Detection efficiency

Marie-Christine LépyLaboratoire National Henri Becquerel

CEA Saclay, F-91191 Gif-sur-Yvette cedex, FranceE-mail: marie-christine.lepy@cea.fr

IAEA - ALMERA Technical Visit – July 2010

Detector efficiency

• Full energy peak efficiency

– Definition

– Experimental calibration

– Curve fitting

• Total efficiency

– Definition

– Experimental calibration

• Monte Carlo simulation

FULL ENERGY PEAK EFFICIENCY

Ratio of the number of counts in full-energy peak corresponding

to energy E (NP(E)), by the number of photons with energy E

emited by the source (F(E))

εP(E) depends on the source-detector geometry

and on the energy

Full-energy Peak Efficiency (FEPE): εP(E)

)(

)()(

EF

ENE

P

P =ε

NP(E)

Energy

Counts per channel

εp(E) depends on the geometrical conditions and on the energy

Measurement

conditionsDetector

characteristics

εP(E) = εG. εI(E)

Geometrical

efficiency

Intrinsic

efficiency

Sample

Distance

Shielding

Full-energy Peak Efficiency (FEPE): εP(E)

Geometrical efficiency

πε

4G

Ω=

d

r

Point source

Detector

Ω

Ω = solid angle between source and detector (sr)

For a point source :

)1(222 rd

d

+−=Ω π

εG depends only on the source-detector geometry

Ratio of the number of photons emitted towards the detector

by the number of photons emited by the source

εI(E) depends on the energy of the

incident photons:

transmission

absorption

full-energy deposition

εI(E) = Ratio of the number of counts in

full-energy peak by the number of impiging photons

Difficulty: exact composition badly known

Intrinsic efficiency

Ge dead layer

Be window

Capvacuum

vacuum

HPGe

Electrical contact(Au, Ni, ... thickness?)

Incoming photons

Interacting photons

Transmission probability through material i

with thickness xi : (Beer-Lambert law) :

Thus

Interaction probability in the same material:

PT (E, xi) = exp (-µi(E) . xi)

PI (E, xi) = 1 - PT(E, xi)

Calculation of the detector FEP efficiency

To result in a count in the FEP peak:

The photon must :

- be emitted in the Ω solid angle,

- cross the screens (air,n window, dead layer,…) without being absorbed,

- and be totally absorbed in the detector active volume.

GeHP

Ω

Source

Transmission

through screens

Interaction in the

detectorr volume

( )( ) ( ) Ω⋅⋅⋅−−⋅⋅−= ∫∏Ω

dEPxExEER Pdd

i

ii

P )()(exp(1)(exp)( µµ

For the low-energy range:

Probability of total absorption in the detector

For higher energies:

Total absorption is due to succesive effects : multiple scattering

Thus the calcul is not possible

Pp(E) ≈ τd(E)/µd(E)

Calculation of the detector efficiency

Many difficulties for an accurate calculation

Radiography of a HPGe detector:

Rounded crystal, axially shifted, tilted …

- Exact knowledge of the detector parameters:

materials (composition)

geometry (thickness, position …)

- Data used in the calculation :

Attenuation coefficients

Material density

- Semi-conductor effects: parameters and physical

interaction :

Electrical field

Electrodes

Calculation of the detector efficiency

Experimental FEP efficiency calibration

This is performed using standard radionuclides with standardized activity A (Bq) with

photon emission intensities, Iγ well known

NP(E) : peak net area

Energy (keV)

Counts

per

channel

100 200 300 400 500 600 700

1

2

10

100

1000

10000

20000

50000

NP(E)Spectrum 137 Cs

εP(E) Full-energy peak (FEP) efficiency depends on the energy and

on the source-detector geometrical arrrangement

)E(F

)E(N)E(

P

P =ε)E(IA

)E(N)E(

P

P

γ

ε⋅

=

Associated standard uncertainty

)E(IA

)E(N)E(

γ

ε⋅

=Efficiency calibration

2222

+

+

=

)E(I

)E(I

A

A

N

)E(N)E( ∆∆∆

ε

ε∆

33 101)(

)(105

)(

1

)(

)(

)(

)( −− ⋅=∆

⋅=∆

==∆

EI

EI

A

A

ENEN

EN

EN

EN

γ

γ

Influence of the peak area :

if N = 104 ∆N/N = 10-2 -> ∆ε/ε = 1,1 10-2

if N = 105 ∆N/N = 3,1 10-3 -> ∆ε/ε = 6 10-3

FEP efficiency calibrationTo get an efficiency values at any energy : energy calibration over the whole

energy range

1. Use different radionuclides to get energies regularly spaced over the range of interest

Single gamma-ray emitters : 51Cr (320 keV), 137Cs (662 keV) 54Mn (834 keV) : one efficiency value per one measurement

Multigamma emitters : 60Co, 133Ba, 152Eu, 56Co : several efficiencies values per one

measurement , but coincidence summing effects !

For each energy, discrete values of the FEP efficiency ε(E1), ε(E2), … ε (En)

2. Computation of the efficiency for

1. Local interpolation

2. Fitting a mathematical function to the experimental values

!

Efficiency calibration for different source-to-detector distances

0.01

0.10

1.00

10.00

10 100 1 000 10 000Energy(keV)

D = 10 cm

D = 15 cm

D = 5 cm

D = 4 cm

D = 3 cm

D = 2 cm

D = 1 cm

D =7 cm

FEP efficiency calibration

Efficiency calibration - Interpolation

Local interpolation

E1 -> ε1

E2 -> ε2

To get ε(E) for E1 < E < E2

Solution 1 : linear interpolation )()EE(

)EE()E( 12

12

11 εεεε −⋅

−

−+=

Solution 2 : logarithmic interpolation

: valid only for close energies!

)ln(ln)ElnE(ln

)ElnE(ln)ln())E(ln( 12

12

11 εεεε −⋅

−

−+=

Efficiency calibration - InterpolationLocal interpolation : example

Actual value (at 662 keV) : ε(E) = 1.90 10-3

Conclusion : approximation to be used only if high uncertainties (> 10 %) are acceptable

Determine efficiency for E = 662 keV (137Cs) knowing :

1. E1 = 569 keV (134Cs) ε1 = 2.21 10-3

E2 = 766 keV (95Nb) ε2 = 1.67 10-3

Linear interpolation : ε(E) = 1.96 10-3

Logarithmic interpolation : ε(E) = 1.92 10-3

2. E1 = 122 keV (57Co) ε1 = 8.22 10-3

E2 = 1 173 keV (60Co) ε2 = 1.13 10-3

LInear interpolation : ε(E) = 4.57 10-3

Logarithmic interpolation : ε (E) = 1.87 10-3

Determination of the best fitted function to a given set of experimental data (energy, efficiency)

In the logarithmic scale , the shape is smoother than in the linear scale.

Efficiency calibration : mathematical fitting (1)

Experimental efficiency calibration of a

HPGe detector (100 cm3) for a point source

at 12 cm between 10 and 1500 keV (linear scale)

Energy (keV)

0 200 400 600 800 1000 1200 1400 1600 1800

Effic

iency (

%)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Experimental efficiency calibration of a

HPGe detector (100 cm3) for a point source

at 12 cm between 10 and 1500 keV (logarithmic scale)

Energy (keV)

10 100 1000

Effic

ien

cy (

%)

0.2

0.3

0.4

0.5

0.6

0.70.80.9

0.1

1

Functions frequently used:

Polynomial fitting in the log-log scale: i

n

0ii )E(lna)E(ln ∑

=

⋅=ε

in

0ii Ea)E(ln −

=∑ ⋅=ε

Efficiency calibration : mathematical fitting

Remarks :

-ai coefficients are determined using a least-squares fitting method

- experimental data must be weighted

- the polynomial degree (n) must be adjusted depending on the number of experimental data (p) : n << p

- in some case two different functions can be used with a cross point

- check the resulting fitted curves !

Efficiency calibration : mathematical fitting

Example : 40 experimental values

in the 122-to-1836 keV range

Fitting function :

Adjusted coefficients :

in

ii EaE )(ln)(ln

0∑=

⋅=ε

fitting 122 to 1836 keV

deg0 -34,11961

deg1 48,16797

deg2 -25,89215

deg3 5,80219

deg4 -0,40503

deg5 -0,01632

Efficiency fitting must be

visually checked

Efficiency calibration : mathematical fitting

In the case of cross

points be carefull :- Avoidzones with

important inflexion - Avoid highdegreepolynomials

Spline functions

Simply a curve

Special function defined piecewise by polynomials

A piecewise polynomial f(x) is obtained by dividing of X into contiguous intervals, and representing f(x) by a separate polynomial in each interval

The polynomials are joined together at the interval endpoints (knots) in such a way that a certain degree of smoothness of the resulting function is guaranteed

Slides from Dr. Oliver Ott - PTB

Piecewise defined polynomials

constraints

K1 ≤ xmin < K2 < … < Kk < xmax < Kk+1

Series of knots

xmin: minimum x-value of the data points

xmax: maximum x-value of the data points

Ki ( i = 1, …, k+1)

• 2 consecutive knots establish an interval [Ki, Ki+1) where a polynomial Pi(x) of order n (degree n-1) is defined.

• In total k intervals are fixed by the knots.

• Outside the interval i the polynomial Pi(x) is not defined.

• At the joining points K2 …Kk they have to fit smoothly – continous up to the (n-2) derivative

Spline functions mathematically

Condition that a spline function solution must satisfy:

For the derivatons of neighboured polynomials at all knots:

( ) ( ) 2-n m 0 ,01

)(

11

)(≤≤=− +++ i

m

ii

m

i KPKP

k polynomials in k intervals: ( )∑=

=k

i

i xPxF1

)(

where

( ) =xP( )∑

=+

−≤≤−⋅

n

j

i

j

iij KxKxa1

1i

1Kfor

0 else

Efficiency calibration at 25 cm

Efficiency calibration at 10 cm

Multigamma emitters -> Coincidence summing effets

• Efficiency calibration for reference geometry

– For point source : relative uncertainty 1-2 %

• Corrective factors needed if differentmeasurement geometry

FEP Efficiency calibration : remarks

TOTAL EFFICIENCY

Ratio of the total number of counts in the spectrum (NT(E)), by the

number of photons with energy E emitted by the source (F(E))

εT(E) depends on the source-detector geometry

and on the energy

Total Efficiency (TE): εT(E)

)(

)()(

EF

ENE T

T =ε

NT(E)

Energy

Counts per channel

( )( ) ( ) Ω⋅⋅−−⋅⋅−= ∫∏Ω

dxExEET dd

i

ii )(exp(1)(exp)( µµε

xi : i shield thickness

xd : detector thickness

Total efficiency depends on:

- Solid angle Ω

- Attenuations in absorbing layers (air, window, dead layer, etc)

- Interaction in the detector active volume (any effect)

And on the environment ! (scattering)

where

Attenuation in screens Detector interaction

Calculation of the total efficiency

Experimental total efficiency calibration εT(E)

Energy (keV)

Co

un

tsp

er

ch

an

nel

100 200 300 400 500 600 700

1

2

3

57

10

20

30

5070

100

200

300

500700

1000

2000

3000

50007000

10000

20000

30000

50000

NT (E)

Spectrum 137Cs

Total spectrum minus background

background

)(.

)()(

EIA

ENTET

γ

ε =

1. Using mono-energetic radionuclides

241Am (60 keV), 109Cd (88 keV), 139Ce (166 keV-T1/2=138 d), 51Cr (320 keV- T1/2= 28d),

85Sr (514 keV- T1/2=65 j), 137Cs (662 keV), 54Mn (834 keV - T1/2=302 d)

2. For the low-energy range, if P-type detector, 57Co can be used (mean energy and sum of emission intensities)

3. For the high energy range, 88Y and 60Co can be used... But complex decay scheme

Experimental total efficiency calibration εT(E)

Approximation of the total efficiency with 2-photons emitters:

NT = A (Iγ1. εT 1 + Iγ2

. εT 2 - Ιγ1. εT 1

. P12. Iγ2

. εT 2)

88Y : E1 =898 keV and E2 = 1836 keV (or 65Zn : 511 and 1115keV)

E1 = 898 keV → εT1 can be extrapolated from previous data

(54Mn – 834 keV)

P12 = 1

( )112

11

21 T

T

TIIA

IANT

ε

εγε

γγ ⋅−⋅⋅

⋅⋅−=

Sr88

γ1

2γ

1β

0

2 734

Y88

1 836γ3

EC3

EC4+,

( ) A

N

A

AN T

T

TTT −−=

−⋅

⋅−= 11

1 ε

εε

Approximation of the total efficiency with 2-photons emitters:

NT = A (Iγ1. εT1 + Iγ2

. εT2 - Ιγ1. εT1

. P12. Iγ2

. εT2)

Co

60

γ1

2γ

1β

2β

2 505

1 332

0

60

Pβ2: 0,001

Pβ1: 0,999

60Co :

E1 = 1 173 keV Iγ1 = 0,99

E2 = 1 332 keV Iγ2 = 0,98

P12 = 1 Iγ1 ≈ Iγ2 = 1

εT1 ≈ εT2 = εT

• Efficiency calibration for reference geometry

– Difficult, even with point sources:

• relative uncertainty 5-10 %

• But limited influence on the coincidence correction factor

• Monte Carlo simulation ?

Total Efficiency calibration : remarks

MONTE CARLO SIMULATION

Monte Carlo simulation

• Difficulties: bad knowledge of the detector internal

parameters

• Accurate description is time-consuming:

– Radiography (external dimensions)

– Collimated beam (hole - dead layer)

– Window to crystal distance (source at different distance)

– Comparison with some experimental values

• Different energies

• Different geometries

X-ray tube ->

Geometrical dimensions

Cristal shape (rounding)

60Co source ->

Hole dimensions

Comparison with experimental data

10 100 1000Energy (keV)

0.001

0.01S

r-85

Co-5

7 Am

-241

Cs-1

37

Ce-1

39

Am

-241

Eu-1

55

Cd-1

09

Eu-1

55

Eu-1

52

Co-5

7

Co-5

7

Ce-1

39

Cr-

51

Eu-1

52

Sn-1

13

Sr-

85

Cs-1

37

Nb-9

4E

u-1

52

Mn-5

4

Nb-9

4Y

-88

Zn-6

5C

o-6

0E

u-1

54

Na-2

2

Co-6

0E

u-1

52

Eu-1

54

Y-8

8

-1

0

1

Z-s

core

Experimental

Calculated

Extended-range coaxial HPGe with carbon-epoxy window (61 x 61 mm)Point sources at 16 cm from the detector window

Picture from V. Peyres - CIEMAT

• Can provide

– FEP efficiency

– Total efficiency (very dependent on the

environment)

– Absolute calculation should be compared with

accurate experimental data

– Strong interest for efficiency transfer calculation

Monte Carlo simulation