Relativistic Impulse Approximation Analysis of Unstable Nuclei: Calcium and Nickel Isotopes

Transcription

1 Relativistic Impulse Approximation Analysis of Unstable Nuclei: Calcium and Nickel Isotopes Kaori Kaki Department of Physics, Shizuoka University 21 February 29

2 Relativistic Impulse Approximation (RIA) optical potentials Dirac equation for a projectile proton scattering from a target nucleus given by the optical model: [ p/ m Uˆ ( r)] ψ ( r) =, μ μ p/ = γ μ p, p = ( E, p) the momentum space Dirac equation 1 3 γ E γ p' m ψ( p') d puˆ ( p', p) ψ( p) = 3 ( ) ( 2π )

3 by relativistic analog of non-relativistic multiple scattering theory, optical potential in coordinate space: Uˆ () r =< Φ t Φ > i i + <Φ tgt Φ > i j j i j A 1 <Φ ti Φ > G<Φ tj Φ > A 2 nd order term i 1 st order term j

4 the generalized RIA optical potential in the momentum space for the 1 st order term 3 1 ˆ dk ˆ q q U( p', p) = Tr M (, ', ) ˆ (, ) 3 pp p k p k+ ρ p k q 4 ( 2π ) dk ˆ q q Tr (, ', ) ˆ (, ) 3 M pn pk p k + ρn kq 4 ( 2π ) 2 2 ( k = ) optimal factorization : simple ˆ 1 ( ', ) Tr ˆ q q (, ', ) ˆ U p p = M pp p p ρ p ( q) Tr Mˆ q q (, ', ) ˆ pn p p ρn( q) tρ-form

5 density matrices ) ) ρ( q) = dre 3 iqr ρ( r) ) ) ( kq, ) = dre 3 iqr ρ( rk ( )) ρ each nuclear density distribution i ρ = ρ + γ ρ α r ) ) ρ 2 () r S() r V() r T() r scalar vector tensor

6 the 2 nd order optical potential in the momentum space 3 ˆ dk 1 ˆ qb qb U( p', p) = Tr ˆ 2 2 M( k, p', ) ρ( qb) (2 π ) ˆ 1 ˆ qa qa C( q, ) ( ) Tr ˆ b qa G k 3 M( p, k, ) ρ( qa) propagator correlation part q = p k a q = k p' b

7 propagator G( k) = ( k/ m γ E + i ) p k 1 A ε p ' ˆM ˆM q a 2 q a 2 q b 2 q b 2 correlation function ˆ ρ( q ) Cˆ ( q, q ) ˆ ρ( q ) b b a a

8 correlation function ˆ ρ( q ) Cˆ ( q, q ) ˆ ρ( q ) b b a a 3 3 irq a irq b r r ˆ ρ r ˆ ρ r a b a b a b = dr dre e f( ) ( ) ( ) 3 f() r = f e α α = 1 α r R 2 2 the same parameters as in J.D.Lumpe & L.Ray, Phys.Rev.C35(1987)14

9 .1 (fm 3 ) proton.1 (fm 3 ) neutron 4 Ca 4 Ca density distributions for Ca isotopes.5 ρ s.1 5 r(fm) 48 Ca 6 Ca 68 Ca 74 Ca 4 Ca.5 ρ s.1 5 r(fm) 48 Ca 6 Ca 68 Ca 74 Ca 4 Ca relativistic mean field theory (rmft).5 ρ v 5 r(fm) 48 Ca 6 Ca 68 Ca 74 Ca.5 ρ v 5 r(fm) 48 Ca 6 Ca 68 Ca 74 Ca for 6-74 Ca private communication with L.S.Geng in RCNP.1.2 ρ t.3 5 r(fm) radius(fm).1 4 Ca 48 Ca 6 Ca.2 68 Ca 74 Ca ρt.3 5 r(fm) radius(fm) 4 Ca 48 Ca 6 Ca 68 Ca 74 Ca

10 proton neutron density distributions for Ni isotopes (fm 3 ).8.4 ρ Ni 54 6 Ni Ni 7 76 Ni Ni s s ρ Ni 54 6 Ni Ni 7 76 Ni Ni.8 relativistic mean field theory (rmft) (fm 3 ).4 ρ ρ v v TMA code :Y.Sugahara & H.Toki NPA579 (1994) 557 (fm 3 ).1.2 ρ ρ t t radius(fm) radius(fm)

11 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ca at Ep =2 MeV d σ/ d Ω (mb/str) Ca at Ep =3 MeV d σ/ d Ω (mb/str) Ca at Ep =4 MeV 4 Ca nd Ay Ay Ay 1st med exp. data Q Q Q from global optical potential fittings

12 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ca at Ep =2 MeV d σ/ d Ω (mb/str) Ca at Ep =3 MeV d σ/ d Ω (mb/str) Ca at Ep =5 MeV 48 Ca nd Ay Ay Ay 1st med exp. data Q Q Q A.E.Feldman et al. G.W.Hoffman et al. 1 c) 1 c) 1 c)

13 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni at Ep =2 MeV d σ/ d Ω (mb/str) Ni at Ep =3 MeV d σ/ d Ω (mb/str) Ni at Ep =4 MeV 58 Ni nd A y A y A y 1st med exp. data Q Q Q H.Sakaguchi et al. PRC57(1998)

14 Relativistic Impulse Approximation 6,68,74Ca at 2 MeV 6,68,74Ca at 3 MeV 6,68,74Ca at 4 MeV 6,68,74Ca at 5 MeV

15 Relativistic Impulse Approximation 48-64Ni at 2 MeV 48-64Ni at 3 MeV 48-64Ni at 4 MeV 48-64Ni at 5 MeV 66-82Ni at 2 MeV 66-82Ni at 3 MeV 66-82Ni at 4 MeV 66-82Ni at 5 MeV

16 potential depth scalar potential 4Ca 6Ca 74Ca

17 potential depth vector potential 4Ca 6Ca 74Ca

18 relation between A & Δ = 2 2 Δ < > < r n > r p 4.5 proton neutron 4 Ca r rms (fm) 4 58 Ni A

19 Ca isotopes reaction cross sections & 1 st dip position

20 reaction cross sections & 1 st dip position Ni isotopes 2 MeV σ r (fm 2 ) IA2 1st IA2 eff. IA2 2nd.5 Δ (fm) IA2 1st IA2 eff. IA2 2nd.5 Δ (fm) st dip(deg.) 1 IA2 1st IA2 eff. IA2 2nd IA2 1st IA2 eff. IA2 2nd 16 σ r (fm 2 ) st dip(deg.) A A

21 study for neutron distribution 1. K.Kaki & S.Hirenzaki, int.j.mod.phys. E, 2(1998) K.Kaki, int.j.mod.phsy.e, 13(24) Ca 28 Pb normalized by ρ( r) = 1+ ρ exp{( r r ) / a} A Z = 4π ρ( r) r 2 dr radial parameter diffuseness parameter *proton distributions are fixed to the charge or rmf density

22 Fourier transformation ρ ( ) 4 ( ) 1 q = π j qr + e ( r r )/ a r dr 4 πρ q 3 π qa sinh( π qa) ρ { πqa coth( πqa)sin( qr ) qr cos( qr )} 2

23 d d differential cross section the 1 st Born approx. t-ρ form σ 2 2 μ 2 2 f ( θ) tq ( ) ρ ( q) B 2 = = Ω π h 1 st dip position ρ ( q) =

24 mean-square radius r = 4 π r ρ( r) r dr 4 π ρ( r) r dr 1 = 7( π a ) r 2 analytic function of the parameters

25 contour map of msr & dip with respect to r ( fm ) 2 2 r & a analytic cal. θ ( deg) r(fm) r(fm) a(fm) a(fm)

26 contour map of rcs & dip with respect to r & a σ r 2 ( fm ) obserevables θ ( deg) r(fm) 7 8 r(fm) a(fm) a(fm)

27 to determine parameters a(fm) Ca rmft σ r θ = = (fm 2 ) (deg.) r(fm) r a = = (fm) (fm)

28 obtained density distribution for neutron ρ (fm 3 ) a.1 rmft(n) 6 Ca c rmft(p) 1 r(fm)

29 summary observables of proton-elastic scattering from 4,48,6-74 Ca nuclei and Ni nuclei incident energies : 2, 3,4 & 5 MeV Relativistic Impulse Approximation IA2 parameters Relativistic Mean Field Theory nuclear densities medium effects reaction cross section a little bit smaller multiple scattering effect (2 nd order potential) contributions at rather low energy & larger angle reaction cross section a little bit larger dip positions slightly different but not significant

30 conclusion to determine the neutron distribution of unstable nuclei RIA with IA2 parameter k= for incident proton energies : 2-5 MeV medium & multiple scattering effects : not significant role both in reaction cross section and dip positions near future target nucleus ; Ni isotopes, Sn isotopes energy range : 2-5 MeV 1 st order calculations of RIA with OF contour maps for the WS parameters: r, a

31 Relativistic Impulse Approximation 6,68,74 Ca 2 MeV 2nd 1st med.

32 Relativistic Impulse Approximation 6,68,74 Ca 3 MeV 2nd 1st med.

33 Relativistic Impulse Approximation 6,68,74 Ca 4 MeV 2nd 1st med.

34 Relativistic Impulse Approximation 6,68,74 Ca 5 MeV 2nd 1st med.

35 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni Ni nd A y 1st med Q E p =2 MeV

36 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni Ni nd A y 1st med Q E p =3 MeV

37 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni Ni nd A y 1st med Q E p =4 MeV

38 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni Ni nd A y 1st med Q E p =5 MeV

39 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni 7-82 Ni nd A y 1st med Q E p =2 MeV

40 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni 7-82 Ni nd A y 1st med Q E p =3 MeV

41 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni 7-82 Ni nd A y 1st med Q E p =4 MeV

42 Relativistic Impulse Approximation d σ/ d Ω (mb/str) Ni Ni Ni 7-82 Ni nd A y 1st med Q E p =5 MeV