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I'm trying to understand the first nuclear reactor, the Chicago Pile-1, specifically the math Fermi did to figure out when the reactor would go critical. There's a nice report available from Fermi, where he tracks the value of $R_{eff}^2/A$, where $R_{eff}$ is the effective radius of the pile, and A is the measured neutron intensity at the center of the pile, some screenshots below.

Fermi then claims that when $R_{eff}^2/A$ reaches 0, the pile will become critical. This is where I get lost - I read through the report and some other sources, but I don's see where this math is coming from -> why should the pile become critical when $R_{eff}^2/A=0$?

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    $\begingroup$ Clearly, criticality is associated to $A$ becoming divergent, which kind of makes sense. I guess the question really is why he prefers to plot $R_{eff}^2/A$ instead of just $A$, right ? $\endgroup$
    – Adam
    Commented Feb 14, 2023 at 13:14
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    $\begingroup$ At some volume of the pile ($R_{eff}$ estimated from an ellipse as they built up towards the spherical pile end state) (see page 4), neutron losses to the outside world become irrelevant, and you go critical. When you go critical, the neutron activity (A) at the center of the pile goes to infinity. $\endgroup$
    – Jon Custer
    Commented Feb 14, 2023 at 13:25
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    $\begingroup$ physics.stackexchange.com/q/696091 may be of some interest. $\endgroup$
    – Jon Custer
    Commented Feb 14, 2023 at 15:37

4 Answers 4

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This kind of plot is typically called a "1/M Plot", where M is the reactor multiplication. To measure the multiplication, you need a neutron source and a detector. As you add more and more material to the reactor, the detector signal increases. When the reactor goes critical (or more precisely supercritical), the detector signal (M) will grow to infinity. If you plot the inverse of the signal (1/M), it will become zero when the reactor goes critical.

If you plot 1/M as a function of the reactor size (or control rod position, or number of rods, etc.), you can do a curve fit to estimate where the point of criticality is. In Fermi's plots, they were adding material (layers) and the plot shows the reactor when critical at layer 57.

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  • $\begingroup$ Thanks you for your response. I've come across the "1/M" method in a few places, my challenge is that it's not quite what Fermi did, and if I modify Fermi's approach to look like 1/M I get a different answer. Notebook here: colab.research.google.com/drive/… $\endgroup$
    – Stephen W.
    Commented Mar 21 at 9:50
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I cannot be sure I understand it correctly, but this is how it looks to me at the moment. The report that you quote says:

In a spherical structure having the reproduction factor 1 for infinite dimensions the activation of a detector placed at the center due to the natural [my emphasis] neutrons is proportional to the square of the radius.

@Adam says in a comment:

I guess the question really is why he prefers to plot $R^2_{eff}/A$ instead of just $A$

According to the quote from the report, if you just have $A\to\infty$ when the radius increases, it is possible that you just have natural neutrons. If, however, $R^2_{eff}/A\to 0$, it means that the number of natural neutrons becomes negligible compared to the number of neutrons from chain reaction, therefore, the reactor is getting critical.

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Fermi wrote:

In a spherical structure having the reproduction factor 1 for infinite dimensions the activation of a detector placed at the center due to the natural neutrons is proportional to the square of the radius.

I cannot exactly say what does mentioned by "natural neutrons" but it is looked as the most realistic case to be spontaneous fission neutrons produced by the reactor nuclear materials. Then, in terms of the point kinetic, the ratio $N_{\text{source}}/N_{\text{total}}$ is actually limiting to $0$ like $R^2_{\text{eff}}/A$ in the report if multiplication coefficient $k_m = \left(N_{\text{total}} - N_{\text{source}}\right)/{N_{\text{total}}}$ is limiting to $1$ on fixed $N_{\text{source}}$. So, the main question is about nature of relation $R^2_{\text{eff}}$ and $N_{\text{source}}$.

Accordingly to the neutron diffusion theory, solution of one-speed transport equation for point source on distance $l$ with intensity $S_0$ in infinity media with coefficient of diffusion $D$ and macroscopic cross-section of absorption $\Sigma_a$ (see, for example, here) is $$ \Phi\left(l\right) = \frac{S_0 e^{-l\sqrt{\frac{\Sigma_a}{D}}}}{4\pi l D} $$

Note, that we will systematically overestimated flux in the sphere center if we will use it for a spherical model with leakage relates to a problem of single reactore core. To be exact, we can use this expression for representation of flux in the middle of spherical source of radii $R$ located in infinite scattering homogeneus media of same material what can be related to a graphite reactor core of radii $R$ covered by thick graphite reflector.

For that case, total flux at the reflected reactor core center will be represented by an integral sum of concentric elemental spherical layers volumes $4\pi r^2 \mathrm{d}r$ over sphere of radii $R$: $$ \Phi\left(0, R\right) = \int_{0}^{R} \frac{S e^{-r\sqrt{\frac{\Sigma_a}{D}}}}{4\pi r D} 4\pi r^2 \mathrm{d}r = \frac{S}{D} \int_{0}^{R} r e^{-r\sqrt{\frac{\Sigma_a}{D}}} \mathrm{d}r = \frac{S}{\Sigma_a} \left( 1 - \left( R\sqrt{\frac{\Sigma_a}{D}} + 1 \right) e^{-R\sqrt{\frac{\Sigma_a}{D}}} \right) $$

As we can see now, growing rate of primary neutron flux is much more less than $R^2$. Moreover, on large $R$ function $\Phi\left(0, R\right)$ becomes asymptotic with constant $S/\Sigma_a$ what is confirmed by computation experience when every space-strengh source has a plateau in flux or dose rates distribution at plots.

For small values of $R$ the exponent can be expanded by the Taylor series about the point $R=0$: $$ \exp\left( {-R\sqrt{\frac{\Sigma_a}{D}}} \right) \approx 1 - R\sqrt{\frac{\Sigma_a}{D}} + O\left(R^2\right) $$

Substitution it to the expression gives the following expression for spontaneous fission neutron flux in the center of radii $R \rightarrow 0$ emitting sphere $$ \Phi\left(0, R\right) \approx \frac{S}{\Sigma_a} \left( 1 - \left( 1 - R \sqrt{\frac{\Sigma_a}{D}} \right) \left( 1 + R \sqrt{\frac{\Sigma_a}{D}} \right) \right) = \frac{S R^2}{D} $$

It is proportional to $R^2$.

So, we can conclude that "the rule of $R_{\text{eff}}^2$" relatively quite approximates changing of $N_{\text{source}}$ for initial stages of the experiment with small sizes of the reactor core and high rate of neutron leakage to the reflector so it can be used in approximation of the $N_{\text{source}}/N_{\text{total}}$ ratio changing. For example, the first two lines in the table 1 shows growing of flux in almost 2 times — from 42 to 78. However, the main reason of that is not growing of neutron multiplication but quantity of nuclear material in the system what is seen from changing of $R^2_{\text{eff}}/A$ just in 1,2 times — from 390 to 320. On the last subcritical stages, the $R^2_{\text{eff}}/A$ rate get closer to the commonly used today the $1/A$ ratio ("$1/M$ Plot") for reactor startups with constant sources.

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  • $\begingroup$ Interesting! Will review this when I can find a little time! $\endgroup$
    – Stephen W.
    Commented Mar 16 at 10:27
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I have made a video response here: https://youtu.be/m-LwNJmicmU

EDIT: in response to comments I have included a typed explanation below, exclusively for stack exchange...

There are 2 key quantities here:

  • $R_{eff}^{2}$ = square of the effective radius of the reactor.
  • A = induced activity in the indium foil from the neutron irradiation (counts/minute), measured the next day.

Unfortunately it's a bit of conjecture from here on, but my understanding is that Fermi used $R^{2}$ as a surrogate for the cross sectional area of his reactor, and A as a surrogate for the 'neutron intensity' (presumably, neutron intensity = number of neutrons per unit area).

Fermi was likely most interested in how the value of A changes over time.

  • Increasing A = chain reaction (i.e. nuclear bomb)
  • Constant A = sustainable reaction that continues until you run out of fissile material.
  • Decreasing A = reaction that will create a small bit of heat but die out.

Fermi could have just plotted A against the number of layers and found where his trendline shoots off to infinity, and determined that to be the point of criticality. But A isn't the only influence factor, there's an interplay effect between A and the cross sectional area. Since he's using effective radius ($R_{eff}$) it mightn't be obvious how that behaves, and the result will also be very specific to his setup.

Either way we can see that $R^{2}$ will increase with the square of the number of layers, and we know A will increase exponentially. Hence A grows faster than $R^{2}$, so when we take a ratio of $R^{2}/A$ it trends to $0$ for criticality. Importantly since both cross sectional area and neutrons/area are strictly positive and non-zero, the ratio will never truly reach zero.

There are a few aspects of the experiment that my (rather naive) video solution cannot speak to. For example, I assumed a perfectly spherical reactor with radius '$R$' when in reality it wasn't spherical and Fermi used the effective radius '$R_{eff}$' instead. There is also uncertainty surrounding the indium foil detector, I have assumed that the quantity Fermi was actually interested in was something similar to neutron fluence and his measurements were aimed at evaluating it through a proxy-quantity (induced activity in the foil) since he was unable to measure it directly.

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