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Sources such as AmBe take advantage of Beryllium's $(\alpha,n)$ process to produce high energy neutrons.
Sources such as Cf utilize spontaneous fission.
Why not combine the two?

Such a source would have:

  1. Spontaneous fission neutrons
  2. $\alpha$ from Cf, which would make $(\alpha,n)$ with Be
  3. $\gamma$ from Cf, which would utilize (the mostly unused) $(\gamma,n)$ in Be

This seems like a very high yield source, one which would "waste" less of Cf's decays (>95% of its decays are alpha)
Should a smaller yield be required, it seems that it would require less Cf for the same yield, and thus, would also be more cost-efficient.

Yet, when searching for this source, I found zero results. Nobody even discusses this (As far as I know), and I don't understand why.

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The very flexibility that you mention in your post is a bit of a problem in an experimental context. In order to understand the signal that a Cf-Be calibration source would generate in your detector and tease useful information out of it, you're going to have to model all three channels that generate neutrons and the gammas that escape the source. This means that the response of your detector to such a source is going to be very complicated. (As an aside, it's already pretty complicated when you use AmBe or just a plain Cf-252 source, but the community has a long history with these things so there are resources for modelling them.)

So you have to balance the costs with the gains. As I see it there are two gains: more neutrons and more features in the neutron spectrum.

In most experimental situations, the number of neutrons generated by more common sources is more than sufficient. When I was a calibration guy on KamLAND, it was routine to collect 15-20 thousand events per calibration point with our AmBe source. That was more than sufficient to make the systematics of the machine a larger source of uncertainty than the counting error.

Even the Poloniun-210/Carbon-13 source (a reaction characterized by extremely low neutron yield) had a neutron activity of about $28 \,\mathrm{Bq}$ when it was new (about about $21 \,\mathrm{Bq}$ by the time we got it cleaned, characterized, shipped, through customs, and deployed), which meant that 1% statistics was a matter of 500 seconds per calibration point.

So you're looking at the proposed source for neutron lines in places where you can't get them with more common sources. But we already use the direct Cf-252 lines, and the $\mathrm{Be}(\alpha,n)$ lines are pretty well excited by AmBe source. So the only gain is in the $\mathrm{Be}(\gamma,n)$ lines. Are they worth it? Are they worth designing, fabricating, cleaning, characterizing, and testing a one-of-a-kind source? Even with the extra hassle of teasing those results out of a complicated data set?

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