Basic detector principles and counting statistics

Experimental Techniques for

Nuclear and Particle Physics

Statistical models of signal generation –short review

The binomial distribution

( )∑ −=−=

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛=−

−= −−

)1()(

)1()1(!)!(

!)(

22 pnpxPxx

pnx

ppxn

ppxxn

nxP xnxxnx

σ

Roll a die Success=3,4,5 or 6

P=4/6 N=10

The Poisson distribution

( )

xpnx

xexxPxx

=

=

=−

2

)(!

)(

σ

Birthday today P=1/365 N=1000

The distribution of times between successive events

A detector detects a reaction rate, rAn event has ocurred at time t = 0 Then the distribution function to describe the time intervals between adjacent random events is:I(t)dt = P(0)rdtThen the probability to have no events during the time t is:

P(0) =rt( )0 e−rt

0!= e−rt

The probability to have the next event in the time period dtI(t)dt = P(0)rdt = re−rtdt

The Gaussian distribution

2

2

2

2)(

2

2

2

2)()()( ;

:form Normalised

2)(

) mean, fromdeviation offunction a (as :form Continuous )(

21)(

t

x

xxx

eGdtdGtGt

ex

G

xpnx

ex

xP

−

−

−−

===≡

=

==

=

πσε

εε

σε

πε

ε

σ

π

ε

The Chi-squared distribution χ 2 =

1xe

xi − xe( )2=

(N −1)s2

xei=1

N

∑

(where xe =1N

xi and s2 =1

N −1i=1

N

∑ xi − xe( )2

i=1

N

∑ )

The degree to which χ 2 differs from N -1 is a measure of the departure of the data from thepredictions of the Poisson model. Set χ 2 = u, then it can be shown that:

P(u)du =u / 2( ) ν /2( )−1 exp(−u / 2)

2Γ(ν / 2)du

µ =ν, σ 2 = 2ν

The Chi-squared distribution χ 2 =

1xe

xi − xe( )2=

(N −1)s2

xei=1

N

∑

(where xe =1N

xi and s2 =1

N −1i=1

N

∑ xi − xe( )2

i=1

N

∑ )

The degree to which χ 2 differs from N -1 is a measure of the departure of the data from thepredictions of the Poisson model.

The reduced χ 2 is defined as:

χ 2red =

χ 2

ν(where ν = N-1 is the number of degress of freedom)which should be close to 1 for a good fit

Maximum likelihood method - how to estimate the parameters of a probabilistic

model from measured data

L(θ, x) = f (θ, x1) f (θ, x2 ) f (θ, x3) f (θ, x4 )... f (θ, xn )is the probability for observing the sequence of valuesx1,...xnThe maximum likelihood principle states that this probability is maximumfor the observed values. Then:dLdθ

= 0 ⇒ θ̂, the estimator for the parameter θ.

Alternatively one can solve d(lnL)dθ

= 0

The uncertainty in θ̂ is given by σ 2 (θ̂ ) = − d 2 lnLdθ 2

#

$%

&

'(

−1

The method works if the form of the theoretical distribution is known. If the theoretical distribution is f(θ,x), where x is the observable for which independent measurements are made then:

Error Propagation

Standard deviation follows the Error propagation formula:

x, y, z… independent variables

u: quantity to be obtained from the data:

Mean value of multiple measurements (Poisson statistics)

Σ = xi σ Σ2 = σ xi

2

i=1

N

∑i=1

N

∑

σ xi= xi ⇒σ Σ

2 = xi = Σ ⇒i=1

N

∑ σ Σ = Σ

x = ΣN⇒ σ x =

NxN

=xN

The standard deviation expected for the sum of all the counts is the same as if the measurement had been carried out by performing a single count extended over the entire period.

σ xi= xi

σ x =NxN

=xN

The mean value based on N independent counts will have an smaller error compared with any single measurement

Combination of multiple measurements with unequal errors

What is the best value for a given measured quantity if N measurementshave been made, each with a different uncertainty?Each measurement is given a weight, ai.

Then the weighted mean becomes: xopt=

aixii=1

N

∑

aii=1

N

∑

aj =1σ x j

21σ xi

2i=1

N

∑"

#$$

%

&''

−1

(here aj are normalized)

1σ x

2 =1σ xi

2i=1

N

∑ Each data point should be weighted inversely as the square of its own error.

ai should be chosen to minimize the error

Optimization of Counting experiments The principle of error propagation can be applied in the design of counting experiments to minimize the associated statistical uncertainty.

Example: S: counting rate due to the source alone without background B: counting rate due to background

N1 and N2 are the total counts in each measurement

Optimum time:

Detector energy resolution: Response function for different detectors

Full Width at Half Maximum of the peak (FWHM) = 2.35 σ

Resolution = ΔE/E

H0 = KN σ = KN1/2

K: proportionality cte

Detector energy resolution: The Fano factor

When the energy of incoming radiation is measured in a detectorthe signal is typically formed by a large number of charge carriers or photons.If the energy is deposited by a particle passing through the detector the fluctuationsare given by the Poisson distribution: i.e. σ 2 = J, where J is the number of charge

carriers or photons (E /w) and the energy resolution is: R = 2.35 JJ

But if the full energy is absorbed the fluctuations can be limited since the ionisation eventsthen are not independent. Then σ 2 = FJ, where the Fano factor F is a number less than 1.Different materials have different Fano factors.The energy resolution is given by:

R = 2.35 FJJ

= 2.35 FwE

,

where w is the energy needed to produce one charge carrier or photon.For germanium detectors, F is typically around 0.1 whereas scintillators have F ≈1.

Detector energy resolution: The Fano factor

The Fano factor for Ge detectors is about 0.12 and the energy needed to create a pair e-h at 77K is 2.96 eV. Which will be the resolution for 1.33 MeV γ rays?

R = AEE

= 2.35 FwE

= 2.35 0.12*2.961.33*106

= 0.12%

Detector energy resolution:

NaI ≈ 9% Ge ≈ 0.1 %

Scinitillator & HPGe detector

Detector dead time

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Limits of Detectability Detection limit: Smallest signal that can be detected reliably

Minimum detectable activity (MDA) is defined as three standard deviations of the background count rate where the sample is counted for the same period of time

Activity: Rate at which a source of unstable nuclei decays measured in decays per second (1Bq=1 decay per second)

Rb: backg. count rate (cpm) Tb: backg. counting period (m) γ: correction factor

Limits of Detectability

Fukhusima accident: 11/03/2011

The radioactivity of the aerosol particles after the Fukushima Daiichi nuclear power plant accident in Tsukuba, Japan

Limits of Detectability

Fukhusima accident: 11/03/2011

Gamma ray spectra of particles in Seattle´s air (31/03/2011)

Examples

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

Solution:

E.g. solve graphically à n = 2.0 MHz or 120 kHz

Example: pulse shaping of a semiconductor detector

0 1000 2000 3000 4000 50000

0.2

0.4

0.6

0.8

1Preamp response

-1000 0 1000 2000 3000 4000 5000-1

-0.5

0

0.5

1

Amplifier response

0 500 1000 1500 20000

5

10

15

20x 10-3 Raw signal

0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

time, ns

Preamp signal

0 1000 2000 3000 4000 5000 6000-400

-200

0

200

400

600

800Amplifier signal

ADC methodologies Peak sensing – maximum voltage

Charge sensing – integrated current

Wave-form digitization

ADC techniques - Ramp / Wilkinson: time-based techniques (capacitor based)

- Successive approximation: comparator-DAC system (reference voltages)

- Flash ADC (direct conversion)

“Digitizing” the magnitude of a detector pulse:

Analogue to digital Converters (ADCs)

Direct conversion (Flash) ADC

“Digitizing” the time of arrival of detector pulse wrt a reference signal:

Time to digital Converters (TDCs)