Shape and angular distribution of the 4.438-MeV line from proton inelastic scattering off ¹²C

The formalism for inelastic scattering reactions proceeding through an intermediate CN resonance, or CN state, has been taken from the monograph of Ferguson Ferguson (1965). Taking a CN state with spin $b$, its decay by particle emission with angular momentum $L$ to an excited state of the target nucleus with spin $c$, followed by a deexcitation $\gamma$ ray with multipolarity $L_{\gamma}$ to target state $d$, is described as a cascade with angular momenta:

The most general formulation of the angular correlation is given by equation (2.96) in Ferguson, implying a summation over 20 indices. In the present case, the deexcitation of the $c$ = 2⁺, 4.439-MeV state to the ¹²C ground state with $d$ = 0⁺ fixes $L_{\gamma}$ = $c$ = 2, which reduces considerably the formula. For the line shape calculations, in the case of an unobserved deexcited recoil nucleus, equation (2.96) leads to:

where the tensor $t_{kq}(b)$ describes the CN state, the $\epsilon^{\star}_{kq}$ are efficiency tensors for particle and $\gamma$-ray counters, $f_{c}$ is a factor containing the angular couplings, and $M$ are reduced matrix elements. Summation is over $k,q,k_{L},q_{L},k_{L_{\gamma}},q_{L_{\gamma}},L,L^{\prime}$. The different terms are detailed in the appendix of the present paper, the general expression of $t_{kq}(b)$ is given in equation (2.98) in Ferguson.

The tensor $t_{kq}(b)$ is the result of the reaction in the ingoing channel, the formation of the CN state with angular momenta:

Here, $s$ = $\frac{1}{2}^{+}$ is the proton spin (spin of ¹²C_g.s. = 0⁺), which results in a unique orbital angular momentum $l_{i}$ for a given $b$ of the CN state. Assuming an unpolarized proton beam along the laboratory system z axis, only $q$ = 0 elements of $t_{kq}(b)$ are nonzero, and $t_{kq}(b)$ can be reduced to:

where $\hat{j}$ = $\sqrt{2j+1}$, $<l_{i}0,l_{i}0\mid k0>$ is a Clebsch-Gordan coefficient and $W(l_{i}bl_{i}b;ak)$ a Racah coefficent. $k$ runs from 0 to 2$l_{i}$, and $k$ = even, selected by the Clebsch-Gordan coefficient.

Equation 2 is thus completely defined by the angular momenta and their couplings, with the exception of the relative values of the reduced matrix elements $M(cLb)~M(cL^{\prime}b)$. For the outgoing proton, $L$ is the result of spin-orbit coupling $\vec{L}$ = $\vec{l}$ + $\vec{s}$. Since it can be supposed that $M(cLb)$ does not depend on the proton spin ($M(cLb)=M(clb)$), the probability for decay of the CN state by proton emission with orbital angular momentum $l$ is given by $W(l)\propto M(clb)M(clb)^{\star}$. In the present studies, 2 different values of $l$ were sufficient to describe the CN reactions at all energies. Let $l_{0}$ be the smallest possible angular momentum in the decay channel, the branching ratio

is the only free parameter. For the line shape calculations, equation 2 was calculated as a function of $\theta_{p}$, separately for $l_{0}$ and $l_{0}+2$. The differential cross section ${d\sigma}/{d\Omega}(\theta_{p})$ can be obtained from equation (2.96) in Ferguson, by assuming unobserved $\gamma$ ray and excited nucleus (see Appendix). Finally, the line shapes and $\gamma$-ray angular distributions of the CN state were taken as an incoherent sum of both angular momenta with the respective weight factors $W_{l_{0}}$ and (1 - $W_{l_{0}}$).

The formalism has been taken from the monograph of Satchler Satchler (1983). It is similar to the above described one for CN resonances, but with a different division of the treatment of the reaction sequence. The first two steps in the CN case, equations 3 and 1(1), leading to the excited state of ¹²C by inelastic scattering, is here contracted to one step with spins

where $a$ and $b$ are the ¹²C ground and excited state, respectively, $s$ is the proton spin and $l_{i}$ and $l_{f}$ the incoming and outgoing angular momenta.

The orientation of $b$ is the result of coherent summing of all contributions with possible combinations of $l_{i}$, $l_{f}$ and $s$. It is calculated in the framework of nuclear reactions in the coupled-channels formalism. Practically, the program Ecis97 Raynal (1994) was used for these calculations.

The angular distribution of the emitted $\gamma$ ray is given by equation (10.132) of Satchler:

where the polarization tensors $t_{kq}(b,\theta_{p})$ describe the state of orientation of the excited state $b$ after proton inelastic scattering with angle $\theta_{p}$, and $R_{k}(\gamma)$ are the gamma-radiation parameters, detailed by equation (10.154) in Satchler (see Appendix).

The polarization tensors $t_{kq}(b,\theta_{p})$ can be constructed from the scattering (or transition) amplitudes $T$ (equation (10.32b) in Satchler):

where $\alpha$, $\beta$ are the projections of spins $a$ and $b$, respectively and $\sigma_{i,o}$ the projections of the proton spin in the ingoing and outgoing channels. Summation in the present case ($\alpha=a=0$) is only over $\beta,\sigma_{i},\sigma_{o}$.

Scattering amplitudes are either obtained directly from the nuclear reaction codes, or have to be assembled from other output parameters. The latter is the case for the code Ecis97 Raynal (1994), where the amplitudes can be derived from the scattering-matrix elements (see Appendix.)

The results of the parameter search for the CN and the used nuclear potential for the direct component to reproduce the measured line shapes and $\gamma$-ray angular distributions are summarized in table 1. Good agreement with measured data could be obtained at each energy with the potential of Meigooni et al. Meigooni et al. (1985) for the direct reaction component and one single resonance $J^{\pi}$ for the CN component. At all energies except the lowest, the $\chi^{2}$(C) obtained with the calculated angular distributions are similar to and sometimes even smaller than the $\chi^{2}$(L) of Legendre-polynomial fits to the angular distribution data. The $\chi^{2}$ are in both cases obtained with 3 free parameters ($W_{l_{0}}$, $W_{CN}$ and an overall normalization factor for the calculated distributions, 3 coefficients for the Legendre-polynomial fit of an electric quadrupole transition 2⁺ $\rightarrow$ 0⁺). Error bars on the experimental data are of the order of 4 - 6 %.

The $J^{\pi}$ of the best adjustment corresponds in nearly all cases to a resonance state in ¹³N whose energy is within $\Gamma_{tot}$, the total width of the state, to the excitation energy $E_{x}$ in the CN reaction. At $E_{p}$ = 12.0 MeV ($E_{x}$ = 13.0 MeV), calculations with $J^{\pi}$ = $\frac{3}{2}^{+}$ corresponding to the most probable state at 13.50(20) MeV ($\Gamma_{tot}$ $\sim$6500 keV) could not reproduce the data. They were reproduced by a $J^{\pi}$ = $\frac{7}{2}^{-}$, that may be attributed to the state at 12.937(24) MeV ($\Gamma_{tot}$ $>$ 400 keV) whose $J^{\pi}$ is not known. No nearby $\frac{5}{2}^{+}$ state in ¹³N was found at $E_{p}$ = 16.25 MeV, but the measured data could also be reasonably reproduced without a CN component.

As expected, the proportion of the CN component $W_{CN}$ is high for proton energies on the three narrow peaks at 9.1, 10.4 and 10.9 MeV in the cross section excitation function, see Fig. 1. For the peak at 9.1 MeV, the closest measurement is at $E_{p}$ = 9.2 MeV, where, however, no line shapes were available. Because of the particular importance of the cross section here, the result of the angular distribution fit was nonetheless included in table 1. There, equivalent $\chi^{2}$ were obtained for $\frac{1}{2}^{+}$ and $\frac{7}{2}^{-}$ resonances, but $\frac{1}{2}^{+}$ is more probable because the width of the ¹³N state with $J^{\pi}$ = $\frac{7}{2}^{-}$ ($E_{x}$ = 10.360 MeV, $\Gamma_{tot}$ = 75 keV) seems barely compatible with the apparent peak width of $\sim$200 keV in the cross section data of Dyer et al. Dyer et al. (1981).

Below $E_{p}$ = 12 MeV with significant CN component, the values of $W_{l_{0}}$ and $W_{CN}$ are typically constrained to within $\pm$0.15. For $\frac{1}{2}^{-}$ resonances, decay angular momenta $l$ = 1 and $l$ = 3 give exactly the same line shapes and $\gamma$-ray angular distributions, such that $W_{l_{0}}$ could not be constrained. For $\frac{1}{2}^{+}$ resonances, only $l$ = 2 is possible. Above that energy, the CN component is weak, reasonable adjustements could also be obtained throughout with $W_{CN}$ = 0, and the constraints on $W_{l_{0}}$ are weaker ($\pm$ $\sim$0.3). At the highest energy with the collodion target, $E_{p}$ = 15.2 MeV, the ¹⁶O($\alpha$,2$\alpha$)¹²C reaction contributes and has been added in the calculations.

Calculated and measured line shapes and angular distributions at 4 proton beam energies are shown in Figs. 2 and 3. It includes proton energies with largely dominating CN or direct reaction components, at $E_{p}$ = 11.4 MeV and $E_{p}$ = 16.25 MeV, respectively, and energies with a more equal mixture of both components at $E_{p}$ = 8.6 MeV and at $E_{p}$ = 10.0 MeV. These examples represent also the typical degree of agreement between calculation and measured data, with very good adjustments of the $\gamma$-ray angular distribution and reasonably well reproduced line shapes with some deviations of the order of 20% at a few angles.

The results of the parameter search to reproduce measured line shapes and $\gamma$-ray angular distributions in the energy ranges $E_{p}$ = 5.44 - 8.4 MeV and $E_{p}$ = 20.0 - 25 MeV, are listed in table 2. Good agreements with measured data could be obtained with the direct reaction and one single resonance $J^{\pi}$ component. In the range $E_{p}$ = 5.44 - 8.4 MeV, the angular distribution adjustments are good to excellent, considering error bars on the measured data of 1.5 - 2.5 %. For the three highest energies, experimental error bars are of the order of 5 - 15 % due to lower count statistics, and adjustments are good.

The potential of Meigooni et al. Meigooni et al. (1985) gave good results in the $E_{p}$ = 20-25 MeV range but was not able to reproduce several of the experimental data below 8 MeV. Attempts with modified strengths of the surface imaginary potentials and different coupling schemes could in some cases improve the agreement with data, but were not conclusive. Also attempts with a low-energy extension Weppner of another global nucleon-nucleus optical model Weppner et al. (2009), did not give a better description of the data. Finally, for the sake of simplicity, it was decided to use optical model potentials listed in the compilation of Perey and Perey Perey and Perey (1976) for similar proton energies (see table 2), and without modification in the coupled-channels calculations. This could significantly improve the agreement with the experimental data for 4 proton energies.

For the CN component below $E_{p}$ = 7.6 MeV, the best agreements were obtained with $J^{\pi}$’s corresponding to the most probable ¹³N state. At $E_{p}$ = 8.0 - 8.4 MeV, calculations with $J^{\pi}$ = $\frac{9}{2}^{+}$ and $\frac{3}{2}^{-}$ of the closest ¹³N states at 9.000 and 9.476(8) MeV, respectively, were unable to describe the data. They could, however, be reproduced with a $\frac{3}{2}^{+}$ component, probably corresponding to the $\frac{3}{2}^{+}$ state at 7.900 MeV whose total width is $\Gamma_{tot}$ $\sim$1500 keV. Above $E_{p}$ = 20 MeV, the data could best be described with a $\frac{5}{2}^{-}$ CN component. A corresponding state with known spin parity could only be found for $E_{p}$ = 20 MeV. Here, line shapes are better reproduced with a negligible CN component, while the angular distributions are better fitted with $\sim$30% CN component ($\chi^{2}$(C) = 0.3-2.1). The indicated values $W_{CN}$ represent a compromise and are also in better agreement with adjustments of other angular distribution data in this energy range, see below.

Again, similar to the results for the Orsay-1997 data, there is a sizeable CN component at lower proton energies, while the data above $E_{p}$ = 20 MeV are dominated by the direct reaction component. $W_{CN}$ and $W_{l_{0}}$ are typically constrained within $\pm$0.15, except at $E_{p}$ = 20 - 25 MeV, where the uncertainties on $W_{l_{0}}$ are of the order of $\pm$0.3.

Calculated and measured line shapes and angular distributions at 4 proton beam energies are shown in Figs. 4, 5. There is an excellent agreement between the calculated laboratory $\gamma$-ray angular distributions and the measured data for the 3 lower proton energies. As mentioned above, the angular distribution at $E_{p}$ = 20 MeV would be better fitted with about twice as much of the quasi-isotropic CN component, however with a degrading fit for the line shapes. The calculated line shapes at $E_{p}$ = 5.44 MeV and 20 MeV reproduce nicely the experimental data, while for $E_{p}$ = 7.0 MeV and 8.23 MeV, the agreement is generally good, except at 157.5^∘.

The most comprehensive data set of 4.438-MeV $\gamma$-ray cross sections in proton inelastic scattering off ¹²C has been obtained at the Washington tandem accelerator from threshold up to $E_{p}$ = 23 MeV by Dyer et al. Dyer et al. (1981). No line shapes are available from that experiment, but angular distributions are available at some selected energies Dyer et al. . They agree well with the Orsay-1997 and 2002 data up to $E_{p}$ = 18 MeV, and thus support the results presented in Tables 1, 2. At higher energies, the Washington and the Orsay-1997 data have slightly more pronounced minima and maxima than the Orsay-2002 data and clearly agree with negligible CN components. This is illustrated on Fig. 6 where Orsay and Washington data scaled to the same integrated cross section, are shown together for some proton energies.

Kolata, Auble & Galonsky Kolata et al. (1967) published partial $\gamma$-ray spectra around the second escape peak of the 4.438-MeV line at 11 different angles in the range $\theta_{lab}$ = 20 - 160^∘ for $E_{p}$ = 23 MeV. These data, after subtracting an estimated background and applying an energy shift of 40 keV for all detectors (in the spectra of Fig. of Ref. Kolata et al. (1967), only channel numbers are shown, and the energy dispersion is indicated in the caption), are well reproduced by the direct reaction component with the potential of Meigooni et al. Meigooni et al. (1985) and with a negligible CN component. A comparison with calculated line shapes at 6 different angles is shown on Fig. 7.

For $E_{p}$ = 40 MeV, a line shape at $\theta$ = 90^∘ and the $\gamma$-ray angular distribution have been published by Lang et al. Lang et al. (1987). The line shape is well reproduced by a pure direct reaction component, calculated with the potential of Meigooni et al. Meigooni et al. (1985), but there is an obvious disagreement between the calculated and the mesured angular distribution, see Fig. 8. However, the difference of 17% between the differential cross section data at 70^∘ and 110^∘ is much larger than the possible assymetry with respect to 90^∘ in the laboratory system, having furthermore the wrong sign. This may indicate an underestimation of the error bars, presented to be of the order of 5%.

Calculations with the nuclear reaction code Talys Koning et al. (2008) predict a non-negligible contribution of ¹¹B at $E_{p}$ = 40 MeV, by deexcitation of its second excited state at 4.445 MeV. Using for the ¹²C(p,2p)¹¹B reaction the method of Kiener, de Séréville and Tatischeff Kiener et al. (2001) for the calculation of the 4.439-MeV line in the ¹⁶O(p,p$\alpha$)¹²C reaction, the 4.445-MeV line at 90^∘ presents a flat profile in the range 4350 - 4530 keV and an isotropic angular distribution. The deep valley at the nominal line energy of the measured 4.439-MeV line shape (Fig. 8) leaves practically no space for a sizeable ¹¹B contribution. This contribution was consequently neglected for the line shape calculations in the next section.

$\gamma$-ray emission from the Sun during strong solar flares is regularly observed by orbiting spacecraft since more than 40 years. The high-energy photon spectrum often features some strong, relatively narrow nuclear $\gamma$-ray lines sitting on a quasi-continuum composed of thousands of other, weaker or broader, nuclear lines and a smooth spectrum from electron bremsstrahlung (see e.g. Verstrand et al. (1999)). This emission is thought to be produced in impulsive solar flares by energetic particles that are accelerated in the solar corona up to GeV energies and trapped in magnetic loops on the solar surface. The interactions leading to electron bremsstrahlung and nuclear $\gamma$-ray line emissions happen then essentially in the solar chromosphere and photosphere, where the energetic particles are eventually stopped.

Important questions in solar flares are the composition, the energy spectrum and the directional distribution of the energetic particles, that are linked to the mechanism that accelerates and propagates them in the solar atmosphere. Composition and directional distribution, and to a lesser extent the energy spectrum, can be deduced from the observed shapes of the strongest nuclear $\gamma$-ray lines. The 4.438-MeV line is often, together with 2.225-MeV neutron capture line on ¹H and the 511-keV annihilation line, the strongest line from nuclear interactions. The narrow component of the 4.438-MeV line in solar flares is mainly produced in interactions of energetic protons and $\alpha$ particles with ¹²C and ¹⁶O of the solar atmosphere Ramaty et al. (1979). The line is also produced in reverse kinematics by energetic ¹²C and ¹⁶O interacting with ambient ¹H and ⁴He, but with a very large width and it is hardly detectable.

The line-shape calculations are done with a solar flare loop model of the energetic particle interactions as detailed in Refs. Hua et al. (1989); Murphy et al. (1990), that describes many features of the observed solar flare $\gamma$-ray emission. Accelerated particles are injected isotropically at the top of a magnetic loop, consisting of an arc in the solar corona connected to two straight portions along a solar radius through the chromosphere down to the loop footpoints in the photosphere, with field strength B constant in the corona and depending on the pressure in the other regions. Particle transport along the magnetic field lines is calculated, including pitch-angle scattering on MHD turbulence until the particle is absorbed in a nuclear interaction or stopped. The interactions of the energetic particles in this model are predominantly induced by downward-directed particles inside the chromosphere.

For the spectrum and composition of the incident, accelerated particle population the values of Ref. Kiener et al. (2006) are used, which were deduced from the analysis of the October 28, 2003 solar flare. This flare was observed by the gamma-ray spectrometer SPI onboard the INTEGRAL satellite Gros et al. (2004) and features the most precise data available for the 4.438-MeV line in solar flares. In the described solar flare loop model for this case, slightly more than half of the 4.438-MeV $\gamma$ rays accounting for the narrow component are produced by proton inelastic scattering off ¹²C. Also, more than 90% of these proton inelastic scattering reactions take place at energies lower than $E_{p}$ = 25 MeV, completely covered by the calculations, validated by experimental data, of the present study.

A detailed analysis of the 4.438-MeV ¹²C and 6.129-MeV ¹⁶O lines observed from that flare was made in Ref. Kiener et al. (2006), based on the line shape calculations of Ref. Kiener et al. (2001). There, both observed lines could be very well reproduced with relatively narrow downward-directed energetic particle distributions and, for a simultaneous fit of the 4.438-MeV and 6.129-MeV lines, with an $\alpha$-to-proton ratio of $\sim$0.1. In particular, it could be shown that a narrow particle distribution around the flare axis as resulting from pitch-angle scattering (PAS) with a small mean free path, defined as the PAS($\lambda$=30) distribution in the model of Hua et al. (1989); Murphy et al. (1990), could perfectly describe both lines, with the flare axis being perpendicular to the solar surface at the flare coordinates.

Taking the parameters of Ref. Kiener et al. (2006) for the carbon-to-oxygen ratio in the solar atmosphere and the energy spectra of the incident particles, the 4.438-MeV line shape in the present study was calculated for the PAS($\lambda$=30) directional distribution and a range of energetic $\alpha$-to-proton ratios ($\alpha/p$). The calculation of proton and $\alpha$-particle reactions with ¹⁶O was done following the method described in Kiener et al. (2001). For the ¹⁶O(p,p$\alpha$)¹²C reaction, isotropic emission of proton, $\alpha$ particle and $\gamma$ ray perfectly reproduced measured angular distributions and line shapes below $E_{p}$ = 20 MeV. Extrapolation to higher energies and to the ¹⁶O($\alpha$,2$\alpha$)¹²C reaction should be quite accurate in this case.

For proton inelastic scattering off ¹²C, the previous line shape calculations Kiener et al. (2001) and the present ones have been used. The best fit $\alpha/p$ and the resulting line shapes, compared with the observed data, are shown in Fig. 10. A nearly perfect fit could be obtained with both line shape calculations with, however, significantly different $\alpha/p$. This result is in line with the findings of Ref. Kiener et al. (2006) about the directional distribution of the interacting energetic particles, but shows also the sensitivity to small differences in the line shape calculations for inelastic proton scattering. It illustrates clearly the need for accurate line-shape calculations in a wide particle energy range, as it is done for the first time in the present work.

It is worth mentioning also that in this example, $\alpha$-particle inelastic scattering plays a non-negligible role. Its contribution to the 4.438-MeV emission is for example about 30% for $\alpha/p$ = 0.22. Still higher $\alpha/p$ values of $\sim$0.5 have been proposed to prevail in impulsive solar flares Murphy et al. (1991). For the future, an improvement in the line shape calculations for $\alpha$-particle inelastic scattering would therefore be very appreciable for such studies.

Similar to solar flares, the 4.438-MeV line in human tissue during proton radiotherapy is essentially produced by inelastic scattering off ¹²C and reactions with ¹⁶O. Incident proton energies are in the range of about 60 MeV to 200 MeV. For an incident proton energy of $E_{p}$ = 68 MeV, typical for eye cancer treatment, there are 2.7$\cdot$10^-3 4.438-MeV $\gamma$ rays emitted per incident proton due to inelastic scattering off ¹²C, and 5.6$\cdot$10^-3 per incident proton due to proton reactions with ¹⁶O and a negligible part ($\sim$10^-4 per incident proton) is from reactions with ¹⁴N. The calculated angular distribution and shapes of the 4.438-MeV line are shown on Fig. 11. The calculations were done with the average composition of the human body Emsley (1983), the experimental cross sections for the 4.438-MeV line production in proton inelastic scattering off ¹²C, the cross sections of table I in Ref. Kiener et al. (2001) for proton reactions with ¹⁶O and from the cross section compilation of Ref. Murphy et al. (2009) for proton reactions with ¹⁴N. The 4.438-MeV line yield from proton reactions with ¹⁶O is reduced by about 25% if one uses the compilation, Ref.Murphy et al. (2009) for cross sections with ¹⁶O instead of Ref. Kiener et al. (2001). The energy loss of protons in human tissue is calculated with the tables of SRIM Ziegler .

The resulting 4.438-MeV line exhibits characteristic shapes at the different observation angles. The knowledge of these shapes is essential for the extraction of the 4.438-MeV line integral in $\gamma$-ray spectra eventually measured by radiation monitor detectors. Inelastic scattering modulates the $\gamma$-ray angular distribution, with a difference of about 10-15% between the maximum at 45^∘ and the minima. At this incident proton energy, slightly more than half of the $\gamma$-ray emission in reactions with ¹²C is produced for proton energies above $E_{p}$ = 25 MeV, and about 80% in reactions with ¹⁶O. The calculations are thus strongly based on extrapolations, but line shapes and angular distributions should nonetheless be quite accurate, as explained above. The present study should therefore allow a more precise monitoring of the radiation dose, independent of the position of the radiation monitor detector with respect to the beam direction. This may not hold for proton energies well above $E_{p}$ = 100 MeV, where other potentials may be needed for the inelastic scattering off ¹²C. More importantly, cross sections of the 4.438-MeV line emission in proton reactions with ¹⁶O have only been measured up to $E_{p}$ = 50 MeV, and there is a strong discrepancy at $E_{p}$ = 40 MeV between the data of Refs. Lang et al. (1987) and Lesko et al. (1988).

In equation 2, the efficiency tensor $\epsilon^{\star}_{k_{L}q_{L}}$ of the emitted proton in the CN resonance decay for a counter with axial symmetry, can be derived from equations (2.52) and (2.78) in Ferguson Ferguson (1965). In the present calculations, the scattering plane is taken as the y - z plane of the laboratory system, thus $\phi_{p}$ = 0, and ideal counters for all particles are supposed. For a beam of unpolarized protons:

where $L$ results from the vector sum of proton spin $s$ and orbital angular momentum $l$.

The efficiency tensor for the $\gamma$-ray counter $\epsilon^{\star}_{k_{L}{{}_{\gamma}}q_{L}{{}_{\gamma}}}$, with $L_{\gamma}$ = $L^{\prime}_{\gamma}$ = $c$ for the $2^{+}$ $\rightarrow$ $0^{+}$ transition and no polarization and parity mixture (eqs. (2.57) and (2.78) in Ferguson), is:

The term $f_{c}$ with angular couplings is:

where $W_{9}$ means the Wigner 9-j coefficient with indices written line by line.

The probability for inelastic scattering with scattering angle $\theta_{p}$ can be obtained from equation 2 with the efficiency tensor of unobserved $\gamma$ ray and excited nucleus

and integration over $\theta_{\gamma}$ and $\phi_{\gamma}$:

with $\epsilon_{kq}(LL^{\prime})$ of equation 9, and the sum running over $k$, $q$, $L$ and $L^{\prime}$. The differential cross section $d\sigma/d\Omega(\theta_{p})$ is proportional to $W(\theta_{p})$.

In equation 7, Satchler’s gamma-radiation parameters (eqs. (10.153) (10.154) in Satchler (1983)) are, with $\mid g_{l}\mid$ = 1; $k$, $l,l^{\prime}$ = even and $I_{c}$ = 0:

When using the nuclear reaction code Ecis97 Raynal (1994) for OM calculations, the scattering amplitudes $T_{\beta,\sigma_{o},\alpha,\sigma_{i}}(\theta_{p})$ have to be constructed from other output, like the scattering matrix elements $S^{j}_{l_{o}j_{o}l_{i}j_{i}}$ (see equation 16). As this construction happened to be among the more laborious tasks in the calculations, an outline is therefore given here. In the spin-orbit coupling scheme with proton spin $s$, and $i(o)$ for incoming (outgoing) channel:

they are given by eq. (5.45) in Satchler, in the scattering plane ($\phi_{p}$ = 0) with the incoming beam in direction of the z-axis ($\lambda_{i}$ = 0) and $a$ = 0:

where $\lambda,\alpha,\beta,\sigma,\eta$ are the spin projections of orbital angular momenta $l$, target ground state spin $a$ and excited state spin $b$, proton spin $s$ and coupled spins $j$, respectively. $\sigma_{ps}$ is the phase (in Ecis97 output: phase with Coulomb) of the scattering matrix $S$. The sum runs over $l_{i}$, $l_{o}$, $j_{i}$ and $j_{o}$; here $j$ = $j_{i}$ for $a$ = 0.