UWFDM-1268 Analysis of an Improved Fusion Reaction Rate Model for Use in Fusion Plasma Simulations

An analysis of reaction rate models from the NRL Plasma Formulary, BUCKY 1-D radiation hydrodynamics code and DRACO 2-D radiation hydrodynamics code are compared to the reaction rate model developed by H. S. Bosch and G. M. Hale. The reaction rates for T(d,n)He, D(d,n)He, D(d,p)T and He(D,p)He fusion reactions were analyzed. A timing analysis was performed to compare computation times of the BUCKY and Bosch-Hale models. It was found that the Bosch-Hale reaction rate equations gave more accurate results at low plasma energies while increasing computational cost by 1% over the reaction rate equation used in BUCKY. Introduction Most modern computer simulations of fusion reaction rates utilize fitting functions based on reaction rates calculated from data that was published almost thirty years ago [1]. During the course of the last thirty years, improved experimental techniques were developed that allowed for the collection of more accurate data at low plasma temperatures [2]. This paper analyzes the new reaction rate model proposed by H. S. Bosch and G. M. Hale [2] and compares their R-matrix reaction rate model with the published values found in the Naval Research Laboratory (NRL) Plasma Formulary [3], the BUCKY 1-D radiation hydrodynamics code developed at the University of Wisconsin – Madison [4] and the DRACO 2-D radiation hydrodynamics code developed at the Laboratory for Laser Energetics (LLE) [5]. NRL Plasma Formulary Reactivity In order to provide a baseline comparison, the reaction rate data for the T(d,n)He, D(d,n)He + D(d,p)T (DDtotal), and He(D,p)He fusion reactions from the NRL Plasma Formulary [3] is included in Table 1. This published set of values is based on data from Duane [6]. Temperature (keV) T(d,n)He (cm/s) DDtotal (cm/s) He(D,p)He (cm/s) 1 5.50E-21 1.50E-22 1.00E-26 2 2.60E-19 5.40E-21 1.40E-23 5 1.30E-17 1.80E-19 6.70E-21 10 1.10E-16 1.20E-18 2.30E-19 20 4.20E-16 5.20E-18 3.80E-18 50 8.70E-16 2.10E-17 5.40E-17 100 8.50E-16 4.50E-17 1.60E-16 Table 1. NRL Plasma Formulary Reaction Rate Data DRACO Reactivity DRACO is a two-dimensional radiation hydrodynamics code developed by LLE to simulate experiments performed on the OMEGA laser [5]. The reaction rate subroutine uses tabulated data from Hively (1977) [1]. Table 2 contains a subset of the tabulated data found in the DRACO reaction rate subroutine. BUCKY Reactivity BUCKY is a one-dimensional radiation hydrodynamics code developed by the University of Wisconsin that models high energy density fusion plasmas [4]. The fusion reaction rate subroutine uses an exponential data fitting function to generate fusion reactivities as a function of plasma thermal energy, σv = exp A1 T r + A2 + A3T + A4T 2 + A5T 3 + A6T 4     . (1) The coefficients for equation (1) as given in the BUCKY source code are given in Table 3. Table 4 shows the fusion reaction rates resulting from this equation for each of the fusion reaction species. Temperature (keV) T(d,n)He (cm/s) D(d,n)He (cm/s) D(d,p)T (cm/s) DDtotal (cm/s) He(D,p)He (cm/s) 1 5.48E-21 6.92E-23 8.30E-23 1.52E-22 3.02E-26 2 2.62E-19 2.60E-21 2.82E-21 5.42E-21 1.42E-23 5 1.30E-17 8.94E-20 8.78E-20 1.77E-19 6.65E-21 10 1.08E-16 6.26E-19 5.81E-19 1.21E-18 2.28E-19 20 4.28E-16 2.73E-18 2.43E-18 5.16E-18 3.79E-18 50 8.64E-16 1.11E-17 9.66E-18 2.08E-17 5.46E-17 100 8.50E-16 2.51E-17 2.13E-17 4.64E-17 1.60E-16 Table 2. DRACO Reaction Rate Data. T(d,n)He DDtotal He(D,p)He A1 -2.1377692E+01 -1.5511891E+01 -2.7764468E+01 A2 -2.5204050E+01 -3.5318711E+01 -3.1023898E+01 A3 -7.1013427E-02 1.2904737E-02 2.7889999E-02 A4 1.9375450E-04 2.6797766E-04 -5.5321633E-04 A5 4.9246592E-06 -2.9198658E-06 3.0293927E-06 A6 -3.9836572E-08 1.2748415E-08 -2.5233325E-08 r 0.2935 0.3735 0.3597 Table 3. Coefficients used in the reaction rate polynomial for BUCKY. Temperature (keV) T(d,n)He (cm/s) DDtotal (cm/s) He(D,p)He (cm/s) 1 5.48E-21 8.52E-23 3.02E-26 2 2.62E-19 2.97E-21 1.42E-23 5 1.30E-17 9.98E-20 6.65E-21 10 1.08E-16 7.53E-19 2.28E-19 20 4.28E-16 4.08E-18 3.79E-18 50 8.64E-16 3.51E-17 5.46E-17 100 6.56E-16 2.92E-16 1.74E-16 Table 4. Reaction Rate Data based on the polynomial equation used for BUCKY. Bosch-Hale Reactivity R-matrix theory [7] [8] was used by Bosch and Hale [2] in conjunction with more recent low-energy experimental cross section data [9] [10] [11] to derive a more accurate fusion reactivity model. Equations (2) through (5) are the result of the R-matrix fit to the experimental data. For the fusion reactions described in this paper, Bosch and Hale compare their R-matrix fit to experimental data (Appendix I, Fig. 2-15). BG = παZ1Z2 2mrc 2 (2) θ = T 1− T C2 + T C4 + TC6 ( ) ( ) 1+ T C3 + T C5 + TC7 ( ( )) 