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Chapter 10: Problem 70

An aluminum antimonide solid-state laser emits light with a wavelength of $730 . \mathrm{nm}$ . Calculate the band gap in joules.

Short Answer

Expert verified
The band gap energy of the aluminum antimonide solid-state laser is approximately \(2.72 \times 10^{-19}\) joules.

Step by step solution

01

Identify the given wavelength

The wavelength of the emitted light from the aluminum antimonide solid-state laser is given as 730 nm.
02

Convert wavelength to meters

To calculate the band gap energy in joules, we need to convert the given wavelength from nanometers to meters. 1 nm = \(10^{-9}\) m So, 730 nm = 730 × \(10^{-9}\) m = \(7.3 \times 10^{-7}\) m
03

Use the Planck-Einstein relation

The Planck-Einstein relation states that the energy of a photon, denoted by \(E\), is directly proportional to its frequency, denoted by \(f\): \(E = hf\) where \(h\) is the Planck constant, given as \(6.63 \times 10^{-34}\) Js.
04

Calculate the frequency of the light

Now, we can use the speed of light and the wavelength to find the frequency of the emitted light. The speed of light, \(c\), is approximately \(3 \times 10^{8}\) m/s. The relation between the speed of light, wavelength, and frequency is given by: \(c = \lambda f\) Where \(\lambda\) is the wavelength of the light. Rearrange the equation and solve for frequency: \(f = \frac{c}{\lambda}\) Plug in the values: \(f = \frac{3 \times 10^{8}\ \text{m/s}}{7.3 \times 10^{-7}\ \text{m}}\) \(f \approx 4.11 \times 10^{14}\ \text{Hz}\)
05

Calculate the band gap energy

Now that we have the frequency, we can use the Planck-Einstein relation to find the band gap energy. \(E = hf\) Plug in the values: \(E = (6.63 \times 10^{-34}\ \text{Js})(4.11 \times 10^{14}\ \text{Hz})\) \(E \approx 2.72 \times 10^{-19}\ \text{J}\) Therefore, the band gap energy of the aluminum antimonide solid-state laser is approximately \(2.72 \times 10^{-19}\) joules.

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