When the conduction electrons are only scattered by thermal vibrations of thermal ions, then τ in the mobility expression refers to the mean time between scattering events by this process. The resulting conductivity and resistivity are denoted by σT and ρt, where the subscript T represents "thermal" vibration scattering". To find the temperature dependence of the mean free time τ, since this determines the drift mobility. An electron moving with a means speed u is the scattered when its path crosses the cross-sectional area S of a scattering centre. The scattering centre may be vibrating atom, impurity, vacancy, or some other crystal defect. Since τ is the mean time taken for one scattering process, the mean free path l of the electron between scattering process is uτ. If Ns is the concentration of scattering centres, then in the volume SI, there is one scattering centre, that is, Ns=1 thus the mean free path is given . The mass speed u of conduction electrons in a metal can be shown to only slightly temperature dependent. Because the atomic vibrations are random the atoms cover a cross-sectional area A= ∏r2 where a is the amplitude of the vibrations. If the electron's path crosses A= ∏r2 it gets scattered. Therefore the mean time between scattering events is inversely proportional to the area that scatters the electron, that is. As the temperature raises, the amplitudes of the atomic vibrations increases thus,
Where C is a temperature independent constant. Substituting the values we obtain
Where A is the temperature independent constant. This shows that the resistivity of a pure metal wire increases linearly with the temperature, and that the resistivity is due simply to the scattering of conduction electrons by the thermal vibrations of the atoms. We term this conductivity lattice-scattering-limited conductivity. The change in resistance of a material per ohm per degree change in temperature is called the temperature coefficient of resistance of that material. The resistance of a conductor changes with temperature according to the law:
Rt = R0 (1+αt)
Where Rt, and R0 are respectively the resistance of the conductor at t degrees and zero degree centigrade's and α, the temperature coefficient of resistance. Based on temperature effects, electrical materials can be classified into two groups:
Positive temperature coefficient of materials: It means that the resistance of some of the metals and alloys increases when their temperature is raised.
Negative temperature coefficient of materials: It means that the resistance of some of the materials, carbon and insulators decreases when their temperature is raised.