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Down To Earth
Around the same time as the discovery of the neutron, a number of completely new particles were also discovered that did not fit into the atomic picture at all. These particles were unstable, existing only fleetingly before disintegrating, and were only seen in sensitive detector experiments that were not possible when the chemical elements were being catalogued. Such particles were therefore completely unknown and of no concern at all to earlier scientists, but added to the list of questions present in the 1930's.
On the theoretical side, the best mathematical description of the electron suggested that it should be possible to have something just like an electron but with opposite electric charge, in other words just like a proton but much less massive and free of any nucleus. Lo and behold, a particle with these exact properties was then discovered, making it the first to be predicted from a mathematical theory. The electric charge of the electron had been labelled as "negative", with that of the proton as "positive", and so the new positive partner to the electron was called the "positron", which was thought of as an anti-electron. This began the notion of "antiparticles" as a general concept.
One by one new particles were found. At first they were discovered coming naturally from space, in so-called "cosmic rays", but it was later realized that they could be artificially created. One of Einstein's great achievements showed that mass is another type of energy, and so, as energy can be converted from one type to another, it was determined that mass can actually be generated. By taking known stable particles and colliding them at high speed, the kinetic energy they carry can sometimes be forced to turn into mass, or in other words forced to turn into new particles. Higher energy collisions allow for the creation of either more particles of lesser mass, or fewer particles of greater mass. As long as the total energy adds up before and after the collision, anything goes. In this way, generation after generation of particle accelerator has been constructed, operating at ever higher energy, and particle after particle has been discovered.