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Ion Collision Theory
954 ION COLLISION THEORY
Ion Collision Theory Anil K Shukla and Jean H Futrell, University of Delaware, Newark, DE, USA Copyright © 1999 Academic Press
Introduction Collisions between an ion and neutral species result in a number of possible outcomes depending upon the chemical and physical properties of the two reactants, their relative velocities, and the impact parameter of their trajectories. These include elastic and inelastic scattering of the colliding particles, charge transfer (including dissociative charge transfer), atom abstraction, complex formation and dissociation of the colliding ion. Each of these reactions may be characterized in terms of their energy-dependent rate coefficients, cross sections and reaction kinematics. This article outlines a theoretical framework for discussing these processes that emphasizes simple models and classical mechanics. We divide the discussion of collision processes into the two categories of low-energy and high-energy collisions. Experiments under thermal or quasi-thermal conditions – swarms, drift tubes, chemical ionization and ion cyclotron resonance – are strongly influenced by long-range forces and often involve ‘capture collisions’ in which atom exchange and extensive energy exchange are common characteristics. High-energy collisions are typically impulsive, involve short-range intermolecular forces and are direct, fast process.
Low-energy collisions For historical reasons, and because the majority of investigations of ion–molecule reactions have involved quasi-thermal collisions of relatively lowmass ions and neutrals, we begin with the simplest case of a point charge interacting with a polarizable neutral. It is recognized that this Langevin model applies only at low collision energies where longrange forces dominate collision dynamics. Despite these restrictions, this simplistic picture provides a near-quantitative rationalization for the rates of thousands of ion–molecule reactions. Ion–molecule collisions at low energies are dominated by the attractive long-range polarization force described in the simplest case by
MASS SPECTROMETRY Theory where the ion is tacitly assumed to be a point charge, q, and the neutral is a point polarizable atom or molecule having a polarizability α at a distance r from the ion. At relative velocities where repulsive forces can be neglected (that is, they are sensed at much closer distances than the impact parameter for orbiting collisions, as discussed below), Equation [1] is a plausible approximation. Conservation of angular momentum in a collision is imposed by expressing the effective potential of the ion–molecule system as the sum of the central potential energy V(r) and the centrifugal potential energy:
where L is the classical orbital angular momentum of the two particles and µ is the reduced mass of the system. Here L is given by µvb where b is the impact parameter and v is relative velocity. The total relative energy of the system is given by
where Etrans(r) is the translational energy along the line of centres of the collision. A plot of Veff(r) versus r at constant Er for several values of the impact parameter is shown in Figure 1 for the collisions of N with N2. For the special case in which the centrifugal barrier height equals Er (at b = bc in Figure 1), Etrans(r) = 0, and the particles will orbit the scattering centre with a constant separation distance rc. For b > bc the distance of closest approach ≥ bc /√2 and for b < bc the particles spiral into small radii before they separate. The cross section for orbiting collisions is
The rate coefficient corresponding to the orbiting or capture cross section is given by
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been developed to rationalize cross sections larger than Langevin and the temperature/energy dependences of cross sections. Considering first ion–dipole interactions, the classical expression for the effective potential is
Figure 1 Plot of effective potential, Veff, versus r for N colliding with N2 at a fixed energy of Er and several values of the impact parameter. Reproduced with permission of Academic Press from Su T and Bowers MT (1979) In Bowers MT (ed) Gas Phase Ion Chemistry, Vol. 1.
which is the commonly cited energy (velocity)-independent Langevin rate constant for ion–molecule reactions. Use of these expressions implicitly assumes there is a reactive cross section σrx for real systems which is smaller than the capture cross section given by Equation [4]. The reaction cross section also sets a conceptual limit of relative velocity beyond which treatments of only long-range critical potentials cannot apply. For a hypothetical example of σrx = 50 Å2, polarizability of 10 Å3 and reduced mass of 50 amu, the capture cross section equals the σrx at a centre-of-mass (CM) collision energy of approximately 0.7 eV. We may infer that collision energies of the order of 1 eV roughly define the boundary between low-energy collisions – e.g., reactions in high-pressure ion sources, drift tubes, ion cyclotron resonance and plasma swarm experiments – and higher-energy collisions, considered later. Considering these limitations, it is remarkable that this simplistic theory accurately correlates collision cross section and rate coefficients for hundreds of reactions of ions with nonpolar molecules.
Ion–dipole and ion–quadrupole interactions Over time, increasingly sophisticated modifications to the Langevin capture cross-section model have
where µD is the dipole moment of the molecule and θ is the angle between the dipole and the line of centres of the collision. The first such treatment by Hamill and co-workers assumed that the dipole ‘locks in’ on the ion, i.e. θ = 0. This upper limit, locked-dipole approximation is obviously unrealistic for reactions at and above room temperature, for which rotation of the polar molecule tends to average out the dipolar potential. Addressing this limitation, Su and Bowers developed the average dipole orientation (ADO) theory in which the orientation angle of the dipole with respect to the line of centres of collision is averaged as a function of ion–molecule separation distance. This concept leads to a ‘locking constant’, c, which is multiplied by the cos θ term; this constant has been parametrized as a function of µD/α½ for the range of temperatures that are of general interest in lowenergy ion–molecule reactions. Further elaboration of this approach by Su and Bowers modified the treatment of angular momentum to formalize angleaveraged dipole orientation (AADO) theory. These workers also considered the interaction with a point charge of the quadrupole moment of molecules with D∞h symmetry and formulated the average quadrupole orientation (AQO) theory for these types of long-range interaction. Barker and Ridge developed a statistical model for ion–polar molecule collisions based on similar concepts. They considered the average interaction energy for a statistical ensemble of ion–neutral pairs as a function of their separation. The effective potential for such an ensemble is
where kB is the Boltzmann constant, TR is the rotational temperature of the neutral, E is the relative translational energy, b is the impact parameter and (X) is the Langevin function defined as (X) = cotanh(X) − 1/X. The momentum transfer collision frequencies calculated from this model or from the
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family of ADO theories described above are in reasonable agreement with experimental measurements of rate coefficients. They rationalize both the larger cross sections found for ion–molecule reactions involving highly polar neutrals and the increase in reaction rate coefficients for polar molecules at low temperatures.
where is the reaction probability. When Jmax(j, Ω) is larger than j, the sum over J in this equation gives the relationship
Quantum-mechanical approaches Quantum-mechanical calculations of ion–dipole capture rate constants have been modelled successfully for low-temperature experiments and correlate well with the classical collision theory and capture cross sections just described. Clary has calculated rate constants of several ion–molecule reactions using the adiabatic capture centrifugal sudden approximation (ACCSA). Since the rotational motion of the neutral is strongly hindered by the ion at low velocity, he chose collinear reaction geometry, leading to a basis set that localizes the rotational motion of the neutral about this collinear configuration. The partial wave Hamiltonian for the entrance reaction channel is given by
where is Planck’s constant divided by 2π, µ is the reduced mass, B is the rotational constant, j is the rotational angular momentum operator, J is the total angular momentum operator, V is the ion– molecule potential energy surface, θ is the orientation angle and r is the position vector of the particle measured from the centre-of-mass. As an approximation, J– j2 is replaced by the diagonal value [J(J+1)2+j2−22Ω2], where Ω is the projection of both rotational and total angular momentum along the Z axis (centre-of-mass vector). Thus for a fixed value of r, the Hamiltonian becomes
The reaction cross section is obtained from partial wave expansion and the capture approximation discussed earlier; that is, the reaction takes place if there is enough energy to surmount the centrifugal barrier. It follows that
with the condition that σ(j) = 0 if the energy is below the centrifugal barrier. Rate constants calculated by this formalism are in excellent agreement with experimental room-temperature constants for such systems as positive and negative ion–molecule reactions of HCN with H−, D− and H ions. Troe has used a statistical adiabatic channel model to calculate thermal ion–molecule capture rates using a pure long-range potential. The adiabatic channel potentials V(r) are calculated from perturbation theory, and analytical expressions for channel threshold energies, activated complex partition functions and capture rate constants are obtained. His results are in good agreement with trajectory calculations at temperatures above 10 K for such systems as H + HCN. Additionally, Sakimoto has developed a time-independent quantum-mechanical approach for collinear triatomic systems A+BC in the energy range from thermal to several eV. Using hyperspherical coordinates, the time-independent Schrödinger equation is solved numerically using the discrete variable representation algorithm for collision energies of 1–6 eV. Partial cross sections for reactive scattering and ion dissociation for He + H (plus isotopes) were calculated that generally agree with limited experimental data for this system.
High-energy ion–neutral collisions: Beam scattering Low-energy ion–neutral collisions considered thus far often result in random angular scattering with extensive mixing of orbital and angular momentum. Beam experiments at the low-energy limit demonstrate that persistent complexes are formed but provide no further information on reaction mechanisms. In contrast, conservation of angular momentum in high-energy collisions leads to specific angular scattering, which can provide detailed information on reaction mechanisms. Deducing this information is a complex exercise and we begin with a description of elastic scattering. After these scattering characteristics have been described – both classically and quantum mechanically – we shall outline briefly how
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reactive scattering is treated in this framework. A collision is better described by CM coordinates, which permit us to describe kinetic energy T as the motion of the centre-of-mass of the system in the laboratory frame and the relative motion of the particles in the CM frame
where M is the sum of the masses, VCM is the velocity of the CM and the reduced mass and relative velocity have been defined previously. Since the motion of the CM is conserved in any collision, we can focus our attention on the relative motion in which the two particles are described as a single particle having mass µ, angular momentum µ(r × v) and kinetic energy µv2. For central potentials considered previously, the angular momentum is constant and the total CM energy at separation r is
where the effective potential
includes the centrifugal potential L2/2µr2 described previously. The centrifugal potential is included in the equation for radial motion to account for angular motion of the system. The observable results of a collision are the velocity of the particle after collision (deduced from its kinetic energy) and its direction of motion. For ordinary collisions between spherical particles, total kinetic energy must be conserved, so the initial and final velocities must be equal at distances large enough that V(r) is negligible. Figure 2 depicts how the deflection angle – the quantity of interest in these elastic and nonreactive collisions – relates to the interaction potential. Here a particle of reduced mass µ approaches parallel to the x axis with kinetic energy µv2 and angular momentum µvb. When the effective potential energy Veff equals the initial kinetic energy, the particle reaches its distance of closest approach, rc, the turning point of the trajectory. The particle then recedes, and at a large distance from the scattering centre no longer senses the
Figure 2 Deflection of particles interacting through a potential V(r ). A single particle of reduced mass µ and velocity v moves toward the stationary scattering centre with impact parameter b and is scattered by a central potential V(r ) through an angle θ. Reproduce with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
interaction potential. The change in direction of motion is the deflection angle θ, and is measured from the negative x axis in Figure 2. We note parenthetically that deflection through +θ and −θ are experimentally indistinguishable and that deflection by ±2πNθ (where N is an integer) is indistinguishable from the deflection θ. Figure 2 illustrates that the deflection is symmetric about the turning point; that is, the deflection of the particle in the outgoing trajectory equals its deflection in the approach trajectory for elastic scattering. Consequently, the polar scattering angle 2φc and the final scattering angle θ is π−2φc (see Figure 2). This leads to the expression for the deflection angle as a function of initial energy and impact parameter:
where rc is the outermost solution of E = L2/ 2µrc2 + V(rc). Note that if b = 0 (head-on collision), then θ = π (unless V = 0). Further, as b increases from zero, θ decreases from π, and θ decreases to zero as b increases without limit. A reasonable central force potential for ion–neutral elastic scattering is the Morse potential given by
where x = α(r − re), with α specifying the curvature of the potential at its minimum. This potential has an attractive well of depth D at the equilibrium
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To illustrate the relationship between potential and scattering, Figures 3 to 5 show respectively the effective potential, deflection function and scattering intensity for the Morse potential, (Eqn [16]), parameterized to describe scattering of low-energy protons by argon atoms. The potential for zero impact parameter (hence zero angular momentum, L = 0) in Figure 3 has the parameters re = 1.28 Å, D = 4.17 eV and α = 1.85 Å3; the Morse potential with these parameters matches the experimentally determined scattering of protons by argon to better than 1% accuracy over the interval from 0 to 5 Å. The curves in Figure 3 are labelled with the angular momentum, L, in units of . At high L, the centrifugal potential overpowers the attractive well and the scattering is always repulsive. At L = 75.95
there is an inflection point at V = 1.153 eV, and for lower angular momenta collisions the effective potential has two regions of repulsion separated by an attractive well. The outermost root is the classical turning point for the potential and a plot of the turning point r as a function of impact parameter (or, equivalently, of L) leads to a singularity where √c = b for all collision energies. This corresponds to the point where the potential V(r) = 0 [∼1 Å for the potential shown in Figure 3] and is a measure of the ‘size’ of the interacting particles, analogous to but different from a hard-sphere radius for the collision. Figure 4 shows the deflection function calculated from Equation [15] for the effective potential given in Figure 3 for a number of collision energies. For zero impact parameter, the deflection angle is π, just as it would be for hard-sphere collisions, with positive scattering angle corresponding to repulsive collisions. The deflection angle goes through zero and becomes negative as the collision senses the attractive potential at longer distances. As b increases, the deflection angle goes smoothly to zero. For moderate energy collisions, the centrifugal potential dominates scattering at all impact parameters. For very low energies, < 1.153 eV for the potential illustrated, there are three roots of the deflection function. (The reader is reminded that positive and negative deflections are experimentally equivalent.) The ‘piling up’
Figure 3 Plot of effective potentials derived using the Morse potential for several values of angular momentum, L, as marked on each curve. See text for the parameters used for the Morse potential. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
Figure 4 The relationship between deflection function and impact parameter for interactions governed by the Morse potential of Figure 3 at the indicated collision energies. 1 bohr = 0.529 Å. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
separation, r = re. For r < re, the repulsive part of the potential rises steeply to a high value, for r > re the attractive part of the potential is dominant. Because the Morse potential goes to zero exponentially, it does not correctly describe the potential at large r. If necessary, it can be modified by adding a term for long-range polarization forces. It nicely describes the short-range behaviour and the balance between attractive and repulsive forces.
Differential scattering
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of intensity at a particular scattering angle is called the rainbow angle. Measurement of rainbow scattering is an important experimental parameter for deducing the scattering potential. For specific L values below this critical energy, two of the roots coincide at the maximum of the effective potential. For these trajectories, orbiting occurs and the deflection angle becomes infinite. This phenomenon is readily detected in careful beam scattering experiments; it corresponds to the capture cross section in our description of low-energy collisions. Figure 5 illustrates the connection of crossed-beam measurements of ion scattering with the interaction potential of the ion–neutral collision, again using the Morse potential. The quantity measured experimentally is the differential scattering cross section for a given CM collision energy. The differential cross section, I(θ), at a specified energy is given by
Figure 5 illustrates for several energies the singularities determined by the two terms in the denominator of Equation [18]. The sin θ term gives infinity at CM angles of zero and π. The rainbow angle gives rise to a singularity when dθ /db goes to zero; this occurs when several impact parameters give the same deflection angle. At higher collision energies this occurs at small angles; with decreasing collision energy the rainbow angle increases smoothly with decreasing energy and, for 1.5 eV is greater than 180°. Numerically it is 254° for this potential, leading to an observed scattering angle of 360° −254° = 106°. Two details distinguish this class of trajectories experiment from high energy collisions – namely, the singularity at 180° and the reversal of the ‘bright side’ of the rainbow. The backscattering of particles corresponding to the sin θ singularity at 180° is the experimental demonstration that particles are scattered by a potential sufficiently attractive at that collision energy to hold them together for at least half a revolution.
Reactive encounters
where dΩ is the differential of solid angle. The cross section calculated for the Morse potential shown in
We have explained in some detail the relationship of measured variables in elastic scattering to the interaction potential between the ion and neutral. This framework is the starting point for describing inelastic and reactive scattering. Inelastic scattering may be introduced as an instantaneous conversion of kinetic energy of motion into internal energy of either or both of the collision partners, while reactive collisions involve both energy exchange and reaction. It is plausible that these physical and chemical conversions occur at or near the distance of closest approach in the respective trajectories. This breaks the symmetry of the elastic scattering formalism and the retreat trajectory of products involves both a different energy inventory in the collision partners and a change in the potential governing the trajectory as they separate from the collision centre. This approach has been reasonably successful in rationalizing the differential scattering cross sections of a variety of proton transfer reactions. The time-dependent quantum-mechanical treatment of collisions described in the next section automatically takes into account energy transfer and reaction – at least in principle.
Newton diagrams Figure 5 Plot of scattering intensity (differential cross section) versus centre-of-mass scattering angle for interactions with Morse potential of Figure 3 using classical treatment. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
As noted previously, describing collision processes in the CM frame is more informative than in the laboratory (LAB) frame. Explicitly the motion of the CM is conserved in collisions and is unavailable to the
960 ION COLLISION THEORY
reactants. The proper framework for displaying collisions is the CM relative velocity diagram, frequently called the Newton diagram. The collision process (for simplicity assumed to occur at right angles) of
with M and M2, having velocities of V1 and V2, is depicted in Figure 6. The collision centre is the origin of the laboratory reference frame. In the CM frame, M and M2 move collinearly towards each other with velocities U1 and U2 and collide at the CM located on their relative velocity vector at the point defined by conservation of momentum such that M1U1 = M2U2. After collision, the two particles recoil from the CM in opposite direction with velocities U1′and U2′. The velocity of the CM is given by
and is constant. The relative kinetic energy of the collision partners, ECM, is available for the reaction and is given by
Similarly, the postcollision relative kinetic energy is given by
The difference ∆T = E – ECM, the translational exoergicity of the collision process, is a direct measure of the energy used in the process. Consequently, experimental measurements and transformation of mass, intensity, energy and angular distributions of product ions as Newton diagrams provides quantitative information on the energetics and dynamics of the collision process. Two limiting cases of ion–molecule reaction dynamics will serve to illustrate the utility of Newton diagrams. If a collision between an ion and a molecule proceeds via an intermediate whose lifetime is larger than several rotational periods (~10–12 s), internal energy redistribution is usually complete and the complex loses its memory of the reactant velocity vectors. The products will separate with equal probability on both sides of a plane passing through the CM and normal to the collision axis, giving rise to the forward–backward symmetry. Such orbiting complexes are usually formed at relative kinetic energies that are of the order of or less than the well depth of the interaction potential governing the collision. As relative kinetic energy increases, the lifetime of the intermediate complex decreases and there is a transition to a direct reaction. The interaction time is too short for the intermediate to rotate and the Newton diagram contour plot becomes asymmetric with respect to the CM. The dependence of reaction probability on impact parameter and the effective potential for the approach and retreat trajectories determine the scattering angle and the form of the Newton diagram.
Semiclassical theory of collisions, the WKB phase shift In three dimensions and for central potentials, the Schrödinger equation separates in spherical coordinates analogously to corresponding classical equations of motion. The incoming particle is described by a wavefunction with a local wavenumber k(r). Repulsive potentials decrease the velocity of the wave packet and the resulting scattered wavefunction has fewer nodes, resulting in a negative phase shift. Conversely, attractive potentials result in an increase in velocity; the wavefunction acquires more nodes and this results in a positive phase shift. The overall phase shift for scattered particles is
Figure 6 Newton diagram showing pre- and postcollision velocity vectors in the laboratory and centre-of-mass reference frames for the collision of M and M2 having velocities of V1 and V2.
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Integrating this expression from the distance of closest approach to infinity (this gives half the total difference in phase for the entire trajectory; this is the convention for phase shift) gives
This makes the precise connection we seek between WKB scattering theory and the classical model developed previously.
Quantum scattering Moving from a semiclassical towards a full quantum treatment we note that, at large distances, the wavefunction for a scattered particle is a combination of an incoming plane wave and a scattered wave:
For both integrals, the lower limit of integration is the (outermost) zero of the associated integrand. Since both integrals are divergent, δ depends on cancellation between the two. This difficulty is repaired by subtracting dr from both terms:
where the z direction is the direction of travel. The first term describes a plane wave and the second term is an outgoing spherical wave whose amplitude is f(θ). This simple form of the wavefunction is valid for r large enough that no interference between the incoming and outgoing waves is important. This applies experimentally to a collimated beam of finite width. Since the wavefunction in Equation [29] obeys the Schrödinger equation, it can be expanded in any complete set of orthogonal functions, for example Legendre polynomials:
This may be recast into a form incorporating the definition of phase shift from Equation [23]:
where Pl is Legendre polynomial. Multiplying by Pl′(cos θ), and integrating over θ we obtain for large values of r the asymptotic result where the integral term gives the phase difference resulting from the interaction potential and a correction for the difference in turning points of the unscattered and scattered particle includes the effect of the centrifugal potential. Finally, it can be shown that the phase shift method leads to expressions for the collision time (different between V(n) ≠ 0 and V(n) = 0) and for scattering angle.
which has solutions ψ1(r) = a1sin(kr − lπ/2 + δl). Here the phase shift, δl, for the lth partial wave contains all the information about the potential, and the term lπ/2 comprises the centrifugal potential. The expression for scattering amplitude is obtained as
The differential cross skection I(θ) is the probability density for observing a particle at an angle θ,f (θ)2 and the total cross section is obtained by integration
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additional terms that couple them together. Sophisticated computer programs are now available to solve these complex equations and calculate the detailed dynamics of a collision process.
List of symbols
Figure 7 Plot of scattering intensity versus centre-of-mass scattering angle at 5 eV collision energy using quantum-mechanical treatment. The calculations were performed using 500 phase shifts and the Morse potential of Figure 3. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (ed) Gaseous Ion Chemistry and Mass Spectrometry.
of I(θ) over all angles as given below.
The phase shifts for calculating scattering as a function of angle are obtained by starting at r = 0 and increasing r until V(r) becomes negligible. Figure 7 illustrates the differential cross section for the Morse potential (Eqn [17]) calculated using Equation [32] and 500 phase shifts that were calculated using Equation [26]. The relative energy is 5 eV. The classical differential cross section from Figure 5 is also shown. The quantum-mechanical form of I(θ) removes the classical singularity. The additional peaks occurring at angles smaller than the rainbow angle are interferences from the three different values of the impact parameter that contribute to scattering into these angles. The quantum result tends to oscillate about the classical mechanical result, which cannot recover the interferences of the quantum result. For noncentral potentials – e.g. molecular scattering – the Schrödinger equation does not separate into partial waves. The solution can still be expanded in partial waves, but then the individual partial-wave radial Schrödinger equations will have
b = impact parameter; B = rotational constant; c = locking constant; D = depth of attractive potential well; E = energy; f() = function; H = Hamiltonian; I(θ) = differential cross section; j = rotational angular momentum operator; J = total angular momentum operator; k = rate coefficient; local wavenumber; kB = Boltzmann constant; = Langevin function; L = classical orbital angular momentum; P = Legendre polynomial reaction probability; q = ionic charge; r = distance; magnitude of position vector; polar coordinate; r = position vector; re = equilibrium distance; T = kinetic energy; TR = rotational temperature; v = relative velocity vector; ν = magnitude of relative velocity vector; V(r) = central potential energy; V, V′, U, U′ = velocity in centre-of-mass frame; VCM = velocity of centre-of-mass; α = atomic/molecular polarizability; curvature of potential energy curve at minimum; δ = phase shift on scattering; θ = angle between dipole and line of centres; final scattering angle; polar coordinate; µ = reduced mass; µD = dipole moment; σ = collision cross section; τ = collision time; ϕc = polar scattering angle; ψ = wavefunction; Ω = projection of rotational and total angular momentum on Z axis; solid angle.
Subscripts c = capture; c = closest approach; CM = centre of mass; e = equilibrium; eff = effective; r = relative; trans = translational See also: Ionization Theory; Quadrupoles, Use of in Mass Spectrometry.
Further reading Chesnavich WJ, Su T and Bowers MT (1979) Ion-dipole collisions: recent theoretical advances. In Ausloos P (ed) Kinetics of Ion Molecule Reactions, pp. 31–53. New York: Plenum. Child MS (1973) Molecular Collision Theory. New York: Academic Press. Fluendy MAD and Lawley KP (1973) Chemical Applications of Molecular Beam Scattering. London: Chapman and Hall. Franklin JL (ed) (1972) Ion Molecule Reactions. New York: Plenum Press. Futrell JH (ed) (1986) Gaseous Ion Chemistry and Mass Spectrometry. New York: Wiley.
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Levine RD and Bernstein RB (1987) Molecular Reaction Dynamics and Chemical Reactivity. Oxford: Oxford University Press. McDaniel EW (1964) Collision Phenomena in Ionized Gases. New York: Wiley. McDaniel EW, Cermak V, Dalgarno A, Ferguson EE and Friedman L (1970) Ion–Molecule Reactions. New York: Wiley-Interscience. Shirts RB (1986) Collision theory and reaction dynamics. In Futrell JH (ed) Gaseous Ion Chemistry and Mass Spectrometry, pp. 25–57.New York: Wiley.
Steinfeld JI, Francisco JS and Hase WL (1989) Chemical Kinetics and Dynamics. Englewood Cliffs, NJ: PrenticeHall. Su T and Bowers MT (1979) Classical ion-molecule collision theory. In Bowers MT (ed) Gas Phase Ion Chemistry, Vol. 1, pp. 83–118. New York: Academic Press. Vestal ML, Wahrhaftig AL and Futrell JH (1976) Application of a modified elastic spectator model to proton transfer reactions in polyatomic system. Journal of Physical Chemistry, 80: 2892–2899.
Ion Dissociation Kinetics, Mass Spectrometry Bernard Leyh, F.N.R.S. and Université de Liège, Belgium
MASS SPECTROMETRY Theory
Copyright © 1999 Academic Press
The ionization process taking place in the source of a mass spectrometer or the activation step of a tandem mass spectrometry experiment leads to ions with a range of internal energies extending usually well above the first dissociation threshold, i.e. the molecular ion has enough energy to dissociate. Dissociation is a dynamic process characterized by its associated lifetime. The whole story of kinetics in general and of ion dissociation kinetics in particular is to measure how fast a dissociation proceeds under given conditions (state selection, well-defined energy or temperature, for example) and to determine the reaction mechanisms. Both aspects will be addressed.
This represents the most simple situation where a single dissociation process occurs for ions possessing a single total energy, E. k(E) is called the unimolecular rate constant. However, in practice, the parent ion can be formed with a broad distribution of internal energies, P(E), and competitive reactions can take place. If the different competitive channels are characterized by rate constants denoted ki(E), then the following equation holds for the rate of production of the jth fragment ion:
Unimolecular rate constant Dissociation kinetics of state-selected reactant ions has been investigated only in a limited number of relatively small systems. However, a large amount of data are now available on ionic reactions investigated with internal energy selection. The dissociation of an energy-selected molecular ion is a unimolecular process, the rate of which, R(E,t) is defined by the following equation, where N0 is the initial number of ions formed:
If further dissociations occur, the rates must be calculated by solving the set of differential equations resulting from the appropriate kinetic scheme. If the dissociating ion is characterized by a welldefined internal energy, i.e. if P(E) reduces to a single value, then each ion of the ionized sample can be considered to be one of the replicas of a microcanonical ensemble. For this reason, k(E) is often referred to as the ‘microcanonical rate constant’. If, however, P(E)
Ion Collision Theory Anil K Shukla and Jean H Futrell, University of Delaware, Newark, DE, USA Copyright © 1999 Academic Press
Introduction Collisions between an ion and neutral species result in a number of possible outcomes depending upon the chemical and physical properties of the two reactants, their relative velocities, and the impact parameter of their trajectories. These include elastic and inelastic scattering of the colliding particles, charge transfer (including dissociative charge transfer), atom abstraction, complex formation and dissociation of the colliding ion. Each of these reactions may be characterized in terms of their energy-dependent rate coefficients, cross sections and reaction kinematics. This article outlines a theoretical framework for discussing these processes that emphasizes simple models and classical mechanics. We divide the discussion of collision processes into the two categories of low-energy and high-energy collisions. Experiments under thermal or quasi-thermal conditions – swarms, drift tubes, chemical ionization and ion cyclotron resonance – are strongly influenced by long-range forces and often involve ‘capture collisions’ in which atom exchange and extensive energy exchange are common characteristics. High-energy collisions are typically impulsive, involve short-range intermolecular forces and are direct, fast process.
Low-energy collisions For historical reasons, and because the majority of investigations of ion–molecule reactions have involved quasi-thermal collisions of relatively lowmass ions and neutrals, we begin with the simplest case of a point charge interacting with a polarizable neutral. It is recognized that this Langevin model applies only at low collision energies where longrange forces dominate collision dynamics. Despite these restrictions, this simplistic picture provides a near-quantitative rationalization for the rates of thousands of ion–molecule reactions. Ion–molecule collisions at low energies are dominated by the attractive long-range polarization force described in the simplest case by
MASS SPECTROMETRY Theory where the ion is tacitly assumed to be a point charge, q, and the neutral is a point polarizable atom or molecule having a polarizability α at a distance r from the ion. At relative velocities where repulsive forces can be neglected (that is, they are sensed at much closer distances than the impact parameter for orbiting collisions, as discussed below), Equation [1] is a plausible approximation. Conservation of angular momentum in a collision is imposed by expressing the effective potential of the ion–molecule system as the sum of the central potential energy V(r) and the centrifugal potential energy:
where L is the classical orbital angular momentum of the two particles and µ is the reduced mass of the system. Here L is given by µvb where b is the impact parameter and v is relative velocity. The total relative energy of the system is given by
where Etrans(r) is the translational energy along the line of centres of the collision. A plot of Veff(r) versus r at constant Er for several values of the impact parameter is shown in Figure 1 for the collisions of N with N2. For the special case in which the centrifugal barrier height equals Er (at b = bc in Figure 1), Etrans(r) = 0, and the particles will orbit the scattering centre with a constant separation distance rc. For b > bc the distance of closest approach ≥ bc /√2 and for b < bc the particles spiral into small radii before they separate. The cross section for orbiting collisions is
The rate coefficient corresponding to the orbiting or capture cross section is given by
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been developed to rationalize cross sections larger than Langevin and the temperature/energy dependences of cross sections. Considering first ion–dipole interactions, the classical expression for the effective potential is
Figure 1 Plot of effective potential, Veff, versus r for N colliding with N2 at a fixed energy of Er and several values of the impact parameter. Reproduced with permission of Academic Press from Su T and Bowers MT (1979) In Bowers MT (ed) Gas Phase Ion Chemistry, Vol. 1.
which is the commonly cited energy (velocity)-independent Langevin rate constant for ion–molecule reactions. Use of these expressions implicitly assumes there is a reactive cross section σrx for real systems which is smaller than the capture cross section given by Equation [4]. The reaction cross section also sets a conceptual limit of relative velocity beyond which treatments of only long-range critical potentials cannot apply. For a hypothetical example of σrx = 50 Å2, polarizability of 10 Å3 and reduced mass of 50 amu, the capture cross section equals the σrx at a centre-of-mass (CM) collision energy of approximately 0.7 eV. We may infer that collision energies of the order of 1 eV roughly define the boundary between low-energy collisions – e.g., reactions in high-pressure ion sources, drift tubes, ion cyclotron resonance and plasma swarm experiments – and higher-energy collisions, considered later. Considering these limitations, it is remarkable that this simplistic theory accurately correlates collision cross section and rate coefficients for hundreds of reactions of ions with nonpolar molecules.
Ion–dipole and ion–quadrupole interactions Over time, increasingly sophisticated modifications to the Langevin capture cross-section model have
where µD is the dipole moment of the molecule and θ is the angle between the dipole and the line of centres of the collision. The first such treatment by Hamill and co-workers assumed that the dipole ‘locks in’ on the ion, i.e. θ = 0. This upper limit, locked-dipole approximation is obviously unrealistic for reactions at and above room temperature, for which rotation of the polar molecule tends to average out the dipolar potential. Addressing this limitation, Su and Bowers developed the average dipole orientation (ADO) theory in which the orientation angle of the dipole with respect to the line of centres of collision is averaged as a function of ion–molecule separation distance. This concept leads to a ‘locking constant’, c, which is multiplied by the cos θ term; this constant has been parametrized as a function of µD/α½ for the range of temperatures that are of general interest in lowenergy ion–molecule reactions. Further elaboration of this approach by Su and Bowers modified the treatment of angular momentum to formalize angleaveraged dipole orientation (AADO) theory. These workers also considered the interaction with a point charge of the quadrupole moment of molecules with D∞h symmetry and formulated the average quadrupole orientation (AQO) theory for these types of long-range interaction. Barker and Ridge developed a statistical model for ion–polar molecule collisions based on similar concepts. They considered the average interaction energy for a statistical ensemble of ion–neutral pairs as a function of their separation. The effective potential for such an ensemble is
where kB is the Boltzmann constant, TR is the rotational temperature of the neutral, E is the relative translational energy, b is the impact parameter and (X) is the Langevin function defined as (X) = cotanh(X) − 1/X. The momentum transfer collision frequencies calculated from this model or from the
956 ION COLLISION THEORY
family of ADO theories described above are in reasonable agreement with experimental measurements of rate coefficients. They rationalize both the larger cross sections found for ion–molecule reactions involving highly polar neutrals and the increase in reaction rate coefficients for polar molecules at low temperatures.
where is the reaction probability. When Jmax(j, Ω) is larger than j, the sum over J in this equation gives the relationship
Quantum-mechanical approaches Quantum-mechanical calculations of ion–dipole capture rate constants have been modelled successfully for low-temperature experiments and correlate well with the classical collision theory and capture cross sections just described. Clary has calculated rate constants of several ion–molecule reactions using the adiabatic capture centrifugal sudden approximation (ACCSA). Since the rotational motion of the neutral is strongly hindered by the ion at low velocity, he chose collinear reaction geometry, leading to a basis set that localizes the rotational motion of the neutral about this collinear configuration. The partial wave Hamiltonian for the entrance reaction channel is given by
where is Planck’s constant divided by 2π, µ is the reduced mass, B is the rotational constant, j is the rotational angular momentum operator, J is the total angular momentum operator, V is the ion– molecule potential energy surface, θ is the orientation angle and r is the position vector of the particle measured from the centre-of-mass. As an approximation, J– j2 is replaced by the diagonal value [J(J+1)2+j2−22Ω2], where Ω is the projection of both rotational and total angular momentum along the Z axis (centre-of-mass vector). Thus for a fixed value of r, the Hamiltonian becomes
The reaction cross section is obtained from partial wave expansion and the capture approximation discussed earlier; that is, the reaction takes place if there is enough energy to surmount the centrifugal barrier. It follows that
with the condition that σ(j) = 0 if the energy is below the centrifugal barrier. Rate constants calculated by this formalism are in excellent agreement with experimental room-temperature constants for such systems as positive and negative ion–molecule reactions of HCN with H−, D− and H ions. Troe has used a statistical adiabatic channel model to calculate thermal ion–molecule capture rates using a pure long-range potential. The adiabatic channel potentials V(r) are calculated from perturbation theory, and analytical expressions for channel threshold energies, activated complex partition functions and capture rate constants are obtained. His results are in good agreement with trajectory calculations at temperatures above 10 K for such systems as H + HCN. Additionally, Sakimoto has developed a time-independent quantum-mechanical approach for collinear triatomic systems A+BC in the energy range from thermal to several eV. Using hyperspherical coordinates, the time-independent Schrödinger equation is solved numerically using the discrete variable representation algorithm for collision energies of 1–6 eV. Partial cross sections for reactive scattering and ion dissociation for He + H (plus isotopes) were calculated that generally agree with limited experimental data for this system.
High-energy ion–neutral collisions: Beam scattering Low-energy ion–neutral collisions considered thus far often result in random angular scattering with extensive mixing of orbital and angular momentum. Beam experiments at the low-energy limit demonstrate that persistent complexes are formed but provide no further information on reaction mechanisms. In contrast, conservation of angular momentum in high-energy collisions leads to specific angular scattering, which can provide detailed information on reaction mechanisms. Deducing this information is a complex exercise and we begin with a description of elastic scattering. After these scattering characteristics have been described – both classically and quantum mechanically – we shall outline briefly how
ION COLLISION THEORY 957
reactive scattering is treated in this framework. A collision is better described by CM coordinates, which permit us to describe kinetic energy T as the motion of the centre-of-mass of the system in the laboratory frame and the relative motion of the particles in the CM frame
where M is the sum of the masses, VCM is the velocity of the CM and the reduced mass and relative velocity have been defined previously. Since the motion of the CM is conserved in any collision, we can focus our attention on the relative motion in which the two particles are described as a single particle having mass µ, angular momentum µ(r × v) and kinetic energy µv2. For central potentials considered previously, the angular momentum is constant and the total CM energy at separation r is
where the effective potential
includes the centrifugal potential L2/2µr2 described previously. The centrifugal potential is included in the equation for radial motion to account for angular motion of the system. The observable results of a collision are the velocity of the particle after collision (deduced from its kinetic energy) and its direction of motion. For ordinary collisions between spherical particles, total kinetic energy must be conserved, so the initial and final velocities must be equal at distances large enough that V(r) is negligible. Figure 2 depicts how the deflection angle – the quantity of interest in these elastic and nonreactive collisions – relates to the interaction potential. Here a particle of reduced mass µ approaches parallel to the x axis with kinetic energy µv2 and angular momentum µvb. When the effective potential energy Veff equals the initial kinetic energy, the particle reaches its distance of closest approach, rc, the turning point of the trajectory. The particle then recedes, and at a large distance from the scattering centre no longer senses the
Figure 2 Deflection of particles interacting through a potential V(r ). A single particle of reduced mass µ and velocity v moves toward the stationary scattering centre with impact parameter b and is scattered by a central potential V(r ) through an angle θ. Reproduce with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
interaction potential. The change in direction of motion is the deflection angle θ, and is measured from the negative x axis in Figure 2. We note parenthetically that deflection through +θ and −θ are experimentally indistinguishable and that deflection by ±2πNθ (where N is an integer) is indistinguishable from the deflection θ. Figure 2 illustrates that the deflection is symmetric about the turning point; that is, the deflection of the particle in the outgoing trajectory equals its deflection in the approach trajectory for elastic scattering. Consequently, the polar scattering angle 2φc and the final scattering angle θ is π−2φc (see Figure 2). This leads to the expression for the deflection angle as a function of initial energy and impact parameter:
where rc is the outermost solution of E = L2/ 2µrc2 + V(rc). Note that if b = 0 (head-on collision), then θ = π (unless V = 0). Further, as b increases from zero, θ decreases from π, and θ decreases to zero as b increases without limit. A reasonable central force potential for ion–neutral elastic scattering is the Morse potential given by
where x = α(r − re), with α specifying the curvature of the potential at its minimum. This potential has an attractive well of depth D at the equilibrium
958 ION COLLISION THEORY
To illustrate the relationship between potential and scattering, Figures 3 to 5 show respectively the effective potential, deflection function and scattering intensity for the Morse potential, (Eqn [16]), parameterized to describe scattering of low-energy protons by argon atoms. The potential for zero impact parameter (hence zero angular momentum, L = 0) in Figure 3 has the parameters re = 1.28 Å, D = 4.17 eV and α = 1.85 Å3; the Morse potential with these parameters matches the experimentally determined scattering of protons by argon to better than 1% accuracy over the interval from 0 to 5 Å. The curves in Figure 3 are labelled with the angular momentum, L, in units of . At high L, the centrifugal potential overpowers the attractive well and the scattering is always repulsive. At L = 75.95
there is an inflection point at V = 1.153 eV, and for lower angular momenta collisions the effective potential has two regions of repulsion separated by an attractive well. The outermost root is the classical turning point for the potential and a plot of the turning point r as a function of impact parameter (or, equivalently, of L) leads to a singularity where √c = b for all collision energies. This corresponds to the point where the potential V(r) = 0 [∼1 Å for the potential shown in Figure 3] and is a measure of the ‘size’ of the interacting particles, analogous to but different from a hard-sphere radius for the collision. Figure 4 shows the deflection function calculated from Equation [15] for the effective potential given in Figure 3 for a number of collision energies. For zero impact parameter, the deflection angle is π, just as it would be for hard-sphere collisions, with positive scattering angle corresponding to repulsive collisions. The deflection angle goes through zero and becomes negative as the collision senses the attractive potential at longer distances. As b increases, the deflection angle goes smoothly to zero. For moderate energy collisions, the centrifugal potential dominates scattering at all impact parameters. For very low energies, < 1.153 eV for the potential illustrated, there are three roots of the deflection function. (The reader is reminded that positive and negative deflections are experimentally equivalent.) The ‘piling up’
Figure 3 Plot of effective potentials derived using the Morse potential for several values of angular momentum, L, as marked on each curve. See text for the parameters used for the Morse potential. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
Figure 4 The relationship between deflection function and impact parameter for interactions governed by the Morse potential of Figure 3 at the indicated collision energies. 1 bohr = 0.529 Å. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
separation, r = re. For r < re, the repulsive part of the potential rises steeply to a high value, for r > re the attractive part of the potential is dominant. Because the Morse potential goes to zero exponentially, it does not correctly describe the potential at large r. If necessary, it can be modified by adding a term for long-range polarization forces. It nicely describes the short-range behaviour and the balance between attractive and repulsive forces.
Differential scattering
ION COLLISION THEORY 959
of intensity at a particular scattering angle is called the rainbow angle. Measurement of rainbow scattering is an important experimental parameter for deducing the scattering potential. For specific L values below this critical energy, two of the roots coincide at the maximum of the effective potential. For these trajectories, orbiting occurs and the deflection angle becomes infinite. This phenomenon is readily detected in careful beam scattering experiments; it corresponds to the capture cross section in our description of low-energy collisions. Figure 5 illustrates the connection of crossed-beam measurements of ion scattering with the interaction potential of the ion–neutral collision, again using the Morse potential. The quantity measured experimentally is the differential scattering cross section for a given CM collision energy. The differential cross section, I(θ), at a specified energy is given by
Figure 5 illustrates for several energies the singularities determined by the two terms in the denominator of Equation [18]. The sin θ term gives infinity at CM angles of zero and π. The rainbow angle gives rise to a singularity when dθ /db goes to zero; this occurs when several impact parameters give the same deflection angle. At higher collision energies this occurs at small angles; with decreasing collision energy the rainbow angle increases smoothly with decreasing energy and, for 1.5 eV is greater than 180°. Numerically it is 254° for this potential, leading to an observed scattering angle of 360° −254° = 106°. Two details distinguish this class of trajectories experiment from high energy collisions – namely, the singularity at 180° and the reversal of the ‘bright side’ of the rainbow. The backscattering of particles corresponding to the sin θ singularity at 180° is the experimental demonstration that particles are scattered by a potential sufficiently attractive at that collision energy to hold them together for at least half a revolution.
Reactive encounters
where dΩ is the differential of solid angle. The cross section calculated for the Morse potential shown in
We have explained in some detail the relationship of measured variables in elastic scattering to the interaction potential between the ion and neutral. This framework is the starting point for describing inelastic and reactive scattering. Inelastic scattering may be introduced as an instantaneous conversion of kinetic energy of motion into internal energy of either or both of the collision partners, while reactive collisions involve both energy exchange and reaction. It is plausible that these physical and chemical conversions occur at or near the distance of closest approach in the respective trajectories. This breaks the symmetry of the elastic scattering formalism and the retreat trajectory of products involves both a different energy inventory in the collision partners and a change in the potential governing the trajectory as they separate from the collision centre. This approach has been reasonably successful in rationalizing the differential scattering cross sections of a variety of proton transfer reactions. The time-dependent quantum-mechanical treatment of collisions described in the next section automatically takes into account energy transfer and reaction – at least in principle.
Newton diagrams Figure 5 Plot of scattering intensity (differential cross section) versus centre-of-mass scattering angle for interactions with Morse potential of Figure 3 using classical treatment. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (eds) Gaseous Ion Chemistry and Mass Spectrometry.
As noted previously, describing collision processes in the CM frame is more informative than in the laboratory (LAB) frame. Explicitly the motion of the CM is conserved in collisions and is unavailable to the
960 ION COLLISION THEORY
reactants. The proper framework for displaying collisions is the CM relative velocity diagram, frequently called the Newton diagram. The collision process (for simplicity assumed to occur at right angles) of
with M and M2, having velocities of V1 and V2, is depicted in Figure 6. The collision centre is the origin of the laboratory reference frame. In the CM frame, M and M2 move collinearly towards each other with velocities U1 and U2 and collide at the CM located on their relative velocity vector at the point defined by conservation of momentum such that M1U1 = M2U2. After collision, the two particles recoil from the CM in opposite direction with velocities U1′and U2′. The velocity of the CM is given by
and is constant. The relative kinetic energy of the collision partners, ECM, is available for the reaction and is given by
Similarly, the postcollision relative kinetic energy is given by
The difference ∆T = E – ECM, the translational exoergicity of the collision process, is a direct measure of the energy used in the process. Consequently, experimental measurements and transformation of mass, intensity, energy and angular distributions of product ions as Newton diagrams provides quantitative information on the energetics and dynamics of the collision process. Two limiting cases of ion–molecule reaction dynamics will serve to illustrate the utility of Newton diagrams. If a collision between an ion and a molecule proceeds via an intermediate whose lifetime is larger than several rotational periods (~10–12 s), internal energy redistribution is usually complete and the complex loses its memory of the reactant velocity vectors. The products will separate with equal probability on both sides of a plane passing through the CM and normal to the collision axis, giving rise to the forward–backward symmetry. Such orbiting complexes are usually formed at relative kinetic energies that are of the order of or less than the well depth of the interaction potential governing the collision. As relative kinetic energy increases, the lifetime of the intermediate complex decreases and there is a transition to a direct reaction. The interaction time is too short for the intermediate to rotate and the Newton diagram contour plot becomes asymmetric with respect to the CM. The dependence of reaction probability on impact parameter and the effective potential for the approach and retreat trajectories determine the scattering angle and the form of the Newton diagram.
Semiclassical theory of collisions, the WKB phase shift In three dimensions and for central potentials, the Schrödinger equation separates in spherical coordinates analogously to corresponding classical equations of motion. The incoming particle is described by a wavefunction with a local wavenumber k(r). Repulsive potentials decrease the velocity of the wave packet and the resulting scattered wavefunction has fewer nodes, resulting in a negative phase shift. Conversely, attractive potentials result in an increase in velocity; the wavefunction acquires more nodes and this results in a positive phase shift. The overall phase shift for scattered particles is
Figure 6 Newton diagram showing pre- and postcollision velocity vectors in the laboratory and centre-of-mass reference frames for the collision of M and M2 having velocities of V1 and V2.
ION COLLISION THEORY 961
Integrating this expression from the distance of closest approach to infinity (this gives half the total difference in phase for the entire trajectory; this is the convention for phase shift) gives
This makes the precise connection we seek between WKB scattering theory and the classical model developed previously.
Quantum scattering Moving from a semiclassical towards a full quantum treatment we note that, at large distances, the wavefunction for a scattered particle is a combination of an incoming plane wave and a scattered wave:
For both integrals, the lower limit of integration is the (outermost) zero of the associated integrand. Since both integrals are divergent, δ depends on cancellation between the two. This difficulty is repaired by subtracting dr from both terms:
where the z direction is the direction of travel. The first term describes a plane wave and the second term is an outgoing spherical wave whose amplitude is f(θ). This simple form of the wavefunction is valid for r large enough that no interference between the incoming and outgoing waves is important. This applies experimentally to a collimated beam of finite width. Since the wavefunction in Equation [29] obeys the Schrödinger equation, it can be expanded in any complete set of orthogonal functions, for example Legendre polynomials:
This may be recast into a form incorporating the definition of phase shift from Equation [23]:
where Pl is Legendre polynomial. Multiplying by Pl′(cos θ), and integrating over θ we obtain for large values of r the asymptotic result where the integral term gives the phase difference resulting from the interaction potential and a correction for the difference in turning points of the unscattered and scattered particle includes the effect of the centrifugal potential. Finally, it can be shown that the phase shift method leads to expressions for the collision time (different between V(n) ≠ 0 and V(n) = 0) and for scattering angle.
which has solutions ψ1(r) = a1sin(kr − lπ/2 + δl). Here the phase shift, δl, for the lth partial wave contains all the information about the potential, and the term lπ/2 comprises the centrifugal potential. The expression for scattering amplitude is obtained as
The differential cross skection I(θ) is the probability density for observing a particle at an angle θ,f (θ)2 and the total cross section is obtained by integration
962 ION COLLISION THEORY
additional terms that couple them together. Sophisticated computer programs are now available to solve these complex equations and calculate the detailed dynamics of a collision process.
List of symbols
Figure 7 Plot of scattering intensity versus centre-of-mass scattering angle at 5 eV collision energy using quantum-mechanical treatment. The calculations were performed using 500 phase shifts and the Morse potential of Figure 3. Reproduced with permission of John Wiley from Shirts RB (1986) In Futrell JH (ed) Gaseous Ion Chemistry and Mass Spectrometry.
of I(θ) over all angles as given below.
The phase shifts for calculating scattering as a function of angle are obtained by starting at r = 0 and increasing r until V(r) becomes negligible. Figure 7 illustrates the differential cross section for the Morse potential (Eqn [17]) calculated using Equation [32] and 500 phase shifts that were calculated using Equation [26]. The relative energy is 5 eV. The classical differential cross section from Figure 5 is also shown. The quantum-mechanical form of I(θ) removes the classical singularity. The additional peaks occurring at angles smaller than the rainbow angle are interferences from the three different values of the impact parameter that contribute to scattering into these angles. The quantum result tends to oscillate about the classical mechanical result, which cannot recover the interferences of the quantum result. For noncentral potentials – e.g. molecular scattering – the Schrödinger equation does not separate into partial waves. The solution can still be expanded in partial waves, but then the individual partial-wave radial Schrödinger equations will have
b = impact parameter; B = rotational constant; c = locking constant; D = depth of attractive potential well; E = energy; f() = function; H = Hamiltonian; I(θ) = differential cross section; j = rotational angular momentum operator; J = total angular momentum operator; k = rate coefficient; local wavenumber; kB = Boltzmann constant; = Langevin function; L = classical orbital angular momentum; P = Legendre polynomial reaction probability; q = ionic charge; r = distance; magnitude of position vector; polar coordinate; r = position vector; re = equilibrium distance; T = kinetic energy; TR = rotational temperature; v = relative velocity vector; ν = magnitude of relative velocity vector; V(r) = central potential energy; V, V′, U, U′ = velocity in centre-of-mass frame; VCM = velocity of centre-of-mass; α = atomic/molecular polarizability; curvature of potential energy curve at minimum; δ = phase shift on scattering; θ = angle between dipole and line of centres; final scattering angle; polar coordinate; µ = reduced mass; µD = dipole moment; σ = collision cross section; τ = collision time; ϕc = polar scattering angle; ψ = wavefunction; Ω = projection of rotational and total angular momentum on Z axis; solid angle.
Subscripts c = capture; c = closest approach; CM = centre of mass; e = equilibrium; eff = effective; r = relative; trans = translational See also: Ionization Theory; Quadrupoles, Use of in Mass Spectrometry.
Further reading Chesnavich WJ, Su T and Bowers MT (1979) Ion-dipole collisions: recent theoretical advances. In Ausloos P (ed) Kinetics of Ion Molecule Reactions, pp. 31–53. New York: Plenum. Child MS (1973) Molecular Collision Theory. New York: Academic Press. Fluendy MAD and Lawley KP (1973) Chemical Applications of Molecular Beam Scattering. London: Chapman and Hall. Franklin JL (ed) (1972) Ion Molecule Reactions. New York: Plenum Press. Futrell JH (ed) (1986) Gaseous Ion Chemistry and Mass Spectrometry. New York: Wiley.
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Levine RD and Bernstein RB (1987) Molecular Reaction Dynamics and Chemical Reactivity. Oxford: Oxford University Press. McDaniel EW (1964) Collision Phenomena in Ionized Gases. New York: Wiley. McDaniel EW, Cermak V, Dalgarno A, Ferguson EE and Friedman L (1970) Ion–Molecule Reactions. New York: Wiley-Interscience. Shirts RB (1986) Collision theory and reaction dynamics. In Futrell JH (ed) Gaseous Ion Chemistry and Mass Spectrometry, pp. 25–57.New York: Wiley.
Steinfeld JI, Francisco JS and Hase WL (1989) Chemical Kinetics and Dynamics. Englewood Cliffs, NJ: PrenticeHall. Su T and Bowers MT (1979) Classical ion-molecule collision theory. In Bowers MT (ed) Gas Phase Ion Chemistry, Vol. 1, pp. 83–118. New York: Academic Press. Vestal ML, Wahrhaftig AL and Futrell JH (1976) Application of a modified elastic spectator model to proton transfer reactions in polyatomic system. Journal of Physical Chemistry, 80: 2892–2899.
Ion Dissociation Kinetics, Mass Spectrometry Bernard Leyh, F.N.R.S. and Université de Liège, Belgium
MASS SPECTROMETRY Theory
Copyright © 1999 Academic Press
The ionization process taking place in the source of a mass spectrometer or the activation step of a tandem mass spectrometry experiment leads to ions with a range of internal energies extending usually well above the first dissociation threshold, i.e. the molecular ion has enough energy to dissociate. Dissociation is a dynamic process characterized by its associated lifetime. The whole story of kinetics in general and of ion dissociation kinetics in particular is to measure how fast a dissociation proceeds under given conditions (state selection, well-defined energy or temperature, for example) and to determine the reaction mechanisms. Both aspects will be addressed.
This represents the most simple situation where a single dissociation process occurs for ions possessing a single total energy, E. k(E) is called the unimolecular rate constant. However, in practice, the parent ion can be formed with a broad distribution of internal energies, P(E), and competitive reactions can take place. If the different competitive channels are characterized by rate constants denoted ki(E), then the following equation holds for the rate of production of the jth fragment ion:
Unimolecular rate constant Dissociation kinetics of state-selected reactant ions has been investigated only in a limited number of relatively small systems. However, a large amount of data are now available on ionic reactions investigated with internal energy selection. The dissociation of an energy-selected molecular ion is a unimolecular process, the rate of which, R(E,t) is defined by the following equation, where N0 is the initial number of ions formed:
If further dissociations occur, the rates must be calculated by solving the set of differential equations resulting from the appropriate kinetic scheme. If the dissociating ion is characterized by a welldefined internal energy, i.e. if P(E) reduces to a single value, then each ion of the ionized sample can be considered to be one of the replicas of a microcanonical ensemble. For this reason, k(E) is often referred to as the ‘microcanonical rate constant’. If, however, P(E)