Periodic boundary conditions lead to a restriction of the possible k values of atomic vibrational states. consider a linear chain of copper atoms. the length of the chain is 1 cm, the lattice spacing is 0.36 nm and the force constant 50 nm calculate the smallest possible finite wave number k. what would be the vibrational angular frequency for this wave number? demonstrate that there is a standing wave at the brillouin zone boundary for the case of waves on such a row of masses. note the dispersion relation for a 1-dimensional chain is given by 141 13] 13] 4,01 . ka where co is the vibrational angular frequency, β is the force constant, a is the lattice spacing, m is the mass of a copper atom and k is the wave number. the atomic mass of copper 63.55 amu).