1 Introduction

High energy X-ray sources are becoming more popular for the non-destructive investigation of large, thick, dense, high atomic number objects. Accurate angular and energy dependent source models are extremely useful for field flatness corrections, beam hardening corrections, detector response corrections, and radiation protection and shielding assessments. MCNP6.2 [1] is an industry standard radiological transport modeling tool; however, licensing availability, high performance computing resources, and expert knowledge are required for accurate results. These hurdles lead to simplifications in source models that may skew results. For instance, Miller [2] utilized a source with a single energy spectrum with no angular dependency for a 9 MVp source. Lewis [3] obtained a single energy spectrum and a radial intensity curve for a 4 MV beam by utilizing a series of tally planes. Hodges [4] went a step further and obtained 200 bin energy spectra for solid angles between 0°–90° in 10° increments for a 15 MVp source. Finally, McClanahan [5] utilized tallying planes rather than solid angles to obtain a series of 44 bin energy spectra for angles between 0°–180° in 1° increments for a 22 MVp source. Finally, Manoharan [6] simulated an electron beam crashing into a converter target to create X-rays. While this does accurately model the energy and angular dependence of the source, it is extremely computationally expensive and inefficient. In response to such shortcomings, this study uses MCNP6.2 to create high definition angular and energy dependent X-ray source definitions for the most commonly used high energy industrial X-ray sources at 450 kVp, 3 MVp, 6 MVp, 9 MVp, and 15 MVp.

2 Methods

2.1 Reflection Target Spectrum and Profiles

The 450 kVp X-ray source was modeled after the Comet MesoFocus 450 source [7] with a tungsten target angled at 20° with respect to the electron beam, a 40° × 40° illumination field, and a 60 mm distance between focal spot and the outside of the 5.0-mm-thick Be window (see Fig. 1). In simulation space, the source was modeled as a 205.212 mm radius vacuum cylinder with the 4.25-mm-thick tungsten target inside. The outside of the 5 mm thick beryllium window was placed 60 mm from the focal spot. This simulation geometry replicated tube parameters and achieved the correct 40° × 40° illumination field. All other surrounding systems contained in the MesoFocus tube such as the copper cooling jacket were not simulated. A 300-µm-thick Gd2O2S:Pr (GOS) scintillator was placed 14.0 cm below the outside of the beryllium window. The space below the beryllium window was simulated as air at STP in order to replicate air scatter. An F2 tally plane with the E-card tally modifier for 450 1 keV bins recorded the energy spectrum at the detector plane, and a (40 × 40 cm) (1000 × 1000 pixels) *FMESH4 energy deposition tally in the GOS captured the illumination field profile relative to a standard flat panel imager. This simulation was computationally intensive and required roughly 25.5 billion electrons over a 30 day period on 64-core nodes of the University of Tennessee Nuclear Engineering Department high performance computing cluster. The fmesh image was then extracted using Python 3.9 [8] converted to multiple text images, and rendered in color with Fiji/ImageJ [9].

Fig. 1
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450 kV simulation geometry. See text for details

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2.2 Transmission Targets Optimization

The Varex Linatron M9A [10] was the model for the 6 and 9 MVp sources, and the Varex K15 [11] was the model for the 15 MVp source. Due to the proprietary nature of the exact target specifications of the linacs, the ideal thicknesses for tungsten-equivalent targets were reverse engineered. A series of optimization simulations in MCNP6.2 were performed, consisting of cylindrical targets with thicknesses ranging from 1.00 to 4.00 mm in 0.25 mm increments. A 30° tally cone was placed in front of the target to calculate X-ray yield, and each simulation utilized one million source electrons. An example simulation geometry for the 2.00 mm target thickness is shown in Fig. 2.

Fig. 2
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2.00 mm target thickness simulation geometry

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2.3 Transmission Target Spectra and Angular Dependence

The angular and energy dependent X-ray spectra from the transmission mode linear accelerator targets were acquired using a series of stacked cones within a sphere to create annular tallying volumes for each solid angle from 0°–180° in 1° increments, as shown in Fig. 3. Each annular tallying volume was further subdivided into up to 10-keV wide energy groups for up to 1500 energy bins as in the 15 MV simulation. The target with the ideal thickness found in Sect. 2.2 was placed in the center of the sphere with an electron beam directed at its center. The target was assumed to be just the tungsten target with no other surrounding support systems such as the copper cooling jacket. Each simulation was computationally intensive, simulating approximately 2.5 billion electrons during roughly 350 h simulations on 64-core nodes of the cluster. Each spectrum was then extracted and a series of curves were fit to the energy spectra at key angles.

Fig. 3
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a Target geometry, b side view of entire tally sphere, c zoomed in side view of 0°–1° to 10°–11° annular tally regions and b–d front view of 0°–1° to 7°–8° annular tally regions

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2.4 Angular and Energy Dependent Source Implementation

The angular probability and energy spectrum for each annular region was then implemented into a source definition for each energy. The annular regions and corresponding angular probabilities were produced using the standard DIR distribution definition. The FDIR function was utilized to make the energy distribution dependent on the direction of the source particle. A sample of the source definition is shown in Fig. 4.

Fig. 4
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Sample angular and energy dependant source definition

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The above source definition creates a photon source originating from the (0, 0, 0) cartesian coordinate position with the source pointed along the positive x-axis. The direction term (DIR = D999) establishes the angular dependency where D999 contains an SI999 card with the cosines of all integer angles from 0° to 180° with respect to the positive x axis and the SP999 card contains the probabilities of a particle originating in each annular region. The original angular probability tally results for each annular region need to be weighted by the surface area of the respective region before being implemented into the SP999 card in the source definition. Without the weighting, the smaller annular regions of the new source definitions will be over represented and vice versa for the larger annular regions. Next, the energy term (ERG FDIR D888) makes the energy spectrum distribution dependent on the originating direction of the particle. The D888 distribution contains a DS888 card where each entry is the naming number of the energy distribution corresponding to the annular region in the DIR distribution that shares the same location on each respective list. The DS888 card is followed by a series of SI and SP cards that represent the energy spectrum for each annular region. For example, a photon originates at (0, 0, 0) and is given an original direction in the 5–6° annular region. MCNP will use the location of the 5–6° annular region from the DIR distribution and find the same location on the DS888 card. This entry on the DS888 card contains the name of the energy distribution that corresponds to the 5–6° annular region. MCNP will then find the SI and SP cards of the same name and use them as the energy distribution.

3 Results and Discussion

3.1 450 kVp Spectrum and Beam Profiles

The fmesh image of the X-ray distribution from the 450 kVp simulation can be found in Fig. 5. The F2,E tallies of the 450 kVp 40° illumination field resulted in the spectrum and curve fit shown in Fig. 6. The simulated 450 kVp spectra unveil the many expected low energy characteristic X-rays and an absorption edge around 70 keV. Table 1 in appendix A shows the polynomial coefficients for the fit shown in Fig. 6. The *FMESH4 tally image segmented orthogonally resulted in both the beam heel profile and the perpendicular beam flatness profile and their respective fits shown in Fig. 7a, b, respectively. Table 2 in appendix A shows the polynomial fits for the beam heel and flatness profiles. The beam flatness curve and the beam heal curve reveal that the beam is not expected to have the same intensity across the entire illumination field. This simulated data can be used to correct detector output.

Fig. 5
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450 kVp source X-ray distribution

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Fig. 6
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450 kVp energy spectrum and fit

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Fig. 7
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450 kVp a beam heel profile and b beam flatness profile

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3.2 Transmission Target Thickness Optimization

A 1.375 mm W-equivalent target yielded the maximum amount of X-rays in the forward 30 degree cone for the M9A operating at 9 MV while 1.750 mm W-equivalent target yielded the maximum amount of X-rays for the K15 in the forward 30° cone. A 1.375-mm-thick target was also used for the 3 and 6 MVp sources. Curves showing X-ray yield versus target thicknesses for the 9 MVp and 15 MVp simulations are shown in Fig. 8a, b

Fig. 8
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a 9 MVp and b 15 MVP target thickness X-ray yields

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3.3 Transmission Target Optimization Energy and Angular Dependence

Each annular region tallied the yield of X-rays in said annular region per source electron. Figures 9 and 10 show the X-ray yield angular dependance of the transmission targets. The polar plots show the increasing yield of X-rays from the transmission targets as the energies increase and how much more forward-scattered the higher energy (9 and 15 MVp) sources are compared to those of lower energies (3 and 6 MVp). The data shows that as the energies decrease, the target is expected to produce many more side- and back-scattered X-rays. In fact, the 6 and 9 MVp sources are expected to have more backscatter than the 15 MVp source, even though the overall yield from the 15 MVp source is much higher. It must be noted though that the backscattered X-rays from the 15 MVp source are expected to be much higher energy than the 6 or 9 MVp backscattered X-rays. These can be important considerations when designing shielding around a source.

Fig. 9
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Transmission targets angular probabilities [/sp/MeV]

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Fig. 10
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Transmission targets angular probabilities [log(/sp)]

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Figure 11 shows the forward spectra for each source {0.45, 3, 6, 9, 15}MVp. This figure displays the greater intensity and average energy of the higher energy beams in the forward direction.

Fig. 11
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Forward 0°–1° energy spectra for {0.45, 3, 6, 9, 15} MVp

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3.4 3 MVp

Key angles were selected to perform curve fits namely 0°, 5°, 10°, 45°, 90° and 180°. The spectrum for each of these angles is shown in Fig. 12. The polynomial piecewise curve fits for each spectrum are shown in Table 3 in Appendix A. This plot clearly shows the backscattered direction shares almost the same intensity as the forward direction, but does have a lower average energy. This information can be useful in shielding designs for lower energy transmission sources.

Fig. 12
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3 MVp key angles energy spectra

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3.5 6 MVp

Key angles were selected to perform curve fits namely 0°, 5°, 10°, 45°, 90° and 180°. The spectrum for each of these angles is shown in Fig. 13. The polynomial piecewise curve fits for each spectrum are shown in Table 4 in Appendix A. This plot shows that bremsstrahlung produced X-rays tend to be more forward scattered as the energy increases. Unlike the 3 MeV key angles plot, the forward directions have a much higher overall intensity than the backscattered angles.

Fig. 13
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6 MVp key angles energy spectra

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3.6 9 MVp

Key angles were selected to perform curve fits, namely 0°, 5°, 10°, 45°, 90° and 180°. The spectrum for each of these angles is shown in Fig. 14. The polynomial piecewise curve fits for each spectrum are shown in Table 5 in Appendix A.

Fig. 14
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9 MVp key angles energy spectra

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3.7 15 MVp

Key angles were selected to perform curve fits namely 0°, 5°, 10°, 45°, 90° and 180°. The spectrum for each of these angles is shown in Fig. 15. The polynomial piecewise curve fits for each spectrum are shown in Table 6 in appendix A.

Fig. 15
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15 MVp Key Angles Energy Spectra

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4 Conclusions

Tables of fit coefficients based on the charts of each spectra at each angle are a valuable reference for calibrations, corrections, and radiation protection and shielding. The simulated data provided will enable high energy X-ray source facilities to accurately model their sources in various simulation spaces. These source definitions enable higher accuracy and higher confidence in 2D/3D radiography and shielding assessments.