Lecture9ForcesPotentialsandtheShellModel_000

Lecture9ForcesPotentialsandtheShellModel_000 - Forces...

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Forces, Potentials, and the Shell Model Recall the Infinite Square Well (1D) Solve Shroedinger’s equation: E H E V dx d 2 2 Result: Consideration of boundary conditions (the behavior of the wavefunction at the walls) results in quantization. Both wavefunctions and eigenstates (energy levels) 2 2 2 8 mL h n E n Notice the dependence of the energy levels on the size of the box, and on the principal quantum number. Harmonic oscillator (1D) Hooke’s law : ) ( 0 x x k F If 0 x x , the system is at equilibrium because there is no force. However if x is different from 0 x there is a force which acts to restore the position to the equilibrium value (Notice the negative sign.) dx dV F Integrating we get, 2 0 ) ( 2 1 x x k V Now solve Schrodinger’s equation using this potential. Solution: Wavefunctions and eigenvalues Eigenvalues: ) 2 1 ( n E n where m k
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Notice the energy spacing for the harmonic oscillator. What is the minimum energy of the harmonic oscillator?
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V. Nuclear Shell Model A. Quantum Properties of Nuclei 1. Discrete Energy Levels 2. Nuclear Spin I a. Experimental Summary e-e : 0 I ALWAYS e-o, o-e: 2 n I , where n is an odd integer (1/2, 3/2, . ..) o-o : n I , where n is an integer (0, 1, 2 . ..) WE'LL USE 1 for our spins b. Implication e-e result implies strong pairing is energetically favorable spins must cancel c. Reason: Nuclear Force is attractive ; in contrast spins are unpaired in a atomic orbitals due to e-e r e p u l s i o n ( Pauli exclusion principle ) 3. Closed Shells – Unusual Stability a. Magic Numbers 2, 8, 20, 28, 50, 82, 126 (neutrons) b. Energetics: (M LD – M), B p , B n , B c. Lifetimes: 82 208 126 Pb 82 209 127 Pb 84 210 Po 126 84 212 120 Po STABLE 22y 138d
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4. Magnetic Moments Moving Charge created a magnetic field with moment = e 2Mc f(I) ; N = nuclear magneton (M = M p ) a. Expect p = N Observe : p = 2.793 N n = 0 n = 1.913 N b. Implication: nucleon has substructure, since one observes charge on periphery of particle. e.g.,
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Lecture9ForcesPotentialsandtheShellModel_000 - Forces...

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