Dependence of Zeff for the exposure buildup factors of 1 MeV gamma rays

DEPENDENCE OF Ze//FOR THE EXPOSURE BUILDUP FACTORS OF 1 MeV GAMMA RAYS

Berkay CAMGÖZ*, Gültekin YEĞİN*, Mehmet N. KUMRU*

(*) Institute o f Nuclear Sciences, Ege University 35100 Bornova-Izmir/Turkey

ABSTRACT

The Zeff dependence of exposure buildup factors for plane incident source gamma rays of IMeV have been investigated by using electron gamma shower Monte Carlo code, EGS4. To prepare materials with different Ze//. homogen mixtures composed of Li, Cu, Fe and Pb were used in different ratios. All the EGS4 calculations were made in one dimensional plane geometry for each mixture. To check the accuracy of EGS4 calculations, an extra work has been performed for the exposure buildup factors of Pb, Fe and H20. Results of these calculations are agreed well with the published data.

In the EGS4 calculations, the exposure was obtained from the photon flux and the energy absorption coefficient of air. The effect of Rayleigh scattering was included but effects of bremstrahlung and fluorescent radiation were not taken into account. All calculations were performed with the exponential transform. The calculated data showed that buildup factors vary smoothly with increasing Z. At deep penetration depths, variation of buildup factors is more rapidly. The value of the buildup factor for an arbitrary atomic number can be obtained by using data resulted from this calculations.

INTRODUCTION

Buildup factors are always useful for practical calculations in gamma-ray shield design. Shielding materials frequently consist of elements not included in standard data or compounds and homogeneous mixtures of a number of elements. Furthermore, there is significant disagreement in the buildup data using the different approximating techniques in the present computational art for analysis of photon transport. Therefore, it is important to predict consistent buildup factors using available data.

The literature contains many empirical formulas for the interpolating of buildup data to elements not calculated. Such as Taylor’s formula [1] contains an exponential attenuation function with three parameter and provides a better fit to data for high Z materials. Berger’s formula [2] consist of two terms. The contribution from the uncollided radiation is provided by the first term and the second term determines the inscattered contribution. Polynomial formula [3] provides reasonable accuracy up to 20 mfp lengths. In addition to the above three forms, many other fitting formulas have been introduced and applied. The values of the parameters for these formulas, experimentally determined only a few materials useful for the shielding design.

So the production of the database of gamma-ray buildup factors from low-Z material to high-Z material is needed.

The objective of the present work is to investigate and provide exposure buildup factors with respect to the atomic number. In the calculations, the gamma-ray sources considered were plane normal to the material. The exposure buildup factor is based on the energy absorption response of air; that is, exposure is assumed to be equivalent to absorbed dose in air as measured by a nonperturbing detector.

METHOD OF CALCULATION

For the simulation of photon transport in the medium a general electromagnetic radiation transport code Electron Gamma Shower version (EGS4), was used [4]. The data up to 10-mfp in a semi infinite medium were estimated from the calculation for a 12-mfp-thick plane medium. The exposure buildup factor B(E0,x) is defined to be

B ( Eq, x) Eq

f

j

V

\

\

(E ) Energy transfer coefficient for air,

(1)

I(x, R) = Energy flux at the position x.

In the EGS4 calculation, the exposure was obtained from the photon flux and the energy transfer coefficient of air. To get good statistics at deep penetration, all calculations were performed with the exponential transform. In the calculations, secondary electrons produced in each reaction were discarded immediately to save computer time.

To cheek the accuracy, the exposure buildup factors calculated by EGS4 for Lead, Iron, Water from 1 MeV plane normally incident gamma-ray are compared with the Angular Eigenvalue Method (AEM) calculations [5] shown in Fig. 1-3. Our calculated data for these materials, are given in Tables 1-3. Slight differences between our and AEM results are resulted from different geometry used for the calculations. The statistical errors of EGS4 results were obtained by dividing the EGS4 runs into ten runs of 106 incident gamma rays each.

MATERIAL DATA AND CROSS SECTIONS

PEGS4, a stand-alone program, used to produce material data for EGS4. In PEGS4, Compton scattering cross sections were taken from Klein-Nishina. Photoelectric reaction and coherent scattering cross sections were taken from the compilation by Storm and Israel [6]. The mass energy transfer coefficients of air were taken from Hubbel’s compilation [7]. Using PEGS4,

constitute the mixtures due to the their atomic numbers. The composition and effective atomic number of each mixture are given in the Table 1.

CONCLUSION

In this paper, the dependence of effective atomic number Zef for the exposure buildup factors was investigated at 1MeV. The dependence of the exposure buildup factors as a function of Zef is plotted in Fig. 4 for various penetration depths in mean free path lengths. Calculated exposure buildup factors for each mixture are given in Table 5. This results are in agreement with an experimental work at Ref. 8. At shallower penetration depths, the curves of buildup factors vary smoothly with increasing Z. Nevertheless, at deep penetration depths, variation is more rapidly. In addition, for low-Z values of materials (Zef f < 35) the value of exposure buildup factor fluctuate rapidly according to the atomic number. This is for the reason that photon scattered to low energies in a Compton scattered tend to accumulate more and more as the energy decreases. Photoelectric absorption competes with the accumulation of scattered photons, however, so that in light elements, the energy flux exhibits a very high peak arising from scattered low energy photons. The energy corresponding to a peak by the multiple scattering of photons in an energy flux at deep penetration decreases with decreasing atomic number.

Table 1 : Mixing Ratios of Elements Constitute Materials Used for Compilation. Fraction of The Elements(%)

The Number of Mixture Z e ff Li Al Cu Pb 1 10 46 48 4 2 2 15 38 34 24 4 3 20 18 52 20 10 4 25 14 6 78 2 5 30 28 34 6 32 6 35 4 18 56 22 7 40 8 32 20 40 8 45 12 14 28 46 9 50 22 14 2 62 10 55 6 26 2 66 11 60 16 0 12 72 12 65 12 4 4 80 13 70 4 10 0 86

Table 2 : Gamma-Ray Exposure-Buildup Factors on Lead at 1 MeV.

Penetration EGS4a AEMb

(mfp) 0,5 1,2137 ± 0,0004 -1 1,3856 ± 0,0004 1,37 2 1,6831 ± 0,0008 1,67 3 1,9442 ± 0,0012 1,95 4 2,1828 ± 0,0023 2,20 5 2,4052 ± 0,0030 2,44 6 2,6173 ± 0,0037 2,68 7 2,8250 ± 0,0054 -8 3,0247 ± 0,0062 3,11 10 3,3984 ± 0,0177 3,52

a : This study (For Plane Normally Incident Source). b : AEM Data (For Point Isotropic Source ) [5].

Table 3 : Gamma-Ray Exposure-Buildup Factors on Water at 1 MeV.

Penetration EGS4 AEM

(mfp) 0,5 1,5121 ± 0,0003 -1 2,1478 ± 0,0009 2,07 2 3,7692 ± 0,0014 3,61 3 5,8318 ± 0,0052 5,51 4 8,3087 ± 0,0113 7,71 5 11,155 ± 0,0255 10,2 6 14,319 ± 0,0241 12,9 7 17,794 ± 0,0411 -8 21,534 ± 0,0923 19,1 10 30,068 ± 0,1382 26,2

Table 4 : Gamma-Ray Exposure-Buildup Factors on Iron at 1 MeV. Penetration (mfp) EGS4 AEM 0,5 1,4070 ± 0,0006 1 1,8324 ± 0,0007 1,85 2 2,7634 ± 0,0020 2,86 3 3,8036 ± 0,0026 4,01 4 4,9520 ± 0,0045 5,32 5 6,1906 ± 0,0040 6,75 6 7,5343 ± 0,0176 8,32 7 8,9691 ± 0,0164 -8 10,517 ± 0,024 11,8 10 13,749 ± 0,082 15,7

Table 5 : Buildup Factors for Mixtures Used in The Calculations Versus Penetration Depths in mfp.

Penetration The Number of Mixture

(mfp) 1 2 3 4 0,5 1,4143 ± 0,0007 1,3888 ± 0,0003 1,3610 ± 0,0003 1,3888 ± 0,0005 1,0 1,8518 ± 0,0005 1,7857 ± 0,0006 1,7154 ± 0,0007 1,7848 ± 0,0007 2,0 2,8110 ± 0,0012 2,6368 ± 0,0015 2,4522 ± 0,0012 2,6358 ± 0,0012 3,0 3,8828 ± 0,0022 3,5643 ± 0,0035 3,2339 ± 0,0039 3,5755 ± 0,0031 4,0 5,0774 ± 0,0040 4,5632 ± 0,0083 4,0541 ± 0,0054 4,5903 ± 0,0054 5,0 6,3490 ± 0,0054 5,6413 ± 0,0143 4,9062 ± 0,0060 5,6536 ± 0,0048 6,0 7,7462 ± 0,0110 6,7659 ± 0,0352 5,7987 ± 0,0078 6,8061 ± 0,0112 7,0 9,2462 ± 0,0220 8,0154 ± 0,0765 6,7424 ± 0,0185 8,0548 ± 0,0237 8,0 10,824 ± 0,041 9,2251 ± 0,1540 7,7094 ± 0,0321 9,3143 ± 0,0306 9,0 12,531 ± 0,034 10,451 ± 0,268 8,6964 ± 0,0279 10,666 ± 0,041 10,0 14,197 ± 0,089 11,592 ± 0,434 9,7298 ± 0,0458 11,452 ± 0,322

Penetration The Number of Mixture (mfp) 5 6 7 8 0,5 1,3045 ± 0,0004 1,3228 ± 0,0005 1,2897 ± 0,0004 1,2841 ± 0,0003 1,0 1,5825 ± 0,0007 1,6247 ± 0,0005 1,5492 ± 0,0006 1,5367 ± 0,0006 2,0 2,1167 ± 0,0015 2,2213 ± 0,0021 2,0368 ± 0,0014 2,0090 ± 0,0013 3,0 2,6414 ± 0,0018 2,8180 ± 0,0016 2,5141 ± 0,0020 2,4666 ± 0,0017 4,0 3,1680 ± 0,0035 3,4250 ± 0,0050 2,9729 ± 0,0036 2,9050 ± 0,0041 5,0 3,6879 ± 0,0055 4,0479 ± 0,0082 3,4270 ± 0,0061 3,3415 ± 0,0060 6,0 4,2153 ± 0,0093 4,6872 ± 0,0062 3,8837 ± 0,0061 3,7689 ± 0,0063 7,0 4,7666 ± 0,0106 5,3305 ± 0,0113 4,3552 ± 0,0101 4,2204 ± 0,0092 8,0 5,2949 ± 0,0210 5,9610 ± 0,0191 4,8057 ± 0,0236 4,6357 ± 0,0186 9,0 5,8016 ± 0,0240 6,6129 ± 0,0221 5,2397 ± 0,0199 5,0497 ± 0,0153 10,0 6,3107 ± 0,0304 7,2717 ± 0,0179 5,6860 ± 0,0233 5,4988 ± 0,0282

Penetration The Number of Mixture

(mfp) 9 10 11 12 13 0,5 1,2561 ± 0,0004 1,2520 ± 0,0003 1,2431 ± 0,0003 1,2339 ± ,0003 1,2282 ± ,0003 1,0 1,4754 ± 0,0005 1,4666 ± 0,0006 1,4460 ± 0,0003 1,4276 ± ,0004 1,4157 ± ,0005 2,0 1,8714 ± 0,0007 1,8535 ± 0,0007 1,8084 ± 0,0010 1,7707 ± ,0007 1,7459 ± ,0007 3,0 2,2392 ± 0,0016 2,2082 ± 0,0014 2,1408 ± 0,0021 2,0794 ± ,0014 2,0420 ± ,0010 4,0 2,5876 ± 0,0022 2,5459 ± 0,0017 2,4466 ± 0,0032 2,3656 ± ,0026 2,3116 ± ,0027 5,0 2,9234 ± 0,0049 2,8669 ± 0,0033 2,7398 ± 0,0073 2,6355 ± ,0020 2,5691 ± ,0031 6,0 3,2495 ± 0,0051 3,1863 ± 0,0060 3,0162 ± 0,0143 2,9026 ± ,0054 2,8188 ± ,0057 7,0 3,5838 ± 0,0109 3,5064 ± 0,0101 3,3137 ± 0,0312 3,1642 ± ,0077 3,0632 ± ,0079 8,0 3,8948 ± 0,0139 3,7971 ± 0,0110 3,6045 ± 0,0233 3,4249 ± ,0106 3,3086 ± ,0093 9,0 4,2221 ± 0,0146 4,0936 ± 0,0165 3,7895 ± 0,0594 3,6512 ± ,0137 3,5071 ±0,0156 10,0 4,5186 ± 0,0168 4,3925 ± 0,0152 3,8036 ± 0,1043 3,8821 ± ,0180 3,7340 ± ,0159

-this study —* — AEM

Figure 1 : Gamma-ray Exposure-Buildup Factors on Lead at 1 MeV.

-this study -AEM

Figure 2 : Gamma-ray Exposure-buildup Factors on Water at 1 MeV.

-this study —A— AEM

Z eff

^X = 5 - A — ^X = 1 0

Figure 4: Buildup Factors Versus Effective Atomic Number Zef

REFERENCES:

1. Taylor J.J. “Application of gamma ray buidup datato shield design, WAPD-RM- 217,(1954).

2. M. J. Berger, “Effect of Boundaries and Inhomgeneities on the Penetration of Gamma Radiation,” Report NBS-4942, U.S. National Bureau of Standards Washington D. C., (1956)

3. M. Metghalchi, “On the Polynomial Form of Gamma-Ray Buildup Factor Functions,” Nucl. Sci. Eng. 70, 207 (1979).

4. W. R. Nelson, H. Hirayama, and D. W. O. Rogers, “The EGS4 Code System,” SLAC- 265, Stanford Linear Accelator Center. (Dec. 1985)

5. S. Akinao, “Development of Angular Eigenvalue Method for Deep Penetration problems of Gamma Rays,” The Wakasawan Energy Research Center, Journal of Nuclear Science and Technology, Supplement 1, p. 454-458 (March 2000).

6. E. Storm and H. I. Israel, “Photon Cross Section from1 keV to 100 MeV for Elements Z=1 to Z= 100,” At. Data Nucl. Dta Tables, 7, 565 (1970)

7. J. Hubbel, “Photon Cross Section , Attenuation Coefficients and Energy Absrbtion Coefficients from 10 keV to 100 GeV,” NSRDS-NBS 29, National Bureau of Standards (1969)

8. Ö. Mersinlioglu, “Gamma Isinlarina Ait Sayisal Yigilma Faktörlerinin Sogurucunun Atom Numarasina Göre Degisiminin Incelenmesi” Ms.C. Thesis, Ege University,