Thread: Fusion Summary?
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sunday
#57
Jul3-07, 02:23 PM
P: 31
In the discussion about the Polywell topic in Wikipedia (http://en.wikipedia.org/wiki/Talk:Polywell), there is an interesting, if intuitive, approximation to velocity distribution in a polywell.

Regrettably, the author is unknown:

(...)
Regarding Maxwellianization, this is how i see it: the well establishes a relationship between the ions distance from the center and it's radial velocity. It's velocity perpendicular to it's radial velocity is essentially an angular velocity. So let's split this up into radial and angular velocities:

angular velocity: Assuming for a moment that the angular veocity for each ion is mantained (inertial), as the ion gets closer to the center the angular velocity's contribution to the kinetic energy decreases, such that when it's in the center it is exactly zero. (because a ball traveling in a circle of radius r at angular velocity 2pi is traveling with linear velocity 2pi*r.) So as ions approach the center, their angular velocity components become non-maxwellian, such that at the center they are all exactly the same: 0.

Now, dropping that assumption, the rule still applies, because it's a law of geometry (translation of coordinate systems). However, we add in maxwellianization of the angular velocity, which will make their angular velocities approach an average, with a normal distribution. That average, since as many will be going clockwise as counterclockwise, is zero. It is only a matter of what the standard deviation of the normal distribution is. Or, more precisely, how fast the standard deviation increases or decreases with respect to the distance from the center. (This is really a crude model, because they do not actually linear velocity perpendicular to their radial velocity as they go away from the center. their angular velocity decreases. the trajectory (in two dimensions) would look more like a polar rose or trefoil knot.) in any case, it seems to me that maxwellianization will tend to reduce the standard deviation of their angular velocities, leading to better focus.

radial velocity: Again, starting with the simpler non-maxwellian case: because of the radial electrostatic gradient, the radial velocity of an ion will be a function of its initial radial velocity and distance from the center. To simplify this, one can combine initial distance from center and radial velocity by finding the distance from the center in which the radial velocity is zero. Thus, if they all start from the same distance from the center at wich the radial velocity is zero (disregarding that this is quasi-spherical rather than spherical), they will all have the same radial velocity at any given distance from the center.

Ideally, they all have zero radial velocity at the very outer edge of the sphere, thus giving them maximum kinetic energy at the center. this is what the polywell attempts to do by using microwaves to ionize the gas. according to mainstream scientific theory, the ionization rate will depend largely on how well matched the magnetic field strength at the point of ionization is to the frequency of the microwave radiation. Thus, since the magnetic field strength decreases as one goes towards the center, one can set the microwave frequency such that it ionizes the most at the very edge of the sphere.

So now lets take into account maxwellianization of the radial velocity component. And here is the key: potential energy does not maxwellianize. Sure, they will maxwellianize on the outside, all to the same low average velocity (and thus low standard deviation (i.e. "temperature"). But, since their initial distance from the center of the ions is all about the same (the very edge, where the gas is ionized), and their velocities (KE) are relatively low in comparison to their potential energy (the huge voltage gradient between the center and the outside), as they go toward the center they will be accelerated at the same rate, and thus their radial velocities relative to each other will stay the same. since maxwellianization is a function of velocities relative to each other, and not absolute velocities(throwing ice in space won't make it melt), maxwellianization will occur at the same rate, and their standard deviation ("temperature") will not increase as they go towards the center, even though their average KE gets much higher.

Now this doesn't take into account the fact that as ions are going towards the center, the same number of ions are going away from the center, at the same average radial velocity. So you have two sets of ions whose relative radial velocities are higher and higher as you approach the center. That is a possible source of additional maxwellianization. As I understand it, if they are flying past each other very fast, the ions going in opposite directions spend so little time at any distance from each other where inter-atomic forces would be significant that tehy don't really affect each other. So maxwellianization between inbound and outbound ions would occur at a higher rate as you get further from the center. - a maxwellianization that leads to zero average radial velocity. This causes the ions' "distance from the center at which their radial velocity is zero" to approach "far from the center" quicker than it approaches "close to the center". Leading towards the ideal condition mentioned earlier.

That's all very rough. Forgive me: I don't really know what I'm talking about. 69.131.30.74 00:59, 8 April 2007 (UTC)
Pity I haven't got the maths to model that intuition. My last battles with multidimensional differential operators were just before the fall of the Berlin Wall...

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