The ones with really long halflives formed way back during the supernova that created all of the atoms that make up our solar system.
The ones with shorter halflives form naturally when cosmic rays hit otherwise stable atoms, altering their nuclei to form radioactive isotopes. For example, when normal atmospheric nitrogen is hit with cosmic rays, it can form carbon-14, a radioactive isotope of carbon-12.
Some don’t form naturally at all. Many of the elements at the very bottom of the periodic table can only be reliably created artificially in particle accelerators.
There are a few ways. As another answer said, the long-lived ones usually form in supernovae and collisions between neutron stars. Ones like uranium-235 and potassium-40 last long enough that some is left over from Earth’s formation.
Some short-lived isotopes like carbon-14 and beryllium-7 are produced by cosmic radiation.
All the elements from polonium to actinium are part of the decay chains of uranium-235, uranium-238, and thorium-232. In a decay chain, the immediate product of radioactive decay is still radioactive. So there will be a series of decays until it reaches something stable. In the case of uranium and thorium, those are isotopes of lead.
As to how they become other elements, it is due to particles leaving the nucleus, and changing the number of protons and neutrons. Changing the number of protons changes, the element, because the identity of the element is defined by the number of protons.
The two most common ways this happens are alpha decay and beta decay. In alpha decay, an atom emits, a particle consisting of two protons and two neutrons. So, for instance, uranium 238 (92 protons, 146 neutrons) emits an alpha particle to become thorium-234 (90 protons, 144 neutrons).
In beta decay an atomic nucleus emits an electron, turning a neutron into a proton. For example, carbon-14 (6 protons, 8 neutrons) goes through beta decay to become nitrogen-14 (7 protons, 7 neutrons).
Other forms of decay exist, such as positron emission and electron capture. In this instance, the nucleus either shoots out a positron (anti-electron) or captures an electron. This turns a proton into a neutron. For example fluorine-18 (9 protons, 9 neutrons) emits a positron to become oxygen-18 (8 protons, 10 neutrons). This kind of decay is less common in nature. In fact, the example I gave, fluorine-18, does not occur naturally.
Elements are made in stars, which begin by fusing hydrogen together into helium
When they start to run out of hydrogen they fuse heavier elements, but this caps at iron because beyond that elements take more energy to fuse together than is released.
Elements heavier than iron are rarer because they are made by more extreme and rarer processes like when a star explodes
Elements decay because the nucleus is not stable from having an imbalance of protons and neutrons. Some elements have a stable form that doesn’t decay, some don’t.
For Francium there is about 1g in the earth at any time, but it is continually produced because it is part of the decay chain for Uranium 235, and U-235 is pretty long lived so there is still plenty of it around
Specifically:
U-235 has a half-life of 700 million years
It gives off an alpha particle to become Thorium 231, which has a half-life of a day before becoming Protactinium 231 by giving off a beta particle
Pa231 has a half-life of 30,000 years or so and emits an alpha particle to become Actinium 227, this has a half life of 22 years and usually decays to another form of Thorium (227) but slightly more than 1% of the time becomes Francium 223
That in turn has a half-life of 22 minutes but as long as there is U-235 around it will continue to be made in tiny amounts
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