Black body radiation - Planck`s law
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Unit - III Quantum Physics Introduction The Classical theories fail to explain the phenomena like Compton effect, photoelectric effect, Zeeman effect, emission of light, absorption of light etc., The failure of these theories lead to the discovery of a new theory, called Quantum Theory of radiation of light. This theory was discovered by Maxplanck and developed by others. Black Body Radiation Definition A perfect black body is one which absorbs radiation of all wavelengths incident on it. A black body can radiate energy in all possible wavelengths when it is heated to a suitable temperature. The radiation emitted from black body is known as black body radiation or total radiation. Characteristics of Black Body Radiation Black body radiations are characterized by Stefan’s law, Wien’s law and Rayleigh-Jean’s law. Stefan – Boltzmann’s Law The radiation energy (E) emitted per unit area of a perfect black body is directly proportional to the fourth power of its absolute temperature ‘T’. πΈ = ππ 4 Where π is a proportionality constant known as Stefan’s constant and its value is 5.78x10-8 Wm-2K-4. Wien’s Displacement Law Wien’s Displacement Law states that the wavelength corresponding to the maximum energy is inversely proportional to absolute temperature ‘T’. ππ π = πΆπππ π‘πππ‘ This law shows that as the temperature increases the wavelength corresponding to maximum energy decreases. This law holds good only for shorter wavelengths and not for longer wavelengths. Rayleigh – Jean’s Law The energy distribution is directly proportional to the absolute temperature and is inversely proportional to the fourth power of the wavelength. πΈπ = 8πππ π4 This law holds good only for longer wavelength regions and not for shorter wavelengths. Planck’s Quantum Theory of Black Body Radiation Planck derived the expression for the energy distribution based on the following hypothesis: Planck’s hypothesis 1. The black body radiation chamber is filled up not only with radiations but also with a large number of oscillating particles. The particles can vibrate in all possible frequencies. 2. The frequency of radiations emitted by an oscillator is the same as that of the frequency of the vibrating particles. 3. The oscillatory particles cannot emit energy continuously. Radiation is emitted (or) absorbed by a body in integral multiples of a fundamental quantum of energy called “photon”. 4. The vibrating particles can radiate energy when the oscillators move from one state to another. The radiation of energy is not continuous, but discrete in nature. The values of the energy of the oscillators are such as 0, hν, 2hν, 3hν…… nhν. Derivation Let us consider the number of vibrating particles in the body as N0, N1, N2…………., Nn. According to Planck’s hypothesis, the energy of the above particles can be written as 0, ε, 2ε, 3ε … nε. Therefore, the total number of vibrating particles is given as, N = N0+N1+N2+………. + Nn -------- (1) Similarly, the total energy of the body is given as, E = 0+ε+ 2ε+3ε +…… +nε -------- (2) Therefore, the average energy of the particle is given as, E* = E/N -------- (3) According to Maxwell’s distribution formula, the number of particles in the nth oscillatory system can be written as, π = π0 π −ππ⁄ ππ -------- (4) Where ε is the energy per oscillator, k is the Boltzman’s constant and T is the absolute temperature. According to Maxwell’s distribution function, the total number of particles N can be written as, π = π0 π Or −π⁄ ππ π = π0 [1 + π + π0 π −π⁄ ππ +π −2π⁄ ππ −2π⁄ ππ + π0 π +π −3π⁄ ππ −3π⁄ ππ +….. +…..] --------- (5) We know, 1+x+x2+x3+……… =1/(1-x). Therefore, eqn (5) can be written as, π0 1−π π= ------- (6) −π⁄ ππ Similarly, the total energy of the body can be written as, E = 0+ε π0 π E = ε π0 π −π⁄ ππ −π⁄ ππ + 2επ0 π [1 + 2π −2π⁄ ππ −π⁄ ππ +3π +3επ0 π −2π⁄ ππ −3π⁄ ππ +……] +…… ---------- (7) We know, 1+2x+3x2+4x3+……… + nxn-1 =1/(1-x)2. Therefore, eqn (7) can be written as, πΈ= −π⁄ ππ −π⁄ ππ (1− π )2 π0 ππ πΈ Average energy(E*) = π =[ ------- (8) −π⁄ ππ −π⁄ ππ (1− π )2 π0 ππ = E* = 1−π ][ −π⁄ ππ π0 ] −π⁄ ππ −π⁄ (1− π ππ ) ππ π ---------- (9) π π ⁄ππ − 1 Substituting the value of ε = hν in eqn (9), We have E* = hν hν (π ⁄ππ ---------- (10) )−1 If ν and ν+dν is the frequency range, the number of oscillations can be written as, π= 8ππ 2 πΆ3 ππ ------------ (11) Therefore the total energy per unit volume for a particular frequency can be obtained by multiplying eqns (11) and (10) as, Eνdν = [ hν ][ hν (π ⁄ππ ) − 1 8ππ 2 πΆ3 ] ππ Or Eνdν = [ 8ππ 3 πΆ3 1 ][ hν (π ⁄ππ ] ππ ----------- (12) )−1 Equation (12) is known as Planck’s equation for radiation law interms of frequencies. It can also be written interms of wavelength as, Eλdλ = [ Or Eλdλ = [ 8πβ πΆ 3 πΆ 3 π3 8πβπΆ π5 1 ][ hC⁄ πππ (π ][ 1 hC⁄ πππ (π πΆ 2 )−1π ] ππ ] ππ [... ν=C/λ] ----------- (13) )−1 Equation (13) represents the Planck’s radiation law of wavelength.
