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Rutherford Backscattering Modeling Algorithms
Shuvan PrashantPC5209 Coursework
Nuclear Instruments and Methods in Physics Research B9 (1985) 344-351
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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
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Final AimCompute Spectra with normalized yield vs. energy
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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
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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
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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
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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
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Energy Location• Assuming elastic scattering,• Eafter prop to Ebefore • Evaluate Kinematic factors for different
elements
2
1
222
1
cossin1MMwhereK
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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 ?
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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
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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% .
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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
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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
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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
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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
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Thanks for your attention
• Q & A
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Si Energy Loss Evaluation
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Pt Energy Loss Evaluation