The discovery of the neutron by James
Chadwick in 1932 finally made the nature and structure of
the atomic nucleus clear. It consisted of a collection of two
non-identical particles, proton and neutron, of almost the same
mass, with the neutron having no charge, bound together by the
strongest known force in nature! When the size of the nucleus,
around 10-13 cm, is compared to the size of the atom
itself, about 10-8 cm, it was realized that typical
nucleon kinetic and binding energies are of the order of MeV,
compared to eV for atomic electrons. Thus processes in which the
strong nuclear force does work liberate a million times the
kinetic energy per unit mass, compared to chemical processes in
which the electromagnetic force does work. Because of the
extreme density of the nucleus, it was assumed that the nucleons
formed a chaotic system of strongly interacting particles, and
that a detailed description of nuclear states might be
practically impossible. But at the end of WW 2, physicists were
able to turn to a detailed study of the excited states of
various nuclei, and an astonishing discovery quickly followed.
Just as the state of an electron in a complex atom can be
understood as an interaction of the single electron with a
charge distribution generated by all the other electrons, plus
the nucleus, the state of a single nucleon in the nucleus can be
understood to a large degree as due to the interaction of that
nucleon with a matter and charge distribution created by all the
other nucleons... this concept is usually known as the
“independent particle model.” The discovery was made
independently by Maria
Goeppert Mayer and Hans Jensen, and earned the Nobel Prize
in 1963. Their first calculations were done with our old pal,
the harmonic oscillator potential.
How about nuclei with an even number of protons and neutrons? Aage Bohr and Ben Mottelson in the 1950s showed that such nuclei display excitations involving every nucleon in the nucleus, in so-called “collective” states. They won the Nobel Prize in 1975, along with another physicist who was a pioneer in this area.
The nucleus is a bound system, confined by a potential that goes to zero at infinity, so as Einstein emphasized, the mass of any nucleus is less than the total mass of the particles of which it is composed. That is, M(Z,N) < Zmp + Nmn. The difference is called the binding energy, and is the work you would have to do to take the nucleus completely apart and leave the pieces at rest an infinite distance apart. The standard way to describe a given nucleus is to write ACh, where A is the total number of nucleons, and Ch is the chemical symbol, which tells you the number of protons. For example. organic matter is made of 12C, in other words carbon, with 6 protons and 6 neutrons. Nuclei with the same Z but different numbers of neutrons have the same chemical symbol, but A varies. So for example carbon also exists as a famous isotope, 13C.
Consider a collision between two
nuclei, A and a, which results in a rearrangement of nucleons.
The nuclear force can do either positive or negative work during
the process, leading to new nuclei B + b. If the total mass of B
+ b is less than the mass of A + a, the result is that B + b
have more kinetic energy than the original A + a, by amount Q,
the so-called Q-value. If the Q-value is negative, the process
is obviously forbidden by conservation of energy, unless the
original kinetic energy of A + a is great enough to make up the
difference.
Rutherford and his collaborators observed the first nuclear
reaction, 4He + 14N leading to p + 17O,
in 1917.
The independent particle model
(nucleons in a central potential generated by all the other
nucleons, plus a spin-orbit interaction) gave a beautiful
explanation of the observed fact that some specific numbers of
protons or neutrons were “magic,” resulting in a nucleus that is
unusually stable and tightly bound. In other words, that
specific number of nucleons fills a shell, creating an inert
core. Again note how similar this is to the atomic case,
where filled electron shells create atoms that don't participate
in chemical reactions, the so-called noble gases, like helium
and argon!
More generally, it was noticed
early on that nuclei with even numbers of both protons and
neutrons were unusually stable. If a nucleus has an odd number
of either protons or neutrons, it is much less likely to be
stable, and if it has an odd number of both protons and neutrons
it is quite unlikely to be stable at all. This suggested that
protons and neutrons in nuclei form stable S = 0 pairs, and that
it takes considerable energy to break a pair. These pairs
behave like bosons, a phenomenon also seen as causing superconductivity
in some metals (electrons bond into so-called Cooper pairs)!
It is a property of all unstable
quantum states that the probability of a decay of the state, per
unit time, is independent of time. This leads to a
simple differential equation, dN/dt = −N/τ, where τ is the average
lifetime of the state. The solution is by inspection N(t)
= N(0)e−t/τ. Since essentially the entire
population of planet earth is mathematically illiterate, and
would never have heard of Napier's number e, the basis of the
natural logarithms, it is more common to write the solution in
terms of powers of 2, as N(t) = N(0)(1/2)t/T1/2,
where the half-life T1/2 = ㏑2 × τ is the time for
half a sample of material to undergo the decay.
Heavy nuclei are often unstable to the penetration of tightly bound, paired clusters of 2 protons and 2 neutrons (the nucleus of the He atom!) through the Coulomb barrier that ordinarily helps to keep nucleons inside the region of the strong nuclear potential energy. Because the probability of barrier penetration depends exponentially on the energy of the particle hitting the barrier, lifetimes for this “alpha decay” cover an enormous range from billions of years to very tiny fractions of a second, such as 10−6 seconds!
The most common type of nuclear decay occurs when a nucleus has too many protons or too many neutrons to be, according to the mass formula, the most tightly bound system with that value of A. An excess neutron is unstable to the weak interaction, one of the four fundamental forces of nature. The weak interaction can change a neutron into a proton, an electron, and an anti-electron-neutrino. Since the neutrino has no charge, and little or no mass, the only particle that is normally observed being emitted is the electron, which before it was recognized was called a β− ray. However, when the energy is favorable to the process, an excess proton can be changed into a neutron, a positron (anti-electron) and an electron neutrino. Before the positron was recognized, it was called a β+ ray.
What is actually going on in these decays is that either an up quark is changing into a down quark, by emitting a weak boson, or a down quark is changing into an up quark, by emitting a weak boson. The boson then creates a particle-antiparticle pair.
The final type of nuclear decay, gamma or γ decay, occurs because, after alpha or beta decay, the resulting nucleus is almost never in its ground state, but rather in one of its excited states. So the excited state or states decay to the ground state by emission of a photon. Because the spacing of the energy levels is of order MeV, the photon energies are of order MeV, a million times the kinetic energy of a photon of visible light.
Suppose you have a system of identical radioactive nuclei, with half lives of one day. So after one day, only 50% of the original nuclei remain. Suppose after 6 years, you have kept careful track of nuclei which have not yet decayed. What is the probability that one of these nuclei, which has not decayed in 6 years, decays in the next day? Answer: 50%! The probability of decay per unit time is a constant!
Example: find a rock that contains Ar but not K . This tells you the natural abundance of Ar. Now you can date a rock that contains both K and Ar, because the excess of Ar in that rock is due to decay of K, and the amount of the excess gives you the time since the rock formed.