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Phonon Dispersion Relations in Crystals

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PHONON DISPERSION RELATIONS IN CRYSTALS Gordon Edward Peckham, Trinity College, Cambridge. Dissertation submitted for the Ph. D. Degree; July 1964. Preface Most of the work described in this dissertation was carried out at A.E.R.E. Harwell, during the tenure of a D.S.I.R. Research Studentship. I wish to thank Drs E. Bretscher and P. A. Egelstaff for providing the experimental facilities, and their staff for much help with the experimental work. I am especially indebted to my fellow research student, Mr S. K. Sinha, with whom I collaborated in building some of the apparatus, and who rendered invaluable assistance at all times. I am grateful to members of the P.N.R. group at Harwell for many informal discussions; to Dr J. R. Hardy for suggesting the choice of magnesium oxide as a substance to study; to Dr W. Cochran for discussions about the Shell Model theory; and to my supervisor, Dr G. L. Sguires, for many helpful discussions and for his encouragement throughout the period of this research. Detailed acknowledgements are given in the text and in Appendix IV. i A note on this digital version of the thesis Optical character recognition was used to prepare digital text from a pho- tocopy of the original thesis. The document was typeset using LATEXwith the American Mathematical Society’s package amsmath. Line drawings were ren- dered into postscript using the package pstricks; other diagrams are from pho- tocopies. Pagination has changed – original page numbers are printed in the outside margins. Because diagrams may be in a different location with respect to the text some captions have been added to aid identification. The list of contents, of course, now refers to the new page numbers. A few typographic and spelling errors in the original have been corrected. G. E. Peckham 2011 ii Contents Preface i A note on this digital version of the thesis ii 1 Introduction 1 2 An Introduction to Lattice Dynamics and the Inelastic Scatter- ing of Neutrons from Single Crystals 2 2.1 Introduction.............................. 2 2.2 The Normal Modes of Vibration of a Crystal Lattice . 2 2.3 The Direct and Reciprocal Lattices . 4 2.4 The Phonon Dispersion Relation . 4 2.5 Some Effects of Crystal Symmetry . 5 2.6 Periodic Boundary Conditions . 7 2.7 Experimental Methods of Investigating the Phonon Dispersion Relation................................ 7 2.8 Methods for Neutron Spectroscopy . 10 2.9 The Neutron Cross Section Formula . 12 3 Cold Neutron Apparatus 13 3.1 Introduction.............................. 13 3.2 The Cold Neutron Apparatus . 13 3.3 Rotor Theory . 19 3.4 The Time Spread at the Sample . 26 3.5 Resolution of the Apparatus . 27 3.6 OptimumDetectorEfficiency . 31 3.7 Choice of Detector . 32 4 Data Processing 39 4.1 Introduction.............................. 39 4.2 Time of Flight Recorder . 39 4.3 Crystal Goniometer and Control Units . 41 4.4 The Magnetic Tape Analyser . 42 4.5 The Computer Programme ‘Fit and Search’ . 42 4.6 The Computer Programme ‘Phonon Reduction’ . 44 5 Phonon Dispersion Relation in Lead 47 5.1 Introduction.............................. 47 5.2 Theory................................. 47 5.3 Experiment .............................. 49 5.4 The Force Constant Model . 51 5.5 Conclusions . 54 iii 6 The Phonon Dispersion Relation for Magnesium Oxide – The- oretical 58 6.1 Introduction.............................. 58 6.2 The Rigid Ion Model . 58 6.3 The Short Range Forces . 58 6.4 The Electrostatic Forces . 62 6.5 TheShellModel ........................... 65 6.6 The Macroscopic Constants . 68 6.7 Reduction of the Number of Arbitrary Parameters . 71 6.8 Evaluation of the Parameters of the Theory in Terms of the Macroscopic Constants . 71 6.9 Degeneracy . 72 6.10 Preparation of the Contour Maps . 75 6.11 The Infra-Red Dispersion Frequency for Magnesium Oxide . 86 7 The Phonon Dispersion Relation of Magnesium Oxide – Exper- imental 88 7.1 Introduction.............................. 88 7.2 Apparatus . 88 7.3 Results................................. 89 7.4 Resolution . 90 7.5 Intensities............................... 107 7.6 Bragg Reflections in the Sample . 109 7.7 Discussion . 110 8 Crystal Spectrometer 112 8.1 Introduction . 112 8.2 The Three Axis Spectrometer . 112 8.3 Focussing ...............................114 8.4 The Locus of the Ends of the Wave Vectors of Bragg reflected Neutrons in the Case of Finite Incident Collimation and Finite Mosaic Spread . 117 8.5 Redesigned Instrument . 118 Appendices I The Circular Arc Approximation for Rotor Slots 121 II Slots not Passing through the Centre of the Rotor 125 IIIAlignment of Apparatus 128 III.1 Goniometer Alignment . 128 III.2 Alignment of the Single Crystal Sample . 129 References 133 iv Chapter 1 Introduction page 1 Inelastic neutron scattering experiments can provide much information about the thermal motions in solids and liquids. Neutrons which have been thermalised by the moderator of a nuclear reactor have energies similar to those associated with these thermal motions. It so happens that they also have wavelengths similar to the interatomic spacing in solids and liquids. Quite large and easily measured changes in energies and wave vectors are apparent when neutrons are scattered from solids or liquids, and these may be related to properties of the thermal motions. The theory of the thermal motions in a crystalline solid is well-known, and was first published as long ago as 1912 (Born, von K´arm´an, 1912). The motion is described in terms of plane waves or normal modes of vibration. At first, the only experimental information came from specific heat measurements, and the success of the very much simpler Debye theory did not encourage development of page 2 the Born–von K´arm´ancrystal dynamics. However, discrepancies were apparent and some more detailed calculations were made (e.g. Kellermann 1940). Since the introduction of nuclear reactors, large fluxes of thermal neutrons have been available for inelastic neutron scattering experiments. The scattering from a single crystal enables us to deduce the phonon dispersion relation, that is the frequency of the normal modes as a function of their wave vectors. These experiments provide a far more direct test of theories of crystal dynamics than do specific heat measurements. The possibility of experimental measurements of the dispersion relation has stimulated theoretical work on crystal dynamics. The theory of ionic crystals is particularly well advanced. The Shell Model has been developed by Cochran and others to give excellent agreement with the experimental dispersion relations for the alkali halides (Cowley, Cochran, Brockhouse and Woods 1963). This dissertation describes experiments to investigate the inelastic scatter- ing of neutrons from a magnesium oxide single crystal. The phonon dispersion page 3 relation is deduced, and is discussed in terms of the Shell Model . The Shell Model is shown to provide a good description of the crystal dynamics of magne- sium oxide. A less extensive experiment of the same type to deduce the phonon dispersion relation for a lead single crystal is also described. 1 Chapter 2 An Introduction to Lattice Dynamics and the Inelastic Scattering of Neutrons from Single Crystals 2.1 Introduction The Born–von K´arm´antheory of crystal dynamics for crystals with more page 4 than one atom per unit cell is summarised to introduce the notation used in this thesis. The notation is similar to that of Born and Huang (1954), but cartesian components are indicated by superscripts, all other indices being writ- ten as subscripts. The principal features of the phonon dispersion relation are discussed, including the effects of crystal symmetry. The value of experimental measurements of this dispersion relation, and the advantages of thermal neu- tron scattering over other methods are emphasised. Two particular methods for neutron spectroscopy are mentioned, and the relevant cross section formulae given. 2.2 The Normal Modes of Vibration of a Crystal Lattice The dynamics of crystals have been discussed at length by Born and Huang (1954). The basis of the Born von K´arm´antheory of crystal dynamics is the page 5 use of a potential function for the crystal which is a function of the nuclear co- ordinates only, and is quadratic in these coordinates. To derive such a potential function, we must make two approximations. The adiabatic approximation is to assume that the electrons accommodate themselves rapidly to the changing nu- clear coordinates, so that the electron coordinates do not enter into the potential function. The electron does not make transitions from one state to another, but the state itself is continuously deformed by the nuclear motion. This is a good approximation if the energies of electron excitations are very much higher than the energies of vibration of the nuclei. This is the case for insulators, but it is more difficult to justify the approximation in the case of metals. The harmonic approximation is to ignore higher powers of the nuclear coordinates than the second. This is a necessary approximation in order to obtain a first solution to the problem, but is at variance with such properties of the crystal as thermal expansion and heat conduction. Having obtained a first solution in terms of normal modes of vibration, the effects of the higher powers may be described by an interaction between the normal modes. page 6 A crystal has translational symmetry. That is, it could be built up from a series of identical cells placed together so as to fill the volume occupied by the crystal. The unit cell is the smallest such cell from which the crystal can be constructed. It is not unique, but all possible unit cells will have the same volume. Consider a crystal with n particles in its unit cell. A particle is referred to 2 by two indices, l denoting the unit cell, and k = 1, 2 ...,n denoting the position in the unit cell.
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