Feasible. (but scary)

With the single edit that these delta rays not only ease fission but they (or another emission) also ease fusion.

During the development of the nuclear bomb, the scientists under Oppenheimer were concerned about the possibility that the temperatures created by the fission bomb might be enough to initiate fusion in the hydrogen on Earth. They were sufficiently worried about it that they sent considerable time calculating the effect, before accepting that the temperature and density simply were not enough to initiate this sort of runaway reaction.

But! If there is a secondary effect in the fission bomb that eases additional fission and fusion, then that is a totally different matter.

As for the scale: If the Little Boy bomb (hiroshima) , using the same components as on our Earth, generated a 1Mt explosion... Yes, that is quite believable. ALL it requires is that the entirety of the fissile material in the bomb get used up, instead of the 1.7% that fissioned in Little Boy. (if you want to be pedantic, the yield would be 880Kt, not 1 Mt)

Now for your question: :"How can I prevent the delta rays from using up all accessible unstable elements on the planet through a runaway reaction?"

You...............cannot. byebye.
End-of-world music start playing, please. .
.
.

Oh. You really want a way out of this? Um... handwavium, erm, technobabble doubletalk, hmm, McGuffin...

AHA!

Make your Delta Rays be a particle that decays very, very^very quickly. Meaning it only has an effect out to a certain radius from the initiating point, before it decomposes into something else that is more harmless. Say the particle decays in under half a nanosecond. That will give it (with relativity giving a helping hand), a range of about 500 meters.

Thus, the intense burst of Delta is enough to trigger fission/fusion in nearby elements in the ground. But those fissions are diffuse enough that the Delta wavefront they propagate is much less concentrated, resulting in much less induced fission/fusion in adjoining material. Pretty soon, it tapers off to nothing.

Unless.. You detonate a nuclear bomb near another bomb. Or near a nuclear reactor. Or on that mountain range that has a really rich Pitchblende ore in it!

Fission that yields energy only occurs with elements heavier than iron. But maybe you could have delta rays catalyze fusion.

As written your delta rays catalyze fission. It is not unstable elements that undergo fission. It is heavy elements.

https://www.physlink.com/education/askexperts/ae659.cfm

Why is uranium fissionable and not, say, aluminum? Asked by: Bill

Answer The short answer is "aluminum's not big enough." Here's why: The nuclear force saturates due to its short range, meaning that heavier nuclei have protons that don't attract each other via the nuclear force but still repel each other electrostatically. This is why heavier nuclei tend to have more neutrons in relation to the number of protons - the neutrons only attract. The binding energy per nucleon, a measure of how tightly a nucleus is bound, peaks at about 60 nucleons. (There's also a sharp peak at 4; the alpha particle, which is a He-4 nucleus, is very tightly bound) So light nuclei require energy to split apart and would release energy only if you can fuse them together. You might expect that anything heavier than 120 nucleons would fission, but these nuclei are still bound together, so the two parts you would get in fission aren't likely to fly apart. It's not until you get into the elements heavier than lead (all of which are radioactive) that you find nuclei whose binding energy per nucleon is low enough that the fission fragments could tunnel apart. Answered by: Tom Swanson, Ph.D., Physicist, US Naval Observatory

Elements heavier than iron are not commonly just lying around. There might be a little copper or arsenic in the soil. However, the people conducting the test may have brought in a lot of elements heavier than iron. Specifically lead, used to shield equipment and viewers. There might be a lot of that.

If you wanted to have your delta rays catalyze fusion, that would release a lot of energy the way you want. There are lots of light elements on the surface which could undergo fusion. If fission produces delta rays that catalyze fusion, that would make your big kaboom but not an actual earthshattering kaboom. Since fusion does not make the rays, the reaction would not be self sustaining. This would also provide a real world identity for delta rays - they are high velocity muons, and they are producing muon-catalyzed fusion