The production of $^{3}$H, $^{7}$Be, and $^{22}$Na by interactions of cosmic-ray particles with silicon can produce radioactive backgrounds in detectors used to search for rare events. Through controlled irradiation of silicon CCDs and wafers with a neutron beam that mimics the cosmic-ray neutron spectrum, followed by direct counting, we determined that the production rate from cosmic-ray neutrons at sea level is ($112 pm 24$) atoms/(kg day) for $^{3}$H, ($8.1 pm 1.9 $) atoms/(kg day) for $^{7}$Be, and ($43.0 pm 7.1 $) atoms/(kg day) for $^{22}$Na. Complementing these results with the current best estimates of activation cross sections for cosmic-ray particles other than neutrons, we obtain a total sea-level cosmic-ray production rate of ($124 pm 24$) atoms/(kg day) for $^{3}$H, ($9.4 pm 2.0 $) atoms/(kg day) for $^{7}$Be, and ($49.6 pm 7.3 $) atoms/(kg day) for $^{22}$Na. These measurements will help constrain background estimates and determine the maximum time that silicon-based detectors can remain unshielded during detector fabrication before cosmogenic backgrounds impact the sensitivity of next-generation rare-event searches.
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