# Detection of multiple hits of neutrons.

1. Apr 30, 2007

### vanesch

Staff Emeritus
Hi all,

I have a question, which sounds somewhat silly, but I'm stuck with it. I'm writing some monte carlo code to simulate the effect of multiple hits on neutron detectors, and I'm confronted with the following issue.
Consider a "multiple hit" of a detector. That means, a neutron impact at positions p1, p2, p3, ... pk. I want to study the behaviour of the detector under uniform irradiation with k multiple hits, so I did some simple counting statistics for this case: I counted the different possible cases of k-hit sets {p1, p2...pk}. This gives me a certain distribution of behaviours of the detector.
However, I considered the case of {p1,p2...} and {p2,p1...} as identical: I was considering a set of k neutrons, hitting different positions, without "numbering" them, as neutrons are in principle indistinguishable particles.
Now, you can write the code differently, and say that I draw k times a uniformly distributed neutron, to form my k-hit. But that changes the statistics in the case of identical hits (which is exactly the kind of statistic I'm interested in).

Indeed, consider the simple case of only two "detection outcomes" possible, "left" or "right". Consider that I look at double hits. In my first approach, I'd say, I have 3 possibilities:
{twice "left"},
{twice "right"}
{once "left", once "right"}

Each of these different situations gets equal weight.

However, in the "independent hit" approach, we would have 4 different possibilities:
(first left, second right)
(first left, second left)
(first right, second right)
(first right, second left)

Each of these now has equal weight.

However, the second and the forth correspond to an identical physical situation of two simultaneous hits of a neutron, one left, and one right.

Clearly, these different statistics correspond to two different particle counting statistics: Maxwell-Boltzmann versus Bose Einstein. (it's funny of course to use B-E for neutrons...)

So, quantum-mechanically, I'd opt for the B-E approach, while "standard" neutron detector considerations would usually lead to the M-B approach. Problem is, for what I want to calculate, this differs quite importantly.

Last edited: Apr 30, 2007
2. Apr 30, 2007

### lpfr

For the two neutron case, I can not make de difference with a two identical coins toss. And I always find that the probability of a head and a tail is twice the probability of two heads. Maybe there is something that I'm missing.

3. Apr 30, 2007

### vanesch

Staff Emeritus
I know, maybe I'm just being silly. It is that with a two coin toss, you can say that the FIRST coin had heads, and the SECOND had tails. But I'm considering simultaneous hits - in as much as this makes physical sense, I don't know (thermal neutrons do behave rather wavy). So should I distinguish between "neutron 1" hitting at A, and, simultaneously, "neutron 2" hitting at B and the case where neutron 1 hit B and neutron 2 hit A (like the two coins), or should I simply say that *A* neutron hit A and *ANOTHER* neutron hit B at the same time ? In which case that there is no such distinction...

4. Apr 30, 2007

### lpfr

Put the two coins in an opaque container and close your eyes while throwing them.
Now you don't know which is A or B. Would the probability change? I don't think so, but with probabilities you must be very careful.

5. Apr 30, 2007

### rewebster

Isn't this a variation/interpretation of Bell?

Last edited: Apr 30, 2007
6. Apr 30, 2007

### vanesch

Staff Emeritus
As long as the coins are *in principle* distinguishable, (say, a small scratch on one of them, or a few dislocations in a crystal somewhere on which you can tag afterwards), the probabilities would remain the same. However, from the moment that they become truely indistinguishable, the probabilities would alter.

The point is, there should be a physically "taggable" distinction between "1 gave heads and 2 gave tails" and "1 gave tails and 2 gave heads" - even if I don't care to look for it. Now, the point is, for a multi-hit on a detector, within (I should verify but I guess so) coherence time of the neutrons, I'm not sure that there is, even in principle, a distinction between neutron 1 hit position A and neutron 2 hit position B on one hand, and neutron 2 hit position A and neutron 1 hit position B, on the other.

7. Apr 30, 2007

### lpfr

If you "me cherchez des pous" with the crystallographic quality of my coins I will do the same with your electronic detector. The two impulsions will never, ever, be simultaneous. There will always be a picosecond or 10^-40 seconds of difference in the detection. There will be always a first and a second.

On the other hand I will never accept that the probability will change from ¼ to 1/3 if there is a dislocation difference between the coins. But this is just my personal problem. We will never agree.

8. Apr 30, 2007

### vanesch

Staff Emeritus
Yes ; however, for that to be meaningful, the time tagging should be a property of the *neutron* and not of the local detection phenomenon. Imagine, for the sake of explanation, that a 2-neutron state hits the detector (maybe it is totally silly to talk about that, but in photon counting statistics this plays a role in the formation of the poisson statistics). This means that one should use the "unordered-list" statistics in this case. However, the detection phenomena at both interaction positions can introduce differences in time, in signal amplitude and I don't know what else. It is not because one has now detected differences in "detection time" or amplitude or I don't know what else, that the statistics of the impact positions changes.
Now, of course, if the two neutron states are slightly different, and one wavepacket hits the detector before the second does, then the time difference might be (partly or entirely) due to the signal, and not due to a local detection phenomenon. In that case, indeed, there are two different 2-neutron states possibly hitting the detector.
However, if the time spread of the wave packets is much larger than the detection separation, then you cannot reasonably say that the tiny time difference corresponds to *different* neutron states.
Now, a thermal neutron has a velocity of about 2000 m/s and we know that there are coherence lengths of several hundred micrometers (otherwise, small angle scattering wouldn't work I guess). So this means, coherence times of hundreds of nanoseconds. We are talking about smaller coincidence times here.

Now, I also think that the "right" statistics in practice are the classical statistics - but I have to say that I'm not 100% sure, when confronted to the above reasoning.

Last edited: Apr 30, 2007