Nuclear physics is the study of atomic nuclei, their constituents, and the interactions that hold them together. Nuclei are the massive cores at the center of atoms and are made up of protons and neutrons (which are called hadrons) which determine the element identity and isotope, and some of the radioactive processes . Nuclei make up most of the visible mass around us, and are critical to the inner workings of stars, the origin of chemical elements, and the early universe. The hadrons themselves are composites of more fundamental particles known as quarks and gluons and their interactions lead to the strong nuclear force that provides the binding force to hold protons and neutrons near each other. This is mathematically described in the theory of quantum chromodynamics (QCD).
More than 99% of the mass of the visible matter in the universe is nuclear matter. Protons and neutrons are the building blocks of atomic nuclei. Exotic forms of nuclear matter were present in the early universe and continue to exist today in neutron stars. Nuclear fusion processes at the core of our Sun are the source of the vast energy flow that sustains life on Earth. Nuclear fusion in stars and nuclear processes at the end of stellar life have formed the rich spectrum of elements we observe in nature.
Experimental nuclear physics drives innovation in scientific instrumentation and has a far-reaching impact on research in other fields of science and engineering. From medicine— x-ray and magnetic resonance imaging, radiation therapies for cancer treatment—to materials science— x-ray lithography and neutron scattering—to propulsion and energy production—nuclear physicists have changed our world. The current research in nuclear physics is not only unraveling fundamental questions about matter and energy but also enabling a host of new technologies in materials science, biology, chemistry, medicine, and national security.
The quantum mechanical operator associated with the total energy of a quantum system (nucleus in this case) is called the Hamiltonian. In classical mechanics, the system energy can be expressed as the sum of the kinetic and potential energies. For quantum mechanics, the elements of this energy expression are transformed into the corresponding quantum mechanical operators. The Hamiltonian contains the operations associated with the kinetic and potential energies. We will deduce the Hamitonian for a nucleus in this post.
There is more complexity to the mass formula of the nucleus than just a simple linear dependence on the total number of nucleons which are protons and neutrons. Readily at first , there seems to be the kinds of energies mentioned below that should play the key role:
Bulk energy
Surface energy
Pauli or Symmetry energy
Coulomb energy.
Each of these energies will be explained in this post.
Each nucleus has a (positive) charge \(Ze\), and integer number times the elementary charge \(e\). This follows from the fact that atoms are neutral
Nuclei of identical charge come in different masses, all approximate multiples of the “nucleon mass”. (Nucleon is the generic term for a neutron or proton, which have almost the same mass, \(m_p= 938.272 \text{MeV}/c^2\), \(m_n= \text{MeV}/c^2\).) Masses can easily be determined by analysing nuclei in a mass spectrograph which can be used to determine the relation between the charge \(Z\) (the number of protons, we believe) vs. the mass.
Electrostatic repulsion and gravitational attraction between masses in the nucleus are two forces that act between nucleons (protons and neutrons). The magnitude of electrostatic repulsion between protons is much greater than the gravitational attraction between the nucleons. In fact, the gravitational attraction between nucleons is so small it can be disregarded in most calculations involving the forces between nucleons
There are two important classes of nuclear models: single particle and microscopic models, that concentrate on the individual nucleons and their interactions, and collective models, where we just model the nucleus as a collective of nucleons, often a nuclear fluid drop. Microscopic models need to take into account the Pauli principle, which states that no two nucleons can occupy the same quantum state. This is due to the Fermi-Dirac statistics of spin 1/2 particles, which states that the wavefunction is antisymmetric under interchange of any two particles.
The simplest of the single particle models is the nuclear shell model. It is based on the observation that the nuclear mass formula, which describes the nuclear masses quite well on average, fails for certain “magic numbers”, i.e., for neutron number \(N=20,28,50,82,126\) and proton number \(Z=20,28,50,82\), as shown previously. These nuclei are much more strongly bound than the mass formula predicts, especially for the doubly magic cases, i.e., when \(N\) and \(Z\) are both magic. Further analysis suggests that this is due to a shell structure, as has been seen for electrons in atomic physics.
Among nuclear models, another and actually older way to look at nuclei is as a drop of “quantum fluid”. This ignores the fact that a nucleus is made up of protons and neutrons, and explains the structure of nuclei in terms of a continuous system, just as we normally ignore the individual particles that make up a fluid.
Once we have started to look at the liquid drop model, we can try to ask the question what it predicts for fission, where one can use the liquid drop model to good effect. We are studying how a nuclear fluid drop separates into two smaller ones, either about the same size, or very different in size. This process is indicated in Figure \(\PageIndex{1}\).