Blackbody Radiation

from Statistical Physics using Mathematica
© James J. Kelly, 1996-2002

Introduction

All bodies at temperature above absolute zero emit electromagnetic energy due to thermal excitation of charges near the surface and the thermal radiation power increases with temperature.  In general the thermal emissivity of a body depends upon the frequency or wavelength of the radiation emitted and upon the nature of the body.  Thus, we define the spectral emittance (Subscript[𝒫, α])[(Overscript[k, ⇀]), 
        T] such that the power emitted per unit surface area in the range of wave vectors ((Overscript[k, ⇀]), 
        (Overscript[k, ⇀]) + d(Overscript[k, ⇀])) with polarization α is given by (Subscript[𝒫, α])[(Overscript[k, ⇀]), 
          T] (d^3) k.  The radiant flux

ϕ = (∑_α)∫(Subscript[𝒫, α])[(Overscript[k,  ⇀]) , 
                  T] (d^3) k

is the total power per unit surface area emitted in any wavelength with either polarization.  In 1879 Stefan proposed the empirical emissivity law ϕ = e σ (T^4) where 0 ≤ e ≤ 1 is the emissivity of the surface and σ=5.67×(10^(-8)) Watt (m^(-2))(kelvin^(-4)) is known as the Stefan-Boltzmann constant.  Conversely, because any body bathed with electromagnetic radiation will absorb a fraction of the incident radiation, we can define an analogous spectral absorbance (Subscript[𝒜, α])[(Overscript[k,  ⇀]) , 
        T] and an integrated property called absorptivity a.  An ideal blackbody absorbs all incident radiation with a = 1.  Using a general thermodynamic argument, Kirchhoff demonstrated that e = a for any body; hence, a blackbody is also the most efficient radiator of electromagnetic energy with e = a = 1.  Thus, the emissivity is a measure of the efficiency of thermal radiation compared with an ideal blackbody.

A good example of an ideal blackbody is provided by a small hole in the surface of a heated enclosure or oven.  The hole absorbs any radiation incident upon it, which is then reflected or absorbed by the interior walls with practically no chance of prompt escape.  Similarly, any radiation that happens to reach the hole from the inside will be emitted.  Therefore, the hole acts as a blackbody with e = a = 1 independent of frequency.  The radiant flux is given by ϕ = u c/4 where u is the energy density in the enclosure, c is the speed of light, and the numerical factor accounts for geometrical effects.  The electromagnetic radiation within an otherwise empty enclosure can be analyzed as a thermodynamic system in equilibrium with the walls of the enclosure at temperature T.

Electromagnetic radiation within an otherwise empty enclosure can be analyzed as a thermodynamic system in equilibrium with the walls of the enclosure, which serve as a reservoir at temperature T.  The distribution of radiation may be described by a spectral intensity (Subscript[ℐ, α])[(Overscript[k, ⇀]), 
        (Overscript[r, ⇀]), T] defined so that the energy density at position (Overscript[r, ⇀]) with polarization α and wave vector within the three-dimensional interval (d^3) k centered upon (Overscript[k, ⇀]) is given by (Subscript[ℐ, α])[(Overscript[k, ⇀]), 
          (Overscript[r, ⇀]), T] (d^3) k.  However, the stipulation that the radiation is in thermal equilibrium with the walls and with any bodies placed within the enclosure severely limits the form the spectral function may take.  Suppose that two different enclosures are constructed and that both are brought into thermal equilibrium with the same reservoir T.  According to the (0^th) law of thermodynamics, the two enclosures must also be in equilibrium with each other.  Now suppose that the two enclosures are brought into contact with each other using a narrow tube which permits electromagnetic energy to be exchanged.  A special filter is placed within the tube which only allows radiation with specified wavelength and polarization to pass, with all other radiation reflected.  If (ℐ^((1))) were to differ in any way from (ℐ^((2))) a net flow of energy from one system to the other could then be arranged, changing the temperatures of both systems.  This hypothetical result clearly violates the thermodynamic definition of equilibrium.  Hence, the spectral function must be independent of the size, shape, or composition of the walls.  By varying the curvature of the tube and placement of its ends, one also concludes that the spectral function must be independent of the position (Overscript[r, ⇀]) or the direction of (Overscript[k, ⇀]).  Therefore, elementary considerations require the spectral function to take a form

(Subscript[ℐ, α])[(Overscript[k, ⇀]), 
        (Overscript[r, ⇀]), 
        T] ⟶ ℐ[λ, 
        T]

that depends only upon wavelength and temperature and is independent of any other characteristics of the enclosure.  Using a more detailed analysis of the interaction between radiation and a wall formed by a movable piston, Wien was able to demonstrate in 1893 that the spectral function should take the form

ℐ[λ, T] = η[λ T]/(λ^5)

where η is a function of the product of wavelength and temperature whose form should be obtained from statistical mechanics.

Originally, Planck hypothesized that the atoms within the walls of the enclosure behaved as quantized oscillators emitting and absorbing electromagnetic energy only in discrete packets.  This hypothesis enabled Planck to derive the first quantitative analysis of thermal radiation.  Later, Einstein explained the threshold behavior of the photoelectric effect by applying the quantum hypothesis directly to the radiation field itself.  Einstein hypothesized that the electromagnetic energy carried by a mode with frequency ω is organized into discrete packets or quanta such that the energy carried by that mode is (Subscript[ϵ, n])(ω) = 
        n ℏω where n is an integer and ℏ is the same constant that Planck used to relate the frequency of atomic oscillations to the electromagnetic energy transfer.  Electromagnetic quanta are now called photons.  Although both models yield the same results, Einstein's hypothesis is much more general than Planck's as it purports to describe a fundamental property of the electromagnetic field, be it in a cavity or in free space.

The quantized normal modes of the electromagnetic field are represented by photons which obey the dispersion relation ω  =  k c, where k is the wave number and c is the propagation velocity.   If we assume that the system is isotropic, the density of states can be expressed as g[k]  dk  =  
        (Subscript[g, 0]) V 4  π (k^2) d(k/(((2  π))^3)) where  (Subscript[g, 0]) = 2 represents the degeneracy of each mode and has the value 2 because there are two independent transverse polarization directions.  The spectral density I[ω] is defined so that the energy per unit volume carried by all modes with frequencies in the interval \(TraditionalForm\`\((ω, \ ω + dω)\)\) is I[ω] dω.  It is also useful to define a dimensionless variable y  =  ℏ(ω/((Subscript[k, B]) T)) and corresponding spectral density I[y] dy.

The statistics of occupation numbers are studied in the notebook occupy.nb and a closely related analysis of phonons in a crystal is developed in debye.nb.

Initialization

Defaults,  packages, and symbols

ClearAll["Global`*"];
Off[General::spell, General::spell1];

$DefaultFont = {"Times", 12};
$TextStyle = {FontFamily -> "Times", FontSize -> 12, 
          FontSlant -> "Italic"};

Needs["Utilities`Notation`"];
Needs["Miscellaneous`PhysicalConstants`"];
Needs["Miscellaneous`Units`"];
Needs["Graphics`Master`"]

Symbolize[(Subscript[k, B])];Symbolize[(Subscript[C, V])];Symbolize[(Subscript[Z, m])_];Symbolize[(Subscript[I, x])_]

Symbolize[\(n\&_\)]

Symbolize[(Subscript[ℐ, Wien])]

SetAttributes[{(Subscript[k, B]), ℏ }, Constant]

FundamentalConstants = {ℏ -> PlanckConstantReduced, 
          c -> SpeedOfLight, 
          (Subscript[k, B]) -> BoltzmannConstant};

Rules for changing variables

βToT  =  {β -> (1/((Subscript[k, B]) T))};
ωToy  =  {ω -> (((Subscript[k, B]) T y)/(ℏ )), β -> 
            (1/((Subscript[k, B]) T))};
ωToλ  =  {ω -> ((2  π c)/λ), β -> (1/((Subscript[k, B]) T))};

Statistical mechanics of blackbody radiation

For simplicity, suppose that the enclosure is a cube of volume V = (L^3) with perfectly conducting walls.  The electromagnetic field can then be represented as a sum over normal modes with wave numbers (k^2) = ((Subscript[k, x])^2) + ((Subscript[k, y])^2) + ((Subscript[k, z])^2) with (Subscript[k, i]) = (Subscript[n, i]) π/L for integers (Subscript[n, i]).  Only transverse modes are present because \(TraditionalForm\`∇\&⇀\(\(·\)\(E\&⇀\)\) = 0\)  ⟹  (Overscript[k, ⇀])·(Overscript[E, ⇀]) = 
        0.  Thus, the density of states can be expressed as g[k]  dk  =  
        (Subscript[g, 0]) V 4  π (k^2) d(k/(((2  π))^3)) where  (Subscript[g, 0]) = 2 represents the degeneracy of each mode and has the value 2 because there are two independent transverse polarization directions.  To satisfy the wave equation, the frequency and wave number are related by the dispersion relation ω = k c.  Therefore, the density of states can be expressed in terms of either wave number or frequency as

g[k]  =  2 V ((4  π (k^2))/(((2 π))^3))

g[ω]  =  
      g[k] (k/ω) /. {k -> ω/c}

where the factor ∂(k/∂)ω = (k/ω) represents the Jacobian of transformation.  It is useful to define a dimensionless energy variable y = ℏ ω/(Subscript[k, B])  T as the ratio between photon and thermal energies.  The spectral intensity can be expressed in terms of ω, y, or λ using \(TraditionalForm\`ℐ[
            y] dy = \(ℐ[ω] dω = ℐ[λ] dλ\)\).

The Rayleigh-Jeans model is based upon the classical equipartition theorem, which suggests that each normal mode of the electromagnetic field should have an average energy of (Subscript[k, B]) T, such that

(Subscript[ℐ, RJ])[ω] =  (Subscript[k, B]) 
        T g[ω] /V

(Subscript[ℐ, RJ])[
        y]  =  (Subscript[ℐ, RJ])[ω] (((Subscript[k, B]) T)/ℏ)  /. ωToy  // Simplify

(Subscript[ℐ, RJ])[λ] = (Subscript[ℐ, RJ])[ω] (ω/λ) /. ωToλ

This model is consistent with Wien's analysis, with the identification η ∝ λ T, and shows that the scaling law, I[y]  ∝  (T^4), arises as a simple consequence of the dimensionality of the density of states.  However, in this classical model the spectral density is found to be proportional to (y^2) (T^4), and hence diverges for large frequencies (small wavelengths).  If one takes equipartition seriously, the power radiated by a classical blackbody should be infinite!  This divergence is sometimes called the ultraviolet catastrophe.

A reasonable fit to the experimental data available at the end of the nineteenth century was provided by a model that is often attributed to Wien even though he apparently never specified the form of η[λ T].  Using the Boltzmann factor to damp high frequencies, one might try an empirical function of the form

(Subscript[ℐ, Wien])[ω]  =  
      Exp[-β ϵ] ϵ g[ω] /
            V  /. { ϵ -> ℏ ω}

(Subscript[ℐ, Wien])[
        y]  =  (Subscript[ℐ, Wien])[ω] (((Subscript[k, B]) T)/ℏ)  /. ωToy  // Simplify

(Subscript[ℐ, Wien])[λ] = (Subscript[ℐ, Wien])[ω] (ω/λ) /. ωToλ

The spectral density is then found to have the form (y^3) (e^(-y)) (T^4) and is exponentially damped for high frequencies.  Although this model is in good agreement with data for short wavelengths, improved data showed that it is damped too strongly for long wavelengths.  This deficiency can be described as an infrared problem, but at least it isn't catastrophic.

Planck knew that the Rayleigh-Jeans law agreed with data for long wavelengths but diverged for short wavelengths, whereas the Wien law was accurate for short wavelengths but was too strongly damped for long wavelengths.  Therefore, in 1900 he constructed an empirical function

\(TextForm\`ℐ[ω]\  = \ \ ℏ\/\(π\^2\ c\^3\)\ ω\^3\/\(\(\ \)\((e\^\(β\ ℏ\ ω\)\  - \ 1)\)\)\)

which interpolates between these two extremes and adjusted the parameter h = 2  π ℏ to fit experimental data.  However, he found it very difficult to devise a theoretical explanation of the interpolation formula that was consistent with nineteenth-century physics.  Eventually he found that the interpolation formula could be explained using the statistical mechanics of quantized oscillators.  Thus, he hypothesized that vibrations of the atoms comprising the wall of the enclosure were quantized in integral multiples of h ν.  However, he regarded his own hypothesis as "an act of desperation" and searched in vain for an alternative explanation which did not require quantization.

In 1905 Einstein explained the photoelectric effect by postulating that the energy carried by each normal mode of the electromagnetic field is quantized, such that ε = 
        n h ω where n is an integer and h is Planck's constant.  Einstein then derived the blackbody spectrum based upon quantization of the electromagnetic field itself, rather than on the characteristics of atoms comprising the walls.  This approach is more general than the Planck model because it is independent of the nature of the enclosure.  Each normal mode of the electromagnetic field is quantized and may carry an arbitrary number of photons.  The average energy in each mode is thus the occupation number times the photon energy for that mode.  The total electromagnetic energy within the enclosure is then obtained by summation over the normal modes according to

\(TextForm\`U = ∫\_0\%∞dω\ g[ω]\ n\&_[ω, T] ℏω\)

where n[ω, T] is the average number of photons (quanta) with frequency ω that are present at temperature T.  Each mode can be treated as an independent thermodynamic system whose states are represented by occupation numbers n = 0, 1, ⋯, ∞.  The thermodynamic average occupancy can be obtained using the canonical partition function

(Subscript[Z, 1])  =  (∑_(m = 0)^∞) 
          Exp[-β m ℏ ω]

such that

\(n\&_ = \ ∑\+\(m = 0\)\%∞ 
            Exp[\(-β\)\ m\ ℏ\ ω]\ m\ /\ Z\_1\)

The spectral intensity then becomes

\(ℐ[ω] = \(\(n\&_\ ℏ\ ω\)\/V\) 
        g[ω]\)

ℐ[
        y]  =  ℐ[ω]  (((Subscript[k, B]) T)/ℏ)  /. ωToy  // Simplify

ℐ[λ] = ℐ[ω] (ω/λ) /. ωToλ

Although this result is identical to that of Planck, Planck never accepted Einstein's hypothesis that the electromagnetic field is quantized and, in fact, never believed in Einstein's photon theory despite its success in explaining both the photoelectric effect and the blackbody spectrum simultaneously.

The total energy density integrated over all wavelengths

u  = (∫_0^∞) ℐ[
            y] \[DifferentialD]y

is found to be proportional to (T^4).  Therefore, the radiant flux (energy/area/time) emitted from the surface of a black body is proportional to (T^4) and Stefan's constant

flux  =  (c/4) u

flux /. FundamentalConstants

is in good agreement with experimental data.  Similarly, the Planck spectral density predicts that the frequency at which the spectral density peaks is proportional to temperature because ℐ is a function of ω/T.  The proportionality constant must be computed numerically:

eq  =  
      D[ℐ[y]/u, y] == 0  // Simplify

PlanckPeak  =  FindRoot[eq, {y, 3}]

Although the Wien model of the spectral density is not quite correct, this linear relationship between (Subscript[y, peak]) and T is called the Wien displacement law because a similar result, with slightly different proportionality constant, was obtained from it first.

The figure below shows the spectral intensity in a universal form by dividing out the total energy density and expressing frequency in terms of reduced energy.  The Rayleigh-Jeans and Wien models are included for comparison.  One finds that the Planck model interpolates smoothly between the classical Rayleigh-Jeans model for large y and the Wien model at small y.  More importantly, the Planck spectral intensity accurately reproduces experimental data for all measured ranges of ω and T.

\(\(Plot[
        Evaluate[{ℐ[y], ℐ\_Wien[y], ℐ\_RJ[y]}\/u], {y, 0, 10}, 
        PlotRange -> {{0, 10}, {0, 0.25}}, GridLines -> Automatic, 
        Frame -> True, 
        FrameLabel -> {\*""ℏω/(Subscript[k, B])T"", "ℐ/u"}, 
        PlotStyle -> {{}, Dashing[{0.02, 0.02}], 
            Dashing[{0.04, 0.01, 0.02, 0.01}]}, 
        PlotLegend -> {"Planck", "Wien", "Rayleigh-Jeans"}, 
        LegendPosition -> {0.3, 0.2}, LegendSize -> {0.7, 0.3}, 
        LegendSpacing -> \(-0.1\)];\)\)

Problems

S  =  (((Subscript[k, B]) (c^3))/(4 G ℏ)) A  =  
        4  π ((G (M^2))/(ℏ c)) (Subscript[k, B])