Characterization of Thin Films and Novel Materials Using Resonant

Home , Mechanical resonance

The Pennsylvania State University

The Graduate School

Department of Physics

CHARACTERIZATION OF THIN FILMS AND

NOVEL MATERIALS USING

RESONANT ULTRASOUND SPECTROSCOPY

A Thesis in

Physics

Joseph Rhea Gladden, III

c 2003 Joseph Rhea Gladden, III

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

August 2003 ii

The thesis of Joseph R. Gladden, III has been reviewed and approved* by the following:

Julian D. Maynard Distinguished Professor of Physics Thesis Advisor Chair of Committee

Peter C. Eklund Professor of Physics

Vincent H. Crespi Associate Professor of Physics

Anthony A. Atchley Professor of Engineering Head of the Graduate Program in Acoustics

Jayanth R. Banavar Professor of Physics Head of the Department of Physics

*Signatures are on ﬁle in the Graduate School iii Abstract

Crystalline thin films have become an important area of research in condensed matter and applied physics, electrical engineering, and materials science. There exists fundamental differences between 2 dimensional and bulk physics for many materials; and ever decreasing element sizes in mechanical and electronic devices requires a practical understanding of these differences. With an increased interest in thin films comes a need for new probes, able to gather information on structures with ever decreasing dimensions. This work describes a new extension of the acoustic method resonant ultrasound spectroscopy (RUS) to the measurement of the elastic tensor of a thin film deposited on a substrate. The thin film RUS technique has been successfully applied to various novel materials such as mats of carbon nanotubes, which exhibit novel attenuation effects on small mechanical resonators; and colossal magneto-resistance (CMR) films, for which a previously unreported phase transition has been detected. Typical film thicknesses studied were 100 - 1,000 nm for the CNT films, and 200 and 400 nm for the CMR films.

In general, films occupying only 1/1,000 of the substrate can be reliably measured. In addition to thin films, traditional RUS was used to determine a correction to the sign of the c14 of α-alumina, previously measured over 40 years ago; and the first application of RUS to biological materials, specifically human dentin. iv Table of Contents

List of Tables ...... ix

List of Figures ...... xi

Acknowledgments ...... xv

Chapter 1. Introduction ...... 1

1.1 Elasticity Measurements in Thin Films ...... 5

1.1.1 Travelling Wave Methods ...... 5

1.1.2 Resonance Methods ...... 6

1.2 Organization and Summary ...... 8

Chapter 2. Theoretical Foundations ...... 16

2.1 Elasticity in Solids ...... 17

2.2 Wave Propagation in Solids ...... 22

2.3 Calculating Resonance Spectra: the “forward” Problem ...... 25

2.3.1 The Rayleigh-Ritz Method ...... 25

2.3.2 Choice of Basis Functions ...... 28

2.3.3 The Visscher Basis Set ...... 30

2.3.3.1 Accuracy of the Solutions ...... 31

2.3.3.2 Cylindrical Shells ...... 32

2.3.3.3 Bilayer Stacks and Thin Films ...... 35 v

2.4 Determining Elastic Constants: the Inverse Problem ...... 39

2.4.1 Levenberg-Marquardt Minimization ...... 40

2.5 Theory of Internal Friction in Elastic Solids ...... 45

2.5.1 Maxwell Model (1867) ...... 46

2.5.2 Kelvin-Voigt Model ...... 47

2.5.2.1 Wave Propagation in a Voigt Solid ...... 49

2.5.3 Zener Model (1948) ...... 52

2.5.3.1 Wave Propagation in a Zener Solid ...... 52

2.5.4 Dissipation Mechanisms ...... 58

2.5.4.1 Material Speciﬁc (Intrinsic) Losses ...... 58

2.5.4.2 Sample Speciﬁc Losses ...... 60

2.5.4.3 Environmental Losses ...... 63

Chapter 3. The Experimental Method ...... 67

3.1 Small Sample RUS Techniques ...... 68

3.1.1 Sample Preparation ...... 68

3.1.1.1 Polishing and Cleaning ...... 69

3.1.1.2 Atomic Force Microscope Characterization of Sur-

face Roughness ...... 76

3.1.1.3 Sample Size Measurements ...... 77

3.1.1.4 Eﬀect of Geometry Errors ...... 80

3.1.2 Cell Design and PVDF Thin Film Transducers ...... 83

3.1.3 Mounting a Sample ...... 88 vi

3.1.3.1 Eﬀects of Transducer Loading ...... 92

3.1.3.2 Frequency Shifts Due to the Film ...... 97

3.1.4 Data Acquisition ...... 101

3.2 Determining Elastic Constants ...... 107

3.2.1 Uncertainties in Fitted Parameters ...... 111

3.3 General Techniques for Thin Film Experiments ...... 112

3.3.1 A Test Case: Aluminium Film on Strontium Titanate . . . . 114

3.4 Conclusions ...... 117

Chapter 4. Carbon Nanotube Films ...... 120

4.1 Structure and Properties of Carbon Nanotubes ...... 121

4.2 CNT Film Sample Preparation ...... 125

4.2.1 Film Deposition ...... 126

4.3 Results ...... 132

4.3.1 The “Q-eﬀect” ...... 133

4.3.2 CNT Film on a Passivated Si Surface ...... 135

4.4 Analysis ...... 137

4.5 Conclusions ...... 142

Chapter 5. Colossal Magnetoresistance Films ...... 145

5.1 Introduction ...... 145

5.2 The CMR Experiment ...... 147

5.3 Analysis and Results ...... 149 vii

Chapter 6. Aluminum Oxide ...... 156

6.1 Introduction and Previous Work ...... 156

6.2 RUS for Rotated Trigonal Crystals ...... 158

6.3 Orienting and Polishing ...... 162

6.4 Analysis and Results ...... 166

Chapter 7. Elasticity of Human Dentin ...... 169

7.1 Introduction ...... 169

7.2 The Eﬀects of Moisture Content ...... 171

7.3 Anisotropy of Dentin ...... 179

Chapter 8. Conclusions ...... 181

Appendix A. Listing of RUS programs ...... 185

A.1 Inverse RUS program for a substrate with a ﬁlm ...... 186

A.2 Calculating Surface Displacements for Normal Modes ...... 211

A.2.1 Visualizing Normal Modes with Mathematica ...... 218

A.2.2 Fitting Resonance Curves with Mathematica ...... 220

Appendix B. Computer Code for the Alumina Experiment ...... 222

B.1 Euler Transformation of the Elastic Tensor ...... 222

B.2 Simulation of X-Ray Back-reﬂection Diﬀraction Pattern ...... 223

Appendix C. Thin Film PVDF Transducer Fabrication ...... 229

C.1 Preparation of PVDF Film ...... 229

C.2 Evaporation ...... 230 viii

C.3 Mounting transducers on the cell ...... 234

References ...... 236 ix List of Tables

2.1 Independent Elastic Constants for Various Crystal Symmetries . . . . . 22

2.2 Comparison of measured and calculated resonances for a stainless steel

cylindrical shell for N=9 and N=12. Due to slight machining errors,

degenerate modes are typically 2 closely spaced and overlapping peaks.

The experimental values were then taken as a weighted average (based

on amplitude)...... 36

2.3 Listing of commonly used dissipation parameters...... 56

3.1 Reproducibility of frequencies and quality factors for a typical strontium

titanate sample...... 96

3.2 Elastic constants in Mbar and shifts in quality factors for two aluminum

ﬁlms deposited on a strontium titanate substrate...... 117

4.1 Change in quality factor for a 500 nm SWNT ﬁlm on a strontium titanate

substrate...... 133

4.2 Modes exhibiting the largest increase in Q for various substrates. . . . . 137

6.1 Comparison of measured resonance frequencies with those generated us-

ing +/- c14 in the forward calculation...... 168

6.2 Results for ﬁtting the alumina elastic constants to our data...... 168

7.1 Time constants for Young’s modulus (E) and Poisson’s ration (σ) for

drying dentin...... 178 x

7.2 Isotropic and hexagonal symmetry results for wet and dry dentin. . . . . 180 xi List of Figures

1.1 Free energy of an atom in a crystal...... 2

1.2 Transverse surface acoustic wave...... 6

1.3 Example of a heterogeneous resonator: a ﬁlm on a substrate...... 7

1.4 Schematic for a RUS thin ﬁlm measurement...... 8

2.1 Simple spring (a) at rest and (b) after a force is applied...... 18

2.2 Small strains of an elastic solid of (a) axial and (b) shear types...... 19

2.3 Selected normal modes of a cylindrical shell...... 34

2.4 Geometry for a bilayer stack parallelpiped with n=2 ...... 38

2.5 Maxwell model for an anelastic solid...... 48

2.6 Kelvin-Voigt model for an anelastic solid...... 50

2.7 Zener model for an anelastic solid...... 53

2.8 Dissipation as a function of frequency for the Zener model ...... 55

2.9 The quality factor of a mechanical resonance...... 57

2.10 Illustration of linear displacement of a point in the bulk (solid), and

nonlinear displacement of a point near a surface crack (dotted)...... 64

2.11 Two schemes for supporting a cylindrical resonator to minimize losses.

Support contacts are made at displacement nodes...... 65

3.1 Cut away view of the polishing jig...... 70

3.2 Polishing disk with right angle blocks for polishing small parallelepiped. 75 xii

3.3 AFM images of polished alumina and strontium titanate samples. . . . . 78

3.4 The Starrett microdimensioner used for accurate sample dimension mea-

surements...... 81

3.5 Several types of possible geometry errors during parallelepiped sample

preparation...... 82

3.6 Schematic of the mounting apparatus in the small sample RUS cell. . . 85

3.7 Schematic of the entire small sample RUS cell...... 86

3.8 Template for metallization of PVDF transducers...... 89

3.9 Thin ﬁlm PVDF transducer fabrication...... 90

3.10 Vacuum tweezers used to mount a sample...... 93

3.11 Sample mounting apparatus to be used in an inert atmosphere glove box. 94

3.12 The eﬀects of transducer loading on natural frequencies and quality fac-

tors...... 98

3.13 Film thickness dependence of resonant frequencies an hypothetical alu-

minum ﬁlm deposited on a strontium titanate substrate...... 100

3.14 Schematic view of data acquisition system for a RUS experiment. . . . . 103

3.15 Circuit diagram for a low noise, high frequency JFET preampliﬁer. . . . 104

3.16 Example of quadrature (a) and in-phase resonance signals with lorentzian

ﬁts...... 108

3.17 Microscope view of 546 nm aluminum ﬁlm on STO substrate mounted

between the transducers...... 118

4.1 Rolling a graphene sheet into a carbon nanotube...... 122 xiii

4.2 Chirality vector map for CNT...... 124

4.3 The energy dispersion over the entire Brillouin zone for a 2D graphene

sheet...... 124

4.4 Buckled CNTs can rebound to their original shape...... 126

4.5 Air brush system for depositing nanotube ﬁlms...... 130

4.6 Schematic of the artists air brush used for CNT ﬁlm deposition...... 131

4.7 AFM calibration data and ﬁt for air brush deposition...... 131

4.8 The eﬀect of air exposure on a 500 nm SWNT/STO ﬁlm sample. . . . . 134

4.9 Shift in quality factor as a function of SWNT ﬁlm thickness for various

substrates...... 136

4.10 The eﬀect of a SWNT ﬁlm on a passivated silicon surface...... 138

4.11 Model for the reinforcement of surface microcracks with carbon nanotubes. 141

5.1 Typical manganite unit cell (inset) and resistivity data illustrating the

CMR eﬀect...... 146

5.2 RUS cryostat used in the CMR ﬁlm experiments...... 150

5.3 Elastic constant and resistance results for 400 nm LCMO (a) strained

and (b) unstrained...... 153

5.4 Elastic constant and resistance results for strained 200 nm LCMO. . . . 154

6.1 The “parallelogram” distortion governed by c14...... 158

6.2 Illustration of an Euler angle transformation for alumina...... 161

6.3 Orientation of the crystal in the x-ray apparatus...... 162 xiv

6.4 Plan view (X, Y plane) (a) and perspective view (b) of the rhombohedral

unit cell for alumina...... 164

6.5 Cutting and polishing details for the 2 alumina samples...... 165

7.1 Anatomy of a human tooth...... 170

7.2 Structure of human dentin...... 172

7.3 AFM images showing the internal structure of dentin...... 173

7.4 Typical resonances for the ﬁrst few modes of a dentin sample over pump-

ing times from 0 (in air) to 28 hours...... 175

7.5 Pumping results for dentin pretreated by water only, Hank’s solution

only, and Hank’s followed by water...... 177

C.1 Template for metallization of PVDF transducers...... 231 xv Acknowledgments

The work required to obtain a doctorate in physics is extraordinary and can not be done without the help and support of many people. I would ﬁrst like to thank my advisor, Jay Maynard, whose creativity in thought and ingenuity in detail never ceased to amaze me. My son Chase, daughters Camille and Josephine, and particularly my wife

Nicole also deserve an enormous thank you for putting up with long work hours. There is no doubt in my mind that without their support, I could never have ﬁnished this work.

Finally, I would like to thank my parents, Joe and Sally Gladden, whose emotional and

ﬁnancial support were also critical.

This thesis is dedicated to the memory of my great-grandfather, Dr. Woolford B.

Baker, Professor of Biology at Emory University. “Dad” Baker taught me from a very young age that the world around us is a remarkable place and deserves to be investigated, regardless of the age of the investigator. 1

Chapter 1

Introduction

When a new material is created, the first important information to be gathered is the location of the constituent atoms, or the crystal structure. This is defined by the minima in the free energy with respect to atomic positions, and often measured by X- ray diffraction methods. Of next importance is the curvature of the free energy near the minimum, as shown in Fig. 1.1, which is represented by the elastic stiffness tensor. The elements of this tensor characterize the linear strain response of a material to an applied stress, and are known as the second order elastic constants. Elastic constants are sensitive probes into the atomic environment; and thus are important not only in engineering and materials science, but also to many areas of fundamental and applied physics. Numerous methods for measuring elastic constants have been devised, including static loads, X-ray scattering under constant stress, Brillioun scattering, and inelastic neutron scattering.

Historically, the most widely used method is based on the result of classical elastic theory that the speed of sound in a material is related to the elastic constants where the sound speed of different wave modes and directions can, in principle, produce all elements of the elastic tensor. [36] [73] This suggests the conceptually simple pulse-echo method in which a high frequency acoustic wave pulse is introduced into the material through a polished face, allowed to traverse a sample and reflect off a parallel face. The time of flight is determined by the returning pulse, making the wave speed straightforward 2

Ener

ro r

Etot

Fig. 1.1. Free energy of an atom in a crystal. The equilibrium position is ro and the elastic constants are a measure of the curvature near the minimum. Acoustic energy displaces the atom from equilibrium.

to calculate. The analysis in a pulse-echo experiment is relatively simple, however the experimental details require great care and are fraught with difficulties. Since many materials of interest have similar acoustic properties to typical transducers, impedances can be well matched, making the beginning and ending of the wave pulse difficult to define. The transducer must be very carefully bonded to the sample and faces polished precisely parallel and aligned to specific crystal orientations to decouple the wave modes.

Very small samples require small transducers and very short pulses, making precise time measurements diﬃcult. Also, for a highly anisotropic crystal, the samples must be recut and polished (perhaps several times) to obtain all independent sound speeds, making the experiments time consuming. Many of these shortcomings can be overcome by using standing rather than travelling acoustic waves.

The resonance method for determining elastic constants in crystals, becoming feasible in the mid 1980’s, utilizes the fact the spectrum of natural frequencies of a solid body depends on the geometry, mass density, and elastic tensor of the sample. The 3 method has come to be known as resonant ultrasound spectroscopy (RUS). The ﬁrst step in such a method is the calculation of the normal modes of an anisotropic solid body.

The fundamental concepts for such a problem were developed over a century ago with classical elastic theory [36]. However, the complexity of the resulting equations of motion limited solutions to very simple cases such as isotropic spheres and certain modes of an isotropic cube. Signiﬁcant progress was made by applying the Rayleigh-Ritz method in which the functions describing the set of normal modes are expanded in a basis set with coeﬃcients determined by minimizing a Lagrangian and transforming the equations of motion into an eigenvalue problem.[14] [22] Early choices of basis sets were sinusoids or

Legendre polynomials because of their ease of manipulation and orthonormality. Until the advent of high speed digital computers, basis sets were limited in size and sophistica- tion, so results were marginal. A significant development in terms of making a resonance methods viable as an experimental technique was that of Visscher et al in 1991.[75] They introduced a basis set consisting of a linear combination of powers of the cartesian coordinates: xpyqzr, where p, q, r are a set of positive integers. This basis can be used to treat a wide variety of geometries, such as arbitrary parallelepipeds, spheroids, cylinders, bells, and even “potatoes” (as defined by ellipsoids) of arbitrary crystal symmetry. The calculation of eigenvalues, given the geometry and material properties of the resonator, is known as the “forward” problem. While this represents a significant development in theoretical mechanics, to be useful as a measurement technique the problem must be inverted so that the elastic constants can be obtained from the (measured) eigenvalues. This is known as the “inverse” problem. Soon after the robustness of the Visscher method was demonstrated, experimentalists (notably J.D. Maynard [45] and A. Migliori 4

[46]) devised analysis techniques which allowed a solution to the inverse problem. Over the past decade, the RUS method has gained in popularity to the point where one can now purchase a complete commercial system. [15] But despite its advantages over other methods, there are still relatively few active research groups practicing RUS today. The superiority of RUS is further evidenced by its ability to treat heterogeneous samples such as thin crystalline ﬁlms deposited on a substrate.

Materials in the form of a thin film deposited on a substrate have become a very important field of study for both fundamental and applied condensed matter physics. It is well known that the physics governing the change in the properties of a sample is altered as its dimensionality shifts from 3D bulk to a 2D film. Some phenomenon such as the quantum Hall effect do not appear until the material approaches a 2D form, and various critical phenomenon drastically change in nature as dimensionality changes. [29] Such phase transitions are readily probed by the elastic constants. Thin films also represent an important component in many technological applications, such as digital data storage devices, shrinking electronic components, and novel computing architectures involving spintronics and quantum computing schemes. It is the goal of this work to develop a new experimental technique, based on the well established method of resonant ultrasound spectroscopy, to measure elastic constants in crystalline films of thicknesses as small as

100 nm deposited on a substrate. 5

1.1 Elasticity Measurements in Thin Films

1.1.1 Travelling Wave Methods

A number of experimental methods currently exist for measuring elasticity in thin

ﬁlms. One such method is a pulse-echo type experiment using a surface acoustic wave

(SAW). SAWs stem from a solution to the wave equation in a semi-infinite medium in which the amplitude of the wave decreases exponentially in the direction pointing into the medium. The characteristic depth of the SAW is on the order of its wavelength, so to keep the wave localized very near the surface requires high frequencies (short wavelengths). Fig. 1.2 shows the displacement near a surface of a film/substrate system as a SAW passes. If the speed of a SAW can be determined, then so can the elastic constants in the film provided the wave can be reasonably localized within the film volume. To make a estimate of typical frequencies required, let us imagine we have a film of (polycrystalline) aluminum with a thickness of 200 nm. The speed of sound is about 5100 m/s, so the frequency range required to localize a SAW in the film is f = v/λ = 25.5 GHz! Transduction at these frequencies is exceedingly difficult, so the

SAW method of determining ﬁlm elastic constants is generally limited to much thicker

films, beyond what a physicist would typically term “thin”. In addition to the limitations in film thickness, surface acoustic wave methods are difficult to apply to anisotropic films and often require permanent changes to the sample such as patterning of electrodes or thermal damage. 6

film d substrate

Fig. 1.2. Transverse surface acoustic wave travelling horizontally. The amplitude decreases exponentially into the material with a characteristic depth of 1 wavelength (λ).

1.1.2 Resonance Methods

The flexibility of the Visscher basis set allows one to solve the normal mode problem for heterogeneous objects such as the important system of a film deposited on a substrate, as shown in Fig. 1.3. For such a system, the resonance spectrum is defined by the material properties of both the substrate (unprimed) and the film (primed). Thus, if the substrate parameters and geometry are well known, the film elastic constants may be extracted from the resonance data. The steps required for calculating normal modes for such a sample are described in Chapter 2 and the FORTRAN code is listed in Appendix

A. An important issue is what thickness of the ﬁlm is required if the elastic constants are to be resolved.

Forward calculations can be used to measure the shifts in eigenvalues as a function of film thickness (or volume fraction). Such an analysis is given in Chapter 3 and shows that films on the order of 1/1,000 of the substrate thickness should be within reach if the resonances are sharp (high quality factor). Typically “thin” films are less 7

z film: c’ij ,r’

substrate:cij,r y

Fig. 1.3. Example of a heterogeneous resonator: a ﬁlm on a substrate.

than a few hundred nanometers in thickness, which implies a substrate thickness of a few hundred microns. However, very thin plate-like resonators introduce other analysis problems such as an increased eﬀect of loading by the transducers, possibly a reduced dependence on some elastic constants (e.g. thickness modes will not be excited until high mode numbers are reached), and lower quality factors. Thus the measurement of a thin ﬁlm sample requires substrates which are small in ALL dimesions. While many practitioners of RUS use samples on the order of several millimeters to centimeters in size, J.D. Maynard’s implementation can measure samples as small as a few hundred microns on a side through the use of thin piezoelectric transducer strips to excite and detect resonances. The smallest RUS sample successfully measured by Maynard’s group was a mere 70 µg! [67] Fig. 1.4 schematically shows the general procedure of a RUS thin

ﬁlm measurement.

The ﬁrst step in a ﬁlm experiment is to fabricate a substrate and perform a standard RUS measurement on it to determine the optimal set of sample parameters

(dimensions, elastic constants, and crystal orientation). The substrate is removed from 8

Fig. 1.4. Schematic for a RUS thin ﬁlm measurement.

the cell and the film is deposited on one of the faces. The sample is remounted and a new resonance spectrum is obtained. All substrate parameters are held constant during the analysis phase, and only film elastic constants are allowed to vary in such a way as to minimize the error between the observed and calculated frequency shifts due to the presence of the film.

1.2 Organization and Summary

This section outlines the various chapters of the thesis and presents some important points of each. No eﬀort is made toward derivations or lengthy explanations here; that is reserved for the referring chapters.

Chapter 2: Theoretical Foundations

This chapter is dedicated to a detailed derivation of the computational tools used in performing a RUS experiment. Basic elastic theory is reviewed, followed by the governing equations for the normal mode problem. Next the application of the

Rayleigh-Ritz method is described and the casting of the normal mode problem into 9 eigenvalue form. The Visscher basis set is presented along with several calculations for diﬀerent geometries: homogeneous parallelepiped, a bi-layer stack parallelepiped which is considered the more general case of a ﬁlm on a substrate, and a homogeneous cylindrical shell (tube), for which a solid cylinder is a special case.

After the “forward” problem is thoroughly addressed, methods for the “inverse” problem are presented. The fundamental idea is to iterate the forward problem, adjusting the elastic constants (or other parameters), until the calculated spectrum best

ﬁts the measured data in a least squares sense. The nonlinear Levenberg-Marquardt

ﬁtting technique is described in some detail along with the singular value decomposition

(SVD) method of inverting matrices. The ﬁnal section presents some fundamental concepts regarding attenuation in oscillating solids. This issue is often ignored in typical descriptions of RUS since most crystals have very low dissipation. The reason for its inclusion here, however, is to lay the ground work for an explanation of the experimental observations of carbon nanotube ﬁlms on small resonators.

Chapter 3: The Experimental Method

This chapter presents many of the details required to perform a thin ﬁlm RUS experiment. It begins with a discussion of one of the most important components of the experiment: sample preparation. The forward calculation on which the measurements are based assumes a perfect geometry (e.g. parallelepiped, cylinder, etc.), so the sample must be very carefully polished to match this geometry as closely as possible. A scheme utilizing a polishing jig, right angle blocks, wax, and diamond polishing paper is used to accomplish this task for samples as small as a few hundred microns. As mentioned above, the ability to measure very small samples is critical to the extension of traditional 10

RUS to thin film measurements, so some time is spent on the cell and transducer design. The sample is held very lightly on opposing corners by the two taught transducer strips to most closely approximate the free boundary conditions assumed in the forward calculation. As in situ film deposition is nearly impossible, it is important to understand how the resonances vary between mountings, so a statistical study is presented in which a sample was remounted 10 times by 2 different people. The results of this study where coordinated with forward film calculations for various film thicknesses to ascertain how thin a film may be reasonably probed. If the set of measured modes is carefully chosen, films as thin as 1/1,000 of the substrate thickness are well within measurement precision. Extracting film elastic constants from the data is perhaps the trickiest part of the experiment, so some time is spent on this procedure. The chapter concludes with a test experiment of an aluminum film deposited by evaporation on a strontium titanate substrate at thicknesses of 1/10,000 and 1/1,000 of the substrate thickness. ∼

Chapter 4: Carbon Nanotube Films

A system of potentially interesting elastic properties are carbon nanotubes (CNTs), a unique form of carbon which can be visualized as a 2D graphene hexagonal sheet of carbon atoms wrapped into a cylindrical shell having typical dimensions of 1-2 nm in diameter and microns in length. These novel structures have perhaps the highest strength to weight ratio of any known material. This chapter begins with an overview of the basic structure and properties of CNTs. Next comes an explanation of how CNT ﬁlms were prepared and deposited on various substrate materials for the RUS experiments. The most notable results are not the eﬀect of the CNT presence on the resonance frequencies, but on the quality factors (Q) which are a measure of energy loss in a resonator. 11

This novel behavior was exhibited for a wide variety of substrates including strontium titanate, silicon, quartz, brass, and alumina. Evidence is presented that the effect is strongly suppressed by chemically passivating the substrate surface. Many technological devices use small resonators for which the precision depends on the sharpness of the resonance; thus this effect may find significant applications in the micro electro-mechanical systems (MEMS) community. The chapter concludes with the description of a model consistent with the energy dissipation data.

Chapter 5: Colossal Magnetoresistance Films

As mentioned earlier, elastic constants are an excellent probe for various phase transitions. An important phenomenon in terms of both fundamental physics and technological applications is the colossal magnetoresistance (CMR) transition, in which there is a strong dependence of electrical resistivity on an externally applied magnetic ﬁeld.

The behavior is dramatically (colossally) enhanced for a particular class of materials known as perovskite manganites. The chapter begins with a brief introduction into the

CMR effect, the structure of perovskites, and current theoretical understanding of the physics behind the effect. In this experiment, 200 and 400 nm CMR films were deposited on strontium titanate substrates and resonances were tracked in a custom designed RUS cryostat from 245 to 285K. Precise temperature control is critical to this experiment, so some time is spent on the design of the cryostat. The elastic constants c11 and c44 for the film were extracted from the resonance data and plotted along with the electrical resistance data as a function of temperature. An anomaly in the cij was observed at the resistance peak temperature. An adidtional, and previously unreported, anomaly occurred about 17K higher, with no corresponding feature seen in the resistance data. 12

Due to lattice mismatch between the CMR film and strontium titanate substrate, the as-grown films are under strain, to which the CMR effect has been shown to be sensitive. The 400 nm sample was annealed such that the residual strain is relaxed. The resulting data is compared to our strained film data and bulk CMR behavior reported in the literature. No speculations are made about the nature of the higher temperature transition or how it may tie in to current CMR theory. It is hoped however that this extra information may provide additional clues into the phenomenon.

Chapter 6: Aluminum Oxide

The mechanical properties of aluminum oxide, a very hard and transparent crystal with rhombohedral (or trigonal) symmetry, are important in many areas of engineering and industry. As one may expect, there have been a number of measurements of alumina’s 6 elastic constants, including one by Ohno and Anderson using an early form of

RUS. Historically, the sign of c14 has been reported as negative, based almost exclusively on the ﬁrst careful measurements done by Wachtman, et al of the National Bureau of

Standards (NBS) in 1960. The elastic distortion governed by c14 is a “parallelogram” distortion, the orientation of which is dictated by the sign. Thus all subsequent studies were either insensitive to the sign of c14, used the same samples as the Wachtman group, or thank the NBS for a reorienting their samples after originally measuring a c14 > 0. A resonance spectrum is entirely degenerate for c unless all crystallographic axes are 14 misaligned with the sample body axes. Recently, a new ab initio method of calculating elastic constants has been developed by Le Page and Saxe, who calculate a positive c14 for alumina, while correctly calculating the sign for other trigonal crystals such as quartz and calcite. [60] The goal of this work is to remeasure the sign of c14 in alumina. 13

The sign of c14 is set by the definition of the crystallographic axes, uniquely defined since 1949 by IEEE (then IRE) standards. Great care must be taken during the sample preparation to keep track of the crystal orientation, so some time is spent discussing the details of these steps and precautions taken. Resonance spectra were obtained and compared with forward calculations using both c for several samples. 14 The agreement for the two samples measured was approximately 10 times better with a positive value than a negative value using only measured parameters (dimension and crystal orientation) and elastic constant magnitudes from Ohno and Anderson. Correct- ing a sign of an elastic constant of a heavily utilized material is certainly a worthwhile endeavor, however the real significance of this work is that it removes the last known discrepancy between existing experimental data and LePage’s method of calculating elastic constants from atomic position data and quantum potential energies.

Chapter 7: Elasticity of Human Dentin

While RUS has been successfully used on a wide variety of materials including single crystals, geologic samples, and even heterogeneous samples, to our knowledge it has not been applied to biological materials. This chapter describes a RUS experiment on human dentin, which is the largest component of calciﬁed tissue in the human tooth. It begins with a description of the structure of dentin, which consists of a matrix of collagen

ﬁbers lying preferentially in a plane, creating a felt-like structure. Perpendicular to these planes is an array of microtubules approximately 1 micron in diameter which carry nutrients to the outer enamel. Due to the small contiguous sample sizes available, other methods of measuring basic elastic properties such as Young’s modulus and Poisson’s 14 ratio have yielded highly discrepant results even over recent years. The sample sizes available however are highly amenable to our implementation of RUS.

The first experiment described was designed to test the effect of moisture type and content. Dentin samples were saturated with water and dried under vacuum while resonances were tracked. The first order elastic parameters Young’s modulus and Poisson’s ratio were extracted from the data and plotted against pumping time. It was found that dentin stiffened as it is dried and that the Young’s modulus values fell within the wide range of previous measurements by other methods. The second experiment reported is a test of the effect of mineral content. Dentin samples were soaked in a calcifying solution and similarly measured. After drying, the dentin was noticeably stiffer than when treated with water only. Also presented is some unexpected behavior of Poisson’s ratio for the calcified dentin.

The chapter concludes with a description of an experiment to investigate the elastic anisotropy of the dentin. The structure suggests that perhaps a planar isotropic (or hexagonal) symmetry, with the c-axis along the tubule direction, would better describe dentin than the typically purely isotropic models previously used. It was found that dry dentin exhibited almost no anisotropy, however wet dentin showed large anisotropy with stiffness in the direction of the collagen planes higher than in the tubule direction. This is important because most previous measurements have been performed on relatively dry samples, however in vivo dentin is saturated with fluids. It also suggests that the collagen fibers dominate the mechanical properties of the material as opposed to the tubules. 15

Appendices

The appendices list useful, but lengthy, information for any reader interested in conducting a RUS experiment. Appendix A lists several computer programs by which the RUS analysis is performed. They include FORTRAN programs for extracting ﬁlm elastic constants from the measured resonance data with arbitrary crystal orientation

(FILMMRQROT.F), a FORTRAN program for reconstructing the surface displacements for the normal modes of a parallelepiped (XYZMA.F) and writing data ﬁles to be read by a Mathematica script which plots the modes (MODE DISP.NB), and another Math- ematica script which ﬁts RUS raw resonance data with Lorentzian lineshapes to extract center frequencies and quality factors (PEAKFIT.NB). The second appendix lists two

FORTRAN programs used in the aluminum oxide experiments which (1) performs an

Euler rotation on the elastic tensor so that arbitrary crystal orientations can be analyzed and (2) simulates the x-ray back reﬂection pattern of a trigonal crystal. The ﬁnal appendix lists the detailed steps required for fabricating the PVDF transducers used in these experiments. 16

Chapter 2

Theoretical Foundations

As stated in the Chapter 1, resonant ultrasound spectroscopy (RUS) is the general name given to the technique of determining elastic constants and acoustic attenuation from a measured set of mechanical resonances of a particular sample. Since the primary focus of this work is to use RUS to measure the elastic constants of a crystalline thin

ﬁlm deposited on a crystalline substrate, the fundamental problem to be solved is to determine the resonance spectrum for an anisotropic and inhomogeneous geometrical solid. This component of the analysis is known as the “forward” problem. Once this solution is obtained, the elastic constants may be determined by iteratively adjusting these parameters until a least-squares ﬁt between the measured and calculated spectrum is obtained.

The contents of this chapter will include a basic overview of relevant elastic theory for crystalline solids, speciﬁc techniques for calculating resonance spectra for inhomogeneous objects, and fundamental concepts on the dissipation of elastic energy in solids.

The latter will become useful in discussing the results of the carbon nanotube ﬁlm experiment. A very good theoretical review of elastic theory as it pertains to general RUS is given in Phil Spoor’s Ph.D. thesis [67]. 17

2.1 Elasticity in Solids

A basic goal for elastic theory in solids is to relate the force a solid experiences to the resulting displacement ﬁeld. The simplest introduction to these concepts is the example of a one dimensional spring lying along the x-axis. Imagine a point on the spring (at rest) is marked at x and another point marked at x + dx, as shown in Fig.

2.1. After a force is applied, the points are displaced by an amount ψ(x) and ψ(x + dx) respectively. The strain () is deﬁned as the amount the spring has stretched per unit rest length and so can be expressed as

ψ(x + dx) ψ(x) dψ − = . (2.1) ≡ dx dx

The compliance of a spring (C) is the inverse of the force constant k, both of which are dependent on the length of the spring (e.g. if the length doubles, C 2C), and so is → not a material speciﬁc parameter. However the compliance per unit length is material speciﬁc. For a spring, Hooke’s law states that the force exerted is linearly related to the amount the spring stretches (dψ), and can be written

1 dx dψ F = kdψ = dψ = . (2.2) − −C − C dx

We can extend these concepts into 3 dimensions by replacing the spring with an elastic solid. In doing so, we introduce many more degrees of freedom with regard to the type of deformation so that the strain becomes a rank II tensor. Figure 2.2 shows both axial and shear strains. The axial strain shown in Fig. 2.2(a) is analogous to the 18

(a) x x+dx

F (b)

y(x) y(x+dx)

Fig. 2.1. Simple spring (a) at rest and (b) after a force is applied.

δψx 1D spring and xx = δx where the double subscripts of indicate a deformation in the x direction (δψx) normalized by the rest length in the x direction (δx). Shear strains characterized by a small angle θ, as shown in Fig. 2.2(b), can be decomposed into a symmetric shear deformation and a solid body rotation. The solid body rotation does not originate from any elastic deformation, and thus has no dependence on the elastic properties of the object. Thus the strain tensor is commonly expressed in such a way as to eliminate the rotation:

1 ∂ψi ∂ψj ij = + (2.3) 2 ∂xj ∂xi ! which is symmetric (ij = ji) and dimensionless. If the angle of shear deformation is δψ small, we can express the shear strain as θ tan θ = x = . The subscripts take ' δy xy values of either (x, y, z), or more commonly (1, 2, 3).

We must now examine the nature of the forces producing these deformations. The force acting on a diﬀerential surface area element dS may have a component normal to 19 the element, producing an axial strain, and/or parallel to the element producing a shear strain. The normal and shear forces can be written respectively as F = t ndS and n · F = t ndS, where n is the unit normal to the surface element and t is the traction, s × the components of which are

ti = σijnj (2.4)

and σij is the stress tensor having units of force per unit area. Here the ﬁrst subscript refers to the direction of the force and second refers to the direction of the normal to the surface element on which it acts as shown in Fig. 2.2. As stated above, we do not want to consider solid body rotations, thus the net torque applied to the object must be 0.

In order for this restriction to be maintained, the stress tensor must be symmetric also

(σij = σji).

dy y s dx x dyx xy x q dy sxx

(a) (b)

Fig. 2.2. Small strains of an elastic solid of (a) axial and (b) shear types. 20

In linear elastic theory, each stress component will be linear combinations of the strains with each term having some constant of proportionality. We can then ﬁnally write Hooke’s Law in its most general form

3 σij = cijklkl (2.5) kX,l=1 where cijkl are the components of the elastic stiﬀness tensor (elastic constants), and the summations over repeated indices will generally be implicit in what follows. As i, j, k, l each range from 1 to 3, both stress and strain are 9 component rank II tensors and the elastic constants comprise a 81 component rank IV tensor. However, we have already stated the symmetry of the stress and strain reducing their independent elements from

9 to 6. This also imposes a symmetry on cijkl such that each i and j can be switched and each k and l can be switched: cijkl = cjikl = cjilk = cijlk. Thus the number of independent components are reduced from 81 to 36. A further reduction of indices can be accomplished by noting that in the linear regime, the elastic potential energy must be quadratic in the strain: 1 U = U + c . (2.6) o 2 ijkl ij kl

So each pair of indices ij and kl can be switched without changing the potential energy.

Thus the ﬁnal number of independent elements is reduced to 21. As a side note, we can compare eqn. (2.6) with the Taylor expansion of the potential energy in strain about equilibrium ( = 0):

1 2 1 3 U() = U + U00 + U000 + . . . (2.7) o 2 o 6 o 21

So it is clear that we may associate the elastic constants with the second derivative of the potential energy with respect to strain. Note that there is no first derivative term which would imply a net force and thus a translation of the object, and that the 3rd order derivatives contain anelastic effects and have been ignored in this discussion. These effects are non-linear and are quantified by the so-called 3rd order elastic constants.

We have so far not taken into account any symmetries imposed by the material itself, so 21 is the number of the most anisotropic crystal class – triclinic. More symmetric crystal structures however impose further symmetries. For example, a cubic crystal has equivalent elastic response to stresses applied in either the 1, 2, or 3 directions; thus c1111 = c2222 = c3333 and likewise for shear strains. So the number of independent elastic constants for a cubic crystal is reduced to 3. Table 2.1 shows the fundamental crystal classes, number, and name of independent elastic constants.

Since our various symmetries have reduced the elements of the stress, strain, and stiﬀness tensors to 6, 6, and 21 respectively, it is possible to devise a reduced system of notation in which all the independent elements of the elastic tensor cijkl are represented by the elements of a 6 by 6 matrix cij which is also symmetric. [44] The standard rules for reducing the indices are:

11 1 23 4 → → 22 2 13 5 → → 33 3 12 6. → →

Some examples include: c c and c c . Most often elastic constants 1122 → 12 2323 → 44 are reported in the literature in the reduced index form, however for computational purposes (either analytic or numerical) it is necessary to use the full index form. More 22 detailed discussions of the relation between the elastic tensor elements and various crystal symmetries is given in Phil Spoor’s thesis [67] and the excellent resource Ultrasonic

Methods in Solid State Physics [73].

Table 2.1. Independent Elastic Constants for Various Crystal Symmetries

Crystal Class Number of cij Elastic Constants Triclinic 21 all possible combinations Monoclinic 13 c11, c12, c13, c16, c22, c23, c26, c33, c36, c44, c45, c55, c66 Orthorhombic 9 c11, c12, c13, c22, c23, c33, c44, c55, c66 Trigonal 6 or 7 c11, c33, c44, c13, c12, c14, (c25) Tetragonal 6 c11, c33, c44, c13, c12, c66 Hexagonal 5 c11, c33, c44, c12, c14 Cubic 3 c11, c12, c44 Isotropic 2 c11, c44

2.2 Wave Propagation in Solids

Up to this point, we have discussed only static deformations. In order to understand dynamic deformations, and thus the propagation of elastic waves in a solid, we must examine the time dependent behavior. Newton’s Law can be used to write a force balance equation of a diﬀerential volume element of density ρ

2 ∂σij ∂ ψ = ρ i . (2.8) 2 ∂xj ∂t 23

Making use of Hooke’s Law (2.5) and the deﬁnition of the strain, we can write

1 ∂ ∂ψ ∂ψ ∂2ψ c k + l = ρ i 2 ijkl ∂x ∂x ∂x 2 j l k ∂t where the summations extend over j, k, l. Making use of the symmetries in cijkl, we can reduce this to the equation of motion for the volume element

∂2ψ ∂2ψ c k = ρ i . (2.9) ijkl 2 ∂xj∂xl ∂t

While (2.9) is certainly reminiscent of the wave equation, the tensor nature obscures what the solutions and wave velocities might be. Indeed, a wave pulse excited along a particular direction in a crystalline solid may propagate along a different direction. A more convenient form of (2.9) are the Christoffel equations, which are derived in the literature [35] and simply stated here. If a volume element undergoes displacements u, v, and w in the x, y, and z directions respectively, and the wave vector is given by the direction cosines p, q, and r; then a distance along this direction can be written as s = px + qy + rz. The Christoffel equations are then

∂2u ∂2u ∂2v ∂2w ρ = α11 + α12 + α13 ∂t2 ∂2s ∂2s ∂2s

∂2v ∂2u ∂2v ∂2w ρ = α21 + α22 + α23 (2.10) ∂t2 ∂2s ∂2s ∂2s

∂2w ∂2u ∂2v ∂2w ρ = α31 + α32 + α33 ∂t2 ∂2s ∂2s ∂2s 24 where the αij are combinations of elastic constants and direction cosines. Determinan- tal equations can be obtained from (2.11) which yield 3 particle velocities and their direction cosines. This is the basis of determination of elastic constants by pulse-echo time of flight experiments which measure sound speed in a crystal along specific crystal directions. The directions must be chosen carefully so that the tensor equations uncou- ple producing particle (volume element) displacements for the three different modes of propagation which are mutually orthogonal. In this way, the wave pulse maintains its original direction.

As an example, consider a cubic crystal with only 3 independent elastic constants: c11, c12 and c44. The Christoffel equations can then be solved to show that the equations only decouple for waves propagating in the crystal directions 100 , 110 , and 111 . The h i h i h i longitudinal wave velocity can then be written vl = (c11 + c12 + 2c44)/ρ, and the two p shear velocities (for the two polarizations of shear waves) can be written vs1 = c44/ρ p and v = (c c )/ρ. With the three measured sound speeds, the three equations s2 11 − 12 p can then be solved for the three elastic constants. The lower symmetry crystals are increasingly difficult to measure using the pulse-echo method as more parallel faces need to be polished perpendicular to specific crystal directions, and only three velocities can be obtained from each direction. On the other hand, a crystalline sample will have a resonance spectrum which is well defined regardless of the crystal orientation. The spectrum has a wide variety of displacement types, and thus contains information about the all the relevant elastic constants and the orientation of the crystal with respect to the sample geometry. The RUS technique is based on the fact that all this information in the spectrum can be extracted, if a proper analysis can be devised. 25

2.3 Calculating Resonance Spectra: the “forward” Problem

A classic problem in mechanics is the calculation of the normal modes of an elastic solid. The traditional method of solution involves writing down the diﬀerential equations of motion, assume a time harmonic solution, and apply appropriate boundary conditions to obtain analytic solutions to the equations of motion and equations for the natural frequencies of oscillation. Unfortunately, there are very few cases for which this is possible, the exceptions being an isotropic sphere and certain modes of a parallelepiped.

These solutions will not be outlined here but can be found in the literature. [[67], [48], and [36]]

2.3.1 The Rayleigh-Ritz Method

One of the most powerful tools of theoretical physics is the Rayleigh-Ritz method of approximating eigenvalues of any Sturm-Liouville problem. The method is based on Hamilton’s principle that the Lagrangian of the system is stationary with respect to small perturbations in the eigenfunctions. Consider an oscillating elastic solid free of dissipation. The mechanical energy density of the solid is steadily changing forms between kinetic (T ) and potential (U), precisely analogous to a simple mass on a spring system. By conservation of energy, the total energy (E = T + U) is a constant. The

Lagrangian is deﬁned as L = T U. One of the most basic statments of classical − mechanics is that the time integral of the Lagrangian, known as the action, is a constant of the motion. Thus any variation of the action over time must be 0, and we can write 26

Hamilton’s variational integral

t2 δ (T U)dV dt = 0. (2.11) − Zt1 ZV

The kinetic energy is maximum when the strain over the volume is 0, and the potential energy is maximized when the kinetic energy is 0 (at turning points in the cycle). The potential energy density is

1 1 ∂ψi ∂ψk U = cijklijkl = cijkl , 2 2 ∂xj ∂xl and kinetic energy density for a normal mode is

1 T = ρω2ψ ψ . 2 i · i

We now have an expression for the Lagrangian in terms of the displacement ﬁeld

(ψ), which is still unkown. The Ritz method suggests approximating the displacements by an expansion of known functions with a set of unknown coeﬃcients. So the displacement of any point in the solid at an equilibrium position ~r can be written

3 N ψ~(~r) a φ (~r)ˆe (2.12) ≈ µ,i µ i iX=1 µX=1 where N is the number of terms in the expansion and functions φµ form the basis set.

The next step is to insert this expansion into the kinetic and potential energy expressions in the Lagrangian. It is useful here to contract the indices so that the 27 eigenfunction expansion becomes

3N ψ~(~r) a φ ˆe(m), φ φ . (2.13) ≈ m m m ∈ µ mX=1

We can then write the Lagrangian as

2 L = U T amanΓmn ω amanEmn (2.14) − ≈ m,n − m,n X X where ∂φ ∂φ Γ = c m n dV (2.15) mn imjknl ∂x ∂x ZV j l and

E = ρφ φ δ dV. (2.16) mn m n imkn ZV

We can then express the set of equations (2.14) in matrix form L aT(Γa ω2Ea) ≈ − where the components of the vector a are the expansion coeﬃcients. Now we impose

Hamilton’s principle that the Lagrangian be stationary to small perturbations in the eigenfunctions, which for this case means perturbations in the expansion coeﬃcients am:

∂L/∂am = 0. This allows us to cast the problem in the eigenvalue form

Γa ω2Ea = 0. (2.17) −

Finally, standard eigenproblem solution techniques (either analytic or numerical) can be employed to obtain a set of natural frequencies and expansion coeﬃcients (eigenvectors) for the elastic solid. 28

2.3.2 Choice of Basis Functions

Some care must be taken with the choice of basis functions φm. The more closely the functions match the true solution, the fewer terms will be required for a reasonable solution for the resonance spectrum. One reason the Rayleigh-Ritz method is so powerful is that the approximate eigenvalues are always larger than the true values, thus any choice in function which lowers the lowest eigenvalue is a better choice. Secondly, if the

(presumably small) deviation of the approximate eigenvectors from the true solution is

, then the errors in the eigenvalue is of order 2, meaning that even mediocre choices for the basis functions can yield good values for the natural frequencies. This is all very good news, but there are still some important criteria in choosing basis functions.

1. Boundary Conditions: The resonating elastic solid may be clamped at certain

edges or faces, or maybe free of all restrictions as if it were ﬂoating in space (free

boundary). The boundary conditions must be deﬁned during the formulation of the

Lagrangian. By a rather fortunate turn of events, if no additional terms are added

to the Lagrangian, then the free boundary conditions are automatically obtained.

The basis functions must be of a form that their sum can reasonably approximate

the boundary conditions of the problem at hand.

2. Linear Independence: Since the approximate solutions are built from a sum of

the basis functions, they must form a linearly independent set so that each of the

terms in the variational integral 2.11 equal 0 if their sum is 0.

3. Integrability: As each element in the E and Γ matrices requires the evaluation

of volume integrals of products of basis functions and their spatial derivatives, it 29

would be very helpful if these integrals could be worked out analytically. In a

practical sense, this means choosing rather simple basis functions.

Before the age of high speed digital computers, manageable basis function choices were quite limited, which in turn limited the geometries of the ﬁnite solid being modelled.

A short history of signiﬁcant developments follows. Holland [22] in 1968 modelled a parallelpiped of size 2a 2b 2c with the origin in the center by a basis set of sine functions × × pπx qπy rπz φpqr = sin 2a sin 2b sin 2c . His expansion included 100 combinations of p, q, r, yet still obtained 10% errors for some modes for which exact solutions were known. Soon after (in 1969), Demarest [14] obtained good results for the solution for a similar geometry by choosing Legendre polynomials as the basis:

φpqr = Pp(x/a)Pq(y/b)Pr(z/c)

which were normalized such that the E matrix was the identity matrix, simplifying the calculation.

A signiﬁcant development came from Ohno in 1976 [52] who used the same expansion basis as Demarest, but realized that by carefully coordinating the symmetries of the expansion basis functions, crystal, and object geometries, the large Γ matrix can be reorganized into a block diagonal form since many of the symmetric integrals will be 0.

Each block corresponds to a particular mode type (dilation, shear, torsion, or ﬂexure).

For example, for a homogeneous solid with orthorhombic or better crystal symmetry, the

Γ matrix can be arranged into 23 = 8 diagonal blocks. Thus calculations (e.g. matrix inversions) need only be performed on 8 small matrices rather than 1 very large one. If an 30 inhomogeneous sample is being analyzed, such as a substrate with a thin ﬁlm deposited in the x y plane, then symmetry is lost in the z-direction and the number of blocks − reduces to 4, and the blocks are larger. Interestingly, if identical ﬁlms were deposited on both sides of the substrate, symmetry in the z-direction is recovered and we may again use 8 small blocks. For lower symmetry crystals (e.g. trigonal), no blocking may be used even for homogeneous samples.

2.3.3 The Visscher Basis Set

The basis functions mentioned above are orthogonal making the E matrix easier to manipulate, however the integrals for most geometries are very diﬃcult. In 1991

Visscher et al. [75] proposed using simple powers of the cartesian coordinates xpyqzr, where (p, q, r) are a set of positive integers with the maximum power set by N p+q+r. ≤ Such a basis requires a larger number of terms and is not orthonormal so that the E matrix is no longer the identity, however it oﬀers much more ﬂexibility in the geometry of the object being modelled. Also, the matrix elements Γmn and Emn are integrals all of a consistent form so that a simple function can be written to calculate the matrix elements.

As the powers p, q, r can be rather large (10 or 12), it is computationally advantageous to normalize the coordinates with the sample dimensions. If m denotes a particular set of powers, m = (p , q , r ), then for a parallelpiped of dimensions 2a 2b 2c the m m m × × basis can be written

x p y q z r φ = m m m = xˆpmyˆqmzˆrm. (2.18) m a b c 31

If this function is inserted into (2.15) and evaluated over the intervals -1,1 by the above { } normalization, we see the matrix elements are all a simple function of the exponents

1 1 1 F (p, q, r) = φdV = abc xˆpyˆqzˆrdxdˆ yˆdzˆ V 1 1 1 Z Z− Z− Z−

8 abc p, q, r even (p+1)(q+1)(r+1) =  (2.19)   0 otherwise.   2.3.3.1 Accuracy of the Solutions

The accuracy of normal mode solutions using the Visscher basis set can be tested by comparison with the few cases where exact analytical solutions are known: (1) an isotropic sphere and (2) Lam´e modes of an isotropic parallelepiped. Upon ordering the eigenvalues in order of increasing frequency, it is expected that accuracy will degrade with increasing mode number as the higher frequency modes will have more “wiggles” and thus will need higher degree polynomials (N) to reasonably model them. It is useful to have some gauge of how high N needs to be to exceed the experimental error in resonance measurements. Phil Spoor has carried out such comparisons with the exact solutions of an isotropic sphere with the results that for p + q + r N = 12, the eigenvalues agree to ≤ better than 0.0005% for modes as high as 200 [67]. It should be noted however that these mode numbers count every mode even though many are degenerate due to the extremely high symmetry of a sphere. Even so, from an experimental point of view, measuring 25 or so resonances is more than adequate to determine even 8 variable parameters (there are only 2 elastic constants for isotropic symmetry, however dimensions could also be varied).

The few Lam´e modes showed similar accuracy for N=12. Another test of the accuracy 32 of the Rayleigh-Ritz method is to run calculations and compare results of increasing N.

The nature of Hamilton’s variational principle is that as the approximate eigenvalues always are higher than the true values. Thus one can track how quickly the eigenvalues decrease with increasing Nand see where the solutions approach their minimum values.

For the case of a parallelepiped, this occurs around N=10 for most modes. A word of caution here: one must remember that these calculations are performed on a digital computer, so at some value of N, machine precision issues will come into play and accuracy will degrade. Over the course of this research, it has found that this generally occurs around N=14 with double precision calculations, although this is not a hard and fast rule. During the analysis phase of an experiment, one can perform fits to the frequency data and check when the root mean square (RMS) fit errors are minimized as a function of N. Also, as computation time goes like N 3, the experimenter should strike a balance between desired accuracy and computational effort, particularly for crystal or sample symmetries which do not allow full blocking of the matrices. To give the reader a rough comparison, with no blocking and varying 6 parameters (typical of a trigonal crystal) and N=12 takes around 5 hours on a 1GHz Pentium III class PC. The more typical case with blocking and adjusting 3 parameters (typical of a cubic crystal) and

N=12 takes about 2-3 minutes on the same machine.

2.3.3.2 Cylindrical Shells

While the evaluation of the Γ integrals is trivial for the above case of a homogeneous parallelepiped, with a little more eﬀort Visscher et al. were able to work out F (p, q, r) for many shapes such as spheroids, bells, cones, sandwiches, and even a 33

“potato” described by ellipsoids. Part of this work involved calculating the integrals for an open ended cylindrical shell and using the code to calculate the normal modes for a carbon nanotube resulting in

p+q+2 (p 1)!!(q 1)!! F (p, q, r) = 1 α 4π − − (2.20) − (r + 1)(p + q + 2)!! where α = rinner/router is the dimensionless measure of the shell thickness. In the limit that α 0, we have the solution for a solid cylinder. For modes which are indepen- → dent of the length of the tube, the results agreed well with Raman spectroscopy data and calculations based on interatomic force constants [4] [32]. Graphical representation of the normal modes are shown in Fig. 2.3 where modes 1 and 14 are independent of tube length. Note that the amplitudes are greatly exaggerated for clarity. These pictures are constructed from the eigenvectors of expansion coeﬃcients for each mode which are used to reconstruct the expansion polynomial and determine the displacement of each point on the surface of the solid from it’s equilibrium position. A separate FOR-

TRAN program,‘CYLSHMA.F’ (analogous to ‘XYZMA.F’ listed in Appendix A) takes the ‘EIGVECS.DAT’ output file from the ‘XYZCYLSH.F’ program as input and writes a series of files containing a grid of mesh points for the surface of the tube at various points during a half period for each mode. The files can then be read by a Mathematica script (also listed in Appendix A) which can be used to generate graphics like those in

Fig. 2.3 and animations of any particular mode. The shading in the ﬁgure indicates the magnitude of displacement of the point from its equilibrium position. The darkest regions are nodes and lightest regions are antinodes. 34

Mode 1 Mode 3 Mode 5 Mode 7

Mode 9 Mode 10 Mode 12 Mode 14

Mode 15 Mode 17 Mode 18 Mode 20

Mode 22 Mode 24 Mode 26 Mode 28

Fig. 2.3. Selected normal modes of a cylindrical shell. The mode numbers which are skipped are degenerate due to the symmetry in the x y plane. Black indicates nodal − regions while lighter shaded regions are antinodes. 35

To test the accuracy of the cylindrical shell calculations, a stainless steel tube was machined and polished to dimensions: c = 5.00 cm (the axial length), a = b = router=

1.60 cm (the outer radius), and d = rinner = 1.27 cm (the inner radius); so α = 0.80. The modes were excited and detected with a commercial RUS unit from Dynamic Resonance

Systems (DRS) [15] and compared with the forward calculations for several values of

N. The results are listed in Table 2.3.3.2. Clearly, ﬁtting a cylindrical geometry with cartesian expansion basis requires more terms than a parallelepiped geometry, but even with N=12 reasonable agreement is achieved. Note that these are the results of a forward calculation so that no elastic or geometrical parameters are being adjusted.

2.3.3.3 Bilayer Stacks and Thin Films

Since the primary focus of this work is extending the RUS technique to charac- terizing thin ﬁlms deposited on a substrate, the analysis of such a system will be given here in some detail. A substrate with a ﬁlm is actually a subset of the more general case of a stack of n bilayers in the z-direction. Figure 2.4 shows such a stack for n = 2. Each bilayer consists of one material (A) of thickness d and a second material (B) of thickness d , with the overall dimensions of the parallelpiped sample being 2a 2b 2c. Since the 0 × × layers are stacked in the z-direction, c = n(d + d0)/2, and the normalized thicknesses of the 2 materials are d d α = , β = 0 . c c 36

N=9 N=12 Mode Fobs (kHz) Fcalc (kHz) Error (%) Fcalc (kHz) Error (%) 1 19.667 19.606 0.31 19.518 0.76 2 19.667 19.606 0.31 19.518 0.76 3 20.296 21.057 3.75 20.152 0.71 4 20.296 21.057 3.75 20.152 0.71 5 24.870 25.777 3.65 25.018 0.60 6 24.870 25.777 3.65 25.018 0.60 7 26.074 25.777 1.14 25.773 1.16 8 26.074 25.777 1.14 25.773 1.16 9 31.037 31.166 0.41 31.166 0.41 10 36.556 40.019 9.47 36.916 0.99 11 36.556 40.019 9.47 36.916 0.99 12 47.815 47.721 0.20 47.701 0.24 13 47.815 47.721 0.20 47.701 0.24 Avg.Error= 2.88 Avg.Error= 0.71 Table 2.2. Comparison of measured and calculated resonances for a stainless steel cylindrical shell for N=9 and N=12. Due to slight machining errors, degenerate modes are typically 2 closely spaced and overlapping peaks. The experimental values were then taken as a weighted average (based on amplitude). 37

Now symmetry is lost in the z-direction, but preserved in the other directions, thus the x and y integrals are identical to (2.18), however the z integrals must be performed over each layer and then summed. Then we have

1 1 1 xˆpmyˆqmzˆrmdxdˆ yˆdzˆ 1 1 1 Z− Z− Z− 4 α 1 (n 1)(α+β) = zrdz + − − zrdz (p + 1)(q + 1) " 1 1 (n 1)(α+β) β Z− Z − − − 1 (n 2)(α+β) 1 + − − zrdz + + zrdz 1 (n 2)(α+β) β · · · 1 β # Z − − − Z − 8 = + (2.21) (p + 1)(q + 1)(r + 1) n 4 [1 (n i)(α + β)]r+1 [1 (n i)(α + β) β]r+1 (p + 1)(q + 1)(r + 1) − − − − − − iX=1 n o if p and q are even and 0 otherwise. Due to the loss of symmetry, r can be either odd or even, producing 4 larger blocks in the Γ matrix. The reader can check for the limiting case that, if β = 0, eqn. (2.21) reduces to the homogeneous case shown in (2.18). The

ﬁrst term in (2.21) is the same F function we had for the homogeneous case, so we can deﬁne the term containing the sum as the G function so that the matrix elements are

Γmn = ci jk lF (p, q, r) + (c0 ci jk l)G(p, q, r) (2.22) m n imjknl − m n where the set of integers (p, q, r) are designated by the matrix indices m, n. The unprimed elastic constants are those of the α layer and primed are those of the β layer. It is most eﬃcient to precalculate the sum in the G function, which can be done with the following

FORTRAN code segment 38

C *******Precalculate GSUM values******** DO 10 IR=0,INT(3*NN) IRP1=IR+1 TMP=0.0 do 8 I=1,INL TMP=(1-(INL-I)*(ALPHA+BETA))**IRP1- & (1-(INL-I)*(ALPHA+BETA)-BETA)**IRP1+TMP 8 continue GSUM(IRP1)=TMP 10 continue ...... C***********G function ******************* G(ip,iq,ir) = 4.0*a*b*c*GSUM(ir)/((ip+1)(iq+1)(ir+1)) C***************************************** where NN is the highest order of the polynomials (N in the previous derivation), INL is the number of bilayers, and IR is the z exponent r. For the case of a thin ﬁlm deposited on a substrate, n = 1 and α β.

z 2c material B: c', r' d' material A: c, r d

Fig. 2.4. Geometry for a bilayer stack parallelpiped with n=2 39

2.4 Determining Elastic Constants: the Inverse Problem

As stated earlier, an accurate method of calculating the normal modes of an elastic solid of almost any shape and crystal symmetry is a truly signiﬁcant development in applied physics. However, to make use of this technique to experimentally determine a set elastic constants from a set of measured resonance frequencies requires an inversion of the problem. Unfortunately, there is no direct way of mathematically accomplishing this task. Also, there are questions of uniqueness of the solutions. For instance, since the frequencies obey the proportionality relationships f c /ρ and f c /l, if ∝ ij ∝ √ ij q the elastic constants are doubled and density doubled the same set of frequencies results, likewise if the dimensions are doubled and elastic constants are quadrupled. However, if these parameters are even roughly known, it is not diﬃcult to ensure one is in the proper parameter space for the correct solution.

As no direct inversion procedure is available, we must perform an iterative solution in which initial values for the parameters in question (elastic constants, dimensions, crystallographic orientation) are given to a routine performing a forward calculation, the results of which are then compared to the measured resonance frequencies. The parameters are then adjusted in such a way as to lower the sum of the squares of the diﬀerences in measured and calculated frequencies. This procedure is repeated until the change in the sum of the squares between successive iterations is lower than some tolerance value. This least squares ﬁtting procedure requires a reasonably good initial guess for the variable parameters to ensure the iterations will terminate at the global, rather than a local, minimum in parameter space. These initial guesses of the elastic constants 40 can come from previous experiments such as pulse-echo, static, or neutron scattering.

Also recent developments in ab initio theoretical calculations of elastic constants from crystallographic data and quantum potentials have resulted in reliable theoretical elastic constants for a wide variety of materials and all crystal classes. [58],[59],[60] This will be of particular help in RUS measurements on crystals of low symmetry where many elastic constants must be determined (13 for monoclinic or 21 for triclinic).

2.4.1 Levenberg-Marquardt Minimization

This section presents the details of the technique used to find the minimum in the sum of the squares of the differences between the calculated resonance spectrum and measured spectrum. It is a cursory treatment at best, more rigorous treatments can be found in the literature [[67] and [63]]. The fundamental concept is the the same as adjusting the slope and intercept parameters of a line to best fit to a set of measured data points. The difference is that the measured frequencies do not form a linear relationship, thus a non-linear fitting routine must be used, the most popular and robust of which is the Levenberg-Marquardt technique. [63] The problem is as follows: we want to find a set of n optimum parameters, tabulated in a vector c, such that

(fobs f (c))2 = (∆f )2 i − i i Xi Xi

obs is minimized, where fi are the observed frequencies and fi(c) are calculated frequencies which depend on the vector c. If the n values in c are close to their optimum values, then

2 the n-dimensional surface of i(∆fi) will be close to parabolic, thus the frequencies P 41 will be linear in c (by a Taylor series expansion), and the calculated and measured frequencies will be close. Then we may write

∂fi ∆fi ∆cj ≈ ∂cj

and multiply each side by ∂fi/∂ck to obtain

∂fi ∂fi ∂fi (∆fi) ∆cj = ∆cj[α] ∂ck ≈ "∂cj ∂ck # where [α] is an n n matrix known as the curvature matrix. We can then write an × expression for the proper adjustment for the jth parameter toward its optimum value

1 ∂fi ∆cj [α]− (∆fi) . (2.23) ≈ ∂ck

If on the other hand, our starting point is not close to the minimum, then the error surface may not be close to quadratic and eqn (2.23) will not work. In this case, the best we can hope to do is take a step in the “down hill” direction. Since the gradient

∂f of the sum of the squares in parameter space is (∆f )2 = 2(∆f ) i , we can write ∇ i − i ∂cj the proper adjustment to the parameters as

∂fi ∆cj K(∆fi) (2.24) ≈ ∂cj where K is a positive constant which controls the step size in the direction of steepest descent. 42

The contribution of Marquardt was to combine the two rules for adjusting variable parameters (eqns (2.23) and (2.24)) into one method which smoothly varies from one case to the other. A parameter γ must be introduced which acts as a control over which type of step is best to take. To introduce this parameter, we note that as α is defined, the diagonal elements are positive definite. We can then define a new matrix

αj0 k = αjk + γαjj (2.25)

so that eqn (2.23) becomes

1 ∂fi ∆cj [α0]− (∆fi) . (2.26) ≈ ∂ck

Now if γ is small [α ] [α] and we have (2.23). If γ is large [α ] is almost diagonal so 0 ≈ 0 its inverse is close to [α ] 1 [diag(1/α )]. Thus the diagonal elements of α play the 0 − ≈ j0 j 0 role of K in eqn (2.24) and their magnitude is controlled by γ. If we are given an intial set of parameters c, then the ﬁtting proceeds as follows:

2 1. Compute the sum of the squares i(∆fi) with the initial parameter set using a P forward calculation outlined in the previous section.

2. Pick a small value for γ (say 0.001), evaluate (2.25), and update the parameter set:

c c + ∆c . j → j j

3. Compute the new sum of the squares using the updated parameters in the forward

calculation. 43

4. If the sum of the squares increases, then increase γ by 10 or so and iterate again

(step 2). This pushes the minimization scheme toward the “down hill” mode given

by (2.24).

5. If the sum of the squares decreases, then decrease γ by a factor of 10 or so and

iterate again. This pushes the solution toward the parabolic mode given by (2.23).

6. Continue the procedure until the sum of the squares stops changing for several

iterations or falls below some tolerance value.

1 7. A ﬁnal iteration can be performed with γ = 0. Then the matrix [C] = [α]− is an

estimate of the covariance matrix of standard errors in the ﬁtted parameters.

Some practical comments on dealing with the [α] matrix are warranted. Since the

∂f analytical relation between the frequencies and parameters is not known, i must be ∂cj calculated from one or more initial forward calculation(s). Finite diﬀerencing can then be used to estimate the derivatives. Secondly, eqn (2.25) requires a matrix inversion of [α] which can be tricky business as the matrix may be (or close to being) singular. Perhaps the safest method of numerical matrix inversion, and that recommended by the authors of Numerical Recipes [63], is singular value decomposition (SVD). In this method, the matrix to be inverted (A) is decomposed into a product of three matrices

T A = U[diag(wj)]V .

where the structure of U and V are described in the above reference, and wj are the singular values of A which are positive or 0. For our purposes, the matrix to be inverted 44 is square, so we may write the inverse as

1 T A− = V[diag(1/wj)]U .

Which will work fine if all wj are not 0 (or close to it), otherwise these elements will dominate the inverse matrix. To avoid this problem, we simply to set 1/wj = 0 for these elements. The condition number of the matrix is defined as the ratio wmax/wmin, so that the larger the condition number, the more poorly conditioned the matrix. Now, let us consider what this means in terms of our particular application of SVD. The matrix we are inverting is composed of derivatives of the resonance frequencies with respect to the variable elastic constants. Thus if a particular mode is very insensitive to a particular elastic constant, then this derivative will be very small. So SVD will not attempt to adjust this parameter for this particular mode by “zeroing it out”, but leave the other elastic constants (which do effect the mode) variable to improve the fit.

In practice, these derivatives can be very useful as they allow one to pick modes which have strong sensitivity to particular elastic constants, or conversely as a check to make sure the set of modes being used in the fit are (as a set) sensitive to all the elastic constants. Appendix A lists a FORTRAN program called ‘FILMMRQROT’ which uses the Levenberg-Marquardt technique with SVD to fit data for a substrate with a film with crystal axes which are arbitrarily rotated with respect to the sample axes. More practical advice on fitting frequencies will be given in Chapter 3. 45

2.5 Theory of Internal Friction in Elastic Solids

With an eye toward the results of the carbon nanotube ﬁlm experiments in Chap- ter 4, it will be useful to include here some background material on energy dissipation in a resonating elastic solid. All of the elastic theory presented thus far has assumed no (or at least very small) dissipation. It is not the aim of this section to re-derive the elastic theory including dissipation, but to provide the reader with a primarily conceptual understanding of the physical sources of energy dissipation in solids and their eﬀects. There are many excellent works detailing both theoretical and practical topics on dissipation available. [ [7], [51], [12], [77], [33], and [62] ]

A perfectly elastic solid, that is one with no dissipation of elastic energy, has the following properties:

The strain at any point in the solid is a function only of the instantaneous stress, •

as indicated by the form of Hooke’s law: σij = cijklkl.

If the stress is removed, the strain is instantaneously and fully recoverable. •

All processes associated with the strain, and its relaxation, are reversible. This • is analogous to thermodynamic pressure and volume changes which must be done

quasi-statically so that the system is constantly in a state of “equilibrium”.

When caused to vibrate in a normal mode, the elastic solid will vibrate indeﬁnitely • – a struck bell would ring forever.

In reality, the equilibrium condition is never satisﬁed exactly, so at any given in- stant in time processes occur which attempt to reestablish the equilibrium state. These 46 are irreversible processes, and thus cause mechanical energy to be converted into some other form (often thermal). These irreversible processes produce a time lag between an applied stress and strain response. This suggests an important parameter in dissipation discussions, the so called characteristic relaxation time τ, and for a perfectly dissipa- tionless solid τ = 0. The nature and relative importance of these processes are typically dependent on the frequency of strain changes and temperature of the solid. Over the history of progress in elastic theory, several phenomenological models of internal friction have been developed. A review of these models provides good insight into the basic principles involved and so will be presented here.

2.5.1 Maxwell Model (1867)

In this model, the deformation of an applied stress is considered to made up of an elastic deformation and a ﬂow of a viscous Newtonian ﬂuid. Let us consider a shear stress in an isotropic medium so from Hooke’s law we have

σ12 = G12

where G = c44 is the shear modulus. And Newton’s law for a viscous ﬂuid is

σ12 = η˙12

where η is the shear viscosity coeﬃcient. Combining the above relations we obtain

σ˙ σ ˙ = 12 + 12 . (2.27) 12 G η 47

For the case of constant strain, we see the solution of eqn (2.27) with ˙12 = 0 will be an exponential decay in the stress with some characteristic relaxation time τ = η/G:

σ = σ exp( t/τ). For the case of constant stress, σ˙ = 0 and the solution for the 12 o − 12 σo strain is 12 = η t+o. Figure 2.5 depicts this case along with the mechanical equivalent model using springs for elasticity and dashpots for viscosity.

2.5.2 Kelvin-Voigt Model

Kelvin (1875) and Voigt (1892) independently suggested an adjustment to Hooke’s law so that the stress is not only related to the strain, but also its time derivative, or rate of strain. For an anisotropic solid, the model is

σij = cijklkl + ηijkl˙kl (2.28)

where ηijkl is the viscosity tensor. For simplicity, we will consider the case of constant shear stress. The the strain solution of (2.28) is

σo t G 1 exp τ while stress is applied (t) =  − − (2.29)  h i  exp t after stress is released. o −τ   The model along with its mechanical equivalent are depicted in Fig. 2.6. If we consider a mass per unit area m attached to a Voigt type solid, we recover the equation of motion for a forced damped harmonic oscillator: m¨+ η˙ + G = σ which has the solution (with

σ = 0) of

(t) = Aexp( δt) cos(ω t + φ) (2.30) − d 48

Fig. 2.5. Maxwell model for an anelastic solid: (a) Constant applied stress, (b) strain response, (c) equivalent mechanical model. 49 where δ = η/2m is the damping coeﬃcient, φ is a phase constant, and ωd is the damped natural frequency given by

G 1/2 1 ω = δ2 = 4Gm η2 d m − 2m − q which reduces to the typical natural frequency in the limit of no damping (δ = 0). The exponentially damped amplitude constitutes a energy loss in the oscillator. The Voigt model is perhaps the most basic realistic model of internal friction in solids, and it has been used with some success on rubber, cork and elasto-electronic interactions in solids at low temperatures.

2.5.2.1 Wave Propagation in a Voigt Solid

Now consider a harmonic disturbance propagating in the x direction of an isotropic

Voigt solid. Then the equation of motion will be

∂2u ∂3u ρu¨ = (λ + 2G) 1 + (χ + 2η) 1 , (2.31) 1 2 2 ∂x1 ∂x1∂t where ρ is the mass density, λ = c12 is the one of the Lam´e elastic moduli, which are often used in place of the full stiﬀness tensor when analyzing isotropic systems, and

χ is the compressional viscosity coeﬃcient. Now since we have assumed an harmonic disturbance, u has the form

u˜ = A˜exp[i(k˜x ωt)] (2.32) 1 1 − 50

Fig. 2.6. Kelvin-Voigt model for an anelastic solid: (a) Constant applied stress, (b) strain response, (c) equivalent mechanical model. 51

˜ where any quantity with a˜is complex, and k = k + iα = ω/cl + iα. Here k is the wave number, α is called the attenuation coeﬃcient, and cl is the longitudinal wave velocity in the medium. Substituting (2.32) into (2.31) and separating real and imaginary parts yields (to lowest order) the following relations

λ + 2G 1/2 c 1/2 c = = 11 (2.33) l ρ ρ and ω2(χ + 2η) α . (2.34) ≈ 3 2ρcl

We see the Voigt model yields a wave speed governed by the elastic constants, whereas attenuation is proportional to the viscosity constants and inversely to the elastic constants

3 through the cl term in the denominator. This is consistent with the common experience of any child living near a railroad track that an oncoming train can be detected long before its arrival by putting ones ear to the stiﬀ steel rail. If the tracks were made of softer plastic however, the attenuation would be much higher and this technique would yield dire consequences! A similar procedure for a shear wave yields the shear velocity and shear attenuation relations

G 1/2 ω2η ct = and α . (2.35) ρ ≈ 2ρc3 t 52

2.5.3 Zener Model (1948)

Zener improved on the Kelvin-Voigt model by including time derivatives of both the stress and the strain in Hooke’s Law which may be written

σ + τ1σ˙ = MR( + τ2˙), (2.36)

where τ1 is the relaxation time of the stress under constant strain, τ2 is the relaxation time of the strain under constant stress, and MR is called the relaxed elastic modulus which is the ratio of the stress to the strain after complete relaxation. The Zener model takes into account the property of many materials that a sudden applied stress results in an instantaneous strain followed by a relaxed approach to the equilibrium state, likewise if the stress is suddenly released. The model and its mechanical equivalent are shown in

Fig. 2.7.

2.5.3.1 Wave Propagation in a Zener Solid

Now let us assume a harmonic stress is applied to a Zener solid, such that σ =

iωt iωt δ σoe resulting in a harmonic strain = oe − where δ is the phase lag between the stress and strain. Substituting these into (2.36) yields

1 + iωτ2 (1 + iωτ1)σ = MR(1 + iωτ2) σ = M∗ where M∗ = MR (2.37) ⇒ 1 + iωτ1 53

Fig. 2.7. Zener model for an anelastic solid: (a) Constant applied stress, (b) strain response, (c) equivalent mechanical model. 54 and M∗ is known as the complex modulus. We may then express the phase lag as

m[M ] ω(τ τ ) tan δ = = ∗ = 2 − 1 (2.38) e[M ] 1 + ω2τ2 < ∗ where τ = √τ1τ2 is the geometric mean of the relaxation times. The physical interpretation of (2.38) for three regimes is as follows.

1. ω 1/τ: here τ τ and the period of the disturbance is long compared to 1 ' 2 the relaxation time. Thus the state of “quasi-equilibrium” is achieved and there is

little dissipation of energy and a small phase lag results.

2. ω 1/τ: here the disturbance period is on the order of the relaxation time. Elastic ∼ energy is readily coupled into the relaxation process and signiﬁcant dissipation

results. tan δ is maximized at ωτ = 1, known as the Debye relaxation peak. This

peak can be used to determine relaxation times for various dissipation processes,

or can be used to identify a process if the relaxation times are known.

3. ω 1/τ: here the disturbance period is so fast compared to the relaxation time that there is insuﬃcient time for signiﬁcant relaxation to occur.

A plot of tan δ and M is adapted from [62] and shown in Fig. 2.8. | ∗| The energy lost over one period can be found by

2π/ω ∆E = σdt˙ = πσoo sin δ. (2.39) Z0

Near resonance (ω ω ), the ratio of the total energy of the oscillator to the energy ≈ d lost per cycle is known as the quality factor Q, the inverse of which is a measure of the 55

Fig. 2.8. Dissipation (solid) and complex modulus (dashed) as a function of frequency for the Zener model. The peak at ωτ = 1 is known as the Debye relaxation peak. 56 internal friction.

1 1 ∆E Q− = = sin δ tan δ (2.40) 2π E ≈ if the loss (and δ) is small. The quality factor is often used as the determination of internal friction in mechanical resonators. For small losses, the quality factor can be related to the width of the resonance curve: Q f /∆f [67]. f is the center frequency ≈ o o of the resonance and ∆f is the full width at half maximum (FWHM) of the real part of the response as shown in Fig. 2.9. Often resonance curves are plotted as the amplitude vs frequency where the amplitude is A = ( e[u])2 + ( m[u])2. In this case, the width is < = q defined at the 1 A (half power) point rather than the 1A point. For our implementation √2 2 of RUS, the individual components of the response are each fit with a Lorentzian line shape, thus the above definition of the Q (FWHM) is correct. Further discussions on Q measurements with RUS experiments and issues encountered will be given in Chapter

3. Thus far we have introduced quite a number of material parameters as measures of internal friction. It may be useful to group them in a table listing some of their relations.

This is done in Table 2.5.3.1.

Name Symbol Units Relations Shear and Compressional Viscosity η, χ [Ns/m2] – Damping Coefficient (Rate) δ [1/s] δ = η/2m Relaxation Time τ [s] τ = η/G 2 ω (visc. coeff.) Attenuation Coefficient α [1/m] α 3 ≈ 2ρc Stress/Strain Phase Lag δ [radians] – 1 1 (Inverse) Quality Factor Q− [no units] Q− tan δ ∆f/fo ≈ 1 ≈ Logarithmic Decrement ∆ [no units] ∆ = πQ− Table 2.3. Listing of commonly used dissipation parameters. The relations listed above may not be applicable in all cases. 57

Fig. 2.9. The real part of the response of a driven damped resonator as a function of frequency around resonance. The quality factor is deﬁned by the full width at half maximum: Q = fo/∆f. 58

2.5.4 Dissipation Mechanisms

Thus far we have discussed various phenomenological models of internal friction in elastic solids and some consequences, however nothing has been said of the sources of these losses. This is a very rich ﬁeld with entire books dedicated to the subject (see the reference list early in this section); only fundamental concepts will be presented here. Our primary objective here is to answer the following basic questions: “Where does the elastic energy go?” and “How does it get there?” More detailed exlpanations along with some theoretical development can be found in the text Systems with Small

Dissipation [7]. Dissipation mechanisms can be grouped in three main categories: (1) material specific (intrinsic), (2) sample specific, and (3) environmental specific.

2.5.4.1 Material Speciﬁc (Intrinsic) Losses

These are losses due to intrinsic processes occurring within the bulk of the sample and are controlled by various physical parameters of the material such as electrical and thermal conductivities, sound speed, heat capacity, thermal expansion, etc. There are three basic processes of this type: (1) thermo-elastic, (2) phonon-phonon, and (3) phonon-electron. An explanation of each follows, along with approximations of their importance (without derivation).

Thermo-elastic Process

Consider a perfect (defect free) crystalline sample being mechanically driven at resonance. By definition, certain regions of the crystal are compressed while other regions are rarefied. In the same way as a gas will heat when compressed and cool when expanded, the crystal will develop temperature gradients between these compressed and 59 rarefied regions. In an attempt to reestablish equilibrium, heat will flow from the warmer to cooler regions. A quarter cycle later, all regions should have returned to a completely unstrained state, however the flow of thermal energy has left a residual temperature gradient and thus a non-zero thermal strain in the material due to thermal expansion. Only after some fraction of a cycle more is thermal equilibrium reached as the reverse heat

flow negates the temperature gradients. This is the source of the phase lag between the stress and strain. The resulting increase in entropy constitutes an irreversible process and thus an energy loss. An estimation of these losses can be derived from fundamental thermo-elastic theory [35]. 2 1 κT α ρω Q− (2.41) th ≈ 9C2 where κ is the thermal conduction coefficient, α is the thermal expansion coefficient, ρ is mass density, ω is the wave frequency, and C is the heat capacity per unit volume. Note that at very low temperatures, this effect becomes small.

Phonon-Phonon Process

Still considering a resonating crystal, the compressed and rareﬁed regions induced by the acoustic (low frequency) phonons locally change the density of the material. Any elastic anharmonicity (deviation from a parabolic potential) in the crystal lattice locally changes the thermal phonon frequencies. The phonon gas then attempts to reach a new equilibrium distribution which is an irreversible process. This loss mechanism can be estimated by 2 CT γˆ ωτph Q 1 (2.42) ph− ph 2 2 − ≈ ρc 1 + (ωτph) 60 where c is the sound speed, τph is the thermal phonon relaxation time, and γˆ is the

Gruneisen¨ parameter - a measure of the anharmonicity in the crystal lattice vibrations.

2 3 1 2 τph can be found by κ = 1/3CvDτph where the Debye velocity is given by 3 = 3 + 3 vD cl ct where cl and ct are the longitudinal and transverse sound speeds respectively.

Phonon-Electron Process

Consider a resonating electrically conducting crystal. The conduction electrons in the metal constitute an electron gas in some equilibrium configuration when the solid is at rest. Under resonance, regions of the material are compressed, increasing the charge density; other regions are rarefied, decreasing the charge density. The charge density gradient produces an electric field, and thus induces a current flow to reestablish equilibrium. There are thermal losses associated the electron conduction, and thus a conversion of elastic energy to thermal energy. Clearly, this loss mechanism is only present in con- ductors. If a free electron model is assumed with a spherical Fermi surface, and the mean

¯ free path of the electrons is smaller than the acoustic wavelength (ω/cle < 1), then these losses can be estimated by

8 Ef meσω Q 1 (2.43) ph− el 2 2 − ≈ 15 ρc e where Ef is the Fermi energy, σ is the electrical conductivity, and e is the electron charge.

2.5.4.2 Sample Speciﬁc Losses

The loss mechanisms in this category are those which result from various defects in the resonating sample due to crystal growth or sample preparation. The experimenter 61 has, of course, some control over these losses, but they can never be completely eliminated. Obviously, the precise nature of these defects will vary from sample to sample and are somewhat random in nature, so theoretical derivations of these losses in bulk samples are difficult. However, making assumptions as to the uniform distributions of the defects and randomness, some progress can be made. As one might expect, there are many specific types of defects which can occur in crystals and all will not be treated here. What follows should give the reader at least a flavor of the relevant issues.

Crystal Lattice Defects

A lattice defect can be generally deﬁned as any location of the lattice that deviates in any way from its periodic counterparts in the rest of the crystal. There are obviously many manifestations of this deﬁnition, however two common types are listed below.

Point Defects: when a lattice site is either empty, occupied by a diﬀerent atom, or • contains an additional atom.

Dislocations: when two large sections of the lattice are misaligned. Under strain, • the (grain) boundary can move or change shape in an attempt to relax.

Generally, acoustic losses due to lattice defects arise from “strain reordering” – when changes in the local strain near a defect site causes an inelastic shift in the atoms in an attempt to achieve a lower energy state. That is, after the strain is released, the arrangement of atoms around the defect will not be the same as their original conﬁgu- ration. The irreversibility of this process causes the energy loss. Obviously these losses depend on the defect densities and explain why, in general, high quality single crystal resonators have lower acoustic losses than their polycrystalline counterparts. 62

Surface Losses

The atomic layers at and near the surface of the crystal are always different than those in the bulk. This is because the half space above the surface (in the vacuum) is entirely different that the half space below (in the bulk). This is true even for a perfectly flat and otherwise defect free surface. In reality, crystal surfaces must always be polished, leaving microscopic cracks, scratches, and pits. Also the polishing medium, typically diamond paste or paper, infuses carbon into the surface layers of the crystal. Finally, the stresses induced during polishing generate higher density of crystal defects (as discussed above) in the surface layers, leaving them in a highly polycrystalline state. All this extra

“structure” at the surface provides many opportunities for relaxation. Perhaps the most important of which is thermo-elastic losses as heat “sloshes” back and forth between the randomly oriented crystallites. Also important is large scale strain reordering in and around micro-cracks and pits. These losses (per unit volume) are greater at the surface than in the bulk for two reasons: (1) there is a much higher density of defects, and

(2) the lack of a supporting crystal in the upper half space allows for larger amplitude displacements and larger changes in the atomic conﬁgurations during each acoustic cycle. Surface losses in any mechanical resonator can be large and various techniques for minimizing such losses are outlined in [7] (e.g. ﬂame polishing and chemical etching).

These losses are even more important for our RUS experiments since our samples are typically quite small ( 0.5 mm per side) meaning the surface to volume ratio is higher ∼ than larger samples. A rough approximation of surface losses for a cylinder of length L 63 and diameter D vibrating in a longitudinal mode (uniform strain over a cross section) is

2 1 4h Y T α ωτT∗ Q− (2.44) surf ≈ D C 2 P 1 + (ωτT∗ ) where h is the depth of the damaged surface layer, Y is Young’s modulus, CP is the speciﬁc heat at constant pressure, α is the thermal expansion coeﬃcient, and τT∗ = 2 a CP /κ is the thermal relaxation time (a is the average size of the crystallites) [7].

The issues presented here will be important in discussing the experimental results of the carbon nanotube ﬁlm experiments in Chapter 4.

In addition to the loss mechanisms listed above, it is highly likely that the displacement of regions very near a crack deviates from a sinusoid and is thus nonlinear.

The restoring force at diﬀerent points in the period is not linearly related to the strain

– Hooke’s Law is violated. With the introduction of such nonlinearities, the Fourier transform of oscillating systems begins to pick up more spectral components. The “oﬀ resonance” frequencies will radiate away from the region, carrying energy with them, and ultimately thermalize in the bulk. Fig. 2.10 illustrates the sinusoidal displacement of a point in the bulk (solid) and nonlinear displacement of a point near a surface crack

(dotted).

2.5.4.3 Environmental Losses

The losses in this category are due to energy leaking from the resonator into its surroundings. The primary mechanisms are acoustic radiation into the ﬂuid medium 64

time

displacement

bulk(linear) crack(nonlinear)

Fig. 2.10. Illustration of linear displacement of a point in the bulk (solid), and nonlinear displacement of a point near a surface crack (dotted).

(usually air or helium) and radiation into the support mechanism. For RUS measurements, the support losses dominate the resonator. For resonance experiments focusing on internal friction, great pains are taken to minimize support losses by carefully supporting resonators on silk thread at a nodal plane for instance. Excellent discussions of these issues are given in [7]. Some support mechanisms are shown in Fig. 2.11 for a cylindrical resonator.

Losses to the Fluid Medium

Any oscillating solid will radiate sound into the surrounding fluid. The power radiated will depend inversely on the impedance mismatch across the solid-fluid boundary (ρscs vs ρf cf ), which is typically very large, meaning the transmission coefficient is small. An 65

Fig. 2.11. Two schemes for supporting a cylindrical resonator to minimize losses. Support contacts are made at displacement nodes.

estimation of these losses for a cylindrical bar is

1/2 1 2P CP µ Q− = , (2.45) med πρ c C RT bar bar V where R is the universal gas constant, µ is the mean molecular weight of the gas, and

P is the gas pressure. As one expects, these losses increase with gas pressure and can be minimized by conducting RUS experiments in vacuum. If the pressure falls below

10 3 Torr, (2.45) is no longer valid and a non-interacting ballistic model for the gas ∼ − must be used. One can readily estimate ﬂuid losses in a RUS experiment by measuring resonance spectra at atmosphere (in air) and at P 10 6 Torr, and compare the Q ∼ − of the resonances. During the course of this work, it has been observed that this yields only a modest (1-5%) decrease in losses.

Losses to the Support

If an experiment is conducted on the earth’s surface, the weight of the sample must be 66 supported in some way. If the points on the surface of the resonator at which the support is in contact are moving (ie. not nodal points), then vibrations and energy will propagate into the support structure. Of course, diﬀerent normal modes have diﬀerent nodal points, so these losses are highly mode dependent. In fact, all loss mechanisms have some dependency on mode type. These losses can be minimized by careful design, as shown in

Fig. 2.11, and even calculated in some cases. For a RUS experiment, a primary concern is detecting as many normal modes as possible. Since the corners of a parallelepiped are never nodes, they are an ideal place for detectors. The cost paid however is larger energy losses through the supporting transducers. In our implementation, the transducers are made of very thin and ﬂexible piezoelectric strips, which are probably better at minimizing support losses than any other method employed today due to the large impedance mismatch between the sample and transducers. It is important however that the sample be held as lightly as possible between the transducers to minimize these losses and match the free boundary conditions assumed in the previous sections. Generally, ANY measurement must take energy away from the system. The relevant question for RUS, however, is how much do the supporting transducers aﬀect the resonance frequencies.

More details on the equipment, mounting, and eﬀects of transducer loading can be found in Chapter 3. 67

Chapter 3

The Experimental Method

This chapter is meant to provide the reader with enough details of the practical techniques to conduct their own small sample or thin-ﬁlm RUS experiment. It is broken into two major sections: a description of the general RUS method followed by additional issues related to thin-ﬁlm experiments. Another presentation of general small sample

RUS measurements is given in Phil Spoor’s thesis. [67] While much of the related equipment and software has been designed in this lab, it should be noted that a commercial

RUS unit is available from Dynamic Resonance Systems (DRS) [15]. A sample unit has been evaluated in our lab and seems to be adequate for basic RUS experiments for a limited set of crystal classes, however most “cutting edge” research is beyond this limited scope and requires knowledge of, and adjustments to, the source code performing RUS calculations and, in some cases, to changes in electronics depending on the particulars of the cell design. Thus, the commercial unit is not recommended for fundamental research.

That being said, the DRS unit does have a convenient interface for data acquisition and visualization, and has been used by our group for preliminary sample measurements. 68

3.1 Small Sample RUS Techniques

3.1.1 Sample Preparation

In regard to sample quality, any RUS measurement follows the law of “garbage in

- garbage out”. Thus sample preparation is a critical part of any RUS experiment. Each practitioner has his or her own techniques (see Migliori [47] for another method), but the goal is always the same: to create a nearly geometrically perfect sample with a minimum of defects. Bulk material from which samples are made should be single crystals (for crystalline samples) and free of internal structural defects and inhomogeneities such as cracks, voids, or contaminants. Often the materials of interest are novel and thus are fabricated in small quantities by other specialists. Before beginning an experiment, it is useful to learn something from the suppliers of the reproducibility of the fabrication technique, quality of the sample, and any characterization by other measurements (eq. resistance, optical, magnetic, X-ray diﬀraction, ...). A commercial supplier should have purity and crystal orientation data available. A cautionary note: when ordering a single crystal, be sure to request all required data before the crystal is grown. After a bulk material is obtained, parallelepiped samples must be cut and polished. The following sections provide instructions which are detailed enough that the reader should be able to prepare a RUS sample themselves. In these sections, “water” refers to distilled water,

“microscope” refers to a Nikon SMZ-10 stereoscope with a ﬁberoptic light ring and variable magniﬁcation, and “Kimwipes” refers to lent-free laboratory tissues. 69

3.1.1.1 Polishing and Cleaning

Sample preparation is a tedious but extremely important part of the experiment.

The crystallographically oriented bulk crystal is secured to a diamond wire saw from

South Bay Technologies (Model 850) [70] with Duco household cement. The saw is used to cut rough parallelepiped shapes. If enough raw material is available, it is useful at this point to cut 3-4 samples. The samples should be about 25% larger than the expected

ﬁnal sample dimensions to allow for polishing. Acetone is used to dissolve the Duco and plastic or teﬂon coated tweezers are used to carefully remove the samples and soak in clean acetone to remove residual Duco.

The goal from this point forward is to produce a near geometrically perfect and scratch free parallelepiped sample, usually less than 1 mm on a side. Achieving this goal is facilitated with the use of a South Bay Model 151 hand polishing jig, shown schematically in Figure 3.1. A stainless steel mounting disk is attached to the jig via a plunger running along the central axis. The weight of the disk is counterbalanced by two springs. The pressure between the sample face and polishing medium may be controlled by adding weights to the top of the plunger. A dial attached to the plunger provides a safety stop, so that speciﬁc thicknesses may be polished oﬀ. To ensure proper vertical orientation, precision machined tool steel shims are used for the feet of the jig.

The polishing medium used in this lab is a thin plastic sheet coated with diamond chips between 9 and 0.1 microns in size (referred to as “diamond paper”) . The paper is backed with a glass plate to aid in producing a ﬂat sample surface. The following steps will produce the ﬁrst polished face. 70

Fig. 3.1. Cut away view of the polishing jig used during sample preparation. The mounting disk screws onto the plunger, the weight of which is counterbalanced by the springs. Extra weight on top provides controlled pressure between the sample and paper. The dial provides a safety stop so too much sample is not polished away. 71

1. If a sample has a prepolished face, it is placed face down on the disk so that the

opposite side will be polished parallel. It is useful to use a ﬁlm of Duco cement

thinned with acetone (approximately 20 parts acetone to 1 part Duco) to secure

the sample to the disk. The disk is then placed on a hotplate with some small

pieces of Quickstick wax from South Bay. Quickstick is a clear, hard wax which

melts at about 55◦C, and is soluble in acetone. Encasing the sample in the wax

supports corners and edges, keeping them sharp during the polishing process. As

the wax melts, it is helpful to carefully spread the wax with a toothpick to ensure

no air bubbles are trapped near the sample surfaces. Once the wax is completely

melted and the sample is well covered, remove the disk from the hotplate with

tweezers and let cool. After the wax hardens, take a close look at the sample under

the microscope to make sure there are no air bubbles, and that the sample face is

entirely ﬂush against the disk face. Then mount the disk on the polishing jig.

2. The jig is spring loaded to counter the weight of the disk. Weights can be added to

the central plunger to provide variable levels of normal force between the sample

and polishing surface. Start with all available weights. Thoroughly clean the glass

base plate with water and Kimwipes to remove any dust grit that would cause

groves in the sample. Use a little water to aﬃx 600 grit emery paper to the glass

and spray some water on top as a lubricant. Polishing proceeds by carefully placing

the jig on the wet emery paper and slowly moving the jig in a ﬁgure 8 fashion.

After every 4-5 turns, rotate the jig a little and avoid running the sample along

the same path more than 1-2 times. At this point, you are simply removing wax 72

on top of the sample. Check your progress with the microscope often to ensure

you do not begin polishing the sample with this paper. Once only a thin layer of

wax remains above the sample surface, switch the paper to 9 µm diamond lapping

paper. The sample should soon “break through” the wax. Continue the polishing

and monitoring until the entire surface is exposed and any major defects (chips,

gouges, ...) are polished out, although many scratches will be left by the 9 µm

grains. Follow similar procedures for diamond grain sizes of 3, 1, and ﬁnally 0.5

microns. If all goes well, each grain size takes about 15-30 minutes. Before and

after switching lapping paper, spray the paper vigorously with water to clean away

dust and bits of wax. The glass baseplate should also be cleaned during paper

switches. If sizable scratches occur, it may be necessary to back up a grain size

to polish it out. Scratches at the 1 micron level are not really visible under the

microscope, although tilting the sample and viewing the reﬂected light can reveal

smaller features.

3. Once a defect free mirror ﬁnish has been achieved, remove the mounting disk and

place at an angle in a small beaker of acetone. After about 15 minutes, all the wax

will be dissolved and sample freed. A size 0 horse-hair artists brush is used to ﬁsh

out the sample and place it in a small aluminum cup of clean acetone.

4. The sample should now have two polished and parallel faces. The remaining 4

(perpendicular) sides must be polished in such a way as to produce faces that are

perpendicular to those already polished. This is accomplished with two precision

machined tool steel blocks, each with three adjacent faces accurately polished at 73

right angles. Such blocks are commercially available from Starrett [71]. Thinned

Duco ( 5 parts acetone to 1 part Duco) is used to attach these to a mounting ∼ disk as shown in Fig. 3.2(a) with a little unthinned Duco along the outside edges

for extra support. Care should be taken that the two blocks are ﬂush against each

other and the disk. This can be checked under the microscope. The disk is then

placed on edge in a vice such that an inner face of the “L” formed by the right-

angle blocks is facing up. Under the microscope, the sample is placed on this face

with a polished side down, and an adjacent face snugly against the adjacent block

also shown in Fig. 3.2(a). A drop of thinned Duco is applied to hold the sample

in place. The disk may then be turned upright and viewed under the microscope

to make sure the sample is square and ﬂush against the blocks. Quickwax is then

melted over the sample and blocks as discussed previously (see Fig. 3.2(b)). Once

the wax hardens, the disk may be attached to the jig and polishing may proceed as

before, until a defect free mirror ﬁnish is achieved (see Fig 3.2(c)). The sample is

again removed using acetone, with some care taken to extract the mounting disk as

soon as the sample is freed; otherwise, the acetone will dissolve the Duco holding

the blocks in place which may fall on the sample and damage it.

5. The remaining three sides may be polished in the same way, by rotating the sample

each time so that the previously polished faces are ﬂush against the adjacent block.

This creates very nearly perpendicular faces.

6. After the ﬁnal face is polished, clean the sample well in acetone followed by

methanol and place in a small container lined with a folded over Kimwipe. The 74

sample should then be carefully examined under the microscope for any scratches or

chips, and for perpendicularity of sides. Some very minor (barely visible) scratches

can generally be tolerated, however defects on corners and edges will perturb the

natural frequencies. See the following section for atomic force microscope (AFM)

images of various crystal surfaces prepared in this manor.

7. A ﬁnal cleaning of the sample may consist of simply soaking in acetone followed

by soaking in methanol. It has been observed that the quality factor is generally

improved slightly if the sample is ultrasonically cleaned in a soft plastic container

of acetone. A ﬁlm canister works well for this. The ultrasonic cleaner needs to

be of very low power and should only proceed for 20-30 seconds. Longer times or

higher powers can damage the sample. Most likely, this step removes microscopic

bits of wax from the surface thereby resulting in slightly higher Q. Any time the

sample is exposed to acetone, it should be followed by a methanol wash to remove

any residual ﬁlm.

Some additional notes on polishing are worth mentioning. The polishing process can be quite tedious, but it is very important to “maintain focus”. It generally takes much more time fixing problems than it takes to do it right the first time. The entire process takes about 4-5 days. The samples are generally quite small and fragile. Take great care to avoid dropping or otherwise mishandling it. Ensure that at any point the sample is likely to fall (e.g. during transfer or mounting on blocks) it will be caught in a soft container. Aluminum sample cups lined with Kimwipes are good for this. Nothing is more frustrating than working on a sample for 5 days only to have it fly off the tip 75

right-angle block sample polishing mount

(a)

wax

(b)

polished surface

(c)

Fig. 3.2. (a) Polishing disk with right angle blocks for polishing small parallelepiped. (b) Quickstick wax is used to protect corners and edges during polishing. (c) Picture of the apparatus during polishing.[Spoor [67]] 76 of a brush and be lost forever. The diamond lapping paper from South Bay is rather expensive ( $40 per sheet), so one should utilize as much of the paper as possible. ∼ Typically several full samples can be completed with one set of paper. However, when reusing paper, be careful to avoid regions where the lapping ﬁlm has been worn oﬀ or saturated with wax. The smaller grain sizes tend to wear out a bit faster than the large grain sizes.

3.1.1.2 Atomic Force Microscope Characterization of Surface Roughness

The preceding procedure can produce a sub-millimeter sample with optically smooth surfaces having only a few minor scratches visible under a (x100) microscope.

For most RUS applications, this is really all the information necessary as any minor surface defects do not eﬀect the results in any meaningful way; remember that the acoustic wavelength is on the order of the sample size. For thin ﬁlm RUS however, the microscopic

(atomic) nature of the surface is of interest as the ﬁlm will then form an interface with this surface. A series of AFM experiments were performed on strontium titanate (STO), alumina, and silicon substrates all prepared according to the steps above including 0.5

µm diamond grit paper as the ﬁnal polishing step. In addition to images with vertical resolution of 1 nm, the software can calculate a mean surface roughness over a selected ∼ area according to the following formula

1 Lx Ly R = f(x, y) dxdy L L | | x y Z0 Z0 77 where Lx,y are the dimensions of the selected area and f(x, y) is a numerical function representing the surface height over the area relative to the mid plane. Two such scans are shown in Fig. 3.3 for alumina (a) and STO (b). Surface roughness was calculated for (a) with the bounding box shown in white. The resulting mean roughness and range in height were R = 2.3 nm, z z =35.4 nm respectively. These numbers are very max − min representative of all measurements performed on all three samples. The white spots represent sparse (but tall) surface features which are most likely bits of diamond from polishing. Occasionally, relatively large cracks can be seen as in Fig. 3.3(b) which shows a 5 5µm scan of the STO surface. The crack is about 500 nm wide and 62 nm deep. The × depth is underestimated due to the ﬁnite curvature of the AFM tip. This is in fact the largest crack seen in these experiments. One may wonder how surface roughness of 2-4 nm can result from 500 nm grit size. The diamond grit is imbedded in an adhesive on the paper, thus only a small percentage is actually in contact with the sample surface. The general conclusion is that this polishing method produces surfaces with mean roughnesses of 2-4 nm with sparse larger features such as cracks and contaminants. These results seem to be mostly independent of the sample material except that high quality single crystals were used.

3.1.1.3 Sample Size Measurements

There are two stages to measuring the sample dimensions. After polishing and cleaning the sample, the microscope maybe used with a eyepiece reticule which has been calibrated with known lengths. For the Nikon SMZ-10 under maximum magniﬁcation and the eyepiece used for these experiments, the smallest lines are calibrated as being 12.5 78

~500nm

~62nm

(a) (b)

Fig. 3.3. AFM images of polished alumina (a) and strontium titanate (b) samples. The mean roughness of the alumina sample was 2.3nm. The surface crack shown in (b) is the largest seen in any of the samples. 79

µm apart. This system can be used to obtain a rough measurement of the dimensions

( 4µm) without “touching” the sample. These measurements, along with existing elastic constant values and mass density, can be used to calculate an anticipated resonance spectrum with the computer program, listed in Appendix A, for solving the “forward” problem described in Chapter 2. This is extremely helpful in determining frequency ranges and mode identiﬁcation during data acquisition.

Accurate dimensions are obtained with a Starrett Model 25-109 microdimensioner, illustrated in Fig. 3.4, only after all resonance data has been obtained. A spring loaded probe used in the instrument could cause small indentations in the sample which may perturb the natural frequencies. The instrument should ﬁrst be “zeroed” by elevating the sample platform until it comes in contact with the plunger, and the needle aligns with the ﬁrst major tick mark, the plunger may then be locked into place with a thumb screw.

The sample is placed on the platform under the plunger which has been extracted by a lever. The plunger is gently released until it comes to rest on the top of the sample. The diﬀerence in needle positions is then used to determine that dimension. The resolution is about 1.0 µm. This should be repeated 3 times at various positions on the sample face. The repetition allows one to check how parallel the opposing faces are and allows some averaging. If any measurement diﬀers by more that 2-3 µm, particularly in a consistent fashion along the length of the sample, there may be a problem in the sample geometry. The process is repeated 2 more times to obtain the other two dimensions.

If the dimensions are larger than 0.5 mm, a stainless steel shim of calibrated thickness must be used to increase the range of the Starrett. During rotation and manipulation of 80 the sample on the platform, it is quite easy for the sample to “pop” oﬀ. Placing a few shims around the sample helps support it an keeps it from becoming lost.

3.1.1.4 Eﬀect of Geometry Errors

An important issue is how perfect the geometry needs to be for the computer code to reasonably model the physical sample. Phil Spoor [67] conducted a detailed error analysis in his thesis in which he employed perturbation theory with small deviations in geometry as the perturbing parameter. Examples of possible errors are dimension lengths and deviations from perpendicular faces in the form of parallel but skewed faces and deviations from parallel faces (tilted). Examples are shown in Fig. 3.5. The eﬀect of such errors (in frequency) were plotted as a function of error magnitude for various mode types. The details of his work will not be repeated here, but his results provide good general guidelines for any experimenter in RUS and so will be summarized.

For simple errors in dimensions and assuming no other errors as shown in Fig. • 3.5(a), the eﬀect is quite straightforward. One may simply run the forward program

for dimensions at the measured lengths and upper and lower bounds for these

measurements. The natural frequencies depend linearly on inverse length, so if

the desired resolution in frequency is 100 ppm (1/10,000), then the lengths should

also be measured to this approximately this level, which is not a trivial matter.

The Starrett microdimensioner mentioned above has a resolution of about 1/1,000

for a sample 1mm on a side. However, when ﬁtting the frequencies, two of the ∼ dimensions can also be variable parameters, so the spectrum itself can be used to

more accurately determine all but one of the dimensions. 81

retraction lever

plunger

sample

Fig. 3.4. The Starrett microdimensioner used for accurate sample dimension measurements. The retraction lever is used to retract the plunger to place or extract the sample. The resolution of the instrument is 1-2 microns. The support assembly has been omitted for clarity. 82

Skewing: If a set of the sample faces are parallel but skewed (as shown in Fig. • 3.5(b)), then the worst case mode is perturbed by 25 ppm for an angle θ 0.5 . ≈ ◦ Such geometry tolerances are quite reasonably met using the polishing scheme

described previously.

Tilting: If a set of faces deviate from being parallel as shown in Fig. 3.5(c) then • the errors are more signiﬁcant. A error of δ 0.3 produces a maximum error in ≈ ◦ the frequencies of about 400 ppm. Fortunately these types of errors are less likely

than the skewing type for our polishing procedure.

Fig. 3.5. Several types of possible geometry errors during parallelepiped sample preparation and measurement. (a) Error in dimension measurements, (b) Skewing: opposing faces are parallel but not at right angles with adjacent faces, (c) Tilting: opposing faces are not parallel.

The magnitude of geometrical errors stated above are be well within the precision of our sample preparation procedure, and can be detected with the microscope and/or the microdimensioner. It should also be noted that the natural frequencies for a good

RUS experiment typically deviate from calculated values by about 0.05% RMS which is 83 larger than any deviations due to geometry errors. On may then wonder why are the errors so large. This issue will be addressed in some detail in following sections on data analysis, but the short answer is the bulk of the error is due to transducer loading eﬀects.

First, the transducers themselves and the RUS cell should be discussed.

3.1.2 Cell Design and PVDF Thin Film Transducers

The RUS cell and transducers used in these experiments have been designed to optimize measurements on small samples. The interior of the cell is shown in Fig.

3.6, while the entire cell is shown in Fig. 3.7. The sample is lightly held in place at two opposing corners by two ﬂexible thin ﬁlm piezoelectric transducer strips (discussed later) which are attached to insulating mounting blocks, one of which is adjustable. The adjustable mounting block may be retracted (advanced) with a lead screw providing more

(less) space between the transducers enabling the sample to be mounted and extracted.

This procedure is fully discussed in the section on “Mounting a Sample.” One of the transducers is driven with an AC voltage from a frequency synthesizer, while the other picks up the response from the sample. The electrical connections are made to the transducers through the conducting mounting tabs and small coaxial cables.

A copper cross-talk shield between the transducers has a small hole about 2 mm in diameter which is aligned with the center of the transducer strips where the sample will be held. The grounded shield minimizes the electromagnetic cross-talk between the drive and pick up transducers. Even with the shield, the cross talk can not be completely eliminated so that any sample resonance signal is superimposed upon the background signal. The interference between the resonance signal and cross talk results 84 in a modification to the usual resonance lineshape as discussed later. A cover to the entire cell has been fabricated (shown in the inset of Fig. 3.7) which helps reduce EM noise from the lab. This is, in fact, a rather important point. Any high (radio) frequency measurements can be very sensitive to environmental noise from florescent lighting, high power lab equipment, cell or cordless phones, etc. When setting up the experiment, it is useful to “play around” with different locations in the lab as well as cabling and grounding schemes to achieve the highest signal to noise ratio.

The cell also may be inserted in a vacuum jacket allowing measurements at low

8 pressures (10− Torr) or in various atmospheres. In addition to the capability of measuring pressure dependence, this system is useful for measuring samples sensitive to ambient gases. This technique was used in the carbon nanotube experiments discussed in Chap- ter 4. The vacuum jacket can also be incorporated in a temperature control system if desired, as discussed in Chapter 5.

Perhaps the key to the small sample RUS technique used throughout this work are the thin transducer strips, particularly for investigating thin ﬁlm systems. The transducers are made of polyvinylidene ﬂuoride (PVDF), also known by its trade name

Kynar [34], which is a piezoelectric material capable of being fabricated in a thin ﬁlm form. The thickness used for these experiments is 9µm, which is robust enough to withstand the fabrication process and produce good signal amplitude, yet ﬂexible enough to be easily manipulated and to protect the sample.

Small and fragile samples require small and ﬂexible transducers. The ﬂexibility of the slightly tensioned transducer strips allows reasonable approximations of the free boundary conditions assumed in the models discussed in Chapter 2. Also important in 85

Fig. 3.6. Schematic of the small sample RUS cell mounting blocks. The beryllium- copper tabs are used for their “springiness”. One insulating block is mounted in a lead screw so it can be retracted and advance during sample mounting. Electrical connections are made via the screws at the back of the mounting tabs. The capacitive active area of the transducers is shown by the hatched region. [Spoor [67]] 86

Fig. 3.7. Schematic of the entire small sample RUS cell. The thumb screw is used to advance the adjustable block when mounting a sample along opposing corners. The copper cross-talk shield provides a ground plane between the transducers, minimizing cross-talk between the drive and pick up transducers. [Spoor [67]] 87 any resonance experiment are transducer resonances which can be confused with sample resonances. PVDF is excellent in this regard since the thickness resonances over the thin (9µm) ﬁlm occur at frequencies much higher than typical RUS resonances of typical sample size 0.5 0.4 0.3 mm, which is generally in the 0.5 - 5 MHz range. Resonances × × of types other than thickness modes may occur in this range, however the quality factors are generally very low ( 10) which allows them to be easily distinguished from RUS ∼ samples (Q > 200). These transducers are unique to this lab and the details of their fabrication are given in Appendix C. A rough sketch of the process will be given here to give the reader a ﬂavor of the process and to provide a summary of the principle of operation.

First, a 1” 2” piece is cut from the PVDF sheet by sandwiching it between ∼ × two pieces of paper and cutting with high quality scissors. The piece is then thoroughly cleaned with Alconox and distilled water. After drying, the piece is fixed to an evaporation mask with a thin film of vacuum grease as shown in Fig. 3.8. The mask is designed with knife-edge masks on either side which overlap along the length by a width of 0.75 0.5mm (Fig. 3.9(a)). Using a Veeco VE-300 vacuum evaporator, a 20 nm ∼ − layer of chromium is deposited followed by a 300 nm layer of gold. The intermediate chromium layer is used to improve the adherence of the gold to the PVDF film. The mask is then flipped, and the process is repeated on the other side. After removal from the evaporator, a small toothpick can be scraped along an outer edge of the metal layer to test for adherence to the film. The thicknesses reported here produce good transducers, but are somewhat arbitrary. There may be a different set of parameters which improve performance. A scalpel or new razor blade is used to cut 0.75 mm strips ∼ 88 from the metallized film perpendicular to the overlap region as illustrated in Fig. 3.9(c).

Two of these strips are then attached to the mounting tabs in the RUS cell with conductive epoxy. Once the epoxy is dried, the strips can be tensioned by inserting small pieces of a pencil eraser between the mounting tab and supporting block. The electronics should be connected to the transducers such that the metallic leads facing each other are held at electrical ground. Since these are the leads in physical contact with the sample, this grounding scheme allows investigation of conducting samples without the worry of shorting the transducers, and reduces the cross-talk between them.

The principle of operation is rather straightforward. The overlapping metal layers

(active area) form a small capacitor sandwiching a piezo-electric material (the PVDF).

For the drive transducer, an AC peak to peak voltage of 5-10 V is applied, inducing an

AC electric field across the active area. The PVDF reacts by alternatively expanding and contracting with a frequency equal to the electric field. As the sample corner is in contact with this region, an AC stress field in the sample is induced. During a resonance, the opposite (receiving) transducer will be strained by the motion of the sample at the corner. The AC strain will then induce an AC electric field across the active area and the resulting voltage is recorded through the metallic leads.

3.1.3 Mounting a Sample

As the reader may guess, mounting a sub millimeter cube lightly by its corners is not a trivial task. The corner mounting is important because the corners of a resonating parallelepiped are always active (never nodes), thus all modes can, in principle, be detected. To aid in the mounting procedure, a microscope, micromanipulator, and set 89

Vacuum Grease

PVDF Film

Fig. 3.8. Template for metallization of PVDF transducers. Vacuum grease is used to secure the ﬁlm to the template. 90

Fig. 3.9. Several steps in the fabrication of the PVDF transducers. (a) Cleaned 9µm thick PVDF film is sandwiched between two knife-edge masks overlapping by about 0.75 mm (b) chromium and gold metallization of both sides (broken hatching on bottom and full hatching on top), (c) a 0.75 mm wide strip is cut from the bulk, (d) side view of the cut strip, overlapping metal layers constitute a 0.75 0.75 mm2 active area. × 91 of “vacuum tweezers” are essential. A schematic of the mounting apparatus designed for use in an inert atmosphere glove box is shown in Fig. 3.11. The vacuum tweezers consist of a hypodermic needle attached to a small cylindrical hand set approximately the size of a pencil. A rubber hose connects the hand set to a small vacuum pump so that air is drawn in the tip of the needle. This suction fixes the sample on the tip of the needle. A finger tip size hole in the hand set allows the user to vent the system and release the sample. One set of vacuum tweezers is attached by two hose clamps to a micromanipulator, which allows precise control over the samples position between the transducers. The needle on these tweezers is ground at a 45◦ angle to position the sample properly for corner mounting between the transducers. A second set of vacuum tweezers is used to pick up the sample and place it on the micromanipulator as shown in Fig. 3.10 [adapted from Spoor [67]]. The cell is secured in a mechanical vise such that the transducers can be viewed through the microscope. The sample on the needle is then positioned so that the opposing corners are aligned with the center of the active area of the transducers, and a corner is just in contact with the fixed transducer. Then the adjustable transducer is advanced until it just contacts the opposite corner. Proper alignment is ensured if the sample does not rotate as the adjustable transducer contacts the sample. If misalignment does occur, retract the adjustable transducer and use a size

0 artists brush to gently rotate the sample on the vacuum needle. The force required to hold the sample securely is not known, but we may estimate that the force is at least as much as the weight of the sample and perhaps twice this value. The eﬀect of transducer loading on natural frequencies and quality factors is an important question which will be addressed in some detail later. This procedure takes a good deal of practice to complete 92 quickly, however once proﬁcient, one can mount a sample in a matter of minutes. It is not uncommon for a sample to fall during mounting, so a container lined with a Kimwipe should be placed below the cell.

The procedure is somewhat more complicated for samples sensitive to atmosphere such as carbon nanotube films. For these samples, the film is first deposited on the

8 substrate and then degassed under high vacuum (10− Torr) at 500◦ C for about 12 hours (see Chapter 4). After cooling, the oven containing the sample is transferred to a sealed He atmosphere glove box and the sample is extracted. The mounting procedure follows as outlined above, except instead of viewing through a microscope eyepiece, everything must be monitored through the output of a CCD camera attached to the microscope. After the sample is mounted, the cell is sealed in its vacuum jacket and is removed from the glove box. At this point, the sample cell is ﬁlled with He at 1 atm; note that the sample has not been exposed to atmosphere since degassing. The He can then be pumped from the cell and measurements taken in vacuum, or various gasses or vapors can be introduced into the cell to check eﬀects of adsorption/contamination.

More details of the equipment introduced here can be found in Chapter 4.

3.1.3.1 Eﬀects of Transducer Loading

The effects of transducer loading on sample resonant frequencies and quality factors is important to understand for any RUS measurement, but even more important for RUS measurements on thin film samples. Due to the design of the RUS cell, it is generally impossible to deposit a film on a substrate while it is mounted between the 93

sample end of stationary needle end of hand-held needle

hose clamps hypodermic needle to vacuum pump

suction control port sample (usually plugged with wax) micromanipulator arm

Fig. 3.10. Vacuum tweezers used to mount a sample. The needle attached to the micromanipulator arm is polished at a 45◦ angle to allow proper orientation of the sample. The ﬁnger holes in the tweezers allow control of the suction at the needle tip. Opposing corners of the sample should be aligned horizontally and perpendicular to the needle axis. [Spoor [67]] 94

To ex ternal m onitor

CCD Cam era

Microscope

Hypoderm ic Micro− RUS Cell Needle Manipulator V ise

to m icro− pum p Safety Plate Cell Stabilizer Base Plate

Fig. 3.11. Sample mounting apparatus to be used in an inert atmosphere glove box. A CCD camera is attached to the microscope and signal is sent out of the box to a CRT monitor allowing monitoring of sample alignment. 95 transducers. Thus it is important to know how frequency and Q measurements vary between mountings. To address this question, 10 mountings of a strontium titanate (STO) sample were conducted by two different experimenters. The results, shown in Table 3.1, indicate that frequencies for this material are reproducible to at worst 100 ppm and quality factors to about 10%. This is the worst case because temperature was not rigor- ously controlled, averaging was performed over mountings by two different people, and data was taken over the period of about 1 month during which time slight changes to the sample surface may have occurred. These statistics can be correlated with a study of how transducer loading effects the data. It should be explicitly noted here that these deviations are the effect of different sample mountings, NOT uncertainties in the frequency and Q measurements for an individual mounting. These uncertainties can be estimated by repeatedly fitting a number of peaks, and are typically on the order of several parts per million for the frequency and 0.1% for the Q. This precision for a single mounting allows one to very accurately detect small changes in the sample as parameters (other than mounting) are changed. This sensitivity is vital to thin film experiments in which the film is undergoing a phase transition, as discussed in Chapter 5.

To better quantify the eﬀect of transducer loading, a sample was mounted nor- mally and measured. The thumb screw was then advanced 1/4 of a turn while the sample and transducers were viewed through the microscope; and new data was acquired. This procedure was repeated for 1/2 and 3/4 turns. The experiment was stopped here due to fears of damaging the thin transducers. Visually, even an inexperienced experimenter would not over tighten the sample by more than 1/4 turn, which should then provide a rough upper limit to shifts in data due to transducer loading. The results are shown 96

Average Deviation of Average Deviation Mode Freq. Freq. (ppm) Q of Q (%) 1 712379 393 459 7 2 959917 166 2639 11 3 1200167 127 1088 9 4 1582167 146 1437 7 5 1661024 102 1703 9 6 2222446 76 3382 8 7 2366013 27 8244 11 8 2384306 68 3906 12 9 2505490 35 6801 9 10 2517668 45 6835 10 11 2671077 142 3152 9 12 2714955 51 5189 10 13 2757673 157 5680 9 14 2817676 71 2849 10 15 3129127 76 2496 11 16 3168801 101 10676 11 17 3642570 50 9769 10 18 3667821 39 11944 10 Average=104 ppm Average=10% (87 w/o 1st ) Table 3.1. Reproducibility of frequencies and quality factors for a typical strontium titanate sample. Deviation in peak frequency is defined as: ppm = (std. dev.)/(avg. f) * 106. 97 in Fig. 3.12 along with the modes tracked (a) and visual estimate of the transducers as viewed from above through the microscope (d). The shifts in the frequency data show that the first mode (almost always torsional) is most strongly effected by loading. This data offers an explanation of the observation of many RUS experimenters that the first mode is generally not fit well during data analysis and is often given a fitting weight of zero. Interestingly, the mode least effected by loading is the one with displacements primarily perpendicular to the transducer plane (mode 2). Discounting the first mode, we see that the magnitude of the shifts in frequencies is on the order of 100 ppm in agreement with the statistical study mentioned above. As one might expect, the effects of transducer loading in the quality factor, as shown in Fig. 3.12(b), is quite dramatic.

Again the first mode is most severely effected, and even a 1/4 extra turn of the screw produces a 30% drop in the Q. This sensitivity is reflected in the statistics above by the 10% variation in Q measurements over different mountings. The loading data ∼ could be used to extrapolate the statistical 10% variation in Q with about 1/10 of a turn uncertainty in sample mounting which is probably quite reasonable. For reference, the advancing screw has 80 threads per inch, so 0.25 turns corresponds to a translation of about 0.0031” = 79µm.

3.1.3.2 Frequency Shifts Due to the Film

With some statistics on reproducibility of frequency measurements due to remounting a sample, we may perform some theoretical calculations for a substrate with a ﬁlm, the thickness of which may be varied in the calculation. In this way, we may obtain an estimate of the thinnest ﬁlm which can be reasonably resolved. In a forward 98

Fig. 3.12. The effects of transducer loading on natural frequencies and quality factors. (a) Displacement plots of the 4 modes measured, (b) shifts in the normalized quality factors due to loading, (c) shifts in the normalized frequencies due to loading, (d) visual depiction of the transducers holding the sample (oriented as in (a)) for the different turns of the thumb screw advancing the adjustable transducer. 0 turns is defined as a “normal” mounting. 99 calculation, discussed in Chapter 2, all material parameters are known and the eigenvalue problem is established by minimizing the Lagrangian with respect to the expansion coefficients of the basis set. The eigenvalues correspond to the natural frequencies of the sample. A forward calculation was performed with a STO substrate (c11 = 3.182, c12 =

1.027, c = 1.236 Mbar, ρ = 5.169g/cm3) with dimensions (0.6 0.5 0.3 mm). The ﬁlm 44 × × 3 was aluminum (c110 = 1.0675, c120 = 0.6041, c440 = 0.2834 Mbar, ρ = 2.697g/cm ) with a thickness which was varied from 1/400 to 1/2,000 of the substrate thickness. The relative frequency shifts, in parts per million (ppm), for the ﬁrst 20 modes were tracked as the

f f 0 film became thinner. Frequency shifts for the ith modes are calculated by i,d− i, 106 fi,0 where subscript zero indicates a bare substrate resonance and d indicates the resonance of the substrate with a film of thickness d. The first eight modes are plotted in Fig. 3.13 along with pictures of the normal modes. Some very useful information can be extracted from the plot. It is clear that modes break into two distinct groups: those that are sensitive to the presence of the film and those that are not. By correlating the data to the mode pictures, it becomes clear that modes with a significant displacement normal to the film plane are sensitive (e.g. those bending the film), and modes which primarily stretch the film laterally are insensitive. Modes 1, 2, 4, and 7 fall into the first category and modes 3, 5, 6, and 8 fall into the latter. This sort of analysis can serve as a guide during a film experiment so that highly sensitive modes are not excluded from the fit.

As long as 20 or so modes are included in the film fit, there should be ample sensitivity for film as thin as 1/1,000 of the substrate. 100

Film

2000 Mode1 1800 Mode2 Mode3 1600 Mode4 Mode5 1400 Mode6 Mode7 1200 Mode8 1000 800 600

Shift(ppm) 400 200

RelativeFrequency 0 -200 400 600 800 1000 1200 1400 1600 1800 2000

Vsub/Vfilm

Fig. 3.13. Film thickness dependence of resonant frequencies an hypothetical aluminum ﬁlm deposited on a strontium titanate substrate. The upper portion shows plots of the normal modes corresponding to the plotted frequency data. The dark grey line in the plot represents the uncertainty in frequency measurements due to remounting samples (100 ppm). Order N=10 polynomials were used for this calculation. 101

3.1.4 Data Acquisition

The ﬁrst issue to address is the number of resonances required to obtain reliable elastic constants. Migliori suggests 5 times as many frequencies as number of variable parameters (cij or dimensions) [47]. Based on experiences in this group, it has been found that accurate ﬁts can be obtained using about 3 frequencies per variable if the

fitting is done carefully, described in detail in the following section. This number can actually be further reduced if the derivatives ∂fi/∂ci,n are calculated and modes with high sensitivities to all elastic constants are selected. This technique is useful when temperature dependencies are being tracked, in which case the entire spectrum is remeasured and fitted for each temperature point. This is the technique used in the colossal magnetoresistance film experiments reported in Chapter 5.

Figure 3.14 depicts the equipment required to acquire resonance data. The setup consists of a Hewlett-Packard model 3325B frequency synthesizer as the drive source, the RUS cell described previously, a low noise voltage preamplifier (developed in this lab and discussed later), and a Stanford Research Systems (SRS) SR844 two phase lock-in amplifier. Control of all equipment is centralized by a Sun workstation through a IEEE bus. The frequency synthesizer generates a peak to peak drive signal of 5-10 V. The corresponding fields are well below the manufacturer listed dielectric breakdown field for the PVDF film which would occur at 50 V. In fact, some consideration has been given to adding an amplifier to the drive side of the cell to boost sensitivity. The trade-off is increasing cross talk and possibly entering a non-linear regime. The latter may in fact be an interesting experiment for estimating 3rd order elastic constants. Most of the data 102 acquired for this work was done so with 10 volt drive levels. It should be noted here that a frequency synthesizer with high resolution and stability is critical to RUS experiments.

The raw voltages coming out of the cell from the pick up transducer are the sum of any sample resonance plus transducer cross talk and noise. The sample signals are on the order of 1 µV while the cross talk is on the order of 50 µV. The signal is processed through a custom designed high frequency and low noise junction ﬁeld eﬀect transistor

(JFET) ampliﬁer with 1MΩ input impedance, to match the capacitive transducer, and

50Ω output impedance, to match the lock-in input. The circuit diagram is shown in Fig.

3.15, some details are as follows. The operational ampliﬁer is a ultra low noise wideband model CLC425 from National Semiconductor; and the JFET is a model 2N5911 from

Calogic Corporation. The bandwidth is deﬁned as the signal frequency at which the gain falls by 3dB, which is 11.5 MHz for this circuit. The gain within the operational frequency range (1kHz 9MHz) is 100. The voltage noise for the entire circuit is 9.32 ∼ nV/Hz1/2 at 11.5 MHz. As with any RF application, care should be taken with solder joints and wiring layout. This circuit is built on a CLC730013-DIP printed evaluation board from National Semiconductor with chips mounted in sockets. Power is supplied through 4 C cell batteries, which provide much lower noise levels than if power is taken from the mains.

The raw signal input into the lock-in ampliﬁer is a sinusoid with frequency equal to the drive frequency and amplitude depending on the cross talk, which varies very slowly with frequency, and sample corner displacement at or near the resonance frequency.

Added to this sinusoid is some background noise. The two phase lock-in then splits the input signal, multiplying one by a reference signal directly from the frequency synthesizer, 103

Fig. 3.14. Schematic view of data acquisition system for a RUS experiment. The drive synthesizer and RF lock-in ampliﬁer are controlled through an IEEE bus to a Sun Microsystems Ultra 5 workstation. The high frequency, low noise preamp is custom designed (see Fig. 3.15) and discussed in the text. 104

Fig. 3.15. Circuit diagram for a low noise, high frequency JFET preamplifier. The bandwidth is 11.5 MHz, gain is 100, voltage noise is 9.32 nV/Hz1/2 at 11.5 MHz. Power is supplied by 4 C batteries. Circuit is housed in die cast project box. 105 and the other by the reference signal phase shifted by π/2. The output from the first branch is called the in-phase channel, and the output from the second branch is called the quadrature channel. The frequencies are the same so the resulting signal for each channel is comprised of a DC component, the amplitude of which is dependent on the signal amplitude and phase difference between the signal and reference, and a high frequency component. The output of the in-phase channel will then be:

in-phase S = S S mix in × ref

= Vin sin(2πft + φin) sin(2πft + φref ) 1 1 = V cos(φ φ ) + V cos(4πft + φ + φ ) (3.1) 2 in in − ref 2 in in ref where the reference signal amplitude is one. The quadrature channel has a similar form, except phase shifted by π/2 (cosine terms become sine terms). Both signals are then passed through a low pass filter to strip away the high frequency component, and the output is then the DC signal in the first term. A more complete theory on lock-in amplifiers can be found in Horwitz and Hill [61] and the SRS Model SR844 manual [68].

Since the cross talk is much larger than the resonance signal, it is helpful to adjust the phasing of the lock-in at a frequency oﬀ resonance so that the entire signal (cross talk only) is loaded onto the quadrature channel, and the in-phase channel is then “zeroed out” until the sample signal is added.

Both the in-phase and quadrature outputs of the lock-in are recorded to disk on the Sun workstation for a series of drive frequencies. Software has been developed to automate data acquisition such that a series of frequency ranges can be scanned and 106 data for each recorded in a separate ASCII text ﬁle. The lock-in data must then be ﬁt with a Lorentzian lineshape which includes an arbitrary phase and background due to cross talk.

2 S(f) = a0 + a1f + a2f 2 (f/f0) cos φ + 1 (f/fo) Q sin φ +A − (3.2) 2 (f/f )2 + 1 (f/f )2 Q2 o − o where the first line fits any background, A is the peak amplitude, f0 is the center frequency, Q is the quality factor, and φ is the phase factor. This phase comes about from the interference between the cross talk and sample signals, and a phase of zero would produce a commonly illustrated Lorentzian resonance curve. From the best fit to the data, the center frequency and quality factor can then be extracted. A nonlinear fitting algorithm (e.g. Levenberg-Marquart) can be employed to perform the task, and a Mathematica script for doing this is given in Appendix A. This technique works reasonably well if a given data set contains only 1 resonance peak. However, it may happen that two closely spaced peaks overlap and thus can not be fit independently.

There are 3 ﬁt parameters for the cross talk background (a0, a1, a2), and 4 for each peak

(fo, Q, φ, A). Thus fitting 2 peaks requires adjusting 11 parameters and the results are marginal at best. Complicating matters are data points resulting from spurious signals in the electronics or noisy data for low amplitude peaks. A better method of fitting data is to use a much more powerful pattern recognition engine: the human eye and brain. The procedure is to plot the data and model curve on the computer screen and adjust the various parameters by hand until a visual best fit is achieved. With practice, 107 this procedure can be done quickly and the results for overlapping peaks are much more reliable than numerical fitting procedures. An example of such a fitting session is shown in Fig. 3.16. Both the in-phase and quadrature channel data can simultaneously be fit by adding an extra π/2 to the phase in (3.2) for the quadrature fit. Fitting both sets provides more confidence in the final set of fitted peak parameters.

3.2 Determining Elastic Constants

After obtaining a set of resonance frequencies from a sample, one is ready to begin the inverse calculation, discussed in Chapter 2, to determine the set of elastic constants providing the best agreement between the observed and calculated frequencies.

In the inverse calculation, some set of the the sample specific parameters such as elastic constants, dimensions, and crystal orientation are varied iteratively until the calculated resonance spectrum most closely matches the measured spectrum. Since the spectrum forms a nonlinear sequence, the Levenberg-Marquardt method is used to adjust the variable parameters. The FILMMRQROT.F program listed in Appendix A performs such a calculation for a thin film / substrate system with arbitrary crystal orientation. The program reads in a file (MRQIN.DAT) containing the starting points for all parameters, a list of parameters which are allowed to vary, density, order of polynomials to be used in the model (N), and the list of measured frequencies. To generate a new calculated spectrum, the forward calculation is called as a subroutine with the current set of parameters.

The calculated and measured frequencies are compared, and the Levenberg-Marquardt subroutine adjusts the variable parameters. This process is repeated until the sum of the squares of the diﬀerences in the frequencies no longer changes. 108

Fig. 3.16. Example of in-phase and quadrature resonance signals with lorentzian fit. The jagged (black) curve is raw voltage data and smooth (red) curve is the lorentzian fit dictated by eqn. (3.2). After the best visual fit is achieved, the relevant peak parameters (A, fo, Q, φ) are written to a file. This particular data is for a carbon nanotube film deposited on a silicon substrate. 109

The exact procedure of fitting elastic constants varies somewhat on a case by case basis. Generalities will be presented here with comments on various cases. The first important issue is listing the frequencies in the correct order. The forward calculation performed before any data was acquired can be used as a guide to recognize missed peaks, which should be entered in the MRQIN.DAT input file as a 0. A negative sign in front of any frequency in this file informs the program to exclude that peak from the fit. This is useful if a particular mode is of low Q, noisy, or otherwise questionable. If reasonably accurate elastic constants are known, it is best not to vary the elastic constants and vary the smallest 2 dimensions until a fit is achieved. The errors at this point should be low

( 0.1%) if the elastic constants really are accurate. Starting points for the dimensions ∼ should of course be the measurements from the Starrett. As a first run, it is sometimes useful to include only the first few frequencies (which are generally widely separated) in the fit to check for missed modes. Another useful trick here is to fix all parameters, in which case a single forward calculation is performed with the starting parameter values and results compared with observed data. Once one is certain the frequencies are in the correct order and missed modes are all accounted for, dimension fits can proceed. If the starting parameters are good, the dimensions should vary less than a few microns from their measured values. After dimension values have been obtained, the calculation is repeated using the new dimensions as starting points and varying only the elastic constants. Ideally, one would vary all elastic constants simultaneously, and this should be done on the first pass. Sometimes, however, the off diagonal elements

(c , i = j) stray rather far from “reasonable” values. The reason becomes clear when ij 6 derivatives are investigated. Typically these elements effect the frequencies less than the 110 diagonal elements and thus are not as well determined by the spectrum. In this case, the off diagonal elements can be fixed while the diagonal elements varied until a fit is achieved. Then switching the elements being varied allows one to vary all parameters while restricting the off diagonal elements. Different algorithms have been considered in which variables can be restricted to a set range, although no working code has been developed.

If no, or unreliable, information is available on elastic constants, then the fitting procedure is a bit different. In this case, the best possible information is used for starting points for elastic constants and dimensions should be fixed. A systematic variation of starting points should be performed so that the convergence of the solution to a true global minimum can be checked. The best solution from this step can be used to fix elastic constants and vary dimensions in a manner similar to that above. The solutions for the dimensions can then be used to refine the solution for elastic constants. This “iterative” approach to finding the set of parameters producing the true global minimum in error generally results in RMS error on the order of 0.05% for 15 - 20 frequencies and a high

Q material (Q & 1000).

There are some further tips and hints for fitting frequencies. Sometimes it happens that one or two modes are not fitting well. The first thing to do is check the order of the modes and that any missed modes have been accounted for. The next thing to do is look at the raw resonance data and see if there might be a reasonable explanation such as low signal to noise or a Q notably lower than the rest. These modes can be remeasured to see if the results are consistent. Some possible sources for discrepancies are changes in the environmental conditions before a mode was measured, such as temperature or ambient 111 pressure changes. If some such explanation can be found, the data can be remeasured or excluded from the fit. Excluding frequencies from the fit must be done with care, and should only be considered if the errors are anomalously large (&0.25%) and/or there is some experimental justification for doing so. In such a case, fits should be obtained with and without the data point to see the effect on the errors for the entire spectrum, which will generally depend on the total number of points measured. Specifically, if excluding the point results in the other frequencies obtaining a better fit, it can be reasonably assumed that the excluded point was perturbing the fit and its exclusion is justified. As a practical note, excluding 1 or 2 (in addition to the first) out of 20 frequencies generally does not effect the final elastic constants in any significant way.

3.2.1 Uncertainties in Fitted Parameters

Unfortunately, there is no rigorous way of determining the uncertainties in the elastic constants from the frequency residuals. A number of schemes have been developed by practitioners of RUS and are, in general, empirical in nature. One scheme utilized by

Albert Migliori [47] is to explore the curvature of the n-dimensional error surface near the minimum and perturb a parameter until the error increases by 2% which is assumed to be larger than all other combined experimental errors. The corresponding perturbation is then taken to be the confidence level of the fitted parameter. Another scheme utilized by Phil Spoor [67] is to randomly perturb the frequency data by an amount equal to the frequency residuals. The standard deviation in the resulting scatter in the fitted parameters can then be used to estimate the confidence levels. Using this technique on a sample of the well characterized material silicon, Spoor estimated uncertainties in the 112

3 elastic constants of 0.06%, 0.16%, and 0.25% for c11, c12, and c44 respectively with an

RMS error in the frequencies of 0.05%, typical of the ﬁts performed in this lab.

3.3 General Techniques for Thin Film Experiments

In the previous sections, the methods outlined apply to any RUS experiment.

However for experiments involving a thin film deposited on a substrate, additional steps are required. In what follows, general techniques for thin film RUS will be presented and specific steps and issues required for particular materials will be outlined in the following chapters. The first consideration in a thin film experiment is the geometry of the substrate. For reasonable results, the film should occupy at least 1/1,000 of the sample volume. Thus if films on the order of a few hundred nm are of interest, then substrates on the order of a few hundred microns are required. While the thickness must be small, one might hope that the other dimensions could be large, making the sample easier to handle. However, there is another limitation in that resonances of plate-like samples are strongly perturbed by transducer loading. It is thought this is caused by the comparatively large displacements at the corners of the plates. Since the sample contacts the transducers at these points, their contribution to the resonating system can no longer be ignored. Thus, the free boundary conditions assumed in the calculations are no longer valid. Therefore substrates of small thickness also must be rather small in the other dimensions. It is for this reason that small sample RUS techniques as outlined in this thesis are required for thin film measurements.

As the reader may guess, the nature of a sample mounted in the RUS cell is such that in situ deposition of a film is nearly impossible. Therefore a substrate must 113 be prepared and variable parameters well fitted before a film is deposited. The sample must then be removed from the cell and film deposited in some way (e.g. molecular beam epitaxy, laser ablation, etc.). It is crucial that during this step, the sample is handled with great care as any slight chips or impurities adhering to the surface may perturb the resonances and obscure the effect of the film. Once the film is deposited, the sample is remounted in the cell. If possible, the sample should be mounted in the same orientation as it was during the original measurement. This seems to provide added consistency in transducer loading effects. Then the same set of resonances is obtained under the same ambient conditions.

The analysis phase of a thin ﬁlm experiment begins by “correcting” the raw thin

ﬁlm frequencies. The corrected frequencies are calculated by

(sub) f f(film) = f(film), where = cal (3.3) corr raw (sub) fraw note that is calculated from the best fit for the bare substrate. This rescaling of the frequencies is required because the typical residual errors in the frequency fits for the substrate are on the order of 0.1%; and, since the film occupies about 0.1% of the sample volume, the shifts of the resonances due to the presence of the film are comparable to the errors resulting from the substrate fit. If the frequencies are not corrected, the computer will essentially be fitting the residual errors from the substrate fit by adjusting the film elastic constants. It was found that the errors for particular modes are consistent through sequential fits to data obtained over a range of temperatures. 114

During the film fitting session, ALL substrate parameters and the film thickness are held fixed. The only parameters allowed to vary are the elastic constants of the film cij0 . As with a homogeneous sample, if reasonably accurate film constants are known, it may be advantageous to vary only the diagonal elements first and then the off diagonal elements. It is very important when fitting the film data that all fitting criteria are exactly the same as those used when obtaining the substrate fits. Such criteria include: the modes included in the fit, and the order of the polynomials in the Visscher basis expansion (N). After an initial fit is obtained, one should again try various starting points of the film constants to test for reproducibility and whether the error minimum is truly global.

Uncertainties in the accuracy of the film constants are an even more difficult problem than in general RUS measurements. The fitted parameter uncertainty estimation methods mentioned previously can be used, however even large variations in the film constants typically produce small changes in the fit. Other possibilities are to perform several measurements on the same sample and check the reproducibility of the results, or measure a thin film with known elastic constants and check the consistency. The section below describes a measurement of elastic constants of a 546 nm aluminum film evaporated on a 500 µm thick STO substrate.

3.3.1 A Test Case: Aluminium Film on Strontium Titanate

As with any new experimental technique, a measurement of a sample whose properties are known should be conducted. To this end, a strontium titanate (STO) substrate was prepared and measured resulting in elastic constants: c11 = 3.2056, c12 = 115

1.0347, c44 = 1.2380 Mbar. 18 modes were included in the ﬁt with an RMS error of

0.056%. The fitted dimensions of the STO substrate were 1.6869 1.8238 0.4741 × × mm. The substrate was then removed from the RUS cell and an aluminum film was deposited by vacuum evaporation to a thickness of 50 nm, or about 1/10,000 of the substrate thickness. The sample was allowed to cool to room temperature and remounted between the transducers in the same orientation and new resonance data obtained. A view through the microscope of this sample mounted between the transducers is shown in Fig. 3.17. The raw film data was corrected using (3.3).

Before ﬁtting the frequencies a decision must be made as to the crystallinity of the ﬁlm. The STO substrate is a high quality single cubic crystal with lattice constant

3.905A.˚ Aluminum is also cubic with a lattice constant of 4.050 Aand˚ is thus mismatched rather strongly with STO at the interface. The STO deposition surface was produced by mechanical polishing to a grain size of 500 nm and not by epitaxial growth, thus the STO surface layers are damaged and polycrystalline. These considerations strongly suggest that the aluminum film is polycrystalline rather than a single crystal. Although the crystallite sizes are not known. If they are small compared to the lateral sample dimensions, the film will be elastically isotropic. In any case, both isotropic and cubic models were used to fit the film resonance data. As the single crystal aluminum elastic constants are well known, c110 and c440 were first allowed to vary while fixing c120 . The new c110 and c440 values were then fixed and c120 allowed to vary. Another possibility is to vary only c110 and c440 and perform a series of fits systematically stepping through values of c120 around the best known value. The errors for each fit can be compared to determine the optimum values for the film constants. The fit was then repeated 116 assuming an isotropic model such that c = (c c )/2. Here the elastic constants 44 11 − 12 (c11 c12)(2c12+c11) have been converted into a value for Young’s modulus E = − and (c11+c12) c12 Poisson’s ratio σ = c11+c12 . The results for each are shown in Table 3.17. The known values obtained from the Landolt-Börnstein Tables [2] are indicated as “L-B”.

The experiment was repeated for a aluminum film thickness of 546 nm or 1/1,000 the ∼ substrate thickness. These results are also shown in Table 3.2. It is clear the 50 nm film is so thin that the shifts in resonance frequencies are overwhelmed by the noise in the data. The results for the 546 nm film however are rather reasonable and robust in the sense that even changing the starting points for the film constants by 75% yield the same solution. The Landolt-Börnstein values for bulk aluminum are listed here for reference, however it is not clear that when a material is fashioned in a film this thin that the elastic parameters should be the same as in the bulk. These results suggest that the elastic constants are a bit higher than their bulk counterparts. This is in fact consistent with reports of a so-called super modulus effect in very thin films and super lattices. [50]

[13] While this effect has been primarily observed in films much thinner than this ( ∼ 1-5 nm), the effect was much stronger ( 150% increase in Young’s modulus). Perhaps ∼ a residual effect is still present in these thicker films. Unfortunately, few studies exist for this phenomenon and no explanation could be found in the literature. The point here is that bulk elastic constant values may in fact not be the same as corresponding thin film values, although they certainly are the most obvious starting points for the

film constants. The film constants for this study are 28% larger than bulk for Young’s modulus and 13% for Poisson’s ratio for the 546 nm aluminum film. This experiment tests the accuracy of thin film RUS, however for many experiments it is the precision 117 which is important. In such experiments, the sample remains mounted between the transducers while another parameter (usually temperature) is varied and changes in the resonance frequencies (and thus elastic constants) are tracked. The precision of this implementation of RUS is much better than the accuracy, and temperature dependence is readily observed for changes of 0.5◦ C. Thus the temperature dependence of elastic constants and phase transitions can explored with a high sensitivity. More discussions on such an experiment are given in Chapter 5 on colossal magnetoresistance films.

L-B Values 50 nm film 546 nm film Cubic c11 1.07 1.59 1.45 c12 0.60 0.64 0.62 c44 0.28 -0.005 0.45 RMS (%) – 0.0048 0.0211 Isotropic E 0.72 0.36 1.00 σ 0.34 0.48 0.39 RMS (%) – 0.00475 0.0207 Qfilm/Qbare – 0.916 0.755 Table 3.2. Elastic constants in Mbar and shifts in quality factors for two aluminum films deposited on a strontium titanate substrate to thicknesses of 1/10,000 and 1/1,000 the sub- ∼ ∼ strate thickness.

3.4 Conclusions

This chapter provided detailed instructions for executing a standard RUS experiment and addressed additional issues related to thin ﬁlm experiments. An important topic was the eﬀect of transducer loading. While this topic is of interest in any RUS 118

Fig. 3.17. Microscope view of 546 nm aluminum ﬁlm on STO substrate mounted between the transducers.

experiment, it becomes critical to thin film experiments as shifts in resonance frequencies due to less than a part in a thousand film begins to approach shifts due to slight differences in the mounting of the sample between the transducers. It was found through a statistical study that frequencies of a sample can be reproduced to less than 87 ppm if care is taken about the sample orientation between the transducers and temperature is stabilized. As in situ deposition of a film on a substrate is nearly impossible, the general procedure for a thin film experiment is to characterize the bare substrate as accurately as possible, the sample is removed from the cell and film deposited, then remounted and measured. Correcting the film frequencies based on the best fit of the substrate yields the best results for the film constants. In addition to transducer loading effects, uncertainties in sample dimensions, densities, and/or crystal orientations may become limiting factors in film constant accuracy. A numerical feasibility study was conducted 119 in which it was found that film volume fractions on the order of 1/1,000 should be ac- cessible through experiment, and that normal modes which bend the film perpendicular to the film plane are much more sensitive to the film presence than those which stretch the film laterally. A test case of an aluminum film deposited on an STO substrate was presented for film volume fractions of 1/10,000 and 1/1,000. The film measurements indicated elastic constants higher than bulk values by about 28% and 13% for Young’s modulus and Poisson’s ratio respectively. 120

Chapter 4

Carbon Nanotube Films

Carbon nanotubes (CNTs) have attracted an enormous amount of attention from researchers in a wide variety of disciplines. An axial Young’s modulus in the terrapas- cal range and very large elastic limits are a few of the remarkable mechanical properties which have helped fuel this intense interest. Such unique properties have prompted many proposed applications including the use of CNT coatings (perhaps in a resin matrix) to increase durability and strength of structures. Thin-film RUS is well suited to probe mechanical effects of a CNT film deposited on a substrate. This chapter describes several experiments on systems consisting of small RUS samples coated on one side with a tangled mat of CNTs. First, it may be advantageous to review some of the basic properties of CNTs.

Carbon is perhaps one of the most studied elemental materials; very few other materials can naturally form such diverse morphologies as diamond and graphite. The hardness of diamond is derived from the strong covalent carbon-carbon bond. The strength of this bond also exhibits itself in the hexagonal plane of carbon atoms comprising a graphene sheet. Graphite is formed from stacks of these sheets, which are very loosely bound, making it useful as a dry lubricant. One of the fruits of carbon research are carbon ﬁbers (CFs), solid rough cylindrical forms of amorphous and polycrystalline carbon with typical dimensions of 75 nm in diameter and perhaps 1 µm in length. These 121

ﬁbers exhibit attractive mechanical properties such as a high Young’s modulus and tensile strength. CFs embedded in a supporting (usually polymer) matrix have found their way into various commercial applications requiring high strength to weight ratios such as aircraft components and sports equipment.

As the fabrication technology advanced (e.g. chemical vapor deposition (CVD)), it was found that improving the crystallinity of the carbon in the fibers improved their strength. [42] Some carbon experts envisioned the perfect carbon fiber: a single perfectly crystalline graphene sheet rolled into a seamless hollow cylinder (1991). [18] These theoretical musings were realized only months later with Iijima’s report in Nature of the first TEM images of carbon nanotubes, discovered while experimenting with C60

(“buckeyball”) synthesis techniques. [26] Since this discovery, the scientiﬁc community has produced an astounding volume of research on this unique from of matter. A search of the Web of Science [1] on “carbon nanotubes” from 1991 to Feb. 2003 produced 4,517 papers on the subject!

4.1 Structure and Properties of Carbon Nanotubes

The basic structure of a carbon nanotube can readily be envisioned by taking a

2 dimensional hexagonal grid and rolling it in such a way that the vertices (representing carbon atoms) all align along the length of the resulting tube, shown schematically in

Fig. 4.1. One may deﬁne a chiral vector (C) along the hexagonal grid representing the circumference and orientation of the tube. A set of two integers (m, n) may then be deﬁned for each of the possible end points of the vector. This scheme is illustrated in

Fig. 4.2. Tubes for which n = 0 are called “zig-zag” tubes and those for which m = n are 122

c zig czig

Fig. 4.1. Rolling a graphene sheet into a “zig-zag” carbon nanotube. The diameter is on the order of 1-2 nm while the length can be up to 10 microns. [Artwork by Nathan Gabor] 123

“armchair” tubes both from the arrangement of the atoms along and around the tube.

Fig. 4.1 shows a “zig-zag” tube. Any tube with a chiral vector for which m = n = 0 is 6 6 known as a chiral tube. Perhaps one of the most fascinating properties of CNTs is that all armchair tubes are metallic for electron transport along the tube axis, while most others are semiconductors with variable band gaps depending on the chiral vector. The reason for this becomes clear when the energy dispersion relation is plotted for the entire

Brillouin zone for a 2D graphene sheet as in Fig 4.3. The conduction and valence bands become degenerate at the 6 symmetric K points in the Brillouin zone. When a sheet is rolled into a tube, wave vectors around the circumference of the tube become quantized due to phase matching requirements, but the wave vector along the length of the tube remains continuous. The 1D energy dispersion curves for the tube then are simply slices of Fig. 4.3. The slice for an armchair tube coincides with the K points and, since the

Fermi levels are occupied at ﬁnite temperatures, the tube is metallic. For intersections not lying in the plane of the K points, a variable energy gap between the conduction and valence bands results, thereby producing a semiconducting tube. This unique electrical property of carbon nanotubes suggests many possibilities for “nanocircuits” and has driven much of the research in this ﬁeld. The CNTs discussed above consist of a single graphene sheet and are known as single wall nanotubes (SWNT). Another morphology of CNTs consists of concentric sets of single wall tubes. These are known as multi-wall nanotubes (MWNT)

CNTs also possess remarkable mechanical properties. The nearly perfectly crystalline network of carbon-carbon bonds produces a structure which is arguably the strongest in nature. Several theoretical investigations and direct measurements of elastic 124

Fig. 4.2. Chirality vector map for a CNT. Armchair tubes (with m = n) are conducting, while most others are semiconducting with diﬀering band gaps. The axis of the tube (z) is perpendicular to the chiral vector C.

K K K K K K

Fig. 4.3. The energy dispersion over the entire Brillouin zone for a 2D graphene sheet. The upper (conduction) and lower (valence) surfaces are separated by the Fermi level. At the 6 K points, the surfaces touch and the sheet behaves like a conductor for electrons with these wave vectors. 125 parameters of individual tubes have been conducted by atomic force microscope (AFM) methods [39] and cantilever resonance techniques [55]. The results show some variability due to diﬀerences in tube diameters and defect concentrations, but some typical values are: c = 10.6 14.6 Mbar, c = 4.4 0.3 Mbar, and bending modulus E = 1.0 12.6 11 − 44 b − Mbar with strong dependence on tube diameter (10Mbar = 1TPa). Single tubes have also been shown to withstand axial strains of 10% with an ultimate tensile strength of

100 MPa. [40] [54] [64] A quick calculation shows that a 1 cm2 ﬁber consisting of a ∼ parallel array of CNTs would be able to withstand almost a 3,000 pound tensile load.

In addition to exceptional stiﬀness and strength properties, CNTs have been shown to be extremely “tough” as deﬁned by a high elastic limit for bending strains. As a tube is bent, it initially behaves like a perfectly elastic tube with linear stress-strain curves, however at some critical strain, the tube will buckle (as shown in Fig. 4.4) allowing a very large bending angle. The remarkable feature is that the tube can rebound to it’s original shape after this high degree of deformation with no loss of elasticity. Such behavior has prompted many proposals for the use of CNTs as strengthening additives for structural applications where high strength to weight ratios are desirable, and as building blocks for future micro electro-mechanical devices (MEMS). Unfortunately, even after an astounding volume of work, many technical details remain unsolved and only modest mechanical improvements have been realized to date. [76] [41]

4.2 CNT Film Sample Preparation

One of the ﬁrst technical problems to be solved for this experiment is how to deposit SWNTs into a uniform ﬁlm, or more accurately a tangled mat, which adheres 126

Fig. 4.4. Buckled CNTs can rebound to their original shape making them extremely “tough”.

to a substrate. An additional concern is adsorption of foreign materials into the mat.

Tangled mats of SWNTs readily adsorb contaminants from the atmosphere such as oxygen and water vapor due to their extremely high speciﬁc surface area. Thus, to ensure that any features observed are due to the CNTs themselves, a scheme must be developed which produces a contaminant (adsorbate) free mat. Details for such a scheme are discussed next.

4.2.1 Film Deposition

The technique developed in this lab for depositing a ﬁlm of nanotubes on a small substrate has been successfully used to fabricate visually (as viewed through a microscope) uniform ﬁlms of relatively well controlled thickness from 100 - 1,000 nm. The nanotubes were fabricated by either arc-discharge (AD) or pulsed laser vaporization 127

(PLV) techniques by P. C. Eklund’s group of The Pennsylvania State University. Typi- cal “as prepared” samples are 20% tubes and 80% amorphous carbon and metals used as a catalyst in the fabrication process. Most of the tubes used in these experiments have been puriﬁed by a HCl acid bath process developed by Eklund’s group such that the material is 70% pure nanotubes by weight. [personal communication] Obtaining high ∼ quality puriﬁed SWNT samples was found to be very important for these experiments.

The general procedure for depositing nanotubes on a substrate is to ultrasonically disperse them in ethanol, then use an artists air brush to spray controlled bursts of the suspension on the substrate. The ethanol evaporates, leaving a uniform film on the substrate surface. The airbrush is a model VL from Paasche Airbrush Company [57], and is illustrated in Fig. 4.6. A compressed gas (helium for these experiments) flows through the airbrush at a high velocity. The flowing gas lowers the pressure above a

fluid inlet port downstream allowing the nanotube suspension to be drawn up into the gas flow and vaporized. The resulting fine mist is then focused through a nozzle at the tip, and a conical pattern is produced. This particular airbrush allowed for independent control of both the compressed gas flow and fluid flow. This feature was found to be quite useful in that a small burst of fluid could be sprayed followed by a burst of dry helium to facilitate evaporation of the ethanol. Repeating these steps produced the most uniform films. The details for depositing SWNT films are outlined as follows:

1. A mass of 2 mg of puriﬁed SWNT soot is added to 15 ml of pure ethanol in a

glass jar ﬁtted with a screw-on aluminum lid, and the dispersion is sonicated in a

high power ultrasonic cleaner for 15 minutes. After this treatment, the suspension 128

should appear uniformly black. Ethanol is the chosen dispersion medium because

it easily wets carbon and evaporates quickly and cleanly.

2. The suspension will begin to settle into a non-uniform density proﬁle along the

liquid height after a period of an hour or so. To keep the suspension well mixed

during deposition, the jar is held in a low power ultrasonic cleaner. Fig. 4.5 shows

a schematic of the airbrush apparatus. A small section of 1/4” thread stock, with

a hole drilled along the axis, was fastened to a hole in the jar lid with two nuts.

One end of a teﬂon tube was then inserted through the hole in the thread stock

into the suspension, and the other end snugly ﬁtted into the input port on the

airbrush.

3. The substrate to be used in the RUS measurement is placed on a clean microscope

slide positioned at the base of a ring stand. The airbrush is secured to the ring stand

such that the height of the nozzle tip is 18 cm above the substrate surface. This

height was found to minimize the lateral ﬂow of the suspension on the substrate due

to the downward pressure while maintaining a reasonable spot size for the spray

(about 3 cm). A small plumbbob and thread is useful to measure this distance and

center the nozzle tip over the sample center. The regulator on the helium cylinder

is set to 10 psi.

4. The operation of the air brush is illustrated in Fig. 4.6. The thumb lever controls

the gas ﬂow by pressing down (toward the brush axis) and the suspension ﬂow by

pulling back. The highest quality ﬁlms are produced by spraying short ( 1 sec) ∼ bursts of suspension, followed by 3 sec burst of dry helium to quickly evaporate ∼ 129

the alcohol. This process is repeated until the desired thickness is achieved. Also

helpful is shining a 100W desk lamp on the sample to warm the substrate and speed

evaporation. This system was calibrated by spraying various numbers of bursts and

using an AFM to measure the resulting ﬁlm thickness. The data showed a linear

relationship between number of sprays and ﬁlm thickness with a (free) intercept at

very nearly 0 thickness for 0 sprays. The data and linear ﬁt are shown in Fig. 4.7.

5. The sample is degassed under in a small high vacuum oven ( 10 8 Torr) at 500 C ∼ − ◦ for 24 hours. This step has been found by P. C. Eklund’s group to eﬀectively drive

oﬀ typical atmospheric contaminants such as water vapor and oxygen from the

nanotubes [personal communication].

After cooling, the oven chamber is valved oﬀ and passed into an inert atmosphere

(He) glove box in which a RUS cell and mounting apparatus are already in place. The sample is removed from the oven and mounted in the RUS cell with care taken not to disturb the ﬁlm. The mounting process is essentially the same as outlined in Chapter 3 with the exception that the image of the cell through the microscope is captured by a

CCD camera and viewed on an external monitor. Once the sample is mounted and sealed in its vacuum jacket, the cell is removed from the glove box and data may be acquired.

The helium gas in the cell may be pumped and replaced by other gases or vapors. It was found that the CNT ﬁlms can be removed from the substrate by submersing in ethanol in a soft plastic container and placing in a low power ultrasonic cleaner for about 20 seconds. Also, the residual CNTs on the microscope slide can easily be scraped oﬀ with a clean razor blade and reused. 130

Fig. 4.5. Air brush system for depositing nanotube ﬁlms. The small ultrasonic cleaner is used to keep the suspension well mixed during deposition. See text for parameter details. 131

Fig. 4.6. Schematic of the artists air brush used for CNT film deposition. The high velocity helium lowers the pressure over the suspension drawing it up into the brush where it becomes vaporized upon hitting the gas flow. The thumb lever can be used to adjust both the helium and liquid flow rates independently.

Fig. 4.7. AFM film thickness calibration data and fit for air brush deposition. The fit was used to calculate film thicknesses between 100 and 1,000 nm for these experiments. 132

4.3 Results

The most interesting effect of CNT films on a resonating substrate is the quality factor. From earlier discussions in Chapter 2, one may expect that adding a tangled mat of nanotubes to a surface of a resonating solid would introduce many more avenues for energy loss, and thus lower the Q. This is in fact typical of films on substrates. To quote from Systems with Small Dissipation [7], “Films adsorbed on resonator surfaces create additional energy losses, and therefore stringent standards are set for the cleanliness of high-Q resonators.” Such avenues would include friction between tubes in the mat and between tubes and the substrate as well as reordering of tubes due to the surface motion of the substrate. The surprising observation however was that the Q increased with the addition of the CNT mat. For a strontium titanate (STO) substrate with a 500 nm thick SWNT film in 1 atm He, the results are shown in Table 4.1. The increase in

Q due to the presence of the ﬁlm is about a factor of 1.36 on average over 11 modes with the largest increase of a factor of 1.75 for the ﬁrst mode. These increases are well above the uncertainties introduced by remounting the sample, which is on the order of

10%. Clearly the nanotubes are suppressing one (or several) of the loss mechanisms in the substrate. Speculations on the origin of this effect will be presented later. This “Q effect” was checked for several different film depositions on several STO substrates with similar results.

As one might expect, the frequencies decreased with the addition of the ﬁlm, most likely due to mass loading. The eﬀect of air exposure was checked by degassing a SWNT/STO sample and tracking resonances as a function of air exposure time. The 133

Qbare QSNW T QSNW T /Qbare Mode 1 1778 3111 1.75 Mode 2 544 822 1.51 Mode 3 2002 2827 1.41 Mode 4 1948 2949 1.51 Mode 5 4469 5398 1.21 Mode 6 5952 6798 1.14 Mode 7 2349 3005 1.28 Mode 8 3163 4924 1.56 Mode 9 4336 5208 1.20 Mode 10 10713 – – Mode 11 7927 10567 1.33 Mode 12 4503 4935 1.10 Average 1.36 Table 4.1. Change in quality factor for a 500 nm SWNT ﬁlm on a strontium titanate substrate. The uncertainty introduced by remounting is about 10%.

frequencies of all observed modes rose until about 40 hours of air exposure, after which all frequencies remained constant for up to 60 hours. The data for 10 modes are shown in Fig. 4.8. The rise in frequencies suggests a stiﬀening of the SWNT mat which may be due to adsorption of polar water molecules, which saturates after 40 hours. The quality factors over this period dropped very modestly by about 5-10% until 40 hours and then remained constant. Since the general increase in Q remains in force even after air exposure, it seems we may be able to skip the degassing step, which is the most time consuming component by far, and still be able to study the phenomenon.

4.3.1 The “Q-eﬀect”

A systematic investigation of the ability of the nanotubes to increase the quality factors of small resonators was undertaken. The primary adjustable parameters were substrate material and ﬁlm thickness. In what follows, the ﬁlm deposition was carried out 134

1.0001

1.0000

0.9999

ba 0.9998

/freq

T 0.9997

W S 0.9996

freq 0.9995

0.9994

0.9993 0 10 20 30 40 50 60 Airexposuretime(hours)

Fig. 4.8. The effect of 1 atm air exposure on a 500 nm SWNT/STO film sample. The sample was degassed as described in the text and resonance frequencies tracked over exposure times up to 75 hours. 135 as outlined above, except the samples were not degassed. After deposition, the samples were mounted in the RUS cell and spectra obtained. For film thickness experiments, the sample was repeatedly removed from the cell and additional layers were typically deposited in 100 nm increments from 100-1,000 nm. The substrates studied were STO, silicon, alumina, quartz, and brass. Perhaps the most important result of this work is that all of these materials exhibited an increase in quality factor with the addition of SWNT films, however to varying degrees. Thus, whatever the mechanism causing this phenomenon, it seems to be general. Fig. 4.9 shows the average increase in Q for the various substrates investigated, and Table 4.2 lists the modes exhibiting the largest increases in Q for each of the materials. In addition to a general increase in the Q, we see some dependence on the thickness of the film. Most materials showed peaks between film thicknesses of 400 - 600 nm, except quartz which peaked at 200 nm. With the exception of the brass, all samples were prepared according to the steps outlined in Chapter 3 with a final polishing using 0.5 µm diamond paper. The brass sample was only roughly polished with emery paper.

4.3.2 CNT Film on a Passivated Si Surface

It seems logical that the ability of SWNTs to increase the Q of a resonator is intimately tied to the interaction between the tubes and the substrate surface. In an attempt to alter this interaction, a silicon 100 sample was prepared in the usual fashion h i and polished with 0.5 µm diamond paper. A SWNT ﬁlm was deposited as described above and quality factors tracked as a function of ﬁlm thickness. This data is shown in

Fig. 4.9(b) and Table 4.3.1. After removing the ﬁlm, the Si sample was exposed to a 136

Fig. 4.9. Shift in quality factor averaged over 10-15 resonances as a function of SWNT ﬁlm thickness for various substrates: (a) alumina, (b) silicon, (c) STO, (d) quartz, (e) brass. The brass surface was much rougher than the rest and showed no particular peak. 137

Substrate Qbare Qﬁlm Ratio Thickness (nm) Mode Alumina 2343 13355 5.70 400 12 STO A 5917 12175 2.06 400 15 STO B 6071 13357 2.20 500 16 STO C 5389 32334 6.00 200 6 Silicon 1827 3043 1.67 600 15 Quartz 1996 8725 4.37 200 6 Brass (rough) 1324 2352 1.78 400 8 Table 4.2. Modes exhibiting the largest increase in Q for various substrates. All sample surfaces were prepared with 0.5 µm diamond paper polishing except the brass which was roughly polished with emery paper.

dilute HF acid bath (1.6%) for 10 minutes at room temperature, followed by boiling in distilled water for 30 minutes. This process has been shown to terminate the Si dangling bonds with hydrogen atoms and thus pacify the surface against oxidation. [19] This treatment had no significant effect on either the resonance frequencies or quality factors for the bare substrate. The SWNT film experiment was then repeated with the data shown (along with polished surface data) in Fig. 4.10. The most dramatic effect was the decrease in the Q with the addition of nanotubes on the passivated Si surface. Also interesting is the film thickness dependence which showed a minimum in the Q at the same thickness for which the Q was maximized for the polished surface.

4.4 Analysis

One of the great joys a scientist can experience is to be confronted with an unexpected result. As stated above, depositing a ﬁlm on a resonator provides many avenues by which extra energy can be converted from acoustic to thermal forms, thus lowering 138

Fig. 4.10. The effect of a SWNT film on a passivated silicon surface. 139 the quality factor; and this is the general behavior for films. Nevertheless, several checks for the effect on the Q where performed for films of other materials. The data shown in

Table 3.17 for the aluminum film showed an average drop in the Q to factors of 0.916 and 0.755 for the 50 nm and 550 nm films respectively. Another check was made by depositing a film of high surface area activated carbon in exactly the same manor as the nanotubes were deposited. For this case, the average drop in Q was even more dramatic at a factor of 0.64. A third check was attempted with a “failed” nanotube batch supposedly containing only amorphous carbon nanoparticles. After depositing a

500 nm ﬁlm, the average Q was found to rise by a factor of 1.96! After reexamining the sample with a transmission electron microscope (TEM), it was found that CNTs were in fact abundantly present. These experiments verify that the “Q-eﬀect” is unique to the structure and properties of carbon nanotubes.

For the quality factor in a resonator to rise with the addition of a film, one of two things must be happening. Either the film is somehow supplying more energy to the resonator and the frequency of this energy is matched to the resonance; or the nanotubes are suppressing one or more of the many avenues by which a bulk resonator losses energy. As no mechanism for the former seems reasonable, we pursue the latter. As previously discussed, there are surely loss mechanisms within the film such as tube-tube and tube-substrate friction as well as tube reordering. Thus, whatever mechanism is being suppressed must over compensate for these additional losses so that the net effect of the film is an overall reduction in energy loss for the system. If we confine ourselves to friction within the film, then all films prepared in a similar way should have similar losses. The fact that increases in Q of a factor of 6.0 have been observed suggests that 140 very little attenuation occurs within the film itself. In other words, the nanotubes in the form of a tangled mat exhibit very little internal friction.

Our attention is immediately drawn to energy loss mechanisms which are confined to the surface of the resonator. As discussed in Chapter 2, microcracks and defects at the surface can be a significant source of energy loss, particularly for very small resonators with high surface area to volume ratios as is typical for the samples used in these experiments. These losses are primarily due to increased thermo-elastic losses caused by the added texture of the material and strain reordering of atoms within the cracks and at crystallite boundaries. Since there is no supporting material in the half space above the surface, the relative motion of two small patches of material in the cracks during 1 cycle of oscillation is larger than in the bulk. Thus it is likely that massive strain reordering is occurring within the cracks. The proposed model is that the nanotubes, being very long and stiff, bridge the cracks, as illustrated in Fig. 4.11, and provide reinforcement such that the relative motion of the substrate around the crack is reduced and suppresses the strain reordering. For this model to be realistic, the tubes must be anchored reasonably well to the substrate around the crack and thus the magnitude of the effect should be dependent on the substrate material, which is supported by our data.

This model is also consistent with the hydrogen terminated silicon surface experiment.

If the interaction energy between the surface and nanotubes is greatly reduced with the

H termination, then the tubes would simply slide over the oscillating substrate surface and provide no reinforcement, but leaving in place the avenues for energy loss in the

ﬁlm. 141

Van der Waals anchoring force

micro−crack CNT rope

Fig. 4.11. Model for the reinforcement of surface microcracks with carbon nanotubes. The stiﬀ nanotube suppresses the large scale strain reordering within the crack. 142

Perhaps more difficult to explain is the film thickness dependence of the effect.

In general, the eﬀect seems to be maximized between thickness of 400 - 600 nm for most materials studied here. One possibility is that the ﬁrst monolayer of tubes reinforcing the cracks become increasingly well anchored to the substrate as more material is added.

However, with additional material comes additional losses within the ﬁlm, which at some point becomes comparable to energy savings and the two competing mechanisms begin to cancel. For almost all of the experiments presented here, Qfilm/Qbare approached unity at a thickness of 1 µm. For some samples the thickness was extended as high as 1.2µm where the Q dropped below bare substrate levels. This is likely a vast oversimpliﬁcation of the problem; a rigorous theoretical treatment could probably occupy an entire thesis.

4.5 Conclusions

The unique electronic and mechanical properties of carbon nanotubes have enticed literally thousands of scientists to investigate and propose many applications for their use.

This research has uncovered yet another unique feature: the ability of carbon nanotubes to decrease attenuation in small solid state resonators. Such high Q resonators are ubiquitous in many technologies such as frequency synthesizers, evaporation thickness monitors, wrist watches, and solid state gyroscopes to name a few. The resolution of these devices depends on the resolution of the resonance frequency, or sharpness of the resonance peak. The ability to increase the quality factor by a factors of 6 or better would be a significant enhancement for such technologies. In particular, micro electro-mechanical systems (MEMS) resonators can be constructed out of quartz in the form of a tuning fork and used as the basis for a gyroscope. If the resonating fork is 143 rotated, energy flows from one type of mode to another, due to the Coreolis forces, with a slightly different frequency. Such devices are commercially available, however they lack the resolution required by the large aviation industry. By increasing the Q of each of the modes so that their components can be better resolved, MEMS gyroscopes may improve enough to enter this market.

The process reported here for Q enhancement is far from being optimized. One area which may yield more impressive results is tuning the interaction between the nanotubes and substrate by chemically functionalizing the surface. This may maximize the effect at much lower film thicknesses, thus eliminating the extra losses produced by the excess film. Also increasing the purity of the nanotubes and devising a deposition technique which provided some control over tube alignment could enhance the effect for particular modes of vibration.

The reader may well wonder why no mention has been made of the elastic constants of a nanotube ﬁlm. Several calculations were performed in an attempt to extract the elastic constants from the shifts in resonance frequencies, however the results were not reasonable. A review of these eﬀorts follows.

There are two possible symmetries for the film. The tube orientation in the film plane is random, however the tubes preferentially lie flat on the substrate surface for thicknesses between 100-1,000 nm. Thus a hexagonal (or planar isotropic) symmetry was employed during fitting with the c axis perpendicular to the film plane. Obviously the bulk elastic properties of a tangled mat of nanotubes are not known, however we may guess that they are rather low since it seems from the attenuation arguments given above there is little friction and cohesiveness within the film. Various starting points for 144 the 5 elastic constants were used during fitting. The optimized set of elastic constants unfortunately tended toward unreasonable values such as slightly negative values for positive definite cij and impossible combinations of cij values. Remember that CNTs were not embedded in any sort of supporting matrix (e.g. epoxy), and so it is probable that the elastic constants are so low for the film as to be beyond the capabilities of this technique. The resonance frequencies always decreased monotonically with the addition of nanotube thickness suggesting that the film was not adding any elastic contribution, but simply mass loading the resonator. 145

Chapter 5

Colossal Magnetoresistance Films

5.1 Introduction

The phenomenon of colossal magnetoresistance (CMR) refers to a strong dependence of electrical resistivity on an externally applied magnetic ﬁeld. [37] While many materials exhibit magnetoresistance to some degree, it is dramatically (colossally) enhanced in perovskite manganite materials. Manganites have the chemical formula

A(1 x)BxMnO3, where A is a trivalent rare earth metal (La,Pb,Nd,Pr), and B is a di- − valent alkali earth metal (Ca,Sr,Ba). The manganite resistivity vs temperature data are shown in Fig. 5.1 with an inset showing the unit cell. [49] [56] A discussion of the crystal symmetry will be presented in Section 5.3. By tuning the constituent atoms (A,B) and their ratios (x), the magnitude of the CMR effect has been extended to changes in resistivity of thin films spanning 7 orders of magnitude. A good deal of interest in thin films of perovskite manganites has been prompted by potential applications in digital data storage and the ability to study effects of strain induced by the underlying substrate.

The ﬁlm studied in this experiment was La0.67Ca0.33MnO3 (LCMO) at thicknesses of

200 and 400 nm on a strontium titanate (STO) substrate, chosen for its close lattice constant match and high quality factor.

The underlying physics of the CMR eﬀect is still not fully understood, but clearly it is rooted in a complicated interplay between electron charge, spin and orbital degrees 146

100 0T 1T 80

cm) 2T

W 60 Manganese Oxygen 4T 40 Tri,Di-valention

Resistivity(m 20

0 0 50 100 150 200 250 300 350 Temperature(K)

Fig. 5.1. Typical manganite unit cell (inset) and resistivity data for several magnetic fields illustrating the CMR effect. The ions used in this study are La and Ca with x = 0.33. [Adapted from [49]] 147 of freedom, external magnetic field, and lattice dynamics. Many standard quantum solid state theories ignore some of these aspects in order to study others, and are thus ill equipped to explain systems simultaneously sensitive to all these influences. Qualita- tively, the external field aligns the core spins at the manganese sites so that the valence electrons flow more easily from one Mn site to another via the oxygen site; this is the so-called double exchange process. The macroscopic effect is a reduction in the resistivity and increase in thermal conductivity of the material. A competing process is the Jahn-

Teller effect in which a Mn valence electron produces a local distortion in the oxygen octahedron around the Mn site. The distortion tends to localize the valence electron and thus increase resistivity. The Jahn-Teller contribution to the CMR effect brings about the sensitivity to lattice dynamics and strain in the crystal. A successful model for CMR would constitute a significant advance in solid state theory and may help explain other strongly correlated systems such as high Tc superconductivity. A good review article on the current state of manganite theory is given by Salamon and Jaime [37].

5.2 The CMR Experiment

The elastic constants of a material are sensitive to changes in lattice dynamics, structure, and magnetic configuration, making thin film RUS a suitable probe to study the CMR transition. Also, crystalline films on a substrate posses some residual strain due to lattice mismatch between the substrate and film. These strains have been shown to alter the transition temperature of the material. [25] [24] [9] The LCMO films were deposited by Qi Li’s group at the Pennsylvania State University on the 100 surface h i of a STO wafer by pulsed-laser deposition. An excimer laser laser of wavelength 248 148 nm, energy density of 2 J/cm2, and repetition rate of 5 Hz was used to vaporize the ∼ raw LCMO material in a 500 mTorr oxygen atmosphere. The substrate temperature during deposition was 780◦C. The film deposition rate was 0.04 nm/pulse; this rate was used to calculate the final film thicknesses used in the analysis of the RUS data. After deposition was complete, the substrate was cooled at a rate of 20◦C/min in 400 mTorr of oxygen. X-ray diffraction measurements showed that the films were epitaxially grown with the c axis perpendicular to the film plane. This deposition procedure has been shown to produce partially strained films at these thicknesses. [24] A RUS sample was then produced by cutting the wafers into parallelepipeds and polishing all sides except the one on which the film was deposited. Final sample dimensions were typically 1.0 × 0.9 0.3 mm with the film occupying one of the larger faces. After initial RUS data was × acquired for the 400 and 200 nm samples, the 400 nm sample was annealed in a quartz

3 tube at 900◦C with an oxygen flow of 50 standard cm /min for 24 hours to relax the strain in the film. The microscopic structure of as-deposited and annealed films have been well characterized. [9] After resonance data was obtained, the film is removed from the substrate by chemical etching; and the substrate resonance data was obtained.

A RUS experiment investigating phase transitions requires precise temperature control. A cryostat was designed and constructed by J. H. So, a schematic of which is shown in Fig. 5.2. The cryostat consists of a stainless steel outer vacuum jacket, and an copper inner jacket in which the RUS cell resides. The inner jacket is ﬁlled helium at 1 atm which provides the thermal contact between the sample and inner jacket wall.

Liquid nitrogen (LN2) is used as the heat sink, and all thermal contact between the

(LN2) and He exchange gas is made through copper contacts consisting of copper braids 149 clamped to the inner jacket, and the other end is attached to a copper plug with a tail protruding through a small hole in the outer jacket into the LN2. The hole is then sealed with solder. The heater is a brass disk wrapped with nichrome heating wire located as close to the sample as possible. Temperature monitoring is provided through a silicon diode sensor embedded in a small copper block located an equal distance from the sample as the heater. Temperature control was provided by a Lakeshore Model 330

(Autotuning) controller with stability of a few mK. The rest of the RUS electronics are the same as described in the previous chapters. The RUS cell is sealed with a vacuum

ﬁtting which usually requires a rubber O-ring. It was found that below about 225 K, the

O-ring became brittle and leaks resulted. The rubber O-ring was replaced by one made from extruded indium which seals well even at liquid He temperatures.

5.3 Analysis and Results

The crystal structure of LCMO is orthorhombic which may have Jahn-Teller dis- tortions to lower symmetry (described below). Determining all 9 elastic constants of an orthorhombic crystal with thin film RUS is possible in principle, however results may be spurious for a film occupying 1/1,000 of the sample. Fortunately, LCMO allows for a reasonable reduction of the 9 cij to the 2 “effectively isotropic” elastic constants c11 and c44 representing the longitudinal and shear moduli of the material respectively. The reasons this simplification is possible are: (a) the fundamental orthorhombic structure only deviates from cubic by 1-2%, (b) in LCMO films of thickness greater than 50 nm, orthogonal orientations occur with equal frequency and a domain size of about 100 nm,

temperature control

drive signal

LN2

silicondiode sensorin copperblock Hegas Ni/Crwire& brassdisk

sample

vacuumjacket

hose copper clamps braid copperplug

Fig. 5.2. RUS cryostat used in the CMR ﬁlm experiments. The internal RUS cell is ﬁlled with 1 atm of He exchange gas and liquid nitrogen is used as the heat sink. The temperature range is approximately 80 - 325 K. The inner jacket is copper and outer jacket is stainless steel. The gas handling system is omitted for clarity. 151 sample (100-500 µm), and (d) previous experiments of LCMO indicate close to isotropic behavior [9]. Similar assumptions have been made in RUS experiments on bulk manganites [74]. Also, what is of interest here is the temperature dependence of the elastic constants, so high precision rather than accuracy is required.

The temperature dependence of the substrate must be well known before film elastic constants may be extracted. There are no known phase transitions for STO in this temperature range, but of course the elastic constants have a general temperature dependence which over the temperature range of interest is essentially linear. The temperature derivatives of the elastic constants for STO are documented in the literature, but more refined measurements were carried out on the substrates after the film was removed. Six to eight resonances were tracked at temperature intervals of 1K from 245

- 285K. The resonances were chosen for their sensitivity to the film elastic constants of interest. The same resonances are used in both the fits to the film and bare substrate data. From the resonance data for the bare substrate, the three cubic elastic constants at each temperature were extracted, and linear fits performed to obtain their temperature derivatives.

The film resonances were fit with only the film constants c11 and c44 as adjustable parameters. The proper substrate elastic constants for each temperature were obtained from the temperature derivatives obtained previously. A point to consider is that the substrate will thermally expand with increases in temperature, so both the dimensions and density must also be adjusted at each temperature. The film frequencies were corrected by the bare substrate fit in the manner outlined in Chapter 3. In addition to the elastic constants, resistance data for the films was obtained by standard four 152 probe measurements in Qi Li’s lab. All measurements were made in the absence of any magnetic field except that of the earth.

The results for the 400 nm strained ﬁlm are shown in Fig. 5.3(a), unstrained

ﬁlm results are shown in Fig. 5.3(b). Strained 200 nm ﬁlm results are shown in Fig.

5.4. The peak in the resistance data of all samples corresponds well to either jumps or breaks in the slopes of a few percent in both the elastic constants at 262K for the 400nm unstrained, 256K for the strained film, and 252K for the 200nm strained film. However an unexpected feature in the elastic constants also occurs for all samples at a temperature of about 17K higher than the resistance peak, and no corresponding feature is seen in the resistance data at that temperature. Higher temperature features can be discerned in previous neutron scattering measurements [30][11], volume change measurements [30], and ultrasound attenuation measurements [10]; however these features were not discussed in the reports. A structural transition in La(1 x)CaxMnO3 has been discussed in the − literature at a temperature different than the magnetic transition for x 0.2, however ≤ no such transition is mentioned for x > 0.2. [17] For this study x = 0.33.

Strained films are known to have resistance peaks lower than their bulk counterparts which decrease with thinner films. This is consistent with our data. Resistance peaks and elastic constant features both decreased with strain (compare the 400 nm strained and unstrained data), and thickness (compare the 200 and 400 nm strained data). While the unstrained film exhibited no hysteretic behavior, there are some features in c11 of the strained films which did not repeat when the sample was thermally cycled. This may be due to irreversible changes in the microscopic structure of the film 153

Fig. 5.3. c11 (squares), c44 (diamonds), and resistance (circles) results for 400 nm LCMO (a) strained and (b) unstrained. The lines are piecewise linear ﬁts to the data. 154

Fig. 5.4. c11 (squares), c44 (diamonds), and resistance (circles) strained 200 nm LCMO. The lines are piecewise linear ﬁts to the data. 155 induced by thermal stress. [74] However, the transitions at the two temperatures re-

ﬂected in the elastic constants for both strained samples do repeat and track changes in the resistance peak.

The CMR eﬀect remains an important problem in the physics of strongly correlated systems, and hopefully the results presented here will prompt further investigations into the nature of the higher temperature transition. The most important results of this study are: (a) there are two features in the elastic constants, one at the resistance peak temperature and one 17K higher for all samples; and (b) no signiﬁcant hysteresis was ∼ observed. The RUS sample preparation and data acquisition for this project was carried out by J. H. So. 156

Chapter 6

Aluminum Oxide

6.1 Introduction and Previous Work

Aluminum oxide forms a clear rhombohedral crystal of very high hardness with chemical formula of Al2O3, and two formula units per unit cell. This material is also known as alumina, corundum, and sapphire. Alumina has several phases with the α phase being the only permanently stable form. α-alumina is a rhombohedral (or trigonal) crystal, and so is completely described by three equivalent unit cell vectors (ar=5.13A)˚ and the angle between any two (α = 55.28◦). However the hexagonal unit cell is often used with the three fold rotational symmetry axis assigned to c=12.99Aand˚ the two fold axis assigned to ah=4.754A.˚ The relationship between these systems is illustrated in

Fig. 6.4. Alumina is used widely in industry as an abrasive, protective surface coating, and in high temperature environments; it is a prominent constituent of the earth’s crust.

There have been a number of experimental and theoretical studies of alumina’s elastic properties by various methods. [23]-[31] Perhaps the most reliable experimental values for the elastic constants are those of Ohno and Anderson [53] who used an early form of

RUS yielding the results shown in Table 6.2. These measurements were also extended from room temperature to 1825 K in a subsequent paper. [21]

A trigonal crystal has 6 independent elastic constants: c11, c33, c44, c12, c13, c14.

The sign of c14 can be obtained by acoustic measurements only if the crystallographic 157 axes are rotated with respect to the body axes of the sample (e.g. crystal axes are not perpendicular to any face of a parallelepiped); c14 was originally reported by Wachtman et al. of the National Bureau of Standards (NBS and now NIST) to be negative. [31]

This is the only study we have found which is sensitive to the sign of c14. Subsequent papers have reported the magnitude but quoted the sign determined by Wachtman or thank the NBS for use of their samples or reorientation of their crystal. Recently the theorists Yvone LePage of the National Research Council of Canada and Paul Saxe of

Materials Design, Inc. have used ab initio methods to calculate elastic constants of many materials. [58][59][60] Their calculations on alumina show a c14 which is positive while correctly calculating the sign of c14 other trigonal crystals such as quartz and calcite.

[60] As only one previous, and rather dated, study was sensitive to the sign, it seemed worthwhile to perform the measurements again using RUS.

At ﬁrst glance, it may seem completely unreasonable for any elastic constant to be negative as it seems to imply a material would lengthen when exposed to a compressive stress. This would be the case for any diagonal element in the tensor (e.g. c11 or c33), however that is not necessarily the case for the oﬀ-diagonal elements. To better understand the sign of c14 , it is helpful to visualize the corresponding distortion. Consider a stress in the 11 direction, that is, one exerted on the x-face in the x direction. One can write Hooke’s Law to examine what sort of strains are associated with the c1123 (c14 in full notation). One will see that the body will have strains such as those pictured in Fig.

6.1 which may be called a “parallelogram distortion”. The sign of c14 simply dictates the orientation of the distortion (e.g. whether it is oriented as pictured, or rotated by 90◦ about the x-axis), so either positive or negative values would be physically reasonable. 158

Fig. 6.1. The “parallelogram” distortion governed by c14, the sign of which dictates the orientation of the distortion.

Also important to note is that the sign of c14 is dictated by the deﬁnition of the crystallographic axes. Before 1949, there were two commonly used conventions for reporting data on trigonal crystals, known as the “obverse” and “reverse” settings. After 1949, the obverse setting was chosen to be the standard by which all trigonal crystals are to be reported. [8] This standard was strictly adhered to in this experiment.

6.2 RUS for Rotated Trigonal Crystals

For this experiment, the RUS code had to be adapted to the symmetry of trigonal crystals which are rotated with respect to the sample axes. For higher symmetry cubic crystals, parity relationships exist which allow blocking of the large E and Γ matrices, providing signiﬁcant savings in computation time. Trigonal crystals however have suf-

ficiently low symmetry that the parity relationships are no longer valid, the matrices can no longer be put into block form, and computation time increases. The only other alterations necessary are the assignment of all 6 independent elastic constants with their tensor indices. For this work, the elastic tensor must be rotated with respect to the 159 sample body geometry. This is accomplished by applying an Euler angle rotation transformation to the elastic tensor to generate an effective elastic tensor for a particular orientation. There can be some ambiguity in the definition of Euler angles; we take the convention of Arfken [5] and group theory where the rotation angles are taken in the order:

1. rotation of α about the Z-axis

2. rotation of β about the new Y -axis

3. rotation of γ about the new Z-axis where the positive sense is given by the right hand rule for all angles. These rotations are illustrated in Figure 6.2 for a sample measured in this work. Throughout this chapter, these angles will be expressed as (α, β, γ). The full rotation matrix is shown in (6.1).

cos γ cos β cos α sin γ sin α cos γ cos β sin α + sin γ cos α cos γ sin β  − −  R = (6.1)  sin γ cos β cos α cos γ sin α sin γ cos β sin α + cos γ cos α sin γ sin β   − − −       sin β cos α sin β sin α cos β     

The elastic constant tensor cijkl must be deﬁned relative to some coordinate system. The IRE standards of the obverse trigonal setting require that the elastic tensor be reported in a coordinate system such that the +X direction lie along the ah axis

(axis of 2 fold rotational symmetry) and +Z direction lie along the c axis (axis of 3 fold rotational symmetry), and the +Y direction be that of a right-handed coordinate system. [8] This standard is also employed in the Landolt-Bornstein Tables. [16] If 160 the above (X, Y, Z) coordinate system does not coincide with the parallelepiped sample, then it must be rotated via the Euler matrix to create an eﬀective elastic tensor, dpqrs, which does: dpqrs = ijkl RpiRqjRrkRslcijkl. Thus a set of euler angles which P rotates the crystallographic coordinate system (X, Y, Z) into the sample coordinate system (X0, Y 0, Z0) must be found. Appendix B shows the portion of the FORTRAN code which performs this transformation of the elastic tensor. This code was be tested by performing a rotation of the tensor by 90◦ about the Z axis, and checking that the unrotated +x-axis now points in the +y direction, and that the unrotated +y-axis no points in the -x direction.

The RUS code can be used to generate spectra of hypothetical samples and study the interplay between the orientation angles and sign of c14 . The angle the c-axis makes with the normal to the polished face will determine β. The ah-axis lies in a plane perpendicular to the c-axis and its orientation will define α and γ. With these definitions in mind, we ran various forward calculations and found that the criteria for c to 14 generate different resonances is that both ah and c must be off the normal to any face of the final sample. An orientation given by (90,30,0) maximized the sensitivity to the sign of c with the differences in the spectra much larger ( 1.5%) than the typical 14 ∼ uncertainty in a RUS calculation of a high Q resonator ( 0.1%). Another important ∼ result is that if the positive sense of the angles is reversed, the spectra for the c are 14 also exactly exchanged. This is equivalent to using the reverse rather than the obverse trigonal setting, and underscores the importance of keeping careful track of the crystal axes during the polishing procedure. 161

Fig. 6.2. Illustration of an Euler angle transformation for alumina sample 2 (see Figure 6.5). The primed coordinate system is oriented to the polished sample geometry, and unprimed system is oriented to the hexagonal lattice vectors. The rotations are taken in the order shown from (b) to (d). 162

reference scratch

crystal filmback

x-ray 3.0 cm beam

Fig. 6.3. Orientation of the crystal in the x-ray apparatus as viewed from behind the film. The large black circles show the set of highest intensity diffraction spots on the x-ray film which were used to orient the ar-axes as shown in Figure 6.4(a).

6.3 Orienting and Polishing

The bulk alumina crystal was supplied by the Goodfellow Corp. [20] as 10x10x1 mm sheets with the large faces polished and oriented in the < 0001 > 1 direction. A ◦ diamond pen was used to place a reference scratch in a corner of the crystal to facilitate proper cutting of the bulk crystal into RUS samples. The Laue X-ray back-reflection technique was used to orient the ah-axes. Figure 6.3 shows a schematic of the crystal and film in the x-ray apparatus. The set of the brightest 3-fold symmetric diffraction spots (shown in the figure) then define the ar-axes in the obverse setting of a trigonal crystal. One of the axes forms a slight (4◦) angle with the bulk crystal edge. Figure

6.4a shows a view of the rhombohedral unit cell looking in the Z,-c direction with the − diﬀraction spots and ah axis superimposed. Had the reverse setting been used, the solid

(rather than dashed) lines would deﬁne the ar-axes which would reverse the orientation 163 of the elastic X and Y axes, and the sign of c14 . All samples in the present work are analyzed carefully adhering to the obverse setting standard.

The two samples were cut with the wire saw from the crystal as shown in Figure

6.5. The bulk crystal was fixed to a rotating stage on the saw so that the orientation could be controlled. Once the parallelepipeds were cut, they were placed on a polishing shim cut at a 30◦ so that the c axis formed a 30◦ angle with the normal to the newly polished face. The sample was flipped to polish a parallel face, and then mounted on the right-angle blocks discussed in Chapter 3 to complete the final parallelepiped. Between each step, diagrams were made to keep track of the lattice vectors. This is very important since the dimensions assigned to the (X0, Y 0, Z0) coordinates for a non-cubic crystal are not arbitrary. The Euler angles shown are accurate to 1 . ◦

The ﬁnal sample dimensions in the coordinate system (X0, Y 0, Z0) and in units of microns are: (728.7, 1068.1, 830.9) for sample 1, and (764.7, 1032.2, 810.2) for sample 2.

The Euler angles were obtained by rotating the crystal coordinate system (X, Y, Z) into the sample coordinate system (X0, Y 0, Z0). This was facilitated by fashioning foam blocks with aspect ratios similar to the samples and sticks pointing out of them representing the elastic coordinate system. Then a right handed triad coinciding with the elastic coordinates could then be rotated through a set of Euler angles until it aligns with the sample coordinate system. This process also produces the correct assignment of dimensions to the (X0, Y 0, Z0) coordinates, and is illustrated in Figure 6.2. 164

c,Z

2/3 2/3 2/3 2/3

a a 1/3 r Y r 1/3 1/3 1/3 a c,Z X,a r a h 1/3 r a r 2/3 2/3 a r X−ray 1/3 Diffraction Y Spot

(a) (b) X,a h

Fig. 6.4. Plan view (X, Y plane) (a) and perspective view (b) of the rhombohedral unit cell for alumina. The c-axis is pointing out of the paper and ar-axes adhere to the obverse setting in (a). The IRE standards are used to deﬁne the X, Y, Z elastic coordinate system relative to the hexagonal lattice vectors (ah and c). 165

Sample 1 c,Z Z' a cut lines reference r a r scratch X' l a t

c s

y X r Y' Y C

k l

u Polishing Y B Shim X,a h (a) (α,β,γ) = (8,−30,180 or 0) (b)

a r Sample 2 c Z' reference scratch a Y X' r a r Y X,Y'

c X,a Polishing h Shim (c) (d) (α,β,γ) ≈ (−86,30,0)

4º a r

Fig. 6.5. Cutting and polishing details for the 2 alumina samples. The hatched region shows the ﬁnal polished samples viewed along the Y 0 axis. 166

6.4 Analysis and Results

The ﬁeld of crystallography is wrought with somewhat arbitrary standards and deﬁnitions, but the positions of atoms in the crystal lattice are dictated strictly by nature.

It is quite useful to visualize the 3-dimensional arrangement of atoms in the unit cell of alumina along with the Laue back reflection pattern produced by that arrangement in order to understand the real x-ray photograph. A FORTRAN program was written to calculate an x-ray pattern produced by an arrangement of atoms. (see Appendix B) A set of (X, Y ) coordinates of constructive interference spots on the film is produced along with intensities. The program was developed based on standard x-ray diffraction theory

[43] with atomic position data taken from the NRL web site [?]. The output from the program can then be imported into a CAD program allowing rotations of the unit cell and diﬀraction pattern until the screen pattern matches the x-ray photograph. Then the arrangement of atoms in the bulk crystal can be determined unambiguously. These results were used to orient the lattice vectors as shown in Figure 6.5 during cutting and polishing. The agreement with calculated and experimental x-ray data was excellent.

After resonance data was obtained for the two samples, it was compared with theoretical spectra calculated by the RUS code with 9th order polynomials. The elastic constants used are those of Ohno and Anderson [21]. Spectra were calculated using both positive and negative values for c14 and then compared with observed spectra. The results for sample 2 are listed in Table 6.1; sample 1 had comparable results. Both samples showed clearly better ﬁts to the data for positive values of c14 . Higher order polynomials did not improve the ﬁts. 167

For typical RUS measurements, the input data would be adjusted until a least- squares ﬁt between measured and calculated spectra is achieved. The RMS errors after such a ﬁtting session is typically less than 0.1%. Thus it should be noted that the RMS errors reported in Table 6.1 are for “forward” calculations based on measured dimensions and Euler angles with no adjustable parameters.

While the magnitudes of the alumina elastic constants are not the primary objective of this work, inverse calculations were performed on the data to check if the error in fits would decrease to more typical RUS levels ( 0.1%). For the rotated trigonal ∼ crystals, there are 12 total parameters which could be adjusted to improve the fits: 6 cij, 3 dimensions, and 3 Euler angles. Given the number of frequencies obtained and the possible degeneracies in fits, not all parameters were allowed to vary at the same time.

The magnitudes of the cij’s from Ohno and Anderson were used as starting points with c14 positive. For the first run, only dimensions and angles were allowed to vary. These parameters were then fixed at the fitted values and the elastic constants were adjusted for the second run. The results for the two samples are shown in Table 6.2 along with

RMS ﬁt errors. The diagonal elastic constants (c11, c33, c44) agree with Ohno and An- derson to within 0.5%, however our “oﬀ diagonal” constants (c12, c13, c14) deviate by

2%. These constants are notoriously difficult to measure accurately and thus typically ∼ vary more between studies. It should be noted that had our primary objective been to accurately measure the elastic constant magnitudes of alumina, many more resonance peaks would have been obtained to improve our confidence in the fitted values. The above fits were performed simply to check if “correcting” for errors in dimension and angle measurements improved the agreement between calculated and observed spectra. 168

Alumina Resonance Frequencies (MHz) Measured Calc. (+c14) % Error Calc. ( c14) % Error 2.707688 2.705586 0.078 2.749469− -1.520 3.672555 3.680767 -0.223 3.649729 0.625 3.835520 3.836131 -0.016 3.867467 -0.826 3.975381 3.977453 -0.052 3.914977 1.543 4.169249 4.155786 0.324 4.122196 1.141 4.797780 4.774484 0.488 4.627694 3.675 4.942362 4.929546 0.260 4.947461 -0.103 5.086565 5.097846 -0.221 5.045944 0.805 5.161420 5.208598 -0.906 5.139197 0.432 5.305400 5.310701 -0.100 5.435047 -2.385 5.323344 5.347387 -0.450 5.475603 -2.781 5.501625 5.512956 -0.206 5.479522 0.403 5.561940 5.576348 -0.258 5.684228 -2.151 5.739862 5.734950 0.086 5.957796 -3.658 5.866725 5.842906 0.408 6.153921 -4.667 6.370460 6.370622 -0.003 6.460665 -1.396 6.488406 6.462084 0.407 6.486284 0.033 RMS % Error= 0.29 2.44 Table 6.1. Comparison of measured resonance frequencies of alumina sample 2 with those generated using +/- c14 in the forward calculation. The sample parameters were: 3 (E1,E2,E3)=(764.7,1032.2,810.2)µm, (α, β, γ)=(-86,30,0), ρ=3.986 g/cm . The elastic constants (in Mbar) were those of Ohno and Anderson: c11 = 4.973, c33 = 5.009, c44 = 1.468, c = 1.628, c = 1.160, c = 0.219 12 13 14

cij (Mbar) Anderson This Study Diﬀerence (%) Theory c11 4.973 4.965 0.2 4.950 c33 5.009 5.034 1.0 4.860 c44 1.468 1.462 0.4 1.480 c12 1.628 1.593 2.2 1.710 c13 1.160 1.190 2.5 1.300 c14 -0.219 0.226 3.0 0.200 RMS Fit Error (%) 0.115 0.072 Table 6.2. Results for ﬁtting the alumina elastic constants to our data. The theory results are those of Lepage and Saxe [60] 169

Chapter 7

Elasticity of Human Dentin

7.1 Introduction

While resonant acoustics has been utilized to study many classes of materials such as single crystals, heterostructures, geologic samples, moon rocks, and even the oscillations of the earth itself; there have been no applications to biological samples.

Such materials rigid enough for a free standing resonance experiment include the various mineralized tissues such as bone and teeth. Knowledge of the mechanical properties of these materials is important in the ﬁelds of kinesiology and bioengineering. The tissue studied in these experiments is human dentin, which forms the central core of the tooth

(see Fig. 7.1) and occupies most of its volume.

Other measurement techniques such as static loads, pulse-echo acoustics, and microindentation have been applied to dentin; however, such basic measurements as

Young’s modulus have resulted in 3 fold discrepancies just in the past 3-4 years. [6][3][27]

Since dentin forms the central core of the human tooth, only small contiguous samples ( ∼ 2-4 mm3) can be obtained. The diﬃculty in reliable measurements is often attributed to the small sample size which makes uniform stress states diﬃcult in static measurements, and the high frequencies required for pulse echo measurements mean high strain rates.

However, such sample sizes are ideally suited to our implementation of RUS; and because the frequencies are much lower for low order resonances, so are the strain rates. The 170 goals for this experiment were (i) to check the applicability of RUS to biological samples,

(ii) investigate the eﬀect of moisture content in dentin, and (iii) study possible elastic anisotropies in the material.

enamel

dentin

root

Fig. 7.1. Anatomy of a human tooth. The hard outer enamel allows for cutting of tough materials while the more compliant inner dentin resists cracking from sudden shocks.

The basic structure of a human tooth is similar to that of a golf ball: a rigid outer shell (the enamel) surrounding a more compliant core (the dentin). This structure gives the tooth enough rigidity to cut through tough materials without easily cracking.

The dentin itself has a great deal of internal structure as well. It is primarily composed of a matrix of collagen ﬁbers lying preferentially in a plane creating a felt-like layered structure. The matrix is perforated by an array of roughly parallel tubules approximately

1 µm in diameter, oriented perpendicularly to the collagen plane, and running from the pulp chamber (near the roots) to the outer enamel. The tubules are surrounded by a region of increased mineral content, the so called hypermineralized region, about 0.5 µm 171 thick known as the peritubular cuff (as in Fig. 7.2). The purpose of the tubules is to carry nutrients to the enamel so that it may be repaired. Much of the fine structure of dentin has been studied using atomic force microscopes, as seen in Fig. 7.3. [69] The tubule array provides a large surface area through which fluids may be readily absorbed by the collagen matrix, thereby changing the mechanical properties of the material.

7.2 The Eﬀects of Moisture Content

An important question from a bioengineering point of view is how much moisture content affects the mechanical properties of dentin. In the “natural” state (in-vivo), dentin would be filled with fluids from the body and thus may behave quite differently than a sample (partially) dried in laboratory air for weeks. Several dentin samples were obtained from John Kinney’s group at the University of California, San Francisco. The samples were cut from healthy third molars into approximately 2 2 2 mm cubes with a × × low speed diamond saw; the tubules were oriented parallel to one axis. Once received in this lab, the samples were polished in the standard way except that the smallest diamond grit size was 3 rather than 0.5 µm. It was anticipated that the quality factors of dentin would be quite low compared with typical single crystals studied in this lab, so some attention was paid to the dimensions. First, cube like geometries result in many closely spaced peaks which, if wide (low Q), would strongly overlap making center frequency determination more difficult. Secondly, a recent RUS study on detecting anisotropy in low Q samples showed that a parallelepiped geometry with aspect ratios (z over x) of approximately 2:1 produced the highest sensitivity to anisotropy. [72] With these issues in mind, the samples were polished to typical dimensions of 0.8 1.1 1.7 mm. × × 172

tubule ~1mm

peritubular cuff

x y

Fig. 7.2. Structure of human dentin. The cylindrical and roughly parallel tubules are about 1 µm in diameter surrounded by a hypermineralized cuﬀ. The collagen matrix layers lie preferentially in the x y plane. Their purpose is to carry nutrients from the − pulp chamber to the enamel for repair. 173

peritubular 5 mm dentin

collagen fiber matrix b

Fig. 7.3. AFM images showing the internal structure of dentin. (a) Several tubule openings are shown with diameters 1µm and separated by 5-8µm. (b) Dentin is ∼ composed of a matrix of collagen ﬁbers lying preferentially in a plane perpendicular to the tubule axis. [Adapted from [69]] 174

Before measurement, the dentin samples were pretreated in 3 ways: (a) soaked in distilled water for 48 hours, (b) soaked in a calcium phosphate remineralizing solution

(Hank’s balanced salt solution), and (c) the Hank’s sample was then soaked in distilled water for 48 hours after drying under vacuum. After each pretreatment, the sample was loaded into the RUS cell and data acquired at 1 atmosphere. The cell was then evacuated by mechanical pumping and resonances monitored over about 30 hours. Typical resonances are shown in Fig. 7.4 for 0 to 28 hours. What can be observed is that the peaks shifted to higher frequencies with drying indicating a general stiffening of the material, and/or a decrease in the mass density. A question arises as to whether the increase in frequencies could be solely caused by the decrease in the mass of the sample due to the evacuation of the water. If the elastic constants are unchanged, the ratio of the squares of the frequencies of wet and dry dentin should be proportional to the fractional mass change between the two states. The mass of the sample (3.2 0.1 mg) did not change enough to be measured by our scale even after 39 hours of soaking in water. However, we may estimate the mass change by calculating the volume fraction occupied by the tubule array and making the assumption that all of the water contained in the sample is confined to this volume. The tubules have a diameter of about 1 µm, and are typically separated by about 5 µm. The volume fraction (Vtube/Vsample) is then about 3%. The frequency ratio for the modes shown Fig. 7.4, which are typical, is 1 f 2 /f2 0.10. − dry wet ' Since for dentin ρ = 2.1g/cm3 and for water ρ = 1.0g/cm3, we may calculate the mass ratio m /m 0.015. Thus, the 10% rise in the square of frequency can not water sample ' be accounted for by a mass change of only 1.5%; and we conclude that the increase in frequencies is primarily due to a stiffening of the material. 175

The quality factors increased somewhat, ranging typically from 30 (wet) to 60

(dry). Indeed, these Qs are quite low compared to typical single crystal resonators with quality factors > 1, 000 and even with the precautions taken during preparing the sample geometry, higher modes did overlap. It was not uncommon in this experiment to have to fit 4 peaks simultaneously. An isotropic model was used to fit the data and the first order elastic parameters Young’s modulus (E) and Poisson’s ratio (σ) were calculated from the c11 and c12 values:

(c c )(c + 2c ) c E = 11 − 12 11 12 and σ = 12 . (7.1) c11 + c12 c11 + c12

28hrs

23hrs

3hrs

0.25hrs

0hrs

0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Frequency(MHz)

Fig. 7.4. Typical resonances for the ﬁrst few modes of a dentin sample over pumping times from 0 (in air) to 28 hours. 176

The results are shown in Fig. 7.5. The solid curves are exponential ﬁts of the form A(t) = A Cexp[ t/τ]. As expected from nanoindentation experiments [38] dry − − and raw resonance data, E increased with drying for all pretreatments with limiting

(dry) values increasing in the order of: water only Hank’s followed by water → → Hank’s only. Our interpretation is that the Hank’s balanced salt solution infuses into the material through the tubules and increases the mineral content. As moisture is extracted into the vacuum, the minerals crystalize and thus stiffen the dentin. The minerals are however water soluble, so when the sample is again exposed to water, some of the mineral content is lost and the stiffness falls somewhere between the pure water sample and pure Hank’s sample. Poisson’s ratio showed more interesting behavior in that the different treatments produced qualitatively different responses. The samples exposed to water (water only and Hank’s followed by water) behaved almost identically:

σ decreased with drying for both pretreatments. However the sample exposed only to the Hank’s remineralizing solution showed a significant increase in σ. Recall that poisson’s ratio is defined as transverse compression to the longitudinal extension due to an longitudinal stress and is thus strongly dependent on c12. The data may suggest that re-exposing the hypermineralized (Hank’s only) sample to water preferentially dissolves the excess mineral content in such a way as to remove any effect on poisson’s ratio, but still influence Young’s modulus; perhaps by removing excess minerals within the tubules, but leaving them in the collagen matrix. A more rigorous explanation of these results could be the subject of future research.

The time constants for drying have little meaning in their absolute values as they are sensitive to the details of the pump and tubing between sample and pump, as well 177

30 (a) 29

sModulus(GPa)

’

oung 26

(b) 0.34

0.32

sRatio

’ 0.30

Poisson 0.28

Hanks>Water 0.26 HanksOnly WaterOnly 0.24 0 2 4 6 8 10 12 14 16 18 20

PumpingTime(hours)

Fig. 7.5. Pumping results for (a) Young’s modulus and (b) Poisson’s ratio of dentin pretreated by water only (squares), Hank’s solution only (+), and Hank’s followed by water (diamond) assuming isotropic symmetry. Solid curves represent exponential ﬁts. 178 as surface area to volume ratios of the sample; however their relative values may be useful as all conditions were the same for each experiment. The results are listed in

Table 7.1. For both elastic parameters, the Hank’s-Water sample dries most quickly

(lowest time constant) while the hypermineralized dentin dries most slowly. The latter observation suggests that the excess minerals crystalize near the surface ﬁrst and impede the evacuation of the water from the interior. The fact that the sample exposed to Hank’s solution and then water dries the most quickly may be an indication that the mineral deposits create small deformations in the material which leave voids after being dissolved in water. This opens up more pathways for the water to escape and allows the sample to dry more quickly. Obviously these are “hand waving” arguments to qualitatively explain the data and more study would be needed to put these conjectures on ﬁrm footing.

Time constants (τ) for drying dentin E Ratio σ Ratio Water only 1.70 1.45 0.662 1.06 Hanks only 2.27 1.88 2.04 3.28 Hank’s Water 1.17 — 0.621 — → Table 7.1. Time constants (in hours) for Young’s modulus (E) and Poisson’s ration (σ) for drying dentin pretreated by three diﬀerent procedures described in the text. The ratios are the maximum τ over the minimum.

The signiﬁcant accomplishment of these experiments is that human dentin can indeed be probed with RUS and that Young’s modulus varies between 29.5 and 24.4 GPa depending on water and mineral content. These values fall within the range historically 179 reported in the literature (19-29 GPa) [6][3][27], and we believe are the most accurate determinations to date for human dentin.

7.3 Anisotropy of Dentin

The pronounce structural anisotropy of dentin would strongly suggest a corresponding elastic anisotropy. While such anisotropies in the mechanical properties of dentin have been previously suggested [27], no reliable measurements have been produced. The most reasonable elastic symmetry is hexagonal with the isotropic planes oriented with the layers of collagen ﬁbers and c axis aligned with the perforating tubules.

To test this anisotropy, the resonance spectra for the water soaked dentin for wet (in air) and dry (in vacuum for 28 hours) was ﬁt with both isotropic and hexagonal models.

The independent elastic constants for isotropic symmetry are c11 and c12 with the additional independent constants of c33, c44, and c13 for hexagonal. The isotropic condition requires that c33 = c11, c13=c12, and c44 = 1/2(c11-c12). The results of these ﬁts are summarized in Table 7.2 along with corresponding ﬁrst order elastic parameters. The results show that dry dentin is highly isotropic with anisotropy factors of only 1-2%.

However wet dentin shows marked anisotropy with c11 being larger than c33 by about

23%. Here the 1,2 directions are in the collagen (xy) plane and the 3 direction (z) aligns with the tubules. Likewise c12 is larger than c13 by about 29% producing a large anisotropy in Poisson’s ratio.

That the stiﬀest direction in dentin is in the plane of the mineralized collagen

ﬁbers rather than the hypermineralized tubules seems somewhat surprising. However, micromechanical arguments have been suggested that the intertubular matrix is in fact 180

Condition Symmetry c11 c33 c12 c13 c44 σ E G dry isotropic 36.5 — 14.5 — — .28 28.3 11.0 dry hexagonal 36.7 36.5 14.7 15.1 11.1 —– —– —– wet isotropic 66.11 —– 49.01 —– —– .43 24.4 8.6 wet hexagonal 42.6 34.6 25.4 19.7 9.4 —– —– —– Gilmore Isotropic 19-29 7-11 Table 7.2. Results for dry and wet human dentin as determined for both isotropic and hexagonal symmetry models. Units are GPa with the exception of the Poisson’s ratio (σ), which is dimensionless. E is the Young’s modulus and G is the shear modulus. For hexagonal symmetry, the Poisson’s ratio, and E and G depend on the direction. The (11) direction is perpendicular to the tubule axis, and the (33) direction is along the tubule axis. For comparison purposes, we include the range of values from Gilmore et al. based on an isotropic ﬁt to sound speed measurements in bovine dentin. [65]

the dominant contributor to dentin’s elasticity and that the tubules play only a secondary role. [28] This would be consistent with our data and suggest the picture that felt-like collagen planes are strongly bound but the interplanar binding is weaker; similar to elasticity in graphite. It is difficult to explain why this anisotropy disappears when the fluid within the matrix is extracted. However, that too is consistent with previous contact stiffness [28] and microhardness [66] studies which detected no anisotropy in dry dentin samples. As noted previously, live dentin is not dry, but filled with fluids which carry minerals to replenish and repair damage to the enamel and interior dentin. Thus it is likely that live dentin also exhibits such anisotropy. This would be an important consideration for bioengineers designing synthetic teeth. 181

Chapter 8

Conclusions

In this thesis, a new experimental method for probing mechanical properties of thin ﬁlms deposited on a substrate has been described. The technique is an extension of the well established acoustic method, resonant ultrasound spectroscopy, for determining the elastic tensor of a crystalline material. The extension required mathematical tools to calculate, given all relevant material parameters, the normal modes of vibration of a heterogeneous parallelepiped. A numerical solution was obtained using the Rayleigh-

Ritz method. To be useful as an experimental technique, this problem must be inverted so that the elastic constants can be obtained from a set of measured frequencies. Since depositing a ﬁlm on the substrate while mounted in the RUS cell is impossible, it is important to understand the uncertainties introduced by remounting the sample. A statistical study showed these measurements to have an uncertainty on the order of 100 ppm for samples repeatedly mounted and measured. This is to be distinguished from the precision of a frequency measurement which is on the order of 2-3 ppm. Given the frequency resolution for repeatedly mounted samples, calculations were performed to estimate the minimum thickness of the ﬁlm which could be resolved. It was found that

ﬁlms as thin as 1/1,000 the substrate thickness could be reliably probed.

Performing a RUS experiment on a thin film system, rather than a bulk sample, requires additional steps in the procedure and analysis. The substrate on which the film 182 is deposited must be very accurately characterized. Specifically, the substrate elastic constants, dimensions, and crystal orientation must be well known before the film elastic constants can be determined. With the addition of the film, the natural frequencies of the system will be perturbed. Since the expected frequency shifts due to a film occupying 0.1% of the system are on the same order of magnitude as the typical errors between calculated and measured frequencies for the substrate, a scheme for correcting the measured film/substrate frequencies was developed. This ensures the film elastic constants are adjusted to fit the frequency shifts due to the film, rather than the residual errors.

Several important and topical systems were investigated using thin film RUS, including mats of carbon nanotubes and colossal magnetoresistance (CMR) films. Carbon nanotubes have been shown to possess an extremely high axial Young’s modulus and elastic limit. Such remarkable mechanical properties suggest novel effects on a resonating system. It was found that the most interesting of these effects is on the quality factor of the resonator, which is a measure of attenuation of acoustical energy in the system. For a wide range of substrate materials, including strontium titanate, silicon, alumina, brass, and quartz, the quality factors increased by factors of up to 6.0 with the addition of the nanotubes. The effect was also found to have some film thickness dependence, peaking between 400 and 600 µm for most of the substrates. This unique effect is expected to have applications in the field of micro electro-mechanical systems

(MEMS) devices.

The phenomenon of CMR, a strong dependence of the electrical resistance on an applied magnetic field, has generated a great deal of interest in the physics community 183 because it represents a highly correlated system, in which dynamics due to electron spin and charge, and crystal phonons are strongly coupled. Such materials fashioned in a thin film also have applications in the data storage industry. Thin film RUS was applied to CMR films of 200 and 400 nm on a strontium titanate substrate. The elastic constants were calculated over a temperature range in which the CMR transition occurs

(245 - 285 K). A shift in the elastic constants for both samples was observed at the electrical resistance peak temperature, however another previously unreported transition was observed about 17K higher in temperature. The nature of this new transition is not known, but it is hoped to be an additional step toward a full understanding of this important phenomenon.

Several traditional RUS experiments were performed on aluminum oxide, and the biological material, human dentin. Aluminum oxide is a very hard material with trigonal crystal symmetry. This symmetry results in an independent elastic constant c14, which has been reported in the literature as being negative since its first careful measurement in 1960 by Wachtman et. al. [31] The sign of this elastic constant depends on the assignment of crystallographic axes with respect to the atoms in the crystal, definitively set in 1949 for trigonal crystals. [8] Recent ab initio theoretical calculations have predicted the sign of this elastic constant to be positive. The objective of this work was to experimentally check these results. After very careful analysis of several samples, it was found that the sign of c14 is, in fact, positive. While correcting an elastic constant of a technologically important material is a worthwhile endeavor, the real significance of this result is that it removes the last known discrepancy between ab initio calculations of elastic constants and experimental observations. 184

Dentin is a calcified tissue comprising the bulk of the human tooth. It has a unique structure in which a matrix of collagen fibers lie preferentially in a plane, and an array of cylindrical hollow tubules run perpendicular to the collagen matrix. Reli- able measurements of elastic properties of dentin have been hampered due to the small contiguous sample sizes available; however, such sizes are quite amenable to the small sample RUS method. This is the first known application of RUS to biological materials.

The first experiment measured the effect of moisture and mineral content on the elastic properties. It was found that dentin had a Young’s modulus of 26.5 GPa when saturated with water, and 28.0 GPa when dried in a vacuum. Soaking dentin in a remineralizing solution instead of water caused the Young’s modulus to increase to about 29.3 GPa upon drying. A separate experiment investigated the elastic symmetry of dentin. Water soaked and dried dentin was modelled using both isotropic and planar isotropic (hexagonal) symmetries, with the hexagonal c axis aligned with the tubule axis. A strong dependence on moisture content was observed; dry dentin was well described by the isotropic model, with anisotropy factors of 1-2%, however wet dentin showed anisotropy factors on the order of 25-30%. This is important because the tubules in live dentin are filled with fluids carrying nutrients to the outer enamel, while many of the previous experiments are conducted on dentin dried in laboratory air.

In conclusion, the ability of RUS to quantitatively characterize thin ﬁlm systems has been established. The elastic constants of a material are sensitive probes into the atomic environment, and thus, it is expected that thin ﬁlm RUS will become a useful tool to explore a wide variety of systems of importance to both fundamental and applied physics. 185

Appendix A

Listing of RUS programs

This appendix lists several computer programs used in RUS experiments. The programs along with a brief description are

FILMMRQROT.F: a FORTRAN program used to calculate the elastic constants • of a thin ﬁlm deposited on a substrate. The ﬁlm and substrate crystal axes may

be arbitrarily rotated with respect to the sample axes. The “forward” problem of

calculating the eigenvalues and vectors of the sample is included as a subroutine

in this program.

XYZMA.F: a FORTRAN program which reads in the eigenvectors from the • ’XYZBLK.F’ program (or its variants) and writes a series of ASCII ﬁles containing

surface meshpoints of the normal modes of oscillation of a parallelpiped at various

points during a half period along with the magnitude of the displacement of the

point from its equilibrium position (used for shading). These ﬁles are to be read

in and rendered by Mathematica using the ’MODE DISP.NB’ script.

MODE DISP.NB: a Mathematica script for reading in mode data produced by • the ’XYZMA.F’ program and graphically rendering each mode. An animation of

each mode can also be produced. 186

PEAKFIT.NB: a Mathematica script for ﬁtting Lorentzian line shapes using the • Levenberg-Marquart algorithm. Such numerical methods work reasonably well for

a single peak, however for closely spaced and overlapping peaks a visual scheme in

which parameters are manually adjusted while data and ﬁt curve are observed on

the computer works much better.

A.1 Inverse RUS program for a substrate with a ﬁlm

Below is listed the FORTRAN code for determining the elastic constants of a cubic

film deposited on a cubic substrate using the xyz film code for the forward calculation and a Levenberg-Marquardt algorithm to fit the frequency data by adjusting the film elastic constants. It reads in the resonance spectrum data and other input parameters from mrqin.dat which has the format

Descriptive header C11, C12, C44, E1, E2, E3 D11, D12, D44, E4

RHO, RHOP, NN, NMDS F_1 F_2 F_3 ... F_NMDS

where Cxx are the fixed elastic constants of the substrate, Dxx are the initial guesses for the film constants, E1-3 are the edge lengths of the parallelpiped, E4 is the film thickness, RHO and RHOP are the densities of the substrate and film respectively, NN is the maximum order of the polynomials in the Visscher basis set, and NMDS is the number 187 of frequencies measured. A rotation option is included in case the crystallographic axes are not aligned with the sample edges, however an assumption is made that the film and substrate crystal axes are aligned.

C**************************************************************** C PROGRAM FILMMRQROT C C*************************************************************** IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) C PARAMETER (MMAX=20,MDAT=60) C C INTEGER*2 JJ1, JJ2 C REAL*4 FREAD C INTEGER IA(MMAX) REAL ALAMDA,SUMSQ,A(MMAX),ALPHA(MMAX,MMAX),COVAR(MMAX,MMAX) REAL SIG(MDAT),X(MDAT),Y(MDAT) REAL*8 RAD,ALPH,BET,GAM,ROT(3,3) C DIMENSION WSAV(MDAT),WTMP(MDAT),FDATA(MDAT) CHARACTER HEADER*50, HEADA(50)*1, USED(MDAT)*3 EXTERNAL FREQS C COMMON /SUBSTR/ E1,E2,E3,C11,C12,C44,DRHO C COMMON /RHO/ RHO,TCORR COMMON /NN/ NN COMMON /NMDS/ NMDS COMMON /NDATA/ NDATA COMMON /IA/ IA COMMON /X/ X COMMON /WSAV/ WSAV,WTMP COMMON /SIG/ SIG COMMON /ROT/ ROT C EQUIVALENCE (HEADER, HEADA(1)) C TOL=1.0E-4 188

WRITE(*,’(A)’) ’ Enter ALPHA, BETA, GAMMA’ READ(*,*) ALPH0, BET0, GAM0

C OPEN(3,FILE=’mrqin.dat’,STATUS=’OLD’,ERR=101) GO TO 102 101 WRITE(*,’(A)’) ’ Error opening mrqin.dat.’ GO TO 9900 C 102 READ(3,’(A50)’) HEADER READ(3,*) C11,C12,C44,E1,E2,E3 READ(3,*) D11,D12,D44,E4 READ(3,*) READ(3,*) RHO, RHOP, NN, NMDS NMDS=0 103 NMDS=NMDS+1 READ(3,*,END=104,ERR=101) FDATA(NMDS) IF(NMDS.LT.51) GO TO 103 104 NMDS=NMDS-1 IF(NMDS.LE.0) GO TO 9000 CLOSE(3,ERR=9900) C

C C Parameter setup C CALL INIT DRHO=(RHOP-RHO)/RHO C D44=(D11-D12)/2.0 C C Here are the settings for substrate with film: C MA=7 A(1)=D11 A(2)=D12 A(3)=D44 A(4)=E4 A(5)=C11 A(6)=C12 A(7)=C44 IA(1)=1 IA(2)=1 IA(3)=1 IA(4)=0 IA(5)=0 189

IA(6)=0 IA(7)=0 C RAD=TWOPI/360. ALPH=ALPH0*RAD BET=BET0*RAD GAM=GAM0*RAD C ROT(1,1)=DCOS(ALPH)*DCOS(BET)*DCOS(GAM)-DSIN(BET)*DSIN(GAM) ROT(1,2)=DCOS(ALPH)*DSIN(BET)*DCOS(GAM)+DCOS(BET)*DSIN(GAM) ROT(1,3)=-DSIN(ALPH)*DCOS(GAM) ROT(2,1)=-DCOS(ALPH)*DCOS(BET)*DSIN(GAM)-DSIN(BET)*DCOS(GAM) ROT(2,2)=-DCOS(ALPH)*DSIN(BET)*DSIN(GAM)+DCOS(BET)*DCOS(GAM) ROT(2,3)=DSIN(ALPH)*DSIN(GAM) ROT(3,1)=DSIN(ALPH)*DCOS(BET) ROT(3,2)=DSIN(ALPH)*DSIN(BET) ROT(3,3)=DCOS(ALPH) C NDATA=0 DO 106 I=1,NMDS IF(FDATA(I).LE.0.0) THEN USED(I)=’EXC’ GO TO 105 ELSE IF(FDATA(I).GT.0.0) USED(I)=’INC’ END IF NDATA=NDATA+1 X(NDATA)=FLOAT(I) Y(NDATA)=FDATA(I) GO TO 106 105 IF(FDATA(I).EQ.0.0) USED(I)=’--’ FDATA(I)=-FDATA(I) 106 CONTINUE

C OPEN(3,FILE=’mrqout.txt’,STATUS=’UNKNOWN’) c OPEN(4,FILE=’mrqout2.asc’,STATUS=’UNKNOWN’) WRITE(3,’(A50)’) HEADER WRITE(*,’(6F12.6)’) C11,C12,C44,D11,D12,D44 WRITE(3,’(6F12.6)’) C11,C12,C44,D11,D12,D44 WRITE(3,’(1H )’) C ALAMDA=-1. SUMSQ=1.E23 ITER=0 190

C 2000 ITER=ITER+1 OLAMDA=ALAMDA CHI0=SUMSQ WRITE(*,’(A20,I2)’) ’ Iteration number’,ITER C CALL MRQMIN(X,Y,SIG,NDATA,A,IA,MA,COVAR,ALPHA,MMAX, & SUMSQ,FREQS,ALAMDA) C CHANGE=ABS((SUMSQ-CHI0)/CHI0) RMS=sqrt(SUMSQ/NDATA) WRITE(*,’(6E12.4)’) OLAMDA,ALAMDA,CHI0,SUMSQ,CHANGE,RMS WRITE(3,’(6E12.4)’) OLAMDA,ALAMDA,CHI0,SUMSQ,CHANGE,RMS C C WRITE(3,’(4F12.5)’) (A(I),I=1,4) WRITE(3,’(3F12.3)’) ALPH0, BET0, GAM0 WRITE(*,’(4F12.5)’) (A(I),I=1,4) WRITE(*,’(3F12.3)’) ALPH0, BET0, GAM0 SUMERR=0.0 DO 8200 I=1,NMDS ERR=0.0 IF(FDATA(I).GT.0.) ERR=100.*((FDATA(I)-WSAV(I))/WSAV(I)) WRITE(3,’(I3,2F10.6,F8.3,A6)’) I,FDATA(I),WSAV(I),ERR,USED(I) IF (I.LE.NMDS) WRITE(*,’(I3,2F10.6,F8.3,A6)’) I,FDATA(I),WSAV(I),ERR,USED(I) IF (USED(I) .EQ. ’INC’) SUMERR=SUMERR+ABS(ERR) 8200 CONTINUE C WRITE(*,*) ’CUBIC MODEL’ C WRITE(*,’(A20,6F12.6)’) ’ANISOTROPY PARAMETER= ’, 1-(2.*A(3)/(A(1)-A(2))) C WRITE(3,’(A20,6F12.6)’) ’ANISOTROPY PARAMETER= ’, 1-(2.*A(3)/(A(1)-A(2))) ERRAVG=SUMERR/NDATA WRITE(*,’(A42,F8.3)’) ’Average error magnitude of included peaks= ’, ERRAVG WRITE(3,’(A42,F8.3)’) ’Average error magnitude of included peaks= ’, ERRAVG C C WRITE(*,*) ’E= ’,(A(1)-A(2))*(2.*A(2)+A(1))/(A(1)+A(2)) C WRITE(*,*) ’SIGMA= ’, A(2)/(A(1)+A(2)) C WRITE(3,*) ’E= ’,(A(1)-A(2))*(2.*A(2)+A(1))/(A(1)+A(2)) C WRITE(3,*) ’SIGMA= ’, A(2)/(A(1)+A(2)) 8210 IF(CHANGE.GT.0.0) GO TO 2000 C c WRITE(4,’(F5.0,3F12.6)’) T, (A(I),I=1,3) c CLOSE(4,ERR=9000) C 9000 CLOSE(3, ERR=9900) 9900 STOP 191

END C C**************************************************************************** C SUBROUTINE XYZBLK(A,MA,WSAV) C C**************************************************************************** IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) C PARAMETER (MMAX=20,MDAT=60) PARAMETER(NRR=256,TWOPI=6.283185307795864769) C REAL A(MA) REAL*8 DG(3,3,3,3),DT(3,3,3,3), CG(3,3,3,3) C DIMENSION LB(NRR),MB(NRR),NB(NRR),IC(NRR),LMN(3) DIMENSION E(NRR,NRR),GAMMA(NRR,NRR),W(NRR),V(NRR,NRR),P(NRR) DIMENSION WSAV(MDAT) DIMENSION C(3,3,3,3), D(3,3,3,3), ROT(3,3) C DIMENSION DT(3,3,3,3),CG(3,3,3,3), DG(3,3,3,3) C COMMON /NN/ NN COMMON /NMDS/ NMDS COMMON /RHO/ RHO,TCORR COMMON /CIJKL/ C, D COMMON /SUBSTR/ E1,E2,E3,C11,C12,C44,DRHO COMMON /ALPHA/ ALPHA COMMON /ROT/ ROT C C Initialize matrices****** DO 12 I=1,3 DO 12 J=1,3 DO 12 K=1,3 DO 12 L=1,3 CG(I,J,K,L)=0.D0 DT(I,J,K,L)=0.D0 DG(I,J,K,L)=0.D0 12 CONTINUE

E4=A(4) C C C11, C22, C33 CG(1,1,1,1)=A(5) CG(2,2,2,2)=A(5) 192

CG(3,3,3,3)=A(5) C C12, C23, C13 CG(1,1,2,2)=A(6) CG(2,2,1,1)=A(6) CG(1,1,3,3)=A(6) CG(3,3,1,1)=A(6) CG(2,2,3,3)=A(6) CG(3,3,2,2)=A(6) C C44 CG(2,3,2,3)=A(7) CG(2,3,3,2)=A(7) CG(3,2,2,3)=A(7) CG(3,2,3,2)=A(7) C C55 CG(1,3,1,3)=A(7) CG(1,3,3,1)=A(7) CG(3,1,1,3)=A(7) CG(3,1,3,1)=A(7) C C66 CG(2,1,2,1)=A(7) CG(2,1,1,2)=A(7) CG(1,2,2,1)=A(7) CG(1,2,1,2)=A(7) C DO 110 I=1,3 DO 110 J=1,3 DO 110 K=1,3 DO 110 L=1,3 C(I,J,K,L)=0.0 DO 100 IP=1,3 DO 100 IQ=1,3 DO 100 IR=1,3 DO 100 IS=1,3 C(I,J,K,L)=C(I,J,K,L)+ & ROT(I,IP)*ROT(J,IQ)*ROT(K,IR)*ROT(L,IS)*CG(IP,IQ,IR,IS) 100 CONTINUE 110 CONTINUE C DG(1,1,1,1)=A(1) DG(2,2,2,2)=A(1) DG(3,3,3,3)=A(1) C DG(1,1,2,2)=A(2) DG(2,2,1,1)=A(2) DG(1,1,3,3)=A(2) 193

DG(3,3,1,1)=A(2) DG(2,2,3,3)=A(2) DG(3,3,2,2)=A(2) C DG(2,3,2,3)=A(3) DG(2,3,3,2)=A(3) DG(3,2,2,3)=A(3) DG(3,2,3,2)=A(3) C DG(1,3,1,3)=A(3) DG(1,3,3,1)=A(3) DG(3,1,1,3)=A(3) DG(3,1,3,1)=A(3) C DG(2,1,2,1)=A(3) DG(2,1,1,2)=A(3) DG(1,2,2,1)=A(3) DG(1,2,1,2)=A(3) C DO 112 I=1,3 DO 112 J=1,3 DO 112 K=1,3 DO 112 L=1,3 DT(I,J,K,L)=0.0 DO 111 IP=1,3 DO 111 IQ=1,3 DO 111 IR=1,3 DO 111 IS=1,3 DT(I,J,K,L)=DT(I,J,K,L)+ & ROT(I,IP)*ROT(J,IQ)*ROT(K,IR)*ROT(L,IS)*DG(IP,IQ,IR,IS) 111 CONTINUE 112 CONTINUE C D(1,1,1,1)=DT(1,1,1,1)-A(5) D(2,2,2,2)=DT(2,2,2,2)-A(5) D(3,3,3,3)=DT(3,3,3,3)-A(5) C D(1,1,2,2)=DT(1,1,2,2)-A(6) D(2,2,1,1)=DT(2,2,1,1)-A(6) D(1,1,3,3)=DT(1,1,3,3)-A(6) D(3,3,1,1)=DT(3,3,1,1)-A(6) D(2,2,3,3)=DT(2,2,3,3)-A(6) D(3,3,2,2)=DT(3,3,2,2)-A(6) C D(2,3,2,3)=DT(2,3,2,3)-A(7) 194

D(2,3,3,2)=DT(2,3,3,2)-A(7) D(3,2,2,3)=DT(3,2,2,3)-A(7) D(3,2,3,2)=DT(3,2,3,2)-A(7) C D(1,3,1,3)=DT(1,3,1,3)-A(7) D(1,3,3,1)=DT(1,3,3,1)-A(7) D(3,1,1,3)=DT(3,1,1,3)-A(7) D(3,1,3,1)=DT(3,1,3,1)-A(7) C D(2,1,2,1)=DT(2,1,2,1)-A(7) D(2,1,1,2)=DT(2,1,1,2)-A(7) D(1,2,2,1)=DT(1,2,2,1)-A(7) D(1,2,1,2)=DT(1,2,1,2)-A(7)

ALPHA=E4/(E3/2.D0) C DO 43 I=1,NMDS WSAV(I)=1.D64 43 CONTINUE C C Normalization factors for the calculation of Gamma C E11=4.D0/(E1*E1) E22=4.D0/(E2*E2) E33=4.D0/(E3*E3) E23=4.D0/(E2*E3) E13=4.D0/(E1*E3) E12=4.D0/(E1*E2) C C Loop for four blocks C DO 8000 L0=1,2 DO 8000 M0=1,2 C IG=0 DO 2 N0=1,2 DO 2 I=1,3 LMN(1)=L0 LMN(2)=M0 LMN(3)=N0 JLMN=1 IF(LMN(I).EQ.1) JLMN=2 LMN(I)=JLMN C 195

DO 2 L=LMN(1),NN+1,2 DO 2 M=LMN(2),NN+1,2 DO 2 N=LMN(3),NN+1,2 IF(L+M+N.GT.NN+3) GO TO 2 IG=IG+1 IC(IG)=I LB(IG)=L-1 MB(IG)=M-1 NB(IG)=N-1 2 CONTINUE C C Calculate Gamma C NR=IG DO 3 IG=1,NR DO 3 JG=IG,NR E(IG,JG)=0.D0 GAMMA(IG,JG)=0.D0 I=IC(IG) J=IC(JG) LS=LB(IG)+LB(JG) MS=MB(IG)+MB(JG) NS=NB(IG)+NB(JG) GAMMA(IG,JG)= & C(I,1,J,1)*DFLOAT(LB(IG)*LB(JG))*F(LS-2,MS,NS)*E11 & +C(I,2,J,2)*DFLOAT(MB(IG)*MB(JG))*F(LS,MS-2,NS)*E22 & +C(I,3,J,3)*DFLOAT(NB(IG)*NB(JG))*F(LS,MS,NS-2)*E33 & +(C(I,1,J,2)*DFLOAT(LB(IG)*MB(JG))+C(I,2,J,1)* & DFLOAT(MB(IG)*LB(JG)))*F(LS-1,MS-1,NS)*E12 & +(C(I,1,J,3)*DFLOAT(LB(IG)*NB(JG))+C(I,3,J,1)* & DFLOAT(NB(IG)*LB(JG)))*F(LS-1,MS,NS-1)*E13 & +(C(I,2,J,3)*DFLOAT(MB(IG)*NB(JG))+C(I,3,J,2)* & DFLOAT(NB(IG)*MB(JG)))*F(LS,MS-1,NS-1)*E23 & +D(I,1,J,1)*DFLOAT(LB(IG)*LB(JG))*G(LS-2,MS,NS)*E11 & +D(I,2,J,2)*DFLOAT(MB(IG)*MB(JG))*G(LS,MS-2,NS)*E22 & +D(I,3,J,3)*DFLOAT(NB(IG)*NB(JG))*G(LS,MS,NS-2)*E33 & +(D(I,1,J,2)*DFLOAT(LB(IG)*MB(JG))+D(I,2,J,1)* & DFLOAT(MB(IG)*LB(JG)))*G(LS-1,MS-1,NS)*E12 & +(D(I,1,J,3)*DFLOAT(LB(IG)*NB(JG))+D(I,3,J,1)* & DFLOAT(NB(IG)*LB(JG)))*G(LS-1,MS,NS-1)*E13 & +(D(I,2,J,3)*DFLOAT(MB(IG)*NB(JG))+D(I,3,J,2)* & DFLOAT(NB(IG)*MB(JG)))*G(LS,MS-1,NS-1)*E23 C GAMMA(JG,IG)=GAMMA(IG,JG) IF(I.EQ.J) E(IG,JG)=F(LS,MS,NS)+DRHO*G(LS,MS,NS) 196

E(JG,IG)=E(IG,JG) 3 CONTINUE C CALL CHOLDC(E,NR,NRR,P) C OOP=1.D0/P(1) DO 21 I=1,NR V(I,1)=GAMMA(I,1)*OOP 21 CONTINUE DO 23 J=2,NR OOP=1.D0/P(J) DO 23 I=J,NR SUM=GAMMA(I,J) DO 22 K=1,J-1 SUM=SUM-V(I,K)*E(J,K) 22 CONTINUE V(I,J)=SUM*OOP 23 CONTINUE GAMMA(1,1)=V(1,1)/P(1) DO 25 I=2,NR SUM=V(I,1) DO 24 K=1,I-1 SUM=SUM-E(I,K)*GAMMA(K,1) 24 CONTINUE GAMMA(I,1)=SUM/P(I) GAMMA(1,I)=GAMMA(I,1) 25 CONTINUE DO 27 J=2,NR DO 27 I=J,NR SUM=V(I,J) DO 26 K=1,I-1 SUM=SUM-E(I,K)*GAMMA(J,K) 26 CONTINUE GAMMA(I,J)=SUM/P(I) GAMMA(J,I)=GAMMA(I,J) 27 CONTINUE C CALL TRED2(GAMMA,NR,NRR,W,P) C CALL TQLI(W,P,NR,NRR,GAMMA) C C Save lowest frequencies. C 60 DO 64 J=1,NR W1=W(J) 197

IF(W1.LT.1.D-4) GO TO 64 DIFMAX=-1.D0 DO 61 I=1,NMDS DIF=WSAV(I)-W1 IF(DIF.LE.DIFMAX) GO TO 61 DIFMAX=DIF IMAX=I 61 CONTINUE IF(DIFMAX.LE.0.D0) GO TO 64 WSAV(IMAX)=W1 64 CONTINUE C 8000 CONTINUE C C Sort frequencies. C DO 8012 I=1,NMDS-1 K=I TMP=WSAV(I) DO 8011 J=I+1,NMDS IF(WSAV(J).GT.TMP) GO TO 8011 K=J TMP=WSAV(J) 8011 CONTINUE IF(K.EQ.I) GO TO 8012 WSAV(K)=WSAV(I) WSAV(I)=TMP 8012 CONTINUE C DO 8020 I=1,NMDS WSAV(I)=DSQRT(WSAV(I)/RHO)/TWOPI 8020 CONTINUE C RETURN END C C***************************************************************** C SUBROUTINE INIT C C***************************************************************** IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) C PARAMETER (MMAX=20,MDAT=60) 198

C REAL SIG(MDAT) C DIMENSION ALST(MMAX),EPS(MMAX) DIMENSION C(3,3,3,3), D(3,3,3,3) DIMENSION DYDAA(MMAX,MDAT) C COMMON /ALST/ ALST,EPS COMMON /CIJKL/ C, D COMMON /DYDAA/ DYDAA COMMON /SIG/ SIG C DO 11 I=1,MMAX ALST(I)=0.D0 EPS(I)=1.D-3 11 CONTINUE C DO 12 I=1,3 DO 12 J=1,3 DO 12 K=1,3 DO 12 L=1,3 C(I,J,K,L)=0.D0 D(I,J,K,L)=0.D0 12 CONTINUE C DO 13 I=1,MMAX DO 13 J=1,MDAT DYDAA(I,J)=0.D0 13 CONTINUE C DO 14 I=1,MDAT SIG(I)=1.D0 14 CONTINUE C RETURN END C C*************************************************************** C FUNCTION F(IP,IQ,IR) C C*************************************************************** IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) C 199

F=0.00D+00 IF((IP.LT.0).OR.(IQ.LT.0).OR.(IR.LT.0)) RETURN IF((MOD(IP,2).NE.0).OR.(MOD(IQ,2).NE.0).OR.(MOD(IR,2).NE.0)) RETURN C IP1=IP+1 IQ1=IQ+1 IR1=IR+1 C F=8.D0/DFLOAT(IP1*IQ1*IR1) RETURN END C C***************************************************************** C FUNCTION G(IP,IQ,IR) C C***************************************************************** IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) COMMON /ALPHA/ ALPHA C G=0.00D+00 IF(ALPHA.LE.0.0) RETURN IF((IP.LT.0).OR.(IQ.LT.0).OR.(IR.LT.0)) RETURN IF((MOD(IP,2).NE.0).OR.(MOD(IQ,2).NE.0)) RETURN C IP1=IP+1 IQ1=IQ+1 IR1=IR+1 C G=(4.D0/DFLOAT(IP1*IQ1*IR1))*(1.D0-(1.D0-ALPHA)**(IR1)) RETURN END C C************************************************************** C SUBROUTINE FREQS(XVAL,A,YMOD,DYDA,MA) C C************************************************************** IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) C PARAMETER (MMAX=20,MDAT=60) C INTEGER IA(MMAX) 200

REAL XVAL,A(MA),YMOD,DYDA(MA),A0,X(MDAT) C DIMENSION WSAV(MDAT),WTMP(MDAT) DIMENSION ALST(MMAX),EPS(MMAX) DIMENSION DYDAA(MMAX,MDAT) C COMMON /ALST/ ALST,EPS COMMON /WSAV/ WSAV,WTMP COMMON /DYDAA/ DYDAA COMMON /NDATA/ NDATA COMMON /X/ X COMMON /IA/ IA C 10 DO 11 I=1,MA IF(A(I).NE.ALST(I)) GO TO 100 11 CONTINUE GO TO 200 C 100 CALL XYZBLK(A,MA,WSAV) C DO 112 I=1,MA IF(IA(I).EQ.0) GO TO 112 A0=A(I) A(I)=A0*(1.D0+EPS(I)) DA=A(I)-A0 CALL XYZBLK(A,MA,WTMP) DO 111 J=1,NDATA JX=NINT(X(J)) DYDAA(I,JX)=(WTMP(JX)-WSAV(JX))/DA 111 CONTINUE A(I)=A0 112 CONTINUE C DO 113 I=1,MA ALST(I)=A(I) 113 CONTINUE C 200 IX=NINT(XVAL) YMOD=WSAV(IX) DO 201 I=1,MA DYDA(I)=DYDAA(I,IX) 201 CONTINUE C RETURN END 201

C C*********************************************************** C FUNCTION FREAD(CHARA,I,I2) C IMPLICIT INTEGER*2 (I-N) REAL*4 FREAD CHARACTER CHARA(1)*1, CHR*1 C c DIMENSION CHARA(1) C C*********************************************************** C FREAD=0.0 FSGN=1.0 J=1 C 1 IF(I.GT.I2) RETURN CHR=CHARA(I) I=I+1 IF(CHR.EQ.’-’) GO TO 2 IF((CHR.EQ.’+’).OR.(CHR.EQ.’.’)) GO TO 3 IF((CHR.GE.’0’).AND.(CHR.LE.’9’)) GO TO 5 GO TO 1 C 2 FSGN=-1.0 3 IF(I.GT.I2) RETURN CHR=CHARA(I) I=I+1 4 IF(CHR.EQ.’.’) GO TO 6 IF((CHR.EQ.’E’).OR.(CHR.EQ.’e’)) GO TO 7 IF((CHR.LT.’0’).OR.(CHR.GT.’9’)) RETURN 5 FREAD=FREAD*10.+FSGN*FLOAT(ICHAR(CHR)-48) GO TO 3 C 6 IF(I.GT.I2) RETURN CHR=CHARA(I) I=I+1 IF((CHR.EQ.’E’).OR.(CHR.EQ.’e’)) GO TO 7 IF((CHR.LT.’0’).OR.(CHR.GT.’9’)) RETURN FREAD=FREAD+FSGN*FLOAT(ICHAR(CHR)-48)/(10.**J) J=J+1 GO TO 6 C 7 IF(I.GT.I2) RETURN 202

IEXP=0 ISGN=1 CHR=CHARA(I) I=I+1 IF(CHR.EQ.’+’) GO TO 8 IF(CHR.NE.’-’) GO TO 9 ISGN=-1 8 IF(I.GT.I2) RETURN CHR=CHARA(I) I=I+1 9 IF((CHR.LT.’0’).OR.(CHR.GT.’9’)) RETURN FREAD=FREAD/(10.**IEXP) IEXP=IEXP+ISGN*(ICHAR(CHR)-48) FREAD=FREAD*(10.**IEXP) GO TO 8 C END C C************************************************************** C************************************************************** C SUBROUTINE CHOLDC(A,N,NP,P) IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) REAL*8 A(NP,NP),P(N) DO 13 I=1,N DO 12 J=I,N SUM=A(I,J) DO 11 K=I-1,1,-1 SUM=SUM-A(I,K)*A(J,K) 11 CONTINUE IF(I.EQ.J)THEN IF(SUM.LE.0.)PAUSE ’CHOLDC FAILED’ P(I)=DSQRT(SUM) ELSE A(J,I)=SUM/P(I) ENDIF 12 CONTINUE 13 CONTINUE RETURN END C SUBROUTINE TRED2(A,N,NP,D,E) IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) 203

REAL*8 A(NP,NP),D(NP),E(NP) DO 18 I=N,2,-1 L=I-1 H=0. SCALE=0. IF(L.GT.1)THEN DO 11 K=1,L SCALE=SCALE+DABS(A(I,K)) 11 CONTINUE IF(SCALE.EQ.0.)THEN E(I)=A(I,L) ELSE DO 12 K=1,L A(I,K)=A(I,K)/SCALE H=H+A(I,K)**2 12 CONTINUE F=A(I,L) G=-DSIGN(DSQRT(H),F) E(I)=SCALE*G H=H-F*G A(I,L)=F-G F=0. DO 15 J=1,L C OMIT FOLLOWING LINE IF FINDING ONLY EIGENVALUES C A(J,I)=A(I,J)/H G=0. DO 13 K=1,J G=G+A(J,K)*A(I,K) 13 CONTINUE DO 14 K=J+1,L G=G+A(K,J)*A(I,K) 14 CONTINUE E(J)=G/H F=F+E(J)*A(I,J) 15 CONTINUE HH=F/(H+H) DO 17 J=1,L F=A(I,J) G=E(J)-HH*F E(J)=G DO 16 K=1,J A(J,K)=A(J,K)-F*E(K)-G*A(I,K) 16 CONTINUE 17 CONTINUE ENDIF 204

ELSE E(I)=A(I,L) ENDIF D(I)=H 18 CONTINUE C OMIT FOLLOWING LINE IF FINDING ONLY EIGENVALUES. C D(1)=0. E(1)=0. DO 24 I=1,N C DELETE LINES FROM HERE ... C L=I-1 C IF(D(I).NE.0.)THEN C DO 22 J=1,L C G=0. C DO 19 K=1,L C G=G+A(I,K)*A(K,J) C19 CONTINUE C DO 21 K=1,L C A(K,J)=A(K,J)-G*A(K,I) C21 CONTINUE C22 CONTINUE C ENDIF C ... TO HERE WHEN FINDING ONLY EIGENVALUES. D(I)=A(I,I) C ALSO DELETE LINES FROM HERE ... C A(I,I)=1. C DO 23 J=1,L C A(I,J)=0. C A(J,I)=0. C23 CONTINUE C ... TO HERE WHEN FINDING ONLY EIGENVALUES. 24 CONTINUE RETURN END C SUBROUTINE TQLI(D,E,N,NP,Z) IMPLICIT INTEGER (I-N) IMPLICIT REAL*8 (A-H,O-Z) REAL*8 D(NP),E(NP),Z(NP,NP) CU USES DPYTHAG DO 11 I=2,N E(I-1)=E(I) 11 CONTINUE E(N)=0. DO 15 L=1,N 205

ITER=0 1 DO 12 M=L,N-1 DD=DABS(D(M))+DABS(D(M+1)) IF (DABS(E(M))+DD.EQ.DD) GOTO 2 12 CONTINUE M=N 2 IF(M.NE.L)THEN IF(ITER.EQ.30)PAUSE ’TOO MANY ITERATIONS IN TQLI’ ITER=ITER+1 G=(D(L+1)-D(L))/(2.*E(L)) R=DPYTHAG(G,1.D0) G=D(M)-D(L)+E(L)/(G+DSIGN(R,G)) S=1. C=1. P=0. DO 14 I=M-1,L,-1 F=S*E(I) B=C*E(I) R=DPYTHAG(F,G) E(I+1)=R IF(R.EQ.0.)THEN D(I+1)=D(I+1)-P E(M)=0. GOTO 1 ENDIF S=F/R C=G/R G=D(I+1)-P R=(D(I)-G)*S+2.*C*B P=S*R D(I+1)=G+P G=C*R-B C OMIT LINES FROM HERE ... C DO 13 K=1,N C F=Z(K,I+1) C Z(K,I+1)=S*Z(K,I)+C*F C Z(K,I)=C*Z(K,I)-S*F C13 CONTINUE C ... TO HERE WHEN FINDING ONLY EIGENVALUES. 14 CONTINUE D(L)=D(L)-P E(L)=G E(M)=0. GOTO 1 ENDIF 206

15 CONTINUE RETURN END C FUNCTION DPYTHAG(A,B) DOUBLE PRECISION A,B,DPYTHAG DOUBLE PRECISION ABSA,ABSB ABSA=DABS(A) ABSB=DABS(B) IF(ABSA.GT.ABSB)THEN DPYTHAG=ABSA*DSQRT(1.0D0+(ABSB/ABSA)**2) ELSE IF(ABSB.EQ.0.0D0)THEN DPYTHAG=0.0D0 ELSE DPYTHAG=ABSB*DSQRT(1.0D0+(ABSA/ABSB)**2) ENDIF ENDIF RETURN END C SUBROUTINE mrqmin(x,y,sig,ndata,a,ia,ma,covar,alpha,nca,chisq, *funcs,alamda) INTEGER ma,nca,ndata,ia(ma),MMAX REAL alamda,chisq,a(ma),alpha(nca,nca),covar(nca,nca), *sig(ndata),x(ndata),y(ndata) EXTERNAL funcs PARAMETER (MMAX=20) C USES covsrt,gaussj,mrqcof INTEGER j,k,l,m,mfit REAL ochisq,atry(MMAX),beta(MMAX),da(MMAX) SAVE ochisq,atry,beta,da,mfit if(alamda.lt.0.)then mfit=0 do 11 j=1,ma if (ia(j).ne.0) mfit=mfit+1 11 continue alamda=0.001 call mrqcof(x,y,sig,ndata,a,ia,ma,alpha,beta,nca,chisq,funcs) ochisq=chisq do 12 j=1,ma atry(j)=a(j) 12 continue endif j=0 207

do 14 l=1,ma if(ia(l).ne.0) then j=j+1 k=0 do 13 m=1,ma if(ia(m).ne.0) then k=k+1 covar(j,k)=alpha(j,k) endif 13 continue covar(j,j)=alpha(j,j)*(1.+alamda) da(j)=beta(j) endif 14 continue call gaussj(covar,mfit,nca,da,1,1) if(alamda.eq.0.)then call covsrt(covar,nca,ma,ia,mfit) return endif j=0 do 15 l=1,ma if(ia(l).ne.0) then j=j+1 atry(l)=a(l)+da(j) endif 15 continue call mrqcof(x,y,sig,ndata,atry,ia,ma,covar,da,nca,chisq,funcs) if(chisq.lt.ochisq)then alamda=0.1*alamda ochisq=chisq j=0 do 17 l=1,ma if(ia(l).ne.0) then j=j+1 k=0 do 16 m=1,ma if(ia(m).ne.0) then k=k+1 alpha(j,k)=covar(j,k) endif 16 continue beta(j)=da(j) a(l)=atry(l) endif 17 continue 208

else alamda=10.*alamda chisq=ochisq endif return END C SUBROUTINE mrqcof(x,y,sig,ndata,a,ia,ma,alpha,beta,nalp,chisq, *funcs) INTEGER ma,nalp,ndata,ia(ma),MMAX REAL chisq,a(ma),alpha(nalp,nalp),beta(ma),sig(ndata),x(ndata), *y(ndata) EXTERNAL funcs PARAMETER (MMAX=20) INTEGER mfit,i,j,k,l,m REAL dy,sig2i,wt,ymod,dyda(MMAX) mfit=0 do 11 j=1,ma if (ia(j).ne.0) mfit=mfit+1 11 continue do 13 j=1,mfit do 12 k=1,j alpha(j,k)=0. 12 continue beta(j)=0. 13 continue chisq=0. do 16 i=1,ndata call funcs(x(i),a,ymod,dyda,ma) sig2i=1./(sig(i)*sig(i)) dy=y(i)-ymod j=0 do 15 l=1,ma if(ia(l).ne.0) then j=j+1 wt=dyda(l)*sig2i k=0 do 14 m=1,l if(ia(m).ne.0) then k=k+1 alpha(j,k)=alpha(j,k)+wt*dyda(m) endif 14 continue beta(j)=beta(j)+dy*wt endif 209

15 continue chisq=chisq+dy*dy*sig2i 16 continue do 18 j=2,mfit do 17 k=1,j-1 alpha(k,j)=alpha(j,k) 17 continue 18 continue return END C SUBROUTINE gaussj(a,n,np,b,m,mp) INTEGER m,mp,n,np,NMAX REAL a(np,np),b(np,mp) PARAMETER (NMAX=50) INTEGER i,icol,irow,j,k,l,ll,indxc(NMAX),indxr(NMAX),ipiv(NMAX) REAL big,dum,pivinv do 11 j=1,n ipiv(j)=0 11 continue do 22 i=1,n big=0. do 13 j=1,n if(ipiv(j).ne.1)then do 12 k=1,n if (ipiv(k).eq.0) then if (abs(a(j,k)).ge.big)then big=abs(a(j,k)) irow=j icol=k endif else if (ipiv(k).gt.1) then pause ’singular matrix in gaussj’ endif 12 continue endif 13 continue ipiv(icol)=ipiv(icol)+1 if (irow.ne.icol) then do 14 l=1,n dum=a(irow,l) a(irow,l)=a(icol,l) a(icol,l)=dum 14 continue do 15 l=1,m 210

dum=b(irow,l) b(irow,l)=b(icol,l) b(icol,l)=dum 15 continue endif indxr(i)=irow indxc(i)=icol if (a(icol,icol).eq.0.) pause ’singular matrix in gaussj’ pivinv=1./a(icol,icol) a(icol,icol)=1. do 16 l=1,n a(icol,l)=a(icol,l)*pivinv 16 continue do 17 l=1,m b(icol,l)=b(icol,l)*pivinv 17 continue do 21 ll=1,n if(ll.ne.icol)then dum=a(ll,icol) a(ll,icol)=0. do 18 l=1,n a(ll,l)=a(ll,l)-a(icol,l)*dum 18 continue do 19 l=1,m b(ll,l)=b(ll,l)-b(icol,l)*dum 19 continue endif 21 continue 22 continue do 24 l=n,1,-1 if(indxr(l).ne.indxc(l))then do 23 k=1,n dum=a(k,indxr(l)) a(k,indxr(l))=a(k,indxc(l)) a(k,indxc(l))=dum 23 continue endif 24 continue return END C SUBROUTINE covsrt(covar,npc,ma,ia,mfit) INTEGER ma,mfit,npc,ia(ma) REAL covar(npc,npc) INTEGER i,j,k 211

REAL swap do 12 i=mfit+1,ma do 11 j=1,i covar(i,j)=0. covar(j,i)=0. 11 continue 12 continue k=mfit do 15 j=ma,1,-1 if(ia(j).ne.0)then do 13 i=1,ma swap=covar(i,k) covar(i,k)=covar(i,j) covar(i,j)=swap 13 continue do 14 i=1,ma swap=covar(k,i) covar(k,i)=covar(j,i) covar(j,i)=swap 14 continue k=k-1 endif 15 continue return END

A.2 Calculating Surface Displacements for Normal Modes

The XYZMA.F program listed below reads in the forward program output ﬁle

’EIGVECS.DAT’ and calculates the surface displacements based on the eigenvector expansion for each mode. The series of output ﬁles can be visualized with the Mathematica script listed in the next section.

PROGRAM XYZMA C*********************************************************** C Reads in ’EIGVECS.DAT’ file and outputs frame files with C points on the surface and displacement from equilibrium in 212

C form: x y z du. C Output files are ’FRXXXX.dat’ To be read in and rendered by C Mathematica program called ’mode_disp.nb’. C*********************************************************** IMPLICIT INTEGER*2 (I-N) IMPLICIT REAL*4 (A-H,O-Z) C PARAMETER (NRR=660,NMDS=40,TWOPI=6.283185307795864769) DIMENSION FSAV(NMDS),VSAV(NRR,NMDS) DIMENSION LMN(3) DIMENSION NRSV(NMDS),LSAV(NMDS),MSAV(NMDS),NSAV(NMDS) CHARACTER FNAME*10, FNAMA(10)*1 C COMMON /EDGES/ E1,E2,E3,E4 COMMON /VEC/ VEC(NRR) COMMON /NDATA/ NR,NFACE COMMON /INC/ DR COMMON /FACE/ FACE(3,3,33,33) COMMON /INDEX/ LB(NRR),MB(NRR),NB(NRR),IC(NRR) C EQUIVALENCE (FNAME,FNAMA(1)) C C*********************************************************** C NFACE=10 C AMPL=.05 C C WRITE(*,’(A)’) ’Enter number of frames per animation (e.g. 8):’ READ(*,*) NFRAMES IF((NFRAMES.LE.0).OR.(NFRAMES.GT.16)) GO TO 9000 IF(NFRAMES.EQ.1) NFRAMES=0 C C Read in NN, NMDS, E1, E2, E3 C 10 OPEN(3,FILE=’EIGVECS.DAT’, STATUS=’OLD’, ERR=11) GO TO 12 11 WRITE(*,’(A)’) ’ Error with file.’ STOP C 12 READ(3,*) NN,MMDS,E1,E2,E3 DO 13 I=1,MMDS READ(3,*) I1,NRSV(I),LSAV(I),MSAV(I),NSAV(I),FSAV(I) DO 13 J=1,NRSV(I) READ(3,*) VSAV(J,I) 13 CONTINUE CLOSE(3, ERR=11) C 213

NNP1=NN+1 NNP3=NN+3 C A=E1/2. B=E2/2. C=E3/2. EMAX=E1 IF(E2.GT.EMAX) EMAX=E2 IF(E3.GT.EMAX) EMAX=E3 C SCALE=FLOAT(LYRSIZ)/EMAX C DR=2.0/FLOAT(NFACE) C 14 WRITE(*,’(A21,I2,A2)’) ’Enter mode number (1-’, MMDS, ’):’ READ(*,*) MODE IF((MODE.LT.1).OR.(MODE.GT.MMDS)) GO TO 9000 C IG=0 DO 15 I=1,3 LMN(1)=LSAV(MODE) LMN(2)=MSAV(MODE) LMN(3)=NSAV(MODE) LMN(I)=LMN(I)+1 IF(LMN(I).EQ.3) LMN(I)=1 C* This section must be modified if anything other than 8 blocks are used** DO 15 L=LMN(1),NNP1,2 DO 15 M=LMN(2),NNP1,2 DO 15 N=LMN(3),NNP1,2 IF(L+M+N.GT.NNP3) GO TO 15 IG=IG+1 IC(IG)=I LB(IG)=L-1 MB(IG)=M-1 NB(IG)=N-1 15 CONTINUE NR=IG C******************************************************************* DO 16 I=1,NR VEC(I)=VSAV(I,MODE) 16 CONTINUE C CALL FACELIFT(FMAX) FMAX=AMPL*EMAX/FMAX C DPHASE=3.14159265/FLOAT(NFRAMES+1) PHASE=-DPHASE 214

FNAME=’FR0000.dat’ FNAMA(3)=CHAR(48+(MODE/10)) FNAMA(4)=CHAR(48+MOD(MODE,10)) C DO 500 JFRM1=0,9 DO 500 JFRM0=0,9 JFRM=JFRM1*10+JFRM0 IF(JFRM.GT.NFRAMES) GO TO 14 PHASE=PHASE+DPHASE COSP=COS(PHASE) 120 FNAMA(5)=CHAR(48+JFRM1) FNAMA(6)=CHAR(48+JFRM0) OPEN(3, FILE=FNAME, STATUS=’UNKNOWN’, ERR=121)

GO TO 130 121 CLOSE(3,ERR=122) 122 WRITE(*,’(A)’) ’Error writing FRAME file.’ READ(*,’(A)’) FNAME GO TO 120 C 130 WRITE(3,*) NFACE,E1,E2,E3 C C XZ-FACE C 200 Z0=1.0 201 Y0=-1.0-DR DO 210 I=1,NFACE+1 Y0=Y0+DR X0=-1.0-DR DO 210 J=1,NFACE+1 X0=X0+DR X1=A*X0+FMAX*FACE(1,1,I,J)*COSP Y1=B*Y0+FMAX*FACE(1,2,I,J)*COSP Z1=C*Z0+FMAX*FACE(1,3,I,J)*COSP du=FMAX*COSP*SQRT((FACE(1,1,I,J))**2+(FACE(1,2,I,J))**2+ &(FACE(1,3,I,J))**2) write(3,*) X1,Y1,Z1,du 210 CONTINUE C C XY-FACE C 300 Y0=1.0 301 Z0=-1.0-DR DO 310 I=1,NFACE+1 Z0=Z0+DR X0=-1.0-DR DO 310 J=1,NFACE+1 X0=X0+DR 215

X1=A*X0+FMAX*FACE(2,1,I,J)*COSP Y1=B*Y0+FMAX*FACE(2,2,I,J)*COSP Z1=C*Z0+FMAX*FACE(2,3,I,J)*COSP du=FMAX*COSP*SQRT((FACE(2,1,I,J))**2+(FACE(2,2,I,J))**2+ &(FACE(2,3,I,J))**2) write(3,*) X1,Y1,Z1, du 310 CONTINUE C C ZY-FACE C 400 X0=1.0 401 Z0=-1.0-DR DO 410 I=1,NFACE+1 Z0=Z0+DR Y0=-1.0-DR DO 410 J=1,NFACE+1 Y0=Y0+DR X1=A*X0+FMAX*FACE(3,1,I,J)*COSP Y1=B*Y0+FMAX*FACE(3,2,I,J)*COSP Z1=C*Z0+FMAX*FACE(3,3,I,J)*COSP du=FMAX*COSP*SQRT((FACE(3,1,I,J))**2+(FACE(3,2,I,J))**2+ &(FACE(3,3,I,J))**2) write(3,*) X1,Y1,Z1, du 410 CONTINUE

C 500 CONTINUE C GO TO 14 C 9000 STOP END

C*********************************************************** C SUBROUTINE FACELIFT(FMAX) C IMPLICIT INTEGER*2 (I-N) IMPLICIT REAL*4 (A-H,O-Z) C COMMON /NDATA/ NR,NFACE COMMON /INC/ DR COMMON /FACE/ FACE(3,3,33,33) C C*********************************************************** C FMAX=-1.0 C C XZ Faces 216

C 200 Z0=1.0 201 Y0=-1.0-DR DO 210 I=1,NFACE+1 Y0=Y0+DR X0=-1.0-DR DO 210 J=1,NFACE+1 X0=X0+DR X=X0 Y=Y0 Z=Z0 CALL DISPL(X,Y,Z,FX,FY,FZ) FACE(1,1,I,J)=FX FACE(1,2,I,J)=FY FACE(1,3,I,J)=FZ IF(ABS(FX).GT.FMAX) FMAX=ABS(FX) IF(ABS(FY).GT.FMAX) FMAX=ABS(FY) IF(ABS(FZ).GT.FMAX) FMAX=ABS(FZ) 210 CONTINUE C WRITE(*,’(A)’) ’ ’ C C XY Faces C 300 Y0=1.0 301 Z0=-1.0-DR DO 310 I=1,NFACE+1 Z0=Z0+DR X0=-1.0-DR DO 310 J=1,NFACE+1 X0=X0+DR X=X0 Y=Y0 Z=Z0 CALL DISPL(X,Y,Z,FX,FY,FZ) FACE(2,1,I,J)=FX FACE(2,2,I,J)=FY FACE(2,3,I,J)=FZ IF(ABS(FX).GT.FMAX) FMAX=ABS(FX) IF(ABS(FY).GT.FMAX) FMAX=ABS(FY) IF(ABS(FZ).GT.FMAX) FMAX=ABS(FZ) 310 CONTINUE C C ZY-FACE C 400 X0=1.0 401 Z0=-1.0-DR DO 410 I=1,NFACE+1 Z0=Z0+DR 217

Y0=-1.0-DR DO 410 J=1,NFACE+1 Y0=Y0+DR X=X0 Y=Y0 Z=Z0 CALL DISPL(X,Y,Z,FX,FY,FZ) FACE(3,1,I,J)=FX FACE(3,2,I,J)=FY FACE(3,3,I,J)=FZ IF(ABS(FX).GT.FMAX) FMAX=ABS(FX) IF(ABS(FY).GT.FMAX) FMAX=ABS(FY) IF(ABS(FZ).GT.FMAX) FMAX=ABS(FZ) 410 CONTINUE C 500 RETURN END C*********************************************************** C SUBROUTINE DISPL(X,Y,Z,FX,FY,FZ) C IMPLICIT INTEGER*2 (I-N) IMPLICIT REAL*4 (A-H,O-Z) C PARAMETER (NRR=660) COMMON /NDATA/ NR,NFACE COMMON /INDEX/ LB(NRR),MB(NRR),NB(NRR),IC(NRR) COMMON /VEC/ VEC(NRR) C C*********************************************************** C D=1.0E-6 C AX=ABS(X) AY=ABS(Y) AZ=ABS(Z) C FX=0.0 FY=0.0 FZ=0.0 DO 100 L=1,NR FAC=VEC(L) IF((LB(L).GT.0).AND.(AX.LT.D)) FAC=0. IF((LB(L).GT.0).AND.(AX.GE.D)) FAC=FAC*X**LB(L) IF((MB(L).GT.0).AND.(AY.LT.D)) FAC=0. IF((MB(L).GT.0).AND.(AY.GE.D)) FAC=FAC*Y**MB(L) IF((NB(L).GT.0).AND.(AZ.LT.D)) FAC=0. IF((NB(L).GT.0).AND.(AZ.GE.D)) FAC=FAC*Z**NB(L) IF(IC(L).EQ.1) FX=FX+FAC IF(IC(L).EQ.2) FY=FY+FAC 218

IF(IC(L).EQ.3) FZ=FZ+FAC 100 CONTINUE C RETURN END

A.2.1 Visualizing Normal Modes with Mathematica

This is a Mathematica script for visualizing normal modes. As written, it requires Mathematica version 4.0 or better, although with a minor modiﬁcation to the line indicated it can be run on earlier versions.

(* Set working directory containing the ’FRXXXXX.dat’ files. *) SetDirectory["/home/wigner/jgladden/work/rus/xyz/trig"]

(*Script for animating a particular mode *) animate[mode_] := Do[ If[mode < 10, fname = StringJoin["FR0", ToString[mode], "0", ToString[frame - 1], ".dat"], fname = StringJoin["FR", ToString[mode], "0", ToString[frame - 1], ".dat"] ]; d = ReadList[fname, {{Number, Number, Number}, Number}]; nface = d[[1, 1, 1]] + 1; d = Drop[d, 1]; (* list[[All]] is not supported for versions below 4.0 *) dumax = Max[d[[All, 2]]];

polys = Table[ Table[ Table[{RGBColor[1, 1, 1], Polygon[{d[[i, 1]], d[[i + 1, 1]], d[[i + nface + 1, 1]], d[[i + nface, 1]]} ]}, {i, 1 + j*nface, (j + 1)*nface - 1}], {j, (k - 1)*nface, k*nface - 2}], {k, 1, 3}];

plt = Graphics3D[polys, Boxed -> False, ViewPoint -> {1, 1, 1}, Lighting -> False ]; pltlst = Append[pltlst, plt]; Show[plt], {frame, 1, 8}]

(* Calls the animation script with a mode number as argument.*) 219

(* Double click a resulting graphics frame to view animation. *) (* Set to run forward to backward to see full cycle. *) animate[1]

(* Script for displaying an array of normal modes *)

modes[mlist_] := { pltlst = {}; Do[ If[mlist[[j]] < 10, fname = StringJoin["FR0", ToString[mlist[[j]]], "00.dat"], fname = StringJoin["FR", ToString[mlist[[j]]], "00.dat"] ]; d = ReadList[fname, {{Number, Number, Number}, Number}]; nface = d[[1, 1, 1]] + 1; d = Drop[d, 1]; dumax = Max[d[[All, 2]]];

polys = Table[ Table[ Table[ {(*Hue[d[[i, 2]]/dumax *.19]*)RGBColor[1, 1, 1], Polygon[{d[[i, 1]], d[[i + 1, 1]], d[[i + nface + 1, 1]], d[[i + nface, 1]]} ]}, {i, 1 + j*nface, (j + 1)*nface - 1}], {j, (k - 1)*nface, k*nface - 2}], {k, 1, 3}];

plt = Graphics3D[polys, Axes -> False, Boxed -> False, Lighting -> False, ViewPoint -> {1, 1, 1}, PlotLabel -> StringJoin["Mode ", ToString[mlist[[j]] ] ] ];

pltlst = Append[pltlst, plt]; ,{j, 1, Length[mlist]}];

pltlst = Partition[pltlst, 4]; modeplt = GraphicsArray[pltlst(*, Background -> RGBColor[0, 0, 1]*)]; Show[modeplt]; }

(*Calls ’modes’ with a selected list of modes to plot.*)

modes[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, 12, 13, 14, 15, 16}]

(* Outputs plots to a postscript file. *) 220

Display["filename.eps", modeplt, "EPS", ImageSize ->576]

A.2.2 Fitting Resonance Curves with Mathematica

This is a Mathematica script for using the built-in Levenberg-Marquart algorithm for fitting RUS resonance data with a phase shifted Lorentzian as in (3.2). Outputs are plots of data and fit curves and fit parameters written to a file.

(* Loads the non-linear fitting package *) << Statistics‘NonlinearFit‘

(* Sets the data directory *) SetDirectory["E:\\Work\\rus\\xyz\\gaas3b\\data2\\"]

(* Defines the fitting function *) fitfn = a0 + a1 f + a2 f^2 + a3 f^3 + A ((f/fo) Cos[phi] + (1 - (f/fo)^2) Q Sin[ phi])/((f/fo)^2 + (1 - (f/fo)^2)^2Q^2)

(* Defines a list of data files to fit *) datfiles = {s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12, s13, s14, s15}

(* Performs the fit on each of the data files. Reports time required, fitted *) (* parameter values (fo and Q), and fit statistics. Results are written to a *) (* text file. *) paraml = {}; Do[ Print[datfiles[[j]]"working on:"]; data = Drop[ReadList[StringJoin[ToString[datfiles[[j]]], ".dat"], Real, RecordLists -> True], 1]; peakfit = Timing[{BestFit, BestFitParameters, ANOVATable} /. NonlinearRegress[ data, fitfn, f, {{a0, 0, -10, 10}, { a1, 0, -10, 10}, {a2, 0, -10, 10}, {a3, 0, -10, 10}, (* First Peak *) {fo, (Last[data][[1]] - data[[1, 1]])/2 + data[[1, 1]], data[[1, 1]], Last[data][[1]]}, 221

{phi, 1, -\[Pi], \[Pi]}, {Q, 2000, 100, 20000}, {A, 1, 0.1, 5}}, AccuracyGoal -> 10, PrecisionGoal -> 10, MaxIterations -> 20000, RegressionReport -> {BestFit, BestFitParameters, ANOVATable}]]; Export[StringJoin[ToString[datfiles[[j]]], ".mprm"], peakfit[[2, 2]], "Table"]; datafit = Show[ {Plot[peakfit[[2, 1]], {f, data[[1, 1]], Last[data][[1]]}, PlotStyle -> RGBColor[0, 0, 1], DisplayFunction -> Identity], ListPlot[data[[All, {1, 3}]], Prolog -> AbsolutePointSize[0.2], DisplayFunction -> Identity, PlotJoined -> True]}, GridLines -> None, PlotRange -> All, PlotLabel -> ToString[datfiles[[j]]], Axes -> False, Prolog -> AbsolutePointSize[0.2], DisplayFunction -> $DisplayFunction];

Print[peakfit[[1, 1]]"Seconds Required"]; Print[NumberForm[peakfit[[2, 2, 5]], 7]]; Print[peakfit[[2, 2, 7]]]; Print[peakfit[[2, 3]]]; paraml = Append[paraml, {datfiles[[j]], NumberForm[fo/10^6, 7] /. peakfit[[2, 2, 5]], NumberForm[Abs[Q], 5] /. peakfit[[2, 2, 7]]}];, {j, 1, Length[datfiles]}]; Export["mdata.txt", paraml, "Table"]; Export["freqs.dat", paraml, "Table"]; Print[TableForm[paraml]] 222

Appendix B

Computer Code for the Alumina Experiment

B.1 Euler Transformation of the Elastic Tensor

The following is the FORTRAN code which rotates the elastic tensor labelled

G(I,J,K,L) into the sample geometry creating an eﬀective elastic tensor labelled C(I,J,K,L).

WRITE(*,’(A)’) ’ Enter ALPHA, BETA, GAMMA:’ READ(*,*) ALPH0, BET0, GAM0 RAD=TWOPI/360.0 ALPH=ALPH0*RAD BET=BET0*RAD GAM=GAM0*RAD C ROT(1,1)=DCOS(GAM)*DCOS(BET)*DCOS(ALPH)-DSIN(GAM)*DSIN(ALPH) ROT(1,2)=DCOS(GAM)*DCOS(BET)*DSIN(ALPH)+DSIN(GAM)*DCOS(ALPH) ROT(1,3)=-DCOS(GAM)*DSIN(BET) ROT(2,1)=-DSIN(GAM)*DCOS(BET)*DCOS(ALPH)-DCOS(GAM)*DSIN(ALPH) ROT(2,2)=-DSIN(GAM)*DCOS(BET)*DSIN(ALPH)+DCOS(GAM)*DCOS(ALPH) ROT(2,3)=DSIN(GAM)*DSIN(BET) ROT(3,1)=DSIN(BET)*DCOS(ALPH) ROT(3,2)=DSIN(BET)*DSIN(ALPH) ROT(3,3)=DCOS(BET) C DO 100 I=1,3 DO 100 J=1,3 DO 100 K=1,3 DO 100 L=1,3 C(I,J,K,L)=0.0 DO 100 IP=1,3 DO 100 IQ=1,3 DO 100 IR=1,3 DO 100 IS=1,3 C(I,J,K,L)=C(I,J,K,L)+ & ROT(I,IP)*ROT(J,IQ)*ROT(K,IR)*ROT(L,IS)*G(IP,IQ,IR,IS) 223

100 CONTINUE

B.2 Simulation of X-Ray Back-reﬂection Diﬀraction Pattern

The following FORTRAN code calculates an x-ray diﬀraction pattern based on the atomic positions in a unit cell of alumina. The output data can then be imported into a CAD program allowing 3 dimensional rotation of the pattern along with the actual atoms.

PROGRAM XRAY C DIMENSION BX(3),BY(3),BZ(3) DIMENSION AO0(10,3) DIMENSION F(10) C COMMON /RECIP/ BX,BY,BZ COMMON /ATOMS0/ AO0 TWOPI=6.2831853 C D=3.0 RCUTOFF=3.0 FCUTOFF=100.0 JKLMAX=9 C DO 1 I=1,4 F(I)=4.6 1 CONTINUE DO 2 I=5,10 F(I)=1.0 2 CONTINUE C 10 OPEN(3,FILE=’WIREFRAME.ASC’,STATUS=’NEW’,ERR=15) GO TO 20 14 CLOSE(3,ERR=15) 15 WRITE(*,’(A)’) & ’ Error opening wireframe file. Type any key to continue.’ PAUSE GO TO 10 C 20 CALL INIT CALL OCALC 224

WRITE(*,*) C DO 190 J=3,JKLMAX DO 190 K=J-2,J+2 DO 190 L=J-2,J+2 IF((K.GT.JKLMAX).OR.(L.GT.JKLMAX)) GO TO 190 IF((J.EQ.K).AND.(J.EQ.L)) GO TO 190 C UH=FLOAT(J) UK=FLOAT(K) UL=FLOAT(L) C C*****NORMALS TO (J,K,L) PLANE******* C XN=UH*BX(1)+UK*BX(2)+UL*BX(3) YN=UH*BY(1)+UK*BY(2)+UL*BY(3) ZN=UH*BZ(1)+UK*BZ(2)+UL*BZ(3) C C*****SCREEN POSITION OF SPOT******* C DENOM=ZN-(XN**2+YN**2+ZN**2)/(2.*ZN) SX=D*XN/DENOM SY=D*YN/DENOM SR=SQRT(SX**2+SY**2) C C*****SPOT INTENSITY******* C SUMCOS=0.0 SUMSIN=0.0 DO 110 I=1,10 PHI=TWOPI*(UH*AO0(I,1)+UK*AO0(I,2)+UL*AO0(I,3)) SUMCOS=SUMCOS+F(I)*COS(PHI) SUMSIN=SUMSIN+F(I)*SIN(PHI) 110 CONTINUE C F2=SUMCOS**2+SUMSIN**2 C IF((SR.GT.RCUTOFF).OR.(F2.LT.FCUTOFF)) GO TO 190 WRITE(*,’(3I4,3F8.2,F11.4)’) J,K,L,SX,SY,SR,F2 C WRITE(3,’(3I8)’) -9,0,1 WRITE(3,’(3I8)’) 16,203,0 WRITE(3,’(3I8)’) 0,0,0 C IX=16384+NINT(3200.*SX) IY=16384+NINT(3200.*SY) IZ=21384 C WRITE(3,’(3I8)’) IX-96,(-IY-1),IZ WRITE(3,’(3I8)’) IX+96,IY,IZ WRITE(3,’(3I8)’) IX,(-IY+95),IZ WRITE(3,’(3I8)’) IX,IY+96,IZ 225

WRITE(3,’(3I8)’) IX,(-IY-1),IZ-96 WRITE(3,’(3I8)’) IX,IY,IZ+96 C 190 CONTINUE C WRITE(3,’(3I8)’) 0,0,0 CLOSE(3,ERR=9000) C 9000 STOP END C C************************************************ C SUBROUTINE OCALC C DIMENSION AX(3),AY(3),AZ(3) DIMENSION AO0(10,3) C COMMON /UNITV/ AX,AY,AZ COMMON /ATOMS0/ AO0 C WRITE(3,’(3I8)’) -9,0,1 WRITE(3,’(3I8)’) 16,3,0 WRITE(3,’(3I8)’) 0,0,0 C DO 190 I=1,10 AOX=AO0(I,1)*AX(1) AOY=AO0(I,1)*AY(1) AOZ=AO0(I,1)*AZ(1) DO 110 J=2,3 AOX=AOX+AO0(I,J)*AX(J) AOY=AOY+AO0(I,J)*AY(J) AOZ=AOZ+AO0(I,J)*AZ(J) 110 CONTINUE WRITE(*,’(I4,3F10.4)’) I,AOX,AOY,AOZ C IX=16384+NINT(1600.*AOX) IY=16384+NINT(1600.*AOY) IZ=16384+NINT(800.*(AOZ-2.0)) C WRITE(3,’(3I8)’) IX-96,(-IY-1),IZ WRITE(3,’(3I8)’) IX+96,IY,IZ WRITE(3,’(3I8)’) IX,(-IY+95),IZ WRITE(3,’(3I8)’) IX,IY+96,IZ WRITE(3,’(3I8)’) IX,(-IY-1),IZ-96 WRITE(3,’(3I8)’) IX,IY,IZ+96 C IF(I.NE.4) GO TO 190 WRITE(3,’(3I8)’) -9,0,1 226

WRITE(3,’(3I8)’) 16,1,0 WRITE(3,’(3I8)’) 0,0,0 C 190 CONTINUE C IX=16384+NINT(1600.*AX(1)) IY=16384+NINT(1600.*AY(1)) IZ=16384+NINT(800.*(AZ(1)-2.0)) WRITE(3,’(3I8)’) 16384,-16385,14784 WRITE(3,’(3I8)’) IX,IY,IZ IX=16384+NINT(1600.*AX(2)) IY=16384+NINT(1600.*AY(2)) IZ=16384+NINT(800.*(AZ(2)-2.0)) WRITE(3,’(3I8)’) 16384,-16385,14784 WRITE(3,’(3I8)’) IX,IY,IZ IX=16384+NINT(1600.*AX(3)) IY=16384+NINT(1600.*AY(3)) IZ=16384+NINT(800.*(AZ(3)-2.0)) WRITE(3,’(3I8)’) 16384,-16385,14784 WRITE(3,’(3I8)’) IX,IY,IZ C RETURN END C C********************************************* C SUBROUTINE INIT C DIMENSION A0X(3),A0Y(3),A0Z(3) DIMENSION AX(3),AY(3),AZ(3) DIMENSION BX(3),BY(3),BZ(3) DIMENSION AO0(10,3) C COMMON /UNITV/ AX,AY,AZ COMMON /RECIP/ BX,BY,BZ COMMON /ATOMS0/ AO0 C C*****ATOMIC POSITIONS FROM NRL******* C A0X(1)=0.256984 A0Y(1)=3.621398 A0Z(1)=3.621398 C A0X(2)=3.621398 A0Y(2)=0.256984 A0Z(2)=3.621398 C A0X(3)=3.621398 A0Y(3)=3.621398 227

A0Z(3)=0.256984 C AO0(1,1)=0.35228 AO0(1,2)=0.35228 AO0(1,3)=0.35228 C AO0(2,1)=-0.35228 AO0(2,2)=-0.35228 AO0(2,3)=-0.35228 C AO0(3,1)=0.14772 AO0(3,2)=0.14772 AO0(3,3)=0.14772 C AO0(4,1)=-0.14772 AO0(4,2)=-0.14772 AO0(4,3)=-0.14772 C AO0(5,1)=-0.0564 AO0(5,2)=0.5564 AO0(5,3)=0.25 C AO0(6,1)=0.0564 AO0(6,2)=-0.5564 AO0(6,3)=-0.25 C AO0(7,1)=0.5564 AO0(7,2)=0.25 AO0(7,3)=-0.0564 C AO0(8,1)=-0.5564 AO0(8,2)=-0.25 AO0(8,3)=0.0564 C AO0(9,1)=0.25 AO0(9,2)=-0.0564 AO0(9,3)=0.5564 C AO0(10,1)=-0.25 AO0(10,2)=0.0564 AO0(10,3)=-0.5564 C S2=SQRT(0.5) S3=SQRT(1./3.) S6=S2*S3 228

C*****ROTATION TO ORIENT UNIT CELL (C=Z)******* C RXX=S2 RXY=-S2 RXZ=0 RYX=S6 RYY=S6 RYZ=-2.*S6 RZX=S3 RZY=S3 RZZ=S3 C*****ROTATED ATOMIC POSITIONS******* DO 10 I=1,3 AX(I)=RXX*A0X(I)+RXY*A0Y(I)+RXZ*A0Z(I) AY(I)=RYX*A0X(I)+RYY*A0Y(I)+RYZ*A0Z(I) AZ(I)=RZX*A0X(I)+RZY*A0Y(I)+RZZ*A0Z(I) 10 CONTINUE C C*****Volume of Unit Cell******* C V=AX(1)*(AY(2)*AZ(3)-AZ(2)*AY(3))+AY(1)*(AZ(2)*AX(3)-AX(2)*AZ(3)) & +AZ(1)*(AX(2)*AY(3)-AY(2)*AX(3)) C*****RECIPROCAL LATTICE VECTORS******* C BX(1)=(AY(2)*AZ(3)-AZ(2)*AY(3))/V BY(1)=(AZ(2)*AX(3)-AX(2)*AZ(3))/V BZ(1)=(AX(2)*AY(3)-AY(2)*AX(3))/V C BX(2)=(AY(3)*AZ(1)-AZ(3)*AY(1))/V BY(2)=(AZ(3)*AX(1)-AX(3)*AZ(1))/V BZ(2)=(AX(3)*AY(1)-AY(3)*AX(1))/V C BX(3)=(AY(1)*AZ(2)-AZ(1)*AY(2))/V BY(3)=(AZ(1)*AX(2)-AX(1)*AZ(2))/V BZ(3)=(AX(1)*AY(2)-AY(1)*AX(2))/V C RETURN END 229

Appendix C

Thin Film PVDF Transducer Fabrication

C.1 Preparation of PVDF Film

Cut a 1 1/16” x 2” piece from the bulk PVDF ﬁlm using clean glass and razor • blade at low angle (a glass microscope slide works well for a straight edge). The

long edge should be in the direction of ﬁlm polarization. Mark (+) side with a

small cut oﬀ a corner.

Thoroughly rinse ﬁlm with distilled water (DW) and place in 200 ml of Alconox • ∼ prepared as per package instructions. Place beaker in ultrasonic cleaner for 3 ∼ minutes.

Remove ﬁlm with tweezers and thoroughly spray with DW ( 7-10 minutes), then • ∼ place in clean beaker with 200 ml of DW, use ultrasonic cleaner again for 1 ∼ ∼ minute, remove ﬁlm with spray with DW. Gently shake dry and place in clean Al

cup and let dry (under vacuum is quick and clean, but pump slowly!).

Mount ﬁlm in template (without rotating frame) •

Make sure template is clean and dry. •

Put thin strip of vacuum grease on template everywhere the ﬁlm will touch the • aluminum EXCEPT leave 2 mm without grease near the evaporation window. ∼ 230

Do this for both halves of the template. Lightly wipe excess grease away with a

Kim-Wipe.

Carefully place film on template using the grease to hold it in place and make sure • the film is flat and edges parallel with edge of evaporation window, that screw

holes are cleared and the ENTIRE window is covered by ﬁlm. A little will extend

on both ends which can be used to gently pull the ﬁlm taught with tweezers. (See

Figure C.1)

Place the top plate on and use calipers to make the overlap gap 1 mm or slightly • less. Since grease is on this half also, try to put it down with the 1 mm gap so you

won’t have to slide it much.

Tighten all screws securely. •

Check and make sure the ﬁlm is ﬂat and there is no grease on any exposed part. •

C.2 Evaporation

Get evaporator (Veeco V-300) up and running with good vacuum ( 10 7 torr). • ∼ −

LN2 tank should be ﬁlled.

Place Chromium coated tungsten wire (Alfa Aesar # 41040) on lower electrodes • (controlled by external power supply) and 5 cm of 0.5 mm gold wire (Alfa Aesar ∼ # 14725) cut into small pieces in tungsten boat on upper electrodes.

TURN ON water to thickness meter (but leave monitor oﬀ). • 231

Vacuum Grease

PVDF Film

Fig. C.1. Template for metallization of PVDF transducers. Vacuum grease is used to secure the ﬁlm to the template. 232

TURN ON water to the cold ﬁnger. •

Use a small, clean C-clamp to mount the template on cold finger with aluminum • filler block in the upper (against cold finger) evaporation window. Block should fit

in easily and be ﬂush with rest of template face.

Place 2 clean glass windows to enclose evaporation area. •

Apply thin coat of vacuum grease to bell jar gasket and place over top, rotate a • little to spread grease. Have the Tesla coil ready and make sure shutter to the

PVDF is OPEN.

Turn lever on evaporator to ROUGH to rough out the bell jar. After about 1 • minute, touch the Tesla coil to the electrode and check for a purplish glow inside

the jar. The glow should spread as the pressure decreases.

At <1 mtorr, switch lever to HI VAC and the glow should disappear. Remove • Tesla coil.

Monitor pressure with ion gauge until it reaches mid 10 7. Switch on BOTH power • − supplies.

Make sure shutters to thickness monitor (bottom) and sample (middle) are CLOSED. • Degass elements by applying enough current that both the boat and ﬁlament glow,

but don’t let the gold begin evaporating yet. ( 40 amps for Cr and 140 A for Au). ∼ Do this for about 10 minutes, and turn down current to Au.

Slowly increase Cr current to 70-80 amps and it begins to evaporate (watch for • glass to begin to darken). Open shutter to thickness monitor. Fine tune current 233

until it increases at a steady rate to 2 nm/second. Open shutter to the ﬁlm and

zero the monitor.

Chromium thickness calibration: the monitor is tuned for aluminum and Cr is 1.93 • times as massive as Al so 400 A˚ of Cr would read 772A˚ on the monitor. BUT there

is also an r2 correction since the monitor is slightly closer to the ﬁlament than the

ﬁlm. The distances to the Cr ﬁlament are: dmon=18 cm and dfilm=22 cm. So the 2 scaling would be (dfilm/dmon) = 1.49, thus 400A˚ of Cr will be deposited when

the monitor reads 1150A˚ .

As the Cr is depositing, warm up the gold to a dull red ( 140 Amps). As the Cr • ∼ approaches 700 A˚ , slowly increase the current to the gold to 280A and time ∼ things so both are evaporating for a few seconds. Then turn the Cr current slowly

down to 0 and switch oﬀ power.

The gold evaporation rate should be 20 nm/sec. Gold is 7.3 times more massive • 2 than Al and the distances are dmon=17 cm and dfilm=19.5 cm. So the r scaling

would be 1.32, thus 3000A˚ of Au will be deposited when the monitor reads 3000 *

7.3 * 1.32 = 28,900A˚ .

When the gold is done, slowly ramp down the current to the gold. •

Wait about 20-30 minutes until everything has cooled down until you vent. •

Open the evaporator and examine the ﬁlm, gently touch a clean tooth pick to an • edge of the ﬁlm extending from the end of the template and see if any gold scrapes

oﬀ. If so, don’t continue with the other side. 234

If all seems well, flip the template, put the filler block in the top side and C-clamp • it to the cold finger.

Repeat steps 9 - 17 to do the other side. •

When your done, carefully remove the ﬁlm from the template and carefully clean • the vacuum grease with methanol and Kim-wipes.

Place your new transducers on a clean microscope slide with tape at the ends. •

C.3 Mounting transducers on the cell

Adjust these procedures to what works for you. Another option is presented in • Phil Spoor’s thesis.

Have these ready: small vise, screwdrivers, two pair of tweezers, scalpel or new • razor blade, all cell hardware.

Install mounting tabs on blocks and place blocks in vise with tabs up. The mount- • ing tabs should be roughed up with a dental tool in the area where the epoxy will

be.

Use ruler to place small cuts in ﬁlm (away from gold) which are 0.5 - 1mm apart. • Use scalpel and clean microscope slide as straight edge to cut transducer strips

from the ﬁlm and place in clean aluminum cup.

Cut very thin strips of sticky tape stuck to a glass slide and peel up for easy access. •

Carefully place one transducer strip centered on tabs with active area centered • between the tabs and use tape strips just next to mounting screws to hold the 235

transducer in place. IMPORTANT: The ground connection must be facing away

from the block.

Use toothpick to apply conductive epoxy to the transducer and tabs. •

Repeat to mount the other transducer on the other block taking care that they • will line up vertically and active areas are directly opposite each other and leads

facing each other are gounded.

NOTE on epoxy: Kurt Lesker (#KL-325k) will fall apart under prolonged exposure • 6 to pressure below 10− torr. Dynaloy works better at low pressures.

Let epoxy set up for 18 hours, then cut the strips just behind the epoxy. • ∼

Mount the blocks in the cell, cross talk shield, and spacers between tabs and block • to tension the transducers as needed. Check everything carefully under microscope

and if all looks well, mount a sample and test them.

Driving at 5 V , the cross talk level should be between about 0.1 and 0.2 mV , • pp pp and signal oﬀ the preamp should look pretty clean on the oscilloscope.

Do some wide frequency scans and check for clear peaks and fairly ﬂat background • (over 500 kHz). Find a very small peak and scan it to check signal to noise - ∼ even the smallest peaks should be pretty clear over the noise. 236 References

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als. Technomic Publishing Company, Inc., Pennsylvania, 1994. Vita

Joseph R. Gladden, III was born in Atlanta, GA in 1969. After graduating

Lovett High School, he attended The University of the South in Sewanee, TN graduating with a B.S. in Physics in 1991. While pursuing a Masters of Science degree from the

University of Montana in Missoula MT, he was awarded a internship at the NEC Opto- electronics Laboratory in Tsukuba, Japan to study the initial oxidation mechanisms of silicon surfaces. It was a wonderful opportunity to do science at a ﬁrst rate facility and with ﬁrst rate collaborators. This work became the basis of his Masters thesis. Also while in Montana, he was fortunate enough to meet his future wife, Nicole Pattee.

Upon completing his degree in 1993, he took a faculty position at Virginia Episco- pal School in Lynchburg, VA teaching courses in Physics, and Computer Science. During his two year tenure at VES, he and Nicole were married and they had their ﬁrst child,

Chase Baker Gladden. In 1996, Joseph took a faculty position at the United World Col- lege in Montezuma, NM as a Physics Instructor. The UWC movement is a system of 10 sister colleges around the world each with a student body covering around 70 nationali- ties. In addition to teaching duties, he designed and implemented an academic computer network to serve students and faculty, was active in the college wilderness search and rescue team, and was Chair of the Faculty Admissions and Academic Technology Com- mittees. In 1999, he returned to school as a student to pursue his doctorate from The

Pennsylvania State University. During their time in State College, Nicole and Josh were blessed with the birth of their daughters Camille Marie in 1999 and Josephine Rose in

2003.