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Nuclear Physics 1

Nuclear Physics 1. Topics. Composition of the Nucleus Ground State Properties Nuclear Structure Binding energy Nuclear Models Summary. Geiger, Marsden, Rutherford expt. (Geiger, Marsden, 1906 - 1911) (interpreted by Rutherford, 1911) get  particles from radioactive source

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Nuclear Physics 1

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  1. Nuclear Physics1

  2. Topics • Composition of the Nucleus • Ground State Properties • Nuclear Structure • Binding energy • Nuclear Models • Summary

  3. Geiger, Marsden, Rutherford expt. • (Geiger, Marsden, 1906 - 1911) (interpreted by Rutherford, 1911) • get  particles from radioactive source • make “beam” of particles using “collimators” (lead plates with holes in them, holes aligned in straight line) • bombard foils of gold, silver, copper with beam • measure scattering angles of particles with scintillating screen (ZnS)

  4. Geiger Marsden experiment: result • most particles only slightly deflected (i.e. by small angles), but some by large angles - even backward • measured angular distribution of scattered particles did not agree with expectations from Thomson model (only small angles expected), • but did agree with that expected from scattering on small, dense positively charged nucleus with diameter < 10-14 m, surrounded by electrons at 10-10 m Ernest Rutherford 1871-1937

  5. Composition of the Nucleus -- 1 From the experiments of Geiger and Marsden in 1911, it became clear that most of the mass of an atom is contained within a nucleus of size ~ 1 fm (=10-15 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 -- 2 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 it contains. N is the number of neutrons in the nucleus and A = N + Z is the mass number. For example, 15O has A = 15, N = 7 and Z = 8.

  7. Composition of the Nucleus -- 3 Neutrons and protons are referred to collectively as nucleons.

  8. Ground State Properties -- 1 • Nuclei with the same atomic number, but which • differ in mass number, e.g., 15O and 16O are • called isotopes. • examples: • deuterium, heavy hydrogen 2D or 2H; • heavy water = D2O (0.015% of natural water) • U- 235 (0.7% of natural U), U-238 (99.3% of natural U), • If nuclei have the same neutron number N, e.g., • 13C and 14N they are called isotones. • Those with the same mass number, e.g., 14C and • 14N are called isobars.

  9. Ground State Properties -- 2 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 electrostatic energy.

  10. Ground State Properties - 3 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 Nov. 9

  11. Ground State Properties - 4 The nucleus 15O is radioactive and decays as follows 15O 15N + e+ + . The energy difference between the nuclei is From numerous measurements of this energy difference it has been deduced that with R0 = 1.2 ± 0.2 fm

  12. Ground State Properties - 5 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 - 6 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 - 7 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 - 8 Nuclear Density – Since the radius of a nucleus is proportional to A1/3, 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 1017 kg/m3. A mere 1 cubic millimeter of nuclear matter would weigh as much as a full supertanker!

  16. Ground State Properties - 9 Nuclear Energies – The electrostatic energy can be written as where α ~ 1/137 and ħc = 197 MeV.fm For 16O, Z = 8, R = 1.2A1/3 = 3 fm, therefore, U = 18.3 MeV

  17. Ground State Properties - 10 Nuclear Pressures – The pressure can be computed using For 16O, U = 18.3 MeV, R = 3 fm, so V = 116 fm3. Therefore, P = (1/3) x (U/V) = (1/3) x (18.3 MeV/116 fm3) = 0.053 MeV/fm3 = 8.4 x 1030 Pa

  18. Nuclear Structure - 1 The neutron number, N, increases faster than the atomic number, Z. Why? The exclusion principle Line of stability

  19. Nuclear Structure - 2 A system with 7 neutrons has a higher overall energy than one with 4 neutrons and 3 protons

  20. Nuclear Structure - 3 Moreover, for large Z, because neutrons are electrically neutral, less energy is needed to add 2 neutrons than to add 1 neutron and 1 proton Therefore, N-Z increases with Z

  21. Atomic mass unit • “atomic number” Z • Number of protons in nucleus • Mass Number A • Number of protons and neutrons in nucleus • Atomic mass unit is defined in terms of the mass of the atom 126C (A = 12, Z = 6): • 1 amu = (mass of 126C atom)/12 • 1 amu = 1.66 x 10-27 kg • 1 amu = 931.494 MeV/c2

  22. Properties of Nucleons • Proton • Charge = 1 elementary charge e = 1.602 x 10-19 C • Mass = 1.673 x 10-27 kg = 938.27 MeV/c2 = 1.007825 u = 1836 me • spin ½, magnetic moment 2.79 eħ/2mp • Neutron • Charge = 0 • Mass = 1.675 x 10-27 kg = 939.6 MeV/c2 = 1.008665 u = 1839 me • spin ½, magnetic moment -1.9 eħ/2mn

  23. Nuclear energy levels: example Problem: Estimate the lowest possible energy of a neutron contained in a typical nucleus of radius 1.33×10-15 m. E = p2/2m = (cp)2/2mc2 x p = h/2  x (cp) = hc/2 = ħc (cp) = hc/(2 x) = hc/(2 r) (cp) = 4.1357x10-15eVs * 3x108 m/s / (2 * 1.33x10-15 m) (cp) = 1.485x108eV = 148.5 MeV E = p2/2m = (cp)2/2mc2 = (148.5 MeV)2/(2*940 MeV) = 11.7MeV

  24. Nuclear Masses, binding energy • Mass of Nucleus  Z(mp) + N(mn) • Mass defect m = difference between mass of atom and mass of constituents • Binding energyEB = energy defect =m c2 = amount of energy that must be invested to break up atom into its constituents Example: mass(7P + 7N + 7e) – mass(147N) = 7(1.00728 + 1.00866 + 0.00055) – 14.003074 = 0.1124 u • binding energy per nucleon = EB /A

  25. Binding Energies Iron is the most tightly bound nucleus. This fact is very important in stellar evolution.

  26. Nuclear (“strong”) force - 1 • atomic nuclei small -- about 1 to 8 fm • at small distance, electrostatic repulsive forces are of macroscopic size (10 – 100 N) • there must be short-range attractive force between nucleons -- the “strong force” • strong force essentially charge-independent • “mirror nuclei” have almost identical binding energies • mirror nuclei = nuclei for which n  p or p  n (e.g. 3He and 3H, 7Be and 7Li, 35Cl and 35Ar); slight differences due to electrostatic repulsion • strong force must have very short range – << atomic size, otherwise isotopes would not have same chemical properties

  27. Strong force -- 2 • range: fades away at distance ≈ 3fm • force between 2 nucleons at 2fm distance ≈ 2000N • nucleons on one side of U nucleus hardly affected by nucleons on other side • experimental evidence for nuclear force from scattering experiments; • low energy p or  scattering: scattered particles unaffected by nuclear force • high energy p or  scattering: particles can overcome electrostatic repulsion and can penetrate deep enough to enter range of nuclear force

  28. N-Z and binding energy vs A • small nuclei (A<10): • All nucleons are within range of strong force exerted by all other nucleons; • add another nucleon  enhance overall cohesive force  EB rises sharply with increase in A • medium size nuclei (10 < A < 60) • nucleons on one side are at edge of nuclear force range from nucleons on other side  each add’l nucleon gives diminishing return in terms of binding energy  slow rise of EB /A • heavy nuclei (A>60) • adding more nucleons does not increase overall cohesion due to nuclear attraction • Repulsive electrostatic forces (infinite range!) begin to have stronger effect • N-Z must be bigger for heavy nuclei (neutrons provide attraction without electrostatic repulsion • heaviest stable nucleus: 209Bi – all nuclei heavier than 209Bi are unstable (radioactive)

  29. Nuclear Models • Liquid Drop Model (Bohr, Bethe, Weizsäcker): • Nucleus described as a drop of incompressible nuclear fluid interacting via the strong force • Predicts spherical shape of nuclei • Qualitative description of fission of large nuclei • Good description of binding energy vs A • Fermi Gas Model • Neutrons and protons described as a free gas • Shell Model (Hans Jensen, Maria Goeppert-Mayer) • Similar to shell description of atoms

  30. Summary • Nuclei are made of protons and nucleons and have radii that scale roughly as A1/3, where A is the mass number. • The density of nuclear matter, 1017 kg/m3, is roughly independent of the size of the nucleus • The nuclear energy scale is of order 10 MeV • High Z nuclei tend to be neutron-rich

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