普朗克公式

概述

德国物理学家M.普朗克在量子论基础上建立的关于黑体辐射的正确公式。19世纪末,经典统计物理学在研究黑体辐射时遇到了巨大的困难:由经典的能量均分定理导出的瑞利-金斯公式在短波方面得出同黑体辐射光谱实验结果相违背的结论。同时,维恩公式则仅适用于黑体辐射光谱能量分布的短波部分。也就是说,当时还未能找到一个能够成功描述整个实验曲线的黑体辐射公式。

The Planck Function (convert from temperature and wavelength to spectral radiance)

The Planck Function:

L(λ,t)=c1λ5(ec2/λt1)L(\lambda,t)= \frac{c_1} {\lambda^5(e^{c_2/\lambda t}-1)}

Where:

L(λ,t)L(\lambda,t)=blackbody radiance (W/m2srum)(W/m^2 \cdot sr \cdot um)
c1=1.191042108(W/m2srum4)c_1=1.191042*10^8(W/m^2 \cdot sr \cdot um^{-4})
c2=1.4387752104c_2=1.4387752*10^4
λ\lambda=wavelength(um)(um)
tt=blackbody temperature(K)(K)

The Planck Function (convert from temperature and wavenumber to spectral radiance)

The Planck Function:

L(v,t)=c1v3ec2v/t1L(v,t)= \frac{c_1v^3}{e^{c_2v/t}-1}

Where:

L(v,t)=L(v,t)= blackbody radiance (mW/m2srcm1)(mW/m^2 \cdot sr \cdot cm^{-1})
c1=1.191042105(mW/m2srcm4)c_1=1.191042*10^5(mW/m^2 \cdot sr \cdot cm^{-4})
c2=1.4387752(Kc_2=1.4387752(K cm)cm)
v=v= wavenumber(cm1)(cm^{-1})
t=t= blackbody temperature (K)(K)

The Inverse Planck Function (convert from spectral radiance and wavelength to temperature)

The Inverse Planck Function:

t(λ,L)=c2λln(c1/λ5L+1)t(\lambda,L) = \frac{c_2}{\lambda\ln(c_1/\lambda^5L+1)}

Where:

t=t = blackbody temperature (K)(K)
L=L = blackbody radiance (W/m2srum)(W/m^2 \cdot sr \cdot um)
c1=1.191042108(W/m2srum4)c_1=1.191042*10^8(W/m^2 \cdot sr \cdot um^{-4})
c2=1.4387752104(Kc_2=1.4387752*10^4(K um)um)
λ=\lambda= wavelength (um)(um)

The Inverse Planck Function (convert from spectral radiance and wavenumber to temperature)

The Inverse Planck Function:

t(v,L)=c2vln(c1v3/L+1)t(v,L)=\frac{c_2v}{ln(c_1v^3/L+1)}

Where:

t=t= blackbody temperature (K)(K)
L=L= blackbody radiance (mW/m2srcm1)(mW/m^2 \cdot sr \cdot cm^{-1})
c1=1.191042105(mW/m2srcm4)c_1=1.191042*10^5(mW/m^2 \cdot sr \cdot cm^{-4})
c2=1.4387752(Kc_2=1.4387752(K cm)cm)
v=v= wavenumber (cm1)(cm^{-1})