PHYS 3446 – Lecture #6Slide Number 2 Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Assignment 4Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 1PHYS 3446 – Lecture #6Monday, Sep. 27 2010Dr. Brandt1. Nuclear Properties-Size-Spin-Magnetic dipole moment-Stability-Instability• Reviewed cgs units. See for example:http://en.wikipedia.org/wiki/Statcoulomb• Problem 1.4 can be turned in for extra credit in HW 2 on Friday• Went through problems 1.1 and 1.10 (next 2 slides) Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 2Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 3Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 4Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 5• For a nucleus of mass M consisting of with Z protons and A-Z=N neutrons, the mass deficitis always negative and is proportional to the nuclear binding energy•How are the BE and mass deficit related? • The binding energy is the minimum energy required to release all nucleons from a nucleusNuclear Properties: Binding Energy( ),M AZ∆=( )2.,BE M AZ c= ∆( ),M AZ( )pnZm A Z m− −−Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 6• BE per nucleon isNuclear Properties: Binding Energy( )2,M AZ cA−∆=( ) ( )( )2,pnZm A Z m M A Z cA+− −=• Rapidly increase with A till A~60 at which point BE~9MeV.• A>60, the B.E gradually decrease For most of the large A nucleus, BE~8MeV.BEA−Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 7• de Broglie’s wavelength:– Where is the Planck’s constant– And is the reduced wavelength• Assuming 8 MeV was given to a nucleon (mn~940MeV), its wavelength is• Makes sense for nucleons to be inside a nucleus since the size is smaller than the nucleus.• What about an electron with 8MeV? – The wavelength is ~25 fm, much larger than a nucleus.Nuclear Properties: Binding Energy=p= =2mT=22cmc T p1971.62 940 8Mev fmfm−≈≈⋅⋅An electron with 120 MeV would fit in nucleus, but energy is too high!Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 9• Scatter very high E projectiles for head-on collisions– As E increases DCA becomes 0.– High E particles can probe deeper into nucleus • Use electrons to probe the charge distribution (form factor) in a nucleus– What are the advantages of using electrons?• Electrons are fundamental particles No structure of their own• Electrons primarily interact through electromagnetic force • Electrons are not affected by the nuclear force– The radius of charge distribution can be regarded as an effective size of the nucleusNuclear Properties: Size2min0'ZZ erE=Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 10• At relativistic energies the magnetic moment of electron also contributes to the scattering– Neville Mott formulated Rutherford scattering in QM and included the spin effects– R. Hofstadter, et al., discovered the effect of spin, nature of nuclear (& proton) form factor in late 1950s • Mott scattering x-sec (scattering of a point particle) is related to Rutherford x-sec:• Deviation from the distribution expected for point-scattering provides a measure of size (structure)Nuclear Properties: SizeMottddσΩ=24cos2RutherfordddθσΩMonday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 11• Another way to probe the nucleus is using strongly interacting particles (π mesons, protons, etc.)– What is the advantage of using these particles?• If the energy is high, Coulomb interaction can be neglected• These particles readily interact with nuclei, getting “absorbed” into the nucleus• Thus, probe strong interactions directly• The size of a nucleus can be inferred from the diffraction pattern (analogous to light diffracted by a disk)Nuclear Properties: SizeMonday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 12• All this phenomenological investigation resulted in a startlingly simple formula for the radius of the nucleus in terms of the number of nucleons or atomicnumber, A: Nuclear Properties: Size130R rA= ≈Does this formula make sense? why 1/3 power?13 1 31.2 10 A cm−×=131.2 fmAConsider a spherical nucleusMonday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 13• Both protons and neutrons are fermions with spin• Since nucleons inside a nucleus have spin they have orbital angular momentum• In Quantum Mechanics orbital angular momenta are integers• Thus the total angular momentum of a nucleus is– Integer: if an even number of nucleons in the nucleus– Half integer: if an odd number of nucleons in the nucleus• Interesting facts are– All nuclei with even number of p and n are spin 0.– Large nuclei have very small spin in their ground state• Hypothesis: Nucleon spins in the nucleus are very strongly paired to minimize their overall effectNuclear Properties: Spin12Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 14• Every charged particle has a magnetic dipole moment associated with its spin• e, m and S are the charge, mass and the intrinsic spin of the charged particle• The constant g is called Landé factor with a value:– : for a point like particle, such as the electron– : Particle possesses an anomalous magnetic moment, an indication of having a substructure (g-2 experiments)Nuclear Properties: Magnetic Dipole Moment2egSmcµ=2g ≠2g =Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt 15• For electrons, µe~µB, where µBis Bohr Magneton• For nucleons, magnetic dipole moment is measured in nuclear magneton, defined using proton mass• Measured magnetic moments of proton and neutron:Nuclear Properties: Magnetic Dipole MomentBµ=2Npemcµ=2.79pNµµ≈1.91nNµµ≈ −2eemc=115.79 10 MeV/T−×Monday, Sept. 27, 2010 PHYS 3446, Fall 2010 Andrew Brandt16• What important information do you get from these magnetic moment measurments?– The Landé factors of the nucleons deviate significantly from 2.• Indication of substructure– An electrically neutral neutron has a significant magnetic moment• Must have extended
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