The shell model Monte Carlo approach to level densities: recent developments and perspectives

0.56 MB·

by  Y. Alhassid

EPJ manuscript No.
(will be inserted by the editor)
The shell model Monte Carlo approach to level densities: recent
developments and perspectives
Y. Alhassid
Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520, USA
Received: 7 June 2015/ Revised: 2 December 2015
Abstract. We review recent advances in the shell model Monte Carlo approach for the microscopic calcu-
6 lationofstatisticalandcollectivepropertiesofnuclei.Wediscussapplicationstothecalculationof(i)level
1
densities in nickel isotopes, implementing a recent method to circumvent the odd-particle sign problem;
0
(ii) state densities in heavy nuclei; (iii) spin distributions of nuclear levels; and (iv) finite-temperature
2
quadrupole distributions.

n
a PACS. 21.10.Ma – 21.60.Cs – 02.70.Ss – 21.60.Ev
J

1
1 Introduction 2 Shell model Monte Carlo method

]
h
t Thenuclearleveldensityisamongthemostimportantsta-
- TheSMMCmethodisbasedontheHubbard-Stratonovich
tisticalpropertiesofnucleiandiskeyinputintheHauser-
l
c Feshbach theory [1] of statistical nuclear reactions. How- transformation[17,18],inwhichtheGibbsensemblee−βH
u ofasystemdescribedbyaHamiltonianH atinversetem-
ever, its microscopic calculation in the presence of cor-
n peratureβ =1/T isexpressedasafunctionalintegralover
relations is a challenging many-body problem. Most ap-
[
proaches are based on empirical models such as the back- one-body ensembles that describe non-interacting nucle-

1 shifted Fermi gas model and the constant temperature ons moving in external auxiliary fields
v formula, and on mean-field theories such as the Hartree-
7 (cid:90)
0 FocTk-hBeocgoonlifiugbuorvat(iHonF-Bin)tearpapcrtoioxnim(CatIi)osnhe[2ll,3m].odelapproach e−βH = D[σ]GσUσ . (1)
1
offers an attractive framework that includes correlations
0
0 beyond mean-field theory and shell effects, but its appli- HereGσ isaGaussianweightandUσ isthepropagatorof
. cations in mid-mass and heavy nuclei have been hindered non-interacting nucleons moving in time-dependent aux-
1
by the combinatorial increase of the dimensionality of the iliary fields σ(τ) (0 ≤ τ ≤ β). The thermal expectation
0
6 many-particle model space with number of orbitals and value of an observable O is calculated from
1 numberofvalencenucleons.TheshellmodelMonteCarlo
: (SMMC) method [4,5,6,7] enables calculations in model Tr(e−βHO) (cid:82) D[σ]W Φ (cid:104)O(cid:105)
v spaces that are many orders of magnitude larger than (cid:104)O(cid:105)= = (cid:82) σ σ σ , (2)
Xi those that can be treated in conventional diagonalization Tre−βH D[σ]WσΦσ
methods,andisapowerfulstate-of-the-artmethodforthe
r
a microscopic calculation of nuclear level densities [8,9,10, where Wσ = Gσ|TrUσ| is a positive-definite weight func-
tion,Φ =TrU /|TrU |istheMonteCarlosignfunction,
11,12]. σ σ σ
and (cid:104)O(cid:105) = Tr(OU )/TrU is the thermal expectation
HerewereviewrecentadvancesintheSMMCmethod σ σ σ
value of the observable for a given configuration σ of the
for calculating level densities. These include (i) a method
auxiliary fields.
to circumvent the odd-particle Monte Carlo sign prob-
lem [13], enabling the first accurate SMMC calculation of Themany-particlespace istheFock spacespannedby
level densities in odd-mass nuclei [14]; (ii) calculation of a set of Ns spherical single-particle orbitals. The many-
state densities in even-mass [12] and odd-mass [15] heavy particle propagator Uσ can then be represented in the
nuclei; (iii) calculation of the spin distributions of level single-particlespacebyamatrixUσ ofdimensionNs×Ns,
densities [10], and (iv) a method to calculate the proba- and the quantities in the integrands of Eq. (2) can be ex-
bility distribution associated with the quadrupole opera- pressedintermsofthismatrix.Thegrandcanonicaltrace
tor [16], which might facilitate the modeling of level den- of Uσ is given by
sitiesasafunctionofdeformation,animportantinputfor
the calculation of fission rates. TrU =det(1+U ), (3)
σ σ
2 Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives
whilethegrandcanonicalexpectationvalueofaone-body 2.2 Circumventing the odd-particle sign problem
observable O =(cid:80) O a†a can be calculated from
i,j ij i j Forgood-signinteractions,thesignΦ (inthegrandcanon-
σ
(cid:20) (cid:21) icalensemble)ispositive.Theprojectiononanevennum-
1
(cid:104)a†a (cid:105) = . (4) ber of particles keeps the sign of the projected partition
i j σ 1+U−σ1 ji function TrAUσ positive. However, the projection on an
oddnumberofparticlescanleadtonegativesignofTr U
A σ
The grand canonical expectation value of a two-body ob- for some of the samples. At low temperatures, this leads
servablecanbesimilarlycalculatedusingWick’stheorem. to the so-called sign problem where the statistical errors
Inthefinitenucleusitisimportanttocalculatetheob- oftheobservablesbecometoolarge.Inparticular,itisdif-
servables at fixed numbers of protons and neutrons using ficult to determine accurate ground-state energies of odd-
the canonical ensemble. This is done using an exact rep- massnuclei.Thisodd-particlesignproblemhashampered
resentation of the particle-number projector as a Fourier applications of SMMC to odd-mass nuclei.
sum.Forexample,thecanonicalpartitionfunctionatfixed Recently, we developed a method to circumvent the
number of particles A is given by [19] odd-particlesignproblem(forgood-signinteractions)and
determineaccurateground-stateenergiesofodd-massnu-
TrAUσ = e−NβµA (cid:88)Ns e−iϕmA det(cid:0)1+eiϕmeβµUσ(cid:1) ,(5) colfetih[1e3s]i.nTglhee-pmarettihcoledGisrebeans’esdfuonnctthioenassiynmimptaogtiincabreyhtaivmioer.
s
m=1 The scalar Green’s functions are given by
where ϕ = 2πm/N (m = 1,...,N ) are quadrature Tr (cid:2) e−βHT (cid:80) a (τ)a† (0)(cid:3)
m s s G (τ)= A m νm νm , (11)
points and µ is a real chemical potential [required for the ν Tr e−βH
A
numerical stabilization of the sum in (5)]. Similar expres-
where ν ≡ (nlj) labels the nucleon single-particle orbital
sionscanbewrittenforcanonicalexpectationvaluesofob-
with radial quantum number n, orbital angular momen-
servables,relatingthemtograndcanonicalvaluesthrough
tum l and total spin j. Here T denotes time ordering and
a Fourier transform.
a (τ) ≡ eτHa e−τH is an annihilation operator of a
In SMMC, we choose configurations σ of the auxil- νm νm
k nucleon at imaginary time τ (−β ≤ τ ≤ β) in a single-
iary fields that are distributed according to W and then
σ particle state with orbital ν and magnetic quantum num-
estimate the observables from
ber m.
(cid:80) (cid:104)O(cid:105) Φ For an even-even nucleus A ≡ (Z,N) with ground-
(cid:104)O(cid:105)≈ k(cid:80) σk σk . (6) state spin 0, the scalar Green’s function has the asymp-
Φ
k σk totic form
G (τ)∼e−∆EJ=j(A±)|τ| , (12)
ν
2.1 State density in SMMC where A± denote the even-odd nuclei (Z,N ± 1) when
ν is a neutron orbital and the odd-even nuclei (Z±1,N)
whenν isaprotonorbital,andthe+(−)subscriptshould
In SMMC, we calculate the thermal energy versus inverse
be used for τ > 0 (τ ≤ 0). ∆E (A ) is the difference
temperature β as the expectation value of the Hamilto- J=j ±
between the energies of the the lowest spin J eigenstate
nian E(β)=(cid:104)H(cid:105), and then integrate the thermodynamic
of the A particle nucleus and the ground state of the A
relation ±
∂lnZ particle nucleus. In this asymptotic regime, we can thus
− =E(β) (7) calculate ∆E (A ) from the slope of lnG (τ). By mini-
∂β j ± ν
mizing ∆E (A ) over all possible values of J, we deter-
j ±
to find the canonical partition function Z(β). The state mine the difference between the ground-state energy of
density is determined from the partition function by an the A nuclei, and the ground state energy of the A nu-
±
inverse Laplace transform cleus.ThislatterenergyandG (τ)arebothpropertiesof
ν
the even-even nucleus and are free of a sign problem for a
1 (cid:90) i∞ good-sign interaction.
ρ(E)= dβeβEZ(β). (8)
2πi
−i∞
3 Application to nickel isotopes
In practice we calculate the average state density by eval-
uating the integral in (8) in the saddle-point approxima-
We discuss a recent application to the calculation of level
tion [20]
densitiesinafamilyofnickelisotopes[14]usingtheHam-
(cid:18) dE(cid:19)−1/2 miltonian of Ref. [8] in the fpg shell. The level density
ρ(E)≈ −2π eS(E) , (9) 9/2
dβ ρ˜ is defined without counting the 2J + 1 magnetic de-
(cid:80)
generacy of levels with spin J, i.e., ρ˜(E ) = ρ (E )
where β is determined as a function of energy E from the x J J x
where ρ is the density of levels with spin J. Recent ex-
J
saddle-point condition E(β) = E, and S is the canonical periments extracted the level densities of the 59−64Ni iso-
entropy
topes through the measurements of proton evaporation
S =lnZ+βE . (10) spectra [21].
Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives 3
6
10
59 61 63
Ni Ni Ni
4
10
2
) 10
1
_
V
e
M
6
10
(
~ ρ 60Ni 62Ni 64Ni
4
10
2
10
0
10
0 5 10 15 20 5 10 15 20 5 10 15 20
E (MeV)
x
Fig. 1. Leveldensitiesρ˜ofnickelisotopes59−64NiversusexcitationenergyE .TheSMMCresults(solidcircles)arecompared
x
withlevelcountingdataatlowexcitationenergies(solidhistograms)[24],neutronresonancedatawhenavailable(triangles)[24]
and level densities determined from proton evaporation spectra [21,22] (open squares and quasi-continuous lines). Taken from
Ref. [14].
InSMMC,wecancalculatetheleveldensitydirectly[23] 4.1 Crossover from vibrational to rotational collectivity
by projecting on the spin component M. Denoting by ρ in the CI shell model approach
M
the density for levels with J =M, we have ρ˜=ρ for
z M=0
even-mass nuclei and ρ˜=ρ for odd-mass nuclei. The various types of collectivity, e.g., vibrational and ro-
M=1/2
tational collectivity, are usually identified by their corre-
In Fig. 1, we show the level densities for a family
sponding spectra. However, SMMC does not provide de-
of nickel isotopes 59−64Ni. The ground-state energies of
tailed spectroscopy; it is more suitable for the calculation
the odd-mass isotopes were determined accurately using
of thermal observables. We have therefore identified an
our solution to the odd-particle sign problem discussed in
observablewhoselow-temperaturebehaviorissensitiveto
Sec. 2.2.
the type of collectivity. Such observable is (cid:104)J2(cid:105) . Its tem-
T
perature dependence at low T is given by [11,12]

(cid:104)J2(cid:105) ≈30(cid:0)1−ee−−EE22++//TT(cid:1)2 vibrationalband (13)
4 Heavy nuclei T
 6 T rotationalband
E2+
whereE istheexcitationenergyofthefirst2+ level.In
2+
Fig. 2 we show (cid:104)J2(cid:105) versus T for the deformed nucleus
Open shell heavy nuclei can be strongly deformed and T
162Dy (left panel) and for the spherical nucleus 148Sm
are characterized by rotational collectivity. An important
(right panel). The solid circles with error bars are the
question we discuss in Sec. 4.1 is whether such deformed
SMMC results. In 162Dy, they are well fitted by the ro-
nucleicanbedescribedinasphericalshellmodelapproach
tationalbandmodel,whilein148Sm,thevibrationalband
using a truncated single-particle space.
model is a better fit. These results validate the rotational
TheSMMCmethodwassuccessfullyextendedtoheavy character of 162Dy and the vibrational nature of 148Sm.
nuclei [11,12]. The single-particle model space we use for The observable (cid:104)J2(cid:105) also describes well the crossover
T
calculationsinrare-earthnucleiiscomposedofthe50−82 from vibrational to rotational collectivity. In Fig. 3, we
shell plus the 1f7/2 orbital for protons, and the 82−126 showbysolidcirclestheSMMCvaluesof(cid:104)J2(cid:105)T asafunc-
shell plus the 0h11/2,1g9/2 orbitals for neutrons. The cor- tion of temperature for a family of samarium isotopes
respondingmany-particlemodelspacefor162Dyisoforder 148−154Sm. We observe a crossover from a “soft” response
∼1029. to temperature in 148Sm characterizing spherical vibra-
4 Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives
20 10
162 148
Dy Sm
T
> T
2 >
J 10 2 5
< J
<
0 0
0 0.1 0.2 0 0.2 0.4
T (MeV)
T (MeV)
Fig. 2. Left: (cid:104)J2(cid:105) in 162Dy. The SMMC results (solid circles) are compared with fits to the rotational model (solid line) and
T
vibrational model (dashed-dotted line). Adapted from Ref. [11]. Right: (cid:104)J2(cid:105) in 148Sm. The SMMC results (open circles) are
T
compared with fits to the vibrational model (solid line) and rotational model (dashed line). Taken from Ref. [25].
20
148 150 152 154
Sm Sm Sm Sm
T
>
10
2
J
<
0
0 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
T (MeV)
Fig. 3. (cid:104)J2(cid:105) forafamilyofsamariumisotopes148−154Sm.TheSMMCresults(opencircleswithstatisticalerrors)areingood
T
agreement with values extracted from experimental data (solid lines; see text). Adapted from Ref. [12].
tional nuclei, to a “rigid” linear response to temperature calculatedbythemethoddescribedinSec.2.1,areshown
in 154Sm characterizing deformed rotational nuclei. inthetoptworowsofFig.4(opencircles)[12,26].Wefind
The solid lines in Fig. 3 are extracted from the exper- good agreement with experimental data including level
imental data using countingatlowexcitationenergies(solidhistograms)and,
(cid:32) N when available, neutron resonance data (triangles) at the
(cid:104)J2(cid:105) = 1 (cid:88)J (J +1)(2J +1)e−Ei/T neutron binding energy [24]. The Hamiltonians used for
T Z(T) i i i the calculations are given in Ref. [12].
i
(cid:90) ∞ (cid:19)
+ dE ρ(E )(cid:104)J2(cid:105) e−Ex/T , (14)
x x Ex
EN
whereZ(T)=(cid:80)Ni (2Ji+1)e−Ei/T+(cid:82)E∞N dExρ(Ex)e−Ex/T Recently,wecalculatedthestatedensitiesinodd-mass
is the experimental partition function. The summations
samarium (149−155Sm) and neodymium (143−149Nd) iso-
in Eq. (14) and in the expression for Z(T) extend over
topes [15]. The corresponding densities are shown in the
a complete set of experimentally known low-lying levels i
bottom two rows of Fig. 4. The odd-mass isotopes has
with excitation energy E and spin J , and ρ(E ) is the
i i x an odd-particle sign problem. The solution discussed in
back-shifted Bethe formula (BBF) with parameters that
Sec.2.2requiresadditionaldevelopmentfortheheavynu-
aredeterminedfromlevelcountingandneutronresonance
clei, in which much larger β values are required to reach
data.
the ground-state energy E . We use experimental data
0
to make a one-parameter fit (E ) of the SMMC thermal
0
energy to the thermal energy calculated from the exper-
4.2 State densities in families of samarium and
imental data. We find that the SMMC densities for the
neodymium isotopes
odd-mass isotopes are consistent with the experimental
The SMMC state densities of the even samarium isotopes data using the same family of Hamiltonians as for the
148−154Sm and the even neodymium isotopes 144−152Nd, even-mass isotopes.
Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives 5
9
10
6
10
3
) 10
-1V 148Sm 150Sm 152Sm 154Sm
1
e
M
) ( 109
x
E
(ρ 106
3
10
144 146 148 150 152
Nd Nd Nd Nd Nd
1
0 4 8 12 4 8 12 4 8 12 4 8 12 4 8 12
E (MeV)
x
9
10
6
10
3
] 10 149 151 153 155
1 Sm Sm Sm Sm
-
V 1
e
M
9
[ 10

ρ
6
10
3
10
143 145 147 149
Nd Nd Nd Nd
1
0 4 8 12 4 8 12 4 8 12 4 8 12
E [MeV]
x
Fig. 4. State densities ρ versus excitation energy E in families of even-mass (top two panels) and odd-mass (bottom two
x
panels) samarium and neodymium isotopes. The SMMC densities (open circles) are compared with level counting data at low
excitationenergies(solidhistograms)andneutronresonancedata(triangles)whenavailable[24].Thesolidlinesareback-shifted
Fermi gas densities extracted from experimental data. Adapted from Refs. [12,15,26].
5 Projection methods transform decomposition for δ(Jˆ −M) [10]
z
Tvaoludeeoteframninobestehrveabdleen,switeyudsiestprriboujetcitoinosnamteathogidvsenbyfirxeepd- TrMUσ = 2J1+1 (cid:88)Js e−iϕkMTr (cid:16)eiϕkJˆzUσ(cid:17) . (15)
s
resenting the corresponding Dirac δ function as a Fourier k=−Js
transform.Belowwediscussprojectionsongoodquantum
Here J is the maximal many-particle spin in the model
s
numbers, e.g., spin (Sec. 5.1), as well as on observables
space and ϕ ≡ πk/(J +1/2) with k = −J ,...J are
k s s s
that do not commute with the Hamiltonian, e.g., the ax-
quadrature points.
ial quadrupole operator Qˆ20 (Sec. 5.2). Using Tr Xˆ =Tr Xˆ−Tr Xˆ (which hold for
J M=J M=J+1
anyscalaroperatorXˆ),wecanalsocalculateρ (E ),i.e.,
J x
the level density for a given spin J.
5.1 Spin distributions
InFig.5weshowthespindistributionρ /ρasafunc-
J
tion of J at different values of the excitation energies E
x
We can project on Jˆ, the spin component along the z for three mid-mass nuclei: the odd-even nucleus 55Fe, the
z
axis, by using (within the HS representation) a Fourier even-evennucleus56Feandtheodd-oddnucleus60Co.We
6 Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives
55 56 60
Fe Fe Co
0.04
E = 4.39 MeV E = 5.6 MeV E = 3.39 MeV
x x x
0.02
0.00
E = 7.06 MeV E = 7.5 MeV E = 6.88 MeV
x x x
0.02
ρ
/ 0.00
J
ρ E = 11.95 MeV E = 11.6 MeV E = 11.89 MeV
x x x
0.02
0.00
E = 17.37 MeV E = 16.1 MeV E = 17.71 MeV
x x x
0.02
0.00
0 4 8 0 4 8 0 4 8 12
J J J
Fig.5.Spindistributionsρ /ρversusspinJ forseveralvaluesoftheexcitationenergyE andforthenuclei55Fe(leftcolumn),
J x
56Fe (middle column) and 60Co (right column). The SMMC results (solid squares with error bars) are compared with the spin
cutoff model with a fitted spin cutoff parameter σ (solid lines) and the spin cutoff model with rigid-body moment of inertia
(dashed lines). Taken from Ref. [10].
comparetheSMMCresultswiththespincutoffmodel[27] The spin cutoff parameter is related to the thermal
moment of inertia I by
ρJ(Ex)=ρ(Ex)(22√J2+πσ13)e−J(2Jσ+21) , (16) σ2 = I¯hT2 . (17)
In Fig. 6 we show the moment of inertia I versus exci-
tation energy E for the same three nuclei as in Fig. 5.
where σ is an energy-dependent spin-cutoff parameter. x
We find that at low excitation energies and for the even-
The spin-projected density ρ (E ) does not include the
J x even nucleus 56Fe, I is suppressed to values that are be-
2J +1 magnetic degeneracy and are thus normalized ac-
(cid:80) lowitsrigid-bodyvaluebecauseofpairingcorrelations.In
cording to (2J +1)ρ (E ) ≈ ρ(E ). The distribution
J J x x contrast, the moments of inertia for the odd-even nucleus
(16) corresponds to a normal distribution in the angular
55Fe and the odd-odd nucleus 60Co are not reduced much
momentumvectorJ.ThesolidlinesinFig.5describethe
below their respective rigid-body values.
spin cutoff model distributions in which σ is determined
by a fit to the corresponding SMMC spin distributions.
We observe that at higher excitation energies the spin 5.2 Quadrupole distributions
cutoff model works quite well and this seems to be the
casealsoatlowenergiesforodd-evenandodd-oddnuclei. The level density as a function of deformation is impor-
However, for the even-even nucleus 56Fe, we observe at tantinputformodelingofshapedynamicssuchasnuclear
low excitation energies an odd-even staggering in spin. fission.Herewereviewthefirststeptowardsuchacalcula-
This phenomenon is generic for even-even nuclei and was tionintheframeworkoftheCIshellmodelapproach[16].
observed in empirical analysis of low-lying energy levels Theaxialquadrupoleoperatorinthelaboratoryframe
across the table of nuclei [28,29]. Qˆ and the Hamiltonian H do not commute, and thus
20
Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives 7
0
1) 55Fe 56Fe 60Co
-V15
e
M
10

2h
_( 5

I
2
0 4 8 12 16 0 4 8 12 16 0 4 8 12 16 20
E (MeV)
x
Fig. 6. MomentofinertiaI versusexcitationenergyE forthenuclei55Fe,56Feand60Co.ThemomentofinertiaI extracted
x
fromσ2usingEq.(17)(solidsquareswitherrorbars)iscomparedwiththerigid-bodymomentofinertia(dashedlines).Adapted
from Ref. [10].
cannot be diagonalized simultaneously. The distribution
of Qˆ at temperature T is then given by 0.0015 T=4.0 MeV
20
PT(q)=(cid:88)δ(q−qn)(cid:88)(cid:104)q,n|e,m(cid:105)2e−βem , (18) q) 0.001
(
n m P
0.0005
where |q,n(cid:105) are eigenstates of Qˆ satisfying Qˆ |q,n(cid:105) =
20 20
q |q,n(cid:105) and similarly |e,m(cid:105) are eigenstates of H. We cal-
n
culate PT(q) by using its Fourier representation 0.0015 T=1.14 MeV
P (q)= 1 (cid:90) ∞ dϕe−iϕqTr (cid:16)eiϕQˆ20e−βHˆ(cid:17). (19) q) 0.001
T Tre−βHˆ −∞ 2π P(
0.0005
togetherwiththeHStransformationfore−βH.Inpractice,
we approximate the quadrupole-projected trace by a dis-
creteFourierdecomposition.Wetakeaninterval[−qmax,qmax] 0.0015 T=0.1 MeV
and divide it into 2L+1 equal intervals of length ∆q =
2qmax/(2L+1). We have q) 0.001
(
P
Tr(cid:16)δ(Qˆ20−qm)Uσ(cid:17)≈2q1 (cid:88)L e−iϕkqmTr(eiϕkQˆ20Uˆσ), 0.0005
max
k=−L 0
(20)
-1200 -600 0 600 1200
where q = m∆q (m = −L,...,L) and ϕ = πk/q
m k max
(k =−L,...,L). q (fm2)
TheSMMCquadrupoledistributionsP (q)areshown
T
in Fig. 7 for the deformed nucleus 154Sm at several tem- Fig.7.AxialquadrupoledistributionsP (q)inthelaboratory
T
peratures.AtthelowesttemperatureT =0.1MeV,weob- frame for the deformed nucleus 154Sm at three temperatures.
serveaskeweddistributionwhichwecomparewiththeax- The dashed line at the low temperature T = 0.1 MeV is the
ialquadrupoledistributionofaprolaterigidrotor(dashed quadrupole distribution of a prolate rigid rotor. Taken from
line). The overall similarity of the distributions is a clear Ref. [16].
signature of deformation. Deformation is an important
concept in our understanding of heavy nuclei but it is in-
troduced in the framework of a mean-field approximation the vicinity of the transition temperature (middle panel
(e.g, Hatree-Fock) that breaks rotational symmetry. Here in Fig. 7). At a high temperature of T =4 MeV, the dis-
we observe a signature of deformation in the quadruple tribution is close to a Gaussian (top panel). In 148Sm, a
distribution in the laboratory frame using a rotationally spherical nucleus, we find that the quadrupole distribu-
invariant CI shell model approach. tions are close to a Gaussian even at low temperatures.
In the finite-temperature HF approximation, 154Sm Our goal is to eventually determine the distribution
undergoesashapetransitionfromadeformedtoaspheri- of the intrinsic quadrupole deformation. To this end, we
cal nucleus. The quadrupole distribution is still skewed in constructcombinationsofthesecondrankquadrupoleop-
8 Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives
erator Qˆ that are invariant under rotations. For exam- The dependence of level densities on intrinsic nuclear
2µ
ple, Qˆ·Qˆ is a second order invariant, and (Qˆ×Qˆ)(2)·Qˆ deformation is important input in the modeling of shape
is a cubic invariant. The expectation values of these in- dynamics and, in particular, nuclear fission. Our recent
variants are independent of the particular frame they are study of deformation in the framework of the CI shell
evaluated in, and in particular, they can be evaluated in model can be viewed as the first step toward the calcu-
the intrinsic frame. Effective values of the intrinsic defor- lation of level densities versus deformation.
mation parameters β and γ can then be calculated from IncomparingthecalculatedSMMCdensitieswithneu-
these invariants tron resonance data, we converted the measured s-wave
√ neutron resonance mean spacing to total densities assum-
β = 5π (cid:104)Qˆ·Qˆ(cid:105)1/2 ing a spin cutoff model with rigid-body moment of iner-
3r2A5/3 tia and equal positive- and negative-parity densities. It
0
(cid:114)7(cid:104)(Qˆ×Qˆ)·Qˆ(cid:105) would be interesting to make a direct comparison of the
cos3γ =− . (21) measuredaverageneutronresonancedensitywiththecor-
2 (cid:104)Qˆ·Qˆ(cid:105)3/2 responding sum of SMMC spin-parity level densities that
areallowedbytheselectionrules.Thisrequirestheuseof
To determine the intrinsic quadrupole shape distribu-
a combined spin-parity projection method in SMMC.
tionP (β,γ),weusethefactthatP (β,γ)isarotational
T T InourSMMCstudiesofleveldensitiesweuseCIshell
invariant, and expand −lnP (β,γ) in the quadrupole in-
T model Hamiltonians that are specific to the mass region
variants. Carrying the expansion up to the fourth order
under consideration. A long-term goal is to derive an ef-
invariant, we have
fective shell model Hamiltonian from a density functional
−lnP (β,γ)=N+Aβ2−Bβ3cos3γ+Cβ4+... , (22) theory.Sincedensityfunctionaltheoriesarevalidglobally
T
across the table of nuclei, this would enable us to derive
whereA,B,C aretemperature-dependentparametersand systematically CI shell model Hamiltonians across differ-
N is a normalization constant. Eq. (22) carries a similar- entmassregions.Preliminaryideasforhowtoconstructa
ity to the expansion of the free energy in the quadrupole map from a density functional theory on a CI shell model
shapeorderparameterintheLandautheoryofshapetran- Hamiltonian were discussed inRefs. [32,33]. It would also
sitions[30,31].TheparametersA,B,C canbedetermined be useful to extend the SMMC studies to heavier mass
from the expectation values of the three quadrupole in- regions (e.g., actinides), and to unstable nuclei.
variants β2,β3cos3γ and β4. In Ref. [16] we showed that
the expectation values of these low-order quadrupole in-
variants can be related to moments of Qˆ in the lab- 6.1 Acknowledgments
20
oratory frame, and can therefore be extracted from the
I would like to thank G.F. Bertsch, M. Bonett-Matiz, L.
distributions P (q). Once the distributions P (β,γ) are
T T
Fang, C.N. Gilbreth, S. Liu, A. Mukherjee, H. Nakada,
known,wecanusethesaddle-pointapproximationtocon-
and C. O¨zen for their collaboration on the work reviewed
vertthemtotheintrinsicshapedistributionsP (β,γ)at
Ex above.ThisworkwassupportedinpartbytheDOEgrant
fixed excitation energy E .
x
DE-FG-0291-ER-40608.Theresearchpresentedhereused
After we determine P (β,γ), the joint level density
Ex resourcesoftheNationalEnergyResearchScientificCom-
distribution ρ(β,γ,E ) at a given intrinsic deformation
x
putingCenter,whichissupportedbytheOfficeofScience
β,γ and excitation energy E can be calculated from
x
oftheU.S.DepartmentofEnergyunderContractNo.DE-
ρ(β,γ,E )=ρ(E )P (β,γ). (23) AC02-05CH11231. It also used resources provided by the
x x Ex
facilities of the Yale University Faculty of Arts and Sci-
ences High Performance Computing Center.
6 Conclusion and prospects
The SMMC method is a powerful method for the calcu- References
lation of level densities in the presence of correlations in
very large model spaces. A recent method to circumvent 1. W. Hauser and H. Feshbach, Phys. Rev. 87, 366 (1952).
theodd-particlesignproblemenabledustocalculateaccu- 2. A.J. Koning, S. Hilaire and S. Goriely, Nucl. Phys. A 810,
ratelyleveldensitiesinodd-massnuclei.Wedemonstrated 13 (2008), and references therein.
3. S. Hilaire, M. Girod, S. Goriely, and A.J. Koning, Phys.
that a spherical CI shell model approach defined within a
Rev. C 86, 064317 (2012).
truncatedsingle-particlespaceiscapableofdescribingmi-
4. G.H.Lang,C.W.Johnson,S.E.Koonin,andW.E.Ormand,
croscopically various types of collectivity in heavy nuclei.
Phys. Rev. C 48, 1518 (1993).
Projection methods enable the calculation of the distri-
5. Y. Alhassid, D.J. Dean, S.E. Koonin, G.H. Lang, and
bution of good quantum numbers such as spin. We can
W.E. Ormand, Phys. Rev. Lett. 72, 613 (1994).
alsoprojectonobservablesthatdonotcommutewiththe
6. S.E.Koonin,D.J.Dean,andK.Langanke,Phys.Rep.278,
Hamiltonian, such as the axial quadrupole operator. Us-
2 (1997).
ing such a quadrupole projection, we showed that nuclear 7. Y. Alhassid, Int. J. Mod. Phys. B 15, 1447 (2001).
deformation can be studied in the rotationally invariant 8. H. Nakada and Y. Alhassid, Phys. Rev. Lett. 79, 2939
framework of the CI shell model. (1997).
Y. Alhassid: The shell model Monte Carlo approach to level densities: recent developments and perspectives 9
9. Y. Alhassid, S. Liu and H. Nakada, Phys. Rev. Lett. 83,
4265 (1999).
10. Y. Alhassid, S. Liu and H. Nakada, Phys. Rev. Lett. 99,
162504 (2007).
11. Y.Alhassid,L.FangandH.Nakada,Phys.Rev.Lett.101,
082501 (2008).
12. C. O¨zen, Y. Alhassid and H. Nakada, Phys. Rev. Lett.
110, 042502 (2013).
13. A. Mukherjee and Y. Alhassid, Phys. Rev. Lett. 109,
032503 (2012).
14. M. Bonett-Matiz, A. Mukherjee and Y. Alhassid, Phys.
Rev. C 88, 011302 (R) (2013).
15. C. O¨zen, Y. Alhassid and H. Nakada, Phys. Rev. C 91,
034329 (2015)
16. Y. Alhassid, C.N. Gilbreth, and G.F. Bertsch, Phys. Rev.
Lett. 113, 262503 (2014).
17. J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).
18. R.L. Stratonovich, Dokl. Akad. Nauk. S.S.S.R. 115, 1097
(1957).
19. W.E. Ormand, D.J. Dean, C.W. Johnson, and S.E.
Koonin, Phys. Rev. C 49 (1994) 1422.
20. A.BohrandB.Mottelson,NuclearStructure,Vol.1,Ben-
jamin, New York (1969).
21. A.V.Voinov,S.M.Grimes,C.R.Brune,T.Massey,andA.
Schiller, EPJ Web of Conferences 21, 05002 (2012).
22. A. V. Voinov (private communication).
23. Y.Alhassid,M.Bonett-Matiz,S.LiuandH.Nakada,Phys.
Rev. C 92, 024307 (2015).
24. R. Capote, M. Herman, P. Oblozinsky, P.G. Young, S.
Goriely, T. Belgya, A.V. Ignatyuk, A.J. Koning, S. Hilaire,
V.A. Plujko, M. Avrigeanu, O. Bersillon, M.B. Chadwick,
T. Fukahori, Zhigang Ge, Yinlu Han, S. Kailas, J. Kopecky,
V.M. Maslov, G. Reffo, M. Sin, E.Sh. Soukhovitskii, and P.
Talou,RIPL - a Reference Input Parameter Library for Cal-
culationofNuclearReactionsandNuclearDataEvaluations,
Nuclear Data Sheets 110, 3107 (2009).
25. Y. Alhassid, M. Bonett-Matiz, A. Mukherjee, H. Nakada
and C. O¨zen, Journal of Physics: Conference Series 580,
012009 (2015).
26. Y. Alhassid Y, C. O¨zen and H. Nakada, Nuclear Data
Sheets 118, 233 (2014).
27. T. Ericson, Adv. Phys. 9, 425 (1960).
28. T.vonEgidyandD.Bucurescu,Phys.Rev.C78,051301
(R) (2008).
29. T.vonEgidyandD.Bucurescu,Phys.Rev.C80,054310
(2009).
30. S. Levit and Y. Alhassid, Nucl. Phys. A 413, 439 (1984).
31. Y.Alhassid,S.LevitandJ.Zingman,Phys.Rev.Lett.57,
539 (1986).
32. Y.Alhassid,G.F.Bertsch,L.Fang,B.Sabbey,Phys.Rev.
C 74, 034301 (2006).
33. R. Rodriguez-Guzman, Y. Alhassid, G. F. Bertsch, Phys.
Rev. C 77, 064308 (2008).