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Energy production in stars occurs mainly when a nucleus absorbs a proton or fuses with another nucleus. Some examples:

(i) $\rm{p}(\rm{p},\rm{e}^+\nu)\rm{d}~$ and $~\rm{d}(\rm{p},\gamma)^3\rm{He}~$ and $~^3\rm{He}(^3\rm{He},2\rm{p})\alpha~$ in the ppI chain in the sun;

(ii) $2\alpha\to\rm{Be}^*$ in the triple alpha process in heavy star cores after H-depletion;

(iii) $^{12}\rm{C}(^{12}\rm{C},\rm{p})^{23}\rm{Na}~$ as one of the possible reactions during carbon burning in heavy stars.

The very first of the above reactions proceeds via quantum mechanical tunneling, because very few protons have enough kinetic energy to penetrate the Coulomb barrier, given the velocity distribution at the temperature of the sun's core.

First question about all nuclear reactions in which a proton is absorbed or two nuclei fuse: do ALL these reactions proceed via quantum tunneling?

Second question: do all "resonant reactions" proceed via tunneling, as is the case in (ii)?

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They all need tunneling; you just need to work out the required temperatures to penetrate the Coulomb barrier.

The "most likely" reaction would be getting two protons together. If you argue that they need to get within $r\sim 10^{-15}$ m of each other to feel the strong nuclear potential then the Coulomb barrier is of height: $$ E_C = \frac{e^2}{4\pi \epsilon_0 r} \simeq 1.4\ {\rm MeV}$$. But if you were to translate this into a temperature $k_BT$, then $T = 1.6 \times 10^{10}$ K.

Now even if you accept that at lower temperatures there is a tail in the Maxwell-Boltzmann distribution, the temperatures in the core of a main sequence stars never come within a factor of 100 of this.

All subsequent reactions have higher Coulomb barriers because although the barrier increases as the product of the charges involved, the radii of the nuclei only grow as something like $A^{1/3}$ where $A$ is the atomic mass.

I would say this also applies to resonant reactions because you still have to get the particles within the range of the strong nuclear potential and hence the resonance still occurs within the Gamow window, which is set by the steep exponential decay of the Maxwell-Boltzmann tail and the exponential increase of the tunneling probability.

This is essentially why you need much higher temperatures even to get these reactions started in the cores of more evolved stars, but temperatures don't exceed a few $10^{9}$ K even in the more advanced stages of nuclear burning in the cores of the most massive stars.

I guess you can't say that fusion wouldn't happen at all without tunneling - it would just proceed at a much. much lower rate unless the stars were able to contract sufficiently to get to much higher temperatures in their interiors. They can't do that because electron degeneracy sets in; but I suppose if you are going to ignore one quantum effect maybe you can ignore others!

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  • $\begingroup$ Thank you. One detail: you give the "average" energy of a particle in a gas as kT while it is often given as 3/2kT, taking into account three degrees of freedom (spatial x,y,z). Is this simply an "order of magnitude" matter without physical significance or is there a specific reason for having the factor of 3/2 (or not having it)? $\endgroup$ – gamma1954 Oct 21 at 19:25
  • $\begingroup$ @gamma1954 a factor 1.5 is not important. $\endgroup$ – Rob Jeffries Oct 21 at 21:54

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