Page 1
Carbon 12 Impacting on Oxygen 16� Wood - Saxon Potential (data from (Hamada Nuclear Physics A 859 (2011) 29-38)
From Hamada, the data for a carbon 12 nucleus hitting an oxygen 16 nucleus
is given as:
v0 = 93.89 MeV
rr = 1.18*10-15
Rr = rr (At
1
3 + Ap
1
3) where At, Ap are target and projectile atomic mass
numbers
ar=0.454*10-15
is the diffuseness of the potential
The principle calculation is the integration
Θ = ExpB 2 Μ V0
h
( -1
1+ExpB r-R
a>+Vc - Λ)]
where
Λ =e
V0
and
Vc =z1 z2 e2
4 Π Ε0 r V0 r>R
Vc =z1 z2 e2
8 Π Ε0 R V0 ( 3 - I r
RM2
) r<R
Page 2
If we write
Η =r
R then if
Η0 =a
R
Θ = ExpB 2 Μ V0 a
h á -1
1+ExpB Η-1
Η0
> - Λ + Νc âΗ where
Νc =z1 z2 e
2
4 Π Ε0 R V0 Η Η < 1
Νc =z1 z2 e
2
8 Π Ε0 R V0
( 3 - Η2) Η > 1
Clear@ΛD;
v0 = H93.89 * 10^6 L * H1.602 * 10^H-19LL;
h = 1.054571726 H47L ´ 10^H-34L;
rr = 1.18 * 10^H-15L;
At = 15.9994;
Ap = 12.011;
Rr = rr * HAp^H1 � 3L + At^H1 � 3LL;
ar = .454 * 10^H-15L;
vs = -v0 � H1 + Exp@Hr - RrL � arDL;
Μ = At Ap * 1.6005 * 10^H-27L � HAt + ApL;
Coulomb potential from Hamada
k = 8.99 * 10^9;
zt = 8;
zp = 6;
q = 1.602 * 10^H-19L;
rc = 1.25 * H10^H-15LL;
Η0 = ar � Rr;
vc11 = k * zt * zp * q^2 � Hv0 Η RrL;
vc22 = Hk * zt * zp * q^2L * H3 - Η^2L � H2 * v0 * RrL;
vcc = Piecewise@88vc11, Η >= 1<, 8vc22, Η < 1<<D;
2 Carbon 12 Impacting on Oxygen 16- Hamada data extended g.nb
Page 3
f3 =
-1
1 + ExpB Η-1
Η0F
- Λ + vcc ;
Plot3D@f3, 8Η, 0, 5<, 8Λ, 0, .1<, PlotRange ® 80, .1<,
AxesLabel ® 8"Η", "Λ", "V-E"<, ColorFunction ® "RustTones",
PlotStyle ® Directive@Yellow, Specularity@White, 20D, [email protected] ,
ExclusionsStyle ® 8None, Red<, Mesh ® None, AxesLabel ® Automatic,
BaseStyle ® 8FontWeight ® "Bold", FontSize ® 16<,
PlotLabel ® "Region of Positive Potential", LabelStyle ® HFontFamily ® "Arial"LD
Region of Positive Potential
0
2
4
Η
0.00
0.05
0.10
Λ
0.00
0.05
0.10
V-E
The above figure shows the region for which the term V - E is positive,
defining the lower limit on Η for a real value for V - E in
the calculation of Θ. The integral then requires the value of Η for
a given Λ for which the value of V - E is zero, as a lower limit on Η.
Clear@ΛD;
factor =
2 Μ v0 ar
h
;
The integration limits are found by solving
-1
1+ExpB Η-1
Η0F - Λ + vcc=0
Carbon 12 Impacting on Oxygen 16- Hamada data extended g.nb 3
Page 4
Λ = .025;
f11 = factor * f3 �. Λ ® %;
NSolve@f11 � 0, Η, RealsD;
Part@%, 1D;
Part@%%, 2D;
Θ = Exp@NIntegrate@Sqrt@f11D, 8Η, Η �. %%, Η �. %<DD;
T = Re@4 � H2 Θ + 1 � H2 ΘLL^2D;
8%%%%%%%, %<data = 880.0005`, 0.02176012479519214`<,
80.001`, 0.07213624355101142`<, 80.002`, 0.16307940033942034`<,
80.003`, 0.2300024746731435`<, 80.004`, 0.2801129596788937`<,
80.005`, 0.3190894141225901`<, 80.006`, 0.350404674923039`<,
80.007`, 0.3762235620795487`<, 80.008`, 0.39795520763321846`<,
80.009`, 0.41655571840976424`<, 80.01`, 0.43269759657240314`<,
80.015`, 0.4899448288544386`<, 80.02`, 0.5255241995191092`<,
80.025`, 0.550156209639818`<, 80.03`, 0.5683651022991135`<,
80.04`, 0.5936299660802987`<, 80.05`, 0.6103213636258508`<,
80.06`, 0.6220478262258062`<, 80.07`, 0.6305710799764025`<,
80.08`, 0.6368984488821532`<, 80.081`, 0.6374385952626274`<,
80.082`, 0.6379643874555523`<, 80.083`, 0.6384763913628796`<,
80.084`, 0.6389751561873208`<, 80.085`, 0.6394612142178806`<,
80.086`, 0.6399350806112528`<, 80.0861`, 0.6399818160279523`<<;
ListLinePlot@data, AxesLabel ® 8"Λ", "T"<, PlotLabel -> "Transmission Coefficient",
LabelStyle ® HFontFamily ® "Arial"L, BaseStyle ® 8FontWeight ® "Bold", FontSize ® 16<D80.025, 0.550156<
0.02 0.04 0.06 0.08Λ
0.1
0.2
0.3
0.4
0.5
0.6
T
Transmission Coefficient
The thermal energy of the carbon nucleus in a 6000 degree plasma has a Λ
of
4 Carbon 12 Impacting on Oxygen 16- Hamada data extended g.nb
Page 5
The thermal energy of the carbon nucleus in a 6000 degree plasma has a Λ
of
ethermal = 1.3806503 * 10^H-23L * 6000 � v0
5.50748´ 10-9
The transmission coefficient in traditional theory is essentially zero for
this input energy.
Carbon 12 Impacting on Oxygen 16- Hamada data extended g.nb 5
