Question

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

(a) Maximum wavelength: 227 nm, in the UV region. (b) Transparent due to high band gap. (c) Impurities introduce colors.

Step-by-step solution

Understand the Energy-Wavelength Relationship

The energy (E) of a photon is related to its wavelength (\( \lambda \)) by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant and \( c \) is the speed of light. To find the maximum wavelength that can excite the electron, we need to isolate \( \lambda \).

Rearrange for Wavelength

To find the wavelength, we rearrange the formula: \( \lambda = \frac{hc}{E} \). This allows us to calculate the wavelength once we know the energy gap, which is given as 5.47 eV.

Convert Energy from eV to Joules

We need to convert the energy from electron volts to joules. The energy gap is 5.47 eV and 1 eV is equal to \( 1.602 \times 10^{-19} \) J, making the energy: \[ E = 5.47 \times 1.602 \times 10^{-19} \text{ J} \approx 8.77 \times 10^{-19} \text{ J}. \]

Input Constants and Calculate Wavelength

Use Planck's constant \( h = 6.626 \times 10^{-34} \) J\(\cdot\)s and the speed of light \( c = 3.00 \times 10^8 \) m/s and substitute them into the equation: \[ \lambda = \frac{(6.626 \times 10^{-34} \cdot 3.00 \times 10^8)}{8.77 \times 10^{-19}} \text{ m}. \] Calculate \( \lambda \): \[ \lambda \approx 2.27 \times 10^{-7} \text{ m} \text{ or } 227 \text{ nm}. \]

Determine the Electromagnetic Spectrum Region

The calculated wavelength of 227 nm lies in the ultraviolet (UV) region of the electromagnetic spectrum, which is not visible to the human eye.

Explain Diamond's Transparency and Colorlessness

Pure diamond is transparent and colorless because its band gap (5.47 eV) is too high to absorb visible light, which is why it doesn't show color under normal circumstances.

Discuss the Effect of Impurities

Impurities, such as nitrogen, can introduce energy levels within the band gap, allowing absorption of certain wavelengths of visible light. This can cause color in diamonds, such as yellow, when blue light is absorbed.

Key concepts

Valence Band

The valence band in a semiconductor is where the electrons are when they are bound to atoms and do not have enough energy to be involved in conduction. It's essentially the outer shell of electrons that can participate in bonding.
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.

Understanding the role of the valence band helps explain the transparency and insulating properties of materials like diamond.

Conduction Band

The conduction band is the range of electron energy, higher than that of the valence band, where electrons are free to move and contribute to conduction. Thus, when electrons gain enough energy, they can move to the conduction band and free themselves from their atomic bonds.
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.

In diamonds, this band gap is relatively large, contributing to its transparency.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, each defined by different wavelengths and frequencies. It ranges from gamma rays with very short wavelengths to radio waves that can be kilometers long.
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.

Understanding the electromagnetic spectrum is crucial in explaining which energies are absorbed or transmitted by different materials.

Photon Wavelength

Photon wavelength is the distance between consecutive peaks of a light wave. This measurement determines the energy of the photon since energy is inversely proportional to 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.

Understanding photon wavelengths helps illustrate why certain materials appear the way they do or why certain photon energies result in specific reactions.

Impurities in Crystals

Impurities in crystals can drastically change their physical properties. These impurities are atoms or molecules that are different from the primary constituent material and get incorporated into the crystal structure.
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.

Understanding how impurities affect crystals supports explanations for variations in color and electrical conductivity, important in both gemology and electronics.