1. Computational Nuclear Structure SNP 20081 Moment (and Fermi gas) methods for modeling nuclear state densities Calvin W. Johnson (PI) Edgar Teran (former postdoc) San Diego State University supported by grants US DOE-NNSA

2. Computational Nuclear Structure SNP 20082 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

3. Computational Nuclear Structure SNP 20083 3 approaches to calculating nuclear level/state density: Tools: mean-fields and moments 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 an incomplete list of names…

4. Computational Nuclear Structure SNP 20084 Overview of Talk Part I: Fermi gas model vs. exact shell model Theme: Testing approximate schemes for level/state densities against (tedious) full-scale CI shell model diagonalization Part II: Moment (spectral distribution) methods -- mean-field (centroids or first moments) -- residual interaction (spreading widths or second moments) -- collective interaction (third moments) -- Single particle energies from Hartree-Fock -- Adding in rotation (new!)

5. Computational Nuclear Structure SNP 20085 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.

6. Computational Nuclear Structure SNP 20086 Input into shell model: set of single-particle states (1s 1/2,0d 5/2, 0f 7/2 etc) many-body configurations constructed from s.p. states: (f 7/2 ) 8, (f 7/2 ) 6 (p 3/2 ) 2, etc. two-body matrix elements to determine Hamiltonian between many-body states: (assume someone else has already done the integrals) Output: eigenenergies and wavefunctions (vectors in basis of many-body Slater determinants) What an interacting shell-model code does:

7. Computational Nuclear Structure SNP 20087 Mean-field Level densities Result: “exact” state density from CI shell-model diagonalization 32 S Okay, let’s start comparing with approximations!

8. Computational Nuclear Structure SNP 20088 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 single-particle levels arise from a mean field! The parameter a reflects the density of single-particle states near the Fermi surface

9. Computational Nuclear Structure SNP 20089 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 Fermi Gas Models approximate integrand by a Gaussian the “saddle-point condition” fixes the value of β 0 for a given energy E

10. Computational Nuclear Structure SNP 200810 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: ε i p,n Single-particle density of states: Partition function : Then apply saddle-point method...

11. Computational Nuclear Structure SNP 200811 Mean-field Level densities

12. Computational Nuclear Structure SNP 200812 Mean-field Level densities These are both spherical nuclei... what about deformed nuclides?

13. Computational Nuclear Structure SNP 200813 Mean-field Level densities

14. Computational Nuclear Structure SNP 200814 Mean-field Level densities

15. Computational Nuclear Structure SNP 200815 Mean-field Level densities Difference is due to fragmentation of Hartree-Fock single-particle energies in deformed mean-field spherical Fermi surface deformed Fermi surface 0d 5/2 0d 3/2 1s 1/2 Smaller s.p. level density smaller nuclear level density

16. Computational Nuclear Structure SNP 200816 Adding collective motion (NEW) Deformed HF state as an intrinsic state: (get moment of inertia I from cranked HF)

17. Computational Nuclear Structure SNP 200817 Adding collective motion (NEW) Z(β) = Z π (sp)  Z  (sp)  (1 + Z rot ) 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

18. Computational Nuclear Structure SNP 200818 Adding collective motion (NEW)

19. Computational Nuclear Structure SNP 200819 Adding collective motion (NEW)

20. Computational Nuclear Structure SNP 200820 Adding collective motion (NEW)

21. Computational Nuclear Structure SNP 200821 Adding collective motion (NEW)

22. Computational Nuclear Structure SNP 200822 Adding collective motion (NEW)

23. Computational Nuclear Structure SNP 200823 Adding collective motion (NEW) Cranking is problematic for odd-odd (I used real wfns and should allow complex)

24. Computational Nuclear Structure SNP 200824 Introduction to Statistical 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 H n Key question: how many moments do we need? Rather than many moments (over the entire space) tr H n, n = 1,2,3,4,5,6,7... compute low moments (n = 1,2,3,4) on subspaces

25. Computational Nuclear Structure SNP 200825 How we do it: a detailed version The important configuration moments Centroid: Dimension Width: Higher central moments Scaled moments Asymmetry (or skewness): m 3 (α) Excess : m 4 (α) - 3 = 0 for Gaussian

26. Computational Nuclear Structure SNP 200826 Primer on moments centroid width centroid width asymmetric Introduction to Statistical Spectroscopy Interpretation of moments: centroid = spherical HF energy width = avg spreading width of residual interaction asymmetry = measure of collectivity

27. Computational Nuclear Structure SNP 200827 Introduction to Statistical Spectroscopy Then we consider the level density as being the sum of individual configuration densities

28. Computational Nuclear Structure SNP 200828 We model the level density as a sum of partial (configuration) densities, each of which are modeled as Gaussians Level densities as a sum of configuration densities

29. Computational Nuclear Structure SNP 200829 What can we do to improve our model? Level densities as a sum of configuration densities Go to third moments: asymmetries Not satisfactory!

30. Computational Nuclear Structure SNP 200830 Level densities as a sum of configuration densities It is (often) important to include much better than 3 rd and 4 th momentsusing only second moments collective states difficult to get“starting energy” also difficult to control

31. Computational Nuclear Structure SNP 200831 Comparison with experiments NB: computed + parity states and multiplied × 2

32. Computational Nuclear Structure SNP 200832 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.

33. Computational Nuclear Structure SNP 200833 Obligatory Summary View nuclear many-body Hamiltonian through lens of moment methods: 1 st (configuration) moments = mean-field 2 nd moments = spreading widths of residual interaction 3 rd moments = collectivity of residual interaction