1. The role of isospin symmetry in medium-mass N ~ Z nuclei
Witold Nazarewicz
UWS, May 28, 2010
Isospin symmetry of atomic nuclei is explicitly broken by the charge-dependent interactions, primarily the Coulomb force. Within the nuclear density functional theory, isospin is also broken spontaneously. We propose a projection scheme rooted in a Hartree-Fock theory that allows the consistent treatment of isospin breaking in both ground and exited nuclear states. Applications of the new technique include excited high-spin states in medium-mass N=Z nuclei, such as superdeformed bands and many-particle-many-hole terminating states.

2. Isospin symmetry Introduced 1932 by Heisenberg
Protons and neutrons have almost identical mass: Dm/m = 1.4x10-3
Low energy np scattering and pp scattering below E=5 MeV, after correcting for Coulomb effects, is equal within a few percent in the 1S scattering channel.
Energy spectra of “mirror” nuclei, (N,Z) and (Z,N), are almost identical.
Tz component conserved!
(charge conservation)
charge independence
T is conserved!

4. The deuteron, being an isospin singlet, is antisymmetric under nucleons exchange due to isospin, and therefore must be symmetric under the double exchange of their spin and location. Therefore it can be in either of the following two different states: (i) Symmetric spin and symmetric under parity; (ii) Antisymmetric spin and antisymmetric under parity.
In the first case the deuteron is a spin triplet, so that its total spin S is 1. It also has an even parity and therefore even orbital angular momentum L ; The lower its orbital angular momentum, the lower its energy. Therefore the lowest possible energy state has S=1, L=0.
In the second case the deuteron is a spin singlet, so that its total spin S is 0. It also has an odd parity and therefore odd orbital angular momentum l. Therefore the lowest possible energy state has S = 0, L=1.
Since S=1 gives a stronger nuclear attraction, the deuterium ground state is in the S=1, L=0 state.

5. T=1 multiplet
Mirror Nuclei

6. W. Satula, J. Dobaczewski, W. Nazarewicz and M. Rafalski
Isospin-symmetry restoration within the nuclear density functional theory
W. Satula, J. Dobaczewski, W. Nazarewicz and M. Rafalski
Phys. Rev. C 81, (2010)
Isospin symmetry of atomic nuclei is explicitly broken by the charge-dependent interactions, primarily the Coulomb force. Within the nuclear density functional theory, isospin is also broken spontaneously. We propose a projection scheme rooted in a Hartree-Fock theory that allows the consistent treatment of isospin breaking in both ground and exited nuclear states. Applications of the new technique include excited high-spin states in medium-mass N=Z nuclei, such as superdeformed bands and many-particle-many-hole terminating states.

7. not affected by isospin projection
Symmetry energy is repulsive!
not affected by isospin projection
isospin projection lifts T=0, T=1 degeneracy

8. SD, becomes yrast 4p-4h about I=12
D. Rudolph et al. Phys. Rev. Lett. 82, 3763 (1999)

9. p-n excitations
across gap 28
p and n excitations
fp→g9/2
Why is only ONE band seen experimentally? HF calculations predict that this band becomes yrast around I=16

10. How to carry out isospin projection and mixing?
Isospin Mixing in Atomic Nuclei
within the nuclear density functional theory
W. Satula, J. Dobaczewski, W. Nazarewicz and M. Rafalski
Phys. Rev. Lett. 103, (2009)
UWS, July 13, 2009
(i) We expend the mean-field wave function in a good-isospin basis:
(ii) To assess the true isospin mixing, the total Hamiltonian (strong interaction plus the Coulomb interaction with the physical charge) is rediagonalized in the space spanned by the good-isospin wave functions:

11. Projection gives rise to a ~1 MeV shift of Band 2
HF Band 1 is a fairly pure T=1 structure; weakly affected by projection
HF Band 2 represents an almost equal mixture of T=0 and T=1 components
Projection gives rise to a ~1 MeV shift of Band 2

12. Description of terminating states (seniority isomers) above 40Ca
These specific states appear to have
almost spherical shapes; hence, the
correlations resulting from the angular-momentum restoration are practically negligible there. Hence, they can be regarded as extreme cases of an almost undisturbed single-particle motion, thus offering an excellent
playground to study effective interactions within the mean-field framework.
M. Lach et al., Eur. Phys. J. A 25, 1 (2005)

13. In N~Z:

14. Zalewski et al., Phys. Rev. C 77, 024316 (2008)
New functional with spin-orbit and tensor terms locally refitted, to reproduce the f7/2-f5/2 spin-orbit splitting in 40Ca, 48Ca, and 56Ni
Zalewski et al., Phys. Rev. C 77, (2008)
the modified functional yields
results that are very consistent with SM

15. Summary
We carried out the theoretical analysis of isospin breaking in nuclei around N=Z based on the density functional theory. We show that the spontaneous breaking of isospin symmetry, inherent to the mean-field approaches, limits the applicability of self-consistent theories, such as Hartree-Fock and density functional theory, to nuclear states with T=0. To remedy this problem, we propose a new isospin-symmetry restoration scheme based on the rediagonalization technique in good-isospin basis.
Applications of the isospin-projected DFT approach include the SD bands in 56Ni and terminating states in N~Z medium-mass nuclei. In both cases, the symmetry restoration remedies previously noted deficiencies of the HF method and significantly improves agreement with experiment.
The examples presented in this work primarily concern high-spin configurations in medium mass nuclei in which pairing correlations are expected to be weak. To be able to address theoretically a variety of low-spin phenomena and observables around the N=Z line, such as ground states of odd-odd nuclei, binding energy staggering, superallowed beta decays, and charge-exchange reactions, isovector and isoscalar pairing correlations must be incorporated and the proton-neutron symmetry of HF must be broken before isospin projection. Those will be subjects of our forthcoming studies.