Variable aggregate cross sections and RMS mean free paths

In a previous paper it was suggested that the area cross sections of nuclei and particles was a variable given by the formula s = (p)b$^2$ where b = [Acos 2(p)ft]$^2$ due to nuclear vibration so that s =(p)[Acos 2(p)ft]$^2$, a variable cross section using a simple oscillator. If the aggregate cross section =n(Area)(s)dx, using the variable nuclear cross section would = n(Area)(p)[Acos2 (p)ft]$^2$ dx. If the maximum value for cos=1, aggregate variable cross section = n(Area)(p)A$^2$. RMS cos$^2$ = (1/2), so that the aggregate variable nuclear cross section has an average value= .5n(Area)pA$^2$. The mean free path also uses the area cross section so that l =1/n(s). Substituting for s the variable nuclear mean free path = 1/n(p)[Acos2(p)ft. If cos max = 1, the nuclear maximum free path = 1/n(p)A$^2$.RMS average mean nuclear free path = 2/n(p)A$^2$. b = the impact parameter, A = the amplitude of nuclear motion and Acos2(p)ft is the nuclear oscillator. In all cases Acos2(p)ft is greater than the nuclear radius.