There are several websites that have deep newspaper archives, like ProQuest Historical Newspapers, LexisNexus, and Newspapers.com. The former is pretty comprehensive for certain major newspapers (like the New York Times, Washington Post, Chicago Tribune, etc.) and some less-major ones; the latter has far wider breadth but is sometimes just a handful of issues for any given newspaper; LexisNexus is mostly more contemporary than 1930. All of these are subscription-based on some (like ProQuest) likely too expensive individuals (they thrive on institutional subscriptions).

I searched the several archives I have access to through my various institutional affiliations. A few have some Providence Journal copies but none had 1930 at all. Sorry about that. I doubt there is an easy solution unfortunately.

Your best bet would probably be to seek out a microfilm copy of the relevant newspaper. The sure-fire place to start would be to ask the Providence Public Library. But copies may be in other libraries with extensive newspaper microfilm archives. Worldcat can sometimes be useful for finding those but it's sometimes hard to figure out exactly which libraries have the microform copies.

EDIT: Is this the article? Science Plans to Dramatize the Atom

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Google News Archive is totally free and has digitized archives of several hundred newspapers from the 18th to 21st centuries. It's not searchable by keyword, and the interface is really clunky, but it's still a valuable resource. I believe Google almost killed it many years ago, but it was saved by public outcry. That said, I don't think they've done any work on it in almost a decade.

Newspapers.com requires a subscription (either 7.99 or 19.99 a month, depending on the plan), but is much more user-friendly.

This may not be 100% accurate, but in my experience, there's no overlap in coverage between the two archives. Either it's on Google or it's on Newspapers.com, or it's not on either one.

Unfortunately, the Providence Sunday Journal doesn't appear to be on either one.

That said, based on the title/content of the article, it sounds like it was written by a wire service, rather than by a local reporter. AP, UPI, and Reuters are well-known examples in the US. A wire service has journalists stationed in a variety of locations. News outlets will subscribe to a wire service in order to have access to their stories/visuals. (Pro tip: Even if a story you're looking at is from a wire service, try to find the article in several different papers. Reporters oftentimes file stories that are quite long, and papers cut them down to fit the space that is available.)

Give me a few minutes, and let me check for your article on Newspapers.com. If it's a wire article, I should be able to find it in another paper that is available.