5 Nuclear Force and its Properties-1

Sanjay Kumar Chamoli

Learning Outcomes

From this module students may get to know about the following:

The knowledge of nuclear force and its properties.
The importance of deuteron in understanding nuclear force.
The deuteron is a loosely bound system of a proton and a neutron.

1. Nuclear Force (Strong interaction)

Nuclear force, the force between nucleons, is one of the four fundamental forces found in nature. The nuclear force is powerfully attractive between nucleons at distances of ~ 1 fm between them, rapidly decreases at distances beyond about 2.5 fm and repulsive at distances less than 0.7 fm. This repulsive component is responsible for the physical size of nuclei, since the nucleons can come no closer than the force allows. Nuclear force can’t be of electrical origin since it acts between charged as well as neutral particles. Also it cannot be of magnetic origin as the interaction between magnetic moments of nucleons is extremely weak. The strong interaction, mediated by particles called gluons, is the attractive force that binds the quarks together to form the nucleons themselves. The strong interaction is the mechanism responsible for the strong nuclear force. In general the nuclear force has many properties which qualify it to be a different kind of force.

Properties of Nuclear Force

Nuclear force is short range force having range of few Fermi.
Nuclear force is attractive in nature (upto certain distance within the nucleus).
Not all the particles are subjected to the nuclear force. For example, electrons are not subjected to the nuclear force.
The nucleon-nucleon force depends on whether the spins of the nucleons are parallel or antiparallel.
Nuclear force has saturation property. i.e., constant B.E./A for most nuclei’
Nuclear force is charge independent (acts between p-p, n-n & p-n)
Since scattering length is a measure of the interaction, so the observation of different scattering lengths of pp, nn & np in the low energy scattering experiments is a confirmation of different strengths of nuclear force between proton –proton, neutron – neutron and neutron -proton.

Nuclear force is repulsive at very short distances as the nucleus has constant central density and a repulsive phase shift for higher energies (E > 300 MeV).

Figure 1 shows the phase-shift plot with energy observed in a typical neutron-proton scattering experiments at medium energies. The plot clearly shows that the s-wave phase shift becomes negative for S > 300 MeV, indicating a change in nature of nuclear force from attractive to repulsive at those energies.

Fig. 1: The phase shifts from neutron-proton scattering at medium energies.

(Phys. Rev. 182, 1714 (1969)

1.1. Understanding nuclear force (Deuteron problem)

A deuteron (2H nucleus) is a loosely bound system consisted of a neutron and a proton. Most of the information about nuclear force among nucleons can be obtained from the study of a simple two nucleon system like deuteron. Like hydrogen, the deuteron also does not have excited states because it is a weakly bound system. The deuteron has got many ground state properties measured over the years

Ground state properties of deuteron: The deuteron is observed to have the following properties in its ground state.

1.1.1. Angular momentum of Deuteron

The total angular momentum I of the deuteron is given by

I = Sn+Sp + l

Here Sn and Sp are the individual spins of the neutron and proton (each equal to ½ ) and l the orbital angular momentum of the nucleons. Since it is an isotope of hydrogen, so the ground state of deuteron is assumed to have zero orbital angular momentum l = 0 (not fully true as experiments show that deuteron has small quadrupole moment in its ground state).

So, if the neutron and proton spins are parallel (Sn || Sp), the total angular momentum is given by

I = Sn+Sp = ½ + ½ = 1

The implication is that two nucleons are not bound together if their spins are anti-parallel, and this explains why there are no p-p or n-n bound states. The nuclear force is thus seen to be spin dependent.

1.1.2 Binding Energy of Deuteron

The binding energy of the deuteron can be determined in three different ways.

a) Mass doublet method:

mass of deuteron = 1875.6 MeV

b) By directly measuring the energy of γ-ray released on the formation of deuteron after bringing a proton and a neutron together:

The observed value of binding energy is

B = 2.2245 ± 0.000002 MeV

c) Photo-dissociation:

In this method a reverse reaction is used in which a γ-ray photon breaks apart a deuteron.

The minimum y-ray energy required is equal to the binding energy. The observed value of binding energy is

B = 2.224 ± 0.002 MeV

As average B.E./A of nuclei is 7 ~ 8 MeV, so low B.E. of deuteron (B = 2.224 MeV) clearly indicates that the deuteron is very weakly bound system.

Quantum mechanical description of weak binding for deuteron:

Fig. 3: A picture of square well potential with a depth V0.

Consider a three-dimensional square-well potential as shown in figure 3.

V (r) = – V0 for r < R

= 0 for r > R

Here r represents the separation between the proton and the neutron and R is a measure of the diameter of the deuteron.

The Schrödinger’s equation is given by

Fig. 4: Square well potential showing the deuteron binding energy V0 = -2.2 MeV.

In figure 5, we can see the deuteron wave function for R = 2.1 fm. The exponential joins smoothly to the sine at r = R, so that both u(r) and du/dr are continuous.

If the nucleon-nucleon force were just a bit weaker the deuteron bound state would not exist at all and our universe would have been quite different.

Fig. 5: The deuteron wave function for R = 2.1 fm.

Summary:

The nuclear force is an important force for the existence of life on earth. This force plays an important role in giving various properties to the nuclei. Understanding the nuclear force and its behaviour within the nuclear dimensions is important not only to explain the various observed properties of nuclei but also to predict many other properties in nuclei which are so far inaccessible. To study the properties of nuclear force, deuteron provides an ideal case as it is a loosely bound system of proton and a neutron. Its quantum mechanical treatment shows that its binding energy is ~ – 2.2 MeV, a much less than ~ – 39 MeV required for a strongly bound 2-nucleon system.

References:

Introduction to Nuclear Physics – by Keneth S Krane.
Introductory Nuclear Physics – by Samuel S M Wong.
Nuclear Physics – by R R Roy & B P Nigam.
Handbook of Physics by Condon and Odishaw, TMH NewYork.
Introduction to Nuclear Physics, 2nd Edition, W.N.Cottingham & D.A. Greenwood.
Concept of Nuclear Physics by B L Cohen, McGraw Hill.
Nuclear Physics ; an Introduction by S.B. Patel.
The Origin of the Concept of Nuclear Force by L.M. Brown and Rechenberg.
Theoretical Nuclear Physics by John M. Blatt and Victor F. Weisskopf.
Experimental techniques in Nuclear Physics by Dorin N. Poenaru & Walter Greiner
Exotic Nuclear Excitation by S.C. Pancholi
Nuclear spectroscopy Part B, by Fay Ajzenberg- Selove
Theory and Problems of modern Physics (Schaum’s outline Series)
Basic Ideas & Concepts in Nuclear Physics – by K Heyde
The “Particles of Modern Physics” by J. D. Stranathan, Philadephia: Blakiston.
Nuclear Physics by Irving Kaplan, Narosa Publishing House.

Web Links

Did you know ?

The strong force is one of the four known fundamental interactions of nature, the others being electromagnet force, weak force and gravitational force.
The nuclear force is a short range forces and acts only within the nuclear dimensions (few femtometer). The strong force obeys a quite different distance-dependent behavior between nucleons, from when it is acting to bind quarks within nucleons.
Despite only operating at a distance of a femtometer, it is the strongest force in nature. It is approximately 100 times stronger than electromagnet force, a million times stronger than weak force and 1038 times stronger than gravitational force at that range.
The strong force acts between protons and neutrons (made of quarks and glueons) within the nucleus.
The strong force is mediated by massless particles called gluons that are exchanged between quarks, antiquarks, and other gluons.
The Gluons, are thought to interact with quarks and gluons through a type of charge called color charge.
Color charge is analogous to electromagnetic charge, but it comes in three types rather than one (+/- red, +/- green, +/- blue) that results in a different type of force, with different rules of behavior.
Quarks and gluons are the only fundamental particles that carry non-vanishing color charge, and hence participate in strong interactions.
The contemporary understanding of nuclear force is described by quantum chromodynamics (QCD), a part of the standard model of particle physics.
All quarks and gluons in QCD interact with each other through the strong force and the strength of interaction is parametrized by the strong coupling constant.
In general the nuclear force ensures the stability of ordinary matter and confines quarks into hadron particles, such as the proton and neutron, the largest components of the mass of ordinary matter.

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