Figure 1

The experimental total cross section for neutrons scattering from ${}^{27}\mathrm{Al}$[92], ${}^{28}\mathrm{Si}$[92], ${}^{40}\mathrm{Ca}$[93], and ${}^{56}\mathrm{Fe}$[93] from 20 MeV to 160 MeV for the laboratory energy of the neutron. They are fit to three different optical potential calculations (KD[1], MD[2], and to this work: WP) which are described in the legend. The experimental data and theory have been offset by constant amounts for multiple comparisons on one graph. The WP and MD calculations have a minimum energy limit of 30 and 50 MeV respectively.Reuse & Permissions

Figure 2

The experimental total cross section for neutrons scattering from the light nuclei of ${}^{12}\mathrm{C}$[93], ${}^{14}\mathrm{N}$[92], ${}^{16}\mathrm{O}$[92], and ${}^{19}\mathrm{F}$[93] from 20 MeV to 160 MeV for the laboratory energy of the neutron. They are fit to two different optical potential calculations (MD[2], and to this work: WP) which are described in the legend. The experimental data and theory have been offset by constant amounts for multiple comparisons on one graph. The WP and MD calculations have a minimum energy limit of 30 and 50 MeV respectively. The KD calculation is not applicable for these light nuclei.Reuse & Permissions

Figure 3

The experimental total cross section for neutrons scattering from ${}^{31}\mathrm{P}$[93], ${}^{32}\mathrm{S}$[93], ${}^{39}\mathrm{K}$[93], and ${}^{48}\mathrm{Ti}$[93] from 20 MeV to 160 MeV for the laboratory energy of the neutron. They are fit to three different optical potential calculations (KD[1], MD[2], and to this work: WP) which are described in the legend. The experimental data and theory have been offset by constant amounts for multiple comparisons on one graph. The WP and MD calculations have a minimum energy limit of 30 and 50 MeV respectively.Reuse & Permissions

Figure 4

The experimental total cross section for neutrons scattering from ${}^{51}\mathrm{V}$[93], ${}^{52}\mathrm{Cr}$[93], ${}^{55}\mathrm{Mn}$[93], and ${}^{59}\mathrm{Co}$[93] from 20 MeV to 160 MeV for the laboratory energy of the neutron. They are fit to three different optical potential calculations (KD[1], MD[2], and to this work: WP) which are described in the legend. The experimental data and theory have been offset by constant amounts for multiple comparisons on one graph. The WP and MD calculations have a minimum energy limit of 30 and 50 MeV respectively.Reuse & Permissions

Figure 5

Neutron-nucleus elastic differential cross section experimental data for the target nuclei of ${}^{28}\mathrm{Si}$[87, 97, 98], ${}^{40}\mathrm{Ca}$[88, 98, 99], ${}^{56}\mathrm{Fe}$[87, 97, 98, 100] at a variety of incident laboratory energies. They are fit to three different optical potential calculations (KD[1], MD[2], and to this work: WP). As in all figures, KD OMP is a blue dashed line, MD OMP is a red dot-dashed line and WP OMP is a green solid line. The experimental data and theory have been offset by constant amounts for multiple comparisons on one graph. The target labels at the top read left to right correspond to the calculations and experimental data read top to bottom. The MD calculations has a minimum energy limit of 50 MeV and is therefore missing from calculations below that energy.Reuse & Permissions

Figure 6

Proton-nucleus total inelastic cross section data for the targets ${}^{12}\mathrm{C}$, ${}^{16}\mathrm{O}$, ${}^{27}\mathrm{Al}$, ${}^{40}\mathrm{Ca}$, ${}^{56}\mathrm{Fe}$, and ${}^{58}\mathrm{Ni}$ from 20 MeV to 160 MeV for the laboratory energy of the projectile proton. The experimental data comes from a variety of sources and are compiled and discussed in Refs. [1, 77, 95]. There are also new higher energy measurements for ${}^{12}\mathrm{C}$ and ${}^{40}\mathrm{Ca}$ found in Ref. [46]. They are fit to three different optical potential calculations (KD[1], MD[2], and to this work: WP) which are described in the legend. The experimental data and theory have been offset by constant amounts for multiple comparisons on one graph. The WP and MD calculations have a minimum energy limit of 30 and 50 MeV respectively.Reuse & Permissions

Figure 7

Proton-nucleus elastic Rutherford reduced differential cross section experimental data for the target nuclei of ${}^{27}\mathrm{Al}$[37, 38, 58, 104, 105], ${}^{28}\mathrm{Si}$[42, 106], ${}^{40}\mathrm{Ca}$[5, 38, 70], at varying proton laboratory energies. Refer to Fig. 5 for details of the legend for the theoretical calculations.Reuse & Permissions

Figure 8

Proton-nucleus elastic Rutherford reduced differential cross section experimental data for the target nuclei of ${}^{54}\mathrm{Fe}$[21, 72, 74, 81], ${}^{56}\mathrm{Fe}$[8, 72, 82, 84], ${}^{58}\mathrm{Ni}$[84, 107, 108, 109], at varying proton laboratory energies. Refer to Fig. 5 for details of the legend for the theoretical calculations.Reuse & Permissions

Figure 9

Proton-nucleus elastic Rutherford reduced differential cross section experimental data for the target nuclei of ${}^{34}\mathrm{Ar}$[66, 110], ${}^{30}\mathrm{S}$[110], ${}^{40}\mathrm{Ar}$[74], ${}^{45}\mathrm{Sr}$[72], ${}^{40,42,44,48}\mathrm{Ca}$[71, 74], ${}^{50,52,54}\mathrm{Cr}$[71], at varying proton laboratory energies. The target labels at the top read left to right correspond to the calculations and experimental data read top to bottom for the two rightmost panels. Refer to Fig. 5 for details of the legend for the theoretical calculations.Reuse & Permissions

Figure 10

Proton-nucleus elastic Rutherford reduced differential cross section experimental data for the target nuclei of ${}^{12,13}\mathrm{C}$[36, 51], ${}^{16}\mathrm{O}$[48, 56], ${}^{18,20,22}\mathrm{O}$[50, 57, 110], ${}^{28}\mathrm{Si}$[106], ${}^{40,42}\mathrm{Ca}$[70, 82], ${}^{40,42,44}\mathrm{Ar}$[68, 69], at varying proton laboratory energies. The target labels at the top read left to right correspond to the calculations and experimental data read top to bottom for the two leftmost panels. Refer to Fig. 5 for details of the legend for the theoretical calculations.Reuse & Permissions

Figure 11

Proton-nucleus elastic analyzing power spin observable experimental data for the target nuclei of ${}^{28}\mathrm{Si}$[74, 106], ${}^{40}\mathrm{Ca}$[43, 74, 113, 114], ${}^{54,56}\mathrm{Fe}$[21, 74, 84, 115], at varying proton laboratory energies. Refer to Fig. 5 for details of the theoretical calculations.Reuse & Permissions

Figure 12

Proton-nucleus elastic analyzing power spin observable experimental data for the target nuclei of ${}^{12,13}\mathrm{C}$[36, 51], ${}^{16}\mathrm{O}$[56], ${}^{28}\mathrm{Si}$[106], ${}^{40,42,44,48}\mathrm{Ca}$[73, 74], ${}^{40}\mathrm{Ar}$[52], ${}^{48,50}\mathrm{Ti}$[52], ${}^{50,52,54}\mathrm{Cr}$[73], at varying proton laboratory energies. The target labels at the top read left to right correspond to the calculations and experimental data read top to bottom. Refer to Fig. 5 for details of the theoretical calculations.Reuse & Permissions

Figure 13

Proton-nucleus elastic analyzing power spin observable experimental data for the target nuclei of ${}^{12,13}\mathrm{C}$[20, 20, 25, 30], ${}^{58}\mathrm{Ni}$[84, 107, 116], ${}^{59}\mathrm{Co}$[83, 84, 115], at varying proton laboratory energies. Refer to Fig. 5 for details of the theoretical calculations.Reuse & Permissions

Figure 14

A figure of theoretical comparisons. Plotted are the isoscalar volume real and imaginary space integrals between the three potentials that are under examination at 50 MeV and 150 MeV projectile energy. The light green, medium blue, and dark red lines represent the WP, KD, and MD OMPs respectively. The light gray shaded area represents typical volume integrals for microscopic $t-\rho$ potentials (as detailed in the text). Each atomic number has a sample nuclei attache to it. These nuclei are: ${}^{13}\mathrm{C}$, ${}^{14}\mathrm{C}$, ${}^{15}\mathrm{N}$, ${}^{16}\mathrm{N}$, ${}^{17}\mathrm{O}$, ${}^{18}\mathrm{O}$, ${}^{19}\mathrm{F}$, ${}^{20}\mathrm{F}$, ${}^{21}\mathrm{Ne}$, ${}^{22}\mathrm{Ne}$, ${}^{23}\mathrm{Na}$, ${}^{24}\mathrm{Na}$, ${}^{25}\mathrm{Mg}$, ${}^{26}\mathrm{Mg}$, ${}^{27}\mathrm{Al}$, ${}^{28}\mathrm{Al}$, ${}^{29}\mathrm{Si}$, ${}^{30}\mathrm{Si}$, ${}^{31}\mathrm{P}$, ${}^{32}\mathrm{P}$, ${}^{33}\mathrm{S}$, ${}^{34}\mathrm{S}$, ${}^{35}\mathrm{Cl}$, ${}^{36}\mathrm{Cl}$, ${}^{37}\mathrm{Cl}$, ${}^{38}\mathrm{Ar}$, ${}^{39}\mathrm{K}$, ${}^{40}\mathrm{Ar}$, ${}^{41}\mathrm{K}$, ${}^{42}\mathrm{Ca}$, ${}^{43}\mathrm{Ca}$, ${}^{44}\mathrm{Ca}$, ${}^{45}\mathrm{Sc}$, ${}^{46}\mathrm{Ti}$, ${}^{47}\mathrm{Ti}$, ${}^{48}\mathrm{Ti}$, ${}^{49}\mathrm{Ti}$, ${}^{50}\mathrm{V}$, ${}^{51}\mathrm{V}$, ${}^{52}\mathrm{Cr}$, ${}^{53}\mathrm{Mn}$, ${}^{54}\mathrm{Fe}$, ${}^{55}\mathrm{Mn}$, ${}^{56}\mathrm{Fe}$, ${}^{57}\mathrm{Co}$, ${}^{58}\mathrm{Ni}$, ${}^{59}\mathrm{Co}$, ${}^{60}\mathrm{Ni}$. These same nuclei will be used in many of the remaining figures.Reuse & Permissions

Figure 15

These are two example depictions at 65 MeV projectile energy of the functional coulomb correction potential for the targets ${}^{40}\mathrm{Ca}$ and ${}^{56}\mathrm{Fe}$ respectively. The solid black line is the magnitude of the exact calculation (the coulomb correction is actually negative), for the potential of this work, using Eq. (37). If the coulomb correction term is applied to the proton-nucleus optical potential then the potential can be considered isovector Lane consistent (described in the text). The solid green line is the approximate function which was determined in this work by fitting the global nucleon-nucleus data sets. This approximate form was determined by Eq. (38). The thinner red dashed line is given as an example Woods-Saxon functional form for comparison.Reuse & Permissions

Figure 16

This is a depiction in projectile energy-target atomic number phase space of the real fraction Lane inconsistency relative to the magnitude of the total moduli of the central potential for sample nuclei. The top panel shows the optical potential of this work without a coulomb correction term assuming only ${\mathcal{R}}_c=1.20\times A^{\frac{1}{3}}$ fm. The bottom panel depicts the same but with a coulomb correction term assuming a dynamic $E$ and $A$ dependent coulomb radius, $\mathcal{R}$, described in the text. The unweighted average for the whole phase space of the top panel is 0.043 Lane inconsistency. The unweighted average for the whole phase space of the bottom panel is 0.034 Lane inconsistency. The fraction of 0.04 is assumed here to be a pragmatic minimum and a measure of sufficient Lane isovector consistency. The sample nuclei for each atomic number were near or on the line of stability and are the same as those in Fig. 14.Reuse & Permissions

Figure 17

This is a depiction in projectile energy-target atomic number phase space of the imaginary fraction Lane inconsistency relative to the magnitude of the total complex central force for sample nuclei which are listed in Fig. 14. Overall the isospin inconsistencies are smaller than its real counterpart until $E>140$ MeV. The unweighted average for the whole phase space is 0.025 Lane inconsistency. The fraction of 0.04 is assumed here to be a pragmatic minimum and a measure of sufficient Lane isovector consistency.Reuse & Permissions

Figure 18

This is a calculation of the difference in the real component of the nuclear central potential integrals (volume and coulomb correction) between a proton and neutron projectile (about twice the actual isovector volume integral) for a representative selection of nuclei using three different global optical potentials. The nuclei were chosen to have nonzero $N-Z$ terms and to be close or on the line of stability. The nuclei are the same as used in Fig. 14. The dark blue semi-filled circular points are the calculation using the KD optical potential [1], the red unfilled circular points use the MD optical potential [2] and the light filled green circular data points are calculated using the WP potential of this work. The lines connecting the data points are there to make it easier for the eye to follow. The left panel used a 50 MeV nucleon projectile and the right panel uses a 150 MeV projectile where the differences in the three calculations become more pronounced. The light gray shaded area is the range of values for microscopic potentials which are described in the text in Sec. 4c.Reuse & Permissions

Figure 19

This is a calculation of the difference in the imaginary component of the central volume integrals between a proton and neutron projectile (about twice the isovector strength) for a representative selection of nuclei using the WP and KD global optical potential. The nuclei were chosen to have nonzero $N-Z$ terms and to be close or on the line of stability, the specific nuclei are listed in Fig. 14. The light green fill represents the range possible for nucleon projectile energies between 50 MeV and 150 MeV for the WP global optical potential of this work. The KD global optical potential (dark blue fill) is smaller and has a diminished range but its values are comparable to its real counterpart. The light gray shaded area is the range of values for microscopic potentials which are described in the text in Sec. 4c.Reuse & Permissions

Figure 20

This plot depicts the total normalized energy difference ($J_{\mathrm{iso}}{\rho }_{\mathrm{nuc}}$) for the proton and neutron projectiles on a representative selection of nuclei using two different global optical potentials using potentials at projectile energy of 150 MeV. The nuclei were chosen to have nonzero $N-Z$ terms and to be close or on the line of stability, the specific nuclei are listed in Fig. 14. These calculations are compared to the difference in experimental mass excess energies for the charge-exchange reactions, $Q(n,p)-Q(p,n)$ from the same targets, the experimental data (dark black circles) are from Ref. [146]. The MD calculations, the unfilled red circles, have multiplied the results of Eq. (49) by $.38\hspace{0.5em}{\mathrm{fm}}^{-3}$, the WP calculations, the light filled green circles, has been multiplied by $.16\hspace{0.5em}{\mathrm{fm}}^{-3}$. The experimental mass excess energies shown here were not used to constrain the OMPs during the fitting procedure. Two other potential calculations along with experiment are shown in Fig.  21.Reuse & Permissions

Figure 21

This plot depicts the total normalized energy difference ($J_{\mathrm{iso}}{\rho }_{\mathrm{nuc}}$) for the proton and neutron projectiles on a representative selection of nuclei using two different global optical potentials using potentials at projectile energy of 150 MeV. The nuclei were chosen to have nonzero $N-Z$ terms and to be close or on the line of stability, the specific nuclei are listed in Fig. 14. These calculations are compared to the difference in experimental mass excess energies for the charge-exchange reactions, $Q(n,p)-Q(p,n)$ from the same targets, the experimental data (dark black circles) are from Ref. [146]. The KD calculations, the filled blue circles, have multiplied the results of Eq. (49) by $.17\hspace{0.5em}{\mathrm{fm}}^{-3}$, the lighter gray circles (following the technique described in Ref. [122]), has been multiplied by 1.14 ${\mathrm{fm}}^{-3}$. The experimental mass excess energies shown here were not used to constrain the OMPs during the fitting procedure.Reuse & Permissions