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3D free electron gas 1

3D free electron gas 1. Want to find:. volume of e-gas. Write:. For a free electron:. so:. Now need:. 3D free electron gas 2. Pauli principle – only 2 electrons of opposite spin can occupy one volume element in phase space. volume of e-gas. Volume element of k -space for two states:.

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3D free electron gas 1

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

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

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

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

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

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

  7. 1D free electron gas 2 PY4007 Nanoscience

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