Aharonov-Bohm effect as a scattering event

The Aharonov-Bohm effect is reconsidered as a scattering event of an electron by a magnetic field confined in an infinite solenoid of finite radius both in the situation where the solenoid is penetrable as well as impenetrable. We next discuss the validity of the Born approximation for the partial-wave scattering amplitudes and explain why for the cylindrically symmetric $\mathrm{}(m=0\mathrm{})\mathrm{}$ partial wave the first Born approximation fails in the long-wavelength limit or as the radius of the solenoid shrinks to zero.