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# 2.7: 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 2.6.12.6.4), 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

$x_n = (6-n) \left(1-e^{-x_n} \right)$

and have the values

$x_1 = 4.965114$

$x_2 = 3.920690$

$x_3 = 2.821439$

$x_4 = 1.593624$

The Wien constants then have the values

$W_1 = 2.8978 \times 10^{-3} \ \text{m K}$

$W_2 = 3.6697 \times 10^{-3} \ \text{m K}$

$W_3 = 5.8790 \times 10^{10} \ \text{Hz K}^{-1}$

$W_4 = 3.3206 \times 10^{10} \ \text{Hz K}^{-1}$

The maximum ordinates of the functions are given by

$M_\lambda (\text{max}) = A_1 T^5$

$N_\lambda ( \text{max}) = A_2 T^4$

$M_\nu (\text{max}) = A_3 T^3$

$N_\nu (\text{max}) = A_4 T^2$

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

$y_n = \frac{x_n^{6-n}}{e^{x_n}-1}$

That is,

$y_1 = 21.20144$

$y_2 = 4.779841$

$y_3 = 1.421435$

$y_4 = 0.6476102$

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}