Advanced Laser Diffraction Theory

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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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Advanced Laser Diffraction Theory

  1. 1. Laser Diffraction Theory Jeffrey Bodycomb, Ph.D. HORIBA Scientific www.horiba.com/us/particle© 2012 HORIBA, Ltd. All rights reserved.
  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.
  3. 3. Diffraction Pattern© 2012 HORIBA, Ltd. All rights reserved.
  4. 4. Diffraction Patterns© 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.
  11. 11. Fraunhofer Diffraction A good approximation to develop intuition© 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.
  18. 18. Fraunhofer: Effect of Particle Size© 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.
  21. 21. Mie Scattering An exact treatment for a spherical particle.© 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.
  29. 29. Effect of RI: Real Term It depends….© 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.
  31. 31. Effect of Distribution Width As distribution becomes wider, the peaks become less distinct.© 2012 HORIBA, Ltd. All rights reserved.
  32. 32. Iterations© 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.
  35. 35. Comparison of Models  Fraunhofer (left) vs. Mie (right)© 2012 HORIBA, Ltd. All rights reserved.
  36. 36. Comparison, Large ParticlesFor large particles, match is good out to through several peaks. © 2012 HORIBA, Ltd. All rights reserved.
  37. 37. Comparison, Small Particles For small particles, match is poor.© 2012 HORIBA, Ltd. All rights reserved.
  38. 38. Practical Application: Glass Beads© 2012 HORIBA, Ltd. All rights reserved.
  39. 39. Practical Application: CMP Slurry© 2012 HORIBA, Ltd. All rights reserved.
  40. 40. What about background (blank)? Even pure liquids scatter. Also, there is often some constant background signal due to stray light. And, the detectors will always show a background signal. These effects are small, but they should be corrected. The background is subtracted before applying any of analysis mentioned before. As the background becomes smaller, the analysis becomes easier.© 2012 HORIBA, Ltd. All rights reserved.
  41. 41. LA-950 Optics© 2012 HORIBA, Ltd. All rights reserved.
  42. 42. Short Wavelengths for Smallest ParticlesBy using blue light source, we double the scattering effect of the particle.This leads to more sensitivity. This plot also tells you that you need to havethe background stable to within 1% of the scattered signal to measure smallparticles accurately.© 2012 HORIBA, Ltd. All rights reserved.
  43. 43. Why 2 Wavelengths? 30, 40, 50, 70 nm latex standards Data from very small particles.© 2012 HORIBA, Ltd. All rights reserved.
  44. 44. Viewing Raw DataAdvanced Function ->Intensity Graph© 2012 HORIBA, Ltd. All rights reserved.
  45. 45. During Data CollectionChoose “Both Graphs”Are there peaks in the background beforeadding sample? You may already have particles andneed to clean.Is the peak position different? The average size changed.Is the peak wider? The distribution has become widerIs the background higher than yesterday? Then you need to clean.© 2012 HORIBA, Ltd. All rights reserved.
  46. 46. Conclusions  It helps to know some of the theory to best use a laser diffraction particle size analyzer  Small particles – wide angles  Large particles – low angles  Look at Intensity curves  Big peaks in your background (blank) mean particles or bubbles  Use Mie theory at all times (default whenever choosing an RI kernel other than Fraunhofer)© 2012 HORIBA, Ltd. All rights reserved.
  47. 47. Q&A Ask a question at labinfo@horiba.com Keep reading the monthly HORIBA Particle e-mail newsletter! Visit the Download Center to find the video and slides from this webinar. Jeff Bodycomb, Ph.D. P: 866-562-4698 E: jeff.bodycomb@horiba.com© 2012 HORIBA, Ltd. All rights reserved.

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