Quantum Black Body Radiation PDF

# Formulation of Black Body Radiation ## Classical Physics - **Thermal Radiation**: Continuous radiation. - **Non-thermal Radiation**: If the radiation does not characterize the system, it is called non-thermal radiation. **B(λ,t)** ## Absorption Coefficient $$ \frac{I}{I_0} \leq 1 $$ **Black...

# Formulation of Black Body Radiation ## Classical Physics - **Thermal Radiation**: Continuous radiation. - **Non-thermal Radiation**: If the radiation does not characterize the system, it is called non-thermal radiation. **B(λ,t)** ## Absorption Coefficient $$ \frac{I}{I_0} \leq 1 $$ **Black body radiation:** a=1 $$ a + r + t = 1 $$ - absorption - refraction - transmittance ## Kirchoff's Law $$ \frac{a_\lambda (\lambda)}{\epsilon_\lambda (T)} = \frac{a_b (\lambda)}{\epsilon_b (\lambda)} $$ $$ \frac{a_\lambda (\lambda)}{\epsilon_\lambda (T)} = \frac{a_b (\lambda,T)}{\epsilon_b (\lambda, T)} = \frac{a_b (\lambda,T)}{\epsilon_b (\lambda ,T)} = 1 $$ Thus, black body is an absorber as well as good emitter, $$ \epsilon_b (\lambda,T) = 0.6 \cdot \epsilon_b (\lambda, T) $$ For maintaining this, two bodies must emit radiation as much as they absorb radiation. Black body does not depend on material only depends on temp. and for this reason, black body is a universal property. ## Perfect Black Body For a perfect black body, at all λ, a(λ)=1 Perfect black body is impossible. Black body has inversal behavior as this is independent on the material characteristics, only depends on temperature. ## Stefan-Boltzmann's Law - **1879**: Stefan - **Total emissive power = R(T)= σT^4** - **σ = 5.67 × 10^(-8) Wm^2 K^(-4)** - **Material independent property** - **1884**: Stefan-Boltzmann’s law - R(T) = ∫ R(λ,T) dλ - Accurately calculated by: - O. Lummer - E. Prinshein - 1888 ## Wein's Law - **A_max T = b (constant)** - **b = 2.858 × 10^(-3) mK** - **S(λ,T) α (λ,T)** - **S(λ,T) = ±R(T,T)** - **Formulation of S(λ,T)** - **S(λ,T) = ±R(λ,T)!!!!** - **Restriction of calculation for S(λ,T)** ## Calculation for Determining the Number of Modes per Unit Volume - **E(t) = E_0 e^(i(kr-wt))n** - **V_E = ∫ (1/8π^3) e^(2(kr-wt))n d^3k** = (1/8π^3) ∫ d^3k - **V_E = (1/8π^3)(4πk^2dk) = k^2dk/2π^2** - **kc=w (k,T)** = k - **V=(π/ω^2)(dk/dt)(dε/dω)** = (dk/dt)(dε/dω)/π - **k^2= (2π/λ)^2 = (2π/λ)^2E/(ħ^2/2m)** = (2π/λ)^2E/(ħ^2/2m) - **dε/dω = ħω** = ħ^2 - **V=(π/8π^3)(1/ħ^2)dk = dk/8π^2ħ^2** = (dk/dt)(dε/dω)/π - **dω = 2πdk/λ** = (dk/dt)(dε/dω)/π - **dk = λ dω/2π** - **dω = 2πdk/λ** = (dk/dt)(dε/dω)/π - **dk = λ dω/2π** **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **E = (ħ^2/2m)(kx^2+ky^2+kz^2) + V(t)** = ħ^2 - **(d/dx)(d/dx) + (d/dy)(d/dy) + (d/dz)(d/dz)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **E = (ħ^2/2m)(kx^2+ky^2+kz^2)+V(t)** = ħ^2 - **(d/dx)(d/dx) + (d/dy)(d/dy) + (d/dz)(d/dz)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **E = (ħ^2/2m)(kx^2+ky^2+kz^2) + V(t)** = ħ^2 - **(d/dx)(d/dx) + (d/dy)(d/dy) + (d/dz)(d/dz)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **E = (ħ^2/2m)(kx^2+ky^2+kz^2) + V(t)** = ħ^2 - **(d/dx)(d/dx) + (d/dy)(d/dy) + (d/dz)(d/dz)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **E = (ħ^2/2m)(kx^2+ky^2+kz^2) + V(t)** = ħ^2 - **(d/dx)(d/dx) + (d/dy)(d/dy) + (d/dz)(d/dz)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 - **(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **E = (ħ^2/2m)(kx^2+ky^2+kz^2) + V(t)** = ħ^2 - **(d/dx)(d/dx) + (d/dy)(d/dy) + (d/dz)(d/dz)** = ħ^2 - **ω^2(d^2/dx^2) + ω^2(d^2/dy^2) + ω^2(d^2/dz^2)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2 - **E = (ħ^2/2m)(kx^2+ky^2+kz^2) + V(t)** = ħ^2 - **(d/dx)(d/dx) + (d/dy)(d/dy) + (d/dz)(d/dz)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **E = (ħ^2/2m)(kx^2+ky^2+kz^2) + V(t)** = ħ^2 **V=(d^2/dx^2) + (d^2/dy^2) + (d^2/dz^2)** = ħ^2 - **V=(λ/8 π^2 ħ^2)(dω/2π) = λdω/16π^3ħ^2** - **No. of points in k and k+dk.** - **(1/8π^3)(4πk^2dk)(2π/λ)^3** = λ^3dk/4π^2 - **No. of modes per unit volume** - **(1/8π^3)(4πk^2dk) = k^2dk/2π^2** - **For any electromagnetic radiation, there exist two components.** - **(λ^3/4π^2)dk = (λ^3/4π^2)(2π/λ)dω = (λ^2/2π)dω** ## Rayleigh Jeans Law - **Average energy of an oscillating dipole = ∫e^(−ε/k_BT)Pr dε = k_BT** - **S(λ,T) = 8πk_BT/λ^4** ## Rayleigh Jeans Formula This formula leads to the ultraviolet catastrophe. ## Planck’s Quantum Hypothesis **E = E_0e^(i(kr-wt))** - **ħf = k** - **ħω = ω** - **If f changes, then frequency ω will change, which is not allowed, so any value of f is not possible. So f is a constant. B = 1** - **λ = λ(T+4)** - **dλ = -λ^2ds/s** - **ds = - dλ/λ^2** - **S(λ,T) = (8πh/c^3)(k_BT)(S(λ,T))** ## Determining h and k_B **A_max T = b = (hc/4.96k_B)** - **R(T) = σT^4** - **R(T) = ∫ R(λ,T) dλ** - **R(λ,T) = c/4 S(λ,T)** - **R(T) = (c/4)∫ S(λ,T) dλ** - **R(T) = (c/4)∫ (8πh/c^3)(k_BT)(1/(e^(hc/λk_BT)-1) dλ** - **x = hc/λk_BT** - **λ = hc/xk_BT** - **dx/dλ = -hc/λ^2k_BT** - **dλ = -λ^2k_BT/hc dx** - **R(T) = (8πh/c^3)(k_BT)∫ (x^3/(e^x-1))dx** - **R(T) = (2π^5k_B^4T^4)/(15ħ^2c^2)** - **σ = (2π^5k_B^4)/(15ħ^2c^2)** - **k_B = 1.38 × 10^(-23) J/K** - **ħ = 6.63 × 10^(-34) Js** - **S(λ,T) = (8πħ/λ^4)(k_BT/(e^(hc/λk_BT)-1))** ## Another Formulation of Black Body Black body is just full of gas of photon. ## Quantum of Action - **ħ = 6.62 × 10^(-34) Js** - **ħ = length x momentum ** - **ħ is called the fundamental quantum of action** - **Action** = ∫p dq ## Energy of a Particle in a Box - **m = 10^(-2)kg** - **v = 0.1 m/s** - **k_0 = 10^(-2)m** - **E = p^2/2m + k_0^2/2** - **p = √2m(E-k_0^2/2)** - **ω_0 = ±√(k_0/m)** - **S = ∫√(2E/m-k_0^2/m) √mE √(1-K_0^2/2E) dω** - **Let cosθ = √(k_0^2/2E) / √(2E/m-k_0^2/m)** - **sinθ = √(1-k_0^2/2E) / √(2E/m-k_0^2/m)** - **dω = -√(2E/m-k_0^2/m)sinθdθ** - **Since ω = ±√(k_0/m), θ = π** - **S = -2√(2E/m-k_0^2/m) √mE ∫[√(k_0^2/2E) / √(2E/m-k_0^2/m)]sinθ dθ** - **S = 2√(2E/m-k_0^2/m) √mE ∫(1-cos2θ)dθ** - **S = 2π√(2E/m-k_0^2/m) √mE = E.T** = energy × time = constant - **T = 2π√(m/k_0)** - **S = E.T = 5 × 10^(-5) s** - **T = 2π√(m/k_0) = 0.628 s** - **k_0 = (mv_0^2/2)/E = 10^(-2) s^2** - **ε = 10^29 = very large! Energy quantization is insignificant here. Because ħ_0 = ħs = ħ/2≈10^(-34) Js and ħ_0/ε =10^(-29)→0** If action is quantized, energy is also quantized. ## Plane Wave Modes in a Linear Cavity - **Electric field:** E = E_0 cos(kx-wt+φ)= E_0[cos(kx)cos(wt-φ)+sin(kx)sin(wt-φ)] - **a_v(t) = E_0 cos(wt-φ)** - **p(t) = da_v(t) = -E_0 sin(wt-φ)** - **E = ωa_v cos(kx) - p sin(kx)** - **Total energy in the cavity due to electric and magnetic field:** - **U = ∫ E_0^2cos^2(kx)+p^2sin^2(kx) dx** - **U = (E_0^2ω^2/2 + p^2/2)** - **Total energy in the cavity due to external force:** - **U = (ω^2m/2)x^2 + p^2/2m** - **(m/2)x^2 = p^2/2m** - **P= (d/dt) = p_v/m** ## Specific Heat of Solids - **1818:** Dulong and Petit’s measured the atomic heat of solid - **Atomic heat of all solid elements = 6.1 cal** - **Atomic heat = specific heat × atomic weight** ## Microscopic View of Dulong and Petit’s Law - **c_p - c_v = α N T** - **c_v = c_p - α = (k/αT)** - **c_v = 5.9 cal** - **c_v is linear for of temp.** ## Vibrational Modes - **No. of vibrational modes = 3N_0** - **Average energy = ∫ e^(−ε/k_BT)ε dε = k_BT** - **Thermal energy U = 3N_0k_BT = 3RT** - **c_v = 3R ~ 6 cal** - **Average energy is the same and no. of oscillators is also the same. Thus, the universal behaviour arises.** ## Molecular Heat - **Molecular heat = atomic heat = specific heat × molecular weight** ## Application Study of the Specific Heat of Solid - **In case of specific heat of a solid element = 0.057** - **Atomic weight = (6.4 cal)/0.057 = 112** - **Atomic weight = 38** - **Valency = 3.** ## Molecular Formula - **Mercury Chloride:** Specific heat = 0.069 - **Molecular heat = 0.069 × molecular weight of mercury Chloride** - **Molecular weight of mercury Chloride = 2×200.6 + 35.5 × 2 = 286.4 g/mol** - **Number of atoms = 2 + 3 = 5** ## Heat Capacity of a Monoatomic Solid - **U = 3N_0ħω/(e^(ħω/k_BT)-1)** - **c_v = dU/dT = 3N_0k_B (ħω/k_BT)^2(e^(ħω/k_BT))/(e^(ħω/k_BT)-1)^2** - **At large T, (ħω/k_BT) < 1** - **c_v = 3R (ħω/k_BT)^2(e^(ħω/k_BT))/(e^(ħω/k_BT)-1)^2~ 3R** - **At T→∞, (ħω/k_BT)→0** - **c_v = 3R (ħω/k_BT)^2e^(ħω/k_BT)/(e^(ħω/k_BT)-1)^2 ≈ 3R(ħω/k_BT)^2** - **c_v = 3R (ħω/k_BT)^2 ≈ 3R** - **e^(ħω/k_BT) = 1 + (ħω/k_BT) + (1/2!)(ħω/k_BT)^2 + (1/3!)(ħω/k_BT)^3 +…..** - **e^(ħω/k_BT)-1 = (ħω/k_BT) + (1/2!)(ħω/k_BT)^2 + (1/3!)(ħω/k_BT)^3 +…..** ## Debye Model - **s_v = (1+s^2/(c_L^2) + s^2/(c_T^2) d^3s** - **No. of modes for longitudinal vibration = 4πV (s^2/(c_L^2))ds** - **No. of modes for transverse vibration = 8πV (s^2/(c_T^2))ds** - **Total no. of modes** = 4πV(1/c_L^2 + 1/c_T^2)s^2ds - **∫g(s)ds = 3N_0** - **4πV √(1/c_L^2 + 1/c_T^2) = (3N_0/V_m)** - **4πV √(1/c_L^2 + 1/c_T^2) = (9N_0/V_m)** - **g(s)ds = (9N_0/V_m)(s^2/Θ_D^3)ds** - **Let ∫_0^Θ_D s^3 g(s)ds = ∫_0^Θ_D s^3[(9N_0/V_m)(s_2/Θ_D^3)ds]** - **U = ∫ (ħω/(e^(ħω/k_BT)-1)) g(s)ds = ∫ (ħω/(e^(ħω/k_BT)-1)) [(9N_0/V_m)(s^2/Θ_D^3)ds]** - **Let x = ħω/k_BT, dx = ħdω/k_BT, s = (k_BT/ħ)x** - **U = (9N_0/V_m)(ħω/Θ_D^3)∫(ħω/(e^(ħω/k_BT)-1))[ (k_BT/ħ)^3x^2 (k_BT/ħ)dx / (k_BT/ħ)^3 ]** - **U = (9R/Θ_D^3) T^4 ∫(x^4/(e^x-1)) dx** - **Θ_D = ħω_D/k_BT** - **At high T, x→small, x = ħω_D/k_BT** - **U = (9R/Θ_D^3)T^4∫ (x^4/(e^x-1)) dx = (9R/Θ_D^3)T^4 ∫x^4 dx = (9R/Θ_D^3)T^4(x^5/5)** - **U= (9R/5)(ħω_D/k_B)^3T** - **c_v= dU/dT = (36R/5)(ħω_D/k_B)^3T^3 = 3RT** - **At low T, x→∞** - **U = (9R/Θ_D^3)T^4∫ (x^4/(e^x-1)) dx = (9R/Θ_D^3)T^4∫ x^3e^(-x) dx = (9R/Θ_D^3)T^4(π^4/15)** - **c_v = (36R/15)(π^4/Θ_D^3)T^3 = (46.4/Θ_D^3)T^3** - **c_v is valid at low temperatures.** ## Photoelectric Effect - **Extended the work of Hertz.** - **Additional spark is generated.** - **Charged particle is generated.** - **Used cross-field by J.J. Thompson.** - **Intensity is proportional to the number of ejected electrons.** - **If frequency does not cross the threshold frequency, then electron emission will not occur.** - **Stopping potential is intensity independent.** - **As intensity increases, energy transfer will increase, therefore, stopping potential increases.** - **Ejected electrons kinetic energy varies linearly with threshold frequency and depends on intensity.** - **Electron is ejected immediately from the wave. A certain time is required for energy transfer, therefore, there should a time delay. (n-1)ħω, nhω. Energy transfer will be instantaneous for particle-particle interaction.** - **ħω = p_L^2/2m + W** - **ħω = p_L^2/2m + W** - **W = work function** - **The energy of the photon is fixed at a particular frequency.** - **The energy of the photon is fixed at a particular frequency.** - **ΔK = - ΔU = -(-e)(-V_0) = -eV_0** - **(1/2)mv^2_max = eV_0** - **eV_0 = ħω-W** - **The threshold frequency** = **ω = W/ħ.** - **I = nħω = number of photons × ħω** - **I = current** - **I = n × ħω = current** - **Confirmation of Einsteir's ear experimentally** = **V_0= ħω/e - W/e** - **h = 6.56 × 10^(-34) Js** - **This value of h supports Einsteir's experiment.** ## Compton Effect - **Bransden** - **Discovered by Rontgen in 1895** - **The frequency and wavelength of the scattered wave are the same** - **J.J. Thomson (1890), observed anomalous behaviour for hard X-rays.** - **M. van Laue and D.L. Bragg’s X-ray diffraction in 1912** - **nλ = d sinθ** = Diffraction as well as interference. - **Experimental sketch used by compton.** - **hard X-ray is the incident wave. The detector observes the intensities of the scattered waves.** - **Incident angle is 0°** - **The scattered wave is independent of the property of the metal** - **Electron is free** - **If the incident wavelength is high (hard X-ray), the electron is free.** - **If the incident wavelength is low (soft X-ray), the electron is not free.** - **Momentum transfer depends on the angle and direction.** - **This phenomena is due to particle-particle interaction where the light behaves as a particle.** - **Δλ = λ’-λ_0 = (hc/m_ec)sin^2(θ/2)** - **c = 0.048 Å** - **If the incident wavelength is high, then the energy of the incident wave is also high. So the scattered wave energy is also high. We consider the electron as a free electron.** - **If the incident wavelength is low then the energy of the incident wave is also low, hence electron can't be considered as free electron.** ## Compton's Model - **It is an isolated system** - **Linear momentum is conserved. P_0 = p_r + p_e** - **P_0 cosθ = p_r cosφ + p_e cosα** - **P_0 sinθ = p_r sinφ - p_e sinα** - **p_e cosα = P_O cosθ - p_r cosφ** - **p_e sinα = p_r sinφ** - **p_e^2 = P_0^2 + p_r^2 - 2P_0p_r cosθ** - **Energy conservation** - **E^2/c^2 = (1 - v^2/c^2)m^2c^4** - **E^2 = p^2c^2 + m^2c^4** - **p_e^2 = (E/c)^2 - m^2c^2 = (E/c)^2(1-v^2/c^2)** - **E^2 = (p_e^2c^2/((1-v^2/c^2)) + m^2c^4** - **(1/c^2)(p_0^2c^2 - 2p_0p_rc^2cosθ + p_r^2c^2) + m^2c^2 = (p_0^2c^2/((1-v^2/c^2)) + m^2c^4** - **(p_0^2c^2 - 2p_0p_rc^2cosθ + p_r^2c^2) + m^2c^4(1-v^2/c^2) = p_0^2c^2 + m^2c^4(1-v^2/c^2)** - **- 2p_0p_rc^2cosθ + p_r^2c^2 = 0** - **- 2p_0p_rc^2cosθ = -p_r^2c^2** - **p_0p_r = p_r^2/2 = (E_0/c)((E/c)(1-cosθ))** - **(E_0/E)(1-cosθ) = (ħ/λ_0)((ħ/λ)(1-cosθ))** - **λ’ - λ_0= (ħ/m_ec)sin^2(θ/2)** **Therefore, light behave as a particle.** ## Bohr Correspondence Principle **hν = R (1/(n-1)^2 - 1/n^2)** - **ν = [R/(n-1)^2 - (1/n)^2] / (2π^2(ħ(2n-1)))** - **ν = [R/(n-1)^2 - (1/n)^2] / (2π^2(ħ(2n-1)))** - **ν = [R/(n-1)^2 - (1/n)^2] / (2π^2(ħ(2n-1)))**