Most of this course deals with the interaction between electromagnetic radiation (usually ionizing) and matter. Therefore it is important to review the properties of electromagnetic radiation, particularly in terms of its ability to produce effects on matter.
Electromagnetic (EM) radiation can be described as having wave
properties or as having particle properties;
in any given experiment it will display one or other set of properties,
and in another experiment it will display the other set.
Waves can be characterized by their frequency ν
(the inverse of the time interval between successive
peaks in the wave's unulation),
their wavelength λ (the spatial distance between peaks)
and in the case of EM radiation, their energy E.
These three quantities are closely related.
Frequency and wavelength are related by
ν * λ = v
where v is the speed (the magnitude of the velocity)
with which the wave is traveling.
The fundamental principle of Einsteinian relativity is that all
electromagnetic radiation travels in a vacuum at the same speed,
namely, the speed of light, c.
Planck showed that the energy associated with an electromagnetic
wave of frequency ν is proportional to that frequency.
The proportionality constant is termed h:
E = h ν
It has a value 6.6262*10-34 J s, or
4.1346*10-15 eV s. We'll explain the meaning of an electron
volt (eV) in a moment.
Combining these two equations enables us to relate energy to wavelength:
E = hc/λ
The combination of constants hc occurs often enough that its value
is probably worth recording: it is 1.9865*10-25 J-m,
or 1.23984*10-6eV-m.
Electromagnetic (EM) radiation encompasses a wide range of energies, all the way from radio waves to cosmic rays. The following table characterizes a number of the ranges within the overall spectrum of electromagnetic radiation according to energy, wavelength, and frequency:
| Category | Energy, eV | Wavelength, nm | Frequency, Hz |
| Radio | 10-10 - 10-5 | 108 - 1011 | 2*104 - 2*109 |
| Microwave | 10-5 - 10-2 | 105 - 108 | 2*109 - 2*1012 |
| Infrared | 0.01 - 1.6 | 750 - 105 | 2*1012 - 4*1014 |
| Visible | 1.6 - 3 | 400 - 750 | 4*1014 - 7*1014 |
| Ultraviolet | 3 - 1000 | 1 - 400 | 7*1014 - 2*1017 |
| X-rays | 103 - 105 | 10-2 - 1 | 2*1017 - 2*1019 |
| Gamma Rays | 105 - 109 | 10-6 - 10-2 | 2*1019 - 2*1023 |
This would be a good time to explain the use of the electron volt as a
unit of energy. A volt is a unit of electromotive force (EMF) or (uhh)
voltage, and is defined as the amount of EMF required to generate one Joule
of energy in a coulomb of charges; i.e. 1 V = 1 J Coul-1.
The charge on an electron is 1.60264*10-19 Coulomb; to put
it another way, a Coulomb is 6.2397*1018 electronic charges.
Therefore an electron volt is defined as a single electronic
charge multiplied by one volt:
1.60264*10-19 Coul * 1 J Coul-1 =
1.60264*10-19 J.
This turns out to be a useful unit in many calculations involving
atoms and molecules. Another common unit is the MeV, i.e. one
million electron volts, or 1.60264*10-13J.
One of the most significant advances in physics in the second half of the nineteenth century was the James Clerk Maxwell's formulation of the fundamental laws of electrodynamics:
These laws, which can also be written in integral rather than differential forms, in fact are reformulations of four concepts that arose earlier in the 19th century, namely,
Maxwell's contribution was to develop the notions of electric and magnetic fields and then show that the older four laws could be cast into the compact forms listed above.
Maxwell's equations describe the behavior of electromagnetic fields. In the closing years of the nineteenth century it was recognized that a electromagnetic field travels away from its source at a fixed velocity, namely (μ0 ε0)1/2, which turns out to be 2.9979*108 m/sec -- the velocity of light. That suggests that light really is an electromagnetic wave. From this recognition comes the development of the relationship among velocity, wavelength, and energy that we wrote above.
A few years after Maxwell, Max Planck sought to explain one of the conundra
of early twentieth-century physics: the behavior of radiation inside a
cavity. His solution to a problem known to physicists of his day as
the "ultraviolet catastrophe" was to posit that the atoms in the cavity
were electromagnetic oscillators with characteristic frequencies, and
that the oscillators could absorb and emit radiation--but only at certain
specific energies. These energies were proportional to the
frequencies of the radiation, according to
E = (n+1/2)hν
where ν is the frequency, h is the constant we discussed earlier,
and n is some integer.
At any energy other than those satisfying this equation for integer n,
no energy will be emitted.
In the same decade Einstein studied the photoelectric effect,
as shown in the figure. 
Here visible light is entrained upon the surface of a metal and
electrons are emitted by the surface as a result. He was able to
show that no electrons are emitted unless the light has a frequency
above a threshold value νmin, and that for light with
a frequency above the threshold, the maximum kinetic energy of the electrons
escaping from the surface is proportional to the difference between
the frequency employed and the threshold value. That is,
Kmax = Q(ν - νmin)
He then posited the existence of discrete quanta of electromagnetic
radiation called "photons," such that the energy of each photon satisfied
Ephoton = hν = E0 +
Kmax
,
where E0 is an energy that is characteristic of the surface
itself and Kmax is the maximum kinetic energy that can
be imparted to the electron. Clearly
E0 = hνmin
Einstein also contributed to the next round of influences on our
understanding of electromagnetic radiation. Consider an automobile being driven
on top of a large boxcar with a velocity w relative to the boxcar.
The boxcar itself is traveling in a train whose velocity relative to the
ground is v:
Physicists up through the early twentieth century employed Galilean
relativity to compute the velocity u of the automobile relative
to the ground:
Einstein argued that there was a need to modify this formula in order
to preserve the notion that the velocity of light, c, was the
maximum velocity that any object could attain. The formula above
might violate that if, for example, v = 0.7c and w = 0.35c.
Einstein's laws of special relativity said that the time over which
an event occurs and the distance traveled by an object depend on who
is measuring them. An observer at rest with respect to the object will
measure time and distance in one way, and an observer traveling at
a velocity v with respect to the object will obtain different measurements
for time and distance.
The correction for the displacement is known as the Lorentz contraction:
x' = x / γ
where
γ = (1 - v2)-1/2
for displacements in the direction of the observer's velocity.
The correction for time is known as the time dilation formula:
t' = γt
Note that this parameter γ is greater than 1 for all real,
nonzero values of v, so lengths are contracted and times are increased
by motion. The effects are small at any velocities that we can readily
achieve terrestrially, but they become significant as v approaches c:
| v/c | γ | v/c | γ |
| 0.01 | 1.00005 | 0.05 | 1.00125 |
| 0.10 | 1.00504 | 0.30 | 1.04828 |
| 0.60 | 1.25000 | 0.87 | 2.02815 |
| 0.90 | 2.29416 | 0.95 | 3.20256 |
| 0.98 | 5.02519 | 0.99 | 7.08881 |
At the same time, Einstein introduced the notion of
relativistic energy. Here he posits that an object carries an
energy equal to
E = γmc2
so that the rest energy Ez
(the energy at v = 0 or γ = 1) is
Ez = m0c2
where m0 is the rest mass, i.e. the mass at v = 0.
Einstein then introduced the notion of the relativistic mass
mrel = γm0
so that
E = mrelc2
This (apart from a subscript) is the immortal energy-mass relationship.
This analysis suggests that the corrections for relativity are modest
at terrestrial speeds (m = 1.05m0 at v = 0.1c),
but become important if v approaches c
(m = 5.03m0 at v = 0.98c).
Radiation physics often involves analysis of particles traveling at close
to the velocity of light, so these corrections are relevant.
We now turn our attention to the structure of the atom as it was understood around the beginning of the twentieth century. J.J. Thomson developed a model for the atom in 1897 in which the positive charge and most of the mass of the atom were contained in a central nucleus, and the electrons, which carried the negative charge, were revolving around that nucleus. Two decades later Rutherford showed that for this model to be consistent with his scattering experiments, the nucleus had to be much smaller than the atom itself--on the order of 10-5 the size of the atom. A problem with this model was that accelerating charges (e.g. the electrons in these atom) should radiate, according to Maxwell's equations; if they did so all the time, the atom would run out of energy and collapse. Niels Bohr developed a model based on Planck's notion of quantized energy levels under which atoms emit radiation only when electrons change from one state to another, rather than perpetually. The radiation emitted when this happens emerges with an energy equal to the difference between the energy levels of the two states. In formulating this idea, Bohr began the journey toward the development of modern quantum theory, but in fact he used classical energy calculations to derive formulae for the energy levels in his atoms. The Bohr model of the atom explains a prodigious number of experiments, but not all, so it has been supplanted by the more elaborate (and philosophically troubling) quantum model of Heisenberg, Dirac, Schrödinger, Bohr himself, and others.
Bohr's analysis begins with the idea that an electron in an atom has a
quantized angular momentum
mvr = (1/2π)nh
The Coulombic attraction of this electron for the neighboring nucleus
must equal the centripetal force:
F = mv2/r = kZe2/r2,
where k is the Coulomb's law constant 1/(4πε0)
and Z is the charge on the nucleus. We can solve for r:
r = kZe2/(mv2)
But using this notion of quantized angular momentum,
v = nh/(2πmr) = 2πkZe2/nh
so r = n2h2 /
(4π2kZe2m)
For the case of a hydrogen atom with its electron in the ground state,
n = Z = 1,
we find that
r = h2/(4π2ke2m)
has the value 0.529*10-10 m,
a value known as the Bohr radius. This value (~ 0.5 Å)
is shorter than but of the order of magnitude of the lengths of the
covalent bonds between atoms in molecules,
so it is a reasonable estimate of the size of a hydrogen atom.
In the Bohr atom, we can tally the following quantities directly from the above analysis:
For n = Z = 1, the conditions under which we calculated the Bohr radius, these formulas work out to
We then argue that photons emerge from transitions from one value of n to another. For example, if an electron makes the transition from the n = 3 state to the n = 2 state, then the energy of the photon given off will be -2π2k2e4m/ [ (1/9 - 1/4)h2] = 1.89 eV
We have been considering electromagnetic radiation as capable of
behaving as a wave and as a particle.
Can we, then, devise a formula that describes light as having momentum?
de Broglie did so, and this notion was found to have useful consequences.
He expressed the momentum P associated with electromagnetic radiation
to be
P = E / c = hν / c = h / λ
In electron transitions, then, photons carry off this amount of momentum
in the direction in which they are traveling. Since momentum must
be conserved, we can expect the atom to recoil with some modest momentum
in the opposite direction.
Having described electromagnetic radiation as having both wavelike and
particle-like properties, can we also describe ordinary matter as having
both kinds of properties? The answer is again yes, as was shown in the 1920's
when electrons were shown to display interference, an unambiguous
wave effect. In analogy to the equation above, we express the wavelength
of a particle as
λ = h / P = h / mv
This equation may be either nonrelativistic or relativistic, provided that
the mass m we use is mrel in the latter case.
In the nonrelativistic case, we can express the kinetic energy as
KE = mv2/2, so
λ =
h(2(KE)m)-1/2
If we then argue that the angular momentum L = mvr is quantized,
mvr = nh / (2π),
then P = L / r = nh/(2πr) = h / λ, so that
2πr = nλ,
i.e. the circumference 2πr of the electron's orbit around the
nucleus is an integer multiple of the electron's wavelength.
This brings on the image of the electron existing in a stable,
standing-wave arrangement.
From these considerations we can call forth some homework problems: