Table Of ContentNMR
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)
