and the structure factor Hij(p) is the Fourier transform of

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Pair Distribution Functions of Sticky Particles In natural media, the densely packed particles can have adhesive force. They adhere together to form aggregates. Thus the model of sticky particles was developed. The sticky particle has a stickiness parameter T. The smaller the T is, the more sticky the particles are. The Percus-Yevick approximation of the pair distribution function g( r) for the sticky spherical particles can be solved analytically using the factorization method of Baxter [1970]. The calculations of g(r) can be found in 8 of Volume II. We will study the use of dense media with a single particle size a and with a sticky parameter Sticky parameter T. In Fig. 6.4.2, the pair functions for several T'S are plotted. The scattering depends on the structure factor H(p). In Fig. 6.4.3, the structure factor H(p) is plotted as a function of pb, where b = 2a is the diameter of the particles, for several T'S. For non-small T, the interparticle force is small. H(O) is negative indicating that scattering is smaller than independent scattering. For small T, H(O) is positive indicating that scattering is larger than independent scattering. We note in Fig. 6.4.3 that for sticky particles, there are sharp variations of pair functions. These make it difficult to usp the Lorentz-Lorenz law to calculate ImK which is much smaller than ReK. Thus, in implementations, we use Lorentz-Lorenz law to calculate ReI< but use (6.4.41) to calculate "".. and (6.4.15) to calculate ""a' Then 21mI< = ""a + K'8'

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4.3 Numerical R.esults for Sticky and Non-Sticky Particles

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Figure 6.4.2 Pair distribution function of sticky particles.

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Figure 6.4.3 Structure factor H(p) a.'i a function of pb for sticky particles, where b is the diameter of the particles.

6 QUASI-CRYSTALLINE APPROXIMATION IN DENSE MEDIA SCATTERJNG

species diameter (em) fractional volume

1 0.05 0.1

2 0.1 0.12

5 0.4

0.2 0.05

0.3 0.02

Table 6.4.1 Numerical values for the QCA non-sticky extinction rate calculation.

0 0 00

Frequency (GHz)

independent scattering QCA non-sticky

10-' 1':-0----=20:----==30--4':-0--=50:----==60----:----:7"'"0--=80::----=90----:---:0 100

Figure 6.4.4 Extinction rate as a function of frequency for multiple moderate size particles for QCA and independent scattering. Parameters are listed in Table 6.4.1.

Results of Multiple Sizes Non-Sticky Particle Model

To analyze the effect of multiple sizes, we first illustrate the use of a medium with particles of five sizes. The size and the fractional volume of each species are listed in Table 6.4.1. The relative permittivity of particles is assumed to be Ers = 3.2 for all frequencies. In Fig. 6.4.4, the extinction rate is plotted as a function of frequency and compared with the results of independent scattering. For brightness temperatures calculation, we illustrate the use of a medium with particles of two sizes. The size and the fractional volume of each species are listed in Table 6.4.2. The first species is of the small particles which occupy a higher volume fraction while the second species is of the large particles. Assume that the first species has the size of 1 mm in diameter and occupies a volume of 30% while the second species has the size of 4 mm

4.3 Numerical Results for Sticky and Non-Sticky Particles

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80 90

Observation angle II in degree

vertical polarized horizontal polariZed

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80 90

Observation angle II in degree

Figure 6.4.5 Brightness temperatures versus observation angles based on the no interparticle force model. Medium consists of two particles sizes of 0.1 and 0.4 cm in diameter are used with small particles occupying a much larger fractional volume. Parameters are found in Table 6.4.2. (a) 19GHz, (b) 37GHz.

species radius (cm) fractional volume

1 0.05 0.3 572

2 0.2 0.03 0.89

no (cm- 3 )

Table 6.4.2 Numerical values of the physical parameters for the QCA non-sticky particle simulation.

6 QUASI-CRYSTALLINE APPROXIlHATION IN DENSE I'vIEDlA SCATTERING

independent scauering OCA sticky

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10" 10

Frequency (GHz)

Figure 6.4.6 Comparisons of extinction rate as a function of frequency for sticky moderate size particles of QCA and independent scattering. Fractional volume is 30%. the other parameters are found in Table 6.4.3.

in diameter but only occupies the fractional volume of 3%. The number density ratio of the two sizes is 643. The permittivity of particles is assumed to be E = 3.2 + iO.003E o for both frequencies. The brightness temperatures p for 19 and 37 GHz are presented in Figs. 6.4.5a and 6.4.5b as a function of observation angle. The snow pack physical temperature and the ground temperature are both assumed to be 270 K. The thickness of the snow layer is 50cm and the ground permittivity is E2 = (6 + 'iO.1)E o . The model of multiple size particles without interparticle force exhibits a strong frequency dependence similar to independent scattering. We next study the sticky particle model which exhibits a weaker frequency dependence.