# Asymptotic normalization coefficients from transfer reaction and R-martix
analysis of direct capture in 22Ne(p,$\gamma$)23Na reaction

Rajkumar Santra  Saha Institute of Nuclear Physics, 1/AF, Bidhan Nagar,
Kolkata 700064, INDIA  Homi Bhabha National Institute, Anushaktinagar, Mumbai
400094, INDIA  Suprita Chakraborty  Belgachia Aswini Dutta Vidyapith for Girls
High, Belgachia Howrah, INDIA  Subinit Roy<EMAIL_ADDRESS>Saha
Institute of Nuclear Physics, 1/AF, Bidhan Nagar, Kolkata 700064, INDIA

###### Abstract

The 22Ne(p,$\gamma$)23Na reaction in NeNa cycle plays an important role in the
production of only stable sodium isotope 23Na. This nucleus is processed by
the NeNa cycle during hot bottom burning (HBB) in asymptotic giant branch
(AGB) stage of low metallicity intermediate mass stats (4 MO $\leq$ M $\leq$ 6
MO). Recent measurements have addressed the uncertainty in the thermonuclear
reaction rate of this reaction at relevant astrophysical energies through the
identification of low lying resonances at Ep = 71,105, 156.2, 189.5 and 259.7
keV. In addition, precise measurements of low energy behaviour of the non-
resonant capture has also been performed and the contribution of the sub-
threshold resonance at 8664 keV excitation in 23Na has been established. Here,
in this article, we have presented a systematic R-matrix analysis of direct
capture to the bound states and the decay of the sub-threshold resonance at
8664 keV to the ground state of 23Na. A finite range distorted wave Born
approximation (FRDWBA) calculation has been performed for 22Ne(3He,d)23Na
transfer reaction data to extract the asymptotic normalization coeeficients
(ANC-s) required to estimate the non-resonant capture cross sections or
astrophysical S-factor values in R-matrix analysis. Simultaneous R-matrix
analysis constrained with ANC-s from transfer calculation reproduced the
astrophysical S-factor data over a wide energy window. The S${}_{tot}^{DC}$(0)
= 48.8$\pm$9.5 keV.b compares well with the result of Ferraro, et al. and has
a lower uncertainty. The resultant thermonuclear reaction is slightly larger
in 0.1 GK $\leq$ T $\leq$ 0.2 GK temperature range but otherwise in agreeent
with Ferraro, et al..

###### pacs:

21.10.Ma, 24.60.Dr,25.70.Gh

##  I Introduction

Proton capture reaction 22Ne(p,$\gamma$)23Na of the neon-sodium cycle of
hydrogen burning in stars consumes 22Ne, a seed nucleus for neutron production
for $s$-process nucleosynthesis and converts to 23Na, the only stable isotope
of sodium. The reaction occurs in the convective envelop of massive ($M\geq
4M_{O}$) asymptotic giant branch (AGB) stars at temperature $T\sim 1.0\times
10^{8}$K ($T_{9}\sim 0.1$). In more massive stars ($M\geq 50M_{O}$), the
reaction takes place in the surface layer along with the carbon-nitrogen-
oxygen (CNO) and magnesium-aluminium (Mg-Al) cycles of hydrogen burning at
temperature upto $T_{9}\sim 0.8$) cavallo ; cottrell ; pilachowski ; paltoglou
. Since oxygen is destroyed in CNO cycle and Na is produced in Ne-Na cycle,
the reaction 22Ne(p,$\gamma$)23Na is said to be responsible for the observed
anti-correlation in surface oxygen and sodium abundances in galactic globular
clusters Carretta ; Lind ; yong .

The rate of 22Ne(p,$\gamma$)23Na capture reaction at relevant astrophysical
energy domain is dominated by the contributions of several low energy
resonances in 23Na and a slowly varying off-resonant capture contribution.
Precision measurements have been carried out in recent years to identify and
confirm the important low energy resonances and determine the resonance
strengths cavanna1 ; depalo ; cavanna2 ; bremmer ; kelly ; ferraro . The
studies helped resolve the discrepancy that existed between the recommended
value of the reaction rate from the NACRE I compilation angulo  and those
determined in Refs. Hale ; iliadis1 ; STARLIB  in AGB stars in the temperature
window 0.08 GK $\leq T\leq$ 0.25 GK.

The measurement of Ferraro, et al. ferraro , besides providing the stringent
upper limits for the strengths of low energy resonances at proton beam
energies of 71 and 105 keV, also reported a slowly varying non-resonant
component having significant contribution in the same temperature window.
Prior to this work, the non resonant, direct capture processes in
22Ne(p,$\gamma$) reaction to the bound states of 23Na and their contributions
to low energy behaviour of astrophysical $S$-factor have been studied in Refs.
Rolfs ; gorres1 ; kelly . Earlier studies used a constant value of $S_{DC}$(E)
= 62.0 keV.b obtained by G$\ddot{o}$rres, et al. gorres1  from their
measurement of higher energy direct capture data. Kelly, et al. kelly
measured the direct capture cross section at a single proton beam energy of
425 keV and maintained the same value for $S_{DC}$(E=0 MeV). Ferraro, et al.
in a subsequent high statistics and low background measurement extended the
non-resonant capture cross section to 310, 250, 205 and 188 keV proton beam
energies. Their analysis yielded $S_{DC}$(0) = 50 $\pm$ 12 keV.b. However, the
authors observed a distinct rise in the low energy astrophysical $S$-factor
data and showed it to in the low energy non-resonant astrophysical $S$-factor
is shown to be the consequence of capture to the sub-threshold state at 8664
keV excitation in 23Na. Thus the direct capture component along with the
contribution of sub-threshold state actually determines the total off-resonant
component around the temperature of $T_{9}\sim 0.1$.

In the present article, we reported a detailed analysis of the available
$S$-factor data for non-resonance capture to the ground state and 440, 2392,
2982, 6318, 6918 and 8664 keV excited states of 23Na nucleus and the total
non-resonant $S$-factor data within the framework of $R$-matrix model. The aim
is to investigate the energy dependence of the total off-resonance
astrophysical $S$-factor including the contribution of broad, sub-threshold
(-130 keV) state at 8664 keV and to extend the curve to still lower energy of
astrophysical interest. The spectroscopic information required for the
$R$-matrix calculation for the direct capture contribution have been extracted
from the re-analysis of one proton transfer reaction data on 22Ne from the
literature. The total reaction rate is then estimated from the calculated
reaction rate due to off-resonance process and using the measured strengths of
important low energy resonances in 23Na except for the state with resonance
energy of 151 keV. Experimental evidence shows that the state at 8945 MeV
excitation corresponding to resonance energy 151 keV is actually a doublet
with capture of $d$\- and $f$\- waves respectively. A reanalysis of proton
transfer reaction to 8945 MeV unbound state ($Q_{p}$ = 8794 MeV) has been
performed to extract the resonance strengths indirectly. The resultant
reaction rate up to $T$ = 1.0 GK has been compared with the recent
estimations.

Figure 1:  Level scheme of 23Na  Table 1: Presnt status of ${}^{22}Ne(p,\gamma)^{23}Na$ reaction rate. Year |  |  | SDC(0) |  |  |  | Reaction rate(T9$\sim$0.1)
---|---|---|---|---|---|---|---
|  |  | (keV.b) |  |  |  | cm3mol-1s-1
Rolfs 1975 Rolfs  |  |  | 67$\pm$19 |  |  |  | -
Gorres 1983 gorres1  |  |  | 62 |  |  |  | -
Hale 2001 Hale  |  |  | - |  |  |  | 2.1$\times$10-9
Sallaska 2013 STARLIB  |  |  | - |  |  |  | 5.52$\times$10-9
Cavanna 2015 cavanna1  |  |  | - |  |  |  | 6.6$\times$10-8
Depalo 2016 depalo  |  |  | - |  |  |  | 2.7$\times$10-8
Kelly 2017 kelly  |  |  | - |  |  |  | 3.53$\times$10-8
Ferraro 2018 ferraro  |  |  | 50$\pm 12$ferraro  |  |  |  | 2.1$\times$10-8
Table 2: Spectroscopic factors and asymptotic normalization coefficients (ANC-s) for the first seven states of 23Na |  |  |  |  |  |
---|---|---|---|---|---|---
Ex | Jπ | nlj | C2S111Geometry parameters of b.s. potential $a_{0}$=0.6, $r_{0}$=1.26 | C2S | b | ANC
(keV) |  |  | Present | Ref. Power  | (fm-1/2) | (fm-1/2)
g.s. | 3/2+ | 1d3/2 | 0.082$\pm$0.012 | 0.08 | 6.86 | 1.96$\pm$0.5
440 | 5/2+ | 1d5/2 | 0.38$\pm$0.08 | 0.35 | 7.62 | 4.69$\pm$0.8
2392 | 1/2+ | 2s1/2 | 0.26$\pm$0.05 | 0.25 | 17.56 | 8.8$\pm$1.6
2982 | 3/2+ | 1d3/2 | 0.35$\pm$0.04 | 0.32 | 4.38 | 2.59$\pm$0.87
6308 | 1/2+ | 2s1/2 | 0.14$\pm$0.02 | 0.13 | 11.14 | 4.16$\pm$0.79
6917 | 1/2- | 2p1/2 | 0.18$\pm$0.04 | 0.15 | 7.27 | 3.1$\pm$0.7

##  II Analysis

The model calculation progressed in two steps. In the first step, the data
from 22Ne(3He,d)23Na transfer reaction to the bound states of 23Na have been
reanalyzed to extract the Asymptotic Normalization Coefficients or ANCs of the
states. The second part constitutes of R-matrix calculation for the data of
direct capture to those bound states using the ANCs from the first part of the
calculation.

###  II.1 Finite Range DWBA analysis and extraction of ANC

A Finite Range Distorted Wave Born Approximation (FRDWBA) calculation has been
performed for the angular distribution data of 22Ne(3He, d)23Na one proton
stripping reaction from Refs. Hale ; Power . In FRDWBA model, conventionally
the experimental cross section of a transfer reaction $A+a(=b+x)\rightarrow
B(=A+x)+b$ (where $x$ is the transfered particle) is compared with the
calculated cross section by the relation

$\Big{(}\frac{d\sigma}{d\Omega}\Big{)}_{exp}=(C^{2}S)_{bx}(C^{2}S)_{Ax}\Big{(}\frac{d\sigma}{d\Omega}\Big{)}_{mod}$
(1)

where $C^{2}S_{bx}$ is the product of spectroscopic factor $S_{bx}$ and
isospin Clebsh Gordon coeeficient $C^{2}_{bx}$ of $b+x$ configuration in
projectile $a$ and $C^{2}S_{Ax}$ is that for $A+x$ configuration in residual
nucleus $B$. $\Big{(}\frac{d\sigma}{d\Omega}\Big{)}_{mod}$ is the cross
section obtained from model calculation.

To extract the spectroscopic factors from data, a finite range distorted wave
Born approximation (FRDWBA), using the code FRESCO (ver. 2.9) FRESCO , has
been performed. Angular distribution data, measured at 15 MeV incident energy,
for transfer to ground state and 440, 2392, 2982, 6308, 6917 keV exited states
are taken from Ref. Power . The data from Ref. Hale , measured with 20 MeV 3He
beam, have been used for transfer to the sub-threshold state of 23Na at 8664
keV excitation energy.

In the reanalysis of 15 MeV data within the FRDWBA framework, the optical
model potential parameters for entrance and exit channels are taken from Ref.
Power . Standard Woods-Saxon form has been used for the potentials. The shape
parameters for the bound sate potentials related to 22Ne+p and d+p systems are
from Refs. Power  and bem , respectively. The strengths of the bound state
potentials are varied to get the binding energies of the states of the
composite nuclei.

The model calculations reproduced the angular distribuitons for transfer to
ground state and 440, 2392, 2982, 6308, 6917 keV exited states of 23Na quite
well. While extracting the spectroscopic factors of the states of 23Na, the
spectroscopic factor $C^{2}S_{dp}$ for 3He is taken as 1.092, a value that is
derived for the 3He ground state with $d+p$ configuration using the method
reported in Ref.bem . The resultant spectroscopic factors of 23Na states are
shown in Table 2. The values obtained from the present FRDWBA analysis match
well with those reported from zero range DWBA calculation in Ref. Power .

####  II.1.1 DWBA analysis of 22Ne(3He,d) reaction for Ex=8664 keV state

The excited state 8664 keV of 23Na is 130 keV below the proton threshold at
8794 keV. Capture through this sub-threshold resonance controls the low energy
behaviour of the astrophysical S-facor of 22Ne(p,$\gamma$) reaction. To
extract the spectroscopic factor of this state we again performed a FRDWBA
calculation for the transfer reaction 22Ne(3He,d) at 20 MeV with the data from
Ref. Hale . Potential parameters used to obtain the transfer angular
distribution are given in Table 3. It has been observed that unlike the more
deeply bound states in 23Na, a complex remnant term is required to obtain a
very good overall fit to the angular distribution data. The parameters of
d+22Ne core-core potential are also given in Table 3. The resultant fit is
shown by solid red line in Fig. 3. The blue dashed dotted line represents the
FRDWBA calculation without the remnant term. Improvement in the fit is quite
remakable. In Col.5 of Table 4, the extracted spectrosopic factors have been
shown. The value obtained from the present work is very close to the values
reported in Refs. Hale ; gorres1 .

Table 3: Potential parameters for 22Ne(3He,d)23Na (E∗= 8664 keV), $E_{lab}$ = 20 MeV Hale . channel | Vr | rr | ar | Wi | WD | ri=rD | ai=aD | Vso | rso | aso | rc |
---|---|---|---|---|---|---|---|---|---|---|---|---
| (MeV) | (fm) | (fm) | (MeV) | (MeV) | (fm) | (fm) | (MeV) | (fm) | (fm) | (fm) |
3He+22Ne | Ref. Hale  |  |  |  |  |  |  |  |  |  |  |
d+23Na | Ref. Hale  |  |  |  |  |  |  |  |  |  |  |
d+22Ne | 88.0 | 1.17 | 0.73 | 0.24 | 35.8 | 1.33 | 0.73 | 13.85 | 1.07 | 0.66 | 1.33 |
d+p | * | 1.25 | 0.65 |  |  |  |  | 6.2 | 1.25 | 0.65 | 1.30 |
p+22Ne | Ref. Hale  |  |  |  |  |  |  |  |  |  |  |
* Varied to match separation energy. |  |  |  |  |  |  |  |  |  |  |  |
Table 4: Spectroscopic factor and asymptotic normalization coefficient (ANC) of 8664 keV state of 23Na |  |  |  |  |  |
---|---|---|---|---|---|---
Ex | Jπ | lp | nlj | C2S | b | ANC
(MeV) |  |  |  |  | (fm1/2) | (fm1/2)
8664222Subthreshold state. Er=-130 keV | 1/2+ | 0 | 2s1/2 | 0.32$\pm$0.05 | 252333Geometry parameters of b.s. potential $a_{0}$=0.69, $r_{0}$=1.17 | 143.7$\pm$15.2
|  |  |  | 0.29Hale  |  |
|  |  |  | 0.3gorres1  |  |
|  |  |  | 0.42$\pm$0.08ferraro  |  |
|  |  |  | 0.58$\pm$0.08Terakawa  |  |

####  II.1.2 Extraction of ANC

The spectroscopic factor so determined includes the effect of nuclear interior
and measures the many-body effect in the transfer reaction process. It depends
on the choice of the potentials, more sensitively on the geometry parameters
of bound state potential used to describe a particular configuration. In low
energy radiative capture reactions, instead of spectroscopic factor,
asymptotic normalization coefficient or ANC is more relevant a quantity. ANC
measures the amplitude of the tail of the overlap between the bound state
wavefunctions of initial and final nuclei. It is related to the spectroscopic
factor of the two-body configuration as

$C^{2}S_{J_{f}l_{f}}=\big{(}\frac{C_{J_{f}l_{f}}}{b_{l_{f}j_{f}}}\big{)}^{2}$
(2)

where $C^{2}S_{J_{f}l_{f}}$ is the spectroscopic factor of the configuration
in the composite nucleus with total spin $J_{f}$. The relative orbital angular
momentum and spin of the two clusters in the final bound state are denoted by
$l_{f}$ and $j_{f}$. $C_{J_{f}l_{f}}$ is the corresponding ANC and
$b_{l_{f}j_{f}}$ is the single particle asymptotic normalization constant
(SPANC) with $l_{f}$ and $j_{f}$ quantum numbers of the bound state orbital
used in the DWBA calculation. The SPANC ’b’ is expressed in terms of bound
state wave function of the composite nucleus Thomson book  as

$b(r_{0},a_{0})=\frac{u(r,r_{0},a_{0})}{W_{-n,l+1/2}(2\kappa r)}$ (3)

in the asymptotic radial region. W${}_{-n,l+1/2}(2\kappa r)$ is the Whitteker
function, $\kappa=\sqrt{2\mu\epsilon}$ the wave number, and
$\mu,\epsilon,R_{N}$ are the reduced mass, binding energy and the nuclear
interaction radius, respectively for the bound state of the final nucleus.
Both bound state wavefunction and Whitteker function have similar radial fall
of in the asymptotic region. The parameters $r_{0}$ and $a_{0}$ are the radius
parameter and diffuseness of Woods Saxon potential generating the required
bound state wavefunction. In all cases the strength of the bound state
potential is obtained by reproducing the binding energy of the state.

Figure 2: Transfer angular distributions fitted with FRDWBA model calculation.
Figure 3:  DWBA fit to angular distribution data of Power et al. Power  for
the state Ex=8664 keV. The line represents calculated cross section Figure 4:
Variation of spectroscopic factor with single particle ANC for 8664 MeV (top
panel) and 440 MeV (lower panel) states. Figure 5: Variation of ANC (C
$fm^{-1/2}$) as a function of SPANC (b $fm^{-1/2}$) for 440 keV (top panel)
and 8664 keV (bottom panel) states. Figure 6:  Variation of ANC (C
$fm^{-1/2}$) as a function of binding energy for 8664 keV (top panel) and 6917
keV (bottom panel) states for fixed geometry parameters of bound state
potential. Figure 7:  Angular distribution data and DWBA fits to 151 keV
resonance state of ${23}$Na.

In the present work, the value $b_{l_{f}j_{f}}$ associated with each bound
state has been obtained from the best fit ratio value using Eqn.3 for the
region beyond the radius RN = 5.5 fm. Like the spectroscopic factor, SPANC b
also depends on the choice of potential parameters. But the variations of the
two quantities with the geometry parameters of the bound state potential are
opposite in nature. Hence, the product of these two quantities that gives the
required ANC, remains constant with the change of potential parameters for a
peripheral reaction. Thus, for a pure peripheral condition the variation of
spectroscopic factor should be proportional to the inverse square of SPANC
value pang  from Eq.3. In Fig. 4 we have shown the plots of variation of C2S
as a function of SPANC b for the 8664 keV subthreshold state and deeply bound
440 keV state. The error shown in the figure for C2S includes the uncertainty
of DWBA fit to the angular distribution data for a particular SPANC value
obtained for the chosen r0 and a0 parameters of the bound state potential and
the experimental error of individual cross section data. The radius and
diffuseness parameters have been changed in small steps to generate the
corresponding SPANC value. The fits with inverse square function to the
extracted data ensures the pheripheral nature of the process and hence the
correctness of the extraction of ANC value. In case of the state 8664 keV,
which is a subthreshold state, the reproduction of the variation does not
follow a purely inverse square dependence. The mismatch is a result of non-
reproduction of the tail part of this weakly bound state with the asymptotic
radial behaviour of the Whitteker function. The values of SPANC-s and the
corresponding ANC-s of the states of 23Na have been listed in Tables 2 and 4.

####  II.1.3 Uncertianties of extracted ANC values

The uncertainty of the estimated value of ANC has been calculated by
propagating the error of the spectroscopic factor through the relation given
in Eq.2. In Fig. 5, extracted ANC with the estimated error has been shown as a
function of SPANC b. The fit to these secondary data points produce the mean
ANC value along with its uncertainty. Besides, the dependence of the extracted
ANC on the binding energy of the state has also been checked. In Fig. 6, we
have shown the plots of ANC as a function of binding energy keeley  for the
states at 8664 keV and 6917 keV excitation energies of 23Na. Binding energy of
a state is varied keeping the geometry parameters of the bound state potential
corresponding to the mean ANC value for the state fixed. The plots show that
unlike the more bound 6917 keV state, the ANC value of sub-threshold state
8664 keV decreases with the increasing binding energy. Thus for the 8664 keV
state the uncertainty of the ANC due to the $\pm$3 keV Hale  uncertainty in
the binding energy has been estimated graphically from the plot shown in Fig.
6. Uncertainties in the ANC-s for other deeply bound states corresponding to
the error in the binding energies are negligibly small and not considered.

Table 5: Angular momentum transfers, spectroscopic factors and proton widths of 8945 MeV state in 22Ne(3He,d)23Na reaction. Ex | Jπ | lp | nlj | C2S | C2S (Litt.) |  $\Gamma_{p}$ (keV)
---|---|---|---|---|---|---
(keV) |  |  |  | Present | Ref.Hale  |
8944 | 3/2+ | 2 | 1d3/2 | (5.54$\pm$1.41)$\times$10-4 | 8.32$\times$10-4 | (9.99$\pm$2.50)$\times$10-8
8945 | 7/2- | 3 | 1f7/2 | (3.94$\pm$0.9)$\times$10-4 |  $\leq$1.08$\times$10-3 | (9.83$\pm$2.24)$\times$10-10

####  II.1.4 DWBA calculation for Ex=8945 keV resonance state

The state at excitation of 8945 keV in 23Na is a resonance state about 151 keV
above the proton threshold. It has an important contribution in the reaction
rate of 22Ne(p, $\gamma$) reaction at T = 0.1 GK as it falls within the Gamow
window at this temperature. In earlier reports gorres ; Hale , it was
considered that at this excitation a single state exists with generally
adopted spin parity of 7/2-. Later Jenkins, et al.Jenkins  in their $\gamma$
spectroscopic study of 23Na have shown that at this excitation the nucleus has
a doublet of states with about a keV difference in excitation energy. One of
them has a spin parity of $J^{\pi}$ = 7/2- and decays to 9/2+ and 5/2+ states
of 23Na by dipole transitions. The assignment is consistent with a $l$ = 3
angular momentum transfer from a ($d,n$) study Childs . On the otherhand, the
measurement also shows a distinct coincidence of a 3914 keV $\gamma$ ray
depopulating the 5/2+ state at 3914 keV excitation with a 5030 keV $\gamma$
ray that depopulates the relevant 8944 keV state. A spin parity of 3/2+ has
been assigned to this second state from its decay branches and angular
correlation ratio. In a further study, Kelly, et al. kelly  have also observed
a strong primary transition from this 3/2+ state to the 5/2+ 3914-keV state
with branching ratio of 80$\%$ and to the 1/2+ 2391 keV state with 20$\%$.
Authors have also performed zero-range DWBA fits to the data of 22Ne(3He,
$d$)23Na∗ (8945 MeV) reaction Hale  with $l$=1,2,3 angular momentum transfers.
They opted for $l$=2 transfer assigning a 3/2+ spin-parity for the 8945 MeV
state.

Although the number of data points in the angular distribution is small, we
carried out a re-analysis within the zero range DWBA framework for this
unbound state using the code DWUCK4 code DWUCK4 . Same set of potenials from
Ref. Hale  is used. Values obtained by Hale, et al. assuming a $l$=3 transfer
and by Kelly, et al. kelly  assuming $l$=2 are reproduced. Subsequently, we
completed a least square fit to the angular distribution data assuming that
both $l$=2 ($J^{\pi}$=3/2+) and $l$=3 ($J^{\pi}$=7/2-) can contribute and the
calulation yielded the spectroscopic factors shown in Col.5 of Table 5. The
fits obtained are compared in Fig. 7. A normalizing constant N=4.42 has been
used in the zero range DWBA calculation for (3He,d) Hale ; Power . It is
apparent from Fig. 7 that the fit obtained considering the contributions of
both $l$=2 and 3 is a better reproduction of the limited angular distribution
data available.

The partial widths $\Gamma_{p}$ have been estimated for the doublet states
having excitation energy Ex= 8945 MeV from the extracted spectroscopic factors
using the relation

$\Gamma_{p}=(C^{2}S)\Gamma_{sp}$ (4)

where C2S) is the spectroscopic factor of the resonant state of 23Na for the
particular configuration and $\Gamma_{sp}$ is single particle width of the
state. Single particle width $\Gamma_{sp}$ for a pure single particle
configuration depends, like the spectroscopic factor, on the choice of the
nuclear potential used to generate the corresponding wavefunction. To estimate
the systematic uncertainty in extracted $\Gamma_{p}$, we varied the radius and
diffuseness parameters of the bound state potential from 1.125 to 1.375 fm and
from 0.39 to 0.89 fm, respectively keeping the binding energy fixed. It is
observed that partial width $\Gamma_{p}$ is more-or-less independent of the
chosen parameters as C2S and $\Gamma_{sp}$ have opposing trends of dependence
on the parameters. In the last column of Table 5, the extracted particle
widths have been listed. The error shown include the fitting uncertainty as
well as the systematic uncertainty. We retained the individual contributions
of the doublet pair in the estimation of rate of proton capture reaction
within the relevant temperature window.

Table 6: Background pole parameters obtained from R-matrix fits Jπ | Ex | $\Gamma_{P}$ |  $\Gamma_{\gamma}$[E1]
---|---|---|---
| (MeV) | (MeV) | (eV)
|  |  | R$\rightarrow$g.s | R$\rightarrow$0.44 | R$\rightarrow$2.39 | R$\rightarrow$2.98 | R$\rightarrow$6.30 | R$\rightarrow$6.91 | R$\rightarrow$8.66
1/2- | 15 | 5.0 | 589.92 | - | 2.77$\times$103 | 499.10 | 912.64 | - | 118.42
1/2+ | 15 | 5.0 | - | - | - | - | - | 4.41 |
3/2- | 15 | 5.0 | - | 632.97 | - | - | - | - |
Figure 8:  R-matrix fit to the S-factor curve for direct capture to six bound
states of 23Na. The red solid curves are the R-matrix fits with the
contribution of background poles while the dashed curves represent the
caculation without the background poles. The panel showing the S-factor curve
for DC$\rightarrow$8664 keV sub-threshold state also includes the R-matrix fit
(green dashed) with ANC value fixed from transfer reaction calculation. The
solid curve in this panel depicts the R-matrix fit with ANC value of 166
fm-1/2 for the state.

###  II.2 R matrix calculation for direct capture process

The low energy behaviour of off-resonance astrophysical S-factor for
${}^{22}Ne(p,\gamma)^{23}Na$ reaction is determined by the direct capture
process and a broad sub-threshold resonance at 8664 keV in compound nucleus
23Na ferraro . Present work attempts a R-matrix description of the low energy
behaviour of the off-resonance S-factor through the estimation of direct
capture component and the contribution of the sub-threshold state constrained
by the extracted asymptotic normalization coefficients (ANC-s).

The modeling of direct capture in ${}^{22}Ne(p,\gamma)^{23}Na$ is done using
the R-matrix code AZURE2 azure  based on the basic theory developed in the
seminal works of Lane and Thomas lane-thomas  and of Vogt vogt . In R-matrix
modeling, the channel radius ($r_{c}$) divides the radial space into an
external and an internal parts lane-thomas . Accordingly, the capture cross
section is divided into an external capture contribution coming from the
radial region beyond $r_{c}$ and an internal capture contribution from the
region below $r_{c}$. The magnitude of the external capture cross section is
determined by the asymptotic normalization coefficient (ANC) of the final
bound state azure ; angulo ; Kontos . The internal capture component of the
direct or non-resonant contribution, on the other hand, is simulated by the
high energy background states in the composite nucleus. Thus the direct
capture part of the cross section is modelled as a sum of external capture
component and the contribution from high energy background poles in AZURE2.

In the present work, we have fitted simultaneously the direct capture data of
Rolfs et al. and G$\ddot{o}$ress et al.. A detailed experimental study of
nonresonant or direct capture component of the reaction
${}^{22}Ne(p,\gamma)^{23}Na$ by Rolfs et al. Rolfs  reported the measurement
of cross sections for the transitions to six excited states and the ground
state of 23Na for proton energy varying from Ep = 550 keV to 2 MeV. In a
subsequent experiment, G$\ddot{o}$ress et al. gorres1  remeasured the direct
capture cross sections elaborately from Ep=550 keV to 1.6 MeV and also deduced
the spectroscopic factors of the final bound states from the fit to the
capture data. The data from recent low energy direct or off-resonance capture
measurements by Kelly et al. kelly  at Ep= 425 keV and by Ferraro, et al.
ferraro  at Ep=188, 205, 250 and 310 keV beam energies have also been included
in the R-matrix analysis. Ferraro et al. provided S-factor data for off-
resonance capture to ground state including the contribution of decay of sub-
threshold resonance at 8664 keV and the total off-resonance S-factor data. The
new measurements restricted the R-matrix model prediction for low energy
S-factor data for off-resonance capture in ${}^{22}Ne(p,\gamma)^{23}Na$.

The channel radius, $r_{c}$, is fixed at $r_{c}$ = 5.5 fm, a value greater
than nuclear radius of $R_{N}$ = $1.25\times(A_{p}^{1/3}+A_{T}^{1/3}$) = 4.75
$fm$ for 22Ne + $p$ system. Channel radius is not a parameter in R-matrix
modelling. A value of $r_{c}$ = 5.5 fm has been chosen based on $\chi^{2}$
minimization employing a grid search technique keeping the ANC-s fixed but
varying the parameters of the background poles. Search has been performed on
the total off-resonance S-factor data to choose the radius .

Figure 9:  R-matrix fit to the data of direct capture to the ground state of
23Na  Figure 10:  R-matrix fit to the data of total direct capture in 23Na

To fit the data for direct capture to individual states and the total off-
resonant capture S-factor simultaneously, we consider $M1$ transitions to
states with $J^{\pi}$ = 1/2+ ($s$-wave capture), $M1+E2$ transitions to
$J^{\pi}$ = 3/2+ , 5/2+ ($d$-wave capture) and $M1$ transition to $J^{\pi}$
=1/2- ($p$-wave capture) final bound states. A R-matrix calculation without
any background poles or considering only the external contribution with the
ANC-s listed Tables 2 and 4 resulted into S(E) curves shown by dashed lines in
Fig. 8. In Fig. 9, only external capture to the ground state yields the thin
black dashed-dotted line and leads to an extremely low S(0) value. Apparently,
the external contribution becomes more and more prominent as the excitation
energy of the state increases. To account for the missing cross section, we
introduced the high energy background poles in the R-matrix analysis. The
poles having spin parity 1/2+, 1/2-, 3/2- are included and only the $E1$ decay
of the states have been considered. The number of background poles is found to
be minimum to obtain a simultaneous fit to the data set considered. The poles
are placed at an excitation energy of 15 MeV Kontos . The proton partial width
of the poles are fixed at $\Gamma_{p}$ = 5 MeV and it is within the estimated
Wigner limit rolfs book  for particle widths. However, $\Gamma_{\gamma}$
values of the background poles are left as free parameters with intial value
taken from Weiskoff limit for corresponding gamma transitions. The fitted
background pole parameters are shown in Table 6. The resultant R-matrix fits
to the astrophysical S(E) data for direct capture to excited states are shown
in Fig. 8.

In the last panel of Fig. 8, it is observed that a better fit to DC
$\rightarrow$ 8664 keV capture data is obtained for ANC of (166 $\pm$ 4)
fm-1/2 instead of 144 fm-1/2 from transfer calculation. The value and its
uncertainty have been obatined from a simultaneous best fits, with minimum
total $\chi^{2}$, to DC $\rightarrow$ 8664 keV state, 8664 keV $\rightarrow$
GS and the total S-factor data. In the multiparameter fit, the background
$\Gamma_{\gamma}$ values are kept free while a grided search is performed over
the ANC of the state. The corresponding background pole parameters are listed
in Table 6. The condition of simultaneous fitting has reduced the uncertainty
in the ANC value. The enhanced ANC corresponds to spectroscopic factor C2S =
0.43 for the state compared to the value of 0.32 that yielded ANC = 144 fm-1/2
from transfer calculation and corroborates well with the value given by
Ferraro, et al. ferraro .

In Fig. 9, along with the DC $\rightarrow$ GS contribution (black dashed-
dotted line), we have shown the contribution from the decay of 8664 keV sub-
threshold (-130 keV) resonance (blue dashed line). The state decays to the
ground state with a branching of (84$\pm$3)$\%$ firestone  ($\Gamma_{\gamma}$
= 4.7 eV gorres ). The orange solid line in Fig. 9 represents the result from
R-matrix fit to total capture to the ground state of 23Na. The rise in the low
energy S-factor data has been nicely reproduced. No interference effect
between the two transitions is observed as the summed contribution of
individual DC $\rightarrow$ GS and 8664 keV $\rightarrow$ GS (pink dashed
line) coincides with the solid orange line obtained directly. The total S(E)
for off-resonance capture in 22Ne(p,$\gamma$)23Na obtained by summing all the
individual S(E) functions for transitions to the ground and the excited states
is shown in Fig. 10. Excellent overall fits to the data sets are obtained.

The total S${}_{tot}^{DC}$(0) value for direct capture contribution is
48.8$\pm$9.5 keV.b from the present R-matrix calculation. The uncertainty in
the value includes the contributions from the variation in $r_{c}$, energy
location of background poles and uncertainty values of the ANCs added in
quadrature. Dominating contribution comes from the uncertainty in the ANC of
8664 keV sub-threshold state. Also a 10$\%$ variation in $r_{c}$ introduces a
variation of 6.24 keV.b in total direct capture S-factor. The present value is
close to S${}_{tot}^{DC}$(0) = (50$\pm$12) kev.b reported by Ferraro ferraro
but less than the previously adopted value of 62 keV.b gorres ; angulo .
However, the resultant uncertainty from the present estimation is less.

Figure 11:  Ratio of reaction rate from present calculation to the STARLIB rate for direct and resonant captures in ${}^{22}Ne$(p,$\gamma$)23Na reaction. The solid curve represents the ratio with the total capture rate of the reaction.  Table 7: Summary of resonance strengths ($\omega\gamma$) used in reaction rate estimation. Ex |  |  | Er |  |  |  |  $\omega\gamma$(keV)444Resonance strengths of states with $E_{r}$ above 458 keV are taken from STARLIB Compilation STARLIB  |  |  |  |  $\omega\gamma$(keV)
---|---|---|---|---|---|---|---|---|---|---|---
(keV) |  |  | (keV) |  |  |  | Literature |  |  |  | Present
8830 |  |  | 35 |  |  |  | (3.6$\pm$0.2)$\times$10-15 Hale  |  |  |  |
8862 |  |  | 68 |  |  |  |  $\leq$ 6$\times$10-11 ferraro  |  |  |  |
8894 |  |  | 100 |  |  |  |  $\leq$ 7.0$\times$10-11 ferraro  |  |  |  |
8945 |  |  | 151 |  |  |  |  |  |  |  | (2.0$\pm$0.5)$\times$10-7
8944 |  |  | 150 |  |  |  |  |  |  |  | (3.93$\pm$0.9)$\times$10-9
8972 |  |  | 178 |  |  |  | (2.7$\pm$0.2)$\times$10-6 ferraro  |  |  |  |
9000 |  |  | 205.6 |  |  |  |  $\leq$ 2.8$\times$10-8 cavanna1  |  |  |  |
9042 |  |  | 248.4 |  |  |  | (9.7$\pm$0.7)$\times$10-6 ferraro  |  |  |  |
9211 |  |  | 417 |  |  |  | (8.8$\pm$1.02)$\times$10-2 kelly  |  |  |  |
9252 |  |  | 458 |  |  |  | 0.5 Meyer  |  |  |  |

##  III Thermonuclear reaction rate of ${}^{22}Ne(p,\gamma)^{23}Na$.

The thermonuclear reaction rate of 22Ne(p,$\gamma$)23Na is controlled by
several non interfering low energy narrow resonances and the total off-
resonance capture reaction. Reaction rate for narrow resonance is calculated
using the analytical expression

$\displaystyle N_{A}<\sigma\upsilon>=\sqrt[3]{(\frac{2\pi}{\mu
kT}})\hbar^{2}(\omega\gamma)_{r}exp(-\frac{E_{r}}{kT}).$ (5)

The quantity $\mu$ is the reduced mass, k is Boltzmann${}^{,}s$ constant,
$E_{r}$ is resonance energy in the centre of mass frame, and $\omega\gamma$ is
the resonance strength with $\omega$ being the statistical spin factor and
$\gamma=\frac{\Gamma_{p}\Gamma_{\gamma}}{\Gamma}$ where $\Gamma_{p}$,
$\Gamma_{\gamma}$ and $\Gamma$ are the proton partial width, $\gamma$ decay
width and the total width, respectively. Resonance strengths used in the
estimation are listed in Table 7. Only the strengths of the doublet states at
around $E_{r}$ = 151 keV have been determined in the present work and the
corresponding summed contribution is shown by green solid line in the figure.
In estimating the reaction rate, all the resonance strengths have been divided
by the calculated electron screening enhancement factor corresponding to
respective excitation energy and tabulated in Table I of Ref. ferraro . The
rates plotted for Er = 68 and 100 keV are calculated with only the
experimental upper limits of the respective resonance strengths reported by
Ferraro, et al.. The Stot(E), yielded by the R-matrix calculation for total DC
plus sub-threshold contribution to ground state, is used to get the rate for
off-resonant component (black solid line in the upper panel). The uncertainty
limits of the off-resonant contribution (black dashed dot line) is calculated
from the total uncertainty in the off-resonant astrophysical S-factor.
Individual components are shown in the upper panel of Fig. 11. The non-
resonant reaction rates have been determined using the code EXP2RATE V2.1 by
Thomas Rauscher rauscher .

Total rate, which is the sum of all individual components, is indicated in the
figure by a bold red line in the lower panel of Fig. 11. Based on their
estimation of the upper limits of 68 and 100 keV resonance strengths, Ferraro,
et al. assumed that the role of these resonances in the total rate at relevant
temperature is insignificant. To compare our total rate in the same
temperature window with that of Ferraro, et al., Hale, et al. and Cavanna, et
al., we estimated the total rate without the contributions of 68 and 100 keV
resonances. At around T = 0.1 GK, the present rate is about an order higher
than Hale’s rate but only slightly higher compared to the rates determined by
Cavanna as well as Ferraro. For 0.1 GK $\leq$ T $\leq$ 0.2 GK region, the
estimated rate is distinctly higher than both the rates. In the T $\leq$ 0.1
GK, our rate is similar to the rate of Ref. ferraro . The associated upper and
lower uncertainty limits are shown by red dashed lines. While calculating the
limits for the resonant capture rate, we took into account the uncertainties
of the energy locations of states along with respective uncertainties of the
resonance strengths. While the upper limits of the strengths of 68 and 100 keV
resonances are considered in estimating the upper limit of total rate, for
lower limit of the total rate the lower limits of the strengths are set to
zero depalo . Thus the effect of 68 and 100 keV resonances is included in the
uncertainty region of the total rate bounded by the red dashed lines.

##  IV Conclusion

A consistent analysis of direct capture reaction in 22Ne(p,$\gamma$)23Na has
been performed within the R-matrix framework. constrained with the asymptotic
normalization constants of the bound states of 23Na obtained from the transfer
reaction calculation. Asymptotic normalization constants have been extracted
from finite DWBA analysis of 22Ne(3He,d)23Na transfer data.

Astrophysical S-factor data for capture to the bound states of 23Na have been
reproduced from the analysis. Contribution of capture through the sub-
threshold resonance at 8664 keV excitation in the total capture to ground
state of 23Na has been delineated. The observed rise in the ground sate
capture data is reproduced nicely. Total direct capture S-factor at zero
relative energy, $S_{DC}$(0), is found to be 48.8 $\pm$ 9.5 having less
uncertainty.

The total reaction rate obtained as a function of temperature differs from the
recent estimations by Ferraro et a. in the temperature window of 0.1 GK $\leq$
T $\leq$ 0.2 GK. The difference is caused due to slightly higher contribution
from direct plus sub-threshold capture to the ground state. However, the
present uncertainty in the total rate in this region is relatively higher due
to the uncertainty in the resonance strength of the unbound state extracted
from transfer angular distribution data. However, in T $\leq$ 0.1 GK, the
uncertainty in the rate is comparable with the result of Ferraro, et al..

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