Penetration Depth and Scattering Depth

Detlef Smilgies

In order to discuss the scattering depth or information depth, it is useful to take a step back from our beautiful description, and consider the following: We would like to know the scattered intensity from a little test volume dV at a depth z below the surface. An incident wave with amplitude A gets attenuated before it reaches dV which we can write using the penetration depth L_i(a) as

A' = A exp{ -z/L_i(a) }

The generated scattered wave in dV leaves the sample and reaches the detector with a probability given by the scattering probability f(a,y,b) and the escape depth L_f(b):

A" = A' f(a,y,b) exp{-z/L_f(b)}

Hence the scattered wave from the test volume dV is related to the incident wave by

A"/A = f(a,y,b) exp{-z [1/L_i(a) +1/L_f(b)] } = f(a,y,b) exp{-z/L_s(a,b)}

where we introduced the scattering depth or information depth L_s(a,b) by the simple formula

1/L_s(a,b) = 1/L_i(a) + 1/L_f(b)

The information depth is a function of both the incident angle a and the exit angle b. The scattering depth is a very general concept and can be defined for various problems. In a full scattering theory taking into account both refraction and absorption (e.g. DWBA, dynamic theory), the penetration depth effects are automatically included.

A simple integration

If we make the simplifying assumption that we have a constant density of scatterers r within a layer of thickness L, and keep the exit angle of detection fixed, then we get a simple expression for the scattering intensity as a function of incident angle:

I(a) = ∫₀^L r exp(-z/L(a)) dz

The integral can be evaluated easily:

I(a) = r L(a) {1 - exp(-L/L(a)}

We can distinguish two limiting cases:

(i) L → ∞ (semi-infinite medium) :

I(a) = r L(a)

(ii) L << L(a) (thin film) :

I(a) = r L(a) {1 - (1-L/L(a))} = r L

A word of warning: for a nice well-defined thin film, interference effects of the wavefields (so-called waveguide modes) may cause deviation from this simple behavior, and a full scattering calculation may be needed. For relatively thick (more than 200 nm) and rough films the above formula may yield good results.