Disappearance of Mott oscillations in the scattering of ... · Disappearance of Mott oscillations...
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Disappearance of Mott oscillations in the scattering of identical charged particles
M.S. Hussein Universidade de São Paulo
Canto, Hussein, Mittig, Phys. Rev. C89 (2014) 024610 Canto, Donangelo, Hussein, Mod. Phys. Lett. A16 (2001) 1027
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Quantum theory of scattering - discernible particles:
� ~22µ
r2 + V (r)
� (+)(r) = E (+)(r)
asymptotic boundary condition: in(r) + scatt(r)
Cross section:
A [ ]
scatt(r) = f(✓)eikr
r
in(r) = eikz
�(✓) ⌘ d�(✓)
d⌦= |f (✓)|2
(+)(r) !
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identical bosons à symmetric wave function:
Consequence:
And thus,
P � T ⌘ r ! �r =) ✓ ! ⇡ � ✓
f (✓) ! f (✓) + f (⇡ � ✓)
(r) = (�r) =) (r) ! 1p2
[ (r) + (�r)]
symmetry around ⇡/2 !
�(✓) =���f (✓) + f (⇡ � ✓)
���2
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c.m. q z
T
P
p - q
zP
T
q
c.m.
direct: detection of P
exchange: detection of T
f (q)
f (p - q)
�class(✓) =���f (✓)
��� +���f (⇡ � ✓)
���2
The cross sec)on can be wri/en,
Classical term,
Interference (Q.M.) term,
�(✓) = �class(✓) + ⌅int(✓)
⌅int(✓) = 2Ren
f⇤(✓)⇥ f(⇡ � ✓)o
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Symmetry around π/2 à σ (π/2) is a maximum or a minimum
Which one ? Look at data!
Maximum !
Is this a general result ?
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An problem with exact solution: Coulomb scattering
- discernible particles (Rutherford scattering)
fC (✓) = �a
2e2i�0 e�i⌘ ln(sin2 ✓/2)
Rutherford cross section:
�0 is the Coulomb phase shift for l = 0
⌘ = qP qT/~v is the Sommerfeld parameter
a is half the distance of c. a. in a head�on collision
�R(✓) =���fC(✓)
���2
=a2
4
1
sin4 (✓/2)
�
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- identical particles (Mott scattering)
�R(✓) =���fC(✓) + fC(⇡ � ✓)
���2
⌅int(✓) ⌘⌅int(✓
⌅int(⇡/2)=
1
16
2
cos [2⌘ ln (tan(✓/2))]
sin
2(✓/2) cos2(✓/2)
�
�class(✓) ⌘�class(✓)
�class(⇡/2)=
1
16
1
sin
4(✓/2)
+
1
cos
4(✓/2)
�classical term (renormalized) – system & energy independent:
exchange term (renormalized) – system & energy dependent:
�class(⇡/2) = 1
!
⌅int(⇡/2) = 0
!
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Behavior of σM(θ) around π / 2
�class(⇡/2) ! minimum (constant curvature )
�(⇡/2) ! competition : �class ⇥ ⌅int
• minimum for small ⌘
• maximum for large ⌘
⌅int(⇡/2; ⌘) ! maximum (curvature grows with ⌘)
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40 60 80 100 120 140q (deg)
0
1
2
3
4
s (q
) / s
(90o
)
h = 0.2 (sclass dominates) h = 4.0 (Xint dominates)
• minimum at ⌘ = 0.2 !d2�(✓; ⌘)
d✓2
�
✓=⇡/2,⌘=0.2
> 0
• maximum at ⌘ = 0.2 !d2�(✓; ⌘)
d✓2
�
✓=⇡/2,⌘=0.2
< 0
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A transition takes place at η = η0 , such that:
d2�(✓; ⌘)
d✓2
�
✓=⇡/2,⌘=⌘0
= 0
That is:
d2
d✓2
1
sin
4(✓/2)
+
1
cos
4(✓/2)
+ 2
cos [2⌘ ln (tan(✓/2))]
sin
2(✓/2) cos2(✓/2)
�= 0
Very complicated equation ….. but can be solved analytically by programs like Maple
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The solution is ⌘0 =
p2
Thus, for this Sommerfeld parameter, The cross section should be very flat.
40 60 80 100 120 140(deg)
0
1
2
3
4/
= 0.2= 4.0
= 2
(Transverse isotropy - TI)
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Experimental investigation of TI Nucleus-nucleus collisions are good candidates
VN (r) VC (r) Vtot (r)
VN (r) ~ 0
VB E
r
V(r)
Only if E < VB so that
Is this condition satisfied ?
VN ' 0 ) Vtot
' VC
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0 2 4 6 8 10Z
0
5
10
15E 0
/ V
B
E0 / VB = 1
2H4He
6Li8Be
10B
12C
⌘ =Z2e2
~v �! E0 =e4Z4µ
2~2⌘20VB ' Z2e2
2r0A1/3
Ideal system: 4He + 4He, Ec.m. = 397 keV
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4He – 4He elastic scattering
106
105
104
103
102
100
101
102
103
101
100
102103104
101
100
102103104
101
100
102103104
104
103
102
103
102
104
103
102
101
104
103
102
101
101
100
102103104
10-1
101
100
102
103104
10-1
10-3
101
103105
0 40 80 120 0 40 80 120 0 40 80 120
0 40 80 120 0 40 80 120 0 40 80 120
0 40 80 120 0 40 80 120 0 40 80 120
16012080400 16012080400
data at Ec.m. = E↵/2 � 1.0 MeV
• no data at Ec.m. ⇠ 0.4 MeV
• TI at Ec.m. ⇠ 1.5 MeV
Are there other ⌘0 solutions for
d2�(✓; ⌘)
d✓2
�
✓=⇡/2,⌘=⌘0
= 0
Abdullah et al. NPA 775 (2006) 1
• TI at Ec.m. ⇠ 1.0 MeV
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Second derivative for pure Coulomb potential
0 0.5 1 1.5 2 2.5 3 3.5 4Ec.m. (MeV)
-2
-1
0
1
2
s'' (9
0o)
(b
arn/
rad2 ) only VC
0.397 MeV (h0 = ÷2 )
V = VC à TI only for Ec.m. = 0.397 MeV !!
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Effect of nuclear forces?
Akÿuz-Winther interaction (popular in HI collisions)
VC �! VC + V A.W.N =) VB = 0.87 MeV
data at Ec.m. = 1 MeV > VB
Nuclear potential is not negligible !
Must be taken into account in the condition,
d2�(✓; ⌘)
d✓2
�
✓=⇡/2,⌘=⌘0
= 0
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0 0.5 1 1.5 2 2.5 3 3.5 4Ec.m. (MeV)
-2
-1
0
1
2
s'' (9
0o)
(b
arn/
rad2 ) only VC
VC + VN
VB = 0.87 MeV
0.40 MeV1.10 MeV
VC + V A.W.N
• Ec.m. < 0.4 MeV : �00 < 0 ! maximum at 90
o
• Ec.m. = 0.4 MeV : �00 = 0 ! flat(TI)
• 0.4 < Ec.m. < 1.1 : �00 > 0 ! minimum at 90o
• Ec.m. = 1.1 MeV : �00 = 0 ! flat(TI)
• Ec.m. > 1.1 MeV : �00 < 0 ! maximum at 90
o
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Ec.m. = 1.0 MeV Ec.m. = 1.5 MeV
min
max max max
max max
max max max
flat flat
Ec.m. = 1.92 MeV
Ec.m. = 2.63 MeV Ec.m. = 3.24 MeV Ec.m. = 3.74 MeV
Ec.m. = 4.44 MeV Ec.m. = 4.94 MeVmax
Ec.m. = 5.44 MeV
Ec.m. = 5.94 MeV Ec.m. = 6.15 MeV Ec.m. = 7.60 MeV
• 0.4 < Ec.m. < 1.1 : ! minimum
• Ec.m. = 1.1 MeV : ! flat
• Ec.m. > 1.1 MeV : ! maximum
Theory Experiment
• Ec.m. = 0.4 MeV : ! flat(TI)
• Ec.m. < 0.4 MeV : ! maximum
Cannot be checked
About right
Wrong! minimum at 1.92 MeV
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A more realistic potential for 4He + 4He scattering (AW is a Woods-Saxon. For 4He, gaussians are more realistic) The BFWC potential (sum of attractive and repulsive gaussians):
0 0.5 1 1.5 2 2.5 3Ec.m. (MeV)
-2
-1
0
1
2
''(barn/rad2)
VA = 123 MeV, RA = 2.13 fm
VR = 3.0 MeV, RR = 2.0 fmV FBWCN = � VA e�r2/R2
A + VR e�r2/R2R , with
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Ec.m. = 1.0 MeV Ec.m. = 1.5 MeV
min
max max max
max max
max max max
flat flat
Ec.m. = 1.92 MeV
Ec.m. = 2.63 MeV Ec.m. = 3.24 MeV Ec.m. = 3.74 MeV
Ec.m. = 4.44 MeV Ec.m. = 4.94 MeVmax
Ec.m. = 5.44 MeV
Ec.m. = 5.94 MeV Ec.m. = 6.15 MeV Ec.m. = 7.60 MeV
0.5 1 1.5 2 2.5 3
-2
-1
0
1
2
s'' (9
0o)
(b
arn/
rad2 ) VBFWC + VC
s'' small --8 flatweak min.
stron
g m
ax.
Theory (BWFC potential) Experiment
Fully consistent !!
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Concluding remarks • Mott scattering is a good testing vehicle for effects arising from different
contributions, such as vacuum polarization, dipole polarization, color Van der Waals contribution etc. These are very small contributions to the predominantly Coulomb, nucleus-nucleus interaction, but their effects are quite visible in the interference term of the cross section
• The Transverse Isotropy is a purely EM effect, but the correction arising
from the very weak nuclear interaction at sub-barrier energies is found to be very important to accurately account for the available data
• Missing, is a measurement of the 4He + 4He cross section at Ec.m. about 0.4
MeV. We are hoping that the measurement is forthcoming.