What state of matter has atoms that are in a fixed position and only vibrate back and forth?

The state of matter characterized by atoms that are in a fixed position and only vibrate back and forth is a solid. In solids, the particles are tightly packed together, which restricts their movement to primarily vibration around their fixed positions. This structure provides solids with a definite shape and volume. The strong intermolecular forces in solids keep the atoms or molecules in place, allowing only minimal movement. This is in contrast to liquids, where particles are close but can flow past each other, and gases, where particles are far apart and move freely in all directions. Plasma, on the other hand, consists of highly energized particles that are not bound to each other, allowing for a completely different behavior. Thus, the unique arrangement and behavior of particles in a solid clearly define it as the state of matter with fixed positions and vibrational movement.