1. The Semi-empirical Mass Formula
Asymmetry Term Ba = - aa (A-2Z)2 / A
Light nuclei: N = Z = A/2 (preferable).
Deviation from this “symmetry”  less BE and stability.
Neutron excess (N-Z) is necessary for heavier nuclei.
Ba / A = - aa (N-Z)2 / A2.
Back to this when we talk about
the shell model.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

2. The Semi-empirical Mass Formula
Pairing Term Bp = 
Extra Binding between pairs of identical nucleons in the same state (Pauli !)  Stability (e.g. -particle, N=2, Z=2).
even-even more stable than even-odd or odd-even and these are more tightly bound than odd-odd nuclei.
Remember HWc 1\ ….?!
Bp expected to decrease with A; effect of unpaired nucleon decrease with total number of nucleons. But empirical evidence show that:
  A-¾ .
Effect on:
Fission.
Magnetic moment.
Effect of high angular momentum.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

3. The Semi-empirical Mass Formula
Closed Shell Term Bshell = 
Extra binding energy for magic numbers of N and Z.
Shell model.
1 – 2 MeV more binding.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

4. The Semi-empirical Mass Formula
Fitting to experimental data.
More than one set of constants av, as …..
In what constants does r0 appear?
Accuracy to ~ 1% of experimental values (BE).
Atomic masses 1 part in 104.
Uncertainties at magic numbers.
Additional term for deformation.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

5. The Semi-empirical Mass Formula
Variations…….
Additional physics….
Fitting……(Global vs. local)…..
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

6. Work it out …
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

7. Mass Parabolas and Stability
HW 16
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

8. Mass Parabolas and Stability
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

9. Mass Parabolas and Stability
Double  decay!
Both Sides!
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

10. Mass Parabolas and Stability
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

11. Mass Parabolas and Stability
Vertical spacing between both parabolas ?
Determine constants from atomic masses.
Odd-Odd
Even-Even
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

12. Mass Parabolas and Stability
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

13. Nuclear Spin
Neutrons and protons have s = ½ (ms = ± ½) so they are fermions and obey the Pauli-Exclusion Principle.
The Pauli-Exclusion Principle applies to neutrons and protons separately (distinguishable from each other) (Isospin).
Nucleus seen as single entity with intrinsic angular momentum .
Associated with each nuclear spin is a nuclear magnetic moment which produces magnetic interactions with its environment.
The suggestion that the angular momenta of nucleons tend to form pairs is supported by the fact that all nuclei with even Z and even N have nuclear spin =0.
Iron isotopes (even-Z), for even-N (even-A) nuclei =0.
Odd-A  contribution of odd neutron  half-integer spin.
Cobalt (odd-Z), for even-N  contribution of odd proton  half-integer spin.
Odd-N  two unpaired nucleons  large integer spin.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

14. Nuclear Spin - Z A Spin Natural Abundance Half-life Decay 26 54 0.059
0.059
stable
... 55
3/2
2.7y EC 56
0.9172 57
1/2
0.021 58
0.0028 60
1.5My -
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

15. Nuclear Spin Z A Spin Natural Abundance Half-life Decay 27 56 4 ...+ 57
7/2
271d EC 59
1.00
stable 60 5
5.272y -
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

16. Nuclear Magnetic Moment
Remember, for electrons Revise: Torque on a current loop.
Gyromagnetic ratio (g-factor)
Z component ?? Experiment, applied magnetic field.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

17. Nuclear Magnetic Moment
For Nuclei
For free protons and neutrons
Proton: g = ±  3.6 
Neutron: g = ±  3.8 
The proton g-factor is far from the gS = 2 for the electron, and even the uncharged neutron has a sizable magnetic moment!!!
 Internal structure (quarks).
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

18. Nuclear Magnetic Moment
Nuclide
Nuclear spin
Magnetic moment  (in N) n
1/2 p
2H (D) 1
17O
5/2
57Fe
57Co
7/2
+4.733
93Nb
9/2
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

19. Nuclear Parity (r)  (-r) Even. (r)  -(-r) odd.
For a nucleon  is either of even ( = +) or odd ( = -) parity.
For the nucleus  = 1 2 3 … A.
Practically not possible.
Overall  can be determined experimentally.
Overall  for a nucleus (nuclear state).
Transitions and multipolarity of transitions (-emission).
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

20. Electromagnetic moments
Electromagnetic interaction  information about nuclear structure.
Charge  electric; current  magnetic.
Electromagnetic multipole moments.
Field1/r2 (zeroth, L=0) electric monopole moment.
1/r3 (first, L=1) electric dipole moment.
1/r4 (second, L=2) quadrupole moment.
………
1/r2 magnetic monopole (questionable….!).
Higher order magnetic moments, we already discussed the magnetic dipole moment.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

21. Electromagnetic moments
Expectation value of the moment.
Each multipole moment has a parity, determined by the behavior of the multipole operator when r  -r.
Parity of  does not change the integrand.
Electric moments: parity (-1)L.
Magnetic moments: parity (-1)L+1.
Odd parity  vanish.
electric dipole.
magnetic quadrupole.
electric octupole.
…………
Vanishing moments
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

22. Electromagnetic moments
Electric monopole: net charge Ze.
Magnetic dipole: (already discussed).
g-factors.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

23. Electromagnetic moments
The nucleus has charge (monopole moment).
No dipole moment since it is all positive.
But if the nucleus is not spherically symmetric, it will have a quadrupole moment.
Classical moments
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

24. Electric Quadrupole Moment
For a point charge e: eQ = e(3z2 - r2).
Spherical symmetry  x2 = y2 = z2 = r2/3  Q = 0.
For a proton:
In the xy-plane: Q  - r2.
r2 is the mean square radius of the orbit.
Along z: Q  +2 r2.
Expected maximum  er02A2/3.
6x10-30 to 50x10-30 em2.
0.06 to 0.5 eb.
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).

25. Electric Quadrupole Moment
Nuclide
Q (b)
2H (D)
17O
59Co
+0.40
63Cu
-0.209
133Cs
-0.003
161Dy
+2.4
176Lu
+8.0
209Bi
-0.37
Closed shell  Spherically symmetric core.
Test for shell model
Strongly deformed nuclei…..!
Nuclear and Radiation Physics, BAU, First Semester, (Saed Dababneh).