Draft: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

Moving coordinate system

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.

Matrix formalism

Since the equation of motion is linear it's solution which is a particle dynamics can be described in a terms of matrix transformation as following:

${\begin{pmatrix}x(s)\\x'(s)\end{pmatrix}}={\begin{pmatrix}t_{11}&t_{12}\\t_{21}&t_{22}\end{pmatrix}}\cdot {\begin{pmatrix}x(s_{0})\\x'(s_{0})\end{pmatrix}},$

where (x,x') is a vector of coordinate and slope and $T(s_{0}\to s)$ is a transport matrix from position s₀ to s along reference trajectory. Any external fields distribution along the orbit can be expressed in a piecewise-constant way. Thus the beamline could be imagined as a consequence of magnetic elements with constant parameters (curvature and gradient). Each magnet or focusing lens can be described by it's transport matrix. For example the matrix of drift space with length L:

${\begin{pmatrix}1&L\\0&1\end{pmatrix}},$

and matrix of "thick" quadrupole magnet with constant k_x:

${\begin{pmatrix}cos({\sqrt {k_{x}}}L)&{\frac {1}{\sqrt {k_{x}}}}sin({\sqrt {k_{x}}}L)\\-{\sqrt {k_{x}}}sin({\sqrt {k_{x}}}L)&cos({\sqrt {k_{x}}}L)\end{pmatrix}}.$

The consequence of any number of magnetic elements is described with product of corresponding matrices (in a proper order: from right to left!): $T=T_{n}\dots T_{2}T_{1}$. For the circular machine such as a synchrotron the transport matrix along the ring is called one-period matrix $M(s)=T(s\to s+\Pi )$. Due to Liouville's theorem which force transformation to save the phase space all the transport matrices must be symplectic. For one dimensional motion and 2×2 matrices it results in unit determinant: ${\begin{vmatrix}T(s_{1}\to s_{2})\end{vmatrix}}={\begin{vmatrix}M(s)\end{vmatrix}}=1$.

Literature

• Edwards, D. A.; Syphers, M. J. (1993). An introduction to the physics of high energy accelerators. New York: Wiley. ISBN 978-0-471-55163--8.
• Wiedemann, Helmut (2007). Particle accelerator physics (3rd ed.). Berlin: Springer. pp. 158–161. ISBN 978-3-540-49043-2.

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