1 Nuclear Physics 1 Section 11-1,2

2 Electron Pressure From the total energy of a free electron gas we can calculate the electron pressure using

3 Electron Pressure For Cu, N/V = 8.47 x /m 3 E F = 7.06 eV = 1.12 x J therefore, the electron pressure in copper is P = (2/5) x (N/V) x E F = (2/5) x (8.47 x ) x 1.12 x Pa = 3.8 x Pa that is, about 400,000 times the pressure in this room!

4 Topics l Composition of the Nucleus l Ground State Properties l Summary

5 Composition of the Nucleus From the experiments of Geiger and Marsden in 1911 it became clear that most of the mass of an atom was contained within a tiny nucleus of size ~ 1 fm (= m). In 1932, the neutron was discovered by Chadwick, the positron by Anderson and the first nuclear reaction with protons was observed by Cockcroft and Walton.

6 Composition of the Nucleus A nucleus is a quantum system consisting of neutrons and protons, held together by a strong nuclear force. A nucleus of a given species, called a nuclide, is defined by its atomic number Z, that is, by the number of protons. N is the number of neutrons and A = N + Z is the mass number. For example, 15 O has A = 15, N = 7 and Z = 8.

7 Composition of the Nucleus Neutrons and protons are referred to collectively as nucleons.

8 Ground State Properties Nuclei with the same atomic number, but which differ in mass number, e.g., 15 O and 16 O are called isotopes. If they have the same neutron number N, e.g., 13 C and 14 N they are called isotones. Those with the same mass number, e.g., 14 C and 14 N are called isobars.

9 Ground State Properties Nuclear Sizes – The size of nuclei can be inferred in many ways. One way is to use mirror nuclides: those with Z and N numbers switched, for example: 8p + 7n 7p + 8n If we assume that the nuclear force is the same for protons and neutrons, then the energy of these nuclei will differ by their only by their electrostatic energy

10 Ground State Properties Let’s model the positive charge q of a nucleus as a ball of uniform charge of radius R. The potential energy of this ball of charge is given by Extra Credit: Prove this. Hint: consider the potential energy between a sphere of charge of radius r and a thin shell of charge of radius r and thickness dr, then integrate. Due Apr. 4

11 Ground State Properties The nucleus 15 O is radioactive and decays as follows 15 O -> 15 N + e + +. The energy difference between the nuclei is From numerous measurements of this energy difference it has been deduced that with R 0 = 1.2 ± 0.2 fm

12 Ground State Properties Another way to measure nuclear radii is to scatter electrons off nuclei and measure the resulting diffraction pattern of the scattered electrons The first minimum of this pattern occurs at

13 Ground State Properties The electron scattering experiments were first carried about by Robert Hofstadter in the 1950s at SLAC. These experiments gave information about the charge profile of nuclei, as shown in the figure.

14 Ground State Properties The results of Hofstadter’s experiments showed that the charge distribution of a nucleus can be described as a ball of uniform charge density, which, near the surface, decreases to zero over a zone of thickness t = 2.4 ± 0.3 fm. The radius measurements obtained by his team were consistent with those deduced from the mirror nuclei.

15 Ground State Properties Nuclear Density – Since the radius of a nucleus is proportional to A 1/3, the density of nuclear matter is roughly independent of the size of the nucleus. Consequently, nuclear matter behaves rather like a liquid with the enormously high density of kg/m 3. A mere 1 cubic millimeter of nuclear matter would weigh as much as a full supertanker!

16 Ground State Properties Nuclear Energies – The electrostatic energy can be written as where  ~ 1/137 and ћc = 197 Mev.fm For 16 O, Z = 8, R = 1.2A 1/3 = 3 fm, therefore, U = 18.3 MeV

17 Ground State Properties Nuclear Pressures – The electrostatic pressure is given by For 16 O, U = 18.3 MeV, R = 3 fm, so V = 116 fm 3,therefore, P = (1/3) x (U/V) = (1/3) x (18.3 MeV/116 fm 3 ) = MeV/fm 3 = 8.4 x Pa

18 Ground State Properties Line of stability N = Z The neutron number, N, increases faster than the atomic number, Z. Why? The exclusion principle

19 Ground State Properties A system with 7 neutrons has a higher Fermi energy than one with 4 neutrons and 3 protons

20 Ground State Properties Moreover, for large Z, less energy is needed to add 2 neutrons than to add 1 neutron and 1 proton Therefore, N-Z increases with Z

21 Summary l Nuclei are made of protons and nucleons and have radii that scale roughly as A 1/3, where A is the mass number. l The density of nuclear matter, kg/m 3, is roughly independent of the size of the nucleus l The nuclear energy scale is of order 10 MeV

22 PRS Question Which pair of nuclei are mirror nuclei? l 16 Oand 15 O l 15 Oand 15 N l 14 Cand 14 N