The measurement principle of laser particle size
The measurement principle of laser particle size
(Dandong Bettersize Instrument Ltd. Wangyongquan)
Introduction
At present, laser diffraction particle size tester has been applied widely, especially in
abroad, it has been recognized conformably. The remarkable features are: high measurement
precision, fast response speed, good repetitiveness, wide measurable particle diameter range and
touchless measurement etc.
The research and production of the kind of instruments in China is comparatively short.
The need of the instruments is at least 100 every year in home market, but the lowest price in
foreign countries is 50,000 $ every machine, so our country pays foreign exchange 5000,000 $ in
purchasing the kind of instruments at least every year. In latest several years, we have developed
multifold model laser particle size tester successfully, the main performance is alike with foreign
same products.
The measurement principle of laser diffraction particle size tester
The operational principle of laser diffraction particle size tester we studied bases on
Fraunhofer diffraction and Mie scattering theory. We know from physics optics deduction that the
scattering of incidence light vs particles accord with classical Mie theory. Mie scattering theory is
rigorous mathematic root of Maxwell electromagnetic wave equation group, while Fraunhofer
diffraction is only a kind of approximation of Mie scattering theory. Fraunhofer diffraction is
applicable to the situation that particle diameter is far more than the incidence wave length, and is
assumed that the light source and receiving screen are all boundlessly far away the diffracting
screen. Considering from the theory, Fraunhofer diffraction is relatively simpler in application.
The basic device of laser diffraction particle size tester sees attached drawing 1. Low
energy laser sends monochromic light of wavelength 0.635 um, and the light passes through space
filtering and diffusing beam lens to filter miscellaneous light and form Max diameter 10mm
parallel monochromic light beam. The light beam irradiates the particles in measurement area, and
occurs light diffractive phenomenon. The intensity distribution of diffracted light follows to
Fraunhofer diffraction theory. Fourier conversion lens at the back of measurement area is
receiving lens (knowing lens range), scattered light forms far magnetic field diffractive graph on
back focus surface. Multi-ring photoelectric detector on back focus surface of receiving lens can
receive the energy of diffracted light and translate it into electric signal and output. The center
hole of the detector measures the consistency of allowable sample volume. The diffractive graph
of the particles is still and centralizes on light shaft range of the lens. So it does not matter that the
particles are dynamic to pass through analyzing light beam, the diffractive graph is a constant to
any lens distance. The lens conversion is optics, so it is very fast.
According to Fraunhofer diffraction theory, when a spherical particle of diameter d is
within measurement area, its light intensity distribution of any angle is:
I    I 0
 2d 4
16 f 2 2
 J 1  X 
2 X  1


2
In the equation:
f ：the focal length of receiving lens
λ：the wavelength of incidence light
Dandong Bettersize Instruments Ltd.
The measurement principle of laser particle size
J1 ：first order Bessel function
θ：scattering angle
X   d sin  
When the diffracted light intensity distribution of the laser lays upon the No n ring of
photoelectric detector (ring radius is from Sn to Sn+1, corresponding scattering angle is fromθn
toθn+1), the light energy is:
en  
S n 1
Sn
I   2 S dS
n  1,2,3
 2
The equation (1) is substitute into I（θ）, then we get:
en 
d2
4


I 0 J 02  X n   J 12  X n   J 02  X n 1   J 12  X n 1  3
J0 ：zero order Bessel function
If there are N quantity particles of diameter d, the received light energy on No n light
ring is N times（N·en）more than that of one particle. On the analogy of this, if there are Ni
quantity particles of diameter di in the particles, total diffracted light energy in the particles is the
sum of all particles diffracted light energy, that is,
en 
 I0
4
 N d J X   J X   J X
i
2
i
2
0
i ,n
2
1
i ,n
2
0
i , n 1
  J X
2
1
i , n 1
4
If we use W to represent dimension distribution, the relation between W and N is:
Ni 
6Wi
5
d i3
In the equation：ρ is particle density，above equation is substitute into equation（4）,we get：
en 
3I 0
2
W
 d J X   J X   J X
i
2
0
i ,n
2
1
i ,n
2
0
i , n 1
  J X
2
1
i , n 1
6
i
The equation (6) set up the corresponding relation between every ring diffracted light signal on
photoelectric detector and particle diameter and distribution of measured particles.
In particle calculating, there are 96 effective rings on photoelectric detector we use, so
we divide the diameter into 96 sections, the geometric shape of photoelectric detector see attached
drawing 2, radius data of every ring is as follows (unit: mm):
S1  0.079536
S i  1.0977  S i 1
i  2 to 97
Above formula shows inner radius is Sn and outer radius is Sn+1 of No n ring.
Choose particle diameter section according to the following formula calculating:
in the formula：
 Di S i
 1.37
f
i  1 to 97
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The measurement principle of laser particle size
f ：receiving lens of focal length 180mm
λ：semiconductor laser of wavelength 0.635
Above formula shows the section upper limit is Dn and section lower limit is Dn+1 of No n
particle grade in 96 particles grade.
The geometric mean value can be chosen as the representative value of particle diameter in every
particle grade:
d i  Di  Di 1
i  1 to 96
We can work out coefficient matrix by formula (6), once we mensurate light energy
distribution E on 96 effective light rings, work out system of linear equations (6), we can get
weight distribution W of particle dimension. To be convenient, we use least square method to
process data. We assume that weight distribution W accords with some distribution rule (called
distribution function restrictive method), or arbitrary initial value (called free distribution method),
and calculate diffracted light energy of 96 rings on photoelectric detector, compare with real value
one by one, until the error between two values is the least.
The following is the discussion on the solutions of free distribution method and several
distribution function restrictive method, and assuming section weights of 96 particles grade are
W1、W2、W3…W95、W96, light intensity values of all rings are E1、E2、E3…E95、E96.
Free distribution method:
First step: assuming initial value of every particle grade section wight Wi is 1, institute
into formula (6), working out light intensity values of all rings e1、e2、e3…e95、e96, then we
calculate light intensity variance by formula (7):
96
 2   ei  Ei 
2
7 
i 1
The variance is in variable χ, calculating proportionality coefficient between measured
value and calculated value of every ring light intensity according to formula (8):
 i  Ei e
8
i
Update weight values Wi of all particle grade sections according to formula (9):
Wi   i  Wi
9
Second step: updated weight values Wi of all particle grade sections are institute into
formula (6), working out diffracted light intensity of all rings e1、e2、e3…e95、e96, and calculating
light intensity variance by formula (7), comparing this variance with last variance, if σ2 is
greater thanχ, turn to third step; or:
Dandong Bettersize Instruments Ltd.
The measurement principle of laser particle size
Update χ value, calculate proportionality coefficient between measured value and
calculated value of every ring light intensity according to formula (8), update weight values Wi of
all particle grade sections according to formula (9).
Repeat second step.
Third step: the value Wi is our want final all particle grade sections weight. The section
percentage can be calculated by formula (10):
Wi
Wi
fi 
10
The percentage greater than some a particle diameter (screen) is calculated by formula
(11):
R1  f1 Ri  Ri 1  f i
i  2 to 96
11
The normal distribution of distribution function restrictive method: the formula is
f x  
 x  u  2 
d
1

exp 

2
dx  2
 2

 12
in the formula：

 f x dx  1

assume：
 xu
t 

  
then：
 dt  dx
the formula (12) turn into：
 t2 
d
1

exp   
dt
2
 2
so：

 d 
0
x u

 t2 
exp
   2 dt
2 
1
13
Dandong Bettersize Instruments Ltd.
The measurement principle of laser particle size
The formula (13) is a standard normal distribution function; there are integral tables in various
statistics books.
The corresponding points of particle diameters and accumulative percentages on normal
probability coordinate paper should appear like beeline. We can use least square method to fit
percentage, work out Ri and corresponding tI in turn by interpolation in integral table, then solve
coefficients σand u by the following formula.
96

 96  96 
96
x
t

  xi   t i 


i i
i

1
 i 1  i 1 
  
2

 96 2   96 
96  t i     t i 


 i 1   i 1 

96

 96   96  96


  xi   t i2     t i   xi t i 

 i 1  i 1   i 1  i 1

u 
2
96
96




2
96
t

t






i
i

 i 1   i 1 

 14 
Convert coefficients σand u into t value, then work out R by interpolation in integral table.
Rosin-Rammler distribution of distribution function restrictive method:
Assume the rest percentage on screen of hole diameter x is R, Rosin-Rammle educed
from probability theory:

R  100 exp  bx n

After simplification：
 100 
ln  ln
  ln b  n ln x
R 

 15
The above formula is a linear equation; we can count two coefficients referring to formula (14),
and return to get accumulative percentage.
Know about particles
Particle size distribution
To understand the meanings on output result of laser diffraction particle size tester, we
need to explain some basic concepts.
First: The result is basis on volume. For example, result shows the distribution is 11%
within the range 6.97-7.75μm. That means total volume of all particles in this range is 11% of
total volume of all particles in whole distribution. Simply, we assume the sample has two kinds of
particles, their diameters are 1μm and 10μm, every kind is 50%, that is to say, the volume of a
large particle is more 1000 times than that of small, so the volume of large particles is 99.9% of
total volume. Certainly, for a kind of particle size distribution, the diameters of all particles are
same, whether quantity or volume, the distribution is 100%.
Dandong Bettersize Instruments Ltd.
The measurement principle of laser particle size
Second: The result is indicated by equivalent sphere. Assuming the diameter of a
columnar particle is20μm, height is 60μm, the volume is:
V   10  60
2
Convert it into a spherical volume, the diameter is between 20μm and 60μm:
3
6V

 33
Third: The derivation of distribution function. The distribution we analyze is particle
size group indication of a set optical system with best resolution. All distribution parameters are
educed from the basic distribution. In calculating, representative diameter of every particle size
section means geometric mean of two-end values of this particle size section, there is a little
different with arithmetic average.
Data, sample and background
The measuring output is the array of numerical value, including the test results of background and
samples. If we want to get the practical scattering of sample particle its self, we must take out the
background measurement from sample measurement, at the same time, must correct. After
correction, background is:
D j  S j  1  Ob B j
In the formula：
D is the passage No of data j
S is the sample measurement
B is the background measurement
Ob is opacity – definition is：
Ob  1 
Ls
Lb
Ls is measured light intensity by center detector when a sample is put in the sample pool.
Lb is measured light intensity with only pure dispersing medium, but without sample.
Analyzing, deviation and data fit
Analyzing is that the measured data and other experiment parameter are processed with restrictive
square method of iteration course and educed output.
Estimate an initial particle size distribution, and input a scattering matrix to optical model selected
to predict forthcoming light scattering. It is showed in mathematics:
If vector Lj indicates measured data, vector Ri indicates result, and then they can connect by the
following equation:
Lj=AijRi
Aij is scattering matrix included in optics correction, can be calculated according to scattering
theory accurately.
Compare the calculated value with real value, and use a set of designed correct project to modify
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The measurement principle of laser particle size
and adjust, repeat this process, until the calculated value and measured value tally with acceptable
extent, at the time, output the particle size distribution as result.
Comparing the least square error of the calculated value with that of measured value, It is showed
in mathematics:
 L
RESID  100
j
j
 Aij Ri'
L

2
j
j
R' is the result of present analyzing stage, with analyse going on, the present residual error is
decreasing with the reducing of fit, the calculated value AijRi' is more closer to measured value
Lj.
Optical Model
Laser diffraction particle size tester measures that the range of scattering angle is 0.01-15°
generally and the range of particle diameter is over 1μm.
It is known that for such particle diameter and scattering angle, scattering character has no relation
with the internal optical property of sample. Such instruments use usually Fraunhofer scattering
theory, because it doesn’t need any assumption to particle optical property.
Actually, when the particle diameter begins to reduce to below 10μm and particles dip in liquid,
or are transparency in optics, Fraunhofer scattering model occurs errors generally.
In order to measure the particle diameter of least 0.05μm, the measure range of the instrument is
broadened up to 135°. The scattering of small particles has close relation with the optical
property of scattering material under so big angle, and cannot be omitted. Mie Theory on
scattering meets the case, which describes the scattering of optical uniform sphere and makes
some assumptions to optical property of the particle. Mie Theory includes Fraunhofer scattering
and anomalous scattering model entirely and they are conformable in proper range.
Choose the appropriate optical model for experiment from the instrument, and set it as “standard
value”. Because of adopting the optical model, you needn’t think of optical property of the sample
when you use the instrument. However, to ensure high precise sample, the optical model is
considered probably.
Derivative diameter and distribution statistics
The analysis outcome is the relative volume distribution of the particle among a series of particle
size groups, and the distribution D[m,n] statistics is calculated on basis of the outcome, which is
the recognized method by international. The particle outcome within two boundaries of particle
size groups can be worked out by means of inner interpolation.
The definition of derivative diameter:
1
  Vi d im 3  m  n
Dm, n  
n 3 
  Vi d i 
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The measurement principle of laser particle size
In the formula:
Vi are the relative volume among the groups, average diameter of the group is di. m and n is
integer，indicating the type of derivative diameter. D[4，3]—volume weighted average diameter，
D[3，2]—surface weighted average diameter，D[1，0]—arithmetic average diameter.
Distribution statistics
Average diameter
d
d V
V
i
i
i
Standard deviation
V d  d 
V
2

i
i
i
Skew
V d  d 
Skew 
 V
3
i
i
3
i
Kurt
V d  d 
Kurt 
 V
4
i
i
4
3
i
Specific surface area
weight of the particle.
Specific surface area（SSA）is the ratio of total area and gross
Vi
di
SSA 
  Vi
6
Span and Uniformity
The definition of distribution span：
Span 
d v,0.9  d v,0.1
d v,0.5
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The measurement principle of laser particle size
Span is the distribution width and has no relation with middle particle diameter; uniformity is
distribution pattern, also relation with middle particle diameter, shows the extent of distribution
deviating from the middle. The definition is:
Uniformity 
V d v,0.5  d
d v,0.5V
i
i
i
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