NMR: Basic Principles and Progress Grundlagen und Fortschritte PDF
Preview NMR: Basic Principles and Progress Grundlagen und Fortschritte
NMR Basic Principles and Progress Grundlagen und Fortschritte Volume 3 Editors: P. Diehl E. Fluck R. Kosfeld With 73 Figures Springer-Verlag Berlin· Heidelberg· New York 1971 Professor Dr. P. DIEHL Physikalisches Institut der Universitat Basel Professor Dr. E. FLUCK Institut fUr Anorganische Chemie der Universitat Stuttgart Professor Dr. R. KOSFELD Institut fUr Physikalische Chemie der Rhein.-Westf. Technischen Hochschule Aachen ISBN 3-540-05392-1 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05392-1 Springer-Verlag New York' Heidelberg· Berlin The use of general descriptive names, trade marks, etc. in this publication, even if the former are not especially identified, is not be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro duction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 70-94160. Printed in Germany. Type-setting and printing: Druckerei Dr. A. Krebs, Hemsbach/Bergstr. und Bad Homburg v. d. H., Bookbinding: Briihlsche Universitatsdruckerei, GieBen. NMR Basic Principles and Progress Grundlagen und Fortschritte Volume 3 Editors: P. Diehl E. Fluck R. Kosfeld With 73 Figures Springer-Verlag New York· Heidelberg· Berlin 1971 Professor Dr. P. DIEHL Physikalisches Institut der Universitat Basel Professor Dr. E. FLUCK Institut fur Anorganische Chemie der Universitat Stuttgart Professor Dr. R. KOSFELD Institut fUr Physikalische Chemie der Rhein.-Westf. Technischen Hochschule Aachen ISBN-13: 978-3-642-65182-3 e-ISBN-13: 978-3-642-65180-9 DOl: 10.1007/978-3-642-65180-9 The use of general descriptive names, trade marks, etc. in this publication, even if the former are not especially identified, is not be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro duction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 0 by Springer Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 70-94160. Softcover reprint of the hardcover 1s t edition 1971 Preface Nuclear magnetic resonance spectroscopy, which has evolved only within the last 20 years, has become one of the very important tools in chemistry and physics. The literature on its theory and application has grown immensely and a comprehensive and adequate treatment of all branches by one author, or even by several, becomes increasingly difficult. This series is planned to present articles written by experts working in various fields of nuclear magnetic resonance spectroscopy, and will contain review articles as well as progress reports and original work. Its main aim, however, is to fill a gap, existing in literature, by publishing articles written by specialists, which take the reader from the introductory stage to the latest development in the field. The editors are grateful to the authors for the time and effort spent in writing the articles, and for their invaluable cooperation. The Editors Contents o. Kanert and M. Mehring Static Quadrupole Effects in Disordered Cubic Solids 1 F. Noack Nuclear Magnetic Relaxation Spectroscopy 83 Static Quadrupole Effects in Disordered Cubic Solids O. KANERT and M. MEHRING Physikalisches Institut der Universitat MUnster, BRD Contents I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 II. Fundamentals .............................................. 4 1. Zero Field Spectra ........................................ 4 2. High Field Spectra ........................................ 6 3. Transformation of the Electric Field Gradient Tensor .......... 7 III. The Influence of the Quadrupole Perturbation on the NMR Signal . 8 1. General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 a) The Free Induction Decay ............................... 8 b) The Wide-Line Signal ................................... 11 c) The Spin Echo Signal ................................ :.. 11 2. Dependence of Signal Parameters on the Mean Quadrupole Distor- tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3. Effect of an External Axial Stress on the Signal Parameters. . . . . . 19 4. Double Resonance Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 IV. Field Gradients ............................................. 30 1. General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2. Homogeneous External Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3. Field Gradients Caused by Stress Fields of Lattice Defects ...... 37 a) Point Defects .......................................... 37 b) Dislocations ........................................... 39 c) Dislocation Dipoles and Tilt Boundaries . . . . . . . . . . . . . . . . . . . 47 4. Point Charge Effects in Non-Metallic Crystals. . . . . . . . . . . . . . . . . 48 5. Field Gradients Caused by the Redistribution of Conduction- Electrons around Defects .................................. 49 6. Calculations of Gradient-Elastic Constants ................... 53 V. Quadrupole Distribution Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1. Basic Theory ............................................. 55 2. Distribution Function of Several Defects ..................... 58 a) Point Defects .......................................... 58 b) Dislocation Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 c) Dislocations ........................................... 62 2 o. KANERTand M. MEHRING d) Tilt Boundaries ........................................ 65 e) Comparison of the Distribution Functions ................. 66 VI. Effect of the Defect Density on the NMR Signal . . . . . . . . . . . . . . . . . 68 I. Point Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2. Dislocations ............................................. 71 VII. Acknowledgments....................... . . ......... ......... 79 VIII. References ................................................. 79 Static Quadrupole Effects in Disordered Cubic Solids 3 I. Introduction Nuclei with spin 1 > 1/2 possess an electrical quadrupole moment Q which may interact with the gradient {JI;k} of an electric field. If the electric field gradient (EFG) {JI;k} at the site of the nucleus is caused by a surrounding with cubic symmetry it follows that Vxx = Vyy = Vzz, where (X, Y, Z) is the principal coordi nate system of the electric field gradient tensor. Because of the Laplace equation Vxx + Vyy + Vzz = 0 the electric field gradient vanishes. In crystals the individual ions are assumed to have spherical symmetry in a first approximation. Thus the electric field gradient due to their own electron cloud vanishes. Therefore the electric field gradient at a nucleus in the lattice originates from neighbouring ions. In a perfect cubic crystal the EFG at each nucleus caused by the neigh bouring atoms is zero due to the cubic symmetry. However, lattice defects in the crystal can destroy the cubic symmetry pro ducing local field gradients which would cause a quadrupolar interaction. The corresponding quadrupole Hamiltonian Jr in an arbitrary coordinate system Q (x, y, z) can be written as [1], [2], [3]: (1.1) where Qo = (X' [31;-1' (1 + 1)] Q± I = (X, [IzI ± + I ± Iz] Q±2 = (Xli and Vo = ~z V± I = (Vxz ± j . Vyz) 1(v.:x- v.: V±2 = Vyy) ± j' y with the abbreviation (X = eQ/(4/(2J -1)) (Iz' I±' operators of the nuclear spin I). Transitions between the magnetic energy levels Em = (mIJrQlm) can be meas ured in the "zero-field case" by means of the NQR-technique. The corresponding transition frequencies are derived in chapter II, 1. On the other hand in the "high-field case" the Hamiltonian Jr acts as a Q perturbation of the Zeeman-Hamiltonian Jrz of the spins in the external magnetic field H 0 chosen in the z-direction of the laboratory frame. Transition frequencies are calculated up to second order perturbation in chapter 11,2. Such transitions can be observed by irradiation with an r J-field HI perpendicular to the magnetic field Ho. Two kinds of rf irradiation are discussed in chapter III, 1: Continuous wave and pulse irradiation. The corresponding NMR signals (wide line signal and transient signal) are correlated by a Fourier transform mechanism. In the case of cubic solids treated in this work the shape of the observed NMR signal is determined by the magnetic dipole-interaction of neighbouring spins (VAN VLECK [4]) and the electric quadru pole interaction between the quadrupole moment of the nucleus and the local electric field gradient caused by lattice distortions. The dependence of the NMR signal parameters on the quadrupole distortions is investigated in sections 2, 3 4 o. KANERT and M. MEHRING of chapter III. A very sensitive experimental method for observing quadrupole frequencies was introduced by HARTMANN and HAHN [5], SLUSHER and HAHN [6], HARTLAND [7]. The method shows the usefulness of the quadrupole moment as a probe of local electric fields in solids as examined in III, 4. In chapter IV of this article, electric field gradients caused by external stress (section 2), by the stress field in the neighbourhood of a lattice defect (section 3) and by charge effects (sections 4, 5) are calculated, where the host lattice is treated as an isotropic homogeneous continuum. Supposing a random distribution of the defects one can derive the distribution function of the quadrupole frequencies. This has been performed in chapter V for the following types of defects: point defects, dislocation d,ipoles, dislocations and tilt boundaries. In the case of point defects and dislocations, an experimental verification of the computed line shapes is presented. . In the last chapter, NMR measurements will be reported showing the depend ence of the NMR signal on the mean density of lattice defects. By using the theory derived in the preceding sections it is possible to calculate the defect concentration from the measured signal parameters or vice versa. Values thus obtained are in ·agreement with data evaluated from "solid state experiments". Furthermore the thermal annealing of defects can be observed by a recovery of the NMR signal. The activation energy of the annealing process is calculated from an Arrhenius plot of the observed data. Evidently the various NMR methods can be used with advantage in the investi gation of lattice disorder. (A lot of references are reported by EBERT and SEIFERT [8].) II. Fundamentals 1. Zero Field Spectra An experimental and theoretical discussion of this matter in detail is given by DAS and HAHN [9]. Starting from Eq. (1.1) and choosing a principal axis system (X, y, Z) so that Vxz = VYZ = Vxy = 0 for the symmetrical tensor {V;k} the quadrupole Hamiltonian .n"Q becomes: [/2_ /2)] .n". = a . 1(1 + 1) + .!L(/2 + (2.1) Q 0 z 3 6+- with the quadrupole frequency 3·e·Q (2.2) ao = 41(21-1) 11 . Vzz and the asymmetry parameter Vxx- Vyy I'{= Vzz Evidently Eq. (2.1) is valid for single crystals as well as for powdered samples. In the case I'{ = 0 Eq. (2.1) yields for the energy levels .1f[ Em = ao m2_ 1(1 : 1)] . (2.3)
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