Quantum-state Specific Scattering of Molecules from Surfaces

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Publisher :
ISBN 13 :
Total Pages : 169 pages
Book Rating : 4.:/5 (931 download)

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Book Synopsis Quantum-state Specific Scattering of Molecules from Surfaces by :

Download or read book Quantum-state Specific Scattering of Molecules from Surfaces written by and published by . This book was released on 2014 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: In my work, I investigated the quantum-state resolved scattering of three different diatomic molecules (NO, CO, N2) from different surfaces, including Au(111) and Pt(111). I focused on measurements of the energy transfer between the various degrees of freedom available using both state-of-the-art and new methods developed in the course of this work. I strove to investigate a few simple model systems with the goal of discovering generally valid rules for the coupling between different degrees of freedom of these simple model systems. As a first system, I investigated vibrationally inelastic ...

Dynamics of Gas-Surface Interactions

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Publisher : Springer Science & Business Media
ISBN 13 : 3642329551
Total Pages : 439 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis Dynamics of Gas-Surface Interactions by : Ricardo Diez Muino

Download or read book Dynamics of Gas-Surface Interactions written by Ricardo Diez Muino and published by Springer Science & Business Media. This book was released on 2013-02-26 with total page 439 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a representative survey of the state of the art of research on gas-surface interactions. It provides an overview of the current understanding of gas surface dynamics and, in particular, of the reactive and non-reactive processes of atoms and small molecules at surfaces. Leading scientists in the field, both from the theoretical and the experimental sides, write in this book about their most recent advances. Surface science grew as an interdisciplinary research area over the last decades, mostly because of new experimental technologies (ultra-high vacuum, for instance), as well as because of a novel paradigm, the ‘surface science’ approach. The book describes the second transformation which is now taking place pushed by the availability of powerful quantum-mechanical theoretical methods implemented numerically. In the book, experiment and theory progress hand in hand with an unprecedented degree of accuracy and control. The book presents how modern surface science targets the atomic-level understanding of physical and chemical processes at surfaces, with particular emphasis on dynamical aspects. This book is a reference in the field.

Effects of Surface Temperature in Gas-surface Interaction

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ISBN 13 :
Total Pages : 246 pages
Book Rating : 4.:/5 (71 download)

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Book Synopsis Effects of Surface Temperature in Gas-surface Interaction by : Shengyuan Zhang

Download or read book Effects of Surface Temperature in Gas-surface Interaction written by Shengyuan Zhang and published by . This book was released on 2010 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: The scattering of H2 from Si(100) has been studied using pulsed molecular beam techniques and quantum state-specific detection methods. These studies can be used to test theoretical calculations and give insight into new theories of molecule-surface interactions, a fundamental study in a diverse field of science and technology. In this work, time-of-flight (TOF) spectra of the elastic scattering of H2(v=1. J=1) and H2(v=0, J=1) from clean Si were recorded over a wide range of surface temperatures. Two data processing strategies were developed to extract rich kinematic information from the scattering experiments, e.g., mean translational energy exchange, absolutely survival probability, and angular and speed distribution of the scattered molecules. No such set of quantitative results has been reported before for this system. Compared with close packed metal surfaces, these scattering experiments from a covalently bonded semiconductor surface showed a completely distinct dynamics, e.g. the finding of energy gain instead of loss from the substrate, much broader angular distribution and some counterintuitive surface temperature effects. From the studies of molecules/surface scattering experiments, the thermal excitation on Si(100) surface which depends on surface temperature can substantially alter the adsorption barrier and its distribution, and therefore changes the kinematics of scattered molecules. As a result, even the most basic understanding of the dynamics has to include phonon excitation and deexcitation of the silicon substrate.

Dynamics of Gas-Surface Scattering

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Publisher : Elsevier
ISBN 13 : 0323154611
Total Pages : 352 pages
Book Rating : 4.3/5 (231 download)

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Book Synopsis Dynamics of Gas-Surface Scattering by : Frank O. Goodman

Download or read book Dynamics of Gas-Surface Scattering written by Frank O. Goodman and published by Elsevier. This book was released on 2012-12-02 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dynamics of Gas-Surface Scattering deals with the dynamics of scattering as inferred from known properties of gases and solids. This book discusses measurements of spatial distributions of scattered atomic and molecular streams, and of the energy and momentum which gas particles exchange at solid surfaces. It also considers two regimes of scattering, both of which are associated with a lower range of incident gas energies: the thermal and structure scattering regimes. Comprised of 10 chapters, this book opens with a brief historical overview of the early experiments that investigated the dynamics of scattering of gases by surfaces. The discussion then turns to some elements of the kinetic theory of gases; intermodular potentials and interaction regimes; and classical-mechanical lattice models used in gas-surface scattering theory. The applications of molecular beams to the study of gas-surface scattering phenomena are also described. The remaining chapters focus on experiments and theories on scattering of molecular streams by surfaces of solids, with emphasis on thermal and structure regimes of inelastic scattering; quantum theory of gas-surface scattering; and quantum mechanical scattering phenomena. This text concludes with an analysis of energy exchange processes that may occur when a solid surface is completely immersed in a still gas. This monograph will be a valuable resource for students and practitioners of physics, chemistry, and applied mathematics.

Dynamics at Surfaces: Understanding Energy Dissipation and Physicochemical Processes at the Atomic and Molecular Level

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Publisher : Frontiers Media SA
ISBN 13 : 2832548466
Total Pages : 120 pages
Book Rating : 4.8/5 (325 download)

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Book Synopsis Dynamics at Surfaces: Understanding Energy Dissipation and Physicochemical Processes at the Atomic and Molecular Level by : Marco Sacchi

Download or read book Dynamics at Surfaces: Understanding Energy Dissipation and Physicochemical Processes at the Atomic and Molecular Level written by Marco Sacchi and published by Frontiers Media SA. This book was released on 2024-04-26 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: Energy release to solid interfaces following chemical reactions is ubiquitous in a vast range of phenomena. Energy dissipation and dynamical disorder (surface entropy), surface friction and molecular diffusion control the rates of heterogeneous catalytic reactions, the efficiency of novel technology, lubrication as well as materials growth including self-assembly and nano-structures. Yet we understand little about the underlying nature of these mechanisms. Fundamentally, energy dissipation including interactions with phonons and electron-hole pairs determines the lifetime of molecular vibrations and rotations as well as the decoherence rate of quantum states. These processes form a central point for many aspects in physical chemistry, are embedded in the mechanisms that control surface dynamical processes and are critical factors in catalysis. They are equally relevant for physicochemical processes in the Earth's atmosphere and astrochemistry occurring on cosmic dust grains.

From Diatomic to Polyatomic Quantum-state-resolved Molecule-surface Scattering

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Publisher :
ISBN 13 :
Total Pages : 215 pages
Book Rating : 4.:/5 (1 download)

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Book Synopsis From Diatomic to Polyatomic Quantum-state-resolved Molecule-surface Scattering by : Bastian Christopher Krüger

Download or read book From Diatomic to Polyatomic Quantum-state-resolved Molecule-surface Scattering written by Bastian Christopher Krüger and published by . This book was released on 2017 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work, molecule-surface scattering is studied for the diatomic molecule nitric oxide (NO) and the polyatomic molecule formaldehyde. Following the approach of "molecular surface science", elementary steps in the interaction of molecules with surfaces are revealed using a molecular beam surface scattering apparatus which permits the examination of the collision process under well-defined conditions. This requires that the initial molecular degrees of freedom are well controlled and that the final distribution over the molecular degrees of freedom can be obtained resolving single quantu...

Molecular Quantum Dynamics

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Publisher : Springer Science & Business Media
ISBN 13 : 3642452906
Total Pages : 281 pages
Book Rating : 4.6/5 (424 download)

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Book Synopsis Molecular Quantum Dynamics by : Fabien Gatti

Download or read book Molecular Quantum Dynamics written by Fabien Gatti and published by Springer Science & Business Media. This book was released on 2014-04-09 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on current applications of molecular quantum dynamics. Examples from all main subjects in the field, presented by the internationally renowned experts, illustrate the importance of the domain. Recent success in helping to understand experimental observations in fields like heterogeneous catalysis, photochemistry, reactive scattering, optical spectroscopy, or femto- and attosecond chemistry and spectroscopy underline that nuclear quantum mechanical effects affect many areas of chemical and physical research. In contrast to standard quantum chemistry calculations, where the nuclei are treated classically, molecular quantum dynamics can cover quantum mechanical effects in their motion. Many examples, ranging from fundamental to applied problems, are known today that are impacted by nuclear quantum mechanical effects, including phenomena like tunneling, zero point energy effects, or non-adiabatic transitions. Being important to correctly understand many observations in chemical, organic and biological systems, or for the understanding of molecular spectroscopy, the range of applications covered in this book comprises broad areas of science: from astrophysics and the physics and chemistry of the atmosphere, over elementary processes in chemistry, to biological processes (such as the first steps of photosynthesis or vision). Nevertheless, many researchers refrain from entering this domain. The book "Molecular Quantum Dynamics" offers them an accessible introduction. Although the calculation of large systems still presents a challenge - despite the considerable power of modern computers - new strategies have been developed to extend the studies to systems of increasing size. Such strategies are presented after a brief overview of the historical background. Strong emphasis is put on an educational presentation of the fundamental concepts, so that the reader can inform himself about the most important concepts, like eigenstates, wave packets, quantum mechanical resonances, entanglement, etc. The chosen examples highlight that high-level experiments and theory need to work closely together. This book thus is a must-read both for researchers working experimentally or theoretically in the concerned fields, and generally for anyone interested in the exciting world of molecular quantum dynamics.

Surface Dynamics

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Publisher : Elsevier
ISBN 13 : 0080498345
Total Pages : 393 pages
Book Rating : 4.0/5 (84 download)

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Book Synopsis Surface Dynamics by :

Download or read book Surface Dynamics written by and published by Elsevier. This book was released on 2003-11-07 with total page 393 pages. Available in PDF, EPUB and Kindle. Book excerpt: While much of traditional surface science has been concerned with equilibrium properties and simple kinetics, there is a growing effort in the area of dynamical processes at surfaces. This book draws together a series of chapters written by acknowledged experts in the field, which describe progress in a range of specific topics. The emphasis is on chemical reaction dynamics, including both theoretical and experimental approaches and covering work on low index single crystal surfaces, on stepped surfaces and on supported metal clusters. Other processes, such as surface diffusion are also addressed. Further chapters discuss dynamical processes in electronically-induced desorption, and in surface diffusion on semiconductors and metals.- Presents considerable advances in surface science field - Collection of expert reviews in surface dynamics

Scattering Theory of Molecules, Atoms and Nuclei

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Publisher : World Scientific
ISBN 13 : 9814329843
Total Pages : 646 pages
Book Rating : 4.8/5 (143 download)

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Book Synopsis Scattering Theory of Molecules, Atoms and Nuclei by : Luiz Felipe Canto

Download or read book Scattering Theory of Molecules, Atoms and Nuclei written by Luiz Felipe Canto and published by World Scientific. This book was released on 2013 with total page 646 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the book is to give a coherent and comprehensive account of quantum scattering theory with applications to atomic, molecular and nuclear systems. The motivation for this is to supply the necessary theoretical tools to calculate scattering observables of these many-body systems. Concepts which are seemingly different for atomic/molecular scattering from those of nuclear systems, are shown to be the same once physical units such as energy and length are diligently clarified. Many-body resonances excited in nuclear systems are the same as those in atomic systems and come under the name of Feshbach resonances. We also lean heavily on semi-classical methods to explain the physics of quantum scattering OCo especially the interference seen in the angle dependence of the cross section. Having in mind a wide readership, the book includes sections on scattering in two dimensions which is of use in surface physics. Several problems are also included at the end of each of the chapters.

Dynamics

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Publisher : Elsevier
ISBN 13 : 0080931200
Total Pages : 1037 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Dynamics by :

Download or read book Dynamics written by and published by Elsevier. This book was released on 2008-10-09 with total page 1037 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume of the Handbook of Surface Science covers all aspects of the dynamics of surface processes. Two dozen world leading experts in this field address the subjects of energy exchange in gas atoms, surface collisions, the rules governing dissociative adsorption on surfaces, the formation of nanostructures on surfaces by self-assembly, and the study of surface phenomena using ultra-fast lasers. The chapters are written for both newcomers to the field as well as researchers.• Covers all aspects of the dynamics of surface processes • Provides understanding of this unique field utilizing a multitude of accurate experiments and advanced microscopic theory that allows quantum-level comparisons • Presents the concepts and tools relevant beyond surface science for catalysis, nanotechnology, biology, medicine, and materials

Chemical Bonding at Surfaces and Interfaces

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Publisher : Elsevier
ISBN 13 : 0080551912
Total Pages : 533 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Chemical Bonding at Surfaces and Interfaces by : Anders Nilsson

Download or read book Chemical Bonding at Surfaces and Interfaces written by Anders Nilsson and published by Elsevier. This book was released on 2011-08-11 with total page 533 pages. Available in PDF, EPUB and Kindle. Book excerpt: Molecular surface science has made enormous progress in the past 30 years. The development can be characterized by a revolution in fundamental knowledge obtained from simple model systems and by an explosion in the number of experimental techniques. The last 10 years has seen an equally rapid development of quantum mechanical modeling of surface processes using Density Functional Theory (DFT). Chemical Bonding at Surfaces and Interfaces focuses on phenomena and concepts rather than on experimental or theoretical techniques. The aim is to provide the common basis for describing the interaction of atoms and molecules with surfaces and this to be used very broadly in science and technology. The book begins with an overview of structural information on surface adsorbates and discusses the structure of a number of important chemisorption systems. Chapter 2 describes in detail the chemical bond between atoms or molecules and a metal surface in the observed surface structures. A detailed description of experimental information on the dynamics of bond-formation and bond-breaking at surfaces make up Chapter 3. Followed by an in-depth analysis of aspects of heterogeneous catalysis based on the d-band model. In Chapter 5 adsorption and chemistry on the enormously important Si and Ge semiconductor surfaces are covered. In the remaining two Chapters the book moves on from solid-gas interfaces and looks at solid-liquid interface processes. In the final chapter an overview is given of the environmentally important chemical processes occurring on mineral and oxide surfaces in contact with water and electrolytes. - Gives examples of how modern theoretical DFT techniques can be used to design heterogeneous catalysts - This book suits the rapid introduction of methods and concepts from surface science into a broad range of scientific disciplines where the interaction between a solid and the surrounding gas or liquid phase is an essential component - Shows how insight into chemical bonding at surfaces can be applied to a range of scientific problems in heterogeneous catalysis, electrochemistry, environmental science and semiconductor processing - Provides both the fundamental perspective and an overview of chemical bonding in terms of structure, electronic structure and dynamics of bond rearrangements at surfaces

State Selected and State-to-State Ion-Molecule Reaction Dynamics, Volume 82, Part 1

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Publisher : John Wiley & Sons
ISBN 13 : 0470141921
Total Pages : 702 pages
Book Rating : 4.4/5 (71 download)

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Book Synopsis State Selected and State-to-State Ion-Molecule Reaction Dynamics, Volume 82, Part 1 by : Cheuk-Yiu Ng

Download or read book State Selected and State-to-State Ion-Molecule Reaction Dynamics, Volume 82, Part 1 written by Cheuk-Yiu Ng and published by John Wiley & Sons. This book was released on 2009-09-09 with total page 702 pages. Available in PDF, EPUB and Kindle. Book excerpt: State-Selected and State-to-State Ion-Molecules Reaction Dynamics details the recent experimental and theoretical accomplishments in the field to date by some of its foremost researchers and theorists. Divided into two parts, each of which separately describe the experimental and theoretical aspects of the field, State-Selected and State-to-State Ion-Molecule Reaction Dynamics is an accessible, well organized look at a highly useful and emerging chemical specialty. Part 1, "Experiment," contains eight in-depth studies, which illustrate the key experimental work being done in the field today: Chapter 1 provide a comprehensive review of the theory and application of inhomogeneous rf fields for the study of the dynamics of low-energy ion-molecules processes Chapter 2 describes the application of multiphoton ionization (MPI) for the preparation of reactant ion states Chapter 3 reviews the application of MPI schemes for state specific cross-section measurements involving transition metal cations Chapter 4 describes the development of the threshold photoelectron secondary ion coincidence (TESICO) method Chapter 5 presents the conceptual and practical aspects of a multicoincidence technique Chapter 6 details the experimental results obtained using the photoionization and differential reactivity methods Chapter 7 reviews the several recent crossed beam studies of charge transfer and collision-induced dissociation systems involving atomic and molecular ions Chapter 8 is a survey of 15 years of high resolution crossed beam scattering of protons with atoms, diatoms, and poly-atomic molecules State-Selected and State-to-State Ion-Molecule Reaction Dynamics, Part 1: Experiment offers professionals a true state-of-the-science look at this fascinating and increasingly influential subject.

Scattering, Two-Volume Set

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Publisher : Elsevier
ISBN 13 : 0080540732
Total Pages : 1831 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Scattering, Two-Volume Set by : E. R. Pike

Download or read book Scattering, Two-Volume Set written by E. R. Pike and published by Elsevier. This book was released on 2001-10-09 with total page 1831 pages. Available in PDF, EPUB and Kindle. Book excerpt: Scattering is the collision of two objects that results in a change of trajectory and energy. For example, in particle physics, such as electrons, photons, or neutrons are "scattered off" of a target specimen, resulting in a different energy and direction. In the field of electromagnetism, scattering is the random diffusion of electromagnetic radiation from air masses is an aid in the long-range sending of radio signals over geographic obstacles such as mountains. This type of scattering, applied to the field of acoustics, is the spreading of sound in many directions due to irregularities in the transmission medium. Volume I of Scattering will be devoted to basic theoretical ideas, approximation methods, numerical techniques and mathematical modeling. Volume II will be concerned with basic experimental techniques, technological practices, and comparisons with relevant theoretical work including seismology, medical applications, meteorological phenomena and astronomy. This reference will be used by researchers and graduate students in physics, applied physics, biophysics, chemical physics, medical physics, acoustics, geosciences, optics, mathematics, and engineering. This is the first encyclopedic-range work on the topic of scattering theory in quantum mechanics, elastodynamics, acoustics, and electromagnetics. It serves as a comprehensive interdisciplinary presentation of scattering and inverse scattering theory and applications in a wide range of scientific fields, with an emphasis, and details, up-to-date developments. Scattering also places an emphasis on the problems that are still in active current research. The first interdisciplinary reference source on scattering to gather all world expertise in this technique Covers the major aspects of scattering in a common language, helping to widening the knowledge of researchers across disciplines The list of editors, associate editors and contributors reads like an international Who's Who in the interdisciplinary field of scattering

Chemistry and Physics of Solid Surfaces V

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Publisher : Springer Science & Business Media
ISBN 13 : 3642822533
Total Pages : 569 pages
Book Rating : 4.6/5 (428 download)

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Book Synopsis Chemistry and Physics of Solid Surfaces V by : R. Vanselow

Download or read book Chemistry and Physics of Solid Surfaces V written by R. Vanselow and published by Springer Science & Business Media. This book was released on 2013-11-21 with total page 569 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains review articles which were written by the invited speak ers of the Sixth International Summer Institute in Surface Science (ISISS), held at the University of Wisconsin-Milwaukee in August 1983. The objective of ISISS is to bring together a group of internationally recognized experts on various aspects of surface science to present tutorial review lectures over a period of one week. Each speaker is asked, in addi tion, to write a review paper on his lecture topic. The collected articles from previous Institutes have been published under the following titles: Surface Science: Recent Progress and Perspectives, Crit. Rev. Solid State Sci. 4, 124-559 (1974). Chemistry and Physics of Solid Surfaces, Vol. I (1976), Vol. II (1979), Vol. III (1982) (CRC Press, Boca Raton, FL), and Vol. IV (1982), Springer Ser. Chern. Phys. , Vol. 20 (Springer-Verlag Berlin, Heidelberg, New York 1982) No single collection of reviews (or one-week conference for that matter) can possibly cover the entire field of modern surface science, from heter ogeneous catalysis through semiconductor surface physics to metallurgy. It is intended, however, that the series Chemistry and Physics ofSolid Sur faces as a whole should provide experts and students alike with a comprehen ve set of reviews and literature references on as many aspects of the subject as possible, particular emphasis being placed on the gas-solid interface. Each volume is introduced with a historical review of the devel opment of one aspect of surface science by a distinguished participant in that development.

Theory and Application of Quantum Molecular Dynamics

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Publisher : World Scientific
ISBN 13 : 9789810233884
Total Pages : 390 pages
Book Rating : 4.2/5 (338 download)

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Book Synopsis Theory and Application of Quantum Molecular Dynamics by : John Z. H. Zhang

Download or read book Theory and Application of Quantum Molecular Dynamics written by John Z. H. Zhang and published by World Scientific. This book was released on 1999 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an updated and detailed presentation of modern quantum-mechanical treatments and practical computational methods for dynamical processes of small molecular systems. The main emphasis is on the recent development of successful theories and computational methods for the reactive scattering process. Specific applications are given in detail for a number of benchmark chemical reaction systems in the gas phase and gas surface. Differing from traditional physics books focusing on abstract collision theory for elastic collisions, the book has been written in a fashion in which the development of general reactive or rearrangement scattering theory is accompanied by practical applications for realistic reaction systems.

Molecular Scattering

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Publisher : John Wiley & Sons
ISBN 13 : 0470144157
Total Pages : 552 pages
Book Rating : 4.4/5 (71 download)

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Book Synopsis Molecular Scattering by : K. P. Lawley

Download or read book Molecular Scattering written by K. P. Lawley and published by John Wiley & Sons. This book was released on 2009-09-08 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Advances in Chemical Physics series provides the chemical physics and physical chemistry fields with a forum for critical, authoritative evaluations of advances in every area of the discipline. Filled with cutting-edge research reported in a cohesive manner not found elsewhere in the literature, each volume of the Advances in Chemical Physics series serves as the perfect supplement to any advanced graduate class devoted to the study of chemical physics.

The Transfer of Molecular Energies by Collision: Recent Quantum Treatments

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Publisher : Springer Science & Business Media
ISBN 13 : 3642931227
Total Pages : 337 pages
Book Rating : 4.6/5 (429 download)

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Book Synopsis The Transfer of Molecular Energies by Collision: Recent Quantum Treatments by : F. A. Gianturco

Download or read book The Transfer of Molecular Energies by Collision: Recent Quantum Treatments written by F. A. Gianturco and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: I I These Lecture Notes are intended as an introduction to the theoretical formulation and computational aspects of the molecular energy transfer processes which take place in an increasingly sophisticated range of molecular scattering experiments. They are directed to chemistry graduate students and emphasize the quantum mechanical approach, with little or no attention to classical and semi classical treatments or to formal presentations. Several Sections of the first Chapters are based on lectures given at the Graduate School of Physics of the University of Genoa a few years ago and I thank the students for their sense of duty in following to the end all those no tation-filled blackboards and transparencies. The kind patience of my wife Carolyn in reading the whole manuscript and improving its form is gratefully acknowledged. Franco A. Gianturco Bari, September 1978 CON TEN T S FOREWORD I NTRODUCTI ON Page 1. A RESUME OF QUANTUM MECHANICAL POTENTIAL SCATTERING 1. 1. General formulation of the problem Page 5 1. 2. Solutions of the radial equation 10 " 1. 3. The method of partial waves 13 1. 4. Some properties of 61. The Born appro~imation 18 1. 5. Properties of the S-matrix: bound states and resonances 23 1. 6. Classical and semiclassical scattering,a set of defi- tions 34 References 44 2. POTENTIAL ENERGY HYPERSURFACE CALCULATIONS FOR SIMPLE SYSTEMS 2. 1. Kinematic considerations 45 2. 2. General development of a priori method 52 2. 3. Some approximate treatments 68 2. 4.