346. Thermonuclear fusion. Nuclear reaction rates – second part

Posted by Dmitry

April 10, 2009

Save This Post as PDF

Last time we have figured out that two factors determine effective rates of nuclear reactions – the probability of quantum mechanical tunnelling through the Coulomb barrier and the probability of nuclear transformation. Let us talk today about the second factor a bit.

While the first factor given in the simplest case by the Gamov exponent which is universal, the second factor really depends on the reaction you consider. For example, for reactions involving production of ${}^4He$ (maximally bound nucleus) it is large, with resonant dependence on energy. In particular, this is the case for one of the most perspective reactions $p+{}^{11}B\to{}3^4He$ (no outgoing neutrons).

On the other hand, for reactions that proceed due to weak interaction (first three from the table in the previous post) this factor is extremely small. For example, the reaction like $p+p\to{}d+e^++\gamma$ , extremely important for the nuclear fusion in Sun, was never observed in laboratory.

That’s, I am afraid, pretty much all I know about the second factor in the rate , so let me explain how rate of the reaction depends on thermodynamic quantities describing plasma. Let us start with densities of nuclei. The main contribution into collision integral for plasma comes from one-to-one collisions between nuclei, so we can really estimate it as

$n_1n_2\langle\sigma{}v\rangle$ ,

where n_1 and n_2 are densities of nuclei of the type 1 and 2 correspondingly. If nuclei in the plasma are of the same type, we should write

$\frac{1}{2}n^2\langle\sigma{}v\rangle$

instead.

Dependence of the rate on temperature is defined by the factor $\langle\sigma{}v\rangle$ in the collision integral. When temperatures are not very high (so that the Coulomb barrier is penetrated due to QM tunnelling), cross-section can be estimated using Gamov exponent:

$\sigma\sim{}v^{2}\exp\left(-\frac{2\pi{}Z_1Z_2e^2}{\hbar{}v}\right)$ .

Averaging the product $\sigma{}v$ over Maxwell distribution we have

$\langle\sigma{}v\rangle\sim{}T^{-2/3}\exp\left(-\frac{3}{2}\left(\frac{4\pi^2{}3{}Z_1^2Z_2^2e^4\mu}{\hbar^2kT}\right)^{1/3}\right)$ .

Note that the temperature dependence

$\langle\sigma{}v\rangle\sim\exp\left(-\frac{\rm Const.}{T^{1/3}}\right)$

is relatively weak (say, compared, to $exp(-{\rm Const.}/T)$ that holds for chemical reactions). That’s basically what allows thermonuclear reactions to run at so (relatively) low temperatures.

Entered Apprentice, Nuclear physics

If you liked the post, please kindly consider to leave a comment, subscribe to the RSS feed or get new posts sent directly to your Inbox. If you want to chat with me in real time, you can find me on Twitter. The posts below are probably related to the subject of this one:

379. Thermonuclear fusion: list of posts

344. Thermonuclear fusion. Coulomb barrier and reaction rates

355. Introduction into thermonuclear reactors

358. Thermonuclear reactors. Inertial confinement

361. NEQNET: first two weeks of April