IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

DIRECT STATISTICAL MATHIMATICAL MODEL TOCALCULATE THE
FULL ENERGY PEAK EFFICIENCY OF HPGe DETECTOR
Younis S.Selima, Mohamed S. Hussiena,b, Mohamed A. Fawzya, Ahmed M. El
khatibeaa
Department of Physics, Faculty of Science, Baghdad St., Moharrum Bey,
Alexandria 21511, Egypt.
b
Department of Chemistry & Chemical Engineering, Royal Military College ofCanada P.O. Box 17000, Station Forces, Kingston, Ontario,K7K 7B4
E-mail: mohamed.hussein@rmc.ca
Keywords: gamma spectroscopy systems; HPGe detector efficiency; full energy
peak efficiency; photopeak coefficient; direct statistical mathematical model

Abstract
A direct statistical mathematical model was implemented to calculate
the full energy peak efficiency (FEPE) of HPGe detectors over gamma rayenergy range of 20 keV to 3 MeV. This mathematical model can be applied at
any height from the detector face of an axial point source. The idea of themodel depends of on tracking of successive interactions of gamma ray
photons in the energy range under consideration and uses the physics ofthese interactions with the geometry information to calculate the photo peakattenuation coefficient, and consequently the photo peak efficiency.

All calculations were carried out for different cylindrical detector sizes
over distances of 0 to 25 cm from the detector face of the axial point source.
The relative efficiencies of different sizes of HPGe detectors to NaI detectorsthat were calculated by the present model have an excellent agreement withpublished work.
The calculations of FEPE carried out by the present model werecompared with results from other methods such as experimental, semi
empirical and Monte Carlo calculations. The results of the present model arein excellent agreement with published FEPE results from these methods, andprovide the best match of experimental results than other theoretical methods.

(1) Introduction
Gamma spectrometry is one of the tools commonly used for themeasurement of various environmental radionuclides. Where, the absoluteactivity of different gamma peaks in a wide energy range can be determined,

P210-1IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Gamma ray radioactive isotopes can be identified, measuring the absorbed
doses and determination of the interaction cross sections.
To calculate the absolute activity, the sample to detector full-energy
peak efficiency should be known. The calculation of sample to detector fullenergy peak efficiency using semi-empirical, Monte Carlo approaches andexperimental measurements have been treated by several authors. The semiempirical method has been used by Hoste [1], Moens et al. [2] , Lippert [3],Mihaljevic et al. [4], Wang et al.,[5] and Wang et al., [6] . In this method, thefull-energy peak efficiency is calculated by combining point source
measurements with computer calculations using empirical formulae. It
involves some approximations and simplifications so the application isrestricted to limited source–detector configurations. While Wainio and Knoll[7], and Overwater et al. [8] and B.Lal et al. [9] used Monte Carlo method inwhich, the history of each individual photon, is simulated in an analog step-bystep process in the detector active zone. There are no approximations orlimitations to the source–detector configurations, but it is computationally time
consuming more discussions about these two theoretical methods are foundin Ref. [1,…17]. On the other hand the experimental method has been used to
calculate FEPE by T.Paradellis et al. [18] and A.Owens [19]. The relativeadvantages and disadvantages of each of this methods are discussed in[1,…19].
The aim of this work is introducing a new direct mathematical Model tocalculate full energy peak efficiency of cylindrical HPGe detector for an axial
point sources emitting photon of energies up to 3 MeV. To calculate the FEPE(εp) the photopeak coefficient (μp) must be calculated accurately that could beachieved through the direct statistical calculation and using spherical
trigonometry technique. This technique based on the determination of theaverage path length

covered by a photon inside the detector active

medium and the geometrical solid angle (Ω) which represent the angle
subtended by the detector at the source point.
The arrangement of this paper is as follows. Sections (2) and(3) present direct statistical mathematical formulae for the total efficiency andpeak efficiency for axial point source. Sections (4) and (5) cover thedefinitions and calculations of full energy peak attenuation coefficient (μp) andP210-2

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

partial attenuation factor (fm). Sections (6) and (7) enclose the method of
calculations of both average scattered energies and the average cosine angleof scattering respectively. Section (8) contains the longitudinal and laterallimitation of the scattered ray in the finite detector medium.

Section (9)presents an example of calculating full energy peak efficiency, comparisons
between the calculated peak efficiency using the formulae derived in this workand the published results that proving the validity of the present mathematicalformulae. Finally, the Conclusion is presented in Section (10).

(2) The direct statistical mathematical method
The direct statistical mathematical approach, present work, provides oneof the simplest, potentially and most accurate methods for predicting the
FEPE. Selim and Abbas [20,21,22] derived mathematical expressions to be
used directly by the substitution of geometrical parameters of the sourcedetector system and the total attenuation coefficient (μ) of the incident photon
corresponding to its energy (Eγ) to find the total efficiency, for any source
shape, for any cylindrical detector and at any geometrical locations. The ideaof this method is illustrated in Ref. [20], where the total attenuation coefficient(μ) is the summation of all the coefficients that attribute to consideration
efficiency μ=τ+σ+. Selim and Abbas formula (1) is used for different source
shape at different places with respect to cylindrical detector. In this work, weinterested in full energy peak efficiency for the axial point source so that the
total attenuation coefficient (μ) will be replaced with the photopeak coefficient(μp) which calculated accurately through new technique discussed in somedetails in the following sections.

(3) Axial Point Source
By using spherical coordinate, the integration limits change in steps inaccordance to the values of the effective traversed distanceand the polar

angle (θ). The geometry of coaxial point source to the cylindrical detector(2R×L) is given in figure (1).
Axial point sourceθ 2 θ1hFigure (1): The geometry of coaxial
point source to thecylindrical detector (2R×L)
L

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IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Therefore the total efficiency (T) can be expressed as
θ
θ2
 μL
 μR
μh





1  1
cos
θ
sin
θ
cos
θ 
 sin  dθ   1  e
 εT    1  e
sin

dθ




2  0 
θ1 




R
h L
R
 2  tan 1h(1)

θ1  tan 1

Where;(2)Equation (1) is applied to calculate the total efficiency due to an axial point
source to a bare detector. The most germanium detectors use a thin cap
window so the attenuation of this window must to be taken underconsideration. For capped detector the equation (1) multiplied byeμ t tcos θandbecomesθθ 2   tt
 t
 μL
 μR
μh





1  1 cost 
cos
θ
cos

sin
θ
cos
θ 
 sin  dθ  e 1  e
 εT    e 1  e
sin

dθ





20


θ1





(3)

Where;
μ: is the total attenuation coefficient of the detector material;
L: is the detector height
R: is the radius of detector faceh: is the distance between the source and the detector facet: is the window thickness, and

μ
tt
1
1

2


e

c
o
s
θ

e


μ
t
t 2


μ
t
t
nn






e

c
o
s
θ

(4)

c
o
s
θ

(μt) is the total attenuation coefficient of the window material which is usually
Aluminum unless stated. If there is n-number of caps or shield we multiply thewhere; (μtn) is the coefficient of the nth. element; And, (tn) is the thickness ofnth. element. The dead distance (ho), the distance between the detector active
medium and the window, has to be added to (h) for the source –detector
separation distance.

(4) Full Energy Peak Attenuation Coefficient (μp)
The full energy peak efficiencies defined as the ratio between the numberof photons that are recorded in the detector under certain peak and theP210-4

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

number of photons that are emitted from the source with energy relates to thispeak.”
To calculate the full energy peak efficiencies, we consider only the part ofenergy deposited in the detector and which only contributes to the peak, sayphotopeak attenuation coefficient (μp) then, the total attenuation coefficient (µ)
in equation (3) is replaced by the photo peak coefficient (µp) remembering thatthe photoelectric effect leads always to electrons of maximum energy, we
have to consider the incident gamma rays energies is less than 3 MeV; Sothat the coefficient (μp) given by; p   o  f m o(5)where, m=1,2,…..
(τo) is the photoelectric coefficient,(σo )is the Compton scattering coefficients and(fm) is a fraction allowing for successive Compton scattering .
Where, the final scattered photon after Compton scattering is terminated by aphotoelectric absorption. This fraction is primarily depends on the detectors
dimensions and incident gamma ray energy.
The index (m) is integer numbers 1,2, ..Confining ourselves to gamma rayenergy less than about 3 MeV; we take only these two terms where theprobability of pair production in this energy range is very small. One of themain objects of this work is the determination of the fraction (fm) with respect
to the detector’s dimensions, which will maintain in next section.

(5) The Partial Attenuation factor (fm)Infinite Size Ge detector:In order to fully understand the exact behavior of the factor (fm) in theattenuation coefficient, We preferably start with an infinite size detector withinteracting gamma-rays energy hνo less than 3 MeV where the major
predominant interactions Photoelectric interaction and Compton scattering.
So, we have only two possible allowances to be followed:Photoelectric interaction (τo): where, an electron produces in the detector
medium with a kinetic energy equal (hν0-B.E.), thus, being recorded under
the peak energy spectrum.

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IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Compton interaction (σo): The result of which is an electron set in motion witha kinetic energy To and a scattered photon with energy hν1hν1 = hνo – To

(6)

Consequently; the last scattered photon has one of two fates: First fate; a photoelectric effect takes place, thus, an electron with a kinetic
energy equal to (hν1-B.E.) is set in motion.
Now, this electron and the previous scattered one; take place in rapid
succession that they appear as one electron with a total energy hνo that isregistered at the maximum peak position of the detector. The relative
probability of this to happen is:- 
f m  f1   1 
 1 

(7)

μ1 the total attenuation coefficients of the scattered gamma rays with energyhν1.
 Second fate; If the previous interaction does not take place, the rest
fraction of the original beam σ1/μ1 propagates, and will be scattered to anelectron with a kinetic energy T1 and a scattered gamma ray with energyhν2. Letting this last scattered photon undergoes a photoelectric effect τ2 
with a probability   . These three electrons are again emitted promptly
 μ2 
and they appear as one with a total kinetic energy equals to original
energy of incident ray hνo, Since range of the electrons is few microns
only up to about a millimeter thus appear again in the photo-peak region.

 σ1   τ 2 
The relative probability of this to happen is:   .   μ1   μ 2 
Of course, only one event of either of the two fates takes place. Then the
average probability will be:-

1 
  
f 2  . 1  1 . 2 
2  1 1  2 (8)Therefore; for an infinite medium and allowing for further Compton scattering;
the average effective fraction of the Compton components is given by:-fm 

1   1 1  2
  
  
   .      1 . 2  3     m 1 m  ; m  1,2,3,...
(9)m  1  1 2
 1 2 3
m 1  m 
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IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Where, m is a total number of effective collisions.
Using equations (5,9) and tables in (24) one can calculate the photopeak
efficiency for relation (3) after substituting total attenuation coefficient (µ) withphotopeak coefficient µp then the photopeak efficiency given by for axial point
source.
θ μ pLθ 2   tt
 μ pR μph
 t





1  1 cost 
cos θ cos   ε p   e1e
sin  dθ   e
1  e sin θ cos θ  sin  dθ  (10)




2 0

θ1






Now, the problem is calculating the partial coefficient corresponding to theenergies of scattered photons τ1 , τ2 τ3….. τm and σ1, σ2, σ3, …..σm-1. So, theaverage scattered energies should be calculated.

(6) Calculation of Average Scattered Energies
In order to find the partial attenuation fractions, we have to determine
the scattered energies of the gamma rays. A statistical process governs thisphenomenon, similar to neutron scattering process. Beforehand, thekinematics of the interaction is controlled by the famous Compton’s set of
equations. According to Compton scattering this interaction, as the photon
falls with energy (hνo), its momentum is (hνo/c). From the conservation of
momentum, where the scattering is elastic, the scattered photon must bechanged from the incidence direction by angle (θ) and the electron removes
with angle (υ) but the three directions, the incidence, the scattered and theremove recoil must lie in the same plane i.e. un-polarized scattered radiationis considered.
From the conservation of energy, the photon loses part of its energy, which
the electron acquires as a kinetic energy.
Energy of the scattered photon is given by:

hν1 

hν o
1  ( 1  cos θ)

(11)

hνo
(12)
2
me .c energy for the scattered photon
The energy expectation value of the average



is

h
1 F.ho

(13)

d o

h 1  h d d1d o d dP210-7

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC](14)
where,d2sindIn order to calculate the average scattered photon energy, it is required toknow the angular distribution of the scattered photon to determine thedeposited energy. So, The probability of Compton scattering at the angle (θ)
can be determined through quantum mechanical calculation of the process.
Klein and Nishina [25] derived a formula for the angular distribution to the unpolarized scattered radiation as:-

 
d o
ro2 
1
 2 (1  cos  ) 2 
2
(15)


 1  cos  2 d2  1   (1  cos  )  
1   (1  cos  )To calculate the expected scattered photon energy < hν1>, we integrate thevalues of hν1 over all the possible angles θ.
πh o f θ  sin θ 1  ( 1  cos θ) dθ
3

hν1 

where;(16)0πf θ  sin 0 1  ( 1  cos θ)2 dθ

 2 (1  cos  ) 2
f ( )  1  cos  
1   (1  cos  )2(17)Using equations (13,14,15,16) and (17). The reduction factor (F) given by1 2( 1  cos θ ) 2 2
0 1   (1  cos θ )3  1  cos θ  1   (1  cos θ )  d(  cos θ )
hν1
F
π
(18)
hνo

1
 2( 1  cos θ ) 2 
2

1

cos
θ

d(

cos
θ
)

0 1   (1  cos θ )2 
1   (1  cos θ ) πBy, Carrying out the integration of the different factors term by term ofequation (18) the reduction factor (F) of the original energy leading toaveraged energy for the scattered beam or photon in any medium, could be
expressed as:-P210-8IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]F

3(1  2 ) 3 ln(1  2 )  32 5  12 4  36 3  30 2  63(1  2 )  (1  2 ) 3 [ 2  2  2]  ln(1  2 )  2 4  18 3  16 2  4



(19)

Where, () is stated as before in equation (12). Figure (2) indicates thevariation of reduction factor (F) with the photon energy (hν). It is important to
notice that, for higher energies of the incident gamma rays the sharp
reduction of energies is more than that for the incident lower energies.

Figure (3): The variation of
average cosine <cos θ> with (hν)

Figure (2): The variation of
reduction factor F with the photon

energy hν(7) Calculation of The Average Cosine Angle Of Scattering
The average cosine angle of the scattered photon is given byπdσ o cos θ dΩ dΩcos θ  0

Where;

π

(20)

dσ o
0 dΩ dΩdσ o
as is defined in equation (15) therefore
dΩ


cos θ
 2 (1  cos θ) 2 
2

1

cos
θ

 sinθ dθ0 1   (1  cos θ)3 
1   (1  cos θ) cosθ  π1α 2 (1  cos θ) 2 
2

1

cos
θ


 sinθ dθ0 1   (1  cos θ)2 
1   (1  cos θ) π(21)

P210-9IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

By solving these two integrations, Carrying out the integration term by term,
we finally get;cos  (1  2 ) (    6  3)  ln(1  2 )  6  16  46  30  6 (22) (1  2 ) (  2  2)  ln(1  2 )  2  18  16  4 
2

3

2

2

5

2

4

4

3

3

2

2

Where () is as defined before in equation (12). One can notice that, theaverage cosine depends also on the incident photon energy. The variations of<cos θ> with the photon energy (hν) is presented in figures (3) where theenergetic gamma rays (MeVs’) are scattered forwards, while softer rays
(KeVs’) are scattered sideward. The correlated understanding between
Figures (2) and (3) can be explained as when an incident photon withrelatively high energy is scattered then much of its energy is reduced i.e. (F)becomes small and penetrates more forward i.e. (θ) is small then <cos θ>
becomes big till the photon energy is much reduced and the situation isinterchanged. For smaller energies, the energy reduced more slowly i.e. (F)becomes big and tend to be lose to 1 and spreads more side–wards i.e. (θ)
increases.
By using equation (13) for every incident gamma energy (hνo ) , theaverage energies of the successive Compton-scattered rays <hν1> , <hν2>,…. could be deduced. From table (1) and equations (9,13 and 18) and datafrom tables [24] the components of the partial attenuation coefficient could becomputed1 1  2  1  2  3  m1  m, . ,  , .     
1  1 2  1 2 3 m1 m(23)

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IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Figure(4): Variation of total attenuation, peak, andphotoelectric Coefficients with the photon energy for
an infinite Germanium detector

Figure (4) shows the variation of coefficients (μ, μp and τo) with the photon
energy for infinite size germanium detector. One can notice that the difference
between (μp) and (τo) at high energy explain the feasibility of this work.
The partial attenuation coefficient in expression (23) consider the infinite
number of scattering or successive interactions that could be happened onlyin the infinite detector medium but actually the detection medium is finite so
the longitudinal and lateral limitation should be considered.

(8) Finite Size Ge Detector
In this case, we consider the assumption that all electrons from thephotoelectric effect or/and Compton scattering are completely absorbed within
the detector volume. Since range of the electrons is few microns only up to
about a millimeter. Therefore, the photoelectric coefficient (τo) would not beaffected. Only Compton-components affected because of the scatteredgamma ray photons that may be escape before absorption in the detector
medium and contributing to previous mentioned coefficient (ƒ m). So thelongitudinal and lateral limitation should be considered to determine thelimitation number of (m) successive scattering.

(8-a) Longitudinal Limitation
Let a gamma ray of energy (hνo) to be incident axially and normally tothe detector’s surface. Its path length is (L). The absorption probability in thedetector is:-




oL 
0L
e
e
.eoL

(24)

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IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

The mean free path of the photoelectric effect is (1/τo); and that of Compton
scattering for the first elastic collusion is (1/σo). The average energy of thescattered photon becomes (hν1) with a scattering angle (1).
Figure (5) illustrates the mean free paths and the scattering angles for
successive scattered photon inside the cylindrical detector of dimension(2R×L). The component in the direction of incidence is clearly

x1 

1

1

cos 1(25)

Further, let this ray be scattered further to let its energy be hν2 with a spacescattering θ2 making an angle (β) with the original direction of incidence and arotation angle υ2, where υ2 is the angle of rotation of the third mean free pathwith respect to the plane of the first two paths.

Figure (5): The mean free paths andthescattering
anglesfor
successive scattered photon
inside the cylindrical detector of
dimension (2R×L).
Following the same steps as followed by [25] for the scattering of neutronscos   cos 1 . cos 2  sin 1 . sin 2 . cos  2 (26)

Since all values of (υ2) are equally probable for un-polarized rays, then the
average value of the second term in the last equation is zero. Accordingly,

x2 

1

2

. cos  2 .cos 1(27)

P210-12IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

And so on, for further scatterings. One condition to determine the suitablevalue of index number (m) in the fraction (ƒm) is

L

1 1
1
1

cos 1 cos 1 cos  2       
cos 1    cos  m
σ o σ1σ2σm(28)

In effect, for an axial isotropic point source being situated at a distance (h)from the detector’s surface the average photons ray may not fall normally on
the detector surface.
Therefore, the gamma ray photons may be inter to the detector medium fromany point on the upper surface with equally probability, so the averagecovered path of a ray is not (L) but ( ) as shown in the sketch figure (6).

Figure (6): The average covered distance of a ray from an axial point
source at distance (h) from the upper detector face

The average covered distance of a ray from an axial point source at height(h) from the detector surface could be calculated. The allowed photon
distance ( ) is the average distance which represents a straight line
connecting the photon incident point on the detector and the photon outgoing
point from the detector. Therefore ( ) is considered as a function in solid
angle and due to the statistical nature of the radiation, the allowed photon
distance ( ) can be expressed, form of isotropic emission by formulaP210-13

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC](

This integration is divided into two parts each of them is depending on thevariation of ( ) with (θ1) and (θ2)

Where,

So from figure (6) and equations (29, 30 and 31) the average covered path
could be given by equation (32).

d






 R hLhR  (h  L). ln   h. ln R  tan 1    tan 1 
 2
2 
2
2 

h
(
h

L
)




h

R
(
h

L
)

R








h
1 



h2  R2 (32)

By plotting this average path distance ( ) with the source height distance (h)for different detector dimensions (2RxL) or volumes sizes as shown in figure(7) one can conclude thata) For axial point source, just on the detector face, the average pathdistance covered in the detector is increase by increase the detector
volume.

Figure (7): The variation of the average covered path withthe source height for different detector volumes
with dimensions (2RΧL)

P210-14IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]b) As the height (h) increase the average distance start to decrees until
minimum value. This minimum values ofis depend on the detector

volume and height of the source so the significant conclusion of theminimum average distance of the gamma ray photon in the detector iscorrelated with the distance of the source from the detector face and thevolume of the detector. Then, by replacing (L) in equation (28) with ( ) the
longitudinal limitation becomes:-

d

1 1
1
1

cos θ1 cos θ1 cos θ2      
cos θ1    cos θ m (33)σ o σ1σ2σm(8-b) -Lateral Limitation

The other condition for the maximum value of (m) to be considered forcontributing to eligible successive Compton scatterings is the so-called lateralcondition. Namely, for an axial ray falling along the detector’s axis i.e. the rayenter from the central point of the upper detector face, it is straight forward to
get:-

R

1

1

sin 1 12

sin(1   2 )       1msin(1   2  ..... m )(34)

In fact, considering the effective averaged ray entering the detector from any
point of the upper detector circular face, then the initial entering lateral
distance or entrance angle (θ0) should be considered as shown in figure (8-a)
and the lateral condition becomes:R  (h. tan 0 1
1
1.sin  0 )  .sin( 0  1 )  .......  .sin( 0  2  .....m ) (35)
0
1
m

Where, (θo) is the angle of incidence of the averaged ray in the detector. It is
computed from the relation:

d

h
R

cos  0 sin  0(36)

The initial angle of scattering (θo) can be calculated graphically as shown infig. (8)P210-15

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Figure (8-a): calculating laterallimitation condition

Figure (8-b): calculating the initial
angle of scattering θo graphically

After (fm) is calculated the photopeak coefficient can be calculated fromp ofmo(37)(9) Results(9-1)Calculating Photo Peak Efficiency at Discreet Gamma Energy
Table(1) reproduces the peak coefficient (μp) for a finite size Ge
detector, allowing for (m) successive Compton collision. The first column isthe energy of gamma rays in MeV. The second represents the photoelectriccoefficient alone (τo). The third column gives the values of (μp) for the casem=1 to m=6. All units are in cm-1. In practice for customary detectors, thecolumn m=1 is to be taken for low energies till about 0.3 MeV, while values of
m=2 and 3 are used for higher energies.

P210-16

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Table (1): The peak coefficient μp for a finite size Ge detector, allowing for (m)successive Compton collision.
hv
(MeV)

μp

τo

m=1
m=2
m=3
m=4
m=5
0.015 481.1992 482.467 482.2300 482.1507 482.1110 482.087m=6482.0710.020 220.3722 220.911 220.6423 220.5523 220.5073 220.480

220.4620.030

71.3282

74.588874.291174.190774.1404 74.1103

74.09020.040

31.0863

31.7043

31.3998

31.295431.2431 31.2118

31.1908

0.050

16.1819

16.7999

16.4995

16.3938

16.3408 16.3090

16.2878

0.060

9.474910.07489.78869.68449.63219.60069.57970.080

4.00294.56254.30734.20704.15604.12544.10500.100

2.04402.54012.33152.23862.19012.16092.14140.150

0.5962

0.97230.84230.76990.72770.70150.68400.200

0.25180.49440.45270.4036

0.36910.34610.33040.300

0.07670.20570.20590.19340.17350.15610.14320.400

0.03410.11240.12010.12190.11520.10330.09270.500

0.01880.06440.08240.08950.08780.08040.07170.600

0.01190.04300.05980.06750.07000.06690.06010.800

0.00610.02300.03290.04520.05170.0521

0.04801.000

0.00380.01440.02240.03340.03990.04170.04021.250 0.002433

0.00950.01450.02310.03000.03400.03401.500 0.001751
2.000 0.0011
3.000 0.0006

0.00670.00400.00200.01070.00650.00340.01660.01110.00580.02410.01770.01010.02870.02200.01400.02970.02350.0165Figures (9a and 9b) are showing the variation of the photopeak attenuation
coefficient with number of successive scatterings at different energy for infinite
detector size (Ge-Detector). From these figures one can conclude that at
small energies the photo peak coefficient (μp) is come to maximum after firstor second collision where the photons interactions terminated by photoelectricinteraction. But, as the gamma ray energy going up photo peak coefficient (μp)
increases gradually and reaches to maximum after 4; 5 or 6 successive
collisions. Since these graphs for infinite detector medium then one can notice
the number of collisions until 13. But, in real conditions the number ofsuccessive collisions or scattering are limited by escape from the finite
detection medium or, terminated by photoelectric absorption.

P210-17

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Figure(9a-9b):show the variation of the photopeak attenuation coefficient withnumber of successive scatterings at different energy for infinite
detector size(Ge-Detector).
As an example, consider a photon with energy 1.332 MeV emitted from axial point source atheight 25 cm taking the cap distance 0.5 Cm under consideration from Ge detector. The
calculations are traced as follows where the initial angle of entrance θo could be calculated
graphically from figure (8-b)θo=2.62o
and fm can be found as followshν(MeV)m μt (cm-1)

σ(cm-1)τo(cm-1)

F

cos 

1.332
0.693875
0.424401
0.289441
0.212802

0
1
2
3
4

0.258651
0.354817
0.433781
0.498451
0.55026

0.002209
0.009177
0.030394
0.095143
0.229357

0.520927
0.611640.681997
0.735220.775783------0.387788
0.3236
0.271173
0.228331

0.049524
0.069712
0.090567
0.118718
0.159489hν(MeV)

m

θ

fm

μp

0.693875
0.424401
0.289441
0.2128020.165089

1
2
3
4
5

67.18
71.12
74.26
76.81

0.024726
0.042503548
0.071514422
0.099470924

0.008606616
0.013205668
0.020710781
0.027943128

μp=0.002209045+(0.099470924× 0.258651)= 0.027943128 Cm-1
P210-18

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC](9-2) Relative FEPE of Germanium detector to sodium iodide detector
for axial point source.
The (IEEE) Standard test procedure for germanium gamma ray
detector [28] defines the relative efficiency ofgermanium spectrometerreferenced against a ( 3  3 ) NaI (Tl) detector which has an absolute
efficiency of 1.2 X10-3 at 1.332 MeV of 60Co, where the source to detector
distance is 25 cm when point source used. We use our direct method to
calculate the FEPF of ( 3   3  ) NaI (Tl) at 25 cm at photon energy 1.332
MeV. We obtained good agreement with the result of IEEE [28].
1-

Table (2) compare between the relative FEPE of germanium detector

to that of NaI (Tl) detector calculated by direct method and that from otherreferences [9,18,21,28,29,30,30] for different dimensions of HPGe and at thesame energy and conditions. It is easy to noticing that the excellent
agreement of the results obtained by direct method and that from of otherreferences.
Table (2): Comparison between the relative FEPE of germanium detector tothat of NaI (Tl) detector calculated by direct method and that from otherreferences [9,18,21,28,29,30,30]
Present

Reference

Dimension

Volumework

result

2RΧL Cm

Cm

8.908%

8.8%

4.3Χ3.75

54.46

Ref.[ 9]

11.0%

10.8%

4.04Χ4.4

57.53

Ref.[9](Tl))

24.514%24.5%

5.4Χ4.7

108

Ref.[18],[28]of 1.332 MeV.

36.407%36%

6.05Χ5.88

169.04

Ref.[29]10.012%

10.1%

5Χ4.4

86.39

Ref.[21]

Relative
Efficiency
Of (Ge) to3

Efficiency (NaI

Referenceand [30]

At 25 cm form
the face of thedetector

2-

Table(3) represents the relative percentage of intrinsic Photopeak
efficiency for axial point source for the energy range (0.2 MeV up to 1.2
MeV) with two detector of geometrical arrangements of volume of 2 cm 3and volume 0.39 cm3 at height h=5 cm. The comparison between therelative intrinsic FEPE calculated by direct method and that carried out bysemi empirical formula and that measured experimentally by [19]. The

P210-19

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

agreement between these results is very good through the energy range
under consideration.
Table (3): The comparison between the relative intrinsic FEPE calculatedby direct method and that carried out by semi empirical formulaand measured experimentally by [19].

Detector Volume
(Cm3)

2.0

0.39

Relative Intrinsic
Efficiency [19]PhotonEnergy
(MeV)
0.2
0.3
0.5
0.8
1.00
1.2
0.2
0.3
0.5
0.8
1.00
1.2

ExperimentCalculated byally
Semi-empirical
20.5Χ0.9
21.0
8.10
8.0 Χ 0.6
2.60
2.55 Χ 0.09
1.09
1.1 Χ 0.05
0.76
0.76 Χ 0.04
0.56 Χ 0.02
0.55
27.20
26.0 Χ 2.0
9.00
8.6 Χ 0.4
2.45
2.50 Χ 0.40
0.88
0.87 Χ 0.05
0.58
0.56 Χ 0.08
0.39 Χ 0.07
0.41

Direct
Method(Present
Work)
20.29088.52582.9303
1.0566
0.6631
0.484420.0778
8.94152.89681.04440.65550.47883- Table (4) gives the comparison of intrinsic photopeak efficiency for an axial
point source calculated by direct method at for detector (4.4cm×5.0cm) andthe distance between the source and detector is 25.7 cm for energy range
from 0.2234 MeV up to 3.2536 MeV A.OWENS [19] that measure the
FEPE experimentally and also compared with the result carried out by
HAJNAL and KLUSEK [30] that use semi empirical method. One can seethat the agreement of direct method results with the experimental andsemiempirical results is very good.

P210-20

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Table (4:) The comparison of intrinsic photopeak efficiency for an axial point
source calculated by direct method at for detector (4.4cm×5.0cm)
and the distance between the source and detector is 25.7 cm forenergy range from 0.2234 MeV up to 3.2536 MeV A.OWENS[19]Intrinsic Photopeak Intrinsic Photopeak
Measured Intrinsic
PhotonEfficiency εip using
Efficiency εip
Photopeak Efficiency
Energysemiempiricalcalculated by (Direct
εip by Owens(MeV.)
formulamethod)[ 21 ][32]
Present Work
0.223430.24192
0.29520.30309
0.352
0.35626
0.384090.60930.6616
0.76870.84678
1.04
1.173221.23829
1.27452
1.33224
1.360211.764
1.77133
2.204
2.59852
3.25361

0.575
0.537
0.46
0.46
0.392
0.393
0.36250.223
0.19150.158
0.14450.12
0.11080.10590.109
0.09920.098
0.079
0.080310.061
0.05330.04250.56161
0.53623
0.464440.45437
0.396758
0.392157
0.363746
0.216859
0.196914
0.165495
0.148378
0.120841
0.109870
0.106034
0.104117
0.101229
0.099889
0.080301
0.079933
0.063088
0.0532940.04

0.64090.60020.47780.45800.40280.39740.36280.22500.19710.16230.14510.11220.11480.10790.1046
0.0997
0.09730.07450.07420.05840.0472
0.05(9-3) Graphical Comparisons of FEPE calculated by direct method

and other method
1- Figures (10,11) give the comparison between the intrinsic full energy
peak efficiency with the photon energy range (0.2 MeV up to 1.4 MeV)
calculated by our direct method and that calculated by semi empiricalformula and experimental method [12] for volume 2 cm3 and 0.39 cm3 thetwo represents the excellent agreement between direct method and semiempirical method and experimental method.

P210-21

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

Figure (10,11): The comparison between the intrinsic full energy peak efficiency withthe photon energy range (0.2 MeV up to 1.4 MeV) calculated by ourdirect method and that calculated by semi empirical formula andexperimental method
Figure (12) present Comparison of variations of Full Energy Peak Efficiency
for the energy range from 0.1 MeV and 2.5 MeV. These comparison werecarried out between the present Direct Method and, that calculation by
WAINIO and KNOLL, ref [7], that using Monte Carlo calculation by B.LAL et
al. ref.[9] and by experimental value of CLINE ref. [31] for detector dimensionof radius R=0.9 cm and depth of L=0.8 cm and the axial point source distanceis 0.8 cm.

Figure (12): Comparison of the Calculated FEPE by direct method andother methods.

P210-22

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]One can easy notice that the values of the 4-Methods are very closely toeach other but the values calculated by direct method is the closest one tothe experimental measurements compatibility.

(10) Conclusion
One can conclude that, an exact mathematical model to calculate
directly photopeak efficiency of HPGe detector with an axial point source at
different distances from the detector surface is derived successfully. The
model is applicable for gamma ray energy range up to 3 MeV where, thepredominant reactions considered are Compton scattering and photoelectric
absorption.
The geometrical and mathematical treatment has been done tocalculate the average path of the gamma ray in the detector and consequently
its the lateral and longitudinal limitations in the finite detector size.
Consequently,
The term photo peak coefficient was calculated accurately.
The full energy photo peak efficiencies calculated by direct
mathematical model, for different detector sizes, found in an excellent
agreement with other accurate published works by other methods and morecloser to experimental measurements than other theoretical calculations.
Finally, one can say that this work gives a good support and enhancecalculations of absolute activity of γ-sources with different geometry inaddition to improving the calibration of HPGe detectors.

P210-23

IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC](11) References[1] Hoste, 1981 J. Hoste, Calculation of the absolute peak efficiency of
gamma-ray detectors for different counting geometries, Nucl. Instrum.
Methods 187(1981), p. 451.
[2] Moens et al., 1983 L. Moens and J. De Donder et al., Calculation of thepeak efficiency of high-purity germanium detectors, Int. J. Appl. Radiat.
Isot. 34(1983), p. 1085.
[3] Lippert, 1983 J. Lippert, Detector-efficiency calculation based on pointsource measurement,[4] Mihaljevic et al., 1993 N. Mihaljevic and S. De Corte et al., J. Radioanal.
Nucl. Chem. 169 (1993), p. 209.
[5] Wang et al., 1995 T.K. Wang and W.Y. Mar et al., HPGe detector
absolute-peak-efficiency calibration by using the ESOLAN program, Appl.
Radiat. Isot. 46(1995), p. 933.
[6] Wang et al., 1997 T.K. Wang and W.Y. Mar et al., HPGe detector
efficiency calibration for extended cylinder and Marinelli-beaker sources
using the ESOLAN program, Appl. Radiat. Isot. 48 (1997), p. 83.
[7] K. M. Wainio And G.F.Knoll, Nucl. Inst. And Meth. 122 (1966) 213.
[8] Overwater et al., 1993 R.M. Overwater and W.P. Bode et al., Gamma-ray
spectroscopy of voluminous sources corrections for source geometry andself-attenuation, Nucl. Instrum. Meth. A 324 (1993), p. 209.
[9] B.Lal and K.V.K.Iyengar (1970) Nucl. Instr. and meth. 79, 19.
[10] Marc Décombaz, Jean-Jacques Gostely and Jean-Pascal Laedermann,(1992) Nucl. Instr. and meth. A312, 152.
[11] Finckh and Geissörfer and et.al (1987) Nucl. Instr. and meth. A262, 441.
[12] G.GAGGERO (1971) Nucl. Instr. and meth. 94, 481.
[13] M.Korun et.al (1997)Nucl. Instr. and meth. A 390, 203.
[14] Gerhard Hasse, David Tait and Arnold Wiechen (1993)Nucl. Instr. andmeth A329,483[15] H.Seyfarth, A. M. Hassan, B. Hrastnik, P. Gottel W. Delang (1972) Nucl.
Instr. and meth. 105,301[16] Naim.M.A Isotope & Rad.Res.25,2,65-73(1993)[17] G.B. Beam, L.Wielopolski , R.P.Gardner, and K.Verghese, (1978) Nucl.
Inst. Method 154,501.
[18] T.Paradellis And S. Hontzeas (1969) Nucl. Instr. and Meth. 73,210[19] A.Owens (1989) Nucl. Instr. and meth. A(274),297[20] Selim and Abbas Radiat. Phys. Chem. Vol. 48, No. 1, pp. 23-27, 1996[21] Y.S. Selim, And M.I. Abbas, M.Fawzy, Rad.Phys.Chem. 53 (1998) 589.
[22] Y.S. Selim, And M.I.Abbas, Rad.Phys.Chem. 58 (2000) 15.

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IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC][23] Evans, R.D.(1955) The Atomic Nucleus, Tata McGraw-Hill Press, New
Delhi[24] Hubbell J.H., Gimm, H.A. (1980) Journal of Physical and Chemical
References Data[25] Lamarch, J.R. (1960) Introduction to Nuclear Reactor Theory, AddisonWesley Press, New York[26] (IEEE) Standard test procedure for germanium gamma ray detector St,
325-1971.
[27] M.KORUN et.al (1997)Nucl. Instr. and meth. A 390, 203.
[28] Instructional Manual “Aptec Germanium Spectrometer” Type “Coaxial”
Model No. “G220a31gl” Institute Of Higher Studies And Researchs
Alexanderia University.
[29] Hubbell, J.H. And Seltzer S.M. (1995)NISTIR-5623[30] F.Hajnal And C.Klusek, Nucl. Instr. and Meth. 122(1974)559.
[31] J.E. Cline, IEEE Trans. Nucl. Sci. NS- VOL. 15,No.3 (1968) 198.

Authors
1- Dr. Younis S.Selim,Prof. Emeritus. of Nuclear Physics
Department of physics , Alexandria University, Egypt
For sorry, He passed away 2007
2- Mr. Mohamed S. Hussien ,-PhD, Candidate in Royal Military Collage of Canada-Master of Physics and Technology of Nuclear Reactor Birmingham university,
UK, 2008.
-Master of theoretical radiation physics, Alexandria university, Egypt 2003.
Email: Mohamed.hussein@rmc.ca
3- Dr. Mohamed A. Fawzy,Prof. of Radiation Physics
Department of physics , Alexandria University, Egypt.
Email: salem_adelfawzy@yahoo.com
Tel: 002-03- 3911794
4- Dr. Ahmed M. El-Khatib,
Prof. of Experimental Physics
Former hear of the Department of physics, Alexandria University, Egypt.
Email: elkhatib50@yahoo.com
Tel: 002-03- 3911794

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IEEE NPSS (Toronto), UOIT, Oshawa, ON, 2-3 June, 2011
International Workshop on Real Time Measurement, Instrumentation & Control [RTMIC]

P210-26