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Deriving the Rayleigh-Jeans Radiation Law

  • Page ID
    92744
  • The Rayleigh-Jeans Radiation Law was a useful, but not completely successful attempt at establishing the functional form of the spectra of thermal radiation. The energy density \(u_ν\) per unit frequency interval at a frequency \(ν\) is, according to the The Rayleigh-Jeans Radiation,

    \[u_ν = \dfrac{8πν^2kT}{c^2}\]

    where \(k\) is Boltzmann's constant, \(T\) is the absolute temperature of the radiating body and \(c\) is the speed of light in a vacuum.

    This formula fits the empirical measurements for low frequencies, but fails increasingly for higher frequencies. The failure of the formula to match the new data was called the ultraviolet catastrophe. The significance of this inadequate so-called law is that it provides an asymptotic condition which other proposed formulas, such as Planck's, need to satisfy. It gives a value to an otherwise arbitrary constant in Planck's thermal radiation formula.

    The Derivation

    Consider a cube of edge length \(L\) in which radiation is being reflected and re-reflected off its walls. Standing waves occur for radiation of a wavelength \(λ\) only if an integral number of half-wave cycles fit into an interval in the cube. For radiation parallel to an edge of the cube this requires

    \[\dfrac{L}{λ/2} = m\]

    where \(m\) is an integer or, equivalently

    \[λ = \dfrac{2L}{m}\]

    Between two end points there can be two standing waves, one for each polarization. In the following the matter of polarization will be ignored until the end of the analysis and there the number of waves will be doubled to take into account the matter of polarization.

    Since the frequency \(\nu\) is equal to \(c/λ\), where \(c\) is the speed of light

    \[ν = \dfrac{cm}{2L}\]

    It is convenient to work with the quantity \(q\), known as the wavenumber, which is defined as

    \[q = \dfrac{2π}{λ} \]

    and hence

    \[q = \dfrac{2πν}{c}\]

    In terms of the relationship for the cube,

    \[q = \dfrac{2πm}{2L} = π\left(\dfrac{m}{L}\right)\]

    and hence

    \[q^2 = π^2\left(\dfrac{m}{L}\right)^2\]

    Another convenient term is the radian frequency \(ω=2πν\). From this it follows that \(q=ω/c\).

    If \(m_X\), \(m_Y\), \(m_Z\) denote the integers for the three different directions in the cube then the condition for a standing wave in the cube is that

    \[q^2 = π^2\left[ \left(\dfrac{m_X}{L}\right)^2 + \left(\dfrac{m_Y}{L}\right)^2 + \left(\dfrac{m_Z}{L}\right)^2\right] \]

    which reduces to

    \[m_X^2 + m_Y^2 + m_Z^2 = \dfrac{4L^2ν^2}{c^2}\]

    Now the problem is to find the number of nonnegative combinations of (\(m_X\), \(m_Y\), \(m_Z\)) that fit between a sphere of radius \(R\) and and one of radius \(R+dR\). First the number of combinations ignoring the nonnegativity requirement can be determined.

    The volume of a spherical shell of inner radius R and outer radius \(R+dR\) is given by

    \[dV = 4πR^2\,dR\]

    If

    \[R = \sqrt{m_X^2+m_Y^2+m_Z^2} \]

    then

    \[R = \sqrt{\dfrac{4L^2ν^2}{c^2}} = \dfrac{2Lν}{c}\]

    and hence

    \[dR=\dfrac{2L\,dν}{c}.\]

    This means that

    \[dV = 4π(2Lν/c)^2(2L/c)\,dν = 32π(L^3ν^2/c^3)dν\]

    Now the nonnegativity require for the combinations (\(m_X\), \(m_Y\), \(m_Z\)) must be taken into account. For the two dimensional case the nonnegative combinations are approximately those in one quadrant of circle. The approximation arises from the matter of the combinations on the boundaries of the nonnegative quadrant. For the three dimensional case the nonnegative combinations constitute approximately one octant of the total. Thus the number dN for the nonnegative combinations of (\(m_X\), \(m_Y\), \(m_Z\)) in this volume is equal to (1/8)dV and hence

    \[dN = 4πν^2\,dν\]

    The average kinetic energy per degree of freedom is ½kT, where \(k\) is Boltzmann's constant. For harmonic oscillators there is an equality between kinetic and potential energy so the average energy per degree of freedom is kT. This means that the average radiation energy E per unit frequency is given by

    \[\dfrac{dE}{dν} = kT\left(\dfrac{dN}{dν}\right) = 4πkT\left(\dfrac{L^3}{c^3}\right)ν^2\]

    and the average energy density, uν, is given by

    \[\dfrac{du_ν}{dν} = \left(\dfrac{1}{L^3}\right)\left(\dfrac{dE}{dν}\right) = \dfrac{4πkTν^2}{c^3}\]

    The previous only considered one direction of polarization for the radiation. If the two directions of polarization are taken into account a factor of 2 must be included in the above formula; i.e.,

    \[\dfrac{du_ν}{dν} = \dfrac{8πkTν^2}{c^3}\]

    This is the Raleigh-Jeans Law of Radiation and holds empirically as the frequency goes to zero.