# 2.7: Wien's Law

- Page ID
- 8014

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

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

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}