Quadrupole ion trap

Not to be confused with Quadrupole mass analyzer.

Scheme of a Quadrupole ion trap of classical setup with a particle of positive charge (dark red), surrounded by a cloud of similarly charged particles (light red). The electric field E (blue) is generated by a quadrupole of endcaps (a, positive) and a ring electrode (b). Picture 1 and 2 show two states during an AC cycle.

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.

Overview

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.

Theory

The 3D trap itself generally consists of two hyperbolic metal electrodes with their foci 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]

Linear Ion Trap at the University of Calgary

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.

Equations of motion

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

{\frac {d^{2}u}{d\xi ^{2}}}+[a_{u}-2q_{u}\cos(2\xi )]u=0\qquad \qquad (1)\!

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

{\frac {d^{2}u}{dt^{2}}}={\frac {\Omega ^{2}}{4}}{\frac {d^{2}u}{d\xi ^{2}}}\qquad \qquad (2)\!

Substituting Equation 2 into the Mathieu Equation 1 yields

\frac{4}{\Omega^2}\frac{d^2u}{dt^2} + \left[a_u - 2q_u\cos (\Omega t) \right]u = 0 \qquad\qquad (3) \!

Reorganizing terms shows us that

m \frac{d^2u}{dt^2} + m \frac{\Omega^2}{4}\left[a_u - 2q_u\cos (\Omega t) \right]u = 0 \qquad\qquad (3) \!

By Newton's laws of motion, the above equation represents the force on the ion. This equation can be exactly solved using 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

F_{x}=ma=m{\frac {d^{2}x}{dt^{2}}}=-e{\frac {\partial \phi }{\partial x}}\qquad \qquad (4)\!

Here, $\phi \,$ is the quadrupolar potential, given by

\phi ={\frac {\phi _{0}}{r_{0}^{2}}}{\big (}\lambda x^{2}+\sigma y^{2}+\gamma z^{2}{\big )}\qquad \qquad (5)\!

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

\lambda +\sigma +\gamma =0\,

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

Transforming Equation 5 into a cylindrical coordinate system with ${x}={r} \,\cos\theta$ , ${y}={r} \, \sin\theta$ , and ${z}={z}\,$ and applying the pythagorean trigonometric identity $\sin ^{2}\theta +\cos ^{2}\theta =1\,$ gives

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

\phi _{{r,z}}={\frac {\phi _{0}}{r_{0}^{2}}}{\big (}r^{2}-2z^{2}{\big )}.\qquad \qquad (6)\!

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

\phi _{0}=U+V\cos \Omega t.\qquad \qquad (7)\!

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

Substituting Equation 7 into Equation 5 with $\lambda =1$ gives

{\frac {\partial \phi }{\partial x}}={\frac {2x}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}.\qquad \qquad (8)\!

Substituting Equation 8 into Equation 4 leads to

m{\frac {d^{2}x}{dt^{2}}}=-{\frac {2e}{r_{0}^{2}}}{\big (}U+V\cos \Omega t{\big )}x.\qquad \qquad (9)\!

Comparing terms on the right hand side of Equation 1 and Equation 9 leads to

a_{x}={\frac {8eU}{mr_{0}^{2}\Omega ^{2}}}\qquad \qquad (10)\!

and

q_{x}=-{\frac {4eV}{mr_{0}^{2}\Omega ^{2}}}.\qquad \qquad (11)\!

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

a_{z}=-{\frac {8eU}{mr_{0}^{2}\Omega ^{2}}}\qquad \qquad (12)\!

and

q_{z}={\frac {4eV}{mr_{0}^{2}\Omega ^{2}}}.\qquad \qquad (13)\!

The trapping of ions can be understood in terms of stability regions in $q_{u}$ and $a_{u}$ 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]

Linear ion trap

LTQ (Linear trap quadrupole)

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]

Cylindrical ion trap

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.

Combined radio frequency trap

A combined radio frequency trap is a combination of a Paul ion trap and a Penning trap.^[18] One of the main bottlenecks of a QIT 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 QIT.

References

↑ 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. "Electromagnetic traps for charged and neutral particles". Rev. Mod. Phys. 62: 531–540. Bibcode:1990RvMP...62..531P. doi:10.1103/RevModPhys.62.531.
↑ S. Seidelin; et al. "Microfabricated Surface-Electrode Ion Trap for Scalable Quantum Information Processing". Phys. Rev. Lett. 96: 253003. arXiv:quant-ph/0601173. Bibcode:2006PhRvL..96y3003S. doi:10.1103/PhysRevLett.96.253003.
↑ 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. 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. doi:10.1002/mas.20004. PMID 15389865.
↑ Quadrupole ion trap mass spectrometry: a view at the turn of the century, Raymond E. March, International Journal of Mass Spectrometry 200, 2000, pp.285-312
↑ 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
↑ 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". The Review of scientific instruments. 78 (7): 074102. 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". The Review of scientific instruments. 78 (1): 015107. Bibcode:2007RScI...78a5107C. doi:10.1063/1.2403840. PMID 17503946.
↑ J. Walz, S. B. Ross, C. Zimmermann, L. Ricci, M. Prevedelli and T. W. Hansch (1996). "Confinement of electrons and ions in a combined trap with the potential for antihydrogen production". Hyperfine Interactions. 100: 133. Bibcode:1996HyInt.100..133W. doi:10.1007/BF02059938.

Bibliography

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
Pradip K. Ghosh, Ion Traps, International Series of Monographs in Physics, Oxford University Press (1995), http://web.archive.org/web/20111102190045/http://www.oup.com/us/catalog/general/subject/Physics/AtomicMolecularOpticalphysics/?view=usa

Patents

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
GB 773689 "Improved arrangements for separating or separately detecting charged particles of different specific charges", W. Paul, claims priority of a German application filed on December 23, 1953
US 2939952 "Apparatus for separating charged particles of different specific charges", W. Paul and H. Steinwedel, claims priority of a German application filed on December 23, 1953

External links

Nobel Prize in Physics 1989

Mass spectrometry

Mass m/z Mass spectrum MS software Acronyms

Ion source	APCI APLI CI DAPPI DART DESI DIOS EESI EI ESI FAB FD GD IA ICP LAESI MALDI MALDESI MIP PTR SS SSI SELDI TI TS

Mass analyzer	Sector Wien filter Time-of-flight Quadrupole mass filter Quadrupole ion trap Penning trap FT-ICR Orbitrap NEMS

Detector	Electron multiplier Microchannel plate detector Daly detector Faraday cup Langmuir–Taylor detector

MS combination	MS/MS QqQ Hybrid MS GC/MS LC/MS IMS/MS CE-MS

Fragmentation	BIRD CID ECD EDD ETD HCD IRMPD NETD SID

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