The photoelectric effect

Key concepts

1. The basic ideas behind the photoelectric effect are visualized schematically in this page. A very schematic experimental setup can be studied at this other page.
2. 
   A rough picture of the energetics in a photoelectron experiment is indicated in the figure at the right. The brown region represents the kinetic energy of the electrons inside the potential-energy well V(z) generated by the solid. The distribution of "band" energies in an ordinary metal at "low" temperature drops very fast at some upper value (the upper boundary of the brown region), the chemical potential µ.
3. If the incoming photon energy hν is smaller than the difference W in energy between the potential outside the crystal and the energy position of the electrons with the largest energy inside the crystal, no electron can be extracted, no matter how large the number of incoming photons is.
4. If instead, as drawn in the figure, the incoming photon energy hν is larger than W, then photoemission can take place: electrons are emitted from the solid.
5. The photoemitted electrons have an energy distribution reflecting their energy distribution inside the crystal. The maximum kinetic energy of these "photoelectrons" is the residual T_max=hν - W.
6. W is often called "work function" (lavoro di estrazione), and it is a characteristic of the material. It represents energy threshold for the photons to emit electrons. Sometimes W is given by the equivalent minimum frequency ν_min=W/h or even the maximum wavelength λ_max=ch/W of a photon whose energy is at threshold.
7. The photoemission efficiency of some metal surface is the number of emitted electrons divided by the number of incoming photons (for a given time). The photoemission efficiency depends on the material, on the detailed characteristics of the surface, on the photons incident direction and polarization, and on the photons frequency (for example, obviously it goes to zero as h ν approaches W from above).
8. If a slowing potential V is turned on between the sample and the electron collecting electrode, then the fraction of collected electrons to the emitted ones is reduced, until the energy q_e·V matches exactly the kinetic energy of the fastest emitted electrons T_max (q_e is the elementary charge): at this point the collected electronic current stops. This value of V=T_max/q_e is therefore called stopping potential.

Exercises

1. Light of λ=200nm falls on an aluminum surface. In aluminum W=4.2eV. What is the kinetic energy of
   a) the fastest and
   b) the slowest emitted photoelectrons?
   c) What is the stopping potential?
   d) What is the cutoff λ for Al?
   e) If the intensity of incident light is 2.0 W/m² and the photoemission efficiency of this surface is 2.5 %, what is the average number of electrons emitted per unit time and area?
2. The radiation from a 500 K blackbody strikes a metal surface, whose work function is 0.214 eV. Determine the wavelength for which the blackbody peak occurs, and determine the longest wavelength in the spectrum capable of ejecting photoelectrons from the surface. What portion of the blackbody's total emittance is effective in producing photoemission from the metal surface? Express the result in terms of a dimensionless integral over the Planck distribution.
   RESULT: λ_max = 5.79551 µm; λ_threshold = 5.79366 µm; indicating with y=hc/(k_BT λ_threshold), we have R_eff/R_tot = [∫_y^∞ dx x³/(exp(x)-1) ] / [∫₀^∞ dx x³/(exp(x)-1) ] ≃ 25 %

Trivial(?) questions

1. What sign has the slowing potential of the collecting electrode with respect to the sample? What happens if the sign of this potential is reversed? Is the photo-current affected? Why?
2. Why do the photoelectrons come out of a metal with a spread in kinetic energy, even though the irradiating electromagnetic field is perfectly monocromatic?
3. How would you exploit photoelectric effect to build a device sensitive to infrared light? Between the aluminum of the first problem above and the metal alloy of the second problem which one would you choose for building your sensor?

Comments and debugging are welcome!