This note give detailed calculation on

• what’s the low-energy description of the ferromagnetic impurities to itinerate electrons? — the famous s-d model
• how to explain the lifting tail of the $\rho$-$T$ curve in low temperature? — spin-flip scatterings give logarithmic correction
• Such logarithmic correction brings in an even more serious problem in the limit $T\rightarrow0$K: : the resistivity of dopped alloys now will divergence!. SO how to reconcile this puzzle? — Anderson’s pool man scaling