Why does the resistivity of metals increase with an increase in temperature?

(N/A) The resistivity of a metal is given by the formula $\rho = \frac{m}{ne^2\tau}$,where $m$ is the mass of the electron,$n$ is the number density of free electrons,$e$ is the charge of the electron,and $\tau$ is the average relaxation time.
In metals,the number density of free electrons $(n)$ is independent of temperature.
When the temperature of a metal increases,the thermal vibrations of the positive ions in the lattice increase.
Due to these increased vibrations,the frequency of collisions between the free electrons and the positive ions increases.
This leads to a decrease in the average relaxation time $(\tau)$.
Since $\rho \propto \frac{1}{\tau}$,a decrease in $\tau$ results in an increase in the resistivity $(\rho)$ of the metal.