What is the required density for nuclear fusion without the use of heat?

[BrianConlee]

How high a density would need to be generated to initiate nuclear fusion for a d-d reaction to occur?

I know as we compress, temperature increases, but for a moment consider we can't heat the atoms, we can only compress them.

What kind of density would would need to be reached? (Or is it impossible to ignore the temperature here?)

 

[GeorgeRaetz]

How high a density would need to be generated to initiate nuclear fusion for a d-d reaction to occur? ...

D-D fusion requires about 1000x density compression and a minimum 100 million K temperature at least for complete "combustion".
(Experiments with plasmas can initiate fusion at much lower densities but still require the 100 million K temperature. Also they don't generate enough power output to offset the power loss)

 

[BrianConlee]

D-D fusion requires about 1000x density compression and a minimum 100 million K temperature at least for complete "combustion".
(Experiments with plasmas can initiate fusion at much lower densities but still require the 100 million K temperature. Also they don't generate enough power output to offset the power loss)

I see what you mean with 1000x density compression, but what I'm wondering exactly is what if you kept the temperature constant and just kept increasing the density.

The more I think about it, the more I'm thinking I'd have to calculate the density of 2 atomic masses within range for the strong force to take over. Something tells me that density level is incredibly large, but technically, would bringing the density to this level mean you don't need the 100 million K to achieve fusion??

Is this density approaching that of a neutron star here?

 

[GeorgeRaetz]

I see what you mean with 1000x density compression, but what I'm wondering exactly is what if you kept the temperature constant and just kept increasing the density.

The more I think about it, the more I'm thinking I'd have to calculate the density of 2 atomic masses within range for the strong force to take over. Something tells me that density level is incredibly large, but technically, would bringing the density to this level mean you don't need the 100 million K to achieve fusion??

Is this density approaching that of a neutron star here?

we can infer from muon catalyzed fusion experiments that a 200 fold linear compression leads to a fusion rate of ~10^11 sec^-1 . That would lead to a fantasically high power of ~10^23 watts per mole (at room temp)!
The corresponding density compression would be 200^3=8E6.
That's far short of the density of neutron stars however, they exist at about 10^15 density compression.

 

[sophiecentaur]

I'm not sure that keeping the temperature constant is really worth considering. After all, once the fusion starts, there will be a lot of energy about!

 

[BrianConlee]

george,

so, in theory, if we could attain and maintain that kind of compression level, we could keep a high power fusion reaction going.

I know this goes without saying fuel in and waste helium "ash" out. (imagine a continuous reaction... or perhaps a pulsed)


Sophie, you make a wonderful point about the resulting heat, lol. But what I was imagining was we merely apply compression and retrieve heat for energy only. Of course, added heat would surely increase the fusion rate.

 

[GeorgeRaetz]

I see what you mean with 1000x density compression, but what I'm wondering exactly is what if you kept the temperature constant and just kept increasing the density...?

Here's a definite answer to the question: Will high density make up for low temperature in a fusion reaction? Short answer NO: at least a million degrees K is needed to obtain a significant fusion reaction rate at any reasonable density.Short explanation is that the fusion reaction rate varies as the product of two functions: the first one is the square of the density; the second function is independent of density but varies strongly (~exponentially) with temperature.
The reaction rate for given number density nD of Deuterium at a given temperature is:
RD=nD² <sigma(v) * v>.
Here sigma(v) is the fusion cross section as a function of velocity (or center of mass energy as it is usually tabulated), v is the relative velocity of the deutrons, and <...> denotes the average over the Maxwell distribution at a given temperature.
For deuterium at 500 times normal density (80 gm/cm^3) and at a million K temperature, there will be about 1E24 reactions per cubic meter per second. Using the rough figure of 10Mev energy per DD fusion reaction, the power output is about 1.6E12 watt per cubic meter. (when referring to watts per cubic meter etc, I am referring to the compressed volume). However, at 100 thousand K at the same density, the reaction rate drops to only 250 reactions m^-3 sec^-1 with a useless power output of ~4E-10 watt/m^3. (It's surprising that the reaction rate drops 22 orders of magnitude with only one order of magnitude drop in temperature, but that's because the cross section drops as ~exp(-1/v)) while the number of particles with a given velocity drops ~exp(-1/T) ).
At ten thousand K,even a density compression of 8 million yields only ~3E-50 watt/m^3. The figure I gave in the last post of 10^11 reactions/sec was very misleading--it applies only to Deutrons in a muon bound molecule, not free deutrons.That's because the nuclei bound in a molecule don't sit still. The ground state kinetic energy of the muon bound hydrogen ion is about 312 electron volts; the equivalent temperature is 3.6 million K.
The physical reason for the importance of high temperature is that fusion relies on quantum tunneling. They say the fusion barrier is about 200kev and kinetic energy is needed to break that barrier.

Needless to say, these numbers are subject to correction. I believe the orders of magnitude are OK but factors of two errors are hard to avoid.

References
J.D. Jackson: Catalysis of Nuclear Reactions between Hydrogen Isotopes by mu- mesons, PhysRev V106 n2 p330 (1957)
Ya B Zel'dovich & S.S. Gershtein: Nuclear Reactions in Cold Hydrogen, Soviet Physics V3 n4 p593(1961)

 

[sophiecentaur]

Could we sum up by saying that two particles won't interact, however close they are, unless they are traveling towards each other with sufficient energy (i.e.temperature)? Hence, pressure / density is not enough.

 

[GeorgeRaetz]

Could we sum up by saying that two particles won't interact, however close they are, unless they are traveling towards each other with sufficient energy (i.e.temperature)? Hence, pressure / density is not enough.

In chemistry they call it activation energy; molecules have to bump into each other with a certain minimum kinetic energy to initiate a reaction. This energy is of the order of an electron volt for ordinary chemical reactions (one electron volt per molecule ~= 100 kJ/mole). A Catalyst can lower this energy considerably.
The activation energy of fusion reactions is a hundred thousand electron volts or more(10 GJ/mole); that's an equivalent temperature of about a billion degrees K. Quantum tunneling together with the existence of a few high kinetic energy particles in the Maxwell distribution allows fusion down to maybe a million degrees.
The only known "catalyst" for fusion is the muon, but that catalyst is very quickly "poisoned" by the reaction products and is believed useless.

 

[BrianConlee]

I know our sun has a much higher density than any tokamak we've yet built, but it's temperature is a lot lower.

I was wondering why we don't attempt higher densities in our fusion projects to more closely model the sun? Is it because it has a relatively low fusion rate? (due to the low temperatures -regardless of the incredible density)

The order of magnitude is really a great way to explain this all, thank you!

 

[BrianConlee]

I thought of another brief thought experiment you can use to illustrate your point...

We "contain" our sun in a huge container. We then "quench" it, and cool it down to room temperature, while preventing it from collapsing or expanding. Density stays constant.

Then we take the sun out of the container and turn off our forcefield that is keeping the density constant.

What happens? Does is start collapsing and heat back up until fusion begins?

Does it's density slowly get the fusion furnace going again? maybe a collapse increases the density and slowly starts up the fusion.

Does it collapse into a neutron star with the lack of radiation pressure? Never reaching the heat for fusion again?
From what you've told me, I'm thinking it will begin to collapse until it gets enough heat to get the fusion process to begin. Then the radiation pressure will expand it back out to it's original size. Just a guess

 

[GeorgeRaetz]

...I was wondering why we don't attempt higher densities in our fusion projects to more closely model the sun? Is it because it has a relatively low fusion rate?...

The sun supplies energy at a remarkably low power density: less than 300 watt/cubic meter. Say a man eats 4000(big) Calories per day. That's roughly 200 watts. Now divide that by a typical human volume of less than a tenth cubic meter and you will see the Sun is actually a gentle giant. It operates at biological power density. The Sun puts out enormous energy only because it is enormous. If someone managed to duplicate the Sun's fusion process in a tank the size of a garbage can, he/she could just as well replace that fusion reactor with fresh compost. The biological composting process will produce similar power. (see Wikipedia: http://en.wikipedia.org/wiki/Sun#Core)
The reason for the low power density is the exceedingly low cross section of the proton fusion. That cross section is estimated at 10^-23 barn; an inconceivably small value which renders that fusion reaction totally impractical by any known process on Earth (Deuterium Fusion cross sections are typically a few mili-barns. DT maybe 200 times higher than DD).
By the way, when a researcher in "controlled fusion" (or spokesperson) claims they will "Harness the energy of the sun" or "put a star in a jar" etc., that's pure PR flack. It's rubbish in other words(actually that's being unfair to rubbish, some of it is useful in a compost heap, but not for energy production).

 

[GeorgeRaetz]

...
We "contain" our sun in a huge container. We then "quench" it, and cool it down to room temperature, while preventing it from collapsing or expanding. Density stays constant.
What happens? Does is start collapsing and heat back up until fusion begins?
...

We can roughly estimate the gravity pressure from the energy U, of a self gravitating sphere of volume V, containing particles of mass m at density of n particles per unit volume.
U=-Cn²*V^5/3
Where the constant
C==(36pi)^1/3mG/5
Divide this Energy by V to obtain a rough estimate of the (negative) gravity pressure which tends to hold the particles together. Because the negative gravity pressure varies as ~V^2/3 it only holds a very large object together. I get a pressure of ~-70 million atmosphere for an Earth sized object with 1000 x density at 300K. Now the pressure that makes the thing fly apart is harder to calculate, at least for room temperature but it is independent of volume. The literature gives figures of roughly 600 million atmosphere for that kind of density. So, even an Earth size chunk would fly apart. Smaller ones don't stand a chance.