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Understanding the connection between a photon's energy and its wavelength is crucial in the study of optics. The energy of a photon, denoted as \( E \), is calculated using the equation \( E = \frac{hc}{\lambda} \). Here, \( h \) represents Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \), \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \), and \( \lambda \) is the wavelength of the photon.
This formula reveals a fundamental relationship: energy and wavelength are inversely proportional. This means that shorter wavelengths have higher energy and vice versa. For instance, ultraviolet light has shorter wavelengths and thus, more energy compared to infrared light.
When photon energy is less than a material's band gap energy, the photon isn't absorbed, allowing it to pass through. This principle explains why certain materials are transparent to specific wavelengths.

Silicon, a widely used semiconductor material, has a band gap of \( 1.14 \, \mathrm{eV} \). This implies that photons with energy lower than \( 1.14 \, \mathrm{eV} \) will not be absorbed by silicon, making it transparent to those photons.
To find out which wavelengths silicon is transparent to, we need to calculate the maximum wavelength of light with an energy of \( 1.14 \, \mathrm{eV} \). Using the formula \( \lambda = \frac{hc}{E} \), where energy \( E\) is first converted to joules, the maximum wavelength is determined to be approximately \( 1090 \) nm.
Thus, silicon remains transparent at wavelengths greater than \( 1090 \) nm, which are primarily in the infrared region of the electromagnetic spectrum. This property is significant for infrared applications where unobstructed light passage through silicon is required.

The electromagnetic spectrum consists of various regions distinguished by their wavelengths and frequencies. These regions include radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
Each region possesses distinct characteristics:

• Radio Waves: Longest wavelengths, used for communication.
• Microwaves: Used in cooking and radar technology.
• Infrared: Experienced as heat, often used in night-vision devices.
• Visible Light: The spectrum detectable by the human eye, ranges from about 400 nm to 700 nm.
• Ultraviolet: Beyond visible light, with applications in sterilization and fluorescent lighting.
• X-rays: Used in medical imaging.
• Gamma Rays: Shortest wavelength, produced by radioactive materials.

Wavelengths longer than \( 1090 \) nm fall into the infrared region, where silicon remains transparent, highlighting the importance of understanding different electromagnetic spectrum regions for technological applications.

Transparency in materials, like window glass, depends on their band gap in relation to visible light. Visible light spans wavelengths from roughly \( 400 \text{ nm} \) to \( 700 \text{ nm} \).
For a material to be transparent throughout this visible range, it must have a band gap that exceeds the energy of the shortest visible wavelengths, which is around \( 450 \text{ nm} \). Solving for this wavelength gives a band gap of at least \( 2.75 \, \mathrm{eV} \).
When glass exhibits a band gap greater than \( 2.75 \, \mathrm{eV} \), it allows visible light to pass without absorption, resulting in transparency. This is why glass is commonly used in windows and optical instruments, as it transmits visible light effectively, enabling clear vision and permitting natural illumination.