Photon source term after single collision in targets of silicon,copper and lead for 50-500 keV X-ray beams

Abstract

Single-scattered X-ray doses at 1 m from silicon, copper and lead targets were calculated using an analytical point-kernel method considering the self-absorption, and the calculated values were compared with detailed results of a Monte Carlo calculation with respect to the emission angle. In the calculations, a slab slanted at 3° to the beam axis was used for silicon in addition to the cylindrical targets for the three materials, and the slab geometry showed the largest doses. The analytical calculations were underestimated compared with the Monte Carlo calculations by less than 24% for silicon and 40% for copper, particularly at large-angle scattering, which was attributable to the buildup effect of the single-scattered X-rays in the targets. By considering the buildup effect, the difference from Monte Carlo results decreased to less than 20%. For lead, the influence of fluorescent X-rays produced by the source beam was dominant in the backward direction, which was also calculated analytically. The simple analytical program can be applied to any target size and shape by considering self-absorption and the buildup effect, both of which inform the simple dose estimation method.

Keywords

X-ray beam self-absorption buildup effect Monte Carlo point-kernel method

1 Introduction

X-ray scattering influences the X-ray imaging techniques such as computer tomography (CT) and radiography. Furthermore, scattered X-rays are important for radiation safety and, at high intensity sources, for the prevention of radiation damage to weak-radiation-tolerance materials such as semiconductors. An accuracy in estimating the degree of X-ray scattering, particularly in the case of the beam, determines the accuracy of calculating the total doses because the targets are irradiated only by the scattered X-rays. Scattering strength can be accurately estimated by using X-ray transport codes, e.g. Monte Carlo (MC) codes, which are used to trace multiple scattering; however, the method requires extensive time to prepare the input file and to execute the code. In the present study, in which the single-collision term is mostly dominant for the above-mentioned beam incidence conditions, analytical calculations such as the point-kernel method become possible, drastically reducing the estimation time. The program is easily written, which also results in a higher level of transparency.

In fact, point-kernel single-collision codes have been extensively used for the investigation of improved radiation shielding and imaging techniques. Regarding radiation shielding, a G33 gamma-ray-scattering program [1] has long been used in nuclear facilities. In imaging and radiography, scattering programs have also been proposed [2, 3]. In such codes, however, the volume of the scattering region needs to be divided in the mesh manner for broad irradiation. For narrow-beam X-rays, however, such division is not necessary, and simplicity is improved.

The scattered X-ray flux depends on the target material and size as well as on the beam energy and intensity in the form of self-absorption and the buildup effect. This dependence, while not explicitly discussed, is important for estimating the flux.

The purpose of this study was to demonstrate the usefulness of a point-kernel single-collision code simplified for narrow beam by comparing its results with those from MC calculations for three different scattering materials. The degree of usefulness is predicted on achieving a satisfactory level of accuracy in flux and dose in most practical applications. In the comparisons, the influence of multiple collisions and fluorescent X-ray production in the targets was also examined. Thus, the dependence of the scattered doses on the target geometry and medium was clarified for single-collision calculations.

2 Calculations

2.1 Analytical point-kernel method

A program (target.java) was written in the object-oriented Java programming language specialized in narrow-beam applications. When an X-ray beam is incident to a target, the uncollided flux φ arriving at the scatter point is $φ = φ_{0} e^{- μ l},$ (1) where φ ₀ is the source flux, μ is the linear attenuation coefficient of the target material and l is the penetrating length through the target, i.e. the distance from the target surface to the scatter point. Rayleigh and Compton scattering were considered for unpolarised X-rays. Rayleigh scattering is important at small scattering angles for narrow beams. The differential Rayleigh scattering cross section is $\frac{d σ_{R} (θ)}{d Ω} = \frac{r_{0}^{2}}{2} (1 + {cos}^{2} θ) {[F (x, Z)]}^{2},$ (2) where θ is the scattering angle, r₀ is the classical radius, F(x, Z) is the atomic form factor, and Z is the atomic number of the target material. For the momentum-transfer parameter x, $x = \frac{sin (θ / 2)}{λ},$ (3) where λ(Å)=12.398520/E(keV) [4]. On the other hand, the differential Compton scattering cross section is expressed using the Klein-Nishina cross section as $\frac{d σ_{C} (θ)}{d Ω} = \frac{r_{0}^{2}}{2} {(\frac{E^{'}}{E})}^{2} (\frac{E^{'}}{E} + \frac{E}{E^{'}} - \sin^{2} θ) S (x, Z),$ (4) where E and E’ are the source and scattered X-ray energies, and S(x, Z) is the incoherent scattering function. The number of electrons associated with the scatter volume for the narrow beam is $N = N_{A} \frac{ρ Z}{A} a Δ l,$ (5) where N_A is Avogadro’s number, ρ the density, A the mass number of the target, a the beam area and Δl the length element of the target. As a result, the Rayleigh-scattered flux at the distance d from the scattering point incident by the flux φ₀ is obtained as $φ_{R, Δ l} = 4 π φ_{0} N (\frac{d σ_{R}}{d Ω}) \frac{1}{4 π d^{2}} e^{- μ l} e^{- μ^{'} l^{'}} = n_{0} N_{A} \frac{ρ Z}{A} Δ l (\frac{d σ_{R}}{d Ω}) \frac{1}{d^{2}} e^{- μ l} e^{- μ^{'} l^{'}},$ (6) where φ ₀ equals n₀/a, n₀ is the total photon number and l’ is the travelling distance of the scattering X-rays in the target. Subsequently, the Rayleigh-scattered flux φ _R from the entire target of length L is obtained by the integration $φ_{R} = n_{0} N_{A} \frac{ρ Z}{A} (\frac{d σ_{R}}{d Ω}) \frac{1}{d^{2}} \int_{0}^{L} e^{- μ l} e^{- μ^{'} l^{'}} dl$ (7)

Here, when $f_{ab} = \frac{1}{L} \int_{0}^{L} e^{- μ l} e^{- μ^{'} l^{'}} dl$ (8) is substituted into (7), the following expression is obtained: $φ_{R} = n_{0} N_{A} \frac{ρ Z}{A} L (\frac{d σ_{R}}{d Ω}) \frac{1}{d^{2}} f_{ab} .$ (9)

In the same manner, the Compton-scattered flux φ _C is expressed as $φ_{C} = n_{0} N_{A} \frac{ρ Z}{A} L (\frac{d σ_{C}}{d Ω}) \frac{1}{d^{2}} f_{ab} .$ (10)

The self-absorption f_ab depends on the scattered-X-ray energy and target geometry. For the cylindrical target, because the scattered X-rays exit from the side or bottom of the cylinder, depending on cylinder size and emission angle, the integration needs to be divided into two terms. Thus, the magnitudes are given by the following expressions for the forward, vertically and backward scattering, respectively: ${Lf}_{ab} = \int_{0}^{L - r / tan θ} e^{- μ l} e^{- μ^{'} r / sin θ} dl + \int_{L - r / tan θ}^{L} e^{- μ l} e^{- μ^{'} (L - l) / cos θ} dl,$ (11)

${Lf}_{ab} = \int_{0}^{L} e^{- μ l} e^{- μ^{'} r} dl = \frac{1 - e^{- μ L}}{μ} e^{- μ^{'} r},$ (12)

${Lf}_{ab} = \int_{0}^{r / tan θ} e^{- μ l} e^{- μ^{'} l / (- cos θ)} dl + \int_{r / tan θ}^{L} e^{- μ l} e^{- μ^{'} r / sin θ} dl,$ (13) where r is the target radius and μ’ is the linear attenuation coefficient for the scattered X-rays. With increasing L, the forward scattering flux starts increasing, reaches a peak, and decreases thereafter owing to X-ray attenuation. For the calculations, the target sizes at the maximum intensity were chosen. On the other hand, the vertical and back-scattered X-ray fluxes increase with length and are saturated as shown in (12) and (13). In large target sizes, multiple scattering cannot be neglected. As a result, the target lengths were determined as follows: 1 and 5 cm at and above 50 keV, respectively, for silicon; 1 cm for copper and 0.1 cm at 50 and 100 keV and 0.5 cm at 200 and 500 keV for lead. The radii r of the cylinders were all fixed at 0.5 cm. For the slanted slab, L is replaced with t/sinθ _s in (9) and (10) where t is the slab thickness and θ _s is the angle of inclination to the beam axis. In the calculations, t = 1 cm and θ _s =3° were used.

Buildup factors are ordinarily used in considering the effectiveness of radiation shielding around the targets. Without the shields, the buildup factors were considered only for the X-rays scattered incoherently in the targets because Rayleigh-scattered X-rays are redirected intensely at narrow angles. That is, the effective dose was obtained as $D = f_{D} (E) φ_{R} + f_{D} (E^{'}) B (E^{'}, r) φ_{C}$ (14) where f_D(E) is the flux-to-dose conversion factor at energy E and B(E,r) is the effective dose buildup factor for the point isotropic source at energy E over distance r. Here, the radius r was used for the buildup factor because the flux from the short cut in the radial direction was assumed to be influential.

In the program, the attenuation coefficients [5] and flux-to-dose conversion factors [6] were obtained by log-log interpolation of the data. For F(x, Z) and S(x, Z), simple expressions were used for the numerical values; namely, expressions (9) and (10) for F(x,Z) and (29) for S(x, Z) in [7]. Silicon, copper and lead were used as targets. For silicon and copper, the effective dose buildup factors for concrete and iron, respectively, were used.

2.2 MC calculation

For the MC calculations, the PENMAIN program in the PENELOPE2011 code [8] was executed on the Windows 8.1 Pro platform using NAG Fortran Compiler Release 6.0. Photon pencil beams of 50, 100, 200, and 500 keV were directed at the centre of the bottom of cylindrical silicon, copper and lead targets, and the scattered photons were scored with impact detectors and shown as an energy spectrum. The target sizes were the same as those used in the analytical point-kernel method. The detectors were defined as the intersections of the sphere and cones as shown in Fig. 1(a); namely, the spherical zones. The target was set in the centre of the sphere, which had a radius of 1 m, and the half opening angles of the cone were centred at 3°, 10°, 30°, 60°, 90°, 120°, 150° and 170°; the respective angle bin widths were 1.0°, 1.6°, 2.3°, 1.3°, 0.3°, 1.3°, 2.3° and 1.6°. The effective dose was obtained by multiplying the flux in each energy bin of 5 keV by conversion factors [6]. The statistical fluctuations were less than 2% for silicon, 5% for copper except at 50 keV and 12% for lead except at 90°. The fluctuations for backscattering were larger than those for forward scattering.

For the slanted silicon slab, the detectors had to be set only in the upper side as shown in Fig. 1(b) and (c). Thus, the scoring region was defined as the overlap of the spherical zones and the V-shaped region opening at 6° in Fig. 1(a). The statistical fluctuations were almost the same as those for the cylindrical silicon target.

3 Results and discussion

Figures 2–4 show the doses computed using the MC and analytical methods at 50, 100, 200 and 500 keV for silicon, copper and lead as a function of emission angle. The Rayleigh- and Compton-scattered fluxes calculated using the analytical method without taking the buildup effect into account are also indicated in Figs. 5 and 6 for silicon and copper, respectively. Of the three target materials, silicon showed the largest doses, except for 3° at 200 and 500 keV, at which the heavier targets gave the larger doses because of the Rayleigh-scattering components, as shown in Figs. 5 and 6. Compared with silicon, copper showed considerably smaller dose values at 50 and 100 keV because self-absorption had a much larger effect in the low-energy region. In the forward direction, the doses at 50 keV were smaller than the minimum in the ordinate region. At 90°, the self-absorption sharply increased with decreasing energy through the radius, compared with the scattering in the shallow penetrating depth, and a valley occurred for copper and lead.

For the slanted silicon slab, Fig. 7 represents the doses calculated using the MC and analytical methods without considering the buildup effect between 10° and 170°. The doses were found to be larger than those of the cylindrical target because the path after the scattering can be shorter. The ratios of Figs. 2 and 7 were less than 2.0 above 50 keV and 2.4 at 50 keV at all angles.

As shown in Figs. 2, 3 and 7, without considering the buildup effect even in the small target, the analytical calculations underestimated the dose values. The ratios of the doses calculated using the MC code to those calculated using the analytical method for silicon and copper are indicated in Fig. 8. Upon increasing the angles, the ratios decreased and became constant above 90°; the ratios ranged from 0.74 to 1.06 for silicon and from 0.61 to 1.00 for copper except at 3° and 10° for 100-keV X-rays. At 100 keV, the copper target may be too long for the single-scatter calculations; therefore, when the length was shortened to 0.5 cm, the ratio decreased to 1.12 at 3°. The buildup effect was considered only at 60° and above because of the relatively isotropic scattering compared with the strongly anisotropic scattering in the forward direction. Consideration of the buildup effect improved the agreement with the MC values, with the ratios being between 0.93 and 1.10 for silicon and between 0.89 and 1.18 for copper, as shown in Fig. 8.

For lead, fluorescence X-rays were expected to be dominant. The flux of K series X-rays were calculated using the photoelectric cross section μ_ph [5]; the fraction of all the photoelectric interactions that occur in the K-shell P_K, the fluorescence yield Y_K [9] and the branch ratio ω _i for α₂, β₃ and β₁ [10]: $φ_{K} = n_{0} μ_{ph} P_{K} Y_{K} L \frac{ρ}{4 π d^{2}} f_{ab} \sum_{i} ω_{i} f_{D} (E_{i}) .$ (15)

The doses were also obtained from the flux given by MC calculations at 200 and 500 keV and were confirmed to agree with those obtained by multiplying the result of (15) by the flux-to-dose conversion factors. As shown in Fig. 4, the backscattered X-rays at 100 and 200 keV were nearly dominated by the K-series X-rays, while the magnitudes remained almost the same as those of silicon and copper at 150° and 170°. At 200 and 500 keV, in the forward direction, the analytical single-scattering calculations yielded the values in agreement with those of MC calculations at 3° and 10° within 1% and 13%, respectively.

Figures 5 and 6 roughly indicate that consideration of Rayleigh scattering is necessary below 3° at 500 keV, 10° at 200 and 100 keV and at all angles at 50 keV for silicon. For copper, the intensity remains above 5% of the total flux below 10° at 500 keV, 30° at 200 keV and at all angles at 100 keV. For silicon, the values of S(x,Z) started to deviate from Z at 90°, 60°, 30° and 10° at 50, 100, 200 and 500 keV for silicon, and 90°, 45° and 10° at 100, 200 and 500 keV, respectively, for copper. In general, consideration of S(x,Z) is necessary, except in the forward direction.

4 Conclusions

By considering the buildup effect, a point-kernel single-scattering program yielded a scatteredX-ray dose from cylindrical silicon and copper targets in agreement within 10% and 18%, respectively, with those of MC calculations. Even without considering the buildup effect, the differences were only 26% and 39%, respectively. The simple program used was found to contribute to the flux and dose predictions for X-ray beams, generally with enough accuracy to positively influence the improved radiation dose estimation.

The doses were found to be larger from silicon than from copper and lead and larger from the slanted slab shapes than from cylindrical targets. The doses in the backward direction were almost the same as those yielded by fluorescence X-rays from lead at 100 and 200 keV.

When the target is divided into a one-dimensional mesh, the expression for the self-absorption becomes simpler, comprising yet another calculation method. If the target is furthermore divided into three dimensions, multiple scattering can be traced directly instead of being predicted using buildup factors. Thus, the simple program described can be extended into a very versatile approach for calculating doses.

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