Thermonuclear Fusion The Basic process
The binding energy curve of shows that energy can be released if two light nuclei combine to form a single larger ucleus, a process called nuclear fusion. That process is hindered by the Coulomb repulsion that acts to prevent the two positively charged particles from getting close enough to be within range of their attractive nuclear forces and thus “fusing.” The beight of this Coulomb barrier depends on the charges and the radii of the two interacting nuclei. We show in that, for two protons (Z = I), the barrier height is 400 keY. For more highly charged particles, of course, the barrier is correspondingly higher. o generate useful amounts of energy, nuclear fusion must occur in bulk matter. The best hope for bringing this’ about is to raise the temperature of the material until the particles have enough energy-due to their thermal motions alone-to penetrate the Coulomb barrier. We call this process thermonuclear fusion.the vernacular studies temperature are reported in the kinetic energy K is interesting particle via as relation.
in which K is the kinetic energy corresponding to the most probable speed of the interacting particles, k is the Boltzmann constant, and the temperature Tis in kelvins. Thus, rather than saying, “The temperature at the center of the Sun is 1.5 X 107 K,”it is more common to say, “The temperature at the center of the Sun is 1.3 keV.” Room temperature corresponds to K = 0.03 eV; a particle with only this amount of energy could not hope to overcome a barrier as high as, say, 400 keY. Even at the center of the Sun, where kT = 1.3 keV, the outlook for thermonuclear fusion does not seem promising at first glance. Yet we know that thermonuclear fusion not only occurs in the core of the Sun but is the dominant feature of that body and of ll other stars.The puzzle is solved when we realize two facts: (I) The energy calculated with Eq. 44-9 is that of the particles with the most probable speed, as defined in Section20-7; there is a long tail of particles with much higher ~ and, correspondingly, much higher energies. (2) The barrier heights that we have calculated represent the peaks of the barriers. Barrier tunneling can occur at energies considerably below those peaks, as we saw in the case of alpha decay in sums things up. The curve marked n(K) in this figure is a Maxwell distribution curve for the protons in the Sun’s core, drawn to correspond to the Sun’s central temperature. This curve differs from the Maxwell distribution curve given in Fig. 20-7 in that here the curve is drawn in terms of energy and not of speed.Specifically. for any kinetic energy K, the expression n(K) dK gives the probability hat a proton will have a kinetic energy lying between K and K + dK. The value of kT in the core of the Sun is indicated by the vertical line in the figure; note that many of the Sun’s core protons have energies greater than this value .. The curve marked p(K) in Fig. 44-10 is the probability of barrier penetration by two colliding protons. The two curves in Fig. 44- IO suggest that there is a particular proton energy at which proton-proton fusion events occur at a maximum rate. At energies much above this value, the barrier is transparent enough but too few protons have these energies, so the fusion reaction cannot be sustained. At energies much below this value, plenty of protons have these energies out the Coulomb barrier is too formidable.
Assume a proton is a sphere of radius R = J frn. Two protons are fired at each other with the .same kinetic energy K.
(a) What must K be if the particles are brought to rest by their mutual Coulomb repulsion when they are just “touching” each other? We can take this value of K as a representative measure of the height of the Coulomb barrier.
SOLUTION: The Key Idea here is that the mechanical energy E of the two-proton system is conserved as the protons move toward each other and momentarily stop. In particular, the initial mechanical energy E is equal to the mechanical energy Ef when they, stop. T initial energy E; consists only of the total kinetic: energy ].K of that wo protons. v,:J!en the protons stop~ energy Ef consists only of the electric potential energy U of the system, as given by Bq, 25-43 U = QIQ2/47TF.Or). Here the distance r between the protons when they .stop is their center-to-center distance-‘W, and their.charges -ill ‘and q2 are both e. Then we can write the conservation of energy