1 / 68

R. F. Casten WNSL, Yale Lecture 2 July, 2009

Proton-neutron and pairing interactions A titanic struggle of good vs. good to determine what nuclei do. R. F. Casten WNSL, Yale Lecture 2 July, 2009. Next 2-ish lectures. Preliminaries

nettieg
Télécharger la présentation

R. F. Casten WNSL, Yale Lecture 2 July, 2009

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Proton-neutron and pairing interactionsA titanic struggle of good vs. good to determine what nuclei do R. F. Casten WNSL, Yale Lecture 2 July, 2009

  2. Next 2-ish lectures Preliminaries The independent particle model – nucleons in a box, confinement and quantization, Pauli Principle, and magic numbers. Residual interactions – the key to answering the riddle of the universe. Pairing vs the p-n interaction: what, how, why? Relation to nuclear shapes. How to measure p-n interaction strengths What to do with those numbers once you have them. Answer: LOTS !!! (See, for example, Dennis’ Martin’s theses.)

  3. TINSTAASQ While I don’t mind hearing myself talk, these lectures are actually for YOU So, please ask questions if stuff isn’t clear.

  4. . . Simple Observables - Even-Even (cift-cift) Nuclei 1000 4+ For Odd-A nuclei, also the angular momentum of the ground state. 400 2+ Masses 0 0+ Jπ E (keV)

  5. First, some background – The independent particle model and the magic numbersWithout them, we would be very sadBasic idea is that nucleons move around the nucleus in orbits determined by a common force or potential acting on all of them

  6. 25 23 21 19 17 S(2n) MeV 15 13 Sm 11 Hf 9 Ba Pb 7 Sn 5 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132 Neutron Number Energy required to remove two neutrons from nuclei (2-neutron binding energies = 2-neutron “separation” energies) N = 82 N = 126 N = 84

  7. Broad perspective on structural evolution: R4/2 Note the characteristic, repeated patterns

  8. Ui Vij r = |ri - rj|  r Independent particle model: magic numbers, shell structure, valence nucleons.Three key ingredients First: Nucleon-nucleon force – very complex One-body potential – very simple: Particle in a box ~ This extreme approximation cannot be the full story. Will need “residual” interactions. But it works surprisingly well in special cases.

  9. Second key ingredient: Quantum mechanics Particles in a “box” or “potential” well Confinement is origin of quantized energies levels 3 1 2 Energy ~ 1 / wave length n = 1,2,3 is principal quantum number E up with n because wave length is shorter

  10. Two quantum numbers, n and l Total angular momentum of a nucleon is j = l + - ½ Such a state has 2 j + 1 magnetic substates, m.

  11. Third key ingredient Pauli Principle • Two fermions, like two protons or two neutrons, can NOT be in the same place at the same time: can NOT occupy the same orbit. • Orbit with total Ang Mom, j, has 2j + 1 substates, hence can only contain 2j + 1 neutrons or protons. This, plus the clustering of levels in simple potentials, gives nuclear SHELL STRUCTURE

  12. The nuclear potential: a rounded square well (Wood-Saxon shape) works quite well in reproducing the magic numbers provided we add in a spin-orbit force that lowers the energies of the j = l + ½ orbits and raises those with j = l – ½

  13. The empirical magic numbers near stability • 2, 8, 20, 28, (40), 50, (64), 82, 126 • These numbers, and a couple of “R4/2” values, are the only things I will ask you to memorize.

  14. How to use the Independent Particle Model • Put nucleons (protons and neutrons separately) into orbits. • Special case: Put 2j + 1 identical nucleons (fermions) in an orbit with angular momentum j. Each one MUST go into a different magnetic substate. Remember, angular momenta add vectorially but projections (m values) add algebraically. • So, total M is sum of m’s M = j + (j – 1) + (j – 2) + ...+ 1/2 + (-1/2) + ... + [ - (j – 2)] + [ - (j – 1)] + (-j) M = 0.If the only possible M is 0, then J= 0 Thus, a full j- shell, and hence a full major shell of nucleons, always has total angular momentum 0. This simplifies things enormously !!! It allows us to often consider only the valence nucleons!

  15. Let’s do 91 40Zr51 Reduces an 91 nucleon problem to a 1 nucleon problem. Saves about 1000 orders of magnitude in complexity !!

  16. What happens when we have more than one “valence“ nucleon? Ind. Part. model too crude. Need to add extra interactions among valence nucleons. H = H0 + Hresid.These dominate the evolution of structure

  17. Coupling of two angular momenta j1+ j2 All values from: j1 – j2 to j1+ j2 (j1 =j2) Example: j1 = 3, j2 = 5: J = 2, 3, 4, 5, 6, 7, 8 BUT: For j1 = j2: J = 0, 2, 4, 6, … ( 2j – 1) (Why these? m-scheme) /

  18. How can we predict these energies? • Need only two ingredients: • Nuclear force is short range and attractive • Pauli Principle • Lets assume a d force that acts ONLY when the two particles are in contact. • Later, we will talk about the pairing interaction, which is very similar.

  19. x

  20. This is the most important slide: understand this and all the key ideas about residual interactions will be clear !!!!!

  21. R4/2< 2.0 Another way to look at the results. Remember, everything depended on the angle between the angular momentum vectors (orbital planes) of the two particles. So, it turns out that the results we have can be expressed in terms of a nice simple formula in terms of those angles. What about the data?? Nuclei with 2 identical particles outside closed shells

  22. “Magic plus 2”: Characteristic spectra Yaaaaaaaaaaaaaaaaaaay !!!! This, ultimately, is why all e-e nuclei have 0+ ground states !!!!!!! ~ 1.3 -ish

  23. Broad perspective on structural evolution: R4/2 Note the characteristic, repeated patterns

  24. Pairing interactionAn alternate short range forcePairing vs. dspectra

  25. Evidence for “Pairing” • Ground states of even-even nuclei with 2 particles outside a doubly magic nucleus have J=0+ with energy much lower than other states. • Related: Ground states of ALL even-even nuclei have J = 0+ • Even-even nuclei are more bound than odd-A nuclei. • (Note: the above are related to the energies of 0+ states relative to states of OTHER spins in the same configuration, and are similar to the effects of any short range interaction. The next relates to the energy of the lowest 0+ state relative to all other states) • Nuclei far from closed shells show a large spacing (~ 2 MeV) between the gs and other non-collective states: the “pairing gap”. • Note: These two sets of experimental features of nuclei are very different. How can we understand them by the concept of pairing?

  26. Pairing Gap in even-even nuclei g.s. ~1800 keV Collective states “pairing gap”

  27. Pairing gap in Sn

  28. Pairing: what it is and what it does / Short range force between identical nucleons. As with any short range force, it favors coupling two nucleons in identical orbits to J = 0. Pairing force drives nuclei towards spherical shapes – J = 0 has no preferred direction in space. That is a “diagonal” effect on the energies. There are also strong mixing effects that are extraordinarily important. -- Pairing produces the well-known “pairing gap” in e-e nuclei.

  29. Effects of Pairing – the pairing gap Pairing couples each pair of particle to J = 0. What is the energy of the first excited 0 + state? Suppose on average the single particle levels are separated by ~ 300 keV (typical for heavy nuclei) Then one would expect a 0+ state at ~ 600 keV, and another 1200 keV, right? But even-even nuclei show a clear gap between the ground state and simple excited states (we are ignoring collective states like rotations or vibrations here). That gap is ~ 2 MeV. WHY ?? 0 0 0 2 2 2 Occ.

  30. Mixing induced by the pairing force • Look at the expression for the pairing force: • What does it do? Suppose all the particles on both sides of the interaction are in the same orbit, that is, j1 =j2=j3=j4 and these particles couple in pairs to J = 0. • Then < |V| > = -G( j + ½) for J = 0. That is, the energy of the 0+ state is lowered. We have seen that. Many people think that that is what pairing is all about. However, that is only part of the story. What else does the matrix element do? • It allows a pair of particles coupled to J = 0 in an orbit j1 (= j2) to scatter into a different orbit j3 ( = j4), still with angular momentum zero. But what is that effect? It is mixing !!!! The final wave function of the ground state will have a mixture of two particles in j1 and two in j2.

  31. Effects of Pairing – partial occupanciesConcept of quasi-particles Now, the occupation of levels is spread out over several levels. ~0.2 ~0.6 ~0.8 ~1.0 ~1.6 ~1.8 Occ.

  32. Single particle energies are replaced with quasi-particle energies involving a “gap” parameter related to the strength of the pairing matrix element. In even-even nuclei the ground state is lowered producing a “pairing gap. Making an excited 0+ state means creating two quasi-particles and so is > 2 D > 2 D In odd-even nuclei the excited states are produced by substituting one quasi-particle for another, so they can have very low energies ! So: e-e nuclei – pairing gap: o-e nuclei – energy compression !

  33. Effects of Pairing – partial occupanciesConcept of quasi-particles Now, the occupation of levels is spread out over several levels. ~0.2 ~0.6 ~0.8 ~1.0 ~1.6 ~1.8 Occ.

  34. Effects of mixing in a simple toy model V Notice the production of a gap?! This is the basic reason the mixing gives a pairing gap.

  35. I’ll stop here and discuss the p-n interaction next time along with ways of actually measuring it, and lots of examples, some taken from ISOLTRAP data (!!!!), of what you can learn from it.

  36. Proton-neutron interaction • Also short range. How do its effects differ? • Favors couplings to both J = 0 and to high J states. Hence drives nuclei toward non-spherical shapes. • Induces configuration mixing in the shell model. • It is, by far, the most important residual interaction in determining collectivity and deformation in nuclei

  37. Sn – Magic: no valence p-n interactions Both valence protons and neutrons Two effects Configuration mixing, collectivity Changes in single particle energies and shell structure

  38. Collectivity Correlations, configuration mixing Crucial for structure Crucial for masses

  39. Microscopic mechanism of first order phase transition (Federman-Pittel, Heyde) Gap obliteration (N ~ 90 ) 1-space 2-space Monopole shift of proton s.p.e. as function of neutron number

  40. Effects of short range p-n interactions in 2 valence particle nuclei – contrast with like-nucleon force that we considered above For like nucleons, here we invoked the Pauli Principal to show that the lowest spin state, 0+, came lowest. For unlike particles, protons and neutrons, there is no such restriction and the maximum angular momentum can also be low.

  41. Like, unlike nucleon d force Like nucleons Some states of unlike nucleons

  42. A practical example

More Related