physics 492 lecture 22 - michigan state universitylynch/lecture_wk9.pdf · physics 492 lecture 24....
TRANSCRIPT
![Page 1: Physics 492 Lecture 22 - Michigan State Universitylynch/lecture_wk9.pdf · Physics 492 Lecture 24. Collective excitations • Rotations. Interactions • Gauge invariance is an important](https://reader035.vdocuments.us/reader035/viewer/2022071019/5fd30270fa56d5083449ad66/html5/thumbnails/1.jpg)
• 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
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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)
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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
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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:
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Constants of motion
• Lz, Sz don’t commute with VLS
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Solutions of S.O. term
• Rewrite the S.O. operator:
• Hints for H.W.
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Solutions of H.O. plus S.O.
• Single particle Wavefunctions
• Multiparticle Wavefunctions.
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• 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
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Multiparticle wavefunctions
• Spin and parities of Shell model states are straightforward:– Closed shells have even parity and J=0
• Examples:– 7Be
– 46Ca
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Magnetic Moments
• There are orbital and spin contributions to the magnetic moments– Orbital contribution:
– Units: Magnetons
– Orbital g factor:
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• There are orbital and spin contributions to the magnetic moments– spin contributions:
• Estimation of moments:
Magnetic Moments
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Accuracy of simple shell model magnetic moment estimations
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Excited states
• Excited single particle states
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• Excited single particle states
Simple I.P.M
Excited states
Measured states
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Excited states
• Excited single particle states
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Collective states
• Nuclear vibrations
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• 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
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Collective excitations
• Rotations
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Interactions
• Gauge invariance is an important theoretical requirement:– For E-M fields:
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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
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Exchanged particle
• Aμ is a vector field and corresponds to exchange of spin one particle (photon).
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Perturbation series expansion for wavefunction
• To understand Feynman diagrams, you need to recall the pertubationseries expansion for the wavefunction.
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
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Perturbation series expansion for wavefunction
• To understand Feynman diagrams, you need to recall the pertubationseries expansion for the wavefunction.
![Page 34: Physics 492 Lecture 22 - Michigan State Universitylynch/lecture_wk9.pdf · Physics 492 Lecture 24. Collective excitations • Rotations. Interactions • Gauge invariance is an important](https://reader035.vdocuments.us/reader035/viewer/2022071019/5fd30270fa56d5083449ad66/html5/thumbnails/34.jpg)
Perturbation series expansion for wavefunction
• To understand Feynman diagrams, you need to recall the pertubationseries expansion for the wavefunction.
![Page 35: Physics 492 Lecture 22 - Michigan State Universitylynch/lecture_wk9.pdf · Physics 492 Lecture 24. Collective excitations • Rotations. Interactions • Gauge invariance is an important](https://reader035.vdocuments.us/reader035/viewer/2022071019/5fd30270fa56d5083449ad66/html5/thumbnails/35.jpg)
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What simple models for nuclear shells?Coulomb pot. Square well. V=kr2 pot.
Atomic pot. Nuclear potential.
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• 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
( ) ( ) ( )! 2/! 2/! ,
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
mult.Emss
,...3 ,2 ,1 ,0 ;21
=⎟⎠⎞
⎜⎝⎛ += nnE ωh