2 Backscattering Spectroscopy : Rutherford Back Scattering

Dr. Ajit K. Mahapatro

Learning Objectives:

From this module students may get to know about the following:

i. Scattering Crossection

ii. Energy Straggling

iii. Concepts of Rutherford Back Scattering

1. Inroduction

There are four physical concepts govern the back scattering spectrometry. These are kinamatic factor, scattering cross section, stopping cross section and energy straggling. Kinematic factor gives information on the elastic energy transfer between the projectile and the target. The likelihood of occurrence of scattering leads to the concept of scattering cross section. The concept of stopping cross section gives an idea of the energy loss when the incident particle moves through a dense medium. Finally, statistical fluctuations in the energy loss of the scattered particle can be estimated through the concept of energy straggling. We would focus on the above mentioned concept in this session.

2. Scattering Cross Section

So far we have established the relation between the energy E0 of the incident particle of mass M1 and the energy that this particle possesses after an elastic collision with another particle of mass M2 scattered at an angle θ. Note that M2 is in a stationary state before and after the collision. In order to estimate the frequency of such a collision that really occurs at an angle θ quantitatively, concept of differential scattering cross section ???? is important and will be described in detail below. A narrow beam of incident particles falls on a target which is thin and uniform. If we place a detector at an angle of θ from the direction of incidence, then it can count each particle scattered in the differential solid angle dΩ (see Fig.1). The differential scattering cross section dσ/dΩ is defined as

??/??=(1/??)[(??/?Ω)/?]

Figure 1: Schematics of simple scattering exeriment

Where Q is the total number of particles that have hit the target and dQ is the number of particles recorded by the detector. N is the volume density of atoms in the target and t is its thickness. Thus Nt is the number of target atoms per unit area which is also called areal density. It follows that the solid angle dΩ is so small that the scattering angle θ is well defined. Note that the thickness t is assumed to be minimal and, therefore, the energy loss of the particles in the target is so small that the energy of the particles is virtually the same at any depth in the target. It is worth noting that Q must be large enough to have the ratio dQ/Q has a well-determined value. The dimension of the differential scattering cross section dσ/dΩ is that of an area (“cross section”) whose meaning is based on the geometrical interpretation of the probability that the scattering will yield a signal at the detector. We suppose that each nucleus of an atom expose an area dσ/dΩ to the beam of incident particles. We also assume that this area is small and they are randomly distributed inside the target and they do not overlap with each other. Let S be the surface area of the target hit uniformly by the beam. Then the total number of atoms for a scattering event in the target is SNt. The ratio of the total cross-sectional area of all such atoms SNt dσ/dΩ to the area S actually participated can be defined as the probability that the scattering event will be recorded by the detector; this ratio is equal to (l/dΩ)dQ/Q. Here, the dΩ-1 in the above expression is due to the fact that incresing the solid angle dΩ two fold would double the number of counts dQ. By dividing dQ with dΩ, this geometrical contribution to the number of counts dQ is eliminated. The unit of cross section is a value per unit of solid angle; hence the term differential scattering cross section arises. The number of scattering events falling within a finite solid angle Ω can be described by the integral scattering cross section.

The above equation shows that any parameter can be estimated provided the other factors are known to satisfy this equation. The ability of backscattering spectrometry to provide quantitative information on the number of atoms present per unit area of a sample stems from Eq. (4) and the fact that the average scattering cross section σ of the elements is known quite accurately

Figure 2: Schematics of a back scattering event.

Differential scattering cross section for elastic collision is given by,

where the subscript c indicates the center-of-mass frame of reference system that is considered here. Here Z1 is the atomic number of the incident particle with mass M1 and Z2 is the atomic number of the target atom with mass M2, e is the electronic charge (e = 4.80 x 10-10 statC), and E is the energy of the incident particle just before scattering. This formula is valid also for values in the laboratory frame of reference, but only when M1 ˂˂ M2. For the general case, the transformation of this formula from the center-of-mass to the laboratory frame of reference is given by

where, the first omitted term is of the order of (Ml/M2)4. The last expression shows the functional dependence of the Rutherford differential scattering cross sections:

• ??/?Ω is proportional to square of the atomic mass of the target atom (z1)² and accordingly the yield would be proportional to the atomic mass number.

• ??/ ?Ω is proportional to square of the atomic mass of the projectile atom (z2)^2. Hence heavier projectile would yield more compared to the lighter ones.

• ???Ω is inversely proportional to the square of the energy of the projectile and the yield of back scattered particle would be proportional accordingly.

• ???Ω depends only on θ and hence axially symmetric.

• ???Ω is inversely proportional to the fourth power of the Sinθ/2 when M1 ˂˂ M2. Hence the reduced scattering angle yields more back scattered particle.

3. Energy loss and stopping cross section

The large-angle Rutherford scattering collision occurs rarely and hence the energetic incident particle will enter into it due to the low angle scattering behavior. Backscattering spectrometry is an analytical method to the secondary process. Note that the first-order process is the implantation of the beam particles into the target.
Penetration of energetic particle into the matter can be estimated from the energy profile of the incident beam. As the particle impinges through the target, it slows down and its kinetic energy E = 1/2mv2 decreases. The amount of energy ΔE lost per distance Δx traversed depends on the velocity of the projectile, on the density and composition of the target. This can be realized by performing a simple experiment by taking a very thin target of thickness Δx and of known composition. A beam of particles impinge on this target (see Figure below). The energy difference ΔE of the particles before and after transmission through the target is recorded. The energy loss per unit length, also called the specific energy loss, and frequently abbreviated dE/dx loss, at the energy E of the incident beam is then defined as

Figure 3: Schematics of energy loss process by a thin target.

    lim

Δ?→0 Δ?/Δ? ≡ ????(?)                                         (7)

For backscattering spectrometry, it is the energy loss of ⁴ He in the elements at energies between 0.5 and 3 MeV that is of chief concern, because beams of ⁴He in that energy range are most frequently used. Typical dE/dx values for ⁴He of that energy range lie between 10 and 100 eV/A⁰.

4. Stopping Cross Section Ɛ

The energy loss dE/dx accounts for the energy that the incident particle loses when it comes in contact with the target atom during scattering. The value of dE/dx can be viewed as an average over all possible energy dissipative processes activated by the projectile on its way past a target atom. Thus dE/dx can be calculated as the resultant of independent contributions of every atom exposed to the beam. This number is SN Δx if Δx is the thickness of the target, S is the target area illuminated by the beam, and N the atom density in the target. The projection of all these atoms on the area S produces a surface density of atoms SN Δx /S = N Δx. This quantity increases linearly with Δx, as does the energy loss ΔE = (dE/dx) Δx. We therefore set ΔE proportional to N Δx and define the proportionality factor as the stopping cross section Ɛ:

Ɛ = (l/N)(dE/dx).

The conventional unit for Ɛ is electron volts·square centimeters per atom (eV cm2).

5. ENERGY STRAGGLING

In a scattering process, the incident particle that interacts with a target during the scattering loses energy via many individual encounters, which (energy loss) is subject to statistical fluctuations. For this identical particles, having the same initial velocity, do not have exactly the same energy after passing through a thickness Δx of a homogeneous medium. The energy loss ΔE is subject to fluctuations. The phenomenon, sketched in Fig. 4, is called energy straggling. Energy straggling places a finite limit for the precision with which energy losses, and hence depths can be estimated by backscattering spectrometry. The ability to identify masses is also impaired, except for atoms located at the surface of the target. The reason is that the beam energy E before a collision with a specific mass M2 at some depth within the target is no more monoenergetic, even if it was so initially, so that the ratio E1/E0 and hence the identification of M2, become uncertain as well. For these reasons, it is important to have quantitative information on the magnitude of energy straggling for any given combination of energy, target material, target thickness, and projectile.

6. Concept of Rutherford backscattering

The aim of this topic is to give a basic idea about the concept of Rutherford backscattering with the help of basic concepts developed early in this chapter. It will give a sound knowledge on how the scattering works and the related parameters are estimated.

Figure 5: The schematic diagram of a backscattering system

The schematic diagram of a backscattering system is shown in Fig. 5. A beam of collimated and monoenergetic particles of energy E0 is generated by a source as shown in Fig. 5. Typically a current of 10 to 100 nA of 2.0-MeV He+ ions in a 1-mm2 area is generated. The beam of monoenergitic particle falls on the target. After it hit the target, particles much less than one in 104 are scattered back out of the sample. A small fraction of this incident is captured by the aperture of an analyzing system. In this way, we get analog signal as the output of that system. This signal is taken and fed back to a multichannel analyzer, which subdivides its magnitude into a series of equal increments. Each subdivision is referred to as a channel and is given a unique number which increases by a fixed increment. Modern multichannel analyzers contain thousands of channels. A count is defined as an event whose magnitude falls within a particular channel. Each channel registers a certain number of counts. The output of the multi channel analyzer is thus a series of counts contained in various channels.

Figure 6: Basic content of a back scattering spectrum and recording the data.

A segment of such series from channels 132-134 is shown in Fig. 6. The corresponding counts contained in channel i is represented as Hi. The qualitative analysis of the spectrum can be done by the graphical display. Digital outputs are used for numerical analysis. Such a series of counts versus channel number constitutes a backscattering spectrum. In the graphical display, the ordinate is frequently labeled yield or backscattering yield. The analog signal recorded by the analyzer can be analyzed which contains quantitative information on different parameters of the detected particle including energy, momentum etc. The backscattering spectrum obtained with such an analyzer is a backscattering momentum spectrum. A semiconductor particle detector produces an analog signal proportional to the energy of the backscattered particle. Correspondingly, a spectrum obtained with such a detector is a backscattering energy spectrum

Figure 7: Schematics of particle analyzing system.

The main component of the analyzing system consists of energy dispersive analyzer, amplifiers and a multi channel analyzer, and schematized in Fig. 7. Multichannel analyzer can be replaced by a single channel whose position is changed sequentially so as to scan the range of the parameter measured. In general, there is a correspondence between the channel number and the magnitude of the particle parameter to be measured by the analyzer. The most desirable property of this relationship is that it should be exactly linear and stable in time. Other important points regarding the designing of the detector is a separate issue which can be optimized to get fast acquisition of data. The semiconductor surface-barrier detector and a charge-sensitive preamplifier are used almost universally in backscattering spectrometry.

Summary:

i. The four basic physical concepts which govern the back scattering spectrometry including kinamatic factor, scattering cross section, stopping cross section and energy straggling were discussed in detail. Concept of Rutherford back scattering was introduced. When the target is single-crystalline, the cross sections are clustered along sets of lines in space.

ii. If the incident particles move in a direction parallel to such lines, and if the flux of these particles is concentrated in the voids (“channels”) surrounding these lines, the probability of a scattering collision is reduced. This is the situation commonly referred to as “channeling” which will be discussed later.

iii. Laboratories engaged in backscattering analysis have developed computer programs to calculate backscattering spectra. Most of these programs are tailored to meet the specific needs of the respective laboratories. Mostly FORTRAN programme is suitable for this purpose.

iv. To calculate the differential cross section for an elastic collision, the principles of conservation of energy and momentum are complemented by a specific model for the force that acts during the collision between the projectile and the target masses. In most cases, this force is very well described by the Coulomb repulsion of the two nuclei as long as the distance of closest approach is large compared with nuclear dimensions, but smaller compared with the Bohr radius a₀ = 0.53 A⁰.

References:

1.Leighton, R. B. (1959). “Principles of Modern Physics.” McGraw-Hill, New York.

2.Lindhard, J. (1969). Proc. Roy. Soc. London A311, 11.

3.Lindhard, J., and Scharff, M. (1953). Mat. Fys. Medd. Dan. Vid. Selsk. 27, No. 15.

4.Lodhi, A. S., and Powers, D. (1974). Phys. Rev. A 10, 2131.5.Marion, J. B., and Young, F. C. (1968). “Nuclear Reaction Analysis, Graphs and Tables.” Wiley, New York.

6.Matteson, S., Chau, E. K. L., and Powers, D. (1977). Phys. Rev. A15, 856.

7.Mayer, J. W., Eriksson, L., and Davies, J. A. (1970). “Ion Implantation in Semiconductors.” Academic Press, New York.

8.NorthclifTe, L. C. (1960). Phys. Rev. 120, 1744.

9.Northcliffe, L. C. (1963). Ann. Rev. Nucl. Sei. 13, 67.

10.Powers, D., Chu, W. K., Robinson, R. J., and Lodhi, A. S. (1972). Phys. Rev. A 6, 1425.

11.Powers, D., Lodhi, A. S., Lin, W. K., and Cox, H. L. (1973). Thin Solid Films 19, 205.

12.Robinson, M. T., (1974). Oak Ridge Rep. ONRL-4556.

13.Rousseau, C. C , Chu, W. K., and Powers, D. (1971). Phys. Rev. A 4, 1066.

14.Sauter, G. D., and Bloom, S. C. (1972). Phys. Rev. B 6, 669.15.Schi^tt, H. E. (1973). Interaction of Energetic Charged Particles with Solids, p. 6. Brookhaven Nat. Lab. Rep. No. BNL-50336