physics 492 lecture 22 - michigan state universitylynch/lecture_wk9.pdf · physics 492 lecture 24....
TRANSCRIPT
• Main points of today’s lecture:– Nuclear models.
• Spin-orbit term• Filling single particle
orbits• Nuclear Spins and
parities• Magnetic moments
• Main points of last lecture:– Nuclear models.
• Shell splitting central potential
• Spin-orbit term
Physics 492 Lecture 22
Energy Eigenvalues
• The result is what you expect for adding the energies of H.O. along x, y and z axes. Note ½ kr2= ½ k(x2+y2+z2)
spectroscopic notation: nLJ :L=1,2,3..;→S,P,D...z
What’s needed to get correct magic numbers?
• Spin-orbit interaction is the major missing term
Spin orbit potential
• Mayer and Jensen received Nobel Prize for showing that the nuclear Spin orbit potential gives the correct magic numbers.
• The nucleon-nucleon potential has a spin orbit. This means that the nucleon-nucleus spin orbit potential vanishes in the nuclear interior:
Constants of motion
• Lz, Sz don’t commute with VLS
Solutions of S.O. term
• Rewrite the S.O. operator:
• Hints for H.W.
Solutions of H.O. plus S.O.
• Single particle Wavefunctions
• Multiparticle Wavefunctions.
• Main points of today’s lecture:– Nuclear models.
• Nuclear Spins and parities
• Magnetic moments– Excited states
• Deformed nuclei• Rotations and vibrations
• Main points of last lecture:– Nuclear models.
• Spin-orbit interaction• Filling single particle
orbits
Physics 492 Lecture 23
Multiparticle wavefunctions
• Spin and parities of Shell model states are straightforward:– Closed shells have even parity and J=0
• Examples:– 7Be
– 46Ca
Magnetic Moments
• There are orbital and spin contributions to the magnetic moments– Orbital contribution:
– Units: Magnetons
– Orbital g factor:
• There are orbital and spin contributions to the magnetic moments– spin contributions:
• Estimation of moments:
Magnetic Moments
Accuracy of simple shell model magnetic moment estimations
Excited states
• Excited single particle states
• Excited single particle states
Simple I.P.M
Excited states
Measured states
Excited states
• Excited single particle states
Collective states
• Nuclear vibrations
• Main points of today’s lecture:– Excited states
• Deformed nuclei• Rotations and vibrations
– Interactions• Gauge invariance• Perturbation series
expansion of the wavefunction
• Main points of last lecture:– Nuclear models.
• Nuclear Spins and parities• Magnetic moments
– Excited states • vibrations
Physics 492 Lecture 24
Collective excitations
• Rotations
Interactions
• Gauge invariance is an important theoretical requirement:– For E-M fields:
Gauge invariance and w and interactions
• The Haniltonian must not depend on the gauge fields
• Implies that perturbation potential must couple exchanged particle (photon) to a conserved current– EM interaction potential is inner product of Aμ field with
conserved EM current
Exchanged particle
• Aμ is a vector field and corresponds to exchange of spin one particle (photon).
Perturbation series expansion for wavefunction
• To understand Feynman diagrams, you need to recall the pertubationseries expansion for the wavefunction.
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
Perturbation series expansion for wavefunction
• To understand Feynman diagrams, you need to recall the pertubationseries expansion for the wavefunction.
Perturbation series expansion for wavefunction
• To understand Feynman diagrams, you need to recall the pertubationseries expansion for the wavefunction.
What simple models for nuclear shells?Coulomb pot. Square well. V=kr2 pot.
Atomic pot. Nuclear potential.
• Main points of today’s lecture:– particle in a three
dimensional box.
– harmonic oscillator
• Multiplicity function for set of N spinors:
• Probabilities:
• Main points of last lecture:• Single particle orbitals:
– spinor in magnetic field:
Physic 492 Lecture 1
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sNsNNsNg
−+=
( )
.... 4 3, 2, 1,,,2
),,( 2
22222
=
++=
zyx
zyxzyx
nnnmL
nnnnnnE
πh
( ) ( ) ( ) nNn ppnNn
NnW −−−
= 1! !
!
1-mB-1/21/2
1mB1/21/2
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,...3 ,2 ,1 ,0 ;21
=⎟⎠⎞
⎜⎝⎛ += nnE ωh