PHYSICS AND CHEMISTRY
OF
ATMOSPHERIC AEROSOLS
A COURSE GIVEN AT THE DEPARTMENT OF METEOROLOGY,
UNIVERSITY OF STOCKHOLM, FALL SEMESTER, 1976.
BY
RUDOLF B HUSAR
1
1. AEROSOL PHYSICAL PROPERTIES
Aerosols.
The term aerosol refers to liquid or solid particles suspended in the air. Aerosol
particles may be liquid droplets, aggregates of odd shape, or single crystals of regular,
say cubical, or irregular shape. Their chemical composition may also vary from dilute
water solution of acids or salts, organic liquids, to earth's crust materials (sand) and toxic
heavy metals (Pb, Hg, As).
Depending on their origin and visual appearance aerosols have acquired different
names in the everyday language. Dust generally refers to solid airborne material,
dispersed into aerosol from grainy powders. Fumes are produced by various industrial
processes such as in a foundary, welding etc.. Combustion processes produce smoke
particles, but the incombustible residue of coal is also called flyash. Mist is formed when
a vapor condenses to form fine liquid droplets, which may then grow to larger fog
droplets. In the early days, air pollution had the appearance of both smoke and fog, so it
was appealing to create a new word for it: smog. In the open atmosphere, the visibility
may often be reduced by haze, originating from natural or anthropogenic sources.
It is obvious that in the context of physical and chemical treatment of atmospheric
aerosol behavior, it is advantageous to abandon the above set of rather poorly defined
terms. Instead, the generic term aerosol is used to express all types of airborne matter in
the 10  to 100 m size range. If further classification is needed, then it is beneficial to
perform the subdivision based on physical or chemical grounds. In case of atmospheric
aerosols it is useful to divide the aerosol population into fine particles, having a diameter
less than about 2 m and course particles with size  2 m. The two aerosol populations
have generally distinctly different physical characteristics (shape, volatility) and chemical
composition. Furthermore, the two populations have different sources and more
important - different effects. A scientific rationale for the separate consideration of fine
and coarse particles is given later.
2
Table 1. Characteristics of particles as a function of their size.
3
Particle size.
The size of atmospheric aerosols ranges over five orders of magnitude. The smallest
observed particles are of the order of 10  (1  = 10-8 cm) in diameter and they could well
be called molecular clusters. In Table 1, the typical size of diatomic molecules is
indicated to be between 3 and 6 . On the other extreme, large dust particles of 100 m
(1 m = 10-4 cm) may be kept suspended in the air for extended period of time. Within
the five decade ranges, there are "windows" in which the particles are comparable in size
to the mean free path of atmospheric gases (0.06 m), wavelength of the visible solar
radiation (0.4-0.1 m), the wavelength of the outgoing infrared radiation from the earth
(6-40 m) (8-14 m - atmospheric window- spechrae region over which thermal
emission from ground is radiated directly to space. Paetridge - Plalt, 8, Fig. 7.1) and
other important micro-physical length scales.
This is of importance, because the interaction of the particle with the surrounding
gaseous medium depends primarily on the ratio of the mean free path, 1, to the particle
diameter, Dp, called the Knutsen number Kn. = 1/Dp. For Dp <<1 the gaseous molecules
that are bombarding the particle arriving from far distance and each collision causes a
substantial change in the direction or speed of the particle. Accordingly, such a particle
behaves as if it were of giant molecules and its motion (drag force) and Brownian
diffusion may be described by the rigorous kinetic theory of gases. Such an aerosol is
said to be in the free molecular regime. On the other hand, for a particle which is large
compared to 1, i.e. Kn <<1, a change in its momentum and only the combined effect of
many particles yields an effective change. From the point of view of the particle, it
"feels" as if it were suspended in a continuum and for this reason such a particle is said to
be in the continuum regime.
The interaction of a particle with electromagnetic, including visible, radiation depends
largely on the ratio of the particles diameter to the wavelength, , of the incoming
radiation. This dimensionless ratio, the optical size parameter,  = Dp/, determines
whether the interaction is purely dipole (<<1) or it is governed by geometrical optics i.e.
diffraction, refraction and reflection (>>1).
In addition to the particle size, the behaviour of atmospheric aerosols also depends on
their shape and chemical composition.
Particle shape.
Atmospheric aerosols exhibit a variety of shapes. Most commonly, they are spherical,
liquid droplets. The water solution of sulfuric aced or other salts that constitutes a large
fraction of the fine particle mass (e.g. Brosset, 1976) is believed to be in droplet form.
Photochemically produced aerosol, such as the Los Angeles smog aerosol consists of
droplets formed around a nucleus (e.g. Husar et al. 1976).
Several of the common aerosol salts ((NH4)2SO4, NaCl) exhibit deliquescence (June,
1963; Covert et al. 1974). This behaviour is manifested by a rapid transition from
crystalline form at low relative humidity to a droplet at high RH. Dry sodium chloride
crystals are almost cubical; ammonium and potassium sulfate crystals are also generally
cubical but with more rounded edges.
4
Combustion and recondensation of weakly volatile substances lead to the formation of
chain aggregates of solid spherical spheres. The aggregation is generally taking place via
coagulation in the vicinity of the combustion zone and the individual spheres in the chain
are held together by molecular forces. When suspended in the air, a chain aggregate may
bend, twist and rotate due to asymmetric molecular bombardment of fluid shear.
A sample of atmospheric fine particles impacted onto a grid and viewed through
electron microscope is shown in Figure 2. The chain aggregate in the center, probably
originated from an automobile exhaust. Unfortunately, atmospheric liquid particles lose
their identity when deposited onto a grid, inserted into the vacuum of the electron
microscope and exposed to the heating due to the absorbed electrons. However, such
particles leave a flat, circular residue as evidence of their original form.
Size segregated samples of coarse atmospheric particles are shown on the scanning
electron micrograph (Figure 3). They are generally non-spherical with major fraction
resembling soil dust or fibrous matter.
The characteristic size of spherical particles is their diameter. Both the light scattering
and mechanical (e.g. fluid resistance) properties of nonshperical particles depends
strongly on their particular shape. However, the definition of a characteristic size is
difficult. The method of their measurement is often used as the definition of their size.
For example, in case of a cascade impactor, particles are sorted according to their
aerodynamic (or Stockes) diameter which encompasses all the particles having the same
trajectory in a curved flow field. A small but dense particle may have the same
aerodynamic size as a large particle with low density or an aggregate with large
aerodynamic resistance. Sorting or sizing of particles with optical single particle
counter/spectrometers, yields an "equivalent optical diameter". Here again, a multiplicity
of physical sizes may correspond to the same equivalent optical diameter if their
refractive indexes are different. The above discussion points to the need to specify the
method of measurement whenever particle size or size distribution are presented.
It is sometimes useful to express the shape of an aerosol by shape factors. For a
particle with given geometrical shape, the shape factors for volume, v, and shape factor
for surface, s, are defined as:
V = v Dp3
s = s Dp2
For volume, v and surface, s, respectively. Dp refers to a typical length scale of a
particle. The shape factors of several common shapes are given in Table 1.
5
Figure 1. Transmission electron micrograph of size classified submicron aerosols.
The white area in the center is where the collodion film is broken and the chain
agglomerate aerosol is directly exposed to electron beam. Most of the submicron
aerosol volume is made up from liquid droplets (which leave circular residues)
and of chain agglomerates.
Figure 2. Scanning electron micrograph of size classified (greater than 2 m)
atmospheric aerosol collected by an impactor. The collection substrate is a
collodion coated electron microscope grid. Note the irregular shap of the
particles.
6
Table 1. Shape factors for surface and volume.
v
s
Sphere

/6 = 0.52

Cube
1
6
Crushed sand
0.28
2
Calcite
0.32
2.5
Particle concentration.
The aerosol concentration is defined in a similar manner as the density: the number
N or mass M of particles per unit volume, V, of air (N/V). the concentration will
vary, depending on the size of the volume element V. At large V, the concentration,
say the total number of particles N, will vary from one location to another because of
macroscopic gradients. With decreasing size of V the number of particles may decrease
such that it may exhibit statistical fluctuations because of the finite number of particles.
Mass concentrations of atmospheric particulate matter may reach several hundred
micrograms per cubit meter. This corresponds to a mass ratio of the order of 10-7 grams
of particulate matter per gram of air. Thus the contribution of the aerosol to the total
mass, even in highly polluted atmospheres, is negligible. For most applications, the
effect of the particles on the fluid motion can be neglected.
The total number concentration may range from 102 particles/cm3 in a very clean
atmospheric air to 106 particles/m3 near intensive combustion aerosol sources such as
roadways. Even at 106 cm-3 the average distance between individual particles is about
100 m, which is several orders of magnitudes larger than the characteristic size of
combustion aerosols (0.01-0.1 m). Therefore the motion of aerosols is free from mutual
interaction, except when they approach each other and adhere due to molecular forces
(coagulate).
Size distribution.
The most important physical characteristic of an aerosol population is the particle size
distribution. In principle, the aerosol distribution function should have two independent
aerosol variables, viz. particle size and chemical composition, as well as the usual space
and time co-ordinates, x, y, z and t. in practice, it is convenient and adequate to treat the
chemical distribution as discrete set of species, rather tan as a continuos variable. Thus
we may define a distribution function ni; such that
ni (r, x, y, z, t)
represents the number concentration of chemical species I, at position x, y, z and time
t, and in the range of particle radii between r an r + dr. At times it is instructive to
consider higher weightings of the number distribution function such as the distribution of
aerosol surface and volume (or mass) with respect to size. The surface, volume and mass
distribution functions, si, vi and mi respectively, are related to the number distribution as
follows:
si = 4 r2 ni weighting function
7
vi = 4/3 r3 ni
mi = 4/3 i r3 ni
where i is the mass density of species i. The total aerosol distribution function in
each case may be expressed as the summation over all chemical species. Thus, for
example, the total mass distribution function is defined as
m(r, x, y, z, t)   m i (x, y, z, t, r)
i
Typical total distribution functions for number n, surface s, and volume v, for the Los
Angeles smog is given in Figure 4.
Figure 3. Average size distribution of the Los Angeles smog aerosol (Whitby, Husar
& Liu, 1972).
Integral Moments of the Distribution Function.
The effects of a polydisperse aerosol population are best described in terms of integral
moments of the size spectrum i.e. integrals of the size spectrum weighted by a function of
particle size. If the weighting function is an integer power law (many people confuse
power law (xa) with exponential functions e-x) function of r, then the v-the integral
moment, Iv, is defined as

I v   r ν n(r)dr
0
8
Integral moments have physical significance describing the behaviour of a
polydisperse system as described by the following examples.
The zeroth moment is the total number concentration, N, defined as

I 0   n(r)dr  N
0
The first moment

I1   rn(r)dr
0
gives the total length of a chain if particles were lined up next to each other. A
physically more meaningful parameter is the size (radius) of the average particle, r,
defined as

I1

Io
 rn(r)dr
0

r
 n(r)dr
0
The second moment is proportional to the total aerosol surface area per unit volume,

4I 2  4π  r 2 n(r)dr  S
0
S = 4I2
The surface area of the average particle is given by 4I2/Io = S/N.
The third moment is proportional to the total volume, V, of the aerosol suspension per
volume of air is,
4
3
πI 3 
4
3

π  r 3 n(r)dr  V
0
The total mass, M, is

M
4
π ρ(r)r 3 n(r)dr
3 0
9
where ( r ) is the size dependent particle density.
For  = const., M = V.
The volume of an average particle, V
4
πI 3
3
VN
I0
The fifth moment is proportional to the mass flux, or deposition rate, D, of material
sedimenting from air (g/cm2, sec)

8π 2 gI 5
4
D   v s (r)(ρ πr 3 )n(r)dr 
3
27μ
0
where the sedimentation velocity, vs = 2r2g/9, and  is the dynamic viscosity of the air.
There are several other important moments of the size spectrum for which the
weighting function can not be conveniently expressed as integer power-law of particle
size. In that case the integral moment is defined as

I   f(r)n(r)dr
0
Notable examples for such weighting functions include the light scattering crossection,
which yields the total light scattering of a polydispersed system.
The relative contribution of the different part of the size spectrum to a given moment
is determined by the weighting functional. In a polluted urban atmosphere the number
concentration or zeroth moment is dominated by particles in the 0.01-0.1 m size range;
the volume or mass concentration is contributed from both the 0.1-1.0 m and 0-30 m
size range as shown in Fig. 4.
Relationship Between Different Forms of Size Distribution.
The numbersize spectrum of atmospheric particles extends over five orders of
magnitude in particle size and up to ten orders of magnitude in concentration (Junge,
1963; Clark & Whitby, 1967). This extremely wide range coupled with the different
roles of small and large particles lead to the use of several forms of distribution functions.
In this section, attempt is made to facilitate a convenient intercomparison of several
commonly used distribution functions.
The extent of the aerosol size spectrum over several orders of magnitude necessitates
the use of a logarithmic scale for particle size presentation. It is also advisable, that the
area under a distribution function, when displayed graphically, is proportional to the
integral (i.e. a moment) of that function. (Junge, 1963; Berry, 1967; Whitby et al., 1972)/
10
If that criterion is satisfied, then simple visual impection of a graph reveals the value
of that integral (total number, mass) as well as the contributions of different size ranges to
the integral. Such presentation is facilitated by the semi-log plot of a properly defined
spectrum function (linear) vs. log of particle size (Berry, 1967; Whitby, 1972). If the
distributed variable is. For example the total number such distribution function, n(log r),
is defined as,
dN = n(log r) d(log r)
where dN is the number of particles in the radius range between log r and log r + d (log
r). Conventionally decimal logarithms are used. Unfortunately, practical problems
impose another problem of presentation. The measurements of aerosol size or size
distribution is most frequently performed in terms of particle diameter, Dp, rather than
radius. The corresponding number distribution function is n(log Dp). In this case the
numerical values of the two distribution functions are identical as shown below.
Following the definition of distribution functions, that the number of particles, dN, in an
infinitesimal range of the independent variable, dr, and d(log r) must be identical. So we
may write
dN = n(r) dr = n(log r) d(log r)
Thus the transformation from one independent variable, x, to another, z, follows the
equation
y(z) = y(x)  J
where the Jacobian transformation function J = dx/dy. The Jacobian of the
transformation from r to log r is J = dr/d(log r) = r/log e. Thus n(r)dr is related to the
distribution function n(log r) as follows:
n(log r) 
r
n(r)dr
log e
11
Table 2.
Following similar transformation, the functional relationship between the most
commonly used distribution functions are given in Table 2. The functions in the table are
the multipliers of f2 to yield f1. For example a transformation of the number radius
distribution n(r) to the volume distribution with (log Dp) is:
4r 4
n(r)
3log e
For practical reasons, in the following text, we shall almost exclusively be using (log
Dp) as the size variable.
v(log Dp) 
Commonly Used Distribution Functions.
There is no universal distribution function that adequately describes the majority of
aerosol populations, including their various weightings. It is often desirable, however, to
approximate the form of an aerosol spectrum with an analytical function. In the past , the
log-normal distribution, the power-law (Junge) distribution, the gamma function and the
"bimodal" distribution have been used most frequently.
A. Log-normal distribution.
12
Crushing and atomization processes often give skewed distributions over broad size
range as shown schematically in Figure 4.
Figure 4. Features of the log-normal distribution.
When such distribution are plotted on semi-log paper their shape resembles that of the
normal Gaussian distribution. Functionally, the only difference from the Gaussian
distribution is that the independent variable, r, is replaced by (lnr) or (log r). Thus the
log-normal distribution is defined as
n log r d log r  
  (log r  log rg ) 2 
1
exp 
 d log r 
2(log  g ) 2
2 log  g


where rg is the geometrical mean size and g is the geometrical mean deviation.
In analogy to the calculation of the mean radius (p. 12) the geometrical mean radius rg
is obtained as follows:

log rg   log r n  log r  d  log r 
0
Similarly, the logarithmic standard deviation is defined as


 log g    log r  log rg
2
 nlog r dlog r 
2
0
13
The mode of a distribution is at size r, where the maximum of the distribution occurs.
The median of the distribution is defined as the size which divides the total population
into two equal halves (see Figure 5 a). by definition, the median radius, rm is such that
rm

0
rm
 f log r  d log r  
 f log r  d log r 
where f(log r) may be any number surface or other distribution function. The mass
median radius has often been used to indicate the characteristic size at which the bulk of
the aerosol mass is contained.
The median is best determined from normalized cumulative plots of the quantity
(number, surface, volume or mass) vs. log r. Such plots are also referred to as logprobability graphs.
Figure 5. Normalized cumulative distributions. Log-probability plot.
A convenient property of this function is that of n(r) is log-normal, then all the other
weighting, rn( r ) are also log-normal with the same g. In fact, it can be shown that the
geometric mean radius log rg of a higher (-th) weighting, can be calculated from the
geometric mean size of the zeroth weighting log rg0
log rg = (log g)2 + log rg0
As an example, if geometric number mean radius is rg = 0.1m and g = 2.0 then the
volume mean diameter is
log rg,3 = 3(log 2)2 + log(0.1) = -0.728
rg,3 = 0.187 m
It is worth pointing out that the integral of the log-normal distributions unity. Such a
distribution is defined by three parameters, the mean size, rg, standard deviation and a
scaling factor for the concentration.
Power-law distribution.
In the early stages of atmospheric aerosol science, (Junge, 1963) atmospheric aerosol
spectra have been fitted with a power-law function
14
n(r) = ar-
a = constant
where the exponent ranges from 3 <  < 5. Such fits were arising as straight lines in
plots of n(log r) on log-log paper. The power-law fit brought to the researcher's attention
some of the regularities exhibited by the size spectrum of atmospheric aerosols
(Friedlander, 1951; Clark & Whitby, 1961). More detailed examination of aerosol size
spectrum (Whitby et al., 1972), in particular of its higher weighting (surface ad volume),
revealed that simple functions such as the power-law, are inadequate to describe the
complex and dynamic behaviors of atmospheric aerosols. For this reason, the use of
power-law distributions is no recommended. A direct comparison of different
distribution functions is given later.
The modified gamma function.
The calculation of radiation transfer through have layers, on a prior assumption needs
to be made about the aerosol size spectrum. For such radiation calculations, a modified
gamma function has been proposed by Diermendian (1969) and used extensively (e.g.
Dave, 1971). The form of the function is
n(r) = a1ra2 exp(-b1rb2)
where a1, a2, b1,b2 are positive constants.
a1
a2
b1
b2
Haze M
5.33104
1
8.94
1/2
Haze L
4.97106
2
15.11
1/2
Cumulus C
2.38
6
3/2
1
1.0810-2
8
1/24
3
Corona
cloud C
B. Bimodal distribution.
Measurements of atmospheric aerosol size distributions in urban areas indicate a
frequent occurrence of a bimodal distribution for the aerosol volume (mass)
concentration. The two modes of the volume distribution are illustrated in Figure 4.
There is accumulating evidence that the aerosol populations in the two mass modes are
formed by different mechanisms, come from different sources and have drastically
different chemical compositions. The two modes are separated by approximately one
decade of particle diameter or more, while the width of the two distributions generally
does not exceed g = 2.5. This allows a separation such that the overlap of the two
modes is less than 10-20%. Since the two modes of aerosol mass are contributed by
different sources, they are practically decoupled from each other in their spatial and
temporal pattern.
The existence of bimodal aerosol volume distributions was first pointed out by Whitby
et al, (1972), while some of the physical mechanisms that shaped the bimodal distribution
were discussed by Husar et al, (1972). They have compiled a number of their own size
spectrum measurements obtained in Los Angeles by electrical mobility analyzer and an
15
optical counter and compared whit data of other investigators obtained in variety of
locations (Figure 9).
The data shown in Figure 9 indicate the existence of two aerosol populations, the fine
and the coarse particles. It should be noted, however, that the relative magnitudes of the
two modes (the integral under the two modes) may vary drastically from location to
location.
At present there is no generally accepted expression (formula) which would
adequately describe the bimodal distribution. However, visual inspection of volume or
mass distributions such as shown in Figure 4 and 9 suggests that the two modes of the
volume distribution can be reasonably well approximated by two log-normal
distributions.
We have pointed out previously that the size and the size distribution of a
polydispersed aerosol population depends on the method of measurement. It is,
therefore, important to test whether the bimodal mass distribution emerges when the size
distribution is measured by other methods.
Recently, Kadowaki, (1976), reported mass distribution data obtained by a cascade
impactor. His data are shown in Table 10 for comparison. The normalized cumulative
form of the mass distribution function is shown in Table 11. The mass measurements of
the individual stages of the cascade impactor, the size ranges for each stage, and the
calculated mass distribution function are given in Table 3. The above data clearly
indicate the existence of the bimodal distribution when the aerosol is characterized in terms
of its aerodynamic size.
Figure 6. Comparison of volume distributions measured by several investigators in
different locations: Additional data are given in Table VIII. Note the universal
bimodal nature of all of these data and that the data obtained by Clark, Peterson,
16
and from the more recent Los Angeles and Colorado studies, were obtained
under pollution-free conditions such that it may be assumed that a background
aerosol was being measured, is rising sharply at 10 m. (Whitby, Husar, & Liu,
1972).
Figure 7. Histogram and size distribution curve of total aerosols in Nagoya (16-22
May 1974. Conc 131 gm3).
Figure 8. Typical cumulative particle size distribution of total aerosols in Nagoya.
(Kadowaki, 1976).
17
Stage
Weight of total
Concentration
logp
of
total
aerosols
aerosols M (mg)
Dp
m (g m3)
m logp Dp(g
m)
3
0
2.18
12.2
1.00
12.2
1
1.81
10.2
0.451
22.5
2
2.51
14.1
0.398
35.4
3
2.62
14.7
0.350
42.0
4
1.86
10.5
0.451
23.2
5
1.70
9.55
0.646
14.8
6
2.27
12.8
0.525
24.3
7
2.76
15.5
0.412
37.6
Backup
filter
5.59
31.4
1.68
18.7
Total
23.30
131
Table 3. Determination of histogram size distribution curve of total aerosols.
Sampling period: 16-22 May, 1974; Air volume sampled: 178 m3 (Kadowaki,
1976).
C. Comparison of different common distributions.
The primary criterion for evaluation the quality of the fit by a model distribution
function and real data is that the fit is good in the size range where a give effect
(visibility, health, soiling, etc.) is of importance. For example, for light scattering
calculations in the visible range the model function needs to be best in the 0.1 - 1.0 m
range since that subrange contributes most to the light scattering. On the other hand for
the calculation of soiling effects (deposition flux) the model distribution and the actual
data have to compare well in the 5 - 30 m range. It is, therefore not likely that a single
distribution function of a simple form (log-normal, power law, gamma function) will
satisfy the above set criteria over the size range of five decades.
It is instructive to compare graphically the shape of the above discussed distribution
functions when plotted in different coordinate systems. In Table 12 we show the lognormal, power law ( = 4), and modified gamma function (haze L as defined by
Diermendian). For comparison the data for the number distribution function of the Los
Angeles smog aerosol are also shown. It is evident that in such a log-log plot the power
law distribution is a fair approximation over a more than a decade of particle size, while
the other two distributions deviate strongly both at the upper and the lower end of the
spectrum.
The same distribution functions and aerosol data are shown on a semilog plot of v (log
Dp) in Table 13. As seen there the power law distribution with  = 4 is a straight line
with v(log Dp) = const. It is evident that the power law is an inadequate fit for the
volume distribution in the 0.1 - 1.0 m range. A log-normal distribution with g = 2.24
and Dp = 0.3 um is a reasonably good fit for the finer particle volume mode. The shape
18
of the modified gamma function is quite similar to the log-normal distribution. However,
with the constants given by Diermendian the mean size of the widely used haze L
distribution is by factor of three higher than that of the Los Angeles smog aerosol.
Figure 9. Grand average number size distributions for Los Angeles and the 1966
Clark study in Minneapolis: Also shown are a few data obtained by peterson in
Minneapolis in 1967 under inversion conditions. The Colorado data was
obtainde in Ft. Collins during the summer of 1979 under conditions where it is
believed to represent pollution-free contintetal background. For comparison, the
Junge distribution fitted to Clark's 1966 Minneapolis data is also shown.
(Whitby, Husar, & Liu, 1972).
19
Figure 10. Grand average volume distribution from Los Angeles. The grand average
volumes for subranges V3- (all particles less than 1.05 m) and for subranges
V4+ (all particles greater than 1.05 m) are also shown. Also shown ist the best-fit
log normal distribution to subranges V3-. The geometric mean size and geomtric
standard deviation shown are for the fitted log normal distribution. (Whitby, Husar,
& Liu, 1972).
In conclusion we should note that for calculations involving model aerosol size
distributions one should avoid using "generally accepted" distributions functions. Rather,
it is best to examine the available experimental data for the particular size range of
interest and arrive at a best fit function in that manner.
20
REFERENCES
W. Clark & K.T. Whitby. J. Atm. Sci. (1967).
D.S. Covert, R.J. Charlson & N.C. Ahlquist, "A Study of the Relationship of Chemical
Composition and Humidity to Light Scattering by Aerosols". J. Appl. Meteor. 11,
968 (1972).
J.V. Dave & P. Holpern, "Effect of changes in ozone amount of UN radiation received at
sea level of a model atmosphere." Atm.Env. 10 547-555 (1976).
D. Diermendian, "Electromagnetic scattering on spherical polydispersions". American
Elsevier, N.Y. (1969).
S.K. Friedlander, "On the particle size spectrum of atmospheric aerosols". J. of Meteor.
17, 373-374; 479-483 (1960).
S.K. Friedlander, "Chemical Element Balances and Identification of Air Pollution
Sources". Env. Sci. Technol. 7, 235-240 (1973).
G. Gartrell & S.K. Friedlander, " Relating Particulate Pollution to Sources: The 1972
California Aerosol Characterization Study". Atm. Env. 9, 279-299 (1975).
R.B. Husar, K.T. Whitby & B.Y.H. Liu, "Physical mechanisms Governing the Dynamics
of the Los Angeles Smog Aerosol". J. Coll. Interface Sci. 39, 211-224 (1972).
R.B. Husar, W.H. White & D. Blumenthal, "Direct evidence of heterogeneous aerosol
formation in the Los Angeles smog ". Evn. Sci. &Technology 10, 490-491 (1976).
C.E. Junge, "Air chemistry and Radioactivity", Academic Press (1963).
Kadowaki, Atm. Env. (1976).
K.T. Whitby, R.B. Husar & B.Y.H. Liu, "The Aerosol Distribution of the Los Angeles
Smog", J. Coll. Interface Sc. 39, 177-204 (1972).
21
HOMEWORK PROBLEMS
OCT. 8, 1976
PHYSICS AND CHEMISTRY OF ATMOSPHERIC AEROSOLS
(Homework to be returned Mon. Oct. 18
1.
Complete the conversion of distribution functions in the table on page 23 of class
notes
2.
Take the mass distribution function of Kadowaki (1976) (Table 3, Fig 10,11).
a.
Plot the mass distribution on log-probability paper (supplied) and fit the fine
particle mode and the coarse particle mode each with a log-normal
distribution.
b.
Estimate the mass median diameter and g for each mode from cumulative
distribution plots.
c.
Assume density of 2 g/cm3 for coarse particle mode, 1 g/cm3 for fine particle
mode. Calculate and plot (semilog paper) the distribution of number, surface
volume and deposition flux for each size range.
d.
Calculate the total number N, cm-3; Surface area, S, m2/cm3; volume, V,
m3/cm3; and deposition flux, D, g/cm2, sec.
22
AEROSOL MECHANICS
1. Introduction
2. Fluid Mechanical Regimes
3. Drag Force
4. Settling
5. Impaction
6. Brownian Motion
23
1. AEROSOL MECHANICS
Introduction
The mechanical interaction of particles with gas molecules, bounding surfaces, and
with each other is described by aerosol mechanics. Particles suspended in air are
constantly subjected to bombardment by N2, O2, H2O and other air molecules. The forces
associated with this molecular bombardment are in general in excess of the gravitational
force, hence fine particles remain suspended, such that their settling velocity is small
compared to their velocity caused by other forces. When a particles is in motion relative
to the surrounding gas molecules there is an asymmetry of the molecular bombardment
intensity: molecules impinging near the leading edge of the particle exchange more
momentum (due to higher relative impact velocities) than those impinging at the trailing
edge of the particle. The overall effect is a drag force directed opposite to the direction of
motion. A major component of aerosol mechanics is concerned with the determination of
the drag force as a function of particle size.
Motion relative to the surrounding gas molecules may be caused by a variety of
forces; gravitational, inertial, electrical, and "phoretic" forces caused by gradiance of
temperature, vapor pressure, etc. (Fuchs, 1964, Davies, 19
). Two major forces
acting on atmospheric aerosols are gravitational and inertial, the former being responsible
for aerosol removal by settling and the latter for the inertial deposition by impaction onto
ground surfaces, cloud and rain droplets.
A peculiar feature of aerosol behavior is that the particles have a relative motion to the
gas molecules even in the absence of external forces. Such, Brownian motion is caused
by the asymmetry of the molecular bombardment at any instant of time. For instance in a
short time interval the number of molecules impinging on the particle on the left side is
greater than from the right side than the particle will tend to move toward the right. The
net motion caused by such movement is random in stochastic sense and the entire process
of random walk is referred to as Brownian diffusion.
Brownian diffusion is in many respects analogous to molecular diffusion and leads to
collisions between the particle and other surfaces or to collisions among particles
themselves. At this point, however, the analogy between gases and particles ceases.
When a small particle approaches another solid or liquid surface within a distance of
several molecular sizes the particle and the surface will adhere due to the strong van der
Waol's forces. For example, the strength of this attraction may easily hold a 30 m
particle against the gravity. For fine particles this force exceeds gravity by several orders
of magnitude, making the separation of a deposited fine particle almost impossible.
Collision of particles among themselves and the subsequent adhesion is called
coagulation. In case of coagulation solid particles collision leads to an aggregate, while
collision of droplets leads to their merging into a single droplet or coalescence.
Another area of aerosol mechanics is resuspension. This process refers to the
mechanical separation of particles from the earth surface by high air velocities (shear
near the surface. Atomization is the mechanism by which bulk liquid meter, e.g.
24
seawater is broken up into spray droplets. In the following sections the above processes
and mechanism will be discussed in more detail.
Fluid Mechanical Regimes
The fluid mechanics of aerosols describes the momentum exchange between a particle
and suspending gas, the overall force that arises due to molecular impingements, as well
as the particle velocities and trajectories as a result of the combined interaction of all
forces. The qualitative description of the gas particle interaction may be aided by the sue
of a length scale characteristic for the gaseous molecules: their mean free path, 1. The
mean free path is the average distance traveled by a molecule before collision with
another "air molecule" (We should note here that "air molecules" refer to proper mixture
of N2, O2, H2O and other molecules, each having its own mean free path depending o n
molecular mass.). For air at normal temperature and pressure 1 = 0.066 m and it is
inversely proportional to pressure (Fuchs, 1964).
The gas-particle interaction may be divided into several regimes depending whether
the particle size is much smaller, much greater or comparable to the mean free path of air.
The dimensionless parameter which is used to define these regimes is the Knudsen
number
Kn = 1/Dp = mean free path/particle diameter
The flow regimes as defined by the Knudsen number are listed in Table 1, which also
contains expression for the drag force for each of the regimes.
Table 1. Flow regimes defined by the Knudsen number.
Regime
Kn range
Drag Force
Dp range, m
Continuum
0 - 0.1
0.6
6  rv
Transition
0.1 - 7
0.01 - 0.6
6  rv/C
Free molecules
7
0.01
6  r2v/1.66 1
which is used to define these regimes is the Knudsen number,
Kn = 1/Dp = mean free path/particle diameter
The flow regimes as defined by the Knudsen number are listed in Table 1, which also
contains expressions for the drag force for each of the regimes.
Table 1
Flow Regimes Defined by the Knudsen Number
Regime
Kn range
Drag Force
Dp range, m
Continuum
0 - 0.1
0.6
6 rv
Transition
0.1 - 7
0.01 - 0.6
6 rv/C
Free molecules
7
0.01
6 r2v/1.66 1
 = 1.83x10-4 poise, kinematic viscosity of air;
r, particle radius; v, particle velocity,
25
C, Cunningham slip correction factor.
The molecular flow field in the vicinity of the particle is illustrated schematically in
Fig. 1a, b and c, for the three flow regimes.
Free Molecular Regime.
In the free molecular regime where the mean free path is much greater than the
particle diameter, the molecules that collide with the particle come from a far distance
where their velocity distribution function is not perturbed by the presence of the particle.
The velocity distribution of such molecules is Maxwellian, and hence the momentum
exchange with the particle can be readily calculated by the known laws from kinetic
theory of gases (Epstein 19 ). Epstein's expression for the drag force, FD in the free
molecular flow is
FD  
4 2
r c nmv
3
Where  is a factor characterizing the surface ( = 1.44 for diffuse reflection); r2 is
the cross sectional area of the particle; n, m and <c> are the number, mass and mean
speed of gas molecules impinging on particles. Here we should note that the gas-particle
collision frequency is proportional to the cross sectional area of the particle and the
number concentration of the gas molecules. In this regime, the particles behave like giant
molecules, subject to the laws of the kinetic gas theory.
Continuum regime.
For particles with Dp >>1 the surrounding gas medium appears as if it were a
continuum. In this regime, the particles severely perturb the surrounding gas molecules,
both by displacement and by changing their velocity field (Fig. 1c). The flow field can
be calculated from classical fluid dynamics by solving the properly simplified NavierStokes equation (Schlichting, 1968). The simplifications arise from the fact that the
inertial terms associated with the fluid displacement rate are small compared to the
viscous terms and they can e omitted from the Navier-Stokes equation. The resulting
expression for the drag force in "creeping " flow or inertialess flow is given by
FD = 6  r v
where  is the air viscosity.
This expression is frequently referred to as Stokes law and its main feature is that the
drag force is proportional to the particle radius. The derivation of Stokes law also
requires that the air is incompressible, particles are rigid spheres and that no interfering
objects are in the vicinity of the particle.
26
Figure 11.
Stokes law, which rests on the assumption of slow, creeping motion is only applicable
CD 
24

Re
4
1
(Re) 3
for Reynolds number less than one. For Re = (VDp/) > 1 inertial terms in the Navier-
CD 
Fd
2
( ρ p v )( 14 πD p )
1
2
2
Stokes equation become important and subsequently boundary layer separation will
occur. For an arbitrary flow regime, it is customary to express the drag in terms of the
drag coefficient CD, where v is the relative particle velocity and p the air density. The
value of CD depends on the Reynold number as shown in Figure 3. For Re  1 CD =
24/Re, and for 1  Re  400 an approximate formula holds within 2% of experimental
results (Shlichting, 1964).
For atmospheric aerosols, with Dp < 100 m, the characteristic velocities are such that
the Stokes law is seldom violated.
27
Transition regime.
The gap between the free molecular and continuum regime is characterized by particle
size which is comparable to the mean free path of the suspending gas. In this regime, the
particle perturbs sufficiently the velocity distribution function of the surrounding gas
particles, such that the Maxwellian velocity distribution function and kinetic theory of
gases is no longer applicable. On the other hand , the molecular concentration in the
particle vicinity is sufficiently low, such that rarefaction effects cannot be ignored and
hence the continuum is not established. In principle, the complete molecular velocity
field in the vicinity of the particle and the resulting drag force can be calculated by
solving the Boltzman equation with proper boundary conditions (Hapman and Cowling,
1959). Unfortunately, complete solutions of the Boltzman equation for the entire
transition regime is practically not feasible due to the difficulties arising from
determining the "collision integral".
Figure 12.
There are several approximate methods for the determination of the drag force in the
transition regime that are reviewed by Hidy and Brach, 1970. For our purposes here, it is
convenient to utilize the empirical Cunningham slip correction factor, C, which is applied
to the continuum regime to correct for rarification
FD = (FD) continuum/C = (6  r v)/C
28
The Cunningham slip correction, C, is an empirical expression based on Millikan's
data for oil droplets (Fuchs, 1964).
C = 1 + 2.492 Kn + 0.84 Kn e-0.44/Kn
For small Kn, C  1; For Kn>>1, C 3.333Kn. For PO = 1, Kn = 0.666/Dp. At
pressure P, which is different from PO, the mean free path is l= PO/PO, hence Knudsen
number is
(Kn) p 
Po
(Kn) Po
P
The effect of pressure on the flow regime is most notable in the upper atmosphere
where the pressure is P = PO exp (-…H). For example, at the atmospheric tropopause, at
H = 20 km, the air pressure is P = …atm. With mean free path l = …m. Hence, a 0.1 m
particle with Kn =
is in the near free molecular regime at the tropopause,
while it is in the near continuum regime at sea level. The functional form of C is such
that for Kn  0 and Kn  , the corrected drag force assymtotically approaches the
expressions valid in continuum and free molecular regimes, respectively. The value of C
is graphed in Figure 2.
Mobility, Relaxation Time, Stopping Distance.
Generally, particles follow the mean motion of the air molecules, i.e. they travel along
fluid streamlines. This is due to the intense suspension forces caused by molecular
bombardment from all directions. If, however, external forces act on a particle, such a
force field may cause a deviation from the fluid environment and a particle will acquire a
velocity relative to the fluid. As we have noted before, any relative motion is
counteracted by a drag force due to impinging molecules, directed opposite to the
direction of motion.
Mobility. The relative velocity of a particle due to a force field is determined by the
strength of this force as well as by the properties of the particle and the suspending air.
These include the particle size, shape, viscosity, density, and mean free path of air, etc.
For particles in the Stokes regime, it may be stated that the particle relative velocity will
be proportional to the force that causes the relative motion. The proportionality constant,
B, is called the mobility.
Vp = BF
or
B= V/F
Mechanical Mobility is the proportionality constant between the particle velocity and
the drag force, i.e.
B = V/Fd = C/(6r)
For Kn < 0.1, B r-1 and for Kn > 7, B r-2. Thus, small particles are more “mobile” than
larger ones.
Particles that carry positive or negative electric charges are subject to electrical forces.
The external electrical force exerted on such a particle is F = npeE, where np is the
29
number of elementary charges on a particle (1, 2, 3, etc.), e is the electronic charge 4.8 x
10 –10 esu, and E is the electric field dV/dx (V/cm). For a particle which is in a steady
state motion, the electric force equals the resistance force. Hence, npeE=
And
V = npeB/E = ZpE
where
Zp = (npeC)/(6r)
Is the electrical mobility of a particle.
As noted above, Zp is proportional to the mechanical mobility and also increases with
increasing quantum units of electric charge per particle. The electrical mobility is a
useful parameter for the description of aerosol removal in charged clouds and for the size
distribution measurement by electrical mobility analyzers (e.g. Whitby et al., 1972).
There are several other forces, including the “foretic” forces (Davies, 1967), which
may cause a relative particle motion and for which appropriate mobilities may be defined.
Relaxation time. Consider a particle projecting into quiescent air with the initial
velocity VO. Due to the drag force, the particle velocity will diminish with time. The
rate of change of velocity is dV stated by the equation of motion, which arises from the
force balance. At any instant of time, the inertial force given by Newton’s law m(dV/dt),
is balanced by the fluid resistance force:
m (dV/dt) = -Fd = -V/B
where m = 4/3  r3 p
Is the particle mass. Integration with initial condition V = VO at t = 0 yields an
expression for velocity decay,
V = VO e –(t/mB) = VO e –(t/)
where  = mB is the relaxation time.
A similar expression may be obtained if the air is suddenly accelerated to velocity VO,
such that
V – VO (l – e –(t/))
Hence, after a time lag, 3t, the particle will attain the fluid velocity. The relaxation
time for a 10 m particle is ~300sec.
Stopping distance. Consider again a particle projected into a quiescent air. After
sufficient time, the particle will stop and the total distance traveled, the stopping distance,
s, may be obtained by integrating the equation of motion one more time.
s


0
0
 Vdt   V
O
e
 t 
 
 
dt  VO
The stopping distance for a 10m particle, for VO = 10 cm/sec is s = 30m.
Thus we may immediately conclude that for atmospheric particles with Dp < 10m, the
relaxation times and stopping distances are rather short. Thus it may be stated that any
change in fluid velocity will cause an instantaneous response in the appropriate particle
30
velocity. In such a case, the particle velocity may be determined from the quasi steady
state equation of motion,

 
V  BF  u
where F is the external force and u is the air velocity vector.
Settling velocity. A particle in a gravitational or centrifugal force field will be
accelerated until the drag force equates the external body force. The steady state velocity
is called the terminal velocity or the settling velocity, VS. For gravitational settling, VS
may be obtained by equating the gravitational force, Fg, with the drag force, Fd. In the
Stokes law regime
(4/3)  r3 (p - a)g = 6  r VS
where a is the air density and g the gravitational constant. Since a << p, the air
density may be neglected. The settling velocity is then
VS = (2 r2 p g C)/(9)
The main feature of VS is that it is proportional to the square of the particle size. For
example, for Dp = 10 m, VS = 0.5 cm/sec, while for 100 m, VS = 50 cm/sec. The
settling velocity is also proportional to particle density.
For large particles settling at high speed (Re > 1)
(4/3)  r3 p g C = ½ a VS2  r2 Cd
 8rρpg 
VO  

 3 ρaCd 
1
2
In this case, the drag coefficient, Cd, itself depends on VS and hence the calculation of
VS requires a numerical solution, e.g. by successive approximation. This method is only
required, however, for settling particles with Dp > 100 m (VS = 50 cm/sec).
2. AEROSOL OPTICS
Aerosol optics is concerned with the propagation of visible or near visible
electromagnetic radiation through aerosol clouds and with the processes associated with
the radiation transfer. From the point of view of radiation transfer, and aerosol system
consists of randomly suspended discrete mass centers, which interacts with the radiation
by absorption and scattering.
Radiation
The Physics of Radiation
Radiant energy may be alternatively envisioned as being transported either by
electromagnetic waves or by photons. Neither point of view completely describes the
nature of observed phenomena. Nevertheless, these separate concepts have considerable
utility. For example, the scattering from a single particle within a radiation wavefront
31
may be predicted through use of electromagnetic theory, whereas quantum theory is
employed in determining the properties of gaseous absorption or multiple scattering
media.
Radiation travels at the speed of light. Thus, from the viewpoint of electromagnetic
theory, the waves travel at this speed. Alternatively, from a quantum point of view,
energy is transported by photons, all of which travel at the speed of light. (This differs
from molecular transport in that all the photons have the same speed.) There is, however,
a distribution of energy among the photons. The energy associated with each photon is
h, where h is Planck’s constant and  is the frequency of the radiation. Each photon
also possesses a momentum h/c, where c is the velocity of light within the medium
through which the radiation travels.
Three parameters may be employed in characterizing radiation. They are the
frequency , the wavelength , and the wave or photon speed c. Of these, only two are
independent, since they are related by c=. The choice as to whether to employ  or 
as a characteristic parameter is somewhat arbitrary, although  has one advantage in that
it does not change when radiation travels from one medium to another. The speed of
light c within a given medium is related to that in a vacuum, co, by
c = co/m
m = (cos)/(cos')
m = n - in',
where m is the index of refraction. In vacuum, the index of refraction is unity while in
gases it is approximately one. For common liquids and solids it is between 1 and 3.
The wavelength range encompassed in aerosol optics which is of concern to us is the
visible and the near-visible radiation i.e. 0.3 <  < 1.0 m. The various radiation
subranges are illustrated in Fig. 1
Figure 13. Classifications of Radiation.
Absorption and Scattering
The basic problem of aerosol optics is the change in radiation due to its interaction
with the suspended particulates. The effect of visible radiation i.e. radiation pressure,
absorption, etc., on the particle behavior may in general be neglected. When a particle is
irradiated with light of a given wavelength, two different physical processes will occur.
The incoming radiation can be transformed into other forms of energy, such as heat,
energy of chemical reactions, or radiation at a different wavelength. In such a case the
32
energy transformation is called absorption. The absorption of radiation is associated with
transitions of the energy levels of the atoms or molecules that constitute the aerosol.
Absorption terminates the path of a photon. In visible range very few gases, such as
NO2, have spectral absorption bands. The most important gaseous absorption bands are
in the ultraviolet and infrared regions.
In addition to absorption, a medium may also scatter photons. Scattering is defined as
any change in the direction of propagation of the photons. This process is physically due
to local inhomogeneities within the medium, and such inhomogeneities may result from
suspended solid particles or liquid droplets within the gas. In addition, scattering can also
be produced by the gas molecules. When radiant energy is scattered with no change in
frequency, the scattering is referred to as coherent scattering.
If the scattering of radiation within a gas is strictly molecular scattering (i.e., there are
no foreign particle present), it is designated as Rayleigh scattering. The Rayleigh theory
predicts that the spectral intensity of the scattered radiation will vary as the fourth power
of the frequency; that is, the scattering is predominantly at the shorter wavelengths. This
accounts for the fact that the sky appears blue, for the preferential scattering in the
atmosphere involves the short wavelength blue light. This is also the reason why sunsets
are red, for the long wavelength red light suffers less attenuation in traversing the large
atmospheric path length. Although Rayleigh scattering is an important mechanism in
global atmospheric phenomena, it is usually unimportant for microclimatological
applications due to the short path lengths involved in the latter.
Scattering can, however, play an important role in radiation energy transfer when
foreign particles are present. Typical examples include clouds, fogs, and air pollution
particulates. In these cases scattering may encompass the combined and interactive
effects of reflection, refraction, and diffraction. A theory that is pertinent to such
situations is Mie scattering, which is concerned with electromagnetic scattering from
spherical particles with sizes on the order of the wavelength of the incoming radiation.
The foregoing was a discussion of the physics of absorption and scattering. Attention
will now be turned to the formulation of the processes of absorption and scattering in
terms of defined radiation properties.
Intensity of Radiation
To characterize the amount of radiant energy that departs from a surface along a
certain path, the concept of a single ray is inadequate. The amount of energy passing in a
given direction is described in terms of the intensity of radiation, which is denoted by i.
With reference to Fig. 2, the intensity of radiation is the radiant energy leaving a surface
per unit area normal to the pencil or rays, per unit solid angle and per unit time, where the
differential solid angle is d.
To illustrate the use of intensity, let d represent the radiant energy per unit time and
unit area leaving a given surface in the direction  and contained within a solid angle d.
33
Figure 14. The intensity of radiation.
Then clearly
i
dΦ
dω cosθ


(1)
The energy flux  passing from the surface into the hemispherical space above the
surface is then obtained by integration
Φ   i cosθ dω
Δ
(2)
where the symbol  denotes integration with respect to solid angle over an entire
hemisphere.
As illustrated in Fig. 3, the differential solid angle d may be expressed in terms of the
angles  and  of a spherical coordinate system centered on the surface. Upon recalling
that the differential
Figure 15. Integration of intensity over solid angle.
34
solid angle is the surface element on the hemisphere divided by the square of the
hemisphere radius, it follows that d = sin d d(area on a unit sphere). Consequently,
integration over the entire hemisphere yields
Φ
2π
π 2
0
0
  i cosθ sin θ dθ d












(3)
There are several instances in which the intensity of radiation is independent of
direction. For such cases equation (3) reduces to
=  i
(4)
In the foregoing definition of the intensity i, specific reference has been made to
radiation leaving a surface. When dealing with the radiation concerned with the net rate
at which energy is locally transferred within the medium. The intensity of radiation will,
in this instance, be designated by the symbol I and defined as the local net transfer of
radiant energy per unit area normal to the pencil of rays, per unit solid angle, and per unit
time.
Although the foregoing discussions of intensity have been concerned with the total
energy (over all wavelengths), the formulations apply on a monochromatic basis by
appending the subscript . Thus the spectral (monochromatic) intensities are i and I,
while the expressions

i   id

I
0
 I d

0
(5)
Relate the total and spectral quantities.
Absorption, Scattering, Extinction
Consider a monochromatic beam of radiation with intensity I, as illustrated in Fig. 4.
As the beam traverses a path length ds it will undergo partial attenuation as the result of
local absorption within the medium. The amount of absorption is assumed to be directly
proportional to both the thickness ds and the incident intensity I. Thus the amount of
monochromatic absorption per unit time, per unit area normal to the pencil of rays, and
per unit solid angle may be written as
kIds
(6)
35
Figure 16. Absorption and scattering of incident radiation.
The constant of proportionality k is the monochromatic absorption coefficient (this is
actually a volumetric absorption coefficient, as distinct from the often-employed mass
absorption coefficient.) for radiation of wavelength . Since the volume per unit surface
area of the crosshatched element in Fig. 16 is ds, the monochromatic absorption within
the medium per unit time, per unit volume, and per unit solid angle is
kI
(7)
By integration of equation (2) over all possible values of the solid angle ( = 4), the
local monochromatic absorption per unit time and per unit volume within the medium
due to all incident beams is expressed as
k
 Id
4
(8)
where it has been assumed that the medium is isotropic (i.e., k is independent of
direction).
A parallel development can be carried out for scattering. Again, with reference to Fig.
16, the monochromatic beam of intensity I will further be attenuated due to scattering.
The monochromatic energy that is scattered per unit of time, per unit area
normal to the pencil of rays, and per unit solid angle may be characterized as
 I d
(9)
where  is the monochromatic scattering coefficient. Correspondingly, on a unit
volume basis, the expressions





γλ

I
λ
 I
(10)
dω
4π
(11)
36
respectively characterize the scattered energy due to a single incident beam and to all
incident beams.
The analogous description of radiation attenuation due both to absorption and to
scattering leads to the following representation of the combined effects of both processes
k I ds +  I ds =  I ds
(12)
where = k+  is the monochromatic extinction coefficient. The parallelism
between absorption and scattering ceases, however, at this point. Once photons have
been absorbed, no further consideration need be given to them. On the other hand,
scattered photons continue to transport energy throughout the medium, and these must be
taken into account.
To facilitate the description of the radiant energy flux leaving a volume element as the
result of scattering of incident radiation, consider the schematic representation as shown
in Fig. 17, where s is a scalar distance measured along a pencil of rays. A portion of the
incident beam I (s, ’, ’) will be scattered by the crosshatched volume element in
accord with equation (3). It will be assumed that the scattering is coherent.
Figure 17. Scattered energy leaving a volume element.
The directional distribution of the scattered energy is characterized by the scattering
function P (’, ’: ,) such that
P (’, ’: ,) (d / 4)



(13)
represents the probability that radiation incident in the direction (’, ’) will be
scattered in the direction (,). Since the magnitude of the scattered radiation
corresponding to the incident beam I (s, ’, ’) is described by equation (3.4), the
monochromatic energy per unit time, per unit solid angle, and per unit area normal to I
(s, ’, ’) that is scattered in the direction (,) is
I (s, ’, ’) P (’, ’: ,) (d / 4)ds (14)
37
The integral of the foregoing quantity over all values of the solid angle must coincide
with equation (3). From this, it follows that
1
4
 P ( ' ,  ':  ,  )d  1
4
(15)
which is consistent with the definition of P as a probability. An often-used (though
not necessarily often-realistic) assumption is that of isotropic scattering. In this case,
radiation is scattered uniformly in all directions so that
P (’, ’: ,) = 1
(16)
Scattering and Absorption by Single Particles
In this section we shall consider the scattering characteristics of single particles and
provide the information required for the calculation of radiative transfer and visibility in
the hazy atmosphere.
The problem is to relate the properties of the scatterer, its shape, size and refractive
index, to the intensity and angular distribution of the scattered light.
It is of interest to note that scattering is not restricted to the visible part of the
electromagnetic radiation spectrum and that the waves by satellites, the scattering of
microwaves by raindrops, scattering of thermal radiation by cloud droplets, scattering of
light by small particles, and electron scattering by molecules are similar phenomena,
since in each case the wavelength is of the same magnitude as that of the scatter. Thus,
as a natural scaling fact or for scattering is the wavelength of the incoming radiation and
it is used in the dimensionless parameter, called the optical size parameter, .
= 2  r/ 
where r is characteristic particle size; In the case of a sphere it is the radius. In the
following discussion, the incident radiation will be assumed to be monochromatic.
Role of Particle Size
The primary role of particle size is that it determines the regime of interaction with
radiation. For particles much smaller than the wavelength << 1, the presence of the
particle does not perturb the wavefront of the electromagnetic radiation. The physical
process of scattering in such a regime is described by dipole interaction with
electromagnetic radiation and it is also referred to as Rayleigh scattering.
When the particle size becomes comparable to the wavelength, then the wavefront
suffers a substantial distraction in the vicinity of the particle. Waves scattered by a
particle will tend to interfere positively and negatively with the surrounding radiation
wavefront, which leads to a peculiar interference pattern in the vicinity range and
particles in this size range are called Mie particles.
For a particle with diameter at least ten times the wavelength, the interaction with the
wavefront again becomes relatively simple. Such a particle will reflect, refract (the lens
effect), and diffract certain amounts of radiation. These particles fall in the regime of
geometrical optics or ray tracing.
38
Beyond these qualitative changes in the radiation interaction regime, the particle size
has a crucial role in determining the amount of light each particle will scatter.
Cross Section and Efficiency
When the electromagnetic wavefront passes a particle, the radiation in the vicinity of
the particle will be abstracted from its original path, by scattering and absorption. This
fraction of incoming radiation is extinct from the original beam. Hence, the extinction
cross section of a particle, Cext, is the area in the particle where all the incoming radiation
changes directions or gets absorbed. The scattering cross section Cscat and the absorption
cross section, Cabs, are defined analogously and follow the relation:
Cext = Cscat + Cabs
Intuitively, the extinction cross section is the cross sectional area of the particle  r2,
i.e. all the energy impinging upon the particle cross sectional area is extinct with unit
efficiency factor. However, in reality this efficiency factor Qext, is either less than or
greater than unity depending on  and the refractive index m. Hence,
Cext = Qext  r2
The efficiency factors for scattering, Qscat, and absorption, Qabs, are defined in a
similar manner. In analogy of the cross sections, it also holds that
Qext = Qscat + Qabs
One would expect that for very large spheres, certainly for large absorbing spheres, the
extinction cross section would approach the physical cross section i.e. Qext ~ 1.
Paradoxically, this is not so. A large scattering sphere extincts twice its geometrical cross
section. The cause for this discrepancy is explained by the “diffraction paradox.”
Efficiency per Unit Mass
It is often desirable to define and compute the efficiency of extinction per unit mass of
aerosol. The mass extinction efficiency Mext is defined for spherical particles as
M ext 
C ext
C
Q
 4ext 3  ext1
M ρ p 3 πr
ρp 3 r
The mass scattering and absorption efficiency factors are defined analogously and
satisfy the relationship
Mext = Mscat + Mabs
The utility of the mass efficiency factors lies in that they are identically potency
functions (see pp (8) for scattering and absorption).
Albedo for Single Scatter
In the general case, the photons may be scattered and absorbed by the particles, thus
only a fraction of the incident photons will be leaving the particle. The albedo for single
scatter, , is defined as the fraction of light lost from the incident pencil due to scattering,
while ( 1 –  ) represents the absorbed fraction of the energy. Thus, the albedo  is
defined as
39
 = Cscat / Cext = Qscat / Qext = Mscat/ Mext
The single scatter albedo is particularly useful in multiple scattering calculations.
Scattering Diagram and Phase Function, Asymmetry Factor
The scattered wave of any point in the distant field has the character of a spherical
wave in which energy flows outward from the particle. The direction of scattering is
characterized by the scattering angle  and azimuth angle .
The most important property of the incident and scattered wave is the intensity, I. The
intensity of electromagnetic radiation is the rate of energy flow across a unit area
(erg/cm2sec) perpendicular to the direction of propagation. In optics this is also called
irradiance. The intensity is occasionally also referred to as illuminance, i.e. luminous
flux per unit area (lumens/m2 = lux). Both the incident and scattered waves are
unidirectional, i.e. each confined to a narrow solid angle. The term I is the total energy
flux in this narrow solid angle d.
The waves are also assumed to be monochromatics, i.e. confined to a narrow
frequency interval. Here we should recall that it is advantageous to use frequency rather
than wavelength, since  is independent of the refractive index of the medium. However,
for practical reasons, the intensity is commonly defined in terms of wavelength increment
d.
If Io and I are the intensities of incident and scattered light respectively, and R is the
distance from the particle, then I must be proportional to Io and R –2, and we may write
I
λ2
F( ,  )
4π 2 R 2
Figure 18.
The total amount of incident energy that changes direction of propagation is the
amount confined to the scattering cross-section, Cscat. The same energy is now
distributed in all directions as given by the dimensionless angular function called the
scattering diagram, F (, ). From conservation of energy, we get
Cscat 
1
k
2
 F(θ,  )d
4
40
Wave number k = 2 / 
where d = sin  d d is the infinitesimal solid angle and the integral is taken over all
directions, 0 <  < ; 0 <  < 2.
The scattering diagram, when divided by Cscat k2 yields the phase function P (,):
1
4
 P(θ, )  1
4
P (,) = F (,) / Cscat k2
which is the fraction of the total amount of light scattered into direction 2 and . The
integral of P (,) over all directions is unity.Hence the phase function represents the
probability of scattering in any given direction. The probabilistic interpretation of P (,
) will be useful for the photon tracing in multiple scattering problems. The value of P
(, ) also depends on the size parameter, and the refractive index m. For spherical
particles in natural nonpolarized light, the light scattering phase function has circular
symmetry coinciding with the axis of incident beam propagation such that
P(,) = P()
Generally, the forward scattering (with  > goo) dominates the backscattering and this
asymmetry is quantified by the asymmetry parameter cos , defined as
cos θ 
1
4
 P(θ, )cos θ d
4
The asymmetry factor increases with increasing particle size.
Polarization
Scalar waves, like sound, are fully described by the intensity. However, neither the
incident light nor the scattered light is completely characterized by intensity. The
transverse nature of light waves allows the phenomenon of polarization to occur. The
additional parameters required for the full description are polarization and phase.
Light consists of many simple waves with frequency 1014 sec –1 and with the duration
of coherent wave trains 10 –8 sec. The simple waves are all monochromatic and
completely (elliptically) polarized. Light that is commonly measured is the net effect of
many simple waves and in general is partially polarized. Natural light, such as direct
sunlight, is a mixture of uncorraleted simple waves. Over a time period of usual
measurements ( t > 10 –8 sec) the electric vector exhibits no preferential vibration i.e. it is
unpolarized.
An arbitrary beam of light, of intensity I, consists of an unpolarized part and a totally
polarized part.
I = Iunpol + I pol
The degree of polarization is defined as Ipol / I. The polarized part of the beam is in
general elliptically polarized, and it can be further separated into a linearly polarized part,
Ilp, and a circularly polarized part, Icp, where Ipol = ( Ilp2 + Icp2 ) (1/2).
41
The intensity of incident radiation complete with polarization may be represented by
Stockes vector II = { Io, Qo, Uo, Vo}, where the quantities in the bracket are the four
Stockes parameters. The scattered intensity at distance R in the far field (i.e. R >> ) is
II = (1 / k2 R2) F IIo
where the quantity describing the single scattering properties of scattering medium is F,
the transformation matrix, four columns, and four rows of dimensionless numbers.
For unpolarized natural incident light, the Stockes vector simplifies to Io = { Io, 0, 0,
0}. Hence, the intensity is adequate to describe this type of radiation. In the framework
of the following discussion, we shall confine the discussion to unpolarized incident light.
Unfortunately we cannot make that simplification for the radiation leaving the particle.
The scattered radiation is in general polarized. For the present discussion it will suffice
to say that the scattered intensity will have components vibrating perpendicularly (i 1) and
parallel (i2) to the plane through the directions of propagation of the incident and
scattered beams. The degree of polarization is (i1 – i2)/(i1 + i2).
The scattered wave in the distant field from the particle is a spherical outgoing wave
with an amplitude, S(), inversely proportional to distance. Any point in space is
traversed by two wave systems, the incident and the scattered wave. The intensity of the
scattered wave for perpendicular polarization is
I = (i1 / k2R2) Io
for parallel polarization
I = (i2 / k2R2) Io
and for incident natural light
I = ((1/2) (i1 + i2) / k2R2) Io
The intensity functions are related to the amplitude functions by
i1 = | S1 () |2
i2 = | S2 () |2
The wave phenomena involved in light scattering are illustrated schematically in Fig.
19.
42
Figure 19. Schematics of the spherical wave of linearly polarized light leaving the
particle.
Much of the light scattering theory is concerned with the understanding of the intricate
interference relationship between the scattered (diffracted, refracted, and reflected) and
the incident light. In the next section, consideration is given to the basic physical
processes, which cause light scattering to occur.
Diffraction, Refraction, and Reflection
The physics of light scattering by small particles can be described by the combined
and mutually interfering effects of diffraction, refraction, and reflection. Reflection is a
well known phenomenon and generally contributes < 5% of the scattering. Therefore, no
further consideration will be given to its role at this point.
Role of Diffraction
Diffraction is an edge effect. It arises from the incompleteness of the wavefront
passing the sphere.
Figure 20.
43
A fraction of the radiation near the edge of the particle is bent towards the particle to
“fill its shadow.” On rigorous physical grounds it can be shown that the amount of
radiation diffracted is precisely the cross sectional area of the particle. This holds for any
shape particle. Hence, the extinction cross section of a large sphere consists of the
refracted (or absorbed) cross section, r2, and of the diffracted cross section, which is
also r2. This explains the “diffraction paradox” that Qext = 2 (Van de Hulst, 1957) for
large particles, and the polarization of diffracted light is the same as that of the incident
light.
Role of Refractive Index
The refractive index plays three major roles in extinction: 1) phase shift, due to
difference of speed of wave propagation inside and outside the particle, 2) bending of the
wavefront direction (the lens effect), 3) dissipation (absorption).
The speed of a wavefront entering the particle with n > 1 is reduced by the amount c =
co / n, where co is the speed of light in a vacuum. Since the frequency remains constant,
this leads to reduction of the wavelength within the particle. Consequently a phase shift
develops between the wave within and outside the particle. Hence, the wave leaving the
particle (i.e. center of the particle) may be in or out of the phase with its surroundings.
This then may lead to positive or negative interference. We shall note later that it should
also be stressed that phase shift is a necessary condition for scattering to occur, i.e. the
refractive index difference at the particle interface needs to be large. This is satisfied
when 2(m-1) > 1. (van de Hulst, 1957).
Figure 21. Phase shift.
The refraction index is also responsible for bending the arriving wavefront in a similar
manner as observed in the well known convex lens effect. For small refractive index, the
bending is weak, while at high n, the dispersion is strong as illustrated schematically in
Fig. 22. The refracted light from a particle contributes most of the polarization.
44
Figure 22. The "lease effect" of refractive index
For absorbing media such as metal oxides and elemental carbon, the refractive index,
m, is complex. The refracted wave now becomes inhomogeneous, with decaying
intensity (amplitude) as it penetrates the particle, Fig. 23
Figure 23.
The refractive index, m, is now expressed as a complex number
m = n – in’
where n and n’ are the real and imaginary parts of m. The radiation intensity in the
particle decreases by exp (-4  n’) , and the amplitude by exp (-4 n’).
Refractive Index for Shortwave Radiation
The value of the refractive index for dry atmospheric aerosol particles ranges from 1.4
< n < 1.6. For water, the value is 1.33. For a hygroscopic or deliquescent particle, the
refractive index will approach that of water as the particle grows at high humidities.
Since the growths of particles with humidity is only known for a few pure substances, the
RH dependence of refractive index of atmospheric aerosols is not predictable on
theoretical grounds.
Barnhardt & Strate (1970) quote an empirical formula
N = 1.54 + 0.03 ln (1-RH/100).
Hanel (1971) proposed a linear interpolation between the dry particle refractive index
and water, based on the volume fraction of the dissolved matter.
The refractive indices of crystalline aerosol materials have been compiled by Bullrich
(1964) and are shown in Table 1. These crystalline compounds have 1.48 < n < 1.64 in
45
dry state and all f them are transparent in the visible window (i.e. the imaginary part of
their refractive index is negligible).
The imaginary part of the refractive index is responsible for the absorption (heating) of
particles, but its value for atmospheric aerosols is not well established. For urban
aerosols it is considered to range between 0.005 to 0.03.
Absorption in the visible region may be due to organic aerosols or metal (ferrous)
oxide, but is believed to be dominated by carbon (soot) particles emitted from
combustion sources. Twetty & Weinman (1971) have reviewed the real and imaginary
refractive indices of graphites, soot, coals, etc., and gave an average value of m=1.8-0.5 i.
Bergstrom (1972) proposed 0.63 < n' < 0.69 in the visible window, but he also stressed
the wavelength dependence of both n and n'. Bergstrom's n, n' values for the 0.3-3m
range are given in Table 2.
Figure 24. Real (nr) and imaginary (ni) parts of the refractive index of quartz
(silica). (After Conel, 1969).
46
Figure 25.
Particle
Refractive Index
Particle
Refractive Index
NH4Cl
1.64
CaSO4
1.57
NH4NO3
1.60
KCl
1.49
(NH4)2SO4
1.52
Na2SO4
1.48
MgCl2
1.54
SiO2
1.49
NaNO4
1.59
K2SO4
1.49
Table 1. The refractive index of atmospheric aerosol materials which have a
crystalline structure (Bullrich, 1964).
The calculation of the effective refractive index of a mixture of particles such as the
atmospheric aerosol, is possible if the size, chemical composition distribution function is
known. In that case, the average real and imaginary refractive indices may be extracted
from properly calculated scattering and extinction coefficients.
The real and imaginary parts of the refractive index of atmospheric aerosols have been
examined by a variety of experimental techniques, some requiring aerosol deposition on
filter. In other methods, n and n’ were inferred from in situ optical measurements. Table
3 summarizes n and n’ data for soot, fly ash, atmospheric aerosol, and dry dust. Prudent
users of such tables will carefully compare their own application with the conditions that
the above data were taken.
47
Refractive Index for IR Radiation
In the wavelength range 0.7 < < 14 m, the real and imaginary refractive indices of
materials cannot be taken as constant. Most materials exhibit absorption bands
particularly in the 1-10 m region, in which n’ will vary drastically. If this variation of n’
occurs in the “transparency window” of the atmosphere (8-14 m) the analysis needs to be
with care (in spectral bonds).
The extinction is generally dominated by particle sizes comparable to the wavelength
of radiation. Hence, for the transparency window, the particle sizes 8-14 m are of
particular importance, which is dominated by earth’s crust materials: silicates and metal
oxides. Conel (1969) reviewed the optical properties of silicates and gave n and n’ (Fig.
24) from measurements of Spitzer & Kleinman (1961).
The imaginary part of refractive index for propane soot, chimney soot, soot from
precipitation dust, coarse dust, and fine dust, has been measured (Fig. 25) by Volz (1972)
over a 2-30 m wavelength range. While soot is an efficient absorber, dust has been
shown to be quite absorbing (n’ ~ 0.1) in the 8-14 m band. Further discussion on the
subject is given by Paltridge and Platt (1976, pp 287).
MIE Solution
In principle, the interaction of the particle with the incoming electromagnetic radiation
may be determined by solving Maxwell’s equations for the boundary conditions
corresponding to the particle shape. Solutions, however, are available only for few
shapes: spheres, spherical shells, ellipsoids, and infinite cylinders (van de Hulst, 1957).
The first solution of the Maxwell equations for dielectric (transparent) and absorbing
spheres was given by Gustav Mie (1908). The equation of concern is the wave equation:
+ ( 2m /  )2 = 0
which is satisfied by both the electric and magnetic field strength. This equation is
solved for within and outside the particle, such that the solutions are matched at the
particle boundary. The method of separation of variables yields a formal solution that
may be worked out with the use of Bessel functions.
Mie’s solutions of the spherical wave equation for a homogeneous sphere are given in
terms of the intensity functions i1 and i2, vibrating perpendicularly and parallel through
the plane of incident wave propagation and scattering direction.
i1  S1  
2
  2n  1
a n  n b n τ n 
 
 n 1 m(n  1)

  2n  1

i 2  S2 ( )  
(a n τ n  b n  n ) 
 n 1 n(n  1)

2
2
2
48
The polarized intensity functions are computed from the above infinite series where n
and n are angular functions derived from Legendre polynomials.
polynomials are easily computed from recursion relations. The first two are:
1 () = 1
2 () = 3 cos ()
(m)
Legendre
1 () = cos 
2 () = 3 cos 2
n2
k2
0.3
1.84
0.70
0.4
1.88
0.69
0.5
1.94
0.66
0.6
1.99
0.64
0.7
2.03
0.63
0.8
2.07
0.61
0.9
2.09
0.60
1.0
2.12
0.59
1.5
2.14
0.65
2.0
2.17
0.75
2.5
2.21
0.86
3.0
2.26
0.98
0.42
1.41
2.53
0.589
1.79
3.33
0.75
2.19
4.36
1.0
2.63
5.26
2.25
3.95
9.20
(a) Carbon
(b) Nickel
Table 2. Indices of refraction (2 = n2 - ik2) for carbon and nickel at various
wavelengths. The actual index of refraction of quartz in the solar spectrum is
about  = 1.52--0.0 I (Peterson and Weinman, 1969); however, the value of 1.50
was used to be consistent with that comomnly employed for the natural aerosol
(Quenzel, 1970; Yamamoto and Tanaka, 1969; and others).
49
ni
1.8
1.55
nr
Type
Twitty and Weinman (1971)
0.5
soot and coal dust
Grams et al. (1972)
0.044
fly ash
Fischer (1970)
0.01
atmospheric aerosol
Ivley and Popova (1973)
1.65
0.005
atmospheric aerosol
Lin et al. (1973)
0.01
atmospheric aerosol
Lindberg and Laude (1974)
0.007
dry land dust
Table 3. Values of the real (nr) and imaginary (ni) parts of the refractive index of
atmospheric particles for the short-wave region observed from various
experimental measurements. (Paltridge and Pratt, 1976).
Figure 26. The two sets of functions of the scatttering angle , which occur in the
Mie formulae, for n = 1 to 6. (van de Hulst, 1957).
The heart of the Mie scattering problem is the computation of the series coefficients an
and bn, arising from separation of variables. These functions depend on the size
parameter, , and the complex refractive index m = n – in’. The expressions an and bn
involve Bessel functions (van de Hulst, 1957) and can also be computed with recursion
relations. Appropriate methods of numerical computation have been discussed by
Kattawar & Plass (1967) and Dave (1969).
50
The scattering and extinction efficiency factors also follow from the coefficients an,
bn (van de Hulst, 1957).
as does the asymmetry factor.
4
cosα  2
x Q scat
 n(n  2)

2n  1
Re(a n a *n 1  b n b *n 1 ) 
Re(a n b *n )
n(n  1)
n 1  n  1



The values of an and bn rapidly approach zero when n  . Therefore, the total
number of terms that should be calculated in these series (Hansen & Travis, 1974) is n 
. The infinite series can be physically interpreted as a multipole expansion of the
scattered light (Mie, 1908). The coefficients a1, a2, and a3, i.e. specify the amount of
electric dipole, quadrupole, and octupole, while the bn are coefficients of magnetic
multipole radiation. For particles which are small and do not have large refractive index,
only the electric dipole radiation is significant and the well-known Rayleigh scattering
results.
Size Distribution.
In nature a distribution of particle sizes is usually encountered. The scattering and
extinction coefficients are
r2
r2
r1
r1
γ   C scat (r)n(r)dr   πr 2 Q scat (r)n(r)dr
r2
r2
r1
r1
   C ext (r)n(r)dr   πr 2 Q ext (r)n(r)dr
(2.48)
where n(r)dr is the number of particles per unit volume with radius between r and r + dr,
r1 and r2 are the smallest and largest particles in the size distribution, and under the
assumption of independent scattering the transformation matrix for a unit volume is given
by
r2
F ( )   F ij ( , r) n(r) dr
ij
r1
51
(2.47)
Fij(, r) is one of the elements of the transformation matrix for a particle of radius r.
The normalized (dimensionless) phase matrix follows from
P ij ( ) 
4 ij
F ( )
k 2
(2.49)
And the single scattering albedo from



(2.50)
It is straightforward to make computations for any size distribution of spheres.
However it is important to have systematic computations which allow an understanding
of the effect of the size distribution. This is required if measurements are to be inverted
to yield properties of scattering particles.
To facilitate inversion of radiation measurements the size distribution must be
described with the minimum number of parameters. Clearly the first parameter should be
some measure of the mean particle size. The arithmetic mean is
r2
r
 rn(r)dr
r
1 2
  r n(r)dr
N r1
r1
r2
 n(r)dr
r1
(2.51)
Where N is the total number of particles per unit volume. But since each particle
scatters an amount of light proportional to Cscat = r2 Qscat, the "best" single parameter
describing the scattered light is the mean radius for scattering.
r2
rscat 
r π r
2
Q scat ( , n r , n i ' ) n(r)dr
r1
r2
π r
2
Q scat ( , n r , n i ' ) n(r)dr
r1
(2.52)
The appearance of Qscat (, nr, ni') in (2.52) makes rscat an inconvenient parameter.
However, if rscat is larger than the wavelength a parameter which is almost as "good" can
be obtained by omitting Qscat in (2.52).
Thus we define the effective radius as
r2
reff 
r π r
r1
r2
2
n(r)dr
2
 π r n(r)dr
r1
r

1 2
r π r 2 n(r)dr

G r1
52
(2.53)
Where G is the geometric cross-sectional area of particles per unit volume. Similarly,
as a measure of the width of the size distribution, we define the effective variance,
v eff 
1
Gr eff2
r2
 (r - r
eff
) 2 π r 2 n(r)dr
r1
(2.54)
Where reff2 in the denominator makes veff dimensionless. As a measure of the
departure of the distribution from symmetry we define the effective skewness,
s eff 
r2
1
3
eff
Gr
v
 (r - r
3/2
eff r1
eff
) 3 π r 2 n(r)dr
(2.55)
These definitions are analogous to characteristics used in statistics to describe
frequency distribution (e.g., Kendall and Stuart, 1963), with r2n(r) corresponding to the
frequency distribution.
It is useful to have a standard analytic size distribution for theoretical computations.
We employ the distribution used by Hansen (1971b),
n(r)  constant r (1  3b)/b e  r/ab
(2.56)
As a standard distribution, because it has the simple properties
a = reff
b = veff
for the size distribution (2.56)
(2.57)
As may be verified by substitution into (2.53) and (2.54) with r1 = 0 and r2 = . The
standard distribution (2.56) is a variation of the gamma distribution; other forms of the
gamma distribution have been used extensively for cloud particles, e.g., by Khrgian
(1961) and Deirmendjian (1964). Figure 29 illustrates the standard distribution for
several values of a and b. This distribution, including the normalization constant, is
defined for 0  b = veff< 0.5 and has an effective skewness seff = 2b. Larger values of veff
can be obtained by adding a third parameter to (2.56) or by using the log-normal or
power-law described in the following subsection.
Numerical Examples. Mie scattering computations are a simple task for modern
computers and there now exist several books and reports containing extensive tables of
numerical results. Nevertheless it is useful to have graphical examples which clearly
illustrate the effect of the size distribution and refractive index on the scattered light.
53
The first graphs which we show are for quantities independent of the scattering angle,
Qscat, , and cos. Qscat, the efficiency factor for scattering, is defined for a size
distribution as
r2
Q scat 

G

π r
2
Q scat (r) n(r)dr
r1
r2
π r
2
n(r)dr
r1
(2.58)
Scattering Efficiency Factor, Qscat
The value of Qscat as a function of and refractive index m is shown in the
tridimensional plot, Fig. 27. At m = 0, the value of Qscat is zero. This is due to the
observation made earlier, that in order for scattering (phase shift) to take place, a
difference in m at the particle interface is necessary. The Qscat curves for m > 1 are
characterized by a series of major maxima and minima. The major maxima and minima
are due to interference of waves diffracted and transmitted by the particle. Since the
phase log, 2(n-1), of transmitted light is greater at high n, increasing n (Fig. 27). At
large n, the amplitude of the interference oscillations also increases.
Figure 27. Three-dimensional view of smoothed values of Qsca as a function of  and
ml. The grid lines designate constant values of . (Kerker, 1969)
The behavior of Qscat is shown in more detail in Fig. 28. In addition to the major
oscillations, a ripple on the Qscat curve also arises from the last few significant terms in
the Mie series. Physically the ripple arises from light waves grazing the sphere. These
rays set up surface electromagnetic waves, which travel around the sphere spewing off
energy in all directions. At = 180o, when the surface waves splash and they give rise to
enhanced back scattering intensity. The ripple is due to the interference of the diffracted
and surface waves (van der Hulst, 1957). Fig. 28 also shows Mie computations of Qscat
for polydispersed aerosols with varying width. The size distributions having different
54
spread factor, b, are shown in Fig. 29. It is demonstrated (Fig. 28) that even slightly
polydispersed aerosols do not exhibit the secondary interference peaks and ripples.
However, the first maximum persists even for large values of b.
Figure 28. Efficiency factor for scattering, Qscat as a function of the effective size
parameter, 2a/. The standard size distribution (2.56) was used with four
values of the effective variance b. For the case b = 0, 2a/ = 2r/  x. The
refractive index is nr = 1.33, ni = 0. (Hansen and Travis, 1974)
55
Figure 29. Standard size distribution (2.56) for 2 values of a and three values of
b. The size distribution is normalized so that the integral over all sizes is N = 1.
(Hansen and Travis, 1974)
For absorbing particles, the major maxima and minima and ripples are damped. For
large particles (>> 1) the intensity of light waves traversing the diameter of the sphere
is decreased by a factor exp (-4 n’), and thus the major maxima and minima are
significantly damped for 4 n’  1.
56
Figure 30. Efficiency factor for scattering Qsca, as a function of the size parameter x
2r/. The refractive index nr 1.33, with results shown for four values of m.
Mass Extinction Coefficients
The efficiency of scattering and absorption for unit mass of aerosol is a useful
parameter for the estimation of the contributions of various sources to the total light
scattering and absorption.
The light extinction coefficient, , of a polydispersed aerosol population, consisting of
a mixture of i different species is the sum of the contributions of all species.
r2
    M ext,i (r, n, n' )mi (r)dr
i
r1
where Mext,i (r, n, n’) is the mass extinction efficiency factor (see p. 60) for species I
having real and imaginary refractive indices n and n’. The mass distribution with size of
species I is defined on p. 10 and an example is shown on p. 21b and 38a.
57
A set of calculations for the efficiency factors per unit aerosol volume are shown in
Fig. 19 (Patterson and Wagman, 1976). Another set of computations was performed by
Foxwg (1976), who showed that the commonly found aerosol species have quite different
Mext. While the absorbers, Fe and C have their peak at   1, the nonabsorbers H2O and
SiO2 scatter most efficiently at   5.
The mass scattering and absorption efficiencies were computed by Bergstrom (1973)
for a set of real and imaginary refractive indices. His results for a wavelength of 0.55 m
and indices of refraction of 1.5-0.02i, 1.5-0.05i, and 2-0.66i (carbon) are shown in Fig.
33. For nonabsorbing particles with n = 1.5, the scattering per unit mass peaks at r  0.4
m (Dp  0.8 m). For very small particles, i.e. Dp < 0.1 m the mass scattering function
approaches Dp3 , and the efficiency curves become very small. For large particles Mext ~
r-1 as Qext approaches 2.
For absorbing particles with  << 1, the extinction is equal to the absorption and the
absorption per unit mass is constant. Such particles will behave as gaseous absorbers,
Bergstrom (1973). Bergstrom also performed a set of calculations for the IR wavelength
of 9 m and n’ up to 2.0. In this spectral region the small particle limit extends to about
Dp = 1.0 m. In this regime, r < 1.0 m, the mass absorption coefficient does not depend
on the size distribution, which was also noted by Waggoner et. al. (1973). The increase
in the small particle absorption is almost linear with increasing n’ up to n’ = 1.5.
Figure 31.
Light scattering per unit volume (100
of aerosol. Note that for  =
3
3
3
3
1gr/cm , 1m /cm = 1g/m . The light scattering/volume is ~ double for m = 1.65
compared to m = 1.33. However, the density of materials at m = 1.65 is perhaps
double that of water (1.33). Hence, the scattering/ mass is to be calculated as
shown in Fig 32.
m3/cm3)
58
Figure 32. Optical scattering cross section per unit mass ofr single particles of
various materials at wavelengths at wavelenght of 0.6328 m. (The refractive
indices and densities used in the calculations are: iron (m = 1.51-1.63i p = 7.86);
carbon (m = 2.5-0.75i, p = 2.25); water (m = 1.33, p = 1.0); and silicon dioxide (m
= 1.55, p = 2.66).4 The wavelength used is = 0.6328 m. (Foxwog 1975)).
59
Figure 33. Extinction and absorption coefficients per unit mass as a function of
particle radius for four different refractive indices at a solar wavelength of 0.55
m.
Figure 34. Absorption coefficient per unit mass of a log-Gaussian size
60
distribution of carbon particles asa function of the raidus of maximum
concentration r0 for different values of the spread parameter A at a wavelength
of 0.55 m.
The mass absorption efficiency for a polydispersed carbon aerosol distribution is
shown in Fig. 34. For carbon the efficiency is practically constant from small sizes up to
Dp = 0.2 m, and then it falls linearly with increasing size. The numerical studies of
Bergstrom shown above have clearly pointed to the importance of small particles in
absorption, even for sizes below the “optical window,” 0.2-0.7 m. If we note that most
of the elemental carbon is emitted from combustion sources and is concentrated in the
0.05-0.3 m size range, then these particles obviously deserve most of the attention when
studying the absorption and atmospheric heating effects.
Asymmetry Parameter, cos
This parameter is a measure of the skewness of the phase function toward forward
scattering. A symmetric phase function such as that of Rayleigh scattering has an
asymmetric parameter, cos  = 0, as shown in Fig. 35. On the other hand, for large
optical size parameter, cos  approaches the result for the geometrical optics phase
function, cos  = 0.87 (van de Hulst, 1957). In between it has a major peak and smaller
oscillations for the same reason as the oscillations of Qext: phase lag in the particle and
interferences with diffracted waves. An increase in the width of the size distributions
tends to smooth out these fluctuations.
The asymmetric parameter is highest for small refractive index and systematically
decreases with increasing n (Fig. 36). Physically this may be explained by the strong
forward scattering at small refractive index and the increase of the wave bending away
from forward direction at high n. When n  , and large , cos   0.5, because half of
the scattered radiation is diffracted in the forward direction and half is reflected
isotropically (Hansen and Travis, 1974). The Figures 34 and 36 clearly show that for the
size parameter  > 3, the bulk of scattered light is contained in the forward scattering
lobe with  < 45 (cos  > 0.7).
61
Figure 35. Asymmetry parameter, <cos >, as a function of the effective size
parameter, 2a/. The standard size distribution (2.56) was used with four
values of the effective variance b. For the case b = 0, 2a/ = 2r/  x. The
refractive index is nr = 1.33, ni = 0.
62
Figure 36. Asymmetry parameter, <cos >, as a function of effective size
parameter, 2a/. Results are shown for five values fothe real refractive index,
nr, all with ni = 0. The standard size distribution (2.56) was used with b = 0.07.
Phase Function
The directional dependence of the light scattered by a particle is determined by the
phase function. For homogenous spheres (uniform refractive index inside the particle)
the phase function is determined by the optical size parameter, , the refractive index m =
n-in-1, and the scattering angle . For isotropic homogeneous spheres the scattering
S ( ) 0 
S 1

 0 S2 ( )
matrix has the simple form
(2.34)
63
It follows that the Stokes parameters, Equation (1.4) of the incident and scattered
radiation are related by
I=
1/ (k2R2) FI0
(2.35)
where, following van de Hulst (1957), we use the symbol F for the four-by-four
transformation matrix
 1 (S 1 S 1*
 12
 (S S *
F  2 1 1



 S 2 S *2 ) (S 1 S 1*  S 2 S *2 )
1
0
0
 S 2 S *2 ) (S 1 S 1*  S 2 S *2 )
2
1
i
*
(S 1 S 2  S 2 S 1* ) (S 1 S *2
0
0
2
2
i
1

(S 1 S *2  S 2 S 1* ) (S 1 S *2
0
0
2
2
2
1
0


0

* 
 S 2 S1 )

*
 S 2 S 1 )

(2.36)
The transformation matrix is proportional to the phase matrix,
F = cP
(2.37)
The proportionality constant follows from the normalization condition on P, Equation
(2.5), which yields
c   F11
4
d
4
(2.38)
and the definition of the scattering cross-section
 sca   IR 2 d Io 
4
1
F11d
2 
k 4
(2.39)
Thus
C = (k2sca) / 4
(2.40)
and the specific relations between the matrix elements are
F11 = ((k2sca) / 4) p11 = (1/2) (S1S1* + S2S2*),
F21 = ((k2sca) / 4) p21 = (1/2) (S1S1* - S2S2*),
F33 = ((k2sca) / 4) p33 = (1/2) (S1S2* + S2S1*),
F43 = ((k2sca) / 4) p43 = (i/2) (S1S2* - S2S1*),
64
S1* -- conjugate complex of S1
S2* -- conjugate complex of S2
(2.41)
The scattering functions S1 () and S2 ( ) were defined previously (p. 71). P gives the
angular distribution and polarization of scattered light for any polarization of incident
radiation. The element in the first row and first column, p11 is the phase function and
gives the probability of scattering of unpolarized incident light. The ratio p21/p11 is the
linear polarization for unpolarized incident light. The significance of the other two
matrix elements is indicated in Table 4. For unpolarized incident light p33 and p43 vanish,
i.e. only linear polarization occurs. As noted earlier in such a case the intensity functions
i1 and i2 completely describe the angular distribution of scattering.
The intensity functions i1 an i2 are shown in Fig. 25 for m = 1.33, 1.55, and 2.0, and
for the size parameters x = 1, 1.5, 2, 2.5, 3, 2.6, 4, 5, and 6 (van de Hulst, 1957, p. 152).
The intensity functions i1 and i2 are shown in a polar diagram for  = 1 and for  = 10
(Fig. 26). The dominant lobe is confined to  < 45, and it is also called the diffraction
lobe. With increasing other, smaller lobes occur such that the total number of lobes is
approximately .
Polarization
For small size parameters,  << 1, Rayleigh scattering occurs, thus there is a strong
positive linear polarization
(i1 – i2) / (i1 + i2) > 1
with the maximum polarization at scattering angle  = 90o. With increasing  the degree
of polarization generally decreases and exhibits fluctuations with the major peak shifting
towards larger . For  ~ 1 m for instance, the peak of the polarization is at  ~ 150o
(Fig. 38). Since polarization is a ratio measurement, it can be determined accurately. As
pointed out by Coulson (1974), it has not been explored adequately in the measurement
of atmospheric aerosol parameters. Some aspects of aerosol polarization are discussed by
Harris (1972), Ward et al (1973), and White (1975).
Aureole Scattering
For large size parameters the phase function approaches that for geometrical optics.
At small , the phase function is large and the linear polarization vanishes (i1 = i2),
because of the predominance of the unpolarized diffracted light. The scattered light
contained in the first (forward scattering) lobe is referred to as aureole scattering.
Aureole scattering by aerosols is the cause of circumsolar radiation and plays a major role
in atmospheric optics. As seen in Fig. 26, the aureole scattering becomes narrower as the
particle size increases. For that reason, the aureole scattering function with the sun as the
light source, may be used as a measure of the effective aerosol size and the size
distribution (Gorchakov et al 1970; Green et al, 1971; Ward et al. 1973; Angstrom,
1974a; Angstrom, 1974b; Gorchakov & Isakov, 1974; Twitty et al. 1976). The
circumsolar light scattering may also introduce errors in the turbidity measurements as
pointed out by Grassel (1971).
65
Fgure 37. A sample of the scattering diagrams computed by means of the
rigorous formulae; m = refractive index, x = 2a/. In all graphs the logarithms
of the intensities (1 division = a factor 10) are plotted against the scattering angle
(1 division = a factor 10) are plotted against the scattering angle (1 division = 30.
Spheres (formulae in sec.9.31): Solid curves i1, dotted curves i2. The values for 
= 0 and 180 are indicated in the margin. For m = 2, 1.55, and 1.33 they have
been obtained by squaring the moduli tabulated by Rayleigh (Sci. Papers 344,
1910). (Yan de Hulst, 1957)
66
Figure 38. Phase fucntion, P11, and percent polarization. -100 P21/ P11, for single
scattering of unpolarized incident light. Results are shown for the four size
distributions illustrated in the inset, all of which have the same value for reff (1 )
and veff (0.25), where reff is the effective radius and veff the effectie variance. The
calcualtions are for the real refractive index nr = 1.33 and wavelength L.
(Hanson and Travis, 1974).
67
Figure 39. Model size distriibutions for a Junge (v = 3); an equivalent
distribution of four log-Gaussian components; and a Diermendjian Haze.
(Harris and McCormics 1972).
68
Figure 40. a.)
Average scattering function per particle as a fucntion of angle
for parallel polarization and individual log-Gaussian components numbered
from 1 (small) to 4 (large) for Mainz aerosol, m = 1.500 - 0.010i at  = 0.53 m.
b.) Average scattering function per particle as a function of angle for
perpendicular polarization and individual log-Gaussian comoponents numbered
from 1 (small) to 4 (large) for Mainz aerosols, m = 1.500 - 0.010i at  = 0.53 m.
Glory Scattering
The enhanced intensity in the backscattering direction  ~ 180o, is the so-called
“Glory.” This is caused specifically by the spherical shape of the scatterers which serves
to focus certain rays at  ~ 180o. There are essentially two origins for these rays: edge or
grazing rays which set up surface waves on the sphere, and noncentral rays which emerge
at  ~ 180o after internal reflection. The surface waves are not included in the
formulation of geometrical optics, and their contribution decreases as the particle size
increases. For refractive indices in the range 2 < n < 2, a noncentral ray can emerge at
= 180o after just one internal reflection; this gives rise to the intense glory. Bryant &
Jarmie (1974) give a good detailed discussion of the glory.
Backscattering has received considerable interest recently because of its unique role in
remote probing of atmospheric aerosol vertical structure by lidar. An interesting feature
of the glory is that it is absent for nonspherical particles and thus it can be sued to
distinguish spherical and nonspherical particles. Other features of the phase function are
discussed by van de Hulst (1959) and Hansen & Travis (1974).
In atmospheric optics we always have to consider the role of aerosol distribution.
Numerical work of Hansen & Travis (1974) showed that the effective phase function
and the percent linear polarization is similar for aerosol size distributions described by
gamma function, log-normal, bimodal, and power law distributions provided that their
effective radius (reff) and effective variance (veff) is the same. Their sensitivity analysis
was performed for reff = 1 m and veff = 0.25 (Fig. 27).
69
The effect of the size distribution and percent polarization was also studied by Harris
& McCormick (1972). They compared the power law distribution, Junge (1963), with 
= 4, Deirmendjian (1969), haze L distribution, and four log-normal distributions with logmean radius of 0.08, 0.24, 0.64, and 2.0 m (Fig. 28). The phase functions arising from
the four log-normal distributions (labeled 1 to 4 with increasing size) are shown for the
parallel and the perpendicular polarization components in Fig. 29. The smallest size
components (No. 1) approaches Rayleigh scattering, while the largest size (No. 4) clearly
illustrates the dominance of the forward scattering lobe.
It is instructive to compare the log-normal distributions 1-4 to the two modes of the
bimodal distribution (Fig. 10, p. 29). The distribution No. 1 with log-mean radius rgo=
0.08 m for the distribution function corresponds to rg3 = 0.15 m for the volume
distribution function. This compares well with the measured volume mean diameter
Dp=0.3 m for the fine particle fraction of the Los Angeles smog (Fig. 13, p. 31) and
Dp=0.45 for the Nagoya aerosol (fine particle mode). Hence, the phase function for size
distribution No. 1 by Harris & McCormick is probably representative for the fine particle
mode of urban aerosols. For such a distribution the backward scattering intensity  =
180o is 1/10 of the forward intensity and the polarization ratio (i1-i2) / (i1+i2) ~ 10 at
~100o.
The distribution No. 4 of Harris & McCormick with volume-mean diameter Dp=7.5 m
is illustrative for the coarse particle mode (Fig. 10, p. 29). For such a distribution the
bulk of the light scattering is confined to the aureole scattering ( < 10o) while the
backscattering intensity is three orders of magnitude smaller.
Nonspherical Particles
The theory of nonspherical particles is only developed for a few well defined shapes,
such as ellipsoids and cylinders, illuminated perpendicularly to their axes (van de Hulst,
1954; Kerker, 1969). The theory scattering by irregularly shaped randomly oriented
particles is inhibited by the mathematical problems of satisfying the Maxwell equations,
the boundary conditions of the individual nonspherical particle. However, recently a set
of experiments on such particles was reported which deserves attention.
Measurements of angular scattering from nonspherical particles comparable in size to
the wavelength have been primarily on polydispersed systems of particles. Powell et al.
(1967) showed that such measurements on magnesium oxide cubes and needlelike
fourling of zinc oxide could be approximated with Mie theory for some suitable size
distributions of spheres of the same refractive index. Holland’s & Gagne’s (1970)
measurements of light scattered by platelike particles of silicon dioxide show good
agreement with the Mie theory in the first 50o from forward scattering, but the
backscattered radiation is generally less than the theory predicts for an equivalent size
distribution of spheres. Scattering measurements by Napper & Ottewill (1962) on
monodispersed sols of octahedral and cubic silver bromide exhibit good agreement with
Mie theory for the octahedral sols, but generally poor agreement for the cubic sols.
These measurements were made only for scattering angles 60-120o from the direction of
forward scattering. Scattering measurements on monodispersed Escherichia Coli cells
(Cross & Letimer, 1972), which are prolate ellipsoids, and measurements on
monodispersed particles of barium sulfate (Peters & Dezelic, 1975) show fair agreement
70
with Rayleigh-Debye theory, providing the real refractive index is close to 1. Less
definitive measurements of angular scattering by nonspheres have been made by Quiney
and Carswell (1972), by Phillips & Wyatt (1972), and by Hodkinson (1962).
Pinnick et al. (1976) measured the polarized light scattered from monodispersed
nonspherical randomly oriented aerosol particles and compared with Mie calculations for
spheres of approximately the same cross sectional area. They find that for slightly
nonspherical particles of sodium chloride and potassium sulfate with size parameter
greater than about five, the intensity of light scattered is generally more than as predicted
by Mie solutions in the forward scattering lobe less than five, the MIE results more
closely match the measurements. Measured angular scattering patterns for randomly
oriented particles are smoother than the MIE calculation and are nearly the same for NaCl
and potassium sulfate particles of the same size.
The angular scattering by polydispersed crystals of potassium chloride particles was
measured by Chyler et al. (1976). Their main conclusion is that backscattering is
significantly reduced for nonspherical particles as shown in Fig. 31. Chyler et al. (1976)
has also proposed a somewhat dubious “theory” for nonspherical particles by truncating
the Mie series at specified number of terms. This approach appears to be rather weak on
physical grounds.
The above experiments consistently show a reduction of the backscattering for
nonspherical particles. The primary importance of these observations is that the
backscattering of the lidar signals in dry, dusty areas needs to be corrected for this effect.
Also, the hemispherical backscattering is less for such particles.
71