The behavior of particles in the Stern-Gerlach apparatus can be understood by considering their intrinsic angular momentum, known as spin. In the case of spin-1/2 particles, such as electrons, they have two possible spin orientations along a particular axis, often referred to as "up" and "down" spins.
When a beam of spin-1/2 particles passes through a Stern-Gerlach apparatus, the magnetic field gradient present in the apparatus interacts with the particles' spins. This interaction causes the beam to split into two distinct beams, corresponding to the two possible spin orientations. The particles with "up" spin are deflected in one direction, while the particles with "down" spin are deflected in the opposite direction.
On the other hand, for spin-1 particles, there are three possible spin orientations along a particular axis: +1, 0, and -1. In this case, when a beam of spin-1 particles passes through the Stern-Gerlach apparatus, it splits into three beams. The particles with spin +1 are deflected in one direction, the particles with spin -1 are deflected in the opposite direction, and the particles with spin 0 remain undeflected.
Regarding your question about the direction of spin and motion, it's important to note that spin is an intrinsic property of particles and is not directly related to their physical motion or rotation. It does not refer to particles spinning around an axis like a classical object. Instead, spin is a quantum mechanical property that represents the particle's angular momentum.
The direction of the spin axis is arbitrary and can be chosen in various ways. In the case of spin-1 particles, the spin can align along any direction in space, including the direction of motion. The force experienced by the particles in the Stern-Gerlach apparatus depends on the interaction between their spin and the magnetic field gradient, rather than the specific direction of the spin axis in relation to the motion.
In summary, the splitting of beams in the Stern-Gerlach apparatus arises from the interaction between the particles' spins and the magnetic field gradient. Spin-1/2 particles split into two beams due to their two possible spin orientations, while spin-1 particles split into three beams due to their three possible spin orientations. The direction of spin does not necessarily align with the direction of motion, and the force experienced by the particles depends on the interaction between their spin and the magnetic field gradient.