Radiative Cooling of Gas-Phase Ions

A Tutorial

Robert C. Dunbar

Case Western Reserve University

Innsbruck Cluster MeetingMarch 18, 2003

Outline

Plan:

I. Overview of molecular cooling

II. Diatomics -- A single mode

III. Polyatomics

IV. Tools for measurement, and examples

AdvertisementI wrote a review covering many of these basic principles:

R. C. Dunbar, Mass Spectrom. Rev. 1992, 11, 309

1. Cooling through thresholds: Monitor reactions2. Cooling through a threshold: Two-pulse experiment3. TRPD thermometry

4. kEmit from radiative association kinetics5. 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.

ElectronicExcitation

Vibrationally ExcitedGround State

I. Overview -- Cooling

Radiative Cooling and Equilibration with Walls

Spontaneous Emission

Photons/sec = A1-0 = 1.25 x 10-7 v2 I1-0

I1-0 = Integrated Infrared Intensity

Induced Emission

Absorption

Equal rate constants

Photons/sec = B1-0 = B0-1

A1-0 = 3

3c

h8

~B1-0

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

1ec

h8kTh

3

3

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:

kEmit Rate of IR photon emission

kCool Rate of relative energy loss = dt

Ed

dt

dE

E

1 intint

int

ln

In the special case of exponential cooling kCool is constant. Then

tko

CooleEE int

If the cooling is purely radiative call it kRCool

If the cooling is purely collisional, call it kCCool

Cooling example: N2H+ a

Mode Freq. (cm-1) I1-0 Calc.

(km/mol)

kEmit(1-0)

Calc. (s-1)

kEmit

Expt. (s-1)

1 3600 600 1000 670

2 800 115 9

3 2600 7 6

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:

Avv-1 = vA10

Exponential cooling of diatomics

For a harmonic diatomic oscillator, the rate constants work out to give exactly exponential cooling (kRCool = 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 vi

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=1Remember 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 modesRemember 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-1

10001500

% of Energy 44% 32% 24%

A simple model molecule:

6 vibrational modes2 x 500 cm-12 x 1000 2 x 1500

Total Eint = 4000 cm-1

Tm = 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 Eint of energy.

Pn,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 temperatureDefine effective (microcanonical) internal temperature Tm

Boltzmann: Pn,v = mkTvh

ne

q

1

Polyatomic emission calculations

vn

01vnvn

1vvvnEmit vAPAPk,

,,

,

n01vn

vnRCool hvAPk ,

,

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

Polyatomic experimental examples

kRCool Values

Ion Energy range (eV) kRCool (s-1)

C6H5Cl+ 2.5-1.2 4.0

C6H5Cl+ 0.5-0.2 0.4

C6H6+ 2.7-1.5 15

C6F6+ 2.7-2.3 25

Fe(C5H5)+ 0.7-0.24 0.28

Polyatomic experimental examples

Ion Energy range (eV) kEmit (s-1)

C3H5– 0.3 50

(CH3CN)2H+ 1.3 40

SF6– 0.5 2500

kEmit 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

Monoenergetic Known Energy

Complexes

Electron impact ionization

No No No

PEPICO Yes Yes No Photoionization No Maybe No Photoexcitation Yes Yes Yes Charge transfer ~ Yes Yes No Exothermic ion-

molecule reaction Maybe Maybe Maybe

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.

Transition kEmit (s-1)

10 110

21 204

32 330

43 400

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

h1

h2

h2

Inte

rnal

Ene

rgy

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

h1

112 kJ/mol

h2

h2

Inte

rnal

Ene

rgy

One-photon threshold

67 kJ/mol

Two-pulse example: Cr(CO)5-

Modeling of the cascade relaxation kinetics gives a kRCool value of 15 s-1 at an internal energy level of ~ 110 kJ/mol.

Kemit is modeled to be 115 s-1.

3. TRPD thermometry: TTBB

h

2

CH

++

+ 3

Calibration Reaction

J. D. Faulk, R. C. Dunbar, J. Phys. Chem., 1991, 95, 6932.

TTBB – Cooling curve

kRCool = 1.1 s-1

Y.-P. Ho, R. C. Dunbar, J. Phys. Chem., 1993, 97, 11474.

TTBB – Pressure effect

As pressure is raised, collisional cooling competes with radiative cooling, and can also be measured.

kColl = 4 x 10-10 cm3 molec-1 s-1

Y.-P. Ho, R. C. Dunbar, J. Phys. Chem., 1993, 97, 11474.

4. kEmit 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 kr, which is the same as kEmit.

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 kRCool = 3.3 s-1. But other measurements and modeling show that the true cooling curve is severely non-exponential, and kRCool 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.