Calculation of ideal ignition temperature for a D-T fusion reaction

I've been trying to calculate the ideal ignition temperature for a 50-50% Deuterium-Tritium (D-T) reaction. In the literature this value is ##4.4##keV and I'm getting ##5.2##keV. Here's how I'm carrying out my calculations.

This value can be calculated by making the alpha particle heating, ##S_{\alpha}=(1/4)n^2 E_{\alpha}\langle \sigma v \rangle##, equal to the Bremsstrahlung losses, ##S_B=C_B n^2 \sqrt{T}##:

$$S_{\alpha}=S_B \Leftrightarrow E_{\alpha}\langle \sigma v \rangle-4C_B \sqrt{T}=0$$

The value I get for ##C_B=(\sqrt{2}/3\pi^{5/2})(e^6/(\varepsilon_0^3 c^3 h m_e^{3/2})## is ##4.22\cdot10^{-29}## ##\sqrt{kg}\, m^4\, s^{-2}##, where ##e## is the electron charge, ##\varepsilon_0## the vacuum permittivity, ##h## Planck's constant and ##m_e## the electron mass. The alpha particle energy is ##E_{\alpha}=3.5##keV. For the reactivity, ##\langle \sigma v \rangle##, which is a function of ##T##, I have been using an equation derived analytically from "Plasma Physics and Fusion Energy book by J.P. Freidberg, 2007":

$$ \langle \sigma v \rangle = \frac{4 \sigma_m}{\sqrt{3}}\sqrt{\frac{2T_m}{m_r}} \left(\frac{T_m}{T}\right)^{2/3}e^{-3(T_m/T)^{1/3}+2} $$

Where ##\sigma_m = 5.03##barns and ##T_m = 296##keV. ##m_r## is the reduced mass of the deuterium and tritium, which is given by ##m_r=(m_D m_T)/(m_D + m_T)##, where ##m_D=2.0141u##, ##m_T=3.01605u## and ##u=1.660\cdot 10^{-28}##kg.

Here are all elements for calculating the ideal ignition temperature, but I feel that I'm messing up somewhere the conversions from keV to Joule or Kelvin.

Can someone help me to troubleshoot what I'm doing wrong? Or, do you indeed get the same value as me?

Thanks in advance!

You can plug all these things in WolframAlpha and let it check your calculations. It will do the unit conversions for you.

freddie_mclair said:

TL;DR Summary: Calculating the ideal ignition temperature for a D-T fusion reaction

In the literature this value is keV and I'm getting keV.

When one refers to the literature, does one mean "Plasma Physics and Fusion Energy book by J.P. Freidberg, 2007", or some other literature. One would need to be sure to include the same assumptions with respect to the losses and definition of ideal temperature.

1 eV is ~ 11604.5 K, 1 keV ~ 11604500 K
One can refer to NIST values/constants
https://physics.nist.gov/cgi-bin/cu...=ev&To=k&Action=Convert+value+and+show+factor

Ideal ignition corresponds to the condition of steady state power balance in the plasma
assuming negligible heat conduction losses and no external heating, considering only Bremsstrahlung losses. Therefore, no losses transport losses due to heat conduction which are normally quantified by ##S_{th}=3nT##.