Thermonuclear Burn

The tnburn package models thermonuclear (fusion) reactions. It evaluates temperature-dependent reaction rates for a user-specified network of two-body reactions, evolves the isotopic composition, and couples the released energy back into the fluid. Reactivities and reaction energetics are supplied through tabulated nuclear data.

Governing Equations

Burn in RIOT is per-material: reactions take place among the isotopes of a single material \(m\), driven by that material’s own isotope densities and phase/volume fractions. The isotope inventory is carried per material (the sparse field ccmat::iso), and the reactions modify per-material cell-volume-averaged densities that then aggregate to the bulk density as in Section Per-Material and Bulk Quantities. The released energy, by contrast, is deposited directly into the bulk energy equation.

Reaction Rate

Within material \(m\), for a two-body reaction \(r\) consuming reactant isotopes \(1\) and \(2\), the volumetric reaction rate is

\[R_r = \frac{\bar\rho_1}{m_1}\,\frac{\bar\rho_2}{m_2}\, \langle\sigma v\rangle_r(T)\,\frac{f_m^2}{f_{\text{vol},m}} ,\]

where \(\bar\rho_1,\bar\rho_2\) are the reactant cell-volume-averaged densities within that material, \(m_1,m_2\) their atomic masses, \(\langle\sigma v\rangle_r(T)\) the temperature-dependent reactivity, \(f_m\) the phase fraction, and \(f_{\text{vol},m}\) the material volume fraction (the \(1/f_{\text{vol},m}\) factor converts cell-volume-averaged densities to physical, material-intrinsic number densities). The reactivity \(\langle\sigma v\rangle_r(T)\) is tabulated against \(\log T\) and interpolated at run time; reactions are gated on the volume fraction exceeding vol_frac_thresh.

Isotopic Composition

Each participating per-material isotope density is sourced by every reaction it takes part in. In flux-conservation form, with \(\nu_{ir}\) the (signed) stoichiometric coefficient of isotope \(i\) in reaction \(r\) (negative for reactants, positive for products),

\[\frac{\partial \bar\rho_i}{\partial t} + \nabla\!\cdot\!\left(\bar\rho_i\vec{v}\right) = \sum_r \nu_{ir}\,m_i\,R_r .\]

Summing the isotope sources within a material gives the source on that material’s cell-volume-averaged density \(\bar\rho_m\), and the bulk density follows by the cell-averaged sum \(\rho=\sum_m\bar\rho_m\); the per-reaction net \(\sum_i \nu_{ir}m_i\) reflects the small mass defect converted to energy.

Energy Release

The reaction energetics enter the total-energy equation as a source that removes the reactant input energy and deposits the product kinetic energies (both tabulated against temperature),

\[\frac{\partial E}{\partial t} + \nabla\!\cdot\!\left[\left(E + p\right)\vec{v}\right] = \sum_r \left[ -R_r\,\varepsilon^{\text{in}}_r(T) + \sum_p R_r\,\varepsilon^{\text{out}}_{r,p}(T)\,\mu_{r,p} \right],\]

where \(\varepsilon^{\text{in}}_r\) is the reactant input energy, \(\varepsilon^{\text{out}}_{r,p}\) the energy carried by product \(p\), and \(\mu_{r,p}\) its multiplicity. A product may be flagged to escape the problem rather than deposit locally (see deposit_locally_), in which case its mass and energy are removed. The mass sources above additionally induce momentum and kinetic-energy sources \(\vec{v}\,(\partial_t\rho)_{\text{TN}}\) and \(\tfrac{1}{2}|\vec{v}|^2(\partial_t\rho)_{\text{TN}}\) so that the burned mass carries the local velocity.

Reaction Count

A per-reaction count density \(Q_r\) is advected as a conserved quantity sourced by the reaction rate,

\[\frac{\partial Q_r}{\partial t} + \nabla\!\cdot\!\left(Q_r\vec{v}\right) = R_r ,\]

providing an integrated diagnostic of the total number of reactions.

Input Parameters

Thermonuclear burn is enabled with the tn toggle in the <physics> block (Section Enabling Physics: the <physics> Block). Sparse physics (Chapter Sparse Physics) is not supported with burn, so sparse_physics must be set false.

Parameters in the `<physics>` block relevant to burn.
Parameter	Type	Default	Description
tn	bool	`false`	Enable thermonuclear burn.
sparse_physics	bool	`true`	Not supported with burn; set `false`.

The reaction network is specified in the <tnburn> block by contiguous, zero-based reaction\(K\) entries in NDI reaction notation. Products may be individually flagged to deposit locally or escape.

Parameters in the `<tnburn>` block.
Parameter	Type	Default	Description
reaction\(K\)	string	—	The \(K\)th reaction, e.g. `d+t->n+a`; numbered from `0`.
deposit_locally_\(Z\)	bool	`true`	Deposit product with ZAID \(Z\) locally; if `false` it escapes the problem.

The nuclear data table (isotope masses, charges, reactivities, and energetics) is selected in the <isotope_data> block:

Parameters in the `<isotope_data>` block.
Parameter	Type	Default	Description
filename	string	`isotope_data.hdf5`	HDF5 nuclear-data file (produced by the `ndi2spiner` tool).

Each burning material lists its isotopes by ZAID in its <material> block via isotope\(K\) and initial mass fractions isotope\(K\)_mfrac (see Chapter Materials and Equations of State).

Registered Fields

The burn package registers the per-reaction count fields in the table below; the isotope densities it evolves (ccmat::iso) are registered by the materials package when isotopes are present. All are sparse and carry the burn flag.

Fields registered by the thermonuclear burn package.
Field	Symbol	Components	Metadata / description
ccmat::tn_reaction_density	\(Q_r\)	\(N_{\text{rxn}}\)	Cell, Independent, Intensive, Conserved, Sparse, FillGhost, Advected, WithFluxes, BurnFlag; per-reaction count density.
cm::tn_specific_reactions	—	\(N_{\text{rxn}}\)	Cell, Derived, OneCopy, Sparse, BurnFlag; specific reaction rate.
ccmat::iso	\(\bar\rho_i\)	\(N_{\text{iso}}\)	Cell, Independent, Intensive, Conserved, Sparse, FillGhost, WithFluxes, Advected, BurnFlag; isotope partial densities (via materials).

Example

A deuterium–tritium material burning via the DT reaction, with neutrons removed from the problem:

riot.input("physics", tn=True, sparse_physics=False)
riot.input("material0", label="DT",
           eos_type="IdealGas", Gamma=1.4, Cv=3.0e12,
           isotope0=1002,   # deuterium
           isotope1=1003,   # tritium
           isotope2=2004,   # helium-4 (alpha)
           isotope3=1)      # neutron
riot.input("tnburn", reaction0="d+t->n+a",
           deposit_locally_1=False)  # neutrons escape
riot.input("isotope_data", filename="isotope_data.hdf5")