Moment theories of ion motion and reaction in ideal and stretched quadrupole ion
traps are extended to the case of linear ion traps. Fortran and
Mathematica computer programs based on these theories are
developed. They are applied to the case of O^{+} ions moving through an Ar buffer gas containing a small
amount of a reactive neutral, N_{2}. The rate
coefficient predicted ab initio for a common set of trap
parameters is 6.4±0.9×10^{−13} cm^{3}/s, which is large enough that it
should be measureable.
ion traps, linear trap, quadrupole trap, gaseous ion transport, numerical simulation1. Introduction
Starting from the Boltzmann equation, two-temperature (2T) and
multitemperature (MT) moment theories for the motion of trace
amounts of ions in devices where there are external fields that
vary with position and time were presented in the first [1]
of a series of papers. The accuracy of the 2T theory was
demonstrated in the second paper [2], which considered
field-asymmetric ion mobility spectrometry (also called
differential mobility spectrometry). The 2T theory was applied in
the third paper [3] to ideal quadrupole ion traps (sometimes
called 3D quadrupole ion traps), where it was shown to be more
directly connected to other theories [4] than is the MT
theory, even though the MT theory is expected to be more accurate.
The final papers of the series [5,6] extended the theories
of the first paper to encompass stretched quadrupole traps
containing molecular ions and neutrals, with the use of
spherical-polar basis functions (the SB theory) corresponding to
the 2T theory for atomic systems and the use of Cartesian basis
functions (the CB theory) corresponding to the MT theory.
In subsequent papers [7,8], we extended these theories to
situations in which a small fraction of the neutral gas is a
species that can undergo infrequent reaction with the ion of
interest. We found that, in the first of a series of successive
approximations, the measured ion-neutral reaction rate
coefficients employing time- and space-dependent electric fields
can be equated to thermal rate coefficients appropriate to high
temperatures in the absence of an electric field. This is the case
even though the velocity distribution function of the ions may
differ substantially from an equilibrium distribution. Further, we
provided differential equations for the position- and
time-dependent moments of ion velocity and energy that are
necessary to convert the actual experimental parameters into the
corresponding elevated temperature.
The papers just summarized were generalized beyond ion traps into
uniform moment theories for charged particle motion in gases
[9]. The first approximation of the uniform theory when
applied to ion traps reproduces the first approximation results of
the other papers. Moreover, consideration of the second
approximation of the uniform 2T theory for drift-tube mass
spectrometers shows (in work to be published) that the successive
approximations are converging. The purpose of the present paper is
to apply the uniform moment theories to linear ion traps
(sometimes called 2D ion traps), which are now more commonly used
than ideal or stretched quadrupole ion traps. Because we are
assuming that only trace amounts of ions are present, space-charge
effects are not considered in the previous papers or in this one.
We begin by presenting the general differential equations
governing the average ion velocity and energy that arise in the
first approximation of the various moment theories. The electric
fields are given in the next section for quadrupole and linear ion
traps. The specific first-approximation moment equations in both
types of ion trap are then given. Tests of two computer programs
based on the equations in the previous section are given for the
special cases where the ion-neutral collisions follow the Maxwell
model of constant collision frequency and the rigid-sphere model
of constant cross section. Also considered is the reaction rate
coefficient for ^{107}Ag^{+}
ions reacting with D_{2} that is immersed in a
much larger amount of He in a quadrupole ion trap; it is based on
an ab initio interaction potential for Ag^{+} with He.
We then consider the ab initio rate coefficient for the reaction
of ^{16}O^{+} ions with N_{2} that could
be measured in a linear trap containing argon buffer gas at 300 K.
Finally, the results are summarized and discussed.
2. First-Approximation Moment Equations
Since there are no magnetic fields in radiofrequency ion traps, it
can be shown that each of the four moment theories leads in first
approximation to an ordinary differential equation of the same
form [6,9]:
ddt𝐯¯u−qm𝐄u+ξu𝐯¯u=0.
This equation governs the time, t, evolution of the component of the average ion
velocity, 𝐯¯u, along the Cartesian axis
u, with u = x, y, z. Here 𝐄_{
u
} is the
component along u of the external electric field, q and m
are the ion charge and mass, and ξ_{
u
} is the collision
frequency for momentum transfer along direction u. As discussed
in Appendix A, the microscopic definition of ξ_{
u
} depends
upon which theory is used. Finally, each of the subscripted
quantities in equation (1) can depend upon position,
𝐫, and time, t, but for simplicity this dependence is not
explicitly indicated.
As pointed out previously [1], the moment equation (1) is
equivalent to the damped Mathieu equation so often used to study
ion motion in traps. The Mathieu equation is obtained from
Newton's equation of motion for a single ion by assuming that the
forces involved in that equation are external electrical forces
and by adding in a damping term (with a constant collision
frequency, ξ_{
u
}) to mimic the effect of the forces
that arise due to collisions. Equation (1), however, is obtained
from the Boltzmann equation (in the first of a series of
successive approximations) in which both external fields and
ion-neutral interactions are treated correctly. It governs the
average velocity of a group of ions in a trap at the same time,
rather than the motion of a single ion.
The various expressions for ξ_{
u
} depend upon the gas
temperature, T, and one or more effective ion temperatures that
can be considerably larger than T because the ions are present
only in trace amounts in the apparatus. Therefore, equation
(1) must be coupled to differential equations governing
these ion temperatures. In the first approximation of the 2T and
SB theories, ξ_{
u
} is independent of the direction, so the
subscript can be dropped. More importantly, it is a function only
of the effective temperature, T_{eff}, that is governed
by the equation [6,9]:
ddt32kBTeff−qμm𝐄·𝐯¯+2μξM32kBTeff−T+MΦm32kBTeff=0.
Here μ is the ion-neutral reduced mass, k_{
B
} is Boltzmann's
constant, M is the neutral mass, and Φ is the dimensionless
ratio of the collision integral for inelastic energy loss to that
for momentum transfer that is discussed in Appendix B. It should
be noted that the collision frequency couples each
𝐯¯u to T_{eff} and hence that
all of the 𝐯¯u are coupled. Finally,
Φ = 0 for atomic ion-neutral systems, where the SB theory
reduces to the 2T theory.
In first approximation of the MT and CB theories, the collision
frequencies depend on ion temperatures, T_{
u,eff}, that
may differ along the various Cartesian axes. These effective
temperatures are governed by the following equation [6,9]:
There are three differential equations represented by equation (3), as opposed to
one represented by equation (2). In addition, the dimensionless
ratios Φ_{
u
} in equation (3) provide for energy partitioning
as well as inelastic energy transfer, as discussed in Appendix B.
The Φ_{
u
} are all zero for atomic ion-neutral systems, where
the CB theory reduces to the MT theory.
The collision frequencies in the SB and CB theories depend not
only upon the effective temperature above but also upon an
internal ion temperature that is equivalent to the average
internal energy of the molecular ions. In both theories, the
differential equation governing T_{
i
n
t
} has the same form, which
is given in [6] and so needs not be repeated here.
The ion number density, n, at all positions in the apparatus and
at all times can be obtained by solving the rate equation of
continuity:
∂∂tn+∇·n𝐯¯=−kn,
where k is the second-order reaction rate coefficient. This equation can be
solved by a forward-time, forward-space version of the finite
difference method [10], provided that we have knowledge of
the ion number density (as a function of position) at time t = 0.
Here, as in previous work [8], we assume that the
initial ion number density is normally distributed along each
Cartesian axis, with standard deviations σ_{
x
}, σ_{
y
},
and σ_{
z
} that may be different. This choice has the
advantage that it should apply when no special experimental
techniques have been implemented. In order to match results in a
particular experiment, however, it may be necessary to work with
other initial ion number density functions.
For the two-temperature theory for atomic ions moving in a
background gas of atomic neutrals and reaction with a trace amount
of a single reactive neutral (whether atomic or molecular), the
reaction rate coefficient is given by the following equation
[7,9]:
where μ_{
R
} is the reduced mass for the ion and reactive neutral with mass M_{
R
}.
In addition,
TR,eff=mTm+MR1−MRM+MR(m+M)M(m+MR)Teff,
and Q_{
R
}^{*} is the total reactive cross sections, that is, a
function of the reactive collision energy, γ^{2}k_{
B
}T_{
R,eff}. Since T_{eff} depends upon
position and time, so does both T_{
R,eff} and k(t).
Equation (5) must be integrated over all spatial positions
in order to determine the rate coefficients for all ions in the
trap at a particular time:
k¯(t)=1n0∫∫∫−∞∞n(x,y,z,t)k(x,y,z,t)dxdydz.
This result must, in turn, be integrated over many cycles of the ac fields that
are present or equivalently over one cycle at large enough times
that steady-state behavior occurs, in order to obtain the single
value, k, that applies to the trap as a whole when no changes in
external parameters are imposed.
The effective temperatures in equation (9) are related to the T_{
u,eff} in the
same way that T_{
R,eff} is related to T_{eff},
that is, by equation (6). The MT expression given here
differs by a factor of 2 from that given previously [7], so
that it reduces to equation (5) in the limit where the
three effective temperatures are all the same.
It should be noted that Q^{*} must be computed for
molecules by adding state-specific cross sections, weighted so as
to take into account the probability of having that particular
state present. Since we will not apply the present work to
situations where either the ion or the buffer gas is molecular, no
further comments about the treatment of reactions in the SB and CB
theories are necessary, other than to note that the reactive
neutrals can be molecular when the 2T or MT theories are used.
3. Electric Fields in Traps
In an ideal quadrupole ion trap, the hyperbolic endcap electrodes
are fixed at ground potential and a time-dependent voltage, U + Vcos(Ω_{
R
F
}t), is applied to the ring
electrode. Here U is the amplitude of the dc voltage and V is
the zero-to-peak amplitude of the ac component that has angular
frequency Ω_{
R
F
}. The resulting electric field is purely
linear; in that at a particular point in the apparatus it has
components that are directly proportional to the position. The
standard way [3] of expressing the field is
Eu=−mΩRF24qau−2qucos(ΩRFt)u,
where
az=−2ax=−2ay=−16qUm(r02+2z02)ΩRF2
and
qz=−2qx=−2qy=8qVm(r02+2z02)ΩRF2.
Here it is assumed that the origin of the coordinate system is at
the center of the ring electrode with internal radius r_{0} and
that 2z_{0} is the shortest distance between the endcap
electrodes.
The fields are more complicated in a stretched or otherwise
nonideal quadrupole trap, even though it is cylindrically
symmetric along the z axis. The endcap electrodes are identical
and arranged symmetrically, with a time-dependent voltage applied
between them in a dipolar fashion, so that one has [U_{
D
} + V_{
D
}cos(ω_{
D
}t + δ_{
D
})]/2 while the
other has a negative value of exactly the same magnitude. Here
U_{
D
}, V_{
D
}, and ω_{
D
} are the equivalents of U, V,
and Ω_{
R
F
}. Note the phase shift, δ_{
D
}, that
may be present between this potential and that applied to the ring
electrode. Then the total electric potential can be expressed
[11] as a multipole expansion. The resulting electric field
can be written as [4]
The signs in some of the entries in Table 1 correct errors in the previous paper
[4]. The ellipses indicate higher-order multipole terms that
are usually assumed to be zero based on the values for the
coefficients, A_{
n
}, that are collected in [6] for several
types of quadrupole traps.
Dimensionless quantities characterizing the moment equations for
stretched or non-ideal quadrupole ion traps.
Quantity
Value for stretched or non-ideal quadrupole ion trap
Turning our attention now to linear traps, we note first that
there are various geometries in use [12]. For example, there
are linear traps with hyperbolic rods or slits in one pair of rods
and others with a stretch in the electrode separation. We leave
moment theories of such traps to later work and study here a
simple geometry in which the linear ion trap has a total length of
2z_{0} between flat endcaps and is made of four identical,
parallel rods, symmetrically arranged, with each pair separated by
a gap of distance 2r_{0}. (Note that r_{0} and z_{0} have
different meanings for linear traps than for quadrupole ion
traps.) There has been some dispute [13] about the best
radius, r, to use for the rods, but here we will use the
relationship [14] r = 1.12590r_{0}.
In order to maintain as close a relationship as possible to the
equations used for quadrupole ion traps, we will use equations
(9)–(10) and (12)–(13) for linear traps, even though researchers
in this area often choose definitions such that the x and y
components in these equations have opposite signs from one
another. As a further comment, we assume that the user of the
computer programs we discuss below will choose values of r_{0} and
z_{0} so that any fringe fields due to the finite size of the
electrodes are negligible.
The electrical potential due to a voltage of the form
U + Vcos(Ω_{
R
F
}t) applied on two opposing rods, while the
other two are held at ground, is independent of z and is given
by the following equation [14]:
The numerical values for the constants in equation (16) are sensitive
to the electrode size [15]. For the case r = 1.12590r_{0} that is used here, they are A_{2} = 1.001462,
A_{6} = 0.001292, and A_{10} = −0.002431, and we assume that all of the higher-order terms can
be neglected. For another common situation [16,17], where
r = 1.14511r_{0}, the constants are slightly different
[18,19]. We also note that [20] sets all of the other
coefficients above A_{2} to zero and incorporates A_{2} = 1/2 into the dimensionless parameters.
The more complicated linear ion traps mentioned above could
require terms like A_{4} and A_{8}, but the straightforward addition of such terms to equation
(16) is left to subsequent papers where such traps are considered.
We also note that if an RF field is applied in a symmetric
fashion, that is, with the same amplitude on all rods but 180
degrees out of phase for neighboring rods, then all that is
involved is a modification of the constants in equation (16).
We assume that fringe fields due to the finite size of the ion
trap are negligible. Then the electrical potential due to
a voltage of the form U_{
D
} + V_{
D
}cos(ω_{
D
}t + δ_{
D
}) applied to the flat endcaps of our model of
a linear ion trap is [20]
φends(x,y,z,t)=κUD+VDcos(ωDt+δD)z2zo2−x2+y22zo2,
where the parameter κ depends on the trap geometry and must be determined
experimentally or by solution of the Laplace equation for a
particular trap. Differentiating equations (16) and (17) with
respect to x gives the electric field in the x direction and
similarly for y and z. After some algebraic manipulation, the
results can be put in a form like equation (13), with the values
of the tilde quantities given in Table 2 in terms of the
quantities without tildes defined above.
Dimensionless quantities characterizing the moment equations for
linear ion traps.
Quantity
Value for Linear Ion Trap
a˜x
axr02+2z02r02A2+A63x4-30x2y2+15y4r04+....
a˜y
ay-r02+2z02r02A2+A63y4-30y2x2+15x4r04+...
a˜z
0
q˜x
qxr02+2z02r02A2+A63x4-30x2y2+15y4r04+....
q˜y
qy-r02+2z02r02A2+A63y4-30y2x2+15x4r04+....
q˜z
0
b˜x
bx-r02+2z022z02κ
b˜y
by-r02+2z022z02κ
b˜z
bz-r02+2z022z02κ
d˜x
dx-r02+2z022z02κ
d˜y
dy-r02+2z022z02κ
d˜z
dz-r02+2z022z02κ
f˜x
x
f˜y
y
f˜z
z
4. Moment Equations for Traps
When equation (13) is inserted into equation (1), the moment
equation for 𝐯¯u in stretched or
nonideal quadrupole traps and in linear traps can be written in
the same form:
The important new thing about this result is that it applies both to linear traps
and (when we set a˜u=au, q˜u=qu, and b˜u=d˜u=0) to ideal
quadrupole ion traps, and in all four fo the moment. theories discussed above.
For the 2T and SB theories, equation (2) is simplified by
introducing the dimensionless temperature:
ε=TeffT.
This quantity is also equal to the ratio of the average energy of
the collision between an ion and a buffer gas molecule to the
thermal energy of the gas. It is always greater than 1 because the
electric fields always act to increase the collision energy. When
equations (2) and (19) are used with equation (13), we get
The important thing is that equations (18) and (20) constitute a set of four coupled,
ordinary differential equations describing the ion motion in ideal
and nonideal quadrupole ion traps, or in linear traps, according
to the first approximation of the 2T and SB theories. To study
ion-neutral reactions in the 2T theory, one must also solve
equation (4), use the ion motion results in equation (5), and then
perform the integral in equation (7).
For the MT and CB theories, we introduce a dimensionless effective
temperature along each direction. Thus
εu=Tu,eff3T,
where the factor of 3 is introduced so that ε_{
x
} + ε_{
y
} + ε_{
z
} becomes equal to ε when
the three ion temperatures are identical, and the MT and CB
theories reduce to the 2T and SB theories, respectively. Then
equations (3), (13), and (21) give
The imporatnt thing is that equations (18) and (22) represent six coupled, ordinary
differential equations completely describing the ion motion in
ideal and nonideal quadrupole ion traps, or in linear traps,
according to the first approximation of the MT and CB theories. To
study ion-neutral reactions in the MT theory, one must also solve
equation (4), use the ion motion results in equation (8), and then
perform the integral in equation (7).
We have written computer programs in Mathematica and in Fortran to
solve the first-order moment equations, using the SB and MT
theories. The advantage of the Mathematica program is that the
numerical details of how the coupled differential equations are
solved are left to the software. In contrast, the advantage of the
Fortran program is that the method of solution is completely
specified (fourth-order Runge-Kutta) and the programmer has more
control over the level of numerical accuracy. The Fortran program
generally runs in less time on a Sun workstation than the
Mathematica program requires on a Dell computer, but each program
takes only on the order to 15–60 minutes, depending upon the
choices made for the parameters of the ion trap, the initial
conditions, and so forth. A copy of either program can be obtained
from the authors.
5. Tests
Before using our new programs for linear traps, however, it is
important that we verify that they reproduce the previous results
for ideal and stretched quadrupole traps. The Maxwell model of
constant collision frequency corresponds to an ion-neutral
interaction potential that varies inversely with the fourth power
of the separation between the colliding particles. Since this
describes the attractive interaction potential at very large
separations between any atomic ion and an atom in an s state,
the Maxwell model is generally regarded as a good one to use for
atomic systems when both the gas temperature and electric field
strength are small. In addition, ξ_{
u
} is the same constant
for all directions u, regardless of whether the 2T or MT theory
is used, while Φ = 0 for the SB theory and
Φu=121−ϵx+ϵy+ϵz3ϵu
for
the MT theory. The results obtained from the new computer programs
have been checked for Maxwell molecules against those given in
Figures 1 and 2 of [3] for the ideal quadrupole trap; they
agree to three significant figures or more.
Momentum-transfer cross section, Q^{(1)}(ε), in
square bohr (1bohr = 0.52917721092×10^{−10} m) as a function of the
collision energy, ε, in hartree (1hartree = 4.35974434×10^{−18} J) for ^{16}O^{+}(^{4}S^{0}) ions interacting with Ar atoms.
Total reaction cross section, Q^{*}(ε), in square
Angstrom (1Angstrom = 1.0×10^{−10} m) as a function of the collision
energy, ε, in eV (1eV = 1.602176565×10^{−19} J) for
^{16}O^{+}(^{4}S^{0}) ions reacting with N_{2} molecules.
We also considered a stretched quadrupole ion trap using the
Maxwell model for collisions between singly charged ions with
m = 100 Da and neutral atoms with M = 4 Da, T = 300 K, and
ξ_{
u
} = 740.9 s^{−1}. We focused on the steady-state
velocities by assuming that the average initial velocities are
zero in all three directions and that the initial value of
T_{
u,eff} is T in all directions. We also assumed that
the ion trap operates with no dc field on the rods
(a˜u=0), no ac or dc field on the endcaps
(b˜u=d˜u=0), and with an ac field on the ring
such that q˜x=q˜y=-0.10, q˜z=0.20, and
Ω_{
R
F
}/2π = 1.0 MHz. These choices are exactly those
used previously [6] for an ideal quadrupole trap, and our
results from both programs agree to 3 significant figures with
those in Figures 1–2 of [6].
A rigid-sphere interaction is often used to model collisions at
very high energy. To test our programs in this case we used m = 100 Da, M = 4 Da, T = 300 K, P = 0.001 torr,
a constant cross section of 50 ^{2}, Ω_{
R
F
}/2π = 1.00 MHz, a_{
z
} = 0.0010, a_{
x
} = a_{
y
} = −0.0005,
q_{
z
} = 0.2500, and q_{
x
} = q_{
y
} = −0.1250. The initial velocities were chosen as in [3], and
we obtained the same values for ideal quadrupole traps shown in
Figures 4–6 of that paper.
We also considered a stretched quadrupole ion trap using the rigid
sphere model for the ion-neutral collisions. Using the same
parameters as in [6], we obtained MT results that matched
those given previously [6]. Specifically, both of our new
computer programs gave values of the ion velocity along z as
3.14 km/s and the effective temperature along z as 4.86 kK (4860 K, to three significant figures)
at the first peak of the ac field, 0.250 μs, after the start of the simulation.
Finally, we considered the reaction rate coefficient between
^{107}Ag^{+} and D_{2} in an ideal quadrupole trap containing He in a much
larger amount than D_{2}. The conditions were
the same as in [8], which reported the first ab
initio calculation of the reaction rate coefficient in a trap;
that is, it was based on an ab initio potential energy
curve for the interaction of the ions and rare gas atoms. The
present values for the rate coefficient averaged over one period
of the ac field and over all positions in the trap were within 4%
of each other and of the previous value [9], 4.78×10^{−15} cm^{3}/s. The
differences are due to the numerical techniques used to evaluate
equation (5); although they can be eliminated by letting the
programs run longer with more elaborate quadrature techniques,
they are smaller already than errors ordinarily expected for
ion-neutral reaction rate coefficients in drift tubes, ion traps,
and so forth. Unfortunately, this predicted value is smaller than
the lower limit for determining reaction rate coefficient in ion
traps, so no experimental values are available to compare with.
The reason for this small value is that the reaction cross section
has a threshold at energies substantially above thermal values.
6. New Applications
As a new application of our computer programs, we consider
^{16}O^{+}(^{4}S^{0})
moving in a linear ion trap through Ar at 300.0 K and 0.01000 torr; four significant figures were used here and in all of the
other parameters cited below, in order to be consistent. It has
been shown previously [21] that the appropriate potential for
the description of the motion of this ion through Ar at the
energies of interest in experiments conducted in drift-tube mass
spectrometers and ion traps is the diabatic potential, neglecting
fine structure. By using an accurate ^{4}Σ^{−} potential between 2.08 and 30.24 bohr for the
O^{+}-Ar system, agreement with the mobility measurements [21]
was obtained for low and intermediate values of E/N,
although small discrepancies remained for high values. Since no
potential energy curves of higher accuracy appear to have been
calculated since 2008, we have used this potential to determine
the momentum-transfer cross section with an accuracy of 0.1% over
energies from 10^{−9} to 10 hartree, using program
PC [22]. The results are shown in Figure 1. Note that
Q^{(1)} is not constant, like it would be for a
rigid-sphere action, nor does it decrease linearly (with a slope
of −1/2) on this log-log plot, like it would for the Maxwell
model.
For the total reaction cross section between ^{16}O^{+}(^{4}S^{0})
ions and N_{2} molecules (in equilibrium with the Ar atoms in which they
compose only 0.1000% of the total particles), we use the values
between 0.3 and 10 eV of Albritton et al. [23]. The
reaction products are NO^{+} and N. The cross section
values are shown in Figure 2.
For the MT applications described here, we used r_{0} = 1.000 cm, which means that the r = 1.126 cm. We assumed that
z_{0} = 5.000 cm, which means that the total length of the
trap was 10.000 cm. The trap was assumed to have κ = 1.000, A_{2} = 1.001462, A_{6} = 0.001292,
and A_{10} = −0.002431. There was
assumed to be an ac field of frequency Ω_{
R
F
}/2π = 1.000 MHz on the rods, but no dc field on the rods and no
electrical field of either type on the endcaps. We used q_{
x
} = −0.1000 and studied 4 positions between
0 and 1.420 mm along both x and y, but 4 positions between 0 and 0.7100 mm along z.
We assumed that the initial ion distributions
had a normal distribution in space with σ_{
x
} = σ_{
y
} = 0.7100 mm and σ_{
z
} = 0.3550 mm. The
simulations generally involved 50 time steps for each
1.000 μs, although we did make a limited number of
calculations using 100 time steps that did not produce significant
differences. Steady-state behavior was established by comparing
the set of ion velocities and temperatures at the end of a cycle
of the rf field with those used at the start of the calculations.
We made calculations with two sets of initial conditions. The
first had initial velocities of 395.3 m/s along x, −395.3 m/s
along y, and 350.0 m/s along z. It had initial temperature
(energy) ratios of 1.000 along each axis, so each ε_{
u
} was set equal to 1/3 initially. Averaging the results over
2.5 cycles of the field (2.5 μs) and all positions in space
gave a reaction rate coefficient of 6.12×10^{−13} cm^{3}/s with the Fortran code and 6.52×10^{−13} cm^{3}/s with the
Mathematica code. The second simulation used initial velocities of
0 but temperature ratios of 1.4527 along x and y and 0.99230 along
z; the average reaction rate coefficients were 6.61×10^{−13} cm^{3}/s and 6.40×10^{−13} cm^{3}/s. These values could be
improved by using more values for the positions, putting more time
steps per cycle, and using simulation times long enough so that
the choice of initial conditions becomes unimportant.
Nevertheless, the values are sufficiently close to conclude that
the average reaction rate coefficient could be measured in linear
traps that currently exist.
Because q_{
x
} is inversely proportional to the ion mass, the value
of -0.1 used in the preceding calculations is not as small as what
might be assumed by researchers more accustomed to studying large
ions in traps. At x = y = z = 1.0 mm, the collision energies in these
calculations reached values as large as ε_{
x
} = ε_{
y
} = 4.9
and ε_{
z
} = 2.0,
equivalent to 0.38 and 0.16 eV, respectively. At x = y = z = 2.00 mm,
the values were as large as ε_{
x
} = ε_{
y
} = 9.8 and ε_{
z
} = 3.8, or 0.76 and 0.29 eV.
We performed an additional set of calculations with q_{
x
} = −0.2000 and found that the collision energies at x = y = z = 1.0 mm
reached values as large as ε_{
x
} = ε_{
y
} = 9.7 and ε_{
z
} = 3.8, equivalent to
0.75 and 0.29 eV, respectively; at x = y = z = 2.00 mm, they were as
large as ε_{
x
} = ε_{
y
} = 19.3 and
ε_{
z
} = 7.3, or 1.50 and 0.57 eV. Larger values
of the dimensionless field parameters would reach collision
energies above those for which the cross section for ^{16}O^{+} reactions with N_{2} is
known.
7. Conclusions
We have extended the previous moment theories for ion motion and
reaction so that they apply to linear traps as well as ideal and
stretched quadrupole traps. Fortran and Mathematica computer
programs have been written to implement the first approximation
moment equations. These programs have been checked against one
another and against previous results obtained for quadrupole
traps. They were then used to predict the reaction rate
coefficients that would be measured when ^{16}O^{+}(^{4}S^{0}) ions move through Ar gas in
which there is 0.1% of N_{2}. Taking into account
possible errors due to using only the first approximation moment
equations and the differences observed with the different computer
programs, the predicted values are 6.4±0.9×10^{−13} cm^{3}/s when the trap parameters
are as given above. The error estimate would have to be increased
if the reactive cross sections used here were found to be
inaccurate. The important point, however, is that this value is
sufficiently large that the average rate coefficient should be
measurable in a linear ion trap.
The computer programs described here can be used for many
ion-neutral systems. The first requirement is that information
should be available about the energy dependence of the
momentum-transfer cross section between the ion of interest and
some inert, buffer gas (typically He or Ar). Ab initio
results are available from the authors for just over 50 atomic
ions and a few molecular ions; eventually we hope to add these
values to the on-line database [24] that already contains the
potentials from which they can be determined and the transport
coefficients that depend upon them (and other transport cross
sections). The second requirement is information about the
dependence of the total reaction cross section as a function of
the energy of collision between the ion and some reactive
molecule. Abinitio calculations of these cross
sections are becoming feasible, although it may be necessary to
work with model cross sections, particularly if one is interested
in large molecules. The third requirement is precise information
about the values of the many quantities (e.g., ac frequencies,
electric field strengths, and trap parameters) that are used in
the experiments.
In most cases, the least well known of the three things listed
above are the reaction cross sections. These are the fundamental
quantities, as opposed to the reaction rate coefficients that can
change considerably with even slight changes in experimental
conditions. We look forward to working with an experimental group
to combine theoretical information about the momentum-transfer
cross section with experimental information about ion traps in
order to infer total reaction cross sections.
Acknowledgments
We are thankful for the comments and advice of Dr. Douglas
Goeringer. We are grateful for some preliminary work for this
paper done by Shristi Kharel as part of her undergraduate thesis.
Appendix A: Collision Frequencies
In the MT theory, the collision frequencies, ξ_{
u
}, depend
upon three effective temperatures, one along each Cartesian axis
u. They are given by eqsuations (74)–(76) of [1] as
(It should be noted that there are two factors of 3
missing in the form of these equations given by equation (A.4) of
[6], which arose from oversight of the factor of 3 in equation
(13) of that paper.) The momentum-transfer cross section, Q^{(1)}, is one of a family of transport cross sections that
can be accurately calculated [22] from knowledge of the
interaction potential energy curve for the atomic ion-atom pair.
For quadrupole ion traps, the strict cylindrical symmetry of the
apparatus guarantees that the x and y effective temperatures
are the same. This means that (A2) may be expressed in cylindrical
polar coordinates and then simplified to a two-dimensional
integral. This simplified form was implemented in the computer
program used in previous calculations [7,8]. It is important to
note that this simplification guarantees that the computer
programs will run much faster for quadrupole traps than for linear
traps.
The essence of the 2T theory is to assume that all three effective
temperatures are the same. In this case, equation (A1) simplifies
to
Extensions of eqsuations (A1) and (A3) that apply to
the SB and CB theories of molecular ion-neutral systems are given
in the appendix of [6] and so need not be repeated here.
For classical-mechanical collisions between rigid spheres of
diameter d, the momentum-transfer cross section is independent
of energy. Thus
Q(1)=πd2.
Although the collision frequencies still depend upon
the effective temperature(s), the constant cross section
simplifies equations (A1) and (A4) and greatly increases the
speed of both MT and 2T computer programs—at the cost of a
poorer description of the physics of the collisions.
A different simplification arises for the Maxwell model, where
gQ^{(1)}(g^{2}k_{
B
}T) is constant. In this
case, the collision frequencies are constant, as assumed in the
Mathieu equation for ion traps. This model is perfect for
ion-neutral interaction potentials that vary as the inverse-fourth
power of the separation, which correctly describes most
ion-neutral interactions at very large separation. This in turn
makes it a valid model at low temperature, where most collisions
occur at large separations. Unfortunately, we now know that such
temperatures must be below (and sometimes far below) 1 K. In
short, the Maxwell model can also greatly increase the speed of MT
and 2T computer programs, but again at the cost of a poorer
description of the physics of the collisions at the usual
temperatures employed with ion traps.
Appendix B: The Φ
The Quantities
The Φ quantities that arise in the various moment theories
of ion traps are dimensionless ratios of collision integrals. They
account for energy partitioning among the three Cartesian
directions and, in the case of molecular ions and neutrals,
between translational and internal degrees of freedom. For the 2T
theory, Φ = 0. For the MT theory, it has been shown
[6] that
For the other theories, the expressions are given in
the Appendix of [6].
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