Rutherford Backscattering Modeling Algorithms

Shuvan PrashantPC5209 Coursework

Nuclear Instruments and Methods in Physics Research B9 (1985) 344-351

Motivation

• RBS analysis algorithms though accurate but computationally intensive

• Takes a lot of time on small computers• Rapid simulation with good assumptions can fit

the RBS spectra with reasonable accuracy

Computers are useless. They can only give you answers. Pablo Picasso

Final AimCompute Spectra with normalized yield vs. energy

Assumptions• Sample stack of sublayers• Sublayer uniform composition and fixed

energy loss function(E dependent only)• Sublayer should not be too thick • Elastic Scattering• Screening for low energies can be incorporated • Detector resolution Gaussian convolution• Straggling intensive but possible

Formation of a brick

• Each contribution is known as a brick

• Brick energy location energy lost by beam after scattering on its outward path through different sublayers.

Yiel

dEnergy

Area Q

eb,yb ef,yfef

eb

E0

Energy Loss Evaluation• Geometry => Angle• Beam Energy Loss

ef

eb

E0

)(EdadE

• Stopping Cross-section ε(E) • 5th order polynomial fit from elemental data • Bragg rule for compounds

• a - Path length into material in areal density units

To calculate E(Ntsecθ)Expand using Taylor Series

...61

21)0()(

03

33

02

22

0

daEda

daEda

dadEaEaE

Surface Approximation( Upto first order)

...)'''(')( 223612

21

0 aaaEaE

Using the definition of ε(E) and evaluating higher differential terms ε’ and ε’’,

)(EdadE

Energy Location• Assuming elastic scattering,• Eafter prop to Ebefore • Evaluate Kinematic factors for different

elements

2

1

222

1

cossin1MMwhereK

Building Spectrum • Superpose contributions from each isotope in

sublayer in the sample• Spectrum Calculation involves– Energy Loss evaluation in each sublayer– Final Interpolation of the spectrum

• Shape of the brick Trapezoidal bricks may have kinks if the sublayers are thick

• Area not accurate• How can we solve this ?

Solution• Assume parabolic top profiles

• Rutherford Scattering Cross-section for a small solid angle

sec

0

))(( AreaNt

daaE

22

4

22221

sin1cos

cossincoscos

2)(

where

EeZZE

2)( generalin nuclidesFor CEE

order thirdof polynomial a using

edapproximat is E(a) where)( Area 2-sec

0

2 Nt

daaEC

Height Estimation

)()()]([ )()(

cos)]([)(

EKEKEwhereEE

EExy

AAAi

layers out

in

Ai

iAA

Screening Effects can be accounted

keVwhereaEpEE R 4/3

21Z Z0.049 a

p )1)(()(

For high Z elements and 2MeV beam, the deviation of cross-section is about 2% .

Virtual MCA

• Using values of eb, ef, yb, yf and Q , evaluate the coefficients A, B and C

• Virtual MCA evaluates the expression at boundary points of the channels and substracts to get the yield per channel.

32

2

32

CeBeAedeheightYield

CeBeAheight

Computation• Stage 1 – Calculate energy on inward path and Rutherford

Integrals prop to # of sublayers• Stage 2 – Outward energy loss for each nuclide present at

interface – Interface.nuclide.depth # of sublayers

• Stage 3 – Stopping cross-sections

Straggling

• Occurs because of the statistical nature of energy loss

• Energy loses monochromaticity and becomes gaussian in profile

• Limiting in resolution• Bohr’s formula used

for calculating the amount of straggling

Finally,

Pros1. Simple and fast2. AccurateCons3. Resonance calculations are

not possible4. Nuclear reaction analysis is

not possible.5. Screening effects are

accounted only upto first order

6. Channeling effects

Thanks for your attention

• Q & A

Si Energy Loss Evaluation

Pt Energy Loss Evaluation