h^atibef:,/.i0usflio/0/rhe0rnto/0/52 Applied Physics 2m3, ^ 0 journa| 0f Theoretical and Applied Physics

http://www.jtaphys.com/content/7/1/52 . "" '

a SpringerOpen Journal

RESEARCH Open Access

Calculation of full-energy peak efficiency of Nal (Tl) detectors by new analytical approach for parallelepiped sources

Ahmed M El-Khatib1, Mohamed S Badawi1*, Mona M Gouda1, Slobodan I Jovanovic2,3, Aleksandar D Dlabac3, Nikola N Mihaljevic3,4, Sherif S Nafee1 and Ekram A El-Mallah1

Abstract

A new analytical approach is presented for the calculation of full-energy peak (FEP) efficiency of NaI (Tl) detectors. The self-attenuation of the parallelepiped source matrix, the attenuation by the source container, and the detector housing materials were considered in the mathematical treatment. The efficiency values calculated using the presently suggested analytical approach are compared with those measured values obtained by two different sizes of NaI (Tl) detectors. The calculated and the measured full-energy peak efficiency values were in a good agreement.

Keywords: NaI (Tl) scintillation detectors; Parallelepiped sources; Full-energy peak efficiency; Self-attenuation

Introduction

One of the most important parameters in the calculation of the gamma activity of environmental radioactive sources with respect to emitted gamma energy is the detection efficiency which is usually determined using calibrated standard sources [1]. Standard radioactive sources, if available, are costly and would need to be renewed, especially when the radionuclides have short half-lives [2]. In addition to the influence of the source matrix on the counting efficiency which can be expressed by the self-attenuation factor, the fraction of the gamma rays not registered in the full-energy peak is due to the scattering or absorption within the sample [3]. An effective tool to overcome these problems could be the use of computational techniques [4-9] to complete the calibration of the gamma spectrometry system. Also, Selim and Abbas [10-14] solved these problems by deriving direct analytical integrals of the detector efficiencies (total- and full-energy peak) for any source-detector configuration and implemented these analytical expressions into a numerical integration computer program. Moreover, they [15-18] introduced a new theoretical approach based on that direct statistical method to determine the detector

* Correspondence: ms241178@hotmail.com

Vhysics Department, Faculty of Science, Alexandria University, Alexandria 21511, Egypt

Fulllist of author information is available at the end of the article

efficiency for an isotropic radiating point source at any arbitrary position from a cylindrical detector, as well as the extension of this approach to evaluate the volumetric sources.

In a large extent, we introduced a new analytical approach for the calculation of full-energy peak efficiency of the coaxial detector with respect to point and volumetric sources (cylindrical and spherical) [19-21]. In the present work, we deal with parallelepiped source. The basic idea of this approach is the separate calculation of the intrinsic and the geometrical efficiencies, and the factors which related to the photon attenuation in the detector end cap, inactive layer, source container, and the self-attenuation of the source matrix. The calculations depend on two main factors. First is the accurate analytical calculation of the average path length covered by the photon in each of the following: the detector active volume, the source matrix, the source container, the inactive layer, and the end cap of the detector. Second is the geometrical solid angle D.

Basics of the efficiency computation

The case of a non-axial point source

Consider a right circular cylindrical (2R x L) detector and an arbitrarily positioned isotropic point source located at a distance h from the detector top surface and at a lateral distance p from its axis, as shown in Figure 1.

ringer

© 2013 El-Khatib et al.; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Figure 1 A diagram of a cylindrical-type detector with a non-axial point source (p > R).

£point — fatt eg £i;

O — J J sin0 d0 d(p.

There are two main cases to be considered for calculating the intrinsic and geometrical efficiencies of the detector with respect to point source, namely, (1) the lateral displacement of the source is smaller than or equal the detector circular face's radius (p < R), and (2) the lateral distance of the source is greater than the detector circular face's radius (p > R). The two cases have been treated by Abbas et al. [16]. The values of the polar and the azimuthal angles based on the source-to-detector configuration are shown in Table 1.

The attenuation factor (fatt)

The attenuation factor fatt is expressed as:

ftt fay fc

The efficiency of the detector with respect to point source is given as follows [16]:

where fay and f,ap are the attenuation factors of the detector inactive layer and end cap material, respectively, and they are given by:

fay —

./Cap — e

cap 5caP

where ei and eg are the intrinsic and the geometrical efficiencies which are derived by Abbas et al. [16]. fatt is the attenuation factor of the detector inactive layer and end cap material. In 'The attenuation factor fatt)' section, this factor will be recalculated by a new method.

where ^lay and ^cap are the attenuation coefficients of the detector inactive layer and end cap material, respectively. 5lay and 5cap are the average path length traveled by a photon through the detector inactive layer and the end cap material, respectively, and they are represented as follows:

The intrinsic (e) and the geometrical (eg) efficiencies

The intrinsic and geometrical efficiencies are represented by Equations 2 and 3, respectively.

gi — i_e-^

£g — '

(2) (3)

J d(0, 9) dQ. J J d(0, 9) sin0d0d9

J J i'(0,9) sin0 d0 d9 J J t(0,9) sin0 d0 d9

5lay J J sin0 d0 d9

J J i"(0,9) sin0 d0d9 J J i"(0,9) sin0d0d9

5cap — J J sin0 d0 d9

where ^ is the attenuation coefficient of the detector material. d is the average path length traveled by a photon through the detector, and Q is the solid angle subtended by the source-detector; they are represented by Equations 4 and 5, respectively. These will be discussed in detail according to the source-detector configuration as shown below:

where d and (p are the polar and the azimuthal angles, respectively. d (ftp) is the possible path length traveled by the photon within the detector active volume.

where i'(0, p) and t"(0, p) are the possible path lengths traveled by the photon within the detector inactive layer and the end cap material, respectively.

Consider that the detector has an inactive layer covering its upper surface with thickness tDF and its side surface with thickness tDS, as shown in Figure 1. The possible path lengths and the average path length traveled by the photon within the inactive layer for cases (p < R) and (p > R) are shown in Table 2, where t'1 and t'2 represent the photon path length through the upper and the side surface of the inactive layer, respectively.

Consider that the thickness of the upper and side surfaces of the detector end cap material is ta and tw, respectively, as shown in Figure 1. The possible path

Table 1 The values of the polar and the azimuthal angles based on the source-to-detector configuration [16]

The polar angles

The azimuthal angles

01 = tan"

02 = tan"

03 = tan"

04 = tan"

R—p|

h + L R—p h

R + p'

h + L R+p

(ffiffi

eT = tan—1 ( ffiffi

V max = cos

2ph tanfl

Jp2—R2 + (h + L) tan20 i

V max = cos1'

2p(h + L) tan£

Vc = sin ' p

( V max V max)

-R2 h2 tan

flc = tan

0c ' = tan-1

lengths and the average path length traveled by the photon within the detector end cap material for cases (p < R) and (p > R) are shown in Table 3, where t" i and t"2 represent the photon path length through the upper and the side surface of the detector end cap material, respectively. From Table 3, it was observed that the case in which (p > R) has two sub-cases which are (R < p < Ra) and (p > Ra), where Ra is the inner radius of the detector end cap. There is a very important polar angle (0cap) which must be considered when we study the case in which p > Ra, and this is given by the equation below:

0cap = tan

-l(P-Ra

where k is the distance between the detector end cap and the detector upper surface.

The case of a parallelepiped (block) source

Consider a block source with dimensions A (major side), B (minor side), and H (height) as illustrated in Figure 2, then the efficiency is given by the equation below:

Table 2 Possible path lengths and their average traveled by the photon within the inactive layer [p < R and p > R]

f — JDL.

S = cosfl

f — -IDF—

S = cosfl

P cosv +

]/(R + fDS)2—P2 sin2V

p cosv + y (R) —p2 sin2v

5lay = t

M 1 + 2R2 sin2v

5lay = T?

Z1 = J J f'1 sin

V maxfl4

+ J J f'1 sin

fl V0 max fl0 c V0 max

Z3 = J J f'2 sinfldvde + J J f'1 sin

fl1 0 fl2 0 flc Vc fl4V max

+ J J f'1 sin 0 dv de + J J f'1 sin

fl c 0 flc 0

, (02 - 0'c)

flcV' max fl2 Vc

Z3 = J J f 2 sin0dvdfl + J J f'2 sin

fl1 0 fl c 0

flcvc fl4Vmax

+ J J f'1 sinfldvdfl + J J f'1 sin

fl2 0 flc 0

, < 0/c)

Table 3 Possible path lengths and their average traveled by the photon within the detector end cap [p < R and p > R]

R < p < Ra

p > Ra

f" = f'

1 gos0

f" = f'

1 gos0

GOsV + ^(«a + fw)2-P2 Sir

p GOsV +

p2 sin2V

,2v fw " + ^T7 sin V

— - 2-

"G'P — /2

—-i "G'P — /4

— - i "G'P — /4

Z = uf;si

1 00 '

V m.x04

+ I If" sin

0 02 '

0c V max

Z = I I f'' sin

3 0' 0 ' 0c Vc

+ I I f" sin

0 c0 ' 04V max

+ I I f" sin

0 c V max 0c Vc

Z3 — I I f" sin0 dv d0 + I I f" si

3 0' 0 ' 0 c 0 ' 04Vmax

+ I I f" sin

0c 0 "

0GapV max

z3 = I 1 f2 si

3 0" 0 2

0 c Vmax

+ I I f" sin

0cap 0

+ I I f" sin0dvd0 + I I f" sin

0" > 0ca

0c V max

Z3' = I m f sin

3 0" 0 2

+ I If"" sin

0capVc

+ 11 ^2 si

0 c 02

04V max

+ I I f"" sin

0 c V max 0c Vc

Z3 — I m f" sin0dVd0 + I If" si

3 0" 0 2 0 c 0 2

0GaPVmax 04 Vmax

+ I I f'2sin0dVd0 + I I f" si

0 0 2 0GaP 0 "

04 > 0cap > 0G

0 V max

Z3' = I m f sin

3 0" 0 2 04Vmax

+ I I f2 sin

+ I I fsi

0'c > 0cap

0cap > 0 'c

0cap > 04

^block —

4 Sself Sscftt £i £g

detector and the solid angle will have new forms due to (10) the geometry of Figure 2. The average path length is expressed as follows:

H+h„ / aT § N

J J Nb1 da + J Nb2 da dh

H+h„ / aT 2 \

J J Nb3 da + J Nb4 da dh

where V is the volume of the block source (V = ABH), Sself is the self-attenuation factor of the source matrix, and Ssc is the attenuation factor of the source container material. h V0 aT )

The intrinsic and geometrical efficiencies are defined before in Equations 2 and 3, respectively, but the average

path length d traveled by the photon through the The geometrical efficiency eg is given by

H+h0 / aT 2 \

J J Nb3 da + J Nb4 da dh

ho \ 0 aT y

" 2n ;

J11(p < R) p dp

J11 (p < R)p dp + J 1x(p > R) p dp

(p1 < R) (p1 > R)

where a is the angle between the lateral distance p and the detector's major axis, as shown in Figures 3 and 4. The factor 4 is introduced in Equation 10 to cover the values of a from 0 to 2n. The values of Nb2, Nb3, and Nb4 are given as follows:

Nb2 —

J I1(p < R) p dp

Jl1(p < R) p dp + J11 (p > R)p dp

(p2 < R) (p2 > R)

Nb3 —

rj/2(p <R)pdp (p <R)

J12 (p < R)p dp + J I2(p > R) p dp (p1 > R)

Nb4 —

rjl2(p < R) p dp (p < R)

J12 (p <R)pdp + J12 (p > R)pdp (p2 > R),

a= x ; b = — 2 2

aT = tan 11 — a

(17) _

^cap —

H+ho / ar 2 \

J J NC1 da + J NC2 da dh

ho \ 0 ar J

H+ho i ar ! \

J J Nb3 da + J Nb4 da dh

I1 and I2 are the numerator and the denominator, respectively, of d in the equation obtained by Abbas et al. [16] for non-axial point source, and ho is the distance between the source active volume and the detector upper surface.

The new forms of the average path length traveled by the photon through the detector inactive layer and the detector end cap material are given by Equations 18 and 21, respectively.

^lay =

H+ho Í ar 2 \

J J NL1 da + J NL2 da dh

ho 0 ar

H+ho i ar f \

J J Nb3 da + J Nb 4 da dh

ho 0 ar

J Zi p dp

JZi p dP + J Z3 P dP

J Zi P dP

JZi p dP + J Z3 P dP

(Pi * R) (Pi > R)

(P2 * R)

(P2 > R);

J Z' i p dp

JZ' i p dp + J Z' 3 p dp

(Pi * R) (Pi > R)

N C2 = <

J Z' i p dp

JZ' i p dp + J Z' 3 p dp

(P2 * R)

(P2 > R);

where Z'i and Z'3 are as identified before in Table 3.

If Ah is the source container bottom thickness, Ay is the source container wall thickness from the minor axis, and Ax is the source container wall thickness from the major axis. Table 4 shows the possible path lengths traveled by the photon within the source and the source container, while Table 5 shows the values of the polar and azimuthal angles of the source; the source container also shows the maximum angle of the photon to enter the detector from the face and the minimum angle of the photon to enter the detector from the side which are labeled by 0max and 0min, respectively.

The self-attenuation factor of the source matrix is given by

where Z1 and Z3 are already defined in Table 2.

Sself = e

Table 4 The possible path lengths traveled by the photon within the source and the source container

The possible path lengths

The source

The source container

To exit from the base To exit from side I To exit from side II To exit from side III To exit from side IV

t — h=ha

'0 = cose

h = sine sin/

f2 = a-x

sine cos/3

f3 = b-y I

sin/ sine i

f4 = a+x

sine cos/

t _ Ah

'0c = -cose

fl - Ay

1c sine sin/3 Ax

sine cos/ I

where / = a + 9 , x = p cosa

f3c = f1c = f4c = f2c =

y = p sina ,

sine sin/ ' Ax

sine cos/1

a = tan"1®

Table 5 Values of the polar and the azimuthal angles of the source and the source container

The source

The source container

The polar angles

The azimuthal angles

The polar angles

The azimuthal angles

01 = tan-

02 = tan-

03 = tan"

04 = tan

(h-ho) sinß

(h-ho) cosß

-1 b-y

(h-ho) sinß ^ a + x

(h-ho) cosß

p cos9

+ TR2-

■p2 sin29

pcos^^^/R2-p2 sin29

91 = 2 -a + tan"

9 2 = n-a + tan 1 (-

-1 b-y

9 3 = 2n-a- tan 1 -

lo + x

2n-a + tan

-if b + y

b + y ' a + x

(a>aT ) (a < aT)

01c = tan-1 0 2c = tan-1 0 3c = tan-1 04c tan-1

b + Ay + y

(h-ho) sinß a + Ax-x

(h-ho) cosß b + Ay-y

(h-ho) sinß a + Ax + x

(h-ho) cosß

91 c = 2 -a + tan

92 c = n-a + tan

9 3 c = 2n-a- tan 1

a + Ax-x b + Ay +y b + Ay-y' a + Ax-x b + Ay-y a + Ax + x

2n-a + tan

9 4 c =

b + Ay + y' a + Ax + x

V b + Ay + y^ tan 1 -;- -a

a + Ax + x

(a>aT ) (a < aT)

0 max = tan 1

0 min = tan 1

QJ IÛ

where is the attenuation coefficient of the source matrix, and t is the average path length traveled by a photon inside the source and is given by

t = i(i;-, di, <pt)

H+ho / ar 2 \

J J Mm da + J Mb2 da dh

ho \ 0 aj- y

H+ho /ar § \

J J Mb3 da + J Mb4 da dh

ho V 0 ar /

where j takes the values from 0 to 4, while i takes the values from 1 to 4.

The values of Mb1, Mb2, Mb3, and Mb4 are given as follows:

J to sin9 d9

9l 9 max

J to + J tl

sin9 d9

J to sin9 d9

92 9 max

J to + J t2

sin9 d9

(9l S 9 max) (9 max > 91)

(02 S 9 max) (9max > 92)

Jgbl P dP

Jgb2 P dP + J gb3 P dP

Jgbl P dP

Jgb2 P dP + J gb3 P dP

Jgb4 P dP 0

Jgb4 P dP + J gb5 P dP

(Pl £ R) (Pl > R)

(P2 £ R) (P2 > R)

(Pl £ R) (Pl > R)

to sin9 d9

93 9 max

to + t3

sin9 d9

to sin9d9

94 9 max

to + t4

sin9 d9

(93 S 9 max) (9max > 93)

(94 S 9 max) (9 max > 94)

Jgb4 P dP (P2 £ R)

Jgb4PdP + Jgb5PdP (P2 > R)-

Now, we can find the values of gb1, gb2, gb3, gb4, and gb5 (consequently, we can find the value of t ), see Figures 3 and 4.

Pl P2 P3 P4

JPl + JP2 + JP3 + JP4 + JPl

0 Pl <h P3 P4

P4 Pl P2 P3 2n

J P4 + JPl + JP2 + JP3 + JP4

0 P4 Pl <h P3

(a S aT ) (a < aT)

gb2 = 2 J J to sin9 d9 dp

<Pc 9 m

gb3 = 2 J J to sin9 d9 dp

gb4 = 2J J sin9 d9 dp

Pc 9 m

gb5 = 2J J sin9d9dp.

0 9 min

The attenuation factor of the container material is given by

Ssc = e-^; (39)

where is the attenuation coefficient of the source container material, and tc is the average path length traveled by a photon inside the source container and is expressed as

Table 7 Parameters of the sources

Items Source volume (mL)

100 200

Dimension of the cross-section 59.23 X 37.88 60.48 X 60.48

body (mm2)

Height (mm) 51.02 61.4

Wallthickness (mm) 1.5 1.52

Activity (Bq) 5,048 ± 49.98 5,048 ± 49.98

tc = tj die Vic) ■ (40)

Experimental setup

The FEP values were measured by employing two different NaI (Tl) detectors called as Det. 1 and Det. 2. The manufacturer parameters and the setup values are shown in Table 6.

The sources used are polypropylene (PP) plastic vials of volumes 100 and 200 mL filled with an aqueous solution containing 152Eu radionuclide which emits y-ray in the energy range from 121 to 1408 keV. Table 7 shows the dimensions of the sources. The efficiency measurements are carried out by positioning the sources over the end cap of the detector. In order to minimize the dead time, the activity of the sources is prepared to be 5,048 ± 49.98 Bq. The measurements are carried out to obtain statistically significant main peaks in the spectra that are recorded and processed by winTMCA32 software made by ICx Technologies (ICx Technologies GmbH, Solingen, Germany). The acquisition time is high enough to get a number of counts at least 20,000, which makes the statistical uncertainties to be less than 0.5%. The measured spectra were analyzed with Genie 2000 software (CANBERRA Industries Inc., Meriden, CT, USA) using its automatic peak search and peak area

Table 6 The manufacturer parameters and the setup values

Items Det. 1 Det. 2

Manufacturer Canberra Canberra

Serialnumber 09L 654 09L 652

Detector model 802 802

Type Cylindrical Cylindrical

Mounting Vertical Vertical

Resolution (FWHM) at 662 keV 8.5% 7.5%

Cathode-to-anode voltage +1,100 Vdc +1,100 Vdc

Tube base Model2007 Model2007

Shaping mode Gaussian Gaussian

Crystal type NaI(Tl) NaI (Tl)

Crystalvolume (cm3) 103 347.64

calculations, along with changes in the peak fit using the interactive peak fit interface when necessary to reduce the residuals and error in the peak area values. The peak areas, the live time, the run time, and the start time for each spectrum are entered in the spreadsheets that are used to perform the calculations necessary to generate the efficiency curves.

The full-energy peak efficiency values for the two NaI (Tl) detectors were measured as a function of the photon energy and calculated using the following equation:

e(E) = N (E) , TTCi, (41)

v ; TASP(E)11 n v ;

where N(E) is the number of counts under the full-energy peak that is determined using Genie 2000 software, T is the measuring time (in seconds), P(E) is the photon emission probability at energy E, AS is the radio-nuclide activity, and Ci are the correction factors due to dead time and radionuclide decay. All the sources were measured on the detector entrance window as an absorber to avoid the effect of |3- and X-rays, so no correction was made for X-gamma coincidences. This is because, in most cases, the accompanying X-ray was soft enough to be absorbed completely before entering the detector and also the angular correlation effects can be negligible for the low source-to-detector distance. It must be noted that gamma-gamma coincidences were not taken into account, which can induce deviations of the peaks' area. In these measurements of low activity sources, the dead time is always less than 3%. So, the corresponding factor is obtained simply using the ADC live time. The statistical uncertainties of the net peak areas are smaller than 0.5% since the acquisition time is long enough to get a number of counts of at least 20,000. The background subtraction is done. The decay correction Cd for the calibration source from the reference time to the run time is given by

C d = eXAT, (42)

where A is the decay constant, and AT is the time interval over which the source decays corresponding to the run time. The main source of uncertainty in the

efficiency calculations is the uncertainties of the activities of the standard source solutions. Once the efficiencies have been fixed by applying the correction factors, the overall efficiency curve is obtained by fitting the experimental points to a polynomial logarithmic function of the fifth order using a nonlinear least square fit. In this way, the correlation between the data points from the same calibrated source has been included to avoid the overestimation of the uncertainty in the measured efficiency. The uncertainty in the full-energy peak efficiency aE is given by

o, = e

where ffA, op, and on are the uncertainties associated with the quantities AS, P(E), and N(E), respectively.

—1—1—1—«—1-

* - ^Sv

-4 > *

■ 100 mL (Calculated without Sa!||) «

• 100 mL (measured) ---100 mL (Calculated with Swll)

200 mL (Calculated without Ssi||)

< 200 mL (measured) ..... 200 mL (Calculated with SM)

Photon energy (keV)

Figure 6 The full-energy peak efficiencies of Det. 2. Measured, calculated with Sself, and calculated without Sself for different parallelepiped sources placed at the end cap of the detector as functions of the photon energy.

Results and discussion

Figures 5 and 6 show the full-energy peak efficiencies for both NaI (Tl) detectors (Det. 1 and Det. 2) which include measured, calculated with Sself, and calculated without Sself for parallelepiped sources (100 and 200 mL) placed at the end cap of the detector as functions of the photon energy. The percentage deviations between the calculated (with and without Sself) and the measured full-energy peak efficiency values are calculated by

. ny ecal-without Sself "emeas

A2 % = -—- X 100,

ecal-without Sse]f

where fical-with Sself, fical-without Sself, and fimeas are the calculated with/without self-attenuation factor and experimentally measured efficiencies, respectively. Table 8 shows the

A % = ecal-with Sself- emeas x 100 ecal-with Sse]f

Table 8 Comparison between A-,% and A2% for sources placed at the end cap of the detectors

Photon energy (keV)

Figure 5 The full-energy peak efficiencies of Det. 1. Measured, calculated with Sself, and calculated without Sself for different parallelepiped sources placed at the end cap of the detector as functions of the photon energy.

Detector Energy (keV) Source volume (mL)

100 200

û-,% û2% û-,% û2%

Det. 1 121.78 0.04 26.54 -0.01 29.18

244.69 0.59 22.51 -0.25 24.17

344.28 -0.61 19.23 0.72 22.39

443.97 0.46 18.51 -0.79 19.45

778.90 0.72 15.06 0.22 16.22

964.13 1.00 14.02 0.25 14.83

1,112.11 -0.85 11.58 -0.55 13.24

1,408.01 0.85 11.76 0.05 12.28

Det. 2 121.78 -0.10 27.71 0.09 35.87

244.69 -0.41 22.83 0.10 30.21

344.28 0.13 20.82 1.36 28.11

443.97 -0.41 18.73 -0.24 24.85

778.90 -1.00 14.35 0.86 20.80

964.13 0.25 14.06 -0.02 18.36

1,112.11 -1.42 11.75 -1.19 16.28

1,408.01 -0.65 11.01 0.51 15.88

comparison between the percentage deviations Aj% and A2% for the different volumes placed at the end cap of the Nal (Tl) detectors. The discrepancies between the calculated with Sself and measured values were found to be less than (2%), while those between calculated without Sself and measured values were found to be less than (35%). Obviously, the non-inclusion of the self-attenuation factor in the calculations caused an increase in the full-energy peak efficiency values. So, to get correct results, the self-attenuation factor must be taken into consideration. Also, Figures 5 and 6, and Table 8 show that the source of self-attenuation is more effective with large sources, where the photon has traveled a larger distance within a source matrix, so the possibility of getting it absorbed will be more; hence, the attenuation will be more. Its effect starts to decrease when the volume decreases (the distance traveled by the photon within the source matrix decreases).

Conclusions

In the present work, the authors introduced separate calculation of the factors related to photon attenuation in the detector end cap, inactive layer, source container, and the self-attenuation of the source matrix. Also, a direct analytical approach for calculating the full-energy peak efficiency has been derived. The examination of the present results as given in the figures reflects the importance of considering the self-attenuation factor in studying the efficiency of any detector using parallelepiped sources.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

The mathematical theoreticalanalysis was carried out by MMG, AME-K, and NNM. The experimentalworks were carried out by MSB and MMG. All authors analyzed and discussed the results. The manuscript was written by MSB and SIJ, and the figures were prepared by MSB, EAE-M, SSN, and ADD. Allauthors read and approved the finalmanuscript.

Acknowledgements

The authors would like to express their sincere thanks to Prof. Dr. Mahmoud I. Abbas, Faculty of Science, Alexandria University, for the very valuable professionalguidance in the area of radiation physics and for his fruitful scientific collaborations on this topic. Dr. Mohamed S. Badawi would like to introduce a specialthanks to The Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig, Berlin, Germany for fruitful help in preparing the homemade volumetric sources.

Author details

1Physics Department, Faculty of Science, Alexandria University, Alexandria 21511, Egypt. 2Department of Physics, Faculty of Mathematics and Natural Sciences, University of Montenegro, Cetinjski put b.b, Podgorica 81000, Montenegro. 3Centre for Nuclear Competence and Knowledge Management, University of Montenegro, Dz. Vasingtona bb, MNE-2000, Podgorica 81000, Montenegro. 4Department of Mathematics, Maritime Faculty, University of Montenegro, Kotor, Montenegro.

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doi:10.1186/2251 -7235-7-52

Cite this article as: El-Khatib et a/.: Calculation of full-energy peak efficiency of NaI (Tl) detectors by new analytical approach for parallelepiped sources. Journa/ of Theoret/ca/ and App//ed Physics 2013 7:52.

Received: 26 July 2013 Accepted: 23 September 2013 Published: 30 September 2013