Life Sciences in Space Research 7 (2015) 27–38

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Life Sciences in Space Research

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3DHZETRN: Neutron leakage in finite objects

John W. Wilson a, Tony C. Slaba b,∗, Francis F. Badavi a, Brandon D. Reddell c, Amir A. Bahadori c

a Old Dominion University, Norfolk, VA 23529, USAb NASA Langley Research Center, Hampton, VA 23681-2199, USAc NASA Johnson Space Center, Houston, TX 77004, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 April 2015Received in revised form 24 August 2015Accepted 30 September 2015

Keywords:Space radiationRadiation shieldingRadiation transportHZETRN

The 3DHZETRN formalism was recently developed as an extension to HZETRN with an emphasis on 3D corrections for neutrons and light ions. Comparisons to Monte Carlo (MC) simulations were used to verify the 3DHZETRN methodology in slab and spherical geometry, and it was shown that 3DHZETRN agrees with MC codes to the degree that various MC codes agree among themselves. One limitation of such comparisons is that all of the codes (3DHZETRN and three MC codes) utilize different nuclear models/databases; additionally, using a common nuclear model is impractical due to the complexity of the software. It is therefore difficult to ascertain if observed discrepancies are caused by transport code approximations or nuclear model differences. In particular, an important simplification in the 3DHZETRN formalism assumes that neutron production cross sections can be represented as the sum of forward and isotropic components, where the forward component is subsequently solved within the straight-ahead approximation. In the present report, previous transport model results in specific geometries are combined with additional results in related geometries to study neutron leakage using the Webber 1956 solar particle event as a source boundary condition. A ratio is defined to quantify the fractional neutron leakage at a point in a finite object relative to a semi-infinite slab geometry. Using the leakage ratio removes some of the dependence on the magnitude of the neutron production and clarifies the effects of angular scattering and absorption with regard to differences between the models. Discussion is given regarding observed differences between the MC codes and conclusions drawn about the need for further code development. Although the current version of 3DHZETRN is reasonably accurate compared to MC simulations, this study shows that improved leakage estimates can be obtained by replacing the isotropic/straight-ahead approximation with more detailed descriptions.

Published by Elsevier Ltd on behalf of The Committee on Space Research (COSPAR).

1. Introduction and historical context

Early nucleon transport code development relied on Monte Carlo (MC) methods using the straight-ahead approximation(Wright et al., 1969) where all particles are assumed to travel along a common direction. Further development of a full 3D MC code with the Bertini intranuclear cascade model was used to test the straight-ahead approximation for which resulting dose and dose equivalent were within the statistical uncertainty of the full 3D MC code (Alsmiller et al., 1965). Such 3D MC codes on computers of that time required vast resources even for small

* Corresponding author at: 2 West Reid St., Mail stop 188E, Hampton, VA 23681-2199, USA. Tel.: +1 757 864 1420.

E-mail address: Tony.C.Slaba@nasa.gov (T.C. Slaba).

http://dx.doi.org/10.1016/j.lssr.2015.09.0032214-5524/Published by Elsevier Ltd on behalf of The Committee on Space Research (CO

sample sizes (Wright, 1966; Alsmiller, 1967) and encouraged the development of more efficient means of radiation exposure eval-uation. A 3D MC code that sampled a Bertini model derived database giving it much greater computer efficiency was extended to high energies at the NASA Langley Research Center (LaRC) (Lambiotte et al., 1971) for atmospheric and space radiation studies but still required extensive computer resources for realistic cal-culations (Foelsche et al., 1974). As a result, NASA LaRC began a search for methods of direct solution of the Boltzmann equa-tion. It was found that proton transport based on direct solution of the straight-ahead Boltzmann equation resulted in an efficient code able to reproduce the straight-ahead MC results using the same database (Wilson and Lamkin, 1974, 1975; Lamkin, 1974;Wright et al., 1969). NASA LaRC subsequently began a long-term investigation of direct solution of the Boltzmann equation as an alternative to MC methods (Wilson and Lamkin, 1974, 1975). The

SPAR).

28 J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38

3D MC codes would play an indispensable role in these develop-ments as a test for this alternate evaluation scheme.

A major element in developing Boltzmann solution methods is the examination of computational methods, as was first taken up in the straight-ahead approximation following the MC studies of Wright et al. (1969) and Alsmiller et al. (1965). Convergence studies on the numerical procedures (Lamkin, 1974) and theoret-ical formalism (Wilson and Lamkin, 1974, 1975) were also sub-sequently commissioned. The error associated with the straight-ahead approximation itself was examined by expressing it as a first order angular asymptotic expansion in the Boltzmann formalism. It was shown that errors were on the order of the square of the ra-tio of the beam divergence (few cm) to the radius of curvature of the object (few meters in human rated space systems) (Wilson and Khandelwal, 1974). Additional analysis using the Fredholm version of the Boltzmann equation demonstrated that the straight-ahead approximation is more accurate than assuming that the High charge (Z) and Energy (HZE) nuclear collision fragments proceed with the same velocity as the projectile ion (Wilson, 1977). Neglect of these higher order terms resulted in reduction of the Fredholm version to a Volterra equation that could be solved using numeri-cal marching procedures (Wilson and Badavi, 1986).

Solving the Volterra equation with perturbation series tech-niques (Neumann series) allowed direct comparison to 20Ne at-tenuation experimental data obtained at the Lawrence Berkeley Laboratory’s Bevatron (Wilson et al., 1984) revealing the need for an improved nuclear fragmentation model. The initiation of the NUCFRG model allowed improved comparison with atmospheric air shower data (Wilson et al., 1987a, 1987b) and with the 20Ne Bevatron data (Shavers et al., 1990, 1993). With these successes in transporting HZE ions, only the treatment of light ions and neu-trons required advancement to provide a formalism compatible with the HZE numerical marching procedures.

Unlike the HZE fragments that are produced at velocities nearly equal to the projectile ion, neutron and light ion induced reac-tions produce a broad spectrum of secondary particles and can-not be adequately simplified. This complication was addressed by Wilson et al. (1988a, 1988b) using perturbative methods wherein a closed form solution to the straight-ahead Boltzmann transport equation is first obtained over a small spatial step h and repeated to a chosen depth x = Nh, where N is the number of steps taken. The approach provided a computationally efficient solution over the whole spatial and energy domain. Such marching procedures were fully compatible with the HZE code, and the integration of the baryon transport code (BRYNTRN) into the HZE code provided the initial HZETRN version (Wilson et al., 1991). Cucinotta (1993)added the remaining light ions (deuteron, triton, helion, and alpha) to BRYNTRN and integrated the revised code into HZETRN. Code verification with analytic solution methods were accomplished by Wilson et al. (1987b), Ganapol et al. (1991), Lamkin (1994), and Shinn et al. (1998). Code validation was summarized by Shinn et al. (1998). Later work completed by Slaba et al. (2010a) greatly improved overall code efficiency and reduced numerical discretiza-tion error associated with the marching algorithms. Blattnig et al.(2004) and Norman et al. (2012, 2013) also integrated a coupled model for pion and muon transport and the associated electromag-netic cascade. These efforts largely completed the development of HZETRN in the straight-ahead approximation. Further evaluation of the accuracy of the straight-ahead formalism requires formal im-provements with added detail to the description that can be taken in two directions: corrections to the straight-ahead approximation or addition to the physical process descriptions within the HZETRN formalism.

Although this early code development led to a highly efficient and useful transport code for space radiation shielding, vehicle op-timization, and astronaut risk analysis, inherent limitations of the

straight-ahead approximation may be important in some cases. In particular, the description of neutron leakage from finite geometry boundaries is not well described. An obvious first correction relates to the assumption that even particles produced in the backward hemisphere travel forward in the straight-ahead approximation. Hence as a correction, the backward hemispheric produced neu-trons were assumed to travel straight backward as the next level of development (bi-directional approximation). Note, this changes the character of the solution as forward and backward propaga-tion is coupled in the nature of the Fredholm equation. Clowdsley et al. (2000, 2001) solved the forward/backward neutron equations in the multigroup approximation and using collocation methods. Heinbockel et al. (2003) compared additional solution methods among themselves. Slaba et al. (2010b) fully coupled the forward and backward neutron transport equations and improved numeri-cal efficiency by casting the system in a linear algebraic equation format (as a discrete version of the Fredholm equation). Extensive comparisons have been made with available MC codes (Heinbockel et al., 2011a, 2011b; Slaba et al., 2011, 2013) and found to agree with the various MC codes as well as the MC codes agree among themselves in some shielding geometries.

The next logical step in code development beyond the bi-directional approximation is to allow fuller angular dependence within the Fredholm formalism leading to, among other things, im-proved description of neutron leakage from finite geometry bound-aries. A first step was implemented in a perturbative approach in finite objects and compared to MC results (Wilson et al., 2014a, 2014b, 2014c, 2015). This evaluation of 3D effects by comparing with MC evaluations met with some of the limitations of the var-ious MC codes which use differing nuclear models and databases resulting in large disparity (Heinbockel et al., 2011a, 2011b) so that leakage and related 3D effects are somewhat obscured (Wilson et al., 2014a, 2014b). The nuclear cross sections enter in two ways; first, they differ in the overall production of neutrons/light ions among the transport models, and second, they differ in attenuation and scattering cross sections as particles move through materi-als. Herein, means of reducing the effects of the differing neutron production terms among various transport models will be used to more unambiguously evaluate neutron leakage and related 3D ef-fects in neutron transport.

2. 3DHZETRN formalism

The 3DHZETRN theoretical formalism has already been dis-cussed in detail in Wilson et al., (2014a, 2014b, 2014c, 2015). A brief summary is given here only to explicitly show important fundamental approximations used in the current formalism. The primary approximation used in the code assumes the neutron pro-duction cross section can be represented as a sum of forward and isotropic components, where the forward component is sub-sequently solved within the straight-ahead approximation. These simplifications are shown later to have a noticeable impact on neutron leakage estimates in certain finite geometries, requiring further improvements.

The linear Boltzmann transport equation within the continu-ous slowing down approximation for the flux (or fluence) density, φ j(x, �, E), of a j type particle is given by Wilson et al. (1991, 2005)

B[φ j(x,�, E)

] =∑

k

∞∫

E

∫

4π

σ jk(

E, E ′,�,�′)φk(x,�′, E ′)d�′dE ′,

(1)

J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38 29

where the differential operator on the left hand side is defined as

B[φ j(x,�, E)

] ≡ � · ∇φ j(x,�, E) − 1

A j

∂

∂ E

[S j(E)φ j(x,Ω, E)

]

+ σ j(E)φ j(x,�, E). (2)

In equations (1) and (2), S j(E) is the stopping power of a type j ion with kinetic energy E (vanishes for neutrons), σ j(E) is the total macroscopic cross section for a type j particle with kinetic energy E , and σ jk(E, E ′, �, �′) is the double differential macro-scopic production cross section for interactions in which a type kparticle with kinetic energy E ′ and direction �′ produces a type jparticle with kinetic energy E and direction �.

The double differential cross sections for neutron ( j = n) pro-duction are found to be closely approximated by a forward com-ponent and an isotropic component given by

σnk(

E, E ′,�,�′) = σnk,iso(

E, E ′)/4π + σnk,for(

E, E ′,�,�′), (3)

where the first term is isotropic and associated with lower energy neutrons, and the second term is associated with higher energy forward peaked neutrons. The isotropic component is computed as

σnk,iso(

E, E ′) = 2∫

2π B

σnk(

E, E ′,�,�′)d�′, (4)

where 2π B represents the backward hemisphere. The forward component is found by rearranging the terms in equation (3). The flux density is similarly separated into forward and isotropic com-ponents and, along with equation (3), substituted into equation (1). This produces a simplified equation for the forward component solved within the straight-ahead approximation, utilizing existing numerical methods (Wilson et al., 1991; Slaba et al., 2010a). After obtaining the forward flux solution, the isotropic neutron source term is computed as

ξn,iso(x,�,�0, E) =∑

k

∞∫

E

σnk,iso(

E, E ′)φk,for(x,�0, E ′)dE ′, (5)

where φk,for(x, �0, E ′) is the forward flux solution along a cho-sen incident direction �0. The isotropic neutron solution along the transport direction � is then obtained using bi-directional trans-port methods previously developed (Slaba et al., 2010b). Once the isotropic neutron solution is obtained, an additional source of light ions is evaluated, leading to a partial 3D treatment for charged particles produced from low energy neutrons.

The aspects of the formalism described in equations (1)–(5)of particular importance in the present study are the following. First, neutron production cross sections are currently represented by only a forward and isotropic components. Second, forward neu-trons are solved within the straight-ahead approximation. Lastly, the isotropic neutron cross sections and source term have no ex-plicit angular dependence. While these approximations may be addressed in the future work, the focus of the present study is to determine limitations on the current formalism in various geo-metric shielding configurations.

3. Simple homogeneous geometry

In this section, neutron fluence spectra computed by 3DHZETRN and three MC codes in simple homogeneous geometries are com-pared, and emphasis is given to the leakage occurring in finite (cube and sphere) objects. A ratio is defined to quantify the frac-tional neutron leakage at a point in a finite object relative to a semi-infinite slab geometry. The leakage ratio de-emphasizes large differences in the neutron production cross sections generated by the codes and instead focuses on angular scattering and absorption

Fig. 1. Aluminum slab geometry (top) and associated cube geometry (bottom). The slab is 40 g/cm2 thick with infinite lateral dimensions. The cube is 40 g/cm2 on all sides. The red circles indicate detector locations at which quantities are evaluated. The green arrows indicate general directions of leakage. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

factors. The ratio is also used to deduce strengths and weaknesses of the 3DHZETRN formalism. Observed variation amongst all the codes also suggests areas needing further theoretical development or experimental efforts.

Simple homogeneous geometry and source orientation are cho-sen in this section to allow reasonably efficient MC simulation for comparison with the 3DHZETRN results (Wilson et al., 2014a, 2014b, 2014c, 2015). For all calculations, the boundary condition, or external source, is the Webber representation of the February 23, 1956 solar particle event (SPE) (Webber, 1966) applied uni-formly down onto the geometry of interest. The Webber 1956 SPE is given by an exponential momentum spectrum with momentum parameter p0 = 100 MV and 109 protons/cm2 above 30 MeV.

Consideration is given first to a 40 g/cm2 homogeneous slab of aluminum with sides extending to infinity. A subset of the slab (cube) is also considered, as shown in Fig. 1. Transport within the slab has leakage from the cube subset compensated by lateral leak-age into the subset as shown by green arrows in the upper figure. However, transport within the cube by itself will have leakage from all sides without compensation. Since infinite lateral dimensions cannot actually be applied either in 3DHZETRN or in MC codes, a convergence test is first performed to determine applicable di-mensions that minimize lateral leakage at the detector locations (red dots in Fig. 1) and therefore closely approximate the idealized infinite scenario. This type of convergence testing is also useful in verifying that the 3DHZETRN formalism generates the correct qual-itative trends as geometric factors such as lateral dimensions are systematically varied.

Convergence testing was performed with 3DHZETRN for N = 22rays (Wilson et al., 2014a, 2014b, 2014c, 2015) and with Geant4 version 9.4.6 (Agostinelli et al., 2003). The MC simulation results are provided in this case only to confirm the qualitative trends generated from 3DHZETRN. Discussion regarding the source of quantitative differences in the results is given later in this sec-tion. Results are shown in Fig. 2 at the detector location depths of 35 g/cm2 and 40 g/cm2. Both 3DHZETRN and Geant4 show converging neutron fluence results for slab lateral dimensions ap-proaching 500 g/cm2 and larger, indicating that lateral leakage is almost negligible. Fig. 2 shows that lateral dimensions of at least 500 g/cm2 sufficiently approximate infinite lateral dimensions in 3DHZETRN and MC computational procedures.

In Fig. 2 below 100 MeV, both 3DHZETRN and Geant4 show in-creasing neutron fluences with increasing lateral dimensions up to 500 g/cm2. Below 0.3 MeV where absorptive processes are min-imal in aluminum, the leakage along the long transport paths parallel or nearly parallel to the front and back slab faces are un-derestimated in 3DHZETRN, resulting in the increase in fluence at these low energies. Improved propagation methods are required to

30 J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38

Fig. 2. Convergence test for solution in slab geometry at depths of 35 and 40 g/cm2.

Fig. 3. Neutron fluence induced by the Webber SPE event in a 40 g/cm2 cube of aluminum at two depths.

correct this problem, but it should be noted that previous studies have shown these lower energy neutrons make only small con-tributions to even the total neutron exposure (Slaba et al., 2011). Further discussion on the differences observed between 3DHZETRN and Geant4 is given below.

The calculated fluence using 3DHZETRN within the 40 g/cm2

cube at 35 g/cm2 and 40 g/cm2 depths is shown in Fig. 3 in com-parison with three MC codes (Geant4 version 9.4.6 (Agostinelli et al., 2003), FLUKA (Fasso et al., 2005; Battistoni et al., 2007) and PHITS (Sato et al., 2006, 2013)). The differences among the MC codes are mainly due to differing nuclear production models/databases. In addition, the current version of 3DHZETRN assumes the cross sections consist of only straight-ahead and isotropic components for the neutron collisional source distributions. Therefore, observed variation between 3DHZETRN and MC codes is a combination of nuclear model differences and the forward/isotropic assumption. Herein, attempts are made to separate the effects of the for-ward/isotropic assumption from that of the nuclear cross section model differences.

This separation is accomplished, in part, by looking at the neu-tron leakage ratio given by

r(x, E) = 1 − φn,object(x, E)

φn,slab(x, E), (6)

where φn,object(x, E) is the neutron flux obtained at a depth in a fi-nite object, and φn,slab(x, E) is the neutron flux obtained at the cor-

responding depth in the semi-infinite slab geometry (500 g/cm2

lateral dimensions). Equation (6) quantifies the fractional leakage for neutrons of energy E from the location x through near lateral boundaries. The leakage defined by equation (6) from the cube is shown in Fig. 4 for 3DHZETRN in comparison with results from three MC codes (Geant4, PHITS, and FLUKA). Whereas the fluence estimates of the four codes can differ significantly due to differing neutron production cross section models (as seen in Fig. 3), the leakage ratio varies only by ten to fifteen percent among the codes as seen in Fig. 4. Also note that the vertical axis of Fig. 4 (and subsequent plots showing leakage factors) is in linear scale instead of log scale. Using the leakage ratio removes some of the depen-dences on the magnitude of the neutron production and brings into clearer focus the transport processes of angular scattering and absorption.

Qualitatively, the shape of the neutron leakage ratios as a func-tion of kinetic energy is the same for all codes. At the lowest energies, where the leakage ratio approaches unity, neutrons are escaping from near lateral boundaries in the finite geometry. At the highest energies, neutrons production is nearly in the straight-ahead direction, and the neutron spectra computed in the finite geometry and slab are nearly identical (i.e. leakage occurs mainly from the back boundary in both geometries), leaving the leakage ratio to approach 0. Similar trends are seen later in this section for an aluminum sphere.

J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38 31

Fig. 4. Comparison of MC evaluated neutron leakage from a cube at 35 and 40 g/cm2 depths.

Fig. 5. Aluminum slab geometry (top) and associated sphere geometry (bottom). The slab is 40 g/cm2 thick with infinite lateral dimensions. The sphere diameter is 40 g/cm2. The red circles indicate detector locations at which quantities are eval-uated. The green arrows indicate general directions of leakage. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The high leakage indicated in the 3DHZETRN results below 0.3 MeV is a consequence of overestimating the neutron fluence in the slab along rays nearly parallel to the front and back slab faces as shown in Fig. 2. This apparent error in 3DHZETRN in dif-fusive losses at low energies results in an overestimate of leakage from the cube and requires improved representation of diffusive losses through the faces in slab geometry as was discussed above in connection to Fig. 2. The adequacy of the straight-ahead ap-

proximation in representing the forward neutron flux component is seen at the highest energies (>100 MeV) for this geometry. At intermediate energies, between 0.3 MeV and 100 MeV, the errors in leakage are mainly associated with the assumed isotropic angu-lar distribution, and improved angular dependence in the neutron source is required.

Another geometry of interest is a spherical subset of the slab as shown in Fig. 5. Unlike the cube, the neutron collisional source within the sphere by itself differs from the distribution evaluated within a sphere embedded in the slab. This occurs since portions of the sphere while embedded in the slab are shielded by the ex-terior portion of the slab. These effects can be seen in Fig. 6, where the neutron source distributions for energies above 1 MeV in the cube and sphere are shown. While leakage effects from the cube are unambiguous (i.e. whether or not the cube is embedded in the slab), leakage from the sphere is obscured by the differences in the neutron source distribution; still, leakage as defined by equa-tion (6) remains a useful concept in the sphere geometry.

The calculated neutron fluence at 35 and 40 g/cm2 from 3DHZETRN is shown in comparison with that of the three MC codes in Fig. 7. It is first noted that 3DHZETRN is in better agree-ment with the MC codes in spherical geometry as seen in com-paring Fig. 7 with Fig. 3. It is also observed that the MC codes show greater discrepancies in the sphere than in the cube, sug-gesting non-trivial differences in angular production cross sections (not just the overall magnitude of neutron production).

Fig. 6. Isotropic neutron source (E > 1 MeV) distribution in units of neutrons/(g-event) induced by the Webber SPE spectrum in cube and spherical geometry.

32 J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38

Fig. 7. Neutron spectra at detector locations 35 g/cm2 (left pane) and 40 g/cm2 from top of aluminum sphere (diameter 40 g/cm2) exposed to Webber SPE spectrum.

Fig. 8. Comparison of MC evaluated neutron leakage from a sphere at 35 and 40 g/cm2 depths.

The leakage results at 35 and 40 g/cm2 within the sphere are shown in Fig. 8. Differing angular neutron production models are the main source of variation observed in Fig. 8. It should be re-called that 3DHZETRN uses the straight-ahead approximation for the forward components at the highest energies for which no lat-eral leakage occurs. This straight-ahead approximation is the pri-mary source of leakage differences between the cube (Fig. 4) and the sphere (Fig. 8) above 100 MeV (Wilson et al., 2014a, 2014b). Based on comparing leakage ratio estimates from 3DHZETRN to MC simulations, the straight-ahead approximation is more appropri-ate along the central ray of the cube and less appropriate in the sphere. As in Fig. 4 for the cube, the overestimate of leakage in the sphere below 0.3 MeV follows from errors in diffusion through the slab faces along rays parallel (or nearly parallel) to the facial plane. Improved treatment of the nuclear models below ∼10 MeV, where quasi-elastic scattering is negligible, and above ∼100 MeV, where the assumed straight-ahead approximation introduces error, will require formal improvements in 3DHZETRN.

To further examine 3D effects, the leakage of the cube and sphere calculated with 3DHZETRN is compared with MC codes that have better representation of low energy scattering and the high-energy angular dependent cross sections (relative to the straight-ahead approximation used in 3DHZETRN). MC (Geant4, FLUKA, PHITS) and 3DHZETRN derived results are shown in Figs. 3, 4, 7–12. Whereas good agreement of the leakage factor for the cube above 100 MeV is obtained (see Fig. 4), the failure of the straight-ahead approximation in the sphere is apparent in Fig. 8 where the

3DHZETRN generated leakage factor is clearly below the MC de-rived estimates. Note that the leakage of the sphere exceeds that of the cube in the MC codes (see Figs. 10–12) and that the main differences are above 100 MeV. The source of this high-energy dif-ference results from the slight angular divergence from the forward direction that is of less importance along the cube centerline com-pared with leakage along the centerline of the sphere.

Surprisingly, the leakage factor for the cube and sphere are largely similar in the 3DHZETRN formalism as seen in Fig. 9. Hence, the geometric shape in this case has only modest effects within similarly sized objects. This is an advantage in that the neutron spectrum depends weakly on geometric detail. An interesting fea-ture of the 3DHZETRN results is that the cube exhibits greater leakage than the sphere at intermediate energies (1 to 100 MeV). This results from the isotropic source term which has the same distribution in the cube but is significantly modified in the sphere as shown in Fig. 6, and places the intense neutron source region closer to the detectors and hence, appears as reduced leakage.

This weak dependence on geometry is somewhat borne out in the Geant4 results shown in Fig. 10, except that the sphere exhibits greater leakage as intuition would suggest. The greater leakage ex-pected for the sphere above 100 MeV is likewise given by the Geant4 code. Similar results are found for the PHITS code as seen in comparing Fig. 11 with Fig. 10. A surprising result of the present benchmark is the results from the FLUKA code seen in Fig. 12. Al-though the excess leakage estimated by 3DHZETRN in the sphere above 100 MeV is due to the straight-ahead approximation, the in-

J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38 33

Fig. 9. Neutron leakage factors for cube and sphere at 35 g/cm2 and 40 g/cm2 evaluated by the 3DHZETRN (N = 22) code.

Fig. 10. Neutron leakage factors for cube and sphere at 35 g/cm2 and 40 g/cm2 evaluated by the Geant4 MC code.

Fig. 11. Neutron leakage factors for cube and sphere at 35 g/cm2 and 40 g/cm2 evaluated by the PHITS MC code.

termediate energy excess leakage is due to the differences within the FLUKA nuclear model.

From the above results and discussion, improvements in the 3DHZETRN transport model in which scattering angular depen-dence is treated as an isotropic source lies in the following: the low energy transport requires higher order corrections to scatter-

ing, and the isotropic source contributions need to be expanded to include more complex angular factors extending up to higher energies than currently represented.

Further complications are added in solving the Boltzmann equa-tion in inhomogeneous objects. The next section examines inho-mogeneity in a tissue sphere shielded by aluminum.

34 J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38

Fig. 12. Neutron leakage factors for cube and sphere at 35 g/cm2 and 40 g/cm2 evaluated by the FLUKA MC code.

Fig. 13. Slab geometry (top) and associated cube geometry (bottom). The slab is 30 g/cm2 (30 cm) of ICRU tissue with 20 g/cm2 (7.41 cm) of aluminum above and below and infinite lateral dimensions. The cube has the same thickness of ICRU tis-sue and aluminum and has lateral dimensions of 44.82 cm. The red circles indicate detector locations at which quantities are evaluated. The green arrows indicate gen-eral directions of leakage. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4. Simple inhomogeneous geometry

In this section, simple inhomogeneous slab geometry is consid-ered by substituting a 30 cm (30 g/cm2) thick layer of ICRU tissue (ICRU, 1993) within 7.41 cm (20 g/cm2) of aluminum shielding above and below. The cube subset becomes a box of 44.82 cm on each side. Both geometries are shown in Fig. 13. The alter-nate spherical geometry will consist of an ICRU tissue equivalent sphere (30 cm diameter) surrounded by an aluminum shell with thickness 7.41 cm (20 g/cm2). As in the case of the homogeneous configurations, convergence of the neutron fluence in the slab is considered first; results are shown in Fig. 14. Particular emphasis is again given to the leakage ratio as defined in equation (6).

The E > 1 MeV isotropic neutron source for the rectangular box and shielded sphere (discussed later in this report) are shown in Fig. 15. One sees clearly that the cube neutron source is distributed along the z-axis only, as one would expect for the slab configura-tion. The abrupt change in crossing from the aluminum shield into tissue is clearly seen near the top interface. A similar transition occurs at the top of the tissue sphere as well.

There are two interesting features shown in Fig. 14. First, the convergence rate of the neutron fluence as a function of lateral dimension is much faster in the inhomogeneous geometry than in the simple homogeneous geometry examined in the previous section. Second, the agreement between 3DHZETRN and Geant4 in

Fig. 14 is improved over all energies than what was observed in the homogeneous geometry in Fig. 2. It will be seen later in this section (see Figs. 16 and 19), that all the codes considered here (3DHZETRN, Geant4, FLUKA, PHITS) agree better in the various in-homogeneous geometries (cube, sphere, slab) than they did in the simpler homogeneous geometries.

Both of these features are mainly attributed to the hydrogen content in the ICRU tissue. It can be seen in Fig. 15 that a rapid transition occurs in the isotropic neutron source very near the aluminum–tissue interface, and that the neutron source is greatly reduced and slowly varying within the tissue. The reduction in the neutron source is a result of transitioning from the neutron-rich aluminum shield to the ICRU tissue where the average number of neutrons per target nucleus is much lower. In addition, at energies below ∼100 MeV, elastic collisions between neutrons and hydro-gen in the ICRU tissue dominate neutron transport. These elastic collisions, on average, transfer half the neutron energy to the tar-get hydrogen, which in turn, very rapidly attenuates the neutron energy spectrum. And although neutron production cross sections show significant variation amongst the codes for aluminum targets, as seen in the previous section, neutron-hydrogen elastic cross sec-tions are more precisely represented in the codes either through evaluated nuclear data files (ENDF) or accurate parameterizations (Wilson et al., 1991).

The calculated neutron fluence generated in the inhomoge-neous cube derived from the 3DHZETRN and MC codes is shown in Fig. 16. Note that the 3DHZETRN tends to have good overall agreement with the MC (Geant4, PHITS, FLUKA) codes at least to the extent the MC codes agree among themselves. The neutron leakage as seen from the detector locations on both the top and bottom of the tissue phantom is shown in Fig. 17. Unlike leakage from the uniform aluminum cube, the tissue phantom of the in-homogeneous cube provides significant absorption so that neutron leakage is limited. In this case, there is reasonable agreement be-tween 3DHZETRN and the MC simulations.

Finally, consider a 30 cm diameter ICRU tissue sphere shielded by a 7.41 cm spherical shell of aluminum, as shown in Fig. 18. In this example, the solution for neutron fluence is evaluated at the top and bottom of the ICRU sphere that serves here as proxy for the astronaut (ICRU phantom, see ICRU, 1993 and Wilson et al.2014c, 2015). The neutron fluence is evaluated in the tissue and aluminum interface at the top and bottom of the tissue sphere us-ing both 3DHZETRN and MC methods (Geant4, PHITS, and FLUKA). In order to control statistical fluctuations, the MC simulations re-quire a relatively large volume over which fluences are scored. This implicitly integrates physical variations found locally near the de-tector volumes. To provide a consistent comparison, the 3DHZETRN

J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38 35

Fig. 14. Convergence test in an inhomogeneous slab composed of a 30 cm ICRU tissue slab shielded above and below by 7.41 cm of aluminum exposed to the Webber SPE spectrum. Results shown at 0 cm (left pane) and 30 cm (right pane), corresponding to the top and bottom detector locations, as shown in Fig. 13, respectively.

Fig. 15. Isotropic neutron source (E > 1 MeV) distribution in units of neutrons/(g-event) induced by the Webber SPE in the rectangular box and shielded sphere. The intensity (color) scale has been adjusted to more clearly show the shadow cast by the tissue sphere. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 16. Neutron spectra at detector locations 0 cm (left pane) and 30 cm (right pane) from top of the tissue box (thickness 30 cm) shielded by aluminum above and below (thickness 7.41 cm) exposed to the Webber SPE spectrum.

fluence is obtained as an average of point results over the same volume (a sphere of 0.25 cm radius). The neutron fluence at top and bottom of the tissue sphere are shown in Fig. 19 where rea-sonable agreement with MC results are obtained. The correspond-ing neutron leakage ratio is shown in Fig. 20.

5. Conclusions

The focus of the present study was neutron production and propagation and leakage through finite geometries. It was shown that the neutron differential fluence solution above 1 MeV in slab

36 J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38

Fig. 17. Neutron leakage from an inhomogeneous cube of aluminum/tissue/aluminum as seen at the center of the top interface (x = 0) and the lower interface (x = 30).

Fig. 18. Slab geometry (top) and associated sphere geometry (bottom). The slab is 30 g/cm2 (30 cm) of ICRU tissue with 20 g/cm2 (7.41 cm) of aluminum above and below and infinite lateral dimensions. The sphere is an ICRU tissue sphere with di-ameter 30 g/cm2 (30 cm) surrounded by 20 g/cm2 (7.41 cm) aluminum. The red circles indicate detector locations at which quantities are evaluated. The green ar-rows indicate general directions of leakage. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

geometry converges well to match the Geant4 simulation in homo-geneous materials. At lower energies, an error is introduced by the bi-directional approximation that assumes no additional diffusive losses for the propagation of the isotropic neutron component. The

losses through lateral boundaries in transport along rays that are parallel or nearly parallel to the slab faces are underestimated (as-sumed zero). As a result, the neutron leakage ratio for the single layer sphere and box is overestimated at low energies. Though this apparent error has a noticeable impact on leakage estimates, it has been shown elsewhere (Slaba et al., 2011), that neutron energies below 1 MeV make small contributions to the neutron exposure and total exposure. Still, a correction for losses along these long paths near lateral boundaries is required.

The convergence and overall code agreement for the solution in all geometries considered herein is greatly improved when a highly absorptive tissue layer is introduced. For the finite cube and sphere and semi-infinite slab, overall code agreement was found to be very good. The improved agreement was attributed to the hydroge-nous tissue, where neutron production is minimal and absorption is increased. Also, the elastic cross sections for neutron–proton collisions which dominate transport processes are relatively well known in all the codes, using either evaluated nuclear data files (ENDF) or highly accurate parameterizations (Wilson et al., 1991).

Despite the improved agreement, it was shown that above 100 MeV, the 3DHZETRN code generally underestimates the leak-age due the straight-ahead assumption and is easily seen in the neutron leakage ratio of the sphere where angular divergence is expected to play an important role. On the centerline of the cube, the straight-ahead approximation is more correct. These errors will

Fig. 19. Neutron spectra at detector locations 0 cm (left pane) and 30 cm (right pane) from top of tissue sphere (radius 15 cm) surrounded by an aluminum shell (thickness 7.41 cm) exposed from above by the Webber SPE spectrum.

J.W. Wilson et al. / Life Sciences in Space Research 7 (2015) 27–38 37

Fig. 20. Neutron leakage from an inhomogeneous sphere of aluminum/tissue at the top interface (left pane) and lower interface (right pane).

be partly corrected by treating the angular dependence in greater detail.

Acknowledgements

This work was supported by the Human Research Program un-der the Human Exploration and Operations Mission Directorate of NASA and by NASA Grant number NNX09AR20A.

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