Question

Suppose you have a very small sample of the orthorhombic crystal (orthorhombic, lattice constants 5.3 Å, 5.4 Å and 5.5 Å).  The crystal is 53 nm x 54nm x 55 nm oriented along the respective axis.

1. [2] Show for this crystal, there are one million atoms.
2. [2] If the crystal has dimensions 53nmx54nmx55nm, calculate the smallest and largest phonon wave vectors for this crystal (state answer in mks).
3. [2] What exactly is a phonon and why do we use them?
4. [2] For the Debye theory of specific heat we note: , where n is the number of atoms per m³, and for this particular solid, where K is the wave vector. Calculate the Debye temperature , for our solid.
5. [2] Use your result from c) to determine the specific heat, , N is the total number of atoms.
6. [4] Apply the quantum mechanical free electron model to the crystal.  In this model, the ground state depends on the longest electron wavelengths: , where k = 2π /λ . From the wave nature of electrons, estimate the ground state energy in eV.
7. [1] If the crystal consists of atoms which contribute 1 valence electron, determine the number of electrons in this solid.
8. Calculate the Fermi energy using  and show it is:  1.24eV
9. [3] Calculate the Fermi wave vector , fermi velocity () and fermi temperature ().
10. [5] We find most materials have bands where electron states exist, and band gaps, where no electron states exist.  Explain why band gaps form.  Using the idea and values of band gaps, explain the difference in conduction between metals, semiconductors and insulators.

Answer 1