At the center of atoms is the nucleus that contains one or more protons and(with the exception of hydrogen) one or more neutrons. The protons have a positive charge which is import for determining the overall charge of the atom which affects its chemical properties and is why elements are determined by the number of protons, this charge also causes protons to repel eachother, the neutrons have no charge and, to massively oversimplify hold the nucleus together. With a relatively low number of protons the nucleus can remain stable with roughly a 1:1 proton/neutron ratio, as you move up the periodic table the nucleus gets bigger and the proton:neutron ratio for stable atoms starts to approach 1:1.5. Eventually you reach a point(lead) where there there are no more stable nuclei and eventually they all undergo some form form of decay. Normally this decay involves minor changes like the nucleus spontaneously ejecting a couple protons/neutrons or a neutron converting to a proton and releasing energy etc, but with some really large nuclei under the right conditions like uranium 235 they can be split roughly in half and this process is called fission.
Splitting smaller nuclei like the carbon in your t-shirt is technically possible, but would require some exotic scenario like using it as a target for a partical accelerator or something. While this would produce a nuclear reaction it would be very very small and wouldn’t be very useful unless you wanted an extremely expensive radioactive t-shirt for some reason.
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