Betatron oscillations

Betatron oscillations are the fast transverse oscillations of a charged particle in various focusing systems: linear accelerators, storage rings, transfer channels. Oscillations are usually considered as a small deviations from the ideal reference orbit and determined by transverse forces of focusing elements i.e. depending on transverse deviation value: quadrupole magnets, electrostatic lenses, RF-fields. This transverse motion is the subject of study of electron optics. Betatron oscillations were firstly studied by D.W. Kerst and R. Serber in 1941 while commissioning the fist betatron. The fundamental study of betatron oscillations was carried out by Ernest Courant, Milton S.Livingston and Hartland Snyder that lead to the revolution in high energy accelerators design by applying strong focusing principle.

Hill's equations
To hold particles of the beam inside the vacuum chamber of 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 with field linearly depending on transverse coordinate returns the particles with small deviations forcing them to oscillate stably around reference orbit. For any orbit one can set locally the co-propagating with the reference particle Frenet–Serret coordinate system. 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 betatron or synchrotron. $$G(s)=\frac{\partial B_z}{\partial x}$$ is a gradient of magnetic field. Prime means derivative over s, path along the beam trajectory. The product of guiding field over curvature radius $$B\rho = B\cdot r_0$$ is magnetic rigidity, which is via Lorentz force strictly related to the momentum $$pc=eZB\rho$$, where $$eZ$$ is a particle charge.

As the equation of transverse motion 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 Twiss beta-function, $$\Psi (s)$$ is a betatron phase advance and $$A$$ is an invariant amplitude known as Courant-Snyder invariant.