Fusor Math

First, a quote from R.W. Hamming: "The purpose of computing is insight, not numbers." Use the math to get a feel for what is going on, but don't expect it to predict everything that will happen.

This page has descriptions and sample calculations for things of interest for fusorites:

Reaction Rates and Cross Sections

Detection Rates


Reaction Rates

Nuclear reaction parameters are usually described in terms of their "cross section" (sigma, σ ), given in barns (1E-24 cm2) (as in "can't hit the broad side of a.."). A barn is roughly the actual physical cross section of a typical nucleus, so, amusing name aside, it is of a scale that is useful. The cross section relates to a simple physical model of a beam of point particles (i.e. infinitely small) travelling through a field of targets, each with an area, perpendicular to the beam. (An analogy describing cross section) The event rate (or reaction rate) is simply

events / time = cross section * Intensity of beam * number density of targets

where the intensity of the beam is specified in terms o fthe number of particles per unit time crossing an area (A) that is the cross section of the beam (i.e. it is particles/second/area)

The number density of targets is the number of targets in a unit volume (for gases, this is easy, PV=nRT).

This equation gives you the number of collisions per unit time. Of course, not every collision results in some specified reaction, so you could add a probability to the equation, and get the real number of events:

events/time = probability * cross section * intensity * numberdensity

As it happens, the actual physical cross section of a nucleus isn't actually measureable (since it would require interacting with something, and that interaction would have a probability that is <1, and quantum effects would bite you, etc.), so we use what you can measure (reaction rates), and calculate back to effective cross section (i.e. area), which folds in the probability and the physical cross section, as well as the relative sizes of target and projectile, etc.


A practical example

Lets say we want to calculate the reaction rate of a beam of deuterons hitting a stationary target of deuterium gas. The beam current will be 1 mA, and it's cross sectional area will be 1 cm^2. The target is a cell of deuterium at 20degrees C at a pressure of 1E-3 torr that is 1 cm thick. Finally, lets assume that the deuteron beam has an energy of 50 keV.

Step1: What is the beam intensity?

We know the current: 1 mA, so we know how many deuterons per second are flowing in the beam: 6.24E15 /second (1 Amp = 1 Coulomb/second = 6.242E18 electrons/second). We've got the area of the beam as well (1 cm^2).

Step 2: What is the target number density?

Use PV=nRT, rearranged as n/V = P/RT.

R = 8314.3 J/(kmol K)

P = 0.133 Pascal (1 torr = 133 Pa)

T = 273+20 = 293K

n/V = 0.133/(8314.3 * 293) = 54.7E-6 mols/cubic meter = 54.7E-12 mols/cc

54.7E-12 * 6.022E23 atoms/mol = 3.29E13 atoms/cc

Step 3: look up the cross section for the reaction of interest

50 keV d(d,n) reaction.. 4.55E-3 barn (= 4.55E-27 sq cm)

Fusor reactions of interest can be found at sigma.htm (which will open a new window in the browser)

Step 4: multiply it out...

reaction rate = 4.55E-27 cm^2 * 6.24E15 /(cm^2*second) * 3.29E13 atoms/cc * 1 cc (target vol)

= 934 reactions/second


Detection rates

There are a LOT of factors that go into detection probabilities. There is first the geometric issue (the particles of interest may go in all directions, but your sensor only covers some), which can be influenced by the angular distribution of the particles. For instance, a beam of deuterons hitting a stationary deuterium target does NOT send all the neutrons off in a nice spherical distribution. And, depending on the particle, the ones that don't head right towards the detector might get scattered off something else and then wind up at the detector anyway.

Finally, there is a detector efficiency issue (i.e. the detector doesn't necessarily detect every particle). Depending on the detector, the sensitivity might vary with particle energy as well.

However for a first approximation, a simple calculation will suffice. Assume that the particles of interest are emerging from a point source, and evenly distributed over a sphere. Then, imagine a sphere at the radius of your detector. All you need to do is calculate the fraction of that sphere that your detector occupies:

Detection rate = reaction rate * (diameter of detector^2) / (16 * distance to detector^2)

A sample calculation

Assume a 2" diameter BC-720 neutron scintillator, at 30 cm from the center of the reaction described in the previous example, which yields about 934 neutrons/second.


detection rate = 934 * 5.08^2 / (16 * 30^2) = 934 * 1.79E-3

= 1.67 potential detections/second.

In practice, the neutron detector might have an efficiency much less than 100%. Note also that if you move the detector closer, the rate gets higher faster: halving the distance will quadruple the detection rate. (Of course, moving it closer might also introduce noise and other artifacts)

nuc/fusormath.htm - 6 January 1999, Jim Lux
revised 13 April 2004 - fixed error in ratio of areas
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