Contents Introduction
When choosing the operating point in the tune space, one carefully avoids resonances driven by the lattice periodicity (structure resonances). However, unavoidable presence of errors in the magnetic field sets restrictions associated with the imperfection resonances. As a result, the condition that the individual particle tune should not be depressed by the space charge to integer or half-integer values is known as the space-charge limit. We note that such a definition of the space charge limit is different from the one used in a special class of circular machines (for example, in cooler rings) where additional efforts are undertaken to compensate for emittance growth. The maximum achievable intensity associated with crossing of the integer or half-integer tunes was first formulated using the single-particle approach. Subsequently, a more accurate treatment of collective beam dynamics gave better understanding of the beam response to such resonances [1]–[2]. Such a coherent resonance condition, corresponding to the half-integer single-particle resonance (, where and is the zero-current tune), is n=\Omega_{2}=2\nu_{0}-\Delta \Omega_{2,s c}, \eqno{\hbox{(1)}}
