Evaluation of reaction rates in thermonuclear processes
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Evaluation of reaction rates in thermonuclear processes. Libor Novák. Nuclear fusion probability is limited by the repulsive Coulomb barrier:. The Coulomb potential which the particles have to overcome in order to fuse is given by:. This relation can be applied at distances greater than:.

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Evaluation of reaction rates in thermonuclear processes

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Evaluation of reaction rates in thermonuclear processes

Evaluation of reaction rates in thermonuclear processes

Libor Novák


Nuclear fusion probability is limited by the repulsive coulomb barrier

Nuclear fusion probability is limited by the repulsive Coulomb barrier:


Evaluation of reaction rates in thermonuclear processes

The Coulomb potential which the particles have to overcome in order to fuse is given by:

This relation can be applied at distances greater than:

At present, nobody knows the exact form of the nuclear potential, but experimentally it was discovered, that the potential well of depth U0is about 30 − 40MeV.

Two particles with relative energy ε < Vb can only approach eachother up to the classical turning point:

—› tunnelling effect


Cross section

Cross section

If we would direct a monoenergetic particle beam ona stationarytarget, the number of collisions among the particles on a small distance dswill be proportional to the uncollided beam particles density n1 and to the target particle density n2:

Cross section is obtained experimentally. It can be established by a theory, which usesa tunnel efect to evaluate σ, but this theory only transfers the problem to theastrophysical S factor, which for many important reactions is a weakly varying function

of the energy and must be obtained experimentally, too.


These 4 reactions were considered

These 4 reactions were considered:


Evaluation of reaction rates in thermonuclear processes

The total cross section in barns (1barn = 10−28m2) as afunction of E, the energy in keV of the incident particle, assuming the target ion at rest, can be fitted by:


Evaluation of reaction rates in thermonuclear processes

When the target particles are at rest and the beam particles move with a constant velocity,the reaction rate per unit volume is defined to be:


Evaluation of reaction rates in thermonuclear processes

In case of reaction between two species of particles, each having Maxwellian distributions,charakterized by m1, T1 and m2,T2, we become:


Evaluation of reaction rates in thermonuclear processes

So, we need to know the reaction rate parameter. There are manny publications, which contain values of this parameterfor various reactions and temperatures. However a lot of modern computer simulationsof fusion reaction rates utilize fitting functions based on data that were published almost thirty years ago.

On this account, for evaluation of reaction rate parameter we will use Bosch and Halefusion reactivity model, which is based on R-matrix theory in conjunction with more recent experimental cross section data:


Evaluation of reaction rates in thermonuclear processes

BUCKY is a one-dimensional hydrodynamics code developed by the University ofWisconsin that models high energy density fusion plasma. It was used to generatereaction rate parameter as a function of plasma thermal energy. Its advantage is the simplicity, however there are no

ranges, over which this formula is valid. Moreover, there are no coefficients for D-D reactions separately:

NRL plasma formulary:


Evaluation of reaction rates in thermonuclear processes

Comparison of BUCKY and Bosch-Hale for the D-T reaction:


Evaluation of reaction rates in thermonuclear processes

Comparison of BUCKY and Bosch-Hale for the D-D reaction:


Evaluation of reaction rates in thermonuclear processes

Comparison of BUCKY and Bosch-Hale for the D-He3 reaction:


Evaluation of reaction rates in thermonuclear processes

System of ordinary differential equations, which will describe tem-

poral progress of particles densities of fusion reaction components for x = const. Wewill assume that in time t = 0 we have plasmacompound from D, T and He3 particles. During the reactions, new particles will be generated:


Evaluation of reaction rates in thermonuclear processes

Fuel particle densities for E = 100 keV and ignition conditions nD(0) =1, nT (0) = nHe3(0) = 0:


Evaluation of reaction rates in thermonuclear processes

Product particle densities for E = 100 keV and ignition conditions nD(0) =1, nT (0) = nHe3(0) = 0:


Evaluation of reaction rates in thermonuclear processes

Fuel particle densities for E = 100 keV and ignition conditions nD(0) =1, nT (0) = 1,nHe3(0) = 0:


Evaluation of reaction rates in thermonuclear processes

Product particle densities for E = 100 keV and ignition conditions nD(0) =1, nT (0) = 1,nHe3(0) = 0:


Evaluation of reaction rates in thermonuclear processes

Fuel particle densities for E = 20 keV and ignition conditions nD(0) =1, nT (0) = nHe3(0) = 0:


Evaluation of reaction rates in thermonuclear processes

Product particle densities for E = 20 keV and ignition conditions nD(0) =1, nT (0) = nHe3(0) = 0:


Evaluation of reaction rates in thermonuclear processes

Fuel particle densities for E = 20 keV and ignition conditions nD(0) =1, nT (0) =1,nHe3(0) = 0:


Evaluation of reaction rates in thermonuclear processes

Product particle densities for E = 20 keV and ignition conditions nD(0) =1, nT (0) = 1, nHe3(0) = 0:


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