The shell model Monte Carlo approach to level densities: recent developments and perspectives
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1 The shell model Monte Carlo approach to level densities: recent developments and perspectives Yoram Alhassid (Yale University) Introduction: the shell model Monte Carlo (SMMC) approach Level density in the SMMC approach Circumventing the odd-particle sign problem in SMMC - Level densities in odd-mass nuclei Microscopic emergence of collectivity in heavy nuclei Level densities in heavy nuclei and collective enhancement factors Spin distributions Can we calculate level densities as a function of deformation (useful for fission models) in a spherical shell model approach? Conclusion and prospects.
2 The shell model Monte Carlo (SMMC) method Most microscopic treatments of heavier nuclei are based on mean-field methods but important correlations can be missed. The configuration-interaction (CI) shell model accounts for correlations but diagonalization methods are limited to ~ configurations. The SMMC method enables microscopic calculations in spaces that are many orders of magnitude larger (~ ). e β H ( β = 1/ T) Gibbs ensemble can be written as a superposition of ensembles of non-interacting nucleons in time-dependent fields U σ e β H = D σ G σ U σ ( ) σ τ The integrand reduces to matrix algebra in the single-particle space (of typical dimension ). The high-dimensional integration over methods. G.H. Lang, C.W. Johnson, S.E. Koonin, W.E. Ormand, PRC 48, 1518 (1993); Y. Alhassid, D.J. Dean, S.E. Koonin, G.H. Lang, W.E. Ormand, PRL 72, 613 (1994). σ is evaluated by Monte Carlo
3 Level densities Level density is the most important statistical nuclear property (Fermi s golden rule, Hauser-Feshbach theory of statistical nuclear reactions, etc.), but its calculation is a difficult many-body problem in the presence of correlations. Most approaches are based on empirical modifications of the Fermi gas formula or on mean-field approximations Level density in the SMMC approach H. Nakada and Y. Alhassid., PRL 79, 2939 (1997) E( β ) =< H > β ln Z/ β = E( β) to find the partition function Z( β ) Calculate the thermal energy versus and integrate The average state density is found from approximation: 1 S( E) ρ ( E) 2 2πT C e Z( β ) in the saddle-point S(E) = canonical entropy; SE ( ) = lnz+β E C = canonical heat capacity. 2 C β E/ = β
4 Circumventing the odd-particle sign problem in SMMC A. Mukherjee and Y. Alhassid, Phys. Rev. Lett. 109, (2012) Applications of SMMC to odd-even and odd-odd nuclei has been hampered by a sign problem that originates from the projection on odd number of particles. A breakthrough was a method we introduced to calculate the ground-state energy of the odd-particle system. Consider the imaginary-time single-particle Green s functions G =Σ Ta a for orbitals ν = nl j ν( τ) m νm( τ) νm(0) The energy difference between the lowest energy of the odd-particle system for a given spin j and the ground-state energy of the evenparticle system can be extracted from the slope of ln G ν ( τ ). Minimize Ej ( A± 1) to find the groundstate energy and its spin j. [ Ej( A 1) Egs( A)] G ()~ e β ± τ ν τ
5 Statistical errors of ground-state energy of direct SMMC versus Green s function method Green s function method Direct SMMC Green s function method direct SMMC Direct SMMC Pairing gaps in iron region nuclei from odd-even mass differences SMMC in the complete fpg 9/2 shell (good agreement with experiments)
6 Application to nickel isotopes: theory versus experiment M. Bonett-Matiz, A. Mukherjee and Y. Alhassid, Phys. Rev. C Rapid Com. 88, (2013) Recent determination of level densities in nickel isotopes from proton evaporation spectra [A. Voinov et al. (Ohio University group) 2012]. We can now calculate accurate ground-state energies and thus microscopic level densities for both even-even and even-odd isotopes. Excellent agreement with experiments: (i) level counting, (ii) p evaporation, (iii) neutron resonance data.
7 Microscopic emergence of collectivity in heavy nuclei Heavy nuclei exhibit various types of collectivity (vibrational, rotational, and their crossovers) that are well described by empirical models. However, a microscopic description (e.g., CI shell model) has been lacking. Can we describe vibrational and rotational collectivity in heavy nuclei using the framework of the CI shell model? The large model space required (e.g., ~ in rare-earth nuclei) necessitates the use of SMMC. The various types of collectivity are usually identified by the corresponding spectra, but SMMC does not provide detailed spectroscopy.
8 !" 2 The behavior of < J > versus T is sensitive to the type of collectivity: 162 Dy!" 2 6 < J > = E Dy T is rotational!" 2 e E 2 + /T < J > =30 (1 e E 2 + /T ) Sm is vibrational Y. Alhassid, L. Fang, H. Nakada, PRL (2008) C. Ozen, Y. Alhassid, H. Nakada, PRL (2013) Single-particle model space: protons: shell plus 1f 7/2 ; neutrons: shell plus 0h 11/2 and 1g 9/2.
9 Crossover from vibrational to rotational collectivity in heavy nuclei C. Ozen, Y. Alhassid, H. Nakada, Phys. Rev. Lett. 110, (2013)!" 2 < J > versus T in samarium isotopes!" 2 J (J +1)(2J +1)e E α J /T < J > = Experimental values are found from α J (2J +1)e E α J /T where are the experimentally known levels. α J E α J Add the contribution of higher levels using the experimental level density to get an experimental values at higher T. SMMC describes well the crossover from vibrational to rotational collectivity in good agreement with the experimental data at low T
10 Level densities in samarium and neodymium isotopes level counting n resonance (E x ) (MeV -1 ) Sm 150 Sm 152 Sm 154 Sm Nd 146 Nd 148 Nd 150 Nd E x (MeV) SMMC Excellent agreement of SMMC densities with various experimental data sets.
11 Collective enhancement factor Collective enhancement factors K are one of the least understood topics in level densities and are usually treated empirically. We define K as the ratio of the SMMC state density to the Hartree-Fock-Bogoliubov (HFB) intrinsic density. The damping of vibrational enhancement is correlated with the pairing transition A regime of rotational enhancement up to the shape transition. The damping of rotational enhancement is correlated with the shape transition.
12 Spin distributions in SMMC Y. Alhassid, S. Liu and H. Nakada, Phys. Rev. Lett. 99, (2007) J / = = Fe Fe Co AFMC 2 = = = = 11.8 Low energy data 0.01 AFMC data scaled to SMMC energy 2 σ E x = 4.39 MeV E x = 5.6 MeV E x = 3.39 MeV J = spin cutoff parameter (increases with excitation energy). Staggering effect (in spin) for even-even nuclei Analysis of experimental data [T. von Egidy and D. Bucurescu, PRC 78, (2008)] confirmed our prediction.
13 Nuclear deformation in a spherical shell model approach Y. Alhassid, C.N. Gilbreth, and G.F. Bertsch, arxiv:1408:0081 (2014) Fission dynamics requires level densities as a function of deformation. Deformation is a key concept in understanding heavy nuclei but it is based on a mean-field approximation that breaks rotational invariance. The challenge is to study nuclear deformation in a framework that preserves rotational invariance. We calculated the distribution of the axial mass quadrupole in the lab frame using an exact projection on (novel in that [Q 20, H ] 0 ). Q 20 P β (q) = δ (Q 20 q) = 1 Tr e βh dϕ e iϕ q Tr (e iϕ Q 2π 20 e βh )
14 Application to rare-earth nuclei Application to 154 to Sm 154 Sm and and 148 Sm 148 Sm 154 At T = 1T =, P 1(q) P is (q) the many-particle Sm eigenvalue (a deformed distribution nucleus) of ˆQ 2,0 ˆQ 2, At low temperatures, Sm (deformed Sm (deformed the in HFB) distribution in HFB) is similar to that of The a prolate many-particle rigid rotor eigenvalues a model-independent are closely spaced, signature giving effectively of deformation. a continuous distribution.! At the HFB At high shape temperatures, transition temperature a Gaussian.! (T=1.14 MeV), the distribution is still skewed. At the mean-field shape transition The distribution (T = 1.14 at MeV), high distribution temperatures is skewed.! is close to a Gaussian At low temperatures, the distribution is similar to that of a prolate rigid rotor, 148 Sma clear (a spherical signature nucleus) of deformation. 148 Sm (spherical in HFB) 148 Sm (spherical in HFB) The distribution P (q) is close to a Gaussian even at low P temperatures. (q) is close to a Gaussian even at low temperatures T=4.0 T=4.0 MeV MeV T=1.14 MeV T=1.14 MeV T=0.1 MeV T=0.1 MeV SMMC Rigid rotor q (fm ) q (fm 2 ) Sm T=0.1 MeV FIG. 3: Probability distributions Sm P β (q) T=0.1 formev 154 Sm at 0.1 MeV,T = 1.14 MeV (shape transition temperature) T =4MeV.Thelow-temperaturedistributioniscomp with the rigid-rotor distribution (dashed line) and reflect 0 strongly deformed character of -600 this nucleus q (fm600 2 ) Sm Sm MeV. The distribution is less q skewed, (fm 2 ) but neverthele
15 4 MeV MeV Intrinsic deformation from lab frame distributions Information on intrinsic deformation can be obtained from the expectation 0 values of rotationally 0 1 invariant combinations of Q 2µ FIG. 4: Q Q vs temperature T for the spherical Example: the lowest order invariant is 148 Sm (left) second order Q Q T=4.0 MeV Sm Sm the transition temperature, in that Q Q continues to The sharp be enhanced shape beyond transition its uncorrelated in HFB mean-field ˆQ 2,µ is value. r 154 Sm at T = The second- and third-order invariants can be used 8 temperature) washed and out SMMC to define in the effective finite-size values of the nucleus intrinsic ˆQ2,0 shape parameters β, γ [27] of T=1.14 the collective MeV Bohr model [28, HFB tioniscompared e) and reflects the Sec. 6B-1a]. The model assumes an intrinsic frame a 2,1 = a 2, 1 =0 in which the quadrupole deformation parameters α 2µ = cos γ 5π = ˆQ2µ 7 (Q Q) Q /3r0A 5/3 are expressed as α 20 = β a a 2,2 = a 2, 2 = p 1 cos γ, α 22 = 0 t nevertheless it α 2 2 = ,0 = 1 2 β2sin cos γ, andα Q Q, 2±1 3/2 =0. Effective β and sinγ can. The HFB excit 25 MeV, much then be determined fromt=0.1 themev corresponding invariants 2 T (MeV) T (MeV) can troscopy The and quadrupole for 5π FIG. 4: Q Q vs temperature T for the spherical 148 Sm (left) β = invariants 60 panel shows the r 2 ˆQ ˆQ 1/2 ; cos3γ can = be 7 calculated ( ˆQ ˆQ) ˆQ, and the deformed A5/3 2 ˆQ ˆQ. from lab frame moments of Q Sm (right).(oblate) 20 3/2 The AFMC results (solid h excitation the circles) (7) are compared with the HFB results (dashed lines) Gaussian. In addition, we can 0 extract 600 a 1200 measure β of the fluctuations the in βjoint using the level second- density 48 γ Sm, which Construct is q (fm 2 and fourth-order distribution invariants ρ(β,e x ) = ρ(e x )P Ex (β) ) y are more symthe transition temperature, in that Q Q continues to [ be enhanced beyond its uncorrelated mean-field 20, consistent with ( β/β) 2 = ( ˆQ ˆQ) 2 ˆQ ˆQ 2] 1/2 where P / ˆQ ˆQ. (8) value. Ex (β) is the intrinsic shape distribution at given excitation energy E oment. FIG. 3: Probability distributions P β (q) for 154 x Sm at T = The second- and 148Sm third-order invariants can be used 0.1 MeV, T = 1.14 MeV (shape transition temperature) and -order invariant The invariants themselves are calculated from the to mo- define effective values of 154Smthe intrinsic shape pa- <Q Q> (10 5 fm 4 ) 8 4 T (MeV) T (MeV) and the deformed 154 Sm (right). The AFMC results (solid circles) are compared with the HFB results (dashed lines). β ( Q Q ) 1/2 <Q Q> (10 5 fm 4 ) 4
16 Conclusion SMMC is a powerful method for the microscopic calculation of level densities in very large model spaces. We have circumvented the odd-particle sign problem in SMMC, enabling the calculation of level densities of odd-mass nuclei. Microscopic description of collectivity in heavy nuclei. Damping of the collective vibrational and rotational enhancement factors of level densities correlates with the pairing and shape phase transitions. Description of nuclear deformation in a rotationally invariant framework. Prospects Other mass regions (actinides, unstable nuclei, ). Level densities as a function of deformation (useful for fission). Derive global effective shell model interactions from density functional theory.
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