1. K. Kaneko Kyushu Sangyo University, Fukuoka, Japan Particle-number conservation for pairing transition in finite systems A. Schiller Michigan State University, USA Collaborator:

2. BCS Theory predicts T c =0.57Δ ～ 0.5MeV Particle-number Projection K. Esashika and K. Nakada PRC72, 044303(2005) Motivation Oslo group A. Schiller et al., PRC63, 021306(R)(2001). ０．５ Question: Is the S-shape a signature of the breaking of nucleon Cooper pairs? ExEx (3He,αγ ）、 (3He,3He ’ γ)

3. Pairing transition at finite temperature Pairing correlation is fundamental for many- fermion systems such as electrons in the superconducting metal, nucleons in nucleus, and quarks in the color superconductivity. Pairing correlation is fundamental for many- fermion systems such as electrons in the superconducting metal, nucleons in nucleus, and quarks in the color superconductivity. Infinite systems show superfluid-to-normal sharp phase transition, which is described by the BCS theory. Infinite systems show superfluid-to-normal sharp phase transition, which is described by the BCS theory. In finite systems, recent theoretical approaches demonstrate that thermal and quantum fluctuations are important. The BCS theory fails to describe the pairing transition. In finite systems, recent theoretical approaches demonstrate that thermal and quantum fluctuations are important. The BCS theory fails to describe the pairing transition.

4. Static path approximation (SPA) with number projection The SPA is a microscopic method for going beyond the mean-field approximation at finite temperature, which avoids the sharp phase transition. The SPA is a microscopic method for going beyond the mean-field approximation at finite temperature, which avoids the sharp phase transition. The SPA is an efficient way compared with shell model calculations, and can be applied to heavy nuclei. The SPA is an efficient way compared with shell model calculations, and can be applied to heavy nuclei. In finite systems such as nuclei, the SPA violates particle-number conservation. We need the particle- number projection in the SPA. In finite systems such as nuclei, the SPA violates particle-number conservation. We need the particle- number projection in the SPA. In this talk, I present the particle-number projection in the SPA, and report the numerical results and discussions. In this talk, I present the particle-number projection in the SPA, and report the numerical results and discussions.

5. Out line [I] Brief Review of the static path approximation [I] Brief Review of the static path approximation and the particle-number projection and the particle-number projection [II] Numerical results and discussions in the heat [II] Numerical results and discussions in the heat capacity and the pairing correlation capacity and the pairing correlation [III] Conclusion [III] Conclusion

6. SPA in monopole pairing model

7. ● Hubbard-Stratonovich transform ● Static path approximation No two-body interaction

9. Effective mean-field equation The SPA avoids the sharp phase transition in the BCS equation. Δ(MeV) T (MeV)Tc BCS

10. Heat capacity Thermal energy T (MeV) CvCv Δ 2 /G (MeV) T (MeV) The S shape is closely related to the drastic decrease of pairing correlation.

11. Particle-number projection Particle-number projection

12. Treatment of thermo field dyamics K. Tanabe and H. Nakada PRC71, 024314(2005)

13. Parameters Model space Numerical results in heat capacity and pairing correlation

14. Heat Capacity ・ The S shape appears in the SPA. ・ The particle-number projection enhances the S shape.

15. ・ The SPA result does not show the S shape. ・ The number projection produces the S shape.

16. ・ The results are a similar to those of 94 Mo. ・ The number projection shows a more substantial increase compared with those of the heavier nuclei, 172 Yb and 94 Mo.

17. Number Projection in no pairing phase transition S shape appears even though there is no pairing phase transition.

18. Pairing gap ・ The SPA curve for 172 Yb drastically drops down around the temperature 0.5 MeV, but for the other lighter nuclei they decrease gradually. ・ The particle-number projection makes it steeper than the slope of the SPA curve.

19. Thermal odd-even mass difference Partition function Shifted thermal energy

20. Shell model calculations sd-shell USD interaction 27,28,29 Mg K. Kaneko and M. Hasegawa NPA740, 95(2004) Tc

21. Derivative of thermal odd-even Mass difference with respect to temperature Odd-even difference of heat capacities Tc The derivative of the thermal odd-even mass difference is identical with the odd-even difference of heat capacities. This peak shows odd-even difference of heat capacity corresponding to the S shape.

22. Thermal odd-even mass difference for the neutron R. Chankova et al., PRC73, 034311(2006). K. Kaneko et al., PRC74, 024325(2006).

23. Conclusion The particle-number projection affects the S shape of the heat capacity in all of the nuclei, 172 Yb, 94 Mo, and 56 Fe. The particle-number projection affects the S shape of the heat capacity in all of the nuclei, 172 Yb, 94 Mo, and 56 Fe. In the heavy nucleus 172 Yb, the particle-number projection enhances the S shape in the SPA, which is regarded as a fingerprint of pairing transition. In the heavy nucleus 172 Yb, the particle-number projection enhances the S shape in the SPA, which is regarded as a fingerprint of pairing transition. However, for the lighter nuclei 94 Mo and 56 Fe, the S shape appears only in the calculation with particle- number projection, but not in the SPA alone. However, for the lighter nuclei 94 Mo and 56 Fe, the S shape appears only in the calculation with particle- number projection, but not in the SPA alone. The effective pairing gap in 94 Mo is in good agreement with experimental thermal odd-even mass difference, which is regarded as a direct measurement of pairing correlations at finite temperature. The effective pairing gap in 94 Mo is in good agreement with experimental thermal odd-even mass difference, which is regarded as a direct measurement of pairing correlations at finite temperature.