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1. 16.451: Yesterday’s news!. Solid line: fit to the semi-empirical formula. almost flat, apart from Coulomb effects. Most stable: 56 Fe, 8.8 MeV/ nucleon. gradual decrease at large A due to Coulomb repulsion. some large oscillations at small mass. very sharp rise at small A. 2.

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16.451: Yesterday’s news!


Solid line: fit to the

semi-empirical formula

almost flat, apart from Coulomb effects

Most stable: 56Fe,

8.8 MeV/ nucleon

gradual decrease

at large A due to

Coulomb repulsion

some large


at small mass

very sharp rise

at small A


16.451 Lecture 22: Beyond the mass formula... 25/11/2003


Semi-Empirical Mass (binding energy) Formula (SEMF) implications:


Stable nuclei have the maximum B for a given A; for constant mass number,

B is quadratic in Z “mass parabolas”, e.g.:

Even A: offset

is the pairing term!



Beyond the SEMF: “Magic Numbers” and the Shell Model

We already noted that there

were some marked deviations

from the SEMF curve at small

mass number, e.g. A = 4.

On an enlarged scale, a systematic

pattern of deviations occurs, with

maxima in B occurring for certain

“magic” values of N and Z, given by:

N/Z = 2, 8, 20, 28, 50, 82, 126

These values of neutron and proton

number are anomalously stable

with respect to the average – the

pattern must therefore reflect

something important about the

average nuclear potential V(r) that

the neutrons and protons are

bound in....

(NB, the most stable nucleus of all is

56Fe, which has Z = 28, N = 28,

“double magic” ...)


Other evidence for “magic numbers” 2-n and 2-p separation energies:


  • Energy required to remove a pair of
  • neutrons or protons from a given nucleus is
  • referred to as S2n or S2p
  • like the ionization energy for atoms, but
  • the pairing force is so strong in nuclei that
  • systematics are more easily seen comparing
  • nuclei that differ by 2 nucleons
  • the same pattern of “magic numbers”
  • appears – large separation energies
  • correspond to particularly stable nuclei:
  • N/Z = 2, 8, 20, 28, 50, 82, 126 ....

A periodic table of nuclei?


Systematics are reminiscent of the periodic structure of atoms, which results from

filling independent single-particle electron states with electrons in the most efficient

way consistent with the Pauli principle, but the magic numbers are different:

“Magic numbers” for atoms:

Z = 2, 10, 18, 30, 36, 48, 54, 70, 80, 86 ....




Single Particle Shell Model for Nuclei:


Self-consistent approximation: assume the quantum state of the ith nucleon can be found

by solving a Schrödinger equation for its interaction via an average nuclear potential VN(r)

due to the other (A-1) nucleons:

Assume a spherically symmetric potential VN(r);

then the eigenstates have definite orbital angular

momentum, and the standard radial and angular

momentum quantum numbers (n,l,m) as indicated.

(Justification: measured quadrupole moments of

nuclei are relatively small, at least near the

“magic numbers” that we are interested in explaining;

midway between the last two magic numbers, ie

around Z or N = 70, 100, the picture changes, and

we will have to use a different approach, but at

least for the lighter nuclei this assumption should

be reasonable.)



Shell Model continued...

If we choose the right potential function VN(r), then the wave function for the whole

nucleus can be written as a product of the single particle wave functions for all A

nucleons, or at least schematically:

oversimplification here... actually, it has to be written as an

antisymmetrized product wavefunction since the nucleons are

identical Fermions – the procedure is well-documented in advanced textbooks in any case!

With total angular momentum given by:

And parity:

Always + for an even

number of nucleons...


What to use for VN(r)? – three candidate potential functions:


Advantage: easy to write down


numerical solutions only

edges unrealistically sharp

Advantage: easy to write down and

can be solved analytically

Disadvantage: potential should

not go to infinity, have to cut off

the function at some finite r and

adjust parameters to fit data.

Advantage: same shape as measured

charge density distributions of

nuclei. smooth edge makes sense


numerical solution needed


Comparison: Harmonic Oscillator versus Woods-Saxon solutions:


  • since both potentials are spherically
  • symmetric, the only difference is in
  • the radial dependence of the wave
  • functions
  • amazingly, when parameters are
  • adjusted to make the average
  • potential the same, as shown in the
  • top panel, there is remarkably
  • little difference in the radial
  • probability densities for these
  • two potential energy functions!
  • this being the case, the simplicity
  • of the harmonic oscillator potential
  • means that it is strongly preferred
  • as a model for nuclei


square of the harmonic

oscillator wave function

for the last proton in 206Pb,

quantum numbers: n = 3, l = 0

--- it works !!!

Evidence that this works: (Krane, Fig. 5.13)


electric charge density, measured via electron scattering:

charge density difference

between 205Tl and 206 Pb is

proportional to the square

of the wave function for

the extra proton in 206 Pb,

i.e. we can actually measure

the square of the wave

function for a single proton

in a complex nucleus this



Various potential shapes lead to similar patterns of energy gaps, e.g.:


But the magic numbers are wrong  !

N/Z = 2, 8, 20, 28, 50, 82, 126

Something else is needed to explain

the observed behaviour...


as in the calculation of magnetic moments, lecture 19, we can write:

  • but there are only two ways the orbital and spin angular momentum can add for a
  • single particle nucleon state:
  • a) “stretched state” j = l + ½ :
  • b)“jack-knife state” j = l – ½:

Solution: the “spin-orbit” potential


  • Meyer and Jensen, 1949: enormous breakthrough at the time because it was the
  • only explanation for the observed pattern of “magic numbers” and paved the way
  • for a “periodic table” of nuclei ... and the Nobel prize in physics, 1963!
  • (
  • – see Maria Goeppert-Meyer’s Nobel Lecture link on this page)
  • simple idea:



Spin-orbit force, continued:

  • The energy shift due to the spin-orbit interaction is between states of the same
  • l but different j;
  • the splitting is proportional to land so it increases as the energy increases for the
  • single particle solutions to V(r)
  • each state can accommodate (2j+1) neutrons or protons, each with different mj
  • empirically, the sign of the spin-orbit term for nuclei is opposite to that for atoms
  • and the effect is much stronger in nuclei– the phenomenon has nothing to do with
  • magnetism, which is the origin of this effect in atoms, but rather it reflects a basic
  • feature of the strong nuclear force.
  • with these features, the spin-orbit potential is the “missing link” required to
  • correctly predict the observed sequence of magic numbers in nuclear physics 

something to read....


see also!