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

Transcription

1 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 SNP

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

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

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

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

6 What an interacting shell-model code does: 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: <(f 7/2 ) 2 J=2, T=0 V (f 5/2 p 3/2 ) 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

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

8 Fermi Gas Models Start with (equally-spaced) single-particle levels and fill them like a Fermi gas (Bethe, 1936): 2π exp ρ BBFG ( E) = 12 (4a) 1/ 4 4a( E Δ) ( E Δ) 5/ 4 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 SNP

9 Fermi Gas Models The thermodynamic method centers around the partition function Z( β ) = de e 0 β E ρ ( E) (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 i βe ρ( E) = 21 π dβ e Z( β ) i approximate integrand by a Gaussian ρ( E) Z( β ) e 2πD ln Z( β0 2 β β0e 2 0 ), E ln Z( β0) =, D = β the saddle-point condition fixes the value of β 0 for a given energy E SNP

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) ( α βε + ) ln Z( α, β ) = dε g( ε )ln 1 e + corrections for rotation, shell structure, etc. Single-particle energies from Hartree-Fock mean-field: ε i p,n Single-particle density of states: g ( ε ) = δ ( ε ε ) Partition function : ln Z ( α, β ) = ln( 1+ exp( α βε i )) Then apply saddle-point method... i i SNP i

11 Mean-field Level densities 32 S state density (MeV -1 ) exact (CI shell model) spherical s.p.e.s E x (MeV) SNP

12 Mean-field Level densities 28 Si 1000 state density (MeV -1 ) exact (CI shell model) spherical s.p.e.s These are both spherical nuclei... what about deformed nuclides? E x (MeV) SNP

13 Mean-field Level densities Mg Mg state density (MeV -1 ) -1 ) exact exact (CI (CI shell shell model) model) deformed deformed s.p.e.s s.p.e.s spherical s.p.e.s EE x x (MeV) SNP

14 Mean-field Level densities 32 S state density (MeV -1 ) exact (CI shell model) spherical s.p.e.s deformed s.p.e.s E x (MeV) state density (MeV -1 ) Si exact (CI shell model) spherical s.p.e.s deformed s.p.e.s E x (MeV) SNP

15 Mean-field Level densities Difference is due to fragmentation of Hartree-Fock single-particle energies in deformed mean-field 0d 3/2 1s 1/2 Fermi surface Fermi surface 0d 5/2 ρ BBFG spherical ( E) = 2π 12 exp 4a( E (4a) 1/ 4 ( E Δ) deformed Δ) 5/ 4 Smaller s.p. level density smaller nuclear level density SNP

16 Adding collective motion (NEW) Deformed HF state as an intrinsic state: Ψ HF aj ΨJ J 2 a J 1 (2J + 1) ( J + 2) exp 2 2σ 2σ J ( J +1) E J 2I 2 2 2σ Ψ HF J Ψ 2 ˆ 2 (get moment of inertia I from cranked HF) HF 2 Z = (2J + 1) a exp( βe ) rot J J J π J, J J ( J, E / 2 rot ( 1+ βe ) 2I rot + 1) = = 3 J 2 SNP

17 Adding collective motion (NEW) Z(β) = Z π (sp) Z υ (sp) (1 + Z rot ) All of the parameters derived directly from HF calculation (SHERPA code by Stetcu and Johnson) using CI shell-model interaction ly very cheap: a matter of a few seconds SNP

18 Adding collective motion (NEW) 24 Mg state density (MeV -1 ) exact (CI shell model) deformed s.p.e.s spherical s.p.e.s deformed s.p.e.s + rotation E x (MeV) SNP

19 Adding collective motion (NEW) 32 S state density (MeV -1 ) exact (CI shell model) spherical s.p.e.s deformed s.p.e.s deformed s.p.e.s + rotation E x (MeV) SNP

20 Adding collective motion (NEW) 28 Si 1000 state density (MeV -1 ) exact (CI shell model) spherical s.p.e.s deformed s.p.e.s deformed s.p.e.s + rotation E x (MeV) SNP

21 Adding collective motion (NEW) 48 Cr 1000 state density (MeV -1 ) exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation E x (MeV) SNP

22 Adding collective motion (NEW) 25 Mg 45 Ti 1000 exact (CI shell model) deformed s.p.e.s, no rotation deformed s.p.e.s + rotation 1000 state density (MeV -1 ) state density (MeV -1 ) P exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation E (MeV) state density (MeV -1 ) E x (MeV) exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation SNP E x (MeV)

23 Adding collective motion (NEW) 26 Al V 1000 exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation Cranking is problematic for odd-odd (I used real wfns and should allow complex) state density (MeV -1 ) 10 exact (CI shell model) prolate s.p.e.s oblate s.p.e.s prolate s.p.e.s + rotation oblate s.p.e.s + rotation E x (MeV) SNP

24 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 SNP

25 How we do it: a detailed version Dimension Centroid: The important configuration moments d = TrP α E 1 TrP α = H α α d Higher central moments n α μ ( α) m 1 d ( ) 1 2 Width: σ = TrP H E = TrP H α α α d α ( E ) n Scaled moments ( ) n n ( α ) = μ ( α) / n σ α α α α Asymmetry (or skewness): m 3 (α) Excess : m 4 (α) - 3 = 0 for Gaussian SNP

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

27 Introduction to Statistical Spectroscopy Then we consider the level density as being the sum of individual configuration densities SNP

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

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

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

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

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

33 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 SNP