^{1}

^{2}

^{*}

The possibility of the description of the available experimental data for cross sections of the neutron capture reaction on
^{10}
B at thermal and astrophysical energies, taking into account the resonance at 475 keV, was considered within the framework of the modified potential cluster model with forbidden states and accounting for the resonance behavior of the scattering phase shifts.

Light radioactive nuclei play an important role in many astrophysical occurrences. Such parameter as a total cross section of the capture reactions as a function of energy is very important for investigation of many astrophysical problems such as primordial nucleosynthesis of the universe, main trends of stellar evolution, novae and super-novae explosions, X-ray bursts, etc. The continued interest in the study of processes of radiative neutron capture on light nuclei at thermal and astrophysical energies is caused by several reasons. Firstly, this pro- cess plays a significant part in the study of many fundamental properties of nuclear reactions; and secondly, the data on the capture cross sections are widely used in a various applications of nuclear physics and nuclear astrophysics, for example, in the process of studying of the primordial nucleosynthesis reactions.

One extremely successful line of development of nuclear physics in the last 50 years has been the microscopic model known as the Resonating Group Method (RGM, see, for example, [

However, the possibilities offered by a simple two-body potential cluster model (PCM) have not been studied fully up to now, particularly if it uses the concept of forbidden states (FSs) [

Particularly, in works of [^{6}Li, including its quadrupole moment in the potential cluster model with tensor forces [^{2}H, p^{2}H, p^{3}H, n^{6}Li, p^{6}Li, n^{7}Li, p^{7}Li, p^{9}Be, n^{9}Be, p^{10}B, p^{11}B, n^{11}B, n^{12}C, p^{12}C, n^{13}C, p^{13}C, n^{14}C, p^{14}C, n^{14}N, n^{15}N, p^{15}N, n^{16}O and ^{2}H^{4}He, ^{3}He^{4}He, ^{3}H^{4}He, ^{4}He^{12}C systems at thermal and astrophysical energies is considered. These calculations of the listed above capture processes are carried out on the basis of the modified variant of PCM, described in [

Therefore, in continuing to study the processes of radiative capture [^{10}B interaction for scattering processes will be constructed based on the reproduction of the spectra of resonance states for the final nucleus in the n^{10}B channel. The n^{10}B potentials are constructed based on the description both of the binding energies of these particles in the final nucleus and of certain basic characteristics of these states; for example, the charge radius and the asymptotic constant (AC) for the bound state (BS) or the ground state (GS) of ^{11}B, were formed as a result of the capture reaction in the cluster channel, which coincides with the initial particles [

The study of the reaction^{11}B is a part of the reaction chains in the so-called inhomogeneous Big Bang models [^{10}B:

Evidently, the considered reaction can play a certain role in some models of the universe [

However, it seems to us that the study of this reaction is also interesting, even though that it has been impossible for us to find any similar theoretical calculations for the reaction ^{10}B is a very good absorber of neutrons that it is used in control rods in nuclear reactors. This property also makes it useful for construction of neutron detectors. Boron is used to make windows that are transparent to infrared radiation, for high-temperature semiconductors, and for electric generators of a thermoelectric type [

We regard the results of the classification of ^{11}B by orbital symmetry in the n^{10}B channel as qualitative, because there are no complete tables of Young tableaux productions for systems with more than eight nucleons [^{10}B.

Furthermore, we will suppose that it is possible to assume the orbital Young tableau in the form ^{10}B; therefore, for the n^{10}B system, we have ^{10}B system (for ^{10}B it is known^{6}Р_{3/2} wave (representation in^{11}B with^{10}B system of −11.4541(2) MeV [^{10}B scattering states and BSs can be mixed by isospin with

The spectrum of ^{11}B for excited states (ESs), bound in the n^{10}B channel, shows that at the energy of 2.1247 MeV above the GS or −9.3329 MeV [^{10}B channel, the first ES can be found, bound in this channel with the moment^{6}F_{1/2} wave with an FS. However, we will not consider it, because of the large value of the angular momentum barrier. The second ES at the energy 4.4449 MeV [^{10}B channel has the moment^{6}P_{5/2} and ^{8}P_{5/2} waves without FSs. Furthermore, the unified potential of such a mixed P_{5/2} state will be constructed, because the model used does not allow one to divide states with different spin clearly. The wave function obtained with this potential at the calculation of the Schrödinger equation and, in principle, consisting of two components for different spin channels, does not divide into these components in the explicit form, i.e., we are using the total form of the WF in all calculations. The third ES at the energy of 5.0203 MeV [^{6}P_{3/2} wave without an FS. The fourth ES at the energy of 6.7429 MeV [^{6}P_{7/2} and ^{8}P_{7/2} waves without an FS. In addition, it is possible to consider the ninth ES at the energy of 8.9202 MeV with the moment 5/2^{−}, i.e., at the energy of −2.5339 MeV relative to the n^{10}B threshold, which can be matched to the mixture of the ^{6}P_{5/2} and ^{8}P_{5/2} states without FSs.

Consider now the resonance states in the n^{10}B system, i.e., states at positive energies. The first resonance state of ^{11}B in the n^{10}B channel, located at the energy 0.17 MeV, has the neutron width of 4 keV and the moment ^{6}S_{5/2} scattering wave with an FS. We have not succeeded in the construction of the potential with such small width; therefore, we will consider this scattering wave as non-resonant, which leads to zero scattering phase shifts. The second resonance state has the energy of 0.37 MeV―its neutron width equals 0.77 MeV and the moment ^{8}S_{7/2} scattering wave with an FS. Because of the large width of resonance (two times greater than its energy), we will use non-resonant values of the parameters for the potential to coincide with the previous ^{6}S_{5/2} potential. The third resonance state has the energy of 0.53 MeV in the laboratory system (l.s.)―its neutron width equals 0.031 MeV in the center-of-mass system (c.m.) and the moment ^{6}P_{5/2} + ^{8}P_{5/2} scattering waves without FSs. These characteristics of the resonance are given in

The next resonance is at the energy above 1 MeV and we will not consider it (see ^{11}B that can be matched to the ^{6}P_{3/2} and ^{6+8}P_{7/2} states [^{6+8}P_{5/2} scattering waves without FSs. The width equal to 1.34 MeV, which is given for the state with^{8}S_{7/2} scattering wave with an FS―it is twice that given in [

Continuing to the analysis of possible electromagnetic E1 and M1 transitions, let us note that we will consider only transitions to the GS and to four (2^{nd}, 3^{rd}, 4^{th}, and 9^{th}) ESs from the S and P scattering waves. One can see

The total radiative capture cross sections

where

The value

Here, ^{10}B particles with intercluster distance

For consideration of the

Here,

No. | E1 transitions | Comments | No. | M1 transitions | Comments |
---|---|---|---|---|---|

1 | As the GS is matched with the ^{6}Р_{3/2} level, it is possible to consider E1 transitions from the ^{6}S_{5/2} scattering wave to the GS of ^{11}B. | 6 | Furthermore, it is possible to consider M1 transitions to the GS from the resonance scattering wave ^{6}Р_{5/2} at 0.475(17) MeV, and from the non-resonance ^{6}Р_{3/2} wave. As will be shown later, the cross section of the transition ^{6}Р_{3/2} scattering potential with zero depth will stay at the level 1 - 2 μb, and the other transitions from the non-resonance waves will not be considered. | ||

2 | In addition, it is possible to consider E1 transitions from the ^{6}S_{5/2} and ^{8}S_{7/2} scattering waves to the second ES of ^{11}B, which is the mixture of two P states. Because here we have transitions from the initial S states that differ by spin to the different parts of one total WF of the BS, the cross section of these transitions will sum up, i.e., | 7 | Furthermore, it is possible to consider M1 transitions to the second ES ^{6}P_{5/2} and ^{8}P_{5/2} from the resonance ^{6}P_{5/2} and ^{8}P_{5/2} scattering waves. Because this is the transition from the mixed-by-spin Р_{5/2} scattering wave to the mixed-by-spin second ES, the cross section will be averaged according to transitions given above, i.e., | ||

3 | The third ES is matched with the ^{6}Р_{3/2} level as the GS, and it is possible to consider the E1 transitions from the ^{5}S_{5/2} scattering waves to this ES of ^{11}B. | 8 | It is possible to consider the M1 transitions to the third ES ^{6}P_{3/2} from the resonance ^{6}Р_{5/2} scattering wave. | ||

4 | Another E1 transition is possible from the ^{6}S_{5/2} and ^{8}S_{7/2} scattering waves of the fourth ES of ^{11}B at | 9 | The M1 transitions are feasible to the fourth ES | ||

5 | The last of considered E1 transitions is the capture from the ^{6}S_{5/2 }and ^{8}S_{7/2} scattering waves to the ninth ES of ^{11}B at | 10 | Finally, we can consider the M1 transitions to the ninth ES |

of which are taken from [

The construction methods used here for intercluster partial potentials at the given orbital moment^{2}.

For all partial waves of the n^{10}B interaction potentials, i.e., for each partial wave with the given

Here, as mentioned before, we will not consider the influence of the first resonance at 0.17 MeV in the ^{6}S_{5/2} wave; therefore, we will use the potential with FSs leading to the zero scattering phase

The ^{6}S_{5/2} scattering phase shift of this potential at energy up to 1.0 MeV is less than 0.5˚. The same parameters we be used for the ^{8}S_{7/2} scattering wave, also ignoring the resonance.

The following parameters were obtained for the third resonance state

Such potential leads to resonance, i.e., the scattering phase shift equals 90.0˚, at 475(1) keV (l.s.) with a width of 193(1) keV (c.m.), which is in good agreement with the data of [

For the potential of the pure-by-spin GS of ^{11}B in the n^{10}B channel, where the 6P_{3/2} wave is used, the following parameters were obtained:

We have obtained the value of the dimensionless AC = 1.53(1) in the range of 3 - 10 fm, the charged radius of 2.44 fm, and the mass radius of 2.39 fm at the binding energy of −11.454100 MeV with the accuracy of the finite-difference method, used for the calculation of the binding energy at

The AC value equal to 1.72 fm^{−1/2} was obtained in Ref. [^{11}B in the cluster channel n^{10}B, where the coefficient of neutron identity was assigned (see expression 83b in [^{−1/2} for the GS; therefore, the AC value equals 1.44 in the dimensionless form. The improved value of 1.82(15) fm^{−1/2} is given in the latest results for this AC (see [

The parameters of the GS potential and any BSs in the considered channel at the given number of the bound, allowed or forbidden states in the partial wave, are fixed quite unambiguously by the binding energy, the charge radius, and the asymptotic constant. The accuracy of the determination of the BS potential parameters is connected with the accuracy of the AC, which is usually equal to 10% to 20%. There are no another ambiguities in this potential, because the classification of the states according to the Young schemes allows us unambiguously to fix the number of BSs in this partial wave, which defines its depth completely, and the width of the potential depends wholly on the values of the charge radius and the AC.

The next parameters were obtained for the parameters of the ^{11}B in the n^{10}B channel with

This potential leads to the binding energy of −7.0092 MeV at

The next parameters were obtained for the potential without FSs for the third ES ^{6}Р_{3/2} pure-by-spin with

These parameters lead to the binding energy of −6.4338 MeV at

The next parameters were obtained for the ^{11}B in the n^{10}B channel with

The binding energy of −4.7112 MeV at

These parameters were obtained for the ^{11}B in the n^{10}B channel with

This potential leads to the binding energy of −2.5339 MeV at

The next experimental data were used for the comparison of the calculation results given in ^{11}B are shown in ^{6}Р_{3/2}, open squares (□) represent the total capture cross section to the second ES ^{6+8}Р_{5/2}, black squares (■) represent the total capture cross section to the fourth ES ^{6+8}Р_{7/2}, and open triangles (D) represent the total capture cross section to the ninth ES ^{6+8}Р_{5/2}. Furthermore, in

The E1 transition

the transition to the fourth ES (11), identified in Section 2 as No. 4. The dot-dot-dashed line, which is almost superimposed with the dotted line, shows the transition from the S scattering waves to the ninth ES with potential (12). The solid line gives the total summed cross section of all the above considered transitions, which largely describes the experimental data for the total summed cross sections from [

Let us note that in the measurements of [

As can be seen from the obtained results, the calculated line for the transition to the fourth ES is in a good agreement with the given black squares (experimental data) [

Because, we do not know the AC value for the second ES, it is always possible to construct the potential correctly describing the capture cross sections to this state, shown in

which leads to the binding energy of −7.0092 MeV, the charged radius of 2.44 fm, and the value of the AC equal to 1.45(1) at the range of 4 - 13 fm. The calculation results of the capture cross sections to this state from the S scattering waves are shown in

At the same time, the other variant of the GS potential that describes the total capture cross sections to the GS correctly, shown in

Allow one to describe reasonably the available experimental cross section measurements of the transition [^{11}B at the radiative neutron capture on ^{10}B should be improved in the future; it will also be interesting to obtain new data in the range of possible resonances from 100 to 600 keV.

Reverting to the calculation results given in

The value of the given constant A = 2123.4694 μb∙keV^{1/2} was determined from a single point of the cross-sec- tions (solid line in ^{1/2}.

Furthermore, the considered M1 transitions to the GS and to the different ESs are shown in ^{6}Р_{5/2} scattering wave for potential (7), identified in Section 2 as No. 6. The dot-dashed line it is the transition from the Р_{5/2} scattering wave (7) to the second ES with potential (9), identified in Section 2 as No. 7. The dot-dot-dashed line shows the cross section of the M1 transition ^{6}Р_{3/2} scattering wave was considered with the potential of zero depth―the second transition under No. 6 in Section 2. The result is shown by the dotted line with closely placed dots in the bottom of ^{10}B from [

The sum of all the E1 and M1 transitions described above is shown in _{5/2} scattering wave of 193 keV.

As can be seen from the listed results, the obvious assumptions about the methods of construction of the n^{10}B interaction potentials, if they have FSs, allow one to obtain acceptable results on the description of the available experimental data for the total cross section of the neutron capture on ^{10}B of [

Thereby, the MPCM again confirms, as already done in 27 reactions from [

In conclusion, the authors express their deep gratitude to Prof. Yarmukhamedov R. (INP, Tashkent, Uzbekistan) for provision of the information on the AC in the n^{10}B channel, and also to Prof. Strakovsky I.I. (GWU, Washington, USA) and to Prof. Blokhintsev L.D. (MSU, Moscow, Russia) for discussions of certain questions touching in the paper.

S. B. Dubovichenko,A. V. Dzhazairov-Kakhramanov, (2015) Astrophysical Radiative Neutron Capture on ^{10}B Taking into Account Resonance at 475 keV. Open Access Library Journal,02,1-13. doi: 10.4236/oalib.1101263