Study of the tensor correlation using a mean-field-type model. Satoru Sugimoto Kyoto University

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1 Study of the tensor correlation using a mean-field-type model Satoru Sugimoto Kyoto University

2 The content. Introduction. Charge- and parity-projected Hartree- Fock method 3. pplication to the sub-closed oxygen isotopes 4. Summary

3 Introduction The correlations induced by the tensor force (tensor correlation) are important for structure of nuclei But the tensor force is not usually included in the mean-field-type calculation. We want to construct a mean-field-type model which can treat the tensor correlation and study the effect of the tensor correlation on structure of nuclei, which is probably different from those of the correlations by the central and LS forces. S [ ] σ σ Δ L = ; ΔS = () Y () r ( )

4 The correlation to be included Hartree-Fock cal. (MV(VC)+G3RS(VT, VLS)) E K V VC VT VLS 4O O O O O Single-particle (H-F) correlation 0d 3/ s / 0d 5/ 0p / 0p 3/ Proton j j V V T Neutron j j < > T < > In the simple HF calculation, the tensor correlation cannot be exploited. We need to include at least p-h correlation to exploit the tensor correlation. beyond mean field model 0d 3/ s / 0d 5/ 0p / 0p 3/ p-h correlation 0d 3/ s / 0d 5/ 0p / 0p 3/ Proton Neutron Proton Neutron 0p-0h p-h j j V V T j j < < T > >

5 Charge- and parity-projected Hartree-Fock method Sugimoto et al., Nucl. Phys. 740 (004) 77 Ogawa et al., PRC 73 (006) 03430

6 Charge- and parity-symmetry breaking mean field method Tensor force is mediated by the pion. Pseudo scalar (σ ) To exploit the pseudo scalar character of the pion, we introduce parity-mixed single particle state. (over-shell correlation) Isovector (τ) To exploit the isovector character of the pion, we introduce charge-mixed single particle state. Projection Because the total wave function made from such parity- and charge-mixed single particle states does not have good parity and a definite charge number. We need to perform the parity and charge projections. Toki et al., Prog. Theor. Phys. 08 (00) 903. Sugimoto et al., Nucl. Phys. 740 (004) 77. Ogawa et al., Prog. Thoer. Phys. (004) 75. cf. Bleuler, Proceeding of the international school of physics Enrico Fermi 36 (966) 464. τσ π V T V T τσ

7 Schematic example ( 4 He;=4,Z=) ( ) ( ( δ ν0 p) ) ( π0s) ( ν0s) α( π0s) β( ν0s) γ π0 p simple (0s) ( π0s) ( ν0s) = 6α β 0p-0h ( s) 0 p 4 ( ) β γ ( ν0s) ( π0 p) βγ ( π0s)( ν0s) ( π0 p) ( p) + 6α δ π0 ν α δ ν0 + 6γ δ ( π0 p) ( ν0 p) 4p-4h ( 0s) ( s) ν0 p ( ν0 ) p p-h ( ) γ( π0s)( s) ( π0 p) + α βδ π ν0 + αβ ν0 p-h ( )( 0 0 ) α ( π0s) ( π0 )( 0 p) + βγ δ s π ν p + γδ p ν 3p-3h (other 60 terms (4n, H, Li, Be)) mixed wave function 4 He, He, 0 - By combining the charge and parity mixings and projections, we can obtain a wave function which includes the correlations induced by the tensor force.

8 Symmetry projected Hartree-Fock. Projected wave ( Ζ ; ± ) Ψ = P mix ψ α jm = c π p ( Z) P ( ± ) π dθe function: Φ ( + τ ) 3 method intr i( Z) θ 0 charge projection intr intr ( Z ; ± ) ( Z ; ± ) ± Pˆ parity projection 3. Variation for single particle wave functions: δ. Intrinsic wave function: δ Ψ intr mix Φ = ψ α jm! α jm Ψ H Ψ Ψ. We take a Slater determinant made from single particle wave functions with parity and charge mixing as an intrinsic wave function.. We perform the parity and charge projections on the mixed parity and charge number wave function. 3. Taking a variation with the projected wave function, the charge-and parity-projected Hartree-Fock equation is obtained.! α jm ψ mix α jm = 0 charge- and parity-projected HF equation The method beyond mean filed model!

9 4π Charge- and parity- projected Hartree- Fock equation π izθ (0) dθe n tˆ x ˆ ˆ a x a a v x b a x b 0 a a v x α α α α αb a α x a α b a b= b= ( ) ψ (, θ) + ψ ( ) ψ ( θ) ψ (, θ) ψ ( ) ψ ( θ) ψ (, θ) ( ± Z) ( ( E E ) ( )) α ( x ) ; 0 θ ψ 0 a a θ ηba θ ψα b a b= ( ) ( ) ( x θ),, ( ) ( ) ( ( ) ( ) ) ( ( ) ) ( P) P P P ± n tˆ x, ˆ a ψ x (, ) a a v x α θ + ψα b a ψα θ ψ x b α a a θ ψα b b= b= ( ) ( ± ; ) ( ( E ) ( )) ( ) ( a ) = ( ) ( ) ( ) ( ) Z P P P P ( ; Z) E x, ± θ ψα θ (, ) a ηba θ ψα x b a θ n εabψα b= b= 3 3 ˆ 0 / ( )/ ˆ ˆ ˆ izθ iθ + τ iθ + τ P p, C θ e, B θ ψ e ψ, B θ ψ pe ˆ ψ θ a= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( P ) ( ) a ab αa αb ab αa αb n θ det B θ, n θ det B θ, E θ Φ HC ˆ ˆ θ Φ, E θ Φ HPC ˆ ˆ ˆ θ Φ ( ) ( ) ( ) vˆ x ψ θ ψ, θ a b ( x ) Sugimoto et al., Nucl. Phys. 740 (004) 77 cf. PPHF equation (Takami et al., Prog. Theor. Phys. 96 (996) 407) We solve the CPPHF equation selfconsistently with the gradient method. ( ) ( ) ( ) ( ) 0 0 P P 0 intr intr P intr intr 3 iθ( + τ )/ 0 0 ψ x; θ e ψ x B θ, η ψ tˆ ψ θ + ψ ψ vˆ ψ θ ψ θ ψ θ ψ θ ( ( ) ) ( ) ab ( ) ( ) α ( ) ( ) ( ) c αc αb αa αb αa αb αa αc αb ba b= c= ( ) ( ) ( ( ) ) ( ) ( ( ( ) ) ) ab ( ) ( P ) ( ) 3 P iθ( + τ )/ P P P P P P P ψ x; θ pe ˆ ψ x B θ, η ψ tˆ ψ θ + ψ ψ vˆ ψ θ ψ θ ψ θ ψ θ ( ) ( ) ( ) ( ) α a a ( P ) ( x ) α ( ) ( ) ( ) ( ) αa αb αa αa αa αc αb αc αc αb ba b= c= H-F part b a

10 The effect of the mixings and the projections ( 4 He) unit: MeV X T X TE <V 0 C > No <σ σv S C > <τ τv T C > <σ στ τv ST C > <V 0 T > <τ τv T T > <V 0 LS > <τ τv T LS > <V coul > PE sum KE E total rms R m (fm) P(-) m=0.6,b=h=0.0 Simple H-F PPHF CPPHF original Volkov Sugimoto et al., Nucl. Phys. 740 (004) 77. PPHF CPPHF ττ τ τ + τ + τ + τ τ π π, π. By performing the parity projection, we can obtain the correlation induced by the tensor force.. Performing the charge projection further, we get much more contribution from the tensor force. It becomes three times larger than the PPHF case. 3. Projection before variation is necessary to get the tensor correlation.

11 pplication to O isotopes H = Δ i + V + V + V + V + V E i= M C 3B T LS Coul CM MV G3RS We calculated sub-shell closed oxygen isotopes, 4 O, 6 O, O, 4 O and 8 O. The spherical symmetry is assumed. Only the couplings between the same j states (s / and p /, p 3/ and d 3/ ) are included. NN potential MV (PTP 64 (980) 608) for the central part. G3RS (PTP 39 (968) 9) for the tensor and LS forces. The attraction part of the 3 E part of the central force and the 3-body force are adjusted to reproduce the biding energy and the charge radius of 6 O. The strength of the τ τ part of the tensor force is changed by multiplying a numerical factor, x T to take into account correlations like <s/ s/ V T s/ d3/> which is not included in the present calculation, effectively. (only <j j V T j j> type correlations are included in the CPPHF method.) The strength of the LS force is multiplied by.

12 mix α jm Single particle wave function (spherical case) Single particle wave function with charge and parity mixing ψ Y Y ljm ljm r ( ) = φα ljπ ( r) Ylj m ( Ω) ζ( π) + φα ljν( r) Yljm ( Ω) ζ( ν) positve parity ( r) Y ( ) ζ( π) + φ ( r) Y ( Ω) ζ( ν) + φ Ω αljπ ljm αljν ljm ( j) = Ylm χ m = σ irˆ Yljm Yl 0 jm negative parity : eigenfunction of the total spin, and Gaussian basis expansion: φ Y ljm l+s have the same j but different parities. r = C N a r exp, ( a = a ( i=,... n)) n i l () ( ) r i αlj αlj l i a i ν i= i We expand each φ by the Gaussian basis with a geometric-series widths.

13 Results for 6 O x T E K V V T V LS HF CPPHF CPPHF By performing the parity and charge projection the potential energy from the tensor force becomes sizable value.

14 BE and R m -BE/ (MeV) Binding Energy per Particle Exp MV xt=.0 MV xt=.5 MV xt=.0 HF R m (fm) Matter Radius Exp MV xt=.0 MV xt=.5 MV xt=.0 HF Mass Number Mass Number

15 V and K per particle Kinetic and Potential Energy (MeV) Kinetic and Potential Energy per Particle K MV x T =.0 K MV x T =.5 K MV x T =.0 HF V C +V Coul +V 3B MV xt=.0 V C +V Coul +V 3B MV xt=.5 V C +V Coul +V 3B MV x T =.0 HF Mass Number V T and V LS (MeV) V T and V LS p / d 5/ d 3/ s / p 3/ V T x T =.0 V LS x T =.0 V T x T =.5 V LS x T =.5 V LS x T =.0 HF Mass number The potential energy of the tensor force behaves differently from those of the central and LS forces. It indicates that the tensor force affects the shell structure differently from the central and LS forces.

16 Mixing of the opposite parity components 0.3 xt=.0 Probability p/ s/ 0s/' 0p3/' 0p/' 0d5/' s/' 0d3/' j=/ states are important for the tensor correlation.

17 Wave function( 6 O, x T =.5) Wave Function (fm -3/ ) Wave Function (fm -3/ ) s / proton dominant Proton + Proton - Neutron + Neutron - P(-): 9.5% P(ν): 7.4% r (fm) 0p 3/ proton dominant Proton + Proton - Neutron + Neutron - P(+): 4.% P(ν): 3.6% r (fm) Wave Function (fm -3/ ) p / proton dominant P(+): 5.6% P(ν): 3.5% r (fm) Proton + Proton - Neutron + Neutron - Opposite parity components mixed by the tensor force have narrow widths. It suggests that the tensor correlation needs high-momentum components. HO wave function (fm -3/ ) Wave function square (fm -3 ) Harmonic oscillator wave function (b=.6 fm) r (fm) 4 He case s / neutorn with original Volkov No. s / proton s / neutron p / proton p / neutron R (fm) 0s 0p 0d s P(-)=6% P(π)=7%

18 Density 0.5 Density 4 O 0.5 Density 6 O density (fm -3 ) MV x T =.0 MV x T =.5 Gogny DS density (fm -3 ) MV x T =.0 MV x T =.5 Gogny DS r (fm) r (fm)

19 Charge Formfactor 6 O F(q) E-3 E-4 E-5 E-6 E-7 E-8 HF x T =.0 CPPHF x T =.0 CPPHF x T = The tensor correlation induce higher momentum component. q

20 Summary We make a mean-field model which can treat the tensor correlation by mixing parities and charges in single-particle states. (the CPPHF method) The opposite parity components induced by the tensor force is compact in size. (high-momentum component) The CPPHF calculation with the spherical symmetry shows that in the oxygen isotopes j=/ states are important for the tensor correlation.