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3.1.6. THE DIFFERENTIAL CROSS SECTION

Because of the importance of the angle of scattering we need to introduce the concept of the differential cross section dσ/d?. This term describes the angular distribution of scattering from an atom. As shown in Figure 3.1.3, electrons are scattered through an angle θ into a solid angle ? and there is a simple geometrical relationship between the θ and ?

? = 2π(1 ? cos θ)

[3.1.8]

and therefore

d? = 2π sin θdθ [3.1.9] differential scattering cross section can be written as dσ dσ 1 = [3.1.10] d? 2π sin θ dθ Now, we can calculate σ for scattering into all angles which are greater than θ by integrating equation 3.1.10. This yields π π dσ σθ = ∫ dσ = 2π∫ sin θdθ [3.1.11] 0 0 d? The limits of the integration are governed by the fact that the values of θ can vary from 0 to π, depending on the specific type of scattering. If we work out the integral we find that σ decreases as θ increases (which makes physical sense). Since dσ/d? is often what we measure experimentally, equation 3.1.11 gives us an easy way to determine σ for an atom in the specimen: σ for all values of θ is simply the integral from 0 to π. From this we can use equation 3.1.4 to give us the total scattering cross section from the whole specimen, which we will see later allows us to calculate the TEM image contrast. So we can now appreciate, through a few simple equations, the relationship between the physics of electron scattering and the information we collect in the TEM.

Our knowledge of the values of σ and h is very sketchy, particularly at the 100-400 keV beam energies used in TEMs. Cross sections and mean free paths for particular scattering events may only be known within a factor of two, but we can often measure θ very precisely in the TEM. We can combine all our knowledge of scattering to predict the electron paths as a beam is scattered through a thin foil. This process is called Monte Carlo simulation because of the use of random numbers in the computer programs; the outcome is always predicted by statistics! The Monte Carlo calculation was first developed by two of the United States' foremost mathematicians, J. von Neumann and S. Ulam at Los Alamos in the late 1940s. Ulam actually rolled dice and made hand (!) calculations to determine the paths of neutrons through deuterium and tritium which proved that Teller's design for the "Super" (Hbomb) was not feasible (Rhodes 1995). Monte Carlo methods are more

often used in SEM image calculations (see, e.g., Newbury et al. 1986, Joy 1995), but they have a role in TEM in determining the expected spatial resolution of microanalysis. Figure 3.1.4 shows two Monte Carlo simulations of electron paths through thin foils.

A B Figure 3.1.4. Monte Carlo simulation of the paths followed by 103 100-keV electrons as they pass through thin foils of (A) copper and (B) gold. Notice the increase in scattering with atomic number.

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