6.1 Collisions and Scattering Theory

Collisions and scattering theory are key to understanding how particles interact at the atomic level. This topic dives into the quantum mechanics behind these interactions, exploring how energy and momentum are exchanged when particles collide.

We'll look at different types of collisions, from elastic to reactive, and learn how to calculate cross-sections and rates. This knowledge is crucial for grasping atomic interactions and processes in the broader context of atomic physics.

Principles of Collision and Scattering Theory

Fundamental Concepts

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• Collision and scattering theory studies interactions between particles (atoms, molecules, and subatomic particles) when they come close or collide
• In atomic physics, collisions and scattering events are governed by quantum mechanics laws describing particle behavior at atomic and subatomic scales
• The outcome of a collision or scattering event depends on initial conditions (energy, momentum, and internal states of the particles involved)
• Collisions and scattering can result in the exchange of energy, momentum, and other physical quantities between interacting particles
• Studying collisions and scattering provides valuable information about properties of atoms, molecules, their interactions, and fundamental forces governing them

Quantum Mechanical Description

• Quantum mechanics is essential for accurately describing atomic and subatomic particle interactions during collisions and scattering events
• The wave-particle duality of quantum mechanics is crucial in understanding the probabilistic nature of particle interactions
• The Schrödinger equation is used to determine the wave function of the system, which contains all information about the particles' states and their interactions
• The Born approximation is often used to calculate scattering cross-sections by treating the incident particle as a plane wave and the target as a perturbation
• The partial wave analysis is a powerful technique for studying scattering processes, decomposing the scattering amplitude into contributions from different angular momenta

Types of Atomic Collisions

Elastic Collisions

• In elastic collisions, the total kinetic energy of the system is conserved, and internal states of particles remain unchanged
• Examples of elastic collisions include billiard ball collisions and the scattering of electrons by atoms in the Rutherford experiment
• The scattering angle and the energy distribution of the scattered particles provide information about the interaction potential between the particles
• The differential cross-section for elastic scattering is determined by the Fourier transform of the interaction potential
• Elastic collisions are essential for understanding the structure of atoms and molecules and for studying the properties of materials

Inelastic Collisions

• In inelastic collisions, the total kinetic energy of the system is not conserved, and some energy is converted into internal energy of the particles (excitation or ionization)
• Examples of inelastic collisions include the excitation of atoms by electron impact and the ionization of molecules by high-energy photons
• Inelastic collisions are crucial for understanding the electronic structure of atoms and molecules and for studying the mechanisms of energy transfer in atomic and molecular systems
• The analysis of inelastic collisions requires the use of more advanced theoretical methods, such as the close-coupling approach and the R-matrix theory
• Inelastic collisions play a vital role in various applications, such as plasma physics, astrophysics, and radiation damage studies

Reactive Collisions

• Reactive collisions involve the rearrangement of the particles' internal structure, leading to the formation of new chemical species or the dissociation of existing ones
• Examples of reactive collisions include chemical reactions, such as the formation of molecules from atoms and the dissociation of molecules by collisions with other particles
• The outcome of a reactive collision depends on the energy of the colliding particles and the nature of the interaction potential between them
• The study of reactive collisions is essential for understanding the mechanisms of chemical reactions and for developing new methods for controlling and manipulating chemical processes
• Reactive collisions play a crucial role in various fields, such as atmospheric chemistry, combustion, and catalysis

Calculating Cross-sections and Rates

Scattering Cross-sections

• The scattering cross-section is a measure of the likelihood of a scattering event occurring, determined by the interaction potential and the energy of the incident particle
• The differential cross-section $\frac{d\sigma}{d\Omega}$ describes the angular distribution of scattered particles and provides information about the nature of the interaction potential
• The total cross-section $\sigma$ is obtained by integrating the differential cross-section over all scattering angles: $\sigma = \int \frac{d\sigma}{d\Omega} d\Omega$
• The total cross-section represents the total probability of a scattering event and is used to calculate collision rates and mean free paths
• Experimental techniques, such as crossed-beam experiments and time-of-flight spectroscopy, are used to measure scattering cross-sections and to study the properties of atomic and molecular collisions

Collision Rates and Mean Free Paths

• Collision rates $R$ can be calculated using the scattering cross-section $\sigma$ and the density of the target particles $n$: $R = n \sigma v$, where $v$ is the relative velocity of the colliding particles
• The collision rate provides information about the frequency of collisions in a given system and is essential for understanding the kinetics of atomic and molecular processes
• The mean free path $\lambda$ is the average distance a particle travels between collisions and is given by $\lambda = \frac{1}{n \sigma}$
• The mean free path is a crucial parameter in various applications, such as gas dynamics, plasma physics, and radiation transport
• The knowledge of collision rates and mean free paths is essential for designing and optimizing various technological processes, such as gas lasers, plasma reactors, and particle accelerators

Potential Energy Surfaces in Collisions

Interaction Potentials

• Potential energy surfaces (PES) are mathematical functions describing the potential energy of a system of interacting particles as a function of their relative positions
• In atomic collisions, the PES governs the motion of the particles and determines the outcome of the collision (scattering angle and energy transfer between particles)
• The shape of the PES depends on the nature of the interaction between the particles, such as the Coulomb potential for charged particles or the van der Waals potential for neutral atoms
• Examples of interaction potentials include the Lennard-Jones potential for neutral atoms and the Morse potential for diatomic molecules
• The accurate determination of interaction potentials is crucial for predicting the outcomes of atomic and molecular collisions and for understanding the properties of materials

Reaction Dynamics

• The topography of the PES, including its minima, maxima, and saddle points, plays a crucial role in determining possible reaction pathways and the formation of intermediate states during a collision
• Reaction dynamics studies the motion of atoms and molecules on the PES during a collision and aims to understand the mechanisms of chemical reactions
• Transition state theory (TST) is a powerful framework for studying reaction dynamics, which assumes that the reaction rate is determined by the flux of particles through a dividing surface (the transition state) on the PES
• Molecular dynamics simulations are widely used to study reaction dynamics by solving the equations of motion for the atoms and molecules on the PES
• The study of reaction dynamics provides valuable insights into the mechanisms of chemical reactions, energy transfer processes, and the formation of new atomic or molecular species in collisions