Daghlian was experimenting with the plutonium core to determine exactly how close the core was to supercriticality by stacking bricks of tungsten carbide (a neutron reflector about twice the density of steel.) He slipped and one of the bricks fell directly onto the core. Daghlian said later he knew at that moment he was dead. He threw the brick off the core, but it was too late. The resulting neutron burst killed him a few weeks later.
Interestingly, this same core also caused the death of a second professor, Loius Slotin. Slotin was attempting to perform the same task, determining the point of supercriticality, but using a different approach.
In this case a beryllium sphere(!), another neutron reflector, cut into two hemispheres were slowly lowered together, and the resulting neutron levels were measured.
Slotin was famously reckless, and in his insanity, he developed the apparatus such that the only thing keeping the two hemispheres from completely joining and producing an immediate reaction was a flat head screwdriver he slowly twisted. Enrico Fermi said several times "He's already dead. It's only a matter of whether others die with him."
Sure enough, one day Slotin slipped, the core went briefly supercritical, and the massive dose of radiation killed him.
A superstition developed around the core, and although it was originally slated for wartime testing, the team left it on the shelf until after the war, when it was eventually detonated in a test at Bikini Atoll.
At that point in time, health problems in nuke workers could have as much to do with chemical exposure, i.e. uranium is toxic in the same way as lead, but with 1/3 the dose.
It seems like an irrational thing to do, putting a valuable scientist in danger doesn't advance the war effort.
Nobody claimed it was rational, and just because they were scientists doesn't make them immune to "patriotically" endangering themselves "for the war effort."
In WWII and even Vietnam you had sons (and even a few daughters) of the elite who were either drafted or volunteered -- this is not often the case anymore.
On the other hand, the modern U.S. military spends around $1 million to train a warfighter to the base level where they are sent into real combat. That's much more than PhD training costs.
Most of all it reminds of Vikings. Not entirely accurate, but accurate enough that you get a feeling of the circumstances.