I suggest you first look at the reaction cross section as a function of energy. For example plot out the DD reaction cross section using the relations given in the NRL Formulary

http://wwwppd.nrl.navy.mil/nrlformulary/NRL_FORMULARY_09.pdf

Then you can calculate the rate coefficient at a given ion temperature by doing an integral of this cross section vs energy with the distribution function at the given temperature.

Your result of 400 MeV is very high.

Thanks for pointers. I'm deriving a complete simulation for a test chamber currently under construction, whose design was given to me.

The quantum tunnelling, reduced velocity, fusion probability graph needs to be calculated, where I wanted to know the minimum temperature at which alpha-alpha fusion would certainly occur.

The cross section calculations and integration are on my list, after those for the temperature. A Lawson's Law 3D graph for time, temp and density will be created.

Planned is a web interface for a web calculator, where various ions can be selected, and sliders for various parameters. Should be done end of February.

I'm looking for similar tools, but few exist. Any URLs? See below (far cry from what I want).

ERRATA

I see I made a typo, and have corrected the units in the question explanation, from 400 Mev to 400 million degrees or 37 Kev.

The Navy Formulas from 2009 to 2011 have 3 corrections listed here:

http://wwwppd.nrl.navy.mil/nrlformulary/changes_from_2009.pdf

and the most recent PDF book can be downloaded here (currently 2011):

http://wwwppd.nrl.navy.mil/nrlformulary/

For those wanting some explanation for some of these equations I've found this page is good:

http://www.plasmaphysics.org.uk/

MORE URLs

Basic Physics Math for many fields:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Generic online equation calculator and facts:

http://scienceworld.wolfram.com/physics/

And of course, Matlab and Octave (I'm doing this in Linux) for 3D graphing.

It will definitely be much higher than for P-P fusion, which can be done with about 4kV on accelerating electrodes.

This ref suggests 100M degrees K...

http://hyperphysics.phy-astr.gsu.edu/hbase/astro/helfus.html

@LFortunato: Thanks for your reply. I've looked into this further, and report what I found. I agree there is a non zero probability at all velocities, including in liquid or solid helium. The quantum tunneling effect I have found out makes fusion around 100 times more likely, and is almost the same probability as H+ + H+ fusion. I plan on using SAGE (open source version of MatLab) to graph the Gamov Peak, thus I get the rate at each temperature. The triple alpha process includes not just the product of Be, which can fuse before it decays with a third He, but also C, if 3 He atoms hit each other at the same time. My interest now includes He fusion with C and O, and calculating those fusions. The same SAGE code can be used. So, software code that does Gamov Peak calculations would be nice to get. I'm coding it, as I have not found any.

There are additional considerations I have found of electron shielding, and a newly published physics that if proves right lowers He fusion temperature, making fusion up to four times more probable. Coding these will be harder.

There are nuclear reactions codes like EMPIRE: http://www.nndc.bnl.gov/empire/ that are used for the simulation of cross sections.

I am not very deep into this, but I think a "web interface for a web calculator, where various ions can be selected, and sliders for various parameters" is very ambitious. To make a meaningful simulation, you might have to adapt a suitable model for the nuclear potential, especially when you are interested in fusion at energies well below the Coulomb barriere. A lot of similar work must exist within the context of fusion research, so I strongly recommend to contact an expert in this field to avoid trap doors that might already be known.

@E. Strub: Thank you for the Empire link. I'm reading it's documentation, as I know I will install it, and run it for my fusion calculations. It has more than other software I have read about. I've come across half a dozen software packages, free ones, and over a dozen commercial ones, and not one of them can emulate my entire experiment, not even half of it. Very disappointing, but not unexpected. As it's Fortran and Ubuntu ready, I'll see about adding what is needed, perhaps integrating it with other software, or with custom SAGE algorithms I will write.

I've found fusion is fairly easy to achieve, even in the home lab, much to my surprise. Lots of DIY online with step by step instructions.

Regarding computer simulation, the Coulomb replusion potential is modified by quantum tunneling, easily done using Gamow Peak graphing (must write your own). The velocity is as much 100 times lower. Add electron screening in a quasi neutral plasma, and fusion probability is raised again. I have found more tricks to increase the fusion rate. None have available simulation software, either free or commercial, and even the basic cross section data needed has not been experimentally determined.

My literature search has not found a published value for He+2 fusion. He3 has been done. No one has published on electron screening for alpha-alpha fusion.

Contacting the expert is the next step, to perhaps confirm I am not duplicating the wheel.

I'll answer my own question, with a first approximation. Numerically, the minimum velocity is "around" 15,000 meters per second. This value comes from calculating the Gamow Window at 1 million degrees, which gives a 'small,' non zero rate of fusion. The Gamow peak is 3.9 keV, and it's effective width is 1.3 Kev. These values are only a rough approximation.

I add the fusion rate is proportional to the area integrated under the Gamow Peak. Therefore, there is a non zero value for even lower temperatures, just not 'much' lower. So, getting the 'minimum' velocity was a poorly problem statement. A more useful problem statement would be what is the equation, as a function of temperature, giving the fusion 'rate' for Alpha-Alpha collisions? The next question would be what temperature (velocity) gives the desired rate for the experiment?

Alpha-Alpha fusion product, 8Be, has a lifetime 6.7e-14 seconds, and releases no energy. This fusion product is only 'useful' when a third alpha particle fuses with Be to form Carbon 12, when considerable energy is released. Fred Hoyle proposed this fusion chain, called the Triple Alpha Process, for star gravity well based fusion power, after H-H fusion creates He. The Carbon then provides star power with the CNO cycle. The fourth energy release fusion cycle is called Alpha Ladder, where Oxygen, Neon, and other even numbered elements are formed, up to Iron.

One method to use Alpha-Alpha fusion for power creation would be to confine a cloud of alpha particles in a reactor, at a temperature of 10 million degrees and a density equivalent to a star's gravity well. Many believe this density is not possible with today's technology. So, publications on Alpha-Alpha fusion are rare.

I did install EMPIRE, and upgraded to EMPIRE 3. It is quite difficult software to learn for Alpha-Alpha fusion as many parameters are required to be entered into the user interface form. EMPIRE is not suitable for lay persons, nor for recent college graduates. It gives 'precise' answers, that require knowing the temperature dependent cross section for the reactants as well as other values. It comes with good graphic generation and a full set of data tables. The documentation assumes the reader has a post graduate degree in atomic collision theory.

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