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An ion trap is a combination of electric and/or magnetic fields used to capture charged particles — known as ions — often in a system isolated from an external environment. Atomic and molecular ion traps have a number of applications in physics and chemistry such as precision mass spectrometry, improved atomic frequency standards, and quantum computing.^[1] In comparison to neutral atom traps, ion traps have deeper trapping potentials (up to several electronvolts) that do not depend on the internal electronic structure of a trapped ion. This makes ion traps more suitable for the study of light interactions with single atomic systems. The two most popular types of ion traps are the Penning trap, which forms a potential via a combination of static electric and magnetic fields, and the Paul trap which forms a potential via a combination of static and oscillating electric fields.^[2]

Penning traps can be used for precise magnetic measurements in spectroscopy. Studies of quantum state manipulation most often use the Paul trap. This may lead to a trapped ion quantum computer^[3] and has already been used to create the world's most accurate atomic clocks.^[4][5] Electron guns (a device emitting high-speed electrons, used in CRTs) can use an ion trap to prevent degradation of the cathode by positive ions.

History

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 television receivers prior to the introduction of aluminized CRT faces around 1958, to protect the phosphor screen from ions.^[7] The ion trap must be delicately adjusted for maximum brightness.^[8][9]

Theory

Any charged particle, such as an ion, feels a force from an electric or magnetic field. Ion traps work by using this force to confine ions in a small, isolated volume of space so that they can be studied or manipulated. Although any static (constant in time) electromagnetic field produces a force on an ion, it is not possible to confine an ion using only a static electric field. This is a consequence of Earnshaw's theorem. However, physicists have various ways of working around this theorem by using combinations of static magnetic and electric fields (as in a Penning trap) or by an oscillating electric field and a static electric field(Paul trap). Ion motion and confinement in the trap is generally divided into axial and radial components, which are typically addressed separately by different fields. In both Paul and Penning traps, axial ion motion is confined by a static electric field. Paul traps use an oscillating electric field to confine the ion radially and Penning traps generate radial confinement with a static magnetic field.

Paul Trap

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 Newton's second law to an ion of charge ${\displaystyle e}$ and mass ${\displaystyle M}$ in this a.c. electric field, we can find the force on the ion using ${\displaystyle \mathbf {F} =e\mathbf {E} }$. We wind up with

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

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 Mathieu equation,

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

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 ${\displaystyle \mathbf {E} =E\mathbf {\hat {e}} _{x}}$Zdroj:https://en.wikipedia.org?pojem=Ion_trap
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