1. Isospin symmetry, Saclay, April 2011 Measuring isospin mixing from E1 and E2 transitions P. Van Isacker, GANIL, France Isospin mixing from E1 transitions Collaboration: York, Argonne, Edinburgh, Kolkata Isospin mixing from E2 transitions Collaboration: G. de France, E. Clément, A. Dijon,…

2. Isospin mixing from E1 or E2 Motivation: Determine (0h  ) isospin mixing in N≈Z nuclei. Measurement of B(E1)s between states in a T=1/2 doublet and a T=3/2 quadruplet. B(E2)s between states in a T=0 singlet and a T=1 triplet. E1: See PRC 78 (2008) 024301. E2: See LoI-5 for SPIRAL-1 (11/03/2010, Dijon & de France). For A=38. Isospin symmetry, Saclay, April 2011

3. Isospin selection rule for E1s Internal E1 transition operator is isovector: Selection rule for N=Z (T z =0) nuclei: No E1 transitions are allowed between states with the same isospin. L.E.H. Trainor, Phys. Rev. 85 (1952) 962 L.A. Radicati, Phys. Rev. 87 (1952) 521

4. Isospin symmetry, Saclay, April 2011 E1 transitions and isospin mixing B(E1;5 -  4 + ) in 64 Ge from: lifetime of 5 - level;  (E1/M2) mixing ratio of 5 -  4 + transition; relative intensities of transitions from 5 -. Estimate of minimum isospin mixing: E.Farnea et al., Phys. Lett. B 551 (2003) 56

5. Isospin analogue E1 transitions Isospin symmetry, Saclay, April 2011

6. Isospin symmetry is exact If isospin is an exact symmetry, the B(E1) values satisfy the following equations: Ten B(E1) values can be expressed in terms of four (J,T)-reduced matrix elements M 11, M 13, M 31, M 33.

7. Isospin symmetry, Saclay, April 2011 Isospin symmetry is broken If isospin is broken, initial and final states in the T z =±1/2 nuclei become and Two additional unknowns: mixing angles for initial and final states.

8. Isospin symmetry, Saclay, April 2011 Isospin symmetry is broken If isospin is broken, the equations for B(E1) values are modified. For example, in T z =+1/2 nucleus: In T z =±3/2 nuclei: Etc.

9. Isospin symmetry, Saclay, April 2011 Elimination of our ignorance There are a possible ten measurable B(E1) values. There are two unknown mixing angles and four reduced matrix elements. Since the E1 matrix elements are difficult to calculate (our ignorance), we treat them as unknowns and try to eliminate them.

10. Isospin symmetry, Saclay, April 2011 Application to A=31 & 35 Five transitions have ‘known’ B(E1) values. The reduced matrix elements can be eliminated to give a relation between the two mixing angles: with

11. Schematic doublets & quadruplets Isospin symmetry, Saclay, April 2011

12. Real doublets & quadruplets Isospin symmetry, Saclay, April 2011

13. Isospin mixing correlation, A=31 Isospin symmetry, Saclay, April 2011

14. Isospin mixing correlation, A=35 Isospin symmetry, Saclay, April 2011

15. Isospin selection rule for E2s E2 transition operator is isoscalar + isovector: Rule for E2 transitions: E2 matrix elements vary linearly with T z within an isospin multiplet.

16. Isospin analogue E2 transitions Isospin symmetry, Saclay, April 2011

17. Isospin symmetry is exact If isospin is an exact symmetry, the B(E2) values satisfy the following equations: Four B(E2) values are expressed in terms of three (J,T)-reduced matrix elements M 0 11, M 1 11, M 1 01.

18. Isospin symmetry, Saclay, April 2011 Isospin symmetry is broken If isospin is broken, initial and final states in the T z =0 nucleus become and One additional unknown: mixing angle between the 2 + states.

19. Isospin symmetry, Saclay, April 2011 Isospin symmetry is broken If isospin is broken, the equations for B(E2) values are modified:

20. Isospin symmetry, Saclay, April 2011 Elimination of our ignorance We treat the reduced matrix elements as unknowns and eliminate them. This leads to with

21. Conclusion E1: Applied to A=31 and A=35 but not enough E1 transitions were measured. E2: See G. de France for status of A=38. Isospin symmetry, Saclay, April 2011