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Blackbody Radiation (Ch Lecture 26. Blackbody Radiation (Ch. 7) Two types of bosons: (a) Composite particles which contain an even number of fermions. These number of these particles is conserved if the energy does not exceed the dissociation energy (~ MeV in the case of the nucleus). (b) particles associated with a field, of which the most important example is the photon. These particles are not conserved: if the total energy of the field changes, particles appear and disappear. We’ll see that the chemical potential of such particles is zero in equilibrium, regardless of density. Radiation in Equilibrium with Matter Typically, radiation emitted by a hot body, or from a laser is not in equilibrium: energy is flowing outwards and must be replenished from some source. The first step towards understanding of radiation being in equilibrium with matter was made by Kirchhoff, who considered a cavity filled with radiation, the walls can be regarded as a heat bath for radiation. The walls emit and absorb e.-m. waves. In equilibrium, the walls and radiation must have the same temperature T. The energy of radiation is spread over a range of frequencies, and we define uS (ν,T) dν as the energy density (per unit volume) of the radiation with frequencies between ν and ν+dν. uS(ν,T) is the spectral energy density. The internal energy of the photon gas: ∞ = , dTuTu νν ()∫ S ( ) 0 In equilibrium, uS (ν,T) is the same everywhere in the cavity, and is a function of frequency and temperature only. If the cavity volume increases at T=const, the internal energy U=u(T) V also increases. The essential difference between the photon gas and the ideal gas of molecules: for an ideal gas, an isothermal expansion would conserve the gas energy, whereas for the photon gas, it is the energy density which is unchanged, the number of photons is not conserved, but proportional to volume in an isothermal change. A real surface absorbs only a fraction of the radiation falling on it. The absorptivity α is a function of ν and T; a surface for which α(ν ) =1 for all frequencies is called a black body. Photons The electromagnetic field has an infinite number of modes (standing waves) in the cavity. The black-body radiation field is a superposition T of plane waves of different frequencies. The characteristic feature of the radiation is that a mode may be excited only in units of the quantum of energy hν (similar to a harmonic oscillators) : ε =n+ 2/1 ν h i ( i ) This fact leads to the concept of photons as quanta of the electromagnetic field. The state of the el.-mag. field is specified by the number n for each of the modes, or, in other words, by enumerating the number of photons with each frequency. According to the quantum theory of radiation, photons are massless Eph = ν h bosons of spin 1 (in units ħ). They move with the speed of light : Eph = cpph The linearity of Maxwell equations implies that the photons do not E ν p =ph =h interact with each other. (Non-linear optical phenomena are ph c c observed when a large-intensity radiation interacts with matter). The mechanism of establishing equilibrium in a photon gas is absorption and emission of photons by matter. Presence of a small amount of matter is essential for establishing equilibrium in the photon gas. We’ll treat a system of photons as an ideal photon gas, and, in particular, we’ll apply the BE statistics to this system. Chemical Potential of Photons = 0 The mechanism of establishing equilibrium in a photon gas is absorption ⎛ ∂ F ⎞ and emission of photons by matter. The textbook suggests that N can be ⎜ ⎟ = 0 ⎝ ∂ N ⎠ found from the equilibrium condition: TV, ⎛ ∂ F ⎞ Thus, in equilibrium, the chemical On the other hand, ⎜ ⎟ = μ μ = 0 ⎜ ⎟ ph potential for a photon gas is zero: ph ⎝ ∂ N ⎠TV, However, we cannot use the usual expression for the chemical potential, because one cannot increase N (i.e., add photons to the system) at constant volume and at the same time keep the temperature constant: ⎛ ∂ F ⎞ ⎜ ⎟ - does not exist for the photon gas ⎝ ∂ N ⎠TV, ⎛ ∂F ⎞ FTV( , ) Instead, we can use GN= μ G= F+ PV P= −⎜ ⎟ = − ⎝ ∂V ⎠T V - by increasing the volume at T=const, we proportionally scale F F - the Gibbs free energy of an G Thus, GF= −V 0 = μ = =0 V equilibrium photon gas is 0 ! ph N For μ = 0, the BE distribution reduces to the Planck’s distribution: 1 1 Planck’s distribution provides the average nph= f ph()ε, T= = ⎛ ε ⎞ ⎛ hν ⎞ number of photons in a single mode of exp⎜ ⎟ −1 exp⎜ ⎟ −1 ⎜ ⎟ ⎜ ⎟ frequency ν = ε/h. ⎝kB T⎠ ⎝kB T⎠ hν ε=n h ν = The average energy in the mode: ⎛ hν ⎞ exp⎜ ⎟ −1 ⎝kB T⎠ cal (assiIn the clhν << kBT) limit: ε =kB T In order to calculate the average number of photons per small energy interval dε, the energy of photons peraverage small energy interval dε, etc., as well as the total average number of photons in a photon gas and its total enerneed to know the we gy, density of states for photons as a function of photon energy. kz Density of States for Photons 1 4( /)π 3kk 3 3 ( volume) k 3 N() k= = G() k= π π π 2 2 8 × × 6π 6π LLLx y z kx ky dG(ε ) ε 3 ε 2 g ε = cpε= c= kε G= g 3D ε = () h () 2 3 ph () 2 3 dε 6π ()ch 2π ()ch 2 2 2 extra factor of 2 due dε (hν ) 8πν 3D 8πν g3Dν()= g3D () ε = h = g ()ν = to two polarizations: ph ph 2 3 3 ph 3 dν π ()ch c c h gν ν nν dν = uν , ν T d Spectrum of Blackbody Radiation ( ) ( ) S ( ) The average energy of photons with frequency photon average number between ν and ν+dν (per unit volume): energy of photons ) ) ν 2 n(ν ) ( ν n 3D 8πν ( 3 ) g ∝ ν g ph ()ν = 3 ν ν c × = ( h g ν ν ν h ν π8 h ν3 - the spectral density of the u)νs T, (= ν h () νg )= f ν 3 ( black-body radiation cexpβ() h ν−1 (the Plank’s radiation law) dε u as a function of theu energy: Tε d(), ε u= ν () T, d ν ()(),, u ν= T εu = h() Tν , T × h dν 8π ε3 u()ε , T = ()hc 3 ⎛ ε ⎞ exp⎜ ⎟ −1 ⎝kB T⎠ u(ε,T) - the energy density per unit photon energy for a photon gas in equilibrium with a blackbody at temperature T. Classical Limit (small f, large λ), Rayleigh-Jeans Law At low frequencies or high temperatures: β hν 1<< exp(β hν )−1 ≅ β hν π8 h ν3 π8 ν2 - purely classical result (no h), can be u()ν , T = ≅ k T s c3 expβ() h ν−1 3 c B obtained directly from equipartition Rayleigh-Jeans Law This equation predicts the so-called ultraviolet catastrophe –an infinite amount of energy being radiated at high frequencies or short wavelengths. Rayleigh-Jeans Law (cont.) u as a function of the wavelength: 3 ⎛ c ⎞ ⎜h ⎟ ⎡ dε hc ⎤ 8π λ hc⎛ ⎞ 8π hc 1 uλ T, d λ= − u, ε T ε d = − uλ, T = ⎝ ⎠ = () )( ⎢ 2 ⎥ () 3 ⎜ 2 ⎟ 5 ⎣λd λ⎦ ()hc ⎛ hc ⎞ ⎝ λ ⎠ λ ⎛ hc ⎞ exp⎜ ⎟ −1 exp⎜ ⎟ −1 ⎝ λkB T⎠ ⎝ λkB T⎠ it of large mical lassiIn the clλ: 8πk T u()λ, T ≈ B 1 largeλ λ4 λ4 High ν limit, Wien’s Displacement Law At high frequencies/low temperatures: β hν 1>> exp()β hν1− ≅ exp(β hν ) 8π h 3 us ()ν , T = νexp()− βh ν Nobel 1911 c3 The maximum of u(ν) shifts toward higher frequencies with increasing temperature. The position of maximum: 3 ⎡ ⎛ hν ⎞ ⎤ ⎢ ⎜ ⎟ ⎥ ⎜ ⎟ ⎡ 2 3 x ⎤ du d ⎢ ⎝kB T⎠ ⎥ 3x x e =const × =const ×⎢ − ⎥ = 0 ⎢ ⎥ x x 2 dν ⎛ hν ⎞ ⎛ hν ⎞ ⎣⎢e −1 ()e −1 ⎦⎥ d⎜ ⎟ ⎢exp⎜ ⎟ −1⎥ k T ⎜k T ⎢ k T ⎥ ν ≈2 . 8B ⎝ B ⎠ ⎣ ⎝ B ⎠ ⎦ max h −3( x) = ex 3 →x 2 . ≈ 8 hν Wien’s displacement law max ≈2 . 8 k T - iscovered exdperimentally B by Wilhelm Wien ) ,T ν ( u - he “most likely”t frequof a photon in a ency blackbody radiation with temperature T Numerous applications (e.g., non-contact radiation thermometry) ν νmax ⇔λmax hν u()ν , T u(λ, T) max ≈2 . 8 - does this mean that kB T hc ≈2 . 8 ? No! ν max λmax kB Tλmax 3 ⎛ c ⎞ ⎜h ⎟ ⎡ dε hc ⎤ 8π λ hc⎛ ⎞ 8π hc 1 uλ T, d λ= − u, ε T ε d = − uλ, T = ⎝ ⎠ = () )( ⎢ 2 ⎥ () 3 ⎜ 2 ⎟ 5 ⎣λd λ⎦ ()hc ⎛ hc ⎞ ⎝ λ ⎠ λ ⎛ hc ⎞ exp⎜ ⎟ −1 exp⎜ ⎟ −1 ⎝ λkB T⎠ ⎝ λkB T⎠ du d ⎡ 1 ⎤ ⎡ 5 (−x−exp2 ) ( 1 x) ⎤ / =const × =const × − − = 0 ⎢ 5 ⎥ ⎢ 6 5 2 ⎥ df dx⎣exp x{}() 1− / x ⎦ 1 ⎣ xexp{}() 1 x− /xexp 1{}() 1 x− /⎦ 1 hc 5 expx{ 1() / x− } 1= ( expx) → 1λmax / ≈ 5kB T T = 300 K → λmax ≈ 10 μm “night vision” devices Solar Radiation The surface temperature of the Sun - 5,800K. hc λmax=0 . ≈ 5μ m 5kB T As a function of energy, the spectrum of umax = hν max2= .k 8 TB ≈ 1 . eV 4 sunlight peaks at a photon energy of - close to the energy gap in Si, ~1.1 eV, which has been so far the best material for solar cells Spectral sensitivity of human eye: Stefan-Boltzmann Law of Radiation The total number of photons per unit volume: 3 3 N ∞ π8 ∞ ν2 8π ⎛k T⎞ x∞ 2 dx ⎛ k ⎞ n≡n = ε()() g ε d= ε dν = ⎜ B ⎟ = 8π ⎜ B ⎟ T 3 2× .
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