relativistic impulse approximation analysis of unstable ... · summary observables of...
TRANSCRIPT
![Page 1: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/1.jpg)
Relativistic Impulse Approximation Analysis of Unstable Nuclei: Calcium and Nickel Isotopes
Kaori KakiDepartment of Physics, Shizuoka University
21 February 2009
![Page 2: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/2.jpg)
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
![Page 3: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/3.jpg)
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
![Page 4: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/4.jpg)
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
![Page 5: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/5.jpg)
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) )
![Page 6: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/6.jpg)
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
![Page 7: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/7.jpg)
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
![Page 8: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/8.jpg)
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
![Page 9: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/9.jpg)
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
![Page 10: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/10.jpg)
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)
![Page 11: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/11.jpg)
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
![Page 12: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/12.jpg)
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.
![Page 13: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/13.jpg)
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
![Page 14: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/14.jpg)
60,68,74Ca at 200 MeV60,68,74Ca at 300 MeV60,68,74Ca at 400 MeV60,68,74Ca at 500 MeV
Relativistic Impulse Approximation
![Page 15: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/15.jpg)
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
![Page 16: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/16.jpg)
potentialdepth
40Ca60Ca74Ca
scalar potential
![Page 17: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/17.jpg)
potentialdepth
vector potential40Ca60Ca74Ca
![Page 18: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/18.jpg)
50 60 70 80
3.5
4
4.5
A
r rms(
fm)
neutronproton
relation between A & Δ22
pn rr= −Δ < >< >
58Ni
40Ca
![Page 19: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/19.jpg)
reaction cross sections & 1st dip position
Ca isotopes
![Page 20: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/20.jpg)
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
![Page 21: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/21.jpg)
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
![Page 22: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/22.jpg)
{ }
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
![Page 23: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/23.jpg)
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 =
![Page 24: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/24.jpg)
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
![Page 25: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/25.jpg)
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.
![Page 26: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/26.jpg)
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
![Page 27: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/27.jpg)
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)
![Page 28: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/28.jpg)
obtained density distribution for neutron
0 100
0.1
r(fm)
ρ (fm−3)
60Caa
c rmft(p)
rmft(n)
![Page 29: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/29.jpg)
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
![Page 30: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/30.jpg)
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
![Page 31: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/31.jpg)
Relativistic Impulse Approximation
60,68,74Ca200 MeV
2nd
1st
med.
![Page 32: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/32.jpg)
Relativistic Impulse Approximation
2nd
1st
med.
60,68,74Ca300 MeV
![Page 33: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/33.jpg)
Relativistic Impulse Approximation
2nd
1st
med.
60,68,74Ca400 MeV
![Page 34: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/34.jpg)
Relativistic Impulse Approximation
2nd
1st
med.
60,68,74Ca500 MeV
![Page 35: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/35.jpg)
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
![Page 36: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/36.jpg)
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
![Page 37: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/37.jpg)
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
![Page 38: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/38.jpg)
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
![Page 39: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/39.jpg)
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
![Page 40: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/40.jpg)
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
![Page 41: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/41.jpg)
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
![Page 42: Relativistic Impulse Approximation Analysis of Unstable ... · summary observables of proton-elastic scattering from . 40,48,60-74. Ca nuclei and . 48-82. Ni nuclei. zincident energies](https://reader033.vdocument.in/reader033/viewer/2022050216/5f627832ff61b02647467724/html5/thumbnails/42.jpg)
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