Mass dependence of fixed-barrier tunnelling ($-\log T\sim \sqrt{m}$)

At fixed incident energy and fixed potential-energy profile, the forbidden-region WKB action is $\int \sqrt{2m(V-E)}dx$. Its leading transmission exponent grows as the square root of mass. Equal electrostatic voltages need not define equal potential-energy barriers for particles with different charges. The leading exponential comparison assumes a sufficiently opaque barrier; prefactors and the WKB turning-point conditions remain relevant.