with the three axes of motion depicted by arrows. Endcap electrodes (not shown) generate a trapping potential along the axial direction (red arrow). A radio frequency electric field is applied to the four rods which confines the ion in the two radial directions (green arrows). A trap requires confining forces in all three spatial directions. Electric and magnetic fields exert forces on
ions, called the
Lorentz force. Due to
Earnshaw's theorem it is not possible to confine an ion using only static electric fields. However, a static magnetic and electric field (a
Penning trap), or the combination of an oscillating electric field with a static electric field (a
Paul trap), can trap ions. The confining fields and the resulting motion of ions in a trap are generally decomposed into one axial and two radial components with respect to the trap geometry. In both Paul and Penning traps, a static electric field provides the axial confinement. Paul traps confine the ion radially with an oscillating electric field whereas Penning traps use a static magnetic field.
Paul trap A Paul trap (also known as a quadrupole ion trap) uses static
direct current (DC) and
radio frequency (RF) oscillating
electric fields to trap ions. Paul traps are commonly used as components of
mass spectrometers.
Wolfgang Paul invented the Paul trap, hence its name. For this work he shared the 1989
Nobel Prize in Physics. The RF field generates an
average radial confining force with an oscillating quadrupole potential. The confining and anti-confining directions of the potential are switched faster than the particle's escape time. Since the field affects the acceleration, the position lags behind (by approximately 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. The
quadrupole is the simplest
electric field geometry used in such traps, though
more complicated geometries are possible and used in 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 chip 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. A typical trap configuration has four parallel electrodes along the z-axis that are positioned at the corners of a square in the xy-plane. Diagonally opposite electrodes are connected and a voltage V = V_0\cos(\Omega t) is applied. The electric field produced by this potential is \mathbf{E} = \mathbf{E}_0\sin(\Omega t). The force on an ion of charge e is \mathbf{F} = e\mathbf{E} which with ion mass M leads to the radial equation of motion : M\mathbf{\ddot{r}} = e\mathbf{E}_0\sin(\Omega t) \! . If the ion is initially at rest, two successive integrations give the velocity and displacement as : \mathbf{\dot{r}} = \frac{e\mathbf{E}_0}{M\Omega}\cos(\Omega t) \! , : \mathbf{r} = \mathbf{r}_0 - \frac{e\mathbf{E}_0}{M\Omega^2}\sin(\Omega t) \! , where \mathbf{r}_0 is a constant of integration and corresponding to an arbitrary starting position. Thus, the ion oscillates with angular frequency \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 z-axis, an analysis of the radial symmetry yields a potential : \phi = \alpha + \beta(x^2 - y^2) \! . The constants \alpha and \beta are determined by boundary conditions on the electrodes and \phi satisfies
Laplace's equation \nabla^2\phi = 0. Assuming the length of the electrodes r is much greater than their separation r_0, it can be shown that : \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 : \mathbf{E} = -\frac{V_0}{r_0^2}\cos(\Omega t)(x\mathbf{\hat{e}}_x - y\mathbf{\hat{e}}_y) \! . Defining \tau = \Omega t/2, the equations of motion in the xy-plane are a simplified form of the
Mathieu equation, : \frac{d^2x_i}{d\tau^2} = -\frac{4eV_0}{Mr_0^2\Omega^2}\cos(2\tau)x_i \! .
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. When written for ion motion in a trap, the equation is {{NumBlk|:| \frac{d^2u}{d\xi^2}+[a_u-2q_u\cos (2\xi) ]u = 0 |}} where u represents the x, y and z coordinates, \xi is a dimensionless variable 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 {{NumBlk|:| \frac{d^2u}{dt^2} = \frac{\Omega^2}{4} \frac{d^2u}{d\xi^2} |}} Substituting into the Mathieu yields {{NumBlk|:| \frac{4}{\Omega^2}\frac{d^2u}{dt^2} + \left[a_u - 2q_u\cos (\Omega t) \right]u = 0 .|}} Multiplying by m and rearranging terms shows us that {{NumBlk|:| m \frac{d^2u}{dt^2} + m \frac{\Omega^2}{4}\left[a_u - 2q_u\cos (\Omega t) \right]u = 0 . |}} 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. 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 this case, the average trajectory is called "secular motion", and is superimposed by an oscillation at frequency \Omega but with small amplitude called "micromotion". The forces in each dimension are not coupled, thus the force acting on an ion in, for example, the x dimension is {{NumBlk|:|F_x = ma = m\frac{d^2x}{dt^2} = -e \frac{\partial \phi}{\partial x} |}} Here, \phi is the quadrupolar potential, given by {{NumBlk|:|\phi = \frac{\phi_0}{r_0^2} \bigl( \lambda x^2 + \sigma y^2 + \gamma z^2 \bigr) |}} 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
Laplace's equation, \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 6 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 {{NumBlk|:|\phi_{r,z} = \frac{\phi_0}{r_0^2} \big( r^2 - 2z^2 \big) . |}} 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 into with \lambda = 1 gives {{NumBlk|:| \frac{\partial \phi}{\partial x} = \frac {2x}{r_0^2} \big( U + V\cos \Omega t \big) . |}} Substituting equation 9 into equation 5 leads to {{NumBlk|:| m\frac {d^2x}{dt^2} = - \frac {2e}{r_0^2} \big( U + V\cos \Omega t \big) x . |}} Comparing terms on the right hand side of equation 1 and equation 10 leads to {{NumBlk|:| a_x = \frac {8eU} {m r_0^2 \Omega^2} |}} and {{NumBlk|:| q_x = - \frac {4eV} {m r_0^2 \Omega^2} . |}} Further q_x = q_y\,, {{NumBlk|:| a_z = -\frac {16eU} {m r_0^2 \Omega^2} |}} and {{NumBlk|:| q_z = \frac {8eV} {m r_0^2 \Omega^2} . |}} 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.
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 \mathbf{E} = E\mathbf{\hat{e}}_x, the force \mathbf{F} = e\mathbf{E} accelerates a positively charged ion along the x-axis. For a uniform magnetic field \mathbf{B}= B\mathbf{\hat{e}}_z, the
Lorentz force causes the ion to move in circular motion with
cyclotron frequency : \omega_c = \frac{eB}{M} \! . Assuming an ion with zero initial velocity placed in a region with \mathbf{E} = E\mathbf{\hat{e}}_x and \mathbf{B}= B\mathbf{\hat{e}}_z, the equations of motion are : x = \frac{E}{\omega_c B}(1-\cos(\omega_c t)) \! , : y = -\frac{E}{\omega_c B}(\omega_c t-\sin(\omega_c t)) \! , : z = 0 \! . The resulting motion is a combination of oscillatory motion around the z-axis with frequency \omega_c and a drift velocity in the y-direction. The drift velocity is perpendicular to the direction of the electric field. For the radial electric field produced by the electrodes in a Penning trap, the drift velocity will precess around the axial direction with some frequency \omega_m, called the magnetron frequency. An ion will also have a third characteristic frequency \omega_z between the two end cap electrodes. The frequencies usually have widely different values with \omega_z \ll \omega_m. ==Ion trap mass spectrometers==