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Physical Processes of Elastic Recoil Detection Analysis
The Basic chemistry of  Forward recoil scattering  process is considered to be  charged particle interaction with matters. To Understand Forward recoil spectrometry we should know the physics involved in Elastic and Inelastic collisions. In Elastic  collision only Kinetic Energy  is conserved in the scattering process and there is no role of particle internal energy. Where as in case of Inelastic collision both kinetic energy and internal energy are participated in the scattering process. Physical concepts of two-body elastic scattering are the basis of several nuclear methods for elemental material characterization.

Fundamentals of Recoil (Back Scattering) Spectrometry
The Fundamental aspects in dealing with recoil spectroscopy involves electron back scattering process of matter such as thin films and solid materials. Energy loss of particles in target materials is evaluated by assuming  that the target sample is laterally uniform and constituted by a mono isotopic element. This allows a simple relationship between that of penetration depth profile and elastic scattering yield

Main assumptions in physical concepts of Back scattering spectrometry

 * Elastic collision between two bodies is the energy transfer from a projectile to a target molecule.This process depends on the concept of kinematics and mass perceptibility.


 * Probability of occurrence of collision provides information about scattering cross section.


 * Average loss of energy of an atom moving through a dense medium gives idea on stopping cross section and capability of depth perception.


 * Statistical fluctuations caused by the energy loss of an atom while moving through a dense medium. This process leads to the concept of energy straggling and a limitation to the ultimate depth and mass resolution in back scattering spectroscopy.

Physical concepts that are highly important in interpretation of forward recoil spectrum are depth profile, energy straggling, and multiple scattering.(Tirira, 1996). These concepts are described in detail in the following sections :

Depth profile and Resolution analysis
A key parameter that characterizes recoil spectrometry is the depth resolution. this parameter is defined as the ability of an analytical technique to measure a variation in atomic distribution as a function of depth in a sample layer. In terms of low energy forward recoil spectrometry, hydrogen and deuterium depth profiling can be expressed  in a mathematical notation (Genzer, J, J, & R, 1993)

Δx =   ΔEtotal/(dEdet/dx) where δEdet defines as the energy width of a channel in a multichannel analyzer, and dEdet/dx is the effective stopping power of the recoiled particles.

Let us consider an Incoming and outgoing ion beams that are calculated as a function of collisional depth, by considering two trajectories are in a plane perpendicular to target surface, and incoming and outgoing paths are the shortest possible ones for a given collision depth and given scattering and recoil angles. Impinging ions reach the surface, making an angle θ1, with the inward-pointing normal to the surface. After collision their velocity makes an angle θ1, with the outward surface normal; and the atom initially at rest recoils, making an angle θ1, with this normal. Detection is possible at one of these angles as such that the particle crosses the target surface. Paths of particles are related to collisional depth x, measured along a normal to the surface. (Tirira. J., 1996)

This Figure is plane representation of a scattered projectile on the target surface, when both incoming and outgoing paths are in perpendicular to target surface

For the impinging ion,length of the incoming path L1 is given by :$$L_{1} = \frac{x}{\cos\theta_{1}}$$ The outgoing path length L2 of the scattered projectile is :$$L_{2} = \frac{x}{\cos\theta_{2}}$$

And finally the outgoing path L3 of the recoil is :$$L_{3} = \frac{x}{\cos\theta_{3}}$$

In this simple case a collisional plane is perpendicular to the target surface,the scattering angle of the impinging ion is θ = π-θ1-θ2 & the recoil angle is φ = π-θ1-θ3.

This Figure is plane representation of a Recoiled ion path on the target surface, when both incoming and outgoing paths are in perpendicular to target surface

Target angle with the collisional plane is taken as α, and path is augmented by a factor of 1/cos α.

For the purpose of converting outgoing particle in to collision depth, geometrical factors are chosen.

For recoil R(φ, α)is defined as sin L3 = R(φ, α)L1 R(φ,α) = ${cosθ_{1}cosα} \over{Sinφ \sqrt{cos^{2}α-cos^{2}θ_{1}} - cosθ_{1}cosφ} For forward scattering of the projectile R((φ,α)by:L2 = R(θ,α)L1 R(θ,α) = cos θ1cosα/Sin θ√(cos2α-cos2θ1)-cosθ1cosθ



Energy Depth Relationship
The energy E0(x) of the incident particle at a depth (x) to its initial energy E0 where scattering occurs is given by  the following Equations. (Tirira. J., 1996)


 * $$ E_{0}(x) = E_{0}- \int_{0}^{(x/cos\theta1)}S(E)\,dL_{1}---Equation(1)$$

similarly Energy expression for scattered particle is:$$ E_{1}(x) = KE_{0}(x)- \int_{0}^{(x/cos\theta2)}S(E)\,dL_{2}---Equation(2)$$

and for recoil atom is:$$ E_{2}(x) = K'E_{0}(x)- \int_{0}^{(x/cos\theta3)}Sr(E)\,dL_{3}---Equation(3)$$

The energy loss per unit path is usually defined as stopping power and it is represented by :$$S = \frac{dE}{dx}$$

Specifically, stopping power S(E) is known as a function of the energy E of an ion.

Starting point for energy loss calculations is illustrated by the expression:
 * $$\Delta E

= \int_{0}^{\Delta x} \frac{dE}{dx}\,dx$$

By applying above equation and energy conservation Illustrates expressions  in 3 cases


 * $$L_{1}

= \int_{E_0(x)}^{E_0} \frac{dE}{S(E)}$$


 * $$L_{2}

= \int_{E_1(x)}^{E_01(x)} \frac{dE}{S(E)}$$


 * $$L_{3}

= \int_{E_2(x)}^{E_02(x)} \frac{dE}{Sr(E)}$$

where E01(x)= KE0(x)and E02(x)=K'E0(x)

S(E)and S_r(E) are stopping powers for projectile and recoil in the Target material

Finally stopping cross section is defined by ɛ(E)= S(E)/N ɛ is stopping cross section factor.

To obtain energy path scale We need to evaluate energy variation δE2 of the outgoing beam of energy E2 from the target surface for an increment δx of collisional depth ,here E0 remains fixed. Evidently this causes changes in path lengths L1 and L3 a variation of path around the collision point x is related to the corresponding variation in energy before scattering : δL1 = δE0(x)/S[E0(x)-  Equation 5

Moreover particles with slight energy differences after scattering from a depth x undergo slight energy losses on their outgoing path. Then change δ L3 of the path length L3  can be written as δL3 =  δ(K’E0(x)]/ Sr[K’E0(x)) + δ(E2)/SrE2) -Equation 6 δ L1 is the path variations  due to energy variation just after the collision and  δ L3 is the path variation because of variation of energy loss along the outward path. Solving equations 5 and 6  by considering δ x = 0 for the derivative dL1/dE2 with L3=R(φα)L1,yields

dL1/dE2 = 1/{Sr(E2)/Sr[K’E0(x)]}{[R(φ,α) Sr[K’E0(x)+K’S[E0(x)]} ---Equation 7

In elastic spectrometry ,the term[S] is called as energy loss factor [S] =  K’S(E(x))/Cos θ1 + Sr(K’E(x))2Cos θ2 -Equation 8

finally stopping cross section is defined by ε(E) ≡S(E)/N where N is atomic density of the target material.

Stopping cross section factor [ε] = ((K^'ε(E(x) ))/cos θ1 )+(εr(K^' E(x) )/cosθ3)Equation 9

Depth Resolution
An important parameter that characterizes recoil spectrometer is depth resolution. It is defined as the ability of an analytical technique to detect a variation in atomic distribution as a function of depth. The capability to separate in energy in the recoil system arising from small depth intervals. The expression for depth resolution is given as

δRx = δET/[{Sr(E2)/SrK'E0(x)}][R(φ,α)SrK'E0(x)+K'SE0(x)]---Equation 10

Where δET is the total energy resolution of the system, and the huge expression in the denominator is the sum of the path integrals of initial, scattered and recoil ion beams.

Practical importance Of Depth Resolution
The concept of depth resolution represents the ability of Recoil spectrometry to separate the energies of scattered particles that occurred at slightly different depths δRx is interpreted as an absolute  limit for determining  concentration profile. From this point of view concentration profile separated by a depth interval of the order of magnitude of δRx  would be undistinguishable in the spectrum, and obviously it is impossible to gain accuracy better than δRx to assign depth profile. In particular the fact that the signals corresponding to features of the concentration profile separated by less than δRx  strongly overlap in the spectrum.

A finite final depth resolution resulting from both theoretical and experimental limitations has deviation from exact result when consider a ideal situation. Final resolution is not coincide with theoretical evaluation such as the classical depth resolution δRx  precisely because it results from three terms that escape from theoretical estimations: (Tirira. J., 1996) Incertitude due to approximations of energy spread among molecules. Inconsistency in data on stopping powers and cross section values

Statistical fluctuations of recoil yield (counting noise)

Influence of Energy Broadening on a Recoil spectrum
Straggling: Energy loss of particle in a dense medium is statistical in nature due to a large number of individual collisions between the particle and sample. Thus the evolution of an initially mono energetic and mono directional beam  leads to dispersion of energy and direction. The resulting statistical energy distribution or deviation from initial energy is called energy straggling. Energy straggling data are plotted as a function of depth in the material. (Tschal"ar, 1970)

Theories of energy straggling : Energy straggling distribution is divided into three domains depending on the ratio of ΔE i.e., ΔE /E where ΔE is the mean energy loss and E is the average energy of the particle along the trajectory.



1.Low fraction of energy loss: for very thin films with small path lengths ,where ΔE/E ≤ 0.01,Landau (Landau L., 1994) and Vavilov (Vavilov, 1957) derived that infrequent single collisions with large energy transfers contributes certain amount of loss in energy.

2. Medium fraction of energy loss: for regions where 0.01< ΔE/E ≤ 0.2. Bohr’s model based on electronic interactions is useful for estimating energy straggling for this case, and this model includes the amount of energy straggling in terms of the areal density of electrons traversed by the beam. (Bohr, 1915) The standard deviation Ω2B of the energy distribution is given by : Ω2B=4π((Z1e2)2NZ2∆x Where NZ2Δx is the number of electrons per unit area over the path length increment Δx.

3. Large fraction of energy loss: for fractional energy loss in the region of 0.2< ΔE/E ≤ 0.8,the energy dependence of stopping power causes the energy loss distribution to differ from Bohr’s straggling function. Hence the Bohr theory can not be applicable for this case. (Tschal"ar, 1970) Various theoretical advances were made in understanding energy straggling in this case. (Tschalar, 1984) (L'Hoir, 1984) An expression of energy for straggling is proposed by Symon in the region of 0.2< ΔE/E ≤ 0.5 is a function of momentums Mi( Mi = M1+M2 where M1 is stopping power,M2 is variation in straggling with depth of a stopping power) (Symon,1969)

Tschalar et al.(Tschalar,1970 and 1968) derived  a straggling function solving the differential equation: d Ω2/dx = S(E) .d Ω2/dE

The Tschalar’s expression which is valid for nearly symmetrical energy loss spectra, is Ω2 T = S2[E(x)]σ2(E) dE/S3(E)

Where σ2(E) represents energy straggling per unit length (or) variance of energy loss distribution per unit length for particles of energy E. E(x)is the mean energy at depth x.

Mass resolution
In a similar way mass resolution is a parameter that characterizes the capability of recoil spectrometry to separate two signals arising from two neighboring elements in the target. The difference in the energy δE2 of recoil atoms after collision when two types of atoms differ in their masses by a quantity δM2   is

δE2/ δM2 = E0 (dK’/dM2)

δE2/ δM2 = 4E0(M1(M1-M2)cos2φ/(M1+M2)2

Mass resolution δMR  (≡ δE2/ δM2).

A main limitation of using low beam energies is the reduced mass resolution. The energy separation of different masses is, in fact, directly proportional to the incident energy. The mass resolution is limited by the relative E and velocity v.

Expression for mass resolution is  ΔM = √(∂M/∂E.∆E)2 + √(∂M/∂v.∆v)2

ΔM = M(√((∆E)/E)2+√(2.∆v/v)2)

E is the Energy, M is the mass and v is the velocity of the particle beam.and ΔM is reduced mass difference.

Multiple scattering scheme in Forward Recoil Spectrometry
When an Ion beam penetrating in to matter, ions undergo successive scattering events and deviates from original direction. The beam of ions in initial stage are well collimated(single direction), but after passing through a thickness of Δx in a random medium their direction of light propagation certainly differs from normal direction. As a result both angular and lateral deviations from the initial direction can occur. These two parameters are discussed below. Hence, path length will be increased than expected causing fluctuations in ion beam. This process is called multiple scattering, and it is statistical in nature due to the large number of collisions.

Lateral displacement Case 1

Ion beam fluctuations because of lateral deviations on Target surface is explained by considering Multiple scattering of an ion beam which is directed in x – direction.



Angular deviation Case 2

In the below figure there is a considerable difference between in the shape area of  Gaussian peak (ideal condition) and angularly deviated peak. (Williams, 1929) and α is an angle due to angular deviation of a penetrated ion beam through matter.



Theory and Experimental work involved in Multiple Scattering phenomena
In the study of Multiple Scattering phenomenon angular distribution of a beam is important quantity for Consideration. The lateral distribution is closely related to the angular one but secondary to it, since lateral displacement is a consequence of angular divergence. Lateral distribution represents the beam profile in the matter. both lateral and angular Multiple scattering distributions are interdependent The analysis of Multiple Scattering was started by Bothe(Bothe, W,1921) and Wentzel (Wentzel, G,1922)in the Nineteen twenties using well-known approximation of small angles. The physics of energy straggling and Multiple Scattering was developed quite far by Williams from 1929 to 1945. (Williams, 1929)Williams devised a theory, which consists of fitting the Multiple Scattering distribution as a Gaussian-like portion due to small scattering angles and the single collision tail due to the large angles. William, E.J., studied beta particle straggling, Multiple scattering of fast electrons and alpha particles, and cloud curvature tracks due to scattering to explain Multiple scattering in different scenario and he proposed a mean projection deflection occurrence due to scattering. His theory later extended to multiple scattering of alpha particles. Goudsmit and Saunderson provided a more complete treatment of Multiple Scattering, including large angles. (Goudsmit, 1940). For large angles Goudsmit considered series of Legendre polynomials which are numerically evaluated for distribution of scattering .The angular distribution from Coulomb scattering has been studied in detail by Molière(Molière,1948) and further continued by Marion and coworkers. Marion, J.B., and Young, F.C., in their Nuclear Reaction Analysis provided Tabular information regarding energy loss of charged particles in matter, Multiple scattering of charged particles, Range straggling of protons, deuterons and alpha particles, equilibrium charge states of ions in solids and energies of elastically scattered particles very detaildly.( (Marion, 1968) Scott presents a complete review of basic theory, Mathematical methods, as well as results and applications( (Scott, 1963). A comparative development of Multiple Scattering at small angles is presented by Meyer, based on a classical calculation of single cross section. (Meyer, 1971) Sigmund and Winterbon reevaluated Meyer’s calculation to extend it to a more general case. Marwick and Sigmund carried out development on lateral spreading by Multiple Scattering, which resulted in a simple scaling relation with the angular distribution.( (Marwick, 1975)