Question

Determine the boundaries of the first Brillouin zone for potassium which has a bcc structure and a lattice constant of 5.32 Å.

Answer 1

Answer on Question #84461 - physics - Thermodynamics

1. Determine the boundaries of the first Brillouin zone for potassium which has a bcc structure and a lattice constant of 5.32 Å.

Answer: Calculate the boundaries of Brillouin zone for potassium as follows:

The direct lattice boundaries are given as:

$$a _ {1} = \frac {a}{2} (1, 1, 1)$$ $$a _ {2} = \frac {a}{2} (- 1, 1, 1)$$ $$a _ {3} = \frac {a}{2} (- 1, - 1, 1)$$

Here $a$ is the lattice constant.

Substitute $5.32A{}^{\circ}$ for a.

$$a _ {1} = \frac {5 . 3 2 A {}^ {\circ}}{2} (1, 1, 1) = (2. 6 6, 2. 6 6, 2. 6 6) A {}^ {\circ}$$ $$a _ {2} = \frac {5 . 3 2 A {}^ {\circ}}{2} (- 1, 1, 1) = (- 2. 6 6, 2. 6 6, 2. 6 6) A {}^ {\circ}$$ $$a _ {3} = \frac {5 . 3 2 A {}^ {\circ}}{2} (- 1, - 1, 1) = (- 2. 6 6, - 2. 6 6, 2. 6 6) A {}^ {\circ}$$

Calculate the reciprocal of lattice vector by the relation as follows:

$$b _ {1} = \frac {2 \pi}{a} (1, 0, 1)$$ $$b _ {2} = \frac {2 \pi}{a} (- 1, 1, 0)$$ $$b _ {3} = \frac {2 \pi}{a} (0, - 1, 1)$$

Here $a$ is the lattice constant.

Substitute 5.32 $A{}^{\circ}$ for a

$$b _ {1} = \frac {2 \pi}{a} (1, 0, 1) = \frac {2 (3 . 1 4)}{5 . 3 2 A {}^ {\circ}} (1, 0, 1) = (1. 1 8, 0, 1. 1 8) / A {}^ {\circ}$$ $$b _ {2} = \frac {2 \pi}{a} (- 1, 1, 0) = \frac {2 (3 . 1 4)}{5 . 3 2 A {}^ {\circ}} (- 1, 1, 0) = (- 1. 1 8, 1. 1 8, 0) / A {}^ {\circ}$$ $$b _ {3} = \frac {2 \pi}{a} (0, - 1, 1) = \frac {2 (3 . 1 4)}{5 . 3 2 A {}^ {\circ}} (0, - 1, 1) = (0, - 1. 1 8, 1. 1 8) / A {}^ {\circ}$$

Answer:

$$a _ {1} = (2. 6 6, 2. 6 6, 2. 6 6)$$ $$a _ {2} = (- 2. 6 6, 2. 6 6, 2. 6 6) A {}^ {\circ}$$ $$a _ {3} = (- 2. 6 6, - 2. 6 6, 2. 6 6) A {}^ {\circ}$$ $$b _ {1} = (1. 1 8, 0, 1. 1 8) / A {}^ {\circ}$$ $$b _ {2} = (- 1. 1 8, 1. 1 8, 0) / A {}^ {\circ}$$ $$b _ {3} = (0, - 1. 1 8, 1. 1 8) / A {}^ {\circ}$$