Moment (and Fermi gas) methods for modeling nuclear state densities

1. Moment (and Fermi gas) methodsfor modeling nuclear state densities Calvin W. Johnson (PI) Edgar Teran (former postdoc) San Diego State University supported by grants US DOE-NNSA SNP 2008

2. We all know that Art is not truth. Art is a lie that makes us realize the truth, at least the truth that is given to us to understand.- Pablo Picasso SNP 2008

3. Tools: mean-fields and moments 3 approaches to calculating nuclear level/state density: an incomplete list of names… Fermi gas / combinatorial: Goriely, Hilaire, Hillman/Grover, Cerf, Uhrenholt Monte Carlo shell model: Alhassid, Nakada Spectral distribution/ “moment” methods: French, Kota, Grimes, Massey, Horoi, Zelevinsky, Johnson/Teran SNP 2008

4. Overview of Talk Theme: Testing approximate schemes for level/state densities against (tedious) full-scale CI shell model diagonalization Part I: Fermi gas model vs. exact shell model -- Single particle energies from Hartree-Fock -- Adding in rotation (new!) Part II: Moment (spectral distribution) methods -- mean-field (centroids or first moments) -- residual interaction (spreading widths or second moments) -- collective interaction (third moments) SNP 2008

5. Comparison against “exact” results How reliable are calculations? If we put in the right microphysics, do we get out reasonably good densities? To answer this, we compare against “exact” calculations from full configuration-interaction (CI) diagonalization of realistic Hamiltonians in a finite shell-model basis. We can then look at an “approximate” method (Fermi gas, spectral distribution) using the same input and compare. SNP 2008

6. What an interacting shell-model code does: Input into shell model: • set of single-particle states(1s1/2,0d5/2, 0f7/2 etc) • many-body configurations constructed from s.p. states: (f7/2)8, (f7/2)6(p3/2)2, etc. • two-body matrix elements to determine Hamiltonian between many-body states: <(f7/2)2 J=2, T=0| V| (f5/2 p3/2) J=2, T=0> (assume someone else has already done the integrals) Output: eigenenergies and wavefunctions (vectors in basis of many-body Slater determinants) SNP 2008

7. Mean-field Level densities Result: “exact” state density from CI shell-model diagonalization 32S Okay, let’s start comparing with approximations! SNP 2008

8. Fermi Gas Models Start with (equally-spaced) single-particle levels and fill them like a Fermi gas (Bethe, 1936): Some modern version use “realistic” single-particle levels derived from Hartree-Fock (Goriely) The parameter a reflects the density of single-particle states near the Fermi surface The single-particle levels arise from a mean field! SNP 2008

9. Fermi Gas Models The “thermodynamic method” centers around the partition function (1) Construct the partition function either from single-particle density of states or (later) from Monte Carlo evaluation of a path integral (2) Invert the Laplace transform through the saddle-point approximation approximate integrand by a Gaussian the “saddle-point condition” fixes the value of β0 for a given energy E SNP 2008

10. Fermi Gas Models Of course, one needs the partition function! Traditionally one derives it from the single-particle density of states g(ε), either from Fermi gas or from Hartree-Fock (Bogoliubov) + corrections for rotation, shell structure, etc. Single-particle energies from Hartree-Fock mean-field: εip,n Single-particle density of states: Partition function : Then apply saddle-point method... SNP 2008

11. Mean-field Level densities SNP 2008

12. Mean-field Level densities These are both spherical nuclei... what about deformed nuclides? SNP 2008

13. Mean-field Level densities SNP 2008

14. Mean-field Level densities SNP 2008

15. Mean-field Level densities Difference is due to fragmentation of Hartree-Fock single-particle energies in deformed mean-field 0d3/2 1s1/2 Fermi surface Fermi surface 0d5/2 Smaller s.p. level density smaller nuclear level density deformed spherical SNP 2008

16. Adding collective motion (NEW) Deformed HF state as an intrinsic state: (get moment of inertia I from cranked HF) SNP 2008

17. Adding collective motion (NEW) Z(β) = Zπ(sp) Z(sp) (1 + Zrot) All of the parameters derived directly from HF calculation (SHERPA code by Stetcu and Johnson) using CI shell-model interaction Computationally very cheap: a matter of a few seconds SNP 2008

18. Adding collective motion (NEW) SNP 2008

19. Adding collective motion (NEW) SNP 2008

20. Adding collective motion (NEW) SNP 2008

21. Adding collective motion (NEW) SNP 2008

22. Adding collective motion (NEW) SNP 2008

23. Adding collective motion (NEW) Cranking is problematic for odd-odd (I used real wfns and should allow complex) SNP 2008

24. Introduction toStatistical Spectroscopy (also known as “spectral distribution theory”) Pioneered by J. Bruce French 1960’s-1980’s other luminaries include: J. P. Draayer, J. Ginocchio, S. Grimes, V. Kota, S.S.M. Wong, A.P. Zuker + many others... Problem: diagonalization is too hard and gives too much detailed information Solution: instead of diagonalizing H, find moments: tr Hn Key question: how many moments do we need? Rather than many moments (over the entire space) tr Hn, n = 1,2,3,4,5,6,7... compute low moments (n = 1,2,3,4) on subspaces SNP 2008

25. How we do it:a detailed version The important configuration moments Dimension Centroid: Width: Higher central moments Scaled moments Asymmetry (or skewness): m3(α) Excess : m4(α) - 3 = 0 for Gaussian SNP 2008

26. Introduction toStatistical Spectroscopy Primer on moments Interpretation of moments: centroid = spherical HF energy width = avg spreading width of residual interaction asymmetry = measure of collectivity centroid centroid asymmetric width width SNP 2008

27. Introduction toStatistical Spectroscopy Then we consider the level density as being the sum of individual configuration densities SNP 2008

28. Level densities as a sum of configuration densities We model the level density as a sum of partial (configuration) densities, each of which are modeled as Gaussians SNP 2008

29. Level densities as a sum of configuration densities What can we do to improve our model? Go to third moments: asymmetries Not satisfactory! SNP 2008

30. Level densities as a sum of configuration densities It is (often) important to include much better than 3rd and 4th moments using only second moments collective states difficult to get “starting energy” also difficult to control SNP 2008

31. Comparison with experiments NB: computed + parity states and multiplied × 2 SNP 2008

32. Obligatory Summary Fermi gas model can work surprisingly well and is computationally cheap Deformed nuclei need to have rotation put in. One can use the single-particle energies and moment of inertia from (cranked) Hartree-Fock – compares well to full CI calculation For larger model spaces may need pairing, shell effects, etc. SNP 2008

33. Obligatory Summary View nuclear many-body Hamiltonian through lens of moment methods: 1st (configuration) moments = mean-field 2nd moments = spreading widths of residual interaction 3rd moments = collectivity of residual interaction SNP 2008