Quadrupole ion trap
A quadrupole ion trap exists in both linear and 3D (Paul Trap, QIT) varieties and refers to an ion trap that uses constant DC and radio frequency (RF) oscillating AC electric fields to trap ions. It is a component of a mass spectrometer that would use such a trap to analyze ions. The invention of the 3D quadrupole ion trap itself is attributed to Wolfgang Paul who shared the Nobel Prize in Physics in 1989 for this work.

# 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, non-static) and DC (non-oscillating, 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-1980's 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 configurations: the three dimensional form described above and the linear form made of 4 parallel electrodes. The advantage of this design is in its simplicity, but this leaves a particular constraint on its modeling. To understand how this originates, it is helpful to visualize the linear form. 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 the Mathieu Equations. These equations can only be solved numerically, or equivalently 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 visa 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 experiences 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. When written for ion motion in a trap, the equation is
where u represents the x, y and z coordinated, \xi is a dimensionles 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 substituting the expression for \xi into Equation 1, it can be shown that
Substituting Equation 2 into the Matthieu Equation 1 yields
By Newton's laws of motion, the above equation represents the force on the ion.
The forces in each dimension are not coupled, thus the force acting on an ion in, for example, the x dimension is
Here, \phi\, is the quadrupolar potential, given by
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 Leplace Condition, \nabla^2\phi_0=0\,, it can be shown that
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
The applied electric potential is a combination of RF and DC given by
where \Omega = 2\pi \nu and \nu is the applied frequency in hertz.
Substituting Equation 7 into Equation 5 with \lambda = 1 gives
Substituting Equation 8 into Equation 4 leads to
Comparing terms on the right hand side of Equation 1 and Equation 9 leads to
and
Further q_x = q_y\,,
and
The trapping of ions can be understood in terms of stability regions in q_u and  a_u space.

# Linear ion trap
Linear ion trap use 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).
Thermo Fisher's LTQ is an example of the Linear ion trap.  LTQ stands for Linear Trap Quadrupole, and not Linear Triple Quadruple  as some might think (Ref. call to ThermoFisher's World Headquarters by JP)

# Cylindrical ion trap
Cylindrical ion traps have a cylindrical rather than a hyperbolic ring electrode. This configuration has been used in miniature arrays of traps.