1. PY4007 Nanoscience 3D free electron gas 1 volume of e-gas Want to find: Write: For a free electron: so: Now need:

2. PY4007 Nanoscience 3D free electron gas 2 Pauli principle – only 2 electrons of opposite spin can occupy one volume element in phase space Volume element of k-space for two states: volume of e-gas Determine number of states dn in k  k+dk (k is +ve so only consider one quadrant of sphere in k-space: kxkx kyky kzkz k dkdk spin

3. PY4007 Nanoscience 3D free electron gas 3 substitute in Now we have everything we need:

4. PY4007 Nanoscience 2D free electron gas 1 Consider free electron part first. Area element in k-space for two states: kyky k dkdk kxkx Determine number of states dn in k  k+dk (k is +ve so only consider one quadrant of circle in k-space: spin

5. PY4007 Nanoscience 2D free electron gas 2 Independent of energy!

6. PY4007 Nanoscience 1D free electron gas 1 Consider free electron part first. Length element in k-space for two states: k dkdk kxkx Determine number of states dn in k  k+dk along positive half of line in k-space: spin

7. PY4007 Nanoscience 1D free electron gas 2