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The original invention of the hydrogen bomb was very early in the Manhattan Project, but no significant work on designing one was done until right after the end of World War 2, when Edward Teller completed his "Classical Super" design (simulation of this design on ENIAC showed by the end of January 1946 that this design would not work).

The invention of what became the Teller-Ulam staged radiation implosion hydrogen bomb came in 1950 from work on improved atomic bomb designs. A design team at Los Alamos came up with the idea of using one atomic bomb to implode another much faster than chemical explosives can, resulting in a device having higher yield while using less fissionable material. One member of this team, the mathematician Stanislaw Ulam decided to consult with Edward Teller on issues of x-ray radiation transport, etc. and how to perform the necessary mathematical analyses to model this on a computer. However Edward Teller quickly realized this was the missing "ingredient" needed to make his hydrogen bomb designs practical and while supporting continuing work on the idea with Stanislaw Ulam he discouraged using the concept for improving atomic bombs with this idea as being "inefficient" compared to making hydrogen bombs.

The US built and tested its first hydrogen bomb using staged radiation implosion in 1952 (designed by Richard Garwin, with suggestions by Edward Teller) in shot Ivy Mike a massive 82 ton cryogenic assembly requiring a separate liquid hydrogen plant to keep its deuterium-tritium fusion fuel mixture cold. This was obviously not a practical bomb for an aircraft to deliver.

The USSR built and tested a limited yield, but deliverable weapon they called a "type of hydrogen bomb" in 1953, causing a brief panic in the US. This device did not use a staged radiation implosion, instead its core consisted of alternating layers of enriched uranium and lithium deuteride, making it more of a "dry boosted fission bomb" than a hydrogen bomb.

The US built and tested several different deliverable hydrogen bombs using staged radiation implosion in 1954 in operation Castle. The device tested in shot Castle Romeo was selected to become the first US hydrogen bomb and was fielded later that year as the EC-17 and could only be carried by the B-36 bomber. "EC" meant Emergency Capability: any bomber delivering one was on a suicide mission as there was no way the airplane could avoid being destroyed by the blastwave from the explosion.

In 1955 the US added a retarding parachute to this bomb which gave time for the bomber to escape undamaged, and changed its name to the MK-17. At about the same time the USSR tested and fielded their first deliverable staged radiation implosion hydrogen bomb.

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Q: When was the hydrogen bomb invented?
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