这是一份liverpool利物浦大学PHYS481的成功案例

This is the general equation of motion for a charged particles in a magnetic field $\boldsymbol{B}$. So far no approximations have been made. For practical use we may separate the individual components and get the differential equations for transverse motion
$$
\begin{aligned}
&x^{\prime \prime}-\kappa_{x} h-\frac{1}{2} \frac{x^{\prime}}{s^{\prime 2}} \frac{\mathrm{d} s^{\prime 2}}{\mathrm{~d} z}=\frac{e}{p} s^{\prime}\left[y^{\prime} B_{z}-h B_{y}\right] \
&y^{\prime \prime}-\kappa_{y} h-\frac{1}{2} \frac{y^{\prime}}{s^{\prime 2}} \frac{\mathrm{d} s^{\prime 2}}{\mathrm{~d} z}=\frac{e}{p} s^{\prime}\left[h B_{x}-x^{\prime} B_{z}\right]
\end{aligned}
$$

PHYS481 COURSE NOTES ：

Chromatic effects originate from the momentum factor $\frac{e}{p}$ which is different for particles of different energies. We expand this factor into a power series in $\delta$
$$
\frac{e}{p}=\frac{e}{p_{0}}\left(1-\delta+\delta^{2}-\delta^{3}+\cdots\right),
$$
where $\delta=\Delta p / p_{0}$ and $c p_{0}=\beta E_{0}$ is the ideal particle momentum. A further approximation is made when we expand $s^{\prime}$ to third order in $x$ and $y$ while restricting the discussion to paraxial rays with $x^{\prime} \ll 1$ and $y^{\prime} \ll 1$
$$
s^{\prime} \approx h+\frac{1}{2}\left(x^{\prime 2}+y^{\prime 2}\right)\left(1-\kappa_{x} x-\kappa_{y} y\right)+\cdots
$$