Serpent 2: Advanced Neutron Transport Code

• Neutron Transport Code Serpent 2 is a Monte Carlo simulation tool that provides high-fidelity reactor core modeling with emphasis on accuracy and computational efficiency.
• It employs advanced constructive solid geometry techniques to accurately represent complex reactor configurations, such as the TRIGA Mark II core.
• The code integrates detailed material definitions and temperature-dependent cross-section treatments to deliver precise criticality and reactivity calculations, benchmarked against MCNP.

Serpent 2 is a Monte Carlo neutron transport code designed for high-fidelity reactor core modeling, featuring advanced geometry handling, robust material definition, and integrated support for temperature-dependent cross-section treatment. It has been applied to the simulation and validation of complex reactor systems, such as the TRIGA Mark II reactor at Pavia, with detailed three-dimensional geometries and comprehensive treatment of operational scenarios encompassing both low and full power conditions. Serpent 2 is capable of accurate criticality calculations, supports sophisticated control-rod modeling and calibration, and enables direct benchmarking against established neutronics codes such as MCNP, exhibiting both accuracy and computational efficiency (Castagna et al., 2017).

1. Three-Dimensional Geometry Representation

Serpent 2 employs a flexible geometry modeling framework with its input language centered on constructive solid geometry (CSG). For the TRIGA Mark II reactor, the geometric origin is positioned at the active core’s center. The primary core is modeled as a right circular cylinder (radius $R=0.223$ m, half-height $H=0.324$ m), bounded axially by two aluminum grids ($z = \pm 0.324$ m), declared as:

surf 1 cz 22.3 surf 2 pz 32.4 surf 3 pz -32.4

The “circular cluster” array structure natively supports the placement of 91 core sites (fuel, dummies, control rods, irradiation thimbles, source channels) in concentric rings. For example:

infcar 10 cluster 91 5 10.0 ...

uses the parameters: total locations, ring count, and pitch. Positioning of each element utilizes rotation ($\theta_i$) and radial offset ($r_j$) transformations. Fuel elements and absorbers are defined through cell cards such as:

cell 100 fuel1 -1.0 -1 2 -3 (trans 0 0 z1) (rot 0 0 θ1)

Movable control rods (Ag–In–Cd absorber in Al) are realized as cylinders with user-defined translations representing insertion steps ($\Delta z = \text{step} \times 0.054$ cm). The 30 cm–thick graphite reflector, as well as the surrounding water pool (radius 150 cm, height 300 cm), are constructed as additional cylinders and cells.

2. Materials, Cross Sections, and Temperature Treatment

Material specification in Serpent 2 is granular, allowing precise assignment of physical and isotopic properties to each system component. For TRIGA core fuel, each element contains zirconium hydride (ZrH) and uranium metal at 8 wt% (20% $^{235}$U enrichment). Each fuel rod is individually characterized to replicate historical U-mass data (from 1965):

mat fuel_A01 -5.85 09040.09c 0.325 % Zr 1001.09c 0.181 % H 92235.09c 0.080 % U-235 92238.09c 0.615 % U-238

Cladding, water, graphite, and control rods use similarly detailed cards; for instance, Al-6061 at 2.70 g/cm³ (cladding), H$_2$O at 0.998 g/cm³ (pool), graphite at 1.70 g/cm³, and Ag-In-Cd control rods at 7.8 g/cm³. Cross-section treatment utilizes JEFF-3.1 data for most reactions; thermal scattering for ZrH and water employs ENDF/B-VII.1 S$(\alpha,\beta)$ data. Material temperature assignments are scenario-dependent: low-power runs use $T=300$ K across all materials; full-power conditions require individualized Doppler-broadened cross sections—generated at variable $T$ with MAKXSF and linked to spatial regions in the fuel. In full-power scenarios, the core is divided into five axial layers and five concentric radial rings, with local temperature $T_{i,j}$ (from coupled neutronics–thermal-hydraulics analysis) directly assigned.

3. Serpent 2 Input Syntax and Key Directives

The Serpent 2.1.19 input for a comprehensive TRIGA model consists of initialization, geometry, materials, and calculation modes. Core elements include:

set acelib JEFF-3.1 set b1 0 set thermal 1 mode k ksrc 1e5 pop 4e5 500 50 tol 1e-5

“mode k” specifies eigenvalue search mode; “ksrc” and “pop” define initial source and population/sample cycle configuration (500 cycles × $4 \times 10^5$ histories, skipping the first 50 cycles for source convergence). The geometry and materials are set as illustrated in previous sections. Energy-integrated detectors can be defined (e.g., “det eflux 4”) for flux tallying, though not obligatory for $k_\text{eff}$ determination.

4. Criticality and Reactivity Calculation Algorithms

Serpent 2 solves the $k$-eigenvalue problem with iterative source convergence. Beginning with a trial source, each cycle simulates $N$ neutrons, advancing the fission source and estimating $k_i$. The mean $k_\text{eff}$ is computed over active cycles:

• Statistical uncertainty: $$\sigma(k_\text{eff}) \approx \sqrt{\frac{\mathrm{Var}(k_i)}{N_\text{eff}}}$$, $N_\text{eff} \simeq 500$
• Reactivity: $\rho = \frac{k_\text{eff} - 1}{k_\text{eff}}$
• Reactivity in dollars: $\rho(\$) = \frac{\rho}{\beta_\text{eff}}, \ \beta_\text{eff} = 0.0073$- Uncertainty propagation:$\sigma(\rho) \simeq \sigma(k_\text{eff}) / k_\text{eff}^2 / 0.0073$$</li> </ul> <p>These metrics are critical for validation and for comparison with benchmark data and alternative codes.</p> <h2 class='paper-heading' id='modeling-power-effects-and-temperature-feedback'>5. Modeling Power Effects and Temperature Feedback</h2> <p>Under full-power operation (250 kW), the TRIGA core exhibits significant thermal feedback, manifested in fuel temperature increases and altered neutron moderation via ZrH. Serpent 2 models this by subdividing the core into 5 axial × 5 radial regions ($$z_1 \ldots z_5$,$B\ldots F$$), matching local fuel temperatures from thermal-hydraulics calculations. Each region is assigned a unique material definition with the corresponding temperature, e.g.:</p> <p>$$

  mat fuel_B2_layer3 -5.85 09040.09c 0.325 1001.09c 0.181 92235.09c 0.080 92238.09c 0.615 htemp fuel_B2_layer3 430

  </p> <p>Serpent’s internal logic selects T-indexed cross-sections in 10 K intervals (generated via MAKXSF), providing accurate Doppler and S(\alpha,\beta)$$treatment. Water and graphite structures remain at baseline 300 K in these models.</p> <h2 class='paper-heading' id='control-rod-modeling-and-calibration'>6. Control Rod Modeling and Calibration</h2> <p>Control rods, constructed as Ag–In–Cd alloy cylinders within aluminum tubes, are modeled as movable absorbers. The spatial positioning uses direct translation via z-axis displacement according to the mechanical step calibration (one step: 0.054 cm). For the Shim rod:</p> <p>$$

  surf 10 cz 1.5 cell 300 shim_in -7.80 10 2 -3 (trans 0 0 z_shim)

  </p> <p>where z_\text{shim} = 32.4 - (\text{step} \times 0.054)$$. Calibration proceeds by running simulations at baseline ($$k_\text{eff} \approx 1$$), then for incremental rod movements, and evaluating reactivity shifts:</p> <p>$$\Delta\rho = \rho(\text{step}+\Delta) - \rho(\text{step})$$</p> <p>These differential worths are compared to experimental in-hour data for validation. Simulation reproduces experimental rod calibration curves within 5–10$$\sim0.1$\$ for the Transient rod).

  7. Benchmarking, Validation, and Code Comparison

  A rigorous benchmark suite was executed against archival TRIGA Mark II data and MCNP models:

  • Low-power benchmarks: 26 critical configurations at $T=300$ K. For each configuration, $k_\text{eff}$ and $\rho(\$)$$were computed. The mean offset across all cases was$$(0.08 \pm 0.02)\$, with standard deviation 0.10\$, which is well within the documented systematic uncertainty of 0.26\$ from MCNP model propagation. - **Control rod calibration:** Stepped simulations in 10–20 increments captured differential and integral worth for all rods, yielding 5–10% agreement with experiment (systematic offset $\sim$0.1\$ for Transient rod). - **Full-power operation:** Four critical configurations at 250 kW (5×5 temperature field) yielded average reactivity $(−0.09 \pm 0.04)\$ versus experimental reactivity loss $(1.36 \pm 0.06)\$$ when transitioning from low to high power.
  • Comparison with MCNP: With geometry, materials, and cross-section libraries matched, Serpent 2 systematically gives $k_\text{eff}$ values $\sim$0.06\$higher than MCNP in low-power, but both results fall within combined uncertainties. Full-power agreement is similarly robust. Notably, Serpent’s Shannon-entropy convergence monitoring and automated interpolation of Doppler/S$(\alpha,\beta)$ cross sections streamline the workflow and reduce computation times by a factor of 2–3 on equivalent hardware (Castagna et al., 2017).

  In summary, Serpent 2 delivers $\sim$0.1\$reactivity accuracy across low and full power, matches MCNP within$\sim$0.06\$, and exhibits workflow and runtime efficiencies suitable for advanced performance benchmarking and integration with multiphysics applications.