Chapter 42: Problem 18
The gap between valence and conduction bands in diamond is 5.47 eV. (a) What is the maximum wavelength of a photon that can excite an electron from the top of the valence band into the conduction band? In what region of the electromagnetic spectrum does this photon lie? (b) Explain why pure diamond is transparent and colorless. (c) Most gem diamonds have a yellow color. Explain how impurities in the diamond can cause this color.
Short Answer
Step by step solution
Understand the Energy-Wavelength Relationship
Rearrange for Wavelength
Convert Energy from eV to Joules
Input Constants and Calculate Wavelength
Determine the Electromagnetic Spectrum Region
Explain Diamond's Transparency and Colorlessness
Discuss the Effect of Impurities
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Valence Band
In the context of energy bands, the electrons in the valence band are tightly bound to the atomic nuclei and have lower potential energy.
When energy is applied, these electrons can jump to the conduction band. The energy needed is equal to the band gap, which, for a diamond, is 5.47 electron volts (eV).
- Electrons here remain stable and attached to their parent atom.
- The valence band itself does not contribute to the flow of electric current.
Conduction Band
In materials like semiconductors, electrons in the conduction band have enough energy to participate in electrical conductivity.
The energy gap between the valence band and conduction band, known as the "band gap," determines the amount of energy needed for an electron to jump to the conduction band.
- Electrons are free to conduct electricity once in this band.
- The minimal energy required to excite an electron into this band is the band gap energy.
Electromagnetic Spectrum
Visible light, the light we can see, is a tiny portion of this spectrum.
In the exercise, you encountered a photon with a wavelength of 227 nm, calculated to lie in the ultraviolet region of the spectrum, which is beyond the visible range for humans.
- The spectrum includes visible light, microwaves, infrared, ultraviolet, X-rays, and gamma rays.
- Each type of radiation in the spectrum has different applications, from communications to medical imaging.
Photon Wavelength
In the context of this exercise, the energy of a photon is calculated using the formula: \[E = \frac{hc}{\lambda}\]where \( E \) is the energy, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.
A smaller wavelength corresponds to higher energy photons, such as those found in the ultraviolet region.
- Ultraviolet light has shorter wavelengths and higher energy than visible light.
- By calculating the maximum photon wavelength that bridges the band gap, we can determine whether a material will absorb certain light fields.
Impurities in Crystals
In the case of diamonds, common impurities include nitrogen, which can absorb certain wavelengths of light and thus impart color, like a yellow hue.
These impurities create additional energy levels within the band gap, allowing electrons to jump at different energies that correspond to visible light frequencies.
- Impurities can be introduced during the crystal's formation or through treatment.
- They often create localized energy states within the band gap, influencing optical and electrical properties.