radiation detection and measurement, ju, first semester, 2010-2011 (saed dababneh). 1 counting...

Radiation Detection and Measurement, JU, First Semester, 2010-2011 (Saed Dababneh).

1

Counting Statistics and Error PredictionPoisson Distribution (p << 1)

xpnxPxx

pnxxPx

xP

x

ex

x

epnxP

n

x

n

x

n

x

xxpnx

0

22

0

0

)(

)(

1)(

!!)(

Success ≡ Birthday today.p = 1/365.n = 1000.

• Low cross section.• Weak resonance.• Short measurement (compared to t1/2).

Appendix C

We need to know only the

product.

HW 25HW 25

Asymmetric


2

Counting Statistics and Error Prediction

xpnxPxx

pnxxPx

xP

ex

xP

n

x

n

x

n

x

x

xx

0

22

0

0

2

)(

)(

1)(

2

1)(

2

Gaussian (Normal) Distribution (p << 1, > 20) xSuccess ≡ Birthday today.p = 1/365.n = 10000.

Symmetric

and

slowly

varyi

ng

• Can be expressed as a function of .• Can be expressed in a continuous form.

HW 26HW 26


3


x

xx

ex

xP 2

2

2

1)(

xex

G 2

2

2)(


4

Counting Statistics and Error PredictionCalculate the percentage of the samples that will deviate from the mean by less than:• one .• two .• etc …

HW 27HW 27


5

2

20 )(2

2

0w

xx

ew

Ayy

x

xx

ex

xP 2

2

2

1)(


Baselineoffset

Total area under the curve above the baseline

22, approximately 0.849 the, approximately 0.849 thewidth of the peak at half heightwidth of the peak at half height

This model describes a bell-shaped curve like the normal (Gaussian) probability distribution function. The center x0 represents the "mean", while ½ w is the standard deviation.

What is FWHM? Resolution? Peak centroid? What is FWHM? Resolution? Peak centroid?

HW 28HW 28


6

Counting Statistics and Error PredictionApplicationsApplications 1- Match experiment to model

N

iei xx

Ns

1

22 )(1

1

0

)(x

e xxFx

need) what we(all exx

N

iei xx

Ns

1

22 )(1

and not

because we set exx

• Assume a specific distribution (Poisson, Gaussian).

• Set distribution mean to be equal to experimental mean.

• Compare variance to determine if distribution is valid for actual data set (Chi-squared test).


7

• We can’t use Gaussian model for this data set. Why?• Qualitative comparison.• Is 2 close to s2?

• Close!? Less fluctuation than predicted! • But quantitatively?• Chi-squared test.


36.7)(1

1

22

N

ii xx

Ns

Only to guide the eye!

8.82 x

Back to our example

HW 29HW 29

By definition:

Thus:

or


8


N

iei

e

xxx 1

22 1

Chi-squaredChi-squared

891.15

8.8

36.719112

222

sN

x

sN

e

2

22

1 s

N

The degree to which 2 differs from (N-1) is a measure of the departure of the data from predictions of the distribution.


9


15.891 (interpolation) or

smaller <= Fluctuation => larger

either gives = 0.6646Conclusion:

no abnormal fluctuation.

Perfect fit

http://www.stat.tamu.edu/~west/applets/chisqdemo.html

http://www.stat.tamu.edu/~west/applets/chisqdemo.html


10



11


xx

xxx

Single measurementSingle measurement

S2 = 2 ≈ x

68% probability that this interval includes the true average value.What if we want 99%..?

Fractional standard deviation

xx

x 1

Need 1%?Need 1%?Count 10000.Count 10000.


12


A series of A series of “single” “single”

measurements.measurements.


13



14


Net counts = Gross counts – Background

MeasuredDerived

Gross counts = 1000Background counts = 400Net counts = 600 37 (not 600 24)

Count Rate = ? ?

• What about derived quantities? (Error propagation).(Error propagation).

Compare to addition instead of subtraction.(Count, stop, count).


15


2222

21

...

...

21 Nxxx

Nxxx

Mean value of multiple independent counts.

• Assume we record N repeated counts from a single source for equal counting times:

• For Poisson or Gaussian distributions:

So that

iX xi

Nxxx ...212


16


x N

x N

N

Nx

N

x x

NStandard error of the mean

• To improve statistical precision of a given measurement by a factor of two requires four times the initial counting time.


17


TSBTB optimum

S B

B

Optimizing counting time.

S = (net) source count rate.B = background count rate.TS+B = time to count source + background.TB = time to count background.

To minimize s :

Low-level radioactivity

Weak resonance

Very strong background

High signal-

to-

background

ratio

How to divide the limited available beam time?

HW 30HW 30


18


• Minimum detectable amount.• False positives and false negatives.

• Background measurement?• Without the “source”.• Should include all sources except the “source”.• Accelerator applications: background with beam.

• Rest of Chapter 3.

radiation detection and measurement, ju, first semester, 2010-2011 (saed dababneh). 1 counting...

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