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Chapter 41: Problem 37

(a) What maximum light wavelength will excite an electron in the valence band of diamond to the conduction band? The energy gap is \(5.50 \mathrm{eV}\). (b) In what part of the electromagnetic spectrum does this wavelength lie?

Short Answer

Expert verified
The maximum wavelength is 225.5 nm, which is in the ultraviolet region.

Step by step solution

01

Understand the Energy Gap

The energy gap (also known as the band gap) represents the minimum energy required to move an electron in diamond from the valence band to the conduction band. It is given as 5.50 eV in this problem.
02

Convert Energy to Wavelength

We use the formula that relates energy and wavelength:\[ E = \frac{hc}{\lambda} \]where \( E = 5.50 \text{ eV} \) is the given energy, \( h = 4.1357 \times 10^{-15} \text{ eV}\cdot\text{s} \) is Planck's constant, and \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light. Rearrange to solve for \( \lambda \):\[ \lambda = \frac{hc}{E} \].
03

Calculate the Wavelength

Plug in the values into the formula:\[ \lambda = \frac{(4.1357 \times 10^{-15} \text{ eV}\cdot\text{s})(3 \times 10^8 \text{ m/s})}{5.50 \text{ eV}} \approx 2.255 \times 10^{-7} \text{ m} \]which converts to approximately \(225.5 \text{ nm}\).
04

Determine the Wavelength Region

The calculated wavelength of 225.5 nm falls within the ultraviolet (UV) region of the electromagnetic spectrum, which typically includes wavelengths from about 10 nm to 400 nm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromagnetic Spectrum
The electromagnetic spectrum is a range of all types of electromagnetic radiation. Radiation is energy that travels and spreads out as it moves. Examples include visible light that we can see and ultraviolet light that we cannot. The spectrum includes many types of waves such as radio waves, microwaves, infrared, visible light, ultraviolet (UV), X-rays, and gamma rays. Each type of wave varies in wavelength and frequency.
  • Radio waves have the longest wavelength, and gamma rays have the shortest.
  • Visible light is the only part we can see with our eyes鈥攕panning roughly from 400 nm (violet) to 700 nm (red).
Understanding the electromagnetic spectrum is crucial for identifying the kinds of light and wave energy, which helps in studying their effects and applications in different technologies.
Wavelength Calculation
Calculating the wavelength of electromagnetic waves is a fundamental step in understanding how different waves interact with matter. The key formula that relates energy and wavelength is: \[ E = \frac{hc}{\lambda} \]Where:
  • \(E\) is the energy of the photon.
  • \(h\) is Planck's constant \( (4.1357 \times 10^{-15} \text{ eV}\cdot\text{s}) \).
  • \(c\) is the speed of light, approximately \(3 \times 10^8 \text{ m/s}\).
  • \(\lambda\) is the wavelength, in meters.
By rearranging the formula to \( \lambda = \frac{hc}{E} \), we can solve for wavelength when energy is provided. This is how we can figure out the wavelength associated with specific photon energies, such as those needed to excite electrons across an energy gap in materials like diamond.
Photon Energy
Photon energy is a measure of the energy carried by a single photon, determined by its wavelength or frequency. The formula \( E = \frac{hc}{\lambda} \) allows us to calculate the energy of photons. This energy is crucial because it determines how photons interact with matter.
  • High-energy photons (e.g., X-rays, UV rays) can ionize atoms, meaning they can remove electrons and potentially cause damage.
  • Lower-energy photons (e.g., infrared, visible light) generally cannot ionize atoms, but they can cause molecules to vibrate or rotate.
The concept of photon energy is vital when discussing phenomena such as photoelectric effects and semiconductor behaviour, where energy levels determine the interaction between light and materials.
Ultraviolet Light
Ultraviolet (UV) light is a type of electromagnetic radiation with wavelengths shorter than visible light, ranging from 10 nm to about 400 nm. It is divided into different regions: UVA, UVB, and UVC, each with specific wavelength ranges and effects.
  • UVA (320-400 nm) 鈥� It penetrates the skin deeply and contributes to aging and long-term skin damage.
  • UVB (290-320 nm) 鈥� It is most responsible for sunburn and has more energy than UVA, affecting outer skin layers.
  • UVC (100-290 nm) 鈥� It has the shortest wavelength and is mostly absorbed by the Earth's atmosphere, particularly the ozone layer.
UV light is both beneficial and potentially harmful. It is used in applications such as sterilization and vitamin D production in the skin but also poses risks like skin cancer from prolonged exposure.

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Most popular questions from this chapter

A certain computer chip that is about the size of a postage stamp \((2.54 \mathrm{~cm} \times 2.22 \mathrm{~cm})\) contains about \(3.5\) million transistors. If the transistors are square, what must be their maximum dimension? (Note: Devices other than transistors are also on the chip, and there must be room for the interconnections among the circuit elements. Transistors smaller than \(0.7 \mu \mathrm{m}\) are now commonly and inexpensively fabricated.)

The Fermi energy for silver is \(5.5 \mathrm{eV} .\) At \(T=0^{\circ} \mathrm{C}\), what are the probabilities that states with the following energies are occupied: (a) \(4.4 \mathrm{eV}\), (b) \(5.4 \mathrm{eV}\), (c) \(5.5 \mathrm{eV}\), (d) \(5.6 \mathrm{eV}\), and (e) \(6.4 \mathrm{eV}\) ? (f) At what temperature is the probability \(0.16\) that a state with energy \(E=5.6 \mathrm{eV}\) is occupied?

What is the probability that a state \(0.0620 \mathrm{eV}\) above the Fermi energy will be occupied at (a) \(T=0 \mathrm{~K}\) and (b) \(T=320 \mathrm{~K}\) ?

A sample of a certain metal has a volume of \(6.0 \times 10^{-5} \mathrm{~m}^{3}\). The metal has a density of \(9.0 \mathrm{~g} / \mathrm{cm}^{3}\) and a molar mass of \(60 \mathrm{~g} / \mathrm{mol}\). The atoms are bivalent. How many conduction electrons (or valence electrons) are in the sample?

At \(T=300 \mathrm{~K}\), how far above the Fermi energy is a state for which the probability of occupation by a conduction electron is \(0.15\) ?
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