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Some of this weeks seminars: Dynamical Studies of the Photodissociation of Ozone: From the Near IR to the VUV February 12 | 4-5 p.m. | Pitzer Auditorium, 120 Latimer Hall Dr. Reinhard Schinke, Max-Planck-Institut fuer Dynamik und Selbstorganisation, Goettingen, Germany Engineering Organic-to-Semiconductor Heterojunctions February 13 | 4-5 p.m. | Pitzer Auditorium, 120 Latimer Hall Professor Thomas F. Kuech, Dept. of Chemical & Biological Engineering, University of Wisconsin - Madison Using Supported Lipid Bilayers as a Separation Matrix February 15 | 4-5 p.m. | 775A Tan Hall Professor Paul Cremer, Dept. of Chemistry, Texas A & M University Absorption/Emission Connections between the rates of stimulated and spontaneous emission: Case a) Thermal equilibrium in a cavity E2,N2,g2 A21 E1,N1,g1 B12W(w) B21W(w) W(w), the energy density and A,B the Einstein A and B coefficients which are the rate constants (per molecule), excepting the energy density for the transition probability, Wif. Also N large so we need not consider statistics. At equilibrium dN1/dt = -dN2/dt = 0 = N2A21-N1B12W(w)+N2B21W(w) Connect A and B to Golden Rule 2 1 Wif if fi 2 6 0 N1Wif N1 Bif fi Bif A fi if 2 6 0 2 gi gf 8h 3 if c3 B 2 A fi if gi 8h 3 3 g f 6 2 c 0 rate / molecule A21 W (w ) N1 B B21 N 2 12 E1 / kT g e N1 1 w / kT g / g e 1 2 N2 E2 / kT g 2e W w A21 /g1 / g 2 e w / kT B12 B21 w 3 dw Planck' s law W w dw 2 3 w / kT c e 1 The two expressions are equal at all T only if: g1 g2 B12 B21 B w 3 2c3 21 A21 Comparing again to Planck’s Law A21 / B21W w e T=300K w / kT 1 w / kT 1 at l=50m, 6THz, 200 cm-1 Longer wavelengths stimulated exceeds spontaneous rate Shorter stimulated emission is slow compared to spontaneous rate Case b) A light source, Now W is not thermal energy density but the energy density of the light source (assumed to be large enough that we can neglect the thermal field). At BW / A 1 the stimulated and spontaneous rates are equal. Consider visible light of frequency 5x1014 Hz, 3x10-19J w 3 W dw A / B dw 2 3 dw 1014 J / m3 c Intensity obtained by ,multiplying by c is 3x10-6 dw W/m2 dw for an ordinary spectroscopic light source is ~1011 Hz The intensity required to equalize spontaneous and stimulated emission rates is ~105W/m2 Some light sources Strong Hg Lamp I (W/m2) 104 E(V/m) n/V(m-3) 103 1014 Photons/mode 10-2 cw laser 105 104 1015 1010 pulsed laser (ns) 1013 108 1023 1018 Footnote: Derivation of relations between A, B, assumed thermal radiation. The relations hold so long as either the radiation field or the molecules are randomly oriented in space—not necessarily for solids interacting with lasers. E2,N2,g2 N2+N1=N A21 B12W(w) B21W(w) absorption negligible E1,N1,g1 dN1/dt = -dN2/dt = N2A21+ (N2-N1) B12W(w) NBW A 2 BW t N 2 t 1 e A 2BW For short time s : A 2BW t 1 Expanding the exponential as 1-(A+2BW)t N 2 t NBW t For long times : A 2 BW t 1 NBW N 2 t A 2 BW N2/N Why this behavior? At short time s N 2 t NBW t At long times NBW N 2 t A 2 BW time N2/N Steady state value What is the limiting ratio? 1 BW/A 4 How does this connect to Beer’s Law? Recall I=I0e-Nsl=I02.30310-Cl dI I 0 N 2 N1 BW / N I 0 Ns I 0 2.303C dx Ac 2 g c 2 g BW / N s 2 3 2 3 8n h 8n h rad g() a normalized lineshape function n index of refraction The square of the transition dipole can be expressed as an integral (over all frequencies) of the absorption crosssection (or molar absorptivity). What happens when we increase [ ] vs. I? (Discussion) Breakdown of Beer’s Law. BW/A>>1 Intensity falls off linearly with distance in absorber not exponentially and is independent of I0 I I 0 NA Oscillator Strength (CH4.4.3) f if 2me if wif 2 3e f if 1 f fif=1; 1 electron allowed transition >1 multiple transitions =0.001-0.01; forbidden transitions