The loss of cooling water was unexpected and was caused by a pump failure. The broken pump was actually part of the power-generating loop, not the reactor core-cooling loop. When everything was working properly, water flowing through a loop that included the reactor core transferred heat to water flowing through the generating plant loop. But when the generating plant loop shut down, the reactor core loop had nowhere to deposit its heat and the water in it boiled. Backup cooling water evidently did not exist, did not work, or was not sufficient to keep the reactor core from over heating. I don't know whether it was poor design or poor maintenance that caused this disaster.
As you noted, this process of sticking together smaller atomic nuclei or nuclear fragments to form larger atomic nuclei is called fusion. Many smaller nuclei release energy when they grow via fusion, so long as the resulting nuclei are no larger than 56Fe (the nuclei of a normal iron atom). Above that size, energy is consumed in the process of sticking the nuclei together. So building carbon nuclei would release energy and building gold atoms would require energy. But while it's possible to construct atomic nuclei up to carbon or even gold, it isn't very practical. It's very difficult to bring atomic nuclei close to one another because they are all positively charged and repel one another fiercely. Because the nuclear energy these nuclei release during fusion only emerges at the moment they actually touch, something must push them together for that to occur. The nuclei can be pushed together by (1) nuclear fission reactors, (2) particle accelerators, (3) thermonuclear weapons, (4) giant lasers, or (5) thermal fusion reactors. None of these systems is ready to synthesize large quantities of normal atoms in a cost effective manner (although nuclear fission reactors do produce useful quantities of radioactive isotopes) and none is ready to produce practical energy from fusion processes.
While the radioactive decays from spent nuclear fuel rods continue to produce thermal energy, the amount of energy released each second isn't enough to make it cost effective to use that energy. Since the power output from a spent fuel rod would only be in the watt range, it wouldn't justify the hazardous job of trying to extract that power without encountering the radiation. Furthermore, the laws of thermodynamics make it much harder to use heat from a warm object than heat from a hot object and spent fuel rods would at best be warm objects.
First, nuclear waste isn't 100% radioactive atoms. Much of it is radioactively contaminated material—normal materials that contain enough radioactive atoms to be considered hazardous. Second, nuclear reactors don't wait for radioactive materials to decay via spontaneous processes, the ones that are responsible for half-lives. Instead, they induce the radioactive decays using chain reactions. In a nuclear fission reactor, the spontaneous decay of one uranium or plutonium nucleus is used to induce decays in other uranium or plutonium nuclei. In this manner, huge fractions of the uranium or plutonium nuclei can be "used up" in only a few years. In fact, in a nuclear fission bomb, many or most of the uranium or plutonium nuclei are consumed in less than a millionth of a second because of these induced fissions. Half-life has almost nothing to do with a fission bomb. It becomes nuclear waste so fast you can't imagine it.
A nuclear reactor operates just below critical mass so that each radioactive decay in its fuel rods induces a large but finite number of subsequent fissions. Since each chain reaction gradually weakens away to nothing, there is no danger that the fuel will explode. But operating just below critical mass is a tricky business and it involves careful control of the environment around the nuclear fuel rods. The operators use neutron absorbing control rods to dampen the chain reactions and keep the fuel just below critical mass.
Fortunately, there are several effects that make controlled operation of a reactor relatively easy. Most importantly, some of the neutrons involved in the chain reactions are delayed because they come from radioactive decay processes. These delayed neutrons slow the reactor's response to changes—the chain reactions take time to grow stronger and they take time to grow weaker. As a result, it's possible for a reactor to exceed critical mass briefly without experiencing the exponentially growing chain reactions that we associate with nuclear explosions. In fact, the only nuclear reactor that ever experienced these exponentially growing chain reactions was Chernobyl. That flawed and mishandled reactor went so far into the super-critical regime that even the neutron delaying effects couldn't prevent exponential chain reactions from occurring. The reactor superheated and ripped itself apart.
The Japanese did stop the chain reactions in the Fukushima Daiichi reactors, even before the tsunami struck the plant. The problem that they're having now is not the continued fissioning of uranium, but rather the intense radioactivity of the uranium daughter nuclei that were created while the chain reactions were underway. Those radioactive fission fragments are spontaneously decaying now and there is nothing that can stop that natural decay now. All they can do now is to try to contain those radioactive nuclei, keep them from overheating, and wait for them to decay into stable pieces.
The uranium atom has the largest naturally occurring nucleus in nature. It contains 92 protons, each of which is positively charged, and those 92 like charges repel one another ferociously. Although the nuclear force acts to bind protons together when they touch, the repulsion of 92 protons alone would be too much for the nuclear force—the protons would fly apart in almost no time.
To dilute the electrostatic repulsion of those protons, each uranium nucleus contains a large number of uncharged neutrons. Like protons, neutrons experience the attractive nuclear force. But unlike protons, neutrons don't experience the repulsive electrostatic force. Two neutron-rich combinations of protons and neutrons form extremely long-lived uranium nuclei: uranium-235 (92 protons, 143 neutrons) and uranium-238 (92 protons, 146 neutrons). Each uranium nucleus attracts an entourage of 92 electrons to form a stable atom and, since the electrons are responsible for the chemistry of an atom, uranium-235 and uranium-238 are chemically indistinguishable.
When the thermal fission reactors of the Fukushima Daiichi plant were in operation, fission chain reactions were shattering the uranium-235 nuclei into fragments. Uranium-238 is more difficult to shatter and doesn't participate much in the reactor's operation. On occasion, however, a uranium-238 nucleus captures a neutron in the reactor and transforms sequentially into neptunium-239 and then plutonium-239. The presence of plutonium-239 in the used fuel rods is one of the problems following the accident.
The main problem, however, is that the shattered fission fragment nuclei in the used reactor fuel are overly neutron-rich, a feature inherited from the neutron-rich uranium-235 nuclei themselves. Midsize nuclei, such as iodine (with 53 protons), cesium (with 55 protons), and strontium (with 38 protons), don't need as many neutrons to dilute out the repulsions between their protons. While fission of uranium-235 can produce daughter nuclei with 53 protons, 55 protons, or 38 protons, those fission-fragment versions of iodine, cesium, and strontium nuclei have too many neutrons and are therefore unstable—they undergo radioactive decay. Their eventual decay has nothing to do with chain reactions and it cannot be prevented.
How quickly these radioactive fission fragment nuclei decay depends on exactly how many protons and neutrons they have. Three of the most common and dangerous nuclei present in the used fuel rods are iodine-131 (8 days half-life), cesium-137 (30 year half-life), and strontium-90 (29 year half-life). Plutonium-239 (24,200 year half-life) is also present in those rods. When these radioactive nuclei are absorbed into the body and then undergo spontaneous radioactive decay, they damage molecules and therefore pose a cancer risk. Our bodies can't distinguish the radioactive versions of these chemical elements from the nonradioactive ones, so all we can do to minimize our risk is to avoid exposure to them or to encourage our bodies to excrete them by saturating our bodies with stable versions.
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