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Lecture 3: Nuclear Structure 1 • Why structure? • The nuclear potential • Schematic shell model

Lecture 3: Ohio University PHYS7501, Fall 2017, Z. Meisel ([email protected]) Empirically, several striking trends related to Z,N. e.g.

B.A. Brown, Lecture Notes in Nuclear Structure Physics, 2005. M.A. Preston, Physics of the Nucleus (1962) Q

B.A. Brown, Lecture Notes in Nuclear Structure Physics, 2005. Adapted from B.A. Brown, C.R. Bronk, & P.E. Hodgson, J. Phys. G (1984)

P. Möller et al. ADNDT (2016) Meisel & George, IJMS (2013) A. Cameron, Proc. Astron. Soc. Pac. (1957)

First evidence compilation 2 by M. Göppert-Mayer Phys. Rev. 1948 …reminiscent of atomic structure

B.A. Brown, Lecture Notes in Nuclear Structure Physics, 2005.

hyperphysics

hyperphysics 54 3 Shell Structure

Atomic Nuclear •Central potential (Coulomb) generated by nucleus •No central object …but each is interacted on by the other A-1 and they’re relatively compact together • are essentially non-interacting •Nucleons interact very strongly …but if nucleons in nucleus were to scatter, Pauli blocking prevents them from scattering into filled orbitals. Scattering into higher-E orbitals is unlikely. i.e. there is no “weak interaction paradox”

•Solve the Schrödinger equation for the Coulomb •Can also solve the Schrödinger equation for energy potential and find characteristic (energy levels) shells: levels (shells) …but obviously must be a different shells at 2, 10, 18, 36, 54, 86 potential: shells at 2, 8, 20, 28, 50, 82, 126

…you might be discouraged by points 1 and 2 above, but, remember: If it’s stupid but it works, it isn’t stupid. 4 Calculating eigenstates of the system, a.k.a single particle levels • The behavior of a quantum-mechanical system is described by the • For a particle in some potential, we can solve for ψ using the Schrödinger equation, ψ (in cartesian coordinates, = + + ) • = a.k.a. + = a.k.a. + = 2 2 2 2 2 𝜕𝜕 𝜕𝜕 𝜕𝜕 ћ 2 2 2 • The solutions ψ are the eigenfunctions and their eigenvalues2 are the corresponding𝛻𝛻 energy𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦 𝜕𝜕𝑧𝑧 2𝑚𝑚 𝐻𝐻ψ 𝐸𝐸ψ 𝑇𝑇ψ 𝑉𝑉ψ 𝐸𝐸ψ − 𝛻𝛻 ψ 𝑉𝑉ψ 𝐸𝐸ψ l • As a bonus, when ψ can be expressed in terms of spherical harmonics, ψ=R(r)Y m(θ,φ) we also get the angular momentum for that particular eigenfunction, 𝐸𝐸 and , since the function is either odd or even • Mathematical challenges aside, to get any traction we obviously need to assume a potential • For a single nucleon in the field of a nucleus, • should approximate the mean-field generated by all other nucleons 𝑉𝑉 •The solutions will be single-particle levels, i.e.𝑉𝑉 discrete states the nucleon can occupy • Since nucleons are indistinguishable, we only need to solve for the single-particle levels for a nucleon and then we can fill those levels (working in terms of increasing ) to generate a model to calculate the properties of our nucleus 5 𝐸𝐸 First stab at the potential, : The Harmonic Oscillator • Based on some evidence (and logic) that nuclei aren’t perfectly constant in density, Heisenberg (Z. Phys. 1935) posited that a parabolic potential could be assumed, conveniently allowing the adoption of the𝑉𝑉 harmonic oscillator solutions (one of the few analytically solved systems!) i.e. only odd or even • This provides evenly spaced energy levels , with = + . functions are allowed • The corresponding angular momenta are = 1, 3, … 1 0. for each oscillator shell 2 • The number of particles per angular momentum𝑛𝑛 is𝐸𝐸 2𝑛𝑛( +𝑛𝑛 1) forℎ υ + 1 projections & 2 spins • So, the number of particles per level is: 𝑙𝑙 𝑛𝑛 − 𝑛𝑛 − ≥ 2𝑙𝑙 2𝑙𝑙

Could the HO potential still be useful for some cases?

…can get the job done for light nuclei (e.g. H. Guo et al. PRC 2017) 6 Loveland, Morrissey, & Seaborg, Modern Nuclear Chemistry (2006) …but need to be careful, because can impact results (B.Kay et al arXiv 2017) Move to an empirical potential: Woods-Saxon Cumulative • Since the nuclear interaction is short-range, a natural improvement would be to adopt a central potential mimicking the empirical density distribution • This is basically a square well with soft edges, 92 as described by the Woods-Saxon potential: 58 R. Woods & D. Saxon, Phys. Rev. (1954) • Using the Woods-Saxon 40 is a good idea because of commitment to 20 22 MeV on Pt reality… but we’re no wiser as to the origin of the magic numbers 8

2 Was this step completely useless? No! It broke the degeneracy in ℓ Loveland, Morrissey, & Seaborg, Modern Nuclear Chemistry (2006) 7 The missing link: the spin-orbit interaction LS

• Due to desperation or genius (or both) Maria Göppert-Mayer [Phys. Rev. February 1949] (and nearly simultaneously Haxel, Jensen, & Suess [Phys. Rev. April 1949]) posited that nucleon spin and orbital angular momentum interacted strongly, making j the good quantum number for a nucleon: = + • Prior to this approach, ⃗ angular momentum was coupled as is typically done𝚥𝚥⃗ for𝑙𝑙 atoms,𝑠𝑠⃗

where = + , = , and = A. Kastberg, Lecture Notes Physique Atomique • This is “LS coupling” jj 𝐽𝐽⃗ 𝐿𝐿 𝑆𝑆⃗ 𝐿𝐿 ∑𝑛𝑛𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑙𝑙⃗ 𝑆𝑆⃗ ∑𝑛𝑛𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑠𝑠⃗ • Positing that the spin-orbit interaction is stronger than spin-spin or orbit-orbit means that instead, = and = + • This is “jj coupling” 𝐽𝐽⃗ ∑𝑛𝑛𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝚥𝚥⃗ 𝚥𝚥⃗ 𝑙𝑙⃗ 𝑠𝑠⃗

8 The missing link: the spin-orbit interaction

• Now, in considering a valence nucleon, we should calculate its • can only take on values: + …so for our nucleons, + , 𝑗𝑗 i.e. and are either aligned ( + ) or anti-aligned ( ) 1 1 2 2 𝑗𝑗 • For = 0: = ; = 𝑙𝑙1−: 𝑠𝑠=≤ 𝑗𝑗 ≤; 𝑙𝑙 𝑠𝑠= 2: = ; = 3:𝑙𝑙 −= ≤ 𝑗𝑗 ≤…𝑙𝑙etc. 𝑙𝑙 𝑠𝑠 1 𝑙𝑙 𝑠𝑠1 3 𝑙𝑙 −3 𝑠𝑠 5 5 7 • Each has + 1 projections (a.k.a. # of protons or , depending which nucleon we’re discussing) 𝑙𝑙 𝑗𝑗 2 𝑙𝑙 𝑗𝑗 2 𝑜𝑜𝑜𝑜 2 𝑙𝑙 𝑗𝑗 2 𝑜𝑜𝑜𝑜 2 𝑙𝑙 𝑗𝑗 2 𝑜𝑜𝑜𝑜 2 • i.e. 2 states for = , 4 states for = , 6 states for = , 8 states for = …etc. 𝑗𝑗 2𝑗𝑗 1 3 5 7 • The spin-orbit interaction2 means there’s a2 -dependent part2 of the nuclear potential,2 so the levels corresponding𝑗𝑗 to different𝑗𝑗 for some will𝑗𝑗 be split in energy.𝑗𝑗 • For nucleons, cases with aligned and are𝑗𝑗 energetically favored, so, for example, = 1, = will be lower𝑗𝑗 in energy𝑙𝑙 than = 1, = 3 𝑙𝑙 𝑗𝑗 1 𝑙𝑙 𝑗𝑗 2 𝑙𝑙 𝑗𝑗 2 • While we’re at it, note the spectroscopic notation: • = 0, 1, 2, 3, 4, 5, … = “s”, “p”, “d”, “f ”, “g ”, “h” …

𝑙𝑙 9 Result: the nuclear potential B.A. Brown, Lecture Notes in Nuclear Structure Physics, 2005.

• Nucleons within a nucleus can be treated as if they are • Attracted by a Woods-Saxon central potential • Repelled by a Coulomb potential from a charged sphere (if ) • Attracted or Repelled if and are parallel or anti-parallel by the spin-orbit force (Peaks𝑙𝑙 𝑠𝑠at surface) • Repelled by a centrifugal barrier (if the nucleon were to exit the nucleus, carrying away angular momentum > 0)

𝑙𝑙

10 Putting it all together: “shells” from the nuclear potential • Considering the nucleus as nucleons interacting in a mean-field potential, generated by the spatial distribution of all other nucleons, and each nucleon having a between its orbital & spin angular momentum, properly predicts the magic numbers. • Note that neutrons and protons are considered separately. • When adding neutrons or protons to a nucleus, the lowest energy state will (generally) consist of filling each orbital as you go upward. • The regions between the large gaps in nucleon energy are referred to as “shells”. • E.g. Between 8 and 20 neutrons (or protons) is the “sd-shell”, between 28 and 40 neutrons (or protons) is the “fp-shell”. • More exotic -rich exist, so typically people are talking about the neutron shell • Nucleons can get excited into higher-lying states, 11 so states above the ground-state are relevant in calculations Loveland, Morrissey, & Seaborg, Modern Nuclear Chemistry (2006) As a heads-up, level ordering doesn’t follow a fixed set of rules For stable, spherical nuclides: For, e.g, n-rich O :

126

40 82

50 T. Otsuka et al. Eur. Phys. J. A (2002) 14 28 20

8 Magic numbers “break down” 2 and new ones can appear for exotic nuclides

12 Bohr & Mottelson, Nuclear Structure, Vol. I (1969) Common form for the nuclear potential • = + + + Bohr & Mottelson, Nuclear Structure, Vol. I (1969) 𝑉𝑉 𝑟𝑟 𝑉𝑉𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟 𝑉𝑉𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑟𝑟 𝑙𝑙⃗ � 𝑠𝑠⃗ 𝑉𝑉𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑟𝑟 𝑉𝑉𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟 • = 1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑟𝑟−𝑅𝑅𝑐𝑐𝑐𝑐 𝑉𝑉 • , 𝑟𝑟 =𝑉𝑉 +1+exp or𝑎𝑎𝑐𝑐𝑐𝑐 , = 𝑁𝑁−𝑍𝑍 𝑁𝑁−𝑍𝑍 • 𝑉𝑉𝑐𝑐𝑐𝑐 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 =𝑉𝑉0 𝐴𝐴 𝑉𝑉1 𝑉𝑉𝑐𝑐𝑐𝑐 𝑛𝑛𝑛𝑛𝑛𝑛𝑡𝑡𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 𝑉𝑉0 − 𝐴𝐴 𝑉𝑉1 1 𝜕𝜕 1 𝑟𝑟−𝑅𝑅𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠 𝑟𝑟 𝜕𝜕𝜕𝜕 𝑉𝑉 𝑟𝑟 𝑉𝑉 1+exp for𝑎𝑎𝑠𝑠𝑠𝑠 2 • = 𝑍𝑍𝑒𝑒 𝑟𝑟 for 𝑟𝑟 ≥ 𝑅𝑅𝑐𝑐 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 2 2 𝑍𝑍𝑒𝑒 3 𝑟𝑟 This is only relevant when a 𝑉𝑉 𝑟𝑟 2 ( ) • =𝑅𝑅𝑐𝑐 2 2𝑅𝑅𝑐𝑐 nucleon is removing/adding𝑐𝑐 −2 orbital𝑟𝑟 ≤ angular𝑅𝑅 momentum ℓ, ћ 𝑙𝑙 𝑙𝑙+1 so you would not use it to 2 calculate single particle levels 𝑉𝑉𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟 2𝜇𝜇𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟 • Typically, = = = / • For = 1. , = = 01. 3 , 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠 𝑐𝑐 0 = 𝑅𝑅 , 𝑅𝑅= 𝑅𝑅 𝑟𝑟, 𝐴𝐴 = 0.44 , get neutron single-particle energies above 13 𝑟𝑟0 27𝑓𝑓𝑓𝑓 𝑎𝑎𝑐𝑐𝑐𝑐 𝑎𝑎𝑠𝑠𝑠𝑠 67𝑓𝑓𝑓𝑓 𝑉𝑉0 −55𝑀𝑀𝑀𝑀𝑀𝑀 𝑉𝑉1 −33𝑀𝑀𝑀𝑀𝑀𝑀 𝑉𝑉𝑠𝑠𝑠𝑠 − 𝑉𝑉𝑐𝑐𝑐𝑐 Filling the shells Loveland, Morrissey, & Seaborg, Modern Nuclear Chemistry (2006) • We can construct a nucleus using our “shell model”: • A nucleon will go in the lowest- which isn’t already filled, i.e. • the largest angular momentum, • for the lowest orbital angular momentum, • for the lowest oscillator shell, n 𝑗𝑗 • + 1 protons or neutrons are allowed𝑙𝑙 per level • Each level is referred to by its 2𝑗𝑗• by the # for the oscillator shell (convention either starts with 0 or𝑛𝑛 𝑛𝑛𝑛𝑛1) • 𝑛𝑛by spectroscopic notation (s=0,p=1,d=2,f=3,…) • by the half-integer corresponding to the spin 𝑙𝑙 • For 𝑗𝑗example: 7Li (Z=3, N=4)

14 Basic properties from the shell model: • Recall that, from the pairing hypothesis, nucleons pair & cancel 𝜋𝜋spins. • So, the unpaired nucleons determine the properties of a nucleus.𝐽𝐽 Unpaired nucleons sum to determine the spin & multiply to determine the parity • Revisiting 7Li:

• The only nucleon without a dance partner is the 1p3/2 proton; i.e. = 3/2, = 1 So, the 7Li ground-state should be = 1 − 𝐽𝐽 𝜋𝜋 − • What’s the lowest energy excitation𝜋𝜋 possible?3 (note pairing is strong) 𝐽𝐽 2 Moving the p3/2 proton up to p1/2 • So, the first excited state of 7Li should be = − • Compare to data: 𝜋𝜋 1 𝐽𝐽 2

15 Basic properties from the shell model: • Now that we’re feeling fat & sassy, let’s try another case: 37Ar 𝜋𝜋 • Based on our shell-model, 𝐽𝐽 we expect the ground-state to be 3/2+ …and it is!

• Now for the first excited state, a logical thought would be the odd d3/2 neutron would pop up to the f7/2 level, creating a state with 7/2- • …but the first excited state is 1/2+ (the 2nd x.s. is 7/2-) • What happened? • We have to keep in mind pairing & energy-costs

• The 2s1/2-1d3/2 gap is smaller than the 1d3/2-1f7/2 gap (for low A)

Bohr & Mottelson, Nuclear Structure, Vol. I (1969) • And, pairing energy increases with the of the level 16

𝒍𝒍 Basic properties from the shell model:

• Looking at a more complicated case, 38Cl (Z=17, N=21) 𝜋𝜋

1f7/2 𝐽𝐽 1f7/2 28 1d 3/2 1d3/2 20 2s 1/2 2s1/2 16

1d5/2 1d5/2 14

1p1/2 1p1/2 8

1p3/2 1p3/2 6

1s1/2 1s1/2 2 π ν

• 2 valence nucleons: one d3/2 proton and one f7/2 neutron • Allowed couplings are + • So for this case: = 2, 3, 4, 5 𝑗𝑗1 − 𝑗𝑗2 ≤ 𝐽𝐽 ≤ 𝑗𝑗1 𝑗𝑗2 • How do we decide which combination has the lowest energy? 𝐽𝐽 Using the descriptively named: “jj coupling rules for Odd-Odd nuclei” from Brennan & Bernstein (Phys. Rev. 1960) 17 Basic properties from the shell model: for odd-odd • For Odd-Z, Odd-N nuclides, need a method to determine which 𝜋𝜋jj-coupling is the lowest energy • An empirically-based set of rules was developed by Brennan &𝐽𝐽 Bernstein (Phys. Rev. 1960) • They noticed that, when coupling , = ± and = ± , •Rule 1: If ( = + and = ) or ( = and = + ), then = 1 1 1 2 2 2 = 2 =𝑗𝑗 𝑗𝑗 𝑙𝑙 𝑠𝑠 𝑗𝑗 = 3𝑙𝑙 + =𝑠𝑠 = = 2 •e.g. for a d3/2 proton and a f7/2 neutron , 38 For this case,𝑗𝑗1 𝑙𝑙1= 𝑠𝑠1 = 𝑗𝑗1𝑠𝑠 1 𝑙𝑙32 −1 𝑠𝑠2= 𝑗𝑗1 𝑙𝑙1 − 𝑠𝑠1 1 𝑗𝑗𝑠𝑠7 𝑙𝑙2 7 𝑠𝑠32 𝐽𝐽 Ex:𝑗𝑗1 −Cl𝑗𝑗2 𝑝𝑝 2 2 𝑛𝑛 2 2 2 2 •Rule 2: If ( = + 𝑗𝑗 and − 2= + 3 ) or ( = 𝑗𝑗 and =𝐽𝐽 − ), then = ± 𝜋𝜋 ∏ 𝜋𝜋𝑖𝑖 − ∗ − − •e.g. for a d5/2 proton = 2 + = and a d5/2 neutron = 2 + = , = + = 5 26 For this case,𝑗𝑗1 𝑙𝑙1= 𝑠𝑠1 = 𝑗𝑗1𝑠𝑠 1 𝑙𝑙52 1 𝑠𝑠2= + 𝑗𝑗1 𝑙𝑙1 − 𝑠𝑠1 1 𝑗𝑗𝑠𝑠5 𝑙𝑙2 −5 𝑠𝑠52 𝐽𝐽 Ex:𝑗𝑗1 Al𝑗𝑗2 𝑝𝑝 2 2 𝑛𝑛 2 2 2 2 •Rule 3: If one odd nucleon𝑗𝑗 has2 been promoted2 (e.g. to𝑗𝑗 an s-orbital 𝐽𝐽to pair with a nucleon), ∏ 𝑖𝑖 leaving behind𝜋𝜋 a𝜋𝜋 “hole”,− and∗ the− other odd nucleon stays a particle, then = + 1 = = = + 1 = 4 •e.g. for a d3/2 proton hole and a f7/2 neutron , 40 For this case, = = 1 3 1 = 7 7 3 𝐽𝐽 𝑗𝑗1Ex: 𝑗𝑗2K− 𝑝𝑝 2 𝑛𝑛 2 2 2 𝑗𝑗 2 3 𝑗𝑗 𝐽𝐽 − *These don’t always𝜋𝜋 work…but∏ 𝜋𝜋𝑖𝑖 − when∗ − they −don’t, this can tell you something: Either there’s more than a single-particle level interaction going on, or your particle(s)/hole(s) don’t occupy the levels we naїvely assumed. 18 (e.g. S. Liddick et al. Phys. Rev. C 2004) Shell-model is pretty good at predicting (among other things)

B. Brown, W.A. Richter, & C. Wrede, PRC(R) 2014 …some major challenges 𝜋𝜋are B. Brown & W. Rae, Nuc. Dat. Sheets (2014) 𝐽𝐽 including all the relevant levels in the calculation

and

choosing a good interaction between nucleons et al. Phys Rev. C (2014) C Rev. Phys al. et Stroberg 19 S.R. What else are predictions good for? Magnetic dipole moments = • Recall that for a single𝜋𝜋 particle, the magnetic dipole moment is: • After some fancy footwork, it can be shown that the Landé g-factor can be expressed as: 𝐽𝐽 𝜇𝜇 𝑗𝑗𝑔𝑔𝑗𝑗𝜇𝜇𝑁𝑁 ( ) ( ) • = ( ) + ( ) 𝑗𝑗 𝑗𝑗+1 +𝑙𝑙 𝑙𝑙+1 −𝑠𝑠 𝑠𝑠+1 𝑗𝑗 𝑗𝑗+1 −𝑙𝑙 𝑙𝑙+1 +𝑠𝑠 𝑠𝑠+1 • Since𝑔𝑔𝑗𝑗 spins cancel2𝑗𝑗 𝑗𝑗for+1 paired𝑔𝑔 nucleons,𝑙𝑙 we2𝑗𝑗 𝑗𝑗 might+1 expect𝑔𝑔𝑠𝑠 the magnetic dipole moment of a nucleus with 1-unpaired nucleon to be determined by that nucleon Loveland, Morrissey, & Seaborg, Expected values of are therefore: Modern Nuclear Chemistry (2006) • = = + for = + 1𝜇𝜇 1 • = = 1 + for = 𝜇𝜇 𝑗𝑗𝑔𝑔𝑗𝑗 𝑙𝑙𝑔𝑔𝑙𝑙 2𝑔𝑔𝑠𝑠 𝑗𝑗 𝑙𝑙 2 • Protons: = 1, = 51.6 , Neutrons:1 = 0, = 3.8 1 𝜇𝜇 𝑗𝑗𝑔𝑔𝑗𝑗 𝑗𝑗 2𝑙𝑙+1 𝑔𝑔𝑙𝑙 − 𝑗𝑗 2𝑙𝑙+1 𝑔𝑔𝑠𝑠 𝑗𝑗 𝑙𝑙 − 2 • These boundaries𝑙𝑙 𝑠𝑠 are the “Schmidt𝑙𝑙 lines”𝑠𝑠 (Th. Schmidt,𝑔𝑔 Z.Phys. (1937))𝑔𝑔 𝑔𝑔 𝑔𝑔 − and nearly all measured fall between these ϖ As with excited state J ’s, deviation𝑗𝑗 between experiment & shell model predictions tell𝑔𝑔 us something interesting is going on. E.g. mixing between single-particle occupations, polarization of the core, corrections to -exchange 20 mediating the (G. Neyens, Rep. Prog. Phys. (2003), J. Booten et al. Phys.Rev.C (1991)) What else are predictions good for? Isomers (long-lived x.s.) • Most excited states decay𝜋𝜋 via γ-emission in a matter of femto-seconds, but some stick around for many nanoseconds, milliseconds, seconds,𝐽𝐽 or even universe lifetimes. • These are meta-stable states, a.k.a. isomers • The reason is γ-emission is suppressed, since it would require large angular momentum transfer • So, where do we expect low-lying high- excited states? • Where a large exists between neighboring levels (thanks to the spin-orbit interaction) 𝑗𝑗 that are near the last single particle orbit. ∆𝑗𝑗 • Namely, below magic #’s 50, 82, 126 • For these cases, we expect a parity change • Where multiple are possible for the ground-state (but one is favored by the Brennan-Bernstein rules) and high- single𝑗𝑗-particle levels are involved • Namely, odd-odd nuclei • For these𝑗𝑗 cases, we don’t expect a parity change

21 Loveland, Morrissey, & Seaborg, Modern Nuclear Chemistry (2006) Isomers on the Nuclear Chart L. van Dommelen, for Engineers (2012) Even-Z, Odd N Odd-Z, Even N Odd-Z, Odd N

Special cases exist (mostly for higher-A nuclides) where even-even nuclei have isomers (e.g. M. Müller-Veggian et al., Z.Phys.A (1979)) 22 Impact of Isomers (selected examples) e.g. 26mAl complicating nova calculations Medical Imaging e.g. mapping blood flow in the brain with SPECT using 99mTc

NASA J. José, Stellar Explosions (2015)

Nuclear Energy Storage Controlled energy storage and release using isomers and lasers K. Ogasawara et al. American Journal of Neuroradiology (2001) was a hot topic for a while …but it turns out to be exceedingly difficult, to the point of maybe not being possible

P. Walker & J. Carrol, Physics Today (2005) 23 What else are predictions good for? • Note that our methods to𝜋𝜋 determine didn’t depend on whether we were working with protons or neutrons 𝜋𝜋 • If we interchange N for𝐽𝐽 Z, we will get the𝐽𝐽 same answer • Such pairs are called “mirror nuclei” • When we examine the levels for mirror nuclei, correcting for the different Coulomb energy, we see a remarkable similarity • E.g.

Loveland, Morrissey, & Seaborg, Modern Nuclear Chemistry (2006) C. Yuan et al., Phys. Rev. C (2014) • Such mirror is evidence for charge-independence of the nuclear force and a justification for the concept of isospin It’s also handy when you need estimates for a nucleus you can’t access when you can access its mirror (e.g. C.Akers et al. PRC 2016) 24 The Shell Model: It slices, it dices, it makes julienne fries! What can’t it do!?

Shell Model 25 Shell Model Limitations: L. van Dommelen, Quantum Mechanics for Engineers (2012) • You’ll often find the shell model description isn’t as good as you would hope • The detailed reasons for failures are varied, however, they mostly indicate the basic premise of the calculation is incorrect • Shell Model calculations (generally speaking) assume • Minimally-interacting nucleons (i.e. mostly independent, except pairing) • Spherical inert cores of nucleons • Solutions to these problems are: • Modify the shell-model calculation to get the added level of realism • Use a different model

26 Loveland, Morrissey, & Seaborg, Modern Nuclear Chemistry (2006) Shell Model Limitations: • For example, nuclei have series of states that are spaced in energy and linked via transitions that can be described by a collective rotation/vibration of the nucleus. • The rotational bands of even-even nuclei link the ground state to 2+,4+,6+,8+, etc., excited states

• Another example is the inability to predict nuclear masses. • Shell model potentials must be adjusted to reproduce the ground-state • Some other approach, such as a collective model (e.g. the liquid drop model) is needed instead

27 Further Reading

• Chapter 6, Appendix E: Modern Nuclear Chemistry (Loveland, Morrissey, Seaborg) • Chapters 7: Nuclear & Particle Physics (B.R. Martin) • Chapters 6-8: Lecture Notes in Nuclear Structure Physics (B.A. Brown) • Chapter 14, Section 12: Quantum Mechanics for Engineers (L. van Dommelen) • Chapter 1, Section 6: of Stars (C. Iliadis) • Chapter 11: The (R. Evans)

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