Happy New Year. We at FAS are a serious, hard-working lot but I thought I would start the year with a blog somewhat less Earth-shattering than we normally do. The following is the result of some research made possible by free time over the holidays.
It is with a combination of despair and delight that we discover that what we thought to be true is not. I guess it is all part of learning. Thus with mousetraps and ping-pong balls.
I often get science questions from film-makers. Most recently from a production group doing a piece on nuclear energy and nuclear weapons. I was explaining chain reactions and mentioned the famous demonstration using mousetraps and ping-pong balls. If you are a techie sort reading this blog, you have probably seen one demonstration or another; from my childhood, I can remember one featured in Disney’s 1957 Our Friend the Atom. (The cloud chamber demo is also very cool, and check out the extremely trendy lab coat the female lab tech is wearing; the 50’s were, indeed, the good old days).
After this discussion with the film team, I got online and started looking for other examples of the demonstration. I expected to find some but not many. I was particularly curious to find how large was the biggest layout that had ever been done. Doing this demonstration is a lot of work after all, setting all those mousetraps and carefully placing the ping-pong balls on top. And one mistake and…start all over. (And what do you do afterwards with hundreds of mousetraps and ping-pong balls?)
In my search, I discovered a few very slow motion videos that show that the demonstration is entirely misleading. The mousetraps-with-ping-pong-balls is a terrible analog for a nuclear chain reaction.
The basic idea is simple: In a nuclear fission chain reaction, one neutron splits an atomic nucleus, or induces the fission of the nucleus. (I think the latter, while fancy, is actually a better term. “Splitting” the nucleus suggests to me shooting a bowling ball with a rifle bullet and breaking it in two. That isn’t really what happens. The neutron may be fast or it may be very slowly passing by when it falls into the nucleus and finds an unmated neutron. The binding energy of those neutrons releases so much energy into the nucleus that it comes apart. The kinetic energy of the neutron can be negligible compared to the binding energy released. That is why slow moving neutrons are more likely than fast neutrons to cause the fission of a susceptible nucleus: the kinetic energy of the neutron is unneeded and the slower neutrons are easier to catch.) The nucleus splits into two big chunks, which will become the nuclei of new, lighter atoms, but in the process it releases a couple of neutrons. These neutrons go on to cause additional fissions, which produces more neutrons, which produce more fissions, and so on. If the reaction is allowed to go as fast as possible, the number of neutrons and the rate of fission increase exponentially and very rapidly and an explosion results. In a nuclear reactor, the rate of reaction is carefully controlled to keep a steady rate of fission.
So the macroscopic model is the array of mousetraps with each trap topped by a pair of ping-pong balls. One ball is dropped onto the array and trips a trap. It is the energy of the trap, not the energy of the ball that flings the other two balls up in the air. These come down, setting off two more traps, releasing four balls. These four balls set off four traps releasing eight balls, and so forth and, in a few seconds, ping-pong balls are flying everywhere and all the traps have been tripped.
Or so we were lead to believe. (We can’t even believe Walt Disney?) But some of the videos are slow motion and bear careful study. This one in particular (be sure to have your sound turned on) shows that most of the traps are tripped, not by balls, but by flying traps. Note that balls often hit traps and knock the balls free without tripping the traps. This is the equivalent of the (n,3n) nuclear reaction, which would be very rare because it requires breaking apart a pair of fermions, and fermions just love their pairs. A much more common nuclear reaction is (n,2n), which is equivalent to having one ball come in and knock just one ball off. With nuclei, this is not common enough to sustain a chain reaction.
I have not done a careful counting but my uncareful counting shows that most of the ball impacts occur without setting traps off. That is OK, it is the equivalent of a collision in which no reaction takes place, which is, at least for some energies, the most likely thing to happen when a neutron hits a nucleus. In the video above, keep your eye on one trap about a third of the way up from the bottom and a quarter of the way in from the right hand side. It is hit by several balls during the whole chain reaction and at the end remains unsprung.
Of course, eventually a ball will set off a trap but then something interesting happens: the trap goes flying. (If we really wanted to demonstrate fission, the trap would have to split in two, but we will overlook that.) And it is the big, heavy trap that sets off neighboring traps when it hits the ground. The analog would be that the fission products, not the neutrons, induce further fissions, say, a big heavy cesium nucleus runs into the nearby uranium nucleus and caused it to split. Of course, that never happens.
Another interesting case is here; at time 2:05, note right along the center, a column of traps snaps in sequence when clearly the balls go flying straight up and do not have time to set off the next trap. What is happening is that the hammer (that is the bar that actually whacks the mouse on the back of his neck; I had to look it up) swings from one side to the other and, through conservation of momentum, the base of the traps scoots along in the opposite direction. The trap then whacks the next trap in line, setting it off, and that hits the next, and so on. The video shows clearly that the ping-pong balls have nothing to do with the process in that particular case. (Before you get all huffy about conservation of momentum, remember that, because these demonstrations are taking place in a gravitational field with the traps sitting on the floor, there is an asymmetry, for example, when the hammer is going up, it pushes the trap against the floor and doesn’t move it but when the hammer is on the way down, it can lift the trap up and scoot it in the opposite direction.)
Obviously, some balls do trip some traps; I am just saying that that is not the dominant process, it is traps tripping traps. To see a nice example of traps tripping traps, watch the first video above, right in the center, at 3:08, and see one trap land on another, be actually trapped by it, and then the two go off in a nice 2001: A Space Odyssey-style weightless waltz. I guess this is the equivalent of a fusion that might be seen in a heavy ion accelerator.
So, what would be a good analog? If you meet the following criteria:
(1) You want to demonstrate several aspects of neutron-induced nuclear fission chain reactions,
(2) You have several hundred mousetraps and twice as many ping-pong balls handy, and
(3) You have WAY too much time on your hands,
then you may want to construct the following demonstrations. I know I am not going to be doing this. (And don’t forget to video tape everything and post it on YouTube and let me know.)
First, and most important, the mousetraps have to be glued down. I think it might work to get small pieces of plywood and glue traps to them in sets of 20 or so. Then the number of traps can be varied (see below) and the big plywood base isn’t going to move. You can see one example of fixed traps here, but note the perfect reflector (see below).
But it gets worse. Note from the video that the balls hitting the traps routinely knock the resting balls free without setting off the trap. This is equivalent to the (n,3n) reaction cited above, which is quite rare. Even the (n,2n) reaction is not common enough to sustain a chain reaction. To make our model faithful, we need to stop this reaction. Perhaps a small spot of glue could be used to fasten each ball to the trap. The glue would have to be strong enough to hold when hit by another ping-pong ball, but the bond would have to be weak enough to allow the balls to go flying when the trap was tripped. I think it would just require some experimenting with different types of glue. If the hammer is steel, then perhaps a small magnet poked inside each ping-pong ball would be enough to hold it on and stop the (n,2n) reactions. (I tested a Victor mouse trap. The hammer looks to be copper but must be just copper coating over steel because the hammer is attracted by a magnet.) The effects of having lots of magnetic ping-pong balls flying around might be interesting.
Note that in all the demonstrations, the traps are enclosed by walls (often mirrors to fool you into thinking there are more traps than there actually are). This is equivalent to having a perfect neutron reflector, which doesn’t exist. The height of the walls could be adjusted to show the effect of having some balls escape.
Ironically, the more energetic balls are good analogs for the slowest neutrons. By flying high, they come down with the greatest force and are more likely to “induce a fission,” that is, set off a trap, just as slow neutrons have a greater likelihood, described by a larger “cross-section,” of inducing a fission. In addition, by flying high, they take a long time to come back down just as slow neutrons take longer to get from one nucleus to another. So just lowering the walls of the cage has the effect of letting the slow neutrons out and keeping the fast neutrons in. Hmmm, that doesn’t seem right. Maybe we need to put holes in the sides rather than simply lower the walls or just make a picket fence, with slats and gaps between.
With leaky walls that allow some balls to escape, we will discover the mousetrap equivalent of a critical mass. Below some number, mousetraps within a lower wall, the will not be able to sustain a chain reaction because each reaction will result in two new balls being released but if, on average, more than one ball escapes, then the reaction will die out. That is, it will not “go critical.” But by increasing the number of traps, just increasing the floor space covered by traps, we reduce the surface to volume ratio or, since this is really a two dimensional model, the circumference to area ratio, and the chain reaction should be sustaining because the probability is increased that any given ping-pong ball will induce another reaction before jumping over the wall. This is how a gun-assembled nuclear bomb works, taking two masses of fissionable material and bringing them together quickly to form a mass large enough to sustain a chain reaction.
We could also show how criticality depends not just on the number of traps but on their density. If we placed the traps, not side-by-side, but with some separation, then some of the balls would land between the traps and not set another trap off. The ball would bounce and might bounce over the wall, being lost. If balls bounced out frequently enough, the reaction would not be sustaining. But the same number of traps, simply packed closer, could sustain a reaction. (This is how an implosion bomb works, the amount of mass is constant but the compression increases the density.)
We could also demonstrate the difference between critical masses of different nuclei. The critical mass of plutonium-239 is much less than uranium-235. Why? Two big reasons: (1) When a plutonium nucleus splits, it emits slightly more neutrons on average than when a uranium nucleus does. So we could load the traps with different numbers of ping-pong balls, some with two, some with only one. A lower average number would represent uranium and a higher average would represent plutonium. We would find that we need fewer traps loaded with more balls to sustain a chain reaction. (2) A plutonium nucleus is more likely to be split by a neutron than a uranium nucleus is. The part of the trap where you put the cheese is called the catch (I had to look that up, too). Most traps have just a little metal tab for the cheese (I find peanut butter actually works better) but some traps have a larger plastic tab that the mouse can step on. Such a tab would be equivalent to a larger neutron cross section. We would find that fewer large-catch traps would be needed to sustain a reaction than small-catch traps, just as less plutonium is needed compared to uranium.
So there is some hope for the mousetrap analog but it would be a lot more work. I suspect we will continue to see the simple mousetrap demonstrations and I will just pretend not to have all the above objections because I do think it is very cool.
If you really want to impress me, you could do all of the above using rat traps and billiard balls. (But stand back.)
I hope our readers are not devastated by having a cherished childhood image crushed but we are, after all, the Federation of American Scientists and sworn to the relentless pursuit of truth.
So much for the holiday break. Back to saving the world.