• 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

( ) ( ) ( )! 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