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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