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

Kaori KakiDepartment of Physics, Shizuoka University

21 February 2009

Relativistic Impulse Approximation (RIA)

• optical potentialsDirac equation for a projectile proton scattering from a target nucleus given by the optical model:

the momentum space Dirac equation

ˆ[ ( )] ( ) 0,, ( , )

p m Up p p Eμ μ

μ

ψ

γ

− − =/= =/

r rp

( )( )

0 33

1 ˆ' ( ') ( ', ) ( ) 02

E m d p Uγ ψ ψπ

− − − =∫γ p p p p p

ˆ ( ) | |

| |

1 | | | |

ii

i ji j j

i ji j

U t

t Gt

A t G tA

≠

=< Φ Φ >

+ < Φ Φ >

−− < Φ Φ > < Φ Φ >

∑

∑∑

∑ ∑

r

by relativistic analog of non-relativistic multiple scattering theory, optical potential in coordinate space:

1st order term

2nd order term

the generalized RIA optical potential in the momentum space for the 1st order term

optimal factorization : simpletρ-form

( )

( )

3

3

3

3

1ˆ ˆ ˆ( ', ) Tr ( , ', ) ( , )4 2 22

1 ˆ ˆTr ( , ', ) ( , )4 2 22

pp p

pn n

d kU M k

d k M k

ρπ

ρπ

⎧ ⎫⎪ ⎪= − − → +⎨ ⎬⎪ ⎪⎩ ⎭⎧ ⎫⎪ ⎪− − → +⎨ ⎬⎪ ⎪⎩ ⎭

∫

∫

q qp p p k p k q

q qp k p k q

( 0)=k1ˆ ˆ ˆ( ', ) Tr ( , ', ) ( )4 2 21 ˆ ˆTr ( , ', ) ( )4 2 2

pp p

pn n

U M

M

ρ

ρ

⎧ ⎫= − − →⎨ ⎬⎩ ⎭⎧ ⎫− − →⎨ ⎬⎩ ⎭

q qp p p p q

q qp p q

density matrices

3( ) ( )id r eρ ρ⋅= ∫ q rq r) )

each nuclear density distribution

0( ) ( ) ( ) ( )2S V T

iρ ρ γ ρ ρ⋅= + −

α rr r r r)

)

scalar vector tensor

3( , ) ( ( ))id r e r kρ ρ⋅= ∫ q rk q) )

the 2nd order optical potential in the momentum space

3

22

3

1ˆ ˆ ˆ( ', ) Tr ( , ', ) ( )(2 ) 4 2 2

1ˆ ˆ ˆ( , ) ( ) Tr ( , , ) ( )4 2 2

'

b bb

a ab a a

a

b

d kU M

C G M

ρπ

ρ

⎧ ⎫= − →⎨ ⎬⎩ ⎭⎧ ⎫× − →⎨ ⎬⎩ ⎭

= −

= −

∫q qp p k p q

q qq q k p k q

q p kq k p

propagatorcorrelation part

p 'p

2a−q

k

2b−q

2aq

2bq

M̂ M̂

propagator0 1( ) ( )AG m E iγ ε −= / − − +k k

correlation function ˆˆ ˆ( ) ( , ) ( )b b a aCρ ρq q q q

correlation function

2

2

3 3

3

1

ˆˆ ˆ( ) ( , ) ( )

ˆ ˆ(| |) ( ) ( )

( )

a b

b b a a

i ia b a b a b

rR

C

d r d r e e f

f r f e αα

α

ρ ρ

ρ ρ⋅ ⋅

−

=

= −

=

∫ ∫

∑

r q r q

q q q q

r r r r

the same parameters as in

J.D.Lumpe & L.Ray, Phys.Rev.C35(1987)1040

density distributions for Ca isotopes

relativistic mean field theory (rmft)

for 60-74Ca private communication with L.S.Geng in RCNP

0 50

0.05

0.1

r(fm)

ρρρρ

(fm−3)

40Ca

s

48Ca60Ca68Ca74Ca

0 50

0.05

0.1

r(fm)

ρρρρ

40Ca

v

48Ca60Ca68Ca74Ca

0 5−0.03

−0.02

−0.01

0

r(fm)

ρρρρ

40Ca

t

48Ca60Ca68Ca74Ca

0 50

0.05

0.1

r(fm)

ρρρρ

(fm−3)

40Ca

s

48Ca60Ca68Ca74Ca

0 50

0.05

0.1

r(fm)

ρρρρ

40Ca

v

48Ca60Ca68Ca74Ca

0 5−0.03

−0.02

−0.01

0

r(fm)

ρρρρ

40Ca

t

48Ca60Ca68Ca74Ca

radius(fm)radius(fm)radius(fm)

proton neutron

density distributions for Ni isotopes

relativistic mean field theory (rmft)

TMA code :Y.Sugahara & H.Toki

NPA579 (1994) 557

0 4 80

0.04

0.08

ρρρρ

(fm

−3 )

47−52Ni

s

54−60Ni62−68Ni70−76Ni78−82Ni

0 4 80

0.04

0.08

ρρρρ

(fm

−3 )

v

0 4 8−0.03

−0.02

−0.01

0

ρρρρ

(fm

−3 )

t

0 4 8

ρρρρ

47−52Ni

s

54−60Ni62−68Ni70−76Ni78−82Ni

0 4 8

ρρρρv

0 4 8

ρρρρt

proton neutron

radius(fm) radius(fm)

Relativistic Impulse Approximation

40Ca 0 20 40 6010−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

40Ca at Ep=200 MeV

0 20 40 60

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

40Ca at Ep=300 MeV

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

40Ca at Ep=400 MeV

2nd

1st

med.

exp. data

from global optical potential fittings

48Ca 0 20 40 6010−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

c)

Q

angle(deg.)

48Ca at Ep=200 MeV

0 20 40 6010−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

c)

Q

angle(deg.)

48Ca at Ep=300 MeV

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

c)

Q

angle(deg.)

48Ca at Ep=500 MeVRelativistic Impulse Approximation

2nd

1st

med.

exp. data

A.E.Feldman et al. G.W.Hoffman et al.

Relativistic Impulse Approximation

58Ni2nd

1st

med.

0 20 40 60

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

58Ni at Ep=200 MeV

0 20 40 60

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

58Ni at Ep=300 MeV

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

58Ni at Ep=400 MeV

exp. data

H.Sakaguchi et al. PRC57(1998)1749

60,68,74Ca at 200 MeV60,68,74Ca at 300 MeV60,68,74Ca at 400 MeV60,68,74Ca at 500 MeV

Relativistic Impulse Approximation

48-64Ni at 200 MeV48-64Ni at 300 MeV48-64Ni at 400 MeV48-64Ni at 500 MeV

66-82Ni at 200 MeV66-82Ni at 300 MeV66-82Ni at 400 MeV66-82Ni at 500 MeV

Relativistic Impulse Approximation

potentialdepth

40Ca60Ca74Ca

scalar potential

potentialdepth

vector potential40Ca60Ca74Ca

50 60 70 80

3.5

4

4.5

A

r rms(

fm)

neutronproton

relation between A & Δ22

pn rr= −Δ < >< >

58Ni

40Ca

reaction cross sections & 1st dip position

Ca isotopes

reaction cross sections & 1st dip position

0 0.5

60

80

100

ΔΔΔΔ

IA2 eff.IA2 2nd

IA2 1st

σr

(fm

2 )

(fm)

50 60 70 80

60

80

100

A

IA2 eff.IA2 2nd

IA2 1st

σr

(fm

2 )

0 0.5

8

12

16

ΔΔΔΔ

1st dip(deg.)

IA2 eff.IA2 2nd

(fm)

IA2 1st

50 60 70 80

8

12

16

A

1st dip(deg.)

IA2 eff.IA2 2nd

IA2 1st

200 MeV300400500

Ni isotopes

study for neutron distribution

}/)exp{(1)(

0

0

arrr

−+=

ρρ

1. K.Kaki & S.Hirenzaki, int.J.Mod.Phys. E, 2(1998) 167-178

2. K.Kaki, int.J.Mod.Phsy.E, 13(2004) 787-799

diffuseness parameterradial parameter

normalized by

drrrZA ∫=− 2)(4 ρπ

60Ca208Pb

*proton distributions are fixed to the charge or rmf density

{ }

0

200 ( )/0

03

0 0 0

( ) 4 ( )1

4sinh( )

coth( )sin( ) cos( )

r r aq j qr r dre

qaq qa

qa qa qr qr qr

ρρ π

πρ ππ

π π

∞

−=+

≈

× ⋅ − ⋅

∫

Fourier transformation

differential cross sectionthe 1st Born approx.t-ρ form

22 2 2( ) ( ) ( )

2Bd f t q qdσ μθ ρ

π⎛ ⎞= = ⎜ ⎟Ω ⎝ ⎠h

1st dip position ( ) 0qρ =

mean-square radius

2 2 2 2

2 20

4 ( ) 4 ( )

1 7( ) 35

r r r r dr r r dr

a r

π ρ π ρ

π

=

⎡ ⎤= +⎣ ⎦

∫ ∫

analytic function of the parameters

a(fm) a(fm)

r0(fm

)

r0(fm

)

( )degθ( )2 2fmr

3.53.0

4.0

4.5

5.0

6.0

5.5

6.511

12

13

14

15

16

contour map of msr & dip with respect to &r0 a analytic cal.

r0(fm

)

r 0(fm

)

a(fm) a(fm)

( )degθ( )2fmrσ

100

90

80

70

60

11.5

12.0

12.5

13.0

13.5

obserevablescontour map of rcs & dip with respect to &r0 a

to determine parameters

3.8 4 4.2 4.4 4.6 4.8 50.2

0.4

0.6

0.8

1

1.2

3.8 4 4.2 4.4 4.6 4.8 50.2

0.4

0.6

0.8

1

1.2

60Ca rmft

22.12

34.75

=

=

θσ r

(fm2)

(deg.)

64.0

32.40

=

=

ar (fm)

(fm)

a(fm)

r0(fm)

obtained density distribution for neutron

0 100

0.1

r(fm)

ρ (fm−3)

60Caa

c rmft(p)

rmft(n)

summary

observables of proton-elastic scattering from 40,48,60-74Ca nuclei

and 48-82Ni nuclei

incident energies : 200, 300,400 & 500 MeV

Relativistic Impulse Approximation IA2 parameters

Relativistic Mean Field Theory nuclear densities

medium effects

reaction cross section a little bit smaller

multiple scattering effect (2nd order potential)

contributions at rather low energy & larger angle

reaction cross section a little bit larger

dip positions slightly different but not significant

conclusionto determine the neutron distribution of unstable nuclei

RIA with IA2 parameter

k=0 for incident proton energies : 200-500 MeV

medium & multiple scattering effects : not significant role

both in reaction cross section and dip positions

target nucleus ; Ni isotopes, Sn isotopes

energy range : 200-500 MeV

1st order calculations of RIA with OF

contour maps for the WS parameters:

near future

0 ,r a

Relativistic Impulse Approximation

60,68,74Ca200 MeV

2nd

1st

med.

Relativistic Impulse Approximation

2nd

1st

med.

60,68,74Ca300 MeV

Relativistic Impulse Approximation

2nd

1st

med.

60,68,74Ca400 MeV

Relativistic Impulse Approximation

2nd

1st

med.

60,68,74Ca500 MeV

Relativistic Impulse Approximation

48-64Ni2nd

1st

med.

0 20 40 60

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

48Ni

0 20 40 60

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

52Ni

0 20 40 60

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

64Ni

Ep =200 MeV

Relativistic Impulse Approximation

48-64Ni2nd

1st

med.

0 20 40 6010−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

48Ni

0 20 40 6010−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

52Ni

0 20 40 60

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

64Ni

Ep =300 MeV

Relativistic Impulse Approximation

48-64Ni2nd

1st

med.

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

48Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

52Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

64Ni

Ep =400 MeV

Relativistic Impulse Approximation

48-64Ni2nd

1st

med.

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

48Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

52Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

64Ni

Ep =500 MeV

Relativistic Impulse Approximation

70-82Ni2nd

1st

med.

0 20 40 60

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

70Ni

0 20 40 60

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

76Ni

0 20 40 60

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

82Ni

Ep =200 MeV

Relativistic Impulse Approximation

70-82Ni2nd

1st

med.

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

70Ni

0 20 40 6010−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

76Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

82Ni

Ep =300 MeV

Relativistic Impulse Approximation

70-82Ni2nd

1st

med.

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

70Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

76Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

82Ni

Ep =400 MeV

Relativistic Impulse Approximation

70-82Ni2nd

1st

med.

0 20 40 60

10−5

100

105

(a)

dΩ

σ/d

(mb/

str)

0 20 40 60

−1

0

1

(b)

Ay

0 20 40 60

−1

0

1

(c)

Q

angle(deg.)

70Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

76Ni

0 20 40 60

10−5

100

105

(a)

0 20 40 60

−1

0

1

(b)

0 20 40 60

−1

0

1

(c)

angle(deg.)

82Ni

Ep =500 MeV