Chapter 10 Wave-Particle Duality P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 10: Wave-Particle Duality There is nothing new to be discovered in physics now. All that remains is more and more precise measurement. – Lord Kelvin, 1900 P. J. Grandinetti Chapter 10: Wave-Particle Duality Why do objects glow when they get hot? P. J. Grandinetti Chapter 10: Wave-Particle Duality Why do objects glow when they get hot? As object is heated to higher temperatures it begins to glow, first as red, then orange, yellow, and finally a blue white, as temperature keeps increasing. Heating an object increases motion of all its microscopic constituents. Since these constituents have electric charge (electrons, cations, anions), or electric dipoles, etc, then their motion generates electromagnetic waves, i.e., light. All kinds of microscopic motion can give off electromagnetic radiation: accelerating charge, rotating dipole, vibrating dipole, ... + _ + _ e– tumbling vibrating electron dipole dipole changing orbits P. J. Grandinetti Chapter 10: Wave-Particle Duality Black body radiation Spectrum of radiation emitted is plot of light intensity or light energy density as function of light frequency or wavelength. Spectrum depends somewhat on object’s composition. A class of hot bodies, called black bodies, have universally the same thermal spectrum. Black bodies are non-reflective–they absorb all light incident upon them–and thus appear black. Í Light absorptivity, 훼휆, is fraction of incident light (power) absorbed by real body when it is emitting and absorbing in thermodynamic equilibrium. Í Light emissivity, "휆, ratio of the emissive light power of real body to emissive power of an otherwise identical black body. Í Kirchoff’s law say 훼휆 = "휆 at thermal equilibrium. Perfect black body has 훼휆 = "휆 = 1 for all wavelengths. Only when black bodies are hot enough do they give off visible light–becoming self-luminous. Engineers often make the gray body assumption that "휆 is independent of wavelength. For most non-metallic surfaces we find "휆 ù 1. P. J. Grandinetti Chapter 10: Wave-Particle Duality Emissivity of selected surfaces in gray body assumption. Material Emissivity Charcoal: powder 0.96 3M, black velvet coating 9560 series optical black 1.00 Skin, human 0.98 Polyethylene Black Plastic 0.92 Magnesium Oxide White Paint 0.90 Snow 0.80 Copper, Thick Oxide Layer 0.78 Water 0.95 Glass, Smooth 0.94 Cement 0.54 Copper, Polished 0.023 Silver, Polished 0.020-0032 Aluminum Foil 0.03 P. J. Grandinetti Chapter 10: Wave-Particle Duality Black bodies Black bodies at equilibrium with surroundings emit characteristic spectrum. with increasing temperature... most intensity light frequency shifts to higher frequencies (color changes) 1.5 2000 K area under curve increase (gets brighter). 1.0 Definition 1500 K ? 0.5 The radiant emittance, j , comes from area under curve: 1000 K ∞ 0.0 ? 휋 휈; 휈; 0 1 2 3 4 5 j .T/ = Ê B휈. T/d 0 B휈.휈; T/ is spectral radiance of black body. i.e., the power emitted per unit area of black body surface P. J. Grandinetti Chapter 10: Wave-Particle Duality Stefan-Boltzmann law Experimentally found that j?.T/ of black body increases rapidly with temperature, j?.T/ = 휎T4 Stefan-Boltzmann law, 휎 is Stefan-Boltzmann constant, 휎 = 5:670367 × 10*8 W/(m2⋅K4). Real objects are not perfect black bodies and total light intensity emitted is ? 4 j .T/ = "휆휎T ; "휆 is emissivity of object’s surface, which varies between 0 and 1. P. J. Grandinetti Chapter 10: Wave-Particle Duality Wien displacement law Familiar color change with increasing T comes from shifting of distribution mode (maximum) according to 휆 maxT = b휆 Wien displacement law b휆 is Wien displacement constant, b휆 = 0:0028977729 m⋅K. Color temperature of light bulb is temperature where most intense wavelength of ideal black body radiator matches most intense wavelength of light emitted by bulb. Optical Pyrometry: For example, Wien displacement law is used by astronomers to 휆 determine surface temperature of stars from their max. Í 휆 Sun has max = 510 nm, so T = 5682 K. Í 휆 North star has max = 350 nm, so T = 8279 K. P. J. Grandinetti Chapter 10: Wave-Particle Duality Stamp commemorating Fraunhofer’s measurement of Sun’s spectrum in 1817. Intensity as function of 휆 follows black body distribution for T = 5682 K. Line Assignment wavelength/nm Other interesting aside... C H훼 656.2849 D Na 589.5923 Sun’s spectrum also reveals number of dark lines. 1 D 2 Na 588.9953 Fraunhofer saw that the solar D lines were in same E Ca 527.0276 location as bright yellow doublet present in E Fe 526.9541 F dispersed light of a salted (NaCl) flame. H훽 486.1327 G Fe 430.7906 Other solar lines have been assigned... G Ca 430.7741 H Ca 396.8468 P. J. Grandinetti Chapter 10: Wave-Particle Duality Ideal blackbody Good model for ideal blackbody is small hole into cavity with walls opaque to radiation. Any light that enters hole is absorbed by inner walls and never escapes, i.e., hole absorbs all light and reflects none. At thermal equilibrium walls emit thermal radiation and fill cavity. Hole also acts as ideal emitter of thermal radiation at T and has blackbody spectrum. P. J. Grandinetti Chapter 10: Wave-Particle Duality Raleigh & Jeans’ attempt to predict blackbody distribution Assume walls of ideal black body cavity are metallic. Classic E&M says electric field of light must be zero at walls. Assume standing electromagnetic waves inside box. Recall the normal modes in 3D box. How many normal modes in frequency interval 휈 to 휈 + d휈? Raleigh and Jeans showed that number of normal modes in this interval is given by 8휋V N.휈/d휈 = 휈2d휈 3 c0 V is volume of box. P. J. Grandinetti Chapter 10: Wave-Particle Duality Raleigh-Jeans’ attempt to predict blackbody distribution To predict B휈.휈; T/ R&J needed average energy contained in each normal mode with frequency 휈. E Classical E&M theory says energy of wave is proportional to square of amplitude, i.e., 0. Taking each normal mode as harmonic oscillation with 2 degrees of freedom, R&J assumed thermal equilibrium between normal modes and walls and used equipartition theorem— 1 k T to 2 B 휖 each motional degree of freedom, or = kBT for each normal mode. This leads to the Raleigh-Jeans formula for black body radiation, N.휈/d휈 8휋휈2d휈 u.휈; T/d휈 = k T = k T V B 3 B c0 where u.휈; T/ is the energy per unit volume or the spectral energy density. Spectral radiance, B휈.휈; T/, is related to spectral energy density, u.휈; T/, by 2 c0 2휈 d휈 B휈.휈; T/ = u.휈; T/ = k T 4휋 2 B c0 P. J. Grandinetti Chapter 10: Wave-Particle Duality The ultraviolet catastrophe Plot of R&J expression with black body curve at T = 1500K 1.5 Raleigh-Jeans 1500 K 1.0 0.5 Experiment 0.0 0 1 2 3 4 5 R&J formula was an epic failure. Called the ultraviolet catastrophe as formula predicts cavity would emit ultraviolet and higher frequency light at room temperature. No one at the time, 1900, could find anything wrong with derivation. What prevents higher frequency standing waves from filling cavity? P. J. Grandinetti Chapter 10: Wave-Particle Duality Max Planck started thinking about problem in 1894. Max Planck (1858 -1947) P. J. Grandinetti Chapter 10: Wave-Particle Duality Planck black body distribution Planck thought something wrong with equipartition of energy theorem. Somehow average energy of each normal mode depended on its frequency. At low frequencies, where R&J formula worked 휖 ; lim = kBT 휈→0 but somehow at high frequencies lim 휖 = 0: 휈→∞ Planck could achieve these limits if he assumed that each electromagnetic oscillation produced could only have discrete energy values of 휖.휈/ = nh휈 where n = 0; 1; 2; § h was proportionality constant, now called planck constant, h = 6:62607004 × 10*34J⋅s P. J. Grandinetti Chapter 10: Wave-Particle Duality Planck black body distribution With this assumption Planck followed Boltzmann’s theory of statistical mechanics and got 휈 휖 휈 h . / = 휈 eh _kBT * 1 This expression had correct limiting behaviors in the low frequency limit with 0 1 휈 h휈 h휈 lim eh _kBT = lim 1 + + 5 ù 1 + 휈→ 휈→ 0 0 kBT kBT so that 휈 휖 h lim = ( ) = kBT 휈→0 휈 1 + h _kBT * 1 and in the high frequency limit with 휈 lim eh _kBT = ∞ 휈→∞ so that h휈 lim 휖 = = 0 휈→∞ ∞ * 1 P. J. Grandinetti Chapter 10: Wave-Particle Duality Planck black body distribution Back to Raleigh-Jeans’ expression, instead of N.휈/d휈 N.휈/d휈 ( ) u.휈; T/d휈 = 휖 = k T → Raleigh-Jeans V V B Planck came up with 0 1 휈 휈 휈 휈 휈 휈; 휈 N. /d 휖 N. /d h → u. T/d = = 휈 Max Planck V V eh _kBT * 1 And then using N.휈/d휈 8휋 = 휈2d휈 V 3 c0 Planck finally obtained 0 1 휋휈2 휈 휈; 8 h u. T/ = 휈 Planck black body distribution 3 eh _kBT * 1 c0 He published this expression in 1900 as it perfectly matched all observed curves for black body radiation. ...and so began the revolution of quantum physics. P. J. Grandinetti Chapter 10: Wave-Particle Duality The Photoelectric Effect P. J. Grandinetti Chapter 10: Wave-Particle Duality Photoelectric Effect In 1887 Heinrich Hertz discovered the photoelectric effect—electric discharge occurred more readily when UV light is shined on a metal electrode.