1. Results of Midterm 2 0 102030405060708090 points # of students GradePoints A>76 B+B+ 66-75 B51-65 C+C+ 46-50 C30-45 D,F  29 100

2. Nuclei Parameters of nuclei Strong Interaction Binding Energy Stable and Unstable Nuclei Liquid-Drop Model Numerous Applications: nuclear power applications in medicine, biology and chemistry evolution of stars and the Universe nuclear weapons

3. Size of Nuclei and Rutherford Scattering R – the fitting parameter Calculations were strictly classical. However, because of the Coulomb interaction between alpha-particles and nucleus, the result miraculously coincides with the exact quantum- mechanical one (recall the success of the Bohr model for atoms). Geiger, Marsden, Rutherford,1910 - depends weakly on A (number of nucleons in the nucleus)  -particles: bare He nuclei Scattering pattern was consistent with that expected for scattering of  particles by pointlike objects having a charge of +79e (the charge of the gold nucleus). This allowed Rutherford to put an upper limit on the size of the nucleus (<3  10 -14 m for gold). To measure the size of a nucleus, one has to use more energetic  particles (or electrons, which are more commonly used these days) that get close enough to get inside the nucleus.

4. Nuclear Mass 6 protons + 6 neutrons Atomic Mass (the mass of a neutral atom): - attraction between nucleons in the nucleus and between electrons and the nucleus Mass Unit chemical symbol for the element number of protons in the nucleus (atomic number of the element) number of nucleons in the nucleus (mass number of the nucleus) The neutron number: - we can neglect U el.-nucl. and introduce a convenient mass unit:

5. Nuclear Density The density of neutron stars is comparable with that of nuclei. !

6. The Need for a “Strong Force” Which interaction controls the size of nucleons? This cannot be electromagnetic interaction: protons have the same electric charge (they would repel each other) and also there are attractive forces between protons and electrically neutral neutrons. Strong Interaction: binds protons to protons, neutrons to neutrons, and protons to neutrons with roughly the same force does not affect certain other kinds of particles (specifically electrons) is short-ranged (the range ~ 2 fm). Nucleons separated by a larger distance exert no strong forces on each other. These observations are explained by the quark model of nucleons. Nucleons are the combination of quarks that are strong-interaction-neutral (like an electrically-neutral atom). Two nucleons interact only if they are close enough that the distances between various pairs of quarks are significantly different. g u d u quarks and gluons ~ 10 -15 m

7. Structure of Matter Atom is almost an empty space (the nuclear volume is ~10 -15 of the atomic volume) g u d u quarks and gluons ~ 10 -10 - 10 -9 m ~ 10 -15 - 10 -14 m Protons & Neutrons (nucleons) are almost an empty space (the quark size is <10 -18 m, the rest mass: U - 2.4MeV, D – 4.8MeV) El.-mag. interaction determines the size of atoms ~ 10 -15 m Strong interaction determines the size of nuclei and nucleons

8. Nuclear Mass and Binding Energy M(Z,N) = mass of a nucleus with Z protons and N neutrons, m p = proton mass, m n = neutron mass, K+U = kinetic+potential energies of all nucleons (negative for bound particles ), E B = the binding energy (positive, E B =-(K+U)) M(Z,N)c2M(Z,N)c2 Energy EBEB For a H atom, the binding energy is 13.6 eV (the ground state energy with sign “minus”). A bound system has a lower total energy than its constituents ! (outer) electrons in atoms: E B ~ eV nucleons in nuclei: E B ~ MeV

9. Binding Energy (cont’d) mass deficit We can compute the binding energy if we know masses of a system and its constituents: The binding energy is the energy required to dissasemble a nucleus into protons and neutrons. Binding energy: add E B p n p p n n n

10. Examples: A deuterium atom consists of one proton, one neutron, and one electron. The measured masses of the constituents as free particles are (see Appendix: Atomic Masses): The mass of the deuteron (also an experimentally measured): The mass defect: The binding energy: The binding energy per nucleon: For comparison, the energy required to strip a carbon atoms of all its electrons is ~ 1keV, 10 4 times smaller than the nuclear binding energy.

11. Binding Energy vs. Mass Number Because of the short-range character of strong interaction (basically, between nearest and next-to-nearest neighbors), the interaction energy per nucleon with increasing Z saturates at the level ~ (Z/2)(# of neighbors). The binding energy ~ 10MeV/nucleon is ~1% of the nucleon’s rest energy: we can consider the nucleus as a system of individual nucleons The decrease of the binding energy with increasing Z is caused by the long-range Coulomb repulsion of protons: Binding energy per nucleon (E B /A), MeV Mass number, A

12. Liquid-Drop Model (Gamow, 1929, Weizsacker, 1935) A “semi-classical” model of the nucleus: describes reasonably well the dependence E B (A):  - charge density

13. Liquid-Drop Model (cont’d)

14. Limitations of Liquid-Drop Model Maria Goeppert-Mayer, J.H.D.Jensen

15. Stable Nuclei Isotopes: all nuclei that have the same number of protons (Z) but different number of neutrons (N). Since the chemical properties of an atom are determined by the number of its electrons, isotopes of the same element have almost identical chemical properties. Example: naturally occurring isotopes of oxygen Related questions: - What makes unstable nuclei unstable? - What are the mechanisms by which they transform themselves into stable nuclei? -Why do light stable nuclei tend to have N  Z? - Why do heavier nuclei tend to have more neutrons than protons? - Why are there no stable nuclei with Z>83?

16. What makes unstable nuclei unstable? - If a nucleus is allowed to decrease its energy by transforming “excessive” protons (neutrons) into neutrons (protons), it will do it! The potential experienced by nucleons is a 3D potential well. The ground-state configuration of the carbon-16 nucleus : protons neutrons energy Both protons and neutrons are fermions (they obey the exclusion principle). Nuclei are two-component Fermi systems. Each nuclear energy level can contain four particles: two protons (s=  ½) and two neutrons (s=  ½). The processes responsible for these transformations are driven by weak interaction (the fourth fundamental interaction): The weak interaction (unlike the strong interaction) affects both quarks and leptons, (unlike the el.-mag. interaction) can affect electrically neutral particles, and (unlike gravity) does not affect photons. The effective range of the weak interaction is ~ 10 -18 m. Some important transformation processes driven by weak interaction: 0 r

17. Why N  Z for light nuclei If the electrostatic repulsion of protons can be neglected (this is the case of light nuclei: recall that the positive electrostatic energy  Z 2 ), the nucleus tends to keep approximately equal numbers of protons and neutrons. protonsneutrons energy protonsneutrons energy protonsneutrons energy protonsneutrons energy Even in this case, the nucleus can still lower its total energy: the rest energy of neutron is slightly more than the rest energy of a proton and an electron.

18. Why N > Z for heavy nuclei In the heavy nuclei, the electrostatic energy cannot be neglected. As a result, the protons’ energy levels are “pushed up” with respect to the neutrons’ levels. In the “otherwise stable” 44 Ti, two protons undergo the transformation into neutrons, the end product is stable 44 Ca. protonsneutrons energy.......... protonsneutrons energy.......... The proton-neutron disbalance becomes more pronounced with increasing Z.

19. Questions: 1. The main reason than an alpha particle can be stopped by a single sheet of paper while an electron of the same energy might not be stopped by hundreds of sheets of paper is that the alpha particle: A.is a nucleus B.moves slower C.participates in strong interactions 2. The measured binding energy per nucleon for 7 15 N is 7.70MeV, while that for 8 15 O is 7.46MeV. Why is the average nucleon in the oxygen nucleus is less tightly bound than in the nitrogen nucleus? A.it has fewer nucleons overall B.it has more nucleons on the surface C.electrostatic repulsion is greater on the average D.the nucleus is more asymmetrical 3. The measured binding energy per nucleon for 10 20 Ne is 8.03MeV, while that for 10 21 Ne is 7.97MeV. Why is the average nucleon in the larger nucleus is less tightly bound than in the smaller nucleus? A.it has more nucleons on the surface B.electrostatic repulsion is greater on the average D. has more mass E. has more charge C. the nucleus is more asymmetrical D. other (specify)

20. HW 9 Homework # 9: Beiser Ch. 11, Problems 2, 3, 4, 9 (except of (c)), 10, 12, 15, 17, 18, 20