Constants of nature, atomic masses and some definitions...
Constants of nature
Energy factors
Table of Atomic masses
Astrophysical S-factor
Gamow energy
References
The constants of nature used in this compilation are those provided by the latest published report of the Particle Data Group [PA96]:
Speed of light | c = 299792458 m/s |
Avogadro number | NA = 6.0221e23 mol-1 |
Fine structure constant | 1/a = hbar c/ e2 = 137.036 |
kBT = 0.08617 T9 MeV = T9/11.605 MeV |
hbar c = 197.327 MeV fm |
Some useful energy equivalent factors:
1 amu = 931.494 MeV/ c2 |
1 eV = 1.60218e-19 J |
For the reaction rate calculations, atomic masses have been taken from the 1997 compilation by G. Audi et al. [AU97]. The table below gives the recommended values for some of the atomic masses involved in the present compilation.
Element |
Atomic mass |
Used value |
n | 1.008664924(2) | 1.0086649 |
1H | 1.007825032(1) | 1.0078250 |
2H | 2.014101778(1) | 2.0141018 |
3H | 3.016049268(1) | 3.0160493 |
3He | 3.016029309(1) | 3.0160293 |
4He | 4.002603250(1) | 4.0026033 |
6Li | 6.015122306(509) | 6.0151223 |
7Li | 7.016004073(506) | 7.0160040 |
7Be | 7.016929269(506) | 7.0169292 |
8Be | 8.005305095(38) | 8.0053051 |
8B | 8.024606727(1188) | 8.0246067 |
9Be | 9.012182248(405) | 9.0121821 |
9B | 9.0133288919(1047) | 9.0133288 |
10B | 10.012937097(349) | 10.0129370 |
11B | 11.009305514(405) | 11.0093055 |
12C | 12.000000000 | 12.0000000 |
13C | 13.003354838(5) | 13.0033548 |
13N | 13.005738584(289) | 13.0057386 |
14C | 14.003241991(4) | 14.0032420 |
14N | 14.003074007(2) | 14.0030740 |
14O | 14.008595287(80) | 14.0085953 |
15N | 15.000108973(12) | 15.0001089 |
15O | 15.003065460(540) | 15.0030645 |
16O | 15.994914622(3) | 15.9949146 |
17O | 16.999131501(223) | 16.9991315 |
17F | 17.002095238(266) | 17.0020952 |
18O | 17.999160413(851) | 17.9991604 |
18F | 18.000937665(636) | 18.0009377 |
19F | 18.998403205(75) | 18.9984032 |
19Ne | 19.001879726(612) | 19.0018798 |
20Ne | 19.992440176(3) | 19.9924402 |
21Ne | 20.993846744(43) | 20.9938467 |
21Na | 20.997655100(751) | 20.9976551 |
22Ne | 21.991385500(252) | 21.9913855 |
22Na | 21.994436633(482) | 21.9944366 |
23Na | 22.989769657(262) | 22.9897697 |
23Mg | 22.994124828(1353) | 22.9941249 |
24Mg | 23.985041874(258) | 23.9850419 |
25Mg | 24.985837000(261) | 24.9858370 |
25Al | 24 990428531(760) | 24.9904286 |
26Mg | 25.982592999(264) | 25.9825930 |
26Al | 25.986891675(268) | 25.9868917 |
27Al | 26.981538407(238) | 26.9815384 |
27Si | 26.986704124(264) | 26.9867041 |
28Si | 27.976926494(216) | 27.9769265 |
29Si | 28.976494680(219) | 28.9764947 |
29P | 28.981801337(807) | 28.9818014 |
30Si | 29.973770179(221) | 29.9737702 |
30P | 29.978313768(482) | 29.9783138 |
31P | 30.973761487(269) | 30.9737615 |
At low energies where E<<ECou, the probability that incoming
particles penetrate the Coulomb barrier can be approximated by the simple
expression:
P = exp(-2ph).
The quantity h is called the Sommerfeld parameter.
In numerical units the exponent is 2ph = 0.9895 Z1Z2(A/E)1/2, where the center of mass energy E is given in MeV and the reduced mass A is in amu. This approximate expression for the tunneling probability is commonly referred to as the Gamow factor.Due to the exponential behaviour for tunneling, P, the cross section of charged-particle-induced nuclear reactions drops rapidly for energies E<<ECou (astrophysical energy range). As a quantum-mechanical interaction between particles, the nuclear reaction probability is proportional to a geometrical factor pl2, which is proportional to E-1. The two factors, P and E-1, represent explicitely well-known energy dependences of non-nuclear nature. The cross section can be then described as:
s(E) = S(E) exp(2ph(E))/E.
The intrinsic nuclear part of the reaction probability is the so-called astrophysical S-factor, S(E), defined by this equation.The extrapolation of experimental data to the energies of astrophysical interest, the S-factor is a much more convenient function than the cross section. This is, for example, the case of reaction involving light nuclides, when no resonances are present in the stellar energy range [CL68, RO88]..
For a given stellar temperature T, nuclear reactions take place in a relative narrow window DE0 around the effective burning energy E0. In numerical units these quantities are given by:
E0 = 0.1220 (Z12Z22AT92)1/3 MeV
and
DE0 = 0.2368(Z12Z22AT95)1/6 MeV,
where the symbol T9 represents the temperature T in units of 109 K [CL68, RO88].
[CL68]: D. D. Clayton, Principles of Stellar Evolution and Nucleosynthesis, New York, MacGraw-Hill, 1968.
[RO88]: C. E. Rolfs and W. S. Rodney, Cauldrons in the Cosmos, The University of Chicago Press, Chicago and London, 1988.
[PA92]: Particle Data Group, Review of Particle Properties, Phys. Rev. D45, (1992) 1.
[AU97]: G. Audi, O. Bersillon, J. Blachot, and A.H. Wapstra, Nucl. Phys. A624, 1 (1997).