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Lectures 4 & 5

Lectures 4 & 5. The end of the SEMF and the nuclear shell model. 4.1 Overview. 4.2 Shortcomings of the SEMF magic numbers for N and Z spin & parity of nuclei unexplained magnetic moments of nuclei value of nuclear density values of the SEMF coefficients 4.3 The nuclear shell model

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Lectures 4 & 5

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  1. Lectures 4 & 5 The end of the SEMF and the nuclear shell model

  2. 4.1 Overview • 4.2 Shortcomings of the SEMF • magic numbers for N and Z • spin & parity of nuclei unexplained • magnetic moments of nuclei • value of nuclear density • values of the SEMF coefficients • 4.3 The nuclear shell model • choosing a potential • L*S coupling • Nuclear “Spin” and Parity • Shortfalls of the shell model

  3. 4.2 Shortcomings of the SEMF

  4. (10,10) (N,Z) (6,6) (2,2) (8,8) 2*(2,2) = Be(4,4) Ea-a=94keV 4.2 Shortcomings of the SEMF(magic numbers in Ebind/A) • SEMF does not apply for A<20 • There are systematic deviations from SEMF for A>20

  5. Neutron Magic Numbers Z Proton Magic Numbers N 4.2 Shortcomings of the SEMF(magic numbers in numbers of stable isotopes and isotones) • Magic Proton Numbers (stable isotopes) • Magic Neutron Numbers (stable isotones)

  6. 4.2 Shortcomings of the SEMF(magic numbers in separation energies) • Neutron separation energies • saw tooth from pairing term • step down when N goes across magic number at 82 Ba Neutron separation energy in MeV

  7. Z=82 N=126 N=82 Z=50 N=50 iron mountain 4.2 Shortcomings of the SEMF(abundances of elements in the solar system) • Complex plot due to dynamics of creation, see lecture on nucleosynthesis no A=5 or 8

  8. 4.2 Shortcomings of the SEMF(other evidence for magic numbers, Isomers) • Nuclei with N=magic have abnormally small n-capture cross sections (they don’t like n’s) First excitation energy • Close to magic numbers nuclei can have “long lived” excited states (tg>O(10-6 s) called “isomers”. One speaks of “islands of isomerism” [Don’t make holydays there!] • They show up as nuclei with very large energies for their first excited state (a nucleon has to jump across a shell closure) 208Pb

  9. 4.2 Shortcomings of the SEMF(others) • spin & parity of nuclei do not fit into a drop model • magnetic moments of nuclei are incompatible with drops • actual value of nuclear density is unpredicted • values of the SEMF coefficients except Coulomb and Asymmetry are completely empirical

  10. 4.2 Towards a nuclear shell model • How to get to a quantum mechanical model of the nucleus? • Can’t just solve the n-body problem because: • we don’t know if a two body model makes sense (it does not make much sense for a normal liquid drop) • if it did make sense we don’t know the two body potentials (yet!) • and if we did, we could not even solve a three body problem • But we can solve a two body problem! • Need simplifying assumptions

  11. 4.3 The nuclear shell model This section follows Williams, Chapters 8.1 to 8.4

  12. 4.3 Making a shell model(Assumptions) • Assumptions: • Each nucleon moves in an averaged potential • neutrons see average of all nucleon-nucleon nuclear interactions • protons see same as neutrons plus proton-proton electric repulsion • the two potentials for n and p are wells of some form (nucleons are bound) • Each nucleon moves in single particle orbit corresponding to its state in the potential •  We are making a single particle shell model • Q: why does this make sense if nucleus full of nucleons and typical mean free paths of nuclear scattering projectiles = O(2fm) • A: Because nucleons are fermions and stack up. They can not loose energy in collisions since there is no state to drop into after collision • Use Schroedinger Equation to compute Energies (i.e. non-relativistic), justified by simple infinite square well energy estimates • Aim to get the correct magic numbers (shell closures) and be content

  13. infin. square Coulomb harmonic 4.3 Making a shell model (without thinking, just compute) desired magic numbers • Try some potentials; motto: “Eat what you know” 126 82 50 28 20 8 2

  14. R ≈ Nuclear Radius d ≈ width of the edge 4.3 Making a shell model (with thinking) • We know how potential should look like! • It must be of finite depth and … • If we have short range nucleon-nucleon potential then … • … the average potential must look like the density • flat in the middle (you don’t know where the middle is if you are surrounded by nucleons) • steep at the edge (due to short range nucleon-nucleon potential)

  15. 4.3 Making a shell model (what to expect when rounding off a potential well) • Higher L solutions get larger “angular momentum barrier”  • Higher L wave functions are “localised” at larger r and thus closer to “edge” • Rounding the edge affects high L states most because they are closer to the edge then low L ones. • High L states drop in energy because • can now spill out across the “edge” • this reduces their curvature • which reduces their energy • So high L states drop rounding the well!! Radial Wavefunction U(r)=R(r)*r for the finite square well

  16. 4.3 Making a shell model(with thinking) • Harmonic is bad The “well improvement program” • Even realistic well does not match magic numbers • Need more splitting of high L states • Include spin-orbit coupling a’la atomic • magnetic coupling much too weak and wrong sign • Two-nucleon potential has nuclear spin orbit term • deep in nucleus it averages away • at the edge it has biggest effect • the higher L the bigger the split

  17. Dimension: Length2 compensate 1/r * d/dr 4.3 Making a shell model(spin orbit terms) • Q: how does the spin orbit term look like? • Spin S and orbital angular momentum L in our model are that of single nucleon in the assumed average potential • In the middle the two-nucleon interactions average to a flat potential and the two-nucleon spin-orbit terms average to zero • Reasonable to assume that the average spin-orbit term is strongest in the non symmetric environment near the edge 

  18. 4.3 Making a shell model(spin orbit terms) • Good quantum numbers without LS term : • l, lz & s=½ , sz from operators L2, Lz, S2, Sz with Eigenvalues of l(l+1)ħ2, s(s+1)ħ2, lzħ, szħ • With LS term need operators commuting with new H • J=L+S & Jz=Lz+Sz with quantum numbers j, jz, l, s • Since s=½ one gets j=l+½ or j=l-½ (l≠0) • Giving eigenvalues of LS [ LS=(L+S)2-L2-S2 ] • ½[j(j+1)-l(l+1)-s(s+1)]ħ2 • So potential becomes: • V(r) + ½l ħ2 W(r) for j=l+½ • V(r) - ½(l+1) ħ2 W(r) for j=l -½ • we can see this asymmetric splitting on slide 16

  19. 4.3 Making a shell model(fine print) • There are of course two wells with different potentials for n and p • We currently assume one well for all nuclei but … • The shape of the well depends on the size of the nucleus and this will shift energy levels as one adds more nucleons • Using a different well for each nucleus is too long winded for us though perfectly doable • So lets not use this model to precisely predict exact energy levels but to make magic numbers and …

  20. 4.3 Predictions from the shell model (total nuclear “spin” in ground states) • Total nuclear angular momentum is called nuclear spin = Jtot • Just a few empirical rules on how to add up all nucleon J’s to give Jtot of the whole nucleus • Two identical nucleons occupying same level (same n,j,l) couple their J’s to give J(pair)=0 Jtot(even-even ground states) = 0 Jtot(odd-A; i.e. one unpaired nucleon) = J(unpaired nucleon) Carefull: Need to know which level nucleon occupies. I.e. more or less accurate shell model wanted! |Junpaired-n-Junpaired-p|<Jtot(odd-odd)< Junpaired-n+Junpaired-p there is no rule on how to combine the two unpaired J’s

  21. 4.3 Predictions from the shell model (nuclear parity in groundstates) • Parity of a compound system (nucleus): • P(even-even groundstates) = +1 because all levels occupied by two nucleons • P(odd-A groundstates) = P(unpaired nucleon) • No prediction for parity of odd-odd nuclei

  22. 4.3 Shortcomings of the shell model • The fact that we can not predict spin or parity for odd-odd nuclei tells us that we do not have a very good model for the LS interactions • A consequence of the above is that the shell model predictions for nuclear magnetic moments are very imprecise • We can not predict accurate energy levels because: • we only use one “well” to suit all nuclei • we ignore the fact that n and p should have separate wells of different shape • As a consequence of the above we can not reliably predict much (configuration, excitation energy) about excited states other then an educated guess of the configuration of the lowest excitation

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