Study on Tissue Dose Field of Delayed Gamma Emitted from Fission Products under Effects of Blast Wave

HU Jiaqi¹, SHANG Peng¹, ZHU Jinhui¹, ZUO Yinghong¹, LIU Li¹, NIU Shengli¹, WANG Xuedong¹

¹State Key Laboratory of Intense Pulsed Radiation Simulation and Effect, Northwest Institute of Nuclear Technology, Xi'an 710024, China

Abstract

[Background] Major nuclear accidents (severe reactor accidents, radioactive material terrorist attacks, etc.) are typically accompanied by violent explosions, and the surrounding air of leaked fission products is inevitably disturbed by blast waves. To reasonably evaluate the delayed nuclear radiation environment formed by fission products, it is essential to investigate the transport of delayed gamma from typical fission nuclides under the influence of blast waves. [Purpose] This study aims to analyze the effects of blast waves on the transport of delayed gamma and to calculate the tissue dose field of delayed gamma emitted from fission products of ²³⁵U, ²³⁹Pu, and ²³⁸U under such conditions. [Methods] The Low Altitude Multiple Burst Revised (LAMBR) model, based on the method of images, was employed to simulate the air density distribution around the delayed gamma source under blast wave disturbance. The mass thicknesses of air were calculated using the LAMBR model, and subsequently, the attenuation law based on mass thickness was applied to study the blast wave effects on delayed gamma transport. By combining this attenuation law with the LAMBR model, a fast simulation method for delayed gamma transport under blast wave effects was proposed to calculate tissue doses. The Monte Carlo method and cavity method were also used for comparative simulations. Finally, empirical formulas for ground-level tissue doses were developed to illustrate relationships between dose fields from ²³⁵U, ²³⁹Pu, and ²³⁸U fission products. [Results] Ground-level tissue doses calculated by the fast simulation method, Monte Carlo method, and cavity method show that considering blast wave effects, the maximum relative deviation of tissue doses reaches 45% at an altitude of 1000 m for 500 t TNT equivalent. This maximum relative deviation increases from 4% to 45% as altitude increases from 100 m to 1000 m. The calculated results are broadly consistent with Monte Carlo simulations, but computational time remains under one minute—orders of magnitude shorter than Monte Carlo simulations. The delayed gamma dose rates calculated by the fast method are on average two times higher than cavity method results. Furthermore, average relative deviations between empirical formula calculations and simulations remain within 13%. [Conclusions] Results indicate that compared to calculations without blast waves, delayed gamma transport is significantly enhanced as the source altitude increases within 1000 m. The fast simulation method agrees well with Monte Carlo simulations and yields higher results than the cavity method. Moreover, proportional relationships exist between ground-level tissue doses from ²³⁵U, ²³⁹Pu, and ²³⁸U fission products.

Keywords: Delayed gamma, LAMBR model, Tissue dose rates, Mass thickness, Monte Carlo method

1. Methods

1.1 Calculation Method for Blast Wave Reflection Flow Field

When a blast wave encounters a solid wall, regular and irregular reflections occur sequentially. The flow field distribution is typically calculated rapidly using the method of images. With the solid wall as a symmetry plane, real and virtual explosions are established on both sides. The flow field parameters are determined based on one-dimensional spherical explosion free-field parameters, and the reflected flow field distribution is obtained through linear or nonlinear superposition. Previous studies have shown that the LAMBR model based on the method of images offers significant advantages in calculating physical quantities such as density and velocity in blast wave flow fields. Therefore, this study employs the LAMBR model to analyze air density distributions near the fission delayed gamma source.

Let the pressure, density, and velocity of the real explosion free field at point P be $p_1$, $\rho_1$, and $u_1$, respectively, and those of the virtual explosion free field be $p_2$, $\rho_2$, and $u_2$. According to the LAMBR model, the flow field pressure at point P is expressed as:

$$
p = p_1 + p_2 - p_0 \quad (1)
$$

where $p_0$ and $\rho_0$ represent the pressure and density of the undisturbed flow field before wave incidence. The flow field density at point P is:

$$
\rho = \rho_1 + \rho_2 - \rho_0 \quad (2)
$$

where $g$ denotes the specific heat ratio. The flow field parameters of the real and virtual explosions at any time can be obtained based on free-field propagation laws using geometric approximation methods. Calculations show that shock waves from an isotropic explosion point source cause stratified air density distribution around the source, with air of the same density approximately distributed in spherical shells centered on the explosion source. The air density is low near the source center and gradually increases outward to a peak.

1.2 Mass Thickness Equivalent Attenuation Law

Blast waves cause intense and complex changes in air density near the fission delayed gamma source, increasing the difficulty and computational cost of Monte Carlo modeling. If two calculation regions have the same medium composition and satisfy geometric similarity conditions, it can be inferred that at positions with the same mass thickness, particle transport fluence and dose are only related to the inverse square of distance. This allows particle transport in regions with different densities to be inferred from a reference density region, greatly improving computational speed.

Assume two different calculation regions $F_1$ and $F_2$, where the spatial material composition is consistent and material densities satisfy a proportional relationship $\rho_1(r) = k\rho_2(r')$, with $r$ and $r'$ being spatial coordinates. In region $F_2$, there exists a corresponding point $r'_0$ such that $r' = kr$, and the mass thickness satisfies $\int_0^r \rho_1(r)dr = \int_0^{r'} \rho_2(r')dr'$. Therefore, if the source term $S$ is a point source located at the same coordinate position in both regions, the radiation field total intensity and total dose in region $F_2$ satisfy:

$$
\frac{D_2(r')}{D_1(r)} = k^2 \quad (4)
$$

This shows that for geometrically similar calculation spaces with the same source term, the radiation field parameters at corresponding positions follow an inverse square ratio of geometric scaling.

Research on ionizing radiation protection demonstrates that under specific conditions, gamma ray transport in media follows the above conclusion. The intensity of gamma rays at distance $d$ from the source in a medium is:

$$
N(d) = N_0 B e^{-\mu d} \quad (5)
$$

where $B$ is the buildup factor representing deviation from exponential attenuation, $N_0$ is the initial gamma intensity, and $\mu$ is the mass attenuation coefficient. The mass thickness $m_d$ is the integral of medium density over transport distance. For uniform media, the buildup factor can be expressed as a function of mass thickness, meaning gamma ray intensity is the same when transport occurs in media with identical mass thickness, source, and composition.

For non-uniform media with $N$ layers, where layer 1 is closest to the source and layer $N$ is outermost, the buildup factor for an isotropic point source is:

$$
B = B_N + \sum_{n=1}^{N-1} (B_n - B_{n+1}) \quad (6)
$$

where $B_n$ is the buildup factor for the $n$-th uniform medium layer with corresponding shielding thickness. The right side represents the sum of buildup factor boundary differences from layer 1 to $N-1$ added to the $N$-th layer's buildup factor. Therefore, for media with identical composition and microscopic cross-sections, the buildup factor depends only on the sum of mass thicknesses of each layer.

In summary, shock waves from an isotropic explosion point source create stratified air density distribution around the source, with equal-density air approximately distributed in spherical shells. If the air density of each spherical shell is known, the mass thickness between source and measurement point can be calculated. Using a sphere with uniform air density as a benchmark scenario, the relationship between mass thickness and fission delayed gamma ray intensity is established through Monte Carlo particle transport simulation. Based on the mass thickness equivalent attenuation law (4), fast calculation of delayed gamma transport in air with complex density distribution can be achieved.

2. Results

2.1 Shock Wave Enhancement Effects

An open scenario consisting of ground and air was established, with measurement points located 1 m above ground level. The radiation source was positioned in the air with isotropic emission, coinciding with the blast wave source. Reactor accidents typically involve explosions from hydrogen or nuclear energy release with yields less than 100 tons (t) TNT equivalent. In radioactive material terrorist attacks, conventional explosives may disperse radioactive sources. The Tianjin Port accident involved approximately 500 t TNT equivalent. Therefore, this study sets the upper limit of explosion yield at 500 t TNT equivalent, source height at 1000 m, and maximum ground projection distance at 4000 m. ²³⁵U, ²³⁹Pu, and ²³⁸U are typical fission nuclides in nuclear reactors, with delayed gamma source energy spectra and time spectra taken from literature [12,13,14].

Fast calculation of delayed gamma dose fields for ²³⁵U, ²³⁹Pu, and ²³⁸U was implemented based on the mass thickness equivalent attenuation law. First, Monte Carlo simulation of the benchmark scenario established the relationship between gamma intensity and mass thickness. The LAMBR model then calculated air density distributions under blast wave disturbance. Finally, mass thickness between source and measurement points was computed to rapidly predict gamma dose rates and doses.

For ²³⁵U fission with source height of 100 m and blast wave yields of 100 t, 300 t, and 500 t, calculated ground-level tissue doses are shown in FIGURE:1, where solid lines include blast wave effects and dashed lines exclude them. Relative deviations are shown in FIGURE:1, defined as |(dose without blast) - (dose with blast)| / (dose with blast). The blast wave effect is minor, with maximum relative deviation of only 4%.

For source height of 500 m ([FIGURE:2]), the blast wave effect becomes more significant, with maximum relative deviation of approximately 23%. At 1000 m source height ([FIGURE:3]), the effect is substantial, with maximum relative deviation approaching 45%, and stronger blast wave yields producing more pronounced enhancement.

The ²³⁵U results demonstrate that blast wave effects become increasingly significant with source height, particularly within 1000 m ground projection distance. Blast waves alter air density near the source, affecting delayed gamma transport. For 300 t yield at 100 m height, air density distributions are shown in [FIGURE:4]. The blast wave creates an approximately spherical low-density region near the source that expands over time. At 0.1 s, the shock reaches the ground; at 0.2 s, the reflected wave propagates near the source; at 0.6 s, it reaches approximately 200 m altitude while maintaining high density at the wave front.

For 500 m source height and 300 t yield ([FIGURE:5]), the shock reaches ground at 1 s, but the reflected wave is weak at 2 s due to attenuation. At 1000 m height, the shock attenuates significantly before reaching ground, with almost no reflected wave. The low-density region enhances gamma transport, while the high-density wave front region attenuates it. At 100 m height, the low-density region enhances transport before 0.1 s, but after 0.1 s the reflected wave's high-density front attenuates transport. The net effect yields doses similar to non-blast scenarios ([FIGURE:1]). With increasing source height, ground-reflected shock waves weaken considerably and no longer hinder gamma transport, making the enhancement effect more pronounced over the total time interval.

The evolution of air mass thickness between source and ground measurement points for 500 m height and 300 t yield is shown in [FIGURE:6]. Mass thickness increases gradually until reaching a constant value, rising significantly before 3 s (corresponding to low-density region formation and rapid air backfill) and slowly stabilizing after 3 s (as air density gradually recovers). Assuming unit-intensity delayed gamma emission at all times, the calculated dose rates are shown in [FIGURE:7], where dose rates decrease with increasing mass thickness due to faster attenuation.

The time spectrum of ²³⁵U delayed gamma is shown in [FIGURE:8]. Combining this with geometric corrections from equation (4) yields dose rates in [FIGURE:9], peaking at approximately 0.5 s then decreasing steadily as air density recovers. The time integral gives the doses in FIGURE:2. ²³⁹Pu and ²³⁸U show similar time evolution trends.

2.2 Monte Carlo Simulation Results

The Monte Carlo method is widely used for particle transport problems, obtaining approximate values through extensive sampling. Blast wave effects create different air density distributions at each moment, requiring time-dependent geometric models. For ground projection distances exceeding 3000 m, deep penetration problems arise. Since gamma rays undergo multiple scatterings in air, direct simulation yields few particles reaching detectors, causing large statistical fluctuations. This study employs cell weight windows combined with density approximation iterative variance reduction methods [25] to guide particle splitting and roulette, enabling effective simulation of long-distance gamma transport. Monte Carlo modeling details will be discussed in future work.

For ²³⁵U at 100 m height and 300 t yield, ground-level delayed gamma dose rates from 0.01 s to 15 s are shown in [FIGURE:10], where solid lines represent Monte Carlo results and dashed lines represent mass thickness attenuation law calculations. The results agree well at all times, with Monte Carlo values slightly higher near the source and slightly lower at far distances, indicating that while air density distribution is not perfectly symmetric under blast waves, the mass thickness attenuation law remains well applicable.

2.3 Equivalent Cavity Method Results

Blast waves concentrate air mass primarily in narrow regions near the wave front, forming low-density spherical regions behind the front. The early equivalent cavity method approximated this by assuming a spherical vacuum cavity around the source with unchanged external air density. For ²³⁸U at 100 m height and 500 t yield, dose rates calculated by both methods are shown in [FIGURE:11]. The mass thickness method yields consistently higher results, with differences decreasing as ground range increases. The cavity method shows a peak near 0.5 s followed by gradual decline, corresponding to the cavity radius reaching maximum then decreasing to zero. The mass thickness method better accounts for complex air distribution, providing more realistic simulation of blast wave effects.

Ground-level doses calculated by both methods are compared in [FIGURE:12]. The mass thickness method produces larger doses with similar trends, converging at ground ranges above 3000 m. The equivalent cavity radii for 100 t, 300 t, and 500 t yields are shown in [FIGURE:13], peaking before 0.5 s then decreasing. The maximum cavity radius for 300 t is about 60 m, decreasing to 46 m at 2 s. However, actual low-density regions persist with radii near 100 m at 2 s ([FIGURE:5]), indicating the cavity method underestimates enhancement effects. At far distances, both methods converge, suggesting the cavity method is a reasonable approximation for remote regions.

2.4 Empirical Formula for Fission Delayed Gamma Dose Field

Gamma intensity approximately follows an exponential relationship with transport distance, with buildup factors representing deviations from pure exponential attenuation. Considering geometric attenuation, the empirical formula for the delayed gamma dose field is:

$$
D(r) = A \cdot \frac{e^{-r/\lambda}}{r^2} \cdot Q \quad (7)
$$

where $Q$ is blast wave intensity in t TNT equivalent, $r$ is slant distance from source to measurement point in meters, and $A$ and $\lambda$ are adjustable parameters.

Fitting ²³⁵U results yields $A = 11.02 \times 10^5$, $\lambda = 300.41$ m. The empirical formula reproduces numerical calculations well within 4000 m slant range, with average relative deviation of 12.67% ([FIGURE:14]). Parameters for ²³⁹Pu and ²³⁸U are $A = 9.82 \times 10^5$, $\lambda = 300.41$ and $A = 22.05 \times 10^5$, $\lambda = 300.41$, respectively, with average deviations of 12.47% and 12.61%.

The identical exponential attenuation coefficient $\lambda$ across all three nuclides arises because their delayed gamma energy spectra are very similar, allowing the same spectrum to be used at all times [11,13]. $\lambda$ represents the mean free path of gamma transport in air. The mass attenuation coefficient of 1 MeV gamma in dry air is $6.359 \times 10^{-3}$ m²/kg, giving a mean free path of about 128 m in standard atmospheric conditions. The fitted $\lambda$ of 300.41 m indicates significantly increased mean free path under blast wave effects, demonstrating transport enhancement. Parameter $A$ correlates with the delayed gamma time spectrum; after 0.001 s, ²³⁸U gamma intensity exceeds ²³⁵U, which exceeds ²³⁹Pu [12,13], reflected in the $A$ values.

[TABLE:1] summarizes the empirical formula parameters. The conversion relationships between dose fields are simple multiples: the ²³⁸U to ²³⁹Pu dose ratio is 2.25, and the ²³⁵U to ²³⁹Pu ratio is 1.12.

3. Conclusions

A fast simulation method for delayed gamma transport under blast wave effects was established based on the mass thickness equivalent attenuation law and LAMBR model. Applied to ²³⁵U, ²³⁹Pu, and ²³⁸U delayed gamma transport, the method calculates air density distributions and analyzes enhancement effects. Ground-level dose fields were computed and empirical formulas were developed, yielding conversion relationships between the three nuclides. Comparisons with Monte Carlo and cavity methods show that blast wave enhancement effects increase with source altitude, raising ground-level gamma doses by up to 45% at 1000 m height for 300 t yield. Monte Carlo simulations validate the mass thickness method, while the cavity method underestimates enhancement in near regions but provides reasonable far-field approximations. The empirical formulas accurately reproduce numerical results, with fitted mean free paths significantly larger than in uniform air, confirming transport enhancement. This method provides a reference for radioactive environment assessment in major nuclear accidents involving explosions.

References

1. Mourogov V M. Role of nuclear energy for sustainable development[J]. Progress in Nuclear Energy, 2000, 37(1–4): 19–24. DOI: 10.1016/S0149-1970(00)00018-4.
2. Liu L M, Guo H, Dai L H, et al. The role of nuclear energy in the carbon neutrality goal[J]. Progress in Nuclear Energy, 2023, 162: 104772. DOI: 10.1016/j.pnucene.2023.104772.
3. Grishanin E I. The role of chemical reactions in the Chernobyl accident[J]. Physics of Atomic Nuclei, 2010, 73(14): 2296–2300. DOI: 10.1134/S1063778810140073.
4. Fujisawa N, Liu S, Yamagata T. Numerical study on ignition and failure mechanisms of hydrogen explosion accident in Fukushima Daiichi Unit 1[J]. Engineering Failure Analysis, 2021, 124: 105388. DOI: 10.1016/j.engfailanal.2021.105388.
5. Pakhomov S A, Dubasov Y V. Estimation of explosion energy yield at Chernobyl NPP accident[J]. Pure and Applied Geophysics, 2010, 167(4): 575–580. DOI: 10.1007/s00024-009-0029-9.
6. Fu Y, Shang P, Zuo Y, et al. Monte Carlo simulation of the effect of rainfall on neutron atmospheric transport[J]. Modern Applied Physics, 2022, 13(2): 49–54, 60. DOI: 10.12061/j.issn.2095-6223.2022.020207.
7. Liu L, Zuo Y, Niu S, et al. A variance reduction method for simulating the long-distance transport of neutrons and secondary γ in high-altitude atmosphere by Monte Carlo method[J]. Modern Applied Physics, 2022, 13(1): 30–37, 59. DOI: 10.12061/j.issn.2095-6223.2022.010202.
8. Li X, Niu S, Zhu J. Calculation of space distribution of X-ray energy fluence from nuclear detonation in near space[J]. Modern Applied Physics, 2020, 11(1): 13–17. DOI: 10.12061/j.issn.2095-6223.2020.010201.
9. Liu L, Niu S, Zuo Y, et al. Monte Carlo simulation of delayed γ-rays ionizing the atmosphere based on debris motion model[J]. Nuclear Techniques, 2024, 47(11): 110501. DOI: 10.11889/j.0253-3219.2024.hjs.47.110501.
10. Liu L, Zuo Y, Niu S, et al. Application of global variance reduction methods for the calculation of γ radiation field in large space[J]. Nuclear Techniques, 2024, 47(2): 020602. DOI: 10.11889/j.0253-3219.2021.hjs.47.020602.
11. Editorial Committee of Military Training Textbooks of the General Armament Department of PLA. Introduction to the physics of nuclear explosion[M]. Beijing: National Defense Industry Press, 2003.
12. Fisher P C, Engle L B. Delayed gammas from fast-neutron fission of Th²³², U²³³, U²³⁵, U²³⁸, and Pu²³⁹[J]. Physical Review, 1964, 134(4B): B796–B816. DOI: 10.1103/physrev.134.b796.
13. Walton R B, Sund R E. Delayed γRays between 2 and 80 μsec after U²³⁵(n, f) and Pu²³⁹(n, f)[J]. Physical Review, 1969, 178(4): 1894–1903. DOI: 10.1103/physrev.178.1894.
14. Popeko L A, Val’skii G V, Kaminker D M, et al. Delayed gamma rays from U²³⁵ fission[J]. Soviet Atomic Energy, 1965, 19(2): 1082–1085. DOI: 10.1007/BF01126435.
15. Jia L, Wang S, Tian Z. A theoretical method for the calculation of flow field behind blast reflected waves[J]. Explosion and Shock Waves, 2019, 39(6): 94–102. DOI: 10.11883/bzycj-2018-0167.
16. Zhu J, Li X, Zuo Y, et al. Similarity theory of near-ground gamma-ray transport under different atmospheric density[J]. Modern Applied Physics, 2024, 15(6): 37–43. DOI: 10.12061/j.issn.2095-6223.202403003.
17. Gusev H F. Protection against ionizing radiation (Vol. 1)[M]. Beijing: Atomic Energy Press, 1998.
18. Chan P C, Klein H H. A study of blast effects inside an enclosure[J]. Journal of Fluids Engineering, 1994, 116(3): 450–455. DOI: 10.1115/1.2910297.
19. Kong B, Lee K, Lee S, et al. Indoor propagation and assessment of blast waves from weapons using the alternative image theory[J]. Shock Waves, 2016, 26(2): 75–85. DOI: 10.1007/s00193-015-0581-4.
20. Wu Z, Guo J, Yao X, et al. Analysis of explosion in enclosure based on improved method of images[J]. Shock Waves, 2017, 27(2): 237–245. DOI: 10.1007/s00193-016-0655-y.
21. Xie Z, Deng L. Numerical calculation method of neutron transport theory[M]. 2nd ed. Xi'an: Xi'an Jiaotong University Press, 2022.
22. International Nuclear Safety Advisory Group. INSAG-7 the Chernobyl accident: updating of INSAG-1[R]. INSAG-7, 1992.
23. U. S. Atomic Energy Commission Idaho Operations Office. IDO report on the nuclear incident at the SL-1 reactor[R]. IDO-19302, 1962.
24. Fang Q, Yang S, Chen L, et al. Analysis on the building damage, personnel casualties and blast energy of the "8.12" explosion in Tianjin port[J]. China Civil Engineering Journal, 2017, 50(3): 12-18. DOI: 10.15951/j.tmgcxb.2017.03.002.
25. Zuo Y, Niu S, Shang P, et al. Weight window variance reduction method for simulating long distance γ-ray transport[J]. Modern Applied Physics, 2020, 11(1): 37–42. DOI: 10.12061/j.issn.2095-6223.2020.010205.
26. Xia Y. Advanced course of ionizing radiation protection[M]. Harbin: Harbin Engineering University Press, 2010.