now we can calculate the number of atoms per unit area:
Number of atoms per unit area = (Number of atoms per unit cell) / (Area of the unit cell)
= 4 / ((0.4049 nm)^2 / 2)
Next, we need to know the atomic radius of aluminum. The atomic radius of aluminum is approximately 0.143 nm.
Now we can calculate the planar packing fraction:
Planar packing fraction = (Number of atoms per unit area) x (Atomic radius)^2
= (4 / ((0.4049 nm)^2 / 2)) x (0.143 nm)^2
≈ 0.4038
Therefore, the planar packing fraction of the (110) plane of aluminum at 300K is approximately 0.4038.
Moving on to the second part of the question, the linear packing density refers to the number of atoms per unit length in a specific crystal direction.
To calculate the linear packing density in the [100] direction of aluminum at 300K, we first need to know the lattice constant of aluminum.
The lattice constant of aluminum (a) is 0.4049 nm.
The [100] direction of aluminum in an FCC structure has one atom per unit cell.
Now we can calculate the linear packing density:
Linear packing density = (Number of atoms per unit length) x (Atomic radius)
The number of atoms per unit length in the [100] direction is equal to the inverse of the lattice constant:
Number of atoms per unit length = 1 / a
Linear packing density = (1 / a) x (Atomic radius)
= (1 / (0.4049 nm)) x (0.143 nm)
≈ 2.47 x 10^9 atoms/m
Therefore, the linear packing density in the [100] direction of aluminum at 300K is approximately 2.47 x 10^9 atoms/m.