Vibrations in one-dimensional crystal—two atoms per unit cell

3 Vibrations in one-dimensional crystal—two atoms per unit cell 

The next level of complexity is the one dimensional chain, but now having two atoms per unit cell, the same spacing and for simplicity the same force constants on the springs. The atoms have different masses, m and m2. (A system of the same atoms. A, but with alternating force constants,. .. A-A... A-A. gives similar results.) 

The equations of motion, not presented here, are more complicated and their solutions are given in Eq. (4.48). 

When plotted against k, these roots comprise two modes, or branches, (i.e.y-superscript now has two values, 1 and 2) see Fig. 4.10. The m. root is proportional to A as - 0 (similar to the solutions for the monoatomic chain). This is the acoustic mode (or branch) since it is analogous to the 

We summarise, without proof, relevant properties of the amplitude by taking as an example a linear chain with two atoms of masses m2/wi=3 and solving the corresponding equations of motion. The acoustic modes. Fig. 4.11, near to the centre of the Brillouin zone are the oscillations of 

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