Accepted Manuscript

Title: Direct mathematical method for calculating photofraction and intrinsic efficiency of 4^NaI(Tl) Borehole cylindrical detectors

Author: Salam. F. Noureldine Richard R. Nader

PII: DOI:

Reference:

S1658-3655(15)00052-7

http://dx.doi.org/doi:10.1016/j.jtusci.2015.03.001 JTUSCI161

To appear in:

Received date: Accepted date:

8-3-2015 15-3-2015

Please cite this article as: Sm.F. Noureldine, R.R. Nader, Direct mathematical method for calculating photofraction and intrinsic efficiency of 4rmpz NaI(Tl) Borehole cylindrical detectors, Journal of Taibah University for Science (2015), http://dx.doi.org/10.1016/j.jtusci.2015.03.001

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'Manuscript

Direct mathematical method for calculating photofraction and intrinsic efficiency of 4nNaI(Tl) Borehole cylindrical detectors

Salam. F . Noureldinea and Richard R. Naderb

a Physics Department, Faculty of Science, Lebanese University, Beirut, Lebanon b Physics Department, Faculty of Science, Lebanese University, Beirut, Lebanon

Abstract:

A direct mathematical method for calculating the photofraction and intrinsic efficiency of a borehole cylindrical detector is derived by the use of a direct mathematical method. This method depends on the photon path length inside the detector active volume and the geometrical solid angle Q subtended by the source to the detector. The comparisons with the experimental and Monte Carlo method data reported in the literature indicated that the present method is useful in the efficiency calibration of the borehole detector.

Keywords: borehole scintillator detector, geometrical efficiency, photofraction, direct mathematical method.

Introduction:

There are many important and useful applications exploring the borehole Nal (Tl) cylindrical detectors, due to the relative simplicity, high mass number and low cost of crystal preparation. NaI(Tl) scintillators are widely used in different detecting systems (well type, parallelpiped, cylindrical,..) in environmental radioactivity, low level radioactive waste, prompt gamma-ray neutron activation analysis some nuclear physics experiments, geology, etc. So achieving a high efficiency in the gamma-ray detection is an important and crucial issue for the low-level gamma activity measurements. To meet this efficiency requirement a 4nNaI(Tl) Borehole cylindrical detector (30.4x30.4 cm ) has been developed with a central circular hole of radius 1.75cm at Dynamitron Tandem Laboratory (DTL) at the Ruhr-Universitat Bochum

(Fig.1). The detector walls around this bore hole are made of aluminum with a

thickness of only 5x 10- cm to reduce y -ray absorption [1]

The present work is mainly concerned with introducing a straight forward theoretical approach to calibrate the upgraded 4nNaI(Tl) gamma-ray detector for isotropic radiating gamma-ray (point, plane and volumetric) sources. This approach is based on the direct mathematical method reported by Selim and Abbas [2-10] and has been used successfully before to calibrate point, plane and volumetric sources with cylindrical, well-type, and parallelepiped detectors. The fact that efficiency can be precisely determined by calculation makes the method absolute (or direct). The work described below involves the use of straightforward formulae for the computation of the photofractoin and intrinsic efficiency of a disc source which is extended to cylindrical source of radius 1.5cm and height 3cm. In this study, we introduce a new method for determination of the path length d(0,9) covered by a photon inside the detector active volume and the geometrical solid angle Q (the angle subtended by the detector at the source point). The path length d(0,9) is derived not only as a function in the polar angle 0, but also as a function in the azimuthal angle 9. This will reduce the mathematical formulae to an easiest and compact shape. The validity of the present work is presented by comparing our results with the published experimental and Monte Carlo simulations ones.

Mathematical view point

In the following, direct analytical expression for the photofraction and intrinsic efficiency of a cylindrical borehole detector is derived using a disc and cylindrical radiating sources. The location of the isotropic disc source is defined by the quantities (p,h) and the direction of the photon incidence by the polar (0) and the azimuthal 9 angle [12]. There are eight cases to be considered to find the photon path length d through the cylindrical detector medium, as shown in Fig.1. The incident photon may enter from the inner side of the cylindrical detector and emerge from:

(a) Lower base one (LB 1) and Lower base three(LB3) L R1 + 2pcos(^)

co s(0) 2 s in(0) co s(0) (b) Side one and side three

d = ( R -R)

2sin( 0)cos(^)

(c) Lower base two (LB2) and lower base four (LB4)

cos(0) 2 sin(0) sin(^) sin(0)

(d) Side two and side four

( Ro - R )

2 sin(0) sin(^)

The geometrical notations L, R0, R and p are shown in Fig.1. The photofraction is the ratio between the number of photons that are recorded under certain peak and the number of photons that are recorded in the spectrum at the same energy. It is given by

p ==£-p-

Where Dp is the full energy peak efficiency and DT is the total efficiency [10].

_ _2_ _L

S n/20-

S n 0-

J J Jf pddd^dp+J J Jfpddd^dp + J J Jfpddd^dp + J J Jf pd6d$dp +

o o o o 0- o n/ 2 0- o n/2 0-

S 3n/202+ S 3n/2n/2 S 2n 0 S 2n n/2

J J Jfi Pd0d$dp+J J Jf pddd^dp + J J Jf pddd^dp+J J Jf pdOd^dp

o n 0+

o 3n/20+

o 3n / 2 0+

The direction of the incidence photon is defined by the polar (0) and azimuthal (□) angles, where the radioactive source is an axial point source; the azimuthal (□) angle takes the value from o to 2n, while the polar angle (0) takes the following steps:

Sn/2n/2

S n n/2

o n 0+

g> = tan-(R * 2^cos(0)) (7)

2L cos(0)

0 = tan -. (Ro * 2P co s(0))

2L cos(0) (8)

= tan -(R * 2pS'n(V;)) 2L cos(0)

^ „-VRo + 2Psin(0)>

64+ = tan-i( 0 ' ^-

2L cos(0) (jo)

f =(i-e"M )sin( 9). e ~sui(9) i=i,2,..4 (ii)

where, ^ is the attenuation coefficient of the detector crystal without the coherent part [ii].

The intrinsic efficiency of 4nNaI(Tl) borehole cylindrical detector can be calculated by indirect method, i.e., by calculating the total efficiency (□t) and geometrical efficiency □ It is the ratio between the number of photons that are recorded in the detector with any energy and the number of photons that actually enter the detector. In other words, it is the fraction of photons being recorded in the detector with any possible energy per photon of the considered energy entering the detector's material. This efficiency is given by equation i2:

£' = 7 (i2) bg

The geometrical efficiency □ is equal to Q/4n [ii]; where Q is the solid angle subtended by a 4nNaI(Tl) borehole cylindrical detector from an isotropic radiating axial disc source placed inside the detector hole. It is given by:

Q = JJ sin 6d0d6 (i3)

The intrinsic efficiency for 4nNaI(Tl) Borehole cylindrical detector is also derived for cylindrical source placed at different heights and is given by

„ cylindrical _

hn -h, , 2 1 h 1

jsT dh

Results

The photofraction and intinisic efficiency are calculated using the present expressions, and compared with those obtained by Monte Carlo simulation for borehole scintillation detectors using a circular disk source of radius 1.5 cm [1]. In the present work the height is taken from zero which is considered as the center of borehole detector. The simulated photofraction of a disk source placed at the center of the bore hole at different energies is summarized in Table 1 together with the direct theoretical ones (present work) while The simulated and theoretical (present work) intrinsic efficiency for a disc source is summarized in Table 2. The geometrical efficiency of the detector determined by Eq. (13) at different heights starting from the center to the surface, is shown in Fig. 2. The percentage differences between the calculated values and the measured ones, which are given by equation (15), are shown also in tables 1 and 2.

the discrepancies (in case of photofraction and intrinsic efficiency) between the simulated and the calculated (present theoretical method) photofraction values for the investigated geometry ranged between 0.2% and 2.1%. Also the geometric efficiency shown in table 2 shows a constant value for different energies due to its independency on the energy of the photon. Since the discrepancies between the simulated and the calculated values is very small the calculations for the disc source is extended to cylindrical source at different heights and at energies 0.2, 0.5 and 1 MeV as shown in Fig. 3.

Conclusions

Direct mathematical expressions to calculate the photofraction and intrinsic efficiency of 4nNaI(Tl) bore hole detector have been derived in the case of

disk and cylindrical sources. The photofraction and intrinsic efficiency of the detector is studied as a function of the energy of the incident photon when the disk source is placed at the center of detector. In addition, the geometric efficiency and intrinsic efficiency are studied at different heights and different energies. The results show a good agreement between the present and the published values, the high discrepancies being less than 2. i %.

References

[1] M. Mehrhoff, M. Aliotta, I.J.R. Baumvol, H. W. Becker, M. Berheide, L. Borucki, J. Domke, F. Gorris, S. Kubsky, N. Piei, G. Roters, C. Rolfs, W.H. Schulte, Gamma-ray detection with a 4n Nal spectrometer for material analysis, Nuclear Instruments and Methods in Physics Research B i32 (i997) 67i.

[2] M.I. Abbas, Analytical formulae for well-type NaI(Tl) and HPGe detectors efficiency computation, Applied Radiation and Isotopes, 55 (200i) 245.

[3] M.I. Abbas, Analytical calculations of the solid angles subtended by a well-type detector at point and extended circular sources. Applied Radiation and Isotopes 64 (9) (2006) i048.

[4] M.S. Badawi, I. Ruskov, M.M. Gouda, A.M. El-Khatib, M.F. Alotiby, M.M. Mohamed, A.A. Thabet,, M.I. Abbas, A numerical approach to calculate the full-energy peak efficiency of HPGe well-type detectors using the effective solid angle ratio, Journal of Instrumenation 9 (7) (20i4) P07030.

[5] M.S. Badawi, A.M. El-Khatib, S.M. Diab, S.S. Nafee, E.A. El-Mallah, An approach to evaluate the efficiency of y-ray detectors to use it for

determining radioactivity in environmental samples, Chinese Physics C 38 (6) (2014) 066203.

[6] M.S. Badawi, M.E. Krar, A.M. El-Khatib, S.I. Jovanovic, A.D. Dlabac, N.N. Mihaljevic, A new mathematical model for determining the full energy peak efficiency (FEPE) for an array of two y-detectors counting rectangular parallelepiped source, Nuclear Technology and Radiation Protection 28 (4) (2013) 370.

[7] M.I. Abbas, A new analytical method to calibrate cylindrical phoswich and LaBr3(Ce) scintillation detectors, Nuclear Instruments and Methods in Physics Research Section A 621 (2010) 413.

[8] M.I. Abbas, Analytical formulae for borehole scintillation detectors efficiency calibration, Nuclear Instruments and Methods in Physics Research Section A 622 (2010) 171.

[9] S. Noureldine, Y. Ajeeb, A direct mathematical method to calculate the efficiencies of 4nNaI (Tl) scintillation detector, American International Journal of Research in Science, Technology, Engineering & Mathematics, 7 (2) (2014) 132.

[10] A.M. El-Khatib, M.E. Krar, M.S. Badawi, Studying the full energy peak efficiency for a two y-detectors combination of a different dimensions, Nuclear Instruments and Methods A.719 (2013) 50.

[11] M.I. Abbas, M.S. Badawi, I.N. Ruskov, A.M. El- Khatib, D.N. Grozdanov, A.A. Thabet, Yu.N. Kopatch, M.M. Gouda, V.R. Skoy, Calibration of a single hexagonal NaI(Tl) detector using a new

numerical method based on the efficiency transfer method, Nuclear Instruments and Methods in Physics Research A 771 (2015) 110.

[12] M.I. Abbas, Analytical approach to calculate the efficiency of 4nNaI(Tl) gamma-ray detectors for extended sources. Nuclear Instruments & Methods in Physics Research A, Accelerators, Spectrometers, Detectors and Associated Equipment, 615(1) (2010) 48.

Figure

Fig(1): A schematic drawing of a Borehole Cylindrical detector with a coaxial radioactive

disc source.

o c a)

E o a) O

0.950.900.850.800.750.700.650.600.55-

0 2 4 6 8 10

Height in cm

Figure (2): plot of the geometrical efficiency versus the height of disc source

h in cm

Figure (3): plot of the intrinsic efficiency versus the height of cylindrical source at different

energies.

Table1 : Comparison between calculated (present work), and simulated (GEANT computer code) photofraction of bore hole cylindrical detector for a disk source gamma radiation source. The source is placed at the center of the detector (h=0).

The disk source is at the center of the bore hole scintillator detector h=0

E (MeV) □p [M.C] □ t [M.C] [1] P = □p/^t [M.C] [1] P = Dp/Dt (present work) A1

0.2 0.659 0.889 0.741282 0.728982 1.687289089

0.3 0.712 0.949 0.750263 0.754237 0.52687039

0.5 0.729 0.968 0.753099 0.767368 1.859504132

0.7 0.715 0.961 0.744017 0.745568 0.208116545

1 0.676 0.941 0.718385 0.720682 0.318809777

1.2 0.665 0.932 0.713519 0.718919 0.751072961

1.3 0.668 0.925 0.722162 0.729258 0.972972973

1.4 0.623 0.914 0.681619 0.691454 1.422319475

1.5 0.609 0.913 0.667032 0.678174 1.642935378

1.7 0.59 0.893 0.660694 0.674286 2.015677492

2 0.571 0.879 0.649602 0.659353 1.478953356

2.5 0.545 0.871 0.625718 0.635939 1.607347876

3 0.533 0.861 0.619048 0.628538 1.509872242

3.5 0.514 0.848 0.606132 0.612634 1.061320755

4 0.501 0.847 0.591499 0.602163 1.770956316

5 0.499 0.845 0.590533 0.602657 2.01183432

7 0.481 0.85 0.565882 0.577431 2

8 0.468 0.855 0.547368 0.557143 1.754385965

10 0.466 0.864 0.539352 0.546948 1.388888889

15 0.359 0.882 0.407029 0.409817 0.680272109

Table2: Comparison between calculated (present work), and simulated (GEANT computer code) intrinsic efficiency of bore hole cylindrical detector for a disk source gamma radiation source. The source is placed at the center of the detector (h=0).

The disk source is at the center of the bore hole scintillator detector h=0

E (MeV) sg [M.C] [1] 8i, present work a2

0.2 0.943737 0.95966 1.659292

0.3 1.007431 1.002123 -0.52966

0.5 1.027601 1.008493 -1.89474

0.7 1.02017 1.018047 -0.20855

1 0.998938 0.995754 -0.31983

1.2 0.989384 0.981953 -0.75676

1.3 0.942 0.981953 0.972399 -0.98253

1.4 0.970276 0.956476 -1.44284

1.5 0.969214 0.953291 -1.67038

1.7 0.947983 0.928875 -2.05714

2 0.933121 0.919321 -1.50115

2.5 0.924628 0.909766 -1.63361

3 0.914013 0.900212 -1.53302

3.5 0.900212 0.890658 -1.07271

4 0.899151 0.883227 -1.80288

5 0.897028 0.878981 -2.05314

7 0.902335 0.884289 -2.04082

8 0.907643 0.89172 -1.78571

10 0.917197 0.904459 -1.40845

15 0.936306 0.929936 -0.68493