What happens when an atom's nucleus undergoes: (a) positron emission. (b) electron capture. (c) gamma emission.

Positron emission represents an intriguing facet of nuclear physics, where balance within the atom's nucleus shifts in a bid to achieve stability. Here's a closer look:

In nature's quest for balance, a proton within the nucleus undergoes transformation into a neutron. This process isn't as simple as a change of identity; it's a matter of particle exchange. The proton sheds its positive charge in the form of a positron (denoted as \( \beta^+ \)), which is an antiparticle of the electron, carrying the same mass but with a positive charge. Accompanying the positron, a subtle particle called an electron neutrino (denoted as \( u_e \) ) is also released.

The emission of the positron effectively decreases the atomic number (noted as \( Z \) ) by one, yet the mass number (notated as \( A \) ) remains untouched. To visualize this, picture carbon-11; one of its protons decides to become a neutron, leading to positron emission and the formation of boron-11. While the mass is constant, the fundamental nature of the element shifts — a transformation from one element to another.

Less like an emission and more of an intimate merger, electron capture is an alternate decay route where the nucleus draws in one of its own. This close encounter of a subatomic kind has significant consequences:

Within the heart of the atom, a proton reaches out and captures an inner shell electron, usually from the K or L shell. This extraordinary internal interaction results in the fusion of the proton and electron into a neutron. An electron neutrino, a byproduct of the process, silently slips away from the nucleus.

With the atomic number (\( Z \) ) reduced by one, yet the mass number (\( A \) ) preserved, the atom morphs into a new element. The absence of the captured electron leaves a vacancy that triggers a cascade within the electron shells. An electron from a higher shell falls to fill the void, emitting energy in the guise of an X-ray or, alternatively, an Auger electron — a sophisticated energy-balancing act.

Gamma emission is nature's way of calming an overexcited atomic nucleus. Think of it as the nucleus taking a deep breath and relaxing, releasing a burst of energy as it settles:

An atomic nucleus, agitated and brimming with surplus energy, seeks a return to tranquility, its ground state. This excess energy often follows a nuclear decay process such as alpha or beta decay. The pathway back to a serene state involves emitting a photon far more energetic than any part of visible light — a gamma-ray (denoted as \( \gamma \)).

This phenomenon doesn't involve the electron cloud surrounding the nucleus but engages the nucleus itself, distinguishing it from typical atomic emissions. What comes after the gamma flash is a nucleus that's either found its peace in the ground state or, at the very least, stepped down to a lesser degree of excitement. Importantly, this radioactive release doesn't alter the atomic number (\( Z \) ) or mass number (\( A \) ), leaving the element's identity intact. It's an atomic adjustment internal to the nucleus, a shift from a higher to a lower energy level without a change in form.