Black body radiation|What is Black body radiation ?|Quantum Mechanics

Black-Body Radiation : Various workers attempted to offer theoretical interpretation of the spectral distribution of energy in the spectrum of black-body radiation on the basis of classical mechanics and electromagnetic theory with partial success.

Ultimately in 1901 Max Planck proposed Quantum Theory of Radiation and cloud offer a satisfactory explanation of the spectrum of black-body radiation.

Contents hide

1 Theories of Black Body Radiation

1.1 Wien’s Theory of Black Body Radiation

1.2 Wien’s Displacement Law

1.3 Rayleigh-Jeans Law of Black Body Radiation

1.4 ultra-violet Catastropy

1.5 Planck’s law of Radiation

Theories of Black Body Radiation

Wien’s Theory of Black Body Radiation

Wien in 1893 from the thermodynamical and considerations, proposed an empirical relationship between $\displaystyle E_{\lambda }$

and $\displaystyle \lambda$

for a given temperature T as,

$\displaystyle \displaystyle E_{\lambda }(T)d\lambda =\frac{A}{\lambda ^{5}}f(\lambda T)d\lambda\ \ \ ...(1)$

Where A is a constant and $\displaystyle f(\lambda T)$

is a function of $\displaystyle \lambda T$

We can show that, Stefan-Boltzmann law and wien’s displacement law follow from wien’s distribution law.

Putting $\displaystyle x=\lambda T$

in above equation, we get,

$\displaystyle E=\int_{0}^{\infty }E_{\lambda } d\lambda =A\int_{0}^{\infty }\frac{f(\lambda T)}{\lambda ^{5}}d\lambda$

$\displaystyle \Rightarrow E=AT^{5}\int_{0}^{\infty }\frac{f(\lambda T)}{(\lambda T)^{5}}d\lambda$

$\displaystyle \Rightarrow E=AT^{4}\int_{0}^{\infty }\frac{f(x)}{(x)^{5}}dx$

$\displaystyle \because \int_{0}^{\infty }\frac{f(x)}{(x)^{5}}dx$

being a definite integral, is a constant, $\displaystyle E=\sigma T^{4}$

, where $\displaystyle \sigma$

is a constant. This relation is called Stefan-Boltzmann law.

Wien’s Displacement Law

Differentiating Equation (1), we get,

$\displaystyle \frac{dE_{\lambda }}{d\lambda }=-\frac{5A}{\lambda ^{6}}f(\lambda T)+\frac{AT}{\lambda ^{5}}f'(\lambda T)$

If,

$\displaystyle E_{\lambda }=(E_{\lambda })_{max}$

at $\displaystyle \lambda =\lambda _{m}$

then $\displaystyle (\frac{dE_{\lambda }}{d\lambda })_{max}=0$

or, $\displaystyle Tf'(\lambda _{m}T)-\frac{5f(\lambda _{m}T)}{\lambda _{m}}=0$

or, $\displaystyle \lambda _{m}f'(x_{m})-5f(x_{m})=0$

where, $\displaystyle x_{m}=\lambda _{m}T$

. The above equation is in a single variable $\displaystyle x_{m}$

and can have only one solution.

$\displaystyle \therefore\ x_{m}=\lambda _{m}T=constant$

This is known as Wien’s displacement law which explains the shift of emission peaks towards lower value of $\displaystyle \lambda$

and the increase of T.

Wien’s distribution formula explains the experimental results fairly will for low values of $\displaystyle \lambda T$

but for higher values, it gives $\displaystyle E_{\lambda}$

lower than the experiment values.

Rayleigh-Jeans Law of Black Body Radiation

Applying the law of equipartition of energy to the electromagnetic vibrations, Lord Rayleigh with a contribution from J.H Jeans (in 1900) obtained a formula for energy density inside an enclosure with perfectly reflecting walls.

According to this formula the energy density in the frequency range $\displaystyle \nu$

to $\displaystyle \nu +d\nu$

at temperature T is,

$\displaystyle U_{\nu }d\nu =\frac{8\pi \nu ^{2}kT}{c^{3}}d\nu\ .......(2)$

Where, $\displaystyle U_{\nu }$

is the energy per unit volume per unit frequency range at $\displaystyle \nu$

, k is the Boltzmann constant and c is the velocity of light in vacuum,

Since,

$\displaystyle c=\nu \lambda\ or\ \nu =\frac{c}{\lambda }\ and\ d\nu =-\frac{c}{\lambda ^{2}} d\lambda$

$\displaystyle U_{\lambda }d\lambda =-U_{\nu }d\nu =-\frac{8\pi }{c^{3}}(\frac{c}{\lambda })^{2}kT(-\frac{c}{\lambda ^{2}})d\lambda$

$\displaystyle \Rightarrow U_{\lambda }d\lambda =\frac{8\pi kT}{\lambda ^{4}}d\lambda$

Equation (2) gives relief jeans law in terms of wavelength. This law can explain the experimental results in the long wavelength side but fails in the short wavelength site since $\displaystyle U_{\lambda } \to \infty\ as\ \lambda \to \infty$

for a constant temperature. This is contrary to experimental findings.

ultra-violet Catastropy

The total energy of radiation per unit volume of the enclosure for all wavelengths from 0 to $\displaystyle \infty$

is,

$\displaystyle U=\int_{0}^{\infty }U_{\lambda }d\lambda =\int_{0}^{\infty }\frac{8\pi kT}{\lambda ^{4}}d\lambda =8\pi kT[-\frac{1}{3\lambda ^{3}}]\rightarrow 0$

Hence, for a given quantity of Radiant energy all the energy will be confined to vibration of small wavelength. But experimental results show that $\displaystyle U_{\lambda }d_{\lambda } \to 0\ as\ \lambda \to 0$

These serious disagreement between theory and experiment is known as ultraviolet Catastropy. This indicates the limitations of the classical mechanics on the basis of which the principle of equipartition is deducted.

Planck’s law of Radiation

Max Planck in Germany (in 1901) put forward a new postulate regarding the nature of vibration of the linear harmonic oscillator (LHO) which are in thermal equilibrium with the electromagnetic radiation within the radiation enclosure. According to Planck :

An oscillator has a discrete set of energies which are integral multiplies of a finite quantum of energy equals to new where, h is Planck’s Constant and new is the frequency of the oscillator. So, the energy of the oscillator can have only values like,

$\displaystyle E=n\epsilon =nh\nu$

Where n=0,1,2,3………. This shows that the lowest state (Ground state) energy is zero.

2. So long the oscillator has energy $\displaystyle E=nh\nu$

,it can not emit or, absorb energy. The emission or absorption of energy occurs only when the oscillator jumps from one energy state to another. If the oscillator jumps from state of quantum number $\displaystyle n_{2}$

to lower state of quantum number $\displaystyle n_{1}$

then the Energy Limited is,

$\displaystyle \Delta E=E_{2}-E_{1}=\left ( n_{2}-n_{1} \right )h\nu \ ,if\ n_{2}=n_{1}+1.$

When an oscillator observes a quantum of energy $\displaystyle h\nu$

, it jumps to the next highest rate of energy. Planck estimated the value of $\displaystyle h=6.62618\times 10^{34}\ Js$

which is a Universal constant. If in a enclosure the total number of oscillators in N and the total energy is E then the average energy per oscillator is,

$\displaystyle \left \langle \epsilon \right \rangle=\frac{E}{N}$

If $\displaystyle N_{0}$

, $\displaystyle N_{1}$

, $\displaystyle N_{2}$

,………, $\displaystyle N_{r}$

,… be the number of oscillators with energy 0, $\displaystyle \epsilon$

, $\displaystyle 2\epsilon$

,…., $\displaystyle r\epsilon$

,….respectively, then

$\displaystyle N=N_{0}+N_{1}+N_{2}+\cdot \cdot \cdot \cdot \cdot +N_{r}+\cdot \cdot \cdot$

and $\displaystyle E=0+\epsilon N_{1}+2\epsilon N_{2}+\cdot \cdot \cdot \cdot \cdot +r\epsilon N_{r}+\cdot \cdot \cdot$

$\displaystyle \Rightarrow E=\epsilon (N_{1}+2N_{2}+\cdot \cdot \cdot \cdot \cdot +rN_{r}+\cdot \cdot \cdot)$

According to the Maxwell-Boltzmann distribution formula,

$\displaystyle N_{r}=N_{0}e^{-r\epsilon /kT}$

$\displaystyle N_{1}=N_{0}e^{-\epsilon /kT}$

$\displaystyle N_{2}=N_{0}e^{-2\epsilon /kT}$

……………………………………………………………..

Using these values, we get,

$\displaystyle N=N_{0}+N_{0}e^{-\epsilon /kT}+N_{0}e^{-2\epsilon /kT}+\cdot \cdot +N_{0}e^{-r\epsilon /kT}+\cdot \cdot$

$\displaystyle \Rightarrow N=N_{0}(1+e^{-\epsilon /kT}+e^{-2\epsilon /kT}+\cdot \cdot +e^{-r\epsilon /kT}+\cdot \cdot )$

and

$\displaystyle E=\epsilon (N_{0}e^{-\epsilon /kT}+N_{0}e^{-2\epsilon /kT}+\cdot \cdot \cdot \cdot \cdot +N_{0}e^{-r\epsilon /kT}+\cdot \cdot \cdot)$

$\displaystyle \Rightarrow E=N_{0}\epsilon (e^{-\epsilon /kT}+e^{-2\epsilon /kT}+\cdot \cdot +e^{-r\epsilon /kT}+\cdot \cdot )$

putting $\displaystyle y=e^{-\epsilon /kT}$

, we have from above equations,

$\displaystyle N=N_{0}(1+y+y^{2}+\cdot \cdot \cdot +y^{r}\cdot \cdot )=\frac{N_{0}}{1-y}$

and

$\displaystyle N=N_{0}\epsilon (y+y^{2}+\cdot \cdot \cdot +y^{r}\cdot \cdot )=\frac{N_{0}\epsilon }{(1-y)^{2}}$

Now, the average energy per oscillator is,

$\displaystyle \left \langle E \right \rangle=\frac{E}{N}=\frac{\epsilon e^{-\epsilon /kt}}{1-e^{-\epsilon /kt}}$

$\displaystyle \left \langle E \right \rangle=\frac{\epsilon }{e^{\epsilon /kT}-1}=\frac{h\nu }{e^{\epsilon /kT}-1}$

But, $\displaystyle h\nu =\frac{h}{2\pi }2\pi \nu =\frac{h}{2\pi }\omega$

where, $\displaystyle \omega =2\pi \nu$

If $\displaystyle h\nu < < kT$

, then,

$\displaystyle e^{h\nu /kT}\simeq 1+\frac{h\nu }{kT}$

So that,

$\displaystyle \left \langle \epsilon \right \rangle=\frac{h\nu }{1+\frac{h\nu }{kT}-1}=kT$

Hence, if $\displaystyle h\nu < < kT$

, then $\displaystyle \left \langle \epsilon \right \rangle$

reduces to the classical limit. It can be shown that the number of oscilator per unit volume in the frequency range $\displaystyle \nu$

to $\displaystyle \nu +d\nu$

$\displaystyle N(\nu )d\nu =\frac{8\pi \nu ^{2}}{c^{3}}d\nu$

Where, c is the velocity of light in vacuum.

Now the energy per unit volume of the radiation enclosure in the frequency range $\displaystyle \nu$

to $\displaystyle \nu +d\nu$

$\displaystyle U_{\nu }d\nu =N(\nu )d\nu\times \left \langle \epsilon \right \rangle =\frac{8\pi \nu ^{2}}{c^{3}}\times\frac{h\nu }{h\nu /kT}d\nu$

or, $\displaystyle U_{\nu }d\nu =\frac{8\pi \nu ^{2}}{c^{3}}\times\frac{1}{e^{h\nu /kT}-1}d\nu$

This relation is called Planck’s radiation law which explains the distribution of energy in spectrum of black-body radiation over whole range of wavelength.

Theories of Black Body Radiation

Wien’s Theory of Black Body Radiation

Wien’s Displacement Law

Rayleigh-Jeans Law of Black Body Radiation

ultra-violet Catastropy

Planck’s law of Radiation

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