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Bosch-Hale Methods

These methods are still kept in physics_functions.py but outside the FusionReactionRate class.

Bosch-Hale Constants | BoschHaleConstants

The BoschHaleConstants class is a data structure designed to hold the constants required for the Bosch-Hale calculation for a given fusion reaction. The values for each of the given reactions are given in the original paper1.

Attributes

  • bg (float): Represents the Gamow energy parameter.
  • mrc2 (float): Represents the reduced mass energy term.
  • cc1 (float): Coefficient for the first term in the Bosch-Hale polynomial.
  • cc2 (float): Coefficient for the second term in the Bosch-Hale polynomial.
  • cc3 (float): Coefficient for the third term in the Bosch-Hale polynomial.
  • cc4 (float): Coefficient for the fourth term in the Bosch-Hale polynomial.
  • cc5 (float): Coefficient for the fifth term in the Bosch-Hale polynomial.
  • cc6 (float): Coefficient for the sixth term in the Bosch-Hale polynomial.
  • cc7 (float): Coefficient for the seventh term in the Bosch-Hale polynomial.

Volumetric Fusion Rate | bosch_hale_reactivity()

This function calcualtes the relative velocity fusion reactivity \langle \sigma v \rangle for each point in the plasma profile based on the temperature.

Input Variable Variable Name
Array of temperature values for the plasma profile temperature_profile
Bosch-Hale constants for the specific reaction reaction_constants
\theta = \frac{\text{T}}{\left[1-\frac{\text{T(C2+T(C4+TC6))}}{1+\text{T(C3+T(C5+TC7))}} \right]}
\xi = \left(\frac{\text{B}_\text{G}^2}{4\theta}\right)^{\frac{1}{3}}
\langle \sigma v \rangle = \text{C1} \times \theta \times \sqrt{\frac{\xi}{m_{\text{r}}\text{c}^2\text{T}^3}} \times e^{-3\xi}

This will output a numpy array for of the relative velocity fusion reactivity \langle \sigma v \rangle for each point in the temperature profile in units of [\text{m}^3\text{s}^{-1}] After calculation each value is multiplied by 10^{-6} as the original Bosch-Hale calculation1 give the output in [\text{cm}^3\text{s}^{-1}]

Fusion Rate Integral | fusion_rate_integral()

Input Variable Variable Name
PlasmaProfile object plasma_profile
Bosch-Hale constants for the specific reaction reaction_constants

This function calculates the integrand for the fusion power integration by evaluating the number of fusion reactions per unit volume per particle volume density [\text{m}^3\text{s}^{-1}]. It scales the ion temperature profile by the ratio of the volume-averaged ion to electron temperature and normalizes the density profile by the volume-averaged density. The resulting integrand is used to compute the volume-averaged fusion reaction rate, which can be scaled with the volume-averaged ion density.

  1. Scale Ion Temperature Profile:

    • Scale the ion temperature profile by the ratio of the volume-averaged ion to electron temperature.
    \mathtt{ion\_temperature\_profile} = \frac{\langle T_{\text{i}} \rangle}{\langle T_{\text{e}} \rangle} \\ \times \mathtt{plasma\_profile.teprofile.profile\_y}

  2. Calculate Fusion Reactivity:

    • Calculate the number of fusion reactions per unit volume per particle volume density using the bosch_hale_reactivity function.
    \langle \sigma v \rangle = \mathtt{bosch\_hale\_reactivity( \\ ion\_temperature\_profile, reaction\_constants)}

  3. Normalize Density Profile:

    • Normalize the density profile by the volume-averaged density.
    \mathtt{density\_profile\_normalised} = \frac{1}{\langle n_{\text{i}} \rangle} \\ \times \mathtt{plasma\_profile.neprofile.profile\_y}

  4. Compute and return the Fusion Integral:

    • Calculate the volume-averaged fusion reaction integral.
    \mathtt{fusion\_integral} =2 \int \langle \sigma v \rangle \times \\ \mathtt{plasma\_profile.teprofile.profile\_x} \times \mathtt{density\_profile\_normalised}^2

    The above is returned.

  1. H.-S. Bosch and G. M. Hale, “Improved formulas for fusion cross-sections and thermal reactivities,”Nuclear Fusion, vol. 32, no. 4, pp. 611–631, Apr. 1992,doi: https://doi.org/10.1088/0029-5515/32/4/i07. ↩↩