Junior Comprehensiv

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Chuna

Bremsstrahlung Interactions that occur after Beta Minus Decay

Vocabulary• Beta Minus Decay

• End Point Energyo Difference in Rest Mass between

Parent and Daughter particles

• Bremsstrahlung Interaction

• E[F]- Expectation value function E[F] =

Introduction

Probability

Equation

Understanding The Equation

Probability Distribution Equation

P()d=(1+−)[ln() ]• The “Extremely Simplified” Probability Distribution of

emitting a gamma with energy between and when the electron starts with an initial energy of

-Comprehensive Writer-

Introduction

Probability

Equation

Understanding the

Equation

1000 2000 3000 4000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006Probability

Energy

P()d=(1+−)[ ln ()− ] - electron initial energy - classical electron radiusα - fine structure constant, Z - atomic number of the medium in which the bremsstrahlung is taking placeK - material constant

Substituting Variables ε = , where E is the energy after one bremsstrahlung event

= −E → E =

Probability

Equation

Understanding the

Equation

Formatting

Equation

Note: Specifics – Equation above is for general Bremsstrahlung interactions, it must be formatted to beta decay.Caution – There are multiple Decay Types for beta decay

Energy (keV)

End-point energy

(keV)

Intensity (%)

Dose ( MeV/Bq-s )

   866.8 10   2058.02 22       0.0082 % 6    7.1E-5 5   2481.5 10   5390.86 8      99.9913 % 8    2.4813 10   3286.2 10   7024.53 8       5E-4 % 5    1.6E-5 16 

Beta-:

Understanding Equation

Formatting

Equation

Formatting

Equation

Note : SpecificsBeta Decay of emits an Electron spectrum

To understand the spectrum of Consider = ¾, ½, and ¼ of the endpoint. = 4554.14 keV = .75 * 5390.86 +511.0034 →{F}= 3206.43 keV = .50 * 5390.86 +511.0034 →{G}= 1858.72 keV = .25 * 5390.86 +511.0034 →{H}

(I have included the rest mass of electron)

Formatting

Equation

Formatting

Equation

Normalizing the

Equation

Normalization• F=(-k*4*(Z^2)(re^2)a*/((4.8793×10-61)x))

(1+ (((Eoa-x)/(Eoa))^2)-(((2/3)(Eoa-x))/(Eoa)))(Log[(2*(Eoa)*((Eoa-x)))/(me*(c^2)*(x))]-.5)

• = 4.8793×10-61

• = 4.64323×10-61 • = 4.25774×10-61

Formatting Equation

Normalizing Equation

Probability Distributio

n

Normalizing the

Equation

Probability

Distribution

Expected Photon Energy

Probability Distribution of Photon EnergyF=(-k*4*(Z^2)(re^2)a*/((4.8793×10-61)x))(1+ (((Eoa-x)/(Eoa))^2)-(((2/3)(Eoa-x))/(Eoa)))(Log[(2*(Eoa)*((Eoa-x)))/(me*(c^2)*(x))]-.5)

Plot[ {F,G,H}, {x, 10 , Eoa}]

1000 2000 3000 4000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006Probability

Energy of Emitted Photon (keV)

= 4554.14 keV → {F}

= 3206.43 keV →{G}

= 1858.72 keV →{H}

Mathematica

Expected Photon Energy

Univariate Continuous Distribution of a Random Variable

• E[F] = • E[G] = • E[H] =

<F˃ = 672.763 keV<G> = 502.016 keV<H> = 321.218 keV

Probability

Distribution

Expected Photon Energy

Application to

Experiment

Transmission/Attenuation

Photons’ have 0% transmission rate

photons’ have >0% transmission rate

All the energy given off by Bremsstrahlung interactions trapped inside the crystal.

If no other energy escapes due to particle interaction than detector measures all other initial energy

Some photons escape with some of the electrons initial energy

The detector measures less than initial energy

Expected Photon Energy

Application to

Experiment

F Transmissio

n

F Transmission• 672.763 keV - Average Energy of Photon

given off due to Bremsstrahlung interaction• RESULTS:• Attenuation [%]: 88.0411

Transmission [%]: 11.9589 – Likelihood of photon escaping the medium

• Interaction Probability [%]:Photo absorption: 9.6587Compton Scattering: 75.3758Rayleigh Scattering: 3.0066Nuclear Field Pair Production: 0.0000Electron Field Pair Production: 0.0000

Application to

Experiment

F Transmissio

n

G Transmissio

n

G Transmission• 502.016 keV - Average Energy of Photon

given off due to Bremsstrahlung interaction• RESULTS:• Attenuation [%]: 92.9071

Transmission [%]: 7.0929 -Likelihood of photon escaping the medium

• Interaction Probability [%]:Photo absorption: 16.3393Compton Scattering: 72.0658Rayleigh Scattering: 4.5019Nuclear Field Pair Production: 0.0000Electron Field Pair Production: 0.0000

F Transmissi

on

G Transmissio

n

H Transmissio

n

G Transmissi

on

H Transmissio

nResults

H Transmission• 321.218 keV - Average Energy of Photon

given off due to Bremsstrahlung interaction• RESULTS:• Attenuation [%]: 98.5999

Transmission [%]: 1.4001 - Likelihood of photon escaping the medium

• Interaction Probability [%]:Photo absorption: 35.1417Compton Scattering: 56.3727Rayleigh Scattering: 7.0855Nuclear Field Pair Production: 0.0000Electron Field Pair Production: 0.0000

“Observed” Beta Spectrum

1” NaI Crystal 100” NaI Crystal

H Attenuation

Results Understanding Results

Counts

Energy (MeV) Energy (MeV)

ResultsUnderstanding

ResultsBibliograp

hy

Calculated Efficiency Ratio

1” NaI Crystal 100” NaI Crystal

Ratio

Energy (MeV) Energy (MeV)

Bibliography• http://web-docs.gsi.de/~stoe_exp/

web_programs/x_ray_absorption/index.php• http://www.wiikipedia.com – used to develop

general understanding and look up coefficient values.

H Attenuatio

n

Understanding Results

Bibliography