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UT Arlington PHYS 3446 - PHYS 3446 LECTURE NOTES

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PHYS 3446 – Lecture #8AnnouncementsSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Monday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu1PHYS 3446 – Lecture #8Monday, Oct. 2, 2006Dr. Jae Yu1. Nuclear Models•Shell Model •Collective Model•Super-deformed Nuclei2. Nuclear Radiation•Alpha decay•Beta decayMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu2Announcements•First term exam–Date and time: 1:00 – 2:30pm, THIS Wednesday, Oct. 4–Location: SH105–Covers: Appendix A (special relativity) + CH1 – CH3•Workshop was very successful–We’ve all learned tremendously–We know what we want to do at the next workshopMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu3•Exploit the success of atomic model–Uses orbital structure of nucleons–Electron energy levels are quantized–Limited number of electrons in each level based on available spin and angular momentum configurations•For nth energy level, l angular momentum (l<n), one expects a total of 2(2l+1) possible degenerate states for electronsNuclear Models: Shell ModelMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu4•Orbits and energy levels an electron can occupy are labeled by–Principle quantum number: n•n can only be integer–For given n, energy degenerate orbital angular momentum: l•The values are given from 0 to n – 1 for each n–For any given orbital angular momentum, there are (2l+1) sub-states: ml•ml=-l, -l+1, …, 0, 1, …, l – l, l•Due to rotational symmetry of the Coulomb potential, all these sub-states are degenerate in energy–Since electrons are fermions w/ intrinsic spin angular momentum , •Each of the sub-states can be occupied by two electrons–So the total number of state is 2(2l+1)Atomic Shell Model Reminder2hMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu5•Exploit the success of atomic model–Uses orbital structure of nucleons–Electron energy levels are quantized–Limited number of electrons in each level based on available spin and angular momentum configurations•For nth energy level, l angular momentum (l<n), one expects a total of 2(2l+1) possible degenerate states for electrons•Quantum numbers of individual nucleons are taken into account to affect the fine structure of spectraNuclear Models: Shell ModelMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu6•Nuclei have magic numbers just like inert atoms–Atoms: Z=2, 10, 18, 36, 54–Nuclei: N=2, 8, 20, 28, 50, 82, and 126 and Z=2, 8, 20, 28, 50, and 82 –Magic Nuclei: Nuclei with either N or Z a magic number  Stable–Doubly magic nuclei: Nuclei with both N and Z magic numbers  Particularly stable•Explains well the stability of nucleus Nuclear Models: Shell ModelMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu7•To solve equation of motion in quantum mechanics, Schrödinger equation, one must know the shape of the potential– –Details of nuclear potential not well known•A few shapes of potential energies tried out–Infinite square well: Each shell can contain up to 2(2l+1) nucleonsShell Model: Various Potential Shapes( )( )( )2220mE V r rj� �� + - =� �� �rrhMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu8Nuclear Models: Shell Model – Square well potential caseNMn l=n-1Ns=2(2l+1) NT2 1 0 2 28 2 0,1 2+6 820 3 0,1,2 2+6+10 1828 4 0,1,2,3 2+6+10+14 3250 5 0,1,2,3,4 2+6+10+14+18 5082 6 0,1,2,3,4,5 2+6+10+14+18+22 72Monday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu9•To solve equation of motion in quantum mechanics, Schrödinger equation, one must know the shape of the potential– –Details of nuclear potential not well known•A few models of potential tried out–Infinite square well: Each shell can contain up to 2(2l+1) nucleons•Can predict 2, 8 and 50 but no other magic numbers–Three dimensional harmonic oscillator: •Predicts 2, 8, 20, 40 and 70  Some magic numbers predictedShell Model: Various Potential Shapes( )V r =2 212m rv( )( )( )2220mE V r rj� �� + - =� �� �rrhMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu10•Central potential could not reproduce all magic numbers•In 1940, Mayer and Jesen proposed a central potential + strong spin-orbit interaction w/ –f(r) is an arbitrary empirical function of radial coordinates and chosen to fit the data•The spin-orbit interaction with the properly chosen f(r), a finite square well can split•Reproduces all the desired magic numbersShell Model: Spin-Orbit PotentialTOTV =Spectroscopic notation: n L jOrbit numberOrbital angular momentumProjection of total momentum( )V r( )f r L S- �rrMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu11•Spin-Parity of large number of odd-A nuclei predicted well–Nucleons are Fermions so the obey Pauli exclusion principle  Fill up ground state energy levels in pairs–Ground state of all even-even nuclei have zero total angular momentum •The shell model cannot predict stable odd-odd nuclei spins–No prescription for how to combine the unpaired proton and neutron spinsPredictions of the Shell ModelMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu12•Magnetic Moment of neutron and proton are•Intrinsic magnetic moment of unpaired nucleons contribute to total magnetic moment of nuclei–What does a deuteron consist of?•Measured value is–For Boron (10B5) , the 5 neutrons and 5 protons have the same level structure: (1S1/2)2(1P3/2)3, leaving one of each unpaired proton and neutron in angular momentum l=1 state  •Measured value is•Does not work well with heavy nuclei Predictions of the Shell ModelDm =2.79p Nm m�1.91n Nm m�-Dm =Bm =1.80B Nm m=m=2.79 1.91N N Nm m m- + =pm +2.79Nm1.91Nm- =0.88Nmnm =0.86Nm2Nelm c=hNm12Nem c�=hp nm m+ +orbitm =1.88NmMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu13•For heavy nuclei, shell model predictions do not agree with experimental measurements–Especially in magnetic dipole moments•Measured values of quadrupole moments for closed shells differ significantly with experiments–Some nuclei’s large quadrupole moments suggests significant nonspherical shapes–The assumption of rotational symmetry in shell model does not seem quite right•These deficiencies are somewhat covered through the reconciliation of liquid drop model with Shell model–Bohr, Mottelson and Rainwater’s collective model, 1953Collective ModelMonday, Oct. 2, 2006 PHYS 3446, Fall 2006Jae Yu14•Assumption–Nucleus consists of


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