Advances in Quantum Chemistry (eBook)

Theory of the Interaction of Swift Ions with Matter, Part 2

Remigio Cabrera-Trujillo, John R. Sabin (Herausgeber)

eBook Download: PDF

2004 | 1. Auflage
354 Seiten
Elsevier Science (Verlag)
978-0-08-054408-3 (ISBN)

Infos

Advances in Quantum Chemistry presents surveys of current developments in this rapidly developing field that falls between the historically established areas of mathematics, physics, and chemistry. With invited reviews written by leading international researchers, as well as regular thematic issues, each volume presents new results and provides a single vehicle for following progress in this interdisciplinary area.

The intention of this volume, as with the previous volume in this series is to present the latest developments in the field of energy deposition as it is actually viewed by many of the major researchers working in this area. It is not possible to incorporate all of the important players and all of the topics related to energy deposition in the limited space available, however the editors have tried to present the state of the art as it is now.

Inhalt

Cover 1
Contents 6
Contributors 10
Preface 12
Chapter 1. Density Functional Theory-based Stopping Power for 3D and 2D Systems 14
1. Introduction 15
2. Linear theory of stopping power 17
3. Density functional theory 23
4. Final remarks and conclusions 39
Acknowledgements 39
References 40
Chapter 2. Friction Force for Charged Particles at Large Distances from Metal Surfaces 42
1. Introduction 42
2. Theoretical background 45
3. Specular reflection model 49
4. Time dependent density functional theory 61
5. Comparison between SRM and TDDFT 69
6. Improvement of TDDFT at large distances 71
7. Conclusions 75
Acknowledgements 75
References 75
Chapter 3. Resonant-Coherent Excitation of Channeled Ions 78
1. Introduction 79
2. Theoretical framework 85
3. Dynamical mixing of electronic states 89
4. Resonant-coherent excitation to the continuum 91
5. Full calculation and comparison with experiment 92
Acknowledgements 96
Appendix A. Coupled channel equations for the relevant bound states 96
References 99
Chapter 4. The Barkas-Effect Correction to Bethe–Bloch Stopping Power 104
1. Historical background 104
2. Overview and perspective 110
References 129
Chapter 5. Molecular Stopping Powers from the Target Oscillator Strength Distribution 134
1. Introduction 135
2. Précis of oscillator strength based stopping theory 137
3. Oscillator strength distributions 139
4. The polarization propagator 145
5. Some examples 152
6. Remarks and conclusions 160
Acknowledgements 162
References 162
Chapter 6. Chemical and Physical State Effects in Electronic Stopping 166
1. Introduction 166
2. Bragg’s rule 168
3. Definition of PSE and CSE 169
4. Phenomenological description of PSE and CSE 169
5. Velocity dependence of CSE and PSE 170
References 175
Chapter 7. Calculation of Cross-Sections for Proton and Antiproton Stopping in Molecules 178
1. Introduction 179
2. Theoretical model 181
3. Results and discussions 187
4. Conclusion 204
References 205
Chapter 8. Advances in the Core-and-Bond Formalism for Proton Stopping in Molecular Targets 208
1. Introduction 208
2. The Cores-and-Bond formalism 210
3. Mean excitation energy and the LPA 216
4. Advances in CAB studies of molecular stopping 219
5. Conclusions 249
References 250
Chapter 9. Aspects of Relativistic Sum Rules 254
1. Introduction 254
2. Origin of sum rules 257
3. Review of early work 261
4. Recent advances 265
5. The trouble with relativity 273
6. Conclusion 277
Acknowledgements 277
References 277
Chapter 10. Stopping Power of an Electron Gas for Heavy Unit Charges: Models in the Kinetic Approximation 280
1. Introduction and motivations 281
2. The target model 281
3. The microscopic model of stopping 282
4. Screening 286
5. Results 290
6. Summary and remarks 301
Acknowledgements 302
References 303
Chapter 11. High Z Ions in Hot, Dense Matter 306
1. Introduction 306
2. Semi-classical statistical mechanics 308
3. Green–Kubo relations at small velocities 310
4. Impurity ion in an electron gas 312
5. Summary and discussion 317
Acknowledgements 317
References 318
Chapter 12. Interferences in Electron Emission from H2 Induced by Fast Ion Impact 320
1. Introduction 320
2. Bethe–Born approximation 323
3. Wave optical treatment 325
4. Quantum-mechanical treatment 330
5. Final remarks and conclusions 338
Acknowledgements 339
References 339
Chapter 13. Thoughts About Nanodosimetry 342
1. Introduction 342
2. Interactions of charged particles with matter 343
3. Calculated Energy Loss Spectra ('Straggling Function') f (d x)
4. A simulated energy deposition spectrum g (d,x) 347
5. Realistic relation of energy loss to radiation effect 348
6. Conclusions 350
References 350
Index 352

Textauszug

Density Functional Theory-based Stopping Power for 3D and 2D Systems

A. Sarasola1; R.H. Ritchie2; E. Zaremba3; P.M. Echenique1,4    1 Departamento de Fίsica de Materiales, Facultad de Quίmica, Universidad del Paίs Vasco, Apartado 1072 and Centro Mixto, C.S.I.C.-UPV/EHU, 20080 San Sebastián, Spain
2 Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6123, USA
3 Department of Physics, Queen's University, Kingston, Ont., Canada K7L 3N6
4 Donostia International Physics Center, Paseo Manuel de Lardizabal, No. 4, 20018 San Sebastián, Spain

Abstract

We have reviewed the theoretical and experimental results on linear and non-linear screening and stopping power for 3D and 2D systems. In the linear case (Z1/v1), dielectric formalism has been used to characterize such systems, while the non-linear aspects are described within the density functional theory (DFT) framework. Differences between proton and antiproton probe particles are discussed, both in the quadratic response function theory and in the DFT. For heavy ions, the main results on Z1 oscillations are revised.

1 Introduction

The interaction of particles of charge Z1 entering a solid medium with velocity v has been extensively studied for nearly a century. Understanding of the dynamical properties of microscopic matter must come from measurements at the macroscopic level. It is the task of theory to interpret these results to yield information about the complex microworld. Physicists approach this task by building models. Here we review various models of energy loss for 2D and 3D electron gases, including the state-of-art density functional theory (DFT).

The simplest but widely used model for the description of the valence band electrons of a metal is the homogeneous electron gas. A useful parameter characterizing it is the average electronic density n0, or equivalently, the average electron radius rs (in units of the Bohr radius), where

n 0 = 4 3 ⁢ π ⁢ r s 3 .

An n-dimensional electron gas is defined by the number of spatial coordinates (n) in which electrons can move freely. The 3D case has been studied extensively since Drude constructed his theory of electrical and thermal conduction by applying the kinetic theory of gases to a metal. In the late 1970s, the 2D electron systems began to be studied. Among the properties of the 2D systems that have attracted active theoretical and experimental interest are phase-transitions and long-range order or transport and percolation properties of disordered 2D systems. A 2D Coulomb gas, where electrons are restricted to move in a 2D space, can be applied as an approximation to metal–insulator semiconductor structures or on the surface of liquid helium [1].

Lindhard [2] used linear response theory to calculate the stopping power of charged particles within a 3D electron gas model. He used quantum perturbation theory, treating the perturbing potential to the lowest order. The self-consistent response of the electrons to this perturbation was considered within a mean-field theory, each electron moving in a field determined by all others. At the same time, Bohm and Pines [3] constructed a quantum theory of the electron gas, describing (1) the long-range part of the Coulomb interaction in terms of collective fields that gives rise to plasma oscillations and, (2) individual electron interaction with the other ones and with collective fields via a screened short-range interaction.

In the 2D case, Stern [4] studied the polarizability of a 2D electron gas (2DEG) within the random phase approximation (RPA). Later on, exchange and correlation effects [5] were included providing a 2D equivalence of the Thomas–Fermi–Dirac method. Horing et al. [6] calculated the energy loss of a fast charged particle moving parallel to a 2D electron gas. This was extended by including the effect of finite temperature [7], local field corrections [8] and recoil [9].

Although the dielectric function formalism provides a qualitative description of dynamic screening over the whole range of projectile velocities, it is not quantitatively accurate, especially at low velocities where at metallic densities the charged particle represents a strong perturbation to the system even for the smallest charge Z1=1. In general, linear response theory will remain valid as long as Z1/v1 condition is satisfied, whereas non-linear effects will become significant when it is not.

Different approximations have been tried in order to go beyond the linear description of screening and stopping power. A systematic expansion in powers of the projectile charge Z1 is the basis for a natural extension of linear theory. This first option is based on the calculation of the induced density to second order in Z1 and thus, of the induced potential at the Hartree level. It is done using the quadratic response function within the RPA. As in the linear case, it essentially describes the mean field of all electronic interactions. However, being an expansion in Z1, the theory is limited to those situations in which the corrections to the linear results are relatively small. In practice, for low projectile velocities, the theory is limited to the lowest possible value of Z1 (Z1=1) and high electron densities. Applied to the stopping power, this theory provides a Z13 correction, the well-known Barkas term. Since the work of Ashley et al. [10] on the classical oscillator model (an extension of Bohr's classic treatment) many groups have used quadratic response approaches [11–14], but is worthwhile to mention the application of this concept to the full range of velocities done by Pitarke et al. [15,16] and the complement to that work done by Arbó et al. [17] for the electron gas model.

A second alternative to describe the non-linear effects is to use a hydrodynamic description of the electron gas. In this approach, the electron gas is considered as a charged fluid that interacts with the external particle [18–20] and is essentially described by the electronic charge density and the velocity of the fluid. In the static limit, a Thomas–Fermi–von Weizsäcker [21] description of electronic screening is obtained whereas, for higher velocities a numerical solution of coupled hydrodynamic equations is needed.

The third possible approximation to calculate the non-linear screening of a static external impurity, particularly useful at low ion velocities, can be achieved by using the DFT. The theory is based on the theorems of Hohenberg and Kohn [22]. Its practical implementation, based on the standard orbital picture, results in the so-called Kohn–Sham equations [23] providing a tool to calculate a self-consistent potential to all orders in Z1. Popovic and Stott [24] and Almbladh and von Barth used the DFT to study the problem of non-linear screening of a static proton in a 3D electron gas. The KS method, by construction, gives stationary-state phase shifted one-particle states. These phase shifts, at the Fermi level, are natural inputs in the scattering formulation of the retarding force, the stopping power of an electron gas for slow intruders.

An important self-consistency condition on the scattering potential is provided by the Friedel sum rule (FSR) [25]. Often one-parameter model potentials have been used to determine the screening parameter in a consistent way applying this rule [26–30].

But as the ion velocity increases the standard DFT method ceases to be valid and some generalization or alternative approach becomes necessary to investigate the importance of non-linearities for velocities up to the Fermi velocity. Lifschitz and Arista [31,32] developed an extension to the FSR to finite velocities and a self-consistent optimization of the scattering potential and related phase shifts. Zaremba et al. [33] extended non-linear screening theory making an accurate calculation of dynamic screening, but instead of treating the full cylindrical symmetry of the problem, they made an spherical average in the interaction potential. Salin et al. [34] extended this to account for full cylindrical symmetry.

In this review, Section 2 is devoted to the linear theory of 3D and 2D electron gases in the dielectric formalism, and prove the equivalence between linear response formalism and scattering theory in the Born approximation for slowly moving ions. An extensive description of non-linear effects within the DFT is completed in Section 3: relevant results on non-linear static screening and stopping at low ion velocities in 3D and 2D electron gases for proton and antiproton projectiles and the effective charge treatment for heavier ion are discussed. A brief comparison with second-order perturbation theory results and non-linear dynamic screening effects on slowly moving ions is also presented.

Atomic units are used unless otherwise stated.

2 Linear theory of stopping power

2.1 Dielectric formalism in a 3D electron gas

The scalar electric potential created by an external...

Produkt-Details

• Erscheint lt. Verlag: 15.7.2004
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Naturwissenschaften ► Biologie
• Themenwelt: Naturwissenschaften ► Chemie ► Physikalische Chemie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Atom- / Kern- / Molekularphysik
• Themenwelt: Technik
• ISBN-10: 0-08-054408-8 / 0080544088
• ISBN-13: 978-0-08-054408-3 / 9780080544083