Theory of Molecular Collisions: Volume 7 (Theoretical and Computational Chemistry Series) - Hardcover

Gabriel G Balint-Kurti

Hardcover

ISBN 10: 1849738300 ISBN 13: 9781849738309

Publisher: Royal Society of Chemistry, 2015

Synopsis

Written for postgraduates and newcomers to the field, this book presents aspects of scattering processes and the experimental methods used to determine them. The quantum theory of elastic scattering and reactive scattetrng are also described.

About this title

About the Authors

Gabriel G. Balint-Kurti is Professor of Theoretical Chemistry at the University of Bristol.

Alexander P. Palov is Senior Researcher at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University (LMSU). He has taught the course "Theory of Molecular Collisions" at Department of Physics at LMSU since 2012.

From the Back Cover

Almost 100 years have passed since Trautz and Lewis put forward their collision theory of molecular processes. Today, knowledge of molecular collisions forms a key part of predicting and understanding chemical reactions.

This book begins by setting out the classical and quantum theories of atom-atom collisions. Experimentally observable aspects of the scattering processes; their relationship to reaction rate constants and the experimental methods used to determine them are described. The quantum mechanical theory of reactive scattering is presented and related to experimental observables. The role of lasers in the measurement and analysis of reactive molecular collisions is also discussed.

Written with postgraduates and newcomers to the field in mind, mathematics is kept to a minimum, and readers are guided to appendices and further reading to gain a deeper understanding of the mathematics involved.

From the Inside Flap

Almost 100 years have passed since Trautz and Lewis put forward their collision theory of molecular processes. Today, knowledge of molecular collisions forms a key part of predicting and understanding chemical reactions.

This book begins by setting out the classical and quantum theories of atom-atom collisions. Experimentally observable aspects of the scattering processes; their relationship to reaction rate constants and the experimental methods used to determine them are described. The quantum mechanical theory of reactive scattering is presented and related to experimental observables. The role of lasers in the measurement and analysis of reactive molecular collisions is also discussed.

Written with postgraduates and newcomers to the field in mind, mathematics is kept to a minimum, and readers are guided to appendices and further reading to gain a deeper understanding of the mathematics involved.

Excerpt. © Reprinted by permission. All rights reserved.

Theory of Molecular Collisions

By Gabriel G. Balint-Kurti, Alexander P. Palov

The Royal Society of Chemistry

Copyright © 2015 Gabriel G. Balint-Kurti and Alexander P. Palov
All rights reserved.

ISBN: 978-1-84973-830-9

Contents

Chapter 1 Scattering Experiments and Classical Theory of Atom–Atom Scattering, 1, 
Chapter 2 Quantum Theory of Atom–Atom Elastic Scattering, 19, 
Chapter 3 Inelastic Scattering: Basic Theory, 46, 
Chapter 4 Inelastic Scattering: Exact and Approximate Solutions, 64, 
Chapter 5 Rate Constants, Cross Sections and Reactive Scattering, 86, 
Chapter 6 Time-Independent Quantum Theory of Reactive Scattering, 98, 
Chapter 7 Wavepackets and Time-Dependent Quantum Theory of Reactive Scattering, 115, 
Chapter 8 The Real Wavepacket Method and Time-Independent Wavepackets, 129, 
Chapter 9 Lasers and the Photoloc Method, 141, 
Chapter 10 Polarization, Alignment and Vector Correlation, 153, 
Chapter 11 Collision of Larger Molecules, 165, 
Appendix A Energy Normalization of Plane Wave, 179, 
Appendix B Evaluation of the Phase Shift and the Variable Phase Approach, 183, 
Appendix C Jacobi Coordinates, 191, 
Appendix D Body-Fixed Formulation of Inelastic Scattering Theory, 197, 
Appendix E Integral Equations and Green's Function, 208, 
Appendix F Semiclassical or JWKB Approximation, 213, 
Appendix G Hyperspherical Coordinates and the Schrödinger Equation, 219, 
Appendix H Formalism for Time-Dependent Quantum Dynamics, 231, 
Appendix I Technical Aspects of Time-Dependent Quantum Dynamics, 247, 
Appendix J Technical Aspects of the Real Wavepacket Method, 257, 
List of Symbols, 259, 
Glossary, 266, 
Subject Index, 272, 

CHAPTER 1

Scattering Experiments and Classical Theory of Atom–Atom Scattering

1.1 Crossed Atomic and Molecular Beams

We wish to investigate how atoms and molecules interact with each other in the most fundamental manner. What happens when they closely approach each other or collide? As molecules are far too small to see and the time-scale of their collisions is very short, we resort to making them collide with each other under strictly controlled conditions and examining the results in as much detail as we can. Figure 1.1 shows a schematic of a crossed atomic beam apparatus.

To the left hand side of the diagram there is a sophisticated analyzer which detects the products of the collision. The "quadrapole mass filter" selects products of a defined mass and the chopper permits the determination as to when the products arrive at the detector. From this it will be possible to determine the speed and kinetic energy of the products. The detector assembly remains fixed as it is the most complex component of the apparatus. To the lower right hand side of the diagram are the two atomic or molecular beam sources. These are at some fixed angle, normally 90°, relative to each other and can be rotated with respect to the detector assembly. This permits the measurement of the angular variation of the reaction probability (i.e. the differential cross section).

The heart of the apparatus is the collision region where the two beams cross and the collisions take place. This region is maintained under the greatest attainable vacuum so as to exclude any unwanted molecules.

1.2 Classical Theory of Atom–Atom Elastic Scattering

1.2.1 Hard Sphere Collisions

Let us now consider what happens when two atoms collide. Figure 1.2 shows two atoms traveling towards each other. This is what a collision would look like in the center-of-mass reference frame. We will discuss the difference between the "laboratory frame" and the "center-of-mass" frame a little later. In the center-of-mass frame the collision partners travel directly towards each other. The distance "b" in the figure is called the impact parameter and is the hypothetical distance of what would be the closest approach of the centers of the two atoms if there were no interaction between them. If the atoms were hard spheres then they would either collide or miss each other entirely. They would collide if the impact parameter was smaller than the sum of the radii of the two spheres.

Figure 1.3 shows a collision in which the two atoms just graze each other. If the impact parameter were any larger, then the atoms would miss each other entirely and there would be no collision. The maximum value of the impact parameter for a collision to occur is the sum of the radii of the two atoms, b = r1 + r2. If the projected path of atom 1 lies anywhere within the area of the circle shown in Figure 1.3 then a collision will take place. The area of this circle is called the collision cross section, σ = πR2, where R = r1 + r2.

Let us now consider hard sphere collisions in greater detail. What is the angle of scattering of the atoms after the collision? Figure 1.4 illustrates what happens when two "hard sphere" atoms collide. When their surfaces touch they bounce off each other like two billiard balls. The angle that the direction of impact makes with the normal to the surface (α) is the same as that made by the recoil velocity direction to this normal. This is termed specular reflection. The "scattering angle" is shown as θ in Figure 1.4. From the figure we see that

b = R sin α, (1.1)

where R = r1 + r2, and also

θ + 2α = π, (1.2)

and therefore

b = R sin (π - θ/2) = R cos θ/2. (1.3)

All the atoms colliding with an impact parameter b to db are scattered into scattering angles θ to θ + dθ. The cross section for scattering into polar angles θ to θ + dθ is therefore

2πb|db| = 2πR (cos θ/2) | R (sin θ/2) dθ/2| = 1/2πR2 sin θ dθ. (1.4)

The absolute value signs have been introduced as dθ/db is negative and the cross sectional area is a positive quantity.

The "differential cross section" is defined as the flux of atoms scattered into a given solid angle (dΩ = sin θ dθ dΦ) divided by the incident flux of atoms. In the present case we have not considered the azimuthal angle, Φ. We have in effect integrated over this angle. The solid angle, integrated over all Φ, is 2π sin θ dθ. The differential cross section for scattering into a solid angle dΩ is therefore

dσ/dΩ = R2/4. (1.5)

The integral of the differential cross section over all angles must be equal to the total or integral cross section

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

1.2.2 Scattering Under the Influence of a Potential

Figure 1.2 shows a schematic of two atoms colliding. The position of each atom is described by a vector with three spatial components r1 and r2. Thus the system overall requires six position coordinates to fully describe it. As the forces determining the scattering of the atoms depend on the magnitude of the distance between the two atoms, r = |r2 - r1|, it seems reasonable to change coordinates to r, the position of atom 2 relative to atom 1, and R, the position of the center-of-mass

r = r2 - r1, R = m1r2 + m2r1/m1 + m2 (1.7)

Classical mechanics is based on Newton's laws of motion and these are generally expressed in terms of the Cartesian (i.e. x, y, z) coordinates of each of the particles involved. When other types of coordinates are used or when there are constraints on the motion then it is advantageous to use alternative formulations of classical mechanics. In the present case we will use Lagrange's equations of motion. These equations can be written as

d/dt ([partial derivative]L/[partial derivative][??]i) - [partial derivative]L/[partial derivative]qi = 0 (1.8)

where

L = Kinetic energy - Potential energy

= T - V, (1.9)

qi is a generalized coordinate and [??] = dqi/dt is its time derivative.

Returning now to the specific case of the scattering of two atoms, we can write the kinetic energy in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where

M = m1 + m2,

μ = m1 m2/m1 + m2 (1.11)

and the potential energy V is a function of only r = |r|, so

L = 1/2M[??]2 + 1/2μ[??]2 - V(r) (1.12)

Let us first examine the equations of motion for the motion of the center-of-mass of the system, R = R(X, Y, Z). The derivatives of the Lagrangian with respect to the X coordinate and its time derivative are

[partial derivative]L/[partial derivative][??] = M[??], [partial derivative]L/[partial derivative]X = 0. (1.13)

The equation of motion for X is therefore (see eqn (1.8))

d/dt (M[??] = 0 (1.14)

or

d/dt X = [??] = constant (1.15)

Similar equations apply to the Y and Z components of the center-of-mass vector. We see therefore that the velocity of the center-of-mass is constant. The center-of-mass of the colliding pair moves through space with a constant velocity which is not changed by the collision. The variables X, Y and Z are called cyclic coordinates. The potential function does not depend on them and this leads to the properties just mentioned.

There remain the three coordinates which describe the vector r. As the potential depends only on r = |r| it is natural to use the spherical polar coordinates r, q and f to define the vector r. In terms of these coordinates the expression for the kinetic energy is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.16)

We notice immediately that the azimuthal angle, Φ, does not appear anywhere. It is therefore constant and is not changed by the collision. The polar angle, θ, appears only as a time derivative in the kinetic energy. The Lagrangian equation of motion for θ is therefore

d/dt ([partial derivative]L/[partial derivative][??] - [partial derivative]L/[partial derivative]θ= 0. (1.17)

[partial derivative]L/[partial derivative][??] = constant (1.18)

The quantity [partial derivative]L/[partial derivative][??] is known as pθ, the generalized momentum conjugate to θ

pθ = [partial derivative]L/[partial derivative][??] =μr2[??] (1.19)

This quantity pθ is the orbital angular momentum and in the present case of a collision between two partners without any internal structure it is a conserved quantity. We will, for the moment, denote it by the symbol [??] If the interaction potential was zero, then the closest distance two atoms would approach each other would be the impact parameter, b, and this would be equal to r at this point. The speed of the one atom relative to the other at this point is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As there is no potential in this hypothetical case the relative speed of the two atoms is just the original velocity v. So we can write

[??] = pθ = μr2[??] = bμv (1.20)

Now let us consider the Lagrangian equations of motion for r. From eqn (1.8), (1.9) and (1.16) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)

or, using eqn (1.20), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)

This is the equation of motion for a particle of mass µ in one dimension moving in an effective potential of V + l2/2µr2. The total energy of the system is also conserved. As the energy associated with the motion of the center of-mass is constant (see eqn (1.15)), the energy associated with the relative motion must also be constant. The kinetic energy for the relative motion of the two atoms is given by the second set of curly brackets in eqn (1.16). We see therefore that the energy of relative motion is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)

From eqn (1.22) and (1.23) we see that the one-dimensional equation of motion for r corresponds to a particle of mass µ moving in an effective potential of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The extra term, l2/2µr2, is referred to as the centrifugal potential.

Typically, two atoms attract each other at large distances and repel each other when they approach each other very closely or collide. This leads to a form of potential such as the analytic Lennard-Jones model potential in eqn (1.24) and shown in Figure 1.5

V(r) = 4 ε ((σ/r])12 - (σ/r)6). (1.24)

Let us now look at the scattering trajectories which arise for different impact parameters. In Figure 1.6 we show three characteristic trajectories for different values of the impact parameter. One of the atoms is fixed, and the trajectory of the other atom relative to it is shown. Trajectory (A) shows a head-on collision with an impact parameter of b=0. We see that in this case the incident atom bounces straight backwards and the relative motion of the scattering partners has therefore been deflected by an angle of 180°. Trajectory (B) shows a collision at an intermediate impact parameter. The two atoms initially attract each other and the trajectory is deflected from its original direction towards the other atom. They approach to a distance such that the initial energy of the relative motion (1/2 mv2) equals the effective potential [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. At this point the radial velocity of the colliding atom is reversed and the atom ends up being deflected from its original direction by some positive angle, which we denote here by θ. The point of closest approach of the two atoms is called the "classical turning point". At larger impact parameters (shown in trajectory (C)) the colliding atom never approaches the other atom sufficiently closely to sample their mutual repulsion, but only feels the long distance attraction. This leads to a negative deflection.

The angle of deflection as a function of the impact parameter is called the deflection function χ(E, b). For very large impact parameters the angle of deflection is zero (see Figure 1.6). The polar angle θ on the other hand starts at 180° or π, as this corresponds to the angle of the negative Z axis. For this large impact parameter, straight line trajectory the final polar angle is 0, the angle of the positive Z axis. The change in the angle θ is therefore π when the deflection function χ(E, b) = 0. From Figure 1.6 we see that the change of deflection angle during the course of the trajectory is made up of two equal parts. If we start looking at the trajectory from the point at which the collision partners are closest to each other, i.e. the classical turning point r0, then the total change of angle during the trajectory is twice that which occurs during the final half of the trajectory. Let us call this total change of angle Δθ. The deflection function is therefore χ(E, b) π - Δθ. The most straightforward way to compute the deflection function is to recast eqn (1.20) and (1.23) so as to provide equations for [??] and [??] and to integrate them numerically from the classical turning point outwards.

From eqn (1.20) we obtain

dθ/dt = [??] = bf/r2. (1.25)

The relative velocity is related to the relative collision energy by

E = 1/2μv2. (1.26)

and combining eqn (1.25) and (1.26) we obtain

dθ/dt = b/r2 (2E/μ)1/2. (1.27)

From eqn (1.23) we can obtain an equation for [??]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.28)

From eqn (1.20) we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.29)

and substituting for ?? in eqn (1.28) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.30)

Dividing eqn (1.27) by eqn (1.30) eliminates time from the equations and we can then integrate from the classical turning point outwards to obtain an equation for the deflection function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.31)

where r0 is the classical turning point.

Three deflection functions for a Lennard-Jones model potential calculated using different collision energies are shown in Figure 1.7. At lower energies the deflection function shows deflection angles more negative than -180° or even -360°. This indicates the presence of orbiting type collisions in which the atoms either circle around each other and finally spin off in the direction opposite to their original direction of motion (χ = -180° or completely orbit each other so as to exit the collision moving in their original directions (χ = -360°.

1.2.3 Singularities of a Differential Scattering Cross Section

As in the case of the collision of two hard spheres (see eqn (1.5)) we may now define a differential scattering cross section. This is defined as the number of atoms scattered into an increment of solid angle (dΩ), per unit time and per unit incident flux of the collision partners. For an atom–atom collision, in which the interaction potential is independent of angles, the azimuthal angle φ (i.e. the angle around the initial relative collision velocity) plays no role and we in effect integrate the cross section over this angle. Indeed this is the case in all situations where the target atom or molecule is not in some way oriented in space (see however Chapter 10).

(Continues...)

Excerpted from Theory of Molecular Collisions by Gabriel G. Balint-Kurti, Alexander P. Palov. Copyright © 2015 Gabriel G. Balint-Kurti and Alexander P. Palov. Excerpted by permission of The Royal Society of Chemistry.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Edition details

• Publisher: Royal Society of Chemistry
• Publication date: 2015
• Language: English
• ISBN 10: 1849738300
• ISBN 13: 9781849738309
• Binding: Hardcover
• Number of pages: 294