A Deep Dive into MCNP's Type 3 Mesh Tally: Simulating Energy Deposition

 Following our exploration of various mesh tallies in MCNP, this article provides a specialized examination of the Type 3 mesh tally for energy deposition. Unlike the standard F4: tally, which measures particle flux, this tally (*F4: used with an MESH card and the TOTAL keyword) calculates the energy deposited per unit volume by all particles. This quantity, reported in units of MeV/cm³, is essential for the direct calculation of radiation-induced heating, absorbed dose, and power distribution in complex models. This article will detail the physical principles, precise definition with all specialized keywords, key differences from other tallies, and its applications in nuclear engineering and medicine.

1. Introduction to the Energy Deposition Mesh Tally (Type 3)

The Type 3 energy deposition mesh tally is a powerful tool for calculating the amount of energy stored in a material per unit volume due to particle interactions. This energy is ultimately converted into heat. This tally operates independently of the geometric model and reports results on a user-defined Cartesian, cylindrical, or spherical grid.

Physical Principles: Energy deposition can arise from various sources:

• The slowing down of charged particles (e.g., protons, electrons).

• The kinetic energy of recoil nuclei resulting from neutron interactions.

• Energy from particles that are produced but not tracked due to computational reasons (e.g., energy cut-offs), where their energy is assumed to be deposited locally (e.g., alpha and lithium particles from (n,α) and (n,Li) reactions at low energies).

• The use of response functions to model specific effects.

2. Definition and Critical Keywords

Defining this tally involves two main sections: the Tally Card and the MESH Card, each with its own specific details and keywords.

2.1. Tally Card

To activate this mode, an asterisk (*) is used before F4.

*F4:N  Mesh
FC4 Energy Deposition in MeV/cm^3 per source particle
FM4 1.0

• *F4:N: The * prefix tells MCNP this is an energy deposition tally, not a simple flux tally.

• Mesh: The keyword linking the tally to a mesh definition.

2.2. MESH Card and Specialized Keywords

The general format of the MESH card for this tally is as follows:

MESH4 geom=xyz origin= x0 y0 z0 imesh= i iints jmesh= j jints kmesh= k kints
     TOTAL IEVT SECOL TLEST DELCT MFACT MTERO TRANS

Explanation of Critical Keywords (based on MCNP Manual Table 5-85):

Crucial Normalization Note: Since the mesh is independent of the geometry, a single mesh cell may contain several materials with different densities. Therefore, the output is normalized per unit volume (MeV/cm³ per source particle), not per unit mass (MeV/gram per source particle) as is common with standard volumetric tallies. To convert to mass units, the user must manually apply the density of the materials within each cell during post-processing.

3. Key Differences from Other Tallies

• Comparison with +F6 (Heating Tally): The +F6 tally also reports deposited energy, but it does so for a single cell within the geometric model. The Type 3 mesh tally performs the same calculation but for thousands of virtual cells (the mesh grid), providing a complete 3D map of the heating distribution.

• Comparison with F4: (Standard Flux Tally): The F4: tally reports particle flux (particles per cm²). To obtain heating from flux, a response function (e.g., flux-to-dose conversion coefficients) must be applied. The energy deposition mesh tally performs this conversion internally, considering the physics of particles and their interactions, and directly provides the final result.

4. Practical Applications

1. Power Distribution and Heating in Reactor Cores: Designing reactor cooling systems requires the most accurate possible distribution of heat production in fuel rods, reflectors, and other components.

2. Radiation Dosimetry in Medicine (Radiotherapy): Precise calculation of absorbed dose (in Gray, equivalent to J/kg) in patient tissues. The output (MeV/cm³) must be multiplied by appropriate conversion factors (e.g., converting cm³ to kg using tissue density).

3. Analysis of Radiation Effects on Electronics (Radiation Hardening): Calculating energy deposited in semiconductor chips to predict and mitigate errors caused by ionizing radiation.

4. Estimating Radiation Damage to Materials: Using the SECOL keyword, one can estimate the energy transferred to atoms of a material, leading to atomic displacement.

5. Practical Example: Dose Calculation Around a Cobalt-60 Gamma Source

Objective: Simulate the absorbed dose distribution around a point source of ⁶⁰Co in air.

Sample Input File (abbreviated):

C Example: Energy Deposition Mesh Tally for a Co-60 source
c --- Cell Cards ---
1 0 -1 imp:p=1  $ Air cell
2 0 1 imp:p=0   $ Outside world

c --- Surface Cards ---
1 SO 50  $ A large sphere of air (50 cm radius)

c --- Data Cards ---
MODE P
c --- Material and Physics Cards (omitted for brevity) ---
c --- Source Definition ---
SDEF POS=0 0 0 ERG=D2 PAR=2
SP2 -21 0.5 1.3325 0.5 1.1732  $ Co-60 energy spectrum (approx.)
c --- Tally Definition ---
*F4:P Mesh
FC4 Energy Deposition in MeV/cm^3 from Co-60 source
c --- Mesh Definition ---
MESH4 geom=xyz origin=-10 -10 -10
      imesh=-10 10  iints=20
      jmesh=-10 10  jints=20
      kmesh=-10 10  kints=20
      TOTAL    $ Score total energy deposition from all sources
c ---
NPS 1e6

Results Analysis:

• Output is stored in the meshtal file.

• Values in each cell are reported as MeV/cm³ per source particle.

• To convert to Absorbed Dose (Gray):

  1. Multiply the results by the number of source particles (e.g., for NPS 1e6, multiply by 1e6) to get MeV/cm³.

  2. Convert the value to Joules (1 MeV = 1.602e-13 J).

  3. Calculate the cell volume (e.g., if the mesh is defined from -10 to +10 with 20 intervals, the cell width is 1 cm, thus the cell volume is 1 cm³).

  4. Divide the energy in the cell (J) by the mass in the cell. If the cell is filled with air (density ~0.001225 g/cm³), the mass is Volume * Density = 1 * 0.001225 = 0.001225 g = 1.225e-6 kg.

  5. The final number is the dose in Gray (J/kg).

This post-processing can be automated using scripting (e.g., in Python).

6. Conclusion

MCNP's Type 3 energy deposition mesh tally, despite its complexity in normalization and need for post-processing, is one of the most valuable tools for obtaining the most direct and accurate results in simulations related to radiation-induced heat transfer, dosimetry, and material damage analysis. A deep understanding of its keywords and its differences from other tallies is essential for any advanced MCNP user.

Frequently Asked Questions (FAQ)

Q1: What is the fundamental difference between F4: and *F4:?
A: F4: is a flux tally (particles/cm²). *F4: is an energy deposition tally (MeV/cm³). The first tells you how many particles pass through; the second tells you how much energy they leave behind.

Q2: Why is my energy deposition result in a mesh cell sometimes zero, even though it's near the source?
A: This can happen if the mesh cell is located inside a void or a very low-density material (like air). The energy deposition is normalized per unit volume. Even if a significant amount of energy is deposited in terms of Joules, when divided by the volume of the large cell, the MeV/cm³ value can become very small. Check your geometry and mesh cell sizing.

Q3: How do I convert the output from MeV/cm³ to the standard radiation dose unit, Gray?
A: As shown in the example, you must:
1. Multiply by the number of source particles (NPS).
2. Convert MeV to Joules (1.602e-13 J/MeV).
3. Determine the mass in the mesh cell (Volume * Density).
4. Divide the energy (J) by the mass (kg) to get Gray.
This conversion is a manual post-processing step.

Q4: When should I use a +F6 heating tally in a cell versus a *F4 mesh tally?
A: Use +F6 when you need the total energy deposited in a specific, well-defined geometric component (e.g., total heat in a fuel assembly). Use *F4 with a mesh when you need a detailed 3D spatial map of the energy deposition (e.g., dose distribution across an organ in medical physics or heating gradient across a shield).

Q5: The TOTAL keyword is listed as default. Do I always need to write it on the card?
A: It is considered good practice to explicitly include it to ensure clarity and intentionality in your input deck. MCNP will assume it by default if no other energy deposition keyword (IEVT, SECOL, etc.) is present.