Exact solution of the envelope equations for a matched quadrupole-focused beam in the low space-charged limit
The Kapchinskij-Vladimirskij equations are widely used to study the evolution of the beam envelopes in a periodic system of quadrupole focusing cells. In this paper, we analyze the case of a matched beam. Our model is analogous to that used by Courant and Snyder [E.D. Courant and H.S. Snyder, Ann. Phys. 3, 1 (1958)] in obtaining a first-order approximate solution for a synchrotron. Here, we treat a linear machine and obtain an exact solution. The model uses a full occupancy, piecewise-constant focusing function and neglects space charge. There are solutions in an infinite number of bands as the focus strength is increased. We show that all these bands are stable. Our explicit results for the phase advance sigma and the envelope a(z) are exact for all phase advances except multiples of 180o, where the behavior is singular. We find that the peak envelope size is minimized at sigma = 90o. Actual operation in the higher bands would require very large, very accurate field strengths and would produce significantly larger envelope excursions.