Question

A square lattice has a side of 3.2 Å. Calculate the momentum and energy of an electron whose wave terminates at the boundary of the first Brillouin zone.

Answer 1

Answer on Question #84469 - physics - Thermodynamics

1. A square lattice has a side of 3.2 Å. Calculate the momentum and energy of an electron whose wave terminates at the boundary of the first Brillouin zone.

Answer:

Calculate the energy of the free electron by the relation as follows:

$$E = \frac {h ^ {2} \pi^ {2}}{m a ^ {2}}$$

Here, $h$ is the plank constant, $m$ is the mass of electron and $a$ is the side of lattice.

$$E = \frac {(6.62 \times 10^{-34})^2 (3.14)^2}{(9.1 \times 10^{-31}) (3.2 \times 10^{-10})^2}$$

$E = 4.63 \times 10^{-17}$ joule

Calculate the momentum of electron by the relation as follows:

$$P = \sqrt {2 m (E)}$$ $$P = \sqrt {2 (9.1 \times 10^{-31}) (4.63 \times 10^{-17})}$$ $$P = \sqrt {126.4 \times 10^{-48}}$$ $$P = 11.24 \times 10^{-24} \ \mathrm{kg \cdot m/s}$$

Answer:

$E = 4.63 \times 10^{-17}$ joule

$P = 11.24 \times 10^{-24} \ \mathrm{kg \cdot m/s}$

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