Betatron oscillations

File:Référentiel mobile.png

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:{{Cite journal | last1 = Courant | first1 = Ernest D. | author-link1 = Ernest Courant | last2 = Snyder | first2 = Hartland | author-link2 = Hartland Snyder | title = Theory of the alternating-gradient synchrotron | journal = Annals of Physics | volume = 3 | issue = 1 | pages = 1–48 | date=Jan 1958 | doi = 10.1016/0003-4916(58)90012-5 | bibcode = 1958AnPhy...3....1C| url = https://zenodo.org/record/1229408 }}

\begin{cases}

x'' + k_x(s)x = 0 \\

y'' + k_y(s)y = 0 \\

\end{cases}.

Here $k\_x(s)\; =\; \backslash frac\{1\}\{r\_0^2\}\; +\; \backslash frac\{G(s)\}\{B\backslash rho\}$, $k\_y(s)=-\backslash frac\{G(s)\}\{B\backslash rho\}$ are periodic functions in a case of cyclic accelerator such as betatron or synchrotron. $G(s)=\backslash frac\{\backslash partial\; B\_z\}\{\backslash partial\; x\}$ is a gradient of magnetic field. Prime means derivative over s, path along the beam trajectory. The product of guiding field over curvature radius $B\backslash rho\; =\; B\backslash cdot\; r\_0$ is magnetic rigidity, which is via Lorentz force strictly related to the momentum $pc=eZB\backslash rho$, 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 $x(s)=\; A\backslash sqrt\{\backslash beta\_x\; (s)\}\; \backslash cdot\; cos(\backslash Psi\_x\; (s)\; +\; \backslash phi\_0)$, where $\backslash beta(s)$ is Twiss beta-function, $\backslash Psi\; (s)$ is a betatron phase advance and $A$ is an invariant amplitude known as Courant-Snyder invariant.{{cite journal |last1=Qin |first1=Hong |last2=Davidson |first2=Ronald C. |title=Symmetries and invariants of the oscillator and envelope equations with time-dependent frequency |journal=Physical Review Special Topics - Accelerators and Beams |date=22 May 2006 |volume=9 |issue=5 |doi=10.1103/PhysRevSTAB.9.054001 |url=https://journals.aps.org/prab/pdf/10.1103/PhysRevSTAB.9.054001 |language=en |issn=1098-4402|doi-access=free }}{{additional citation needed|date=June 2024}}

{{Reflist}}

• {{cite book |last1=Edwards |first1=D. A. |last2=Syphers |first2=M. J. |title=An introduction to the physics of high energy accelerators |date=1993 |publisher=Wiley |location=New York |isbn=978-0-471-55163--8}}
• {{cite book |last1=Wiedemann |first1=Helmut |title=Particle accelerator physics |date=2007 |publisher=Springer |location=Berlin |isbn=978-3-540-49043-2 |pages=158–161 |edition=3rd}}

Category:Accelerator physics

{{accelerator-stub}}