Question

3. Consider an infinite linear chain of atoms, as in class. All masses are equal to M. However, there are two different types of springs with two different spring constants C1 and C2 coupling the atoms and their nearest-neighbors. The springs alternate C1, C2, C1, C2 ....in the chain. A crude diagram looks roughly like: This is a one-dimensional model of a solid with two atoms per unit cell, where the two atoms are the same, but they are geometrically in-equivalent (like C, Si, Ge). Let the cell size be a. a. Calculate the normal mode frequencies ("phonon dispersion relations") for this one dimensional solid. (Hint: The results will be similar to those of the diatomic chain. You will find acoustic and optic branches.) b. Show that the acoustic and optic branches of the phonon dispersion relations behave appropriately at the center of the Brillouin Zone.

          3. Consider an infinite linear chain of atoms, as in class. All masses are equal to M. However,
there are two different types of springs with two different spring constants C1 and C2
coupling the atoms and their nearest-neighbors. The springs alternate C1, C2, C1, C2 ....in
the chain. A crude diagram looks roughly like:

This is a one-dimensional model of a solid with two atoms per unit cell, where the two atoms
are the same, but they are geometrically in-equivalent (like C, Si, Ge). Let the cell size be a.
a. Calculate the normal mode frequencies ("phonon dispersion relations") for this one
dimensional solid. (Hint: The results will be similar to those of the diatomic chain.
You will find acoustic and optic branches.)
b. Show that the acoustic and optic branches of the phonon dispersion relations behave
appropriately at the center of the Brillouin Zone.

3. Consider an infinite linear chain of atoms, as in class. All masses are equal to M. However,
there are two different types of springs with two different spring constants C1 and C2
coupling the atoms and their nearest-neighbors. The springs alternate C1, C2, C1, C2 ....in
the chain. A crude diagram looks roughly like:

This is a one-dimensional model of a solid with two atoms per unit cell, where the two atoms
are the same, but they are geometrically in-equivalent (like C, Si, Ge). Let the cell size be a.
a. Calculate the normal mode frequencies ("phonon dispersion relations") for this one
dimensional solid. (Hint: The results will be similar to those of the diatomic chain.
You will find acoustic and optic branches.)
b. Show that the acoustic and optic branches of the phonon dispersion relations behave
appropriately at the center of the Brillouin Zone.

Added by Juan Antonio H.

Question

Please give Ace some feedback