Z 4π
0
Φ s α
ð Þ dΩ s ¼ 1:
ð2:72Þ
Many particles show two peaks in the angular scattering function. The forward
scattering peak is well modelled by the single H-G function but the backscattering
peak (at low phase angles ¼ high scattering angles) is ignored in this function. The
double H-G function is intended to account for this and takes the form
Φ s α
ð Þ ¼
1 þ c
2
1 À ξ 1
2
1 þ 2ξ 1 cos α þ ξ 1
2
À
Á 3=2 þ
1 À c
ð
Þ
2
Â
1 À ξ 2
2
1 þ 2ξ 2 cos α þ ξ 2
2
À
Á 3=2
ð2:73Þ
where c is a variable. Hapke suggests reduction of the number of parameters from
three to two is adequate in many cases by setting ξ ¼ ξ 1 ¼ ξ 2 . An example of how the
double H-G function generates the backscattering peak is also shown in Fig. 2.17.
The correction for macroscopic roughness, introduced into his scheme by Hapke
in 1984, is based on defining a mean slope angle for the surface under study. The
slope distribution function, a s (θ), is normalized so that
Z π=2
0
a s θ
ð Þdθ ¼ 1
ð2:74Þ
and the mean slope angle is then
tan θ ¼
2
π
Z π=2
0
a s θ
ð Þ tan θ dθ
ð2:75Þ
Fig. 2.17 Examples of
Henyey-Greenstein
functions to describe single
particle scattering. Solid: A
single parameter (two-term)
H-G function (ξ ¼ 0.6).
Dashed: A double H-G
function (ξ ¼ 0.6, c ¼ 0.5).
Note the backscattering
peak in the double H-G
function
60
2 The Nucleus