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title : Foundations of Vacuum Science and Technology author : Lafferty, J. M.
publisher : John Wiley & Sons, Inc. (US) isbn10 | asin : 0471175935
print isbn13 : 9780471175933 ebook isbn13 : 9780585339368
language : English
subject Vacuum, Kinetic theory of gases.
publication date : 1998
lcc : QC166.F68 1998eb ddc : 621.5/5
subject : Vacuum, Kinetic theory of gases.
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Page iii
Foundations of Vacuum Science and Technology Edited By
James M. Lafferty
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Page iv
This book is printed on acid-free paper.
Copyright © 1998 by John Wiley & Sons, Inc. All rights reserved.
Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (508) 750-8400, fax (508) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-mail: PERMREQ @ WILEY.COM.
Library of Congress Cataloging in Publication Data:
Foundations of vacuum science and technology/edited by J. M. Lafferty.
p. cm.
"A Wiley-Interscience publication."
Includes bibliographical references and index.
ISBN 0-471-17593-5
1. Vacuum. 2. Kinetic theory of gases. I. Lafferty, J. M.
(James Martin), 1916.
QC166.F68 1997
621.5′5dc21 96-29895 Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
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Page v
Preface
The industrial and scientific importance of vacuum technique has continued to grow during the past 30 years, even with the demise of television and radio receiving tubes and many gas-filled tubes. The replacement of vacuum and low- pressure gas discharge tubes by semiconductors and integrated circuits has placed an even greater demand and more stringent conditions on vacuum technique for the processing and manufacture of these devices. This has led to the development of a number of "dry" vacuum pumps to produce a "clean vacuum" free of hydrocarbons. The space programs, high-energy accelerators, analytical instruments, freeze-drying of foods and drugs, and the manufacture of color television picture tubes, incandescent and metal vapor lamps, high-power vacuum, and x-ray tubes all continue to require the need for vacuum technique.
While the material in this book is totally new, it follows in the tradition set by Scientific Foundations of Vacuum Technique by Saul Dushman, published in 1949. That book enjoyed unprecedented success and is now a classic in its field. By 1960 it was badly in need of revision. This editor had the privilege of participating as editor of the revised edition, which was published in 1962. This second edition was brought up to date by a number of contributors with specialized knowledge in the disciplines involved. An attempt was made to introduce the new developments made in vacuum technique but keep the original plan of the book and retain much of the material that was still of current interest.
The editor was encouraged by Leonard Beavis of the American Vacuum Society Education Committee and the publishers of the previous editions to undertake the publication of the present volume. The advances made in vacuum science and technology during the past three decades has required a complete reworking of the material in the previous volume. However, every effort has been made to follow the unique style of the original bookthat is, to present a survey of fundamental ideas in physics and chemistry that would be useful to both scientists and engineers dealing with problems associated with the use, production, and measurement of high vacuums. This volume is a critical survey of important developments in vacuum technique with many references for those who seek a better understanding and more detailed information in the field. It is not a vacuum handbook, many of which are listed in the Appendix of this book.
Every effort was made to select on a worldwide basis a number of outstanding vacuum specialists who were willing to take time to contribute to this volume. With
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Page vi the curtailment of vacuum research in the major industrial laboratories, one now only finds vacuum experts as
independent consultants or in companies manufacturing vaccum systems and components, a few educational institutions, and government laboratories.
While the basic laws of kinetic theory of gases have not changed over the years, a better understanding of gas flow over a wide range of pressures has necessitated an expanded chapter on the subject. It now encompasses all flow regimes from free molecular flow to atmospheric pressure. It treats compressible flow through tubes and orifices under choked and nonchoked conditions as well as turbulent flow in ducts of any cross section.
Many topics that were only mentioned in the second edition of Scientific Foundations of Vacuum Technique now have full chapters devoted to them. The progress made in vacuum pumps over the past three decades, for example, is remarkable. Three chapters are now devoted to this subject. Detailed information is given for the first time on liquid ring pumps, dry pumps, turbo pumps, getter pumps, and cryopumps.
The subject of leak detection, which had only a few paragraphs devoted to it in the old edition, now has a full chapter describing leak detectors as a rugged industrial tool for everyday use capable of quantitative measurements.
Information on the design of high-vacuum systems has been expanded to help the reader in selecting pump sets for various system applications and in predicting their performance.
Pressure measurements continue to be important on every vacuum system. This subject is fully covered in the chapter on vacuum gauges. While the ionization gauge continues to be the principal pressure sensor for measuring total pressure in high- and ultrahigh-vacuum systems, modern solid-state electronics has simplified its use. The accuracy of this device in measuring pressure depends on a knowledge of the composition of the gas being measured. The partial
pressure analyzer has become a far more sophisticated way to measure pressure and give the vacuum system operator an insight of what is occurring within the system. The invention of the quadrupole mass spectrometer with solid-state electronics has done much to make partial pressure measurements relatively simple and inexpensive. A full chapter is devoted to this subject.
In discussing pressure measurements, a word about pressure units seems appropriate. While use of the pascal, the ISO unit of pressure, has been encouraged in this book, many of the European contributors strongly preferred using the millibar (mbar). The mbar falls in a class of units that are temporarily accepted for use by the ISO. You will find both units in this book. The advantage of the mbar is that it is nearly equal in magnitude to the Torr or mmHg found in earlier publications and is familiar to many readers (1 mbar = 0.75 Torr). When several orders of magnitude of pressure are plotted on a log scale, the mbar and Torr plots are nearly indistinguishable. Some of the figures in this book that have been copied from earlier publications may still have the pressure plotted in Torr.
Ultrahigh-vacuum technique had its infancy in midcentury. Today it is a matured procedure used in a great variety of applications and in commercially available equipment. Researchers do not appear to have reached a limit yet in their quest to produce and measure a perfect vacuum. This work is described in the chapter devoted to ultrahigh and extreme high vacuum. Considerable progress has been made in this area by pushing vacuum techniques to their limit and gaining a better
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Page vii understanding of the gassurface interactions and diffusion in solids as described in the chapter on this subject. While pressures as low as 1011 Pa have been measured in the laboratory, this is still at least three orders of magnitude higher than that in interstellar space.
The final chapter is devoted to calibration and standards. It describes the physical background and state of the art of today's primary vacuum standards in the various national laboratories. It should be useful reading not only for those involved in calibration and quality control but for those interested in the accuracy limitations of various vacuum instruments.
The editor is indebted to several people for suggestions concerning this volume. Special mention is made of the late Hermann Adam for helpful discussions and for suggesting a number of German contributors for the book. John Weed, a member of the American Vacuum Society Education Committee, solicited suggestions for the volume from a number of A.V.S. members. Nigel Dennis coordinated chapters three and four on vacuum pumps, and Benjamin Dayton and Paul Redhead have made many helpful suggestions.
J. M. LAFFERTY
SCHENECTADY, NEW YORK
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Page ix
Contributors
Helmut Bannwarth, LEDERLE GmbH, Gundelfingen, Germany Benjamin B. Dayton, Consultant, East Flatrock, North Carolina, USA
Nigel T. M. Dennis, Edwards High Vacuum International, Crawley, West Sussex, England Johan E. de Rijke, Vacuum Technical Services, Morgan Hill, California, USA
Robert E. Ellefson, Leybold Inficon, Inc., East Syracuse, New York, USA Bruno Ferrario, SAES Getters S.p.A., Lainate (Milano), Italy
Werner Grosse Bley, Leybold Vakuum GmbH, Cologne, Germany Hinrich Henning, Leybold Vakuum GmbH, Cologne, Germany Jörgen Henning, intervac Henning, GmbH, Kreuzwertheim, Germany
John B. Hudson, Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, New York, USA
Karl Jousten, Physikalisch-Technische Bundesanstalt, Berlin, Germany
R. Gordon Livesey, Edwards High Vacuum International, Crawley, West Sussex, England R. Norman Peacock, MKS Instruments, HPS Division, Boulder, Colorado, USA
Paul A. Redhead, Institute for Microstructural Sciences, National Research Council, Ottawa, Ontario, Canada Wolfgang Schwarz, Leybold Systems GmbH, Hanau, Germany
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Page xi
Contents
Preface v
Contributors ix
Acronyms xxiii
1. Kinetic Theory of Gases Benjamin B. Dayton
1
1.1. Ideal Gas Law 2
1.2. Avogadro's Number 6
1.3. Molecular Collisions; Mean Free Path; MaxwellBoltzmann Distribution Laws 8
1.3.1. Relation Between Molecular Velocities and Velocity of Sound 16
1.3.2. Determination of Avogadro's Constant from Distribution of Particles in Brownian Motion 17
1.4. Gas Pressure and Rate at Which Molecules Strike a Surface 18
1.5. Rate of Evaporation and Vapor Pressure 22
1.6. Free Paths of Molecules 26
1.7. Relation Between Coefficient of Viscosity, Mean Free Path, and Molecular Diameter 29
1.7.1. Viscosity at Low Pressures 37
1.7.2. Molecular Diameters 39
1.7.3. Application of the van der Waals Equation 39
1.7.4. From the Density of the Solid or Liquid 40
1.7.5. Cross Section for Collision with Electrons 41
1.8. Heat Conductivity of Gases 41
1.9. Thermal Conductivity at Low Pressures 44
1.9.1. Free-Molecule Conductivity (Knudsen) 46
1.9.2. Temperature Discontinuity (Smoluchowski) 50
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Page xii
1.10. Thermal Transpiration (Thermomolecular Flow) 53
1.11. Thermal Diffusion 57
1.12. Theory of Diffusion of Gases 62
1.12.1. MaxwellLoschmidt Method for Determination of Diffusion Coefficients 65
1.12.2. Effect of Pressure of Gas on Rates of Evaporation of Metals 67
1.13. Random Motions and Fluctuations 69
1.14. Scattering of Particle Beams at Low Gas Pressures 71
References and Notes 73
2. Flow of Gases Through Tubes and Orifices R. Gordon Livesey
81
2.1. Flow Conductance, Impedance, and Gas Throughput 83
2.2. Molecular Flow 85
2.2.1. Conductance of an Aperture 86
2.2.2. General Considerations for Long Ducts 87
2.2.3. General Considerations for Short Ducts 87
2.2.4. Uniform Circular Cross Section 88
2.2.5. Duct of Uniform Rectangular Cross Section 90
2.2.6. Tube of Uniform Elliptical Cross Section 92
2.2.7. Cylindrical Annulus (Flow Between Concentric Cylinders) 93
2.2.8. Uniform Triangular Section (Equilateral) 94
2.2.9. Other Shapes 94
2.2.10. Combinations of Components 96
2.2.11. Cases of Unsteady Flow 102
2.3. Continuum Flow 105
2.3.1. Viscous Laminar Flow 108
2.3.2. Turbulent Flow 112
2.3.3. Compressible Flow 116
2.3.3.1. Flow through an Aperture or Short Duct 119
2.3.3.2. Approximation for Flow Through an Aperture 121
2.3.4. Corrections for Flow Obstructions 121
2.3.5. ApproximationsEntrance Correction Model 122
2.3.6. ApproximationsKinetic Energy Model 124
2.3.7. Long Duct Criteria 126
2.4. Transitional Flow 128
2.4.1. Transitional Flow in Long Ducts 129
2.4.3. Transitional Flow through Apertures and Short Ducts 135
Symbols 137
References 139
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Page xiii
3. Positive Displacement Vacuum Pumps 141
Part I. Oil-Sealed Vacuum Pumps Nigel T. M. Dennis
143
3.1. Oil-Sealed Vacuum Pumps 143
3.1.1. Pump Design 143
3.1.2. Gas Ballast 144
3.1.3. Pump Oil 147
3.1.4. Oil Suckback 148
3.1.5. Power Requirements and System Protection 148
3.1.6. Accessories 149
Part II. Liquid Ring Pumps Helmut Bannwarth
151
3.2. Liquid Ring Pumps 151
3.2.1. Mechanism 151
3.2.2. Single-Stage Liquid Ring Vacuum Pumps 152
3.2.3. Two-Stage Liquid Ring Vacuum Pumps 153
3.2.4. The Operating Liquid 154
3.2.5. Operating Ranges of Liquid Ring Gas Pumps 154
3.2.7. Types of Operation; Conveyance of Operating Liquid 156
3.2.8. Materials of Construction 157
3.2.9. Sealing 157
3.2.10. Drives 157
3.2.11. Accessories 158
Part III. Dry Vacuum Pumps Nigel T. M. Dennis
159
3.3. Dry Vaccum Pumps 159
3.3.1. Roots Pump 159
3.3.2. Claw Pump 162
3.3.3. Screw Pump 164
3.3.4. Scroll Pump 167
3.3.5. Piston and Diaphragm Pumps 169
References 170
General References 171
4. Kinetic Vacuum Pumps 173
Part I. Diffusion and Diffusion-Ejector Pumps Benjamin B. Dayton
175
4.1. Diffusion Pumps 176
4.1.1. History of Development 176
4.1.2. Diffusion Pump Design 181
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Page xiv
4.2. Diffusion-Ejector Pumps 183
4.3. Performance of Vapor-Jet Pumps 185
4.3.1. Pumping Speed 185
4.3.2. Limiting Forepressure for Maximum Speed 187
4.3.3. Influence of Nozzle and Entrance Chamber Design on Speed 190
4.3.4. Ultimate Pressure 192
4.3.5. Backstreaming and Back Migration of Pump Fluid 194
4.3.6. Throughput 198
4.4. Theory of Pump Performance 202
4.4.1. Speed 202
4.4.2. Limiting Forepressure 204
4.4.3. Vapor-Jet Flow Pattern 205
4.4.4. Ultimate Pressure 221
Part II. Molecular Drag and Turbomolecular Pumps Jörgen Henning
233
4.5. Molecular Drag Pumps 233
4.5.1. Theoretical Considerations and Performance Data 234
4.5.2. Design Considerations 237
4.5.3. Typical Performance Data of Commerical Pumps 237
4.5.3.1. Compression 238
4.5.3.2. Pumping Speed 238
4.5.3.3. Ultimate Pressure 238
4.6. Turbomolecular Pumps 238
4.6.1. Theoretical Considerations and Performance Data 239
4.6.2. Design Considerations 241
4.6.2.1. Rotor and Stator Geometry 241
4.6.2.2. Rotor Suspension 242
4.6.2.3. Lubrication of Mechanical Bearings 242
4.6.2.4. Magnetic Rotor Suspension 242
4.6.2.5. Balancing and Vibration 243
4.6.2.6. Rotor Materials 243
4.6.2.7. Drive Systems 243
4.6.3. Applicational Considerations 243
4.6.3.1. Venting 243
4.6.3.2. Baking 244
4.6.3.4. Operation in Magnetic Fields 244
4.6.3.5. Pumping Corrosive Gases 245
4.6.3.6. Pumping Toxic or Radioactive Gases 245
4.6.3.7. Turbomolecular Pumps in Combination with Other Pumps 245
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4.6.4. Performance Data of Commercial Pumps 245
4.6.4.1. Compression 246
4.6.4.2. Pumping Speed 246
4.6.4.3. Ultimate Pressure 247
4.7. Combined Molecular Drag and Turbomolecular Pumps 247
4.7.1. Design Considerations 248
4.7.2. Typical Performance Data for Commericial Combined Molecular Drag and Turbomolecular Pumps
248
4.7.2.1. Compression 248
4.7.2.2. Pumping Speed 248
4.7.2.3. Ultimate Pressure 248
4.8. Backing Pumps 248
Part III. Regenerative Drag Pumps Nigel T. M. Dennis
251
4.9. Regenerative Drag Pumps 251
4.9.1. Mechanism 251
References 254
5. Capture Vacuum Pumps 259
Bruno Ferrario
5.1. Types of Gas Surface Interactions 261
5.2. Basic Concepts of Getter Materials 262
5.3. Adsorption and Desorption 263
5.4. Bulk Phenomena 265
5.4.1. Diffusion 265
5.4.2. Solubility 267
5.5. Equilibrium Pressures 268
5.6. Getter Materials 269
5.6.1. Basic Characteristics of Getter Materials 269
5.6.2. Sorption Speed and Sorption Capacity 269
5.6.3. Principal Types of Getter Materials and Their General Working Conditions 271
5.6.4. Interaction of Getters with Common Residual Gases 275
5.6.5. Evaporable Getters 275
5.6.5.1. Ba Getters 276
5.6.5.2. Titanium Sublimation Getter Pumps 291
5.6.6. Nonevaporable Getters 297
5.6.6.1. Ternary Alloys 305
5.6.6.2. Other Ternary and Multicomponent Alloys 310
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Page xvi
5.7. Getter Configurations 310
5.8. Getter Applications 313
5.8.1. Nonevaporable getters Versus evaporable getters 314
5.8.2. Start-up and Working Conditions of Getters 315
Part II. Sputter Ion Pumps Hinrich Henning
317
5.9. Gas Discharge Vacuum Pumps 317
5.10. The Penning Discharge 319
5.10.1. Pump Sensitivity 321
5.10.2. Ion Motion 323
5.10.3. Electron Cloud 324
5.10.4. Secondary Electrons 326
5.10.5. Transition from HMF Mode to HP Mode 327
5.10.6. Transition from LMF Mode to HMF Mode 328
5.10.7. Sputtering 329
5.11. SIP Characteristics 329
5.11.1. Gettering 329
5.11.2. Ion Burial 330
5.11.3. Volume Throughput 331
5.11.4. Pumping Mechanism 335
5.11.5. Bakeout 338
5.11.6. Types of SIPs 338
5.11.7. Starting Properties 342
5.11.8. Memory Effect 343
5.11.9. Ultimate Pressure 343
5.11.10. Magnets 345
Part III. Cryopumps Johan E. de Rijke
347
5.12. AdsorptionDesorption 348
5.13. Cryotrapping 352
5.14. Pumping Speed and Ultimate Pressure 353
5.15. Capacity 355
5.16. Refrigeration Technology 357
5.17. Pump Configuration 359
5.18. Regeneration 363
5.20. Sorption Roughing Pumps 364
References 368
General References 373
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6. Vacuum Gauges R. Norman Peacock
375
6.1. Pressure Units Used in Vacuum Measurements 377
6.2. Liquid Manometers 378
6.3. McLeod Gauge 379
6.4. Piston Pressure Balance Gauge 381
6.5. Bourdon Gauge 382
6.6. Capacitance Diaphragm Gauges 384
6.6.1. Sensitivity of the Capacitance Method 385
6.6.2. Deflection of a Thin Tensioned Membrane 386
6.6.3. Accuracy of Commercial Gauges 387
6.6.4. Thermal Transpiration 388
6.6.5. Conclusions 388
6.7. Viscosity Gauges 389
6.7.1. Spinning Rotor Gauge 391
6.7.1.1. Theory 391
6.7.1.2. Commercial Gauges 394
6.7.1.4. Secondary or Transfer Standard 399
6.7.1.5. Use Precautions 401
6.7.1.6. Advantages and Disadvantages 401
6.7.2. Oscillating Quartz Crystal Viscosity Gauge 402
6.7.2.1. Advantages and Disadvantages 402
6.8. Thermal Conductivity Gauges 403
6.8.1. Theory 404
6.8.2. Calibration 406
6.8.3. Lowest Useful Pressure 408
6.8.4. Constant Pressure Pirani 409
6.8.5. Calibration Dependence Upon the Gas 410
6.8.6. Upper Pressure Limit 410
6.8.7. Ambient Temperature Compensation 411
6.8.8. Comparison of Pirani and Thermocouple Gauges 412
6.8.9. Stability 412
6.8.10. Thermistor Pirani Gauges and Integrated Transducers 412
6.8.11. Commercial Gauges and Applications 413
6.9. Ionization Gauges 414
6.9.1. Hot-Cathode Gauge Equation 414
6.9.2. Geometric Variations in the BayardAlpert Gauge 419
6.9.3. Modulated BayardAlpert Gauge 421
6.9.4. Extractor Gauge 422
6.9.5. Helmer Gauge 423
6.9.6. Long Electron Path Length Gauges 424
6.9.7. Secondary Standard Hot-Cathode Gauges 425
6.9.8. High-Pressure Ionization Gauges 426
6.9.9. Cold-Cathode Gauges 427
6.9.10. Ionization Gauge Accuracy 435
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6.9.11. Gauge Constant Ratios for Different Gases 438
6.9.12. Ionization Gauge Controllers 439
References 441
7. Partial Pressure Analysis Robert E. Ellefson
447
7.1. Ion Sources 447
7.1.1. Electron-Impact Ionization Process 448
7.1.2. Open Ion Source 449
7.1.3. Closed Ion Source 452
7.2. Ion Detection 454
7.2.1. Faraday Cup Ion Detection 454
7.2.2. Secondary Electron Multiplier Detection 454
7.2.3. Microchannel Plate Detector 456
7.3. Mass Analysis 456
7.3.1. Quadrupole Mass Analyzer 456
7.3.2. Magnetic Sector Analyzer 460
7.3.3. Time-of-Flight Mass Analyzer 464
7.3.4. Trochoidal (Cycloid) Mass Analyzer 465
7.3.5. Omegatron 466
7.4. Optical Measurement of Partial Pressures 467
7.4.1. Photoionization Measurement of Partial Pressure 468
7.4.2. Infrared Absorption Measurement of Partial Pressure 469
7.5. Computer Control, Data Acquisition, and Presentation 470
7.6. Residual Gas Analysis 471
7.7. Pressure Reduction Sampling Methods for Vacuum Process Analysis 474
7.8. Calibration of Partial Pressure Analyzers 475
References 477
8. Leak Detection and Leak Detectors Werner Grosse Bley
481
8.1. Principles of Vacuum Leak Detection 482
8.1.1. Types of Leaks and Leak Rate Units 482
8.2. Total Pressure Measurements 484
8.3. Partial Pressure Measurements 486
8.4. Measurement of Leakage Rates with Helium Leak Detectors 486
8.5. Helium Leak Detection of Vacuum Components 487
8.7. Special Methods and Other Tracer Gases 493
8.8. Mass Spectrometer Leak Detectors 493
8.8.1. Mass Spectrometer System for Helium Leak Detection 494
8.8.2. Direct-Flow Helium Leak Detectors 494
8.8.3. Simple Counterflow Helium Leak Detectors 496
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8.8.4. Advanced Counterflow Helium Leak Detectors 498
8.8.5. Oil-Free and Dry Helium Leak Detectors 499
8.9. Specifications of Mass Spectrometer Leak Detectors 500
8.10. Quantitive Leakage Rate Measurements 502
8.11. Mass Spectrometer Leak Detectors for Other Tracer Gases and Future Developments in Leak Detection
504
References 505
9. High-Vacuum System Design Wolfgang Schwarz
507
9.1. Calculations of Vacuum Systems 507
9.1.1. Basic Pumpdown Equations 508
9.1.2. Process Pressure 511
9.2. Gas Loads in High-Vacuum Systems 513
9.2.1. Outgassing 513
9.2.2. Leaks 516
9.2.3. Permeation 516
9.2.4. Process Gas 518
9.3. Design of High-Vacuum Pump Sets 519
9.3.1.1. Fore-Vacuum Pumps 519
9.3.1.2. Roots Combinations 520
9.3.2. High-Vacuum Pump Sets 524
9.3.2.1. Turbomolecular Pump Sets 526
9.3.2.2. Diffusion Pump Sets 528
9.3.2.3. Pump Sets with Cryosurfaces 531
9.3.2.4. Cryopump Sets 535
9.4. Calculation Methods for Vacuum Systems 537
9.4.1. Analytical Approximations 538
9.4.2. Numerical Methods 541
9.4.2.1. Dedicated Software 541
9.4.2.2. Network Approach 542
General References 546
10. GasSurface Interactions and Diffusion John B. Hudson
547
10.1. Adsorption 548
10.1.1. Basic Equations 548
10.1.2. Adsorption Isotherms 551
10.1.3. Heat of Adsorption 567
10.1.4. Observed Behavior 568
10.1.5. Adsorption Kinetics 572
10.1.6. Chemisorption Kinetics 575
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Page xx
10.1.7. Kinetic Measurements 582
10.1.8. Capillarity Effects 584
10.2. Absorption 589
10.2.1. Equilibrium Solubility 590
10.2.2. Diffusion Rates 590
10.2.3. Kinetics of Absorption and Permeation 591
10.2.4. Steady-State Permeation 592
10.2.5. Transient Permeation 595
10.2.6. Effect of Desorption Kinetics on Permeation 600
10.3. Surface Chemical Reactions 606
10.4. Outgassing Behavior 614
10.4.1. Desorption of Adsorbed Gases 614
10.4.2. Dissolved Gases 616
10.4.3. Overall Pumpdown Curves 616
10.4.4. Mitigation of Outgassing 619
10.4.5. Surface Treatments During Construction 619
10.4.6. In Situ Surface Treatments 620
10.4.7. Bakeout Processes 620
References 622
11. Ultrahigh and Extreme High Vacuum Paul A. Redhead
625
11.1. Limits to the Measurement of UHV/XHV 628
11.1.1. Residual Currents 629
11.1.2. Effects at Hot Cathodes 636
11.1.3. Gauges with Long Electron Paths 639
11.1.4. Comparison of UHV/XHV Gauges 641
11.2. Limits to Pumps at UHV/XHV 642
11.2.1. Kinetic Pumps 642
11.2.2. Capture Pumps 643
11.2.3. Comparisons of Pumps for UHV/XHV 646
11.3. Leak Detection at UHV/XHV 647
11.4. Outgassing 648
11.4.1. Reduction of Outgassing Rates 648
11.5. UHV/XHV Hardware 652
12. Calibration and Standards Karl Jousten
657
12.1. Primary Standards 658
12.1.1. Liquid Manometers and Piston Gauges 659
12.1.2. Static Expansion 661
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12.1.3. Continuous Expansion 665
12.1.4. Molecular Beam Expansion 670
12.2. Calibration by the Comparison Method 673
12.3. Calibration of Vacuum Gauges and Mass Spectrometers 676
12.3.1. Capacitance Diaphragm Gauges 676
12.3.2. Spinning Rotor Gauges 680
12.3.3. Ionization Gauges 683
12.3.4. Mass Spectrometers 686
12.4. Calibration of Test Leaks 689
12.5. Measurement of Pumping Speeds 692
References 695
Appendix 701
Graphic Symbols for Vacuum Components 701
Conversion Factors for Pressure Units 708
Vapor Pressure of Common Gases 709
Vapor Pressure of Solid and Liquid Elements 711
General Reference Books on Vacuum Science and Technology 714
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Page xxiii
Acronyms
Acronyms used in the text are listed under the chapter where they occur.
Chapter 5
Part I. Getters and Getter Pumps
bcc Body-centered cubic (crystal structure) CCRT Color cathode ray tube
CRT Cathode ray tube
fcc Face-centered cubic (crystal structure) FED Field emission display
hcp Hexagonal close-packed (crystal structure) HPTF High-porosity thick film (getter)
HT High temperature
HV High vacuum
LN2 Liquid nitrogen
PDP Plasma display panel
rf Radio frequency
RT Room temperature
SEM Scanning electron microscope
UHV Ultrahigh vacuum
Chapter 5
Part II. Sputter Ion Pumps
BA BayardAlpert (gauge)
DI Diode sputter ion pump with two cathode materials FEM Finite element method
HMF High magnetic field (mode) HP High pressure (mode) LMF Low magnetic field (mode) SIP Sputter ion pump
TM Transition (mode)
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Page xxiv
UHV Ultrahigh vacuum
VTP Volume throughput
Chapter 5
Part III. Cryopumps
BET BrunauerEmmettTeller (adsorption model) GM GiffordMcMahon (thermodynamic cycle)
Chapter 6 Vacuum Gauges
BAG BayardAlpert gauge CCD Cold-cathode gauge
CDG Capacitance diaphragm gauge ESD Electron-stimulated desorption FS Full scale
HCG Hot-cathode gauge
JHP Jauge haut pression (Choumoff gauge)
NIST National Institute of Standards and Technology (USA) PTB Physikalisch-Technische Bundesanstalt (Germany) QBG Quartz helix Bourdon gauge
SCR Silicon controlled rectifier SRG Spinning rotor gauge UHV Ultrahigh vacuum XHV Extreme high vacuum
Chapter 7
Partial Pressure Analysis
CRDS Cavity ringdown spectroscope
IP Ionization potential
IR Infrared
MCP Microchannel plate
MS Mass spectrometer
PPA Partial pressure analyzer
QMS Quadrupole mass spectrometer
rf Radio frequency
RGA Residual gas analyzer
SEM Secondary electron multiplier
TOF Time of flight
TOFMS Time-of-flight mass spectrometer
UHV Ultrahigh vacuum
XHV Extreme high vacuum
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Page xxv Chapter 11
Ultrahigh and Extreme High Vacuum
BA BayardAlpert (gauge)
BAG BayardAlpert gauge
ESD Electron stimulated descorption
EXB Cycloidal mass spectrometer
GIP Getter-ion pump
MG Magnetron gauge
IMG Inverted magnetron gauge
MS Mass spectrometer
MBAG Modulated BayardAlpert gauge
NEG Nonevaporable getter
RGA Residual gas analyzer
SIP Sputter-ion pump
TMP Turbomolecular pump
TSP Titanium sublimation pump
UHV Ultrahigh vacuum
XHV Extreme high vacuum
Chapter 12
Calibration and Standards
AVS American Vacuum Society CDG Capacitance diaphragm gauge DIN German industry standard DKG German calibration service IG Ionization gauge
IMGC Institutodi Metrologia ''G. Colonnetti" (Italy) ISO International Organization for Standards
NIST National Institute of Standards and Technology (USA) NPL National Physical Laboratory (England)
NPL National Physical Laboratory (India)
PTB Physikalisch-Technische Bundesanstalt (Germany)
QBS Quartz Bourdon spiral manometer
SRG Spinning rotor gauge UHV Ultrahigh vacuum XHV Extreme high vacuum
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Page 1
1
Kinetic Theory of Gases Benjamin B. Dayton
For a proper understanding of phenomena in gases, more especially at low pressures, it is essential to consider these phenomena from the point of view of the kinetic theory of gases [1]. This theory rests essentially upon two fundamental assumptions. The first of these postulates is that matter is made up of extremely small particles, which in the gaseous state at moderate temperatures are monatomic or polyatomic molecules and at higher temperatures may be entirely dissociated into atoms or even into positive ions and electrons to form a "plasma." The second postulate is that the molecules of a gas are in constant motion, and this motion is intimately related to macroscopic properties known as the temperature and pressure, which characterize the state of the gas in a given small region.
The center of mass of the molecule is assumed to move in a straight line (neglecting the force of gravity) with a constant velocity between collisions with other molecules, and the forces between molecules are negligible except when the molecular centers approach within a distance known as the mean molecular diameter. Velocity is a vector which must be measured with respect to some "fixed" reference frame. The magnitude of the velocity vector is called the speed of the molecule. In the "laboratory reference frame" the motion of the particles may be divided into bulk or fluid motion due to pressure or concentration gradients which generate mass flow or diffusive flow and the random velocity components associated with the concept of temperature as measured in a reference frame moving with the fluid flow velocity. Kinetic energy is proportional to the square of the molecular velocity and is a scalar quantity.
Foundations of Vacuum Science and Technology, Edited by James M. Lafferty.
ISBN 0-471-17593-5 © 1998 John Wiley & Sons, Inc.
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Page 2 Temperature is defined as a quantity proportional to the average (kinetic) energy of translation of the particles in a reference frame moving with any fluid flow at a given small region. In the case of monatomic molecules (such as those of the rare gases and the vapors of most metals), the effect of increased temperature is evidenced by increased
translational energy of the molecules. In the case of diatomic and polyatomic molecules, an increase in temperature also increases through intermolecular collisions the rotational energy of the molecule about one or more axes, as well as vibrational energy of the constituent atoms with respect to mean positions of equilibrium. However, in the following discussion, only the effect on translational energy will be considered.
1.1
Ideal Gas Law
According to the kinetic theory, a gas exerts pressure on the enclosing walls because of the impact of molecules on these walls. Since the gas suffers no loss of energy through exerting pressure on the stationary solid wall of its enclosure, it follows that each molecule is thrown back from the wall with the same speed as that with which it impinges, but in the reverse direction with respect to the normal; that is, the impacts are perfectly elastic.
Suppose a molecule of mass m to approach a flat wall surface lying in the x, y plane with velocity component vz perpendicular to the wall. Since the molecule rebounds with the same speed, the change of momentum per impact is 2mvz. If ν molecules strike unit area in unit time with velocity component vz, the total impulse exerted on the unit area per unit time is 2mvzν. But the pressure, P, on a wall is defined as the rate at which momentum is imparted to a unit area of surface. Hence,
where the summation is over all values of vz ν and it is assumed that all molecules have the same mass m.
It now remains to calculate ν. Of all the molecules within a volume ∆V extending outward from a small area, ∆x∆y, of the wall by a distance |vzdt|, where dt is a short interval of time, at equilibrium only one half of the molecules will be moving with velocity components vz toward the wall. Let nz denote the number of molecules per unit volume in the gas within ∆V that have a velocity component of either vz or vz. Then the flux rate against the wall will be ν = nzυz/2, and Eq. (1.1) becomes
The total speed of a molecule with velocity components vx, vy, and vz is the square root of the quantity
Defining the averages
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Page 3 where n is the total number of molecules per unit volume regardless of velocity components, we have by symmetry
and
Then from Eq. (1.2) the basic equation for the pressure is
where ρ denotes the gas density, Eq. (1.6) can be expressed in the form
which shows that, at constant temperature, the pressure varies directly as the density, or inversely as the volume. This is known as Boyle's law.
Now it is a fact that no change in temperature occurs if two different gases, originally at the same temperature, are mixed. This result is valid independently of the relative volumes. Consequently, the average kinetic energy of the molecules must be the same for all gases at any given temperature, and the rate of increase with temperature must be the same for all gases. We may therefore define temperature in terms of the average kinetic energy per molecule, and this suggestion leads to the relation
for each of the three degrees of freedom of translational motion, where T is the absolute temperature (degrees Kelvin), defined by the relation T = 273.15 + t (t = degrees Centigrade), and k is a universal constant, known as the Boltzmann constant. The total mean translational energy is then
where υ r is known as the root-mean-square velocity. The total kinetic energy of the molecules in a volume V will be
Combining Eq. (1.8) with (1.5) it follows that Boyle's law can be expressed in the form
where the units chosen must be consistent with either the cgs system or the SI system. When n is expressed as molecules/cm3 and k is in erg/K, then P will be the pressure in
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Page 4 dyne/cm2. Also, from Eq. (1.6) and Eq. (1.11), it follows that
which is known as Charles' law or Gay-Lussac's law.
Lastly, let us consider equal volumes of any two different gases at the same values of P and T. Since P and V are respectively the same for each gas, and is constant at constant value of T, it follows from Eqs. (1.10) and (1.11) that n must be the same for both gases. That is, equal volumes of all gases at any given values of temperature and pressure contain an equal number of molecules. This was enunciated as a fundamental principle by Avogadro in 1811, but it took about 50 years for chemists to understand its full significance.
On the basis of Avogadro's law the molecular mass, M, of any gas or vapor is defined as that mass in grams, calculated for an ideal gas, which occupies, at 0°C and 1 atmosphere, a volume [2]
V0 = 22,414.10 cm3.
This is therefore designated the molar volume, and the equation of state for an ideal gas can be written in the form
where W is the mass in grams, M is the molecular mass in grams, and R0 is a universal constant in units which depend on the choice of units for the volume V, the pressure P, and the absolute temperature T. It is then convenient to express Eq. (1.14) in the form
where nM denotes the number of moles (corresponding to M in grams) in the volume V under the given conditions of temperature and pressure.
Excellent summaries of the various proposals prior to 1967 regarding the unit of pressure to be used in vacuum science and technology have been given by Thomas and Leyniers [3]. When the units of length [l], mass [m], and time [t] are chosen for the three basic units of a coherent system, the pressure unit is expressed by
[p] = [l]1 [m][t]2 or by
[p] = [F][l]2,
where F is the force with dimensions [l][m][t]2. In the cgs system the unit of force is the dyne and the pressure unit is 1 dyne/cm2. In the MKS or SI system the unit of
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Page 5 force is the newton (abbreviated N), defined as 1 kg · m · s2 and the pressure unit is 1 N · m2. In 1962 the French government approved the term "pascal" (abbreviated Pa) for the newton per square meter [4], and this term was
officially adopted by the International Standards Organization in 1978 while deprecating the use of older pressure units such as mmHg and the torr (or Torr) [5]. Decimal multiples and submultiples of the Pa are accepted, and some have special nomenclature such as
1 bar = 105 Pa,
1 millibar (mbar) = 100 Pa.
The previously used unit of 1 microbar (µbar) = 106 bar for the dyne/cm2 equals 101 Pa or 7.5 × 104 Torr. The main reason for now using the mbar in vacuum technology is that its value is close to that of the Torr and the mmHg, which were used for many years in the vacuum industry and the scientific literature. In 1954 both the British [6] and American Committees on Vacuum Nomenclature [7] recommended replacing the mmHg by the Torr, defined as exactly
1,013,250/760 dyne/cm2, thus making it independent of the changing values for the measured density of mercury and the acceleration of gravity. The term torr had previously been used in the German literature as a substitute for mmHg, and at the First International Congress on Vacuum Techniques in Belgium in 1958 the newly defined Torr was favored by the Germans and the Americans while the pascal was favored by the Belgians and the French representatives [8].
The Torr then became widely used throughout the world except in France until 1978, when the International Standards Organization deprecated the use of Torr and defined the standard atmosphere as exactly 101,325 Pa and recommended the pascal. Thus
In practical use the conventions of 1 Torr = 133.322 Pa and 1 Torr = 1333.22 µbar are sufficient. It may be noted that the Torr is not identical to the unit mmHg used prior to 1955 which is based on the equation
where ρ = 13.5951 g/cm3 (at 0°C) is the density of mercury, g = 980.665 cm/s2 was the standard acceleration of gravity accepted at that time, and h is the height (in mm) of a mercury column in a tube above some reference point such as the surface of the mercury in the reference tube of a U-tube manometer or a McLeod gauge. One bar = 750.06 mmHg.
Another unit no longer used is the micron = 103 mmHg.
The basic unit of volume in the cgs system is 1 cm3, and that in the MKS or SI system is 1 m3. Following the Twelfth Conférence Général des Poids et Mesures in 1964 the liter (or litre) is defined as 1 dm3 or 103 cm3 or 103 m3. The preferred abbreviation for the liter is L, but l and the script l have been used. In the United States the volume unit
associated with industrial vacuum chambers and gas flow through mechanical pumps is commonly the cubic foot, which equals 28.316847 liter. The basic unit of mass in the cgs system is the gram (g), and that in the MKS system is
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Page 6 the kilogram (kg). The pound avoirdupois is seldom used for mass, but the pound force (poundf) is used in industrial engineering in connection with the pressure unit pound per square inch (lb/in2.). 1 lb/in2. = 0.0689476 bar or 6.89476 kPa.
Units of temperature are the Celsius degree (°C) (also known as the degree centigrade) and the degree Kelvin (K) on the absolute or thermodynamic scale. The zero of the Kelvin scale is 273.15°C, so that temperature T (in K) = temperature t (in °C) + 273.15. The Fahrenheit temperature scale is seldom used in vacuum physics.
From the equation of state for an ideal gas we obtain the following values for the universal gas constant R0:
R0 = 760 · 22.41383/273.15 = 62.3632 Torr · L · K1 · mol1
= 1013250 · 22414.10/273.15 = 8.314511 · 107 erg · K1 · mol1
= 8.314511 joule · K1 · mol1
= 8.314511/4.1840 = 1.9872 cal (thermochemical) K1 · mol1.
In dealing with gases at low pressures, it is convenient to express the quantity of gas (Q = PV) in Pa · m3 at standard room temperature 23°C. Note that 1 Torr · L = 0.133322 Pa · m3. The quantity of gas thus defined flowing per unit time through a given cross section of a pipe has been called the throughput in the British and American Glossaries of terms used in vacuum technology [9].
The unit of throughput is then
1 Pa · m3 · s1 = 1 N · m · s1 = 1 J · s1 = 1 W.
Objections have been raised against the use of pressurevolume products for quantity of gas on the basis that the
temperature of the walls of a vacuum system is not always clearly specified, and the suggestion is made that flow of gas should preferably be expressed in mol/s to avoid any ambiguity due to unspecified temperatures [10]. The term
throughput should be reserved for flow in Pa · m3 · s1 at some specified temperature, such as a "standard" room temperature of 23°C, and a term such as molar flow rate should be used for flow in mol · s1. Then
1 mol · s1 = 8.31451 × 296.15 Pa · m3 · s1 = 2462.342 Pa · m3 · s1 at 23°C.
1.2
Avogadro's Number
Avogadro's law states that the number of molecules per gram-molecular mass is a constant, which is designated NA.
Although a number of different methods have been used for the determination of this constant, the most accurate method depends upon the determination of both the Faraday (F) and the charge of the electron (e).
For the deposition of one gram-equivalent the most accurate value [2] is F = 96,485.309 absolute coulombs.
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Page 7 Assuming that each univalent ion has a charge equal in magnitude to that of one electron,
NA = F/e.
The value of the charge of the electron is e = 4.8032068 · 1010 absolute esu
= 1.60217733 · 1019 absolute Coulomb
since 1 absolute Coulomb = 1/10c times the charge in esu absolute coulombs, where c = velocity of light = 2.99792458
· 108 m · s1. Hence
NA = 6.0221367 × 1023 mol1.
From Eqs. (1.11) and (1.15) it follows that for one mole of gas in the volume V, nM = 1 and n = NA/V so that
From Eq. (1.9) it also follows that the average kinetic energy per molecule is given by
The mass per molecule is evidently
From Eq. (1.11) it follows that the number of molecules per cubic centimeter is given by
where Pµb is the pressure in microbars and PPa is the pressure in pascal units, while
where Pτ is the pressure in Torr.
Table 1.1 gives values of n for a series of values of T, Pµb, PPa and Pτ (Torr). The value n = 2.687 × 1019 cm3 is
known as the Loschmidt number.
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Page 8
Table 1.1. Number of Molecules per Cubic Centimeter
T(°K) Pµb PPa Pτ n
273.15
1.01325 × 106 1.01325 × 105 760 2.687 × 1019
298.15
1.01325 × 106 1.01325 × 105 760 2.461 × 1019
273.15
1.333 × 103 1.333 × 102 1 3.535 × 1016
298.15
1.333 × 103 1.333 × 102 1 3.239 × 1016
273.15
10 1 7.50 × 103 2.652 × 1014
298.15
10 1 7.50 × 103 2.429 × 1014
273.15
1 0.1 7.50 × 104 2.652 × 1013
298.15
1 0.1 7.50 × 104 2.429 × 1013
1.3
Molecular Collisions; Mean Free Path; MaxwellBoltzmann Distribution Laws
It is evident that there must be a nonuniform distribution of velocities among all the molecules in a given volume because of the constant occurrence of collisions. In an elementary treatment of the collision process the two molecules are assumed to be solid spheres of mass m1 and a well-defined diameter δ1 for one molecule and m2 and δ2 for the second molecule. The collisions are assumed to be elastic; that is, no translational kinetic energy is lost by excitation of molecular rotation or atomic vibrations or by excitation of molecular vibrations when one of the molecules is part of the wall of the enclosure. While the molecular diameters on impact can only be defined in terms of the fields of attractive and repulsive force around each molecule and the relative momenta, it is assumed for simplicity that δ1 and δ2 have fixed values.
The distance between centers on impact will then be
and the mutual collision cross section is defined as
If we consider a single molecule of diameter δ1 (cm) moving at high speed, v, through a gas composed only of molecules of diameter δ2 (cm) moving relatively slowly and having a concentration of n molecules per cm3, then it is obvious that the average distance (in cm) between collisions, known as the mean free path, will be given by
and the collision rate will be v/Lc = nσcv.
In an actual gas at equilibrium where all molecules have random velocities with an average speed va, the mean speed of a first molecule relative to one of the other molecules depends on the angle between their respective directions of motion and the distribution law for molecular velocities and will be equal to 21/2va in a homogeneous
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Page 9 gas as calculated by Maxwell [11]. The collision rate is therefore
and the mean free path in a homogeneous maxwellian gas is
where in this case δ 1 = δ 2 = δ and σ c = πδ 2. This quantity is of fundamental importance in vacuum physics since the Knudsen number defined by the ratio
where Dc is a characteristic length in the vacuum system, such as the diameter of a cylindrical tube, determines the physics of the gas flow through tubes and ducts. From Eq. (1.11) and Eq. (1.26) in cgs units we obtain
and for nitrogen [12] we have δ = 3.78 × 108 cm, giving
At T = 298 K and Pτ = 103 Torr, this gives a mean free path for nitrogen L = 4.92 cm. At this temperature and Pµb = 1 µbar = 0.1 Pa = 7.5 × 104 Torr, L = 6.56 cm. This pressure is regarded as the upper limit of the high vacuum region [13]
in which the molecules may collide with the walls of the vacuum device more frequently than with other molecules in the gas phase. We therefore must also consider the case in which a molecule of the gas collides with a molecule which is part of the wall.
In the laboratory reference frame the velocity vector for the first molecule before the collision may be represented as
and for the second molecule it may be expressed as
where u1 and u2 are the velocity components perpendicular to the line of centers at the time of impact while v1 and v2 are the velocity components along this line of centers before collision as shown in Fig. 1.1.
Similarly, after the collision the velocity vectors for the first and second molecules, respectively, may be represented by
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Page 10
Fig. 1.1 Velocity components
in elastic collision between spheres.
where V1 and V2 are the respective velocity components along this line of centers after the collision, while U1 and U2 are the respective velocity components perpendicular to this line of centers after the moment of collision. For a collision to take place, v1 and v2 must be antiparallel with vectors directed toward each other; but if they are parallel, one
velocity component must be greater than the other, and the relative positions, must be such that the molecule with the larger component overtakes the other molecule in this direction.
The conservation of linear momentum in elastic collisions requires
Also, for the momentum components we require
The conservation of translational energy requires
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Page 11 From Eqs. (1.36), (1.37), (1.38), and (1.39) we have
Also, from Eqs. (1.34) and (1.36) we obtain
Dividing Eq. (1.40) by Eq. (1.41) gives
Combining Eq. (1.42) and Eq. (1.43), we obtain
and similar calculations give
Equations (1.44) and (1.45) together with Eq. (1.36) give the vector components after the collision in terms of the components before the collision. If m1 = m2, then V1 = v2 and V2 = v1 so that the velocity components along the line of centers are interchanged. The resultant speeds after the collision are then
If the initial velocity component v2 was greater than v1, then the collision will cause the first molecule to increase in velocity and the second molecule will suffer a decrease in velocity.
To illustrate how some molecules can acquire very high velocities while others are brought to near zero velocity as a result of collisions, we consider a special class of orthogonal collisions between like molecules in which the first molecule has an initial velocity u1 orthogonal to the line of centers at impact while the second molecule has an initial velocity v2 parallel to this line of centers. In this special case we have v1 = 0, u2 = 0, V1 = v2, V2 = 0, U2 = 0 and the resultant velocity of the second molecule is zero while the speed of the first molecule after the collision has been increased from u1 to W1 as given by Eq. (1.46). Note that we could have assumed the second molecule to have any value of a velocity component u2 other than zero orthogonal to
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Page 12 the line of centers, but this component would have no effect on the final velocity of the first molecule as given by Eq.
(1.46).
If we consider only orthogonal collisions in which the first molecule is repeatedly struck in successive collisions by one of the other molecules with a velocity component v2 which is a constant equal to the most probable velocity, vm, and assume that the initial u1 = vm also, then N successive orthogonal collisions would give a final speed
By a process of reasoning similar to that for calculating the rate of absorbtion of a molecular beam by scattering in a gas [14], it can be shown that the probability of N + 1 or more successive orthogonal collisions, without collisions of
another type, is
which is the probability that the speed of m1 is equal to or greater than W1(0 < W1 < ∞), where b is a constant to be determined. When N is large, the velocity W1 can be much greater than vm but the probability is quite small. This resembles the actual distribution law for molecular speeds as determined by J. C. Maxwell and L. Boltzmann.
The form of the distribution law differs according to the particular type of velocity distribution of interest. If we let designate the components, along the three coordinate axes, of the randomly directed velocity v, then
and the distribution function, with respect to, say, is given by the relation
where N = number of molecules in the volume under consideration.
The distribution function for all three components of v has the form
With respect to the polar coordinates θ and φ, we have
The most important distribution function is that with respect to v in a random direction, which is given by the relation
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Page 13
Hence α corresponds to the value of the most probable velocity, which is given by the relation
In terms of c = v/ α , we can express Eq. (1.53) in the form
where c varies from 0 to ∞ , and dy (= fcdc) corresponds to the fraction of the total number of molecules which have values of c ranging between c and c + dc. Hence
From Eq. (1.57) we derive the value of the arithmetical average velocity, va = αca, where
and, using Eq. (1.55), we obtain
The root-mean-square velocity, vr, corresponds to the square root of the average value of v2 as derived from Eq. (1.9) and is therefore given by the relation
The second column in Table 1.2 gives values of fc for a series of values of c, and Fig. 1.2 shows a plot of these data, on
which the values of fc are indicated for the values c = 1, 1.1284, and 1.2247, respectively.
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Page 14
Fig. 1.2
Plots illustrating Maxwell-Boltzmann distribution laws. Plot fc shows distribution function for random velocity, c expressed in terms of the most probable velocity α ; plot fx shows distribution function for energy, E, in terms of x = E/(kT); y corresponds
to the fraction of the total number of molecules for which the random velocity (expressed in terms of α) is less than or equal to a given value c.
gives the fraction of the total number of molecules which have a random velocity equal to or less than that
corresponding to the value c, or to v = αc. The third and fourth columns in Table 1.2 show values of y and of ∆y, where
∆y gives the fraction of the total number which have velocities (in terms of α as a unit) ranging between c and the immediately preceding value of c. Thus, 8.35% of the molecules have velocities between c = 1 and c = 1.1, and 42.76%
have velocities equal to or less than the most probable value. The values in parentheses are those of (1 y). A plot of y versus c is shown in Fig. 1.2. It is evident that y corresponds to the area under the curve for fc from the origin to the given value of c.
From Eq. (1.53) the distribution formula for translational energy (E) can be derived. It has the form
Substituting the variable x = E/(kT), Eq. (1.61) becomes
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Page 15
Table 1.2. Values of fc, y, and fx, Illustrating Application of Distribution Laws
c fc y ∆y x fx
