Snell's Law Atomic Level
Date: Spring 2014
I am trying to understand Snell's Law at a molecular level. On a molecular level what causes glass to refract a given photon of a given wavelength of light at one angle and another wavelength of light at a different angle?
Refraction is all about light interacting with electrons. (There is also an interaction with nuclei, because they are also charged, but nuclei are so massive that they are relatively unaffected by light, so most of the interesting physics is in the interaction with electrons.) This interaction can be thought of in two ways: (1) as bound electrons being excited to higher energy states by absorbing photons, and then almost immediately returning to their original states, emitting new photons of the same wavelength but a slightly different phase; or (2) as bound electrons being vibrated by light waves, and emitting light of a slightly different phase by vibrating.
Maybe it is best to think of bound electrons being vibrated, because that doesn't get us into quantum mechanics; we can just use regular old mechanics to understand the phase shift. In this picture, "bound" means that if you push an electron away from its resting spot, and then let it go, it will vibrate at some resonant frequency, emitting light, and eventually settle down, because emitting light takes energy from the electron's motion. It is a resonant system - a damped oscillator - like a bell, or a guitar string.
Here is a thing you have to understand about resonant systems: if a system is heavily damped, its resonance is broad, and you don't have to be right near the resonant frequency to excite the resonance. The resonances we are interested in here are very broad.
When you drive a resonant system with a periodic excitation, the motion of the driven system generally has a different phase from that of the driving excitation. For example, if you take a spring with a mass suspended from it, and move your hand up and down very, very slowly, the mass will move in phase with your hand. If you move your hand up and down very, very fast, the mass will move up while your hand moves down, and vice versa. If you go up and down exactly at the resonant frequency, the mass will be 90 degrees out of phase with your hand.
Now that you know how resonant systems behave, what happens when light drives a resonant electron? The electron vibrates, and its phase depends on how the incident light frequency compares to the electron's resonant frequency. As it happens, the relevant resonant frequencies of electrons in many transparent materials are in the ultraviolet; we're far from resonance, but different frequencies of light are differently far from resonance. Blue light is closer to resonance, so it will have one phase shift; red light is further from resonance, so it will have a different phase shift.
The phase shift determines the speed of light through the material. Light excites electrons, which vibrate, reradiating light of a different phase, which excites electrons downstream, which vibrate, reradiating light of yet a different phase, and so on - light accumulates a larger and larger phase difference as it travels through the material, so it looks like it moved way slower than light moves in vacuum. This is how light propagates through a transparent material. Now the speed of light is just proportional to the refractive index, so you can see that the refractive index must depend on frequency, because the phase shift depends on frequency. This is where Snell's law enters the picture, because Snell's law is all about the refractive index. This is why different wavelengths are refracted at different angles.
Thanks for the question. I am going to present a simplistic picture of what happens. The true picture involves quantum theory as well as the Kramers-Kronig relations--which are best left for specialized college classes on spectroscopy. On a molecular level, the photon is scattered by a molecule. In the scattering process, the molecule gets hit by the photon and gains some angular momentum. In the collision, the photon lost some angular momentum. This loss of angular momentum results in the change in the angle of the photon trajectory. The photon trajectory has changed and this change is called refraction. You can demonstrate this type of effect by throwing a piece of clay at a piece of pipe.
I hope this helps.
When considering things at the molecular lever, remember that what we see on a larger scale is actually an average effect. A molecule that absorbs a photon of light can emit it at any angle. Many molecules within a piece of glass emit in many different directions. Along one direction (i.e. the refraction angle), the photons from the many molecules support each other. They are in phase with each other, producing constructive interference. If the glass large is enough to eliminate any awareness of the molecules, then only that angle of constructive interference produces light strong enough to measure.
Kenneth E. Mellendorf
Illinois Central College
We may assume that light is at different frequencies and a material is composed of electrons residing in orbitals at different energy levels. Recall that these orbitals are forever changing due to input energy, but the molecular bonds vibrate or stretch according to frequency.
So if a glass made of fused borosilicate is encountered white light (photons of various frequencies): SiO2 vibrates at several specific frequencies, so does Na2O, CaO and Bo2O3 and AlOx. These vibrational frequencies are absorbing photon energy and then releasing it as the light propagates, So the photons interact with the electrons of these fused materials and thus slow down, causing a refraction or bending dependent on frequency.
It should be noted that although the net speed of light has decreased, the photons that move from molecule to molecule still move at c, the speed of light in that medium.
Because varied frequencies tend to group, the net result is what we visualize as a spectrum. There is a lot of overlap, which is why a rainbow seems to blend the colors. For scientific instruments, we use monochrometers to single out a very small selection of frequencies.
Peter E. Hughes, Ph.D. Milford, NH
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Update: November 2011