16.451: Yesterday’s news! 1

16.451 Lecture 22: Beyond the mass formula... 25/11/2003 2

Most stable: 56Fe,8.8 MeV/ nucleon

gradual decreaseat large A due toCoulomb repulsion

very sharp riseat small A

almost flat, apart from Coulomb effects

Solid line: fit to thesemi-empirical formula

some largeoscillationsat small mass

123/13/2 )2()1(),( AZAaAZZaAaAaAZB ACSV 123/13/2 )2()1(),( AZAaAZZaAaAaAZB ACSV

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: offsetis the pairing term!

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

We already noted that therewere some marked deviationsfrom the SEMF curve at smallmass number, e.g. A = 4.

On an enlarged scale, a systematicpattern of deviations occurs, withmaxima 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 stablewith respect to the average – thepattern 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 is56Fe, which has Z = 28, N = 28,“double magic” ...)

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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 isreferred to as S2n or S2p

• like the ionization energy for atoms, butthe pairing force is so strong in nuclei thatsystematics are more easily seen comparingnuclei 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 ....

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Systematics are reminiscent of the periodic structure of atoms, which results fromfilling independent single-particle electron states with electrons in the most efficientway 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 ....

A periodic table of nuclei? 6

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:

)()()(2

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inlmnlinlmN rErrV

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 shouldbe reasonable.)

Q

A

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Shell Model continued...

If we choose the right potential function VN(r), then the wave function for the wholenucleus can be written as a product of the single particle wave functions for all Anucleons, or at least schematically:

A

i inlmNucleus rr1

)()(

A

i inlmNucleus rr1

)()(

oversimplification here... actually, it has to be written as anantisymmetrized product wavefunction since the nucleons areidentical Fermions – the procedure is well-documented in advanced textbooks in any case!With total angular momentum given by:

)(,,2

1

1

ssjjJ iii

A

ii

)(,,2

1

1

ssjjJ iii

A

ii

And parity:

A

ii

1)1(

A

ii

1)1( Always + for an even

number of nucleons...

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

Advantage: easy to write downDisadvantages:

numerical solutions onlyedges unrealistically sharp

Advantage: easy to write down and can be solved analyticallyDisadvantage: 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

Disadvantage: numerical solution needed

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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

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

Z

ii rer

1

2|)(|)(

electric charge density, measured via electron scattering:

charge density differencebetween 205Tl and 206 Pb isproportional to the squareof the wave function forthe extra proton in 206 Pb,i.e. we can actually measurethe square of the wavefunction for a single protonin a complex nucleus thisway!

Theory: square of the harmonicoscillator wave functionfor the last proton in 206Pb,quantum numbers: n = 3, l = 0 --- it works !!!

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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 explainthe observed behaviour...

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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!

(http://www.nobel.se/physics/laureates/1963/index.html – see Maria Goeppert-Meyer’s Nobel Lecture link on this page)

• simple idea:

srVrVrV soNN

)()()(

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

)1()1()1(2

12222

1 ssjjsjs

• 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:

2

ss

2

)1()1(

ss

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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 l and 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

ln,2/1, jn

2/1, jn

E

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something to read.... http://www.nobel.se/physics/laureates/1963/

see also!

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