STUDIES ON THE THEORETICAL MODEL OF ASTROPHYSICAL  FACTOR FOR DIRECT RADIATIVE CAPTURE REACTION

A dissertation submitted to Gauhati University in partial fulfilment of the requirements for the Degree of Master of Science in Physics

By                                                                                              

Payal Saha                                                                                       

M.Sc. 4^th Semester                                                                                Roll No.: 118/14                                                                                

Reg. No.: 065655 of 2011-2012

Department of Physics                                                                          

Gauhati University                                                                               

August, 2016

Under the Supervision of                                                                        

Dr. Kushal Kalita                                                                                 

Assistant Professor (Stage 3)                                                                    

Department of Physics                                                                          

                                                                           Gauhati University                                                 

DECLARATION

               I hereby declare that the dissertation entitled   “STUDIES ON THE

THEORETICAL MODEL OF ASTROPHYSICAL  FACTOR FOR

DIRECT RADIATIVE CAPTURE REACTION”   has been submitted for the partial fulfilment of the degree of Master of Science in Physics, Gauhati University.  The work has been carried out by me under the supervision of Dr. Kushal Kalita, Assistant Professor (Stage 3), Department of Physics, Gauhati University.  The work has not been submitted in part or in full by me for any other degree or diploma to this or any other university.

                                                                    (Payal Saha)

                                                                    Roll No.: 118/14

Date:                                                                Reg. No.: 065655 of 2011-2012

Place: Guwahati                                                         Gauhati University 

CERTIFICATE

              This is to certify that Payal Saha bearing Roll Number: 118/14 (G.U. Registration Number: 065655 of 2011-2012) has carried out this dissertation

work    entitled    “STUDIES    ON    THE    THEORETICAL    MODEL     OF

ASTROPHYSICAL  FACTOR FOR DIRECT RADIATIVE CAPTURE

REACTION”   under my supervision,  which is being submitted to the Department of Physics,  Gauhati University for the partial fulfilment of the degree of Master of Science in Physics.  This work reported in this dissertation has been carried out by the candidate herself and has not been submitted to any other university for any other degree or diploma.

(Dr. A. Gohain Barua)                                              (Dr. Kushal Kalita)              

Professor & Head                                                      Supervisor          

Department of Physics                                              Assistant Professor (Stage 3)

Gauhati University                                                    Department of Physics

Guwahati-781014, Assam                                         Gauhati University      

Date:                                                                          Date:                            Place: Guwahati                                                        Place: Guwahati

ACKNOWLEDGEMENT

              I am deeply indebted to my dissertation supervisor, Dr. Kushal Kalita, Department of Physics, Gauhati University, whose guidance, instruction and encouragement throughout the course of this study were instrumental in the progress of my work.

              To the honourable HOD, Department of Physics, Gauhati University, Dr. A. Gohain Barua and to the all other faculty members of physics department, Gauhati University, I sincerely acknowledge their support during the course of my dissertation work.

              To my parents, I owe a special debt of gratitude. As in all endeavours, their encouragement and selfless support gave me the strength to continue the work. Last but not the least, I wish to thank all my classmates and my seniors for being so supportive of my work.

                                                                                   Payal Saha

                                                                             M.Sc. 4^th Semester

                                                                         Department of Physics

                                                                            Gauhati University

ABSTRACT

            Among the various theoretical models for determining the Astrophysical  factor, one method is the Asymptotic Normalization Coefficient (ANC) method. In this work a study has been performed how the ANC is used to extract the value of the Astrophysical  factor. The  factor has been calculated using the relation (E) =  exp(2πη) σ , for different energies and considering different transitions in order to study the contribution of different multipole transitions to the reaction cross sections of   reaction at different energies. It is useful to extract the Astrophysical S-factor induced by radioactive ion beam despite the laboratory experiment is almost impossible.

CONTENTS

1.   Introduction............................................................................................ 1

  1.1   Thermonuclear fusion in stars........................................................................................... 1

1.2   Thermonuclear fusion cross section................................................................................... 2

    1.2.1   Definition of Astrophysical S- factor........................................................................... 3

        1.3   Importance of Astrophysical S-factor......................................................................... 5

2.     Methodology.......................................................................................... 7

        2.1   Determination of Astrophysical S-factor.................................................................... 7

                2.1.1   Direct radiative capture reaction...................................................................... 8

                2.1.2   Asymptotic normalisation coefficient (ANC) method................................... 10

         2.2   Advantage of ANC Method over other methods..................................................... 14

3.     Observation......................................................................................... 15

          3.1   Calculation carried out by K. H. Kim et al............................................................. 16

          3.2   Analysis of the data taken from section 3.1............................................................ 17

4.     Results and Summary......................................................................... 21

          4.1   Result of the analysis carried out in section 3.2...................................................... 21

          4.2   Summary................................................................................................................. 22

Bibliography............................................................................................... 24

Chapter 1

Introduction

             The Astrophysical S-factor is defined as the cross section for a particular nuclear reaction which is independent of energy. Generally, reactions are performed at high energy to extract cross section. However, the reactions induced in stars are far below the Coulomb barrier. In laboratory these experiments cannot be performed, therefore one has to do these experiments at slightly higher energy, and then try to extrapolate their cross section at far low energy to get the exact cross section of a particular reaction which gives information of Astrophysical S-factor.

             For instances, the    + p →  +   reaction at solar energies plays an important unique role in the “solar neutrino puzzle”, since the high energy neutrinos from subsequent  decay of   provide about 75% of the flux detectable in the chlorine experiment and they are the only source to which the Kamiokande experiment is sensitive [1].

         As the rare termination of the proton-proton chain

                                     +  p →  + 

results in energetic neutrinos, it is therefore important to have an accurate estimate of the rate or the related zero energy Astrophysical S-factor,  (0), of this reaction in the solar interior, since solar neutrino flux measured by a   detector is almost proportional to this reaction cross section.

1.1   Thermonuclear fusion in stars

              In stars energy is released by nuclear reactions. During fusion, the positive charges of two nuclei will strongly oppose each other by forming a Coulomb barrier between them. Classically, for two nuclei to fuse, their relative kinetic energy must be greater than this Coulomb barrier [2,3]. As in stars, the kinetic energy of the nuclei is generated by gravitational contraction of the star, so we can say that thermonuclear fusion in stars is activated by gravitational collapse. Since the fusion of nuclei is strongly hindered by Coulomb repulsion, the first nuclear fuel to ignite is composed of light nuclei with low charge [4]. The energy generated by this fuel temporarily stops further contraction of the star. But when this particular fuel is exhausted, contraction resumes and the internal temperature then increases until the next available fuel, consisting of heavier nuclei, is ignited. These thermonuclear hang-ups not only prolong the life of a star, they also play a constructive role in the synthesis of heavier nuclei [4].

1.2   Thermonuclear fusion cross section

The probability of thermonuclear fusion is generally expressed in terms of a fusion cross section. The cross section is a scalar that only quantifies the intrinsic rate of an event [5]. Let us consider, a hot ionised gas containing nuclei of types A and B with concentrations and  which can fuse with a fusion cross section σ. If we assume that all the A nuclei move with speed „v‟ and the B nuclei are at rest, then in unit volume of the gas, we have nuclei of type A which fuse at a rate  = σv per second [4].

             But since both types of nuclei move, so we have to consider the average value of the product of the fusion cross section and the relative speed „‟ of the nuclei. Thus the A-B fusion rate per unit volume becomes ----

                                                          = < σ>

             If f(E)  is the distribution function, i.e., number of particles between energy E and

E+dE , then reaction rate per particle pair can be expressed as -----

                                                  <σv> =  v(E) f(E) dE  

1.2.1   Definition of Astrophysical S- factor

              Theoretically, at low energies the thermonuclear fusion cross section is proportional to the probability for tunnelling through the Coulomb barrier keeping the nuclei apart.

Gamow energy,  gives a measurement of the height of the Coulomb barrier [3].

                                                       = 2  where,            is the reduced mass and                      α is the fine structure constant =

,

we can write the fusion cross section for nuclei with relative energy E as -----

                                                    σ (E) =  S(E) exp      .   .   .   .    .   .   .   .   .   .   (1) where, S(E) is the Astrophysical S-factor and it varies much more slowly with energy, but it may peak when the energy is near a nuclear resonance.

The Coulomb barrier causes the cross section to have a strong exponential dependence on the energy E. But, the S-factor remedies this by factoring out the Coulomb component of the cross section [6].

Now,                                               S(E) = E σ(E) exp (2πη)

where,                   η is the Sommerfeld parameter =  

              Gamow factor, P =  , is the probability of an S-wave proton to penetrate the

Coulomb barrier. Gamow factor is a probability factor for two nuclear particles‟ chance of overcoming the Coulomb barrier in order to undergo nuclear fusion [6].

              The factor of    has been introduced because nuclear cross sections at low energies are often proportional to the square of the de-Broglie wavelength for the relative motion of the nuclei before fusion [4].

The expression for reaction rate per particle pair is given below [3] ----

We can see that   is small at low energies, while   is small at higher

energies. Dependence of reaction cross section for charged particle reaction as a function of energy is shown in Fig. 1.2 [3]. Nuclear reaction occur mainly in the energy region straddled by the energy window defined by -----

                                                            E =   

In general, this energy is too low to measure the reaction cross section directly in the laboratory. So, one can measure S (E) over a range of available lab energies and then extrapolates down to the region around  .

Figure 1.2:  Dependence of reaction cross section for charged particle reactions as a function of energy [3].

1.3   Importance of Astrophysical S-factor

The energy generated at the core of the Sun mainly comes from the proton-proton cycle. All the three reaction paths for proton-proton fusion lead to the production of alpha particles, each of which liberates neutrinos. None of the other particles involved can penetrate out of the Sun to be directly observed, so considerable effort has been devoted to detect the solar neutrinos. In 1964, S.N. Bahcall predicted a solar neutrino flux of  5×  neutrinos/sec from solar modelling.

             In an early experiment consisted of huge tank of perchloroethylene buried deep in the earth, the neutrinos detected were only about a third of the expected values from the best solar models. As we have accurate measurements of the amount of energy released by the Sun, a factor of three changes in the rate of the main production reactions is difficult to explain. More recent experiments at Super Kamiokande, the SAGE and GALLEX detectors, and the Sudbury Neutrino Observatory all get about half the expected neutrino flux, so the neutrino deficiency persists.

              As the    + p →  +   reaction at solar energies plays an important unique role in the “solar neutrino puzzle”, it is therefore important to have an accurate estimate of the rate or the related zero energy Astrophysical S-factor,  (0), of this reaction in the solar interior, since solar neutrino flux measured is almost proportional to this reaction cross section.

              Again, during core He burning, 3α and   (α,  reactions compete to determine the helium burning timescale and the relative abundances of oxygen and carbon prior to core C burning. So, it is highly desirable to know the rate or Astrophysical S-factor of the   (α,  reaction with an accuracy comparable [7] to that of the 3α process in order to provide adequate constraints on stellar evolution and the synthesis of elements, e.g., the yield of the neutrino-process isotopes in core-collapse supernovae [8,9] and the production of the important radioactive nuclei [10].

Chapter 2

Methodology

2.1   Determination of Astrophysical  factor

The nuclear physics of thermonuclear fusion is hidden in the astrophysical factor S(E). Even though the Coulomb barrier plays a dominant role in shaping the properties of all thermonuclear reactions, the actual rate depends on the interactions that bring about the fusion [4]. Nuclear strong, electromagnetic and nuclear weak interactions may be involved. The overall effect is summarised by the nuclear factor S(E).

              In practice it is very difficult to measure fusion cross sections at energies relevant to astrophysics, i.e., at energies well below the Coulomb barrier. One of the methods for obtaining the nuclear reaction rates at low energies is to extrapolate the high energy experimental results to solar energies by using a certain theoretical model as a basis for extrapolation. The second method involves theoretical calculation of the S-factor of a thermonuclear reaction on the basis of certain nuclear models [11].

              The aim of using nuclear models and theoretical methods of calculation of thermonuclear reaction characteristics is that if a certain nuclear model describes correctly the experimental data of the astrophysical S-factor in the energy range for which the data exist, e.g., a few hundred keV, then it is quite reasonable to think that this model will describe correctly the form of the S-factor at the lowest energies (a few tens of keV) too [11].

              In this report, we have discussed two indirect methods to extract the Astrophysical

 (0) factor -----

(i)                 Through direct radiative capture reaction [12],

(ii)              By Asymptotic Normalisation Coefficient (ANC) method.

2.1.1   Direct radiative capture reaction

If one of particles in the exit channel is electromagnetic radiation (i.e., a -ray), the reaction is known as a radiative capture reaction [3]. For example

                                                              + p →  +  

              These types of reactions are very common in stellar interior and in general they have positive Q-values and thus can contribute to the energy liberation of the star. These reactions also play an important role in the production of heavy elements.

              R.G.H. Robertson, in 1973 had carried out this indirect method for determining the S-factor. Capture of proton by   involves the radiative transition of a proton to the  ground state of    , bound by 137.2 keV. As the spin and parity of   are  , capture from the s and d- partial waves leads to E1 radiation and from the p-wave, M1 radiation. So, only dipole radiation is of importance at this energy. Higher partial waves cannot contribute to dipole radiation [12].

              The total cross section for dipole capture in the reaction A (ɑ,) B is -----

                 σ =  

where,        is the photon energy,                                                                                   is the mass of the incident particle,                                                                     is the lab energy of the incident particle.                                              

                   , s,    are the total angular momenta of particles A, ɑ and B, and                       are the projections of particles A, ɑ  and B respectively [12].

             The electromagnetic transition matrix element between initial and final states can be written as ------

where,       is the relative wave vector of particles A and ɑ,                                                        is the separation of particles A and ɑ.                                                         are the single-particle operators which describe electric or magnetic radiation of multipolarity λ, polarization  by the i th nucleon.

              The initial state wave function, ψ  includes an incoming distorted wave which is an eigenfunction of the Schrödinger equation for a proton moving in the Coulomb and nuclear potential of A.  A real Woods-Saxon potential for the nuclear case with a Thomas spin-orbit term is assumed to be [12] ----

                                                     V(r) = f(r)  

                                              where,      f(r) =   and the Coulomb potential is assumed to be that of a uniform sphere of radius R.

              Usually one potential is chosen for the bound state, so that the correct separation energy results and another for the incident wave, to meet low-energy scattering properties. The well depth for the incoming wave for each J is adjusted to produce a resonance at the observed energy. To permit a  J dependence in the optical potential the initial wave function is expanded as follows:

Where,         and     M+ m+.

In the above expression,   and    are intrinsic wave functions for the target and projectile respectively, and   is the distorted-wave radial function. Making a fractional parentage expansion of states of B in terms of those of A coupled to a singleparticle state b of angular momentum  , if there is no J dependence in the s and d wave optical potentials in case of E1 capture, one finds by straight forward angular momentum reduction that ----

              (E1) =  

              In this expression, b=1,2,  .  .  . indexes the single-particle states of the captured proton in B, and   is the spectroscopic amplitude connecting those states to the ground state of  B ( where,    indexes the complete set of states of A ( [12].

              Here, they had calculated the bound and continuum radial wave functions with the program DWUCK72 and reduced matrix elements of the second kind were evaluated with the program EMSPME. Well depths for  were chosen to give the observed bound-state and resonance energies. The optical model parameters, which were used, are listed in Table I [13,14]. They had used certain values of spectroscopic amplitudes, which yields a low-energy S-factor, evaluated at the Gamow peak (20 keV), of 31 eV b. This result is consistent with other theoretically obtained values.

2.1.2   Asymptotic normalisation coefficient (ANC) method

At solar energies, the   + p →  +   capture proceeds through the tail of the nuclear overlap function. The shape of this tail is determined by the Coulomb interaction, so the capture rate can be accurately calculated if one knows its amplitude. The asymptotic normalization coefficients for the above reaction specify the amplitude of the tail of the     wave function in the two-body channel when the  core and the proton are separated by a distance large compared to the nuclear radius [15].

              In 1994, H.M. Xu et al., derived this ANC method. Here, they deal with the issue of the normalization of the S-factor. In order to do this, they re-examine the overlap wave function for the virtual decay of    into   + p.  The normalisation of the (0) factor for 

+ p →   reaction can be determined by a single parameter, viz., the ANC or the Nuclear Vertex Constant (NVC) of the overlap wave function, unlike the conventional theoretical approaches which require knowledge of more than two independent parameters

[1].

              Let us consider, the general case for radiative capture reactions  b + c   +   at 

0. Here, we use the concept of the NVC‟s   ( → b + c ). These constants are the fundamental nuclear constants for the amplitudes of the virtual or real decay of a nucleus „‟ into two fragments „b‟ and „c‟ [16-18]. Here,  is related to the ANC, , of the overlap wave function for nucleus  in channel b + c  by [16-18]  -----

                                              .   .   .   .   .   .   .   .   .   .   .   (1)

where,           is the orbital angular momentum,                                                         

                    S is the channel spin,                                                                                                   is the reduced mass and                                                                                          η is the Sommerfeld Coulomb parameter, for the bound state of  b + c.

              In order to check whether we can use the NVC information to determine the solar Sfactor indirectly, we know that the direct radiative capture reaction for b + c →  +  has an amplitude or transition matrix element ----

                                   M =    .   .   .   .   .   .   .   (2)

                                .   .   .   .   .   .   .   .   .   .   .   .   .   .   .    (3)

where,        is the wave function for the bound state of particle „i‟,                                          is the internal coordinate for the bound state of particle „i‟,                                    is the relative coordinate between b and c,                                        

                  is the electromagnetic operator and in the case of   + p → , the  E1                    operator,                                                                                       

                   )  is the distorted wave in the initial channel b + c , which is the regular

                                Coulomb function in this case and                                                                 is the overlap wave function for   → b + c.

                                           .   .    .   .   .   .   .   .   .   (4)

Here,      is the spin or projection of particle „i‟,               is the Clebsch-Gordon coefficient,

              is the radial part of the overlap wave function with the asymptotic behaviour.

                                         ,            r    .   .   .   .   .   .   .   .    (5)

where,       is the nuclear interaction radius between the proton and  ,                                is the Whittaker function,                                                  

                   is the asymptotic normalisation coefficient (ANC).

              In some standard potential models,    is approximated by the product of two factors, the spectroscopic factor   and the bound-state radial wave function   , as follows [13,19] ----

                                                  .   .   .   .   .   .   .    (7)

where,      is the normalisation coefficient of the asymptotic part of the bound-state wave function  .

                                relates to the ANC,      by ----

                                                     .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   (8)

If the protons are captured both inside and outside of the core nucleus, then the entire overlap function, both    and   , is required [1].

From equation (6) and (7), we get ----

              Now, the specific value of ANC,  , can be deduced using equation (1) from the NVC,   , values predicted for the virtual decay of  As the value of   is

now known to us, so we can calculate the value of transition matrix element „M‟ by using the above equations. From this value of M, we can determine the radiative capture reaction cross section,  for   + p →  +   reaction, and thus we can estimate the astrophysical Sfactor by using the relation ----

                                                S =  

              We can see from equation (3), that the transition matrix element  M completely depends on the parameters   and . Here,  and ) are well known, thus the value of M, and, therefore the (0) factor, is determined by the overlap wave function, . H.M. Xu et al., in their calculation found that it is the tail of the overlap wave function, or, more precisely, the normalisation constant of the tail, , that solely determines the value of (0) factor for + p →   reactions [1].

              Weiping Liu et al., in 1996 reported that the conventional method to extract the empirical spectroscopic factor (  or  ) from an experiment uses the relation [20] ----                                                .   .   .   .   .   .   .   .    (9)

where,         is the value of experimental differential cross section in the peak region,               is the value of calculated differential cross section in the peak region,                    is the compound nucleus cross section.

               From equation (9), we can find out the value of spectroscopic factor   by using the values of measured cross sections, and then the value of transition matrix element, M can be calculated. But, the spectroscopic factor  has a large uncertainty, which strongly depends on the geometry parameters of the Woods-Saxon potential  and  (the radius and diffuseness parameters) used for calculating the wave function of a single particle bound state [19,21]. Such kind of uncertainty can be removed by introducing the ANC of the overlap functions, which is related to the spectroscopic factor by equation (8).

2.2   Advantage of ANC Method over other methods

The advantage of the ANC approach is that it provides a method to determine direct capture S-factors at zero energy from measurements of nuclear reactions, such as peripheral nucleon transfer, which have cross sections orders of magnitude larger than the direct capture reactions themselves [15]. The ANC method allows new possibilities, such as transfer reactions, to estimate the (0) factor indirectly with cross sections similar to those in Coulomb breakup reactions. These new possibilities measure the absolute value of   at zero energy directly, and they have the advantage that they are free from complications due to three-body effects in Coulomb breakup reactions in the exit channels [22].

             Unlike the direct radiative capture measurements or Coulomb breakup measurements which measure the E1 and/or E2 transition matrix elements (through cross section measurements) at higher energies, the ANC method measures the overlap wave functions, and then uses the measured wave functions and the well-known electromagnetic operators to calculate the matrix elements at astrophysical energies [1]. Again, with the change of the geometry parameters of the Woods-Saxon potential  and , the variation of the ANC values would be small, although the spectroscopic factor would change dramatically [20]. The ANC method can be used for precise measurements of other (p,)  S-factors involving short-lived nuclei, where direct capture measurements may be very difficult [23].

Chapter 3

Observation

In a simultaneous measurement of the electron neutrino flux and the sum of three active neutrino fluxes, the Sudbury Neutrino Observatory (SNO) group discovered that a large fraction of the high energy electron neutrinos emitted in the   decay of     in the Sun is transformed into other active neutrino flavors on their way to detectors on the Earth [24]. To ensure more accurately if there is a transformation of solar electron neutrinos into other active neutrinos the precision of both experimental measurements and theoretical predictions of the    neutrino flux should be improved [25]. For this, the rate or more precisely the astrophysical S-factor of the   + p →  +   reaction that produces    in the Sun must be better determined. The energy dependence of the S-factor seems moderately well established. However, there are still large uncertainties concerning the absolute normalisation of the S-factor both experimentally and theoretically.

              Theoretically, the studies of the S-factor for   + p →  +   reaction can be divided into two approaches, one is potential models of direct radiative capture [13,19] and the other one is models based on the microscopic resonating group method [26,27]. F.C. Barker showed in his paper that in potential models, significant uncertainties in the overall normalisation of the S-factor exist mainly due to uncertainties in the spectroscopic factor and in the bound-state wave functions of   + p  in    which normally were determined independently [19]. Similarly, T. Kajino et al., showed that significant uncertainties also exit in the normalisation of the calculated S-factor from the resonating group method due to different choices of nucleon-nucleon potentials [26].

              Till now many experiments were carried out and many theoretical predictions have been made in order to determine the correct value of astrophysical S-factor. Although some experimental and theoretical values are in well agreement among themselves, but there are some calculations which are significantly different from each other. The challenging situation surrounding the  (0)  factor calls for further experiments to reduce the uncertainties of the  (0) factor.

3.1   Calculation carried out by K. H. Kim et al.

K. H. Kim, M. H. Park and B. T. Kim in 1987 calculated the differential cross sections as well as the total cross sections for the   + p →   reaction at low energies in the radiative direct capture model which includes the  E1, M1 and E2 transitions from s, p, d and f waves. They found out the astrophysical S-factor to be 0.024 keV b, at the solar temperature of 20 keV. Here, they took into account the resonant part as well as the non-resonant part in the radiative capture description by considering E1, M1 and E2 transitions due to   captures. They had calculated the angular distributions of the differential cross sections in order to investigate how each multipole transition (or each partial wave capture) contributes to the reaction cross sections at different energies [28].

              They had plotted the contributions of E1, M1 and E2 transitions to the total cross sections for the reaction in a graph, which is shown below.

     Figure 3.1:  Multipole and partial wave contributions to the total capture cross section

3.2   Analysis of the data taken from section 3.1

In our study, we have calculated the value of astrophysical  factor at different energies using the values of total cross sections of the reaction + p →    from the graph shown in Fig. 3.1. Here, we have used the formula -----

                                                          S (E) =  σ(E)   where,                            Sommerfeld parameter, η =  .

The value of relative velocity, v of the particles is calculated using the relation------

where,                                        is the reduced mass in g/mol,                 

                                                  is the mass of proton and                                                                           is the mass of  .

              The calculated values of astrophysical  factor at different energies for different multipolarities are listed below.

Table 1:  Data for    factor in case of E1 ( ) transition

Table 2:  Data for   factor in case of E1 () transition

Table 3:  Data for   factor in case of M1 ( 1) transition

Using these calculated values of   factor at different energies, we have plotted a graph for different multipole transitions. In this graph energy  is taken along X-axis and astrophysical   factor is taken along Y-axis. The graph is shown in the next page. 

From this graph we can conclude that ------- 

1.      In the energy range less than 200 keV, the   factor mainly depends on the E1 transition due to s-wave capture.

2.      But we cannot neglect the contribution from d-wave capture in low energy region, it is also significant. As the energy increases, the d-wave capture becomes a dominant process.

3.      There is a resonance peak at about 600 keV in case of M1 transition due to p-wave capture. This may be due to low binding energy of    which enhances the capture from p and d partial waves.

Chapter 4

Results and Summary

4.1   Result of the analysis carried out in section 3.2

The astrophysical   factors which are estimated in our study are listed in the table shown below. It is found that the  factor is coming out to be more consistent from the E1 transition than the rest.

Considering only the E1 transition due to s-wave capture, the average value of astrophysical   factor in our work is found to be 16.04 eV b. This result is consistent with some other theoretical works, which are carried out earlier by others.

Comparison of astrophysical   factor values:

4.2   Summary

A number of great theoretical approaches are there for the study of the astrophysical (0) factor. For different kinds of nuclear reactions, such as direct radiative capture reaction, Coulomb dissociation reaction, transfer reaction; theoretical approaches for extraction of   factor are also different. In our work, we have discussed the Asymptotic Normalisation Coefficient (ANC) method which was carried out by H. M. Xu et al., in 1994 for direct radiative capture reaction. In this method they introduced a relation between the Asymptotic Normalisation Constant (ANC) and the overall normalisation of the  (0) factor. From the ANC, we can calculate the value of transition matrix element and if we could determine the transition matrix element accurately, then from this we are able to calculate the cross section of the reaction at low energy. Now, by using the relation between astrophysical  (0) factor and the cross section of the reaction, we can calculate the value of required  (0) factor. If the initial distorted wave function and the tail of the final state overlap wave functions for the radiative capture reaction are accurately measured, then this method will provide an indirect way to determine the  (0) factor accurately. The Woods-Saxon well depths are adjusted for each channel so that the model can reproduce properties such as the proton separation energy and the location of known resonances. The most uncertain parameters are the potential parameters and the (0) factor is sensitive to this parameter. But with the introduction of ANC , all the dependence becomes apparent.

              In our work, we have analysed the values of total cross sections of the direct radiative capture reaction  + p →  +, which were calculated by K. H. Kim et al., in 1987 at different energies considering different multipole transitions, and from this we have calculated the value of the astrophysical  (0) factor corresponding to these energies using the relation  S(E) =  σ(E) . We have got the average value of astrophysical

(0) factor to be 16.04 eV b, considering only the E1 transition due to s-wave capture, which is consistent with some of the references. It is interesting to study the low-energy behaviour of the multipole contributions to the astrophysical  (0) factor. We have found from this analysis that the  (0) factor mainly depends on the E1 transition due to s-wave capture in the energy region less than 200 keV. But the contribution from the d-wave capture in this energy region cannot be neglected; it is also have a significant value. As the energy increases, the d-wave capture becomes a dominant process. A resonance peak is also found at about 600 keV in case of M1 transition due to p-wave capture. The small binding energy of  results in a spatially extended wave function, which increases the capture from the p and d-partial waves [12]. The E2 contribution in negligible in the low-energy region, so we do not consider this transition in our analysis. From this analysis, we are able to know the contribution of different multipole transitions to the astrophysical  (0) factor at different energies.

                 As a future outlook an attempt may be made to extract  (0)  factor using IUAC,

New Delhi accelerator facility using  RIB. The method adopted by J.J. Das [23] with   is somewhat different from what we have performed. This will be useful for

exploration of further development of RIB facility (ISOL type or In-Flight) in India which will boost information to the scientific community in near future. 

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