Radiative Cooling of Gas-Phase Ions A Tutorial Robert C. Dunbar Case Western Reserve University Innsbruck Cluster Meeting March 18, 2003

Outline Plan: I. Overview of molecular cooling II. Diatomics -- A single mode III. Polyatomics IV. Tools for measurement, and examples Advertisement I wrote a review covering many of these basic principles: R. C. Dunbar, Mass Spectrom. Rev. 1992, 11, Cooling through thresholds: Monitor reactions 2.Cooling through a threshold: Two-pulse experiment 3.TRPD thermometry 4.k Emit from radiative association kinetics 5.Deceptive cooling curves from various techniques

Introduction Consider an ion, (or a population of ions,) having internal energy higher than that of the surroundings. It may lose energy, and thus cool its internal degrees of freedom, by two means: 1.Collisions with neutrals 2.Radiative energy loss We will focus on the theory and measurement of the second of these two possibilities.

I. Overview of cooling: Hot ion preparation Initial preparation of hot ions can involve initial electronic excitation. Most ions rapidly convert this electronic energy to vibrational internal energy by internal conversion. Electronic Excitation Vibrationally Excited Ground State

I. Overview -- Cooling Radiative Cooling and Equilibration with Walls  Spontaneous Emission Photons/sec = A 1-0 = 1.25 x v 2 I 1-0 I 1-0 = Integrated Infrared Intensity  Induced Emission  Absorption Equal rate constants Photons/sec = B 1-0  = B 0-1  A 1-0 = B 1-0 Relations for an individual vibrational mode: Exchange of radiation with walls is governed by three processes, with their Einstein coefficients.

Overview – Background Radiation Black-body radiation field

Cooling rate constants Many “cooling rate constants” are reported in literature. Their quantitative meaning is often obscure! We should be careful about definitions. Two clearcut quantities can be defined: k Emit Rate of IR photon emission k Cool Rate of relative energy loss = In the special case of exponential cooling k Cool is constant. Then If the cooling is purely radiative call it k RCool If the cooling is purely collisional, call it k CCool

Cooling example: N 2 H + a ModeFreq. (cm -1 )I 1-0 Calc. (km/mol) k Emit (1-0) Calc. (s -1 ) k Emit Expt. (s -1 ) a P. Botschwina, Chem. Phys. Lett. 1984, 107, 535; W. P. Kraemer, A. Kormornicki, D. A. Dixon, Chem. Phys. 1986, 105, 87.

II. Diatomics Radiative cascade down vibrational ladder Harmonic approximation: A v  v-1 = vA 1  0

Exponential cooling of diatomics For a harmonic diatomic oscillator, the rate constants work out to give exactly exponential cooling (k RCool = constant)

Uniform cooling of diatomics Thermal energy Arbitrary initial distribution

Uniform cooling of diatomics The preceding picture does not take thermal spreading into account. Actually the final population distribution is a Boltzmann distribution. Arbitrary initial distribution

III. Polyatomics  A polyatomic molecule looks like a collection of 3N-6 (or 3N-5) vibrational normal modes, each having a frequency i and each having a value of its vibrational quantum number v i  In the harmonic approximation, the normal modes are independent of each other. Absorption and emission of radiation is treated individually for each mode.  In the weakly coupled harmonic picture, they are still essentially independent, but they are coupled together sufficiently strongly so that excitation energy flows from one normal mode to another somewhat rapidly (IVR).

Polyatomics: Cooling is much slower Many modes are excited no higher than to v=1 Remember that radiation  v Much of the energy is stored in excitation of dark “reservoir” modes Much of the energy is stored in weak low-frequency modes Remember that radiation  2 Three factors combine to make the cooling of a polyatomic much slower than for a diatomic, even if the oscillator strengths of some normal mode vibrations are comparable to the oscillator strength of the diatomic.

Energy storage in polyatomics 500 cm % of Energy44% 32% 24% A simple model molecule: 6 vibrational modes 2 x 500 cm-1 2 x x 1500 Total E int = 4000 cm -1 T m = 1800 K

Polyatomics:Vibrational distributions Two approaches: 1. Direct statistical count The task is to find the probability of finding a molecule in quantum number v of mode n given that the molecule contains E int of energy. P n,v = Number of states of molecule excluding mode n and excluding energy contained in mode n Number of states of entire molecule with full energy 2. Microcanonical temperature Define effective (microcanonical) internal temperature T m Boltzmann: P n,v =

Polyatomic emission calculations Frequencies and IR intensities can be calculated ab initio, so the cooling properties can be predicted theoretically.

Polyatomic experimental examples k RCool Values IonEnergy range (eV) k RCool (s -1 ) C 6 H 5 Cl C 6 H 5 Cl C6H6+C6H C6F6+C6F Fe(C 5 H 5 )

Polyatomic experimental examples IonEnergy range (eV) k Emit (s -1 ) C3H5–C3H5– (CH 3 CN) 2 H SF 6 – k Emit Values

IV. Techniques for measurement A cooling experiment will usually involve Initial preparation of hot ions A variable delay time t for cooling An experimental probe of the amount of internal energy remaining in the ion after time t (thermometry).

Hot ion preparation Some approaches to making hot ions

Probe methods Thermometry by time-resolved photodissociation (TRPD) TRPD branching ratio Cooling through TRPD threshold Cooling through ion-molecule reaction threshold(s) (Monitor reactions) Ion-molecule reaction branching ratio Radiative association kinetics Calculation from IR absorption intensities (experimental or theoretical)

1. Cooling through thresholds: Monitor reactions Monitor reactions: Reaction is endothermic unless reactant has at least v quanta of vibrational excitation. This provides a thermometer function to observe the radiative cooling of reactant ions. Transitionk Emit (s-1) 101 12 23 343 400 NO + Cooling Beggs, Kuo, Wyttenbach, Kemper, Bowers, Int. J. Mass Spectrom. Ion Proc. 1990, 100, 397 Feinstein, Heninger, Marx, Mauclaire, Yang, Chem. Phys. Lett., 1990, 172, 89.

2. Cooling through a TRPD threshold Two-pulse pump-probe technique 1. Excitation laser pulse at t = 0 Raises ions above the one-photon threshold 2. Delay time t A fraction of the ions cool below the one-photon threshold 3. Probe laser pulse after time t Dissociates ions still lying above one-photon threshold Thermal ions Dissociation threshold h 1 h 2 Internal Energy One-photon threshold

Two-pulse example: Cr(CO) 5 - B. T. Cooper, S. W. Buckner, JASMS 1999, 10, 950. Thermal ions 30 kJ/mol Dissociation threshold 142 kJ/mol h kJ/mol h 2 Internal Energy One-photon threshold 67 kJ/mol

Two-pulse example: Cr(CO) 5 - Modeling of the cascade relaxation kinetics gives a k RCool value of 15 s -1 at an internal energy level of ~ 110 kJ/mol. K emit is modeled to be 115 s -1.

3. TRPD thermometry: TTBB h 2 CH Calibration Reaction J. D. Faulk, R. C. Dunbar, J. Phys. Chem., 1991, 95, 6932.

TTBB – Cooling curve k RCool = 1.1 s -1 Y.-P. Ho, R. C. Dunbar, J. Phys. Chem., 1993, 97,

TTBB – Pressure effect As pressure is raised, collisional cooling competes with radiative cooling, and can also be measured. k Coll = 4 x cm 3 molec -1 s -1 Y.-P. Ho, R. C. Dunbar, J. Phys. Chem., 1993, 97,

4. k Emit from radiative association kinetics At low pressure, the association of an ion with a neutral molecule proceeds with emission of an infrared photon according to the following kinetic scheme: In the McMahon analysis a plot of association rate constant against pressure gives a slope and intercept which yield k r, which is the same as k Emit. P. Kofel, T. B. McMahon, J. Phys. Chem. 1988, 92, 6174

5. Deceptive cooling curves This cooling curve for Cr(CO) 5 - was made by plotting a reaction branching ratio as a function of time. It looks like a simple exponential with k RCool = 3.3 s -1. But other measurements and modeling show that the true cooling curve is severely non- exponential, and k RCool varies from 15 s -1 at t=0 to ~3 s -1 at t=0.4. This branching ratio thermometer is severely non-linear, and was not calibrated. B. T. Cooper, S. W. Buckner, JASMS 1999, 10, 950.