Absolutely Small - Michael D. Fayer [20]
In his work, Planck introduced the formula that related the frequency of the electrons to their energy, E = hν, where ν is the frequency as discussed in Chapter 3, and h is called Planck’s constant. In the equation, h = 6.6 × 10-34 J-s, J is the unit of energy Joule, and s is seconds. In the formula, the units of ν are Hz or 1/ s; so h times ν gives the units of energy, J. In his description of black body radiation, Planck postulated that the energy E could only change in discreet steps. It could be hν, of 2hν, or 3hν, and so on, but the energy could not have values between these integer step changes. The recognition that energy changes in discreet quanta at the atomic level marked the beginning of quantum mechanics.
Einstein proposed that Planck’s formula also applied to photons, so that the energy of a photon was determined by its frequency ν as E = hν. Using this formula, Einstein explained the reason that red light generates slower electrons than blue light. Red light is lower frequency than blue light. Therefore a red photon is lower energy than a blue photon. In the pool ball analogy, a blue photon hits the electron harder than a red photon, and therefore, the blue photon produces an electron that has a higher speed than a red photon. With this picture, it is clear why using redder and redder light produces slower and slower electrons emerging from the metal.
VERY RED LIGHT DOES NOT EJECT ELECTRONS
The one observation left to explain is why do the electrons stop coming out of the metal when the light is tuned far enough to the red? Einstein resolved this as well. When an electron is ejected from a metal by a photon, it has a certain kinetic energy. Kinetic energy means the energy associated with its motion. The higher the energy, the faster the electron moves. The kinetic energy is Ek where the subscript k stands for kinetic. The formula for kinetic energy is given by where m is the mass and V is the velocity. Then the velocity of an electron that emerges from a metal is related to its energy, which in turn is related to the energy of the photon that knocked it out of the metal. A higher energy photon will give the electron more kinetic energy, and the electron will move faster (have a larger V). As mentioned, electrons are held in a metal by a binding energy, call it Eb, where the subscript b stands for binding. Therefore, some of the energy that is carried by the photon has to go into overcoming the binding energy. The kinetic energy of the electron that comes out of the metal is just the photon energy, E = hν, minus the binding energy, Eb. Thus, the electron’s kinetic energy is Ek = hν - Eb. For an electron to be ejected from the metal, the photon energy hν must be larger than the binding energy Eb. As the light is tuned further and further to the red (longer wavelength, λ), ν becomes smaller and smaller because ν = c/λ, where c is the speed of light. At some red enough color, hν is less than Eb, and electrons are no longer ejected from the metal. Turning up the intensity causes more photons to impinge on the metal, but none of these photons has enough energy to eject an electron.
The fact that electrons stop coming out of the metal when the photon is tuned far enough to red (has low enough energy) can be understood by thinking of the child’s game, Red Rover. In Red Rover, a line of kids stands across a field holding hands. A kid on the other team runs at the line. If he runs very fast (high energy), he breaks through the line and keeps going, although he is slowed down. If he runs somewhat slower, he will still break through the line. However, if he runs slow enough, he will not break through the line because his energy is insufficient to overcome the binding energy of the hands holding the line together.