2.6: Wien's Law
- Page ID
- 8014
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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}