Light Scattering Theory

1. Book 5: p.40-81 Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (Plenum Press, 1985) Edited by Robert Pecora Light Scattering Theory Mar. 27, 2007 0

2. LARSON’S BOOK FIG.1. An incoming beam of radiation with wavevector ki is deflected by an angle θ, so that its outgoing wavevector is ks The difference in the phase of the two beams is: Inverse relationship between θ and the structural length scale producing the scattering 1

3. Signal-To-Noise Ratio: Effects Due To Finite Intensity A1 2r • Scattering Geometry: Scattering volume VS: A cylinder of radius r and length 2r/sinθ Lens Aperture Aperture FIG.1. Scattering geometry 2r 2r/sinθ θ愈小則散射體積VS愈大 VS VS Incident beam Incident beam 2r 2r Scattered light Scattered light A1 2r 2

4. Calculation of the Power Incident on the Detector: Lens Aperture Aperture FIG. 1. Scattering geometry Point 3 Should be EMPHASIZED! The Power Pd Incident on the Detector: A 50-mW laser focused into a cell with a 10-cm focal length lens is no more effective than a 5-mW laser focused with a 1-cm focal length lens The cost of the second option is a small fraction of the cost of the first • The noise may be reduced by: • Increasing the incident power P0 or solute concentration Cn • (2) Scattering at a smaller angleθ • (3) Focusing the laser beam to a small diameter r 3

5. Data Analysis • Selecting the Theoretical Form: TABLE I: Theoretical Forms of Correlation Functions Corresponding to Various Physical Processes E.g., Electrophoretic light scattering 4

6. Use of the χ2 Test: If the selected form Gtheo(τ) is where A, B, and C are fitting parameters These parameters are determined by minimizing χ2defined as A histogram showing the number of times each values of χ2 is obtained in a REAL experiment should reproduce the curve shown in Fig. 2 FIG. 2.P(χ2)dχ2 gives the probability that χ2 lies between χ2 and χ2+dχ2 if the form of the equation chosen is correct 5

7. Fabry-Perot Interferometer（法布里-珀羅干涉儀） • Direct Spectral Measurement with a Fabry-Perot Interferometer: • The F-B interferometer is preferred under one or both of two conditions, • namely, • The time scale of interest becomes too short to be measured by a fast • CORRELATOR • An impractically long signal averaging time is required FIG. 3. Fabry-Perot interferometer The condition for survival in the cavity of a wave of wavelength λthat an integral number of half wavelengths must fit in the cavity, that is, By varyingd one can detect the spectral density of the scattered light as a function of frequency 6

8. FIG. 3. Fabry-Perot interferometer EXAMPLE: If d is 0.05 cm and λ=500 nm, then m=2,000 Free spectral range δυ(FSR): Finesse F: <40in practice FIG. 4. Response of a Fabry-Perot interferometer to light of different frequencies 7

9. Comparison between the Instrumental Width of a Fabry-Perot Interferometer Δt And the Minimum Sample Time of a Correlator Δt Q:Under what situation would the use of a Fabry-Perot interferometer with Δf=107Hzbe more appropriate than a correlator with ΔT=100ns? A: For a frequency f>>107 Hz one would use the interferometer. On the contrary, if f<<1.6 x 106 Hz, using a correlator with ΔT=100nsis appropriate Explanation: We may say approximately that if Δf=1/(2π ΔT)the two instruments are of equal utility. The estimated values (greencolor) for each instrument are included in Table II TABLE II Correlator Fabry-Perot interferometer ~1.6 x 106 107 Δf (Hz) ~16 100 ΔT (ns) 8

10. Dynamic Depolarized Light Scattering • The depolarized scattered light is a rich source of dynamic and structural information which is often not readily obtainable by other techniques • Depolarized signals are often much less intense than polarized signals and are thus obscured by (1) interfering signals due to optical imperfections in lenses and cells (2) Multiple scattering (3) Solvent scattering In addition, the depolarized time correlation functions often decay on a fast time scale (~μs) • Common combinations of μand φ are as follows: (1) Polarized scattering (IVV) (2) Depolarized scattering (IVH) (3) Horizontal-Horizontal scattering (IHH) z 雷射 散射體積 y x 偵檢器 FIG. 5. Scattering geometry 9

11. Physical Principles: Q: What reason motivates us to employ the depolarized light scattering experiment? A: The correlation function of IVHusually depends solely on the optical anisotropy of the scattering centers (particles, molecules, or parts of molecules) in the scattering medium and NOT on the AVERAGE molecular polarizability Position of the center of mass of element yx component of the polarizability tensor Assumptions Fourier Transform 10