Scattering from Small Particles
Evaluation of the scattering and attenuation characteristics of particulate random media requires accurate calculation of the scattering amplitude and the total cross section of individual particles. Once these quantites have been found an appropriate single or multiple scattering approximation may be chosen to model the behaviour of an ensemble of such particles. The effect of the ensemble reflects an average over the spectrum of particle shapes, sizes and orientations.
For the case of spherical particles a size distribution function is sufficient
to describe this summation.
Considering a scalar wave 
with unit amplitude,
incident on a semi-infinite slab of identical
spherical particles, where back-scattering is negligible when compared
to forward scattering the symmetry of the problem yields a coherent field
at a distance z into the slab of,
where r is the density of particles in a region of the slab (this representing a size distribution of only one particle radius but number density varying in space), 
is the scattering amplitude in the direction of the incident beam and 
represents the position of a scatterer.
The solution
takes the form,
The expression for the coherent intensity becomes,
and by reference to the forward scattering theorem,
it is clear the exponential decay of the coherent transmitted signal is proportionnal to the sum of the absorption and scattering crossections,
For a homogeneous distribution of nonidentical randomly orientated scatterers
the scattering amplitude must be averaged
over particle shapes and orientations. In an inhomogeneous distribution
the spatial integration must be made with respect to the regional average of the scattering amplitude.
For small dielectric scatterers where 
, a representing the largest dimension of a particle, the Rayleigh approximation may be used to calculate
. This assumes a homogeneous static field approximation for the incident field in the region of the scatterer.
Precise forms of the scattering amplitude for any size of spherical scatterer
are given by the Mie solution. General spheroids may be solved by appeal to
point matching or Fredholm integral techniques (also applicable to ellipsoids
and other less symmetric bodies). These solutions however yield cumbersome
expansions and so where appropriate the simpler Rayleigh approximation can
significantly reduce computation times for an ensemble of particles.
In addition the Rayleigh approximation can be applied to irregularily shaped
objects where a depolarization tensor may be found from it's geometric
properties. This facilites
rapid calculation of the far field of an individual scatterer and hence of the
total scattered field of a collection of particles. The validity of the approximation degrades if we are considering particles with low loss tangents. This is
clear on inspecting the value of the total cross secton derived from the vector forward scattering theorem.
Here
where 
is a unit vector in the direction of polarization of the incident wave. Taking the example of a small sphere with a forward scattering amplitude the same as that for an elemental dipole,
where 
is the integral of the polarization over the sphere,
From the form of and it is clear that in a lossless medium where the imaginary component of 
is vanishing
the forward scattering theorem will suggest a total cross section of zero, when
in fact there is clearly a scattered field. When scattering attenuation predominates over absorption attenuation the approximation for the total cross section
is unreliable and we may assume the same for the prediction of the scattered
field in this case. It is therefore desireable to extend the range of
validity of the Rayleigh approximation into the low loss region and in
improve its accuracy for medium loss scatterers.
This can be done by appeal to the standard integral equation for the field
in the region of a scatterer given in dyadic form as,
representing the standard free space scalar Green's function 
the unit dyad and 
represents the
field inside the scatterer.
In the Rayleigh appoximation the field inside the scatterer is taken as
where 
is the polarizability tensor given
by ,
representing the demagnetizing factors of the scatterer, functions of
its size and shape.
These expressions reflect the non dyadic form,
where
for the field impinging on an ellipsoid with its principal axis a,b and c corresponding to x,y and z axis. The expressions for y and z components are symmetrical to that of x. The 
are equal to the elements
and for general ellipsoids may be derived by satisfy the boundary conditions
for the object when surrounded by a uniform field. For more complex shapes
where approximation to a spheroid is not sufficient, values of must be
calculated numerically.
Significantly the Rayleigh approximation does not satisfy the boundary conditions for the scatterer. Examining the integro-differential equation the singularity is removed from the integraton by excluding a region of the scatterer at the point r of the same shape as the scatterer but of vanishing size. The reaction of the medium at this point results in a correction term which gives the field in the scatterer as
which reduces to,
The first term on the right hand side gives the Rayleigh approximation,
and the second term denotes the scattered field inside the scatterer. The key
step is how we elect to deal with the field inside the scatterer
so as to obtain a solveable form of the above equation.
One recent method [] is to assume negligible phase variation over the scatterer. The form of the inner field is then asssumed to be,
Substituting this field approximation and factoring out 
produces
the equation
where
To carry out the above integration the Weyl's expansion
of the scalar Green's function is used.
The volume of the scatterer is divided into portions above and below the z coordinate of
r where the integrand can be expressed as series expansion in powers of the small quantity 
. ie for the region above the the point r,
Retaining only the dominant term in the integration a simple expression for
is obtained.
and completing the integration and substituting the value of into the tensor equation,
This yields the equivalent polarizability tensor
which directly provides the
solution for the scattered field and the extinction cross section
using the above approximations. It can be shown [] that the resulting extinction
crossection derived from the forward scattering theorem will now be equal to
the sum of the total scattering cross section and the integral of the ohmic
losses or absorption cross section in the scatterer. Numerical assessments indicate a better approximation to the rigorous Mie and Fredholm Integral solutions
for spheres and ellipsoids using this equivalent polarizability tensor than for the basic Rayleigh approximation. This tensor contains frequency dependent
terms accounting for the multiply scattered field inside the scatterer.
This model may consider frequency dependence but it would be of interest to
know of it may be extended to take into account variations of phase over the
length of the scatterer. If the third term in the expression for 
can be neglected is may be possible to derive a spatially
dependent term from the first two.
Here as in most rain situations Raleigh will suffice. Explain the two requiremets. The correction method depends on corrections to the form of the inner fioeld. How can this be improved and applied to psarticles of totally arbitrary shape. via the polarizabilitty tensor.