The formation of atoms after the Big Bang is a process known as Big Bang nucleosynthesis. During the early stages of the universe, a few minutes after the Big Bang itself, the conditions were extremely hot and dense. At this point, the universe consisted primarily of a hot, ionized plasma composed of protons, neutrons, and electrons.
As the universe expanded and cooled, the temperature and density decreased. When the temperature dropped to around one billion degrees Kelvin (approximately three minutes after the Big Bang), the conditions became favorable for the formation of light atomic nuclei through a process called nucleosynthesis.
During nucleosynthesis, protons and neutrons combined to form atomic nuclei. The most abundant nuclei produced during this phase were hydrogen-1 (proton), helium-4 (two protons and two neutrons), and a small amount of lithium-7. This process was primarily governed by the strong nuclear force and the balance between the expansion of the universe and the decay of neutrons.
At this stage, however, the universe was still too hot for electrons to stably combine with atomic nuclei to form neutral atoms. The high energy and density of the plasma caused frequent interactions between photons and charged particles, preventing the formation of stable atoms.
It was only when the universe cooled further, to a temperature of approximately 3,000 degrees Kelvin (around 380,000 years after the Big Bang), that the process of recombination occurred. During recombination, the electrons combined with protons to form neutral hydrogen atoms (one proton and one electron) and other neutral atoms.
This transition from a hot, ionized plasma to a cool, neutral gas allowed photons to travel freely through space, creating the cosmic microwave background radiation, which we can detect today.
In summary, atoms were formed after the Big Bang through the process of Big Bang nucleosynthesis, during which light atomic nuclei, primarily hydrogen and helium, were synthesized. The formation of stable, neutral atoms occurred later during recombination, when the universe cooled down enough for electrons to combine with atomic nuclei.