Binding energy differences of even-even nuclei
as pairing indicators

Nobuo HinoharaCenter for Computational Sciences, University of Tsukuba, Japan

Jun. 26, 2017

China-Japan collaboration workshop on“Nuclear mass and life for unravelling mysteries of r-process”

Outlinep Introduction
p Binding energy differences
p gauge symmetry breaking and pairing rotation
p Theoretical frameworkp Pairing rotations in single-closed nuclei
p Pairing rotations in doubly-open-shell nuclei
p Constraining pairing EDF using pairing rotational MOI
p SummaryBinding energy differences
What does binding energy systematics tell us about?

one-neutron/proton separation energy
Sn (N, Z) = B(N, Z)

B(N1, Z)Sp (N, Z) = B(N, Z)

B(N, Z1)

odd-even staggering (OES)
Sn (N + 1, Z)]
2B(N, Z) + B(N + 1, Z)]( 1)Z[Sp (N, Z) Sp (N, Z + 1)]
2( 1)Z=[B(N, Z 1) 2B(N, Z) + B(N, Z + 1)]2(3)p (N, Z) =Sn(N,Z) (MeV)( 1)N[Sn (N, Z)
2( 1)N=[B(N 1, Z)
2(3)n (N, Z) =1816
14121086420Sn50 55 60 65 70 75 80 85 90
NΔ(3)n,p: size of the pairingBinding energy differences
What does binding energy systematics tell us about?

two-neutron/proton separation energy
S2n (N, Z) = B(N, Z)

empirical shell gaps

B(N2, Z)

S2p (N, Z) = B(N, Z)

2n (N, Z) =S2n (N, Z)

S2n (N + 2, Z)=2, Z) + 2B(N, Z)

B(N

2p (N, Z) =S2p (N, Z)

S2p (N, Z + 2)

=2) + 2B(N, Z)

B(N, Z

B(N, Z2)

B(N + 2, Z)

B(N, Z + 2)

S2n(N+2,Z)/2S2n(N,Z)/2δ2n(N,Z)/2
N neutrons

Lunney et al. Rev. Mod. Phys. 75, 1021 (2003)

δ2n,2p: size of the magic shell gap

Binding energy differences
proton-neutron interaction energy Zhang et al., Phys. Lett. B 227, 1 (1989)
1Vpn (N, Z) = [S2n (N, Z) S2n (N, Z 2)]
41= [S2p (N, Z) S2p (N 2, Z)]
41= [B(N, Z) B(N 2, Z) B(N, Z42) + B(N2, Z2)]

Stoitsov et al., Phys. Rev. C 98,132502 (2007)neutron
B(N2, Z2) = Encore + Epcore + Vncore pcore

B(N, Z2) =B(N2, Z2) + E2n + V2n ncore + V2n pcore

B(N2, Z) =B(N2, Z2) + E2p + V2p ncore + V2p pcore

B(N, Z) =proton

B(N2, Z

δVpn(N,Z)

2) + E2n + E2p + V2n 2p+ V2n ncore + V2n pcore + V2p ncore + V2p pcore

Vpn (N, Z) =1V2n 2p4

N-2 neutrons
Z-2 protons in the coreδVpn: interaction energy of two neutrons and two protons in the valence space

Pairing in mean-field theory
pairing (superconductivity) : nucleon pairs get additional binding

odd-even staggering (OES)
1)N[B(N2((3)
n (N, Z) =1exp
n (N, Z) = [21, Z)

2B(N, Z) + B(N + 1, Z)](3)n (N + 1, Z) +(3)n (N1, Z)]

Kortelainen et al., Phys. Rev. C 82, 024313 (2010)

theoretical pairing gap (average pairing gap)

neutron pairing gap (MeV)n (⇢) =3Trh̃n [⇢, ⇢˜n ]⇢nTr⇢n⇢) =n (˜2mixedvolumesurfaceexp(AME2012)

PbSn

2.5
2Trh̃n [⇢, ⇢˜n ]˜⇢nTr˜⇢n
1.5

1.5

110.5
0.5005060

7080

90N100110120

100 110 120 130 140 150 160 170 180
NProblems
p pairing gap: not an experimental observable
p multiple definitions both in OES and pairing gaps
p time-reversal symmetry: broken in OES, pairing gap is from even-system

Spontaneous symmetry breaking
continuous symmetry of the system S:[Ĥ, Ŝ] = 0SSB: ground state does not have the symmetry of the system

infinite degeneracy of ground state
zero-energy mode of excitation (Nambu-Goldstone modes/bosons)appears as a transition between degenerated ground states.
Nambu (1960), Goldstone (1961)
systembroken symmetry

ground state

Nambu-Goldstone
mode (boson)magnetrotationspin alignment

spin-wave (magnons)

crystaltranslation/rotation

crystalvibration (phonons)

superconductor

gauge symmetry

BCS state(massive gauge field due toAnderson-Higgs mechanism)

QCD

chiral symmetry(approximate)quark-antiquark
condensation

pions(with small mass)

Gauge symmetry breaking
pairing (superconductivity) : gauge symmetry breaking, NG mode(pairing rotation)
superconducting statein gauge space

NG mode excitation (pairing rotation)(particle number projection)(N mixed)

E(N ) =1
(N2JN

symmetry-restored state(Laboratory frame)
N=N0±6
N=N0±4
N=N0±2
N=N0N0 ) 2

pairing rotational moment of inertia: magnitude of pairing collectivity
deformed state
in coordinate space

NG mode excitation (rotation)(angular momentum projection)x’x

J=6z’y’symmetry-restored state(Laboratory frame)(J mixed)~J(J + 1)
2Jrot2E(J) =

J=4
J=2
J=0y

moment of inertia (or E(21+)): magnitude of quadrupole collectivityzPairing observables: pairing rotational MOI
Pairing rotation in isotopes
E(N +moment of inertia

pairing rotational energy( N )2N, Z) = E(N, Z) + n (N, Z) N +2Jnn (N, Z)

2E(N, Z) + E(N 2, Z)
2n (N, Z)=44
Brink and Broglia “Nuclear Superconductivity”
pairing rotation measured from 116Sn
Jnn1 (N, Z) =E(N + 2, Z)

review: Broglia et al., Phys. Rep. 335, 1(2000)

pairing rotational moment of inertia
p quantity from gauge symmetry breaking
p double binding-energy differences (inverse of empirical shell gaps(δ2n,2p))
p involves even-even systems only
p change with particle number

Theoretical description of NG mode
Symmetry-broken state
Density Functional Theory (DFT)Skyrme HFB (HFBTHO, UNEDF1-HFB)
p 20 harmonic oscillator basis
p axial deformation
p pairing superconductivity
p UNEDF1-HFB: Schunck, J. Phys. G 42,034024 (2015)

NG mode excitation
Quasiparticle Random-Phase Approximation (time-dependent DFT)
22[ĤQRPA , P̂NG ] = i~ !NG
JTV Q̂NG = 0[ĤQRPA , Q̂NG ] =moment of inertia from QRPA: Thouless-Valatin inertiaiJTVP̂NG

PNG: particle number operator

Thouless and Valatin, Nucl. Phys. 31 (1962)211

RRI
I
JTV = 2PNG
(A + B) 1 PNG+ 2PNG(A B) 1 PNG
efficient solution based on linear response theory: Finite-amplitude method (FAM)
Nakatsukasa et al., Phys. Rev. C 76, 024318 (2007)
NH, Phys. Rev. C 92, 034321 (2015)

Pairing rotation in single-closed shell
NH and Nazarewicz, Phys. Rev. Lett. 116, 152502 (2016)broken symmetry: neutron U(1), Δn≠0 or proton U(1) Δp≠0pairing density functional (strength fitted to 120Sn and 92Mo gaps)

η=0 (volume pairing) 0.5(mixed pairing) and 1 (surface pairing)1 (⌧ )
1
⌧ (r) = V0
2⇢0 (r)⌘|˜⇢⌧ (r)|2⇢c

p experimental MOI: sensitive to the symmetry breaking
p pairing rotational assumption is not valid at next to magic numbers (experimental values)
p contribution from residual Coulomb suppresses the MOI (ph Coulomb force ~ Z2)

Shell gap and pairing rotational moment of inertia
Experimental pairing rotational moment of inertia
Jnn (N ) =E(N + 2)

42E(N ) + E(N2)empirical shell gap: difference of 2n/2p separation energies
2n (N, Z) = E(N + 2, Z)

2p (N, Z) = E(N, Z + 2)

2E(N, Z) + E(N 2, Z) = S2n (N, Z)
2E(N, Z) + E(N, Z 2) = S2p (N, Z)

Jnn1 (N ) =12n (N )4shell model picture: weak pairing(before symmetry breaking)Lunney et al., Phys. Rep. 75, 1021(2003)
S2n (N + 2, Z)
S2p (N, Z + 2)

Jpp1 (N, Z) =12p (N, Z)
4collective pairing picture: strong pairing(after symmetry breaking)

S2n(N+2,Z)/2
S2n(N,Z)/2δ2n(N,Z)/2
Epairrot (N ) =size of magic shell gap

1( N )22Jnn (N )measure of gauge symmetry breaking

Pairing NG modes in doubly open-shell nuclei
broken symmetry: neutron and proton U(1), Δn≠0 and Δp≠0

[ĤHFB , N̂n ] 6= 0

[ĤHFB , N̂p ] 6= 0

NG modes as QRPA eigenmodes are neutron-proton mixed
formal theory: Marshalek, Nucl. Phys. A 275,416 (1977)
first calculation: NH, Phys. Rev. C 92,034321 (2015)[ĤHFB , N̂1 ] 6= 0

[ĤHFB , N̂2 ] 6= 0N̂1 = N̂n cos ✓ + ↵N̂p sin ✓N̂2 =N̂n sin ✓ + ↵N̂p cos ✓( N1 ) 2( N2 ) 2E(N, Z) = E(N0 , Z0 ) + 1 (N0 , Z0 ) N1 + 2 (N0 , Z0 ) N2 ++2J1 (N0 , Z0 ) 2J2 (N0 , Z0 )( N )22( N )( Z)( Z)2Z) =++2Jnn (N, Z)
2Jnp (N, Z)
2Jpp (N, Z)

Epairrot (N +N, Z +Jnp1 (N, Z) =E(N + 2, Z + 2)=

E(N + 2, Z)
4Vpn (N + 2, Z + 2)

E(N, Z + 2) + E(N, Z)

note: no neutron-proton pairing

Pairing rotation in doubly open-shell nuclei
NH and Nazarewicz, Phys. Rev. Lett. 116, 152502 (2016)
N=100

Z=68

Stoitsov et al (2003)

Wang et al., PRC90,014312(2014)

p simultaneous agreement of three pairing rotational moments of inertia
in prolately deformed open-shell nuclei
evidence for mixing of neutron and proton NG modes
p some sensitivity to density dependence in neutron-rich (proton-deficient) nuclei

δVpn and pairing rotational moment of inertia
Experimental pairing rotational moment of inertia
Jnp1 (N, Z) ==E(N + 2, Z + 2)

E(N + 2, Z)
4

E(N, Z + 2) + E(N, Z)

Vpn (N + 2, Z + 2)

Stoitsov et al., Phys. Rev. C 98,132502 (2007)

shell model picture(before symmetry breaking)neutroncollective pairing picture(after simultaneous symmetry breaking)protonδVpn(N,Z)
Epairrot (N, Z) =interaction energy ofvalence 2n and 2p( N )22( N )( Z)( Z)2++
2Jnn (N0 , Z0 ) 2Jnp (N0 , Z0 ) 2Jpp (N0 , Z0 )measure of simultaneous
gauge symmetry breaking of neutron and proton

Pairing picture of binding energy differences
magic nuclei
signature of magic number
valence proton-neutroninteraction energy

pairing collectivity

neutronprotonE(21+)

doubly magic nuclei

208Pb: E(2+)=4.085 MeV

proton h9/2-h11/2ph excitationδVpn(N,Z)

δ2n(N,Z)/2E(2+)

superconducting nuclei

deformed nuclei

excitation of two NG modes

excitation of NG mode

Epairrot (N, Z) =( N )22( N )( Z)( Z)2++2Jnn (N0 , Z0 ) 2Jnp (N0 , Z0 ) 2Jpp (N0 , Z0 )1
Jnp1 (N, Z) =2n (N )
41
Jpp1 (N, Z) = 2p (N, Z)
4
Jnn1 (N ) =Vpn (N + 2, Z + 2)

magnitude of pairing collectivity(gauge symmetry breaking)E(2+1)⇠

3Jrot

quadrupole collectivity

δVpnempirical shell gap

NH and Nazarewicz, Phys. Rev. Lett. 116, 152502 (2016)

magnitude of quadrupole collectivity(deformation)

Extending pairing functional
NH, arXiv:1706.06547

we have used simple pairing functionals because of lack of observables
conventional pairing functional
general pair-density-bilinear form of the isovector pairing functional
Perlinska et al., Phys. Rev. C 69, 014316 (2004)

constraints on the coupling constants from local gauge invarianceomitting the tensor-pairing terms for simplicity,

one new coupling constantExtending pairing functional
NH, arXiv:1706.06547

coupling constant (Cρ) fitted to the pairing rotational MOI in 120Sn
for selected values of CΔρ

SnPb

improvement of the pairing rotational MOI with pair-density derivative terms

Pairing gaps
Trh̃n [⇢, ⇢˜n ]⇢nn (⇢) =Tr⇢n

SnNH, arXiv:1706.06547

Trh̃n [⇢, ⇢˜n ]˜⇢n⇢) =n (˜Tr˜⇢nPbp pairing gaps increase with pair-density derivative terms
p two average pairing gaps are not consistent with pair-density derivative terms

OESNH, arXiv:1706.06547

E(Nodd ) ⇡1
[EHFB (Nodd21) + Eqp (Nodd1) + EHFB (Nodd + 1) + Eqp (Nodd + 1)]

1(3)(N)⇡[Eqp (Neven + 2) + 2Eqp (Neven ) + Eqp (Nevenevenn4Sn2)]

Pbp OES from qp energies is consistent with the experimental dataSummaryfirst systematic calculation of pairing rotational MOI in doubly open-shell nuclei
Evidence of neutron and proton mixed pairing Nambu-Goldstone modes

p Double-binding energy differences of even-even nuclei
p pairing rotational MOI: double binding-energy differences
p shell gap δ2n and proton-neutron interaction energy δVpn
p Pairing functionals with derivatives of local pair densities
p improve the systematic values of pairing rotational MOI
p pairing gap becomes large and does not correspond to OES

Collaborator: Witek Nazarewicz (MSU/FRIB)
References: NH and Nazarewicz, Phys. Rev. Lett. 116, 152502 (2016)
NH, Phys. Rev. C 92, 034321 (2015)
NH, arXiv:1706. 06547, submitted to J. Phys. G