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Dr. Jeff Bodycomb of HORIBA Scientific discusses the principles which make particle size analysis by laser diffraction possible.

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2. 2. When a Light beam Strikes a Particle  Some of the light is: Reflected  Diffracted Refracted  Reflected  Refracted Absorbed and  Absorbed and Reradiated Reradiated Diffracted Small particles require knowledge of optical properties:  Real Refractive Index (bending of light)  Imaginary Refractive Index (absorption of light within particle)  Refractive index values less significant for large particles Light must be collected over large range of angles © 2012 HORIBA, Ltd. All rights reserved.
5. 5. Using Models to Interpret Scattering Scattering data typically cannot be inverted to find particle shape. We use optical models to interpret data and understand our experiments.© 2012 HORIBA, Ltd. All rights reserved.
6. 6. The Calculations There is no need to know all of the details. The LA-950 software handles all of the calculations with minimal intervention. This talk is to improve your understanding of the results.© 2012 HORIBA, Ltd. All rights reserved.
7. 7. Laser Diffraction Models Large particles -> Fraunhofer More straightforward math Large, opaque particles Use this to develop intuition All particle sizes -> Mie Messy calculations All particle sizes© 2012 HORIBA, Ltd. All rights reserved.
8. 8. Light Expressed in just in y-direction E  E0 sin( ky  t ) Oscillating electric field H  H 0 sin( ky  t ) Oscillating magnetic field (orthogonal to electric field) Cheat by using vector notation (for 3 dimensions) and real part of complex numbers (because it makes the trig easier). E  E 0 exp(ik  x  it ) H  H 0 exp(ik  x  it ) Complements of Lookang @ weelookang.blogspot.com© 2012 HORIBA, Ltd. All rights reserved.
9. 9. Light: Interference Look at just the electric field. E  E0 sin( kx  t ) Oscillating electric field E  E0 sin(kx  t   ) Second electric field with phase shift We will use interference to develop an intuitive understanding later.© 2012 HORIBA, Ltd. All rights reserved.
10. 10. Microscopic View of Scattering Anti‐Stokes scattering:  0+i Vibrational mode  frequency : i Incident light:  Rayleigh Scattering: 0 0 Stokes scattering: 0‐i Today we discuss the scattered photons that do not change wavelength. Also, we treat everything as a continuum. About 1 scattered photon in a million has a wavelength shift that depends on the molecule. These extra photons are the basis for Raman spectroscopy. Horiba makes Raman instruments too!© 2012 HORIBA, Ltd. All rights reserved.
12. 12. Fraunhofer Diffraction Path length difference is s sin(  k = 2/ s  r0 Detector (far away) Electric field at target from each position s: exp(ik (r0  s sin( ))  t )© 2012 HORIBA, Ltd. All rights reserved.
13. 13. Fraunhofer Diffraction We want to treat a hole because it is easy. We need a trick. R R hole -R = -R disc Babinet’s principle: Scattering from an aperture is the same as scattering from its complement© 2012 HORIBA, Ltd. All rights reserved.
14. 14. Fraunhofer Diffraction R Path length difference is s sin(  hole s  r0 -R Detector (far away) R E ( )  R  A exp(iks sin  )ds© 2012 HORIBA, Ltd. All rights reserved.
15. 15. Fraunhofer Diffraction Need to figure area of each strip of aperture to integrate. For a slit it is easy (and we usually ignore length): A  wds w s For a circle, it is a bit messy: w A  2 R 2  s 2 ds s© 2012 HORIBA, Ltd. All rights reserved.
16. 16. Fraunhofer Diffraction Path length difference is s sin(  s  r0 Detector (far away)Now integrate over s from –R to R R  R 2 R 2  s 2 exp(iks sin  )ds dimensionless size parameter  = D /  2 E0 2 J1 ( sin  ) E R r  sin © 2012 HORIBA, Ltd. All rights reserved.
17. 17. Fraunhofer Approximation dimensionless size parameter  = D/; 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  >>1)* If =650nm (.65 m), then 40 x .65 = 26 m If the particle size is larger than about 26 m, then the Fraunhofer approximation gives good results. Rephrased, results are insensitive to refractive index choice.© 2012 HORIBA, Ltd. All rights reserved.
19. 19. Diffraction Pattern: Large vs. Small Particles  LARGE PARTICLE:  Peaks at low angles  Strong signal Wide Pattern - Low intensity  SMALL PARTICLE:  Peaks at larger angles  Weak Signal Narrow Pattern - High intensity© 2012 HORIBA, Ltd. All rights reserved.
20. 20. Poll How many of you work with particles with sizes over 1 mm? How many of you work with particles with sizes over 25 microns and less than 1 mm? How many of you work with particles with sizes over 1 micron and less than 25 microns? How many of you work with particles with sizes less than 1 micron?© 2012 HORIBA, Ltd. All rights reserved.
22. 22. Maxwell’s Equations  Solve these and the universe’s secrets are yours.  Notation from Bohren and Huffman Absorption and Scattering of Light by Small Particles E  0 Net charge is zero H  0 No magnetic monopoles   E  i  H Time varying magnetic field gives electric field and vice   H  i  E versa. No current flows through system.© 2012 HORIBA, Ltd. All rights reserved.
23. 23. Boundary Conditions  Properties of particle different from surrounding medium  Tangential components of E and H are continuous at boundary n = 2-0.05i Goodn = 1 (for air) Bad © 2012 HORIBA, Ltd. All rights reserved.
24. 24. Overview of Solution  Sum of incoming plane wave and scattered waves.  We also get information on extinction and so on.© 2012 HORIBA, Ltd. All rights reserved.
25. 25. Mie Scattering I0  I s (m, x, )  2 2 S 2  S1 2k r 2 2  Use an existing computer program for the calculations! 2n  1  Pn1 (cos  ) S1 (m, x,  )   an n  bn n  n  1 n( n  1) sin   2n  1 S 2 (m, x,  )   an n  bn n  1 n(n  1) n  d 1 d  Pn (cos  )  an  m n (mx) n ( x)  n ( x) n (mx) : Ricatti-Bessel m n (mx) n ( x)   n ( x) n (mx) functions Pn1:1st order Legendre  n (mx) n ( x)  m n ( x) n (mx) Functions bn   n (mx) n ( x)  m n ( x) n (mx)© 2012 HORIBA, Ltd. All rights reserved.
26. 26. MieThe equations are messy, but require just three inputs which areshown below. The nature of the inputs is important. x  D Decreasing wavelength is the same as  increasing size. So, if you want to measure small particles, decrease wavelength so they “appear” bigger. That is, get a blue light source for small particles. We need to know relative np m refractive index. As this goes to 1 nm there is no scattering.  Scattering Angle© 2012 HORIBA, Ltd. All rights reserved.
27. 27. Effect of Size As diameter increases, intensity (per particle) increases and location of first peak shifts to smaller angle.© 2012 HORIBA, Ltd. All rights reserved.
28. 28. Effect of RI: imaginary term As imaginary term (absorption) increases location of first peak shifts to smaller angle.© 2012 HORIBA, Ltd. All rights reserved.
30. 30. Mixing Particles? Just AddThe result is the weighted sum of the scattering from each particle.Note how the first peak from the 2 micron particle is suppressed since itmatches the valley in the 1 micron particle.© 2012 HORIBA, Ltd. All rights reserved.
33. 33. Pop Quiz As particle size increases: Peaks shift to smaller angles Peaks shift to larger angles Nobody said there would be a quiz!© 2012 HORIBA, Ltd. All rights reserved.
34. 34. Pop Quiz As particle size increases: Peaks shift to smaller angles Peaks shift to larger angles Nobody said there would be a quiz! ½ credit© 2012 HORIBA, Ltd. All rights reserved.