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Beam focusing

Figure 2.2: Field lines in a magnetic quadrupole. For an ion moving in the s-direction, the force components are focusing in the $x$-direction and defocusing in the $y$-direction.
\begin{figure}\begin{center}\parbox{90mm}{
\epsfig{file=quad.eps,width=90mm}
}\end{center}\end{figure}

In the present section we discuss focusing in quadrupole fields. A magnetic quadrupole field is described by


\begin{displaymath}
B_y=B_0\frac{x}{a},\quad B_x=B_0\frac{y}{a}
\end{displaymath} (2.1)

Such a field is produced by a magnet configuration with hyperbolic pole shapes, as shown in Fig. 2.2. The equations of motion in a magnetic quadrupole are


\begin{displaymath}
\ddot{x}+\frac{qv_sB_0}{\gamma ma}x=0
\end{displaymath} (2.2)


\begin{displaymath}
\ddot{y}-\frac{qv_sB_0}{\gamma ma}y=0
\end{displaymath}

We eliminate the time $t$ and write the above equations as trajectory equations. With $s=v_0t,\quad v_s\approx v_0=\mbox{const},\quad d^2/dt^2=v_0^2(d^2/ds^2)$ we get


\begin{displaymath}
x^{''}(s)+\kappa(s)x=0
\end{displaymath} (2.3)


\begin{displaymath}
y^{''}(s)-\kappa(s)y=0
\end{displaymath}

with the focusing strength for a magnetic quadrupole
\begin{displaymath}
\kappa=\frac{qB_0}{\gamma mav_0}
\end{displaymath} (2.4)

for electric quadrupole one obtains

\begin{displaymath}
\kappa=\frac{qE_0}{\gamma ma v_0^2}
\end{displaymath}

We can write the solution in matrix form as

\begin{displaymath}
\left( \begin{array}{c} x \\ x^{'}
\end{array} \right)=M
\left( \begin{array}{c} x_0 \\ x_0^{'}
\end{array} \right)
\end{displaymath}

where $x_0$ and $x_0^{'}$ are the initial coordinates and $x$ and $x^{'}$ are the final values. The $2\times 2$ matrix $M$ is called a transfer matrix. For building beam transport lines we are interested in quadrupole magnets that give transverse focusing when $\kappa>0$ and defocusing when $\kappa<0$. Furthermore we have field-free drift spaces with $\kappa=0$. Assuming force functions $\kappa (s)$, which are piecewise constant inside the elements and change step-like at the element boundaries, we can obtain the following results for the corresponding transfer matrices

Focusing quadrupole ($\kappa>0$) of length $l$:
\begin{displaymath}
M_F=\left( \begin{array}{cc}
\cos\sqrt{\kappa}l & \displays...
...}\sin\sqrt{\kappa}l & \cos\sqrt{\kappa}l
\end{array} \right)
\end{displaymath} (2.5)

Defocusing quadrupole ($\kappa<0$) of length $l$:
\begin{displaymath}
M_D=\left( \begin{array}{cc}
\cosh\sqrt{\kappa}l & \display...
...sinh\sqrt{\kappa}l & \cosh\sqrt{\kappa}l
\end{array} \right)
\end{displaymath} (2.6)

Drift space ($\kappa=0$) of length $l$:
\begin{displaymath}
M_O=\left( \begin{array}{cc}
1 & l \\
0 & 1
\end{array} \right)
\end{displaymath} (2.7)

The matrix for the passage through the whole transfer line is then obtained by multiplication of the matrices of all the transfer line elements $M=M_1\cdot M_2\cdot M_3\dots$.


next up previous contents
Next: Periodic Focusing Channel Up: Linear Beam Dynamics without Previous: Beam particle coordinates   Contents
Oliver Boine-Frankenheim 2001-07-09