Mixed congurations and intertwined quantum phase transitions in odd-mass nuclei N. Gavrielov1 2A. Leviatan2yand F. Iachello1z 1Center for Theoretical Physics Sloane Physics Laboratory

2025-05-02 0 0 613.18KB 6 页 10玖币

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Mixed conﬁgurations and intertwined quantum phase transitions in odd-mass nuclei

N. Gavrielov,1, 2, ∗A. Leviatan,2, †and F. Iachello1, ‡

1Center for Theoretical Physics, Sloane Physics Laboratory,

Yale University, New Haven, Connecticut 06520-8120, USA

2Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel

(Dated: December 7, 2022)

We introduce a new Bose-Fermi framework for studying spectral properties and quantum phase

transitions (QPTs) in odd-mass nuclei, in the presence of conﬁguration mixing. A detailed analysis

of odd-mass Nb isotopes discloses the eﬀects of an abrupt crossing of states in normal and intruder

conﬁgurations (Type II QPT), accompanied by a gradual evolution from spherical- to deformed-core

shapes within the intruder conﬁguration (Type I QPT). The pronounced presence of both types of

QPTs demonstrates, for the ﬁrst time, the occurrence of intertwined QPTs in odd-mass nuclei.

Structural changes induced by variation of parame-

ters in the Hamiltonian, called quantum phase transi-

tions (QPTs) [1,2], are salient phenomena in dynamical

systems, and form the subject of ongoing intense exper-

imental and theoretical activity in diverse ﬁelds [3]. In

nuclear physics, most of the attention has been devoted

to the evolution of structure with nucleon number, ex-

hibiting two types of phase transitions. The ﬁrst, de-

noted as Type I [4], is a shape-phase transition in a single

conﬁguration, as encountered in the neutron number 90

region [5]. The second, denoted as Type II [6], is a phase

transition involving an abrupt crossing of diﬀerent con-

ﬁgurations, as encountered in nuclei near (sub-) shell clo-

sure [7]. If the mixing is small, the Type II QPT can be

accompanied by a distinguished Type I QPT within each

conﬁguration separately. Such a scenario, referred to as

intertwined QPTs (IQPTs), was recently shown to occur

in the Zr isotopes [8,9].

Most studies of QPTs in nuclei have focused on systems

with even numbers of protons and neutrons [5,7,10–12].

The structure of odd-mass nuclei is more complex due to

the presence of both collective and single-particle degrees

of freedom. Consequently, QPTs in such nuclei have been

far less studied. Fully microscopic approaches to QPTs

in medium-heavy odd-mass nuclei, such as the large-scale

shell model [13] and beyond-mean-ﬁeld methods [14],

are computationally demanding and encounter diﬃcul-

ties. Alternative approaches have been proposed, includ-

ing algebraic modeling (shell-model inspired [15–17] and

symmetry-based [18–24]) and density functionals-based

mean-ﬁeld methods [25–28], involving particle-core cou-

pling schemes with boson-fermion or collective Hamilto-

nians. So far these approaches were restricted to Type I

QPTs in odd-mass nuclei without conﬁguration mixing.

The goals of the present Letter are twofold. (i) To in-

troduce a framework for studying spectral properties and

QPTs with conﬁguration mixing, in odd-mass nuclei.

This is motivated by a wealth of new experimental data

∗noam.gavrielov@yale.edu

†ami@phys.huji.ac.il

‡francesco.iachello@yale.edu

on shape-coexisting states in such nuclei near shell clo-

sure [29,30], whose awaiting interpretation necessitates

multiple conﬁgurations. (ii ) To apply the formalism and

show evidence for concurrent types of QPTs exemplify-

ing, for the ﬁrst time, IQPTs in odd-mass nuclei.

Odd-Anuclei are treated in the interacting boson-

fermion model (IBFM) [16], as a system of monopole (s)

and quadrupole (d) bosons, representing valence nucleon

pairs, and a single (unpaired) nucleon. We propose to ex-

tend the IBFM to include core excitations and obtain a

boson-fermion model with conﬁguration mixing (IBFM-

CM), employing a Hamiltonian of the form,

H=ˆ

Hb+ˆ

Hf+ˆ

Vbf .(1)

The boson part ( ˆ

Hb) is the Hamiltonian of the conﬁg-

uration mixing model (IBM-CM) of [31,32]. For two

conﬁgurations (A,B), it can be cast in matrix form [6],

Hb=ˆ

b(ξ(A))ˆ

Wb(ω)

Wb(ω)ˆ

b(ξ(B)).(2)

Here, ˆ

b(ξ(A)) represents the normal A conﬁguration

(Nboson space) and ˆ

b(ξ(B)) represents the intruder

B conﬁguration (N+ 2 boson space), corresponding to

2p-2h excitations across the (sub-) shell closure. Stan-

dard forms, as in Eq. (4) of Ref. [9], include pairing,

quadrupole, and rotational terms, and a mixing term

Wb=ω[(d†d†)(0) + (s†)2] + Hermitian conjugate (H.c.).

Such IBM-CM Hamiltonians have been used extensively

for the study of conﬁguration-mixed QPTs and shape

coexistence in even-even nuclei [8,9,31–39].

The fermion Hamiltonian ( ˆ

Hf) of Eq. (1) has the form

Hf="Pj(A)

jˆnj0

0Pj(B)

jˆnj#,(3)

where jis the angular momentum of the occupied orbit,

ˆnjthe corresponding number operator, and (i)

j(i= A,B)

are the single-particle energies for each conﬁguration.

The boson-fermion interaction has the form

Vbf =ˆ

bf (ζ(A))ˆ

Wbf (ωj)

Wbf (ωj)ˆ

bf (ζ(B)).(4)

arXiv:2210.00441v2 [nucl-th] 6 Dec 2022

Here, ˆ

V(i)

bf (i= A,B) involve monopole, quadrupole, and

exchange terms with parameters ζ(i)=(A(i)

j,Γ(i)

jj0,Λ(i)j00

jj0).

Using the microscopic interpretation of the IBFM [16],

these couplings can be expressed in terms of strengths

(A(i)

0,Γ(i)

0,Λ(i)

0) and occupation probabilities (uj, vj).

The term ˆ

Wbf (ωj)=Pjωjˆnj[(d†d†)(0)+(s†)2+H.c.], con-

trols the mixing for each orbit.

The Hamiltonian of Eq. (1) is diagonalized numerically.

The resulting eigenstates |Ψ; Jiare linear combinations

of wave functions ΨAand ΨB, involving bosonic basis

states in the two spaces |[N], α, Liand |[N+ 2], α, Li.

The boson (L) and fermion (j) angular momenta are cou-

pled to J,|Ψ; Ji=Pα,L,j C(N,J)

α,L,j |ΨA; [N], α, L;j;Ji+

Pα,L,jC(N+2,J)

L,j |ΨB; [N+ 2], α, L;j;Ji. The probabil-

ity of normal-intruder mixing is given by

a2=X

α,L,j |C(N,J)

α,L,j |2, b2=1 −a2=X

α,L,j |C(N+2,J)

α,L,j |2.(5)

Operators inducing electromagnetic transitions of type σ

and multipolarity Lcontain boson and fermion parts,

T(σL) = ˆ

Tb(σL) + ˆ

Tf(σL).(6)

For E2 transitions, ˆ

Tb(E2) = e(A) ˆ

Q(N)

χ+e(B) ˆ

Q(N+2)

χ,

where the superscript (N) denotes a projection onto the

[N] boson space and ˆ

Qχ=d†s+s†˜

d+χ(d†˜

d)(2). For

M1 transitions, ˆ

Tb(M1)=Piq3

4πg(i)ˆ

L(Ni)+˜g(i)[ˆ

Q(Ni)

χ×

L(Ni)](1), where i= (A,B), NA=N, NB=N+2. The

fermion operators ˆ

Tf(σL) have the standard form [16]

with eﬀective charge effor E2 transitions, and gs

quenched by 20% for M1 transitions. In what follows,

we apply the above IBFM-CM framework to the study

of QPTs in the odd-mass Nb isotopes.

The A

41Nb isotopes with mass number A=93–105 are

described by coupling a proton to their respective 40Zr

cores with neutron number 52–64. In the latter, the

normal A conﬁguration corresponds to having no ac-

tive protons above the Z= 40 subshell gap, and the in-

truder B conﬁguration corresponds to two-proton excita-

tion from below to above this gap, creating 2p-2h states.

The parameters of ˆ

Hb(2) and boson numbers are taken

to be the same as in a previous calculation of these Zr

isotopes (see Table V of Ref. [9]), except for χ=−0.565

at neutron number 64.

For 41Nb isotopes, the valence protons reside in the

Z= 28–50 shell. Using as an input the empirical single-

proton energies (taken from Table XI of Ref. [40]) and a

pairing gap ∆F= 1.5 MeV, a BCS calculation yields the

single quasiparticle energies (j) and occupation prob-

abilities (v2

j) for the considered 1g9/2,2p1/2,2p3/2,1f5/2

orbits, assuming, for simplicity, the same parameters for

both conﬁgurations. The derived jand v2

j, and the com-

mon strengths (A0,Γ0,Λ0), obtained by a ﬁt, are listed

in Table I. As seen, the monopole term (A0) vanishes for

neutron number 52–56 and corrects the quasiparticle en-

ergies at neutron number 58–64. The quadrupole term

TABLE I. Parameters in MeV of the boson-fermion in-

teractions, ˆ

V(i)

bf of Eq. (4), obtained from a ﬁt assuming

A(i)

0=A0, Γ(i)

0= Γ0and Λ(i)

0= Λ0,(i)

j=j, where

(i= A,B). From a BCS calculation, j= 1.639, 1.524, 2.148

and 2.519 MeV and v2

j=0.299, 0.589, 0.858, and 0.902, for the

1g9/2,2p1/2,2p3/2,1f5/2orbits, respectively, and Fermi energy

λF=2.024 MeV.

Neutron number 52 54 56 58 60 62 64

A00.00 0.00 0.00 −0.11 −0.20 −0.20 −0.20

Γ01.00 1.00 1.00 1.00 1.00 1.00 1.00

Λ01.00 1.00 3.00 3.00 3.80 3.80 3.80

(Γ0) is constant for the entire chain. The exchange term

(Λ0) increases towards the neutron midshell [16]. Alto-

gether, the values of the parameters are either constant

for the entire chain or segments of it and vary smoothly.

We take ωj=0 in the ˆ

Wbf term of Eq. (4), since for equal

ωjit coincides with the ˆ

Wbterm of Eq. (2).

In the present Letter, we concentrate on the positive-

parity states in Nb isotopes, postponing a discussion of

both parity states to a longer paper. Such a case reduces

to a single-jcalculation, with the π(1g9/2) orbit cou-

pled to the boson core. Figure 1shows the experimental

and calculated levels of selected states, along with assign-

ments to conﬁgurations based on Eq. (5). Open (solid)

symbols indicate a dominantly normal (intruder) state

with small (large) b2probability. In the region between

neutron number 50 and 56, there appear to be two sets of

levels with a weakly deformed structure, associated with

conﬁgurations A and B. All levels decrease in energy for

52–54, away from the closed shell, and rise again at 56

due to the ν(2d5/2) subshell closure. From 58, there is a

pronounced drop in energy for the states of the B conﬁg-

uration. At 60, the two conﬁgurations cross, indicating a

Type II QPT, and the ground state changes from 9/2+

1to

5/2+

1, becoming the bandhead of a K= 5/2+rotational

band composed of 5/2+

1,7/2+

1,9/2+

1,11/2+

1,13/2+

1states.

The intruder B conﬁguration remains strongly deformed

and the band structure persists beyond 60. The above

trend is similar to that encountered in the even-even Zr

isotopes with the same neutron numbers (see Fig. 14 of

Ref. [9]).

A possible change in the angular momentum of the

ground state (J+

gs) is a characteristic signature of Type II

QPTs in odd-mass, unlike even-even nuclei where the

ground state remains 0+after the crossing. It is an

important measure for the quality of the calculations,

since a mean-ﬁeld approach, without conﬁguration mix-

ing, fails to reproduce the switch 9/2+

1→5/2+

1in J+

gs for

the Nb isotopes [46]. Figure 2(a) shows the percentage

of the wave function within the B conﬁguration for J+

and 7/2+

1, as a function of neutron number across the Nb

chain. The rapid change in structure of J+

gs from the nor-

mal A conﬁguration in 93−99Nb (small b2probability),

to the intruder B conﬁguration in 101−105Nb (large b2)

is clearly evident, signaling a Type II QPT. The conﬁg-

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Mixedcongurationsandintertwinedquantumphasetransitionsinodd-massnucleiN.Gavrielov,1,2,A.Leviatan,2,yandF.Iachello1,z1CenterforTheoreticalPhysics,SloanePhysicsLaboratory,YaleUniversity,NewHaven,Connecticut06520-8120,USA2RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,Israel(Dated:Decembe...

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Mixed congurations and intertwined quantum phase transitions in odd-mass nuclei N. Gavrielov1 2A. Leviatan2yand F. Iachello1z 1Center for Theoretical Physics Sloane Physics Laboratory

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