aa r X i v : . [ nu c l - t h ] A ug Nuclear deformation at finite temperature
Y. Alhassid, C.N. Gilbreth, and G.F. Bertsch Center for Theoretical Physics, Sloane Physics Laboratory,Yale University, New Haven, CT 06520 Department of Physics and Institute of Nuclear Theory, Box 351560University of Washington, Seattle, WA 98915 (Dated: October 2, 2018)Deformation, a key concept in our understanding of heavy nuclei, is based on a mean-field de-scription that breaks the rotational invariance of the nuclear many-body Hamiltonian. We presenta method to analyze nuclear deformations at finite temperature in a framework that preservesrotational invariance. The auxiliary-field Monte-Carlo method is used to generate the statisticalensemble and calculate the probability distribution associated with the quadrupole operator. Ap-plying the technique to nuclei in the rare-earth region, we identify model-independent signatures ofdeformation and find that deformation effects persist to higher temperatures than the spherical-to-deformed shape phase-transition temperature of mean-field theory.
PACS numbers: 21.60.Cs, 21.60.Ka, 21.10.Ma, 02.70.Ss
Motivation .— Mean-field theory is a useful method forstudying correlated many-body systems. However, it of-ten breaks symmetries, making it difficult to compare itsresults with physical spectra that preserve these symme-tries. In addition, although mean-field theory often pre-dicts sharp phase transitions at finite temperature, theyare washed out in finite-size systems. The challenge isto find tools to study the properties of finite-size systemswithin a framework that preserves the underlying sym-metries while also allowing calculation of the quantitiesthat describe symmetry breaking in mean-field theory.In nuclear physics, this issue is especially importantin the understanding of heavy deformed nuclei, whichare of wide experimental and theoretical interest. Thecurrent theory of these nuclei is based on self-consistentmean-field (SCMF) theory, which predicts both spher-ical and deformed ground states [1] depending on thenucleus. SCMF is a convenient tool to study their in-trinsic structure but it breaks rotational invariance, aprominent symmetry in nuclear spectroscopy. The oc-currence of large deformations in the ground state andat low excitations gives rise to rotational bands and largeelectric quadrupole transition intensities between stateswithin the bands. At higher excitations, much less isknown experimentally. Characterization of this part ofthe spectrum is needed for accurate calculation of thenuclear level density, which is very sensitive to deforma-tion and other structure effects; observed level densitiesin rare-earth nuclei at the neutron evaporation thresholdvary by more than an order of magnitude [3]. In addi-tion, nuclear fission is a phenomenon of shape dynamics,and calculation of fission rates for excited nuclei requirestheir level densities as a function of deformation [4].Here we investigate nuclear deformation at finite tem-perature using the auxilliary-field Monte Carlo (AFMC)method, which is well suited to the study of the evolu-tion of nuclear properties with excitation energy whilepreserving rotational invariance. In particular, we cal- culate the distribution of the quadrupole operator inthe lab frame and demonstrate that it exhibits model-independent signatures of deformation. We use momentsof this distribution to calculate rotationally invariant ob-servables, which allow us to extract effective values of theintrinsic deformation and its fluctuations. Deformationshave been studied previously by the AFMC method, butwith an ad hoc prescription to extract the intrinsic-frameproperties [5]. The methods presented here should be ap-plicable to other finite-size systems in which correlationsbeyond the mean field are important.
Methodology .— Formally, we can examine the sta-tistical characteristics of nuclei at finite excitations bycalculating the thermal expectation values of observ-ables ˆ O associated with the property of interest, h ˆ O i =Tr ( ˆ Oe − β ˆ H ) / Tr e − β ˆ H . Here β − is the temperature andˆ H is the Hamiltonian, which we assume to be rotation-ally invariant. We denote operators in the many-particlespace with a circumflex, to be distinguished from op-erators in the single-particle space, which are ordinarymatrices, denoted by bold-face symbols. Also, we de-note the trace over the full many-particle Fock space asTr and the trace of matrices in the single-particle spaceby tr. The probability distribution of an operator ˆ O , P β ( o ) = Tr( δ [ ˆ O − o ) e − β ˆ H ] / Tr e − β ˆ H can be calculated us-ing the Fourier representation of the δ function: P β ( o ) = 1Tr e − β ˆ H Z ∞−∞ dϕ π e − iϕo Tr (cid:16) e iϕ ˆ O e − β ˆ H (cid:17) . (1)Eq. (1) is well-known for one-body observables ˆ O thatcommute with the Hamiltonian, e.g., the number opera-tor and the z -component of the angular momentum [6].Nuclear shape is different in that the relevant oper-ators, e.g., the quadrupole operators, do not commutewith the Hamiltonian. Nevertheless, it is possible withEq. (1) to define the distribution of quantum-mechanicalobservables that carry information about deformation aswell as energy. The distribution (1) can be expressed interms of the many-particle eigenstates of ˆ O and ˆ H as P β ( o ) = X n δ ( o − o n ) X m h o, n | e, m i e − βe m . (2)Here | o, n i are eigenstates of ˆ O satisfying ˆ O | o, n i = o n | o, n i and similarly for | e, m i . Eq. (2) is valid whetheror not the operators ˆ O and ˆ H commute. When they docommute, they share a common basis of eigenstates suchthat h o, n | e, m i = δ m,n and the distribution (2) reducesto its more familiar form P β ( o ) = P n δ ( o − o n ) e − βe n .Note that in a finite model space the eigenvalues o n forma discrete set and P β ( o ) is a finite sum of δ functions.In this work we consider the observable ˆ O to bethe spectroscopic mass quadrupole operator ˆ Q = P i (cid:0) z i − x i − y i (cid:1) where the sum is taken over all nucle-ons. The probability distribution P β ( q ) of ˆ Q is definedas in Eq. (1) with ˆ O = ˆ Q and o = q .As we will show, this distribution can be accuratelycomputed by the AFMC method. However, the intrinsic-frame properties are not directly accessed by the operatorˆ Q , which is a laboratory-frame observable. We shalldemonstrate in this work that nevertheless the distribu-tion P β ( q ) is sensitive to deformation effects and that themain properties of the deformation in the intrinsic framecan be recovered from moments of this distribution.Intrinsic frame quantities may be defined in terms ofthe expectation values of rotationally invariant combi-nations of the quadrupole tensor operator ˆ Q µ ( µ = − , . . . ,
2) [7, 8]. The lowest-order invariant is quadratic,ˆ Q · ˆ Q = P µ ( − ) µ ˆ Q µ ˆ Q − µ . There is one third-orderinvariant defined by coupling three quadrupole oper-ators to angular momentum zero, ( ˆ Q × ˆ Q ) · ˆ Q = √ P µ ,µ ,µ (cid:18) µ µ µ (cid:19) ˆ Q µ ˆ Q µ ˆ Q µ . The fourth-and fifth-order invariants are also unique [9] and we de-fine them as ( ˆ Q · ˆ Q ) and ( ˆ Q · ˆ Q )(( ˆ Q × ˆ Q ) · ˆ Q ), respectively.When the invariant is unique at a given order, its expec-tation value can be computed directly from the lab-framemoments of ˆ Q , defined by h ˆ Q n i β = R q n P β ( q ) dq . Theconversion factors are given in Table I. n 2 3 4 5invariant 5 − / / − (11 / / / rotor 1/5 2/35 3/35 4/77TABLE I: First line: the ratio of the expectation value ofthe invariant of order n (see text) to the n -th moment of ˆ Q .Second line: the n -th moment of ˆ Q for the rigid rotor inunits of q n ( q is the rotor’s intrinsic quadrupole moment). AFMC .— We shall use the AFMC to evaluate the dis-tribution in Eq. (1) for ˆ O = ˆ Q . AFMC is arguably themost powerful computational tool for finding the ground states and thermal properties in large-dimension many-particle spaces. It is based on the Hubbard-Stratonovichrepresentation [10] of the imaginary-time propagator, e − β ˆ H = R D [ σ ] G σ ˆ U σ , where D [ σ ] is the integration mea-sure, G ( σ ) is a Gaussian weight, and ˆ U σ is a one-bodypropagator of non-interacting nucleons moving in aux-iliary fields σ . Practical implementations require thatthe Hamiltonian be restricted to one- and two-bodyterms, and that the two-body terms have the so-calledgood sign [11]. The method has been applied to nucleiin the framework of the configuration-interaction shellmodel [12–14], where it is called the shell-model MonteCarlo (SMMC). It has been particularly successful in cal-culating statistical properties of nuclei such as level den-sities [15]. The distribution of ˆ Q is obtained from theMonte Carlo sampling of fields σ as a ratio of averages P β ( q ) = * Tr h δ ( ˆ Q − q ) ˆ U σ i Tr ˆ U σ Φ σ + W h Φ σ i − W , (3)Here h X i W = R D [ σ ] W σ X σ / R D [ σ ] W σ , where W σ = G σ | Tr ˆ U σ | is used for the Monte Carlo sampling andΦ σ = Tr ˆ U σ / | Tr ˆ U σ | is the Monte Carlo sign function.For a given ˆ U σ , we carry out the ˆ Q projection us-ing a discretized version of the Fourier decomposition inEq. (1). We take an interval [ − q max , q max ] and divide itinto 2 M + 1 equal intervals of length ∆ q = 2 q max / (2 M +1). We define q m = m ∆ q , where m = − M, . . . , M , andapproximate the quadrupole-projected trace in (3) byTr (cid:16) δ ( ˆ Q − q m ) U σ (cid:17) ≈ q max M X k = − M e − iϕ k q m Tr( e iϕ k ˆ Q ˆ U σ ) , (4)where ϕ k = πk/q max ( k = − M, . . . , M ). Since ˆ Q is a one-body operator and ˆ U σ is a one-body propaga-tor, the Fock space many-particle traces on the r.h.s. ofEq. (4) reduce to determinants in the single-particle spaceTr (cid:16) e iϕ k ˆ Q ˆ U σ (cid:17) = det (cid:0) e iϕ k Q U σ (cid:1) . Here Q and U σ are the matrices representing, respectively, ˆ Q andˆ U σ , in the single-particle space. In practice, projectionsare carried on the neutron and proton number operatorsas well to fix the Z and N of the ensemble [14].We found the thermalization of ˆ Q n to be slow with thepure Metropolis sampling. This can be overcome by aug-menting the Metropolis-generated configurations by ro-tating them through a properly chosen set of N Ω rotationangles Ω. In practice, it is easier to rotate the observ-ables, i.e., we replace h e iϕ ˆ Q i σ by N Ω P j h e iϕ ˆ Q (Ω j ) i σ .Here ˆ Q (Ω) = ˆ R ˆ Q ˆ R − with ˆ R being the rotation op-erator for angle Ω. Details will be given elsewhere.We next discuss a few simple examples that can betreated analytically or nearly so. Rigid rotor .— As a first simple example, we consider anaxially symmetric rigid rotor with an intrinsic quadrupole P g . s . ( q ) q (units of q ) FIG. 1: The ground-state distribution P g . s . ( q ) vs q/q for aprolate rotor with intrinsic quadrupole moment q . moment q in its ground state. The distribution ofits spectroscopic quadrupole operator in the laboratoryframe Q = q (3 cos θ − / q > P g . s . ( q ) = ( (cid:16) √ q q qq (cid:17) for − q ≤ q ≤ q . (5)This distribution is shown in Fig. 1. The oblate rotor( q <
0) distribution is obtained from (5) by replacing q with − q and q with | q | . The moments of the dis-tribution (5) can be calculated from a simple recursionrelation; their values for 2 ≤ n ≤ Ne .— As a simple illustration in nuclear spec-troscopy, we consider the light deformed nucleus Ne.The orbital part of the single-particle wave functions aretaken to be the states of the N = 2 harmonic oscillatorshell, i.e., the sd -shell. The single-particle eigenvalues of Q are -2, 1, and 4 (in units of b [16]) with degeneraciesof 6, 4 and 2, respectively. The many-particle eigenvaluesof ˆ Q for Ne in the valence sd -shell thus range from − P β ( q )at β = 0 is just the distribution of these eigenvalues.We have used this nucleus as a simple test of theAFMC. Here we take the single-particle energies accord-ing to the USD interaction [17] and consider an attrac-tive quadrupole-quadrupole interaction − χ ˜ Q · ˜ Q , with˜ Q µ = P i r i Y µ (ˆ r i ) and χ = π . A / MeV /b [18]. InFig. 2 we show the quadrupole distribution of the Neground state. The discrete nature of the many-particleeigenvalues of ˆ Q is evident; the distribution is a set δ functions at integers − , − , . . . , ,
16. The envelope ofthe strengths has the skewed shape that looks qualita-tively similar to the prolate rigid-rotor distribution.
SCMF .— It is instructive to compare our results withthose of the thermal SCMF, e.g., the finite-temperatureHartree-Fock-Bogoliubov (HFB) approximation. TheHFB solution is characterized by temperature-dependentone-body density matrix ρ β and pairing tensor κ β . Ingeneral, two types of phase transitions can occur vs P g . s . ( q ) q (b ) FIG. 2: The AFMC ground-state quadrupole distribution P g . s . ( q ) for Ne. The sharp δ -like peaks demonstrate thediscrete nature of the spectrum of ˆ Q and their enveloperesembles the prolate rigid-rotor distribution in Fig. 1. temperature, a pairing transition and a deformed-to-spherical shape transition [19–21]. A shape phase tran-sition is also the generic result of a Landau theory inwhich the order parameter is the quadrupole deforma-tion tensor [22]. The vast majority of deformed HFBground states are axially symmetric [23], i.e., h ˆ Q µ i = 0for µ = 0. The second-order invariant h ˆ Q · ˆ Q i may becalculated in HFB by using Wick’s theorem h ˆ Q · ˆ Q i = Q + X µ ( − ) µ tr [ Q µ ( − ρ β ) Q − µ ρ β ]+ X µ ( − ) µ tr (cid:2) Q µ κ β Q T − µ κ ∗ β (cid:3) , (6)where Q ≡ tr( Q ρ β ) is the intrinsic axial quadrupolemoment. The remaining terms on the r.h.s. of (6) repre-sent the contributions due to quantal and thermal fluc-tuations. We shall compare our AFMC results for rare-earth nuclei with the HFB theory in the next section. Rare-earth nuclei .— Here we present results for rare-earth nuclei. The single-particle orbitals are taken from aWoods-Saxon potential plus spin-orbit interaction; theyspan the 50 −
82 shell plus 1 f / orbital for protons andthe 82 −
126 shell plus 0 h / , g / orbitals for neutrons.We use the same interaction as in Refs. [24, 25]. Thequadrupole moments are scaled by a factor of 2 to ac-count for the model space truncation.We first examine Sm, a strongly deformed nucleuswith an intrinsic quadrupole moment of Q ∼ , as determined experimentally from in-band electricquadrupole transitions [26]. AFMC P β ( q ) distributionsare shown in Fig. 3 at three temperatures. The distribu-tions appear continuous because the many-particle eigen-values of ˆ Q are closely spaced. At the lowest temper-ature of T = 0 . e − β ˆ H effectivelyprojects out the ground-state band. We observe the char-acteristic skewed distribution of the prolate rotor. Thedashed line is the rotor distribution (5) with q taken atthe HFB value of Q . The middle panel is the distribu-tion at the HFB shape transition temperature, T = 1 . P ( q ) T=4.0 MeV P ( q ) T=1.14 MeV P ( q ) q (fm ) T=0.1 MeV
FIG. 3: Probability distributions P β ( q ) for Sm at T =0 . T = 1 .
14 MeV (shape transition temperature) and T = 4 MeV. The low-temperature distribution is comparedwith the rigid-rotor distribution (dashed line) and reflects thestrongly deformed character of this nucleus. MeV. The distribution is less skewed, but nevertheless itretains some trace of a prolate character. The HFB exci-tation energy at this temperature is about 25 MeV, muchhigher than energies of interest for spectroscopy and forthe neutron-capture reaction. The top panel shows thedistribution at T = 4 MeV. At this high excitation thedistribution is featureless and close to a Gaussian.We have also calculated P β ( q ) for Sm, which isspherical in its HFB ground state. They are more sym-metric and change less with temperature, consistent withthe absence of a coherent quadrupole moment.
Invariants .— Fig. 4 shows the second-order invariant h ˆ Q · ˆ Q i vs temperature T for Sm and
Sm. TheAFMC results (circles) are compared with the HFB re-sults (dashed lines) of Eq. (6). In HFB, h ˆ Q · ˆ Q i for Smcan be entirely attributed to the fluctuation terms in (6).There is a small kink at T = 0 . h ˆ Q · ˆ Q i in Sm is very different at low temperatures. In HFB,the intrinsic quadrupole moment Q is large, and it per-sists up to a temperature of the order of 1 MeV, closeto the spherical-to-deformed phase-transition tempera-ture. The AFMC results are in semiquantitative agree-ment at the lowest temperatures showing that the coher-ent intrinsic quadrupole moment is not an artifact of theHFB. The sharp kink characterizing the HFB shape tran-sition [19, 20] is washed out, as is expected in a finite-sizesystem. Nevertheless a signature of this phase transitionremains in the rapid decrease of h Q · Q i with tempera-ture. In AFMC deformation effects survive well above
0 1 2 3 4T (MeV)
Sm04812 0 1 2 3 < Q ⋅ Q > ( f m ) T (MeV) Sm FIG. 4: h Q · Q i vs temperature T for the spherical Sm (left)and the deformed
Sm (right). The AFMC results (solidcircles) are compared with the HFB results (dashed lines). the transition temperature, in that h Q · Q i continues tobe enhanced beyond its uncorrelated mean-field value.The second- and third-order invariants can be usedto define effective values of the intrinsic shape pa-rameters β, γ [27] of the collective Bohr model [28,Sec. 6B-1a]. The model assumes an intrinsic framein which the quadrupole deformation parameters α µ = √ π h ˆ Q µ i / r A / are expressed as α = β cos γ , α = α − = √ β sin γ , and α ± = 0. Effective β and γ canthen be determined from the corresponding invariants β = √ π r A / h ˆ Q · ˆ Q i / ; cos 3 γ = − r h ( ˆ Q × ˆ Q ) · ˆ Q ih ˆ Q · ˆ Q i / . (7)In addition, we can extract a measure ∆ β of the fluctu-ations in β using the second- and fourth-order invariants(∆ β/β ) = h h ( ˆ Q · ˆ Q ) i − h ˆ Q · ˆ Q i i / / h ˆ Q · ˆ Q i . (8)The invariants themselves are calculated from the mo-ments of P β ( q ) using the relations in Table I. As ex-pected, the deformed Sm has a larger deformation β than Sm (0 .
232 vs 0 . γ (13 . ◦ vs 21 . ◦ ) that is closer to an axialshape. The deformed nucleus is more rigid in that it hasa smaller ∆ β/β , 0 .
51 for
Sm vs 0 .
72 for
Sm.