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W. Udo Schröder, 2009
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Principles of Measurement
Basic Counting System
Detector
Pulse Height Analysis/Digitization
unipolar
bipolar
0
Amplifier/Shaper: differentiates (1x or 2x)
Final amplitude 2-10V
Binary data to computer
PreAmp
Amp
ShaperRadiationSource
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Slow
Fast
Produce logical signal
Fast-Slow Signal Processing
CFTD
PreAmp
Amp
Gate Generator
Data Acquisition System
Energy
Gate
0
0t
t
CFTD Output
CFTD Internal
t
CFTD Input
Principle of a Constant-Fraction Timing Discriminator:
t independent of E
here f = 0.5
Source
Produce analog signal Binary
data to computer
W. Udo Schröder, 2009
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Trace Element Analysis: X-Ray Fluorescence
Irradiate sample, measure characteristic fluorescence
CFTD
PreAmp
Amp
Gate Generator
Data Acquisition System
Energy
Gate
Slow
Fast
Binary data to
computer
Ring-Source/Apertur
e
Sample
Si detecto
r
Water sample from treatment plantSource: 25 mCi 55Fe (2.60 a) Mn-X rays Ka 5.898 keV, Kb 6.490 keV
Off-line n activation analysis: irradiate with 252Cf neutrons, measure g-rays
CHANNEL NUMBER Lab. Invest. Nucl. Sci,The Nucleus/Tennelec, 1988
W. Udo Schröder, 2009
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Coincidences: Absolute Emission Rate (Activity)
N1
N2
N12
1 1 2 2
12 1 2 12 12
1 2 1 2
12 12
/
N A P N A P individual rates
P P P N A P coincidence rate
N N A P A PA singles coinc
N A P
Activity A [disintegrations/time], radiation types i =1,2 detection probabilities Pi =ei
DWi
No angular correlations between rad’s 1,2.
CFTDGate Generator
Pre Amp
Pre Amp
CFTDGate Generator
Coin
cid
en ce
Counter
Coinc.
Counter 1
Counter 2
1
2
Activity A
W. Udo Schröder, 2009
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Time Measurement
CFTDGate Generator
PreAmp
Amp Energy 1
Time
PreAmp
CFTDGate Generator
Amp
Time to Amplitude Converter
Energy 2
Start
Stop
Data Acquisition System
prompt coincident events
t
cou
nt
s
time spectrum
Delay var.
delayed eventscalibrate time axis with
variable known delays (cables)
W. Udo Schröder, 2009
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Pulse Shape Analysis
TimeDAQ
Different signal decay times for 2 radiation types are translated into different amplitudes
Slow Amp
Energy
DAQ
NeutronsGammas
CFTD
Fast Amp
DDAmp
Time-to-AmplitudeConverter
Zer-CrossDisc.
IntegrAmp Delay
Td
StartStop
W. Udo Schröder, 2009
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2-Dimensional Measurement
coincident signals
CFTDGate Generator
PreAmp
Amp Energy 1
Gate
PreAmp
CFTDGate Generator
Amp
Gate Generator
Energy 2
Coin
cid
en
ce
Data Acquisition System
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Example
159
1037660
33
0
241Am
237Np
5.378 MeV
5.433 M
eV
5.47
6 M
eV
5.50
3 M
eV
5.54
6 M
eV
380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550
30
40
50
60
70
80
90
100
110
120
a Energy (keV) - 5 MeV
g E
nerg
y
(keV
)
a-Decay of 241Am, subsequent g emission from daughter
Find coincidences
(Ea, Eg)
9 g-rays, 5 a
W. Udo Schröder, 2009
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0
Example
159
1037660
33
0
241Am
237Np
5.378 MeV
5.433 M
eV
5.47
6 M
eV
5.50
3 M
eV
5.54
6 M
eV
380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550
30
40
50
60
70
80
90
100
110
120
a Energy (keV) - 5 MeV
g E
nerg
y
(keV
)
W. Udo Schröder, 2009
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1
Example
159
1037660
33
0
241Am
237Np
5.378 MeV
5.433 M
eV
5.47
6 M
eV
5.50
3 M
eV
5.54
6 M
eV
380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550
30
40
50
60
70
80
90
100
110
120
a Energy (keV) - 5 MeV
g E
nerg
y
(keV
)
W. Udo Schröder, 2009
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2
Example
159
1037660
33
0
241Am
237Np
5.378 MeV
5.433 M
eV
5.47
6 M
eV
5.50
3 M
eV
5.54
6 M
eV
380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550
30
40
50
60
70
80
90
100
110
120
a Energy (keV) - 5 MeV
g E
nerg
y
(keV
)
W. Udo Schröder, 2009
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3
Example
159
1037660
33
0
241Am
237Np
5.378 MeV
5.433 M
eV
5.47
6 M
eV
5.50
3 M
eV
5.54
6 M
eV
380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550
30
40
50
60
70
80
90
100
110
120
a Energy (keV) - 5 MeV
g E
nerg
y
(keV
)
W. Udo Schröder, 2009
Pri
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4
Example
159
1037660
33
0
241Am
237Np
5.378 MeV
5.433 M
eV
5.47
6 M
eV
5.50
3 M
eV
5.54
6 M
eV
380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550
30
40
50
60
70
80
90
100
110
120
a Energy (keV) - 5 MeV
g E
nerg
y
(keV
)
No -a g coincidences !
W. Udo Schröder, 2009
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Example
159
1037660
33
0
241Am
237Np
5.378 MeV
5.433 M
eV
5.47
6 M
eV
5.50
3 M
eV
5.54
6 M
eV
380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550
30
40
50
60
70
80
90
100
110
120
a Energy (keV) - 5 MeV
g E
nerg
y
(keV
)
W. Udo Schröder, 2009
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6 The End
Basic Counting Statistics
Uncertainty and Statistics
Nucleus is a quantal system described by a wave function y(x,…;t)
(x,…;t) are the degrees of freedom of the system and time.
Probability density (e.g., for x, integrate over others)
2
2
212
21
( , )| , |
( , )( , ) | , | 1
1 2, | | ( )2
( , )| | 1
t t
dP x tx t
dxNormalization
dP x tP x t dx dx x t
dx
Transition between states M E
dP x tx e e state disappears
dx
1
2
Probability rate l for disappearance (decay of 1) can vary over many orders of magnitude no certainty
W. Udo Schröder, 2009
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Experimental Mean and Variance
1
2 2
1
2 2
1
1( )
1( )
1
1( )
( 1)
N
i truei
N
ii
N
n ii
Average samplecount n
n n n unknownN
Variance of an individual count
n nN
Variance of the sample average
n nN N
n n-<n> (n-<n>)2
36076 129.6 16796.1635753 -193.4 37403.5635907 -39.4 1552.3636116 169.6 28764.1635884 -62.4 3893.7636136 189.6 35948.1635741 -205.4 42189.1635640 -306.4 93880.9636124 177.6 31541.7636087 140.6 19768.3635946 -1.5E-12 3463.76
<n> <n-<n>> n
What can be measured: ensemble (sampling) averages (expectation values) and uncertainties
236U (0.25mg) sample counted a particles emitted during N =
10 time intervals (@1 min). l=??
1: (35496 59)min
. : 59 /n
Result n
std deviation n N
W. Udo Schröder, 2009
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0
Moments of Transition Probabilities23
170
4 114 1
170
14 1
1/ 2
6.022 100.25 0.25 6.38 10
236
3.5946 10 min5.6362 10 min
6.38 10
( ) :
(5.6362 0.009) 10 min
" " (2.34 0.004)
n mg mgg
np
n
Probability for decay decay rate per nucleus
p
corresponds to halflife t
710 a
Small probability for process, but many trials (n0 = 6.38·1017)
n0·l < ∞
Statistical process follows a Poisson distribution: n=“random” Different statistical distributions: Binomial, Poisson, Gaussian
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