Quadrupole ion trap

Ions in a quadrupole field experience restoring forces that drive them back toward the center of the trap. The motion of the ions in the field is described by solutions to the Mathieu equation.[7] When written for ion motion in a trap, the equation is

${\displaystyle {\frac {d^{2}u}{d\xi ^{2}}}+[a_{u}-2q_{u}\cos(2\xi )]u=0}$

(1)

where ${\displaystyle u}$ represents the x, y and z coordinates, ${\displaystyle \xi }$ is a dimensionless variable given by ${\displaystyle \xi =\Omega t/2}$, and ${\displaystyle a_{u}\,}$ and ${\displaystyle q_{u}}$ are dimensionless trapping parameters. The parameter ${\displaystyle \Omega }$ is the radial frequency of the potential applied to the ring electrode. By using the chain rule, it can be shown that

${\displaystyle {\frac {d^{2}u}{dt^{2}}}={\frac {\Omega ^{2}}{4}}{\frac {d^{2}u}{d\xi ^{2}}}}$

(2)

Substituting Equation 2 into the Mathieu Equation 1 yields

${\displaystyle {\frac {4}{\Omega ^{2}}}{\frac {d^{2}u}{dt^{2}}}+\left[a_{u}-2q_{u}\cos(\Omega t)\right]u=0.}$

(3)

Multiplying by m and rearranging terms shows us that

${\displaystyle m{\frac {d^{2}u}{dt^{2}}}+m{\frac {\Omega ^{2}}{4}}\left[a_{u}-2q_{u}\cos(\Omega t)\right]u=0.}$

(4)

By Newton's laws of motion, the above equation represents the force on the ion. This equation can be exactly solved using the Floquet theorem or the standard techniques of multiple scale analysis.[8] The particle dynamics and time averaged density of charged particles in a Paul trap can also be obtained by the concept of ponderomotive force.

The forces in each dimension are not coupled, thus the force acting on an ion in, for example, the x dimension is

${\displaystyle F_{x}=ma=m{\frac {d^{2}x}{dt^{2}}}=-e{\frac {\partial \phi }{\partial x}}}$

(5)

Here, ${\displaystyle \phi }$ is the quadrupolar potential, given by

${\displaystyle \phi ={\frac {\phi _{0}}{r_{0}^{2}}}{\bigl (}\lambda x^{2}+\sigma y^{2}+\gamma z^{2}{\bigr )}}$

(6)

where ${\displaystyle \phi _{0}}$ is the applied electric potential and ${\displaystyle \lambda }$, ${\displaystyle \sigma }$, and ${\displaystyle \gamma }$ are weighting factors, and ${\displaystyle r_{0}}$ is a size parameter constant. In order to satisfy Laplace's equation, ${\displaystyle \nabla ^{2}\phi _{0}=0}$, it can be shown that

${\displaystyle \lambda +\sigma +\gamma =0\,.}$

For an ion trap, ${\displaystyle \lambda =\sigma =1}$ and ${\displaystyle \gamma =-2}$ and for a quadrupole mass filter, ${\displaystyle \lambda =-\sigma =1}$ and ${\displaystyle \gamma =0}$.

Transforming Equation 6 into a cylindrical coordinate system with ${\displaystyle x=r\cos \theta }$, ${\displaystyle y=r\sin \theta }$, and ${\displaystyle z=z}$ and applying the Pythagorean trigonometric identity ${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$ gives

Diagram of the stability regions of a quadrupole ion trap according to the voltage and frequency applied to the ion trap elements.

${\displaystyle \phi _{r,z}={\frac {\phi _{0}}{r_{0}^{2}}}{\big (}r^{2}-2z^{2}{\big )}.}$

(7)

The applied electric potential is a combination of RF and DC given by

${\displaystyle \phi _{0}=U+V\cos \Omega t.}$

(8)

where ${\displaystyle \Omega =2\pi \nu }$ and ${\displaystyle \nu }$ is the applied frequency in hertz.

Substituting Equation 8 into Equation 6 with ${\displaystyle \lambda =1}$ gives

${\displaystyle {\frac {\partial \phi }{\partial x}}={\frac {2x}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}.}$

(9)

Substituting Equation 9 into Equation 5 leads to

${\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {2e}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}x.}$

(10)

Comparing terms on the right hand side of Equation 1 and Equation 10 leads to

${\displaystyle a_{x}={\frac {8eU}{mr_{0}^{2}\Omega ^{2}}}}$

(11)

and

${\displaystyle q_{x}=-{\frac {4eV}{mr_{0}^{2}\Omega ^{2}}}.}$

(12)

Further ${\displaystyle q_{x}=q_{y}\,}$,

${\displaystyle a_{z}=-{\frac {16eU}{mr_{0}^{2}\Omega ^{2}}}}$

(13)

and

${\displaystyle q_{z}={\frac {8eV}{mr_{0}^{2}\Omega ^{2}}}.}$

(14)

The trapping of ions can be understood in terms of stability regions in ${\displaystyle q_{u}}$ and ${\displaystyle a_{u}}$ space. The boundaries of the shaded regions in the figure are the boundaries of stability in the two directions (also known as boundaries of bands). The domain of overlap of the two regions is the trapping domain. For calculation of these boundaries and similar diagrams as above see Müller-Kirsten.[9]

• DE 944900 "Verfahren zur Trennung bzw. zum getrennten Nachweis von Ionen verschiedener spezifischer Ladung", W. Paul and H. Steinwedel, filed on December 24, 1953
• GB 773689 "Improved arrangements for separating or separately detecting charged particles of different specific charges", W. Paul, claims priority of the above German application filed on December 24, 1953
• US 2939952 "Apparatus for separating charged particles of different specific charges", W. Paul and H. Steinwedel, claims priority of the above German application filed on December 24, 1953