From the experimentally measured (p,n) cross-sections for 92,94Zr,93Nb,95,96,98Mo,103Rh,107,109Ag,110Pd,115In,117,122Sn nuclides, for proton energies below 7 MeV, thermonuclear reaction rates in the temperature range 1≤T9≤5(T9=109 K) have been extracted. These reaction rates have been fitted to a three parameter empirical expression proposed by Fowler

Kailas, S.

English

Publications of the IAS Fellows

J. Astrophys. Astr. (1986) 7, 53–57  
 
 
 
 
Thermonuclear Reaction Rates from (p, n) Reactions on Targets
with A = 92–122  
 
S. Kailas Nuclear Physics Division, Bhabha Atomic Research Centre, Trombay,
Bombay 400085  
 
Received 1985 May 8; accepted 1985 December 3
 
 
Abstract. From the experimentally measured (p,n) cross-sections for
92,94Zr, 93Nb, 95,96,98Mo, 103Rh, 107,109Ag, 110Pd, 115In, 117,122Sn nuc- 
lides, for proton energies below 7 MeV, thermonuclear reaction rates in the  
temperature range 1   T9    5(T9=109 K) have been extracted. These  
reaction rates have been fitted to a three parameter empirical expression  
proposed by Fowler.  
 
Key words: (p,n) reaction: A = 92122 targets—thermonuclear reaction
rates  
 
 
1. Introduction 
 
The astrophysical calculations concerning the synthesis of elements require as input 
temperature-dependent expressions for thermonuclear reaction rates (TNRR). These 
TNRR values are obtained starting from the cross-section data of interest, measured 
experimentally or calculated theoretically. Recently, an excellent review on the topic of 
charged particle reaction cross-section and nucleosynthesis has been published 
(Sargood 1982,1983). Extensive tabulations of the TNRRs have been made for proton 
induced reactions covering the nuclei with Ζ up to 36. (Woosley et al. 1975,1978). These 
TNRRs have been obtained employing theoretical models for the calculation of the 
cross-sections. Experimentally only 5060 per cent of these cross-sections have been 
measured so far. For nuclei with Ζ > 36, TNRRs have been determined recently from 
measured (p,n) and (p,y) cross-sections (Roughton et al. 1979). In the present work, 
starting from the recently measured (p,n) cross-section data for nuclei with A 
= 92–122, the TNRRs have been computed. Most of the nuclei considered here have
not been included in the work of Roughton et al. (1979). In Section 2, the procedure for 
the calculation of TNRR is discussed. In Section 3, the results obtained are given. 
General conclusions are given in Section 4.
 
 
2. Procedure 
 
The TNRR is the rate at which a specific nuclear reaction takes place in the stellar 
environment. The factors contributing to this rate are the cross-section for the reaction 
and the temperature of the stellar environment which determines the energy 
 
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54  S.  Kailas 
 
distribution of the colliding particles. Excellent reviews are available on this topic. 
(Burbidge et al. 1957; Clayton 1968). The reaction rate between two different nuclei 1 
and 2 in a high temperature (Τ ) environment depends upon the densities of 1 and 2, the 
temperature, and the appropriate nuclear properties of the system. It is given by the 
expression 
 
(1) 
 
where n1, n2 are the number densities of nuclei 1 and 2. Generally, the nuclei are 
assumed to possess a Maxwell-Boltzmann distribution of velocities at Τ and hence 
〈σ  〉 is averaged over this distribution. Converting to energy space,
 
(2) 
 
where k is the Boltzmann constant, µ the reduced mass, Ε the centre of mass energy, Q is 
the Q value of the reaction. A more usual form of quoting the TNRR is as follows: 
 
(3) 
where N
A
 is the Avogadro number.
It is assumed that the target nucleus is in the ground state; otherwise nuclear partition 
functions must be included. Once the cross-section σ (E) is given as a function of E, it is 
easy to compute P12. For the convenience of the calculation, the σ (Ε) values measured 
experimentally are fitted with an expression (Vlieks, Morgan & Blatt 1974)
 
(4) 
 
This expression is introduced in Equation (3) to determine P12. In the present work, we 
have kept the parameter b = Ζ — 0.5 where Ζ is the atomic number of the target.  
Earlier, we have reported TNRR calculation for several medium-weight nuclei using  
the above technique (Kailas & Mehta 1976). Starting from the experimentally 
measured (p,n) reaction cross-sections (Johnson et al. 1977; Johnson, Galonsky &
Kernell 1979, Hershberger et al. 1980; Flynn, Hershberger & Gabbard 1979) for nuclei 
with A in the range 92 to 122 and for proton energies below 7 MeV, the TNRRs have 
now been calculated in the temperature range 1    T9     5,where T9= T/109 K. 
 
3. Results 
 
The TNRRs have been calculated by the procedure discussed in Section 2. The upper 
limit of integration (expression 3) has been kept equal to ~7 MeV, the highest energy 
up to which cross-section data are available. The TNRRs are listed in Table 1. The 
TNRR is estimated to have an uncertainty of ±10 per cent, mainly due to error on σp.n 
 
For comparison, we have included the values obtained by Roughton et al. (1979) from 
thick target activation measurements. They have also made measurements for 95,96 Mo 
and 117,122 Sn targets. For Mo targets, they have listed two sets of cross-sections, one
where the final state is the ground state and the other where the low-lying isomer, is the 
final state. For95,96 Mo targets the TNRR values obtained by Roughton et al. (1979)— 
the ground state or the isomer—agree reasonably well with the present results within  
their respective uncertainties. Similarly, the TNRR values from both these works for  
 
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v
Thermonuclear reaction rates  55
 
56 S. Kailas 
 
the 122Sn target agree with each other. However, the results of Roughton et al. for the
117Sn target differ considerably from that of the present work. Roughton's (1979)  
values are a factor of 2 to 3 higher as compared to the present results. The reason for this 
discrepancy is not clear and it needs to be explored. It may be noted that the TNRR  
rates given by Roughton et al. (1979) and calculated here refer to only 'laboratory' rates. 
Thus at high temperatures they do not include the effect of having a thermal population  
of accessible excited states in the target nucleus.
In order to enhance the usefulness of these TNRR values in astrophysical  
calculations, we have fitted them to an empirical form suggested by Fowler (1973). The 
TNRR is given as  
(5)  
 
where α, β and g are constants to be found by fitting the P12 values at various T9 values  
for each target. Even though the above functional form was tried for all the nuclides 
discussed in the present work, only half the number of them could be adequately fitted 
by expression (5) to yield α, β and g  values. The results are listed in Table 2. A careful 
examination of these parameters (Table 2) or the TNRR values (Table 1) does not 
reveal any systematic behaviour as a function of A or Ζ of the targets considered here.
 
4. Conclusion 
 
Starting from the measured (p,n) cross-sections, we have computed the TNRR values 
for 13 nuclides in the mass range A = 92122. The TNRR values extracted in the 
present work agree with those obtained by Roughton et al. (1979)in the case of95,96Mo 
and 122Sn nuclei. The two results differ considerably for 117Sn target. Roughton et al. 
(1979) have quoted the TNRR values with errors ranging from 15 to 50 per cent. The 
present results are estimated to have uncertainties of not more than 10 per cent. The 
compilation of the TNRR values for A = 92–122 does not seem to indicate the presence
of any kind of simple systematics. It is hoped that the experimentally determined values 
of TNRR will be helpful in improving the theoretical models developed to compute 
them, so that these theoretical calculations can be extended with greater confidence to 
many other nuclei for which measurements do not exist.
 
Acknowledgement 
 
The author thanks Professor Ramadurai for his helpful comments.
Table 2.  Parameters fittings the TNRR data (1   T9     5).
⋝ ⋝ 
Thermonuclear reaction rates 57
 
References 
 
Burbidge, E. M., Burbidge, G. R, Fowler, W. Α., Hoyle, F. 1957, Rev. Mod. Phys., 29, 547.
Clayton, D. D. 1968, Principles of Stellar Evolution and Nucleosynthesis, McGraw Hill,  
New York.  
Flynn, D. S., Hershberger, R. L, Gabbard, F. 1979, Phys. Rev., C20, 1700.  
Fowler, W. A. 1973, in Explosive Nucleosynthesis, Eds D. N. Schramm & W. D. Arnett, Univ. 
Texas press, p. 297.  
Hershberger, R. L., Flynn, D. S., Gabbard, F., Johnson, C. Η. 1980, Phys. Rev., C21, 896.  
Johnson, C. Η., Bair, J. Κ., Jones, C. Μ., Penny, S. Κ., Smith, D. W., 1977, Phys. Rev., C15,196.  
Johnson, C. H., Galonsky, Α., Kernell. 1979, Phys. Rev., C20, 2052.  
Kailas, S., Mehta, M. K. 1976, Pramana, 7, 6.  
Roughton, Ν. Α., Fritts, M. J., Peterson, R. J., Zaidins, C. S., Hansen, C. J. 1979, Atomic Data and 
Nuclear Data Tables, 23, 177. 
Sargood, D. G. 1982, Phys. Rep., 93, 61. 
Sargood, D. G. 1983, Phys. Rep., 98, 396. 
Vlieks, A. E., Morgan, J. F., Blatt, S. L. 1974, Nucl. Phys., A224, 492.  
Woosley, S. E., Fowler, W. A., Holmes, J. Α., Zimmermann, B. A. 1975, Orange aid preprint
No. 422, California Inst. Technology.  
Woosley, S. E., Fowler, W. A., Holmes, J. Α., Zimmermann, B. A. 1978, Atomic Data and Nucl. 
Data Tables, 22, 371.  


Thermonuclear reaction rates from (p, n) reactions on targets with A = 92-122

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Thermonuclear reaction rates from (p, n) reactions on targets with A = 92-122

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