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 2008 1
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 2
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 3
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 4
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 5
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 2008 6
Mean-field Level densities Result: exact state density from CI shell-model diagonalization 32 S Okay, let s start comparing with approximations! SNP 2008 7
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 2008 8
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 2008 9
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 2008 10 i
Mean-field Level densities 32 S 10000 state density (MeV -1 ) 1000 100 10 exact (CI shell model) spherical s.p.e.s 1 0 5 10 15 20 25 30 E x (MeV) SNP 2008 11
Mean-field Level densities 28 Si 1000 state density (MeV -1 ) 100 10 exact (CI shell model) spherical s.p.e.s These are both spherical nuclei... what about deformed nuclides? 1 0 5 10 15 20 E x (MeV) SNP 2008 12
Mean-field Level densities 24 24 Mg Mg 10000 10000 state density (MeV -1 ) -1 ) 1000 1000 100 100 10 10 exact exact (CI (CI shell shell model) model) deformed deformed s.p.e.s s.p.e.s spherical s.p.e.s 1 0 5 10 10 15 15 20 20 25 25 30 30 EE x x (MeV) SNP 2008 13
Mean-field Level densities 32 S 10000 1000 state density (MeV -1 ) 100 10 exact (CI shell model) spherical s.p.e.s deformed s.p.e.s 100 1 0 5 10 15 20 25 30 E x (MeV) state density (MeV -1 ) 1000 10 28 Si exact (CI shell model) spherical s.p.e.s deformed s.p.e.s 1 0 5 10 15 20 E x (MeV) SNP 2008 14
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 2008 15
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 2008 16
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 2008 17
Adding collective motion (NEW) 24 Mg 10000 state density (MeV -1 ) 1000 100 10 exact (CI shell model) deformed s.p.e.s spherical s.p.e.s deformed s.p.e.s + rotation 1 0 5 10 15 20 25 30 E x (MeV) SNP 2008 18
Adding collective motion (NEW) 32 S 10000 state density (MeV -1 ) 1000 100 10 exact (CI shell model) spherical s.p.e.s deformed s.p.e.s deformed s.p.e.s + rotation 1 0 5 10 15 20 25 30 E x (MeV) SNP 2008 19
Adding collective motion (NEW) 28 Si 1000 state density (MeV -1 ) 100 10 exact (CI shell model) spherical s.p.e.s deformed s.p.e.s deformed s.p.e.s + rotation 1 0 5 10 E x (MeV) 15 20 SNP 2008 20
Adding collective motion (NEW) 48 Cr 1000 state density (MeV -1 ) 100 10 exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation 1 0 5 10 15 E x (MeV) SNP 2008 21
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 ) 100 10 state density (MeV -1 ) 100 31 10 P exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation 0 2 4 6 8 10 12 14 E (MeV) state density (MeV -1 ) 100 10 1 0 2 4 6 8 10 E x (MeV) exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation SNP 2008 22 1 0 5 10 15 E x (MeV)
Adding collective motion (NEW) 26 Al 100 10 46 V 1000 exact (CI shell model) deformed s.p.e.s deformed s.p.e.s + rotation 1 0 2 4 6 8 10 12 100 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 1 0 2 4 6 8 10 E x (MeV) SNP 2008 23
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 2008 24
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 2008 25
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 2008 26
Introduction to Statistical Spectroscopy Then we consider the level density as being the sum of individual configuration densities SNP 2008 27
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 28
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 29
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 2008 30
Comparison with experiments NB: computed + parity states and multiplied 2 SNP 2008 31
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 32
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 2008 33