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GATE 2020 PH – Question 45

Solid State Physics · Band theory of solids: nearly free electron model · 2 marks · Multiple choice

A one-dimensional non-magnetic crystal has one atom per unit cell. Valence electrons do not interact with one another and interact weakly with ions. If n is the number of valence electrons per cell, at 0 K

  1. The crystal is metallic for every n.
  2. The crystal is non-metallic for every n.
  3. The crystal is metallic for even n.
  4. The crystal is metallic for odd n.

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Correct answer: (D) The crystal is metallic for odd n.

Explanation

Each band holds two electrons per unit cell because of spin degeneracy. For an odd number n of electrons per cell, the highest occupied band is partly filled, allowing states arbitrarily close in energy and metallic transport. An even number can fill bands completely and permit an insulating gap. Thus the guaranteed metallic case among the options is odd n, D.

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