photon Monte Carlo calculations of X-ray scattering with application to industrial radiography

Coupled electron/photon Monte Carlo calculations of X-ray scattering with application to industrial radiography G. Barnea and C.E. Dick Monte Carlo transport methods are used to simulate the scattering of X-rays in polystyrene and iron slabs. The calculations are made with monoenergetic X-ray sources in the energy region from 30 keV (100 keV for iron) to 20 MeV. This energy range includes the energy regions for diagnostic radiology (0.03-0.15 MeV), nuclear medicine (0.1-2.0 MeV) and industrial radiography (0.2-20 MeV). The slab thicknesses for polystyrene were 53, 100, and 210 mm and for iron 7 and 14 mm. The present calculations include the effects of secondary electron/ positron radiation which become quite important at high energies. As a function of the incident photon energy, the ratio of the scattered to the total radiation (scatter fraction) was found to have a characteristic 'N' shape. Increasing the atomic number of the scattering media has the effect of 'squeezing the N'.

Keywords: Monte

Carlo radiation, polystyrene, iron

calculations,

In ordinary transmission radiography, information about an object structure is obtained by irradiating the object and by recording the transmitted radiation. The transmitted fluence consists of an unscattered and a scattered component. In the conventional technique, only the unscattered radiation carries information about the object structure. The scattered component provides background noise which can be comparable to or even greater in magnitude than the unscattered component and therefore can reduce the detail and contrast in the radiograph considerably. The effect of the scattered radiation on the quality of the radiographic image is conventionally expressed in terms of the scatter fraction F, defined as F = Ns/(Nu +

Ns)

(1)

where No and Ns are the unscattered and the scattered photon fluences (number per unit area) at a specified point in the image plane. The scatter fraction depends on: the incident photon energy, E0; the diameter, W, of the incident photon beam ('the beam width' or 'the field size'); the thickness, L, of the object under investigation; the distance, H, between the image plane and the body; and the density and the atomic number of the object's material. The dependence o f f on these variables for low atomic number materials (polystyrene and water) and in the energy region relevant for diagnostic radiology and nuclear medicine (30-660 keV) has been the subject of many experimental and theoretical studiesl~-sl. To the best of our knowledge there are no accurate published data in 0308-9126/87/020111-05

X-ray

scattering,

electron/positron

the energy region most often used for industrial radiography (0.2-20 MeV). At the higher energies, the secondary electrons and positrons resulting from Compton scattering or pair production can produce additional photons through bremsstrahlung or annihilation radiation. Furthermore, the more energetic electrons and positrons can also exit the object and reach the detector, thus contributing to the scattered component. These effects are included in the present calculations and the scatter fraction is calculated for incident photon energies of up to 20 MeV. Two materials are considered for scattering media: polystyrene and iron. Above the threshold energy (18.7 MeV for carbon and 11.2 MeV for iron) photonuclear absorption may occur; however, even at the resonance peak the magnitude of the corresponding cross-section is less than 3% of the competing 'electronic' cross-sectionlg,161. Photonuclear absorption is not included in the present work. Different antiscatter techniques may be used for photons and for charged particles. It is, therefore, important to differentiate between the photon fluence and the electron/ positron fluence. We define two scatter fractions: one for scattered photons and one for scattered photons and charged particles (electrons and positrons): F(~) = N (~) s / ( N u + N (st))

(2)

F (7+e)

(3)

-~-

(#St) + N (Se))/(Nu + N (S~') + N (e))

where Nu is the unscattered photon fluence, and N (sv) and

$3.00 © 1987 Butterworth f~ Co (Publishers) Ltd

NDT International Volume 20 Number 2 April 1987

111

N(se) are the scattered photon and charged particle fluences respectively. Obviously, Fir+e) > F(~').

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Method of calculation In our calculations we use a combinatorial coupled electron/photon transport code (ACCEPT)[gl, which utilizes Monte Carlo techniques to solve the transport problem. The generation and transport of the electron/ photon cascade are calculated according to the ETRAN modelh0,111. The main modification of the A C C E P T code is the use of combinatorial geometry which makes it possible to deal with complicated multi-material, threedimensional geometries. The versatility and accuracy of the code was demonstrated in previous work on X-ray scattering in transmission diagnostic radiologylsl. The geometry for all the following calculations is given in Figure I. The scattering medium is a slab (250 mm × 250 mm and thickness L) of polystyrene or iron. The source is assumed to be point-like and monoenergetic. The source energy was varied from 30 keV for polystyrene (from 100 keV for iron) to 20 MeV. These lower limits were chosen to keep computing times at a reasonable level and since radiography at energies lower than these is seldom done. The distance from the source to the slab was kept constant at 2 m. The diameter (W) of the incident photon beam at the image plane is 100 mm. A disc of diameter 20 mm located at a distance h from the exit side of the slab serves as a surface detector of efficiency one. As in our previous worklSl, three thicknesses were considered for the polystyrene slab (L = 53, 100 and 210 mm). In the case of the iron slab, L = 7 mm and 14 mm. Between 0.4 MeV and l0 MeV the 7 and 14 mm iron slabs have roughly the same effective absorption thickness as the 53 and 100 mm polystyrene slabs, respectively.

Results Dependence of the scatter fraction on the incident X-ray energy The dependence of FW) and F (~'+e) on the incident X-ray energy is shown in Figures 2 and 3 for polystyrene and iron respectively. The behaviour is very much the same in all cases considered. As the energy is increased from the lowest value considered (30 keV for polystyrene, 100 keV for iron) the scatter fraction increases at first until a maximum is reached at about 60 keV for polystyrene (300 keV for iron), followed by a sharp decrease. At about

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Fig. 1 Geometry for the calculations. The values of the slab thickness, L, the beam width, W, and the object-to-detector distance, H, are given in the text

10 MeV for polystyrene (2-3 MeV for iron) the scatter fraction reaches a minimum followed by a very sharp increase. This behaviour is explained as follows. In the energy region considered, the mass attenuation coefficient, ~(E), for polystyrene decreases with increasing energy and therefore the unscattered fluence, Nu, increases. In iron,/-KE) has a minimum (and therefore Nu has a maximum) at about 9 MeV. The variation of the scattered fluences N (sv) and N (sO)are more complex and may be visualized only for singly scattered photons. We first consider the energy region below 1 MeV, where we may neglect N tse). Three factors govern the energy dependence of N(sY) in this region.

0.3

(1) For the energy range considered here, the scattering coefficient,/xs, decreases with increasing energy and, therefore, the intensity of the scattered photons originating from any volume element in the scattering object decreases. (2) The scattered photons originating from any volume element (excluding volume elements which lie on the line between the source and the detector) must undergo an angular deflection in order to reach the detector. The differential angular scattering crosssection is peaked in the forward direction. The higher the energy the sharper is the peak in the angular distribution and, therefore, the smaller the portion of photons that are deflected into the direction pointing towards the detector. For the geometry of Figure 1, only about 4% of the source photons are emitted in the direction of the detector.

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(3) From its point of origin to the detector, the scattered radiation is attenuated. The higher the energy, the lower the attenuation and the higher the probability for scattered photons to reach the detector. The first two factors causeN ~s ~) to decrease with increasing incident photon energy. The last factor has the opposite effect. At intermediate energies, the scattering medium behaves like a pure Compton scatterer and the first two factors dominate:N ~s r) decreases with increasing energy. Towards low energies, photoabsorption becomes important and the third factor dominates. N ~s ~') decreases with decreasing energy. At first, the decrease i n n ~s v) is compensated by the decrease in N U and Ft~) becomes independent of the energy but, as the energy is lowered further, N ~s v) decreases much faster than Nu and the scatter fraction falls off. At higher incident photon energies (E0 > 1.02 MeV), a fourth factor, namely the production of electron-positron pairs, becomes effective. These secondary particles can produce additional photons via bremsstrahlung or annihilation (the code treats positrons as if they were electrons but includes annihilation when the positron comes to rest). The more energetic charged particles can exit the object and reach the detector (positrons are recorded as electrons). As the incident photon energy increases more and more pairs are produced with higher and higher energies and b o t h F tr) a n d F ~r+e) increase with increasing energy. In iron, Nu decreases above about 9 MeV and the increase in the scatter fraction is accelerated considerably. To demonstrate the importance of the secondary radiation, in Figures 2 and 3 we also give the results of calculations in which the secondary particles (electrons and positrons) are not followed (or detected). The scatter fraction, in this case, continues to decrease as a function of the energy.

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Fig. 4 Dependence of the scatter fractions on the object-to-detector distance for 2 MeV photons incident on iron (E0 = 2 MeV, W = 100 ram)

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Dependence of the scatter fractions on the atomic number

The higher the atomic number, Z, of the scattering media, the shorter is the energy region in which the Compton effect is dominant: the photoelectric effect dominates at higher energies and pair-production dominates at lower energies as compared with low-Z materials. This explains the 'squeezing' effect on the characteristic 'N' shape of the energy dependence of the scatter fraction when Z is increased.

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H (rnm) Fig. 5 Dependence of the scatter fractions on the object-to-detector distance for 8 MeV photons incident on iron (Eo= 8 MeV, W = 100 mm)

Dependence of the scatter fraction on the objectto-image-plane separation

B(E) is the build-up factor:

where

B(E) = (1 - F(E)) -1

An increase in the distance, H, between the object and the detector causes a decrease inNu as well as i n N tsr)a n d N ~se). However, since the scattered particles (photons, electrons and positrons) originate closer to the detector, N ~v) and N ~s e) decrease faster than Nu and both Fry) and F (yq-e) fall off. This effect is basically geometrical in nature and reflects the difference in the effective solid angle for the scattered and unscattered components.

and

(6)

Nu(E) is proportional to the unscattered fluence:

Nu(E ) = exp(-/x(E) pL)

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The dependence of F~r) and F ty+e) on H for iron is shown in Figures 4 and 5 for incident photon energies of 2 MeV and 8 MeV respectively. The rate of decrease of FtV) and Fry+e) is approximately the same.

Error analysis Random errors are inherent in all Monte Carlo calculations. The result of the calculation is a sample mean of the total number of photons or electrons (scattered and unscattered) that reach the detector. To obtain an estimate of the statistical fluctuations, the relative standard deviation (RSD) of ten independent runs was calculated. In some cases (low energies and thick test objects) a total of 106 to 2 × l& incident photon histories was required to obtain adequate statistics, that is, RSDs of less than 2%.

I C 0

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Systematic errors originate from uncertainties in the photon cross-sections and, in the high-energy region, also from uncertainties in the bremsstrahlung cross-section. In the relevant energy regions, the relative errors in the crosssections are estimated as followsll2-~51: photoelectric cross-sections incoherent (Compton) scattering crosssections pair-production cross-sections bremsstrahlung cross-sections.

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Equation (4) applies to both FW) and Ft,+e). Using (4) we estimate AF/F to be about 3% for largeF and about 10% for small F.

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Polyenergetic incident photon beams The scatter fraction in this study has been calculated for a monoenergetic incident photon beam. However, the scatter fraction for polyenergetic beams, Fp, can be calculated if the scatter fraction as a function of the energy, F(E), and if the number spectrum of the incident beam, ~b(E), are known. The following formula is derived from Equation (1):

y x Nu(E) dE

[

10-4

where X is the mean free path number (/AoL), A#//z is the relative (systematic) uncertainty in the mass attenuation coefficient and ANs/Ns is the relative (systematic and statistical) uncertainty of the total number of scattered photons (or electrons) that reach the detector.

=

10-3

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The relative error of the scatter fraction (AF/F) can be estimated by using the following equation, derived in Reference 8: AF/F = (1 -- F) (X Ap./p. + ANs/Ns)

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The dependence of the scatter fractions on the incident photon energy has a characteristic 'N' shape, that is, a maximum at low energies and a minimum at higher energies followed by an increase at very high energy.

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An increase in the atomic number, Z, of the scattering media causes a 'squeezing' effect on the characteristic 'N' shape: the maximum appears at higher energies and the minimum at lower energies.

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Acknowledgements

0-'~ 0 Photon energy (MeV)

Fig. 8 Spectral distribution of the scattered photons in 7 mm iron slab (E0 = 10 MeV, W = 100 mm). (a) Secondary electrons/positrons are not followed. (b) Full electron/positron/photon cascade followed

Spectral distribution of the scattered photons Figures 6, 7 and 8 show the spectral distribution of the scattered photons for the 7 m m iron slab and for incident photon energies of 1, 4 and 10 MeV, respectively. The d-function-like spikes in Figures 7 and 8 are the positron annihilation quanta at 511 keV. The energy bin width around the spikes is 2 keV. In Figures 7a and 8a we show the results of calculations where the secondary electrons (and positrons) are not followed (except for the annihilation of positrons). In Figures 7b and 8b, the full cascade is followed. The difference in the results of the two cases is attributed to the bremsstrahlung contribution, which is missing in 7a and 8a.

Conclusions We have calculated the ratio of the scattered to the total radiation fluence (the scatter function) for X-rays transmitted through polystyrene and iron slabs. The incident photon energy region covered in this work is 30 keV to 20 MeV for polystyrene and 100 keV to 20 MeV for iron. This range includes the energy region for diagnostic radiology and nuclear medicine as well as the energy region for industrial radiography. It is shown that for high incident photon energies it is important to include the secondary radiation of electrons and positrons. These charged particles can produce additional photons via bremsstrahlung or annihilation. The more energetic electrons and positrons can also exit the object under investigation and reach the recording system. Since different antiscatter techniques may be used for photons

The author would like to thank J.W. Motz for suggesting this work and S.M. Seltzer for many helpful discussions and suggestions. One of us (GB) would like to acknowledge the hospitality of the National Bureau of Standards during the course of this work. This work was supported in part by the Office of Naval Research.

References 1

Reiss, K.H. and Steinle, B. Phys Med Biol 18 (1973) p 746

2 3

Motz,J.W. and Dick, C.E. Med Phys 2 (1975) p 259 Stargardt, A. and Angerstein, W. Fortschr Geb Rbntgenstrahl 123 (1975) p 364 Dick, C.E., Soares, C.G. and Motz, J.W. Phys Med Biol 23 (1978) p 1076 Bureekner,K.A. (1977) unpublished Kalender,W. Phys Med Biol 26 (1981) p 835 Heang-Ping Chan and Kunio Doi Med Phys 12 (1985) p 152 Barnea, G. and Dick, C.E. Med Phys 13 (1986) p 490 Halbleib St, J.A. 'ACCEPT: a three-dimensional electron/ photon Monte Carlo transport code using combinational geometry"Sandia Report SAND 79-0415 Berger, M.J. and Seltzer, S.M. 'ETRAN Monte Carlo code system for electron and photon transport through extended media" Radiation Shielding Information Center. Computer Code

4 5 6 7 8 9 10

Collection, Oak Ridge National Laboratory Report CCC-107

(1968) 11 Berger, M.J. 'Monte Carlo calculation of the penetration and diffusion of fast charged particles' Methods in Computational Physics Academic Press, New York, vol 1 (1963) 12 Biggs,F. and Lighthill, R. 'Analytical approximations for X-ray cross-sections IF Sandia National Laboratories Report SC-RR-710507 (1971) 13 Hubbell, J.H. "Photon cross sections, attenuation coefficients, and energy absorption coefficients from 10 keV to 100 GeV' National Bureau of Standards Report NSRDS-NBS 29 (1969) 14 Storm, E. and Israel, H. Nuclear Data Tables A 7 (1970) p 565 15 Seltzer, S.M. and Berger, M.J. Nucllnstrum Methods B (1985) in press 16 Berman,B.L.Atlas of Photoneutron Cross Sections Obtained with Monergetic Photons UCRL-75694, 2nd edn (1974)

Authors The authors are with the Center for Radiation Research, National Bureau of Standards, Gaithersburg, MD 20899, USA. G. Barnea is on sabbatical leave from Technion: Israel Institute of Technology, and Rafael Laboratories, Haifa, Israel.

Paper received 18 March 1986. Revised 22 August 1986