The tricky history of Black Body radiation

There are three ways to get the Planck’s formula for Black Body radiation The Planck’s solution The Bose-Einstein quantum statistics The Einstein invention of stimulated emission All of them tell something new In order to appreciate everything, it is firt necessary to recall the Boltzmann classical statistics.

How to get Boltzmann statistics We look for distribution of particles among the energy levels of a material system. Each distribution will be more or less probable The most probable is assumed to define the thermodinamic equilibrium

Probability of 1 particle in layer gs = gs 6 4 g6 1 g5 g4 2 g3 3 g2 7 g1 5 Probability of 1 particle in layer gs = gs Probability of 2 particles in layer gs = gs gs = gs2 Probability of ns particles in layer gs = gsns

Probability of n1 particles in layer g1 and n2 particles in layer g2… nz particles in layer gz There are n! modes to have this same distribution by permutation of the n particles but n1! only change particles in the same box 1, n2! only change particles in the same box 2… Probability W of different modes for having the same occupation numbers of the first distribution with different particles When is this probability maximum, preserving the constraints ?

Because max (W)=max(lnW) =0 =0 z+2 equations define the z occupation numbers ni and the constants  and  Each derivative has the form

Stirling’s approximation for large n, ns

The value of  What is this? Total variation of the number of particles =0 Energy variation because of changes in the number of particles per energy level Energy variation because of changes in energy levels due to deformation pressure volume heat

But we know from thermodynamics that  = Integrating factor of Q Exact differential But we know from thermodynamics that Entropy 1/T = Integrating factor of Q Exact differential The unicity of the integrating factor imposes that  is proportional to 1/T and that S is proportional to ln(W)

For systems where g(e) is a constant A system with 2 degrees of freedom has g(e) constant

Black Body

General formulas for Black Body spectral density of energy un Stefan’s Law Wien’s (Displacement) Law For a system of 1D classical oscillators (2 degrees of freedom) Rayleigh and Jeans’ Ultraviolet Catastrophe . But good for low 𝜈.

The Planck’s solution: e = ne0 Continuous distribution of energy The Planck’s solution: e = ne0 kT e0→0

If we assume e0 proportional to the frequency n This obeys the Wien and the Stefan laws. It approaches the Rayleigh and Jeans formula for low n and works very well with experiments.

z1 a1a2z2 a3a4a5a6 z3 z4a7……. The Bose-Einstein Quantum Statistics Let us go back to statistics and introduce one single variation with respect to classical statistics: Particles are undistinguishable This changes the way for calculating the probability W of a distribution Let us consider the s-th level, with energy es, a total number of states gs and a total number of particles ns occupying those states. Let us imagine the «states» in this level as identical boxes, called z1, z2 ,…, zgs z1 a1a2z2 a3a4a5a6 z3 z4a7……. We can write a string as It thells that we have 2 particles in box 1, 4 in box 2, none in box 3, etc. How many different ways we have to get this distribution?

z1 a1a2z2 a3a4a5a6 z3 z4a7……. The string is made of gs + ns elements, and always starts with a «z» We have gs ways to choose a «z», and (gs + ns -1) ! ways to write the following part. Among these ways, ns! are undistinguishable, because the particles are. But also the boxes are identical, and then there are gs ! undistinguishable ways to exchange boxes among themselves. The number of distinguishable ways to get a distrinution in that s-th level is then This means that the probability of n1 particles in layer g1 and n2 particles in layer g2… nz particles in layer gz is

If we now use this new W to repeat what we did for the classical statistics, we get: If this expression must describe the radiation from a black body, the ratio n/T must appear, and T is only present in the exponential term (g does not depend on T)

the total number of undistinguishable particles is allowed to change Planck Bose-Einstein We can neglect the explicit form of g. The important point is that there is  inside the exponential.  Is the constant that multiplies the constraint of conservation of the total number of particles The Bose-Einstein quantum statistics leads to the Black Body formula provided the total number of undistinguishable particles is allowed to change (destruction and creation).

A quick recall of the Einstein’s treatment of Black Body radiation (1905) Search for the spectral power density u𝜈 at equilibrium Fermi’s Golden rule not yet discovered No Fermi-Dirac or Bose-Einstein statistics available: only Boltzmann. No exclusion principle No quanta. Planck (1901) not sure of their existence. Einstein going to explain the phototelectric effect on the same year (1905) Classical results from Thermodynamics Stefan’s Law Wien’s (Displacement) Law Rayleigh and Jeans’ Ultraviolet Catastrophe . But good for low 𝜈.

2-level system with N0 particles Energy Density of states Population E2 g2 Level 2 E1 g1 Level 1 Rate of spontaneous 2→1 transitions Rate of stimulated 2→1 transitions Rate of stimulated 1→2 transitions Coefficients A and B can depend on 𝜈 but not on T At equilibrium 2→1 = 1 →2

At high temperature: Increase with T4 Do not change with T Go to unity Wien’s (Displacement) Law Do not include T

Must be (Rayleigh and Jeans): Planck’s Law

To achieve the same formula Planck imagined the quantization of Energy Bose-Einstein statistics requires particle unistinguishability and non-conservation of the total number of particles Einstein invented the stimulated emission All cases have to harmonize. All solution are dramatically non-classical Light is made of photons. They behave as particles identical and undistinguishable, that can be created and destroyed. Destruction of a photon transfers its energy to the material system. It is called absorption Emission of a photon may be a spontaneous release of energy by the materal system But a photon can also stimulate the system to release its energy by emitting a second photon