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346. Thermonuclear fusion. Nuclear reaction rates – second part

Posted by Dmitry

at April 10, 2009

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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

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

345. Lagrangian turbulence: video of the day

Posted by Dmitry

at April 9, 2009

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A simulation by Guido Bofetta, U. of Torino. Recall that Lagrangian description of hydrodynamics is when you pick a liquid particle and keep track of its motion. Here it is shown how particles are transported by a turbulent flow in the presence of a vortex.

Entered Apprentice, Hydrodynamics, Turbulence

347. Numerical simulation of vortices: video of the day

319. Turbulence. Dynamical approach

35. Lectures on AdS/CFT by Maldacena

65. Simulations of decaying turbulence

344. Thermonuclear fusion. Coulomb barrier and reaction rates

Posted by Dmitry

at April 9, 2009

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This post is the next in the series devoted to the discussion of our main energy source in the 22 century – thermonuclear fusion

Today let us talk a bit about reaction rates. Somehow, it is accustomed that we estimate these rates in terms of the maximal effective cross-section of the reaction. Here are some important and most common reactions that happen in Sun (as well as their cross-sections):

reaction	energy released	$\sigma_{\rm max}$ , $E<1{\rm MeV}$	energy of incoming particle, corr. to $\sigma_{\rm max}$
$p+p\to{}d+e^++\nu$	2.2 MeV	$10^{-23}$ barn
$p+d\to{}{}^3He+\gamma$	5.5 MeV	$10^{-6}$ barn
$p+t\to{}^4He+\gamma$	19.7 MeV	$10^{-6}$ barn
$d+d\to{}t+p$	4.0 MeV	0.16 barn	2.0 MeV
$d+d\to{}^3He+n$	3.3 MeV	0.09 barn	1.0 MeV
$d+d\to{}^3He+\gamma$	24.0 MeV
$d+t\to{}^4He+n$	17.6 MeV	5.0 barn	0.13 MeV
$t+t\to{}^4He+2n$	11.3 MeV	0.10 barn	1.0 MeV

Note that reactions involving light particles like p, d ( ${}^2He$ nucleus) and t ( ${}^3He$ nucleus) are rather low. The last column basically shows how much energy you need to pump into the system in orde to start nuclear reaction – recall that 1 eV is about 10000 K.

The picture below shows how the effective cross-sections for different reaction behave with increasing the projectile energy:

Effective cross-section for nuclear reactions

How to calculate the total cross-section of a nuclear reaction? There are two factorized contributions into $\sigma_{\rm max}$ . First, you need to overcome the Coulomb barrier (estimation of the associated probability is an easy task for any person who is familiar with quantum mechanics). Second, the probability that nuclear transformation actually happens should also be taken into account – this one is much harder to estimate, so I postpone discussion of this contribution till tomorrow.

The probability to overcome the Coulomb barrier is estimated as follows. The height of the barrier is given by

$E\sim\frac{Z_1Z_2e^2}{R}$ ,

where Z_1e and Z_2e are electric charges of nuclei. Even for smallest Z_1 and Z_2 possible (equal to 1 as in the reaction d+d), this height is about 200 keV.
On the other hand, the temperature of plasma in the center of a star (like our Sun) is about $10^7\div{}10^8$ K, which corresponds to $1\div{}10$ keV. So, we have to conclude that in stars Coulomb barrier is overcomed because of the quantum tunnelling

When the energies of particles participating in the reaction are much lower than the height of the Coulomb barrier, the probability of tunnelling is given by Gamov exponent (first instanton discovered, whether you want it or not)

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

where $v=\sqrt{2E/\mu}$ and $\mu=m_1m_2/(m_1+m_2)$ .

Interestingly, in contemporary experiments this simple expression ceases to describe physics properly. The reason is the presence of nuclei in the beam with energies higher or comparable with the height of the Coulomb barrier.

Entered Apprentice, Nuclear physics

346. Thermonuclear fusion. Nuclear reaction rates – second part

355. Introduction into thermonuclear reactors

361. NEQNET: first two weeks of April

358. Thermonuclear reactors. Inertial confinement

343. Followup: BumpTop

Posted by Dmitry

at April 8, 2009

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Just wanted to let you know that BumpTop finally went public – that is, you don’t need to be invited to the private beta to have fun with you desktop. The version 1.0 can be downloaded for free on the BumpTop website.

Entered Apprentice, Scientist's gadgets

361. NEQNET: first two weeks of April

179. Followup on ekpyrosis and phoenix universe

332. NEQNET: last two weeks of March

133. Multi-Field Inflation on the Landscape

342. Thermal equilibrium in special relativity

Posted by David Cubero

at April 8, 2009

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David Cubero is professor at the Department of Applied Physics of the University of Sevilla. Dmitry.

Special relativity, despite being more than a hundred years old, still shows an intriguing capacity to surprise us in very fundamental issues, such as thermal equililbrium. In this post, we will review a recent controversy about the proper velocity distribution of dilute gases at thermal equilibrium.

In 1911, just six years after Albert Einstein had presented his theory of special relativity to the world, Ferencz Juettner formulated [1] a generalization to relativistic particles of the celebrated formula that describes the one-particle momentum distribution of a gas in equilibrium in the non-relativistic limit, namely, the Maxwell-Boltzmann distribution:

$\phi({\bf p})=Z^{-1}\exp[-\beta E({\bf p})], \quad (1)$

where $\beta=1/(k_BT)$ , k_B is the Botlzmann constant, the temperature, a normalization constant and the particle energy. In the non-relativistic case, this energy is simply given by the kinetic energy of the particle: E=p^2/(2m) . Using a maximum entropy derivation, Juettner proposed to replace this non-relativistic expression with its relativistic counterpart:

$E=c\sqrt{m^2c^2+p^2}. \quad (2)$

The corresponding momentum distribution, i.e. Eq. (1) with (2), is called since then the Juettner distribution. The velocity distribution $f_\mathrm{J}({\bf v})$ can be obtained by transforming to the particle velocity using the relativistic formula of the momemtum ${\bf p}=m{\bf v}\gamma(v)$ , where $\gamma(v)=1/\sqrt{1-(v/c)^2}$ .

Despite lacking a rigorous microscopic derivation due to the intrinsic difficulties introduced by special relativity (and more specifically due to the difficulty of formulating a relativistically consistent Hamiltonian mechanics of interacting particles), the Juettner distribution had been widely accepted among theorists for most of the 20th century. Starting from the 1980s, a number of theorists began to question the validity of the Juettner distribution, based on “manifestly covariant” theories [2], or the lack of explicit compatibility with relativity of the original maximum-entropy principle used by Juettner [3]. Using a maximum-entropy principle in combination with Lorentz symmetry, the following “modified” Juettner distribution was proposed [3]:

$\phi_\mathrm{MJ}({\bf p})=Z^{-1}\exp[-\beta E({\bf p})]/E({\bf p}), \quad (3)$

thus differing in a 1/E prefactor.

Evidently, identifying the correct relativistic equilibrium distribution is essential for the proper interpretation of experiments in high energy and astro-physics. This controversy was resolved in a simple one-dimensional system in Ref. [4] by resorting to computer simulations. Generally, fully relativistic simulations are very hard to carry out because interactions at a distance are very difficult to implement in special relativity, ussually requiring the introduction of fields. This difficulty was avoided in Ref. [4] by considering a two-component gas made of impenetrable point-particles with elastic point-like binary collisions in 1D. The results were conclusive, favoring the Juettner distribution as the only distribution able to account for the common equilibrium of the two-component gas.

One particle velocity distributions

One-particle velocity distributions using lab time (Juettner) and proper-time (Modified Juettner) parameterizations of a 1D two-component gas with 5000 light particles of mass m_1 and 5000 heavy particles of mass m_2=2m_1 .

Later on, it was found that the modified Juettner distribution could be generated from a Monte Carlo simulation in which collisions are not consistent with the laws of Special Relativity. More specifically, the modified Juettner distribution could be obtained with a uniform random pairing technique, which is known to violate the relativistic invariance of the number of collisions in a space-time element [5].

This would have settled the controversy, had not it been for the fact that the modified Juettner function actually represents a physical distribution, as it was realized first in Ref. [6] from the analysis of relativistic Brownian processes. In our recent manuscript [7], using similar fully-relativistic molecular dynamics simulations of a two-component gas as in [4], we demonstrate that the modified Juettner distribution is obtained as the stationary one-particle distribution when a proper-time parameterization is used instead of the laboratory time, both when ensemble and time averages are carried out. The proper-time paramatrization is the natural mathematical choice when one tries to extend the theory of stochastic processes to general relativity. But also, it plays a crucial role in the description of particle creation/annihilation processes. These processes are a consequence of the equivalence of mass and energy, and thus, one of the baffling features attributable to the theory of Special Relativity. If we want to address questions like, for example, what is the typical energy distribution at the end of a particle’s life-time? Then we are compelled to consider a proper-time parameterization, which yields the modified Juettner distribution.

Last but not last, our work also reveals a close connection between time parameters and entropy in special relativity, thus, providing a better understanding of Juettner’s original derivation.

Some literature

[1] F. Juettner, Ann. Phys. (Leipzig) 34, 856 (1911).

[2] L. P. Horwitz, W. C. Schieve, and C. Piron, Ann. Phys. (N.Y.) 137, 306 (1981); L. P. Horwitz, S. Shashoua, and W. C. Schieve, Physica (Amsterdam) 161A, 300 (1989).

[3] J. Dunkel, P. Talkner, and P. Hanggi, New J. Phys. 9, 144 (2007).

[4] D. Cubero, J. Casado-Pascual, J. Dunkel, P. Talkner, and P. Hanggi, Phys. Rev. Lett. 99, 170601 (2007).

[5] F. Peano, M. Marti, and L. O. Silva, Phys. Rev. E 79, 025701(R) (2009).

[6] J. Dunkel, P. Hanggi, and S. Weber, Phys. Rev. E 79, 010101(R) (2009).

[7] D. Cubero and J. Dunkel, arXiv:0902.4785.

Astrophysics, Condensed Matter, Journal club, Master Mason