What is so difficult about developing nuclear weapons that makes some countries incapable of making them?
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First, I will tell you two things that are not difficult:
1. Obtaining enough uranium to make a bomb. Uranium is a fairly common element in the earth’s crust, and there are significant deposits on six continents. It is not hard to get.
2. Handling uranium’s radioactivity. Natural uranium is more than 99% U-238 (half-life of 4.5 billion years), and most of the rest is U-235 (half-life of 700 million years), plus a small trace amount of U-234 (half-life of 250,000 years). In other words, uranium is not very radioactive at all, and since its primary decay mode is alpha emission, the radiation is very easy to shield against.
Plutonium can be used in implosion weapons, but it does not occur in nature (except in absolutely miniscule trace amounts). You have to make plutonium in a nuclear reactor, which is, of course, a bit of a challenge. You also have to extract Pu-239 from the other plutonium isotopes that are produced in the reactor, which can be a challenging reactor design problem, as well as a difficult post-processing engineering problem.
The hardest part, historically, has been constructing a bomb that doesn’t weigh a ton, like gun-type weapons (important if you would like to transport the bomb to another country instead of exploding it in your own facility). It’s technically challenging to make implosion work well, and there is a lot of practical knowledge about the actual manufacture of the bombs that is hard to come by.
Well, there is one other difficult part: convincing other nations not to prevent you from developing a weapons program. It’s not easy to cover up an effective weapons program these days. The big military powers mostly don’t like other nations developing nuclear weapons, and they use a variety of techniques to prevent it from happening. Diplomacy is the most popular, but it isn’t the only option that has been used.
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