When an electron emits a photon, the direction of emission is not random but rather determined by the specific circumstances of the interaction. The emission of a photon from an electron occurs through a quantum mechanical process known as spontaneous emission.
In the case of a lone, isolated electron in free space, the emitted photon will be radiated in a specific direction. The direction of emission depends on factors such as the initial state of the electron, the energy and momentum of the emitted photon, and any external influences or surrounding fields. These factors determine the probability distribution of the emission direction, but the exact direction is not predetermined.
It's important to note that the emitted photon does not expand as a sphere from the electron at the speed of light (C) in the way you described. The photon itself travels as a wave or particle, depending on how it is observed, and its propagation is governed by the laws of electromagnetism. The emission process is typically represented as an electromagnetic wave spreading out from the source, but it is not a literal expansion of a sphere from the electron's location.
In more complex scenarios involving multiple particles or in the presence of external fields, the emission properties can be further influenced, leading to phenomena such as beaming or directionality. Overall, the direction of emission and the subsequent behavior of the photon depend on the specific conditions and interactions involved in the emission process.