Used for MS Short Course at Tsinghua by R. Graham Cooks, Hao Chen, Zheng Ouyang, Andy Tao, Yu Xia and Lingjun Li

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1 k(e) and RRKM Direct Dissociation and Predissociation: Predissociation: Electronic, vibrational, or rotations delayed dissociation (i.e metastable ions) Predissociation described by transistion state or phase space statistical theory Unimolecular rate constant k(ε) is a function of internal energy, ε, by RRKM or QET Vibrational Pre-Dissociation: (main type of unimolecular fragmentation) Rate depends on internal energy of activated complex Transition state theory (in contrast to phase space theories) postulates equilibrium between activated molecule and activated complex hence quasi-equilibrium theory Also, postulates internal energy is completely randomized over all internal states (compare strong and weak forms) so fragmentation occurs from high vibrational levels of the lowest electronic state All microscopic rate constants k ij from initial statei, to final state j, are equal, so for (a) final states, (b) initial states. 1

2 k ij is just a rate barrier traversal so: Now all transition states of internal energy E ++ Σ are positive since they have one degree of freedom which is not in the activated molecule, hence: K bt Rate Barrier Crossing In quantum collision theory, probability of state-to-state transistion is: Uncertainty Principle = Where T is the energy width of the initial state (cf laser time widths) Now the probability of transition (k bt ) = rate constant = T/h 2

3 Wahrhaftig Diagram: Unimolecular kinetics and concsequences for dissociation Metastable Ions: example of delayed dissociation --- time range corresponding to k(e) = 10 6 s -1 Fragmentation of metastable ion (m 1 + ): m 1 + m m 3 From energy and momentum conservation: 3

4 m*: metastable peak position m 2 : mass of fragment ion m 1 : mass of fragmenting ion Aston bands (CID) Magnetic Field Scan m* in each graph highlights diffuse metastable peaks of different shapes 4

5 Ion Kinetic Energy Spectra (IKES): Display of all metastable peaks for a population of ions gnereate from a particular sample. The setup: Displays mass change (m 2 /m 1 ratio) and charge changing collisions 5

6 Metastable Ions: Kinetic Energy Release: Mass-select m 2 + and scan kinetic energy (accelerating voltage): Peak shape due to the transition C 7 H C 4 H 4 high voltage (HV) scan technique. + C 3 H 3 + in toluene plotted by the m 1 + m m T: Kinetic energy release e: charge on ion V: accelerating voltage ΔE: difference in kinetic energy between fragments E: kinetic energy of fragmenting complex ΔKE lab : C 3 H 3 + [CH=CH + -CH=CH] T = 4.8 ev 3 Å intercharge distance Compare C 6 H C 5 H 3 + CH H 3 C.. C=C=C=C- CH 3 T = 2.8 ev 5.14 Å intercharge distance Zeit. Naturforsch. 20a (1965) 180 J H Beynon, R A Saunders and A E Williams Trans. Faraday Soc., 1966, 62, W. Higgins and K. R. Jennings 6

7 Kinetic Energy Release Derivation Consider an ion m 1 +, mass m 1, velocity v 1, which fragments to give m 2 + and m 3. If no energy is converted from kinetic to internal modes, then from energy conservation: (1) (2) From momentum conservation: Thus kinetic energy of the fragment ion is expressed as: This forms the basis of ion kinetic energy spectrometry Now consider the general case in which internal energy is converted into translational energy of separation (kinetic energy release). The fragment velocities will not equal each other or v 1. From conservation of energy: From conservation of momentum: In the center of mass system analogous equations can be written: (3) (4) 7

8 From (4) (in CM opposing velocity vectors): (5) Substituting for u 3 in (3) gives (6) If m 2 + is observed in the lab frame, it has a velocity given by the vector sum: When u 2 and v 2 are collinear, (forward + backward energy release): and respectively Thus, the range of labe velocities due to kinetic energy release is 2u 2. The corresponding lab kinetic energies vary from: and hence the range of K(E) s of m 2 + is: to Substituting from (6), the range lab KE is With multiplication factor! i.e. the cross velocity term. (7) Now this is due to kinetic energy release T. So defining an amplification factor A as the ratio of lab energy range to T i.e.: But: 8

9 (8) i.e. maximizes A. A is inverselty related to T, and directly related to (ev 1/2 ) Determination of Kinetic Energy Release: The relationship between u 2, hence v 2 and T mean that T can be determined by measuring v 2 or something related to it, viz kinetic energy. Still dealing only with axial dissociation from (7) But (fractional peak width, equivalent to any other measurement of fractional spread) But 9

10 Collision Processes Types: Elastic, inelastic, reactive scatterning Inelastic scattering Ar + on Ar to give Ar +2 Expt: measure energy loss of fast ion Disproportionation rule means for small angle, high energy this loss ΔE m1+ is equal to endothermicity Q Nominal zero angle Non-zero angle 10

11 Maximum energy available for Internal Excitation Q max = KE cm = KE rel = m target /[m target + m projectile ]KE lab This energy is seldom made available, usually fraction is much smaller although cases in which the target is captured by the projectile approach it. Relationship between Reactive and Inelastic Collisions Reactive collisions typically involve colllision complexes; these complexes can dissociate leading to an excited projectile (collisional activation) or dissociate to give (excited) products (ion/molecule reaction). At low collision energies where collision complexes are more common their co-occurrence is common and peaks in the MS/MS spectrum due to ion/molecule reactions can be seen e.g. for metal ion reactions in ion traps.. Scattering Angle and Interaction Potential Small impact parameter (b) samples repulsive wall of potential energy surface rather than the attractive ion/induced dipole portion. This is likely to lead to larger excitations as b gets smaller In addition, 1/b prop nl to reducted scattering angle KE cm θ cm For the common fast projectile, stationary target case, 1/b is prop nl to KE lab θ lab It will also be seen that in at least some cases Δε changes quadratically with 1/b i.e. Δε is prop nl to (1/b) 2 so Δε is prop nl to (KE lab θ lab ) 2 or E 2 θ 2 Dependence of Cross Section on Thermochemistry Ignoring other factors, transitions with larger endo/exothermicities (absolute values of Q change in internal energy and also equal to the total change in kinetic energy for the system) will be less likely than resonant (Q = 0) or near-resonant (Q small) transitions. This is known as the Hasted Resonance Criterion Note: this reln. was developed from observations on atomic systems for charge transfer The explicit relationship is σ = c e - Q k where c, k are constants One consequence is that thermoneutral charge transfer (including accidental resonance) has the highest cross section Dependence of Energy Deposition P(ε) on Thermochemistry Highly endothermic processes will likely not occur with even more energy deposition than the large amounts needed..so P(ε) for these processes tend to fall rapidly with ε Highly exothermic processes will tend to have flat P(ε) distributions 11

12 Dependence of Energy Deposition on Collision Energy (hence cross section on Collision Energy) Energy deposition of a certain amount (Δε) in the ion will tend to be maximized when the interaction time (often expressed as a distance and a velocity) is equal to the characteristic period of the mode being excited. This is known as the Massey adiabatic criterion. When the two times are very different the transition is unlikely. When equal the probability is maximized. The transition time is given by a characteristic interaction distance a divided by the velocity v and the characteristic period of the motion being excited is given by the energy change divided by Planck s constant h Hence a/ v = h/ (Q) units seconds LHS: E = hν RHS: distance/velocity v max (for a particular value of Q) = a (Q)/ h from expts on atomic charge exchange a taken to have a value of ca 7A Typically values: Q max = 7 m -1/2 ev (at 8keV) Consequences: (i) small excitations are favored, esp. in large ions (ii) ions already excited (on formation, thermally, etc) will be most likely to dissociate (iii) structural conclusions best based on higher energy excitations e.g. dissociative charge stripping ( ε = 10eV) 12

13 Operation of Hasted (adiabatic) and Massey (resonance) criteria 13

14 Mechanisms of Collisional Activation 1. Complex formation Most efficient T-V conversion process; E cm is entirely converted into internal energy of a sticky complex (multiple rotations before going on to products); when it dissociates a rel large fraction of E cm often is used to form products Complex formation is favored by low collision energies capture collision cross section. Samples attractive part of interaction potential (large cross section but only at low KE, so not very effective, even though the fraction of Ecm transferred is high. Long complex life-times mean that forward/backward isotropic CM scattering is observed. 2. Impulsive collisions Common mechanism, perhaps most common for large polyatomics. Part of the projectile ion undergoes an elastic collision with the target gas; the remainder is a spectator (origin molecular beam expt was spectator stripping ). After this rapid collision, a lot of energy must be redistributed (up to the CM energy of part of the projectile and the target) and some of this recoil energy goes into vibration.then, as usual, there is delayed fragmentation. Repulsive part of potential is sampled (small cross section) so large angle scattering; the larger the angle, the smaller the impact parameter and the larger the energy transfer. At large energy (kev), forward scattering is favored. Large and small collision energies ok and a fraction of a E cm is converted into internal energy. Only a single collision occurs (multiple little collisions in the collision complex) but large multiple ev energy transfer for kev projectiles. 3. Polarization (Russek) mechanism Direct excitation of vibration by rapidly changing polarization forces. Long range so no momentum transfer. Energy transfer just a few quanta. 4. Direct Electronic excitation Vertical electronic excitation (prototype case is H 2 + /He) No scattering. Large excitations. Forward scattering. No momentum transfer. Must be fast otherwise system adjusts adiabatically (hence the 10keV prototype case of H 2 + /He).Likely unimportant for polyatomics since Massey criterion requires much higher kev for heavy ions to get right life time. 5. Curve crossing (between electronic states) Now known at both low and high energies. Not large cross section but important process. Angular scattering too. 6. Oblique (curve crossing and impulsive) Likely very common. 14

15 1 (EΘ reduced scattering angle) Small angles, stationary target Θ in radians Ditto Q is inelasticity b: impact parameter Stationary target Elastic collisions Small Angle D / / Cross section (σ), Single collision limit, 80% adiabatic criterion, Δε increases with ν Hasted resonance criterion, P(Δε) falls with Δε 15

16 Center of Mass Coordinate System: Collision Processes Simpler than laboratory system because momentum is zero by definition (and of course conserved in this or any system). Consider scattering (elastic and inelastic) in a plane between M 1 and M 2, of mass m 1 and m 2. Collision occurs at the origin in the lab frame. Prior to collision, the lab coordinates of m 1 and m 2, are vectors R 1 and R 2, ie. M 2 ~ R 2 M 1 ~ R 1 Lab origin Now the center of mass at this (unit) time prior to collision is given by and it moves at constant velocity along the line Rcm. (Rcm is its position in the lab system) CM origin M 2 (moves) (2 ) T 2 R 2 (1) Rcm ~ M 1 R 1 In the cm system, the coordinates of M 1 and M 2 are by definition of the center-of-mass, vectors: = m 2/ /(m 1 +m 2 ) (R 1 -R 2 ) (1) r 1 = - m 1 /( m 1 +m 2 ) (R 1 -R 2 ) (2) r 2 Where the term in brackets is the vector M 2 to M 1 ie. M 2 ~ M 1 16

17 ~ and the mass ratio comes from the definition of center of mass, viz /r 1 m 1 /=/r 2 m 2 / If velocities in the lab are v 1 and v 2 and in cm u 1 and u 2, time differentiation of (1) and (2) gives U 1 = m 2 (v 1 -v 2 ) (3) m 1 + m 2 U 2 = - m 2 (v 1 -v 2 ) (4) m 1 + m 2 But sum cm momenta is zero by definition m 1 u 1 + m 2 u 2 =0 (m 1 m 2 ) (v 1 -v 2 ) = - (m 1 m 2 ) (v 1 -v 2 ) (m 1 +m 2 ) (m 1 +m 2 ) ie. Now the sum of the center of mass kinetic energies of M1 and M2 ( the energy available for conversion from translation to internal) is FIGURE ½ m 1 (m 2 ) 2 g 2 + ½ m 2 (m 1 ) 2 g 2 (m 1 +m 2 ) 2 (m 1 +m 2 ) 2 = ½ m 1 m 2 g 2 (m 2 + m 1 ) m 1 +m 2 m 1 +m 2 m 1 +m 2 = ½ ug 2 Where g=v 1 -v 2 =u 1 -u 2 u is reduced mass g is relative velocity When m 2 is stationary in the lab g = v 1 and KEcm = ½ u 1 v = ½ m 1 m 2 v 1 =m 2 KE lab m 1 +m 2 m 1 +m 2 17

18 KEcm is the relative KE, total energy available for conversion to internal energy. Elastic Collisions Translational energy is conserved ½ ug 2 = ½ ug 2 ie. g=g or /v 1 -v 2 /=/v 1 -v 2 / ie. relative velocity is conserved and from (3) + (4) cm velocities are conserved (u 1 =u 1, u 2 =u 2 ) Products must lie in the elastic circles shown in the Newton diagram. Notethat both species must be scattered through the same cm angle for cm velocity to be conserved. In the stationary target case (high energy collision) vcm is small compared to v1, and small lab (and cm) scattering is most likely (FORWARD SCATTERING - BUT THIS IS A DYNAMICAL NOT A KINEMATIC CONSIDERATION) 18

19 Inelastic Collision Fast projectable, small angle scattering is particularly interesting because the inelasticity Q is given by an observable, E, the energy change of the fast species in the lab system. E = Q Inelastic collisions Fast projectable case From conservation of energy: 19

20 In fact, there can be defined as Disproportional factor, D, the ration in which the kinetic energies of projectable and target change upon collision (for case of negligible lab scattering) (see collision appendix) c.f. Ar/Ar+ collisions also organics Example: Typical charge changing collision What error is IE(M++) if measured as threshold (min KE loss) in a collision with O 2 at 7keV, 0 0 /a and M=W(CO)6 (m/z352)? Q=IE II -IE I +15eV 20

21 Newton Diagram Note the inelasticity in the backward scattering process. Forward means forward in CM system! 21

22 Eg. H 2 +* Ar H + +H +Ar Three mechanisms : 1. Almost dissociating ions undergo large impact parameter (gentle) collisions 2. Stable (low E) ions undergo small impact parameter collisions ( head-on ) 3. Direct electronic excitation Courtesy of Jean Futrell 22

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Used for MS Short Course at Tsinghua by R. Graham Cooks, Hao Chen, Zheng Ouyang, Andy Tao, Yu Xia and Lingjun Li

Used for MS Short Course at Tsinghua by R. Graham Cooks, Hao Chen, Zheng Ouyang, Andy Tao, Yu Xia and Lingjun Li Lecture 4 Ion Thermochemistry Reminmder of terms and quantities Thermochemical parameters AE Appearance Energy EA Electron Affinity IE Ionization Energy PA Proton Affinity (enthalpic) GB Gas Phase Basicity