Springer heitjans p krger j (eds) diffusion in condensed matter methods materials models (springer 2005)(ISBN 3540200436)(971s)

Diffusion in Condensed Matter Paul Heitjans · Jörg Kärger Diffusion in Condensed Matter Methods, Materials, Models With 448 Figures ABC Editors Professor Dr Paul Heitjans Professor Dr Jörg Kärger Universität Hannover Institut für Physikalische Chemie und Elektrochemie Callinstr 3–3a D-30167 Hannover, Germany Email: heitjans@pci.uni-hannover.de Universität Leipzig Institut für Experimentelle Physik I Linnéstr D-04103 Leipzig, Germany Email: kaerger@physik.uni-leipzig.de Library of Congress Control Number: 2005935206 ISBN -10 3-540-20043-6 Springer Berlin Heidelberg New York ISBN -13 978-3-540-20043-7 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the authors and S Indris using a Springer LATEX macro package Cover design: Cover design: design &production GmbH, Heidelberg Printed on acid-free paper SPIN: 10816487 56/3141/jl 543210 To Maria and Birge Preface Diffusion as the process of migration and mixing due to irregular movement of particles is one of the basic and ubiquitous phenomena in nature as well as in society In the latter case the word “particles” may stand for men or ideas, and in the former for atoms or galaxies In this sense diffusion is a truly universal and transdisciplinary topic The present book is confined, of course, to diffusion of atoms and molecules As this process shows up in all states of matter over very large time and length scales, the subject is still very general involving a large variety of natural sciences such as physics, chemistry, biology, geology and their interfacial disciplines Besides its scientific interest, diffusion is of enormous practical relevance for industry and life, ranging from steel making to oxide/carbon dioxide exchange in the human lung It therefore comes as no surprise that the early history of the subject is marked by scientists from diverse communities, e.g., the botanist R Brown (1828), the chemist T Graham (1833), the physiologist A Fick (1855), the metallurgist W.C Roberts-Austen (1896) and the physicist A Einstein (1905) Today, exactly 150 and 100 years after the seminal publications by Fick and Einstein, respectively, the field is flourishing more than ever with about 10.000 scientific papers per year From the foregoing it is evident that a single volume book on atomic and molecular diffusion has to be further restricted in its scope As the title says, the book is confined to diffusion in condensed matter systems, so diffusion in gases is excluded Furthermore, emphasis is on the fundamental aspects of the experimental observations and theoretical descriptions, whereas practical considerations and technical applications have largely been omitted The contents are roughly characterized by the headings Solids, Interfaces, Liquids, and Theoretical Concepts and Models of the four parts under which the chapters have been grouped The book consists of 23 chapters written by leading researchers in their respective fields Although each chapter is independent and self-contained (using its own notation, listed at the end of the chapter), the editors have taken the liberty of adding many cross-references to other chapters and sections This has been facilitated by the common classification scheme Further VIII Preface help to the reader in this respect is provided by an extended common list of contents, in addition to the contents overview, as well as an extensive subject index The book is a greatly enlarged (more than twice) and completely revised edition of a volume first published with Vieweg in 1998 Although the first edition was very well received (and considered as a “must for students and workers in the field”), it was felt that, in addition to the broad coverage of modern methods, materials should also be discussed in greater detail in the new edition The same applies to theoretical concepts and models This, in fact, is represented by the new subtitle Methods, Materials, Models of Diffusion in Condensed Matter The experimental Methods include radiotracer and mass spectrometry, Mă oòbauer spectroscopy and nuclear resonant scattering of synchrotron radiation, quasielastic neutron scattering and neutron spin-echo spectroscopy, dynamic light scattering and fluorescence techniques, diffraction and scanning tunneling microscopy in surface diffusion, spin relaxation spectroscopy by nuclear magnetic resonance (NMR) and beta-radiation detected NMR, NMR in a magnetic field gradient, NMR in the presence of an electric field, impedance spectroscopy and other techniques for measuring frequency dependent conductivities Materials now dealt with are, among others, metals and alloys, metallic glasses, semiconductors, oxides, proton-, lithium- and other ion-conductors, nanocrystalline materials, micro- and mesoporous systems, inorganic glasses, polymers and colloidal systems, biological membranes, fluids and liquid mixtures The span from simple monoatomic crystals, with defects in thermal equilibrium enabling elementary jumps, to highly complex systems, exemplarily represented by a biomembrane (cf Fig 12.3), is also indicated on the book cover Models in the subtitle stands for theoretical descriptions by, e g., correlation functions, lattice models treated by (approximate) analytical methods, the theory of fractals, percolation models, Monte Carlo simulations, molecular dynamics simulations, phenomenological approaches like the counterion model, the dynamic structure model and the concept of mismatch and relaxation Despite the large variety of topics and themes the coverage of diffusion in condensed matter is of course not complete and far from being encyclopedic Inevitably, it reflects to a certain extent also the editors’ main fields of interest Nevertheless the chapters are believed to present a balanced selection The book tries to bridge the transition from the advanced undergraduate to the postgraduate and active research stage Accordingly, the various chapters are in parts tutorial, but they also lead to the forefront of current research without intending to mimic the topicality of proceedings, which normally have a short expiry date It is therefore designed as a textbook or refer- Preface IX ence work for graduate and undergraduate students as well as a source book for active researchers The invaluable technical help of Dr Sylvio Indris (University of Hannover) in the laborious editing of the chapters, which in some cases included extensive revision, is highly acknowledged We also thank Jacqueline Lenz and Dr T Schneider from Springer-Verlag for accompanying this project As ever, the editors have to thank their wives, Maria Heitjans and Birge Kă arger, for their patience and encouragement Hannover, Germany Leipzig, Germany August 2005 Paul Heitjans Jă org Kă arger Contents Overview Part I Solids Diusion: Introduction and Case Studies in Metals and Binary Alloys Helmut Mehrer The Elementary Diffusion Step in Metals Studied by the Interference of Gamma-Rays, X-Rays and Neutrons Gero Vogl, Bogdan Sepiol 65 Diffusion Studies of Solids by Quasielastic Neutron Scattering Tasso Springer, Ruep E Lechner 93 Diusion in Semiconductors Teh Yu Tan, Ulrich Gă osele 165 Diffusion in Oxides Manfred Martin 209 Diffusion in Metallic Glasses and Supercooled Melts Franz Faupel, Klaus Ră atzke 249 Part II Interfaces Fluctuations and Growth Phenomena in Surface Diffusion Michael C Tringides, Myron Hupalo 285 Grain Boundary Diffusion in Metals Christian Herzig, Yuri Mishin 337 NMR and β-NMR Studies of Diffusion in InterfaceDominated and Disordered Solids Paul Heitjans, Andreas Schirmer, Sylvio Indris 367 XII Contents – Overview 10 PFG NMR Studies of Anomalous Diusion Jă org Kă arger, Frank Stallmach 417 11 Diffusion Measurements by Ultrasonics Roger Biel, Martin Schubert, Karl Ullrich Wă urz, Wolfgang Grill 461 12 Diffusion in Membranes Ilpo Vattulainen, Ole G Mouritsen 471 Part III Liquids 13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems Dieter Richter 513 14 The Molecular Description of Mutual Diusion Processes in Liquid Mixtures Hermann Weingă artner 555 15 Diffusion Measurements in Fluids by Dynamic Light Scattering Alfred Leipertz, Andreas P Fră oba 579 16 Diffusion in Colloidal and Polymeric Systems Gerhard Nă agele, Jan K G Dhont, Gerhard Meier 619 17 Field-Assisted Diffusion Studied by Electrophoretic NMR Manfred Holz 717 Part IV Theoretical Concepts and Models 18 Diffusion of Particles on Lattices Klaus W Kehr, Kiaresch Mussawisade, Gunter M Schă utz, Thomas Wichmann 745 19 Diffusion on Fractals Uwe Renner, Gunter M Schă utz, Gă unter Vojta 793 20 Ionic Transport in Disordered Materials Armin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer 813 21 Concept of Mismatch and Relaxation for Self-Diffusion and Conduction in Ionic Materials with Disordered Structure Klaus Funke, Cornelia Cramer, Dirk Wilmer 857 950 List of Contributors Prof Wolfgang Grill Universită at Leipzig Institut fă ur Experimentelle Physik II Linnestr 04103 Leipzig Germany grill@uni-leipzig.de Prof Reinhold Haberlandt Universită at Leipzig Institut fă ur Theoretische Physik Augustusplatz 10/11 04109 Leipzig Germany Reinhold.Haberlandt @physik.uni-leipzig.de Prof Paul Heitjans Universită at Hannover Institut fă ur Physikalische Chemie und Elektrochemie Callinstr 3-3a 30167 Hannover Germany heitjans@pci.uni-hannover.de Prof Christian Herzig Westfăalische Wilhelms-Universităat Mă unster Institut fă ur Materialphysik Wilhelm-Klemm-Straòe 10 48149 Mă unster Germany herzig@uni-muenster.de Dr Manfred Holz Universită at Karlsruhe Institut fă ur Physikalische Chemie Kaiserstr 12 76128 Karlsruhe Germany Manfred.Holz @chemie.uni-karlsruhe.de Dr Myron Hupalo Iowa State University Dept of Physics and Astronomy Ames, Iowa 50011 U.S.A hupalo@ameslab.gov Dr Sylvio Indris Universită at Hannover Institut fă ur Physikalische Chemie und Elektrochemie Callinstr 3-3a 30167 Hannover Germany indris@pci.uni-hannover.de Prof Jan W Kantelhardt Martin-Luther-Universită at HalleWittenberg FG Theoretische Physik von Senckendor-Platz 06099 Halle Germany kantelhardt@physik.uni-halle.de Prof Jă org Kă arger Universită at Leipzig Institut fă ur Experimentelle Physik I Linn´estr 04103 Leipzig Germany kaerger@physik.uni-leipzig.de Dr Ruep E Lechner Hahn-Meitner-Institut Glienicker Straße 100 14109 Berlin Germany lechner@hmi.de Prof Alfred Leipertz Universită at Erlangen Lehrstuhl fă ur Technische Thermodynamik Am Weichselgarten 91058 Erlangen Germany sek@ltt.uni-erlangen.de List of Contributors Prof Philipp Maass Technische Universită at Ilmenau Fachgebiet Theoretische Physik II Weimarer Straße 25 98684 Ilmenau Germany philipp.maass@tu-ilmenau.de Prof Ole G Mouritsen University of Southern Denmark Physics Department Campusvej 55 5230 Odense M Denmark ogm@memphys.sdu.dk Prof Manfred Martin RWTH Aachen Institut fă ur Physikalische Chemie I Landoltweg 52074 Aachen Germany martin@rwth-aachen.de Dr Kiaresch Mussawisade Forschungszentrum Jă ulich GmbH Institut fă ur Festkă orperforschung 52425 Jă ulich Germany k.mussawisade@fz-juelich.de Prof Helmut Mehrer Westfăalische Wilhelms-Universităat Mă unster Institut fă ur Materialphysik Wilhelm-Klemm-Straòe 10 48149 Mă unster Germany mehrer@nwz.uni-muenster.de Dr Gerhard Meier Forschungszentrum Jă ulich GmbH Institut fă ur Festkă orperforschung 52425 Jă ulich Germany G.Meier@fz-juelich.de Dr Martin Meyer Justus-Liebig-Universităat Gieòen Institut fă ur Theoretische Physik III Heinrich-Bu-Ring 16 D-35392 Gieòen Germany Prof Yuri Mishin George Mason University School of Computational Sciences 4400 University Drive, MSN 5C3 Fairfax, VA 22030-4444 U.S.A ymishin@gmu.edu Prof Gerhard Nă agele Forschungszentrum Jă ulich GmbH Institut fă ur Festkă orperforschung 52425 Jă ulich Germany G.Naegele@fz-juelich.de Dr Klaus Ră atzke Universită at Kiel Lehrstuhl fă ur Materialverbunde Kaiserstr 24143 Kiel Germany kr@tf.uni-kiel.de Dr Uwe Renner Wissenschaftszentrum LeipzigFă orderverein Goldschmidtstr 26 04103 Leipzig Germany Prof Dieter Richter Forschungszentrum Jă ulich GmbH Institut fă ur Festkă orperforschung 52425 Jă ulich Germany D.Richter@fz-juelich.de 951 952 List of Contributors Dr Andreas Schirmer Strahlenmessstelle der Bundeswehr Humboldtstraße 29633 Munster Germany Dr Martin Schubert Universită at Leipzig Institut fă ur Experimentelle Physik II Linnestr 04103 Leipzig Germany Prof Gunter M Schă utz Forschungszentrum Jă ulich GmbH Institut fă ur Festkă orperforschung 52425 Jă ulich Germany G.Schuetz@fz-juelich.de Prof Bogdan Sepiol Universită at Wien Institut fă ur Materialphysik Strudlhofgasse 1090 Wien Austria bogdan.sepiol@ap.univie.ac.at Prof Tasso Springer Geigerstr 82166 Gră afeling Germany Box 90300 Hudson Hall Durham, NC 27708-0300 U.S.A ttan@acpub.duke.edu Prof Michael C Tringides Iowa State University Dept of Physics and Astronomy Ames, Iowa 50011 U.S.A tringides@ameslab.gov Prof Ilpo Vattulainen Helsinki University of Technology Laboratory of Physics P.O Box 1100 02015 HUT Helsinki Finland ilp@fyslab.hut.fi Prof Gero Vogl Universită at Wien Institut fă ur Materialphysik Strudlhofgasse 1090 Wien Austria gero.vogl@ap.univie.ac.at Prof Gă unter Vojta Donndorfstr 20 01217 Dresden Germany Dr Frank Stallmach Universită at Leipzig Institut fă ur Experimentelle Physik I Linnestr 04103 Leipzig Germany stallmac@physik.uni-leipzig.de Prof Hermann Weingă artner Ruhr-Universită at Bochum Lehrstuhl fă ur Phys Chemie II Universită atsstr 150 44780 Bochum Germany Hermann.Weingaertner @ruhr-uni-bochum.de Prof Teh Yu Tan Duke University Department of Mechanical Engineering & Materials Science Dr Thomas Wichmann Weilerswisterstr 50968 Kă oln Germany List of Contributors Dr Dirk Wilmer Westfăalische Wilhelms-Universităat Mă unster Institut fă ur Physikalische Chemie Corrensstraòe 30/36 48149 Mă unster Germany wilmer@uni-muenster.de Dr Karl Ullrich Wă urz PBS Softwareberatung Schwanheimer Str 144a 64625 Bensheim Germany 953 Index acoustic microscopy 466 activation energy 19, 344, 813 activation enthalpy 18 activation volume 19, 40, 268 adsorbate-adsorbate interactions 288 Ag/Ag(111) 320 0.5Ag2 S·0.5GeS2 877 0.3Ag2 SO4 ·0.7AgPO3 884 alloys binary 49, 53 ordered 78 aluminium 39, 77 amino acid 729 amorphous alloys 259 anomalous diffusion 417–421, 428, 434, 441, 448, 479, 488, 731, 797, 803, 806, 867, 904 anticorrelation in anomalous diffusion 801 antiphase boundaries 77 antistructure defects 71, 72, 78 antistructure-bridge mechanism 46, 81 Arrhenius law 18, 289, 746, 753, 813, 925 association degree 220 asymmetric double well potential (ADWP) model 836, 866, 888 Auger electron spectroscopy (AES) 14 autocorrelation function 817, 920, see also correlation function electric field 625 light intensity 625 orientation 636 average ensemble 919, 923 time 923 β-NMR method 380–384 relaxation 370, 383, 384, 403, 407, 409 spectrometer 383 β-radiation asymmetry 383 β-relaxation 520 β-titanium 77 B2 structure 71, 72, 76, 78 backbone of percolation cluster 900 BaF2 394 ballistic diffusion 867 binary intermetallics 42 binary non-electrolyte mixtures 573 blocking factor 104, 150 Bloembergen-Purcell-Pound (BPP) behaviour 369, 819 body-centered cubic metals 34 Boltzmann-Matano method 49, 294, 498 bond percolation 901 Boson peak 520 Bragg equation 108 Brillouin lines 581, 604 Brownian dynamics simulation method 687 Brownian motion 472, 632, 717, 794 CaF2 391 caloric glass transition temperature 255 capacitance 907 capillary electrophoresis 738 cation sublattice 209 central limit theorem 420, 423, 625, 799 charge diffusion coefficient 126, 127, 148 956 Index charge of transport 222, 238 chemical diffusion 6, 49, 556 chemical diffusion coefficient 231 chemical potential 8, 226, 291, 918, 941 Chudley-Elliott model 69, 71, 102, 130, 149, 150 cobalt oxide 217 CoGa 82 coherent diffuse scattering 153 coherent quasielastic scattering 149 coherent scattering function 102, 150 collective diffusion 495, 641, 648, 747, 772 coefficient of 289, 646, 771, 778 colloidal rods 666 colloidal spheres 676 colloidal systems 619 collective diffusion 641 interdiffusion 651 rotational diffusion 658 self-diffusion 637 component diffusion coefficient 10 compressibility, isentropic 607 computer simulations see simulations concept of mismatch and relaxation (CMR) 867, 874 conductivity electrical 219, 905, 906 frequency dependence 768, 861 thermal 597 conductivity spectroscopy 861 continuity equation continuous time random walk model 824 continuum percolation 902 copper 77 correlated jumps 118, 124, 127, 858 correlation effects 127 correlation factor 19, 222, 773, 816, 848 correlation function 369, 515, 525, 527, 582, 915, 916, 919–921, 923 long-time tail 633 of coverage 291 of current density 864 Van Hove 66, 96, 98, 921, 935, 937 velocity auto- 928, 930 correlation length 897 correlation technique 582–587 correlation time 369, 586, 832 correlator 594 Coulomb interaction 215, 819, 830 Coulomb lattice gas 820, 840 Coulomb trap 835 counterion model 835, 866 coupled diffusion in B2 intermetallics 44 coupling concept 866 coverage 290 critical fluctuation 295 fluctuation 290, 293 step 299 Co-Zr glasses 268 critical concentration 896, 901, 902 critical dimension 808 critical exponent 897, 905 critical percolation path 832 critical slowing down 600, 650, 703 critical-path approach 763 cross coefficients 236 crossover time 905 Cu3 Au rule 47 current density autocorrelation function 864 D03 structure 84 Darken equation 51, 296, 433, 557, 562, 573, 941 Darken-Manning relation 52 Darwin width 110 dc conductivity plateau 862 de Broglie wavelength 99, 919 Debye length 720 Debye-Hă uckel-Onsager-Falkenhagen effect 859 Debye-Waller factor 67, 70, 74, 99, 113, 119, 138 defect chemistry 210 defect cluster 226 defect structure 210, 215 demixing 234, 236, 240, 703 density distribution 931 detailed balance 71, 767, 786 dichalcogenides 386 dielectric loss 835, 863 Index differential effective-medium theory 849 diffusant diffuse scattering 76 diffuser diffusion ambipolar 227 anomalous 417–421, 428, 434, 441, 448, 479, 488, 731, 797, 803, 867 cation 217, 224 chemical 226 collective 495, 641 continuous jump 371 correlated diffusion anisotropy 431–432 effective diffusivity 428 experimental methods 10 field-assisted 717 in aluminium 39 in amorphous alloys 262 in B2 intermetallics 44, 78 in bcc metals 35 in D03 intermetallics 84 in fcc metals 33 in gallium arsenide 184 in germanium 183 in L12 intermetallics 49 in lead 41 in liquids 251 in membranes 471 in nickel 31 in niobium 28, 29 in polymers 447, 519, 531, 675, 733, 843 in regular pore networks 427 in semiconductors 165 in silicon 166 in silver 38 in zeolites 427, 925 interstitial-substitutional 168 intracrystalline self-diffusion 429, 430, 445 isotope effects 13, 30, 253, 265, 272 lateral 477, 493 long-range 133 low-dimensional 371 molecular mechanism 132 multicomponent 427 957 mutual 556, 569, 600 normal 417–420, 426, 435, 450, 479, 555, 777, 798, 799 on percolation clusters 904 oxidation-enhanced 174 oxidation-retarded 174 oxygen 222, 223 pressure dependence 18 proton diffusion 131 reactive 54 rotational 477, 491, 635 single-file diffusion 434–437, 775 solute diffusion 35 solvent diffusion 35 surface diffusion 285, 302, 339 through membranes 500 transport diffusion 432–434, 943 diffusion coefficient 904, 921, 929, 931, 941 chemical 6, 8, 151, 227, 244 collective 289, 646, 780 distinct-diffusion 570, 571 foreign atom frequency dependence 762 impurity in grain boundaries 338 interdiffusion 654 self-diffusion Stokes-Einstein 630 thermodynamic 561 tracer 7, 219, 286, 478 transport 941 vacancy 231 diffusion coefficient tensor 4, 439, 676 diffusion entropy 18 diffusion equation 5, 289, 494, 632, 752, 798, 928, see also Fick’s second law error function solution source solution 230 thin-film solution diffusion length 5, 479 diffusion mechanisms 23 diffusion-limited reaction 806–808 diffusional line broadening 70, 72, 94 diffusivity effective 41, 170, 348, 420, 429, 453 thermal 597 958 Index diffusivity tensor 4, 439, 676 direct current NMR (DCNMR) 718, 725 disorder models 753, 820 disordered solids homogeneously 402 inhomogeneously 367, 390 disordered systems 372, 746, 753, 813, 857, 895 dispersive transport 822 dissociative mechanism 26 distribution of site energies 257 divacancy mechanism 25 dopant diffusion 172, 235 Doppler drive 111 double differential scattering crosssection 95 drift flux 230, 232, 236 drift velocity 237, 717–720 dynamic conductivity 817 dynamic light scattering (DLS) 579, 589–597, 624 coherence 593 dynamic percolation 846 dynamic rotational disorder 153 dynamic structure factor 94, 521–526, 531, 538, 545, 568, 626, see also scattering function distinct 628 self- 628 dynamic structure model 840 effective activation energy 273 effective charge 236, 239 effective diffusivity 41, 170, 348, 420, 429, 453 effective-medium approximation 746, 758, 762, 912 self-consistency condition 761, 787 Einstein diffusion coefficient Einstein relation 227, 420, 434, 555, 717, 903, 920 Einstein-Debye relation 635 Einstein-Smoluchowski relation 19, see also Einstein relation elastic incoherent structure factor (EISF) 98, 119, 138, 139, 141 electric potential 226, 236 electric potential gradient 228 electro-osmosis 726, 737 electrochemical potential 226 electrokinetic potential 720 electrolyte solution 566, 574 electron hole 211, 212 electron microprobe analysis (EMPA) 14 electrophoresis 718 electrophoretic NMR (ENMR) 717 elementary diffusion step 65, 66, 68 encounter model 124, 128, 370 energy resolution 98 ensemble canonical 917 microcanonical 917 entanglements 513, 531, 543 enthalpy of migration 22 entropic forces 529 entropy of migration 22 equivalent circuit model 906 error function solution escape rate 121 eucryptite 405 excess charge 210, 228, 231 excess volume 269 exchange mechanism interstitial-substitutional 26 ring 127 face-centered cubic metals 32 fast solute diffusion 40 FeAl 78 Fe3 Al 80 Fermi level effect 172 Fe3 Si 84 Fick’s first law 4, 494, 559, 798 non-local 643 Fick’s second law 5, 6, 165, 494, 752 field-assisted diffusion 717 Fisher model 337, 338 five-frequency model 36, 233 fixed-window method 113 fluctuation-dissipation relation 632 fluorescence correlation spectroscopy (FCS) 669 fluorescence recovery after photobleaching (FRAP) 481, 497, 661 forced Rayleigh scattering 613 Index Fourier time window 100, 106 fractal 793 chemical kinetics 806 fractal dimension 446, 793, 897, 898, 904 fractional Brownian motion 794 fragile glass 254 fragile supercooled melt 880 Frank-Turnbull mechanism 26, 168 free induction decay (FID) 376, 400 free-volume model 253, 486 Frenkel defects 211 equilibrium 210, 212 Γ -space 918 generalized force 919 Gibbs free energy of activation 19 Gibbs free energy of binding 24 Gibbs free energy of migration 22 glass electrolyte 403 ion-conducting 402, 819 metallic 259 oxide 403, 405, 884 glass transition 255, 272, 519, 638 Gorski effect 16 grain boundary 337 diffusion coefficient 338 in nanocrystalline materials 352, 390, 391 large-angle 345, 346 segregation 339, 353, 357, 361 segregation energy 353 segregation factor 339, 353 small-angle 345, 346 width 338, 343, 911 grain boundary diffusion 337 activation energy 344 anisotropy 345 Arrhenius law 344 atomistic mechanisms 347, 362 comparison with bulk, surface, liquid 344 empirical rules 344 in intermetallic compounds 361 in moving boundaries 362 kinetic regimes 347 orientation dependence 346 959 Green-Kubo relation 495, 568, 638, 920 Grotthuß mechanism 131 growth constant 54 gyromagnetic ratio see magnetogyric ratio Harrison’s classification 348 Hart’s formula 349 Hartley-Crank relation 557 Hausdorff dimension 793 Haven ratio 127, 137, 148, 817, 865 Henry isotherm (of grain boundary segregation) 358 heterodyne technique 587 homodyne technique 587 hopping rate see jump rate H/Si(111) 308 Huang scattering 119, 153, 155 hybrid solutes 41 hydrodynamic function 678 hydrodynamic interaction 637, 659 hydrogen bond 131 hydrogen diffusion 29, 115 immobile (trapped) state 121 impedance spectroscopy 861 impurity diffusion in grain boundaries 339 in metals 35, 77 in oxides 216 in semiconductors 168, 172, 177, 182, 183, 196 impurity diffusion coefficient impurity-vacancy binding 219 impurity-vacancy pair 233, 239 incoherent scattering function 97–99, 102, 133, 141, 154, 818 incoherent structure factor see incoherent scattering function infinite percolation cluster 896 intercalation 367 intercalation compounds 384 graphite 384 titanium disulfide 386 interdiffusion 6, 49, 187, 263, 556, 651, 654 interdiffusion coefficient 6, 8, 191, 654 interfacial region 50, 352, 367, 390 960 Index diffusion in 392, 908 intermediate dynamic structure factor 521 intermediate scattering function 73, 95, 100, 523, 818, 828 internal interfaces 360, 367 density of 367 interstitial cation 213, 217 interstitial diffusion 27, 242, 745 interstitial impurities 165 interstitial mechanism 23 interstitial-substitutional exchange mechanism 26 interstitialcy mechanism 25 intrinsic diffusion coefficient 10 inversion-recovery experiment 378 ion-conducting glasses 402, 819, 861 ion-conducting materials 861 ion-conducting polymers 843 ionic conductivity 126, 224, 817, 861, 908 ionic mobility 717, 729, 737 ionic self-diffusion coefficient 717, 729 irreversible processes 564, 919 isotope effects 13, 30, 253, 265, 272 Jonscher power law 863, see also universal dielectric response jump length 7, 20, 105, 866, 903 jump rate 19–23, 65, 70–73, 124, 233, 373, 763, 866 jump relaxation 823, 866 jump vector 20, 65, 67, 68, 102, 140 kick-out mechanism 26, 168 kink atoms 286 Kirkendall effect 10, 51 opposite 263 Kirkwood-Buff theory 575 K2 O·2BaO·4SiO2 885 Kohlrausch behavior 255, 521, 866 Kohlrausch-Williams-Watts (KWW) behavior see Kohlrausch behavior Kră oger-Vink diagram 214 notation 210 Kubo formula 817, 941 Lamb-Mă oòbauer factor 67 Landau-Placzek ratio 582, 606 Langevin equation 530, 632 Langmuir-Hinshelwood reaction 808 lanthanum gallate 216, 225, 235 Laplace transformation 644, 760 Larmor condition 421 Larmor precession 74, 516 Larmor precession frequency 369, 814 layer-crystalline materials 367 LiC6 , LiC12 384 LiCl·4D2 O 403 LiClã7H2 O 881 line broadening in Mă oòbauer spectroscopy 68 line broadening in neutron scattering 68, 94 line narrowing in neutron scattering 104, 151 line narrowing in nuclear magnetic resonance 374 linear response theory 150, 567, 858, 919 0.3Li2 O·0.7B2 O3 884 lipid 473 lipid bilayer 475 lithium 370 lithium aluminosilicates 405 lithium intercalation compounds 384 lithium ion conductors 390 lithium niobate 394 lithium oxide 399 lithium titanium disulfide 386 local reptation model 538 localized motion 106, 134, 405, 528, 887 Longini mechanism 26, 193 low energy electron diffraction (LEED) 304 low energy electron microscopy (LEEM) 305 macroscopic diffusion methods 11–15, 65, 368 magnetic field gradient 421, 720 magnetic relaxation methods 16 magnetite 217, 231 magnetogyric ratio 369, 375, 517, 720 majority defect 213 Index Mandelbrot dimension 793 Markovian process 70, 749, 802, 804 mass action law 25, 40, 211, 807 master curves of ionic conductivity 863 master equation 102, 748, 772 McLean’s isotherm 358 mean residence time 7, 17, 21, 68, 70–72, 80, 127, 322, 369, 816 mean square displacement 20, 252, 286, 418–449, 478, 516, 531, 555, 629, 751–777, 799, 816, 826, 868, 903, 929 long time tail 761 mean-field theory 808 mechanical relaxation methods 16 mechanical sectioning technique 11 membrane 441, 471, 733 biological 145, 473 lateral diffusion 477 polypropylene 441 transverse diffusion 500 memory effect 117, 127, 632, 643 mesoporous materials 437–439 MCM-41 438 metallic glasses 264 metalorganic chemical vapor deposition (MOCVD) 187 Meyer-Neldel compensation rule 835 micelles 736 microemulsion 737 microporous materials 427, 925 microscopic diffusion methods 15–17, 65, 368 minority defect 212, 213 mismatch relaxation 866 mixed alkali effect 840 mixed conductor 216 mobility of ions 717, 729, 737 mode coupling theory 256, 275, 691 molecular beam epitaxy (MBE) 187 molecular dynamics simulations 118, 480, 534, 916, 922–925 molecular traffic control 437 molten salt mixtures 566 Monte Carlo simulations 325, 491, 759, 815, 821, 830, 836, 840, 909, 917, 922 961 Mă oßbauer spectroscopy 65–68 motional narrowing of NMR lines 374, 393, 395 moving grain boundary 362 multifractal 797 multiphase diffusion 53 multiple scattering 115, 135, 528 muon 30 muon spin resonance (µSR) 31 mutual diffusion 556, 569, 600 nanocrystalline BaF2 394 nanocrystalline CaF2 391 nanocrystalline composites 399, 908 nanocrystalline LiNbO3 394 nanocrystalline LiTiS2 397 nanocrystalline materials 352, 367, 390 Na3 PO4 154 nearly constant loss (NCL) behaviour 814, 835, 863, 888 Nernst field 227 Nernst-Einstein equation 126, 131, 817, 865, 880, 904 neutron backscattering (BSC) spectrometer 106, 109, 111 neutron capture 382 neutron scattering 65, 68, 93, 514, 817 neutron spin-echo (NSE) spectrometer 519, 520 neutron spin-echo (NSE) spectroscopy 74, 516, 518 neutron time-of-flight (TOF) spectrometer 106, 114 NiGa 82 Ni3 Sb 86 NMR 368, 417, 717 correlation function 369, 818 correlation time 369, 373, 830 flow measurements 720 linewidth 373 PFG diffraction pattern 424–425 PFG effective observation time 426 PFG observation time 418, 421, 426, 427, 449, 455 pulsed field gradient (PFG) technique 417, 421–427 relaxation 369, 426, 814 relaxation techniques 375–380 962 Index spectral density 369, 409, 818 spectrometer 379 spin echo 378, 379, 421, 721 spin-alignment echo technique 390 spin-echo attenuation 422, 424, 426, 440, 452 spin-locking experiment 378 static field gradient (SFG) technique 427 non-Arrhenius behaviour 767, 832 non-stoichiometric oxide 210, 219 normal diffusion 417–420, 426, 435, 450, 479, 555, 777, 798, 799 nuclear magnetization 375, 421 methods in diffusion 16, 65, 368 polarization 380, 404 reactions for polarized β-emitters 381 spin-lattice relaxation 369 nuclear magnetic relaxation see NMR relaxation nuclear magnetic resonance see NMR nuclear reaction analysis (NRA) 15 nuclear resonance scattering (NRS) 65 Onsager reciprocity relation 566, 919 Onsager regression hypothesis 579 Onsager transport coefficient 226, 566 O/Si(111) 307 O/W(110) 295 oxygen activity 209 oxygen deficit 211 oxygen excess 211 oxygen ion conductor 216 oxygen partial pressure 209 oxygen potential gradient 228 oxygen sublattice 209 paddle-wheel mechanism 153 pair correlation function 251, 575 particle distribution function 917 partition function 915, 919 Pb/Si(111) 313 percolation bond 901 continuum 902 interface 908 site 895 percolation cluster 897 conductivity of 905 diffusion on 904 percolation model 755, 820, 895 percolation threshold 896 permeation 500 perovskite structure 225 petalite 405 phenomenological coefficient see Onsager transport coefficient phonon dispersion 125 phonon spectroscopy 77 photon correlation spectroscopy 579 plasma parameter 822 Poisson process 803 polarized neutron capture 382 polybutadiene 519, 528 polydimethylsiloxane (PDMS) 440, 443, 453, 531 polyethylene oxide (PEO) 451, 843 polyethylethylene (PEE) 453 polyisobutylene 528 polyisoprene 528 polymer blends 697 diblock copolymer 453 electrolytes 843 reptation 449, 450 triblock copolymer 450, 451 polymeric systems 619 polypropylene membrane 441 polypropylene oxide (PPO) 451 polystyrene (PS) 440 polyvinylether 528 potential 922, 927 chemical 226 electric 226, 236 electrochemical 226 Lennard-Jones 923, 927, 932 square well 922 pre-exponential factor 18 pressure dependence of diffusion 268 principal diffusivities propagator 419, 420, 422, 424, 429, 435, 798, 802, 928 mean 422–424, 935 proton conduction 139 Index proton pump 146 proton tunneling 123 protonic conductor 131 pseudo Fermi level 782 pulsed field gradient (PFG) NMR 417, 421–427, 720 quantum diffusion 30 quasi-vacancies 261 quasielastic coherent structure factor 154 quasielastic helium scattering 293 quasielastic incoherent neutron scattering (QINS) 136 quasielastic incoherent structure factor 141, 154 quasielastic light scattering 620 quasielastic linewidth 103, 105, 124, 151 quasielastic Mă oòbauer spectroscopy 67, 68 quasielastic methods 65, 66, 68, 73 quasielastic neutron scattering (QENS) 67, 68, 93 observation function 100, 105 observation time 100, 105, 133 observation volume 105 resolution function 105 radiotracer sectioning method 217, 340, see also tracer method random barrier model 753, 867 random phase approximation 699 random trap model 754 random walk 747, 802, 806, 858, 865, 903 randomly blocked sites 755, 765 Rayleigh line 581 RbAg4 I5 878, 890 reaction constant 806 reaction rate 120, 143 reactive diffusion 54 reflection high energy electron diffraction (RHEED) 324 renewal theory 847 reptation crossover 543 diffusion 537 model 514, 537–540, 543 963 residence time see mean residence time residual activity method 217 rotational diffusion 138, 491, 635, 658 Rouse model 514, 529–536, 844 generalized 538 rubber-like model 538 Rutherford backscattering spectrometry (RBS) 14 scanning tunneling microscopy (STM) 306 scattering amplitude 622 scattering cross section 95 scattering function 94, 149, 523, 921 scattering length 95, 101 scattering strength 621 scattering vector 93 Schottky defects 211 equilibrium 210, 224 secondary ion mass spectrometry (SIMS) 13, 222 sedimentation 647 sediments 446–447 segregation 234 self-affine fractal 797 self-correlation function 66–76, 97, 102, 419 self-diffusion 31, 121, 127, 136, 144, 152, 370, 555, 557, 637, 943 self-diffusion coefficient 7, 120, 128, 135, 147, 152, 216, 227, 659 self-interstitial charge state 166 self-similarity 795, 897 short-range order 260 Siegert relation 584, 626 Sierpinski fractal 794 silicon self-diffusion 166 simulations molecular dynamics (MD) 925, 941 Monte Carlo (MC) 815, 821, 915, 922 Sinai model 746, 769 single-file diffusion 434–437, 775 site energy disorder 747 exponential distribution 766, 781 site exclusion model 771 964 Index site occupancy 104, 150 site percolation 895 six-jump cycle mechanism 46, 81 Smoluchowski equation 675 Smoluchowski theory 807 Snoek effect 16 soft phonon modes 77, 86 solid rotator phase 156 solid-state protonic conductor 131 solute diffusion 35 solute-vacancy pair 215, 220 solvent diffusion 35 Soret effect 606 sound attenuation 604 sound velocity 461, 604 specific surface area 446 spectral density 921 spin diffusion 382, 400, 403 spin echo neutron 74, 516 NMR 378, 379, 421, 721 spin incoherent scattering 115 spin-lattice relaxation 369, 818 disorder effects 372 frequency dependence 372, 386, 387, 398, 407 homogeneous 404 inhomogeneous 404 laboratory frame 377, 409 low-dimensionality effects 372 rate 370 rotating frame 379, 409 spin-spin relaxation 378, 409 spinel 217 spodumene 405, 406 sputter sectioning technique 12 SrCl2 128 static field gradient (SFG) NMR 427 static structure factor 95, 150 stochastic process 800 stoichiometric point 212, 216, 228 Stokes-Einstein diffusion coefficient 630 Stokes-Einstein equation 254, 601, 660, 719 Stokesian dynamics simulation method 689 strain field 119, 130, 153 stretched polymers 849 structural relaxation 260 structure factor 153, 298, 919 dynamic 521–526, 921, 937 partial 652 static 918, 921 subdiffusive behaviour 417, 421, 801, 867 substitutional impurities 165 Summerfield scaling 863, 866 superlattice disordering 187 surface diffusion 285, 339 equilibrium measurements 297 non-equilibrium measurements 313 step 302 terrace 302 surface exchange coefficient 222 surface light scattering 608 surface tension 608 surface-to-volume ratio 444–447 synchrotron radiation 65, 73 thermal conductivity 597 thermal diffusivity 597 thermal grating 613 thermodynamic factor 10, 227, 243, 562 time correlation function 567, 920, see also correlation function time-dependent correlation factor 867 time-domain interferometry 76 time-temperature superposition principle 863, 866 titanium 65, 77 titanium disulfide 386 topological distance 900 tortuosity 443, 454, 731 total scattering cross section 102 tracer diffusion 7, 230, 745, 771, 773, 816 tracer diffusion coefficient 7, 286 tracer method 11 tracer self-diffusion coefficient transference number 227, 719, 729 transition rate 746, see also jump rate transition state theory 943 transport coefficient 919–921, see also Onsager transport coefficient transport diffusion 432–434, 556, 943 Index traps 121 saturation of 781 trapping rate 123 triple-defect mechanism 46, 81 tunneling of protons 123 two-dimensional diffusion 130, 143, 145, 372, 386, 387 two-state model 121, 122, 752 ultrasonic interferometer 462 ultrasonic wave velocity 462 universal dielectric response 814, 863 universal dynamic response see universal dielectric response vacancy cation 212, 217 charge state 166 oxygen 211 vacancy availability factor 22 vacancy mechanism 23, 65, 124, 169, 232, 347 vacancy-pair mechanism 47 vacancy-wind corrections 52 Van Hove correlation function 66, 96, 98, 419, 689, 921, 935, 937 van Liempt rule 33 vehicle mechanism 132, 148 velocity autocorrelation function 858, 865, 920 965 velocity cross-correlation coefficients 570 Verlet algorithm 924, 928 vesicles 736 viscoelasticity 513 viscosity 601, 607, 608 Vogel-Fulcher-Tammann (VFT) equation 255, 520, 845, 882 volume diffusion 339 Wagner formula 227 waiting time distribution 802, 825 walk dimension 801, 804 water in gels 466 water in living cells 466 Wiener process 794, 804 x-ray photon correlation spectroscopy (XPCS) 76 Zener effect 16 zeolite 426–437, 925, 926 A-type 422, 427, 433 AlPO4 -5 436 LTA-type 926 structure 926 X-type 445 ZK4 926 ZSM-5 type 431 zeta potential 720 zirconia 216, 224, 235 .. .Diffusion in Condensed Matter Paul Heitjans · J rg Kärger Diffusion in Condensed Matter Methods, Materials, Models With 448 Figures ABC Editors Professor Dr Paul Heitjans Professor Dr J rg... 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