I Gaseous Hydrogen Fusion: Requirements & Limitations

[Aidan Davis]

What requirements must a target of gaseous (at STP) meet to participate as the target in beam-target fusion, besides the obvious requirement that the target be a pure (research grade) sample of the desired target material? If a sufficiently fast (hundreds of KeV for the various hydrogen fusion combinations, according to the Gamow factor) beam of hydrogen were to hit a hydrogen target in the gaseous phase, would fusion occur at the rate predicted by the Gamow factor, or would some other process inhibit the fusion? Assume that cooling is sufficient to keep the target in the gas phase or in a supercritical fluid phase, as opposed to creating a plasma.

 

[Khashishi]

It doesn't really matter if the target is gas or plasma. The energies required for fusion (~50 keV for DT, ~1000 keV for DD) are much higher than the energies holding atoms together (~13.6 eV) so the phase of the target is just negligible. Only the density of the target is important, so a solid target could increase your fusion rate.

 

[Aidan Davis]

It doesn't really matter if the target is gas or plasma. The energies required for fusion (~50 keV for DT, ~1000 keV for DD) are much higher than the energies holding atoms together (~13.6 eV) so the phase of the target is just negligible. Only the density of the target is important, so a solid target could increase your fusion rate.

Hmmmm... a solid target sounds really interesting! However, keeping hydrogen above it's melting point of just 14 K while it's undergoing fusion reactions is only possible in some crazy stretch of the imagination. So I assume you mean a solid, hydrogen dense chemical. The best one, objectively, would be the one in which the hydrogen atoms have the highest portion of the collision cross section. Please correct me if I'm completely wrong, but nuclei fuse when they come within a De Broglie wavelength, but I am not at all sure if that means the wavelength of the thermal target (very large) or of the incoming particle. If it is of the incoming particle, then each atom has roughly the same cross section for fusion-potential "flybys" or collision, as the De Broglie wavelength of a deuteron is ~80 fermis at 250 KeV, compared to the, for example, 1.2 fermi radius of the hydrogen nucleus. So if this is the case, the best choice is the molecule with the most hydrogen by molar fraction. A few that came up after some research were: AlH3, ZrH4, etc. But most hydrides tend to be gaseous. It doesn't seem worth it to sacrifice ~20% of potential fusions, so I guess just a gaseus, pure hydrogen target will do fine. Maybe some intense cooling and pressure to provide some added density. Thanks for the help!

 

[mfb]

If the beam is a short pulse, it can hit a solid target and fusion reactions can happen before the target evaporates.

Please correct me if I'm completely wrong, but nuclei fuse when they come within a De Broglie wavelength

There is no minimal distance, and no distance where fusion is guaranteed, and the de Broglie wavelength is not really important here.

 

[Aidan Davis]

If the beam is a short pulse, it can hit a solid target and fusion reactions can happen before the target evaporates.
There is no minimal distance, and no distance where fusion is guaranteed, and the de Broglie wavelength is not really important here.

Thanks. So I guess a pure hydrogen target is the only way without sacrificing a large amount of potential fusions to other elements which are too large and resist fusion, or worrying about changing the phase of the target.

 

[e.bar.goum]

Thanks. So I guess a pure hydrogen target is the only way without sacrificing a large amount of potential fusions to other elements which are too large and resist fusion, or worrying about changing the phase of the target.

Not really. Polyethylene (C₂H₄) targets are quite common if you want to do reactions with hydrogen (or deuterium if C₂D₄), you don't want to bother with a gas target (lower densities, you have to deal with cryogenics, exit/entrance windows, hydrogen is explosive ... ) and your beam intensity is low enough that you don't melt the target too quickly (a few nA for a few hours, even then you can use a rotating target wheel). Since the barrier to fusion is so much higher for ¹²C than for ^1,2H, you generally don't need to worry about background reactions from the carbon, and when you do, you can play tricks to separate out those contributions.

This is a general statement: quite often, it is advantageous to use a target with impurities (e.g. PbS vs Pb for a higher melting point, or using 12C, 27Al backings on fragile targets) and quite often you can remove any contributions from those impurities.

In general, target evaporation/melting is something one only needs to worry about when you are using pretty high beam intensities. In nuclear physics, targets don't instantaneously evaporate unless you're in the business of doing that deliberately. They can melt or thin over time, sure. You can calculate the power delivered by most research beams, and do some basic thermal physics calculations to work out the heating of the target.

 

[Aidan Davis]

Ah. So even if the C 12 wouldn't react, would it inhibit reactions?

 

[Aidan Davis]

And if the C 12 doesn't inhibit the fusion reactions, then would any other isotope? Why use polyethylene when there are hydrogen compounds witn much higher melting points? CaH2 melts at 816 °C, compared to polyethylene's 115-135 °C.

 

[mfb]

Ah. So even if the C 12 wouldn't react, would it inhibit reactions?

The beam particles can scatter there and lose some of their energy, but 2/3 of the atoms are hydrogen, and carbon atoms are so heavy compared to hydrogen that the hydrogen keeps most of its energy in the collision. The small loss of fusion reaction rate can be tolerable if the other advantages are important enough.

 

[Aidan Davis]

The beam particles can scatter there and lose some of their energy, but 2/3 of the atoms are hydrogen, and carbon atoms are so heavy compared to hydrogen that the hydrogen keeps most of its energy in the collision. The small loss of fusion reaction rate can be tolerable if the other advantages are important enough.

Ok. So the incoming hydrogen beam hits the atoms proportional to their relative molar density? For example, a CaH2 would experience 2/3 of (initial) collisions be with hydrogen, while something like LiOH would experience 1/3 of collisions be with hydrogen? Or do the heavier elements have increasing collision cross sections in this? If so, is there a formula? Thanks!

 

[mfb]

Ca has a larger charge, it has a larger cross section (Z² probably) at keV energies. At very high energies, it still has a larger cross section but just from being a bigger nucleus (A^2/3). In between, you get something in between.

 

[Aidan Davis]

Ca has a larger charge, it has a larger cross section (Z² probably) at keV energies. At very high energies, it still has a larger cross section but just from being a bigger nucleus (A^2/3). In between, you get something in between.

Oh, so at KeV energies the coloumb force dominates? The collision of fast positive particle (in this case deuteron) into a stationary positive target (nuclei) is rutherford scattering, or are there other significant effects?

 

[mfb]

Apart from the fusion chance, it is Rutherford scattering, sure.

 

[e.bar.goum]

Ah. So even if the C 12 wouldn't react, would it inhibit reactions?

Well, if the beam hits the 12C, it's very unlikely that it will (or would have otherwise) hit the H nucleus. On a nuclear scale, the atoms are very far apart, and the probability of multiple scattering is in general, tiny. In any experiment, the vast majority of your beam goes unreacted.

And if the C 12 doesn't inhibit the fusion reactions, then would any other isotope? Why use polyethylene when there are hydrogen compounds witn much higher melting points? CaH2 melts at 816 °C, compared to polyethylene's 115-135 °C.

Part of it is always how convenient the target is to manufacture, and polyethylene is pretty simple, but the other consideration is Rutherford scattering.

 

[mfb]

Well, if the beam hits the 12C, it's very unlikely that it will (or would have otherwise) hit the H nucleus. On a nuclear scale, the atoms are very far apart, and the probability of multiple scattering is in general, tiny. In any experiment, the vast majority of your beam goes unreacted.

Depends on your target depth and density.

 

[Aidan Davis]

At a reasonable beam energy, the rutherford cross section reaches tens of barns per Z². Thats plenty to get a few collisions out of the beam in a reasonably sized target before the deuteron escapes.
Is the fusion cross section just the rutherford cross section multiplied by the probability of fusion?

 

[mfb]

Is the fusion cross section just the rutherford cross section multiplied by the probability of fusion?

Only for a very weird definition of "probability of fusion".Fusion is an unlikely result.

 

[Aidan Davis]

Only for a very weird definition of "probability of fusion".Fusion is an unlikely result.

The gamow sommerfeld factor gives the probability of fusion for two distant particles on a collision course. It is derived here in the context of the sun: http://www.astro.princeton.edu/~gk/A403/fusion.pdf
Is that the weird definition your talking about? It seems to fit.

 

[Aidan Davis]

The gamow sommerfeld factor gives the probability of fusion for two distant particles on a collision course. It is derived here in the context of the sun: http://www.astro.princeton.edu/~gk/A403/fusion.pdf
Is that the weird definition your talking about? It seems to fit.

* It seems to fit the data*
(Pressed upload by mistake)

 

[mfb]

The Gamow factor is independent of the Rutherford cross section. They are different things.

 

[Aidan Davis]

The Gamow factor is independent of the Rutherford cross section. They are different things.

Yes, but the gamow gives probably of fusion upon collision while rutherford gives cross section for collision, so their product should be the cross section for fusion.
Thanks for all the help!