Yoram Alhassid (Yale University)

Statistical properties of nuclei by the shell model Monte Carlo method

•  Introduction

•  Shell model Monte Carlo (SMMC) method •  Circumventing the odd particle-number sign problem

•  Statistical properties of nuclei in SMMC •  Applications to mid-mass and heavy nuclei •  Projection on good quantum numbers (e.g., spin and parity) •  Projection on an order parameter (associated with a broken symmetry): level density vs. deformation in the spherical shell model •  Conclusion and outlook

Recent review: Y. Alhassid, arXiv:1607.01870 (2016), in a review book, ed. K.D. Launey

Introduction

Statistical properties, and in particular, level densities, are important input in the Hauser-Feshbach theory of compound nuclear reactions, but are not always accessible to direct measurement.

- Often calculated using empirical modifications of the Fermi gas formula - Mean-field approximations (e.g., Hartree-Fock-Bogoliubov) often miss important correlations and are problematic in the broken symmetry phase.

The calculation of statistical properties in the presence of correlations is a challenging many-body problem.

Statistical properties of nuclei: level density, partition function, entropy,…

The configuration-interaction (CI) shell model takes into account correlations beyond the mean-field but the combinatorial increase of the dimensionality of its model space has hindered its applications in mid-mass and heavy nuclei.

•  Conventional diagonalization methods for the shell model are limited to spaces of dimensionality ~ 1011.

The auxiliary-field Monte Carlo (AFMC method) enables microscopic calculations in spaces that are many orders of magnitude larger (~ 1030) than those that can be treated by conventional methods.

Also known in nuclear physics as the shell model Monte Carlo (SMMC) method.

SMMC is the state-of-the-art method for the microscopic calculation of statistical properties of nuclei.

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).

Hubbard-Stratonovich (HS) transformation

Assume an effective Hamiltonian in Fock space with a one-body part and a two-body interaction :

are single-particle energies and are linear combinations of one-body densities . ✏i ⇢↵

s↵ = 1 s↵ = iv↵ < 0 v↵ > 0for , and for

⇢ij = a†iaj

H =P

i ✏ini +12

P↵ v↵⇢

2↵

G� = e�12

R �0 |v↵|�2

↵(⌧)d⌧ is a Gaussian weight and is a one-body propagator of non-interacting nucleons in time-dependent auxiliary fields

The HS transformation describes the Gibbs ensemble at inverse temperature as a path integral over time-dependent auxiliary fields

e−βH = D σ⎡⎣ ⎤⎦∫ GσUσ

�(⌧)

U� = T e�R �0 h�(⌧)d⌧

β =1/T

h�(⌧) =P

i ✏ini +P

↵ s↵|v↵|�↵(⌧)⇢↵with a one-body Hamiltonian

e−βH

Uσ

Thermal expectation values of observables

hOi = Tr (Oe��H)

Tr (e��H)=

RD[�]G�hOi�Tr U�R

D[�]G�Tr U�

hOi� ⌘ Tr (OU�)/Tr U�where

Grand canonical quantities in the integrands can be expressed in terms of the single-particle representation matrix of the propagator :

Tr U� = det(1+U�)

ha†iaji� ⌘ Tr (a†iajU�)

Tr U�=

h1

1+U�1�

i

ji

U�

•  The integrand reduces to matrix algebra in the single-particle space (of typical dimension ~100).

Canonical (i.e., fixed N,Z) quantities are calculated by an exact particle-number projection (using a discrete Fourier transform).

Quantum Monte Carlo methods and sign problem

The high-dimensional integration is done by Monte Carlo methods, sampling the fields according to a weight

�� ⌘ TrA U�/|TrA U�| is the Monte Carlo sign function.

For a generic interaction, the sign can be negative for some of the field configurations. When the average sign is small, the fluctuations become very large the Monte Carlo sign problem.

H =P

i ✏ini +12

P↵ v↵ (⇢↵⇢↵ + ⇢↵⇢↵)

where is the time-reversed density. We can rewrite

Sign rule: when all , for any configuration and the interaction is known as a good-sign interaction.

v↵ < 0 TrU� > 0 �

⇢↵

)

Proof: when all , we have and the eigenvalues of appear in complex conjugate pairs and .

h� =P

i ✏ini +P

↵ (v↵�⇤↵⇢↵ + v↵�↵⇢↵)v↵ < 0

h� = h� ) U�

{�i,�⇤i } Tr U� =

Qi |1 + �i|2 > 0

W� ⌘ G�|TrA U�|

Good-sign interactions

σ

The dominant collective components of effective nuclear interactions have a good sign.

A family of good-sign interactions is constructed by multiplying the bad-sign components by a negative parameter g

A practical method for overcoming the sign problem

In the calculation of statistical and collective properties of nuclei, we have used good-sign interactions.

H = HG + gHB

Observables are calculated for and extrapolated to . �1 < g < 0

Alhassid, Dean, Koonin, Lang, Ormand, PRL 72, 613 (1994).

g = 1

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.

†( ) ( ) (0)m m mG T a aν ν ντ τ=Σ n l jν =

Gν (τ )~ e−[ E j ( A±1)− Egs ( A)] |τ |

Circumventing the odd-particle sign problem in SMMC Mukherjee and Alhassid, PRL 109, 032503 (2012)

•  We introduced a method to calculate the ground-state energy of the odd-particle system that circumvents this sign problem.

•  The energy difference between the lowest energy of the odd-particle system for a given spin j and the ground-state energy of the even-particle system can be extracted from the slope of . ln ( )Gν τ

( 1)jE A±

Consider the imaginary-time single-particle Green’s functions for even-even nuclei: for orbitals

Minimize to find the ground-state energy and its spin j.

Statistical errors of ground-state energy of direct SMMC versus Green’s function method

Pairing gaps in mid-mass nuclei from odd-even mass differences

direct SMMC Direct SMMC

Green’s function method

•  SMMC in the complete fpg9/2 shell (in good agreement with experiments)

Direct SMMC

Green’s function method

•  The average state density is found from in the saddle-point approximation:

( ) ( )2

1

2

S E

T CE e

πρ ≈

( ) lnS E Z Eβ= + 2 /C Eβ β= − ∂ ∂

ln / ( )Z Eβ β−∂ ∂ = ( )Z β Calculate the thermal energy versus and integrate to find the partition function .

( )Z β

E(β ) =< H > β

Statistical properties in the SMMC

S(E) = canonical entropy C = canonical heat capacity

⇢(E) = 12⇡i

R i1�i1 d� e�EZ(�)

The level density is related to the partition function by an inverse Laplace transform:

⇢(E)

Partition function

Level density

Nakada and Alhassid, PRL 79, 2939 (1997)

Mid-mass nuclei

Bonett-Matiz, Mukherjee, Alhassid, PRC 88, 011302 R (2013)

Excellent agreement with experiments: (i) level counting, (ii) p evaporation spectra (Ohio U., 2012), (iii) neutron resonance data.

CI shell model model space: complete fpg9/2-shell.

Interaction: includes the dominant components of effective interactions

•  pairing: g0=-0.212 MeV determined from odd-even mass differences

•  multipole-multipole interaction terms -- quadrupole, octupole, and hexadecupole: determined from a self-consistent condition and renormalized by k2=2, k3=1.5, k4=1.

Single-particle Hamiltonian: from Woods-Saxon potential plus spin-orbit

Level densities in nickel isotopes

Heavy nuclei exhibit various types of collectivity (vibrational, rotational, … ) that are well described by empirical models.

However, a microscopic description in a CI shell model has been lacking.

Single-particle Hamiltonian: from Woods-Saxon potential plus spin-orbit

Can we describe vibrational and rotational collectivity in heavy nuclei using a spherical CI shell model approach in a truncated space ?

Heavy nuclei (lanthanides)

CI shell model space:

protons: 50-82 shell plus 1f7/2 ; neutrons: 82-126 shell plus 0h11/2 and 1g9/2

Interaction: pairing (gp,gn) plus multipole-multipole interaction terms – quadrupole, octupole, and hexadecupole.

The behavior of versus is sensitive to the type of collectivity: 〈J!"2

〉

is vibrational

T

is rotational 162Dy⇒ 148Sm⇒ 〈J!"2

〉= 6E

2+

T 〈J!"2

〉=30 e−E

2+/T

(1− e−E

2+/T

)2

162Dy

Alhassid, Fang, Nakada, PRL 101 (2008) Ozen, Alhassid, Nakada, PRL 110 (2013)

The various types of collectivity are usually identified by their corresponding spectra, but AFMC does not provide detailed spectroscopy.

Level densities in samarium and neodymium isotopes

•  Good agreement of SMMC densities with various experimental data sets (level counting, neutron resonance data when available).

0 4 8 121

103106109

ρ(E x) (

MeV

-1)

4 8 12 4 8 12 4 8 12 4 8 12

Ex (MeV)

1103106109

144Nd146Nd 148Nd 150Nd 152Nd

148Sm 150Sm 152Sm 154Sm

Level counting SMMCn resonance

Ozen, Alhassid, Nakada, PRL 110 (2013)

Rotational enhancement in deformed nuclei

A deformed nucleus ( ): Hartree-Fock (HF) vs SMMC 162Dy

Alhassid, Bertsch, Gilbreth and Nakada, PRC 93, 044320 (2016)

100104108

101210161020

0 10 20 30 40 50

ρ (M

eV-1 )

Ex (MeV)

100104108

0 2 4 6 8ρ (

MeV

-1 )Ex (MeV)

SMMC

HF

•  The enhancement of the SMMC density (compared with HF) is due to rotational bands built on top of the intrinsic bandheads.

•  Particle-number projection is carried out in the saddle-point approximation

•  The rotational enhancement gets damped above the shape transition.

•  Exact particle-number projection: Fanto, Alhassid, Bertsch, arXiv:1610.08954

Projection on good quantum numbers: spin distributions in mid-mass nuclei [Alhassid, Liu, Nakada, PRL 99, 162504 (2007)]

= spin cutoff parameter

•  Analysis of experimental data [von Egidy and Bucurescu, PRC 78, 051301 R (2008)] confirmed our prediction.

•  Staggering effect (in spin) for even-even nuclei

2σ

0 2 4 6 80

0.01

0.02

0.03

0.04

ρ J/ρ

σ2 = 7.92

σ2 = 10.4

0 2 4 6 8J

σ2 = 7.96

σ2 = 10.9

0 2 4 6 8

σ2 = 8.11

σ2 = 11.8

56Fe55Fe60Co

Ex = 4.39 MeVEx = 5.6 MeV

Ex = 3.39 MeV

Low-energy data

data scaled to AFMC energy

AFMC

2( 1) / 23

2 12 2

J JJ J e σρρ πσ

− ++=Spin cutoff model:

AFMC distributions agree well with an empirical staggered spin cutoff formula based on low-energy counting data (von Egidy and Bucurescu).

•  Good agreement with spin-cutoff (s.-c.) model at higher excitations. •  Odd-even staggering in spin at low excitation energies.

2( 1) / 23

2 12 2

J JJ J e σρρ πσ

− ++=

0

0.004

0.008

0.012

0.016l J

/lEx = 7.9 MeV

0

0.004

0.008

0.012

0.016l J

/lEx = 7.9 MeV Ex = 6.2 MeVEx = 6.2 MeV Ex = 3.9 MeVEx = 3.9 MeV

0

0.004

0.008

0.012

0 4 8 12 16

l J/l

J

Ex = 2.9 MeV

0

0.004

0.008

0.012

0 4 8 12 16

l J/l

J

Ex = 2.9 MeV

0 4 8 12 16J

Ex = 2.5 MeV

0 4 8 12 16J

Ex = 2.5 MeV

0 4 8 12 16 20J

Ex = 2.3 MeV

0 4 8 12 16 20J

Ex = 2.3 MeV

spin-cutoffstaggered s.-c.

0

0.004

0.008

0.012

0.016

l J/l

Ex = 7.9 MeV

0

0.004

0.008

0.012

0.016

l J/l

Ex = 7.9 MeV Ex = 6.2 MeVEx = 6.2 MeV Ex = 3.9 MeVEx = 3.9 MeV

0

0.004

0.008

0.012

0 4 8 12 16

l J/l

J

Ex = 2.9 MeV

0

0.004

0.008

0.012

0 4 8 12 16

l J/l

J

Ex = 2.9 MeV

0 4 8 12 16J

Ex = 2.5 MeV

0 4 8 12 16J

Ex = 2.5 MeV

0 4 8 12 16 20J

Ex = 2.3 MeV

0 4 8 12 16 20J

Ex = 2.3 MeV

spin-cutoffstaggered s.-c.

0 4 8 12 16 20J

Ex = 2.3 MeV

SMMCspin-cutoff

staggered s.-c.experiment

Spin distributions in heavy nuclei: 162Dy Gilbreth, Alhassid, Bonett-Matiz

20 30 40 50 60 70 80 90

0 2 4 6 8 10

I

Ex

Fit to lJ/lFrom <J2>/3

From ratio of state to level densityRigid-body

σ2 = I T / !2

Thermal moment of inertia I vs. excitation energy

Three methods to determine

•  Moment of inertia I is suppressed by pairing correlations below Ex ~ 5 MeV

Σ Jρ J (Ex )= ρM=0(Ex ) = 1

2πσρ(Ex )

calculated from

σ2 =< Jz

2 >=<!J 2 > /3

Iσ 2

(ii) Fit to spin cutoff model (ii) From (iii) From ratio of level to state density

0 5 10Ex(MeV)

0

0.5

1

1.5

ρ − / ρ +

162DyParity ratio vs excitation energy

•  Parity ratio is equilibrated at the neutron separation energy

•  Deformation is a key concept in understanding heavy nuclei but it is based on a mean-field approximation that breaks rotational invariance.

Modeling of fission requires level density as a function of deformation.

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 ). [Q20,H ]≠ 0Q20

Alhassid, Gilbreth, Bertsch, PRL 113, 262503 (2014)

Pβ (q) = 〈δ (Q20 − q)〉 =1

Tr e−βHdϕ2π e

− iϕ qTr−∞

∞

∫ (eiϕQ20e−βH )

Example: nuclear deformation in a spherical shell model approach

Projection on an order parameter (associated with a broken symmetry)

Application to heavy nuclei

(deformed)

•  At low temperatures, the distribution is similar to that of a prolate rigid rotor a model-independent signature of deformation.

154Sm

(spherical)

148Sm

•  The distribution is close to a Gaussian even at low temperatures.

Rigid rotor

0

0.0005

0.001

0.0015

-1000 0 1000

154Sm

P T(q

)

q (fm2)

T = 0.10 MeV

-1000 0 1000q (fm2)

T = 1.23 MeV

SMMC

-1000 0 1000q (fm2)

T = 4.00 MeV

0

0.0005

0.001

0.0015

-1000 0 1000

148Sm

P T(q

)

q (fm2)

T = 0.10 MeV

-1000 0 1000q (fm2)

T = 0.41 MeV

-1000 0 1000q (fm2)

T = 4.00 MeV

at a given temperature T is an invariant and can be expanded in the quadrupole invariants (Landau-like expansion)

•  The expansion coefficients A,B,C… can be determined from the expectation values of the invariants, which in turn can be calculated from the low-order moments of in the lab frame. q20 = q

Intrinsic shape distributions

Information on intrinsic deformation can be obtained from the expectation values of rotationally invariant combinations of the quadrupole tensor . q2µ

β,γ

PT (β,γ )

− lnPT =Aβ2 − Bβ 3 cos3γ +Cβ 4 + ...

lnPT (β,γ )

Alhassid, Mustonen, Gilbreth, Bertsch

q ⋅q ∝β 2 (q × q) ⋅q ∝β 3 cos(3γ ) (q ⋅q)2 ∝β 4To 4th order: ; ;

( are the intrinsic deformation parameters) β ,γ

<q ⋅q>=5<q202 >; <(q×q)⋅q>= −5 7

2 <q203 >; <(q ⋅q)2 >= 353 <q20

4 >

Expressing the invariants in terms of in the lab frame and integrating over the components, we recover in the lab frame.

q2µµ ≠ 0 P[q20 ]

We find excellent agreement with calculated in SMMC ! P[q20 ]

0 0.0005 0.001

0.0015

-1000 0 1000

154Sm

P(q)

q (fm2)

T = 4.00 MeV

-1000 0 1000q (fm2)

T = 1.19 MeV

-1000 0 1000q (fm2)

T = 0.10 MeV

SMMCexpansion

•  Mimics a shape transition from a deformed to a spherical shape without using a mean-field approximation !

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

T = 4.0

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

T = 1.23

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

T = 0.1

154Sm

− lnP(β ,γ )

We divide the plane into three regions: spherical, prolate and oblate.

Integrate over each deformation region to determine the probability of shapes versus temperature

�

���

���

���

���

�

� ��� � ���

�����

� �����

� ������ ��� � ���

�����

� ������ ��� � ���

�����

� �����

����������������������

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

0.08 0.16 0.24

�

0

⇡12

⇡6

3⇡12

⇡ 3

�

prolate

oblate

spherical

β,γ

•  Compare deformed (154Sm), transitional (150Sm) and spherical (148Sm) nuclei

Level density versus intrinsic deformation

�����������������������

� � ��

�����

� ���������

���

� � ��

�����

� � �� ��

�����

����������������������

•  Convert PT( ) to level densities vs Ex,β,γ β,γ

Deformed Transitional Spherical

Ex (MeV)

Conclusion •  SMMC is a powerful method for the microscopic calculation of level densities in very large model spaces; applications in nuclei as heavy as the lanthanides.

•  Microscopic description of rotational enhancement in deformed nuclei.

•  Spin distributions: odd-even staggering in even-even nuclei at low excitation energies; spin cutoff model at higher excitations.

•  Deformation-dependent level density in a rotationally invariant framework (CI shell model).

•  Other mass regions (actinides, unstable nuclei,…). •  Gamma strength functions in SMMC by inversion of imaginary-time response functions.

Outlook