Is criticality in nuclear physics just a statistical concept?

Factual Questions
1

As I understand it, there is a certain probability that an atomic nucleus of a fissile isotope, left alone to mind its own business, will split in a given second from capturing a neutron that happened to come by in the wild. This event will release the binding energy of the nucleus and several more neutrons. If there is a sufficient number of other fissile nuclei around, the neutrons will split those, which will, in turn, release more neutrons to split yet more nuclei, and we have a chain reaction. If, on the other hand, there isn’t this sufficient number of nuclei, the neutrons will fly by without splitting a nucleus, and the chain reaction dies.

Is this correct? Essentially, it would mean, if my understanding is right, that the magic of a critical mass in nuclear physics is a purely statistical concept: It denotes the mass of fissile material that you need so that the probability of a neutron to hit a nucleus and split it becomes sufficiently high to sustain the chain reaction. So theoretical physicists use stochastic methods to compute the critical mass? How reliable are these calculations - is there a possibility of random fluctuations so that even though your lump of uranium is sub-critical, you get a chain reaction because just coincidentally, more neutrons happened to hit fissile nuclei than your probabilistic calculation estimated? Or, conversely, does it mean that as soon as you have a lump of fissile material that exceeds critical mass, a chain reaction will inevitably occur without any need for ignition from the outside?

2

The number of atoms in a possibly critical mass is so great that random fluctuations are negligible.

3

One can and does, of course, have more complicated configurations than a bare sphere of uranium surrounded by nothing. So, while you can do hand calculations, people do in fact utilize Monte Carlo simulations.

4

In a similar way, a lot of physical properties of macroscopic objects are “just statistical concepts”. The density & the energy of the air molecules in the room I’m sitting in are fluctuating slightly from point to point and from moment to moment, as the molecules move around. But because the system is large, the relative size of the fluctuations is small, and so we just see the “average” value unless we’re looking at very small scales and making very precise measurements.

5

Statistical mechanics is one of the pillars of physics. So yes, physicists do use statistics for a large number of problems. Given the huge number of neutrons involved, statistical methods are entirely valid. Even starting from a single neutron.

Detailed analysis and prediction of fission processes depends upon a lot of second order issues. Even the temperature of the metal matters. A lot of the parameters are found experimentally, so you have estimates of the effective cross section of a nucleus, and probabilities for scattering, adsorption and fission as a neutron hit a nucleus.

A nice simplification can be to work out the probable distance a neutron of a given energy will travel before hitting a nucleus. A neutron will bounce around inside the material, and you might assume that scattering directions are random, and thus the actual distance travelled is a random walk - so the average distance in a mass of material is proportional to the square root of the number of collisions. If your system is exactly critical the number of neutrons escaping from the surface of the material (assume a sphere) needs to match the number of neutrons being created internally. It isn’t a huge leap to get an estimate for the radius of a sphere that meets this criterion - which gives you the critical mass. It is more complicated, and doing things like treating the bouncing neutrons like a diffusing gas can get you closer, along with more accurate experimental estimates of the complicating parameters. Famously, and very sadly, one such experiment during the Manhatten project killed the experimenters.

As noted above, Monte Carlo methods are also very useful. Basically running many computations with randomised initial conditions and looking at the spread of results gains you more insight. Straight mathematical approaches and Monte Carlo co-exist and inform one another.

As the the question about how random neutrons affect things. The answer is that every possibility is important, and this is a key part of atomic weapons. A important point about getting a useful weapon is working out how you get the thing to fission a goodly fraction of its mass before it self disassembles in a ball of plasma - a bang or a fizzle. A simple gun type weapon fires a lump of uranium into a larger lump, assembling the critical mass. Even left to its own there will be a random neutron soon enough around that will initiate fission. A plutonium weapon can’t be used this way. Too many random neutrons. Criticality will occur before the two masses even touch one another and the thing will emit a big puff of plasma and spray plutonium all over the place with no useful result. So an implosion device is required. Getting the thing into a compressed mass before it even realises. The explosion creating the implosion actually compresses the plutonium to about double normal density. Very very quickly. (The gap in technical capability needed between gun and implosion weapon is why rogue nations are usually worried about enriching uranium rather than creating plutonium.)

But you still have a problem if you wan to get a big yield. A chain reaction results in an exponential increase in neutrons, but exponential still starts slow, and you really have a fixed amount of time before the thing disassembles itself. How many doubling of neutrons you get is essentially fixed. So you really want to cut out the early phases and kick start the process by flooding the just now critical mass with neutrons at exactly the right moment. Then your exponential ramp up has some meat to work with. Implosion weapons incorporate a source in the centre of the mass that is initiated by the compression (the pit). Gun weapons can use an external neutron source. Tuning the number of neutrons provided at this moment is probably how variable yield weapons were constructed.

Neutrons are precious, and if you want to increase yield you might add something outside the fissionable mass that will bounce some of the neutrons back again. Sometimes called a neutron reflector, it really doesn’t reflect much - it just adds a lump of heavy nuclei outside the that a few of the exiting neutrons will hit and bounce back into the fissioning mass. Usually called a tamper, this add on can be made from heavy metals like tungsten, or even less fissile uranium (adding a whole new bit of complexity and potential yield). Again, model this statistically. It obviously doesn’t live very long either, but if it can bounce a reasonable number of neutrons back before evaporating the yield will increase.

Modelling neutrons as simple billiard ball like particles banging around inside a mess of much bigger hard balls gets you a long way. Statistical mechanics is one of the great ideas in physics. I remember when I first read though the derivation of the ideal gas law from nothing but ideal particles bouncing around along with conservation of energy and momentum. Utterly beautiful. Amongst other things it provided one of the core bits of evidence to cement in place the atomic theory of matter.

6

Everything is a statistical concept. The oxygen diffusing from the air you breathe into your bloodstream in your lungs. The electron bonds holding your molecules together. The very forces preventing your nuclei from undergoing spontaneous fission. Everything.

WAY back in college, my physics professor Dr. Crawford did a demonstration. He had a jar on the table, filled with air.

“Now, take this jar. What if we watched it. All day long. What is the probability that all the air molecules inside will jump one foot to the side, leaving a vacuum?” (this was said very deadpan)

Then he stood there watching the jar for at least a couple minutes in silence :joy:. The air didn’t move.

The thing is, the probability of that happening ISN’T zero. It’s infinitesimally small, much smaller than one over the number of atoms in the universe, But not zero.

7

Excellent summary, but this should be expounded on. Boosted fission weapons use a small amount of fusion to introduce neutrons fairly late in the explosion, not at the very beginning.

8

That’s because someone didn’t have a finite improbability generator.

The principle of generating small amounts of finite improbability by simply hooking the logic circuits of a Bambleweeny 57 Sub-Meson Brain to an atomic vector plotter suspended in a strong Brownian Motion producer (say a nice hot cup of tea) were well understood. It is said, by the Guide, that such generators were often used to break the ice at parties by making all the molecules in the hostess’s undergarments leap simultaneously one foot to the left, in accordance with the theory of indeterminacy.

9

That’s one heck of a post, thank you for it.

10

And in fact, criticality depends not only on the amount of the material, but on its shape.

While we’re at it, there’s another important concept if you want to build a practical nuclear reactor, rather than a bomb. Sometimes, when a fissile nucleus absorbs a neutron, it’ll split nigh-instantly and release more neutrons (“prompt fission”). Sometimes, though, it’ll sit there for a few seconds quivering in an unstable state before splitting (“delayed fission”). And for any given set of conditions, you can calculate the odds of any given nucleus doing either. If your material is packed densely enough, there will be enough neutrons flying around that even the fraction that result in prompt fission will be enough to go critical, and so things get very extreme very quickly. It’s in a “prompt critical” state. This is what you want for a bomb, but it’s not what you want for a power plant. There, you set the conditions so that the prompt fissions, by themselves, are not enough to maintain criticality, but the prompt fissions and delayed fissions combined are enough. The reactor is “delayed critical”. This enables you to control the reaction, using realistically-practical control mechanisms, because they only have to react in seconds, not microseconds.

11

If I recall correctly, criticality calculations were one of the first applications of Monte Carlo methods - if not the very first such application. I’m going to check on that.

12

Yes, Ulam introduced Monte Carlo methods at Los Alamos - and the term was invented at Los Alamos

Despite having most of the necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments.

Being secret, the work of von Neumann and Ulam required a code name.[22] A colleague of von Neumann and Ulam, Nicholas Metropolis, suggested using the name Monte Carlo, which refers to the Monte Carlo Casino in Monaco where Ulam’s uncle would borrow money from relatives to gamble.[20] Monte Carlo methods were central to the simulations required for further postwar development of nuclear weapons, including the design of the H-bomb, though severely limited by the computational tools at the time. Von Neumann, Nicholas Metropolis and others programmed the ENIAC computer to perform the first fully automated Monte Carlo calculations, of a fission weapon core, in the spring of 1948.[23]

13

Not only delayed fission, but daughter products decaying over seconds to hours.

https://www.sciencedirect.com/topics/engineering/delayed-neutron-precursor

14

Gen. Groves wrote in his book “Now It Can Be Told”, early on in the Manhattan project he grilled the physicists about how much refined uranium was going to be necessary to construct a single device. They weren’t too sure, but he thought they would be able to nail it down within a reaonable range by that point. But instead estimates varied by a factor of 10 and caused a lot of consternation about planning for the enormous gas-diffusion plants. It’s akin to having a dinner party, how much food do you prepare when maybe 10 people will show up, or maybe a hundred.