© 2007 HORIBA, Ltd. All rights reserved.
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:DiffractedReflectedRefractedAbsorbed and Reradiated
ReflectedRefracted
Absorbed and
Reradiated
Diffracted
Small particles require knowledge of optical properties:Real Index (degree of refraction)Imaginary Index (absorption of light within particle)Light must be collected over large range of anglesIndex 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 θd =nλ / sin θ
θ
n = order of diffractionλ = wavelength of lightd = distance between slitsθ = angle of diffraction
For a given measurement
system with a fixed wavelength, the
product nλ is constant
© 2007 HORIBA, Ltd. All rights reserved.
Fraunhofer Diffraction: PinholeThe diameter of the pinhole is INVERSELY proportional to angle (θ)
nλ=d sin θd =nλ / sin θ
θ
n = order of diffractionλ = wavelength of lightd = distance between slitsθ = angle of diffraction
For a given measurement
system with a fixed wavelength, the
product nλ is constant
© 2007 HORIBA, Ltd. All rights reserved.
FRAUNHOFER DIFFRACTION - PARTICLEAngle of scatter from a particle is INVERSELY proportional to angle (θ)
nλ=d sin θd =nλ / sin θ
n = order of diffractionλ = wavelength of lightd = distance between slitsθ = angle of diffraction
For a given measurement
system with a fixed wavelength, the
product nλ is constant
θ
© 2007 HORIBA, Ltd. All rights reserved.
Edge Scattering PhenomenonLight Scatter occurs whether from a
slit, a pinhole or a particle. It occurs at the edge of an object. A SLIT and
PARTICLE of the same size produce the same diffraction pattern
θ
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
θLarge ParticleSmall Particle
© 2007 HORIBA, Ltd. All rights reserved.
Diffraction Pattern: Large vs. Small Particles
LARGE PARTICLE:Low angle scatterLarge signal
SMALL PARTICLE:High Angle ScatterSmall Signal
Wide Pattern - Low intensity
Narrow Pattern - High intensity
© 2007 HORIBA, Ltd. All rights reserved.
Angle of Scatter vs. Particle SizeFraunhofer 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
3.24λ2A
1.22λ2A
2.23λ2A 3.24λ
2A
2.23λ2A
1.22λ2A
W
λ = Wavelength of lightA = Radius of ParticleW = Angle with respect
to optic axis
I (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 μmIf 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-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
0.01 0.1 1 10 100
Angle in degrees
Rel
ativ
e In
tens
ity
3.0 µm by Mie3.0 µm by Fraunhofer100.0 µm by Mie100.0 µm by Fraunhofer
100 μm
3 μm
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
0
0.5
1
1.5
2
2.5
3
3.5
4
0.1 1 10 100Particle Size μm
Extin
ctio
n Ef
ficie
ncy
1.46 - 0.0011.46 - 0.021.46 - 0.051.46 - 0.11.46 - 0.2
Legend shows Particle
refractive index
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.1 1 10 100
Particle Size μm
Extin
ctio
n Ef
ficie
ncy
1.39 - 0.0 1.59 - 0.0 1.79 - 0.0 1.99 - 0.0 2.19 - 0.0
Legend shows Particle
Refractive Index
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-041.0E-031.0E-021.0E-011.0E+001.0E+011.0E+021.0E+031.0E+041.0E+051.0E+061.0E+071.0E+081.0E+091.0E+101.0E+11
0.1 1 10 100 1000
Angle degrees
Rela
tive
Inte
nsity
0.10.52510100
Legend Particle Size μm
Mie calculation; wavelength 633 nm; Np = 1,59 – 0,0 i; nm = 1,33
For singleparticles range from0.1 to 100 μmis 1013
Reduce byweighting tovolume
© 2007 HORIBA, Ltd. All rights reserved.
Light intensity scattering patterns for equal particle volumes in relation to size
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
0.1 1 10 100 1000
Angle in degrees
Rel
ativ
e In
tens
ity
0.10.52510100
Legend ParticleSize μm
Mie calculation; wavelength 633 nm; Np = 1,59 – 0,0 ; nm = 1,33
Weightingto thirdpower ofsize
© 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.
IterationsHow many times to calculate new (better) results before reporting final result?For LA-930 standard = 30, sharp = 150For LA-950 standard = 15, sharp = 1000Fewer iterations = broader peakMore iterations = narrow peakToo 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
Small particles – wide anglesLarge particles – low angles
Look at Intensity curves, R parameter & Chi square calculationsUse Mie theory at all times (default whenever choosing an RI kernel other than Fraunhofer)