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Light Scattering Theory LIGHT SCATTERING THEORY Laser Diffraction (Static Light Scattering) © 2007 HORIBA, Ltd. All rights reserved. When a Light beam Strikes a Particle Some of the light is: z Diffracted Reflected Refracted z Reflected z Refracted Absorbed and z Absorbed and Reradiated Reradiated Diffracted Small particles require knowledge of optical properties: z Real Index (degree of refraction) z Imaginary Index (absorption of light within particle) z Light must be collected over large range of angles z Index values less significant for large particles © 2007 HORIBA, Ltd. All rights reserved. Diffraction Pattern © 2007 HORIBA, Ltd. All rights reserved. Diffraction Patterns © 2007 HORIBA, Ltd. All rights reserved. Young’s Experiment © 2007 HORIBA, Ltd. All rights reserved. Single vs. Double Slit Patterns Single slit diffraction pattern Double slit diffraction pattern © 2007 HORIBA, Ltd. All rights reserved. Fraunhofer Diffraction: Young’s Double Slit Distance between slits is INVERSELY proportional to angle (θ) nλ =d sin θ d=nλ / sin θ For a given measurement system with a fixed θ wavelength, the product nλ is constant n= order of diffraction λ = wavelength of light d= distance between slits θ = angle of diffraction © 2007 HORIBA, Ltd. All rights reserved. Fraunhofer Diffraction: Pinhole The diameter of the pinhole is INVERSELY proportional to angle (θ) nλ =d sin θ d=nλ / sin θ For a given measurement system with a fixed θ wavelength, the product nλ is constant n= order of diffraction λ = wavelength of light d= distance between slits θ = angle of diffraction © 2007 HORIBA, Ltd. All rights reserved. FRAUNHOFER DIFFRACTION - PARTICLE Angle of scatter from a particle is INVERSELY proportional to angle (θ) nλ =d sin θ d=nλ / sin θ For a given measurement system with a fixed θ wavelength, the product nλ is constant n = order of diffraction λ = wavelength of light d = distance between slits θ = angle of diffraction © 2007 HORIBA, Ltd. All rights reserved. Edge Scattering Phenomenon Light Scatter occurs whether from a slit, a pinhole or a particle. It occurs at the edge of an object. A SLIT and θ PARTICLE of the same size produce the same diffraction pattern θ θ © 2007 HORIBA, Ltd. All rights reserved. Fraunhofer Diffraction: Particles Large particles scatter light through SMALLER angles θ Small particles scatter light through LARGER angles Large Particle θ Small Particle © 2007 HORIBA, Ltd. All rights reserved. Diffraction Pattern: Large vs. Small Particles LARGE PARTICLE: z Low angle scatter z Large signal Wide Pattern - Low intensity SMALL PARTICLE: z High Angle Scatter z Small Signal Narrow Pattern - High intensity © 2007 HORIBA, Ltd. All rights reserved. Angle of Scatter vs. Particle Size Fraunhofer Diffraction describes forward scatter technology. Scatter angles are relatively small, less than 30° ANGLE OF SCATTER is INVERSELY proportional to particle size. SMALL particles scatter at larger angles than large particles. The AMOUNT OF LIGHT scattered is DIRECTLY proportional to particle size. LARGE particles scatter MORE light than small particles. © 2007 HORIBA, Ltd. All rights reserved. Light Scattering S1(Θ) and S2(Θ) are dimensionless, complex functions defined in general scattering theory, describing the change of amplitude in respectively the perpendicular and the parallel polarized light as a function of angle Θ with respect to the forward direction. Computer algorithms have been developed in order to allow computation of these functions and, thus, of I(Θ). © 2007 HORIBA, Ltd. All rights reserved. Plot of Airy Function I (W) λ = Wavelength of light A = Radius of Particle W = Angle with respect to optic axis 1.22λ 1.22λ 2.23λ 2.23λ 2A 2A 3.24λ 2A 2A 3.24λ 2A 2A W © 2007 HORIBA, Ltd. All rights reserved. Comparison of Models Fraunhofer (left) vs. Mie (right) © 2007 HORIBA, Ltd. All rights reserved. Fraunhofer Approximation dimensionless size parameter α = πx/λ; J1 is the Bessel function of the first kind of order unity. Assumptions: a) all particles are much larger than the light wavelength (only scattering at the contour of the particle is considered; this also means that the same scattering pattern is obtained as for thin two-dimensional circular disks) b) only scattering in the near-forward direction is considered (Q is small). Limitation: (diameter at least about 40 times the wavelength of the light, or a >>1)* If λ=680nm (.68 μm), then 40 x .68 = 27 μm If the particle size is larger than about 50 μm, then the Fraunhofer approximation gives good results. © 2007 HORIBA, Ltd. All rights reserved. Fraunhofer Approximation © 2007 HORIBA, Ltd. All rights reserved. Mie Theory © 2007 HORIBA, Ltd. All rights reserved. Mie vs. Fraunhofer 1.E+05 100 μm 1.E+04 1.E+03 3 μm 1.E+02 1.E+01 1.E+00 1.E-01 Relative Intensity 3.0 µm by Mie 1.E-02 3.0 µm by Fraunhofer 100.0 µm by Mie 1.E-03 100.0 µm by Fraunhofer 1.E-04 0.01 0.1 1 10 100 Angle in degrees Figure A.3 -- Comparison of scattering patterns of non-absorbing particles according to Fraunhofer and Mie calculations (Np = 1,59 – 0,0; nwater = 1,33; wavelength = 633 nm) © 2007 HORIBA, Ltd. All rights reserved. Extinction Efficiencies 4 5 Legend shows Legend shows 4.5 3.5 Particle Particle refractive index 4 Refractive Index 3 3.5 2.5 3 Efficiency 2 2.5 2 1.5 1.46 - 0.001 1.39 - 0.0 Extinction 1.5 Extinction Efficiency Extinction 1 1.46 - 0.02 1.59 - 0.0 1.46 - 0.05 1 1.79 - 0.0 0.5 1.46 - 0.1 0.5 1.99 - 0.0 1.46 - 0.2 2.19 - 0.0 0 0 0.1 1 10 100 0.1 1 10 100 Particle Size μm Particle Size μm Figure A.2 -- Extinction efficiencies in relation to particle size and refractive index (Mie prediction). (np and kp as indicated; nwater = 1,33; wavelength = 633 nm) (Fraunhofer assumes an extinction efficiency of 2 for all particle sizes) Good agreement for transparent particles >50 μm “ “ “ opaque particles > 2 μm © 2007 HORIBA, Ltd. All rights reserved. Scattering intensity pattern for single particles in relation to size 1.0E+11 For single 1.0E+10 Legend Particle 0.1 particles 1.0E+09 Size μm 1.0E+08 0.5 range from 1.0E+07 2 0.1 to 100 μm 1.0E+06 5 is 1013 1.0E+05 10 1.0E+04 100 Reduce by 1.0E+03 weighting to 1.0E+02 volume Relative Intensity 1.0E+01 1.0E+00 1.0E-01 1.0E-02 1.0E-03 1.0E-04 0.1 1 10 100 1000 Angle degrees Mie calculation; wavelength 633 nm; Np = 1,59 – 0,0 i; nm = 1,33 © 2007 HORIBA, Ltd. All rights reserved. Light intensity scattering patterns for equal particle volumes in relation to size 1.E+05 Weighting 1.E+04 Legend Particle 0.1 to third Size μm 0.5 power of 1.E+03 2 size 1.E+02 5 10 1.E+01 100 1.E+00 Relative Intensity 1.E-01 1.E-02 1.E-03 0.1 1 10 100 1000 Angle in degrees Mie calculation; wavelength 633 nm; Np = 1,59 – 0,0 ; nm = 1,33 © 2007 HORIBA, Ltd. All rights reserved. Influence of particle size on angular light intensity scattering patterns Width = 1% (COV) © 2007 HORIBA, Ltd. All rights reserved. Influence of imaginary parts of RI (absorbancies) Width = 1% (COV) © 2007 HORIBA, Ltd. All rights reserved. Influence of distribution width on angular light intensity scattering patterns © 2007 HORIBA, Ltd. All rights reserved. Result Calculation © 2007 HORIBA, Ltd. All rights reserved. Iterations © 2007 HORIBA, Ltd. All rights reserved. Iterations How many times to calculate new (better) results before reporting final result? For LA-930 standard = 30, sharp = 150 For LA-950 standard = 15, sharp = 1000 Fewer iterations = broader peak More iterations = narrow peak Too many iterations = fit to noise, over-resolved © 2007 HORIBA, Ltd. All rights reserved. Viewing Raw Data © 2007 HORIBA, Ltd. All rights reserved. LA-950 Optics © 2007 HORIBA, Ltd. All rights reserved. Why 2 Wavelengths? 30, 40, 50, 70 nm latex standards © 2007 HORIBA, Ltd. All rights reserved. Practical Application of Theory: Mie vs. Fraunhofer for Glass Beads © 2007 HORIBA, Ltd. All rights reserved. Practical Application of Theory: Mie vs. Fraunhofer for CMP Slurry © 2007 HORIBA, Ltd. All rights reserved. Conclusions It helps to know some of the theory to best use a laser diffraction particle size analyzer z Small particles – wide angles z Large particles – low angles Look at Intensity curves, R parameter & Chi square calculations Use Mie theory at all times (default whenever choosing an RI kernel other than Fraunhofer) © 2007 HORIBA, Ltd. All rights reserved..
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