Nuclear Engineering

Rutherford Back Scattering Spectrometry (RBS)(Cont.)

(iii) RBS Cross section:


Based on the classical mechanics formulation, Rutherford performed a detailed calculation to obtain the cross section of this scattering process. This scattering cross section depends on the atomic numbers of target and projectile, back scattering angle and the energy of the projectile and is given by the following relation:

where Z₁, Z₂ are the atomic numbers of projectile and target atoms respectively. The quantity (Z₁Z₂e² / 4E ) can be equated to the distance of closest approach (d) in case of Coulomb scattering. Considering scattering of 2 MeV He ions from Ag atoms, the d value would be equal to 6.8 × 10^-4angstrom, which is much smaller than the Bohr radius. Therefore, an unscreened cross section can be taken for practical purpose. For the same example, the cross section value (for α = 180⁰) is:
σ(α) = (6.8 × 10^-4 angstrom)² /16 = 2.89 × 10^-24 cm² or 2.89 barns.

(iv) RBS Yield:


In order to calculate RBS yield (i.e. the number of detected backscattered species from any element), let us consider the following schematic (Fig. m5.6) of RBS with geometrical factors:

Figure m5.6 Schematic of RBS with geometrical factors
Here, Ω is the solid angle subtended at the detector, N_t is the areal concentration (atoms/cm²) of the element under probed, Q is the total number of incident particles in the beam. Considering an ideal case (100% detector efficiency) the yield Y is given by the following relation:

The total number of incident particles can be determined by the time integration of the current of charged particles incident on the target. Let us now see few practical examples below to obtain the information about elements in any material, their depth information and qualitative way to examine composition.


Case 1:

Let us first take an example of a bilayer [Fig. m5.7a] material with top layer(thin) indicated by 1 and the bottom layer (thick) indicated by 2. If the mass (M₁) of first element > > mass (M₂) of second element the typical RBS spectrum of this bilayer system is represented by Fig. m5.6b. in this spectrum x-axis is represents energy of the backscattered species and y-axis represents the count of backscattered particles in arbitrary scale.

Figure m5.7 (a) Schematic representation of a bilayer system consisting of two elements 1 and 2, (b) representative RBS spectrum of this system.
The height (H) and width (ΔE) of spectrum corresponds to each element provide the information about the total amount of that element present in this material. To the first approximation, the product of these two quantities (H × ΔE) can be considered as the yield of this element. Therefore, using equation m5.9 the areal concentration of the corresponding element can be determined. As a real example we can consider element 1 to be Au and 2 as Si.


Case 2:

Now if M₁ > M₂, whether both elements will show independent spectrum, will depend on the backscattered energy of the last layer of element 1 (E_L1) and the backscattered energy of the first layer of element 2 (E_F2). If E_L1 > E_F2, the two spectrum will be distinct from each other like above, otherwise there will be an overlap between them, and they will be not exactly resolved. As a real example we can consider element 1 to be C and 2 as Be.


Case 3¹:

In this case we will consider a preliminary analysis of a mixed layer by RBS. Let a thin Ni film (100 nm) is deposited on a Si wafer. Without any inter-diffusion of the atoms between the layers, the individual backscattered spectrum of Ni and Si generated by 2 MeV He ions are distinct similar to those as in case 1. If there is an inter-diffusion of atoms between the layers an formation of an intermixed layer like nickel silicide (Ni₂Si). In this case Ni signal will spread little more and ΔE_Ni will increase. Correspondingly, Si signal will exhibit a step due to the presence of Si in Ni₂Si [Fig. m5.8].

Fig. m5.8 Schematic backscattered spectra of Ni (100 nm) on Si wafer (few hundred micron) after inter-diffusion of atoms at Ni-Si interface
Now the ratio of the heights H_Ni/H_Si in the silicide layer gives a qualitative composition of the silicide layer formed at the interface. Approximately, the expression of the ratio of areal concentration can be written as

Now the yields of Ni and Si in the silicide is considered as the product of signal height and energy width and hence a better approximation of the areal concentration ratio of two elements uniformly mixed in a film can be written as:

In practice, the identification of various elements present in any material and determination of accurate composition require proper fitting of experimental data with simulation programme based on the theory of RBS. Some of the popular computer codes with references used for RBS data analysis are listed below:
1. RUMP by M. Thompson, Cornell University, USA, 1983.6 2. SIMNRA by M. Mayer, Max Planck Institute for Plasma physics, Germany, 1996. These codes are spectrum simulators, i.e. they calculate a spectrum for a given target structure and allow fitting of target compositions to measured spectra. 3. WiNDF by N. Barradas and C. Jeynes calculates the depth profiles of different elements from a given RBS spectrum. This is useful if one or several elements have a non-uniform depth distribution, for example due to diffusion.

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Rutherford Back Scattering Spectrometry (RBS)(Cont.)

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