Utilisateur:Plasmaphys/The Learner rule

The Learner rule is an "unexpected law" concerning the number of weak lines in an emission spectrum . Leaner measured a large number of line intensities in the atomic spectrum of iron, over a dynamic range of 1000 and emphasized in 1982 the existence of a remarkable power law for the density of lines versus their intensity: the logarithm $\log(N_{n})$ of the number of lines whose intensities lie between $2^{n}I_{0}$ and $2^{n+1}I_{0}$ ($n$ is an integer) is a decreasing linear function of $n$:

${\begin{aligned}\log _{10}{\frac {N_{n}}{L}}&\approx a_{0}+p\times n\end{aligned}}$

where $L$ is the total number of lines, $a_{0}$ a constant and $p$ the slope. In Learner's example, i.e. the spectrum of neutral Iron (Fe I), the value of $I_{0}$ is chosen in such a way that this law holds for $1\leq n\leq$ 9 (9 octaves) when about 1500 lines within the wavelength range 290 nm $\leq \lambda \leq$ 550 nm are considered. One has $N_{n}=N_{0}.10^{-np}$, where $N_{0}$ is the number of lines in the first octave: the number of lines is divided by $10^{p}$ when the size of the interval is multiplied by two. Furthermore, analysis of astrophysical data for arsenic also revealed that a line with the same slope could fit the observed data. Learner also mentioned that, in fields sufficient to cause ionization potential lowering, selection rules may be relaxed so that a large number of forbidden lines in isolated atoms could become allowed. It would be interesting to perform additional calculations (and measurements) to access that suggestion.

This work has stimulated much discussion.

Investigations and explanatory attempts

Scheeline performed a careful study of Learner's law using hydrogenic electric-dipole transitions. He confirmed the global tendency observed by Learner, but could not provide a precise theoretical interpretation for Fe I since hydrogenic formulas are not relevant.

In contrast, the emission spectrum from arsenic, which has a much more complex electronic structure than hydrogen, shows an intensity distribution closer to the power law, but with a different value of the exponent .

The extended line-by-line calculations carried out by Kurucz yield a 19 % difference in the slope, using the assumption of Boltzmann equilibrium.

Bauche-Arnoult and Bauche investigated Learner's results, on the basis of two assumptions for the level populations: Boltzmann equilibrium and collisional-radiative steady-state . In both cases, Learner's linear law is reproduced, but with different slopes, given by 25 % and 17 % respectively. Bauche-Arnoult and Bauche also pointed out the fact that the theoretical work should be improved: more lines should be added to the set of Fe I lines, second-order configuration-mixing effects should be accounted for and a more sophisticated collisional-radiative model should be applied.

In 2013, Pain reviewed this power law dependence problem and presented a discussion regarding fractal dimension and quantum chaos . According to his discussion, the line strength distribution evaluated under the fully quantum-chaos assumption does not explain Learner's law. As presented in his review , the origin of this power law is still not understood, despite almost fourty years passing since Learner's paper.

Recently, K. Fujii and J. C. Berengut reported that the combination of two statistical models - an exponential increase in the level density of many-electron atoms and local-thermodynamic-equilibrium excited-state populations - produces a surprisingly simple analytical explanation for this power law dependence. They found that the exponent of the power law is proportional to the electron temperature. This dependence may provide a useful diagnostic tool to extract the temperature of plasmas of complex atoms without the need to assign lines.