Chapter 1 Introduction
5.4 Experimental γ-ray spectra and differential cross sections
5.4.2 Differential cross sections for the 754.8 keV line
The differential cross sections were measured as a function of the incident energy for each detector crystal placed at an angle θ with respect to the beam line. The angular distributions corresponding to the γ-ray lines of interest (754.8 keV and 3736.5keV) emitted by the40Canuclei in the40Ca(p, p0γ)40Careaction, are shown for different proton beam energies in Figures 5.36 - 5.47. The indicated error bars on the figures correspond to the uncertainties of the differential cross sections. All data points in each figure are fitted with the angular distribution function, equation 3.9. In Figure 5.36 panel (A) corresponds to the PR239 data at 66 M eV while panel (B) corresponds to the PR255 data at 66 M eV. This allows a comparison to be done between two different experiment at the same beam energy. Note that one of the eight clovers was placed at 168◦ in PR255, shown in Figure 5.36 (B).
This changed the shape of the angular distribution function, from a curve with a minimum at 0◦/180◦ to a curve with a maximum at 0◦/180◦. This change is due to the added data points at 163◦ and 173◦.
Chapter 5. Experimental cross section measurements 69
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 1 2 3 4 5 6
/ ndf
χ2 12.55 / 27
p0 5.412 ± 0.07986 p1 −0.05295 ± 0.02818 p2 −0.1478 ± 0.09285
/ ndf
χ2 12.55 / 27
p0 5.412 ± 0.07986 p1 −0.05295 ± 0.02818 p2 −0.1478 ± 0.09285 = 66 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(a) Results from PR239.
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 1 2 3 4 5 6
7 / ndf
χ2 9.048 / 29
p0 5.458 ± 0.06182 p1 0.02277 ± 0.007614 p2 0.05091 ± 0.0281
/ ndf
χ2 9.048 / 29
p0 5.458 ± 0.06182 p1 0.02277 ± 0.007614 p2 0.05091 ± 0.0281
= 66 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(b) Results from PR255.
Figure 5.36: (A): The differential cross sections from PR239 with Ep = 66 M eV for the 754keV line with detectors positioned at 90◦ and 135◦. (B): The differential cross sections from PR255 with Ep= 66 M eV for the 754keV line
with one of the 8 clover detectors at 168◦.
Data from PR255 show that actually the angular distribution curve should have a maximum at 0◦/180◦. Due to these data it is clear that the angular distribution curves of PR239 do not have correct shape. Thus, we carried out a few tests with the PR239 data, fixing the parameters a0, a2, a4 and a6, called p0, p1, p2, p3 respectively. Figure 5.37 shows (A): a fit of the the angular distribution function for the PR239 data where the coefficient p1 is fixed to the value of p1 = 0.02277 as obtained from the fit of the PR255 data. The coefficient p2 remains a free variable. (B): a fit of the angular distribution function for the PR239 data where the coefficient p2 is fixed to the p2 = 0.05091 as obtained from the fit of the PR255 data. The coefficient p1 remains a free variable; and (C) a fit of angular distribution for the PR239 where both coefficients p1 and p2 are fixed to the values obtained from the PR255 data. This is done in order to check whether varying the angular distibution coefficients p1 and p2 has an effect on the parameter of interest, p0, which represents the total cross section. The values of p0 are very similar for all the fits including fits where p1 and/ or p2 were fix (Figure 5.37) but also when all p0, p1 and p2 coefficients are free (Figure 5.36). This shows that the total cross section is not sensitive to changes of the p1 and p2 coefficients, and although the shape of the fit in Figure 5.36 A may not be exact, the total cross section, p0, value is reasonable.
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 1 2 3 4 5 6
/ ndf
χ2 20.18 / 28
p0 5.56 ± 0.0593 p1 0.02277 ± 0 p2 0.08725 ± 0.03024
/ ndf
χ2 20.18 / 28
p0 5.56 ± 0.0593 p1 0.02277 ± 0 p2 0.08725 ± 0.03024
= 66 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(a)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 1 2 3 4 5 6
/ ndf
χ2 17.33 / 28
p0 5.525 ± 0.06122 p1 0.003741 ± 0.009237 p2 0.05091 ± 0
/ ndf
χ2 17.33 / 28
p0 5.525 ± 0.06122 p1 0.003741 ± 0.009237 p2 0.05091 ± 0
= 66 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(b)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 1 2 3 4 5 6
/ ndf
χ2 21.62 / 29
p0 5.557 ± 0.05923 p1 0.02277 ± 0 p2 0.05091 ± 0
/ ndf
χ2 21.62 / 29
p0 5.557 ± 0.05923 p1 0.02277 ± 0 p2 0.05091 ± 0 = 66 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(c)
Figure 5.37: Angular distribution for the40Ca754.8 keVγ-ray line at 66 MeV proton energy. The solid lines are the fitted curves for the angular distribution function for a L = 2 transition. The data points represent the differential cross sections measured for each crystal versus the detection angle. The error bars on the differential cross sections include statistical error as well as the errors of
target thickness, the efficiency and the number of incident particles.
Chapter 5. Experimental cross section measurements 71
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
/ ndf
χ2 17.03 / 27
p0 1.478 ± 0.02158 p1 −0.04963 ± 0.0275 p2 −0.07429 ± 0.08996
/ ndf
χ2 17.03 / 27
p0 1.478 ± 0.02158 p1 −0.04963 ± 0.0275 p2 −0.07429 ± 0.08996
= 80 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(a)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
/ ndf
χ2 24.33 / 28
p0 1.516 ± 0.01619 p1 0.02277 ± 0 p2 0.1479 ± 0.03016
/ ndf
χ2 24.33 / 28
p0 1.516 ± 0.01619 p1 0.02277 ± 0 p2 0.1479 ± 0.03016 = 80 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(b)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
/ ndf
χ2 19.02 / 28
p0 1.497 ± 0.01651 p1 −0.01375 ± 0.009339 p2 0.05091 ± 0
/ ndf
χ2 19.02 / 28
p0 1.497 ± 0.01651 p1 −0.01375 ± 0.009339 p2 0.05091 ± 0
= 80 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(c)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
/ ndf
χ2 34.63 / 29
p0 1.511 ± 0.01612 p1 0.02277 ± 0 p2 0.05091 ± 0
/ ndf
χ2 34.63 / 29
p0 1.511 ± 0.01612 p1 0.02277 ± 0 p2 0.05091 ± 0 = 80 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(d)
Figure 5.38: Angular distribution for the40Ca754.8 keVγ-ray line at 80 MeV proton energy. The solid lines are the fitted curves for the angular distribution function for a L = 2 transition. The data points represent the differential cross sections measured for each crystal versus the detection angle. The error bars on the differential cross sections include statistical error as well as the errors of
target thickness, the efficiency and the number of incident particles.
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8 χ2 / ndf 22 / 27
p0 0.5906 ± 0.008631 p1 −0.1134 ± 0.02881 p2 −0.2777 ± 0.09557
/ ndf
χ2 22 / 27
p0 0.5906 ± 0.008631 p1 −0.1134 ± 0.02881 p2 −0.2777 ± 0.09557 = 95 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(a)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8 χ2 / ndf 46.52 / 28
p0 0.6181 ± 0.006608 p1 0.02277 ± 0 p2 0.1477 ± 0.03038
/ ndf
χ2 46.52 / 28
p0 0.6181 ± 0.006608 p1 0.02277 ± 0 p2 0.1477 ± 0.03038 = 95 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(b)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8 χ2 / ndf 34.62 / 28
p0 0.6098 ± 0.006717 p1 −0.02026 ± 0.00928 p2 0.05091 ± 0
/ ndf
χ2 34.62 / 28
p0 0.6098 ± 0.006717 p1 −0.02026 ± 0.00928 p2 0.05091 ± 0
= 95 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(c)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8 χ2 / ndf 56.63 / 29
p0 0.6162 ± 0.006581 p1 0.02277 ± 0 p2 0.05091 ± 0
/ ndf
χ2 56.63 / 29
p0 0.6162 ± 0.006581 p1 0.02277 ± 0 p2 0.05091 ± 0 = 95 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(d)
Figure 5.39: Angular distribution for the40Ca754.8 keVγ-ray line at 95 MeV proton energy. The solid lines are the fitted curves for the angular distribution function for a L = 2 transition. The data points represent the differential cross sections measured for each crystal versus the detection angle. The error bars on the differential cross sections include statistical error as well as the errors of
target thickness, the efficiency and the number of incident particles.
Chapter 5. Experimental cross section measurements 73
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
/ ndf
χ2 15.03 / 27
p0 0.2967 ± 0.004397 p1 −0.06343 ± 0.02869 p2 −0.2101 ± 0.09504
/ ndf
χ2 15.03 / 27
p0 0.2967 ± 0.004397 p1 −0.06343 ± 0.02869 p2 −0.2101 ± 0.09504
= 110 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(a)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
/ ndf
χ2 24.64 / 28
p0 0.3058 ± 0.003261 p1 0.02277 ± 0 p2 0.05955 ± 0.0303
/ ndf
χ2 24.64 / 28
p0 0.3058 ± 0.003261 p1 0.02277 ± 0 p2 0.05955 ± 0.0303 = 110 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(b)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
/ ndf
χ2 22.99 / 28
p0 0.3046 ± 0.003382 p1 0.01075 ± 0.009184 p2 0.05091 ± 0
/ ndf
χ2 22.99 / 28
p0 0.3046 ± 0.003382 p1 0.01075 ± 0.009184 p2 0.05091 ± 0
= 110 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(c)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
/ ndf
χ2 24.72 / 29
p0 0.3058 ± 0.00326 p1 0.02277 ± 0 p2 0.05091 ± 0
/ ndf
χ2 24.72 / 29
p0 0.3058 ± 0.00326 p1 0.02277 ± 0 p2 0.05091 ± 0 = 110 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(d)
Figure 5.40: Angular distribution for the 40Ca 754.8 keV γ-ray line at 110 MeV proton energy. The solid lines are the fitted curves for the angular distri- bution function for a L = 2 transition. The data points represent the differential cross sections measured for each crystal versus the detection angle. The error bars on the differential cross sections include statistical error as well as the errors
of target thickness, the efficiency and the number of incident particles.
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25
/ ndf
χ2 20.35 / 27
p0 0.1905 ± 0.002922 p1 −0.1046 ± 0.03163 p2 −0.5465 ± 0.1076
/ ndf
χ2 20.35 / 27
p0 0.1905 ± 0.002922 p1 −0.1046 ± 0.03163 p2 −0.5465 ± 0.1076 = 125 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(a)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25
/ ndf
χ2 38.06 / 28
p0 0.1989 ± 0.002124 p1 0.02277 ± 0 p2 −0.1323 ± 0.03042
/ ndf
χ2 38.06 / 28
p0 0.1989 ± 0.002124 p1 0.02277 ± 0 p2 −0.1323 ± 0.03042 = 125 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(b)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25
/ ndf
χ2 54.9 / 28
p0 0.2013 ± 0.002274 p1 0.06207 ± 0.008836 p2 0.05091 ± 0
/ ndf
χ2 54.9 / 28
p0 0.2013 ± 0.002274 p1 0.06207 ± 0.008836 p2 0.05091 ± 0
= 125 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(c)
Detection angle (radians)
0 0.5 1 1.5 2 2.5 3
(mb/sr)Ω/dσd
0 0.05 0.1 0.15 0.2 0.25
/ ndf
χ2 74.01 / 29
p0 0.1976 ± 0.002113 p1 0.02277 ± 0 p2 0.05091 ± 0
/ ndf
χ2 74.01 / 29
p0 0.1976 ± 0.002113 p1 0.02277 ± 0 p2 0.05091 ± 0 = 125 MeV
proton
Ca E )40 754 keV
Ca(p,xγ
40
(d)
Figure 5.41: Angular distribution for the 40Ca 754.8 keV γ-ray line at 125 MeV proton energy. The solid lines are the fitted curves for the angular distri- bution function for a L = 2 transition. The data points represent the differential cross sections measured for each crystal versus the detection angle. The error bars on the differential cross sections include statistical error as well as the errors
of target thickness, the efficiency and the number of incident particles.
Chapter 5. Experimental cross section measurements 75