The last quarter of the 19th century
saw numerous cases where the data were inexplicable on the basis of classical
physics:
The spectrum of emitted radiation from
heated objects.
The photoelectric effect.
The behavior of specific heats of simple
gases, which were not (as classical physics predicted) independent of temperature,
and many of which disagreed with the classical prediction even at room
temperature.
Emission of X-rays by "cathode ray" tubes.
Radioactivity of certain substances.
The Michelson-Morley experiment.
The first three of these required development
of a new theoretical framework, called quantum theory, for their explanation.
Historically the first application of
the quantum ideas was to the radiation spectrum question, in a paper by
Planck in 1900. That issue is fairly complicated, and its resolution did
not really occur until 1924.
Until 1905 Planck's new idea of the quantum
had very little impact. In that year Einstein used it to explain the basic
features of the photoelectric effect. Two years later he used it again
to explain much of the behavior of the specific heats.
Although some aspects of X-ray emission
can be understood on classical grounds (rapid stopping of a charged particle
produces such radiation) the details required understanding of atomic structure,
which in turn required quantum theory.
To understand atomic sturcture also required
the discovery of the atomic nucleus, which was an outgrowth of Rutherford's
investigations into radioactivity.
The question of why all heated objects
at the same temperature emit light with the same frequency spectrum, and
why that spectrum has its observed form, was pursued by numerous scientists
in the last half of the 19th century.
On general grounds (the laws of thermodynamics)
it was shown by Kirchhoff that the spectral function must have the same
form for all objects, although the actual intensity of the radiation depends
on details such as the nature of the surface of the emitting body.
He also showed that a good emitter is
a good absorber. Thus the surface that perfectly absorbs all radiation
falling on it will also be the perfect emitter. Such an object is called
a blackbody.
A cavity in solid obect, with a
small hole that lets radiation in or out, gives a good approximation of
a blackbody.
Stefan (experimentally) and Boltzmann
(theoretically) showed that the total radiation (power per unit area) from
an object is propotional to T4, where T is the Kelvin temperature.
The essential quantity to be analyzed
is the energy density per unit frequency of the radiation inside the cavity.
This is denoted by u(f,T), where f is the frequency.
An educated guess as to the universal
form of the function was given by Wien:
where A and b
are constants. Within the accuracy of measurements in the early 1890's
this formula fit the data for high frequencies.
Meanwhile, using classical statistical
mechanics, Rayleigh (with a small contribution by Jeans) had derived the
following formula:
This fit the data for low frequencies,
but it is an absurd result because it predicts a total radiation (integrated
over all frequencies) that is infinite.
Planck first showed that the following
formula could be used to interpolate between the low frequency (Rayleigh-Jeans)
formula and the high frequency (Wien) formulas:
where h is a constant. (Wien's constants
can be expressed in terms of h and k.)
Planck worked for years to find a satisfactory
derivation of this formula.
His ultimate argument focused on the
interior surface of the cavity, which he assumed to be composed of microscopic
"oscillators" capable of emitting and absorbing radiation.
In classical electrodynamics such oscillators
can emit or abosorb energy in arbitrarily large or small amounts, at the
frequency of their oscillation. If Planck allowed the amount of energy
to be arbitrarily small, he merely rederived the Rayleigh-Jeans law.
But if he allowed emission or absorption
of energy only in discrete amounts which are multiples of a quantum
of energy E = hf, then he could obtain his formula.
The numerical value of h that fits the
data is very small. In SI units it has the value 6.626 ¥
10-34 J-s.
Planck hoped somehow to remove this "quantum"
restriction, but allowing h to approach zero only gives the Rayleigh-Jeans
law again. So he published his result without being able to give any deeper
explanation of his ideas.
By the time Planck won the Nobel Prize
for this work (1920) it had become clear that the quantum was of revolutionary
importance in our understanding of nature, but a fundamental understanding
of what he had found took several more years to obtain.
We will return to the cavity radiation
problem in our discussion of quantum statistical mechanics later in the
course.
In his famous experiment (1887) verifying
Maxwell's prediction of electromagnetic waves, Hertz discovered that ultroaviolet
light falling on a clean metal surface resulted in charge emanating into
the space above the surface. This was the first notice of the photoelectric
effect.
After the electron had been established
by Thompson (1897), he showed that the charge produced in the photoelectric
effect consisted of electrons.
Lenard made a systematic study of this
effect in 1902. He found some very surprising aspects.
By that time Lorentz had developed a
theory of metallic conduction that assumed the actual moving charges are
electrons, so it was assumed that metals have a supply of electrons free
to move around, but not completely free to leave the surface.
The classical account of liberation of
electrons from a metal surface by light falling on that surface had these
features:
The oscillating E-field of the incident
light gives energy to the electrons, with the amount given per unit time
proportional to the intensity of the light.
This process should occur at all frequencies
of the light.
The number of electrons liberated per
unit time should be proportional to the light intensity.
For low intensities it would take some
time for the electrons to accumulate enough energy to break free from the
surface.
The kinetic energy of the released electrons
should depend on the light intensity but not its frequency.
Lenard's study showed the following:
The number of electrons released per
unit time was proportional to the light intensity. This agrees with #3
above.
The process did not occur at frequencies
below a certain threshhold. This violates #2 above.
Release of the electrons seemed to start
immediately even at very low intensities. This violates #4 above.
The maximum kinetic energy of the electrons
was independent of intensity but varied linearly with the frequency. This
violates #5 above.
In 1905 Einstein gave an explanation
for the strange features of the photoelectric effect, using Planck's quantum
idea.
Although Planck assumed that radiation
in a heated cavity exchanges energy with the walls as quanta, he still
believed that the radiation itself is made of waves as Maxwell's theory
stated.
Einstein, however, suggested that the
radiation
itself is made of quanta, and that exchange of energy with a surface
consists of emission or absorption of highly localized quanta of radiation.
On that model, ejection of an electron
from the surface would be a localized phenomenon, with the energy of the
quantum given entirely to the electron. Following Planck, Einstein assumed
that energy to be E = hf, where f is the frequency.
Eectrons in a given metal are bound to
the surface so that a certain amount of energy (called the "work function")
is required to liberate them.
If an electron absorbs the energy of
a quantum and subsequently loses none of it to other electrons, then it
will leave the surface with the largest possible kinetic energy.
This maximum, assuming energy conservation,
should thus be
where f
is the work function.
Since kinetic energy cannot be negative,
this formula predicts that no electrons will be released for frequencies
below the threshhold
This explains both the existence of the
threshhold and why it varies from one metal to another (since they have
different values of f).
It also explains why the maximum kinetic
energy varies linearly with frequency and is independent of intensity.
Light quanta (now called photons)
are extremely numerous in light of the intensities we normally deal with.
The light intensity is proportional to the volume density of photons, so
the number of electrons released in the photoelectric effect does increase
with intensity.
Since absorption of a photon is a local
event, with the energy delivered in a very short time to the electron,
there is no reason to expect any time delay in ejection of the electrons,
even for very feeble light.
Einstein's photon theory thus explains
all of the unxpected results of Lenard's experiments. In 1916 Millikan
tested every aspect of Einstein's explanation, confirming them all. It
was mainly for this work that Einstein received his Nobel Prize in 1921.
Shortly before
World war I it became possible to measure with some precision the wavelengths
of X-rays, by scattering them from regular crystals. The angle of scattering
depends on the wavelength.
Using these methods
it was also possible to produce beams of X-rays of nearly a single wavelength.
In 1923 Compton
scattered such a "monochromatic" X-ray beam from a graphite crystal and
measured the intensity in the scattered beam as a function of its wavelength.
He found a strong peak in the scattered intensity at the same wavelength
as the incident beam, as expected. But he also found a pronounced peak
at a longer wavelength.
The difference
between this longer wavelength and the original wavelength is the "Compton
shift" Dl. Compton
observed that its value depended on the angle of deflection of the original
beam.
In classical electrodynamics,
scattering by a charged particle (such as an electron in the crystal) comes
about because the incident wave's E-field sets the charge into oscillation
at the incident frequency. This driven oscillation produces radiation of
its own which, superposed on the original wave, gives the scattered wave.
(This is the phenomenon of Rayleigh scattering, which gives rise to the
light we see in the daytime sky,)
Although the intensity
of the scattered wave depends on wavelength (this accounts for the blue
color of the daytime sky) the scattered wave always has the same frequency
as the incident wave. There is no classical mechanism to produce the shift
Compton observed.
Compton accounted
for what he observed by treating the scattering as an elastic collision
between the incident X-ray photon and a free electron in the crystal.
Before the collision
the electron is at rest. Let the (relativistic) momentum and energy of
the incident photon be k and w.
After the collision let the photon have momentum and energy k' and
w',
and let the electron have momentum and energy p and E.
Then momentum and
energy conservation give
Rearranging, we
have
We use E2
- (pc)2 =(mc2)2 to eliminate p
and E. Then we use the facts that for the photon (initial and final) w
= kc and w' = k'c.
Finally we see from the diagram that
After a little
algebra we find
So far this has
been just a relativistic collision problem. Now we introduce quantum theory
by setting w = hf
= hc/l for each photon.
This leads to Compton's formula:
The shift in wavelength
depends on the angle of scattering, but not on the incident wavelength.
The maximum possible shift (for q
= 180°) is 2h/mc, which is only about 0.005 nm. For visible light of
wavelengths from 400 to 700 nm this is a negligible shift. But for X-rays
of wavelength around 0.1 nm it is easily observed.
Why is there also
in Compton's data a peak showing no wavelength shift? The explanation is
that the shift formula above assumed that the electron is essentially free,
like a conduction electron. If the scattering is from a bound electron,
the mass that must be set in motion (which replaces the electron mass m
in the formula above) is the mass of the whole atom, which is very large.
The corresponding shift is negligibly small, giving rise to the "unshifted"
peak.
