A quadrupole ion trap or paul trap is a type of
ion trap
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 phys ...
that uses dynamic
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
s to trap charged particles. They are also called
radio frequency
Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upp ...
(RF) traps or Paul traps in honor of
Wolfgang Paul
Wolfgang Paul (; 10 August 1913 – 7 December 1993) was a German physicist, who co-developed the non-magnetic quadrupole mass filter which laid the foundation for what is now called an ion trap. He shared one-half of the Nobel Prize in Ph ...
, who invented the device and shared the
Nobel Prize in Physics
)
, image = Nobel Prize.png
, alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in 1989 for this work. It is used as a component of a
mass spectrometer
Mass spectrometry (MS) is an analytical technique that is used to measure the mass-to-charge ratio of ions. The results are presented as a ''mass spectrum'', a plot of intensity as a function of the mass-to-charge ratio. Mass spectrometry is used ...
or a
trapped ion quantum computer
A trapped ion quantum computer is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic state ...
.
Overview
A charged particle, such as an atomic or molecular
ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
, feels a force from an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
. 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
Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upp ...
.
The
quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure refl ...
is the simplest
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
geometry used in such traps, though
more complicated geometries are possible for specialized devices. The electric fields are generated from
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
s on metal electrodes. A pure quadrupole is created from
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
electrodes, though
cylindrical
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infini ...
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. 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
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
metal electrodes with their focuses facing each other and a hyperbolic ring electrode halfway between the other two electrodes. The
ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
s 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
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
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.
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.
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 strong focusing or alternating-gradient focusing is the principle that, using sets of multiple electromagnets, it is possible to make a particle beam simultaneously converge in both directions perpendicular to the direction ...
in
accelerator physics
Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
. 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
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation
:
\frac + (a - 2q\cos(2x))y = 0,
where a and q are parameters. They were first introduced by Émile Léonard Mathieu, ...
.
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
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, 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
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
, the above equation represents the force on the ion. This equation can be exactly solved using the
Floquet theorem
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form
:\dot = A(t) x,
with \displaystyle A(t) a piecewise continuous periodic functio ...
or the standard techniques of
multiple scale analysis In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large value ...
. The particle dynamics and time averaged density of charged particles in a Paul trap can also be obtained by the concept of
ponderomotive force
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 oscilla ...
.
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
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
,
, it can be shown that
:
.
For an ion trap,
and
and for a
quadrupole mass filter
The quadrupole mass analyzer, originally conceived by Nobel Laureate Wolfgang Paul and his student Helmut Steinwedel, also known as quadrupole mass filter, is one type of mass analyzer used in mass spectrometry. As the name implies, it consists ...
,
and
.
Transforming Equation 6 into a
cylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
with
,
, and
and applying the
Pythagorean trigonometric identity
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations b ...
gives
:
The applied electric potential is a combination of RF and DC given by
:
where
and
is the applied frequency in
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
.
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.
Linear ion trap
The
linear ion trap
The linear ion trap (LIT) is a type of ion trap mass spectrometer.
In a LIT, ions are confined radially by a two-dimensional radio frequency (RF) field, and axially by stopping potentials applied to end electrodes.
LITs have high injection effic ...
uses a set of quadrupole rods to confine ions radially and a static electrical potential on-end electrodes to confine the ions axially.
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. 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).
Cylindrical ion trap
Ion traps with a cylindrical rather than a hyperbolic ring electrode
have been developed and microfabricated in arrays to develop
miniature mass spectrometer
A miniature mass spectrometer (MMS) is a type of mass spectrometer (MS) which has small size and weight and can be understood as a portable or handheld device. Current lab-scale mass spectrometers however, usually weigh hundreds of pounds and can ...
s for chemical detection in medical diagnosis and other fields.
Planar ion trap
Quadrupole traps can also be "unfolded" to create the same effect using a set of planar electrodes.
This trap geometry can be made using standard micro-fabrication techniques, including the top metal layer in a standard CMOS microelectronics process,
and is a key technology for scaling trapped ion quantum computers to useful numbers of qubits.
Combined radio frequency trap
A combined radio frequency trap is a combination of a Paul ion trap and a
Penning trap
A Penning trap is a device for the storage of charged particles using a homogeneous axial magnetic field and an inhomogeneous quadrupole electric field. This kind of trap is particularly well suited to precision measurements of properties of io ...
. 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
Antihydrogen () is the antimatter counterpart of hydrogen. Whereas the common hydrogen atom is composed of an electron and proton, the antihydrogen atom is made up of a positron and antiproton. Scientists hope that studying antihydrogen may shed l ...
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.
Digital ion trap
The
digital ion trap
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 anal ...
(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 that are generated by switching rapidly between discrete voltage levels. Major advantages of the DIT are its versatility
and virtually unlimited mass range. The digital ion trap has been developed mainly as a mass analyzer.
See also
*
Quadrupole magnet
Quadrupole magnets, abbreviated as Q-magnets, consist of groups of four magnets laid out so that in the planar multipole expansion of the field, the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. ...
References
Bibliography
* W. Paul ''Electromagnetic Traps for Charged and Neutral Particles'' Taken from Proceedings of the International School of Physics <
> 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.
*
*
*
*
*
*
*
* K. Shah and H. Ramachandran, ''Analytic, nonlinearly exact solutions for an rf confined plasma'', Phys. Plasmas 15, 062303 (2008),
* 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
Patents
*
*
*
External links
Nobel Prize in Physics 1989
{{Mass spectrometry
Mass spectrometry
Measuring instruments
German inventions
Particle traps