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Surrey Mini-School, June 2005

Probing single-particle structure in nuclei: Spin-orbit interaction and monopole shifts in stable isotopes. SJ Freeman, University of Manchester. Surrey Mini-School, June 2005. EXAMPLES THROUGHOUT:. A: Single-particle models and how to probe them experimentally.

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Surrey Mini-School, June 2005

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  1. Probing single-particle structure in nuclei: Spin-orbit interaction and monopole shifts in stable isotopes SJ Freeman, University of Manchester Surrey Mini-School, June 2005

  2. EXAMPLES THROUGHOUT: A: Single-particle models and how to probe them experimentally. B: Recent results from proton-stripping reactions on stable Sn isotopes motivated by spin-orbit interactions. Odd-A antimony isotopes51Sbeven

  3. Fundamental models of nuclei: mean field Schrödinger equation for the nucleus: Kinetic energy of each nucleon Two-body forces between nucleons Mean field potential Residual interaction

  4. If the residual interaction is small, rewrite as sum of single-particle one-body Hamiltonians: Single-particle Hamiltonians: Find U(r), solve equation, get ei and φi for each particle.Find nuclear energy and wavefunctions: E=Σei and Ψ constructed from φi EXAMPLE: odd Sb nuclei ~ πφjℓoutside stable Sn core

  5. So how do you find U(r)? If you’re a theorist you probably want to do Hartree-Foch….…but often in shell models simple minded potentials are easy to handle, at the expense of bigger residual interactions!

  6. These need some modification to reproduce experimental shell closures (near stability).SPIN-ORBIT INTERACTION Single-particle orbitals, φjℓ , used to find nuclear wavefunction EXAMPLE: odd Sb nuclei ~ proton in { g7/2d5/2 d3/2h11/2 s1/2 }outside stable Sn core

  7. Beyond the mean field: We solved this!Eigenfunctions: Ψimade up from a single-particle configuration Reality adds this! Eigenfunctions: Φi = ΣajΨjAdmixture of single-particle configurations. Our eigenfunctions are no longer quite right and residual interaction will mix them together!

  8. So what are “single-particle states”? (with reference to the odd-A Sb example) • If ψ≈ core plus one nucleon in φj ℓ • Otherwise, ψ≈Σ aj ℓφj ℓ • In this situation, the strength from any one single-particle orbital φj ℓis spread or fragmented over many different states (strength function).

  9. 41Ca ℓ=1 Neutron Strength Function Spectroscopic strength Excitation energy (MeV)

  10. Quick pause for breath… • Nuclear wavefunctions can be constructed from single-particle orbitals, with the appropriate choice of a mean-field potential • Mean fields require the ad-hoc introduction of a spin-orbit interaction • Residual interactions mix the “single-particle” nuclear wavefunctions. Any state might consist of an admixture of many configurations. “Single-particle” strength becomes spread across many states.

  11. Single-Nucleon Transfer Reactions Examples:(d,p), (α,3he)…(p,d), (3he, α)…(3he, d), (α, t)…(d,3he), (t,α)… Single-step processleads to population of single-particle strength Surface effect: strong absorption means that any deeper penetration leads to more complex processes

  12. Angular Momentum Matching in A(a,b)B In general there is a spin change, IA≠ IB : |IA-IB| < ∆I < IA+IB In in order to conserve angular momentum, ∆I has to come from relative motion of interacting pair (in units of Ћ): ℓin= r x kinℓout= r x kout The difference is then, given surface transfer: L= (kin − kout) x R = q x R Then∆I = L ¤ S Bottom line condition: |L| = [ℓ (ℓ +1)] ½ ≤ q R

  13. kout kin θ q = kin − kout The reaction Q-value, beam energy (kin) and excitation energy kout The direction of koutdefines the scattering angle, θ: q 2= kin2+kout2− 2kin kout cos θ = (kin − kout) 2− 2kin kout (1 − cos θ) Or for particular final state, changing the scattering angle changes q. Larger θ, larger q.

  14. Satisfying L = q x R L=0 Transfers:Need to satisfy 0 ≤ q R (easy!)Easiest for projectile to sail on undisturbed Peak at θ = 0 L>0 Transfers:Need to satisfyL ≤ q RAs R is surface radius, need progressively higher q. Peak at higher and higher θ, corresponding to L ≈ q R Angular distributions indicative of L transfer

  15. 208Pb (d,p) 209Pb NB: Inelastically scattered waves originate from many points on nuclear surface which satisfy: L= q x R

  16. Energetics and matching Peaks occur at L≈ q R q 2= (kin − kout) 2− 2kin kout (1 − cos θ) Go some way to matching using scattering angle, but energetics plays an important role in comparing different reactions. Large Q values mean that linear momenta in entrance and exit channels are very different, i.e. q is naturally large for some reactions. So some reactions naturally favour larger L transfers. So-called “mismatched” reactions. (Could also meet condition by a change in r… dangerous in terms of validity of simple reaction models)

  17. (α,t) have Q values around 15 MeV more negative than (3He,d) ℓ = 4 ℓ = 0 ℓ = 2 ℓ = 5

  18. Distorted-Wave Born Approximation For a particular single-particle state, can calculate the transfer cross section using DWBA. • Ingredients: • Bound-state wavefunction • Interaction causing transfer • Incoming and outgoing waves In Born approximation: optical model can be used to calculate incoming and outgoing partial waves

  19. If not dealing with the FULL single-particle strength, then cross section is less than this calculation: spectroscopic factor is a measure of overlap of the final state with a wavefunction made up of a single particle outside an inert core • pure single-particle state S is large. • otherwise S is related to the amount of the admixture of φj ℓin the wavefunction of the final state. dσ = Sj ℓ dσℓdΩ dΩDWBA NB: need reliable DWBA calculation

  20. Quick pause for breath… • Single-nucleon transfer is a good probe of single-particle strength • Angular distributions indicate ℓ. Overall spin j from model-dependent considerations, polarisation or fine details of distributions in some reactions • Different reactions are matched for different ℓ transfers, dictated by the reaction Q value. • Spectroscopic factor measured the extent to which a state can be represented by a single particle orbital.

  21. Single-particle structure in nuclei: Is the spin-orbit interaction changing with neutron excess? JP Schiffer, SJ Freeman, C-L Liang, KE Rehm S Sinha, JA Caggiano, C Diebel, A Heinz, R Lewis,A Parikh, PD Parker and JS Thomas Argonne National Laboratory, Yale, Manchester and Rutgers Universities Surrey Minischool June 2005

  22. Magic numbers in exotic systems Beginning to see evidence of changes in the familiar sequences of magic numbers. Demise of:N=8, 20, 28Appearance of:N=6, 16, 30, 32 Deformation, π−ν interactions…….. T. Motobayashi et al., PLB 346, 55 (1995)

  23. Changes in spin-orbit interaction Apparent failure to explain r-process rates without significant shell quenching, possibly by changes to the spin-orbit strength N=82 N=126 Chen et al., PLB 355, 37 (1995)

  24. Spin-Orbit Interaction • Ad-hoc introduction has some basis in nucleon-nucleon scattering. • It must be a surface effect. • But microscopic origins are poorly understood: • Non-relativistic calculations with realistic nuclear forces which reproduce experimental splittings indicate only around a half comes from two-nucleon L.S forces • The rest appears to be made up of pion exchange forces between three or more nucleons Steven C Pieper and VR Pandharipande, PRL 70, 2541 (1993)

  25. Various reasons for changing spin-orbit: • Increases in surface diffuseness. • Influence of continuum states. • Changes in the interior nuclear density distribution.

  26. Measurements of Spin-Orbit Strength • Low ℓ orbitals: measure separation of spin-orbit partners, more accurately the difference in the centroids of the single-particle strength Ex/MeV (d,3He) (e,e’p) When splitting is small, even small admixtures make a difference 10.7 6% p3/2 9.9 3% 6.3 81% 59% p1/2 0.0 113% 63% 6.3 MeV 6.8 MeV 15N ~16O−π

  27. Measurements of Spin-Orbit Strength • High ℓ orbitals Splitting is so large that the higher-lying member of the spin-orbit partner lies at high excitation and always suffers severe fragmentation! Compare energies of: ℓ−½ in a particular shell (ℓ+1) + ½ in the next shell up (intruder)

  28. Locate specific single-particle states in a particular nucleus. Energetic separation of h9/2 − i13/2 h11/2 − g7/2 are, in principle, sensitive to the strength of spin-orbit force 126 h9/2 i13/2 82 h11/2 Where to look? Want inert core. Want as much info as possible. g7/2 50 g9/2 Around Sn isotopes…

  29. Energies from odd-Sb isotopes: Potential variation? Lowest 7/2+ and 11/2− states Are these states an inert core plus odd proton? Could varying amounts of mixing be responsible for the variation in energy?

  30. Radial wavefunctions: Any smooth variation in potentials, or filling of specific neutron levels should have similar effects on both states due to similar radial overlap integrals.

  31. Sn core stability:

  32. Why not just use the old (3He,d)? Measure spectroscopic strengths in proton stripping on Sn targets. • Good matching ensures reaction model’s validity. • If poor, small cross sections where higher-order processes are significant and spectroscopic factors less meaningful. • Four previous experiments with different • Energies • Resolution • DWBA → different S, even for same target. Errors on absolute S are 30-40%

  33. Experimental Details 40 MeV α from Yale ESTU tandem with Enge split-pole spectrometer • For absolute cross sections, elastic scattering measured at 9˚. • Left/right count rates monitored at 30˚ • Measurements made at 6, 13 and 25˚ on all stable Sn targets • More detailed distributions made for 112,118,122Sn

  34. Results 112Sn 114Sn Spectra at 6˚40-50 keV resolution 116Sn 118Sn 120Sn 122Sn 124Sn

  35. DWBA: DWUCK and PTOLEMY, various potentials, FR and ZR Absolute S vary by as much as a factor of 2, but relative S good to 15% Cross sections and spectroscopic factors Cross sections (mb/sr) at 6˚accurate to 10%, ratio to 5%

  36. Relative spectroscopic factors Lowest 7/2+ and 11/2− across stable Sb isotopes have constant spectroscopic factors, and appear to be of near single-particle-like πh11/2 and πg7/2 character.

  37. Conclusions for Sb Chain • Lowest 7/2+ and 11/2− states are single-particle states • Consistent with a decrease in spin-orbit strength of almost a factor 2 by 132Sn Protons outside Z=50 WS parameters fixed with A1/3 dependence; well depth adjusted to BE of 7/2+ state

  38. Similar trend in νi13/2 − νh9/2 for N=83 Core structure not as stable(d,p) although limited already indicates some fragmentationSystematic (α,3He) planned for summer 2005 Neutron excess Neutrons outside N=82

  39. Can this be understood by strong monopole shifts due to tensor π−ν interaction? From 24O to 30Si six protons fill the pd5/2 orbital Strongly attractive pd5/2nd3/2 interaction Results in depression of nd3/2 orbital Large matrix elements for spin-flip/isospin-flip processes Large attractive/repulsive j>j</> interaction between protons and neutrons Vστ=σ.στ.τ fστ(r) Otsuka et al. PRL 87,082502 (2001)

  40. nh11/2 filling in Sn cores with increasing A i.e. nj> Repulsive effect with πh11/2i.e. πj> Attractive effect with pg7/2i.e. πj< Are neutrons affecting protons outside Z=50?

  41. pg7/2 filling in N=82 cores with increasing Z i.e. pj< Attractive effect with ni13/2i.e. nj> Repulsive effect with nh9/2i.e. nj< ph11/2 filling next with increasing Z i.e. pj> Repulsive effect with ni13/2i.e. nj>Attractive effect with nh9/2i.e. nj< Are protons affecting neutrons outside N=82?

  42. Conclusions about monopole shifts • Qualitatively they seem suggestive. • Sn nuclei: nh11/2 until 132Sn, then nf7/2 until 140Sn so trend in splitting is the same for a long while • N=83: trends in splitting do appear to reverse as move from a filling of pg7/2 to ph11/2 • BUT we don’t know we are dealing with single-particle states even for stable systems (will check!) • AND switch to ph11/2 only happens for unstable targets! • QUANTITATIVE calculations have yet to appear…

  43. General Conclusions: • Single-particle spacing does appear to be actually changing, even in stable systems! • Is it something connected to changes in the spin-orbit force? • Or is it due to the effective interaction? • Is there actually a difference? • Either way, these changes would suggest that the shell structure in very neutron-rich nuclei is likely to be radically different from that in stable systems……

  44. Spin-orbit term is an empirical addition to the mean field. • Microscopic calculations with realistic forces ascribe ~50% to two-body L.S. forces and ~50% to pion exchange involving three or more nucleons. • Vs tis a two-body effective interaction.

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