how do I calculate volume of atoms in unit cell and volume of unit cell for plane triangular lattice?
To calculate the volume of atoms in a unit cell and the volume of the unit cell for a plane triangular lattice, you'll need to follow these steps:
1. Determine the structure: In a plane triangular lattice, the unit cell is hexagonal.
2. Identify the lattice parameters: The important parameters are the lengths of the two sides of the hexagon, denoted as a and c.
3. Calculate the area of the base: The hexagonal unit cell can be divided into two congruent triangles. The area of each triangle can be calculated using the formula: (1/2) * a * c * sin(60°). Since there are two triangles, the total area of the base is: 2 * (1/2) * a * c * sin(60°) = a * c * sin(60°).
4. Calculate the height of the unit cell: The height of the hexagonal unit cell can be found by multiplying the value of c by the cosine of the angle at one of the vertices, which is 120°. So, the height (h) is given by h = c * cos(120°).
5. Calculate the volume of the unit cell: The volume of the hexagonal unit cell can be calculated by multiplying the area of the base by the height. Therefore, the volume (V) is given by V = a * c * sin(60°) * c * cos(120°).
6. Calculate the volume of atoms in the unit cell: The volume occupied by an atom in the unit cell depends on the atomic packing. A common packing arrangement in a plane triangular lattice is the hexagonal close-packed (HCP) structure. In the HCP structure, each unit cell contains two atoms. The volume of atoms in the unit cell is therefore 2 times the atomic volume.
You can use these steps to calculate the volume of atoms in a unit cell and the volume of the unit cell for a plane triangular lattice. Just substitute the values of a and c into the formulas to get the final result.