The role of isospin symmetry in collective nuclear structure. Symposium in honour of David Warner

Transcription

1 The role of isospin symmetry in collective nuclear structure Symposium in honour of David Warner

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3 The role of isospin symmetry in collective nuclear structure Summary: 1. Coulomb energy differences as a probe of nuclear correlations (Mike Bentley) 2. Wigner energy in N=Z nuclei 3. Neutron-proton pairing

4 Wigner energy Extra binding energy of N=Z nuclei (cusp). Wigner energy B W is decomposed in two parts: B W = "W ( A) N " Z " d( A)# N,Z $ np W(A) and d(a) can be fixed empirically from "V np = 1 B 4 ( N,Z ) # B( N # 2,Z) # B( N,Z # 2) + B( N # 2,Z # 2) and similar expressions for odd-mass and odd-odd nuclei: d( A) " W ( A) # 47 A $1 MeV [ ] P. Möller & R. Nix, Nucl. Phys. A 536 (1992) 20 J.-Y. Zhang et al., Phys. Lett. B 227 (1989) 1 W. Satula et al., Phys. Lett. B 407 (1997) 103

5 Wigner energy and SU(4) symmetry

6 Supermultiplet or SU(4) model Wigner s explanation of the kinks in the mass defect curve was based on SU(4) symmetry. Symmetry contribution to the nuclear binding energy is "K( A)g (#,µ,$) = K( A) ( N " Z) N " Z + 8% N,Z & np + 6 ' [ % ] pairing SU(4) symmetry is broken by spin-orbit term. Effects of SU(4) mixing must be included. E.P. Wigner, Phys. Rev. 51 (1937) 106, 947 D.D. Warner et al., Nature Physics, to be published

7 Evidence for the Wigner cusp

8 Racah s SU(2) pairing model Assume pairing interaction in a single-j shell: $ j 2 JM ˆ J V pairing j 2 JM J = " 1 2 j +1 2 ( )g 0, J = 0 % & 0, J # 0 Spectrum 210 Pb:

9 Solution of the pairing hamiltonian Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell: j n "J n $ V ˆ pairing j n 1 "J = %g n %" 0 4 1#k<l ( )( 2 j % n %" + 3) Seniority υ (number of nucleons not in pairs coupled to J=0) is a good quantum number. Correlated ground-state solution (cf. BCS). G. Racah, Phys. Rev. 63 (1943) 367

10 Nuclear superfluidity Ground states of pairing hamiltonian have the following correlated character: Even-even nucleus (υ=0): S ˆ + + S + = " a ˆ m+ a ˆ m Odd-mass nucleus (υ=1): Nuclear superfluidity leads to Constant energy of first 2 + in even-even nuclei. Odd-even staggering in masses. ( ) n / 2 o, ˆ Smooth variation of two-nucleon separation energies with nucleon number. Two-particle (2n or 2p) transfer enhancement. + a ˆ m ( S ˆ ) n / 2 + o m>0

11 Two-nucleon separation energies Two-nucleon separation energies S 2n : (a) Shell splitting dominates over interaction. (b) Interaction dominates over shell splitting. (c) S 2n in tin isotopes.

12 Pairing with neutrons and protons For neutrons and protons two pairs and hence two pairing interactions are possible: 1 S 0 isovector or spin singlet (S=0,T=1): ˆ S + = $ m>0 a ˆ + + m" a ˆ m # 3 S 1 isoscalar or spin triplet (S=1,T=0): ˆ P + = # m>0 a ˆ + + m" a ˆ m "

13 Neutron-proton pairing hamiltonian The nuclear hamiltonian has two pairing interactions V ˆ pairing = "g ˆ 0 S + # S ˆ " " g ˆ 1 P + # P ˆ " SO(8) algebraic structure. Integrable and solvable for g 0 =0, g 1 =0 and g 0 =g 1. B.H. Flowers & S. Szpikowski, Proc. Phys. Soc. 84 (1964) 673

14 Quartetting in N=Z nuclei Pairing ground state of an N=Z nucleus: ( P ) n / 4 + o cos" S ˆ + # S ˆ + $ sin" P ˆ + # ˆ Condensate of α-like objects. Observations: Isoscalar component in condensate survives only in N~Z nuclei, if anywhere at all. Spin-orbit term reduces isoscalar component.

15 Model with L=0 vector bosons

16 Model with L=0 vector bosons Correspondence: S ˆ + + " b T =1 Algebraic structure is U(6). Symmetry lattice of U(6): ( ) # U T ( 3) ( ) $ U(6) " U 3 S % & SU 4 ' ( ) " SO 3 S # s + P ˆ + + " b T = 0 ( ) # SO T ( 3) Boson mapping is exact in the symmetry limits [for fully paired states of the SO(8)]. # p +

17 Masses of N~Z nuclei Neutron-proton pairing hamiltonian in nondegenerate shells: ˆ H F = #" j n ˆ j $ g ˆ 0 S + % S ˆ $ $ g ˆ 1 P + % ˆ j H F maps into the boson hamiltonian: H ˆ B = ac ˆ 2 SU 4 [ ( )] + b ˆ ( ) [ ] C 1 U S 3 C 1 U( 6) C 2 U( 6) C [ 2 SO T ( 3) ] H B describes masses of N~Z nuclei. + c 1 ˆ [ ] + c 2 ˆ P $ [ ] + d ˆ E. Baldini-Neto et al., Phys. Rev. C 65 (2002)

18 Masses of pf-shell nuclei Root-mean-square deviation is 254 kev. Parameter ratio: b/a 5.

19 Deuteron transfer in N=Z nuclei

20 Deuteron transfer in N=Z nuclei Deuteron-transfer intensity c T 2 calculated in sp-boson IBM based on SO(8). c 2 + T = [ N b +1]" B b TS 2 [ N b ]" A Ratio b/a fixed from masses in lower half of shell.

21 (d,α) and (p,3he) transfer

22 Liquid-drop mass formula Binding energy of an atomic nucleus: Z( Z "1) ( ) 2 B( N,Z) = a vol A " a sur A 2 / 3 " a cou #( N,Z) + a pai A 1/ 2 Fit to AME03 data base: a vol 16, a sur 18, a cou 0.71, a vsym 23, a pai 13 σ rms 2.93 MeV. N " Z " a A 1/ 3 vsym A C.F. von Weizsäcker, Z. Phys. 96 (1935) 431 H.A. Bethe & R.F. Bacher, Rev. Mod. Phys. 8 (1936) 82

23 Symmetry energy and neutron stars A proper description of neutron stars requires the knowledge of the symmetry energy: E (", x) = E (", x = 1 ) + S 2 # (")( 1$ 2x) 2, x = Z / A A consequence of the uncertainty [in S ν (ρ)] is that different models predict up to a factor 6 variation in the pressure of neutron-star matter near ρ 0. [This] accounts for the nearly 50% variation in the prediction of neutron star radii. J.M. Lattimer & M. Prakash, Astrophys. J. 550 (2001) 426 J.M. Lattimer & M. Prakash, Science 304 (2004) 536

24 Deficiencies of Weizsäcker formula Consistency of the Weizsäcker mass formula requires a surface-symmetry term. Derivation relies on the thermodynamics of an asymmetric two-component system: S v ( N " Z) 2 ( N " Z) 2 ( N " Z) 2 # "a 1+ S s A "1/ 3 vsym + a ssym /S v A 4 A 4A 4 / 3 W.D. Myers & W.J. Swiatecki, Ann. Phys. 55 (1969) 395 A. Bohr & B.R. M ottelson, Nuclear Structure II (1975) A.W. Steiner et al., Phys. Reports 411 (2005) 325 P. Danielewicz, Nucl. Phys. A 727 (2003) 233

25 Modified nuclear mass formula (I) Add surface-symmetry energy: Z( Z "1) B( N,Z) = a vol A " a sur A 2 / 3 " a cou ( ) 2 N " Z " a A 1/ 3 vsym A ( N " Z) 2 #( N,Z) + a ssym + a A 4 / 3 pai A 1/ 2 Fit to AME03 data base: σ rms 2.69 MeV.

26 Quantal effects & Wigner cusp The (N-Z) 2 dependence of the symmetry term arises in a macroscopic approximation. Quantal theories yield a dependence of the form [T is to be identified with N-Z /2]: T(T+1): isospin SU(2); T(T+4): supermultiplet SU(4). This suggests a generalisation of the form T(T+r), with r a parameter. N. Zeldes, Phys. Lett. B 429 (1998) 20 J. Jänecke & T.W. O Donnell, Phys. Lett. B 605 (2005) 87

27 Modified nuclear mass formula (II) Add Wigner energy : Z( Z "1) B( N,Z) = a vol A " a sur A 2 / 3 " a cou A 1/ 3 4T( T + r) " a vsym A 4T( T + r) #( N,Z) + a ssym + a A 4 / 3 pai A 1/ 2 Fit to AME03 data base: σ rms 2.39 MeV.

28 Modified nuclear mass formula (II)

29 Dependence on r

30 Conclusions & unresolved questions Back to basics: Wigner cusp from Wigner s supermultiplet model. Search for evidence of isoscalar pairing with deuteron transfer? Consider surface-symmetry and Wigner energy to determine the symmetry energy of nuclear matter.

31 Isospin Empirical observations: About equal masses of n(eutron) and p(roton). n and p have spin 1/2. Equal (to ~1%) nn, np, pp strong forces. This suggests introduction of isospin and the hypothesis of isospin symmetry: n : t = 1 2, m t = ; p : t = 1 2, m t = " 1 2 # t + n = 0, t + p = n, t " n = p, t " p = 0, t z n = 1 2 n, t z p = " 1 2 p W. Heisenberg, Z. Phys. 77 (1932) 1 E.P. Wigner, Phys. Rev. 51 (1937) 106

32 Shell corrections Observed deviations suggest shell corrections depending on n ν +n π, the total number of valence neutrons + protons (particles or holes). Shell corrections via the F-spin formalism of the neutron-proton interacting boson model. A simple parametrisation consists of two terms, linear and quadratic in F max =(n ν +n π )/2. E.D. Davis et al., Phys. Rev. C 44 (1991) 1655

33 Modified nuclear mass formula (III) Add F-spin shell corrections: Z( Z "1) B( N,Z) = a vol A " a sur A 2 / 3 " a cou A 1/ 3 4T( T + r) " a vsym A 4T( T + r) #( N,Z) + a ssym + a A 4 / 3 pai A 1/ 2 Fit to AME03 data base: σ rms 1.28 MeV. 2 " a f F max + a ff F max

34 Modified nuclear mass formula (III)

35 Correlations Volume-symmetry and surface-symmetry terms are strongly correlated: 120 a vsym 140 MeV 200 a ssym 300 MeV No significant improvement between 1995 and 2003 data!