Temperature, density, and power output

[hanelyp]

It occurred to me last night:

For any fusion reactor the power output is ~ n^2*s
where n is density and s is reaction cross section.

For a pressure limited reactor (most magnetic confinement systems, p = F(B)), that means power ~ s(T)/(T^2)
where T is temperature.

Which means that peak power output will be at a temperature less than for max reaction cross section.

Related, can someone point me to a good source for fusion reaction cross section data in tabular form, suitable for entry in a spreadsheet?

[D Tibbets]

I suspect this is over simplified. The fusion cross section is certainly temperature dependent, but what is the temperature? If thermalized there is a temperarture range. In Tokamaks the obtainable temperature may be well below the peak fusion cross section for D-T or especially any other fuel. The high energy thermal tail of the Maxwell distribution thus provides most of the fusions. This effects the picture considerably. If a system has a near monoenergetic distribution of temperature (KE), then things are different. As the temperature increases, so does the input energy and the associated losses like Bremsstruhlung. The fusion cross section slope becomes less steep and there is a cross over point where the input losses start increasing more rapidly than the fusion output rates. This is indeed below the peak. How much lower depends on the fusion cross section slope, the loss slope, and other technical issues at any given temperature range.

In a thermalized plasma, the average temperature needs to be well below the fusion cross section peak, otherwise the high energy thermal tail actually contributes less to fusion while disproportionately increasing losses. There is a compromise. With near monoenergetic (not thermalized distributions) the recipe is different, and the target may be closer to the cross section peak. This is why a target for Polywell D-D fusion reactor may be ~ 80-100 KeV. The D-D fusion cross section curve is still increasing but the slope is shallower past this ~ point, so Bremsstruhlung and other losses increase more rapidly and you are losing ground. This is also why the target for P-B11 may be ~ 400 KeV (200 KeV if you can get beam - beam dominate conditions). The P-B11 fusion cross section is high enough (hopefully), and has not yet leveled out at the peak. The best fusion gain/ input loss compromise is reached. For D-T in a thermalized Tokamak the fusion cross section peak may be at ~ 50 KeV but the best target for the average plasma temperature may be closer to 20 KeV. For a monoenergetic system like the Polywell, the target average temperature may be closer to ~ 40 KeV. This gives the Polywell a distinct advantage over thermalized machines (a larger proportion of the plasma is at useful fusion temperatures without incurring the wrath of the high thermal tail penalties), provided either works at all. This , I believe, is the root reason that thermalized plasmas will have a very difficult time working beyond D-T fuel. DPF has a two pronged work around- invoking quantum mechanics based Bremsstruhlung supression at very high B fields and high x-ray direct conversion efficiencies. I'm uncertain how this applies to FRCs.

Also, considerations about thermalization time versus average temperature in monoenergetic plasmas play a role. In a large Polywell, it may be difficult to prevent full thermalization at lower temperatures (annealing, if it works, will become less effective if the ions substantially thermalize in under a single transit across the machine), but this problem is mitigated by higher temperatures due to the increased MFP (decreased Coulomb cross section) under these conditions. The final recipe may involve multiple consideration which again gives the best compromise between cross section, losses, thermalization issues, etc., etc...

Dan Tibbets

[mattman]

Hello,


The best resource for cross section is Azenti's first chapter of his book on ICF physics from 2004. Here is an example of one his tables from Page 11:




I will PM you, I can send you allot of material.