I've had a scan on the forums for any info on the total number of recirculations that tend to go on in a fusor. A lot of speculation, but nothing concrete/no actual calculations I think.
So let me give you one of my quickie calculations. Out of another calculation I did [which was to do with improving my previous post on beam-fusion viability], I got the number of ~10 to15 scattering events for a formed ion before it dropped off from 30~50keV to below 10keV, at which point its energy essentially nose-dives into next to nothing, and within a distance of about 2 metres (MFP =~0.2 to 0.4m initially) overall. That was an independent, theoretical calculation (which I'll post along with my new beam-fusion equations - need to check it thoroughly first; my calculus is rubbish!).
That meant an ion would only recycle about 15 times in a fusor of some several inches diameter, which seemed intuitively very low to me, so I did the following order-of-magnitude calculation along the same lines as the calculation on John Futter's 12MeV proton beam and using Richard Hull's fusor IV data.
43kV drive == 21.5kV collision energy == fusion cross section (measured) = 0.26milliBarn
43kV == 2E8 cm/s for a deuterium ion
15 microns == 8E14 D/cm3 background density
10ma == 6.25E16 D/s which implies* 3.1E8 D/cm of beam
*(beam current density/beam velocity, and presuming 'what goes in [ions] must come out [electrons = current drawn]')
So the overall reaction rate would be;
0.26E-27 cm2 * 2E8 cm/s * 8E14 D/cm3 * 3.1E8 D/cm =~ 13000 reactions/cm of beam/s
Bear in mind that here the 'length of the beam' means the total flight distance of an 'average' particle whilst at or very close to its maximum, fusing, energy. Richard reported his Fusor IV as showing 1.36million reactions per sec, which implies a beam length of (1.36E6reactions/s)/(13000reactions/cm/s) =~100cm of beam length [whilst at high energy].
So by a different calculation, I have again arrived at about 15 recirculations (presuming some 7-8 cm of high velocity travel per recirculation) over a total distance of the order of 1 to 2m. It's not one-or-two recirculations, nor is it hundreds, so as an order-of-mag calculation, I am very happy with that!
This was not meant to be some predictive calculation, but only that I was testing out this conclusion of a dozen scattering events over a 2m distance that I did not initially believe. However, the tie-up between a calculation (I did not trust) and the reality of an experimental result (that I intended to use to disprove it) was a surprise I thought I'd share.
best regards,
Chris MB.
A calculation on recirculation in a fusor.
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Chris Bradley
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Todd Massure
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Re: A calculation on recirculation in a fusor.
If you didn't catch this thread already, this is the one to check out. Unfortunately nailing down a standard way for calculating the mean free path in a fusor has never really been agreed on, but there's some good shots taken at it here:
viewtopic.php?f=14&t=6614&hilit=mean+free+path#p42297
Todd Massure
viewtopic.php?f=14&t=6614&hilit=mean+free+path#p42297
Todd Massure
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Chris Bradley
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Re: A calculation on recirculation in a fusor.
Yes. That is one I studied at length.
One issue for a direct 'traditional' MFP calculation is the calculation of the effect of electrons, which I think is a very tricky treatment in a fusor. They have tiny masses, but I am not sure they have a tiny effect because they are so easily pulled around by much heavier ions - i.e. the ion energy dissipates only a little, but very effectively, into electrons. Not sure about that...
But anyway, this is a different calculation altogether - reversing what I have done here, the fusion rate gives a total path length, another calculation I'll be posting describes a total number of likely scatters, so MFP is one/other and in fact is arguably a more representative MFP because it also accommodates the ramp-off of energy down to some considered energy level.
But what the scattering-calc side of that equation doesn't do well is anticipate the increased scattering rate (reduction in MFP) as the ion slows and heads on back, which is a major weakness of my approach if you 'really' want to know MFP more than you want to know total-effective beam length (i.e. 'live' duration of an ion).
One way or the other, my calcs are in agreement with Garrett Young's treatment on that post, so I would have to side with his approach.
best regards,
Chris MB.
One issue for a direct 'traditional' MFP calculation is the calculation of the effect of electrons, which I think is a very tricky treatment in a fusor. They have tiny masses, but I am not sure they have a tiny effect because they are so easily pulled around by much heavier ions - i.e. the ion energy dissipates only a little, but very effectively, into electrons. Not sure about that...
But anyway, this is a different calculation altogether - reversing what I have done here, the fusion rate gives a total path length, another calculation I'll be posting describes a total number of likely scatters, so MFP is one/other and in fact is arguably a more representative MFP because it also accommodates the ramp-off of energy down to some considered energy level.
But what the scattering-calc side of that equation doesn't do well is anticipate the increased scattering rate (reduction in MFP) as the ion slows and heads on back, which is a major weakness of my approach if you 'really' want to know MFP more than you want to know total-effective beam length (i.e. 'live' duration of an ion).
One way or the other, my calcs are in agreement with Garrett Young's treatment on that post, so I would have to side with his approach.
best regards,
Chris MB.
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Carl Willis
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Re: A calculation on recirculation in a fusor.
I think the concept of recirculation is important to understanding the hypothetical efficiency benefits of the fusor over, say, a titanium-target neutron generator tube. But when it comes to quantifying the recirculation, either through a mathematical model or experiment, complexities get in the way (as they do for so much other would-be-easy fusor theory), and hence the apparent absence of these numbers anywhere. Here are a few of the "complexities":
1. Particles in the fusor travel in an inverse-square potential well. Like objects orbiting the Sun, they need not recirculate along straight radial paths (although they can). They may have angular momentum, orbiting in highly eccentric fashion or in circular fashion at fixed distance from the center, particularly after collisions, all the while still remaining viable fusion reactants. Obviously, these elliptical path lengths for recirculation are not the same length as a simple diametrical swing through the center, don't involve the same variations in kinetic energy with position, etc. To me, this kind of makes "recirculation" an ambiguous quantity from the math perspective.
2. Kinetic energy and quantities depending on it, such as stopping power, generally vary with position of a particle in the fusor. (The exception is if the particle has a circular orbit.) Since the stopping power and the related cross-sections vary along the path of particles--which may oscillate between 0 keV and some much higher value during recirculation--a simple approach assuming the particles have fixed kinetic energy is totally out the window.
On the experimental front, recirculation itself is hard to measure or detect. What makes an easier measurement conveying much the same kind of information would be (A) a measure of the difference between injected charge and retrieved charge over the period of time it takes the fusor to start and reach steady-state operation; or (B) a measure of the trapped charge once at steady-state. The latter could be accomplished with microwave interferometry, likely involving equipment that is not hard to build from surplus k-band microwave parts. Not been done by the homebrew crowd. Full of tricks to do right.
-Carl
1. Particles in the fusor travel in an inverse-square potential well. Like objects orbiting the Sun, they need not recirculate along straight radial paths (although they can). They may have angular momentum, orbiting in highly eccentric fashion or in circular fashion at fixed distance from the center, particularly after collisions, all the while still remaining viable fusion reactants. Obviously, these elliptical path lengths for recirculation are not the same length as a simple diametrical swing through the center, don't involve the same variations in kinetic energy with position, etc. To me, this kind of makes "recirculation" an ambiguous quantity from the math perspective.
2. Kinetic energy and quantities depending on it, such as stopping power, generally vary with position of a particle in the fusor. (The exception is if the particle has a circular orbit.) Since the stopping power and the related cross-sections vary along the path of particles--which may oscillate between 0 keV and some much higher value during recirculation--a simple approach assuming the particles have fixed kinetic energy is totally out the window.
On the experimental front, recirculation itself is hard to measure or detect. What makes an easier measurement conveying much the same kind of information would be (A) a measure of the difference between injected charge and retrieved charge over the period of time it takes the fusor to start and reach steady-state operation; or (B) a measure of the trapped charge once at steady-state. The latter could be accomplished with microwave interferometry, likely involving equipment that is not hard to build from surplus k-band microwave parts. Not been done by the homebrew crowd. Full of tricks to do right.
-Carl
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Richard Hull
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Re: A calculation on recirculation in a fusor.
Carl echos the feeling of many us old timers here. This question and much discussion has ensued over many postings. The fusor is a messy system with lots of stuff going on and can in no way be handled by even fairly complex mathematical machinations.
The fusor operates at only a technical and not a scientific vacuum level. As vacuum systems go it is a really dirty pig. As an electrostatically controlled scientific beam apparatus goes it is mostly out of control with only a suggestion of directionality. Ionization takes place over 100% of its volume, as does recombination.
If one really looks at the device crtically, it is a more or less a directed energy deuterium lamp that is not good at directing particle energy nor is it a good deuterium lamp. Somewhere in this odd configuartion fusion takes place due to several theoretically predicted processes, probably in differing amounts at different volume points in the system. If one really looks at the net result from operation, it is a 99.998% efficient space heater.
There is definite empirical evidence that the bulk of the fusion occurs outside the inner grid, but still, rather close to it. A strong bit of evidence tend to point towards fast neutral-fast ion collisions as being a significant part of the fusion process in the fusor. This shows that the idealized recirculation of deuterons colliding in the center is NOT the main source of fusion, around which the entire theoretical concept is constructed.
As Carl notes the data needed has never been taken from a working device to allow a fine point to be placed on anything related to beaming or charged particle interaction.
Richard Hull
The fusor operates at only a technical and not a scientific vacuum level. As vacuum systems go it is a really dirty pig. As an electrostatically controlled scientific beam apparatus goes it is mostly out of control with only a suggestion of directionality. Ionization takes place over 100% of its volume, as does recombination.
If one really looks at the device crtically, it is a more or less a directed energy deuterium lamp that is not good at directing particle energy nor is it a good deuterium lamp. Somewhere in this odd configuartion fusion takes place due to several theoretically predicted processes, probably in differing amounts at different volume points in the system. If one really looks at the net result from operation, it is a 99.998% efficient space heater.
There is definite empirical evidence that the bulk of the fusion occurs outside the inner grid, but still, rather close to it. A strong bit of evidence tend to point towards fast neutral-fast ion collisions as being a significant part of the fusion process in the fusor. This shows that the idealized recirculation of deuterons colliding in the center is NOT the main source of fusion, around which the entire theoretical concept is constructed.
As Carl notes the data needed has never been taken from a working device to allow a fine point to be placed on anything related to beaming or charged particle interaction.
Richard Hull
Progress may have been a good thing once, but it just went on too long. - Yogi Berra
Fusion is the energy of the future....and it always will be
The more complex the idea put forward by the poor amateur, the more likely it will never see embodiment
Fusion is the energy of the future....and it always will be
The more complex the idea put forward by the poor amateur, the more likely it will never see embodiment
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Chris Bradley
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Re: A calculation on recirculation in a fusor.
It seems a straightforward comment - too complex to figure out.
It's in the nature of old timers to say something like that. An easy opt out to explain why they've got stuck at where they've got to.
Intellectually, it is reasonable to make such a comment. But this does not constitute evidence.
It is reasonable to observe that there has been plenty of experience that it's been too much of a challenge to figure out what's going on through all the data noise. But this is not evidence either.
What is the evidence that a fusor is too complex to analyse?
Here I have presented a case where it appears to have been easy to arrive at, using theoretical assumptions, the output rate of your Fusor IV run.
That's not evidence either, but it is a deterministic and factual outcome which can be objectively repeated and therefore further tested, unlike the above two outcomes that the old timers rely on for their argument of 'being too complicated'. Maybe it is simpler than you think? What's your objective, testable, evidence otherwise? This is the same point made before about another simple calculation based on these assumptions that also matched reality, and mine is the same answer - I refer you to Occam's Razor...
How many simple calculations need to be made, that give physically reasonable and useful answers, before it goes from 'co-incidence' to 'useful theory'. I am surely not the first person to contemplate this conundrum.
Let me ask it/make the point another way - what piece of evidence would you consider acceptable that would demonstrate a well-operated fusor CAN be analysed in simple terms? If you weight and phrase your own argument to make it impossible to ever prove the contrary, how do you expect me to prove the contrary?
It sounds to me like you would never be prepared to accept any such evidence, because you are so convinced by your own intellectual argument.
Just to cover Carls' two points;
1) If it moves in a dead straight line across the fusor or absolutely fully aroundabout in a circle, the distance travelled only differs by pi. This is intended to be an order of magnitude calculation.
2) of course you are correct, but the calculation was one of total flight distance rather than actual recirculations, MFP, or other. Some assumptions on scattering can be applied where, though the particle slows down and collisions are more likely, the actual energy losses are low because it is going slowly. So the calculation is one of its flight time at high energy (and is therefore not a pure MFP calculation for all speeds), which doesn't seem an assumption 'totally out of the window' to me. Number of recirculations then becomes an inference - given a certain distance at 'full speed' is being covered by the average particle, how many recirculations is that likely to imply?
So, please, rather than just saying 'No - too complicated - don't bother!' I would prefer you to say
'OK, if you think that's right, you should now be reasonably able to prove A/B/C. I don't think you will be able to prove that, so show me I'm wrong if you can.'.
That's the kind of challenge I would like to hear, not just a subjective closed shut-down that I could therefore never disprove.
best regards,
Chris MB.
It's in the nature of old timers to say something like that. An easy opt out to explain why they've got stuck at where they've got to.
Intellectually, it is reasonable to make such a comment. But this does not constitute evidence.
It is reasonable to observe that there has been plenty of experience that it's been too much of a challenge to figure out what's going on through all the data noise. But this is not evidence either.
What is the evidence that a fusor is too complex to analyse?
Here I have presented a case where it appears to have been easy to arrive at, using theoretical assumptions, the output rate of your Fusor IV run.
That's not evidence either, but it is a deterministic and factual outcome which can be objectively repeated and therefore further tested, unlike the above two outcomes that the old timers rely on for their argument of 'being too complicated'. Maybe it is simpler than you think? What's your objective, testable, evidence otherwise? This is the same point made before about another simple calculation based on these assumptions that also matched reality, and mine is the same answer - I refer you to Occam's Razor...
How many simple calculations need to be made, that give physically reasonable and useful answers, before it goes from 'co-incidence' to 'useful theory'. I am surely not the first person to contemplate this conundrum.
Let me ask it/make the point another way - what piece of evidence would you consider acceptable that would demonstrate a well-operated fusor CAN be analysed in simple terms? If you weight and phrase your own argument to make it impossible to ever prove the contrary, how do you expect me to prove the contrary?
It sounds to me like you would never be prepared to accept any such evidence, because you are so convinced by your own intellectual argument.
Just to cover Carls' two points;
1) If it moves in a dead straight line across the fusor or absolutely fully aroundabout in a circle, the distance travelled only differs by pi. This is intended to be an order of magnitude calculation.
2) of course you are correct, but the calculation was one of total flight distance rather than actual recirculations, MFP, or other. Some assumptions on scattering can be applied where, though the particle slows down and collisions are more likely, the actual energy losses are low because it is going slowly. So the calculation is one of its flight time at high energy (and is therefore not a pure MFP calculation for all speeds), which doesn't seem an assumption 'totally out of the window' to me. Number of recirculations then becomes an inference - given a certain distance at 'full speed' is being covered by the average particle, how many recirculations is that likely to imply?
So, please, rather than just saying 'No - too complicated - don't bother!' I would prefer you to say
'OK, if you think that's right, you should now be reasonably able to prove A/B/C. I don't think you will be able to prove that, so show me I'm wrong if you can.'.
That's the kind of challenge I would like to hear, not just a subjective closed shut-down that I could therefore never disprove.
best regards,
Chris MB.
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Frank Sanns
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Re: A calculation on recirculation in a fusor.
Chris Bradley wrote:
> 1) If it moves in a dead straight line across the fusor or absolutely fully aroundabout in a circle, the distance travelled only differs by pi. This is intended to be an order of magnitude calculation.
Chris I you think are a little out of line here. Both Carl and Richard were trying to convey the complexity of the fusor to you. By your statement above and your apparent lack of physics and higher math background you do not have the knowledge or experience of the "old timers" on the board.
A straight line or a circular path does not differ by pi or even necessarily by a factor of ten. Momentum must be conserved as well as energy. A change in direction implies a change in momentum but the energy can be maintained by a particle or it can be lost entirely to its surroundings and everything in between. Even your most fundemental assumptions and algebra totally missed the trivial. If I were you I would bow my head to Richard and Carl's post and be humbled.
Frank S.
> 1) If it moves in a dead straight line across the fusor or absolutely fully aroundabout in a circle, the distance travelled only differs by pi. This is intended to be an order of magnitude calculation.
Chris I you think are a little out of line here. Both Carl and Richard were trying to convey the complexity of the fusor to you. By your statement above and your apparent lack of physics and higher math background you do not have the knowledge or experience of the "old timers" on the board.
A straight line or a circular path does not differ by pi or even necessarily by a factor of ten. Momentum must be conserved as well as energy. A change in direction implies a change in momentum but the energy can be maintained by a particle or it can be lost entirely to its surroundings and everything in between. Even your most fundemental assumptions and algebra totally missed the trivial. If I were you I would bow my head to Richard and Carl's post and be humbled.
Frank S.
Achiever's madness; when enough is still not enough. ---FS
We have to stop looking at the world through our physical eyes. The universe is NOT what we see. It is the quantum world that is real. The rest is just an electron illusion. ---FS
We have to stop looking at the world through our physical eyes. The universe is NOT what we see. It is the quantum world that is real. The rest is just an electron illusion. ---FS
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Chris Bradley
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Re: A calculation on recirculation in a fusor.
I am more than happy to be humble providing I have at least the opportunity (not necessarily the ability or correctness, I do not mind being proved wrong) to respond to the challenge. Yours is a challenge to my simple physics I *can* respond to (whether rightly or wrongly, but it is an objective challenge that can be explored).
However, I fear I do not actually understand you point, so the first thing is to clarify. Perhaps I therefore simply misunderstood Carl's point, by inference. Momentum of an ion in an electric field is preserved in that the particle imparts its momentum to the electrodes that generate that field, and, by physical connection, to the earth. So I'm not sure I understand what you are saying with regards momentum. The particle's momentum will not be preserved as it moves in an electric field unless it is moving along a line of equipotential. There is no loss of energy in the system except in the event of a collision with another particle. Its KE may change - it may slow down, but it gains potential energy in the electric field by whatever it looses.
I understood Carl to be questioning what a 'recirculation' actually is, which is a very fair point, but I thought it was answered by saying it could be any orbit you desire because my calc was purely a distance of flight calculation NOT a sum total of KE changes or MFP calculation. So I'm not sure what that says about the situation, but I'm happy to presume I currently misunderstand the point being made. Could you please explain what point you are raising that demonstrates an apparent lack of physics or maths background?
However, I fear I do not actually understand you point, so the first thing is to clarify. Perhaps I therefore simply misunderstood Carl's point, by inference. Momentum of an ion in an electric field is preserved in that the particle imparts its momentum to the electrodes that generate that field, and, by physical connection, to the earth. So I'm not sure I understand what you are saying with regards momentum. The particle's momentum will not be preserved as it moves in an electric field unless it is moving along a line of equipotential. There is no loss of energy in the system except in the event of a collision with another particle. Its KE may change - it may slow down, but it gains potential energy in the electric field by whatever it looses.
I understood Carl to be questioning what a 'recirculation' actually is, which is a very fair point, but I thought it was answered by saying it could be any orbit you desire because my calc was purely a distance of flight calculation NOT a sum total of KE changes or MFP calculation. So I'm not sure what that says about the situation, but I'm happy to presume I currently misunderstand the point being made. Could you please explain what point you are raising that demonstrates an apparent lack of physics or maths background?
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Carl Willis
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Re: A calculation on recirculation in a fusor.
Hi Chris
>What is the evidence that a fusor is too complex to analyse?
Fusors are most certainly open to analysis. My critique was restricted to the assumptions you made but didn't sufficiently justify. When an assumption is made that, on prima facie consideration, strays from reality, the assumption has to be justified, and that's your "burden of proof." So as lazy as your respondents may be for telling you "No - too complicated - don't bother!," the hard work is duly yours to do. The assumptions you make include
-reactant particles are monoenergetic
-The beam current is due only to ions
-reactant particles' energy is constant along their paths
-"43kV drive == 21.5kV collision energy"
-Fusion is the dominant particle removal mechanism
This is why I like so much to stay on the conceptual level on this board. Math conveniently hides questionable assumptions. The above are all flat-out wrong at the first order. (You could take a pass on the first two if considering a high-vacuum ion-gunned fusor, but Richard's is not). The fourth one is a typo or a booboo. The last one neglects electronic slowing-down, which at these energies stops many more particles than fusion.
Why isn't it just a coincidence that you got a right number (to an order of magnitude anyway) out of your calculation?
-Carl
>What is the evidence that a fusor is too complex to analyse?
Fusors are most certainly open to analysis. My critique was restricted to the assumptions you made but didn't sufficiently justify. When an assumption is made that, on prima facie consideration, strays from reality, the assumption has to be justified, and that's your "burden of proof." So as lazy as your respondents may be for telling you "No - too complicated - don't bother!," the hard work is duly yours to do. The assumptions you make include
-reactant particles are monoenergetic
-The beam current is due only to ions
-reactant particles' energy is constant along their paths
-"43kV drive == 21.5kV collision energy"
-Fusion is the dominant particle removal mechanism
This is why I like so much to stay on the conceptual level on this board. Math conveniently hides questionable assumptions. The above are all flat-out wrong at the first order. (You could take a pass on the first two if considering a high-vacuum ion-gunned fusor, but Richard's is not). The fourth one is a typo or a booboo. The last one neglects electronic slowing-down, which at these energies stops many more particles than fusion.
Why isn't it just a coincidence that you got a right number (to an order of magnitude anyway) out of your calculation?
-Carl
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Steven Sesselmann
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Re: A calculation on recirculation in a fusor.
Chris,
I have been reading this thread, and I agree that it would be hard to calculate collision probabilities in a standard Farnsworth fusor due to all the variables.
When you think about it it is an fascinating device, ions circulate through the center in various paths throughout the whole chamber and continuously mop up neutrals by knocking into them and ionizing them. This process should, I imagine, create a non uniform particle distribution within the chamber, with higher particle density and energy in the center, and better vacuum along the edges.
So with, variable pressure, variable velocities, variable momentum, variable charges, variable path length, your calculus would have to be pretty complex to make any kind of mathematical model of what is going on inside a Farnsworth fusor.
A better approach is the one that Wilfried has taken, which is to run a good Farnsworth device at many different voltages and create a polynomial that fits the observations, which can then be used for predictions.
Steven
I have been reading this thread, and I agree that it would be hard to calculate collision probabilities in a standard Farnsworth fusor due to all the variables.
When you think about it it is an fascinating device, ions circulate through the center in various paths throughout the whole chamber and continuously mop up neutrals by knocking into them and ionizing them. This process should, I imagine, create a non uniform particle distribution within the chamber, with higher particle density and energy in the center, and better vacuum along the edges.
So with, variable pressure, variable velocities, variable momentum, variable charges, variable path length, your calculus would have to be pretty complex to make any kind of mathematical model of what is going on inside a Farnsworth fusor.
A better approach is the one that Wilfried has taken, which is to run a good Farnsworth device at many different voltages and create a polynomial that fits the observations, which can then be used for predictions.
Steven
http://www.gammaspectacular.com - Gamma Spectrometry Systems
https://www.researchgate.net/profile/Steven_Sesselmann - Various papers and patents on RG
https://www.researchgate.net/profile/Steven_Sesselmann - Various papers and patents on RG
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Chris Bradley
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Re: A calculation on recirculation in a fusor.
The 'duty of care' in understanding what has been put on this thread appears to have been thrown back at me, which I think is very unreasonable and I feel very sad about it.
I made it quite clear in my first post that "This was not meant to be some predictive calculation". Did I say this, or are my eyes deceiving me!!??
So to suggest that I am trying to "allow a fine point to be placed on anything related to beaming or charged particle interaction" is clearly a false suggestion.
What I presented was the conclusion of trying to disprove a null-hypothesis. I made it clear I set about disproving something I did not believe, then found that I could not prove it to be wrong.
To suggest that an actual prediction and absolute conclusion is the result of what I put is a misinterpretation or misunderstanding.
The calculation was simple and presented a case that the total distance an average particle would have to travel, whilst at or near the drive voltage, to liberate the number of reactions observed would be of the order of 1m. I have not underplayed the complexities of where that 1m worth of trajectory would actually occur in a fusor because I have not discussed it excepting a cursory examination of why it does not disprove the conclusion of ~15 recirculations. This is quite, quite different to stating this as a prediction. I posted because of the tie-up between the null-hypothesis and the original prediction. In such cases, further examination and discussion are normally the order of the day in a scientific arena to see why this might be so. Co-incidence may be one conclusion.
I have been characterised as having a poor grasp of physics and maths. Yet the issue here appear to be a misunderstanding of what a null-hypothesis actually concludes. It does not conclude the things I have been attributed of concluding.
To then be challenged with suggestions of 'unknowable complexity' (a fundamentally unscientific, and thus subjective, comment) that cannot be themselves objectively challenged is quite galling.
Please study again what I wrote and reconsider your opinion of what I originally said. If you stand by your critique of my presentation of the material, then if so I will make the humble amends. But, is there no-one who sides with my arguments, and my interest to seek the simplest explanations??
I made it quite clear in my first post that "This was not meant to be some predictive calculation". Did I say this, or are my eyes deceiving me!!??
So to suggest that I am trying to "allow a fine point to be placed on anything related to beaming or charged particle interaction" is clearly a false suggestion.
What I presented was the conclusion of trying to disprove a null-hypothesis. I made it clear I set about disproving something I did not believe, then found that I could not prove it to be wrong.
To suggest that an actual prediction and absolute conclusion is the result of what I put is a misinterpretation or misunderstanding.
The calculation was simple and presented a case that the total distance an average particle would have to travel, whilst at or near the drive voltage, to liberate the number of reactions observed would be of the order of 1m. I have not underplayed the complexities of where that 1m worth of trajectory would actually occur in a fusor because I have not discussed it excepting a cursory examination of why it does not disprove the conclusion of ~15 recirculations. This is quite, quite different to stating this as a prediction. I posted because of the tie-up between the null-hypothesis and the original prediction. In such cases, further examination and discussion are normally the order of the day in a scientific arena to see why this might be so. Co-incidence may be one conclusion.
I have been characterised as having a poor grasp of physics and maths. Yet the issue here appear to be a misunderstanding of what a null-hypothesis actually concludes. It does not conclude the things I have been attributed of concluding.
To then be challenged with suggestions of 'unknowable complexity' (a fundamentally unscientific, and thus subjective, comment) that cannot be themselves objectively challenged is quite galling.
Please study again what I wrote and reconsider your opinion of what I originally said. If you stand by your critique of my presentation of the material, then if so I will make the humble amends. But, is there no-one who sides with my arguments, and my interest to seek the simplest explanations??
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Carl Willis
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Re: A calculation on recirculation in a fusor.
Chris...keep your wits about you for a moment here!
You're trying to lawyer yourself out of a critique that applies to your calculation no matter what the hypothesis for which you invoked it. I can follow your calculation as you've set it up. It just neglects enough essential fusor physics to be non-applicable. (And I do stand by that critique.) Much as you'd like to, you can't declare that calculation immune from critique just because I've ignored the conclusion or the hypothesis!
Your "unknowable complexity" charge is a straw-man, by the way. No claim was ever made to this effect by me.
-Carl
You're trying to lawyer yourself out of a critique that applies to your calculation no matter what the hypothesis for which you invoked it. I can follow your calculation as you've set it up. It just neglects enough essential fusor physics to be non-applicable. (And I do stand by that critique.) Much as you'd like to, you can't declare that calculation immune from critique just because I've ignored the conclusion or the hypothesis!
Your "unknowable complexity" charge is a straw-man, by the way. No claim was ever made to this effect by me.
-Carl
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Carl Willis
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Re: A calculation on recirculation in a fusor.
>"43kV drive == 21.5kV collision energy"
I forgot that you like to look up your cross-sections from a CM reference. Provided that's what you did, and under the assumption that this collision is between a full-energy ion and a background gas atom, you can have that one back.
No hard feelings?
-Carl
I forgot that you like to look up your cross-sections from a CM reference. Provided that's what you did, and under the assumption that this collision is between a full-energy ion and a background gas atom, you can have that one back.
No hard feelings?
-Carl
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Frank Sanns
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Re: A calculation on recirculation in a fusor.
Chris Bradley wrote:
> I understood Carl to be questioning what a 'recirculation' actually is, which is a very fair point, but I thought it was answered by saying it could be any orbit you desire because my calc was purely a distance of flight calculation NOT a sum total of KE changes or MFP calculation.
But distance of flight is precisely related to MFP because MFP is related to velocity of the particle being studied. Velocity is in the numerator so the faster the velocity the longer the MFP or the longer distance of flight of the particle. Slow it down a little and it will have a shorter distance of flight.
If an ion starts its trip in the volume of a fusor and is accelerated towards the inner grid, it will get to some velocity. Depending on the distance from the inner grid, the ion will have a velocity inward and a small perpendicular velocity also. This is due to parallax since the grid is not a perfect point charge and also because of the E field near the grid wires. So the ion now passes with constant velocity through the inner grid and emerges on the far side and may be further perturbed from the E field near the grid wires and it less than perpendicular approach to the inner grid. Hold that thought for a minute.
Now the ion emerges from the inner grid with the same energy that it arrived at the inner grid. There are now three (really two but stick with me) possibilities.
1.) The ion is exactly perpendicular to the inner grid so it has a perfect radial trajectory toward the outer shell. Assuming no collisions, the ion will decelerated in the E field until it stops at a point the same distance from the inner grid that it started. It will then will fall back to the inner grid for another cycle. In this case, for one cycle the momentum has gone from mv to -mv and the energy gone from KE to zero to KE (KE is a scaler not a vector so it can not be negative). Since the velocity goes to zero, MFP (if all of the particles are doing this) goes to zero or at least gets much shorter if time at zero is short enough. This greathly shortens the distance of flight.
The equation for this case would be F=ma where force is found from Coulombs law (F=q1q2/4pie0r^2). Sorry for the sloppy equation in simple text but you most likely know it. But to solve the velocity at each point you have to sum up (integrate) all of the velocities as the field drops off so you would integrate from r = inner grid diameter to R = farthest distance from the inner grid that the particle can go (this is the distance where the ion formed).
2.) The ion emerges slightly less than perpendicular to the surface of the inner grid. In this case velocity does not go to zero. Only the radial component of velocity goes to zero so the distance of flight will not go to zero at this point since there is still some perpendicular velocity component. To calculate this use the above integral but include the sine of the angle relative to the inner grid.
3.) Extreme case of #2 where the ion is in a complete circular path that takes it through the inner grid. This can never be because the force required to do this would be infinite but the reality is most likely somewhere between case #2 and #3. In this case the energy (velocity) is constant at its maximum and it will have the longest path before a collsion. No work is done in this scenario and that makes it interesting for the fusor as it maximizes the chance of a fusion for a given energy input.
I hope you can see that the distance of flight is hugely different depending on the path and much more different than a factor of pi. It is a complicated situation and if you read the Univ of Wisconson papers you will see that it is further complicated because the paths and places that fusion occurs even changes with the gas fills.
Frank S.
> I understood Carl to be questioning what a 'recirculation' actually is, which is a very fair point, but I thought it was answered by saying it could be any orbit you desire because my calc was purely a distance of flight calculation NOT a sum total of KE changes or MFP calculation.
But distance of flight is precisely related to MFP because MFP is related to velocity of the particle being studied. Velocity is in the numerator so the faster the velocity the longer the MFP or the longer distance of flight of the particle. Slow it down a little and it will have a shorter distance of flight.
If an ion starts its trip in the volume of a fusor and is accelerated towards the inner grid, it will get to some velocity. Depending on the distance from the inner grid, the ion will have a velocity inward and a small perpendicular velocity also. This is due to parallax since the grid is not a perfect point charge and also because of the E field near the grid wires. So the ion now passes with constant velocity through the inner grid and emerges on the far side and may be further perturbed from the E field near the grid wires and it less than perpendicular approach to the inner grid. Hold that thought for a minute.
Now the ion emerges from the inner grid with the same energy that it arrived at the inner grid. There are now three (really two but stick with me) possibilities.
1.) The ion is exactly perpendicular to the inner grid so it has a perfect radial trajectory toward the outer shell. Assuming no collisions, the ion will decelerated in the E field until it stops at a point the same distance from the inner grid that it started. It will then will fall back to the inner grid for another cycle. In this case, for one cycle the momentum has gone from mv to -mv and the energy gone from KE to zero to KE (KE is a scaler not a vector so it can not be negative). Since the velocity goes to zero, MFP (if all of the particles are doing this) goes to zero or at least gets much shorter if time at zero is short enough. This greathly shortens the distance of flight.
The equation for this case would be F=ma where force is found from Coulombs law (F=q1q2/4pie0r^2). Sorry for the sloppy equation in simple text but you most likely know it. But to solve the velocity at each point you have to sum up (integrate) all of the velocities as the field drops off so you would integrate from r = inner grid diameter to R = farthest distance from the inner grid that the particle can go (this is the distance where the ion formed).
2.) The ion emerges slightly less than perpendicular to the surface of the inner grid. In this case velocity does not go to zero. Only the radial component of velocity goes to zero so the distance of flight will not go to zero at this point since there is still some perpendicular velocity component. To calculate this use the above integral but include the sine of the angle relative to the inner grid.
3.) Extreme case of #2 where the ion is in a complete circular path that takes it through the inner grid. This can never be because the force required to do this would be infinite but the reality is most likely somewhere between case #2 and #3. In this case the energy (velocity) is constant at its maximum and it will have the longest path before a collsion. No work is done in this scenario and that makes it interesting for the fusor as it maximizes the chance of a fusion for a given energy input.
I hope you can see that the distance of flight is hugely different depending on the path and much more different than a factor of pi. It is a complicated situation and if you read the Univ of Wisconson papers you will see that it is further complicated because the paths and places that fusion occurs even changes with the gas fills.
Frank S.
Achiever's madness; when enough is still not enough. ---FS
We have to stop looking at the world through our physical eyes. The universe is NOT what we see. It is the quantum world that is real. The rest is just an electron illusion. ---FS
We have to stop looking at the world through our physical eyes. The universe is NOT what we see. It is the quantum world that is real. The rest is just an electron illusion. ---FS
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Richard Hull
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Re: A calculation on recirculation in a fusor.
I never said unknowable or even uncalculable. I said that a knowledge of what is going on in a fusor would not even be handled by a FAIRLY complex analysis. It would be an extremely complex analysis and involve a lot of variables and probably best handled as a many body problem. How many bodies? You can guess at what number would provide a realistic estimate and then get a Cray to work it out.
What we have learned is there are many theoretically possible things going on in a mix in a device that is a mess mathematically and not answering first pass predictions.
What experimental work has been done and properly reported has not been done by amateurs. What has been reported is largely empirical and not mathematical as most reports realize the complexities involved through the doing and the multi-level dynamics of the system. There is much left to do, but as the system, as applied here, will never lead to a power fusion system, there is no great rush to put fine points on its mathematics which is readily perceived to be a mess.
A certain fraction (unknown) of the fusions in the system are lucky fusions due to a number of predictable processes and would certainly not be part of the theoretical base for the system in the first place and some can ultimately be discounted mathematically. Oddly, there is a suggestion and empirical evidence that points out that the very reason the system was designed (beam collisional fusion) is probably one of the "also ran", discountable processes!
All my protestations about fusor math are aimed at the very device we use and not an idealized fusor or even a real one with ion guns.....Just our little knockabout fusors. We are willing to take all the neutrons from all the fusions no matter where or when they occur including wall based target fusion which surely contributes. Other real fusion processes found in our fusors should include, fast ion-fast ion in grid collisions, fast ion-slow ion in grid collisions, fast ion-neutral in grid collisions, fast ion-fast ion out of grid collisions, fast ion-slow ion out of grid collisions, fast ion-deuterium embeded wall collisions, and finally, the empirically suggested winner, fast ion- neutral collision out of grid fusions. All fusions here certainly exist to differing amounts over differing volumes or surface areas.
Any look at MFP for ion paths is a tiny part of the iceberg here when gauged out against the number of species in varying volumes at varying energies with fusions occuring via a number of different theoretically possible process conditions.
It all integrates out in the end, which is what we see and measure. Assumptions arrived at through theory or math in this device are wide open to questioning as a myriad of theoretical processes exist, presented against a background of rather limited, often qualitative, yet empirical data.
I have considered the massive surface area of the shell as an ionizer zone and a possible target fusion zone. All are very theoretically possible.
Richard Hull
What we have learned is there are many theoretically possible things going on in a mix in a device that is a mess mathematically and not answering first pass predictions.
What experimental work has been done and properly reported has not been done by amateurs. What has been reported is largely empirical and not mathematical as most reports realize the complexities involved through the doing and the multi-level dynamics of the system. There is much left to do, but as the system, as applied here, will never lead to a power fusion system, there is no great rush to put fine points on its mathematics which is readily perceived to be a mess.
A certain fraction (unknown) of the fusions in the system are lucky fusions due to a number of predictable processes and would certainly not be part of the theoretical base for the system in the first place and some can ultimately be discounted mathematically. Oddly, there is a suggestion and empirical evidence that points out that the very reason the system was designed (beam collisional fusion) is probably one of the "also ran", discountable processes!
All my protestations about fusor math are aimed at the very device we use and not an idealized fusor or even a real one with ion guns.....Just our little knockabout fusors. We are willing to take all the neutrons from all the fusions no matter where or when they occur including wall based target fusion which surely contributes. Other real fusion processes found in our fusors should include, fast ion-fast ion in grid collisions, fast ion-slow ion in grid collisions, fast ion-neutral in grid collisions, fast ion-fast ion out of grid collisions, fast ion-slow ion out of grid collisions, fast ion-deuterium embeded wall collisions, and finally, the empirically suggested winner, fast ion- neutral collision out of grid fusions. All fusions here certainly exist to differing amounts over differing volumes or surface areas.
Any look at MFP for ion paths is a tiny part of the iceberg here when gauged out against the number of species in varying volumes at varying energies with fusions occuring via a number of different theoretically possible process conditions.
It all integrates out in the end, which is what we see and measure. Assumptions arrived at through theory or math in this device are wide open to questioning as a myriad of theoretical processes exist, presented against a background of rather limited, often qualitative, yet empirical data.
I have considered the massive surface area of the shell as an ionizer zone and a possible target fusion zone. All are very theoretically possible.
Richard Hull
Progress may have been a good thing once, but it just went on too long. - Yogi Berra
Fusion is the energy of the future....and it always will be
The more complex the idea put forward by the poor amateur, the more likely it will never see embodiment
Fusion is the energy of the future....and it always will be
The more complex the idea put forward by the poor amateur, the more likely it will never see embodiment
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Chris Bradley
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Re: A calculation on recirculation in a fusor.
Carl.
Thanks for the objective critiques. My responses:
> -reactant particles are monoenergetic
Yes, that is partly one assumption. The collision calculations I did suggest there is a 'knee point' at which a particle will very rapidly drop out of 'fusion contention' within just one or two further collisions because it gets to a point where the energy losses, which appear to be 1/E^2 related, rapidly accumulate disproportionately to the onward distance they may travel. I need to review these calculations to support this, but it essentially implies that monoenergetic is not an unreasonable first-order approximation. This also covers the point "> -reactant particles' energy is constant along their paths". Mostly, yes I think I can justfy that, within the scope of the calculation (which considers only that flight distance whilst a given particle is in a 'high fusion-energy state', and not for all of its time).
However, a critique of an assumption of monoenergetic particles AT the drive voltage is a correct one for which I have no comeback. The calculation is based on that assumption, but of course ions may be initiated at any potential in that e-field, so, agreed, it is correct to examine the collision energies of the calculation more thorughly in this regard.
> -"43kV drive == 21.5kV collision energy"
As per your second post. It presumes fast-ion into neutral. I used some data from a dataset measured in a particularly low energy range. These measured values can vary between experiments considerably in this energy range, and some of these low levels are particularly in disagreement with experiments that run up to MeVs, so I picked a representative low-energy experimental result.
> -Fusion is the dominant particle removal mechanism
Not sure what you mean by this as it doesn't seem to apply to my calculation. Perhaps the answer is the same as the first. The coulomb collision losses are quite slight to start with, but as the energy drops the fractional coulomb losses ramp up considerably giving the effect that the fast ion 'suddenly' looses its main energy, and if it hasn't fused by then, then it is lost to that sequence of coulomb scatters.
> -The beam current is due only to ions
I left this 'til last because this one you've caught me. I have no good comeback for this. For every ion formed there is an electron, which is measured on the current, of course. Also, I consider fusors to be operating in the Townsend regime which typically does not exceed a conductance of a few microamps. Therefore I took the current as a measurement of beam density. This may be true, but I did not accommodate thermionic emissions from the grid which I cannot exclude from interfering with that calculation as I don't know how hot it was. Apart from this I think its an OK assumption, but with it I agree it does tend to close the argument down. I would say, however, that the controlled experiements of Wilfred and Noemi showed little such thermonic behaviour and the same calculation holds for their data also, but this is a subjective interpretation of that data, and I must hold true to my own principle of disregarding these subjections as useful facts. Therefore, you have discredited this calculation by (at least) that observation, unless and until the magnitude of the thermionic current can be determined. The calculation fundamentally needs the overall ion-current density and the thermionic current may well be of the order of ma, and surely would be if the grid had been visibly glowing hot at the time (>orange,~2000K), thus it may dominate the calculation.
Thanks for the objective critiques. My responses:
> -reactant particles are monoenergetic
Yes, that is partly one assumption. The collision calculations I did suggest there is a 'knee point' at which a particle will very rapidly drop out of 'fusion contention' within just one or two further collisions because it gets to a point where the energy losses, which appear to be 1/E^2 related, rapidly accumulate disproportionately to the onward distance they may travel. I need to review these calculations to support this, but it essentially implies that monoenergetic is not an unreasonable first-order approximation. This also covers the point "> -reactant particles' energy is constant along their paths". Mostly, yes I think I can justfy that, within the scope of the calculation (which considers only that flight distance whilst a given particle is in a 'high fusion-energy state', and not for all of its time).
However, a critique of an assumption of monoenergetic particles AT the drive voltage is a correct one for which I have no comeback. The calculation is based on that assumption, but of course ions may be initiated at any potential in that e-field, so, agreed, it is correct to examine the collision energies of the calculation more thorughly in this regard.
> -"43kV drive == 21.5kV collision energy"
As per your second post. It presumes fast-ion into neutral. I used some data from a dataset measured in a particularly low energy range. These measured values can vary between experiments considerably in this energy range, and some of these low levels are particularly in disagreement with experiments that run up to MeVs, so I picked a representative low-energy experimental result.
> -Fusion is the dominant particle removal mechanism
Not sure what you mean by this as it doesn't seem to apply to my calculation. Perhaps the answer is the same as the first. The coulomb collision losses are quite slight to start with, but as the energy drops the fractional coulomb losses ramp up considerably giving the effect that the fast ion 'suddenly' looses its main energy, and if it hasn't fused by then, then it is lost to that sequence of coulomb scatters.
> -The beam current is due only to ions
I left this 'til last because this one you've caught me. I have no good comeback for this. For every ion formed there is an electron, which is measured on the current, of course. Also, I consider fusors to be operating in the Townsend regime which typically does not exceed a conductance of a few microamps. Therefore I took the current as a measurement of beam density. This may be true, but I did not accommodate thermionic emissions from the grid which I cannot exclude from interfering with that calculation as I don't know how hot it was. Apart from this I think its an OK assumption, but with it I agree it does tend to close the argument down. I would say, however, that the controlled experiements of Wilfred and Noemi showed little such thermonic behaviour and the same calculation holds for their data also, but this is a subjective interpretation of that data, and I must hold true to my own principle of disregarding these subjections as useful facts. Therefore, you have discredited this calculation by (at least) that observation, unless and until the magnitude of the thermionic current can be determined. The calculation fundamentally needs the overall ion-current density and the thermionic current may well be of the order of ma, and surely would be if the grid had been visibly glowing hot at the time (>orange,~2000K), thus it may dominate the calculation.
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Richard Hull
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Re: A calculation on recirculation in a fusor.
You noted that the low level d-d production fusion or neutron level results are out of line with reactions up into the mevs range.
d-d fusion drops off as you approach 2mev and the neutron numbers start to rise as the oppenhimer-phillips reaction, (stripping), takes over. So any real d-d fusion is just pounding sand after about 1 mev and there is no appreciable cross sectional payback for d-d fusion after about 250kev.
So d-d fusion in the mevs (plural) range is just not happening. You've stopped even pounding sand..... You are falling out of and away from fusion. This assumes you want fusion and not just a lot of neutrons
Richard Hull
d-d fusion drops off as you approach 2mev and the neutron numbers start to rise as the oppenhimer-phillips reaction, (stripping), takes over. So any real d-d fusion is just pounding sand after about 1 mev and there is no appreciable cross sectional payback for d-d fusion after about 250kev.
So d-d fusion in the mevs (plural) range is just not happening. You've stopped even pounding sand..... You are falling out of and away from fusion. This assumes you want fusion and not just a lot of neutrons
Richard Hull
Progress may have been a good thing once, but it just went on too long. - Yogi Berra
Fusion is the energy of the future....and it always will be
The more complex the idea put forward by the poor amateur, the more likely it will never see embodiment
Fusion is the energy of the future....and it always will be
The more complex the idea put forward by the poor amateur, the more likely it will never see embodiment
Re: A calculation on recirculation in a fusor.
Joining this discussion late, I'm afraid....
Buried deep in the exchanges on the thread Todd Massure mentions above, is some reference I made about simulations of the simple spherical fusor.
I have done a lot more since then (2004), and a kind of lengthy recirculation is indicated, when only simple electrostatic influences are modeled.
One thing we should all be aware of, is this: The inner "spherical grid" only begins to approximate a spherical electrode... at a fairly long distance away. In close, there are much more complex fields caused by the wire diameters, and junctions. The actual electrostatic fields that the ions fly through are extremely complex, and certainly on a molecular scale, not symmetric.
I did not notice any mention, in the discussions on this thread, of the radial gas density gradient that is created as soon as ions start to move. This complexity is a voltage and current dependent condition, which is also under no constraint to be radially symmetric. Indeed, it is most unlikely to be. We have evidence of this in jets, bugles and other ion produced structures that form at different pressures and operating conditions.
The only way these sorts of complexities can be dealt with is to borrow a page from the Monte Carlo simulations crowd, and do a special case examples. Make up some conditions, assume all the world is like this and solve for everything you can. When you're finished, file that answer, choose some new conditions, and repeat the same process.
Done enough times, over all the possible variations in pressure, voltage, ion density, injection trajectory and perhaps a few dozen others variables, - including photo ionization UV, VIS, Xray, illumination, secondary emission, and etc. - an ensemble of results will be collected. Appropriately summed and weighted...a picture begins to emerge of the real world of the fusor. The trouble is, this takes a few million cases, to handle something a involved as the fusor.
While I personally enjoy making these sorts of calculations, one becomes impressed at how little of what we assume is actually known, and how difficult it is to do good measurments to fill in the blanks. Thus the numbers tend to reflect more our personal views, than objective science. Still, they are instructive.
So, while all the cautionary remarks made in this thread are certainly valid, they do not actually render invalid attempts at calculation (such as Chris is making) to quantify and understand what is going on.
I like to think that a calculation attempted will at the very least expose the assumptions for analysis. Which usually leads to revisions of assumptions and often the approach. A healthy thing, this.
Our only mistake here would be to "bet the farm" on some simple results.
Do the math, make some predictions, test to see if the predictions are verified, refine and repeat.. Some time and in some way... the results will converge. More than likely, neither immediately, nor with ease... but eventually. It's the scientific method.
.Dave Cooper
Buried deep in the exchanges on the thread Todd Massure mentions above, is some reference I made about simulations of the simple spherical fusor.
I have done a lot more since then (2004), and a kind of lengthy recirculation is indicated, when only simple electrostatic influences are modeled.
One thing we should all be aware of, is this: The inner "spherical grid" only begins to approximate a spherical electrode... at a fairly long distance away. In close, there are much more complex fields caused by the wire diameters, and junctions. The actual electrostatic fields that the ions fly through are extremely complex, and certainly on a molecular scale, not symmetric.
I did not notice any mention, in the discussions on this thread, of the radial gas density gradient that is created as soon as ions start to move. This complexity is a voltage and current dependent condition, which is also under no constraint to be radially symmetric. Indeed, it is most unlikely to be. We have evidence of this in jets, bugles and other ion produced structures that form at different pressures and operating conditions.
The only way these sorts of complexities can be dealt with is to borrow a page from the Monte Carlo simulations crowd, and do a special case examples. Make up some conditions, assume all the world is like this and solve for everything you can. When you're finished, file that answer, choose some new conditions, and repeat the same process.
Done enough times, over all the possible variations in pressure, voltage, ion density, injection trajectory and perhaps a few dozen others variables, - including photo ionization UV, VIS, Xray, illumination, secondary emission, and etc. - an ensemble of results will be collected. Appropriately summed and weighted...a picture begins to emerge of the real world of the fusor. The trouble is, this takes a few million cases, to handle something a involved as the fusor.
While I personally enjoy making these sorts of calculations, one becomes impressed at how little of what we assume is actually known, and how difficult it is to do good measurments to fill in the blanks. Thus the numbers tend to reflect more our personal views, than objective science. Still, they are instructive.
So, while all the cautionary remarks made in this thread are certainly valid, they do not actually render invalid attempts at calculation (such as Chris is making) to quantify and understand what is going on.
I like to think that a calculation attempted will at the very least expose the assumptions for analysis. Which usually leads to revisions of assumptions and often the approach. A healthy thing, this.
Our only mistake here would be to "bet the farm" on some simple results.
Do the math, make some predictions, test to see if the predictions are verified, refine and repeat.. Some time and in some way... the results will converge. More than likely, neither immediately, nor with ease... but eventually. It's the scientific method.
.Dave Cooper