Nuclear Force

Why a new force is needed

The force that determines the motion of atomic electrons is the familiar Coulomb force. It is attractive between the positive nucleus and the negative electrons, it obeys an inverse-square law, and it acts over the comparatively large atomic distances. When we turn from the atom to the nucleus, however, the Coulomb force becomes a problem rather than a solution. The nucleus packs protons and neutrons into a region only a few femtometres across, and the protons, all carrying positive charge, repel one another fiercely through the Coulomb force. Left to themselves the protons would fly apart.

Yet nuclei are stable. From the study of binding energy we know that, for average mass nuclei, the binding energy per nucleon is approximately 8 MeV. This is much larger than the binding energy in atoms, where electron binding energies are measured in electron volts. Therefore, to bind a nucleus together there must be a strong attractive force of a totally different kind. It must be strong enough to overcome the repulsion between the positively charged protons and to bind both protons and neutrons into the tiny nuclear volume.

NIOS frames the same conclusion as a process of elimination. The extremely small size of the nucleus, where protons and neutrons are closely packed, suggests that the binding force should be strong, short range and attractive. It cannot be electrostatic, because the electrostatic force between protons is repulsive; if only that force operated, the nucleons would fly away, which is contrary to experience. Gravitation is attractive between every pair of nucleons, but if the magnitude of the nucleon-nucleon force is taken to be unity, the gravitational force is only of the order of $10^{-39}$, far too weak to account for the binding. The conclusion is that the attractive force between nucleons is of a new type, the nuclear force, with no analogy in classical physics.

Strength of the nuclear force

The first listed feature of the nuclear force in NCERT is its sheer magnitude. The nuclear force is much stronger than the Coulomb force acting between charges or the gravitational force between masses. This is not an incidental property; it is a structural necessity. Inside the nucleus the nuclear binding force has to dominate over the Coulomb repulsive force between protons. That domination is possible only because the nuclear force is much stronger than the Coulomb force over the distances at which nucleons sit. The gravitational force, although it is always attractive, is much weaker than even the Coulomb force, and so plays no role in binding the nucleus.

The strength can be appreciated through the energy scale it produces. A nucleon held inside an average nucleus is bound with roughly 8 MeV, which is about a million times the few electron volts that bind an electron in an atom. A force that delivers binding energies of a few MeV per nucleon, against the Coulomb repulsion of the protons, must be intrinsically powerful. This is the sense in which the nuclear force is described as the strongest of the basic forces over its range.

Short range and saturation

The second defining feature is the very short range. The nuclear force between two nucleons falls rapidly to zero as their distance becomes more than a few femtometres. NIOS sharpens this picture by noting that the nucleons in a nucleus are very densely packed and that the force which holds them together must therefore exist between neighbouring nucleons; it is a short-range force operating over distances of the order of $10^{-15}\,\text{m}$. Once two nucleons are pulled apart by more than a few fm, the force between them is effectively switched off.

This rapid fall to zero has a direct and important consequence called saturation. Because the force reaches only as far as the nearest neighbours, a given nucleon inside a sufficiently large nucleus is under the influence of only some of its neighbours, those that come within the range of the nuclear force. Any other nucleon farther away than the range has no influence on the binding of the nucleon under consideration. NIOS states the property compactly: each nucleon interacts only with neighbouring nucleons instead of with all nucleons from one end of the nucleus to the other.

Saturation explains one of the headline facts of nuclear structure, the constancy of the binding energy per nucleon for nuclei of middle mass number. If a nucleon can have at most $p$ neighbours within range, its binding energy is proportional to $p$, say $pk$ where $k$ is a constant with the dimensions of energy. Adding more nucleons to a large nucleus does not change the binding of a nucleon already buried inside, because the newcomers are out of range. Since most nucleons in a large nucleus reside inside it rather than on the surface, the binding energy per nucleon stays nearly constant at about $pk$.

The potential-energy curve

The richest single piece of information about the nuclear force is the plot of the potential energy of a pair of nucleons against their separation. NCERT presents a rough version of this curve in Fig. 13.2. The potential energy is a minimum at a separation $r_0$ of about $0.8\,\text{fm}$. This minimum is the equilibrium point: it is the distance at which the two nucleons sit most comfortably, neither pulled together more tightly nor pushed apart.

The slope of the curve tells us the direction of the force. For separations greater than $r_0$, the potential energy rises as the nucleons move apart, so the force is attractive, pulling them back toward $r_0$. For separations less than $r_0$, the potential energy climbs very steeply, so the force is strongly repulsive, pushing the nucleons back out to $r_0$. This strongly repulsive core at small separation is what stops nucleons from collapsing into one another and keeps the average separation between nucleons roughly constant, which in turn keeps the nuclear volume proportional to the number of nucleons.

It is worth stating precisely what "attractive for $r > r_0$ and repulsive for $r < r_0$" means physically. Two nucleons brought together from far away first feel almost nothing until they are within a few femtometres. As they approach, attraction grows and draws them toward the minimum at $r_0 \approx 0.8\,\text{fm}$. If something tries to squeeze them closer than $r_0$, the steeply rising repulsive core resists, and the nucleons are pushed back toward the equilibrium separation. This combination of an attractive outer region and a repulsive inner core is exactly what produces a constant average inter-nucleon spacing and the liquid-drop-like constant density of nuclear matter.

Charge independence

The third defining feature is charge independence. The nuclear force between neutron and neutron, between proton and neutron, and between proton and proton is approximately the same. The nuclear force does not depend on the electric charge of the interacting nucleons. A proton and a neutron attract one another through the nuclear force just as strongly as two neutrons or two protons do, once the Coulomb contribution between the two protons is set aside.

NIOS connects this property to the binding-energy evidence. Since the binding energy per nucleon is found to be roughly the same irrespective of the particular mix of neutrons and protons in the nucleus, we are justified in treating the force between any pair of nucleons as equivalent. The nuclear force must account equally for the attraction between a proton and a neutron, between two protons, and between two neutrons. That equality of behaviour across all three pairings is precisely what is meant by charge independence.

Nuclear force vs Coulomb force

Placing the two forces side by side draws out every feature discussed above. The Coulomb force is the long-range, inverse-square, charge-dependent interaction that governs atomic electrons; the nuclear force is the short-range, charge-independent, saturating interaction that binds nucleons. Unlike Coulomb's law or Newton's law of gravitation, there is no simple mathematical form of the nuclear force; its character was pieced together from a variety of experiments carried out during 1930 to 1950 rather than expressed as one neat equation.

The contrast in the last two rows is the heart of the matter for NEET. Because the Coulomb force does not saturate, every proton in a heavy nucleus repels every other proton, and this cumulative repulsion grows quickly with the number of protons. The nuclear force, by saturating, cannot keep up indefinitely, since a nucleon binds only to its neighbours. This is the seed of the eventual instability of very heavy nuclei and the energetics that make fission favourable, themes developed in the binding-energy and nuclear-energy subtopics.