On the Possible Role of the Planck Length in Fitting the Neutron Lifetime

At present, no unified formula is available for estimating the Planck length or the Newtonian gravitational constant in terms of elementary physical constants. In this context, considering our 4G model of final unification, we have noticed a simple relation for fitting the Planck length in terms of nuclear physical constants. The hypothetical distance travelled by photon in a time span equal to the neutron lifetime seems to be: 1) Directly proportional to the proton mass and the nuclear volume; 2) Inversely proportional to the nucleon mass difference, 4G model of weak interaction range and twice the Planck length. It may be noted that, twice the Planck length can be understood as the Schwarzschild radius of the Planck mass. This relation seems to highlight the need and accuracy of the nuclear charge radius and neutron lifetime. Considering our 4G model, nuclear charge radius is 1.2393 fermi and fitted neutron lifetime is 884.2 sec. For a nuclear radius of (1.23 to 1.24) fermi, obtained neutron life time is (864.5 to 885.7) sec. Interesting point to be noted is that, a small reduction in nuclear volume seems to reduce the neutron lifetime significantly. We are working on understanding the reasons in terms of weak interaction and difference in sub-zero cooling temperatures of the bottle and beam methods of neutron lifetime experiments. Proceeding further, without considering any arbitrary coefficients or numbers, the most complicated macroscopic Newtonian gravitational constant can be estimated in a semi empirical approach connected with neutron lifetime experiments and nuclear charge radii experiments.