Fusion in the Sun: Without tunneling, our Sun would fail us. The source of its energy is n...

Fusion in the Sun: Without tunneling, our Sun would fail us. The source of its energy is nuclear fusion, and a crucial step is the fusion of a light-hydrogen nucleus, which is just a proton, and a heavy-hydrogen nucleus, which is of the same charge but twice the mass. When these nuclei get close enough, their short range attraction via the strong force overcomes their Coulomb repulsion. This allows them to stick together, resulting in a reduced total mass/internal energy and a consequent release of kinetic energy. However, the Sun’s temperature is simply too low to ensure that nuclei move fast enough to overcome their repulsion.

(a) By equating the average thermal kinetic energy that the nuclei would have when distant, , and the Coulomb potential energy they would have when 2 fm apart, roughly the separation tit which they stick, show that a temperature of about 109 K would be needed.

(b) The Sun’s core is only about 107 K. If nuclei can’t make it “over the top,” they must tunnel. Consider the following model, illustrated in the figure: One nucleus is fixed at the origin, while the other approaches from far away with energy E. As r decreases, the Coulomb potential energy increases, until the separation r is roughly the nuclear radius rnuc, whereupon the potential energy is Umax and then quickly drops down into a very deep “hole" as the strong-force attraction takes over. Given that E « Umax, the point b, where tunneling must begin, will be very large compared with rnuc, so we approximate the barrier’s width L. as simply b. Its height, U0, we approximate by the Coulomb poten­tial evaluated at b/2. Finally, for the energy E, which fixes b, let us use , which is a reasonable limit, given the natural range of speeds in a thermodynamic system. Combining these approxi­mations, show that the exponential factor in the wide-barrier tunneling probability is

(c) Using the proton mass for m, evaluate this factor for a temperature of 107 K. Then evaluate it at 3000 K, about that of an incandescent filament or hot flame, and rather high by Earth standards. Discuss the consequences.