PARTICLE-NUMBER CONSERVING MICROSCOPIC APPROACH AND APPLICATION TO ISOSPIN MIXING L. Bonneau, J. Le Bloas, H. Naidja, P. Quentin, K. Sieja (CENBG/Université Bordeaux 1) J. Bartel (IPHC/Université Louis Pasteur) J. Libert (IPNO), H. Lafchiev (INRNE Sofia/CENBG/IPNO) April 7-10, 2008
ESNT April 2008 2/18 INTRODUCTION MOTIVATIONS Description of GS and low-lying excited states of even-even nuclei, especially where particle-number conservation is important (weak pairing regimes near magic nuclei, in rotational bands...) Isospin mixing in GS of even-even nuclei (especially N Z ) and consequence for β ± -decay transition, especially superallowed 0 + 0 + Fermi matrix elements
ESNT April 2008 3/18 INTRODUCTION MODEL USED Microscopic, particle-number conserving approach based on self-consistent mean field: Higher Tamm-Dancoff Approach (developped by N. Pillet, P. Quentin and J. Libert, NPA 2002) Time-reversal, axial and parity symmetries not broken Conventional Skyrme interaction in the mean-field channel (SIII, SLy4...) Correlations: pairing: ordinary (T z = ±1) and n p (T z = 0); δ interaction (different strengths in T = 0 and T = 1 channels) RPA : 1p1h excitations; isoscalar Q Q interaction single-reference framework: no GCM-like correlations, no symmetry restoration
ESNT April 2008 4/18 INTRODUCTION PRESENT WORK Exploratory work in progress: restriction here to either pairing or RPA correlations (easier determination of the strength of the residual interaction) Calculation of isospin-mixing parameter assuming mixing of T = T z and T = T z + 1: formal developments and estimates
ESNT April 2008 5/18 HIGHER TAMM DANCOFF APPROACH BEYOND TAMM DANCOFF HTDA multiparticle-multihole configuration mixing based on the Hartree Fock solution: we do our best at lowest order (HF) and hope that the p-h expansion series converges fast enough to be tractable we can keep track of the types of correlations appearing in the solution: usual pairing (n n and p p), n p pairing, RPA... (see N. Pillet s talk)
ESNT April 2008 6/18 HIGHER TAMM DANCOFF APPROACH IMPLEMENTATION 1 Determine the HF solution Φ 0 2 Build the the many-body basis by creating mp mh excitations on top of Φ 0, retaining those corresponding to selected correlations 3 Diagonalize the residual interaction in the many-body basis using Lanczos method to get the GS solution and the low-lying excitations
HIGHER TAMM DANCOFF APPROACH MANY-BODY BASIS: TYPES OF CORRELATIONS n n and p p pairing ( T z = 1 channel, i.e only a part of T = 1 pairing): essentially excited pairs (up to two or three in practice for each type of nucleon) a jπ a jπ Φ 0 i a iν a īν n p pairing: (mpmh) n (m pm h) p (m, m 2) pair excitations; e.g, for m = m = 1: j a ανa βπ a iνa jπ Φ 0 with K i + K j = 0 and K α + K β = 0 RPA : (1p1h) n (1p1h) p excitations ESNT April 2008 7/18
ESNT April 2008 8/18 HIGHER TAMM DANCOFF APPROACH MANY-BODY BASIS: TIME-REVERSAL INVARIANCE mp mh Slater determinants invariant under T kept as they are the others are combined to get a T invariant state: 1 ( ) Φ + T Φ 2 provided that T Φ is orthogonal to Φ (OK for the retained configurations)
ESNT April 2008 9/18 HIGHER TAMM DANCOFF APPROACH RESIDUAL INTERACTION Definition in our approach: H Sk K + v Sk = K + v Sk Φ v Sk Φ + v }{{} Sk v Sk + Φ v Sk Φ }{{} H sm v res v Sk : one-body reduction of v Sk for a given Slater determinant Φ (e.g, HF solution Φ 0 ) Property (by construction): Φ v res Φ = 0 Approximation: v res replaced by w = A δ + 2 µ= 2 χ 2 Q 2,µ Q 2, µ
ESNT April 2008 10/18 HIGHER DANCOFF APPROXIMATION STRENGTHS OF RESIDUAL INTERACTIONS δ for ordinary pairing: V (p) 0 = 0.9V (n) 0 to take Coulomb anti-pairing into account approximately measure of pairing correlations: difference between v res in a full calculation and a blocked one (pairing gap assumed to arise from breaking of lowest-energy Cooper pair) = v res full v res blocked fit of V (n) 0 to experimental odd-even mass difference (3-point formula): χ 2 ( δ (3) ) 2 minimization with nuclei respect to V (n) 0, with δ(3) = (δ (3) + + δ (3) )/2 and δ (+) 3 = 1 ( ) `2E(N +1) E(N +2) E(N), δ 3 = 1 `2E(N 1) E(N) E(N 2) 2 2
ESNT April 2008 10/18 HIGHER DANCOFF APPROXIMATION STRENGTHS OF RESIDUAL INTERACTIONS Q Q for RPA : w = 2 µ= 2 χ 2 Q 2,µ Q 2, µ exploratory stage no attempt to fit simple estimete χ 2 A 7/3 from Kisslinger and Sorensen (1960)
ESNT April 2008 11/18 HIGHER DANCOFF APPROXIMATION RESULTS Isoscalar quadrupole resonance in 40 Ca µ 1 (MeV.fm 4 ) 18000 16000 14000 40 Ca 12000 10000 8000 6000 4000 2000 0 0 10 20 30 40 50 60 70 80 90 100 E n (MeV) m 1 (MeV.fm 4 ) 90000 80000 70000 60000 50000 40000 30000 20000 15 20 25 30 35 40 45 50 X(MeV) P. Quentin, H. Naidja et al., Int. J. Mod. Phys. E (2007)
ESNT April 2008 12/18 HIGHER DANCOFF APPROXIMATION RESULTS Dynamical moment of inertia in 192,194,196 Pb: better behavior than non-projected HFB in weak pairing regime Lafchiev, Libert et al., Phys. Rev. C
ESNT April 2008 13/18 HIGHER DANCOFF APPROXIMATION RESULTS Variation of the GS correlation energy as a function of the ratio x of δ pairing strengths in T = 0 and T = 1 channels for some N = Z even-even nuclei:! E corr (MeV) 0-0.5-1 -1.5 64 Ge 68 Se 72 Kr 76 Sr 80 Zr (a) -2 0 0.5 1 1.5 2 x L. B., P. Quentin and K. Sieja, PRC (2007)
ESNT April 2008 14/18 ISOSPIN MIXING MOTIVATIONS Probe isospin content of GS (isospin-mixing parameter) Test ground for Standard Model of elementary particles: isospin-breaking correction to Fermi matrix element in superallowed 0 + 0 + β transitions
ESNT April 2008 15/18 ISOSPIN MIXING ISOSPIN SYMMETRY BREAKING Sources: interaction (Coulomb, charge-independence breaking), wave function Measure of degree of symmetry breaking: mixing parameter α such that Ψ (1 α) T T z µ + α T +1 T z µ (1) α 2 = Ψ T2 Ψ T (T + 1) 2(T + 1) where T = T z and T z = (N Z )/2. (2)
ESNT April 2008 16/18 ISOSPIN MIXING EXPECTATION VALUE OF T 2 IN THE HTDA GROUND STATE HTDA ground state: Ψ = i χ i Φ i = χ 0 Φ 0 + i 0 Φ i Dominant term in Ψ T 2 Ψ : ph-diagonal contribution Ψ T 2 Ψ diag = A 2 + T 2 z i χ i 2 n p 2 space spin n Φ i p Φ i
ESNT April 2008 17/18 ISOSPIN MIXING EXPECTATION VALUE OF T 2 IN THE HTDA GROUND STATE Estimate of α 2 for N = Z : assuming identical n and p single-particle states and including only excited pairs in Ψ, we get the following upper limit estimate α 2 (T z=0) 1 χ 0 2 In practice, the actual value of α 2 is close to that estimate.
ESNT April 2008 18/18 CONCLUSION PERSPECTIVES Encouraging results in the various applications Need for guidance in the fit of the strengths of residual interactions in T = 0 and T = 1 channels Calculate isospin mixing parameter with all pairing and RPA correlations Proper description of odd-odd nuclei for an accurate calculation of superallowed Fermi 0 + 0 + transition matrix elements