Why is the movement of the atoms in the chain the same when multiples of 2pi/a are added to, or subtracted from, the wave number?
The movement of atoms in a crystal lattice can be described by a wave, known as a phonon, which represents the collective vibrations of atoms. The wave number (k) in this context refers to the spatial frequency of the wave.
When multiples of 2π/a, where a is the lattice constant, are added to or subtracted from the wave number, it is equivalent to shifting the wave in the reciprocal space or the Fourier domain. In this case, it represents a wave vector that corresponds to a different point in the Brillouin zone, which is a unit cell in the reciprocal lattice.
The reason why the movement of atoms in the chain remains the same is due to the quantized nature of wave vectors in a periodic lattice. The Brillouin zone represents the range of wave vectors that span the entire lattice, and any wave vector within the same Brillouin zone will result in the same pattern of atomic displacements.
To understand this concept, you can consider the analogy of a circular race track. Imagine a runner on the inside lane of the track starting a race. Now, if the runner moves at a certain speed and completes one lap, the runner will return to their starting position. If the runner continues running at the same speed, completing additional laps, they will repeatedly cross the same starting point. Similarly, in the case of crystal lattice and phonons, when wave vectors are shifted by multiples of 2π/a, they represent motion in the reciprocal lattice that eventually leads to the same atomic displacements in the real lattice.
In summary, the periodic nature of the crystal lattice enforces the restriction on the movement of atoms, resulting in the same atomic displacements when multiples of 2π/a are added to or subtracted from the wave number.