2.6: Wien's Law

The wavelengths or frequencies at which these functions reach a maximum, and what these maximum values are, can be found by differentiation of these functions. They do not all come to a maximum at the same wavelength. For the four Planck functions discussed in Section 2.6 (Equations 2.6.1- 2.6.4), the wavelengths or frequencies at which the maxima occur are given by:

For Equation 2.6.1:

\[\lambda = W_1/T \label{2.7.1} \]

For Equation 2.6.2:

\[\lambda = W_2 / T \label{2.7.2} \]

For Equation 2.6.3:

\[\nu = W_3 T \label{2.7.3} \]

For Equation 2.6.4:

\[\nu = W_4 T \label{2.7.4} \]

Any of these equations (but more usually the first one) may be referred to as Wien's law.

The constants are

\begin{array}{c c}
W_n = \frac{hc}{kx_n}, & (n=1,2)
\end{array}

\begin{array}{c c}
W_n = \frac{kx_n}{h}, & (n=3,4)
\end{array}

where the \(x_n\) are the solutions of

\[x_n = (6-n) \left(1-e^{-x_n} \right) \nonumber \]

and have the values

\[x_1 = 4.965114 \nonumber \]

\[x_2 = 3.920690 \nonumber \]

\[x_3 = 2.821439 \nonumber \]

\[x_4 = 1.593624 \nonumber \]

The Wien constants then have the values

\[W_1 = 2.8978 \times 10^{-3} \ \text{m K} \nonumber \]

\[W_2 = 3.6697 \times 10^{-3} \ \text{m K} \nonumber \]

\[W_3 = 5.8790 \times 10^{10} \ \text{Hz K}^{-1} \nonumber \]

\[W_4 = 3.3206 \times 10^{10} \ \text{Hz K}^{-1} \nonumber \]

The maximum ordinates of the functions are given by

\[M_\lambda (\text{max}) = A_1 T^5 \nonumber \]

\[N_\lambda ( \text{max}) = A_2 T^4 \nonumber \]

\[M_\nu (\text{max}) = A_3 T^3 \nonumber \]

\[N_\nu (\text{max}) = A_4 T^2 \nonumber \]

The constants \(A_n\) are given by

\begin{array}{c c}
A_n = \frac{2\pi k^{6-n} y_n}{h^4 c^3}, & (n=1,2) \\
\end{array}

\begin{array}{c c}
A_n = \frac{2\pi k^{6-n} y_n}{h^2 c^2}, & (n=3,4) \\
\end{array}

where the \(y_n\) are dimensionless numbers defined by

\[y_n = \frac{x_n^{6-n}}{e^{x_n}-1} \nonumber \]

That is,

\[y_1 = 21.20144 \nonumber \]

\[y_2 = 4.779841 \nonumber \]

\[y_3 = 1.421435 \nonumber \]

\[y_4 = 0.6476102 \nonumber \]

The constants \(A_n\) therefore have the values

\begin{array}{l l}
A_1 = 1.2867 \times 10^{-5} & \text{W m}^{-2} \text{K}^{-5} \text{m}^{-1} \\
\end{array}

\begin{array}{l l}
A_2 = 2.1011 \times 10^{17} & \text{ph s}^{-1} \text{m}^{-2} \text{K}^{-4} \text{m}^{-1} \\
\end{array}

\begin{array}{l l}
A_3 = 5.9568 \times 10^{-19} & \text{W m}^{-2} \text{K}^{-3} \text{Hz}^{-1} \\
\end{array}

\begin{array}{l l}
A_4 = 1.9657 \times 10^{4} & \text{ph s}^{-1} \text{m}^{-2} \text{K}^{-2} \text{Hz}^{-1} \\
\end{array}

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