Higher order tune derivatives due to low-β insertions

In an electron machine the adjustment of the first order derivatives of the tunes with respect to momentum is important to counteract the dipole mode head-tail instability. In the case where low-β insertions are included in the lattice, this first order correction is not sufficient because important higher order tune derivatives appear. The associated strong quadratic variation of the tunes with momentum, results in a linear betatron instability for a small number of standard deviations in energy. This makes the life-time unacceptably small. The origin of tune derivatives of order larger than one is explained. The principle of their correction is recalled