GENERAL I ARTICLE The Story of Nuclear Matter R Rajaraman This article gives an introduction to the Nuclear Matter - a subject to which Professor Bethe had made seminal contributions. Most of the discus­ sion here will be devoted to describing the ideas that led up to postulating the nuclear matter sys­ tem. That will be followed up with a qualitative description of the difficulties faced and the meth­ R Rajaraman started his ods developed in tackling this system from first research work in the area principles. The article will also be used as an of nuclear many-body excuse to explain many basic features of nuclear theory. Subsequently he enlarged his interests to physics to students, using simple arguments. include particle physics phenomenology, quantum Introduction field theory, soliton theory Professor Hans Bethe was one of the founding fathers of and quantum Hall systems. In recent years nuclear theory. Starting from the early thirties, he made he has also been working numerous pioneering contributions to different aspects on technical and policy not only of nuclear physics, but practically every branch aspects of reducing l of physics . Finally, in the mid-fifties, he embarked on nuclear weapon dangers in the task of solving the Nuclear Matter problem. It took South Asia. over a dozen years before he and his co-workers com­ pleted this task. Of course Bethe was doing many other things alongside, both in physics and on serious public policy issues. But the nuclear matter problem was per­ haps his longest project in physics. This was also the period when the author had the invaluable opportunity 1 Some of these contributions of working closely with Professor Bethe. It is therefore have been discussed in other articles in this issue. tempting to say something about this subject here. But nuclear matter happens to be a particularly nasty and intractable many-body system, for reasons we will explain. Correspondingly, the theoretical structure put together for understanding it also turned out to be com­ Keywords plex and elaborate. Therefore, rather than go into the Nuclear forces, nuclear matter. intricacies of nuclear matter theory it may be more use- -8------------------------------~------------R-ES-O-N-A-N-C-E--I--O-ct-o-be-r--2o-o-S GENERAL I ARTICLE ful here to bring out the basic ideas in nuclear physics About a century ago that lead up to and motivated the study of nuclear mat­ the physics changed ter. in a dramatic way to What is nuclear matter? What is its significance to nu­ the extent that even clear theory in general? What was so important about the philosophical this system that Bethe, after decades of tackling so many foundations of science different aspects of nuclei, decided to make its theory his had to be modified in last and longest project in nuclear physics? And what a fundamental sense. makes it such a difficult system to analyze? We wIll try to answer some of these questions here in simple terms and in the process also, hopefully, explain to student readers some basic concepts in nuclear physics. In its heydays in the mid-twentieth century nuclear physics was considered in the popular lexicon to represent the most rarified and abstruse of sciences. Yet it is possible to extract many features of nuclear physics using ex­ ceedingly simple semi-classical arguments, bordering on just commonsense. Let us begin with a basic question whose answer is of­ ten, post facto, taken for granted. Recall that about a century ago, as the frontiers of physics took us from lnacroscopic objects down to atoms and molecules, the physics changed in a dramatic way to the extent that even the philosophical foundations of science had to be modified in a fundamental sense. The time-tested ideas of deterministic classical physics had to be replaced by a probabilistic, far less intuitive, quantum picture. These dramatic changes happened when we went from, say, grains of sand, for which classical physics is still valid to typical molecules which are smaller in size by a factor of 10-6 (a millionth). In the same way, as nuclei and their properties were being discovered, physicists could well have wondered if the same laws of quantum physics which work so spectacularly for atoms and molecules would continue to work equally well for nuclei. After all, &'3 you go down from atoms to nuclei, you are again going down in size by a huge factor of about 100,000 -RE-S-O-N-A-N-C-E-I--O-c-to-be-r--20-0-5----------~-----------------------------9 GENERAL I ARTICLE from Angstroms· (10-8 cm) typifying atomic distances, to Fermis (10-13 cm), the scale of nuclear sizes. Would the basic laws of quantum physics survive this transi­ tion? As it happened, when calculations were done, one after another, in the nascent field of nuclear physics using the same principles of quantum mechanics that worked for atoms, those principles seemed to hold equally well here too. It helped that nucleons (the common name given to both protons and neutrons) in nuclei are also, to a fair approximation, non relativistic, although less so than 2 2 The electron in the H atom, for electrons in atoms . instance, moves, on the aver­ age, at about one hundredth But while general principles of non-relativistic quantum of the speed of light (vic == 11 theory seemed to hold in the nuclear realm as well, most 100) whereas for nucleons vic calculations of nuclear structure, except for the deuteron == 1/10. But relativistic effects depend only on the square of and other few nucleon systems, were 'phenomenological' this fraction, i.e. on (v/c)2 They did not start from first principles, by solving the Therefore up to an accuracy of basic Schrodinger equation for the nucleons inside the a few percent, non relativistic nucleus in terms of their primary forces. Instead they quantum mechanics can still be began at some intermediate level 'model' with names used for nuclei. such as the Shell model, the Liquid Drop model, the Collective models and so on. The absence of first principles calculations should not surprise us. A typical nucleus, say, that of aluminum, which is not an especially large one, has 12 protons and 14 neutrons. But we know that even in classical New­ tonian mechanics it is hard to exactly solve problems in­ volving more than 2 interacting particles. Every particle exerts a force on every other and the problem becomes highly coupled. When only two bodies are involved, such as the Sun and the Earth (the Kepler problem) one can 'separate' the problem into two uncoupled parts by changing to center of mass (c.m.) and relative coordi­ nates. The c.m. travels with uniform velocity and one then needs only to solve the effective one-body problem involving the relative coordinate. The moment you have 1-O-----------------------------~-----------R-E-S-O-N-A-N-CE--I--O-c-to-be-r--20-0-5 GENERAL I ARTICLE three or more bodies such separation is in general not Individual electrons possible. The c.m. coordinate can still be separated can be viewed as from the others but the rest remain entangled with one simply moving in the another forcing us to look for difficult approximations. electric field of the The same is true in quantum mechanics. We can analyse nucleus, which can the hydrogen atom problem relatively easily since it in­ be taken to be at rest volves only a proton and an electron. But when you go at the origin. to larger atoms involving several electrons, the problem becomes impossible to tackle analytically even though the forces between the different particles obey the simple Coulomb's law. Various approximations, like perturba­ tion theory, the variational method, the Thomas Fermi model, etc., need to be employed. Diffculties of Nuclear Theory The corresponding problem in nuclear physics, of several nucleons interacting quantum mechanically with each other, is even more difficult than the one involving sev­ eral electrons in atoms. There are two reasons for this: 1. In an atom, there is great asymmetry between the sta­ tus of the nucleus and that of the electrons. The nucleus is not only much heavier than the electrons, but also car­ ries a large electric charge which acts like a strong point charge at the center. This can be exploited to set up approximations where the individual electrons can be viewed, in the first approximation, as simply moving in the electric field of the nucleus, which can be taken to be at rest (because of its relative heaviness) at the ori­ gin. Of course the electrons are also interacting with one other, but those effects are treated as corrections. This approach has had considerable success in atomic physics. By contrast, in a nucleus all the nucleons are on an equal footing. They all carry roughly the same mass and ex­ ert similar forces on one another, and there is no nat­ ural center to which you can anchor your approximation process. --------~-------- RESONANCE I October 2005 11 GENERAL I ARTICLE The nuclear forces 2. The nuclear forces holding nucleons together are much more complex than the electromagnetic force oper­ holding nucleons ating in atomic phenomena. The force between two nu­ together are much cleons (the so-called Strong force) , obtained by study­ more complex than ing the scattering of nucleons off one another, consists the electromagnetic of several terms whose dependence on distance is a com­ force operating in plicated function, unlike the simple 1/1'2 Coulomb force. atomic phenomena. Furthermore there is also the possibility of intrinsic three­ body and higher many-body forces acting among nucle­ ons.