Asimple model of an ionic crystal consists of a linear array of a great many equally spaced atoms of alternating masses m and m, where m < m. the masses are connected by identical chemical bonds that are modeled as springs of spring constant k. show that the frequencies of the system’s longitudinal modes of vibration either lie in the band 0 to (2 k/m) 1/2 or in the band (2 k/m) 1/2 to [2 k (1/m +1/m)]1/2 . show that, in the long wavelength limit, modes whose frequencies lie in the lower band are such that neighboring atomics move in the same direction, whereas modes whose frequencies lie in the upper band are such that neighboring atoms move in opposite directions. the lower band is known as the acoustic branch, whereas the upper band is known as the optical branch.