Physics 357 : 1998 Lecture 24

Semi-empirical Nuclear Mass Formula

The binding energy of an atom (Z protons & electrons, N neutrons, A=N+Z) is the difference between the summed masses of the separate nucleons and electrons and the atomic mass:

(See F&H Eqns. 16.1-3)

The net binding energy is due to several factors, but the measured binding energy per nucleon is almost constant (B/A~8MeV) for stable nuclei withA>25.

(See F&H Figure 16.1.)

This binding energies can be approximately understood as the sum of several terms.

Coulomb Energy: It takes energy to force protons together into a nucleus. We can estimate this energy by assuming that the nucleus is spherical (radius R) and that the nuclear charge (Ze) is distributed uniformly inside the nucleus. Classically the electrostatic energy the nucleus is then

If we assume nuclei are formed from incompressible nucleons with fixed nuclear bond lengths, then the nuclear volume should be proportional to A, and the nuclear radius (assuming spherical nuclei) should be

and

or

Volume Energy: Nuclear forces are short range. If we assume nuclear matter is uniform, then each nucleon will feel the same attractive nuclear force and have the same nuclear contribution to the binding energy:

Surface Energy: Nucleons at the surface are not surrounded by other nucleons so they should have less nuclear binding energy. (The forces on the surface are similar to surface tension in a liquid drop, so this force favours spherical nuclei.) If the nuclear forces are nearest neighbour forces, then the this surface contribution to the binding energy should just be proportional to the number of surface nucleons. For fixed radius nucleons, the number of nucleons at the surface should just be proportional to the surface area (4pR²~V^2/3~A^2/3) for a spherical nucleus) and the surface energy contribution is:

Symmetry Energy: Nucleons are fermions, so no two protons (or no two neutrons) can be in the same state. As you add nucleons to a nucleus, the nucleons must go into higher and higher energy states, with one neutron and one proton allowed to share each quantum state. Thus the Pauli exclusion principle favours nuclei with equal numbers of neutrons and protons. If we assume the nucleons in a nucleus form a cold degenerate Fermi gas, then the asymmetric nuclei pay an energy penalty:

So the total binding energy of a nucleus can be estimated by the Bethe-Weisäcker semiempirical mass formula

          B = E_coul + E_vol + E_surf+ E_sym

            = - a_c Z²/A^1/3 + a_v A - a_s A^2/3 - a_sym (Z-N)²/A

or (F&H Eqn. 16.10)

It must be remembered that this formula is only approximate. For example:

• not all nuclei are spherical (F&H Section 18.1)
• nucleons are compressible (e.g. F&H Chapter 16.3)
• the strong force is not just a nearest neighbour force
• the nucleus has a shell structure more complicated that a simple Fermi Gas ball (F&H Chapter 17), e.g.
• spin-orbit couplings.
• closed shells are more tightly bound leading to "magic numbers" for stable nuclei with N or Z =2, 8, 20, 28, 50, 82, 126.

 Return to List of Lecture Notes

Copyright 1998 David Bailey, University of Toronto

A BLACK HOLE'S EVENT HORIZON HAS BEEN DETECTED. Ramesh Naryan and his colleagues at the Harvard- Smithsonian. Center for Astrophysics have used the orbiting ASCA x-ray telescope to study x-ray novas, binary systems in which gas from one star is pulled toward an accretion disk and the spherical region surrounding a compact companion. These systems occasionally flash prominently at x-ray wavelengths (hence the name x-ray nova), but Naryan is more interested in what happens during the quiescent intervals between upheavals. His recent theory, called the advection-dominated accretion flow (ADAF) model, suggests that if the accretion rate is slow enough the inspiraling gas will refrain from radiating away its accumulating energy. Instead the gas continues to get ever hotter, reaching temperatures as high as 10^12 K. Eventually this enormous energy buildup is dealt with in one of two ways: if the compact object is a neutron star, the gas will fall onto its surface, where it heats the star, causing it to radiate. In contrast, if the object is a black hole, there is no surface for the gas to fall upon; instead, like a prisoner being led to execution, the gas crosses the black hole's event horizon, never to be seen again. In effect, 99% of the gas energy disappears from the universe. Because of this, x-ray binaries containing a black hole should be dimmer than those with neutron stars. Naryan, speaking at this week's meeting of the American Astronomical Society in Toronto, reported on 9 binaries which fit the ADAF pattern of behavior. Four of these were thought to harbor black holes (because of their higher masses), and indeed these are all dimmer than the five neutron-star binaries. Naryan judges this dimness, and the binaries' x-ray spectra, to be the sign that an event horizon is at work, and that this in turn constitutes the most direct evidence yet for the existence of black holes.