Electromagnetic Radiation

In the last chapter, we used the fact that photons are involved in quantum energy "transitions" (spin flips) to understand Magnetic Resonance Imaging. A photon can also be absorbed by an atomic electron, increasing its energy level (n). In addition, an atomic electron can drop to a lower energy state by emitting a photon. In both cases, the change in energy is

D E = - E0 Z 2 ( 1 / n f 2 - 1 / n i 2 ),

where i and f signify the initial and final energy levels. But how do we observe these photons?

Electromagnetic radiation can travel in "waves" (see Chapter 9). These waves are analogous to ripples on a pond, but they do not need a "pond": they can travel in a vacuum. Their speed in a vacuum is 1 / Sqrt ( e 0 m 0 ), which is numerically equal to the speed of light ( c = 3 x 10 8 m / s). We conclude that light is an electromagnetic wave, and photons are little pulses or "packets" (a good analogy for "quantum") of light. As in the last chapter, their frequency is measured in Hz and represents how often the electromagnetic fields associated with them oscillate each second. Using dimensional analysis, we can compute the "size" of the waves, which in the case of the pond is the distance between the crests, and in the case of light, is the distance between equivalent field configurations. This "wavelength" is

l = c / n.

Wavelengths of transition photons are usually measured in nm (nanometers), with visible light in the range 700 (red) to 400 (blue) nm. Wavelengths above 700 are called "infrared", and those below 400 are "ultraviolet" (those below 10 nm are called "x-rays" or "gamma" rays). A visible object's color is determined by the frequencies which are not absorbed by its atomic electrons. Chalk is white because it absorbs primarily in the infrared and ultraviolet: visible light is almost completely reflected. Copper compounds are blue-green because they absorb red photons and reflect most others. The exact frequencies are often shifted due to the surrounding atoms.

Everything you see is the result of photons which have reflected off of the electrons in an object (or, in the case of florescence, absorbed and re-emitted), only to be absorbed by the electrons in the cells on your retinae. The world is literally alive with energy in the form of these photons. For example, a 40 W florescent light bulb (at 100 % efficiency) emits 40 J worth of photons each second. If we assume these photons have a wavelength of 550 nm (which is convenient as the "middle" of the visible spectrum), the energy in each photon is 3.612 x 10 - 19 J. This means that 1.11 x 10 20 photons are emitted by the bulb every second! Since the energy of a photon is h n, the frequency of a photon which is absorbed or emitted in an electron transition to a higher or lower energy level is

n = ( - E0 Z 2 / h ) ( 1 / n f 2 - 1 / n i 2 ).

As an example, an n = 6 electron in a He atom drops to an n = 4 level, resulting in the emission of a photon of frequency

n = ( - E0 2 2 / h ) ( 1 / 4 2 - 1 / 6 2 )

= 4.569 x 10 14 Hz.

This means that it has a wavelength of 656 nm, and so is red. Some other typical visible transitions for atoms of low Z are:

AtomTransitionWavelength (nm)
H 3 -> 2 656
He 7 -> 4 541
Li 7 -> 5 517
Be 6 -> 5466
Li 5 -> 4 450
H 5 -> 2 434

Note that in this discussion, we have been ignoring angular momentum (l , m and s). In fact, the orbital motion of an electron produces a sort of current loop and therefore a small magnetic field. The energy of other electrons is increased or decreased slightly by the potential energy of their spins in those magnetic fields (as in the last chapter). These "spin-orbit interactions" cause the spectral lines to "split" into several neighboring energies (and therefore wavelengths). Another aspect of angular momentum is very important to electron transitions and electromagnetic radiation: the total angular momentum of the electron and photon must be conserved in an absorption or emission. The spin of a photon can be 1, 0 or -1 (which corresponds to its "polarization", a fascinating topic that is outside the scope of this text). So an atomic electron with s = -1/2 absorbing a photon of spin 1 is left at a higher energy level and with spin +1/2. We have assumed that its orbital angular momentum was not affected; if it increased by one, the spin would then have remained at - 1/2. You can see that things get complicated pretty quickly!

Photon-induced transitions are not limited to individual atomic electrons. Chemical bonds are often disrupted by photons which give the bonding electron too much energy to remain in the molecular orbital:

BondWavelength (nm)
C-C 345
C-N 409
C-O 341
C-H 289
C=C 196
C==C (triple bond)135
O-H260
C=O168
N-H 278
H bond in H 2 O6358
H bond 57222-2384

Problems

The next section is about the spectral sensitivity of the eye.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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