Fusion energy from a D-D reaction – just the facts
By Richard Hull
The reductions below are incredibly easy to follow by any 8th grade student who has passed
math. However, the units and scientific notation used, to be fully understood, might require a
bright student who has taken and retained his high school physics.
D-D fusion is a 50:50 reaction as seen in the diagram. For each fusion only one of these
branched reactions can take place.
Example: If we detect one million, (1e6), neutrons per second in our neutron counting effort
during fusion, this means that two million fusions have occurred in that second. From this we
know that we fused two million different D-D reactions in that second or we produced 2e6
fusions per second. Of these two million reactions, only one million produced a reaction that
produced a neutron.
This still means that in one second we produced 1e6 neutrons, 1e6 protons, 1e6 tritons, 1e6
3He, (helium 3), nuclei.
Side note: You will notice what appears to be a helium 4, (4He) nucleus formed in the center of the diagram.
While this may be, in fact, the case, the reaction is so rapid and energetic that in most every instance, it instantly
decays, (billionths of a second), into the fusion ash or debris seen to the right. You probably noticed that I said,
“in most every instance”. It seems that on extremely rare D-D fusions, a stable helium 4 nucleus is formed with a
huge release of energy. No precise rate is known for these very rare events, but it is generally agreed that for
every 25,000 D-D fusions, only one stable 4He nucleus is formed. Thus, in a million D-D fusions we might expect
only about 40 helium 4 atoms to be formed.
Let us pose a question: How many joules of energy did we really produce should all 4 particles
be absorbed and turned into joule heat. (Crash into an ideal absorber)
It is to be remembered that this is all the energy available from 2 million fusions/second.
We have 1e6 D-D reactions in the neutron branch, each of which, totals 3.27 meV
We have 1e6 D-D reactions in the proton branch, each of which, totals 4.03 meV
Converting to energy by ergs for each branch
1.6e-6 ergs/meV X 3.27 = 5.2e-6 ergs/neutron branch fusion
1.6e-6 ergs/meV X 4.03 = 1.3e-5 ergs/proton branch fusion
We produced 1e6 of each of these separate reactions in a second (joule energy is per second)
Thus:
1e6 X 5.2e-6 =
5.2 ergs of neutron branch energy in a second
1e6 X 1.3e-5 =
13 ergs of proton branch energy in a second
Total energy produced from 2 million D-D fusions per second is thus:
18.2 ergs.
There are 1e7 ergs per joule 18.2/10,000,000 = 1.8e-6
This means for a continuous 1million neutrons per second on our neutron counters, the fusion
energy produced by 2 million D-D fusions, produced a grand total of:
1.8 microwatts or true fusion energy
As this performance in a common fusor might require an average of 40kv X 10 ma or 400 watts
of energy input, the pure energy output to input ratio is: 1.8-6/400 = 4.6e-9
now 1 / 4.6e-9 = 2.18e8
This means we have had to put in ~220 million times more energy that we get out in fusion
energy.
Now, you need not do such calculations constantly
The above may be simply scaled for energy output only.
Let us say we are getting 2e12 neutrons/sec or 6 orders of magnitude better than a fabulous Amateur
fusor. (improved to 1 million times higher output)
we would get 1e6 X 1.8e-6 = 1.8 watts of fusion energy output.
Just for grins, let us say the input energy scaled to the same degree. (an assumption, of course)
To get 1.8 watts out of the fusor we would have to put in:
1e6 X 4e2 = 4e8 watts
we would have to feed 400 million watts into the fusor to get 1.8 watts of fusion energy out. You can
see there is no path to useful fusion energy production in any fusor while doing fusion.
For those new to fusion
Where does all that energy that I put in go if I only get out microwatts of fusion for 400 watts put in?
It goes where all energy loses go in such situations……Waste Heat!
All the electrons slam into the walls producing heat. (the biggest loss). All the accelerated high-speed
deuterons that do not fuse, ( virtually 100%), turn into high speed neutrals and crash into the walls.
(Almost all the remaining energy). X-rays, lots of them, make up the rest of the energy loss. The grid in
the center heats as well, (electron and deuteron collisions), this transfers by conduction to the insulator
and the wall. (oh, the wall is the fusor reaction chamber’s electrically grounded metal shell or exterior.)
This is usually made of stainless steel and with 400 watts fed in, it heats to the point that you can burn
your hand. One might look at the amateur fusor as an efficient space heater.
An explanation of the fusion process (simplified)
Energy is energy you can’t pull it from space. At least not by any process we know of. Fusion is a
quantum tunneling process. As such, it is probabilistic by its very nature. These probabilities are like
throwing dice. Every throw will not produce “box cars”. You will have to figure the odds on the number
of throws that will be needed to repeat a box car throw. This involves statistics. As noted, fusion is
probabilistic. However, we need not involve ourselves with statistics. It has all been worked out for us.
It turns out that in nuclear physics, empirical experiments have derived a probability chart for us. It is
called a “fusion cross section chart” This chart is a battle between two throws of a dice for fusion to
take place just once! It was found that as a particle, (deuteron or deuterium nucleus), travels faster,
(gains more kinetic energy), the fusion rate, (probability), goes up. By imagining that the more energy a
deuteron has, its “cross section”, (area increases), much like a balloon swelling up. This makes, it more
likely to crash into and fuse with another deuteron of equal energy. The unit used to describe cross
section is the “barn”. The barn is equal to an imaginary circular area of 1e-24 square centimeters.
It is to be remembered that the cross section is a mime of probability. As such, even if the “fusion
energy” deuterons approach dead on, Quantum uncertainty rules, based on whether the natural
coulombic repulsion of the positively charged nuclei can be overcome and allow them to fuse. These are
the dice thrown in fusion. You will not fuse every time two deuterons meet.
This fact alone keeps stars burning for billions of years. If every hydrogen atom fused that met in the
core of a star, it would burn its atmospheric fuel up in a short time period. The universe would consume
itself almost instantly!
Thank goodness fusion is such a rare an event. This fact makes us happy we are still here due to the
rules of physics. However, it makes us sad in that fusion is so difficult and nearly impossible to do here
on earth in a controlled manner.
The chart seen below will help explain the realities of fusion. On the x-axis, we plot the matching energy
of two particles colliding in a fusion friendly environment, (perfect vacuum), versus the y axis cross
section that the deuterons have attained at the given energy seen on the x-axis.
From the graph, we see that the green line represents the D-D fusion cross section that is possible
within the amateur fusor. All the other fusion fuels on the chart are intrinsically impossible for the
amateur to work with. The best reaction is D-T, but the amateur cannot secure the NRC license to
obtain it legally as Tritium is highly radioactive. Helium 3 gas can be had, but at tremendous cost.
However, it doesn’t begin to perform better than D-D until you apply an acceleration voltage in excess of
100,000 volts! The P-B11 reaction involves the corrosive and lethal Boron Tri-fluoride gas and, then,
only beats D-D at 300,000 volts applied. D-D fusion is marginally possible at 10,000 volts applied.
However, fusion is only detectable by amateur means at around 20,000 volts. Most great amateur
fusors work at or above 40,000 volts. As seen in the chart, there is little to be gained in D-D fusion after
about 90,000 volts applied. The cross section is leveling off.
Moving into the physics of the near impossible related to fusion.
The reason fusion is so pitifully difficult is due to the quantum limitations placed on matter versus the
coulombic forces. This probabilistic process is called “Quantum Tunneling”. You might ask what if we
just threw the quantum dice more often? We can do this by putting more nuclei in the same space, thus
increasing the number of possible fusions in a specified time period within the denser plasma.
Fusion has a special criterion related to doing it successfully. It is called the “Lawson Criterion”. There is
an equation related to this. From Wikipedia:
The Lawson criterion is a figure of merit used in nuclear fusion research. It compares the rate of
energy being generated by fusion reactions within the fusion fuel to the rate of energy losses to the
environment. When the rate of production is higher than the rate of loss, and enough of that energy
is captured by the system, the system is said to be ignited.
The central concept of the Lawson criterion is an examination of the energy balance for any fusion
power plant using a hot plasma. This is shown below:
Net power = Efficiency × (Fusion − Radiation loss − Conduction loss)
1. Net power is the excess power beyond that needed internally for the process to proceed in
any fusion power plant.
2. Efficiency is how much energy is needed to drive the device and how well it collects energy
from the reactions.
3. Fusion is rate of energy generated by the fusion reactions.
4. Radiation loss is the energy lost as light (including X-rays) leaving the plasma.
5. Conduction loss is the energy lost as particles leave the plasma, carrying away energy.
Thank you, Wikipedia!
Now, what all this means is that all fusion is lossy if you can’t contain the particles at fusion energy.
These particles, (deuterons), are energetic, remember. Can you begin to imagine the speed of a deuteron
at the peek energy of about 800,000eV! In a dense wad of these “fusion energy” particles, they want to
blast out in every direction. (remember, they are all positive nuclei, (deuterons) under extreme coulombic
repulsion. The more densely you try and pack them to increase fusion probabilities, the more they want
out! The more fusions within even a fully contained plasma of fusion ready deuterons, will be disturbed by
the flying hyper energetic fusion debris, (multi-meV tritons, helium3 nuclei, protons). What’s more, all of
these will add to the coulombic repulsion within the plasma as the fusion debris are all positively charged
nuclei!
Can you begin to see that the laws of physics hate fusion and do all they can to fight any attempt to make
it happen!? Thus, we note that only stars with the Quintillions of tons of hydrogen pressing down on the
hydrogen core via gravitation, can barely begin to fuse hydrogen, (protons)! Even this is done with gross
inefficiency in a controlled manner. (thank goodness)
The cross section for P-P fusion is so low that even with the force of gravity, the protons barely make the
thermal energy to fuse at all. Solar energy comes from the higher probability that the protons are
contained and compressed by gravity against their natural electrostatic repulsion. The density of protons
just making fusion energy is so great that quantum tunneling probability increases dramatically. The core
of the star burns continuously, doing fusion with fresh protons replacing the fusion debris and energy
pours out towards the surface of the star.
This is the nuclear fusion process which fuels the Sun and other stars which have core temperatures less
than 15 million Kelvin. A reaction cycle yields about 25 MeV of energy.
Unfortunately, due to several factors, one sees that the probability of fusion within the sun’s core per
collision is ~2e-31!
From the above, one might wonder, why bother with even the best fusion fuels if nature is against us to
such a degree.
I wonder myself as I see the 60 years of continuous, unending, abject failures by all the king’s horses and
all the kings men, spending billions of the money substances in the attempt to do useful, controlled fusion,
and we are not even close!
I hold to the old saying, “ Fusion is the energy of the future and always will be!”
I have coined my one predictive: No child born on this day shall ever live so long as to see one watt of
fusion generated energy issue from his home’s electrical wall outlet.
Richard Hull