Ion trap

The physical principles of ion traps were first explored by F. M. Penning (1894–1953), who observed that electrons released by the cathode of an ionization vacuum gauge follow a long cycloidal path to the anode in the presence of a sufficiently strong magnetic field.^[6] A scheme for confining charged particles in three dimensions without the use of magnetic fields was developed by W. Paul based on his work with quadrupole mass spectrometers.

Ion traps were used in

A Paul trap that uses an oscillating quadrupole field to trap ions radially and a static potential to confine ions axially. The quadrupole field is realized by four parallel electrodes laying in the ${\displaystyle z}$-axis positioned at the corners of a square in the ${\displaystyle xy}$-plane. Electrodes diagonally opposite each other are connected and an a.c. voltage ${\displaystyle V=V_{0}\cos(\Omega t)}$ is applied. Using Maxwell's equations, the electric field produced by this potential is electric field ${\displaystyle \mathbf {E} =\mathbf {E} _{0}\sin(\Omega t)}$. Applying

${\displaystyle e}$

${\displaystyle M}$

${\displaystyle \mathbf {F} =e\mathbf {E} }$

${\displaystyle M\mathbf {\ddot {r}} =e\mathbf {E} _{0}\sin(\Omega t)\!}$ .

Assuming that the ion has zero initial velocity, two successive integrations give the velocity and displacement as

${\displaystyle \mathbf {\dot {r}} ={\frac {e\mathbf {E} _{0}}{M\Omega }}\cos(\Omega t)\!}$ ,

${\displaystyle \mathbf {r} =\mathbf {r} _{0}-{\frac {e\mathbf {E} _{0}}{M\Omega ^{2}}}\sin(\Omega t)\!}$ ,

where ${\displaystyle \mathbf {r} _{0}}$ is a constant of integration. Thus, the ion oscillates with angular frequency ${\displaystyle \Omega }$ and amplitude proportional to the electric field strength and is confined radially.

Working specifically with a linear Paul trap, we can write more specific equations of motion. Along the ${\displaystyle z}$-axis, an analysis of the radial symmetry yields a potential^[10]

${\displaystyle \phi =\alpha +\beta (x^{2}-y^{2})\!}$ .

The constants ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are determined by boundary conditions on the electrodes and ${\displaystyle \phi }$ satisfies Laplace's equation ${\displaystyle \nabla ^{2}\phi =0}$. Assuming the length of the electrodes ${\displaystyle r}$ is much greater than their separation ${\displaystyle r_{0}}$, it can be shown that

${\displaystyle \phi =\phi _{0}+{\frac {V_{0}}{2r_{0}^{2}}}\cos(\Omega t)(x^{2}-y^{2})\!}$ .

Since the electric field is given by the gradient of the potential, we get that

${\displaystyle \mathbf {E} =-{\frac {V_{0}}{r_{0}^{2}}}\cos(\Omega t)(x\mathbf {\hat {e}} _{x}-y\mathbf {\hat {e}} _{y})\!}$ .

Defining ${\displaystyle \tau =\Omega t/2}$, the equations of motion in the ${\displaystyle xy}$-plane are a simplified form of the

${\displaystyle {\frac {d^{2}x_{i}}{d\tau ^{2}}}=-{\frac {4eV_{0}}{Mr_{0}^{2}\Omega ^{2}}}\cos(2\tau )x_{i}\!}$ .

Penning Trap

A standard configuration for a Penning trap consists of a ring electrode and two end caps. A static voltage differential between the ring and end caps confines ions along the axial direction (between end caps). However, as expected from Earnshaw's theorem, the static electric potential is not sufficient to trap an ion in all three dimensions. To provide the radial confinement, a strong axial magnetic field is applied.

For a uniform electric field ${\displaystyle \mathbf {E} =E\mathbf {\hat {e}} _{x}}$, the force ${\displaystyle \mathbf {F} =e\mathbf {E} }$ accelerates a positively charged ion along the ${\displaystyle x}$-axis. For a uniform magnetic field ${\displaystyle \mathbf {B} =B\mathbf {\hat {e}} _{z}}$, the Lorentz force causes the ion to move in circular motion with cyclotron frequency

${\displaystyle \omega _{c}={\frac {eB}{M}}\!}$ .

Assuming an ion with zero initial velocity placed in a region with ${\displaystyle \mathbf {E} =E\mathbf {\hat {e}} _{x}}$ and ${\displaystyle \mathbf {B} =B\mathbf {\hat {e}} _{z}}$, the equations of motion are

${\displaystyle x={\frac {E}{\omega _{c}B}}(1-\cos(\omega _{c}t))\!}$ ,

${\displaystyle y=-{\frac {E}{\omega _{c}B}}(\omega _{c}t-\sin(\omega _{c}t))\!}$ ,

${\displaystyle z=0\!}$ .

The resulting motion is a combination of oscillatory motion around the ${\displaystyle z}$-axis with frequency ${\displaystyle \omega _{c}}$ and a drift velocity in the ${\displaystyle y}$-direction. The drift velocity is perpendicular to the direction of the electric field.

For the radial electric field produced by the electrodes in a Penning trap, the drift velocity will precess around the axial direction with some frequency ${\displaystyle \omega _{m}}$, called the magnetron frequency. An ion will also have a third characteristic frequency ${\displaystyle \omega _{z}}$ between the two end cap electrodes. The frequencies usually have widely different values with ${\displaystyle \omega _{z}\ll \omega _{m}<\ll \omega _{c}}$.^[11]

Ion trap mass spectrometers

An ion trap

A

The Penning Trap was invented by Frans Michel Penning and Hans Georg Dehmelt, who built the first trap in the 1950s.^[19]

Paul ion trap

A Paul trap is a type of

A Kingdon trap consists of a thin central wire, an outer cylindrical electrode and isolated end cap electrodes at both ends. A static applied voltage results in a radial logarithmic potential between the electrodes.^[22] In a Kingdon trap there is no potential minimum to store the ions; however, they are stored with a finite angular momentum about the central wire and the applied electric field in the device allows for the stability of the ion trajectories.^[23] In 1981, Knight introduced a modified outer electrode that included an axial quadrupole term that confines the ions on the trap axis.^[24] The dynamic Kingdon trap has an additional AC voltage that uses strong defocusing to permanently store charged particles.^[25] The dynamic Kingdon trap does not require the trapped ions to have angular momentum with respect to the filament. An Orbitrap is a modified Kingdon trap that is used for mass spectrometry. Though the idea has been suggested and computer simulations performed^[26] neither the Kingdon nor the Knight configurations were reported to produce mass spectra, as the simulations indicated mass resolving power would be problematic.

Trapped ion quantum computer

Some experimental work towards developing quantum computers use

• Laser cooling
• Mass spectrometry
• Quantum jump

References