1. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 1 Nuclear Binding Energy B tot (A,Z) = [ Zm H + Nm n - m(A,Z) ] c 2 B  m  HW 8 B ave (A,Z) = B tot (A,Z) / A HW 8 Krane 3.9 Atomic masses from: http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=ascii&isotype=all Separation Energy Neutron separation energy: (BE of last neutron) S n = [ m(A-1,Z) + m n – m(A,Z) ] c 2 HW 9 = B tot (A,Z) - B tot (A-1,Z)  HW 9 Show that HW 10 HW 10 Similarly, find S p and S . HW 11HW 12 HW 11 Krane 3.13HW 12 Krane 3.14

2. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 2 Nuclear Binding Energy In general X  Y + a S a (X) = (m a + m Y –m X ) c 2 = B X –B Y –B a The energy needed to remove a nucleon from a nucleus ~ 8 MeV  average binding energy per nucleon (Exceptions???). Mass spectroscopy  B. Nuclear reactions  S. Nuclear reactions  Q-value

3. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 3 Nuclear Binding Energy ~200 MeV  Fission Fusion  Coulomb effectSurface effect

4. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 4 Nuclear Binding Energy HW 13 A typical research reactor has power on the order of 10 MW. a) Estimate the number of 235 U fission events that occur in the reactor per second. b) Estimate the fuel-burning rate in g/s.

5. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 5 Nuclear Binding Energy Is the nucleon bounded equally to every other nucleon? C ≡ this presumed binding energy. B tot = ½ CA(A-1) B ave = ½ C(A-1) Linear ??!!! Directly proportional ??!!! Clearly wrong … !  wrong assumption  finite range of strong force, and force saturation.

6. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 6 Nuclear Binding Energy Lead isotopes Z = 82 For constant Z S n (even N) > S n (odd N) For constant N S p (even Z) > S p (odd Z) Remember HW 12 (Krane 3.14). 208 Pb (doubly magic)  can then easily remove the “extra” neutron in 209 Pb. Neutron Number N Neutron Separation Energy S n (MeV) 208 Pb

7. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 7 Nuclear Binding Energy Extra Binding between pairs of identical nucleons in the same state (Pauli … !)  Stability (e.g.  -particle, N=2, Z=2). S n (A, Z, even N) – S n (A-1, Z, N-1) This is the neutron pairing energy. even-even more stable than even-odd or odd-even and these are more tightly bound than odd-odd nuclei.

8. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 8 Neutron Excess Remember HWc 1. Asymmetry

9. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 9 Abundance Systematics Odd NEven NTotal Odd Z Even Z Total Compare: even Z to odd Z. even N to odd N. even A to odd A. even-even to even-odd to odd-even to odd-odd. HWc 1 \

10. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 10 Abundance Systematics

11. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 11 Abundance Systematics NEUTRON NUMBER MASS NUMBER ABUNDANCE NEUTRON CAPTURE CROSS SECTION r s Formation process  Abundance

12. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 12

13. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 13 The Semi-empirical Mass Formula von Weizsäcker in 1935. Liquid drop. Main assumptions: 1.Incompressible matter of the nucleus  R  A ⅓. 2.Nuclear force saturates. Binding energy is the sum of terms: 1.Volume term.4. Asymmetry term. 2.Surface term.5. Pairing term. 3.Coulomb term.6. Closed shell term.

14. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 14 The Semi-empirical Mass Formula Volume Term B v = + a v A B v  volume  R 3  A  B v / A is a constant i.e. number of neighbors of each nucleon is independent of the overall size of the nucleus. The other terms are “corrections” to this term. constant

15. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 15 The Semi-empirical Mass Formula Surface Term B s = - a s A ⅔ Binding energy of inner nucleons is higher than that at the surface. Light nuclei contain larger number (per total) at the surface. At the surface there are: Nucleons. Remember t / R  A -1/3

16. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 16 The Semi-empirical Mass Formula Coulomb Term B C = - a C Z(Z-1) / A ⅓ Charge density   Z / R 3. W   2 R 5. Why ??? W  Z 2 / R. Actually: W  Z(Z-1) / R. B C / A = - a C Z(Z-1) / A 4/3  Remember HW 7 … ?!

17. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 17 The Semi-empirical Mass Formula

18. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 18 The Semi-empirical Mass Formula HW 14 so far Show that from our information so far we can write: For A = 125, what value of Z makes M(A,Z) a minimum? Is this reasonable…??? So …..!!!!

19. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 19 The Semi-empirical Mass Formula Light nuclei: N = Z = A/2 (preferable). Deviation from this “symmetry”  less BE and stability. Neutron excess (N-Z) is necessary for heavier nuclei. Fraction affected = |N-Z| / A Total decrease in BE  fraction x excess. B a / A = - a a (N-Z) 2 / A 2. Back to this when we talk about the shell model. Asymmetry Term B a = - a a (A-2Z) 2 / A

20. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 20 The Semi-empirical Mass Formula Pairing Term B p =  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. HWc 1 \ Remember HWc 1 \ ….?! B p 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.

21. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 21 The Semi-empirical Mass Formula Closed Shell Term B shell =  Extra binding energy for magic numbers of N and Z. Shell model. 1 – 2 MeV more binding.

22. Nuclear and Radiation Physics, BAU, 1 st Semester, 2006-2007 (Saed Dababneh). 22 The Semi-empirical Mass Formula Fitting to experimental data. More than one set of constants a v, a s ….. In what constants does r 0 appear? Accuracy to ~ 1% of experimental values (BE). Atomic masses 1 part in 10 4. Uncertainties at magic numbers. Additional term for deformation.