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Nuclear Level Densities

Edwards Accelerator Laboratory. Nuclear Level Densities. Steven M. Grimes. Ohio University Athens, Ohio. Fermi Gas Assumptions: 1.) Non-interacting fermions 2.) Equi-distant single particle spacing  (U) ∝ exp [ 2√au ] / U 3/2 generally successful Most tests at U < 20 MeV

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Nuclear Level Densities

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  1. Edwards Accelerator Laboratory Nuclear Level Densities Steven M. Grimes Ohio University Athens, Ohio

  2. Fermi Gas Assumptions: 1.) Non-interacting fermions 2.) Equi-distant single particle spacing (U) ∝ exp[2√au] / U3/2 generally successful Most tests at U < 20 MeV For nuclei near the bottom of the valley of stability Nuclear Level Densities

  3. DATA: (1) Neutron resonances U≈ 7 MeV near valley of stability (2) Evaporation spectra U ≈ 3 – 15 MeV near valley of stability (3) Ericson Fluctuations U ≈ 15 – 24 MeV near valley of stability (4) Resolved Levels U ≤ 4 – 5 MeV Most points for nuclei in bottom of valley of stability Nuclear Level Densities

  4. Study of Al-Quraishi, et al. 20 ≤ A ≤ 110 Found a =  A exp[-(Z-Z0)2]  ≈ 0.11  ≈ 0.04 Z -- Z of nucleus Z0 -- Z of  stable nucleus of that Z Reduction in a negligible if ∣Z-Z0∣ ≤ 1 Nuclear Level Densities

  5. Study of Al-Quraishi, et al. Significant effect for ∣Z-Z0∣ = 2 Large effect for | Z-Z0 | ≥ 3 Nuclear Level Densities

  6. Isospin: (N-Z)/2 = TZ T ≥ TZ Nuclear Level Densities 16F 16O 16N T = 0 T = 1 At higher energies, have T=2 multiplets 16C, 16N, 16O, 16F,16Ne (TZ=0) > (TZ=1) > (TZ=2) predicts a =  A exp(  (N-Z)2 )

  7. CONCLUSION Only one analysis finds a lower off of stability line Limited data is the central problem Nuclear Level Densities

  8. Bulk of Level Density information comes from neutron resonances Example: n + 32S → 33S* At low energy (neutrons) find 1/2+ levels at about 7 MeV of excitation  Not feasible for unstable targets  Only get density at one energy  Need  ( = 〈Jz2〉1/2 ) to get total level density Nuclear Level Densities

  9. PREDICTIONS A compound nucleus state must have a width which is narrow compared to single particle width This indicates compound levels will not be present once occupancy of unbound single particle states is substantial. Nuclear Level Densities

  10. Assume we can use Boltzmann distribution as approximation to Fermi-Dirac distribution Since U = a2  = √U/a  is temperature a is level density parameter U is excitation Energy Nuclear Level Densities

  11. If occupancy of state at excitation energy B is less than 0.1 exp[ -B/ ] = exp[ -B√a/U ] ≤ 0.1 a ≈ A/8 exp[ -B √A/8U ] ≤ 0.1 so, -B √A/8U ≤ -2.3 UC ≤ ( AB2 / 42.3 ) Above this energy we will lose states Nuclear Level Densities

  12. Nuclear Level Densities A B(MeV) U(MeV) 20 8 30.3 200 8 303. 20 6 17.04 200 6 170.0 20 4 7.58 200 4 75.8 20 2 1.9 200 2 19

  13. For nucleosynthesis processes, we will frequently have B ~ 4 MeV for proton or neutron For A ~ 20 level density is substantially reduced by 10 MeV if B = 4 For A ~ 20 level density limit is ≈ B if B = 2 MeV Substantial reduction in compound resonances Nuclear Level Densities

  14. Will also reduce level density for final nucleus in capture We also must consider parity In fp shell even-even nuclei have more levels of + parity at low U than - parity Even-odd or odd-even have mostly negative parity states Nuclear Level Densities

  15. Thus, compound nucleus states not only have reduced (U) but also suppress s-wave absorption because of parity mismatch Nuclear Level Densities

  16. Also have angular momentum restrictions Could have even-even target near drip-line where g9/2 orbit is filling Need 1/2+ levels in compound nucleus Nuclear Level Densities

  17. Low-lying states are 0+ ⊗ g9/2 9/2+ or 2+ ⊗ g9/2 5/2+, 7/2+, 9/2+, 11/2+, 13/2+ At low energy 1/2+ states may be missing No s-wave compound nuclear (p,) (or (n,)) reactions Nuclear Level Densities

  18. EXPERIMENTAL TESTS Measure: 55Mn(d,n)56Fe 58Fe(3He,p)60Co 58Fe(3He,d)57Fe 58Fe(3He,n)60Ni 58Ni(3He,p)60Cu 58Ni(3He,)57Ni 58Ni(3He,n)60Zn Nuclear Level Densities

  19. Result: lower a → higher average E →lower multiplicity If compound nucleus is proton rich Al Quraishi term will further inhibit neutron decay Nuclear Level Densities Change in a a (a-a) (Ep) Ep

  20. Nuclear Level Densities

  21. Span range of  Z-Z0  values up to ~ 2.5 Find evidence that a /A decreases with ∣Z-Z0∣ increasing Nuclear Level Densities

  22. Currently calculating these effects Woods-Saxon basis Find single particle state energies and widths Compare  with all sp states with  including only bound or quasibound orbits Nuclear Level Densities

  23. Expect reduction in a if ∣Z-Z0∣ ≈ 2, 3 As ∣Z-Z0∣ increases, new form with peak at 5-10 MeV may emerge (i.e. (20) ≈ 0) Nuclear Level Densities

  24. Calculations and measurements underway 1.) Hope to determine whether off of stability line a drops 2.) is form a = A exp[-(Z-Z0)2] appropriate? 3.) As drip line is approached, we may have to abandon Bethe form and switch to Gaussian Nuclear Level Densities

  25. HEAVY ION REACTIONS Can get to 30-40 MeV of excitation with lower pre-equilibrium component Cannot use projectile with A about equal to target (quasi-fission) Nuclear Level Densities

  26. HEAVY ION REACTIONS Recent Argonne measurements 60Ni + 92Mo 60Ni + 100Mo got only 30-35% compound nuclear reactions Jcontact too high Not enough compound levels Nuclear Level Densities

  27. Projectiles with A < half of target A are better Want compound nuclei with |Z-Z0| ≥ 2 to compare with |Z-Z0| ≈ 0 Nuclear Level Densities

  28. Nuclear Level Densities REACTIONS 24Mg + 58Fe  82Sr* 24Mg + 58Ni  82Zr* 18O + 64Ni  82Kr* Excitation energy  60 MeV 82Kr has Z = Z0 82Sr has Z = Z0 + 2 82Zr has Z = Z0 + 4

  29. Compare: Rohr a A Al Quraishi a A exp[ (Z-Z0)2 ] Nuclear Level Densities

  30. Nuclear Level Densities Decay Fractions Rohr .96 n .04 p .005  Al-Quraishi .93 n .06 p .01  82Kr .67 n .25 p .05  .02 d .61 n .33 p .05  .01 d 82Sr

  31. Nuclear Level Densities Decay Fractions Al-Quraishi .36 n .55 p .07  .02 d Rohr .43 n .49 p .06  .02 d 82Zr

  32. Nuclear Level Densities Rohr: Higher a for  Z-Z0   2 Al-Quraishi: Lower a for  Z-Z0   2 Get softer spectrum with Rohr More 4-6 particle emissions than Al-Quraishi

  33. Nuclear Level Densities Calculations Solve for single particle energies in a single particle (Woods-Saxon) potential Compare level density including all single particle states with level density including only those with  < 500 keV

  34. Nuclear Level Densities Calculations Small effects for Z  Z0 Large effects for ∣Z-Z0∣ ≥ 4 Looking at including two body effects in these calculations with moment method expansions

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  42. CONCLUSIONS Astrophysics has need for level densities off of the stability line for A ≤ 100 Data base in this region ( ∣Z-Z0∣ ≥ 2 ) is poor Model predicts that a decreases with |Z-Z0| Need more reaction data for nuclei in the region of |Z-Z0| 2 Nuclear Level Densities

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