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Anion trap consists of electrodes that produceelectric fields toions (charged particles), which may be atoms, molecules, or large particles such asdust. Atomic and molecular ion traps have a number of applications in physics and chemistry such as precisionmass spectrometry, precise and stableatomic frequency standards, andquantum computing.^[1] In comparison to neutral atom traps, ion traps have deeper trapping potentials (up to severalelectronvolts) that are agnostic to the internal structure of the ions. The two most popular ion traps are thePaul trap which uses static and oscillating electric fields^[2] and thePenning trap, which uses a combination of static electric and also static magnetic fields to trap. Paul traps are often used when manipulating quantum states. They are used intrapped ion quantum computers^[3] and realizingatomic clocks,^[4] including the most precise instrument humankind has ever made.^[5] Penning traps can be used for precise magnetic measurements in spectroscopy.

The physical principles of ion traps were first explored byF. M. Penning, who observed that electrons released by the cathode of anionization 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 byW. Paul based on his work withquadrupole mass spectrometers.

Ion traps were used intelevision receivers prior to the introduction of aluminizedCRT faces around 1958, to protect the phosphor screen from ions.^[7] The ion trap must be delicately adjusted for maximum brightness.^[8][9]

Ions experience a force from electric fields and magnetic fields (when the ion is moving). Ion traps use these forces to confine ions in a small volume so that they can be studied or manipulated. Although any static electric field produces a force on an ion, it is not possible to confine an ion using only static electric fields, which is a consequence ofEarnshaw's theorem. However, physicists have figured out how to evade Earnshaw's theorem with static magnetic and electric fields (aPenning trap) or with oscillating electric fields and a static electric field (aPaul trap). Trapped ion motion and confinement is generally divided into axial and radial components with respect to the trap geometry. In both Paul and Penning traps, axial motion is confined with a static electric field. Paul traps confine the ion radially with oscillating electric fields whereas Penning traps use a static magnetic field.

A Paul trap is aquadrupole ion trap that uses staticdirect current (DC) andradio frequency (RF) oscillatingelectric fields to trap ions. Paul traps are commonly used as components ofmass spectrometers. The invention of the quadrupole ion trap itself is attributed toWolfgang Paul, hence its name, who shared theNobel Prize in Physics in 1989 for this work.^[10][11] It is used as a component of amass spectrometer or atrapped ion quantum computer.

The RF field generates anaverage radial confining force with an oscillating quadrupole potential. The confining and anti-confining directions of the potential are switched faster than the particle's escape time. Since the field affects the acceleration, the position lags behind (by approximately half a period). So the particles are at defocused positions when the field is focusing and vice versa. Being farther from center, they experience a stronger field when the field is focusing than when it is defocusing.

Thequadrupole is the simplestelectric field geometry used in such traps, thoughmore complicated geometries are possible for specialized devices. The electric fields are generated fromelectric potentials on metal electrodes. A pure quadrupole is created fromhyperbolic electrodes, thoughcylindrical electrodes are often used for ease of fabrication. Microfabricated ion traps exist where the electrodes lie in a plane with the trapping region above the plane.^[12] There are two main classes of traps, depending on whether the oscillating field provides confinement in three or two dimensions. In the two-dimension case (a so-called "linear RF trap"), confinement in the third direction is provided by static electric fields.

A typical trap configuration has four parallel electrodes along the${\displaystyle z}$-axis that are positioned at the corners of a square in the${\displaystyle xy}$-plane. Diagonally opposite electrodes are connected and a voltage${\displaystyle V=V_0\cos (\mathrm{\Omega }t)}$ is applied. The electric field produced by this potential is${\displaystyle \mathbf{E}={\mathbf{E}}_0\sin (\mathrm{\Omega }t)}$. The force on an ion of charge${\displaystyle e}$ is${\displaystyle \mathbf{F}=e\mathbf{E}}$ which with ion mass${\displaystyle M}$ leads to the radial equation of motion

${\displaystyle M\ddot{\mathbf{r}}=e{\mathbf{E}}_0\sin (\mathrm{\Omega }t)}$ .

If the ion is initially at rest, two successive integrations give the velocity and displacement as

${\displaystyle \dot{\mathbf{r}}=\frac{e{\mathbf{E}}_0}{M\mathrm{\Omega }}\cos (\mathrm{\Omega }t)}$ ,

${\displaystyle \mathbf{r}={\mathbf{r}}_0-\frac{e{\mathbf{E}}_0}{M{\mathrm{\Omega }}^2}\sin (\mathrm{\Omega }t)}$ ,

where${\displaystyle {\mathbf{r}}_0}$ is a constant of integration and corresponding to an arbitrary starting position. Thus, the ion oscillates with angular frequency${\displaystyle \mathrm{\Omega }}$ 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^[13]

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

The constants${\displaystyle \alpha }$ and${\displaystyle \beta }$ are determined by boundary conditions on the electrodes and${\displaystyle \varphi }$ satisfiesLaplace's equation${\displaystyle {\mathrm{\nabla }}^2\varphi =0}$. Assuming the length of the electrodes${\displaystyle r}$ is much greater than their separation${\displaystyle r_0}$, it can be shown that

${\displaystyle \varphi ={\varphi }_0+\frac{V_0}{2r_0^2}\cos (\mathrm{\Omega }t)(x^2-y^2)}$ .

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

${\displaystyle \mathbf{E}=-\frac{V_0}{r_0^2}\cos (\mathrm{\Omega }t)(x{\hat{\mathbf{e}}}_x-y{\hat{\mathbf{e}}}_y)}$ .

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

${\displaystyle \frac{d^2{x}_i}{d{\tau }^2}=-\frac{4e{V}_0}{M{r}_0^2{\mathrm{\Omega }}^2}\cos (2\tau )x_i}$ .

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 theMathieu equation.^[14] 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${\displaystyle \xi =\mathrm{\Omega }t/2}$, and${\displaystyle a_u}$ and${\displaystyle q_u}$ are dimensionless trapping parameters. The parameter${\displaystyle \mathrm{\Omega }}$ is the radial frequency of the potential applied to the ring electrode. By using thechain rule, it can be shown that

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

2

SubstitutingEquation 2 into the MathieuEquation 1 yields

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

3

Multiplying by m and rearranging terms shows us that

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

4

ByNewton's laws of motion, the above equation represents the force on the ion. This equation can be exactly solved using theFloquet theorem or the standard techniques ofmultiple scale analysis.^[15] The particle dynamics and time averaged density of charged particles in a Paul trap can also be obtained by the concept ofponderomotive force.

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

${\displaystyle F_x=ma=m\frac{d^{2}x}{d{t}^2}=-e\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}}$

5

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

${\displaystyle \varphi =\frac{{\varphi }_0}{r_0^2}(\lambda x^2+\sigma y^2+\gamma z^2)}$

6

where${\displaystyle {\varphi }_0}$ 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 satisfyLaplace's equation,${\displaystyle {\mathrm{\nabla }}^2{\varphi }_0=0}$, it can be shown that

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

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

Transforming equation 6 into acylindrical coordinate system with${\displaystyle x=r\cos \theta }$,${\displaystyle y=r\sin \theta }$, and${\displaystyle z=z}$ and applying thePythagorean trigonometric identity${\displaystyle {\sin }^2\theta +{\cos }^2\theta =1}$ gives

${\displaystyle {\varphi }_{r,z}=\frac{{\varphi }_0}{r_0^2}(r^2-2z^2).}$

7

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

${\displaystyle {\varphi }_0=U+V\cos \mathrm{\Omega }t.}$

8

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

SubstitutingEquation 8 intoEquation 6 with${\displaystyle \lambda =1}$ gives

${\displaystyle \frac{\mathrm{\partial }\varphi }{\mathrm{\partial }x}=\frac{2x}{r_0^2}(U+V\cos \mathrm{\Omega }t).}$

9

Substituting equation 9 into equation 5 leads to

${\displaystyle m\frac{d^{2}x}{d{t}^2}=-\frac{2e}{r_0^2}(U+V\cos \mathrm{\Omega }t)x.}$

10

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

${\displaystyle a_x=\frac{8eU}{m{r}_0^2{\mathrm{\Omega }}^2}}$

11

and

${\displaystyle q_x=-\frac{4eV}{m{r}_0^2{\mathrm{\Omega }}^2}.}$

12

Further${\displaystyle q_x=q_y}$,

${\displaystyle a_z=-\frac{16eU}{m{r}_0^2{\mathrm{\Omega }}^2}}$

13

and

${\displaystyle q_z=\frac{8eV}{m{r}_0^2{\mathrm{\Omega }}^2}.}$

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.^[16]

A standard configuration for aPenning 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 fromEarnshaw'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${\displaystyle \mathbf{E}=E{\hat{\mathbf{e}}}_x}$, the force${\displaystyle \mathbf{F}=e\mathbf{E}}$ accelerates a positively charged ion along the${\displaystyle x}$-axis. For a uniform magnetic field${\displaystyle \mathbf{B}=B{\hat{\mathbf{e}}}_z}$, theLorentz force causes the ion to move in circular motion withcyclotron frequency

${\displaystyle {\omega }_c=\frac{eB}{M}}$ .

Assuming an ion with zero initial velocity placed in a region with${\displaystyle \mathbf{E}=E{\hat{\mathbf{e}}}_x}$ and${\displaystyle \mathbf{B}=B{\hat{\mathbf{e}}}_z}$, 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.^[17]

An ion trapmass spectrometer may incorporate a Penning trap (Fourier-transform ion cyclotron resonance),^[18]Paul trap^[19] or theKingdon trap.^[20] TheOrbitrap, introduced in 2005, is based on the Kingdon trap.^[21] Other types of mass spectrometers may also use a linear quadrupole ion trap as a selective mass filter.

APenning trap stores charged particles using a strong homogeneous axialmagnetic field to confine particles radially and a quadrupoleelectric field to confine the particles axially.^[22] Penning traps are well suited for measurements of the properties ofions and stable chargedsubatomic particles. Precision studies of the electron magnetic moment by Dehmelt and others are an important topic in modern physics.

Penning traps can be used inquantum computation andquantum information processing^[23] and are used atCERN to store antimatter. Penning traps form the basis ofFourier-transform ion cyclotron resonance mass spectrometry for determining themass-to-charge ratio ofions.^[24]

The Penning Trap was invented byFrans Michel Penning andHans Georg Dehmelt, who built the first trap in the 1950s.^[25]

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.^[20] 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.^[26] In 1981, Knight introduced a modified outer electrode that included an axial quadrupole term that confines the ions on the trap axis.^[27] The dynamic Kingdon trap has an additional AC voltage that uses strong defocusing to permanently store charged particles.^[28] The dynamic Kingdon trap does not require the trapped ions to have angular momentum with respect to the filament. AnOrbitrap is a modified Kingdon trap that is used formass spectrometry. Though the idea has been suggested and computer simulations performed^[29] neither the Kingdon nor the Knight configurations were reported to produce mass spectra, as the simulations indicated mass resolving power would be problematic.

Thedigital ion trap (DIT) is a quadrupole ion trap (linear or 3D) that differs from conventional traps by the driving waveform. A DIT is driven by digital signals, typically rectangular waveforms^[30][31] that are generated by switching rapidly between discrete voltage levels. Major advantages of the DIT are its versatility^[32] and virtually unlimited mass range. The digital ion trap has been developed mainly as a mass analyzer.

• Mass spectrometry
• Trapped ion quantum computer
• Laser cooling
• Quantum jump

• VIAS Science Cartoons A cranky view of an ion trap...
• Paul trap

• Mass spectrometry
• Ions
• Particle traps

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