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

2. EXAMPLES THROUGHOUT: Odd-A antimony isotopes 51 Sb even 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.

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 e i and φ i for each particle. Find nuclear energy and wavefunctions: E=Σe i and Ψ constructed from φ i EXAMPLE: odd Sb nuclei ~ πφ j ℓ outside stable Sn core

5. 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! So how do you find U(r)?

6. These need some modification to reproduce experimental shell closures (near stability). SPIN-ORBIT INTERACTION EXAMPLE: odd Sb nuclei ~ proton in { outside stable Sn core EXAMPLE: odd Sb nuclei ~ proton in { g 7/2 d 5/2 d 3/2 h 11/2 s 1/2 } outside stable Sn core Single-particle orbitals, φ j ℓ, used to find nuclear wavefunction

7. Beyond the mean field: We solved this! Eigenfunctions: Ψ i made up from a single- particle configuration Reality adds this! Eigenfunctions: Φ i = Σa j Ψ j Admixture 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 φ j ℓ If ψ ≈ core plus one nucleon in φ j ℓ Otherwise, ψ ≈ Σ a j ℓ φ j ℓ Otherwise, ψ ≈ Σ a j ℓ φ j ℓ In this situation, the strength from any one single-particle orbital φ j ℓ is spread or fragmented over many different states (strength function). In this situation, the strength from any one single-particle orbital φ j ℓ is spread or fragmented over many different states (strength function).

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

10. Quick pause for breath… Nuclear wavefunctions can be constructed from single-particle orbitals, with the appropriate choice of a mean-field potential 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 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. 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, 3 he)… (p,d), ( 3 he, α)… ( 3 he, d), (α, t)… (d, 3 he), (t,α)… Single-step process leads 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, I A ≠ I B : |I A -I B | < ∆I < I A +I B |I A -I B | < ∆I < I A +I B In in order to conserve angular momentum, ∆I has to come from relative motion of interacting pair (in units of Ћ): ∆I has to come from relative motion of interacting pair (in units of Ћ): ℓ in = r x k in ℓ out = r x k out The difference is then, given surface transfer: Bottom line condition: L= (k in − k out ) x R = q x R |L| = [ ℓ (ℓ +1)] ½ ≤ q R Then∆I = L ¤ S Then ∆I = L ¤ S

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

14. Need to satisfy0 ≤ q R (easy!) Easiest for projectile to sail on undisturbed Peak at θ = 0 L=0 Transfers: Need to satisfy0 ≤ q R (easy!) Easiest for projectile to sail on undisturbed Peak at θ = 0 Need to satisfyL ≤ q R As R is surface radius, need progressively higher q. Peak at higher and higher θ, corresponding to L ≈ q R L>0 Transfers: Need to satisfyL ≤ q R As 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 Satisfying L = q x R

15. 208 Pb (d,p) 209 Pb 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 = (k in − k out ) 2 − 2k in k out (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) (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 ( 3 He,d) ℓ = 5 ℓ = 0 ℓ = 4 ℓ = 2

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

19. 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. pure single-particle state S is large. otherwise S is related to the amount of the admixture of φ j ℓ in the wavefunction of the final state. otherwise S is related to the amount of the admixture of φ j ℓ in the wavefunction of the final state. dσ = S j ℓ dσ ℓ dΩ dΩ DWBA If not dealing with the FULL single-particle strength, then cross section is less than this calculation: NB: need reliable DWBA calculation

20. Quick pause for breath… Single-nucleon transfer is a good probe of single- particle strength 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 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. 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. 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? Surrey Minischool June 2005 JP Schiffer, SJ Freeman, C-L Liang, KE Rehm 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

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

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 Chen et al., PLB 355, 37 (1995) N=126 N=82

24. Spin-Orbit Interaction Ad-hoc introduction has some basis in nucleon-nucleon scattering. Ad-hoc introduction has some basis in nucleon-nucleon scattering. It must be a surface effect. It must be a surface effect. But microscopic origins are poorly understood: 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 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 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. Increases in surface diffuseness. Influence of continuum states. Influence of continuum states. Changes in the interior nuclear density distribution. 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 Low ℓ orbitals: measure separation of spin-orbit partners, more accurately the difference in the centroids of the single-particle strength p 3/2 p 1/2 15 N ~ 16 O− π 0.0 113% 63% 6.3 81% 59% 9.9 3% 10.7 6% E x /MeV (d, 3 He) (e,e’p) When splitting is small, even small admixtures make a difference 6.3 MeV6.8 MeV

27. Measurements of Spin-Orbit Strength High ℓ orbitals 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. 50 82 h 9/2 i 13/2 g 7/2 g 9/2 h 11/2 Locate specific single-particle states in a particular nucleus. Energetic separation of h 9/2 − i 13/2 h 11/2 − g 7/2 are, in principle, sensitive to the strength of spin-orbit force 126 Where to look? Want inert core. Want as much info as possible. Around Sn isotopes…

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

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 ( 3 He,d)? Good matching ensures reaction model’s validity. Good matching ensures reaction model’s validity. If poor, small cross sections where higher-order processes are significant and spectroscopic factors less meaningful. If poor, small cross sections where higher-order processes are significant and spectroscopic factors less meaningful. Four previous experiments with different Four previous experiments with different Energies Energies Resolution Resolution DWBA DWBA → different S, even for same target. Errors on absolute S are 30-40% Measure spectroscopic strengths in proton stripping on Sn targets.

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

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

35. Cross sections and spectroscopic factors Target 7/2 + 11/2 − Ratio S 7/2 S 11/2 112 Sn 14.621.41.470.990.84 114 Sn 19.627.31.391.100.93 116 Sn 19.730.91.570.950.97 118 Sn 20.433.51.640.880.99 120 Sn 27.939.41.411.131.12 122 Sn 24.635.51.450.981.00 124 Sn 24.739.21.591.001.12 Cross sections (mb/sr) at 6˚accurate to 10%, ratio to 5% DWBA: DWUCK and PTOLEMY, various potentials, FR and ZR 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% Absolute S vary by as much as a factor of 2, but relative S good to 15%

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 π h 11/2 and π g 7/2 character.

37. Conclusions for Sb Chain WS parameters fixed with A 1/3 dependence; well depth adjusted to BE of 7/2+ state Lowest 7/2 + and 11/2 − states are single- particle states 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 132 Sn Consistent with a decrease in spin-orbit strength of almost a factor 2 by 132 Sn Protons outside Z=50

38. Similar trend in νi 13/2 − νh 9/2 for N=83 Neutron excess Core structure not as stable (d,p) although limited already indicates some fragmentation Systematic ( α, 3 He) planned for summer 2005 Neutrons outside N=82

39. Can this be understood by strong monopole shifts due to tensor π−ν interaction? From 24 O to 30 Si six protons fill the  d 5/2 orbital Strongly attractive  d 5/2  d 3/2 interaction Results in depression of d 3/2 orbital Large matrix elements for spin- flip/isospin-flip processes Large attractive/repulsive j >  j interaction between protons and neutrons V στ =σ.σ τ.τ f στ (r) Otsuka et al. PRL 87,082502 (2001)

40. h 11 /2 filling in Sn cores with increasing A i.e. j > h 11 /2 filling in Sn cores with increasing A i.e. j > Repulsive effect with πh 11/2 i.e. πj > Attractive effect with  g 7/2 i.e. πj < Are neutrons affecting protons outside Z=50?

41.  g 7/2 filling in N=82 cores with increasing Z i.e.  j < Attractive effect with i 13/2 i.e. j > Repulsive effect with h 9/2 i.e. j Repulsive effect with h 9/2 i.e. j <  h 11 /2 filling next with increasing Z i.e.  j > Repulsive effect with i 13/2 i.e. j > Attractive effect with h 9/2 i.e. j Attractive effect with h 9/2 i.e. j < Are protons affecting neutrons outside N=82?

42. Conclusions about monopole shifts Qualitatively they seem suggestive. Qualitatively they seem suggestive. Sn nuclei: h 11/2 until 132 Sn, then f 7/2 until 140 Sn so trend in splitting is the same for a long while Sn nuclei: h 11/2 until 132 Sn, then f 7/2 until 140 Sn 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  g 7/2 to  h 11/2 N=83: trends in splitting do appear to reverse as move from a filling of  g 7/2 to  h 11/2 BUT we don’t know we are dealing with single- particle states even for stable systems (will check!) BUT we don’t know we are dealing with single- particle states even for stable systems (will check!) AND switch to  h 11/2 only happens for unstable targets! AND switch to  h 11/2 only happens for unstable targets! QUANTITATIVE calculations have yet to appear… QUANTITATIVE calculations have yet to appear…

43. General Conclusions: Single-particle spacing does appear to be actually changing, even in stable systems! Single-particle spacing does appear to be actually changing, even in stable systems! Is it something connected to changes in the spin-orbit force? Is it something connected to changes in the spin-orbit force? Or is it due to the effective interaction? Or is it due to the effective interaction? Is there actually a difference? 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…… 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. 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. Microscopic calculations with realistic forces ascribe ~50% to two-body L.S. forces and ~50% to pion exchange involving three or more nucleons. V  is a two-body effective interaction. V  is a two-body effective interaction.