2.6: 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}