Threshold Wavelength: The Photoelectric Gatekeeper
The Photoelectric Effect: A Quantum Puzzle
Before the 20th century, light was thought to behave purely as a wave. This wave theory, however, couldn't explain a curious phenomenon observed with certain metals: when light shines on them, they can emit electrons. This is the photoelectric effect. The classical wave theory predicted that:
- Brighter light (more intense) should eject electrons with higher kinetic energy.
- Any color (wavelength) of light should be able to eject electrons, given enough intensity.
But experiments showed something completely different. A dim blue light could eject electrons, while an extremely bright red light could not eject a single one. This was a major puzzle. Albert Einstein solved it in 1905 by proposing that light is made of tiny particle-like packets called photons. The energy of a single photon is not determined by the intensity of the light, but by its color, or more precisely, its frequency.
To eject an electron, a single photon must transfer all its energy to a single electron. This energy is used for two purposes: first, to overcome the attractive force that holds the electron inside the metal (this is the work function, symbolized by $\phi$), and second, any leftover energy becomes the kinetic energy of the now-free electron.
Defining the Threshold Wavelength
The threshold wavelength, denoted by $\lambda_{0}$, is the specific boundary condition for the photoelectric effect. It is the longest wavelength (which corresponds to the lowest frequency and thus the lowest energy) of light that can cause electron emission.
At the threshold wavelength, the photon has just enough energy to free the electron, but no energy is left to give it any speed. The electron is ejected with zero kinetic energy. This gives us the fundamental equation that defines the threshold wavelength:
This equation shows a simple inverse relationship: the larger the work function of a metal (meaning it holds onto its electrons more tightly), the shorter its threshold wavelength will be. This means only higher-energy light (like blue or ultraviolet) can cause the effect. A metal with a small work function has a longer threshold wavelength, so even lower-energy light (like red or infrared) might be sufficient.
| Metal | Work Function ($\phi$) in eV | Approximate Threshold Wavelength ($\lambda_{0}$) in nm | Corresponding Light Color |
|---|---|---|---|
| Cesium (Cs) | 1.95 | 636 | Red |
| Sodium (Na) | 2.28 | 544 | Green |
| Zinc (Zn) | 4.33 | 287 | Ultraviolet |
| Platinum (Pt) | 6.35 | 195 | Ultraviolet |
A Practical Example: Solar-Powered Calculators
A great real-world application of the photoelectric effect and threshold wavelength is the solar cell in a calculator. The solar cell is typically made of silicon, which has a specific work function. The threshold wavelength for silicon is in the infrared/near-infrared region. This means that visible light (which has shorter wavelengths and higher energy than the threshold) is perfect for ejecting electrons in silicon.
When sunlight or room light, which contains photons of visible light, hits the solar cell, photons with energy greater than silicon's work function are absorbed. They transfer their energy to electrons, which are then ejected and create an electric current. This current powers your calculator. If the light's wavelength were longer than the threshold (like deep infrared heat from your hand), it wouldn't matter how intense that heat was; it could not generate any current because individual infrared photons lack the minimum energy to free an electron.
Common Mistakes and Important Questions
Q: Can increasing the intensity of light with a wavelength longer than the threshold cause the photoelectric effect?
A: No. This is the most crucial point of the quantum explanation. The photoelectric effect is a "one photon, one electron" process. If a single photon does not have enough energy to overcome the work function, it cannot eject an electron, no matter how many such photons (high intensity) are shining on the metal. It's like trying to knock down a brick wall by throwing millions of ping-pong balls at it. No single ping-pong ball has the required energy, so the wall stays up.
Q: What happens if you use light with a wavelength shorter than the threshold wavelength?
A: If the wavelength is shorter, the frequency and photon energy are higher. The photoelectric effect will occur. The extra energy from the photon (beyond what was needed to free the electron) is converted into kinetic energy, so the ejected electron moves away from the metal at a high speed. The kinetic energy (K.E.) of the electron is given by: $K.E. = hf - \phi$.
Q: Is the threshold wavelength the same for all metals?
A: No, absolutely not. As shown in the table above, different metals have different work functions because the strength of the bond holding their outermost electrons varies. Therefore, each metal has its own unique threshold wavelength. Cesium, for example, has a very low work function and a long threshold wavelength in the red part of the spectrum, while platinum requires high-energy ultraviolet light.
Conclusion
Footnote
1 Photoelectric Effect: The emission of electrons from a material when it is exposed to light (electromagnetic radiation).
2 Photon: A quantum, or discrete packet, of electromagnetic energy. It is the fundamental particle of light.
3 Work Function ($\phi$): The minimum amount of energy required to remove an electron from the surface of a solid (usually a metal) to a point just outside the surface.
4 Planck's Constant ($h$): A fundamental physical constant that relates the energy of a photon to its frequency. Its value is approximately $6.626 \times 10^{-34}$ Joule-seconds.