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It is known that one could get convenient and relatively safe source of neutrons by placing Americium-241 source near beryllium target.

Is there a way to convert alpha particles to gamma radiation in a similar fashion? I.e. is there a convenient target, which converts to short-lived gamma-active isotope when bombarded by alpha particles?

Or one have to go through 2-stage process alpha->neutrons->target activation?

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Some reactions that have been used for composition profiling (PIGE - particle-induced gamma emission) include:

$^{7}$Li($\alpha$,$\gamma$)$^{11}$B

$^{10}$B($\alpha$,p$\gamma$)$^{13}$C

$^{14}$N($\alpha$,$\gamma$)$^{18}$F

$^{20}$Ne($\alpha$,$\gamma$)$^{24}$Mg

None of these are either particularly broad (although that is a reason to choose them for depth profiling) nor have a particularly large cross section (mb or so). Depending on the application, you might find a Co-60 source the better way to go.

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In addition to Jon Custer's examples, it's worth noting that most alpha emitters decay to excited states in the daughter nucleus and emit some cascade of gamma rays afterwards. One of the many unusual things about the Litvinenko case was that the isotope used is a "pure alpha" emitter with no associated gamma rays, so that it could be completely shielded by something like a paper envelope. A more common alpha-gamma emitter could be converted to a pure gamma source by ... putting it in a paper envelope.

Likewise, in the AmBe neutron source you mention (which is usually an alloy rather than two separate materials), each neutron is accompanied by a gamma ray. Enclosing your AmBe source in a box of boron-loaded plastic would make it a pretty pure gamma source.

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