Question

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

Short answer

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

Step-by-step solution

Identify the given wavelength

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

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

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.

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: \(f = \frac{3 \times 10^{8}\ \text{m/s}}{7.3 \times 10^{-7}\ \text{m}}\) \(f \approx 4.11 \times 10^{14}\ \text{Hz}\)

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.

Key concepts

Wavelength Conversion

In order to effectively calculate the band gap energy of the light emitted by a solid-state laser, it is essential to convert the given wavelength into a standard unit. Wavelength is often provided in nanometers (nm) which is not the SI unit. Hence, to ease calculations, wavelength is converted to meters. Remember:

• 1 nanometer (nm) is equal to \(10^{-9}\) meters (m).
• Thus, 730 nm equals \(730 \times 10^{-9}\) meters, which simplifies to \(7.3 \times 10^{-7}\) meters.

By converting the wavelength to meters, it aligns with the units of the speed of light, making further calculations seamless.
Wavelength conversion is pivotal when dealing with photon energy calculations as it sets the groundwork for precise computations.

Planck-Einstein Relation

The Planck-Einstein relation is crucial in understanding how energy is related to the properties of light. This equation expresses the energy of a photon in terms of its frequency:

• \(E = hf\)

where \(E\) represents the energy of the photon, \(h\) is Planck's constant \((6.63 \times 10^{-34}\ \text{Js})\), and \(f\) stands for the frequency of the light. This relation highlights a direct proportionality between energy and frequency.
As the frequency increases, so does the energy of a photon, making this relation a cornerstone in quantum mechanics and photonics. It assists in linking the conceptual gap between wave and particle natures of light.

Photon Energy

Photon energy is a foundational concept when calculating energies related to electronic transitions and band gaps in lasers. It can be derived using the Planck-Einstein relation, once we have the frequency:

• The energy of each photon emitted is given by \(E = hf\).
• Using the frequency calculated from a given wavelength, \(f = \frac{3 \times 10^{8}\ \text{m/s}}{\text{wavelength (m)}}\), we can easily find \(E\).

Photon energy directly determines the efficiency and potential applications of a laser.
In this context, it correlates with the band gap energy of the laser material, impacting its emission capabilities. Understanding photon energy is vital for interpreting how different wavelengths correspond to energy levels in solid-state devices.

Solid-State Laser

Solid-state lasers are a type of laser where the gain medium is a solid, such as a crystal or glass doped with rare-earth or transition metal ions. Aluminum antimonide is one such crystal used here, emitting light at a specific wavelength. These lasers are renowned for their durability and efficiency.

• They operate on the principle of stimulated emission, where photon energy matches the band gap energy to create light.
• Light generated is coherent, monochromatic, and capable of precision uses in medicine, communication, and scientific research.

In a solid-state laser like aluminum antimonide, precise calculations of the band gap energy, related to the emitted wavelength, allow accurate predictions of performance and color of light emitted.
This directly influences how effective the laser will be in its intended application.

Frequency of Light

The frequency of light is a fundamental characteristic that ties into understanding the nature of electromagnetic waves. Frequency is directly tied to both wavelength and the energy of light:

• The relationship between the speed of light \((c)\), wavelength \((\lambda)\), and frequency \((f)\) is given by \(c = \lambda f\).
• Rearranging gives us frequency: \(f=\frac{c}{\lambda}\).
• This formula illustrates that as the wavelength decreases, frequency increases, leading to a higher energy output.

By calculating the frequency from a given wavelength, such as the 730 nm from the laser, it's possible to deduce key properties like photon energy.
Understanding frequency helps relate the observations of light in a laboratory or practical setting back to theoretical models based on quantum mechanics.