Microstructure

This model updates microstructural material properties of the UO₂ matrix that are needed by other models, namely the lattice parameter and the theoretical density. The calculation accounts for chromium content through an empirical correlation for the lattice parameter and propagates this change to the density.

Reference

  1. Cardinaels et al., Journal of Nuclear Materials, 424 (2012) 252–260.

Purpose and outputs

The model computes and updates:

  • Lattice parameter (sciantix variable)

  • Theoretical density (sciantix variable)

The same values are also assigned to the UO₂ matrix object:

  • matrices["UO2"].setLatticeParameter(...)

  • matrices["UO2"].setTheoreticalDensity(...)

Inputs

The model uses:

  • Chromium content (sciantix variable)

  • Fuel density (sciantix variable)

  • Uranium isotopic inventories (sciantix variables): U234, U235, U236, U237, U238

and the constants:

  • avogadro_number

  • molar_mass_Oxygen

  • molar_mass_Chromium

Lattice parameter correlation

The lattice parameter is computed from a baseline UO₂ lattice parameter and an empirical chromium-dependent correction:

\[a = \left(a_0 - 1.2\times 10^{-6}\,C_{\mathrm{Cr}}\right)\times 10^{-10}\]

where:

  • \(a_0 = 5.47109\) (baseline value used in the code),

  • \(C_{\mathrm{Cr}}\) is the chromium content from Chromium content.

The result is stored in Lattice parameter.

Uranium molar mass estimate

A molar-mass-like quantity for uranium is computed from the isotopic inventory variables. In the implementation this is obtained via a conversion factor:

\[\mathrm{conv\_fact} = \rho_f\,N_A\,10\,0.8815\,100\]

where \(\rho_f\) is the fuel density and \(N_A\) is Avogadro’s number. The code then combines the isotopic inventories to produce molar_mass_Uranium. (This follows the current implementation and is kept here for traceability.)

Chromium molar fraction term

The chromium contribution is converted into a molar fraction-like term (\(C_{\mathrm{Cr, mol}}\)) used in the density expression:

\[C_{\mathrm{Cr, mol}} = C_{\mathrm{Cr}}^{\ast} \frac{M_U + 2M_O}{C_{\mathrm{Cr}}^{\ast}(M_U - M_{\mathrm{Cr}}) + M_U}\]

where:

  • \(C_{\mathrm{Cr}}^{\ast} = (\text{Chromium content})\times 10^{-6}\),

  • \(M_U\) is the computed uranium molar mass term,

  • \(M_{\mathrm{Cr}}\) is the chromium molar mass,

  • \(M_O\) is the oxygen molar mass.

Theoretical density

The theoretical density is computed using the lattice parameter and the composition-dependent mass per unit cell:

\[\rho_{\mathrm{th}} = 4\, \frac{(1-C_{\mathrm{Cr, mol}})M_U + C_{\mathrm{Cr, mol}}M_{\mathrm{Cr}} + 2M_O} {N_A\,(a\cdot 10^{2})^{3}} \times 10^{3}\]

where:

  • the factor 4 corresponds to the number of formula units per fluorite unit cell,

  • the unit conversions follow the implementation (including the \(10^2\) and \(10^3\) factors).

The result is stored in Theoretical density.

Implementation notes

This model is a property-update routine and does not solve an evolution equation. It is typically called to keep matrix properties consistent with the current fuel composition (e.g., chromium doping).