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

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

J. Kiener Jurgen.Kiener@csnsm.in2p3.fr Centre de Sciences Nucléaires et de Sciences de la Matière (CSNSM), Univ Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, 91405 Orsay, France
July 16, 2019
Abstract

The emission of the 4.438-MeV -ray line in proton inelastic scattering off C has been investigated in detail. For this, two independent descriptions of the correlated scattering and emission process have been developed, one for the direct reaction mechanism and the other for the compound-nucleus (CN) component. Direct reactions were calculated in the framework of the coupled-channels formalism, while the CN component was described as a superposition of separate resonances with definite spin and parity, treated with the angular momentum coupling theory. Both components were incoherently added and compared to a comprehensive data set on measured line shapes and -ray angular distributions in the proton energy range = 5.44 - 25.0 MeV. In the range 14 - 25 MeV, good agreement with data was obtained with a dominating direct reaction component. At lower energy, an important CN component was required to describe line shape and angular distribution data. In general, a good decription of the data could be found for a CN component with spin and parity corresponding to a known nearby resonance. The new calculation improves significantly the agreement with line shape data compared to previous ones. Based on these results, predictions for line shapes and -ray angular distributions up to = 100 MeV were made and applied to -ray emissions in solar flares and proton radiotherapy.

pacs:
23.20En, 24.10Eq, 24.30Gd, 25.40Ep

Þ

I Introduction

The 4.438-MeV line of C is probably the most interesting target for line shape studies of proton-induced reactions for several reasons. The combination of such a relatively high -ray energy with the low mass of the C nucleus, leads to significant line broadening and energy shift even in proton reactions. Moreover, the lifetime of the emitting 2, 4.439-MeV level of C is short enough (t = 61 fs) for dominant in-flight emission of the ray in practically all materials. The emitting level at 4.439 MeV is furthermore the only excited level below the particle emission threshold in C. It results in negligible feeding by higher-lying levels and simplifies considerably the line shape calculations.

It is an important component in astrophysical sites with accelerated particle populations, where it is one of the strongest prompt nuclear deexcitation lines. It is for example regularly observed in strong solar flares Chupp et al. (1973); Murphy et al. (1997); Verstrand et al. (1999); Smith et al. (2003); Gros et al. (2004); Ackermann et al. (2014) and emission of this line from the inner Galaxy due to interactions of low-energy cosmic rays, is one of the primary science objectives of next-generation -ray observatories Benhabiles-Mezhoud et al. (2013); Tatischeff et al. (2016). Its main production in these sites comes from inelastic scattering of protons and particles off C and the O(p,p)C and O(,2)C reactions and reactions of accelerated C and O with ambiant H and He.

The interest of line shape analysis to deduce accelerated-particle properties has early been pointed out Ramaty and Crannell (1976). First detailed studies were applied to -ray spectra obtained with space-borne moderate-resolution scintillation detectors Share et al. (2002), and more recently with high-resolution HP-Ge detectors onboard the Rhessi and INTEGRAL satellites Smith et al. (2003); Kiener et al. (2006); Harris et al. (2007). In the latter in particular, detailed line shape analysis could be used to pin down composition and directional distribution of the accelerated-particle populations. Furthermore, there is significant interest in the monitoring of the dose deposition in radiotherapy with particle beams Durante and Paganetti (2016). A good example is provided by devices that detect the prompt rays resulting from proton interactions in human tissue, rising the demand of detailed and reliable nuclear cross sections Verburg et al. (2012, 2014); Jeyasugiththan and Peterson (2015). In this context, carbon and oxygen make up more than 3/4 of the human body mass, making the 4.438-MeV line the strongest prompt emission line produced.

A first global, astrophysically motivated approach for the calculation of the 4.438-MeV -ray line profile was presented by Ramaty, Kozlovsky and Lingenfelter Ramaty et al. (1979). It is based on measured differential proton scattering cross sections and a parameterization for the magnetic-substate population of the excited C, reproducing a few measured line shapes and -ray angular distributions, that were available at that time. This approach was further developed with new experiment data, but encountered problems to reproduce simultaneously line shapes and the -ray angular distribution Murphy et al. (2009). The agreement with measured data could be improved for proton energies where excitation of the 2, 4.439-MeV state in C proceeds dominantly through the direct reaction mechanism. Werntz, Lang and Kim Werntz et al. (1990) used nuclear reaction calculations in the coupled-channels formalism, expected to work well in this case, to obtain a complete description of the correlated proton inelastic scattering and subsequent -ray emission process.

Kiener, de Séréville and Tatischeff Kiener et al. (2001) adopted both methods to reproduce simultaneously line shapes and -ray angular distributions of an extensive data set, covering a relatively wide angular and energy range in proton irradiations of carbon and oxygen. The data for the 4.438-MeV line in proton inelastic scattering off C above 15 MeV could be relatively well described by nuclear reaction calculations with the coupled-channels formalism for the direct component, similar to the findings of Werntz, Lang and Kim Werntz et al. (1990). At lower energies, however, the CN component becomes important and the direct reaction mechanism failed to reproduce the data. There, measured line shapes and -ray angular distributions could be approximately described with the slightly modified approach of Murphy et al. (2009).

Since then, new -ray data for proton inelastic scattering off C are available Belhout et al. (2007) which, together with older data Kiener et al. (1998), form a comprehensive data set of line shapes and angular distributions for the 4.438-MeV line in the proton energy range from threshold to = 25 MeV. There are now, in particular, line shape data close to the energy of each CN resonance that shows up as a distinct peak in the -ray production cross section in proton reactions with C. With these data, a new method has been developed that improves specifically the description of line shapes and -ray angular distributions in the region of dominating CN resonances, i.e. from threshold to about = 12 MeV. At higher proton energies, coupled-channels calculations with a deformed potential for nucleon scattering off C Meigooni et al. (1985) reproduced fairly well the measured data. Line shape predicitions were then made, employing the same potential, for proton energies up to 100 MeV.

In the following section, the formalism of the calculations for the 4.438-MeV line emission in proton inelastic scattering off C will be outlined. The extended parameter search for these calculations to reproduce measured -ray line shapes and angular distributions is described and the results are discussed in section III. Then the interpolation of the calculations to higher proton energies is discussed and finally applications to solar-flare -ray emission and dose monitoring in proton radiotherapy are presented.

Ii Formalism for angular correlations

The 4.438-MeV line emission results from the deexcitation of the first excited state of C, 2 at 4.439 MeV with a half-life of 61 fs. This level belongs to the ground-state rotational band and is thus strongly coupled to the 0 ground state, which results in relatively high cross sections in proton inelastic scattering off C. The second excited state of C, a 0 state at 7.654 MeV, is already above the -particle emission threshold and has only a very weak -decay branching, as generally for all higher-excited states.

Lets mention for completeness that there are two 1 states at 12.71 and 15.11 MeV, that have a moderate -decay branching to the 4.439-MeV state. The inelastic scattering cross sections to these states are, however, much smaller than to the 4.439-MeV state, and cascades from these levels may be safely ignored for the present calculations. The only potential source of complexity is the 4.445-MeV state in B, whose deexcitaton gives rise to a 4.444-MeV -ray line. Nuclear reaction calculations predict that it may contribute significantly via the C(p,2p)B reaction above 25 MeV. This line shape component will be briefly discussed in the section with results of the line shape calculations.

Proton inelastic scattering to states of a collective band like the 4.439-MeV state of C generally has a strong direct reaction component. In such cases, calculations in the coupled-channels framework are often able to describe satisfactorily total and differential cross sections (see e.g. Satchler (1983) and references therein). This is readily done with nuclear reaction codes like Ecis Raynal (1994), that provide the necessary flexibility in the reaction parameter input to define the ground and excited states and their couplings, potentials etc. A broad outline of the nuclear reaction calculations has already been given in Kiener et al. (2001), but some practical aspects will be given anyhow further below.

At proton energies below about 15 MeV, the CN component is clearly present, evidenced by the distinct peaks in the cross section of C(p,p)C (see Fig. 1). The typical separation of the peaks suggests that they may be formed from isolated CN resonances with definite spin and parity and in fact, a good part of them can be identified with known states in the compound nucleus N. It is therefore tempting to describe the inelastic scattering reaction in the region up to about = 15 MeV by a superposition of the direct reaction mechanism and isolated CN resonances. In the following, the calculations of the line shapes and -ray angular distributions for the CN and direct component will be more detailed.

ii.1 Compound-nucleus resonances

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 , its decay by particle emission with angular momentum to an excited state of the target nucleus with spin , followed by a deexcitation ray with multipolarity to target state , is described as a cascade with angular momenta:

(1)

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 = 2, 4.439-MeV state to the C ground state with = 0 fixes = = 2, which reduces considerably the formula. For the line shape calcutions, in the case of an unobserved deexcited recoil nucleus, equation (2.96) leads to:

(2)

where the tensor describes the CN state, the are efficiency tensors for particle and -ray counters, is a factor containing the angular couplings, and are reduced matrix elements. Summation is over . The different terms are detailed in the appendix of the present paper, the general expression of is given in equation (2.98) in Ferguson.

The tensor is the result of the reaction in the ingoing channel, the formation of the CN state with angular momenta:

(3)

Here, = is the proton spin (spin of C = 0), which results in a unique orbital angular momentum for a given of the CN state. Assuming an unpolarized proton beam along the laboratory system z axis, only = 0 elements of are nonzero, and can be reduced to:

(4)

where = , is a Clebsch-Gordan coefficient and a Racah coefficent. runs from 0 to 2, and = 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 . For the outgoing proton, is the result of spin-orbit coupling = + . Since it can be supposed that does not depend on the proton spin (), the probability for decay of the CN state by proton emission with orbital angular momentum is given by . In the present studies, 2 different values of were sufficient to describe the CN reactions at all energies. Let be the smallest possible angular momentum in the decay channel, the branching ratio

(5)

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

ii.2 Direct reactions

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 C by inelastic scattering, is here contracted to one step with spins

(6)

where and are the C ground and excited state, respectively, is the proton spin and and the incoming and outgoing angular momenta.

The orientation of is the result of coherent summing of all contributions with possible combinations of , and . 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 ray is given by equation (10.132) of Satchler:

(7)

where the polarization tensors describe the state of orientation of the excited state after proton inelastic scattering with angle , and are the gamma-radiation parameters, detailed by equation (10.154) in Satchler (see Appendix).

The polarization tensors can be constructed from the scattering (or transition) amplitudes (equation (10.32b) in Satchler):

(8)

where , are the projections of spins and , respectively and the projections of the proton spin in the ingoing and outgoing channels. Summation in the present case () is only over .

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.)

Iii Line shape calculations

Measured line shapes come from essentially two experiments, both done at the tandem-Van-de-Graaff accelerator at Orsay. Line shapes from the first experiment in 1997 Kiener et al. (1998) have already been used for the line shape studies of Kiener, de Séréville and Tatischeff Kiener et al. (2001). Line shape data are available for selected runs in the range = 8.4 - 15.2 MeV with collodion targets (chemical composition CHNO) and with carbon targets in the range = 16.25 - 19.75 MeV. The energy ranges below = 8.4 MeV and above = 20 MeV were covered in the second experiment in 2002 Belhout et al. (2007) with carbon targets. Angular distribution data are available at some selected energies from the experiments of Dyer et al. Dyer et al. (). An overview of availabe line shape data and measured cross sections is shown together with known resonance states in N in Figure 1. Additionally to that, some line shapes are published in Kolata, Auble & Galonsky Kolata et al. (1967) at = 23 MeV and Lang et al. Lang et al. (1987) at = 40 MeV.

Figure 1: (Color online) Cross section as a fonction of laboratory proton energy for emission of the 4.438-MeV ray in proton inelastic scattering off C. Black dots connected with the black line are the data of Dyer et al. Dyer et al. (1981), small black squares the data of the Orsay-1997 experiment Kiener et al. (1998). Larger squares indicate Orsay data where line shapes are available: green for the 1997- and blue for the 2002-experiment Belhout et al. (2007). The data point at 15.15 MeV does not figure in ref. Kiener et al. (1998). Its large error bar reflects the uncertainty of the contribution of reactions with O present in the collodion target. Vertical lines indicate the position of N states in the range = 6.5 - 18 MeV with known spin and parity. Broad hatched lines indicate states with either large uncertainties on the excitation energy (100 keV or uncertainty not quoted) or very large widths (1 MeV) (values taken from NuDat at NNDC ref. NNDC ()).

In both Orsay experiments, -ray spectra have been obtained with large-volume coaxial HP-Ge detectors equipped with active BGO shielding. In the 1997 experiment Kiener et al. (1998), 8 detectors were used in the angular range = 45 - 145, and 5 detectors in the 2002 experiment Belhout et al. (2007) in the range = 45 - 157.5 in a first phase and after a detector failure, 4 detectors in the range = 67.5 - 157.5 in a second phase. The HP-Ge crystals had typically 7 cm diameter, 7 cm length and were at about 35 cm from the target. Except for the two highest proton beam energies = 22.5 and 25 MeV, high-statistics profiles of the 4.439-MeV line on a relatively small background have been recorded. For the comparison with calculated line shapes, the Compton background below the lines was subtracted by linear interpolation (1997 experiment) or with the help of Geant simulations of the detection setup (2002 experiment).

Line shape calculations for CN resonances and direct reactions were realized with two separate Monte-Carlo type programs, very similar except for the formalism describing the angular correlations. The programs consisted in event-by-event simulation of the reaction sequence from the inelastic scattering reaction to the emission and detection of the ray. The interaction depth in the target, center-of-mass (CM) scattering angle , time of the -ray emission and -ray emission angles , were randomly drawn from cumulative distributions reflecting the different probabilities. For the -ray emission, the probabilities depend on the CM scattering angle. Therefore, a series of cumulative ()-distributions were made for a grid of CM scattering angles with = 5 in the case of CN resonances and = 10 for direct reactions.

Recoil angle and energy of the excited C in the laboratory were determined from nuclear reaction kinematics following the CM scattering angle and incoming proton energy. The slowing down of the recoil C in the target was calculated with the stopping power tables of SRIM Ziegler () until the deexcitation of the 4.439-MeV state and emission of the ray with a half-life of = 61 fs occurred. Angular straggling was ignored as well as the geometrical effect of the finite beam-spot size on the target. Polar angle and energy of the emitted ray in the laboratory were calculated with relativistic kinematics and stored in an energy-angle histogram. Line shapes were then constructed from the energy-angle histogram for the solid angles spanned by the HP-Ge detectors employed in the Orsay experiments. Only full-energy events in the detectors were considered, with the effect of detector energy resolution on the line shapes, typically 4.5 keV FWHM at 4.4 MeV, included. A few million events per proton energy were simulated, resulting in high-statistics calculated line profiles.

For the direct reaction component, differential cross sections and angular correlations were obtained with the nuclear reaction code ECIS Raynal (1994) in the coupled-channels formalism. Rotational coupling between the C 0 ground state and the 2 state at 4.439 MeV with deformation parameters = -0.61 and = 0.05 was used. At energies above = 15 MeV, the 4 state at 14.08 MeV was also included. This coupling scheme was taken from Meigooni et al. Meigooni et al. (1985), who used it in coupled-channels calculations to derive an energy- and channel-dependent potential for nucleon scattering off C in the range = 20-100 MeV. Calculations with this potential were already succesfully used in the previous study of Kiener, de Séréville and Tatischeff Kiener et al. (2001) for the direct reaction component above = 15 MeV. It provided also good results at lower energies, except at a few energies below = 10 MeV. There, other potentials from the compilation of Perey and Perey Perey and Perey (1976) were also tried and sometimes provided a better description of the data.

Line shapes and -ray angular distributions of the CN component at each proton energy were calculated with the spins and parities of close by N states. It provided mostly satisfactory descriptions of data in the region of dominating CN component with a few exceptions discussed below. At some proton energies, where several nearby states exist, calculations with all of the ’s were made and compared to the data. Calculations with the different spin parities showed in general marked differences in the line shapes or the -ray angular distributions, such that the choice was usually evident.

After selection of the CN state , the branching ratio of CN state decay orbital angular momenta and the proportion of CN () and direct reaction component (1-) were scanned, aiming for a simultaneous good reproduction of the line shapes in all detectors and the -ray angular distribution. Practically, for each , varied in steps in steps of 0.05 - 0.2, the parameter was found in a fit of the angular distribution and the resulting line shapes calculated and compared to the data. Usually, the calculated line shapes at most or all detector angles and the angular distribution had their best agreement with experiment for very similar parameter values.

Iv Results

iv.1 Orsay-1997 experiment

The results of the parameter search for the CN and direct component to reproduce the measured line shapes and -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 for the CN component. At all energies except the lowest, the (C) obtained with the calculated angular distributions are similar to and sometimes even smaller than the (L) of Legendre-polynomial fits to the angular distribution data. The are in both cases obtained with 3 free parameters (, and an overall normalization factor for the calculated distributions, three coefficients for the Legendre-polynomial fit of an electric quadrupole transition 2 0). Error bars on the experimental data are of the order of 4 - 6 %.

The of the best adjustment corresponds in nearly all cases to a resonance state in N whose energy is within , the total with of the state, to the excitation energy in the CN reaction. At = 12.0 MeV ( = 13.0 MeV), a = of the most probable state at 13.50(20) MeV ( 6500 keV) could not reproduce the data. They were reproduced by a = , that may be attributed to the state at 12.937(24) MeV ( 400 keV) whose is not known. No nearby state in N was found at = 16.25 MeV, but the measured data could also be reasonably reproduced without a CN component.

As expected, the proportion of the CN component 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 = 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 were obtained for and resonances, but is more probable because the width of the N state with = ( = 10.360 MeV, = 75 keV) seems barely compatible with the apparent peak width of 200 keV in the cross section data of Dyer et al. Dyer et al. (1981).

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

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

Figure 2: (Color online) Measured shapes of the 4.439-MeV -ray line from proton-inelastic scattering off C in the Orsay-1997 experiment (blue symbols) and results of the line shape calculation with parameters of table 1 (red lines) at the proton beam energies indicated on the figures.
Figure 3: (Color online) Measured angular distribution data of the 4.439-MeV -ray line from proton-inelastic scattering off C in the Orsay-1997 experiment are presented by black symbols. The results of the calculations with parameters of table 1 are shown by dotted red, dashed green and solid blue lines for the CN and direct component, and the sum of both, respectively. Proton beam energies are indicated on the figures.

iv.2 Orsay-2002 experiment

The results of the parameter search to reproduce measured line shapes and -ray angular distributions in the energy ranges = 5.44 - 8.4 MeV and = 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 component. In the range = 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 = 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 at 4 proton energies.

For the CN component below = 7.6 MeV, the best agreements were obtained with ’s corresponding to the most probable N state. At = 8.0 - 8.4 MeV, calculations with = and of the closest N states at 9.000 and 9.476(8) MeV, respectively, were unable to describe the data. They could, however, be reproduced with a component, probably corresponding to the state at 7.900 MeV whose total width is 1500 keV. Above = 20 MeV, the data could best be described with a CN component. A corresponding state with known spin parity could only be found for = 20 MeV. Here, line shapes are better reproduced with a negligible CN component, while the angular distributions are better fitted with 30% CN component ((C) = 0.3-2.1). The indicated values 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 = 20 MeV are dominated by the direct reaction component. and are typically constrained within 0.15, except at = 20 - 25 MeV, where the uncertainties on are of the order of 0.3.

Examples of line shapes and angular distribution adjustments at 4 proton beam energies are shown in Figs. 4, 5. There is an excellent agreement between the calculated laboratory -ray angular distributions and the measured data for the 3 lower proton energies. As mentioned above, the angular distribution at = 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 = 5.44 MeV and 20 MeV reproduce nicely the experimental data, while for = 7.0 MeV and 8.23 MeV, the agreement is generally good, except at 157.5.

Figure 4: (Color online) Measured shapes of the 4.439-MeV -ray line from proton-inelastic scattering off C in the Orsay-2002 experiment (blue symbols) and results of the line shape calculation with parameters of table 2 (red lines) at the proton beam energies indicated on the figures.
Figure 5: (Color online) Measured angular distribution data of the 4.439-MeV -ray line from proton-inelastic scattering off C in the Orsay-2002 experiment are presented by black symbols. The different lines have the meaning as in Fig. 3. Proton beam energies are indicated on the figures.

iv.3 Other experiments

The most comprehensive data set of 4.438-MeV -ray cross sections in proton inelastic scattering off C has been obtained at the Washington tandem accelerator from threshold up to = 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 = 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.

Figure 6: (Color online) Measured angular distribution data of the 4.439-MeV -ray line from proton-inelastic scattering off C from Orsay 1997 (black squares), Orsay 2002 (black circles) and Washington (cyan triangles). The different lines have the same meaning as in Fig. 3. Proton beam energies are indicated on the figures. For = 20 MeV, the data and calculated distributions at = 19.75 MeV from the Orsay-1997 experiment have been used.

Kolata, Auble & Galonsky Kolata et al. (1967) published partial -ray spectra around the second escape peak of the 4.438-MeV line at 11 different angles in the range = 20 - 160 for = 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.

Figure 7: (Color online) Blue squares show measured shapes of the 4.439-MeV -ray line from proton-inelastic scattering off C at = 23 MeV (data around 2 escape peak from Kolata, Auble & Galonsky Kolata et al. (1967), energy corrected and shifted by 1.022 MeV, see text) and results of the line shape calculation (red lines).

For = 40 MeV, a line shape at = 90 and the -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 B at = 40 MeV, by deexcitation of its second excited state at 4.445 MeV. Using for the C(p,2p)B reaction the method of Kiener, de Séréville and Tatischeff Kiener et al. (2001) for the calculation of the 4.439-MeV line in the O(p,p)C 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 B contribution. This contribution was consequently neglected for the line shape calculations in the next section.

Figure 8: (Color online) Left: Measured shape of the 4.439-MeV -ray line from proton-inelastic scattering off C at = 40 MeV and = 90 (Lang et al. Lang et al. (1987), blue squares) and results of the line shape calculation (red lines). The right panel shows the -ray angular distribution data of Lang et al. (black symbols) and the calculated curve.

V Discussion and applications

Line shapes and -ray angular distributions are well reproduced at all energies up to = 20 MeV with one single for the CN component, that in many cases can be attributed to a known state in N. There is a strong CN component up to 12 MeV, above that the direct reaction reaction component dominates. For = 20-25 MeV, only the -ray angular distributions of the Orsay-2002 experiment favor a significant CN component, while the angular distributions of the Orsay-1997 and the Washington experiments, as well as the line shapes of Kolata, Auble & Galonsky and of both Orsay experiments favor a negligible CN component. The latter is also expected from nuclear reaction theory and it is thus highly probable that the CN component is negligible for the 4.438-MeV line emission above = 20 MeV.

Although not all line shapes are perfectly reproduced by the calculations, the present study brings a significant improvement with respect to previous line shape calculations. The improvement is particularly obvious in the proton energy range with strong CN contribution, where the full treatment of particle- correlations in the CN component with the angular momentum coupling theory results in much better line shape and angular distribution adjustments than the magnetic-substate population method as proposed in Ref. Ramaty et al. (1979) and extensively used in Ref. Kiener et al. (2001). This is illustrated in Fig. 9 where the line shape data of the Orsay-1997 experiment and calculations are summed with weight factors corresponding to a thick-target interaction probability of an injected beam with energy spectrum , as encountered in solar flares.

Figure 9: (Color online) Weighted sums of measured shapes of the 4.439-MeV -ray line from proton-inelastic scattering off C in the Orsay-1997 experiment (blue squares) and calculated line shapes (red lines) from the present study (left) and the previous study (right) of Ref. Kiener et al. (2001).

At higher energies, the new data of the Orsay-2002 experiment at = 22.5 and 25 MeV and the data of Refs. Kolata et al. (1967); Lang et al. (1987) at = 23 MeV and 40 MeV, respectively, consolidate the results of Refs. Kiener et al. (2001); Werntz et al. (1990) pointing out a largely dominating direct reaction component. For this component, the potential of Meigooni et al. Meigooni et al. (1985) is very successful, predicting correctly the line shapes up to = 40 MeV and the angular distributions up to = 25 MeV with certainty, and very probably up to = 40 MeV. This potential, developed for elastic and inelastic nucleon scattering off C in the range = 20 - 100 MeV should then also give quite reasonable line shapes and -ray angular distributions up to 100 MeV. It is thus a very important ingredient for applications requiring calculations in a large proton energy range. With the present results, the 4.438-MeV line emission for solar-flare conditions and in proton radiotherapy of eye tumors can be calculated. In both sites, it is a thick target interaction where proton reactions below = 100 MeV are dominating the line emission.

v.1 Solar flares

In impulsive solar flares, the 4.438-MeV line is mainly produced in reactions of energetic protons and particles with C and O Ramaty et al. (1979). It is also produced in reverse kinematics by energetic C and O, but with a very large width. Taking a typical incident particle population with an energy spectrum following a power law in energy with spectral index = 3.5 in the range = 5 - 100 MeV and an -to-proton ratio of 0.1, that is stopped in the solar atmosphere with a carbon-to-oxygen abundance ratio of 0.5, slightly more than half of the 4.438-MeV rays is produced by proton inelastic scattering off C. Also, more than 90% of the proton inelastic scattering reactions take place at energies lower than = 25 MeV, completely covered in the present study by calculations that are validated by experimental data.

The most precise solar flare data for the 4.438-MeV line are from the October 28, 2003 event, observed by the gamma-ray spectrometer SPI onboard the INTEGRAL satellite Gros et al. (2004). A detailed analysis of the 4.438-MeV C and 6.129-MeV O lines 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 a relatively narrow downward-directed accelerated particle beam at the flare location with an -to-proton ratio of 0.1. In particular, it could be shown that a relatively narrow distribution around the flare axis as resulting from pitch-angle scattering (PAS) with a small mean free path (=30) could perfectly describe both lines.

Taking the parameters of Ref. Kiener et al. (2006) for the orientation of the flare axis, the carbon-to-oxygen ratio in the solar atmosphere and incident particle spectra with power-law spectral index of = 3.5, the 4.438-MeV line shape was calculated for PAS(=30) directional distribution and a range of energetic -to-proton ratios . The calculation of proton and -particle reactions with O was done following the method described in Kiener et al. (2001). For the O(p,p)C reaction, isotropic emission of proton, particle and ray perfectly reproduced measured angular distributions and line shapes below = 20 MeV. Extrapolation to higher energies and the O(,2)C reaction should be quite accurate in this case. Only the calculations of -particle inelastic scattering off C have not yet been confronted to measured line shapes or -ray angular distributions. Its contribution to the 4.438-MeV emission is about 15% for = 0.1.

For proton inelastic scattering off C, the previous line shape calculations Kiener et al. (2001) and the present ones have been used. The best fit 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, different . 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. In this example, -particle inelastic scattering plays a non-negligible role and still higher values have been proposed to prevail in impulsive solar flares Murphy et al. (1991). For the future, an improvement in the line shape calculations for -particle inelastic scattering would therefore be very appreciable for such studies.

Figure 10: (Color online) Left: Symbols show the observed spectrum around the 4.438-MeV line from the Oct. 28, 2003 solar flare, with Compton and continuum backgrounds subtracted (see Ref. Kiener et al. (2006) for details). The full green and dashed magenta lines represent the best-fit results of calculated line shapes from the present study ( = 0.22) and the previous one ( = 0.38) (Ref. KST Kiener et al. (2001)), respectively. The results of minimization in the spectrum energy range = 4300-4580 keV as a function of is shown on the right figure.

v.2 Proton radiotherapy

Similar to solar flares, the 4.438-MeV line in human tissue during proton radiotherapy is essentially produced by inelastic scattering off C and reactions with O. Incident proton energies are in the range of about 60 MeV to 200 MeV. For an incident proton energy of = 68 MeV, typical for eye cancer treatment, there are 2.710 4.438-MeV rays emitted per incident proton due to inelastic scattering off C, and 5.610 per incident proton due to proton reactions with O and a negligible part (10 per incident proton) is from reactions with N. 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 C, the cross sections of table I in Ref. Kiener et al. (2001) for proton reactions with O and from the cross section compilation of Ref. Murphy et al. (1988) for proton reactions with N. The 4.438-MeV line yield from proton reactions with O is reduced by about 25% if one uses the compilation, Ref.Murphy et al. (1988) for cross sections with O instead of Ref. Kiener et al. (2001). The energy loss of protons in human tissue is calculated with the tables of SRIM Ziegler ().

Figure 11: (Color online) Angular distribution (left panel) and line shapes (right panel) of the 4.439-MeV -ray emitted in reactions of a 68-MeV proton beam stopped in human tissue. Solid blue lines represent the sum of proton reactions with C and O, dashed red lines show the component of reactions with C. Dot-dashed green lines represent the contribution of reactions with O.

The resulting 4.438-MeV line exhibits characteristic shapes at the different observation angles. Knowledge of the shapes should facilitate the extraction of the 4.438-MeV line integral in -ray spectra eventually measured by radiation monitor detectors. Inelastic scattering modulates the -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 -ray emission in reactions with C is produced for proton energies above = 25 MeV, and about 80% in reactions with O. 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 = 100 MeV, where other potentials may be needed for the inelastic scattering off C. More importantly, cross sections of the 4.438-MeV line emission in proton reactions with O have only been measured up to = 50 MeV, and there is a strong discrepancy at = 40 MeV between the data of Refs. Lang et al. (1987) and Lesko et al. (1988).

Vi Summary

A new study for a detailed calculation of the 4.438-MeV -ray emission in proton inelastic scattering off C has been presented. It is based on results of coupled-channels calculations for the direct reaction component and angular momentum coupling theory for CN resonances. Parameters of the calculations were adjusted by comparison with a comprehensive data set of measured line shapes and -ray angular distributions in the proton energy range from threshold to = 25 MeV. All experimental data could be relatively well reproduced by a CN component of definite spin and parity incoherently added to the direct reaction component. With this study, a substantial improvement with respect to previous line shape calculations, in particular in the energy range of dominating CN resonances could be obtained.

At energies above 14 MeV, where the direct reaction dominates, the data are well described by the results of coupled-channels calculations with an energy- and channel-dependent potential, that has been derived for nucleon scattering off C in the range up to = 100 MeV. This enables straightforward extrapolations to higher proton energies, and the scarce data available in this range, one line shape and the -ray angular distribution at = 40 MeV, are also reasonably described by these calculations. With the present results, line shapes and -ray angular distributions can be predicted with accuracy for proton inelastic scattering off C from threshold to at least = 100 MeV.

Applications to solar-flare -ray emission and proton radiotherapy were discussed, and differences with a previous method outlined. The 4.438-MeV -ray emission in solar flares can now be accurately calculated for not too high -to-proton ratios, but would greatly benefit from an improvement in the treatment of inelastic -particle scattering. Applications to proton radiotherapy would also benefit from new experimental data, in particular for proton reactions with C above = 100 MeV and O above = 30 MeV. Fortunately, new experimental data for p + C and p + O reactions up to = 200 MeV from the iThemba LABS cyclotron should soon be available. First results are encouraging Yahia-Cherif et al. (), such that still more accurate predictions in the cited applications can be possible in the near future.

Acknowledgements.
I like to express my gratitude to I. Deloncle and V. Tatischeff for their suggestions and critical reading of the manuscript.

Vii Appendix

vii.1 Compound-nucleus resonances

In equation 2, the efficiency tensor 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 = 0, and ideal counters for all particles are supposed. For a beam of unpolarized protons:

(9)

where results from the vector sum of proton spin and orbital angular momentum .

The efficiency tensor for the -ray counter , with = = for the transition and no polarization and parity mixture (eqs. (2.57) and (2.78) in Ferguson), is:

(10)

The term with angular couplings is:

(11)

where means the Wigner 9-j coefficient with indices written line by line.

The probability for inelastic scattering with scattering angle can be obtained from equation 2 with the efficiency tensor of unobserved ray and excited nucleus

(12)

and integration over and :

(13)

with of equation 9, and the sum running over , , and . The differential cross section is proportional to .

vii.2 Direct reactions

In equation 7, Satchler’s gamma-radiation parameters (eqs. (10.153) (10.154) in Satchler (1983)) are, with = 1; , = even and = 0:

(14)

When using the nuclear reaction code Ecis97 Raynal (1994) for OM calculations, the scattering amplitudes have to be constructed from other output, like the scattering matrix elements . 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 , and for incoming (outgoing) channel:

(15)

they are given by eq. (5.45) in Satchler, in the scattering plane ( = 0) with the incoming beam in direction of the z-axis ( = 0) and = 0:

(16)

where are the spin projections of orbital angular momenta , target ground state spin and excited state spin , proton spin and coupled spins , respectively. is the phase (in Ecis97 output: phase with Coulomb) of the scattering matrix . The sum runs over , , and ; here = for = 0.

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  (C)   (L)   N state    remarks
8.6 0.16 9.8 0.35 1.0   23   9.5   (), 10.25(15)
9.0 0.15 10.2 0.35 1.0   9.2   6.1   (), 10.25(15)
9.2 0.15 10.4 0.66 1.0   5.3   7.0   (), 10.25(15)   no line shapes
9.2 0.15 10.4 0.78 0.7   5.1   7.0   , 10.360   no line shapes
9.6 0.15 10.7 0.53 0.5   3.4   3.8   , 10.833(9)
10.0 0.14 11.1 0.49 0.3   3.0   3.5   , 11.530(12)
10.6 0.14 11.7 0.76 0.2   2.5   3.1   , 11.740(40)
11.0 0.13 12.0 1.00 0.8   7.0   6.4   , 12.130(50)
11.4 0.13 12.4 0.95 0.8   6.7   7.9   , 12.130(50)
12.0 0.12 13.0 0.34 0.4   10.4   10.2   ( ), 12.937(24)
12.6 0.02 13.6 0.08 0.0   13.6   10.6   , 13.50(20)
13.0 0.02 13.9 0.12 0.2   17.8   14.2   , 14.050(20)
13.6 0.02 14.5 0.21 0.4   13.4   10.2   , 13.50(20)
14.0 0.02 14.8 0.33 1.0   14.7   13.3   (), 15.30(20)
14.4 0.11 15.2 0.17 0.0   14.7   11.2   (), 15.30(20)
15.2 0.10 15.9 0.00 0.5   13.8   8.9   , 15.990(30)   15% of O(p,p)
16.25 0.01 16.9 0.13 1.0   13.8   7.6
17.25 0.01 17.9 0.20 1.0   8.2   5.1   , 18.150(30)
18.25 0.01 18.8 0.10 1.0   10.9   6.8   , 19.830
19.75 0.01 20.2 0.10 1.0   11.8   5.8   , 19.830
Table 1: Results of line shape and -ray angular distribution calculations for the Orsay experiment in 1997. and are the beam energy and energy loss in the target in MeV, respectively, and the excitation energy in N corresponding to -/2. and are the spin-parity and proportion of the CN component of the best adjustment. The probable corresponding state in N, with its excitation energy in MeV (values from NNDC ()), is shown in column 9. (C) is the result of the least-squares fit of the measured data with the calculated angular distributions and (L) the result of the corresponding Legendre-polynomial fit. Values in italics for , the proportion of the lowest possible decay angular momentum, are not constraint.
  (C)   (L)   N state    remarks
5.44 0.04 6.9 0.76 1.0   0.3   0.09   , 6.886(8)
5.95 0.04 7.4 0.36 0.9   0.7   0.37   , 7.376(9) pot. PEA72 Pearson et al. (1972)
6.5 0.04 7.9 0.73 0.4   0.4   0.002   , 7.900
7.0 0.04 8.4 0.57 0.2   0.4   0.01   , 7.900 pot. GUR69 Guratzsch et al. (1969)
7.6 0.04 8.9 0.61 0.5   3.5   0.46   , 8.918(11) pot. PEA72 Pearson et al. (1972)
8.0 0.04 9.3 0.44 0.6   0.3   0.18   , 7.900
8.23 0.04 9.5 0.57 0.6   2.7   0.13   , 7.900 pot. PEA72 Pearson et al. (1972)
8.4 0.04 9.7 0.45 0.7   1.3   0.02   , 7.900
20.0 0.02 20.4 0.15 0.4   3.0   0.13   , 20.200
22.5 0.02 22.7 0.15 1.0   2.4   0.01   , 22.4(5)
25.0 0.02 25.0 0.15 1.0   3.1   1.5
Table 2: Results of line shape and -ray angular distribution calculations for the Orsay experiment in 2002. The different columns are explained in table 1. The last column gives the reference of the used optical potential. When no potential is indicated, the one of Meigooni et al. Meigooni et al. (1985) was used for the direct component.
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