Brief Review of the photoelectric effect

General Terms:

• Light is an electromagnetic wave.  It has energy, but zero mass and zero charge.   The photon is the name of light when it acts as a particle.
• Electron is a particle with mass and charge.  It can be represented by a matter wave or as a particle.

Photoelectric Effect

Light falling on a surface can cause the surface to eject electrons, called photoelectrons, under the proper conditions.

• The wavelength of the light must be shorter than the threshold wavelength for the surface.
  • Shorter wavelength means more blue.
  • Another interpretation is that the frequency of the light must be higher than the threshold frequency.
• Light that is of longer wavelength (lower frequency or more red) than the threshold will not cause ejection of photoelectrons - no matter how intense that light may be, or how long one waits for ejection.
• When a photoelectron is emitted, the electron will have a certain amount of kinetic energy.
  • The amount of kinetic energy depends on the wavelength of the light and the surface.
  • Light that is more blue gives a greater kinetic energy to the ejected electron.
  • Light that is more red gives a lower kinetic energy as long as it is shorter than the threshold wavelength.
• If the light is of a wavelength appropriate to cause ejection of photoelectrons,
  • the number of electrons ejected each second is proportional to the light intensity.
  • the ejection is almost instantaneous - it takes about one-billionth of a second.

The threshold wavelength and the nearly instantaneous ejection can not be explained by classical electricity and magnetism.  According to classical physics, any wave should be able to provide sufficient energy to cause emission of electron - however, it would take much longer for a very dim light.  This contradicts the observed data.

Einstein explain the photoelectric effect by assuming:

• The light acted as a stream of particles, called quanta, and that each quanta interacted with one electron in the surface. 
• The energy of each of the quanta is given by planks rule:  E = hf
  • E is the energy (in Joules or in the more convenient unit eV)
  • h is Planck's constant, a very small number.   h = 6.6x10^-34 Joule sec
  • f is the frequency of the wave in Hertz  (recall that v = f l for a periodic wave and v = c for light waves)
• The energy required to cause photoejection is called the Work Function of the material.  
  • Recall that energy is the ability to do work.
  • From the threshold wavelength, one can calculate the threshold frequency using f_t = c/l_t
  • where l_t is the threshold wavelength.
  • Then Planck's equation can be use to find the work function.
  • A shortcut that can be used is the fact that a photon of wavelength 1240 nm has an energy of 1eV
    • hence, 620 nm corresponds to 2 eV, 310 nm corresponds to 4 eV etc
    • so the wavelength (in nm) multiplied by the energy (in eV) equals 1240
• If the photon has enough energy to remove the electron it interacts with, the photon is absorbed, and the electron is ejected in a the very short time observed.  Any excess energy (i.e. energy above the threshold energy) is given to the electron as kinetic energy.
• If the photon does not have enough energy to eject the electron, the photon will, in essence, be reflected (at least for shiny metal surfaces - for dark surfaces it will be absorbed and add to the internal energy of the system).
• More intense beams of light have more energy transported each second. This corresponds to having more photons hit the surface each second. Hence if the photon can eject photoelectrons, more will be ejected by brighter light than a dimmer light during each second.

Possible Experimental Variations

1.  Shine light of known wavelength on a number of materials and see which materials eject photoelectrons.  You have a go/no-go test of the work function.  If ejection of photoelectrons occurs then the work function is less than the photon's energy.
2. Shine light of differing known wavelengths on a single surface in succession.  This way you can 'bracket' the work function by finding the shortest wavelength that doesn't cause emission of photoelectrons and longest wavelength that does.  Normally one would use emission spectra lines from mercury, sodium, or other elements.
3. Sine light that is more blue than necessary on the surface.  This gives the photoelectrons some kinetic energy. One can then use a repelling voltage to convert the electron's kinetic energy into electrical potential energy in much the same way that gravity converts a ball's kinetic energy into gravitational potential energy - bringing it to a stop, and then making it return.  The energy of the electrical potential energy is the same number as the number of volts used.  Hence a stopping voltage of 1.5 volts can convert 1.5 eV of kinetic energy into electrical potential.  If the electron has less than 1.5 eV of kinetic energy, it will be stopped and returned to the metal surface where it rejoins the material.  If the electron has more than 1.5 eV, an electrical current is said to be flowing.  By carefully adjusting the repelling voltage, one can accurately determine the work function for a surface as long as the wavelength of the light is known.
4. This is a variation on step 3).  A fixed repelling voltage is used (because it helps reduce 'noise' in the data) and the wavelength of the photon is adjusted (like using a prism to spread white light into its colors) until a current just begins to flow.   At this point one uses conservation of energy to determine the work function of the surface. 

• E = hf is the energy of the photon
• KE is the kinetic energy of the electron
• WF is the work function energy of the surface
• E = hf + WF  is a statement of conservation of energy for the process. 

The photon has been absorbed, it no longer exists.  Part of its energy has been used to free the electron (using an amount WF) and the remainder is left as kinetic energy of the electron.  Hence, a simple subtraction can be performed to determine the work function.