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\begin{document}

\title{Trapping Two Dissimilar Rigidity Ions with Identical Average Dynamics}
\author{Stephen Brooks}
\date{January 10, 2025}
\maketitle

\section{Introduction}
It may seem obvious that ions with different charge-to-mass ratios ($q/m$) would have different dynamics in a trap or periodic focussing system, leading to different equilibrium bunch shapes and sizes.  However, the ponderomotive acceleration in an oscillating field is proportional to $(q/m)^2$ rather than $q/m$ for the direct field, leading to the possibility of combining the two effects to cancel any differences in the average dynamics between two chosen rigidities.

\section{Ponderomotive Acceleration}
Nonrelativistically, the ponderomotive acceleration is given by
\[ \av_p=-\frac{q^2}{4m^2\omega^2}\nabla|\E_p|^2, \]
for a oscillating electric field $\E(\x,t)=\E_p(\x)\sin(\omega t)$.  Compare this with the direct acceleration
\[ \av_d=\frac{q}{m}\E_d. \]

For two ion species, consider the case where
\[ \nabla|\E_p|^2 = \frac{4\omega^2}{\frac{q_1}{m_1}+\frac{q_2}{m_2}}\E_d. \]
The total (average) acceleration for ion species $i$ is then
\begin{eqnarray*}
 \av_i &=& \frac{q_i}{m_i}\E_d -\frac{q_i^2}{4m_i^2\omega^2}\nabla|\E_p|^2 \\
 &=& \frac{q_i}{m_i}\E_d -\frac{q_i^2}{4m_i^2\omega^2}\frac{4\omega^2}{\frac{q_1}{m_1}+\frac{q_2}{m_2}}\E_d \\
 &=& \left( \frac{q_i}{m_i} -\frac{q_i^2}{m_i^2}\frac{1}{\frac{q_1}{m_1}+\frac{q_2}{m_2}} \right) \E_d \\
 &=& \frac{\frac{q_i}{m_i}\frac{q_1}{m_1}+\frac{q_i}{m_i}\frac{q_2}{m_2} -\frac{q_i^2}{m_i^2}}{\frac{q_1}{m_1}+\frac{q_2}{m_2}} \E_d.
\end{eqnarray*}
One of the terms on the numerator is also $\frac{q_i^2}{m_i^2}$, so it cancels, leaving only the mixed term.  The average acceleration is therefore the same for both ions:
\[ \av_i = \frac{\frac{q_1}{m_1}\frac{q_2}{m_2}}{\frac{q_1}{m_1}+\frac{q_2}{m_2}} \E_d. \]

\section{Quadrupole Field}
Consider the direct field $\E_d=(kx,ky,-2kz)$.  It has the potential $V_d=-\frac{k}2x^2-\frac{k}2y^2+kz^2$ such that $\E_d=-\nabla V_d$.  For equal acceleration, it is required that 
\[ \nabla|\E_p|^2 = \frac{4\omega^2}{\frac{q_1}{m_1}+\frac{q_2}{m_2}}\E_d = \frac{-4\omega^2}{\frac{q_1}{m_1}+\frac{q_2}{m_2}}\nabla V_d. \]
This would be satisfied if 
\[ |\E_p|^2 = \frac{-4\omega^2}{\frac{q_1}{m_1}+\frac{q_2}{m_2}}V_d + C, \]
for any constant $C$.  The solution cannot fill all of space as $|\E_p|^2\ge0$ and $V_d$ can attain unbounded positive and negative values.  $C$ should generally be positive so that the region around the origin is feasible.

Linear electric fields cannot attain the required saddle shape of $|\E_p|^2$, so consider a combined dipole+sextupole electric field with
\[ \E_p(0,0,z) = (0,0,\sqrt{C}-sz^2) \]
on the $z$ axis.  This extends symmetrically about the $z$ axis to the full sextupole field
\[ \E_p = \left(sxz,syz,\sqrt{C}+\frac{s}2x^2+\frac{s}2y^2-sz^2\right) \]
that satsifies Maxwell's equations in free space.  Its modulus squared is
\[ |\E_p|^2 = C + s\sqrt{C}(x^2+y^2-2z^2) + \mbox{terms in $(x,y,z)^4$}. \]
Ignoring the higher order terms that are small near the origin, this has the correct form.  Equating coefficients of $x^2+y^2-2z^2$ gives
\[ s\sqrt{C} = \frac{-4\omega^2}{\frac{q_1}{m_1}+\frac{q_2}{m_2}}\left(-\frac{k}{2}\right)
\qquad \Rightarrow \qquad
s = \frac{2k\omega^2}{\sqrt{C}\left(\frac{q_1}{m_1}+\frac{q_2}{m_2}\right)}.  \]

\section{Alternating Quadrupole Trap}
The field in the previous section gives the two ion species identical accelerations near the origin, but it is not a trap because the effective potential is saddle-shaped.  This can be rectified by modulating the entire potential at a frequency $\omega_a\ll\omega$, yielding an alternating effective quadrupole that is overall focussing by the alternating gradient principle.  This can be done by setting $k(t)=k_0\sin(\omega_a t)$ in the previous section, which is a simple modulation of $\E_d$ but only changes the sextupole part of $\E_p$ and not the dipole.  This idea uses the ponderomotive motion principle \emph{twice} because alternating gradient focussing reduces to ponderomotive focussing for small phase advances.

\section{Accelerator Beamlines}
This method can also be used to transport two species of different rigidity through a particle accelerator with the same optics.  In this case, a 2D effective quadrupole is formed by a rapidly alternating dipole+sextupole field, superimposed with the correct amount of constant quadrupole and then this combination is alternated more slowly to give alternating gradient focussing.

\end{document}