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Planck's law of black body radiation

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In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by Planck's law of black body radiation:

[I(\nu,T) =\frac}\frac}-1}]
where the following table provides the definition and SI units of measure for each symbol:

Symbol Meaning SI units of measure
[I \,] spectral radiance,
energy per unit time per unit surface area per unit solid angle per unit frequency 		J·s-1·m-2·sr-1·Hz-1
[\nu \,] frequency hertz
[T \,] temperature of the black body kelvin
[h \,] Planck's constant joule per hertz
[c \, ] speed of light meter per second
[e \,] base of the natural logarithm, 2.718282... dimensionless
[k \, ] Boltzmann's constant joule per kelvin

The wavelength is related to the frequency by

[\lambda = .]
The law is sometimes written in terms of the spectral energy density

[u(\nu,T) = I(\nu,T) = \frac~\frac}-1}]
which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz).

The spectral energy density can also be expressed as a function of wavelength:

[u(\lambda,T) = -1}]
as shown in the derivation below.

Max Planck originally produced this law in 1900 (published in 1901) in an attempt to improve upon an expression proposed by Wilhelm Wien which fit the experimental data at short wavelengths but deviated from it at long wavelengths. He found that the above function, Planck's function, fit the data for all wavelengths remarkably well. In constructing a derivation of this law, he considered the possible ways of distributing electromagnetic energy over the different modes of charged oscillators in matter. Planck's law emerged when he assumed that the energy of these oscillators was limited to a set of discrete, integer multiples of a fundamental unit of energy, E, proportional to the oscillation frequency ν:

[E=h\nu\,].
Planck made this quantization assumption five years before Albert Einstein hypothesized the existence of photons as a means of explaining the photoelectric effect. At the time, Planck believed that the quantization applied only to the tiny oscillators that were thought to exist in the walls of the cavity (what we now know to be atoms), and made no assumption that light itself propagates in discrete bundles or packets of energy. Moreover, Planck did not attribute any physical significance to this assumption, but rather believed that it was merely a mathematical device that enabled him to derive a single expression for the black body spectrum that matched the empirical data at all wavelengths.

Ultimately, Planck's assumption of energy quantization and Einstein's photon hypothesis became the fundamental basis for the later development of Quantum Mechanics. Both scientists would eventually receive (separate) Nobel prizes in recognition of these major contributions to the advancement of physics.

Contents

Derivation (Statistical Mechanics)

(See also the gas in a box article for a general derivation.)

Consider a cube of side [L] with conducting walls filled with electromagnetic radiation. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The wavelength [\lambda_] in the three directions [i=1\ldots 3] orthogonal to the walls can be:

[\lambda_ = \frac}]
where the [n_] are integers. For each set of integers [n_] there are two linear independent solutions (modes). According to quantum theory, the energy levels of a mode are given by:

[E_,n_,n_}\left(r\right)=\left(r+\frac\right)\frac\sqrt^+n_^+n_^}\mbox]
The quantum number [r] can be interpreted as the number of photons in the mode. The two modes for each set of [n_] correspond to the two polarization states of the photon which has a spin of 1. Note that for [r=0] the energy of the mode is not zero. This vacuum energy of the electromagnetic field is responsible for the Casimir effect. In the following we will calculate the internal energy of the box at temperature [T] relative to the vacuum energy.

According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by:

[P_=\frac]
Here

[\beta\equiv 1/\left(kT\right)].
The denominator [Z\left(\beta\right)], is the partition function of a single mode and makes [P_] properly normalized:

[Z\left(\beta\right)=\sum_^\exp\left[-beta Eleft(rright)right]=\frac]
Here we have defined

[\varepsilon\equiv\frac\sqrt^+n_^+n_^}]
which is the energy of a single photon. As explained here, the average energy in a mode can be expressed in terms of the partition function:

[\left\langle E\right\rangle=-\frac=\frac]
This formula is a special case of the general formula for particles obeying Bose-Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero.

The total energy in the box now follows by summing [\left\langle E\right\rangle] over all allowed single photon states. This can be done exactly in the thermodynamic limit [L\rightarrow\infty]. In this limit, [\varepsilon] becomes continuous and we can then integrate [\left\langle E\right\rangle] over this parameter. To calculate the energy in the box in this way, we need to evaluate how many photon states there are in a given energy range. If we write the total number of single photon states with energies between [\varepsilon] and [\varepsilon + d\varepsilon] as [g\left(\varepsilon\right)d\epsilon], where [g\left(\varepsilon\right)] is the density of states which we'll evaluate in a moment, then we can write:

[U = \int_^\fracg\left(\varepsilon\right)d\varepsilon \mbox]
To calculate the density of states we rewrite equation (1) as follows:

[\varepsilon\equiv\fracn]
where [n] is the norm of the vector [\vec=\left(n_,n_,n_\right)]:

[n=\sqrt^+n_^+n_^}]
For every vector [n] with integer components larger or equal than zero there are two photon states. This means that the number of photon states in a certain region of n-space is twice the volume of that region. An energy range of [d\varepsilon] corresponds to shell of thickness [dn= \fracd\varepsilon] in n-space. Because the components of [\vec] have to be positive, this shell spans an octant of a sphere. The number of photon states [g\left(\varepsilon\right)d\epsilon] in an energy range [d\varepsilon] is thus given by:

[g\left(\varepsilon\right)d\epsilon=2\frac4\pi n^dn=\frac}c^}\varepsilon^d\varepsilon]
Inserting this in Eq. (2) gives:

[U =L^\fracc^}\int_^\frac}d\varepsilon\mbox]
From this equation one easily derives the spectral energy density as a function of frequency [u(\nu,T)] and as a function of wavelength [u(\lambda,T)]:

[\frac = \int_0^u(\nu,T) d\nu]
where:

:[u(\nu,T) = -1}]
[u(\nu,T)] is known as the black body spectrum. It is a spectral energy density function with units of energy per unit frequency per unit volume.

And:

[\frac = \int_0^\infty u(\lambda,T) d\lambda]
where

:[u(\lambda,T) = -1}]
This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithms. In this case, however, it is possible to calculate the integral in closed form. Let's first make the integration variable in Eq. (3) dimensionless by substituting [\varepsilon = k T x]:

[u(T) =\frac}} J]

Here [J] is given by:

[J=\int_^\frac}dx = \frac}]
We prove this result in the Appendix below. The total electromagnetic energy inside the box is thus given by:

:[ = \frac]
where [V=L^3] is the volume of the box. (Note - This is not the Stefan-Boltzmann law, which is the total energy radiated by a black body. See that article for an explanation.) Since the radiation is the same in all directions, and propagates at the speed of light (c), the spectral intensity (energy/time/area/solid angle/frequency) is

[I(\nu,T) = \frac ]
which yields

:[I(\nu,T) = \frac~\frac-1}]

Derivation (Thermodynamics)

The fact that the energy density of the box containing radiation is proportional to [T^] was derived by Ludwig Boltzmann in 1884 using thermodynamics. It follows from classical electrodynamics that the radiation pressure [P] is related to the internal energy density:

[P=\frac]

The total internal energy of the box containing radiation can thus be written as:

[U=3PV]

Inserting this in the fundamental law of thermodynamics

[dU=T dS - P dV]

yields the equation:

[dS=4\fracdV + 3\fracdP]

We can now use this equation to derive a Maxwell relation. We read off that:

[\left.\frac\right|_=4\frac]

And

[\left.\frac\right|_=3\frac]

The symmetry of second derivatives of [S] w.r.t. [P] and [V] then implies:

[4\left.\frac\right)}\right|_= 3\left.\frac\right)}\right|_]

Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume. In the derivative on the r.h.s. the temperature is thus a constant. Evaluating the derivatives gives the differential equation:

[\frac\frac=\frac]

This implies that [u=3P \propto T^ ]

History

Many popular science accounts of quantum theory, as well as some physics textbooks, contain some serious errors in their discussions of the history of Planck's Law. Although these errors were pointed out over forty years ago by historians of physics, they have proved to be difficult to eradicate. The article by Helge Kragh cited below gives a lucid account of what actually happened.

Contrary to popular opinion, Planck did not quantize light. This is evident from his original 1901 paper and the references therein to his earlier work. It is also plainly explained in his book "Theory of Heat Radiation," where he explains that his constant refers to Hertzian oscillators. The idea of quantization was developed by others into what we now know as quantum mechanics. The next step along this road was made by Albert Einstein, who, by studying the photoelectric effect, proposed a model and equation whereby light was not only emitted but also absorbed in packets or photons. Then, in 1924, Satyendra Nath Bose developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck's law.

Contrary to another myth, Planck did not derive his law in an attempt to resolve the "ultraviolet catastrophe", the name given to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is applied to black body radiation. Planck did not consider the equipartion theorem to be universally valid, so he never noticed any sort of "catastrophe" — it was only discovered some five years later by Einstein, Lord Rayleigh, and Sir James Jeans.

Appendix

A simple way to calculate the integral

[J=\int_^\frac}dx ]
is as follows. After multiplying the numerator and denominator of the integrand we can expand the integrand in powers of [\exp(-x)].

[J=\int_^\frac\exp\left(-x\right)}dx=\sum_^\int_^x^\exp\left(-n x\right)dx=6\sum_^\frac}=\frac}]

Here we have used that [\sum_^\frac}] is the Riemann zeta function evaluated for the argument 4, which is given by [\frac}]. This fact can be proven by considering the contour integral

[\oint_}\frac}]

Where [C_] is a contour of radius [R] around the origin. In the limit [R\rightarrow\infty] the integral approaches zero. Using the residue theorem the integral can also be written as a sum of residues at the poles of the integrand. The poles are at zero, the positive and negative integers. The sum of the residues yields precisely twice the desired summation plus the residue at zero. This means that [\sum_^\frac}] equals minus [\frac}] times the coefficient of [x^3] of the series expansion of the series expansion of the cotangent function.

External link and references

 

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