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 of the aluminum antimonide solid-state laser is \(2.721 \times 10^{-19}\) Joules.

Step-by-step solution

Write down given information and constants

We are given the wavelength of emitted light: λ = 730 nm. The following constants are also required: Planck's constant (h) = 6.626 × 10^{-34} Js Speed of light (c) = 3 × 10^8 m/s Note: We will need to convert the wavelength from nm to meters.

Convert the wavelength to meters

To convert the wavelength from nanometers (nm) to meters (m), multiply by 10^{-9}: λ = 730 nm × 10^{-9} nm/m = 730 × 10^{-9} m

Calculate the energy

Now, we will use the formula E = hc/λ: E = (6.626 × 10^{-34} Js) × (3 × 10^8 m/s) / (730 × 10^{-9} m)

Simplify and solve

Working through the math: E = (1.9878 x 10^{-25} Js^2) / (730 × 10^{-9} m) E = 2.721 x 10^{-19} J

Write down the final answer

The band gap of the aluminum antimonide solid-state laser is 2.721 × 10^{-19} Joules.

Key concepts

Solid-State Laser

A solid-state laser is a high-powered laser that uses a solid medium, often a crystal like aluminum antimonide, to generate a coherent beam of light through the process of stimulated emission. Solid-state lasers are widely used in various applications, including cutting and welding materials, medical surgeries, and telecommunications. Unlike gas or liquid lasers, the active medium in solid-state lasers is a solid. To function, these lasers require a source of energy, typically light from a flash lamp or a diode, to excite the electrons in the laser medium. When these electrons return to their ground state, they emit photons, which are funneled into a beam of light.

The color or wavelength of the light produced by a solid-state laser is directly connected to the band gap of the material: the energy difference between the highest valence band and the lowest conduction band. The wavelength emitted when electrons fall from the conduction band to the valence band is what we see as the laser light. Calculating this band gap is crucial for understanding the properties and applications of the laser.

Planck's Constant

Planck's constant, denoted as 'h', is a fundamental constant in quantum mechanics that relates the energy of a photon to its frequency. Discovered by Max Planck, the constant has a value of approximately 6.626 × 10^{-34} joule-seconds (Js). This small number plays a pivotal role in the calculation of energy at the quantum level, and it is crucial in determining the energy of photons emitted by systems such as solid-state lasers.

In the context of solid-state lasers, Planck's constant is used along with the speed of light to calculate the photon energy corresponding to the emitted light's wavelength. The energy is found using the formula E = hc/λ, where E is the energy of a photon, h is Planck's constant, c is the speed of light, and λ is the wavelength of the light. Understanding Planck's constant helps us appreciate the quantized nature of energy in the microscopic world.

Speed of Light

The speed of light, commonly symbolized as 'c', is a fundamental physical constant representing the speed at which light travels in a vacuum. It has a value of approximately 3 × 10^8 meters per second (m/s). The speed of light is not just significant to relativity and cosmology but also to the practical workings of devices like solid-state lasers.

In laser physics, the speed of light comes into play when we calculate the energy of photons based on their wavelength. As light from the laser travels through various mediums, it may slow down; however, the speed of light in a vacuum remains a constant and is crucial for calculations involving energy and frequency of electromagnetic waves. Understanding the speed of light helps us bridge the gap between wavelength and energy, essential for determining the outcomes in laser technology and beyond.

Energy of a Photon

The energy of a photon is a quantum concept describing the amount of energy carried by a single particle of light. Photons are massless particles, which implies that they move at the speed of light and carry energy that is dependent on their frequency. The energy of a photon (E) can be calculated using the equation E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of light.

In the context of solid-state lasers, controlling the energy of a photon is critical; it determines the efficiency and application of the laser. For example, in an aluminum antimonide laser emitting light at 730 nm, we can calculate the band gap energy of the material by finding the energy of the photon associated with that wavelength. The band gap energy tells us the minimum energy required for an electron to transition from a bounded state within an atom to a free, conductive state. Photons with energy equal to or greater than the band gap can stimulate such transitions, leading to the laser action observed in the solid-state medium.