Lecture 12

The Nucleus

In an earlier lecture we have learned how Rutherford's experiment showed that atoms are mostly empty space, with most of the mass concentrated in a central nucleus, and the electrons whizzing around at a distance 5 orders of magnitude larger than the actual nucleus dimensions.

In lectures following we also learnt that practically all of the visible properties of matter (colour, chemical properties, mechanical and electrical properties, etc.) depend only on the orbiting electrons, with the nucleus not playing any visible part (except for providing the positive charge responsible for keeping the electrons attached).

Does the nucleus play any other role in nature ? As you can guess, the answer is yes, and the revelation of such a role is part of our modern history.

When exploring the nuclear realm, we must be aware of Einstein's famous formula relating mass to energy, E = mc². This famous equation was originally derived from first principles, as a necessary consequence of the theory of Special Relativity, and, at the time it was first enunciated, nobody really believed that it could actually be verified in practice. As nuclear science progressed, matter/energy equivalence was experimentally verified, and it was shown to be at the base of most nuclear phenomena.

Taken at face value, the formula tells us that even a tiny amount of matter contains an enormous amount of energy. Unluckily, such energy is normally locked up within the nuclear matter, and it can be released and exploited only under very special (and limited) circumstances.

As a first step towards understanding the nucleus, let us get familiar with its composition. The first nuclear component to be discovered was the proton. Hydrogen is nothing but a single proton in the nucleus with a single electron orbiting about it. It is rather simple to obtain a sample of pure protons: take some Hydrogen, ionize it by thermal or electrical excitation, pull off the electrons with an (even weak) electric field, and you are left with protons....

Protons have the same electric charge as electrons (same in magnitude, but of opposite sign), and are about 2000 times heavier than electrons.

We now know that the total number of protons in the nucleus of an element is equal to its number of orbiting electrons, and therefore the number of protons determines the sequence of elements in the Periodic Table, but this fact was not always clear. After the proton was identified, it became apparent that the weight of an element's atom was much larger than the weight of a number of protons equal to the number of orbiting electrons (i.e. to the atomic number). The most logical explanation was that the nucleus contained an excess of protons and a certain amount of electrons to balance the extra charge.

Example : Carbon atoms were known to be about 12 times heavier than protons, and were also known to contain a total of 6 orbital electrons. It was then assumed that the nucleus of Carbon contained in fact 12 protons and, in addition, 6 electrons , for a total nuclear charge of +6. This +6 charge was then balanced by the 6 orbital electrons.

It was not until the 30's that it was realized that in reality there are no electrons within the nucleus, but that the extra weight is due to a particle similar to the proton but with no electrical charge, the neutron.

Things then fell into place in a more natural way : in any element, the number of nuclear protons equals the number of orbital electrons. This number is what determines the position of an element in the Periodic Table. In addition, the nucleus of each element contains neutrons, in amounts similar, but not necessarily identical, to the number of protons. A given element, of well defined chemical properties, could contain variable amounts of neutrons in its nucleus, without this affecting its chemical properties. And this should not be surprising, since we know by now that chemical properties are purely determined by the outer electrons.

Nuclei of an element with different amounts of neutrons will still occupy the same location in the Periodic Table, therefore they are referred to as isotopes (= same place). For example, Carbon nuclei will always consist of 6 protons (otherwise it wouldn't be Carbon anymore), but can have 6, 7 or 8 neutrons, and would then be identified as ¹²₆C, ¹³₆C and ¹⁴₆C.

The Nuclear Force

As soon as we realize that nuclei are made of (positive) protons and (neutral) neutrons, an immediate question comes to mind. We know that charges of the same sign exert a strong repulsive force on each other; how can it be that protons are held together in the nucleus, rather than flying apart due to electrostatic repulsion?

It would be easy to convince oneself that the electrostatic repulsion among protons in the nucleus is very, very strong. Remembering that the electric force depends upon the inverse square of the distance, and that the nucleus dimensions are 100,000 (10⁵) times smaller than the electron-nucleus typical distances, this implies that the repulsive force between protons in the nucleus is 10¹⁰ times bigger than the force that keeps electrons orbiting around the nucleus !!!

To explain nuclear stability we must therefore assume that another force, strong enough to counteract the electrostatic force, must be at play: this is in fact the nuclear force.

Note: the nuclear force holding nuclei together is nowadays referred to as strong nuclear force; this nomenclature was introduced to distinguish it from another, much weaker, force felt by sub-atomic particles that we will examine later, the so called weak force.

The understanding of the strong force, and its applications, have made considerable progress since its early days, but even so, it should be made clear that, even today, we cannot say we have a complete understanding of it. We cannot for instance write down an equation that describes the strength of the force, as we can do with Newton's or Coulomb's laws. For many years, most of what was known about the strong force was empirical, i.e. based upon experimental data, and no comprehensive theory was available. Only recently we have moved towards reaching a real theory of the strong force, but this progress was due more to advances in the study of elementary particles than to the study of nuclei.

In spite of these limitations, very much is known about the nuclear force, and its well known applications have played a major role in modern history. Here are some of the basic aspects of the force :

• contrary to electricity and gravity, the strong force is only felt over very short distances, i.e. over distances of the order of the nucleus dimensions (10^-15 m). This is why, in spite of its strength, nuclear force effects are not visible in everyday's lfe: the nuclear force is not felt at all is felt between different nuclei in ordinary matter.
• typical energies of the nuclear world are millions of times larger than energies involved in chemical reactions. Let us recall a unit we had introduced to measure energies in the atomic and sub-atomic world, the electron-Volt.

  We had defined the electron-Volt as the energy acquired by a particle with the charge of one electron when being accelerated across the potential of 1 Volt. We had also stated that energies typical of orbital electron transitions (i.e. energies of visible photons) are of the order of a few eV's. We had also mentioned that typical energies of X-rays are around thousands of electron-Volts (keV). When dealing with nuclear phenomena, typical energies involved are at the level of millions of eV (MeV).

• strong nuclear force and nuclear stability can only be understood in the light of Einstein's E=mc². From the early days of nuclear research it was shown that the mass of a nucleus is somewhat less than the sum of the masses of its constituents protons and neutrons, when taken as free particles. In the process of forming a nucleus, protons and neutrons give up a small fraction of their mass and release an equivalent amount of energy. This "missing mass" is what gives the nucleus its stability : a nucleus is stable because it does not have enough mass to allow its constituents to be in a free state..... The "missing mass" or "missing energy", i.e. the difference between the mass of the nucleus and the sum of the masses of its protons and neutrons, is called the binding energy: this is the energy that would need to be provided in order to completely separate the nucleus into an assembly of unbound protons and neutrons.

For practical exploitations of the nuclear forces, even more important than nuclear stability are the consequences of nuclear instability. Experience has shown that not any arbitrary assembly of protons and neutrons can give origin to a stable configuration. Typically, for smaller nuclei, stable configurations correspond to roughly equal numbers of protons and neutrons, with the ratio of neutrons to protons increasing with atomic number. The bigger the nucleus, the harder it is to reach a stable configuration, and in fact, after a certain atomic number, no stable configuration exists for the heavier elements.

If a nucleus is in an unstable configuration, it will spontaneously undergo radioactive decay, and the products of the decay will be detected as radioactivity. Radioactivity was first observed about 100 years ago (1896, same year as the discovery of the electron) by Becquerel, who observed by chance how some Uranium containing minerals could affect a photographic plate. Following studies revealed that radioactivity could manifest itself into three different types that, for lack of a better name, were called $\alpha , \beta$ and $\gamma$. Nowadays we have a very good understanding of what the three types of radiation are and how they do come about.