# Comparison of Monte-Carlo and Einstein methods in the light-gas interactions. 

Jacques Moret-Bailly ∗ 

(Dated: October 30, 2018) 

To study the propagation of light in nebulae, many astrophysicists use a Monte-Carlo computation which does not take interferences into account. Replacing the wrong method by Einstein coefficients theory gives, on an example, a theoretical spectrum much closer to the observed one. 

Transposition into optics of the Monte Carlo compu- tations used in nuclear physics is wrong because it does not take into account the phases of the pilot-waves of the photons. Indeed, this approach denies the existence of in- terferences because the interaction of a neutron with an atom of uranium is multiform, complex and often slow, so that the phase of the incident neutron is lost. On the contrary, for example, when two photons of same fre- quency and same polarization simultaneously strike an atom, whether their phases are same or opposite is im- portant. 

The correct method for studying the propagation of light in a resonant medium has been introduced by Ein- stein [1], but it remains disputed in astrophysics, as it was by most physicists when Townes built the first maser. A sentence of Menzel written in 1931 [2] continues to be ap- plied: “It is easily proved that the so-called “stimulated emissions” are unimportant in the nebulae”. Menzel got this result confusing the radiance of a ray, independent on distance from the source in a transparent medium, with irradiance, which decreases with distance. Some as- trophysicists, for example the 21 authors of the article by Michael et al. [3] use the Monte-Carlo method without considering the phase of the pilot-wave of the photons. 

The method of Einstein coefficients is simpler and can be used in two equivalent ways, depending on whether one uses the absolute radiance of light deduced from the papers of Planck [4], Einstein and Stern [5], or the usual, relative radiance posed null beyond an opaque, cold screen. For a progressive mode, that is a beam of monochromatic light of frequency ν limited by diffrac- tion, the relative radiance is deduced from the absolute radiance by adding hν $^{3}$/c$^{2}$. Using the absolute radiance, coefficient A is zero and the spontaneous emission which corresponds to the amplification of the minimal radiance hν $^{3}$/c2 is considered as coherent because the amplifica- tion preserves the phase of the minimal, stochastic field. 

Based on thermodynamics, statistical theory, the Ein- stein method is valid only if volumes of homogeneous material are large enough for the application of the law of large numbers, but there is no other limitation. Oth- erwise, fluctuations of density produce an opalescence, incoherent interaction of light with matter. At low pres- 

PACS numbers: 42.25.Kb;42.50.Ar 

sure, these fluctuations result from collisions. The num- ber of binary collisions in a given volume, at low pressure, is proportional to the square of pressure, so that incoher- ent interactions are negligible in astrophysics, except, of course, near the stars. 

The simplest coherent emission is Rayleigh coherent scattering, at the incident frequency. Coherence allows for simple addition of the amplitudes of scattered field and incident field. By adding first the fraction of am- plitude scattered in quadrature, refraction is obtained. If it is linear, the modes are preserved. The fraction of amplitude scattered in phase (or antiphase) produces amplification (or absorption). The scattering of a dif- ferent frequency produces waves of generally different wavelength, so that the scattered amplitudes add with different phases, self destroying. Therefore the final con- tribution of scattering is generally negligible. The use of lasers and crystals, for example, can however pro- duce intense effects. G. L. Lamb [6] describes the use of pulses “shorter than all relevant time constants”. With these conditions, the linewidths are larger than the dis- tance between the exciting and a Raman lines which are mixed. An elementary computation shows that the scat- tered wave interferes with the exciting wave to form a sine wave of slightly different frequency, and negligible residual self-destructive waves. Thus, the frequency of laser pulses that transmit information over long distances through optical fibers are lowered enough to cause prob- lems for color selection. These redshifts come from the excitation of quadrupoles whose energy is transferred si- multaneously to thermal radiation by similar scatterings, the whole forming a parametric interaction. 

The time constants considered by Lamb are, in a gas, the collisional time and the period of the quadrupo- lar transition. Using the pulses forming natural time- incoherent light, long of about a nanosecond, a dilute gas and a frequency below 1 GHz must be used. For exam- ple, the frequencies 178 MHz in the 2S 1 / 2 state of atomic hydrogen, 59 MHz in 2P 1 / 2 state, and 24 MHz in 2P 3 / 2 are very convenient while 1420 MHz is not. The effect is very weak, but can accumulate along an astronomical distance. 

The spectrum of SNR1987A that Michael et al. stud- ied results of the generation and propagation of the Ly- man alpha hydrogen line in the outer region of a sphere of radius R, made of hydrogen almost fully ionized by its high temperature. An increase of the distance r from 

