Proton-Neutron Asymmetry in Exotic Nuclei

Proton-neutron asymmetry in exotic nuclei

M. A. Caprio Center for Theoretical Physics, Yale University, New Haven, CT

RIA Theory Meeting Argonne, IL April 4–7, 2006 Collective properties of exotic nuclei Extensive new set of nuclei – Proton-neutron imbalance – Changes in shell structure? Theoretical effort: Anticipate new collective phenomena – Signatures by which phenomena can be recognized – Estimates of where phenomena may occur

Most low-energy collective phenomena essentially isoscalar Similar proton and neutron distributions in the ground state Deformation arises from strong proton-neutron quadrupole interaction, which couples proton and neutron deformations

M. A. Caprio, CTP, Yale University Proton-neutron asymmetry in collective excitations Scissors mode Mixed symmetry states e.g., 156Gd e.g., 94Mo, 96Ru

N. Lo Iudice and F. Palumbo, Phys. Rev. Lett. 41, 1532 (1978). F. Iachello, Nucl. Phys. A 358, 89c (1981). F. Iachello, Phys. Rev. Lett. 53, 1427 (1984). D. Bohle et al., Phys. Lett. B 137, 27 (1984). N. Pietralla et al., Phys. Rev. Lett. 84, 3775 (2000). Dipole resonances Asymmetry in coupling Giant and pygmy Shears and chiral resonances rotation (odd-odd)

M. Goldhaber and E. Teller, Phys. Rev. 74, 1046 (1948). A. Zilges et al., Phys. Lett. B 542, 43 (2002). S. Frauendorf, Rev. Mod. Phys. 73, 463 (2001).

M. A. Caprio, CTP, Yale University Proton-neutron asymmetry in the ground state? Very neutron-rich nuclei Well-separated proton and neutron valence spaces ⇒ Reduced proton-neutron coupling strengths? ⇒ Larger role for proton-neutron asymmetry in ground state?

Nuclear structure Ground state properties, excitation modes, transition radiations (M1)

Mechanisms for triaxiality – Higher-order interactions in one-ﬂuid Hamiltonian (d†d†d†d˜d˜d˜ or cos2 3γ) P. Van Isacker and J. Chen, Phys. Rev. C 24, 684 (1981). – Higher-multipolarity pairs (hexadecapole) K. Heyde et al., Nucl. Phys. A 398, 235 (1983). – Unaligned proton and neutron symmetry axes A. E. L. Dieperink and R. Bijker, Phys. Lett. B 116, 77 (1982). J. N. Ginocchio and A. Leviatan, Ann. Phys. (N.Y.) 216, 152 (1992).

M. A. Caprio, CTP, Yale University The interacting boson model (IBM-1) Truncation to s-wave (J = 0) and d-wave (J = 2) nucleon pairs

s0 d+2 d+1 d0 d−1 d−2 States: Linear combinations of (s†)n(d† )n ···|0 0 +2 √ Operators (H, Lˆ, Tˆ, ...): Polynomials in b†be.g, Lˆ = 10[d† × d˜](1) Algebraic model: Constructed from elements of Lie algebra ( ) † † ... † U 6 : s0s0 s0d+2 d−2d−2

Dynamical symmetry⎧ ⎫ ⎫ ⎪ U(5) ⎪ ⎪ ⎪ ⎭⊃ SO(5) ⎪ ⎪ SO(6) ⎪ U(6) ⊃ ⎪ ( ) ⎪ ⊃ SO(3) ⊃ SO(2) ⎩⎪ SU 3 ⎭⎪ SU(3) Angular momentum – H constructed from Casimir (invariant) operators of subalgebra chain – Eigenstates have good quantum numbers – Problem exactly soluble (energies, eigenstates, transition MEs) – Deﬁnes distinct form of ground state conﬁguration (“phase”)

M. A. Caprio, CTP, Yale University Classical limit of the IBM-1 Quadrupole-deformed liquid drop

0 b

β, γ, θ1, θ2, θ3

M. A. Caprio, CTP, Yale University 2+ + + + + + 6 4 3 2 0 6+ 4+ 3+ 0+ 5+ + 0 4+ 4+ + + + + 3 4 2 0 2+ 2+ 4+ 2+ 6+ 0+

2+ + 2+ 4

+ 0+ 0+ 2 0+ UH5L SOH6L SUH3L

M. A. Caprio, CTP, Yale University x0 n H ˆ d Ehrenfest = .H eg .Gloe n .R en,Py.Rv C Rev. Phys. Deans, R. Lett. S. Rev. and Phys. Gilmore, Iachello, F. R. and Feng, H. Scholten, D. O. Dieperink, L. E. A. eodorder Second order First order First order Second =( d † · d 1 ˜ − Q

χ ξ =( ) h s N 1 † × n ˆ d d ˜ hs iga fteIBM-1 the of diagram Phase − + Landau

d ξ † × N 1 s ˜ 2 ) Q ( ˆ 2

) χ +

· χ 0 Q ( U ˆ

d χ

† H 5 × L d ˜ ) ( 2 23 1 ) ê 5 24(1981). 1254 ,

order First

χ ξ 44 (prolate) : (spherical) : 77(1980). 1747 , order Second x ↔ ↔

( γ -soft) (deformed) SU .A aro T,Yl University Yale CTP, Caprio, A. M. SO ↔

H 3 6 L (oblate) L 1 0 - ÅÅÅÅÅÅÅÅÅÅ è!!!! c 2 7 The proton-neutron interacting boson model (IBM-2) Proton and neutron pairs as separate boson species s† d† d† d† d† d† s† d† d† d† d† d† π,0 π,−2 π,−1 π,0 π,+1 π,+2 ν,0 ν,−2 ν,−1 ν,0 ν,+1 ν,+2 Proton Neutron H = επnˆdπ + ενnˆdν + κππQπ · Qπ + κπνQπ · Qν + κννQν · Qν + ···

Symmetric dynamical symmetries (isoscalar)

Uπν(5) SOπν(6) SUπν(3) SUπν(3) Spherical γ-soft Prolate Oblate

P. Van Isacker, K. Heyde, J. Jolie, and A. Sevrin, Ann. Phys. (NY) 171, 253 (1986). Asymmetric dynamical⎧ symmetries (isovector)⎫ ⎨⎪ ∗ ⎬⎪ SUπ(3) ⊗ SUν(3) ⊃ SUπν(3) Uπ(6)⊗Uν(6) ⊃ ⊃ SOπν(3) ⊃ SOπν(2) ⎩⎪ ∗ ⎭⎪ SUπ(3) ⊗ SUν(3) ⊃ SUπν(3)

A. E. L. Dieperink and R. Bijker, Phys. Lett. B 116, 77 (1982). A. Sevrin, K. Heyde, and J. Jolie, Phys. Rev. C 36, 2621 (1987). N. R. Walet and P. J. Brussaard, Nucl. Phys. A 474, 61 (1987).

M. A. Caprio, CTP, Yale University ∗ The SUπν(3) dynamical symmetry: Proton-neutron triaxial structure 4+ 4+ 3+ 3+ + + + + + + 4 4 4 4 4 4 2+ + 3+ 3+ 1 + + + + 2 2 2 2 H2Np-1, 2Nn-1L 0+ 0+

H2Np-4, 2Nn +2L H2Np+2, 2Nn-4L

6+ 6+ 6+ 6+

5+ 5+ 4+ 4+ 4+ 3+ 2+ 2+ 0+

H2Np,2NnL

M. A. Caprio, CTP, Yale University ∗ The SUπν(3) dynamical symmetry: Electromagnetic transitions + 6+ + 63 4 62 + 61

+ + 52 51

4+ 4+ 4+ 2 3 1 + 4+ + 42 3 41

+ 31 + 31 + 2+ 21 2 E2 + 2+ 21 2 + .1 m2 M1 01 ~ 0 N + 01

M. A. Caprio, CTP, Yale University Essential parameters and coordinates for the IBM-2 Collective coordinates (“order parameters”) βπ γπ θ1π θ2π θ3π βπ γπ βπ γπ ⇒ ϑ1 ϑ2 ϑ3 ⇒ βν γν θ1ν θ2ν θ3ν βν γν βν γν

Coherent state energy surface E (βπ,γπ,βν,γν,ϑ1,ϑ2,ϑ3) Four order parameters: βπ, γπ, βν, and γν Hamiltonian parameters (“control parameters”) êêêêêêêêêêê SU * H3L 1 1 χπ χν χπ χν pn H =( − ξ) (n π + n ν) − ξ (Qˆπ + Qˆν ) · (Qˆπ + Qˆν ) 1 N ˆd ˆd N2 1 χ = (χ + χ ) cV S π ν prolate-oblate tendency êêêêêêêêêêê 2 SU H3L χ = 1(χ − χ ) pn V 2 π ν proton-neutron asymmetry SO pn H6L cS Three control parameters: x ξ χ χ U H5L , S, and V pn SU pn H3L

* SU pn H3L

M. A. Caprio, CTP, Yale University Phase diagram of the IBM-2

êêêêêêêêêêê* SUpn H3L

SO pn H6L Second c order S x£ First order SU H3L Upn H5L pn

M. A. Caprio and F. Iachello, Phys. Rev. Lett. 93, 242502 (2004). Nπ/Nν = 1 M. A. Caprio and F. Iachello, Ann. Phys. (N.Y.) 318, 454 (2005).

M. A. Caprio, CTP, Yale University ∗ 3+ SUπν(3)-SU (3) transition 2+ 3+ πν + 6+ 1 gq 4+ gq 6+ * + 3+ SU H3L SU H3L 4 + -1.2 5+ 2 2+ 1+ 4+ 0+ q 6+ gg gg -1.6 5+ + 6+ 4 -2.0 3+ 2+ g 4+ Transition point 1.5 2+ + 0 cn=+0.4035 1.0 g r b 0.5 2+ 4+ + 1 + 3+ q£ 4 + 3+ 0.0 2+ + 1 2 £ q g 60 4+

L 6+ 2+

g 45 + + 0 e 6 5+ g£ d 30 6+ H 5+ 4+

r 15 4+

g 4+ 3+ 0 2+ 2+ 0+ -1.0 -0.5 0.0 0.5 1.0 cn=+0.8 cn g

M. A. Caprio, CTP, Yale University Proton-neutron symmetry energy (Majorana operator) Difference between proton and neutron deformation tensors † † (k) (k) Mˆ ≡−2 ∑ (dπ × dν) · (d˜π × d˜ν) k=1,3 † † † † (2) (2) +(sπ × dν − sν × dπ) · (s˜π × d˜ν − s˜ν × d˜π) 2 ≈|απ − αν| Major ingredient in realistic Hamiltonian = ε + ε +κ · + κ · + κ · + λ H πnˆdπ νnˆdν ππQπ Qπ πνQπ Qν ννQν Qν Mˆ Pair energy Quadrupole Symmetry Strength λ approximately known C – From scissors and mixed-symmetry energies MSS M – From 1 mixing ratios l λ l ≈ 5 κπν

M. A. Caprio, CTP, Yale University Effect of Majorana operator on phase transition

HaL l ê k pn = 0 HbL l ê k pn = 1 HcL l ê k pn = 10 -1.2

-1.6

-2.0

HdL HeL HfL 1.5

1.0 r b 0.5

0.0 60 HgL HhL HiL

L 45 g e

d 30 H

r 15 g 0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 cn – Phase transition to triaxiality delayed – Proton and neutron equilibrium coordinates values brought together – But also energy minimum at triaxial deformation shallower ∗ SUπν(3) triaxial ⇒ one-ﬂuid triaxial ⇒ one-ﬂuid γ-soft

M. A. Caprio, CTP, Yale University ∗ Effect of Majorana operator on SUπν(3) structure

6+ 6+ 6+

6+ 6+ 6+ 6+ 6+ 6+ 6+ 6+ 5+ 5+ 5+ + 5+ 5+ 6+ 6 + + + 5 5 4 6+ 4+ 4+ 5+ + + + 4 4 4 4+ 6+ 4+ 6+ 4+ + 4 3+ + + 3+ 3 3 + 4 2+ 4+ 2+ + + + 2 2 2 + 2 2+ 2+ 0+ 0+ 0+ 0+ * SU pn H3L l ê kpn = 1 l ê kpn = 10 SO H6L

Majorana strength

M. A. Caprio, CTP, Yale University Proton-neutron triaxiality Main signatures – Low-lying K =2 band but rotational L(L + 1) energy sequence – Unusual B(E2) strength pattern similar to classic rigid triaxial rotor (Davydov) – Anharmonically low K =4 band – Strong M1 admixtures – Orthogonal scissors mode But attenuated by Majorana operator ∗ SUπν(3) triaxial ⇒ one-ﬂuid triaxial ⇒ one-ﬂuid γ-soft

M. A. Caprio, CTP, Yale University + 1500 22 Ru + 41 L

V 1000 e

k 2+ H 1

E 500

0 3.0 L + 1 2 H

E 2.5 Lê + 1 4 H E

2.0 98 Ru 112 Ru 54 56 58 60 62 64 66 68 Neutron number

M. A. Caprio, CTP, Yale University Where might asymmetric structure be expected? Collective structure depends upon underlying single-particle structure – Energy spacing (subshell gaps?) – Ordering of orbitals (low j? high j?) – Radial wave functions (compact? diffuse?) Manifested in effective interactions – Pairing interaction (s-wave, d-wave, ...) – Multipole interaction (quadrupole, ...) – Symmetry energy (Majorana)

Qualitative estimate

Particle-like bosons ⇒ Prolate tendency Hole-like bosons ⇒ Oblate tendency

But very sensitive to underlying shell structure A. van Egmond and K. Allaart, Nucl. Phys. A 425, 275 (1984). T. Otsuka, Nucl. Phys. A 557, 531c (1993). -+c

M. A. Caprio, CTP, Yale University ∗ Prospective regions for SUπν(3) triaxial structure

60 OsêPt

50 Z 40

30 RuêPd

0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 N

M. A. Caprio, CTP, Yale University Conclusions In preparation for exotic beam facility... Have investigated proton-neutron asymmetric collective structure, within framework of IBM-2 Proton-neutron asymmetry – Suppressed by Majorana interaction – But could play role for nuclei far from stability ∗ SUπν(3) dynamical symmetry – Ideal limit, not likely to be reached – Illustrates basic characteristics of proton-neutron triaxiality Full collective analysis of two-ﬂuid system – Phase diagram – Nature of phase transitions – Signatures of asymmetric structure

M. A. Caprio, CTP, Yale University Bose-Fermi system – Odd mass or odd-odd nuclei bosonic core + unpaired nucleons – Odd nuclei will play major role in shell structure studies – Coupling to unpaired nucleon signiﬁcantly inﬂuences collective structure of even-even core (core polarization) – Interacting boson fermion model (IBFM)

M. A. Caprio, CTP, Yale University