Why does the resistivity of a metal increase with an increase in temperature,or why does its conductivity decrease with an increase in temperature?

(N/A) The conductivity of metals is given by the formula: $\sigma = \frac{n e^{2} \tau}{m}$.
Here,$n$ (charge carrier density),$e$ (charge of an electron),and $m$ (mass of an electron) are constants for a given metal.
Therefore,$\sigma \propto \tau$,where $\tau$ is the relaxation time.
With an increase in temperature,the thermal vibrations of the lattice ions increase,which leads to more frequent collisions of electrons with the ions. This causes the relaxation time $(\tau)$ to decrease.
Since $\sigma \propto \tau$,a decrease in $\tau$ leads to a decrease in conductivity $(\sigma)$.
Resistivity is defined as $\rho = \frac{1}{\sigma}$. Since $\rho \propto \frac{1}{\tau}$,as the temperature increases and $\tau$ decreases,the resistivity $(\rho)$ of the metal increases.