Accepted Manuscript

Title: Validation of new mathematical formulae for calculating the well-type gamma-ray detector efficiencies

Author: Eman. M. El-Bayoumi N.S. Aly Mahmoud I. Abbas

PII: DOI:

Reference:

S1658-3655(15)00158-2 http://dx.doi.org/doi:10.1016/j.jtusci.2015.08.010 JTUSCI 245

To appear in:

Received date: Revised date: Accepted date:

9-5-2015

18-8-2015

30-8-2015

Please cite this article as: En.M. El-Bayoumi, N.S. Aly, M.I. Abbas, Validation of new mathematical formulae for calculating the well-type gamma-ray detector efficiencies, Journal of Taibah University for Science (2015), http://dx.doi.org/10.1016/j.jtusci.2015.08.010

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Validation of new mathematical formulae for calculating the well-type

gamma-ray detector efficiencies

Eman. M. El-Bayoumi1, N. S. Aly1, Mahmoud I. Abbas2' * 1) Physics & Chemistry Department, Faculty of Education, Alexandria University, Alexandria, Egypt. 2) Physics Department, Faculty of Science, Alexandria University, 21511 Alexandria, Egypt *) Corresponding author (mabbas@physicist.net)

Abstract

In this work direct mathematical formulae for measuring the full-energy peak efficiency of HPGe welltype detector are found and the value of the measured efficiencies are compared with the published works of the experimental and theoretical old method and having a good agreement. In this new approach the path length d(0, is derived as a function in the polar angle 0, and the azimuthal angle This will reduce the mathematical formulae to an easiest and compact shape.

Keywords: HPGe well-type detectors; full-energy peak efficiency; path length, azimuthal angle, polar angle.

Introduction

Well-type HPGe and Nal (Tl) detectors are a good choice when low gamma activity samples need to be measured with good energy resolution. The gamma detector efficiencies have been measured before [120] which are using the spherical trigonometry technique. Recently, Abbas [21, 22] introduced a new approach involving the determination of the path length d(0,^) covered by a photon inside the detector active volume and the geometrical solid angle Q. In this new approach the path length d(0,^) is derived as a function in the polar angle 0, and also as a function in the azimuthal angle This will reduce the mathematical formulae to an easiest and compact shape, and the integration over the azimuthal angle ^ will be always from 0 to 2n [23, 24]. Most importantly, it calculates the well type detector efficiencies without the need for standard sources, as is the case for experimental methods, nor optimization of detector parameters as for simulation methods .The work described below involves the use of a new direct analytical expression to calculate the efficiencies of well type detector. So we will measure the well-type HPGe detectors efficiencies using this new method for non-axial point and extended circular disk, and compare the results with previous work (the old method).

Mathematical method

The basic formula of the efficiency with respect to point source :

fattCi-e-^)sin6d<pà8 (1)

where, p is the attenuation coefficient of the detector material. d is the path length traveled by a photon through the detector active medium. fatt is the attenuation factor determining the photon attenuation by the source container and the detector end cap materials is expressed as:

energy peak efficiency (or photo peak efficiency: which is the efficiency for producing full energy peak pulses only, rather than a pulse of any size, for the gamma ray), but not to the total efficiency (is the ratio of the total number of counts in the spectrum, by the number of photons with energy emitted by the source).

And the circular disk efficiency is given by the equation:

where S is the disk radius and e is the point source efficiency.

Fig. 1 introduce the distance M(^) that is the distance between the projection of the non-axial point

source and any point lying on the circumferences of inner circle and for outer circle Ma (V) of

the well type detector, where M(cp) is given by: M((p) - p coscp + J R2 — p2 s in2 (p.

Fig. 1: present the distance M(<$).

The source is located inside the detector at position h and the non- axial point efficiency can be calculated based on five different path lengths covering distances (di, and 5) taking into

account for the gamma ray emitted from the source and enter to the detector, as seen in [Figure 2] and the values of these distances:

l,<p) =

Mo (<p) h'

cose sine

Mo(<p)-M sine

-K + ti

04( $)

02(<))ÍÍ;

r 05( $) Ri

d5(e,^) 1 1 \ :

\| K h

d4(e,$) P,h)

d2(e,A)

/1 1 di( e) i i i i i L

Fig 2-a

.05«) 1 Ri

d5(e,<) Va jjŒ

d4(e,<) i K h

d3(e,<M/ / / 03(<)/

1 ! / d1( e) ! 1 / / 1 ! 1 1 1 L

Fig.2.b

Fig.2: illustrating the dimensions of the detector, and the path of the photons from the source to the detector surface.

The polar angle (0) as a function of the azimuthal (9) angle takes the values:

« , x ,Mt>

£?-.(<p)=atan(—-)

8A{<p) = 7i-atan(—-)

9A(p) = TC-atant*^1—) where,

fi= -pCOS<J£? - p2sin2<p

?= —pcos<p +JRI — p2sin2<p

(10) (11) (12)

The important idea of this method is that the azimuthal angle ^ takes the values from zero to 2n, for a certain value of 0. So the resulting of the non -axial point efficiency by using this method will be:

1- In case of < 02 j:

^point

1 r7r / rS3(<p-)

- r" t r°aWJ r

2jr Jo VO ■ 1

f^M f (g

+Oiw +CE +

1 VVj ■

= ' ■ ■ j=1, 3, 4 and 5

2- In case of (ft, > 92 ) Non- axiai —

1 f2 (8, <p)d8 + f^l /4 (8, <p)d8 +

fj{8, <p) = fatt(l - e^Jimfi) , j=1, 2, 4 and 5 (19)

Result and discussion

For a well-type HPGe detector, with parameters reported by Abbas[17], for energies from 0.0595 MeV to 1.524 MeV ,the photo peak efficiencies are calculated using the present work and compared with those obtained by theoretical and experimental data, for circular disk source with radius = 0.62 cm as seen in Figure (3). Table 1 shows the values of the photo peak efficiency of the experimental and theoretical (old method and new one). The deviation percentage between the theoretical new method and experimental efficiency values is less than 0.5%, and the obtained results confirmed that our present method to measure the efficiencies of well-type detector is useful.

0.1 Q_

-Abbas

A Wang — present work

_l_I_I_I_I_

—I_I_I_I_I_I—

Ey(MeV) Fig- (2)

_i_i_i_i_I_i_i_i_

Fig(3): comparison of the experimental [Wang] and the theoretical old method[Abbas]and the theoretical present data of the photo peak efficiency(ep) as function of gamma ray energies(E8).

the full energy peak efficiency values

Photon Energy(MeV) Mp(cm-1) Experimental data Theoretical data

Old method New method

0.0595 5.768429 0.885 0.888 0.889

0.08803 5.7873823 0.945 0.948 0.941

0.1221 1.0583224 0.814 0.807 0.814

0.3201 0.191034 0.294 0.292 0.294

0.6616 0.0825125 0.142 0.141 0.142

0.8348 0.063778 0.112 0.111 0.112

1.1155 0.046532 0.083 0.082 0.083

1.5247 0.0325112 0.059 0.058 0.059

Tablel: introduce the values of experimental and theoretical full energy peak efficiency, and is the attenuation coefficient of photo peak.

Conclusions

We have been calculated the full-energy peak efficiencies by using a new mathematical technique and compared with those obtained by the published experimental data of Wang et al. [7] and the theoretical one of Abbas [17]. The percentage deviation between the experimental and the theoretical( new method) in this work are less than 0.5%.

References

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