A **quadrupole ion trap** is a type of ion trap that uses dynamic electric fields to trap charged particles. They are also called radio frequency (RF) traps or Paul traps in honor of Wolfgang Paul, who invented the device^{ [1] }^{ [2] } and shared the Nobel Prize in Physics in 1989 for this work.^{ [3] } It is used as a component of a mass spectrometer or a trapped ion quantum computer.

A charged particle, such as an atomic or molecular ion, feels a force from an electric field. It is not possible to create a static configuration of electric fields that traps the charged particle in all three directions (this restriction is known as Earnshaw's theorem). It is possible, however, to create an *average* confining force in all three directions by use of electric fields that change in time. To do so, the confining and anti-confining directions are switched at a rate faster than it takes the particle to escape the trap. The traps are also called "radio frequency" traps because the switching rate is often at a radio frequency.

The quadrupole is the simplest electric field geometry used in such traps, though more complicated geometries are possible for specialized devices. The electric fields are generated from electric potentials on metal electrodes. A pure quadrupole is created from hyperbolic electrodes, though cylindrical 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.^{ [4] } 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.

The 3D trap itself generally consists of two hyperbolic metal electrodes with their focuses facing each other and a hyperbolic ring electrode halfway between the other two electrodes. The ions are trapped in the space between these three electrodes by AC (oscillating) and DC (static) electric fields. The AC radio frequency voltage oscillates between the two hyperbolic metal end cap electrodes if ion excitation is desired; the driving AC voltage is applied to the ring electrode. The ions are first pulled up and down axially while being pushed in radially. The ions are then pulled out radially and pushed in axially (from the top and bottom). In this way the ions move in a complex motion that generally involves the cloud of ions being long and narrow and then short and wide, back and forth, oscillating between the two states. Since the mid-1980s most 3D traps (Paul traps) have used ~1 mTorr of helium. The use of damping gas and the mass-selective instability mode developed by Stafford et al. led to the first commercial 3D ion traps.^{ [5] }

The quadrupole ion trap has two main configurations: the three-dimensional form described above and the linear form made of 4 parallel electrodes. A simplified rectilinear configuration is also used.^{ [6] } The advantage of the linear design is its greater storage capacity (in particular of Doppler-cooled ions) and its simplicity, but this leaves a particular constraint on its modeling. The Paul trap is designed to create a saddle-shaped field to trap a charged ion, but with a quadrupole, this saddle-shaped electric field cannot be rotated about an ion in the centre. It can only 'flap' the field up and down. For this reason, the motions of a single ion in the trap are described by Mathieu equations, which can only be solved numerically by computer simulations.

The intuitive explanation and lowest order approximation is the same as strong focusing in accelerator physics. Since the field affects the acceleration, the position lags behind (to lowest order by 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.

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

where represents the x, y and z coordinates, is a dimensionless variable given by , and and are dimensionless trapping parameters. The parameter is the radial frequency of the potential applied to the ring electrode. By using the chain rule, it can be shown that

Substituting Equation 2 into the Mathieu Equation 1 yields

- .

Multiplying by m and rearranging terms shows us that

- .

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

Here, is the quadrupolar potential, given by

where is the applied electric potential and , , and are weighting factors, and is a size parameter constant. In order to satisfy Laplace's equation, , it can be shown that

- .

For an ion trap, and and for a quadrupole mass filter, and .

Transforming Equation 6 into a cylindrical coordinate system with , , and and applying the Pythagorean trigonometric identity gives

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

where and is the applied frequency in hertz.

Substituting Equation 8 into Equation 6 with gives

Substituting Equation 9 into Equation 5 leads to

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

and

Further ,

and

The trapping of ions can be understood in terms of stability regions in and 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] }

The linear ion trap uses a set of quadrupole rods to confine ions radially and a static electrical potential on-end electrodes to confine the ions axially.^{ [10] } The linear form of the trap can be used as a selective mass filter, or as an actual trap by creating a potential well for the ions along the axis of the electrodes.^{ [11] } Advantages of the linear trap design are increased ion storage capacity, faster scan times, and simplicity of construction (although quadrupole rod alignment is critical, adding a quality control constraint to their production. This constraint is additionally present in the machining requirements of the 3D trap).^{ [12] }

Ion traps with a cylindrical rather than a hyperbolic ring electrode^{ [13] }^{ [14] }^{ [15] }^{ [16] }^{ [17] } have been developed and microfabricated in arrays to develop miniature mass spectrometers for chemical detection in medical diagnosis and other fields.

Quadrupole traps can also be "unfolded" to create the same effect using a set of planar electrodes.^{ [18] } This trap geometry can be made using standard micro-fabrication techniques, including the top metal layer in a standard CMOS microelectronics process,^{ [19] } and is a key technology for scaling trapped ion quantum computers to useful numbers of qubits.

A combined radio frequency trap is a combination of a Paul ion trap and a Penning trap.^{ [20] } One of the main bottlenecks of a quadrupole ion trap is that it can confine only single-charged species or multiple species with similar masses. But in certain applications like antihydrogen production it is important to confine two species of charged particles of widely varying masses. To achieve this objective, a uniform magnetic field is added in the axial direction of the quadrupole ion trap.

The digital 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^{ [21] }^{ [22] } that are generated by switching rapidly between discrete voltage levels. Major advantages of the DIT are its versatility^{ [23] } and virtually unlimited mass range. The digital ion trap has been developed mainly as a mass analyzer.

In physics, the **cross section** is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted *σ* (sigma) and is expressed in units of transverse area. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

In classical mechanics, a **harmonic oscillator** is a system that, when displaced from its equilibrium position, experiences a restoring force *F* proportional to the displacement *x*:

In mechanics and physics, **simple harmonic motion** is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.

The **quantum harmonic oscillator** is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, the emission is called cyclotron emission. If the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum, which is also called continuum radiation.

In 1851, George Gabriel Stokes derived an expression, now known as **Stokes law**, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

In physics, the **Hamilton–Jacobi equation**, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

In theoretical physics, the (one-dimensional) **nonlinear Schrödinger equation** (**NLSE**) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

The **Kerr–Newman metric** is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions.

In physics, a **ponderomotive force** is a nonlinear force that a charged particle experiences in an inhomogeneous oscillating electromagnetic field. It causes the particle to move towards the area of the weaker field strength, rather than oscillating around an initial point as happens in a homogeneous field. This occurs because the particle sees a greater magnitude of force during the half of the oscillation period while it is in the area with the stronger field. The net force during its period in the weaker area in the second half of the oscillation does not offset the net force of the first half, and so over a complete cycle this makes the particle move towards the area of lesser force.

**Fourier-transform ion cyclotron resonance mass spectrometry** is a type of mass analyzer for determining the mass-to-charge ratio (*m*/*z*) of ions based on the cyclotron frequency of the ions in a fixed magnetic field. The ions are trapped in a Penning trap, where they are excited to a larger cyclotron radius by an oscillating electric field orthogonal to the magnetic field. After the excitation field is removed, the ions are rotating at their cyclotron frequency in phase. These ions induce a charge on a pair of electrodes as the packets of ions pass close to them. The resulting signal is called a free induction decay (FID), transient or interferogram that consists of a superposition of sine waves. The useful signal is extracted from this data by performing a Fourier transform to give a mass spectrum.

**Elastic Recoil Detection Analysis** (**ERDA**), also referred to as forward recoil scattering, is an Ion Beam Analysis technique in materials science to obtain elemental concentration depth profiles in thin films. This technique is known by several different names. These names are listed below. In the technique of ERDA, an energetic ion beam is directed at a sample to be characterized and there is an elastic nuclear interaction between the ions of beam and the atoms of the target sample. Such interactions are commonly of Coulomb nature. Depending on the kinetics of the ions, cross section area, and the loss of energy of the ions in the matter, Elastic Recoil Detection Analysis helps determine the quantification of the elemental analysis. It also provides information about the depth profile of the sample.

In the mathematical description of general relativity, the **Boyer–Lindquist coordinates** are a generalization of the coordinates used for the metric of a Schwarzschild black hole that can be used to express the metric of a Kerr black hole.

The **Voigt effect** is a magneto-optical phenomenon which rotates and elliptizes linearly polarised light sent into an optically active medium. Unlike many other magneto-optical effects such as the Kerr or Faraday effect which are linearly proportional to the magnetization, the Voigt effect is proportional to the square of the magnetization and can be seen experimentally at normal incidence. There are several denominations for this effect in the literature: the *Cotton–Mouton effect*, the *Voigt effect*, and *magnetic-linear birefringence*. This last denomination is closer in the physical sense, where the Voigt effect is a magnetic birefringence of the material with an index of refraction parallel and perpendicular ) to the magnetization vector or to the applied magnetic field.

In classical mechanics, a **Liouville dynamical system** is an exactly soluble dynamical system in which the kinetic energy *T* and potential energy *V* can be expressed in terms of the *s* generalized coordinates *q* as follows:

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form is said to be **orbitally stable** if any solution with the initial data sufficiently close to forever remains in a given small neighborhood of the trajectory of .

This article is about the **Bowring series of the transverse mercator.** In 1989 Bernard Russel Bowring gave formulas for the Transverse Mercator that are simpler to program but retain millimeter accuracy.

In physics, and especially scattering theory, the **momentum-transfer cross section** is an effective scattering cross section useful for describing the average momentum transferred from a particle when it collides with a target. Essentially, it contains all the information about a scattering process necessary for calculating average momentum transfers but ignores other details about the scattering angle.

In mathematics, the **exponential response formula** (ERF), also known as **exponential response and complex replacement**, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the function is polynomial, sinusoidal, exponential or the combination of the three. The general solution of a non-homogeneous linear ordinary differential equation is a superposition of the general solution of the associated homogeneous ODE and a particular solution to the non-homogeneous ODE. Alternative methods for solving ordinary differential equations of higher order are method of undetermined coefficients and method of variation of parameters.

The **digital ion trap** (DIT) is an quadrupole ion trap driven by digital signals, typically in a rectangular waveform, generated by switching rapidly between discrete DC voltage levels. The digital ion trap has been mainly developed as a mass analyzer.

- ↑ Paul W., Steinwedel H. (1953). "Ein neues Massenspektrometer ohne Magnetfeld". RZeitschrift für Naturforschung A
**8**(7): 448-450 - ↑ DE 944900 "Verfahren zur Trennung bzw. zum getrennten Nachweis von Ionen verschiedener spezifischer Ladung", W. Paul and H. Steinwedel, filed on December 24, 1953, priority December 23, 1953
- ↑ Wolfgang Paul (1990). "Electromagnetic traps for charged and neutral particles".
*Reviews of Modern Physics*.**62**(3): 531–540. Bibcode:1990RvMP...62..531P. doi:10.1103/RevModPhys.62.531. - ↑ S. Seidelin; et al. (2006). "Microfabricated Surface-Electrode Ion Trap for Scalable Quantum Information Processing".
*Physical Review Letters*.**96**(25): 253003. arXiv: quant-ph/0601173 . Bibcode:2006PhRvL..96y3003S. doi:10.1103/PhysRevLett.96.253003. PMID 16907302. S2CID 9424735. - ↑ Stafford, G. C.; P. E. Kelley; J. E. P. Syka; W. E. Reynolds; J. F. J. Todd (7 September 1984). "Recent improvements in and analytical applications of advanced ion trap technology".
*International Journal of Mass Spectrometry and Ion Processes*. Elsevier Science B.V.**60**(1): 85–98. Bibcode:1984IJMSI..60...85S. doi:10.1016/0168-1176(84)80077-4. - ↑ Ouyang Z, Wu G, Song Y, Li H, Plass WR, Cooks RG (August 2004). "Rectilinear ion trap: concepts, calculations, and analytical performance of a new mass analyzer".
*Anal. Chem.***76**(16): 4595–605. doi:10.1021/ac049420n. PMID 15307768. - ↑ March, Raymond E. (1997). "An Introduction to Quadrupole Ion Trap Mass Spectrometry".
*Journal of Mass Spectrometry*.**32**(4): 351–369. Bibcode:1997JMSp...32..351M. doi:10.1002/(SICI)1096-9888(199704)32:4<351::AID-JMS512>3.0.CO;2-Y. ISSN 1076-5174. - ↑ N. W. McLachlan,
**Theory and Applications of Mathieu Functions**(Oxford University Press, Oxford, 1947), p. 20 - ↑ H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (2012), Chapter 17 on Periodic Potentials, ISBN 978-981-4397-73-5.
- ↑ Douglas DJ, Frank AJ, Mao D (2005). "Linear ion traps in mass spectrometry".
*Mass Spectrometry Reviews*.**24**(1): 1–29. Bibcode:2005MSRv...24....1D. doi:10.1002/mas.20004. PMID 15389865. - ↑ March, Raymond E (2000). "Quadrupole ion trap mass spectrometry: a view at the turn of the century".
*International Journal of Mass Spectrometry*.**200**(1–3): 285–312. Bibcode:2000IJMSp.200..285M. doi:10.1016/S1387-3806(00)00345-6. - ↑ Schwartz, Jae C.; Michael W. Senko; John E. P. Syka (June 2002). "A two-dimensional quadrupole ion trap mass spectrometer".
*Journal of the American Society for Mass Spectrometry*. Elsevier Science B.V.**13**(6): 659–669. doi: 10.1016/S1044-0305(02)00384-7 . PMID 12056566. - ↑ Kornienko, O.; Reilly, P.T.A.; Whitten, W.B.; Ramsey, J.M. (1999). "Micro ion trap mass spectrometry".
*Rapid Communications in Mass Spectrometry*.**13**(1): 50–53. Bibcode:1999RCMS...13...50K. doi:10.1002/(sici)1097-0231(19990115)13:1<50::aid-rcm449>3.3.co;2-b. - ↑ Ouyang Z, Badman ER, Cooks RG (1999). "Characterization of a serial array of miniature cylindrical ion trap mass analyzers".
*Rapid Communications in Mass Spectrometry*.**13**(24): 2444–9. doi:10.1002/(SICI)1097-0231(19991230)13:24<2444::AID-RCM810>3.0.CO;2-F. PMID 10589092. - ↑ Patterson GE, Guymon AJ, Riter LS, Everly M, Griep-Raming J, Laughlin BC, Ouyang Z, Cooks RG (2002). "Miniature cylindrical ion trap mass spectrometer".
*Anal. Chem*.**74**(24): 6145–53. doi:10.1021/ac020494d. PMID 12510732. - ↑ Kanawati B, Wanczek KP (2007). "Characterization of a new open cylindrical ion cyclotron resonance cell with unusual geometry".
*Review of Scientific Instruments*.**78**(7): 074102–074102–8. Bibcode:2007RScI...78g4102K. doi:10.1063/1.2751100. PMID 17672776. - ↑ Cruz D, Chang JP, Fico M, Guymon AJ, Austin DE, Blain MG (2007). "Design, microfabrication, and analysis of micrometer-sized cylindrical ion trap arrays".
*Review of Scientific Instruments*.**78**(1): 015107–015107–9. Bibcode:2007RScI...78a5107C. doi:10.1063/1.2403840. PMID 17503946. - ↑ Seidelin, S.; Chiaverini, J.; Reichle, R.; Bollinger, J.; Leibfried, D.; Britton, J.; Wesenberg, J.; Blakestad, R.; Epstein, R.; Hume, D.; Itano, W.; Jost, J.; Langer, C.; Ozeri, R.; Shiga, N.; Wineland, D. (2006-06-30). "Microfabricated Surface-Electrode Ion Trap for Scalable Quantum Information Processing".
*Physical Review Letters*.**96**(25): 253003. arXiv: quant-ph/0601173 . Bibcode:2006PhRvL..96y3003S. doi:10.1103/physrevlett.96.253003. ISSN 0031-9007. PMID 16907302. S2CID 9424735. - ↑ Mehta, K. K.; Eltony, A. M.; Bruzewicz, C. D.; Chuang, I. L.; Ram, R. J.; Sage, J. M.; Chiaverini, J. (2014-07-28). "Ion traps fabricated in a CMOS foundry".
*Applied Physics Letters*.**105**(4): 044103. arXiv: 1406.3643 . Bibcode:2014ApPhL.105d4103M. doi:10.1063/1.4892061. ISSN 0003-6951. S2CID 119209584. - ↑ J. Walz; S. B. Ross; C. Zimmermann; L. Ricci; M. Prevedelli; T. W. Hansch (1996). "Confinement of electrons and ions in a combined trap with the potential for antihydrogen production".
*Hyperfine Interactions*.**100**(1): 133. Bibcode:1996HyInt.100..133W. doi:10.1007/BF02059938. S2CID 7625254. - ↑ Brancia, Francesco L.; McCullough, Bryan; Entwistle, Andrew; Grossmann, J. Günter; Ding, Li (2010). "Digital asymmetric waveform isolation (DAWI) in a digital linear ion trap".
*Journal of the American Society for Mass Spectrometry*.**21**(9): 1530–1533. doi: 10.1016/j.jasms.2010.05.003 . ISSN 1044-0305. PMID 20541437. - ↑ Hoffman, Nathan M.; Gotlib, Zachary P.; Opačić, Bojana; Huntley, Adam P.; Moon, Ashley M.; Donahoe, Katherine E. G.; Brabeck, Gregory F.; Reilly, Peter T. A. (2018). "Digital Waveform Technology and the Next Generation of Mass Spectrometers".
*Journal of the American Society for Mass Spectrometry*.**29**(2): 331–341. Bibcode:2018JASMS..29..331H. doi:10.1007/s13361-017-1807-8. ISSN 1044-0305. PMID 28971335. S2CID 207471456. - ↑ Singh, Rachit; Jayaram, Vivek; Reilly, Peter T.A. (2013). "Duty cycle-based isolation in linear quadrupole ion traps".
*International Journal of Mass Spectrometry*. 343–344: 45–49. Bibcode:2013IJMSp.343...45S. doi:10.1016/j.ijms.2013.02.012.

- W. Paul
*Electromagnetic Traps for Charged and Neutral Particles*Taken from Proceedings of the International School of Physics <<Enrico Fermi>> Course CXVIII “Laser Manipulation of Atoms and Ions”, (North Holland, New York, 1992) p. 497-517 - R.I. Thompson, T.J. Harmon, and M.G. Ball,
*The rotating-saddle trap: a mechanical analogy to RF-electric-quadrupole ion trapping?*(Canadian Journal of Physics, 2002:**80**12) p. 1433–1448 - M. Welling, H.A. Schuessler, R.I. Thompson, H. Walther
*Ion/Molecule Reactions, Mass Spectrometry and Optical Spectroscopy in a Linear Ion Trap*(International Journal of Mass Spectrometry and Ion Processes, 1998: 172) p. 95-114. - G. Werth (2005).
*Charged Particle Traps: Physics and Techniques of Charged Particle Field Confinement (Springer Series on Atomic, Optical, and Plasma Physics)*. Berlin: Springer. ISBN 3-540-22043-7. OCLC 231588573. - John Gillaspy (2001).
*Trapping Highly Charged Ions: Fundamentals and Applications*. Commack, N.Y: Nova Science Publishers. ISBN 1-56072-725-X. OCLC 42009394. - Todd, John F. J.; March, Raymond E. (2005).
*Quadrupole Ion Trap Mass Spectrometry , 2nd Edition*. New York: Wiley-Interscience. ISBN 0-471-48888-7. OCLC 56413336. - Todd, John F. J.; March, Raymond E. (1995).
*Practical aspects of ion trap mass spectrometry - Volume I: Fundamentals of Ion Trap Mass Spectrometry*. Boca Raton: CRC Press. ISBN 0-8493-4452-2. OCLC 32346425. - Todd, John F. J.; March, Raymond E. (1995).
*Practical aspects of ion trap mass spectrometry: Ion Trap Instrumentation, Vol. 2*. Boca Raton: CRC Press. ISBN 0-8493-8253-X. OCLC 32346425. - Todd, John F. J.; March, Raymond E. (1995).
*Practical aspects of ion trap mass spectrometry, Vol. 3*. Boca Raton: CRC Press. ISBN 0-8493-8251-3. OCLC 32346425. - Hughes, Richard M.; March, Raymond E.; Todd, John F. J. (1989).
*Quadrupole storage mass spectrometry*. New York: Wiley. ISBN 0-471-85794-7. OCLC 18290778. - K. Shah and H. Ramachandran,
*Analytic, nonlinearly exact solutions for an rf confined plasma*, Phys. Plasmas 15, 062303 (2008),*http://link.aip.org/link/?PHPAEN/15/062303/1*^{[ permanent dead link ]} - Pradip K. Ghosh,
*Ion Traps*, International Series of Monographs in Physics, Oxford University Press (1995),*https://web.archive.org/web/20111102190045/http://www.oup.com/us/catalog/general/subject/Physics/AtomicMolecularOpticalphysics/?view=usa*

- 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

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.