K. KanekoK. KanekoKyushu Sangyo University, Fukuoka, JapanKyushu Sangyo University, Fukuoka, Japan

Particle-number conservation Particle-number conservation for pairing transition in finite for pairing transition in finite

systemssystems

A. SchillerMichigan State University, USA

Collaborator:

BCS Theory predictsTc=0.57Δ ～ 0.5MeV

Particle-number ProjectionK. Esashika and K. NakadaPRC72, 044303(2005)

MotivationOslo 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?

Ex

(3He,αγ ）、 (3He,3He’γ)

Pairing transition at finite temperaturePairing transition at finite temperature Pairing correlationPairing correlation is is fundamentalfundamental for many- for many-

fermion systems such as electrons in thefermion systems such as electrons in the superconducting metal, nucleons in nucleus, and superconducting metal, nucleons in nucleus, and quarks in the color superconductivity.quarks in the color superconductivity.

Infinite systemsInfinite systems show superfluid-to-normal show superfluid-to-normal sharp phase transitionsharp phase transition, which is described by , which is described by the BCS theory.the BCS theory.

In finite systemsIn finite systems, recent theoretical approaches , recent theoretical approaches demonstrate that demonstrate that thermal and quantum thermal and quantum fluctuations are importantfluctuations are important. The BCS theory fails . The BCS theory fails to describe the pairing transition. to describe the pairing transition.

Static path approximation (SPA) Static path approximation (SPA) with number projectionwith number projection

The SPA is a microscopic method for going beyond The SPA is a microscopic method for going beyond the mean-field approximation at finite temperature, the mean-field approximation at finite temperature, which avoids the sharp phase transition. which avoids the sharp phase transition.

The SPA is an efficient way compared with shell The SPA is an efficient way compared with shell model calculations, and can be applied to heavy model calculations, and can be applied to heavy nuclei.nuclei.

In finite systems such as nuclei, the SPA violates In finite systems such as nuclei, the SPA violates particle-number conservation. We need the particle-particle-number conservation. We need the particle-number projection in the SPA.number projection in the SPA.

In this talk, I present the particle-number In this talk, I present the particle-number projection in the SPA, and report the projection in the SPA, and report the numerical results and discussions.numerical results and discussions.

Out lineOut line

[I] Brief Review of the static path approximation [I] Brief Review of the static path approximation

and the particle-number projectionand the particle-number projection

[II] Numerical results and discussions in the heat[II] Numerical results and discussions in the heat

capacity and the pairing correlationcapacity and the pairing correlation

[III] Conclusion[III] Conclusion

SPASPA in monopole pairing model in monopole pairing model

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ˆˆˆ

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●Static path approximation

treat. todifficult isIt integral. functional

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energy clequasiparti '

operators clequasiparti ,

)ˆˆ(ˆˆ')(ˆ

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,)(2

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k

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/2

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k

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Effective mean-field equation

The SPA avoids the sharp phase transition in the BCS equation. 　

Δ(M

eV

)

T (MeV)Tc

BCS

Heat capacity

Thermal energy

G -

2

0

EHE

dT

daT

dT

dECv

G

2 - 2

T (MeV)

Cv

Δ2/G

(M

eV

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T (MeV)

The S shape is closely related to the drastic decrease of pairing correlation.

20 aTE

Particle-number projectionParticle-number projection

)ˆ(

0

)ˆ(

1

12

1ˆ

:projectionnumber -Particle

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Treatment of thermo field dyamics Treatment of thermo field dyamics

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operator (tilde) fictitious )~,~ (

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K. Tanabe and H. NakadaPRC71, 024314(2005)

Parameters

0Tat difference mass even-odd

alexperiment reproduce toas so chosen:,

int.orbit -spin withpot.SaxonWoods:

pn

k

GG

Fe and Mofor core Ca

Ybfor coreSn

outside taken are energies s.p. Negative

569440

172132

Model space

Numerical results in heat capacity and pairing correlation

Fe Mo Yb 5694172

Heat Capacity

T

Z

T

EC

ln2・ The S shape appears in the SPA.・ The particle-number projection enhances the S shape.

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

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

Number Projection in no pairing phase transition

S shape appears 　even though there is no pairing phase transition.

Pairing gap ln

2/1

nn

n

G

ZTG N

T

・ The SPA curve for 172Yb 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.

Thermal odd-even mass differenceThermal odd-even mass difference

)],1,(

),,(2),1,([2

)1(),,()3(

TNZE

TNZETNZETNZ

s

ss

N

n

TEi

ii

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i

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),()(/)(),,(

Partition function

Shifted thermal energy

Shell model calculations

sd-shell USD interaction

27,28,29Mg

K. Kaneko and M. HasegawaNPA740, 95(2004)

Tc

Derivative of thermal odd-even Mass difference with respect to temperature

))Mg()Mg((2

1)Mg( 292728

)3(

CCCTn

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.

Thermal odd-even mass difference for the neutron

R. Chankova et al.,PRC73, 034311(2006).

K. Kaneko et al.,PRC74, 024325(2006).

ConclusionConclusion The particle-number projection affects the S shape The particle-number projection affects the S shape

of the heat capacity in all of the nuclei, of the heat capacity in all of the nuclei, 172172Yb, Yb, 9494Mo, Mo, and and 5656Fe. Fe.

In the heavy nucleus In the heavy nucleus 172172Yb, the particle-number Yb, the particle-number projection enhances the S shape in the SPA, which is projection enhances the S shape in the SPA, which is regarded as a fingerprint of pairing transition. regarded as a fingerprint of pairing transition.

However, for the lighter nuclei However, for the lighter nuclei 9494Mo and Mo and 5656Fe, the S Fe, the S shape appears only in the calculation with particle-shape appears only in the calculation with particle-number projection, but not in the SPA alone. number projection, but not in the SPA alone.

The effective pairing gap in The effective pairing gap in 9494Mo is in good Mo is in good agreement with experimental thermal odd-even agreement with experimental thermal odd-even mass difference, which is regarded as a direct mass difference, which is regarded as a direct measurement of pairing correlations at finite measurement of pairing correlations at finite temperature. temperature.