Resistivity Temperature

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For metals, resistivity with temperature generally?

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1. **Microscopic View**: As temperature increases, the thermal energy of the positive metal ions in the lattice increases, causing them to vibrate more vigorously.
2. **Collision Frequency**: Increased vibrations lead to more frequent collisions between free electrons and the ions.
3. **Relaxation Time ($\tau$)**: Frequent collisions mean the average time between collisions (relaxation time) decreases.
4. **Resistivity Relation**: $\rho = \frac{m}{n e^2 \tau}$. Since $\tau$ decreases, resistivity $\rho$ **increases**.
5. **Linear Approximation**: For most metals, $\rho_t = \rho_0 [1 + \alpha(t - t_0)]$, where $\alpha$ is a positive temperature coefficient.

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