Constants of nature, atomic masses and some definitions...


orangebu.gif (324 octets) Constants of nature
orangebu.gif (324 octets) Energy factors
orangebu.gif (324 octets) Table of Atomic masses
orangebu.gif (324 octets) Astrophysical S-factor
orangebu.gif (324 octets) Gamow energy
orangebu.gif (324 octets) References


Constants of nature

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

Energy factors

Some useful energy equivalent factors:

1 amu = 931.494 MeV/ c2
1 eV  = 1.60218e-19 J

Atomic masses

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
(amu)

Used value
(amu)

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

 

Astrophysical S-factor

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]..


 Gamow window and Gamow energy

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].


 References

[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).