To hold particles of the beam inside the vacuum chamber of an accelerator or transfer channel, magnetic or electrostatic elements are used. The guiding field of
dipole magnets sets the reference orbit of the beam while
focusing magnets, whose field linearly depends on the transverse coordinates, applies small deviations to particles, forcing them to oscillate stably around a reference orbit. For any orbit one can locally use the
Frenet–Serret coordinate system, which co-propagates with the reference particle. Assuming small deviations of the particle in all directions and after linearization of all the fields one will come to the linear equations of motion which are a pair of
Hill equations: \begin{cases} x'' + k_x(s)x = 0 \\ y'' + k_y(s)y = 0 \\ \end{cases}. Here k_x(s) = \frac{1}{r_0^2} + \frac{G(s)}{B\rho}, k_y(s)=-\frac{G(s)}{B\rho} are periodic functions in a case of cyclic accelerator such as a betatron or synchrotron. G(s)=\frac{\partial B_z}{\partial x} is a magnetic field gradient. Prime means derivative over s, path along the beam trajectory. The product of guiding field and curvature radius B\rho = B\cdot r_0 is the
magnetic rigidity, which is related to the momentum via the
Lorentz force pc=eZB\rho, where eZ is a particle charge. As the equations of transverse motion are independent from each other, they can be solved separately. For one-dimensional motion the solution of Hill equation is a quasi-periodical oscillation. It can be written as x(s)= A\sqrt{\beta_x (s)} \cdot \cos(\Psi_x (s) + \phi_0), where \beta(s) is the
Twiss beta function, \Psi (s) is a
betatron phase advance and A is an invariant amplitude known as
Courant-Snyder invariant. In lattices using high order magnets, such as sextupoles or octupoles, non-linear effects appear leading to
tune shift with amplitude. == References ==