When considering many turns of an accelerator lattice, we observe the behaviour of a beam envelope. Hill's equation's are second order differential equations for a system with periodic focussing properties. The motion has a variable spring constant $k(s)$ which depends on the magnetic properties of the accelerator ring. If the ring has periodicity L, then so does the function $k(s)$:
\begin{equation}k(s+L) ~=~ k(s)\end{equation}
From this we can expect something similar to harmonic oscillation where the frequency and amplitude depend on the location in the accelerator ring, and show periodicity similar to that of the function $k(s)$.
The Courant-Snyder formalism assumes a solution of Hill's equation inspired by this intuition on position dependent amplitude and phase, to provide us with a set of auxiliary functions that allow us to extract the maximum information from Hill's equations. Thus we use the ansatz:
\begin{equation}x(s) ~=~ \sqrt{\epsilon \beta(s) }~ cos ( \psi (s) + \psi_0).\end{equation}
$\beta(s)$ is a position dependent amplitude, $\psi(s)$ is a position dependent phase, and $\epsilon$ is a constant known as the emittance. As Hill's equation is linear the emittance does not appear in it. $\beta(s)$ is the important variable in the Courant-Snyder formalism. Having many names such as the beta function and the beam envelope function, it has the physical meaning of amplitude, which is dependent on the position along the accelerator. It represents the focussing properties of an accelerator lattice, a small $\beta$ represents a tightly focussed lattice. We note that $\beta(s+L) = \beta(s)$.
Taking derivatives of the Courant-Snyder ansatz and substituting into the equation of motion, we get two terms; one proportional to cos and the other proportional to sin. We obtain two differential equations:
\begin{equation}\frac{1}{2} (\beta \beta'' - \frac{1}{2} \beta'^2) - \beta^2 \psi^2 + \beta^2k ~=~0,\end{equation}
\begin{equation}\beta'\psi' + \beta\psi'' ~=~ 0.\end{equation}
The latter may be integrated as:
\begin{equation} \beta'\phi' + \beta\psi'' ~=~ (\beta\psi')'. \end{equation}
We may choose the integration constant to be 1, $\beta\psi'=1$. The result for the phase function is thus:
\begin{equation}
\psi(s) ~=~ \int\limits_0^s \frac{ds}{\beta(s)},
\end{equation}
a position dependent phase, which is related to an integration of the beta function along the beam line. Knowing the beta function means that we may compute the phase function. We can eliminate the phase function from the first Courant-Snyder differential equation, to obtain a differential equation for the beta function:
\begin{equation} \frac{1}{2} \beta \beta'' - \frac {1}{4}\beta'^2 + \beta^2k ~=~1 \end{equation}
Thus we see that $\beta$ is determined by the distribution of focussing strengths along the accelerator, though we do not solve this equation in practice. We define two functions that, along with $\beta$, are called the lattice, or sometimes TWISS functions:
\begin{equation} \alpha(s) ~=~ \frac{-1}{2} \frac{d\beta(s)}{ds}, \end{equation}
and
\begin{equation} \gamma(s) ~=~ \frac{1+\alpha(s)^2}{\beta(s)}. \end{equation}
Once the lattice functions are known, the motion of a single particle through the lattice is completely defined in specifying the emittance and initial phase factor of the particle.
After some algebra we see that
\begin{equation} \epsilon ~=~ \gamma x^2 + 2 \alpha x x' + \beta x'^2. \end{equation}
Original Paper
E.D. Courant and H.S. Snyder, Ann. Phys. 3, 1 (1958)
Adapted from
[1] S. Y. Lee, Accelerator Physics, Second Edition, World Scientific, 2007
[2] H. Weidemann. Particle Accelerator Physics I: Basic Principles and Linear Beam
Dynamics. Springer, 1999
[3] R. Appleby, Beam Dynamics, Cockcroft Lectures 2011
Thursday 19 December 2013
2. Hill's Equations
Hill's equations; linearised second order differential equations for the transverse variables $x$ and $y$ in dipole and quadrupole fields within an accelerator are :
\begin{equation}x(s)'' + \left( k(s) + \frac{1}{\rho(s)^2} \right) x(s) ~=~ 0,\end{equation}
and
\begin{equation}y(s)'' - k(s) y ~=~ 0.\end{equation}
where $\rho = const$ and $k=const$ . Hills equations may be written compactly;
\begin{equation}x''+ K \cdot x ~=~ 0.\end{equation}
Where $K = k, K = (k + \frac{1}{\rho^2})$; if we write K for the constant and assume that it is constant over the accelerator, these equations appear similar to that of the a harmonic oscillator.
\begin{equation}x(s)'' + \left( k(s) + \frac{1}{\rho(s)^2} \right) x(s) ~=~ 0,\end{equation}
and
\begin{equation}y(s)'' - k(s) y ~=~ 0.\end{equation}
where $\rho = const$ and $k=const$ . Hills equations may be written compactly;
\begin{equation}x''+ K \cdot x ~=~ 0.\end{equation}
Where $K = k, K = (k + \frac{1}{\rho^2})$; if we write K for the constant and assume that it is constant over the accelerator, these equations appear similar to that of the a harmonic oscillator.
1. Coordinates
We begin by defining a curved coordinate system, the curvature $\rho$ produced by a local dipole field, and the path length along the design trajectory is $s$. The position of a particle in these coordinates is:
\begin{equation} r ~=~ \rho + x. \end{equation}
These coordinates represent deviations with respect to the ideal design orbit, we assume these deviations to be small; in reality $x$ is in the of order mm. We use the positions $x$ and $y$, and their slopes $x' = \frac{dx}{ds}$ and $y' = \frac{dy}{ds}$.
Circular coordinate system [1]
\begin{equation} \vec{R} ~=~ r \vec{x} + y \vec{z}, \end{equation}
where\begin{equation} r ~=~ \rho + x. \end{equation}
These coordinates represent deviations with respect to the ideal design orbit, we assume these deviations to be small; in reality $x$ is in the of order mm. We use the positions $x$ and $y$, and their slopes $x' = \frac{dx}{ds}$ and $y' = \frac{dy}{ds}$.
Circular coordinate system [2]
References
[1] R. Appleby. Beam Dynamics. Cockcroft Lectures 2011
[2] H. Weidemann. Particle Accelerator Physics I: Basic Principles and Linear Beam
Dynamics. Springer, 1999
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