2. RUTHERFORD BACKSCATTERING SPECTROMETRY Basic Principles

ELASTIC COULOMB SCATTERING Consider an ion of charge z, mass m and initial kinetic energy Eo being scattered elastically from a stationary nucleus of charge Z and mass M purely by the Coulomb force. The final energy E of the scattered ion is a function of the angle of scatter  from the initial direction and the ratio M/m. M, (Eo – E) M, 0 Recoiling nucleus m, Eo  Scattered ion m, E

KINEMATIC FACTOR The scattered E can be derived from the principles of conservation of energy and momentum, and is given by (in laboratory frame of reference): The multiplication factor k on the right hand side of the equation is often referred to as the kinematic factor. For M/m > 1, k is a slow-varying function of , having the maximum value of 1 at  = 0 and the minimum value at  = 180o. For M/m = 1, the value of k is zero beyond 90o.

Plots of k vs  for M/m = 0, 4, 20 M/m = 20 M/m = 4 M/m = 1

Plots of k vs M/m for  = 180o The small gradient at large M/m ratios implies that mass separation is poor for higher M/m ratios.

DIFFERENTIAL SCATTERING CROSS SECTION The theoretical differential cross section for a given scattering angle is given by: The higher order terms are usually negligible for M>m and the differential cross section can be expressed in the following form, which has the unit of mb/sr (millibarns per steradian) when MeV is used as the unit for Eo:

ds/d versus q The dependence of ds/d on sin-4(q/2) means that ds/d has very large values at small forward scattered angles and approaches infinity at 0o. However, for q = 90o to 180o, it decreases rather slowly because sin-4(q/2) drops gradually from a value of 4 at 90o to 1 at 180o. The diagram below shows the plots of ds/d vs q for the Coulomb scattering of 2 MeV 4He+ ions for oxygen and calcium. Ca (Z=20, A=40) O (Z=8, A=16)

ds/d versus Eo ds/d is inversely proportional to the square of the incident ion energy Eo, implying that it increases with decreasing Eo. This also means that for a thick target, more ions are scattered from a greater depth. The diagram below show a plot of ds/d vs Eo for the scattering of 4He+ from an aluminium target at q = 180o. Projectile: 4He+ Target: Al Scatter angle: 180o

RBS YIELD FOR THIN TARGETS The number of backscattered ions detected by a particle detector at an angle of q and subtending a solid angle  at the target is called the yield. Its relation with the differential cross section is as follows: where no is the number of incident ions and nz is the areal concentration of the target atom. The simple relation is only valid for thin targets, namely targets which are so thin that the loss of energy by an incident proton in them is negligibly small.

RBS SPECTRUM OF A THIN Pt-Co ALLOY TARGET The diagram below shows a simulated RBS spectrum at q = 160o for a 200A thick Pt-Co alloy target and 2 MeV 4He+ incident ions. The width of the Pt and the Co peaks is essentially attributable to the intrinsic resolution of the detector plus the electronic noise. The ratio of Pt to Co in the alloy is 1:1. the Pt peak is much higher than the Co peak because the yield for an element is proportional to the square of it atomic number Z (Z=27 for Co and Z=78 for Pt). Thickness = 200A Pt Co

EFFECT OF TARGET THICKNESS If the thickness of the Pt-Co target is increased, the peaks would be broadened due to the loss of energy by the 4He+ ions in the target through the ionization process. The following simulated RBS spectrum depicts the effect for the case where the Pt-Co alloy thickness is 1000A. Thickness = 1000A Pt Pt Co Co

SIMULATED RBS SPECTRUM Target : Pt-Co (1:1) alloy of thickness 6000A Incident ion: 2 MeV 4He+ Scatter angle: 160o Co edge Pt edge Pt Co

SIMULATED RBS SPECTRUM Target : Pt-Co (1:1) alloy of thickness 12000A Incident ion: 2 MeV 4He+ Scatter angle: 160o Pt edge Co edge Co Pt

SPECTRUM ANALYSIS The technique used to extract information from a RBS spectrum is similar to that used to extract information from a PIXE spectrum. Namely, a model spectrum is constructed and fitted to the experimental spectrum. If the fit is not satisfactory, the parameters for constructing the model spectrum are adjusted and fitting is repeated. This is basically an iterative process to find the best values for the parameters. With the RBS software packages available nowadays, the adjustment of parameters cannot be done automatically. The values of the parameters must be changed manually and the goodness of fit inspected visually. Download simulation code SIMNRA from website

MODEL PARAMETERS Two sets of parameters are needed to construct a model RBS spectrum. One set describes the structural and chemical properties of the specimen. It consists of the following parameters: number of layers, thickness of each layer, and chemical composition of each layer. The other set specifies the experimental conditions. It includes: type of ion beam used, total irradiation charge, beam-target angle, beam-detector angle, solid angle subtended by the detector at the target, the energy calibration parameters, and the detector resolution.