A Solid-state Atomic Frequency Standard Citation White, Christopher John (2005) A Solid-state Atomic Frequency Standard. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ATKE-YX40. https://resolver.caltech.edu/CaltechETD:etd-05242005-172142 Abstract The thesis describes a new class of frequency reference. The frequency source uses the same operating principle as a passive atomic frequency standard; however, the device is entirely solid-state, removing many cost and reliability issues associated with gas-phase atomic clocks. More specifically, the "atomic resonance" is derived from zero-field magnetic resonance transitions of the vanadium ion in a cubic crystal lattice. The characteristics of these resonances will be described in detail. The apparatus for measuring the "atomic" resonances uses a radio-frequency resonant cavity and frequency discriminator circuit. Using integrated circuits, the radio-frequency signal processing functions can be implemented at very low cost in a reliable manufacturing process. We discuss the system design and the measurement sensitivity. Advantages of the new frequency reference may include immunity to vibration and reduced aging compared to crystal oscillators. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: atomic clocks; crystal oscillators; frequency control; frequency standards Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Hajimiri, Ali Thesis Committee: Hajimiri, Ali (chair) Weitekamp, Daniel P. Rutledge, David B. Roukes, Michael Lee Weinreb, Sander Bridges, William B. Defense Date: 16 May 2005 Record Number: CaltechETD:etd-05242005-172142 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05242005-172142 DOI: 10.7907/ATKE-YX40 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5197 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 26 May 2005 Last Modified: 16 Dec 2020 23:20 Thesis Files Preview PDF (thesis_ch1.pdf) - Final Version See Usage Policy. 273kB Preview PDF (thesis_ch2.pdf) - Final Version See Usage Policy. 411kB Preview PDF (thesis_ch3.pdf) - Final Version See Usage Policy. 227kB Preview PDF (thesis_ch4.pdf) - Final Version See Usage Policy. 477kB Preview PDF (thesis_ch5.pdf) - Final Version See Usage Policy. 289kB Preview PDF (thesis_ch6.pdf) - Final Version See Usage Policy. 307kB Repository Staff Only: item control page

A SOLID-STATE ATOMIC FREQUENCY STANDARD Thesis by Christopher J. White In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2005 (Defended May 16th, 2005) ii © 2005 Christopher J. White All Rights Reserved iii ACKNOWLEDGEMENTS The author thanks Ali Hajimiri for his continuous support and guidance during the course of this project. Richard Bramley was very generous in discussing zero-field electron magnetic resonance and sharing details of his earlier spectrometer designs. James White oversaw the machining of the loop-gap resonator pieces at MIT Central Machine Shop. Thanks to Philip Feng from Roukes Group at Caltech for annealing the magnesium oxide crystals. Thanks also to William Bridges, David Rutledge, Daniel Weitekamp, Sander Weinreb, Yu-Chong Tai, and Michael Roukes for serving on the thesis committee and offering many helpful criticisms. The author acknowledges the support of an Intel Fellowship during one year of the project. Donhee Ham was a good friend during my years at graduate school, and I should also mention the help and encouragement I received from my brother James in completing the dissertation. Thanks always to my wife Rachel for her love and support, to Olga and Tony for their hospitality, and to my parents Gerald and Sharlene. iv ABSTRACT The thesis describes a new class of frequency reference. The frequency source uses the same operating principle as a passive atomic frequency standard; however, the device is entirely solid-state, removing many cost and reliability issues associated with gas-phase atomic clocks. More specifically, the “atomic resonance” is derived from zero-field magnetic resonance transitions of the vanadium ion in a cubic crystal lattice. The characteristics of these resonances will be described in detail. The apparatus for measuring the “atomic” resonances uses a radio-frequency resonant cavity and frequency discriminator circuit. Using integrated circuits, the radio-frequency signal processing functions can be implemented at very low cost in a reliable manufacturing process. We discuss the system design and the measurement sensitivity. Advantages of the new frequency reference may include immunity to vibration and reduced aging compared to crystal oscillators. v TABLE OF CONTENTS Acknowledgements ............................................................................................iii Abstract ............................................................................................................... iv Table of Contents................................................................................................. v List of Figures ....................................................................................................vii Chapter 1: Introduction........................................................................................ 1 1.1 Summary.................................................................................................. 1 1.2 Example Applications ............................................................................. 2 1.3 High-Accuracy, Low-Power Frequency Standards................................ 4 1.4 Precision Crystal Oscillators ................................................................... 6 1.5 Miniaturized Rubidium Frequency Standards........................................ 7 1.6 Solid-State Atomic Frequency Standard ................................................ 8 1.7 Organization .......................................................................................... 10 Bibliography ................................................................................................ 11 Chapter 2: Paramagnetic Resonance Signals.................................................... 14 2.1 Introduction ........................................................................................... 14 2.2 Transition Metal Paramagnetic Resonance in Solids........................... 15 2.3 Zero-Field Electron Spin Resonance of Vanadium in MgO................ 16 2.4 Hamiltonian for V++ in a Cubic Lattice ................................................ 17 2.5 Magnetic Resonance at Zero-Field ....................................................... 18 2.6 Selection Rules for Parallel and Perpendicular Polarizations .............. 23 2.7 Magnetic Susceptibility......................................................................... 25 2.8 Electrical Resonator Containing Paramagnetic Sample....................... 27 2.9 Numerical Calculation of Eigenvalues and Transition Intensities....... 29 2.10 Comparison to Measurement .............................................................. 32 Bibliography ................................................................................................ 37 Chapter 3: Dipolar Broadening at Zero-Field................................................... 39 3.1 Introduction ........................................................................................... 39 3.2 Dipolar Broadening of Dilute Paramagnetic Ions at High-Field ......... 40 3.3 Dipolar Broadening at Zero-Field......................................................... 42 3.4 Comparison to Measurement ................................................................ 44 Bibliography ................................................................................................ 47 Chapter 4: Prototype Design ............................................................................. 48 4.1 Introduction ........................................................................................... 48 4.2 Tunable Loop-Gap Resonator Design .................................................. 52 4.3 Loop-Gap Resonator Manufacturing.................................................... 55 4.4 Frequency Discriminator....................................................................... 57 4.5 Automatic Frequency Control............................................................... 63 4.6 Modulation Waveforms and Modulation Driver.................................. 64 4.7 Helmholtz Coil Design and Magnetic Field Homogeneity.................. 65 4.8 Voltage-Controlled Oscillator and Phase Noise................................... 67 vi 4.9 Grounding and Shielding ...................................................................... 68 4.10 Sample Preparation and Heat Treatment ............................................ 68 Bibliography ................................................................................................ 71 Appendix...................................................................................................... 73 Chapter 5: Sensitivity ........................................................................................ 78 5.1 Introduction ........................................................................................... 78 5.2 Resonator Noise..................................................................................... 79 5.3 Receiver Noise Figure........................................................................... 84 5.4 Oscillator Phase Noise .......................................................................... 85 5.5 Prototype Sensitivity ............................................................................. 90 Bibliography ................................................................................................ 91 Chapter 6: Frequency Stability.......................................................................... 92 6.1 Frequency Stability of Passive Atomic Frequency Standards ............. 92 6.2 Measurement of Frequency Stability.................................................... 97 6.3 Solid-State Atomic Frequency Standard ............................................ 100 6.4 Frequency Stability – Thermal Noise Limit ....................................... 102 Bibliography .............................................................................................. 103 Chapter 7: Conclusion ..................................................................................... 104 vii LIST OF FIGURES Number Page 1.1. SARSAT Emergency Radiolocation Beacon .................................................... 3 1.2. Historical Trends in the Performance of Frequency Standards ............................. 5 2.1 Numerically Calculated Zeeman Shift for V++/MgO ......................................... 22 2.2 Transition Intensities for Parallel and Perpendicular Polarizations ....................... 24 2.3 Calculated Spectrum for Mn++ in Magnesium Oxide ......................................... 30 2.4 Example Calculated Spectrum for Mn++ and V++ in Magnesium Oxide ................ 31 2.5 Measured Spectrum for Mn++ and V++ in Magnesium Oxide…… ...................... 32 3.1 Numerically Calculated V++/MgO Spectrum Showing Zeeman Shift at 1 Gauss .... 42 3.2 Measured V++/MgO Spectrum at 50 Gauss Zeeman Field .................................. 45 4.1 Zero-Field Electron Magnetic Resonance Spectrometer ..................................... 49 4.2 Shielded Loop-Gap Resonator: Perspective, Side and Top View ......................... 50 4.3 Audio Modulation Waveform and Lock-In Multiplier Waveform........................ 51 4.4 Mechanical Dimensions of Loop-Gap Resonator .............................................. 53 4.5 Side View of Mechanically Tunable Loop-Gap Resonator ................................. 54 4.6 Frequency Discriminator.............................................................................. 57 4.7 Power Variations at Mixer Input Because of Loop-Gap Resonator Mismatch ........ 58 4.8 Measured Loop-Gap Resonator Transmission at 3 Frequencies ........................... 59 4.9 Measured Loop-Gap Resonator Reflection at 3 Frequencies ............................... 60 4.10 Voltage Standing Wave Ratio without Series Matching Capacitor ..................... 61 4.11 Power Variations at Mixer Input Port with Improved Resonator Matching .......... 62 4.12 Frequency Discriminator with Automatic Frequency Control ........................... 63 4.13 Magnetic Field Modulation ........................................................................ 64 4.14 Helmholtz Coil Axial Magnetic Field Variation ............................................. 66 4.15 Measured Voltage-Controlled Oscillator Phase Noise ..................................... 67 4.16 V++/Mn++/MgO Spectrum in Samples as Grown............................................. 69 4.17 Spectrum after annealing 6 hours @ 1100°C in H2/Argon Atmosphere .............. 70 viii 5.1 Noise Sources in the Frequency Discriminator ................................................. 79 5.2 Tunable Loop-Gap Resonator Side View ........................................................ 80 5.3 Lorentz Force Between Helmholtz Coils......................................................... 83 5.4 Background Pickup Signal During Spectrometer Sweep .................................... 84 6.1 Passive Atomic Frequency Standard .............................................................. 92 6.2 Atomic Resonance Signal Showing Instantaneous and Mean Frequency .............. 96 6.3 Direct Frequency Counter Measurement of Allan Variance ................................ 97 6.4 Solid-State Atomic Frequency Standard ....................................................... 100 6.5 Frequency-Locking of Solid-State Atomic Frequency Standard ........................ 101 1 Chapter 1 INTRODUCTION 1.1 Summary Frequency standards are important in nearly all electronic systems, and are widely used in cellular telephones, wireless networks, satellite broadcasting, fiber-optic communication, global positioning systems (GPS), and a variety of instrumentation applications such as frequency synthesizers. Two classes of frequency standards exist; crystal oscillators and atomic clocks, which differ widely on their accuracy, size, cost, and power requirements [1]. Historically, the mainstream applications of crystal oscillators and atomic clocks have been very different. Most atomics clocks are used in laboratories for making highly precise measurements, or maintaining the basic standard of time. On the other hand, crystal oscillators are ubiquitous in consumer electronics, may cost only a few cents, are very small, and very low-power. There is currently a need for low-cost, high-performance frequency standards which the solid-state atomic frequency standard presented in this dissertation seeks to address. Two technologies which are currently used for high-accuracy frequency control outside a laboratory environment are oven-controlled crystal oscillators and rubidium atomic clocks. However, the cost and power consumption of both technologies stands in the way of more widespread application of precision frequency control. Recent trends in frequency standards have been to aggressively shrink the size of rubidium atomic clocks and attempt to develop batch manufacturing techniques to lower the production cost [2,3,4,5,6,7,8,9,10,11,12]. For crystal oscillators, every possible refinement has been applied in the last few decades to improve the long-term accuracy and temperature stability [13,14,15,16,17,18], and to reduce the power consumption of oven-controlled devices 2 [19,20,21,22,23]. New solutions are needed to realize low-cost, portable frequency sources with improved accuracy. 1.2 Example Applications Many developments in frequency standards have been motivated by the requirements of new communications systems or better navigation, which is likewise true of efforts to develop robust, miniature, low-power frequency standards which also have high-accuracy [24]. Two systems which illustrate the needs for improved frequency control are military spread-spectrum communications and commercial emergency radiolocation beacons [25,26]. The spread-spectrum example describes the need for a frequency reference with high-accuracy and low-vibration sensitivity. The radiolocation beacon example demonstrates the need for a frequency reference with good stability over temperature, lowpower consumption, and low-cost. Spread Spectrum Communications In the first case, the frequency accuracy requirements arise because of the narrow channel spacing in spread-spectrum communications systems. Channel spacing tends to be reduced to allow more users to operate in the same radio band and to make communications more difficult to intercept. In practical cases, channel bandwidths are often only a few kHz, and with carrier frequencies extending up to X-band, frequency accuracy from 5-ppm to 0.001 ppm is required [25]. The frequency accuracy specification should be maintained under acceleration, and for a year or more without re-calibration. Also, phase noise caused by vibration sensitivity of the frequency reference is a limiting factor for the system capacity in some cases [25]. Vibration sensitivity is all the more critical for microwave systems, because audio frequency vibrations are multiplied to phase noise in adjacent channels. Therefore, for spread-spectrum communications, a frequency reference which has highaccuracy and low-vibration sensitivity is essential. 3 Emergency Radiolocation A second example related to navigation is a type of Doppler radiolocation system. Emergency radiolocation beacons allow a user to relay an emergency distress signal via satellite to a search and rescue ground terminal. The location of the distress beacon is detected by tracking the Doppler shift of the beacon signal seen by the satellite flying overhead (Figure 1.1): Low-Earth Orbit Satellite 406 MHz Emergency Radio-Location Beacon Ground Station EARTH Figure 1.1 SARSAT Emergency Radiolocation Beacon As a satellite passes over a beacon, the inflection point between the “blue shift” and “red shift” indicates the latitude of the beacon. The slope of the Dopper gradient determines the distance between the satellite and beacon, and with a more accurate measurement to 4 account for the rotation of the earth, allows for a determination of the latitude and longitude in a single pass. This leads to a requirement on the rate of frequency drift during changes in ambient temperature of less than 10-9 per minute [26], which cannot generally be met by temperature-compensated crystal oscillators. The specifications for radiolocation beacons are modest as far as the absolute accuracy of the frequency reference, because the radiolocation system is designed to measure the gradient of the Doppler shift. However, low-power operation is greatly desired to extend the battery life of the distress beacon. 1.3 High-Accuracy, Low-Power, Low-Cost Frequency Standards The preceding examples emphasize the need to develop high-accuracy frequency sources which are also portable and low-cost. The two types of commercially available atomic frequency standards are passive rubidium standards and cesium beam standards. Cesium standards offer the best long-term stability, and accuracy on the order to 10-12, but are also the most bulky, power hungry, and expensive. Passive rubidium frequency standards are the smallest and lowest-cost atomic standards, and compete with very high-precision oven controlled standards in some applications. In all cases, the challenge is to evaluate the performance of the frequency standard in a variable environment [27,28]. For example, if a crystal oscillator has an acceleration sensitivity of 10-8 / g, then turning over the crystal unity causes the frequency to change by 10-8, which may then be the limiting factor for the practical accuracy. Atomic clocks can have very high-accuracy, buy may also have excessive sensitivity to vibrations. Warm-up time is another important problem for both technologies, which prevents greater use of precision frequency control in battery powered systems [30]. The trend in the development of frequency standards is illustrated in Figure 1.2, where although very high-accuracy frequency standards have been established for many years, the push is for portable, robust technologies. 5 Accuracy Cesium Rubidium 10-12 Miniaturized Rubidium 10 -9 OCXO 10-6 10-7 to 10-11 Low-Power, Low-Vibration Tactical Use MCXO TCXO XO Year Quartz – has the features of low-power and portability, but not enough accuracy Classic Atomic Clock – have extremely good stability but not the size, weight, cost, power, robustness for portable use Figure 1.2: Historical Trends in the Performance of Frequency Standards Technology developments take the form of improvements in the accuracy of crystal oscillators, and manufacturing innovations to miniaturize passive rubidium standards, with performance converging in the range of 10-7 to 10-10 frequency accuracy. Again, the performance evaluation has to account for environmental effects, aging, and vibration, for practical use [27,28]. The solid-state atomic frequency standard addresses the need for a robust, low-cost, low-power frequency reference with some of the absolute accuracy and long-term stability characteristics of atomic clocks. The next section looks more closely at some of the techniques used to optimize the performance of crystal oscillators, and recent work to miniaturize rubidium atomic clocks. 6 1.4 Precision Crystal Oscillators In a crystal oscillator, the resonant frequency depends primarily on the precise cut, mass, and dimensions of the crystal resonator. There are various, common techniques for stabilizing frequency over temperature, such as temperature-compensated crystal oscillators (TCXOs), digitally-compensated crystal oscillators (DCXOs), time-compensated clock oscillators (TCCOs), and oven-controlled crystal oscillators (OCXOs). The operating principle, however, is the same in all cases; i.e., the initial frequency is set by the mechanical dimensions of the piezoelectric resonator, and the resonant frequency is stabilized over environmental variations using techniques temperature compensation or temperature regulation. Power Consumption Very good stability over temperature can be achieved for oven-controlled crystal oscillators, in the range of 10-9 for single oven designs and up to 10-10 for double oven standards, but power consumption between 0.5 W and 5W is a major shortcoming for portable applications. A microcomputer compensated crystal oscillator (MCXO) uses a look-up table and A/D converter to exactly compensate the temperature vs. frequency characteristics of the crystal resonator. However, hysteresis in the quartz resonator frequency under temperature cycling limits the temperature stability of MCXO’s to less than 2 x 10-8 [13]. A different technique is direct heating of the quartz crystal for lowpower temperature regulation [20,21,22]. A serious shortcoming of this technique results from thermal stress across the crystal resonator, which shifts the resonant frequency. Aging Aging is an inherent characteristic of crystal oscillators [14,15,16,17]. The mechanisms which cause aging include surface contamination of the quartz resonator and stress relief in the crystal mounting. A good quality crystal may exhibit aging in the range of 10-9 per day, with accuracies in the range of 10-6 to 10-7 over one year [1]. For an application which requires an accuracy of 10-8 and uses an oven-controlled crystal oscillator, there is a 7 significant logistics cost for re-calibrating the oscillator [25]. For comparison, a rubidium frequency standard may have aging on the order of 10-10 per year, which is why atomic clocks must be used to obtain the highest accuracy and long-term stability. However, the low aging rate does not mean than atomic clocks are less sensitivity to vibration than crystal oscillators. 1.5 Miniaturized Rubidium Atomic Frequency Standards Atomic clocks derive a frequency from the quantum mechanical transitions of an ensemble of atoms or ions in the gas phase. Examples range from primary standards such as the cesium beam atomic clock, to smaller clocks such as rubidium standards. Usually, a technique is devised to interrogate the frequency of the magnetic dipole transitions between the hyperfine states of a particular atom or ion vapor, although a standard based on an optical (i.e., electric dipole) transition has been recently implemented [29]. Rubidium standards can be implemented using optical pumping or coherent population trapping to interrogate the hyperfine splitting of the rubidium ions. Regardless of the implementation, the desired frequency is derived from the atomic spectrum of a gas of atoms or ions, rather than from the mechanical dimensions and mass of a piezoelectric resonator. Further details of conventional, miniature atomic clocks may be found in the following references [2-12]. Most recent miniaturized designs use coherent population trapping rather than the traditional optical pumping technique using a rubidium lamp and filter [6,7,8,9,11]. Batch Production Producing a reliable, miniaturized atomic frequency standard at reasonable cost requires innovations in manufacturing techniques to fabricate the alkali metal cell and integrate the required optical components. One technique is to etch cavities in borosilicate glass to integrate the lens and vapor cell [4], another combines bulk etching of silicon with bonding to glass [2,3]. Alkali metals are highly reactive and therefore designing a stable vapor cell 8 is a difficult problem. Other interesting techniques are to fuse hollow-core pyrex fibers [10]. The frequency stability reported for one design is 5 × 10-10 at 1 second integration, and an aging rate of 2 × 10-8 per day [2]. 1.6 Solid-State Atomic Frequency Standard This thesis describes a new class of frequency reference. The frequency source uses the same operating principle as a passive atomic frequency standard; however, the device is entirely solid-state, removing many cost and reliability issues associated with gas-phase atomic clocks. The basis for the solid-state atomic frequency standard is the zero-field electron magnetic resonance of certain transition metal ions doped into dielectric crystals. As Chapters 2 and 3 will describe, the divalent vanadium ion in magnesium oxide has a narrow magnetic resonance spectrum at room temperature. The features of the zero-field magnetic resonance spectra are discussed in detail. Magnesium oxide is a common ceramic, and may be manufactured at very low cost. An important feature of the solid-state atomic frequency standard is that the reference frequency is accuracy defined by the materials system, rather than depending acutely on the manufacturing tolerances as is the case for crystal oscillators. The apparatus for measuring the “atomic” resonances uses a radio-frequency resonant cavity and a frequency discriminator circuit. The cost of radio-frequency electronics has dropped enormously in the last decade, and particularly for complex radio systems implemented at very low-cost using silicon integrated circuits. The prototype system, described in Chapter 4, uses conventional components, all of which are amenable to implementation in an integrated system, and at room temperature. The heart of the frequency source is an electrical resonator containing a paramagnetic sample. A modulated magnetic field applied to the paramagnetic sample results in frequency modulation of the electrical resonator which the interrogating microwave frequency coincides with the atomic 9 resonance. The resulting frequency modulation of the electrical resonator is demodulated using a frequency discriminator circuit, and detected using a lock-in amplifier to measure the atomic resonance. Chapters 5 and 6 analyze the sensitivity of the magnetic resonance measurement and the stability of passive atomic frequency standards. The results show how the achievable frequency instability is related to the width of the zero-field magnetic resonance and the noise floor of the magnetic resonance measurement. The passive atomic frequency standard is accurately modeled as a linear system, and therefore the analysis of the frequency stability is straightforward. The solid-state atomic clock is presented as a low-cost precision frequency source which, compared to oven-controlled crystal oscillators, may have improved aging, immunity to shock, and power consumption. 10 1.7 Organization The organization of the thesis is a follows: The introduction sections gives an overview of the theory of operation and basic design of the solid-state atomic clock, tailored to a general audience. Chapters 2 through 5 treat the theoretical and experimental questions in detail. Chapter 2 introduces the subject of paramagnetic resonance in solids, then develops the specific quantum mechanics which describes the spectrum of the vanadium ion doped in magnesium oxide. In particular, the intensities, frequency, polarization, and Zeeman shift of the zero-field magnetic resonance are calculated, as well as examples of typical spectra. Chapter 3 discusses the resonance widths of dilute paramagnetic solids due to inhomogeneous and homogeneous broadening. We discuss how the magnetic field independent transitions of the low-field spectrum are insensitive to dipolar broadening, and show examples which validate the theory. Chapter 4 is a detailed description of the design and manufacturing of the zero-field magnetic resonance spectrometer which amounts to a proof-of-concept for the solid-state atomic frequency reference. The design and manufacturing of the loop-gap resonator, frequency discriminator, modulation waveforms, and Helmholtz coils are described. Also, heat treatment of vanadium doped magnesium oxide is discussed. Chapter 5 analyzes the noise floor of the spectrometer. Chapter 6 discusses the theoretical and practical frequency stability of the solid-state atomic frequency reference, followed by conclusions and summary of the contributions of the thesis in Chapter 7. 11 BIBLIOGRAPHY, CHAPTER 1 [1] L. L. Lewis, “An introduction to frequency standards,” Proceedings of the IEEE, vol. 79, no. 7, pp. 927-935, July 1991. [2] S. Knappe et al., “A microfabricated atomic clock,” Applied Physics Letters, 85, pp. 1460, 2004. [3] L. Liew et al., “Microfabricated alkali atom vapor cells,” Applied Physics Letters, 84, pp. 2694, 2004. [4] H. C. Nathanson and I. Liberman, United States Patent 6,570,459, “Physics Package Apparatus for an Atomic Clock,” Northrop Grumman Corporation, Los Angeles, CA, May 27, 2003. [5] J. Kitching et al., “Miniature vapor-cell atomic-frequency references,” Applied Physics Letters, 81, pp. 553, 2002. [6] J. Kitching, “Compact atomic clock based on coherent population trapping,” Electronics Letters, vol. 37, no. 24, pp. 1449-1451, November 2001. [7] J. Vanier et al., “The coherent population trapping passive frequency standard [Rb example],” IEEE Trans. on Instrumentation and Measurement, vol. 52, no. 2, pp. 258-262, April 2003. [8] J. Vanier et al., “Atomic Clocks Based on Coherent Population Trapping: Basic Theoretical Models and Frequency Stability,” Proc. IEEE International Frequency Control Symposium, pp. 2-15, 2003. [9] M. Zhu, “High Contrast Signal in Coherent Population Trapping Based Atomic Frequency Standard Application,” Proc. IEEE International Frequency Control Symposium, pp. 16-21, 2003. [10] S. Knappe et al., “Atomic Vapor Cells for Miniature Frequency References,” Proc. IEEE International Frequency Control Symposium, pp. 31-32, 2003. [11] J. Vanier, “Coherent Population Trapping for the Realization of a Small, Stable, Atomic Clock,” Proc. IEEE International Frequency Control Symposium, pp. 424-434, 2002. 12 [12] J. Kitching, S. Knappe and L. Holberg, “Performance of Small Scale Frequency References,” Proc. IEEE International Frequency Control Symposium, pp. 442-446, 2002. [13] J. Kusters and J. Vig, “Hysteresis in Quartz Resonators – A Review,” IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 38, no.3, pp.281-289, May 1991. [14] F. Walls and J. Vig, “Fundamental Limits on the Frequency Stabilities of Crystal Oscillators,” IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 42, no. 4, pp. 576-589, July 1995. [15] R. Filler and J. Vig, “Long-Term Aging in Oscillators,” IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 40, no. 4, pp.387-394, July 1993. [16] Y. Yong and J. Vig, “Modeling Resonator Frequency Fluctuations Induced by Absorbing and Desorbing Surface Molecules,” IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 37, no. 6, pp. 543-550, November 1990. [17] Y. Yong and J. Vig, “Resonator Surface Contamination – A Cause of Frequency Fluctuations?,” IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 36, no. 4, pp. 452-458, July 1989. [18] R. Filler and J. Vig, “Resonators for the Microcomputer Compensated Crystal Oscillators,” Proc. 43rd Ann. Symposium on Frequency Control, pp. 1-15, 1989. [19] M. Bloch et al., “Low Power Timekeeping,” Proc. 43rd Ann. Symposium on Frequency Control, pp. 34-36, 1989. [20] Garland et al., United States Patent 3,431,392, “Internally Heated Crystal Devices,” Hughes Company, Culver City, CA, March 4, 1969. [21] I. Abramson and A. Dikidzhi, “Improvement of Characteristics of Quartz Resonator-Thermostate with Direct Heating Piezoelement,” IEEE Frequency Control Symposium, pp. 499-504, 1992. [22] I. Abramson, “Internal Heated Quartz Resonator with Low Sensitivity to an Acceleration,” IEEE International Frequency Control Symposium, pp. 838-842, 1995. [23] I. Balaz and M. Minarik, “Towards and OCXO with Infrared Heater,” IEEE International Frequency Control Symposium, pp. 674-680, 1996. 13 [24] V. E. Bottom, “A History of the Quartz Crystal Industry in the USA,” Proc. 35th Ann. Freq. Control. Symposium, pp. 3-12, May 1981. [25] J. R. Vig, “Military Applications of High Accuracy Frequency Standards and Clocks,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 40, no. 5, pp. 522-527, September 1993. [26] Specification for COSPAS-SARSAT 406 MHz Distress Beacons, C/S T.001, Issue 3 – Revision 5, October 2003. [27] IEEE Standard 1193TM-2003, IEEE Guide for Measurement of Environmental Sensitivities of Standard Frequency Generators, March 2004. [28] MIL-PRF-55310D, General Specification for Crystal Controlled Oscillator, March 1998. [29] S. A. Diddams et al., “An Optical Clock Based on a Single Trapped 199Hg+ Ion,” Science, vol. 293, pp. 825-828, August 2001. [30] T. J. Lynch and J. R. Vaccaro, “Ultrafast Warm-Up Rubidium References,” Proc. IEEE International Frequency Control Symposium, pp. 993-1001, 1996.  14 Chapter 2 PARAMAGNETIC RESONANCE SIGNALS 2.1 Introduction The “atomic” resonance which is the basis for the proposed solid-state atomic frequency standard is the divalent vanadium ion substituted into a magnesium oxide lattice. The physics of paramagnetic defects in solids is a large topic within the subject of electron spin resonance (ESR), also called paramagnetic resonance (EPR) [1,2,3,4,5]. Paramagnetic resonance is similar to nuclear magnetic resonance, in that it involves magnetic dipole transitions between spin states, but in paramagnetic resonance the interaction is with the spin of unpaired electrons rather than the spin of the nucleus. The most useful introduction to solid-state paramagnetic resonance of transition ions is the book by J. R. Pilbrow (1990) [1]. A good general introduction to electron spin resonance is the book by Wertz and Bolton (1986) [2]. Professional references include an up-to-date handbook by Poole and Farach (1999) [3], the classic treatise on experimental techniques by Poole (1983) [4], and the classic theory of transition metal paramagnetic resonance by Abragam and Bleaney (1970) [5]. The vast majority of electron spin resonance experiments are “field-swept” or “high-field” experiments, meaning that the microwave spectrometer used to detection paramagnetic resonance operates at a fixed frequency (most commonly X-band), and a large magnetic field is applied to the sample to measure the paramagnetic absorption as a function of magnetic field. Much less common is “zero-field” and “frequency-swept” electron spin resonance, where the applied magnetic field is zero and the microwave frequency is varied. The most important articles on zero-field electron spin resonance are Delfts and Bramley (1997) [6,7], Bramley and Strach (1985) [8], and a review article Bramley and Strach (1983) [9]. An older example of zero-field measurements is reference [10], T. Cole et al., 3+ and reports the zero-field spectrum of Cr 15 (which has the same electronic structure as V++) in magnesium oxide. 2.2 Characteristics of Transition Metal Paramagnetic Resonance in Solids Transition metal ions are some of the most important examples of paramagnetic resonance, and magnetism in general, because the unpaired spins of the d-shell electrons result in magnetic properties of many transition metal compounds. For isolated transition metal ions substituting into a diamagnetic crystal lattice, the electronic energy levels are very different from the free ion because of the electric field of the neighboring cations (i.e., ligand fields.) The ligand fields lift the degeneracy of the d-shell, and the resulting energy splitting results in the colors characteristic of transition metal complexes. The specifics of the crystal field splitting strongly depend on the lattice symmetry, which is described in the literature [1, 2, 5]. The spin of the d-shell electrons interacts with the crystal lattice via spin-orbit coupling, which depends on the orbital angular momentum of the d-electrons in the crystal. Generally, the orbital angular momentum in a crystal lattice is not the same as the orbital angular momentum of the free ion. When the spin-orbit coupling is zero, the electron spin states usually have a long relaxation lifetime and the paramagnetic resonance spectrum can be measured at room temperature. For example, in a cubic crystal, ions with the electronic structure 3d3 (e.g., V++, Cr3+) and 3d5 (e.g., Mn++) have orbitally non-degenerate ground states. An orbitally non-degenerate ground state must have zero orbital angular momentum and therefore the spin-orbit coupling is close to zero ([2], p. 263). The result is that the paramagnetic resonance spectrum of these ions in cubic lattices is easily observed at room temperature. The details of thermal relaxation can be complex at low temperature, where concentration depend interactions between the paramagnetic species may be important (e.g., [11]). However, at room temperature, the picture of a single ion interacting with the crystal lattice is usually accurate. 16 For future reference, the temperature dependence of the paramagnetic resonance spectrum of certain ions in various crystal lattices has been studied, as reported in [12]. In general, ionic crystals have the least temperature dependence, whereas more covalent crystals have stronger temperature dependence because of the partial bonding between the d-electrons and surrounding ligands. 2.3 Zero-Field Electron Spin Resonance of Vanadium in Magnesium Oxide The remainder of the chapter develops the specific quantum mechanics needed to describe the zero-field paramagnetic resonance spectrum of divalent vanadium in cubic symmetry, or similar ions where the spin Hamiltonian is given by an isotropic hyperfine interaction. In general, there are many possible choices of ions, which may have anisotropic hyperfine terms and non-zero crystal field terms in the spin Hamiltonian; however, we will show for the example of vanadium some of the interesting features of zero-field paramagnetic resonance such as the existence of narrow magnetic field independent transitions which are insensitive to dipolar broadening. The theory of dipolar broadening and concentration dependence is developed in Chapter 3. The specific results calculated below describe the frequency, intensity, polarization, and Zeeman shift of the zero-field paramagnetic resonance of divalent vanadium in magnesium oxide. The details are different from the high-field approximation used in most electron spin resonance measurements, because the eigenstates at zero-field are those of the coupled basis F, defined in Section 2.5, rather than eigenfunctions of the electron spin angular momentum operator S. The analysis at zero-field is mathematically the same as results for spin-orbit coupling used in atomic physics [13]. The later sections of the chapter also calculate the magnetic susceptibility of a paramagnetic sample and the effect on the frequency and losses of an electrical resonator containing the sample. Finally, several examples of numerically calculated spectra and example measurements are given. 17 2.4 Hamiltonian for V++ in a Cubic Lattice The spin resonance of divalent vanadium results from three unpaired electrons in the dshell (3d3). The spin of the electrons (S = 3/2) couples to the spin of the nucleus (I = 7/2) via the hyperfine interaction. In the most general case, the spin Hamiltonian for divalent vanadium in a cubic crystal lattice with eight equivalent directions, i.e., in octahedral symmetry, is ([5], c.f., Chapter 18): 1 Η = gβS ⋅ H + AS ⋅ I + uβ [S 3x H x + S 3y H y + S 3z H z − (S ⋅ H ){3S ( S + 1) − 1}] 5 1 + U [S 3x I x + S 3y I y + S 3z I z − (S ⋅ I ){3S ( S + 1) − 1}] 5 (2.1) The first term is the Zeeman interaction between the magnetic moment of the electrons gβS and an applied magnetic field H. The factor g is called the electron g-factor, and is usually close to 2 for ions with room temperature spectra [1,2]. The g-factor for the electron spin is 1.98 for vanadium in magnesium oxide [14]. The factor β is the Bohr magneton, and equals 1.4 MHz / Gauss using frequency units rather than energy. The second term is the isotropic hyperfine interaction between the nuclear spin, represented by the angular momentum operator I, and the electron spin, represented by the angular momentum operator S. The hyperfine constant has been determined from high-field measurements, and is approximately 223 MHz, again using frequency rather than energy units [12,14,15]. The parameter u in Equation 2.1 has been measured in the isoelectronic Cr3+/MgO system, which determined u = 10-4 at 300 K [16]. Hence, the terms in u are much smaller than the isotropic Zeeman term proportional to g ~ 2. Similarly, the coefficient U is expected to be much smaller than the hyperfine coupling A. (In principle, higher order quadropole interactions may exist which are not shown in the spin Hamiltonian (2.1), and may add non-zero terms even in cubic symmetry [17,18]. These higher order quadropole interactions 18 are expected to be very small.) Considering the higher order terms proportional to u and U as perturbations, the Hamiltonian is simply: Η = gβS ⋅ H + AS ⋅ I (2.2) The vanadium spin Hamiltonian (2.2) is similar to the spin Hamiltonian which describes the hyperfine interaction for the valence electron of an alkali metal occupying an S-orbital (i.e., an orbital with zero orbital angular momentum). Several important results, such as the existence of magnetic field independent allowed transitions, are features of the spin Hamiltonian (2.2). 2.5 Magnetic Resonance at Zero-Field The eigenvalues of (2.2) at zero-field can be easily calculated in the coupled basis F. The angular momentum operator F is defined as the sum of the electronic angular momentum operator S and the nuclear spin operator I: F = I+S (2.3) The “coupled basis” is defined by the simultaneous eigenfunctions of the angular momentum operators F2, I2, S2, and Fz. The scalar hyperfine term is diagonal in the coupled basis, as shown below: F 2 = (I + S) 2 = I 2 + 2I ⋅ S + S 2 (2.4) ( (2.5) ⇒ I ⋅S = 1 2 2 F − I − S2 2 ) Hence, because F2, I2, and S2 are diagonal in the coupled basis, the matrix representing I⋅S is also diagonal, with eigenvalues: I ⋅S F , mF = 1 [F ( F + 1) − S (S + 1) − I ( I + 1)] F , mF 2 (2.6) 19 The allowed values for F are from I + S to |I – S| in integer steps. For example, for divalent vanadium (I = 7/2, S = 3/2), F takes the values 5, 4, 3, and 2, as shown in Table 2.1. The corresponding energies are calculated using Equation 2.6. There are 2F + 1 degenerate eigenvalues for each value of F, which is summarized in Table 2.1. F Energy Degeneracy 5 21 A 4 11 4 1 A 4 9 3 − 15 A 4 7 2 − 27 A 4 5 Table 2.1: Zero-Field Hyperfine Energy Levels As will be described in greater detail in Section 2.6, zero-field magnetic resonance transitions obey the selection rule ∆F = ±1, which corresponds to transitions with frequencies 5A, 4A, 3A for the vanadium system. The result for the transition energies is also called the Landé interval rule. Therefore, the zero-field magnetic resonance spectrum of vanadium has three resonance lines, with transition frequencies of approximately 669 MHz, 892 MHz, and 1115 MHz. The effect of a static magnetic field, represented by the first term in Equation 2.2, is to lift the degeneracy of each F manifold. If the external magnetic field is small compared to the hyperfine interaction, then the first-order perturbation to the transition frequencies is calculated from the diagonal elements of the operator Sz represented in the coupled basis. 20 Generally, the representation of an angular momentum operator S in the coupled basis will be a product of one factor which depends only on the quantum number F, and a second factor involving both F and mF. The complete calculation of the matrix representing an angular momentum operator in the coupled basis is detailed in Condon & Shortley [13]. For example, the diagonal elements of Sz are: F , mF g S µB S Z F , mF = g F µB hmF (2.7) ⎡ F ( F + 1) + S ( S + 1) − I ( I + 1) ⎤ gF = gS ⎢ ⎥ 2 F ( F + 1) ⎦ ⎣ (2.8) h h where gF is called the Landé g-factor, equal to: For S = 3/2 and I = 7/2, the Landé g-factor is ⎡ F ( F + 1) − 12) ⎤ gF = gS ⎢ ⎥ ⎣ 2 F ( F + 1) ⎦ The Landé factors for the F-states of V++ are shown in Table 2.2 below. (2.9) 21 F gF-factor F(F+1) F(F+1)gF2 5 3 gS 10 30 27 2 gS 10 4 2 gS 10 20 8 2 gS 10 3 0 12 0 1 gS 2 6 15 2 gS 10 2 − Table 2.2: Landé factors to calculate Zeeman shift for the F-states of V++ The Zeeman shift of a resonance line is determined by the difference of the Landé factors and quantum numbers mF of the initial and final states. There are 2F + 1 Zeeman levels in each F manifold corresponding to the eigenvalues mF = {F, F − 1, F − 2, …, −|F − 1|, −F}. The Zeeman shift for each set of F sublevels of V++/MgO is plotted in Figure 2.1: 22 Hyperfine Energy (MHz) 2000 1500 1000 500 0 -500 -1000 -1500 -2000 0 10 20 30 40 50 60 70 80 90 100 Magnetic Field (Gauss) Figure 2.1: Numerically Calculated Zeeman Shift for V++/MgO In the figure, the exact eigenvalues are used from a numerical solution to Equation 2.2 at each value of the applied magnetic field H. The second-order Zeeman shift is evident at magnetic fields above approximately 50 Gauss. For Zeeman fields from 0 to 20 Gauss, the first-order theory is a close approximation to the exact solution. 23 2.6 Selection Rules for Parallel and Perpendicular Polarizations The selection rules and Zeeman shift for transitions between the F-levels can be explicitly calculated. In the following, the direction of the Zeeman field is chosen to define the z-axis. The intensities of the allowed transitions are simply given by the square of the non-zero off-diagonal elements of the operators Sx and Sz represented in the coupled basis. The intensities of z-polarized transitions, which are non-zero for transitions from F to F ± 1, and between states with the same quantum number mF, are {Condon & Shortley § 93(11) and § 103(5), [13]}: 2 F , mF S z F − 1, m F [F − (S − I ) ][(S + I + 1) − F ]h (F − m ) = 2 2 2 2 2 4 F (4 F − 1) 2 2 2 2 F (2.10) The transitions with radio-frequency field perpendicular to the Zeeman field, the intensities are proportional to: 2 F , mF S x F − 1, mF ± 1 = [F − (S − I ) ][(S + I + 1) − F ]h 1 (F m m )(F m m − 1) 2 2 2 4 F (4 F − 1) 2 2 2 2 4 F F Allowed transitions for perpendicular polarization obey the selection rules: F to F ± 1, and mF to mF ± 1. Notice the selection rule for mF is different for parallel and perpendicular polarizations. The intensities Equations (2.10) and (2.11) are tabulated in Tables 2.3 and 2.4 at the end of this Chapter. (2.11) 24 The relative transition intensities are plotted for both polarizations in Figure. 2.2 below: Relative Transition Intensity (Linear Scale) 6 4 2 0 -2 -4 1090 1095 1100 1105 1110 1115 1120 1125 Frequency (MHz) Figure 2.2: Transition Intensities for Parallel and Perpendicular Polarizations The figure uses a numerical solution of Equation (2.2), and assumes that all transition have the same Lorentzian dispersion lineshape and resonance width. The Zeeman field is 10 Gauss in Figure 2.2. The difference between z (blue) and x (red) polarized intensities allows a measurement of the spectrum by modulating the polarization of the magnetic field which defines the quantization axis (or equivalently, the relative polarization of the RF field). The difference between x- and z- polarized transitions at low-field can be used to measure the the mF = 0 to mF = 0 magnetic field independent transition without using a large (100 Gauss or more) modulation field. Of course, the intensity at zero-field cannot have a polarization, i.e., the sum over all xpolarized transition intensities is the same as the sum over all z-polarized transitions. 25 2.7 Magnetic Susceptibility This section calculates the resonant magnetic susceptibility due to magnetic resonance transitions of the type described in Section (4). The transition rate between the zero-field hyperfine levels is given by (c.f. Yariv, 1967 ref. [2]): Wm → k = 1 ' 2 H km g (ν ) 2 h (2.12) where g(ν) is a normalized lineshape function: ∞ ∫ g (ν )dν = 1 k k (2.13) −∞ The transition rate for the z-polarized F = 5 to F = 4 magnetic field independent transition is: 2 5775 ⎛ g S µ B ⎞ Wm → k = H z , RF ⎟ g (ν ) = 0.0583ω12 g (ν ) ⎜ 99000 ⎝ h ⎠ (2.14) where Hz,RF is the magnitude of the RF magnetic field. The magnetic susceptibility is (Yariv 1967, (8.2-2) [2]): 2 ⎛g µ ⎞ χ ' ' (υ ) = 0.0583 ⋅ 2h⎜ S B ⎟ g (ν )[N F =5, M =0 − N F = 4, M =0 ] ⎝ h ⎠ (2.15) The thermal population difference is N F =5, M = 0 − N F = 4 , M = 0 ≅ 1 h 2πν 0 N 32 kT (2.16) 26 where N is the density of paramagnetic ions and ν0 is the resonant frequency. Suppose the g(ν) is Lorentzian with “half-power” width of ∆ν: g ( v − v0 ) = ∆v 2π [(v − v0 ) 2 + (∆v / 2) 2 ] (2.17) 2 π ∆ v (2.18) g (0 ) = Therefore, the magnetic susceptibility due to the z-polarized 0-0 transition is: 2 1 h 2πν 0 ∆v ⎛ gS µB ⎞ N ⎟ 2 2 ⎝ h ⎠ 2π [(v − v0 ) + (∆v / 2) ] 32 kT χ ' ' (υ 0 ) = 0.0583 ⋅ 2h⎜ ∆vν 0 1 (g S µ B ) N χ ' ' (υ 0 ) = 0.0583 ⋅ 16 kT [(v − v0 ) 2 + (∆v / 2) 2 ] (2.19) 2 (2.20) As an example, we calculate the resonant magnetic susceptibility for a magnesium oxide crystal doped with 100 ppm vanadium relative to the anion lattice. The concentration of spin is: N= 3.58 g cc × 100 ppm × (6.022 × 10 23 mol −1 ) = 5.35 × 1018 / cc 16 + 24.3 g mol (2.21) The physical constants are (in CGS units): ( g S µ B )2 kT (2 ⋅ 9.264 × 10 − 21 ergG −1 ) 2 = 8.29 × 10 −27 ergG − 2 (1.38 × 10 −16 ergK −1 ) ⋅ 300 K (2.22) ∆vν 0 1 8.29 × 10 − 27 ergG − 2 5.35 × 1018 / cc 2 2 16 [(v − v0 ) + (∆v / 2) ] (2.23) = χ ' ' (υ 0 ) = 0.0583 ⋅ 27 The numerical expression for the magnetic susceptibility is: χ ' ' (υ 0 ) = 1.62 × 10 −10 ergG − 2 cc −1 ∆vν 0 [(v − v0 ) 2 + (∆v / 2) 2 ] (2.24) Converting to MKS units, the equivalent magnetic susceptibility is: 4πχ ' ' (υ 0 ) = 2.04 × 10 −9 ∆vν 0 [(v − v0 ) 2 + (∆v / 2) 2 ] (2.25) which for a 5 MHz wide resonance at 1110 MHz is equal to 1.81 × 10-6. The total zero-field intensity for 5 MHz and 100 ppm is 6.6*1.81 = 1.2 × 10-5. 2.8 Circuit Model for Electrical Resonator Containing Paramagnetic Sample . A model for an electrical resonator containing a paramagnetic sample is now easily developed using the magnetic susceptibility. The effective inductance of the loop-gap resonator or other resonator containing the paramagnetic sample is modified by the change in magnetic susceptibility: L = L0 (1 + 4πηχ ) (2.26) The fill factor is equal to: ∫ H dv 2 1 η= sample ∫ H dv 2 1 cavity (2.27) 28 In the present experiment, the fill factor is approximately 1/3. (ii) Calculate the frequency modulation of the resonator due to paramagnetic sample The resonant frequency is: (2πν 0 )2 = 1 LC (2.28) and for a small change in inductance, the frequency perturbation is: ∂ν 0 ν0 =− 1 ∂L 1 = − 4πηχ 2 L 2 (2.29) (iii) Calculate the resonator frequency modulation introduced phase modulation to the transmitted carrier. The electrical resonator dispersion at resonance is: 2Q L ∂θ = ∂ν 0 ν0 (2.30) The phase-modulation introduced by the paramagnetic resonance is therefore ∂θ = −4πηQL χ (2.31) The single-sideband power due to phase modulation is: P = 20 log 4πηQ L χ RMS (2.32) For example, if the fill factor is 1/3 and the loaded-Q is 250, the sideband power due to 5 MHz wide transition with intensity of Z, 0-0, is -76 dBc. (iv) Calculate the sensitivity assuming thermal noise limit 29 The thermal noise power kT is equal to -174 dBm. If the receiver has a 3dB noise figure, the noise floor is -171 dBm. The theoretical sensitivity is therefore 95 dB from a 0dBm carrier. 2.9 Numerical Calculation of Eigenvalues and Transition Intensities Another very useful technique is to numerically solve for the eigenvalues and energies of the spin Hamiltonian, then plot the spectrum by generating a matrix with the frequencies and intensities of the magnetic dipole transitions. An example shown below is the spectrum of Mn++ ion at zero field, which is more complex that the V++ spectrum because of the nonzero crystal field terms in the spin Hamiltonian for S = 5/2 in cubic symmetry. The spectrum is calculated using published values for the hyperfine interaction and crystal field term determined from high-field measurements [13,14]. The zero-field magnetic resonance measurements presented in this thesis measure the dispersion introduced by the paramagnetic resonance of the sample (i.e., the real component of the magnetic susceptibility), rather than measuring absorption (i.e., the imaginary component of the magnetic susceptibility.) When comparing experimental results to calculation, the spectrum assumes Lorentzian dispersion lineshapes for all transitions, modeled using the function: g (v − v 0 ) = ∆v(v − v0 ) 2π [(v − v0 ) 2 + (∆v / 2) 2 ] (2.33) The measured spectrum is the difference between the dispersive component of the magnetic susceptibility at zero magnetic field and the dispersive component of the magnetic susceptibility at the peak Zeeman modulation field. These aspects of the magnetic resonance measurement are detailed in Chapter 4. 30 Relative Magnetic Susceptibility (Linear Scale) 4 3 2 1 0 -1 -2 -3 -4 500 600 700 800 900 1000 Frequency (MHz) 1100 1200 1300 Figure 2.3: Calculated Spectrum for Mn++ in Magnesium Oxide A typical calculation is shown in Figure 2.3 above, calculated using the published parameters for the spin Hamiltonian of Mn++/MgO. The numerical model does not account for the variations in dipolar resonance widths, discussed in Chapter 3. The spectrum shows that Mn++ ions in a magnesium oxide sample have zero-field paramagnetic resonance signals in the range of 500 to 1300 MHz. 31 An example of a calculated dispersion spectrum of a sample containing both manganese and vanadium in MgO is shown in Figure 2.4 below, assuming all transitions have the same 10 MHz linewidth: Relative Magnetic Susceptibility (Linear Scale) 5 4 3 2 1 0 -1 -2 -3 -4 -5 950 1000 1050 1100 1150 1200 1250 Frequency (MHz) Figure 2.4: Example Calculated Spectrum for Mn++ and V++ in Magnesium Oxide The large resonance at 1115 MHz is due to the highest frequency zero-field transitions of V++/MgO, and the remaining signals in the 1000 to 1200 MHz range are due to manganese ions. 32 2.10 Comparison to Measurement A measured zero-field paramagnetic spectrum of a sample of magnesium oxide containing both vanadium and manganese is shown below in Figure 2.5. The modulation field was approximately 12 Gauss. Details of the zero-field spectrometer design are described in Chapter 4. The red curve is the measured spectrum and the blue curve is a calculation. We can see that the basic features of the calculated and measured spectrum are the same. The zero-field measurement allows for a more precise determination of the hyperfine coupling which is seen to be approximately 1% lower than the published value determined from high-field measurements [14]. Volts at Spectrometer Output Calculated Relative Magnetic Susceptibility (Linear Scale) 4 2 0 -2 -4 -6 -8 -10 1000 1020 1040 1060 1080 1100 1120 1140 1160 1180 Frequency (MHz) Figure 2.5: Measured Spectrum for Mn++ and V++ in Magnesium Oxide 1200 33 The measured curve is the output in volts from the zero-field magnetic resonance spectrometer described in Chapter 4. The blue curve was calculated similar to Figure 2.4, using published parameters for V++ and Mn++ ions in magnesium oxide known from highfield measurements. The relative intensities of the vanadium and manganese spectrum, as well as the resonance widths, were adjusted to approximately match the measured spectrum. Therefore, the measurement in Figure 2.5 validates the model of zero-field magnetic resonance presented in this chapter. Chapter 3 will discuss the linewidths at zerofield in greater detail. 34 Table 2.3: Transition Intensities and Zeeman Shift for Parallel Polarization between F = 4 and F = 5 States Transition |F,mF> |5,4> -> |4,4> |5,3> -> |4,3> |5,2> -> |4,2> |5,1> -> |4,1> |5,0> -> |4,0> |5,-1> -> |4,-1> |5,-2> -> |4,-2> |5,-3> -> |4,-3> |5,-4> -> |4,-4> Total F only factor 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 F and mF factor 9 16 21 24 25 24 21 16 9 165 Intensity Zeeman Shift 2079 2 h 99000 3696 2 h 99000 4851 2 h 99000 5544 2 h 99000 5775 2 h 99000 5544 2 h 99000 4851 2 h 99000 3696 2 h 99000 2079 2 h 99000 38115 2 h 99000 4 gSh 10 3 gSh 10 2 gSh 10 1 gSh 10 0 1 gSh 10 2 − gSh 10 3 − gSh 10 4 − gSh 10 − 35 Table 2.4: F = 5 to F = 4 Transition Intensities and Zeeman Shift for Perpendicular Polarization. The table calculates the overall intensity for the 15 non-degenerate transitions resolved via the Zeeman shift. In general, more than one allowed transition contributes to each line in the Zeeman spectrum for perpendicular polarization. Transition |F,mF> 5,5 ↔ 4,4 5,4 ↔ 4,3 5,3 ↔ 4,2 5,2 ↔ 4,1 5,1 ↔ 4,0 5,0 ↔ 4,−1 5,3 ↔ 4,4 5,−1 ↔ 4,−2 5,2 ↔ 4,3 5,−2 ↔ 4,−3 5,1 ↔ 4,2 5,−3 ↔ 4,−4 5,0 ↔ 4,1 F only factor F and mF factor Intensity Zeeman Shift 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 90 4 72 4 56 4 42 4 30 4 20 4 2 4 12 4 6 4 6 4 12 4 2 4 20 4 5198 2 h 99000 4158 2 h 99000 3234 2 h 99000 2426 2 h 99000 1733 2 h 99000 1155 2 h 99000 116 2 h 99000 693 2 h 99000 347 2 h 99000 347 2 h 99000 693 2 h 99000 116 2 h 99000 1155 2 h 99000 7 gSh 10 6 gSh 10 5 gSh 10 4 gSh 10 3 gSh 10 2 gSh 10 1 gSh 10 1 gSh 10 0 0 1 gSh 10 1 − gSh 10 2 − gSh 10 − 36 5,−1 ↔ 4,0 5,−2 ↔ 4,−1 5,−3 ↔ 4,−2 5,−4 ↔ 4,−3 5,−5 ↔ 4,−4 Total 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 231 2 h 99000 30 4 42 4 56 4 72 4 90 4 165 1733 2 h 99000 2426 2 h 99000 3234 2 h 99000 4158 2 h 99000 5198 2 h 99000 38115 2 h 99000 3 gSh 10 4 − gSh 10 5 − gSh 10 6 − gSh 10 7 − gSh 10 − Notes on Table 2.4: The Zeeman shift of a particular transition is determined by subtracting the Landé factors of the corresponding energy levels: Z = g F =5 µ B m F =5 − g F = 4 µ B m F = 4 (2.4.1) There are 18 allowed transitions and 15 non-degenerate transitions. Using the selection rules for perpendicular polarization are ∆F ± 1, ∆mF ±1, and the Landé factors from Table 2, we find for a transition from F = 5 to F = 4: Z= 3 2 1 2 g S µ B mF =5 − g S µ B (mF =5 ± 1) = g S µ B mF =5 m g S µ B 10 10 10 10 (2.4.2) For example, a transition from F = 5, mF = 2 to F = 4, mF = 3 has zero first-order Zeeman shift. The largest Zeeman is for the transition F = 5, mF = 5 to F = 4, mF = 4, and similarly for F = 5, mF = -5 to F = 4, mF = -4. 37 BIBLIOGRAPHY, CHAPTER 2 [1] J. R. Pilbrow, Transition Ion Electron Paramagnetic Resonance, Clarendon Press Oxford, 1990. [2] J. E. Wertz and J. R. Bolton, Electron Spin Resonance: Elementary Theory and Practical Applications, Chapman and Hall Ltd., 1986. [3] Charles P. Poole, Jr. and Horacia A. Farach, Handbook of Electron Spin Resonance, Volume 1 and 2, Springer-Verlag New York, Inc., 1999. [4] Charles P. Poole, Jr., Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques, Wiley, 1983. [5] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Clarendon Press, Oxford, 1970. [6] Christopher D. Delfs and Richard Bramley, “Zero-field Electron Magnetic Resonance Spectra of Copper Carboxylates,” J. Chem. Phys., vol. 107, pp. 8840, 1997. [7] Christopher D. Delfs and Richard Bramely, “The Zero-field ESR Spectrum of a Copper Dimmer,” Chemical Physics Letters, vol. 264, pp. 333, 1997. [8] Richard Bramley and Steven J. Strach, “Zero-Field EPR of the Vanadyl Ion in Ammonium Sulfate,” J. of Magnetic Resonance, vol. 61, pp. 245, 1985. [9] Richard Bramley and Steven J. Strach, “Electron Paramagnetic Resonance Spectroscopy at Zero Magnetic Field,” Chem. Rev., vol. 83, pp. 49-82, 1983. [10] Cole, T. Kushida, T., and Heller, H. C., “Zero-Field Electron Magnetic Resonance in Some Inorganic and Organic Radicals,” J. Chem. Phys., vol. 38, pp. 2915, 1963. [11] P. R. Solomon, “Relaxation of Mn2+ and Fe3+ Ions in Magnesium Oxide,” Phys. Rev., vol. 152, no. 1, December 1966. [12] Walter M. Walsh, Jr., Jean Jeener, and N. Bloembergen, “Temperature-Dependent Crystal Field and Hyperfine Interactions,” Phys. Rev., vol. 139, pp. A 1338, 1965. [13] E. U. Condon & G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, 1963. 38 [14] W. Low, “Paramagnetic Resonance Spectra of Some Ions in the 3d and 4f Shells in Cubic Crystalline Fields,” Phy. Rev., vol. 101, pp. 1827, 1956. [15] J.E. Wertz, J. W. Orton, and P. Auzins, “Spin Resonance of Point Defects in Magnesium Oxide,” J. Appl. Phys., vol. 33, no. 1, pp. 322-328, 1962. [16] R. S. De Biasi, “Influence of the S3 Term on the EPR Spectrum of Cr3+ in Cubic Symmetry Sites in MgO,” Journal of Magnetic Resonance, vol. 44, pp. 479-482, 1981. [17] A. J. Freeman, R. B. Frankel eds., Hyperfine Interactions, Academic Press New York, 1967, c.f. Ch 1 by B. Bleaney. [18] Richard Bramley, personal communication [19] A. Yariv, Quantum Electronics, Wiley, 1st ed., 1967.  39 Chapter 3 DIPOLAR-BROADENING AT ZERO-FIELD 3.1 Introduction The measured linewidth of the magnetic resonance transition is one of the critical parameters which determines the frequency stability and practical accuracy of the frequency reference. In solids, the dominant line broadening mechanism is inhomogeneous dipolar broadening due to the random distribution of spins (i.e., magnetic moments) which causes variations in the local magnetic field throughout the sample. The theory of dipolar broadening is well established, with the original work by Van Vleck (1948) [1] and Kittel and Abrahams (1953) [2]. The series of papers by de Biasi et al. compare theory and experiment for transition metal ions in CaO, MgO, and SrO lattices [2,3,4,5,6,7,8]. The theory presented in [3] takes into account the percentage of exchange coupled pairs. If the exchange coupling is anti-ferromagnetic, the magnetic moment of the pair is zero and does not add to dipolar broadening. At very high concentrations, a different effect is “exchange narrowing” resulting from the rapid diffusion of spins. In this regime, the electrons are delocalized and the hyperfine coupling is no longer observed. The high-field theory of dipolar broadening assumes that the eigenstates of Sz are good quantum numbers, which implies that the Zeeman sensitivity of all the transitions is 2.8 MHz / G, assuming the electron g-factor is ~ 2. As was shown in Chapter 2, the Zeeman sensitivity at zero-field depends on the Landé g-factor and the quantum number mF, and for some transitions the first-order magnetic field sensitivity is zero. As a result, the dipolar linewidths at zero-field vary with the individual transition, and in the magnetic field independent case may not be the dominant broadening mechanism. 40 3.2 Dipolar Broadening of Dilute Paramagnetic Ions at High-Field The case of Cr3+ (S = 3/2) in magnesium oxide powder is studied in de Biasi (1983) [4], which compares theory and experiment in the high-field case, and includes the effect of exchange coupled pairs. In the case of powders, the results are: ∆H pp = C1 A( S ) (3.1) The terms in Equation 3.1 are ⎞ µ gβ ⎛ S ( S + 1) ⎟ A( S ) = 0 3 ⎜⎜ 2 −1 ⎟ 4π a ⎝ 1.4 + 0.3( S + S ) ⎠ ( ) 1/ 2 (3.2) C1 = π 7 / 15 f e [ S1 (rc )]3 / 2 [ S 2 (rc )]1 / 2 (3.3) S1 (rc ) = a 6 ∑ r jk−6 (3.4) S 2 (rc ) = a 12 ∑ r jk−12 (3.5) r jk > rc r jk > rc The variables in Equations 3.2 to 3.5 are the lattice constant is a, the distance between j and k lattice sites rjk, and the electron spin angular momentum S. Equation 3.1 calculates the width of the Lorentzian magnetic field distribution to the random distribution of paramagnetic ions in the sample. The summations S1 and S2 exclude the radius of exchange coupling. The concentration fe is the number of paramagnetic ions which are not exchange coupled. For low concentrations (< 1%), exchange coupling is a minor effect and the width of the magnetic field distribution is linearly proportional to concentration. For magnesium oxide, that lattice constant a is 4.2112 Å [3], g = 1.98 at high-field and S = 3/2. The factor A(S) is therefore (SI units): 41 A(3 2) = 10 −7 −24 1.98(9.274 × 10 ) 1.592 = 0.0391 (4.2112 × 10 −10 ) 3 (3.6) 1.98(9.274 × 10 −24 ) 2.470 = 0.0601 (4.2112 × 10 −10 ) 3 (3.7) and for S = 5/2 (e.g. Mn++), A(5 2) = 10 −7 The geometric factors S1 and S2 are (c.f. [3], Table A1), for cubic lattice and assuming the number of the order of each coordination sphere n = 5, (i.e., the exchange interaction n = 5 nearest neighbor sites), are 0.512 and 0.00219, respectively. Therefore, the factor C1 is equal to: C1 = 1.964 f e (0.512) 3 / 2 (0.00219)1 / 2 = 15.4 f e (3.8) The expression for the width of the magnetic field distribution is: ∆H pp = 15.4 f e ⋅ 0.0391 = 0.602 f e (3.9) where fe is the concentration of single paramagnetic ions and the magnetic field units are Tesla. Therefore, for S = 3/2 ions (e.g., V++) with a concentration of 100 ppm (10-4), the calculated dipolar linewidth in MgO powder is 0.602 Gauss. For S = 5/2 ions at 100ppm concentration, the calculated dipolar linewidth is 0.934 Gauss. These calculations can be used to estimate for paramagnetic ion concentration from the measured high-field peak-topeak linewidths. 42 3.3 Dipolar Broadening at Zero-Field Figure 1 shows a calculation of the intensities and Zeeman shift for transitions between the F = 4 and F =5 states of V++ in MgO. The numerical calculation does not take into account variations in the linewidth that results from the variations in the magnetic field sensitivity of individual transitions, but assumes each transition is 100 kHz wide: Relative Magnetic Susceptibility (Linear Scale) 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 1112 1112.5 1113 1113.5 1114 1114.5 1115 1115.5 1116 Frequency (MHz) Figure 3.1 Numerically Calculated V++/MgO Spectrum Showing Zeeman shift at 1 Gauss The figure shows that the dispersion spectrum of the F = 4 to F = 5 manifold consists of 9 transition frequencies. The center transition at 1114 MHz is magnetic field independent. For an external field of 1 Gauss, the outermost transitions are shifted by ±1.12 MHz, i.e., 43 ± 4 1 g S , the inner two transitions are shifted by only ±0.28 MHz or, ± g S . As a 10 10 result, the zero-field dipolar linewidths must account for the different Zeeman sensitivities at zero-field. The analysis of the zero-field Zeeman terms was discussed in Chapter 2. Therefore, the dipolar linewidth of the transitions are 1/10th, 1/5th, 3/10th, and 2/5th, the high-field linewidths, and the center transition is independent of dipolar broadening. Assuming the variation in local magnetic field is the same as calculated in Equation 3.9 for the high-field case, and assuming the vanadium concentration is 100 ppm, the zero-field resonance widths are 0, 168 kHz, 336 kHz, 504 kHz, and 672 kHz for z-polarized transitions. A more subtle consideration is the modification of the magnetic field distribution width calculated in Equation 3.2. The different Landé g-factors at zero-field mean the magnetic moment is different for each F-state. In effect, each group of ions with the same quantum number F can be considered as a separate species in the analysis, and the overall width of the distribution is the sum of the contribution from each species. Thus, the factor gS in Equation 3.2 should be replaced by gF, and the electron spin S by F: ∆H = C1 F =5 ∑ A( F ) 4 F =2 (3.10) and ⎞ µ g β⎛ F ( F + 1) ⎟ A( F ) = 0 F3 ⎜⎜ 2 −1 ⎟ 4π a ⎝ 1.4 + 0.3( F + F ) ⎠ 1/ 2 (3.11) Following the calculation shows that for a given vanadium concentration, the width of the dipolar magnetic field distribution at zero-field is a factor of two smaller than at high-field. An additional consideration is the small crystal field term in the complete Hamiltonian for divalent vanadium in cubic symmetry, Equation 2.1 [9]. The Hamiltonian at zero-field is: 44 1 ⎡ ⎤ H = AI ⋅ S + U ⎢ S x3 I x + S y3 I y + S z3 I z − ( S ⋅ I ){3S ( S + 1) − 1}⎥ 5 ⎣ ⎦ (3.12) Although U is much less than A, the additional zero-field splitting represented by the S3 terms can affect the observed resonance width at zero-field. The measurement of U and a study of the effects of the term is the subject for future research. 3.4 Comparison to Measurement The article by Bramley (1985) [10] which studied the vanadyl ion at zero-field reports line narrowing due to second-order magnetic field dependent transitions, from 2.3 MHz compared to ~ 18 MHz for transitions with first order magnetic field sensitivity. The measurement shown below in Figure 3.2 is for a vanadium doped magnesium oxide sample in a static magnetic field of approximately 50 Gauss. The modulation field used in the measurement was approximately 5 Gauss. The measurement validates that for V++/ MgO system, the dipolar linewidth scales with the Zeeman sensitivity. The measured linewidth of the mF = ±1 transitions is half the linewidth of the mF = ±2 transitions, and the mF = ±3 and ±4 transitions are seen to be even broader, in accordance with the theory. The magnetic field independent mF = 0 is barely observed in this particular measurement because of the small modulation field. 45 Volts at Spectrometer Output -2.8 -2.9 -3 -3.1 -3.2 -3.3 1080 1090 1100 1110 1120 1130 1140 1150 Frequency (MHz) Figure 3.2 Measured V++/MgO Spectrum at 50 Gauss Zeeman Field To resolve the individual transitions, the Zeeman field must be larger than the magnetic field variations in the crystal, e.g., if the random inhomogeneous broadening due to the paramagnetic ions in the crystal is on the order of 1 Gauss, the Zeeman field needs to be several Gauss. This is also a consideration for defining the polarization axis. The magnetic field independent transitions are very insensitive to magnetic fields smaller than the hyperfine constant (~70 Gauss), therefore a modulation field of 100 Gauss or more would be needed to adequately perturb the transition. However, a static field in the range of 50 to 100 Gauss, combined with modulation on the order of 20 Gauss, is one possibility to measure the field independent transition. A much more interesting technique is to use the “polarization rotation,” i.e., to modulate the quantization axis. As is seen in Figure 2.2 of Chapter 2, the magnetic field independent transition is the most intense transition of the manifold when the RF field is polarized in the direction of the quantization axis, whereas the magnetic field independent transition is the weakest in the perpendicular polarization. Measuring the difference of these intensities by 46 modulating the quantization axis and hence polarization allow for measured of the magnetic field independent transition using only small modulation field. However, the modulation field significantly larger than internal field distribution causing dipolar broadening, for the two polarizations to be well defined. 47 BIBLIOGRAPHY [1] J. H. Van Vleck, “The Dipolar Broadening of Magnetic Resonance Lines in Crystals,” Physical Review, vol. 74, no. 9, pp.1168-1183, Nov. 1948. [2] C. Kittel and E. Abrahams, “Dipolar Broadening of Magnetic Resonance Lines in Magnetically Diluted Crystals,” Physical Review, vol. 90, no. 2, pp. 238-239, Apr. 1953. [3] R.S. de Biasi and A A R Fernandes, “The ESR Linewidth of Dilute Solid Solutions,” J. Phys. C: Solid State Phys., 16 (1983) 5481-5489. [4] R. S. de Biasi and A. A. R. Fernades, “Measurement of Small Concentrations of Cr and Mn in MgO Using Electron Spin Resonance,” Communications of the American Ceramic Society, September 1984 pp. C-173 – C-175. [5] R. S. de Biasi and M. L. N. Grillo, “Influence of Manganese Concentration on the ESR Spectrum of Mn2+ in CaO,” Journal of Alloys and Compounds, 282, (1999), pp. 5-7. [6] R.S. de Biasi, M.L.N. Grillo, “Influence of Manganese Concentration on the ESR Spectrum of Mn2+ in SrO,” Journal of Alloys and Compounds 290 (1999) 22-24 [7] R. S. de Biasi, and M. L. N. Grillo, “Influcence of Copper Concentration on the ESR Spectrum of Cu2+ in MgO,” Journal of Alloys and Compounds, 302, 2000, pp. 26-28. [8] R. S. de Biasi and M. L. N. Grillo, “Influence of Gadolinium Concentration on the ESR Spectrum of Gd3+ in SrO,” Journal of Alloys and Compounds, 337, 2002, pp. 30-32. [9] R. S. De Biasi, “Influence of the S3 Term on the EPR Spectrum of Cr3+ in Cubic Symmetry Sites in MgO,” Journal of Magnetic Resonance, vol. 44, pp. 479-482, 1981. [10] Richard Bramley and Steven J. Strach, “Zero-Field EPR of the Vanadyl Ion in Ammonium Sulfate,” J. of Magnetic Resonance, vol. 61, pp. 245, 1985.  48 Chapter 4 PROTOTYPE DESIGN 4.1 Introduction The previous two chapters have described the physics of zero-field paramagnetic resonance with little reference to the experimental techniques used to measure zero-field magnetic resonance, or the practical design of a solid-state atomic frequency standard. The most important articles which described zero-field electron magnetic resonance spectrometer designs are those by Bramley et al. [1,2,3,4]. Some early spectrometers detect the change in attenuation or phase along a non-resonant transmission line containing the paramagnetic sample, however, a resonant design using a tunable resonator is much more sensitive. While apparently a simple problem, designing a broadly tunable, high-Q electrical resonator is non-trivial. One solution is to use a mechanically tuned “loop-gap” resonator, as will be described in this chapter. The loop-gap resonator is especially effective in paramagnetic resonance experiments because of the combination of high-Q and small volume. The loop-gap resonator is used in the proof-of-concept because of the broad tuning range which is useful for measuring different samples and features of the spectrum. For a prototype solid-state atomic frequency standard, smaller resonators such as a coaxial resonator made from magnesium oxide ceramic can be used with varactor tuning over a few percent bandwidth. As described in Chapter 2, paramagnetic resonance may be detected as a change in the resonant frequency and quality factor of an electrical resonator which contains a paramagnetic sample. The interaction between the paramagnetic resonance and the resonant magnetic fields cause a perturbation to both the frequency and losses of the 49 electrical resonator. A diagram of the prototype spectrometer is illustrated in Figure 4.1. The frequency discriminator consists of the electrical resonator, phase detector, hybrid coupler or power splitter, and phase adjuster. Helmholtz Coils Modulation Driver 90O Hybrid Voltage-Controlled Oscillator Electrical Resonator Lock-In Amplifier Phase Adjuster Frequency Sweep Frequency Counter Phase Detector Data Computer Figure 4.1: Zero-Field Electron Magnetic Resonance Spectrometer Fluctuations or modulation of the resonant frequency of the electrical resonator are demodulated by the frequency discriminator to generate a voltage proportional to the frequency deviation. Because the magnetic resonance signals are small, lock-in detection is necessary to isolate the magnetic resonance signal. The Helmholtz coils shown in Figure 4.0 apply an audio magnetic field to the paramagnetic sample inside the electrical resonator, which modulates the magnetic resonance signal because of the Zeeman shift (Chapter 2). The resulting frequency modulation of the electrical resonator is demodulated using the frequency discriminator and detected with the lock-in amplifier. 50 There are two key features of the zero-field magnetic resonance spectrometer design. The first is the tunable loop-gap resonator mentioned above, which is essential for adequate sensitivity (Figure 4.2): Coupling Loops Resonator Inductive Loop Resonator Shield Capacitive Slits Figure 4.2: Shielded Loop-Gap Resonator: Perspective, Side and Top View The loop-gap resonator consists of an inductive loop with two narrow capacitive slits, which form a lumped LC resonant circuit. The resonator is coupled via the mutual inductance between the small coupling loops and the resonator inductive loop Figure 4.2. The second is to use “bidirectional” magnetic field modulation, shown in the upper waveform of Figure. 4.3. This modulation scheme has the advantage of reducing spurious modulation pickup by 50 dB or more, compared to simple ON/OFF modulation. The response of the electron spin resonance transitions is symmetric to the inversion of the 51 magnetic field, and the desired modulation signal is measured at twice the fundamental, driving the lock-in amplifier with the lower waveform shown in Figure. 4.3. Bidirectional Modulation Waveform Square Wave at Second Harmonic Figure 4.3: Audio Modulation Waveform and Lock-In Multiplier Waveform The radio-frequency signal processing detects the frequency modulation of the electrical resonator containing the paramagnetic sample. Most electron spin resonance spectrometers detect the magnetic resonance absorption signal (i.e., change in the Q of the resonator) rather than dispersion (i.e., change in the frequency of the electrical resonator), and measure the reflection from a critically coupled cavity using a circulator. Frequency discriminators can be very sensitive, limited by thermal noise even at large RF carrier powers, and as such are a type of “null instrument” [6,7]. Fundamentally, the sensitivity of the frequency modulation measurement is limited by the intrinsic frequency fluctuations of the electrical resonator. For example, excessive noise from vibrations, causing random modulation of the electrical resonant frequency, can be a limiting factor in the sensitivity. The noise sources are analyzed in Chapter 5. 52 This chapter contains very specific details on the construction of the loop-gap resonator and electronic system, to provide a “recipe” for future researchers, and a list of materials and suppliers in the appendix. 4.2 Tunable Loop-Gap Resonator Design There are many examples of loop-gap resonators in the electron spin resonance literature [8,9,10,11,12.] The use of loop-gap resonators in place of waveguide cavity resonators is becoming more popular, especially for experiments on aqueous samples where L- and Sband spectrometers have less dielectric loss than at X-band. (a) Loop-gap resonator As previously discussed, Figure 4.2 depicts an electrical resonator. The loop-gap resonator comprises an inductive ring with one or more capacitive slits. In the prototype, there are two capacitive slits which allows for mechanical tuning over more than one octave by adjusting the capacitive gap using a flexure stage. The technique realizes a resonator which have both high-Q (in the range of 500) and wideband tuning. The resonant frequency is approximately equal to: 1 1 ⎛ t ⎞ 2πν = = n1 / 2 ⎜ ⎟ 1/ 2 r ( LC ) ⎝W ⎠ 1/ 2 ⎛ 1 ⎞ ⎟⎟ ⎜⎜ ⎝ πεµ 0 ⎠ 1/ 2 (4.1) Equation 4.1 does not account for fringing fields; more accurate semi-empirical equations are found in the literature [11]. The geometric quantities in the equation are illustrated below (Figure 4.4): 53 Resonator Shield Loop-Gap Resonator r Z t W Figure 4.4: Mechanical Dimensions of Loop-Gap Resonator In the prototype, the resonator height Z was approximately 0.5’’, the inner radius r was 0.3’’, the thickness was 0.1’’, and the capacitive gap t was tuned between 10 microns and 25 microns. The resonant frequency tunes from approximately 1-2 GHz. The prototype dimensions are shown in more detail in the manufacturing drawings (Appendix I). Coupling to the resonator is achieved using coupling loops soldered directly to SMA connectors mounted on the resonator shield. In the prototype, the coupling loops (Figure 4.2) are made from silver plated 18-gauge phosphor bronze wire. The inner diameter of the coupling loops is approximately 4 millimeters. The frequency of the loop-gap resonator can be mechanically tuned over more than one octave by varying the thickness of the two capacitive gaps, and therefore the frequency of the LC resonator. The center of the loop-gap resonator includes a paramagnetic sample during measurements, which interacts with the RF magnetic field inside the inductive ring. In addition to the advantages of high-Q, small volume, and broad tuning, the RF magnetic field inside the inductive loop is substantially uniform, and therefore, the paramagnetic resonance saturates uniformly at high carrier power. 54 Minimizing leakage of the RF signal stored in the loop-gap resonator is essential to prevent unwanted mixing between the RF and modulation signals. The best technique is to electroplate the resonator shield (Figure 4.2 and 4.4) of the loop-gap resonator to a thickness of approximately 10 microns, which provides maximum isolation. The audiofrequency magnetic field penetrates this thin layer of metal with very little attenuation, and the RF signal is completely shielded, because the skin-depth is typically much less than 10 microns. Cutting slits in a metal shield to allow the audio magnetic field to penetrate the sample is not a good design because RF isolation is poor. The loop-gap resonator is tuned by mechanically varying the thickness of the capacitive gaps, using a flexure stage as shown in Figure 4.5: Resonator Holder Piezo Adjust Silica Beam Adjustable Capacitive Gap Flexure Stage Figure 4.5: Side View of Mechanically Tunable Loop-Gap Resonator 55 The flexure stage adjusts the capacitance gap between approximately 10 microns and 25 microns, which tunes the resonator from ~ 800 MHz to 2 GHz. The resonator pieces may be coating with insulating material such as parylene to prevent shorting across the gap. A weakness of the present design is the sensitivity of the cantilevered resonator piece to acoustics and vibrations, which modulate the frequency of the loop-gap resonator. Acoustic pickup of the applied modulation signals is also possible, as vibrations are excited by the Lorentz force between the modulation coils or adjacent wires of a single coil. Further details of the design of loop-gap resonators are extensively elaborated in the scientific literature on electron spin resonance, see references [8,9,10,11,12]. Other techniques for implementing a multi-octave tunable loop-gap resonator are described in [1,2,3]. 4.3 Loop-Gap Resonator Manufacturing The ideal design for the resonator uses thin plated metal surfaces to provide complete RF isolation without shielding the audio magnetic field. RF leakage is to be avoided to prevent spurious mixing with the modulation signals. The loop-gap resonator and resonator shield were machined from MACORTM ceramic. Coupling loops are made from phosphor brass (Figure 4.2). Manufacturing drawings for the design used in the spectrometer prototype are provided in Appendix I, along with the part numbers and manufacturers contact information for the components used in the loopgap resonator in Appendix II. The ceramic pieces are coated with a conducting layer using a commercial silvering solution (HE-300) obtained from Peacock Laboratories, Inc. (Philadelphia, PA). Cleaning with red rouge prior to silvering (Mc-Master Carr P/N 4783A3) greatly improves the quality of the silvered film. The silvered MACOR pieces were then electroplated to approximately 10 microns silver by Hudson Plating Works, Inc. (Canoga Park, CA). In the first design, a Rexolite® rod was used to position the loop-gap 56 resonator. However, a silica rod is a better choice because of greater mechanical stability. Custom precision ground silica rods were manufactured by Rayotek Scientific, Inc. (San Diego, CA.) The glass rod is attached to the silvered MACOR piece using a small clamping fixture and a drop of cyanoacetate glue (Superglue). In a further aspect of the design, resonator holders is machined from Rexolite ® (C-Lee Plastics, Inc., Philadelphia, Pennsylvania) (not shown in Fig. 4.2). The precision flexure stage is used to adjust the resonator capacitive gap, which tuned the resonant frequency. The piezo adjuster was model MDE218 Piezo Adjuster (Elliot Scientific, UK), and the 150V piezo driver was Thorlabs model MDT 694. Noise from the piezo driver must be carefully filtered. It was found that a large electrolytic capacitor (330uF, 150V, Mallory) was an effective noise filter in combination with the 50 Ώ output impedance of the piezo driver. 57 4.4 Frequency Discriminator The frequency discriminator circuit is simply a mixer adjusted in quadrature with the electrical resonator in one path (Figure 4.6). Important considerations are the isolation of the power divider and the impedance matching of all paths. Reflection from the mixer or mismatch of the loop-gap resonator impedance cause unwanted power and phase variations as the frequency is swept. Resonator Tuning Voltage 90O Hybrid Loop-Gap Resonator Phase Adjuster Phase Detector Voltage-Controlled Oscillator Figure 4.6: Frequency Discriminator For an instrument, a hybrid coupler is ideal for the power divider because of high isolation between the RF and LO ports, which prevents reflected signals from circulating in the frequency discriminator. The mixer used in the prototype was a double balanced passive diode ring mixer, Mini-Circuits model JMS-5MH. The VSWR of this mixer at the LO and RF ports is approximately 2.9 in the 1000 – 1250 MHz range, equivalent to reflection magnitude |s11| as large as -6.3 dB. In all the measurements which follow, the RF port of 58 the mixer has a 6dB pad following the loop-gap resonator, not shown in Figure 4.6. The measured power variations at the RF port are shown below (Figure 4.7): Power at LGR output (+10dB coupler) -3 -4 -5 dB -6 -7 -8 -9 -10 1 1.05 1.1 1.15 1.2 1.25 Frequency (GHz) Figure 4.7: Power Variations at Mixer Input Because of Loop-Gap Resonator Mismatch The equivalent power at the RF port of the mixer is 4 dB larger, which accounts for the 10dB directional coupler and the 6 dB pad. The 6dB pad is used to improve the matching to the mixer RF port. The power at the LO port is between 11.4 and 12.8 dBm over the same 1000 – 1250 MHz band. Measurements of the prototype resonator s-parameters show that the large power variation in Figure 4.7 is due to mismatch of the loop-gap resonator output impedance. 59 |s21| Loop-Gap Resonator and 12'' Coaxial Leads 0 -5 -10 -15 dB -20 -25 -30 -35 -40 -45 -50 900 950 1000 1050 1100 Frequency (MHz) 1150 1200 1250 Figure 4.8 Measured Loop-Gap Resonator Transmission at 3 Frequencies The transmission characteristics of the prototype loop-gap resonator are shown above for the resonator tuned to ~1000 MHz, 1060 MHz, and 1150 MHz (Figure 4.8). 60 |s11| Loop-Gap Resonator and 12'' Coaxial Leads 0 -2 -4 dB -6 -8 -10 -12 -14 900 950 1000 1050 1100 Frequency (MHz) 1150 1200 1250 Figure 4.9 Measured Loop-Gap Resonator Reflection at 3 Frequencies The |s11| reflection measurements are plotted above in Figure 4.9. The VSWR results from a matching problem between the output impedance of the LGR (approximately 80Ω) and the input impedance of the mixer. 61 A plot of VSWR near the passband is shown below, with a minimum VSWR of 1.5: Loop-Gap Resonator: VSWR Near Passband 10 9 8 7 6 5 4 3 2 1 0 1060 1062 1064 1066 1068 1070 1072 Frequency (MHz) 1074 1076 1078 1080 Figure 4.10 Voltage Standing Wave Ratio without Series Matching Capacitor The VSWR of 1.5 corresponds to |s11| = -14dB, or an equivalent resistance of 75Ω. The power variations observed at the mixer would be 3.5 dB maximum in this case. The mismatched source will also cause phase variations different from the delay along the coax connecting the LGR to the mixer RF port, VSWR 1.5 corresponds to phase variations of ~ ±12º. The matching of the loop gap resonator is greatly improved by adding a series capacitor (6.8pF ATC 600S capacitor) at the coupling loop to cancel the coupling loop series inductance, which improves matching to better than -20 dB. 62 The measured improvement in VSWR with the improved resonator matching and using a hybrid coupler for better isolation (Krytar 4010180) is shown in Figure 4.11, with less than 1dB power variation: Closed-Loop Power at Mixer RF Port - improved LGR matching, used hybrid coupler for better RF/LO isolation 0.2 0 -0.2 dB -0.4 -0.6 -0.8 -1 -1.2 -1.4 1 1.05 1.1 1.15 1.2 1.25 Frequency (GHz) Figure 4.11 Power Variations at Mixer Input Port with Improved Resonator Matching The power variations shown above can be compared to the much larger variations in Figure 4.7 where the matching of the loop-gap resonator is not optimized. 63 4.5 Automatic Frequency Control The loop-gap resonator frequency is adjusted to track the frequency of the voltagecontrolled oscillator using a simple integrator to lock the mixer input in quadrature (Figure 4.12): Noise Filter Piezo Driver Resonator Tuning Voltage 90O Hybrid Integrator and Loop Filter Loop-Gap Resonator Phase Adjuster Phase Detector Voltage-Controlled Oscillator Figure 4.12 Frequency Discriminator with Automatic Frequency Control In the prototype, the bandwidth of the loop is on the order of 1 Hz. The loop is much slower than the modulation frequency (kHz range) used to detect magnetic resonance, and therefore the frequency control does not compensate the desired signals. 64 4.6 Modulation Waveforms and Modulation Driver The bidirectional waveform greatly reduces spurious pickup of the modulation waveform. Also, ac coupled audio power amplifiers may be used to process the bidirectional waveform, because the average is zero. The modulation driver needs to drive currents of several amps to an inductive load of ~ 0.25mH. For an instrument, power consumption is not a critical constraint, and the best design is to reduce the inductance of the modulation coils by using fewer turns and driving the maximum current. One driver design used an audio power op-amp (National Semiconductor LM675) and a current feedback arrangement to drive the modulation coil. The bidirectional waveform was generated from a square-wave using a divide-by-two circuit and multiplier. Uniform Magnetic Field Audio Power Amplifier Helmholtz Coil Pair Modulation Waveform Figure 4.13: Magnetic Field Modulation A second design uses a commercial 200W audio power amplifier (JL Audio Model 1200e monoblock full-range amplifier). The audio power amplifier is powered from a car battery, and has good linearity. The bidirectional waveform is generated using direct-digital synthesis from an arbitrary waveform generator (HP Model 33120A). The audio amplifier can drive up to 10 A at 2 Ohms. Running from a car battery is also useful in preventing large modulation currents from coupling to earth ground and common mode voltage. Another possibility is to pre-distort the audio waveform in software to adjust the driving voltage so that the current through the inductor better approximates a square wave. 65 4.7 Helmholtz Coil Design and Magnetic Field Homogeneity The loop-gap resonator is place between two modulation coils, which are used to apply an audio magnetic field to the paramagnetic sample. Helmholtz coils are a pair of wire coils separated by a distance equal to the radius of the two coils. The reason Helmholtz coils are used in magnetic resonance spectrometers is because the magnetic field inside a Helmholtz coil pair is uniform over approximately 2/3 of the radius. In the prototype spectrometer, the diameter of the coils is approximately 1.8’’ inches, and the spacing of the two coils is 0.9’’ inches. Each coil has approximately 30 turns and the two coils are connected in parallel. The magnetic field at the center of the coils is (in Gauss): H ≈ 0 .9 nI a (4.2) where n is the number of turns, I is in amperes, and radius a in centimeters. For the parameters given above, H ≈ 0 .9 30 I = 11.8 G A 0.9 ⋅ 2.54cm (4.3) The LM675 driver circuit is limited to a peak current of approximately 2A, and if the coils are in parallel, the peak magnetic field is approximately 12 Gauss. A typical sample of magnesium oxide used in the experiment has dimensions of roughly 1 centimeter on a side, and is placed in the center of the loop-gap resonator. The local magnetic field can be calculated from the Biot-Savart law. The magnetic field along the zaxis through the center of the coils is: 66 H= µ 0 nI ⎛⎜ ( a2 2 ⎜⎝ 5 4 a 2 + az + z 2 + 3/ 2 ) ⎞ ⎟ 2 2 3/ 2 ⎟ 5 4 a − az + z ⎠ ( a2 (4.4) ) where z is the distance from the center of the coil pair, and the magnetic field is directed along the z-axis. The calculated field is plotted below (Figure 4.14), and is uniform to within 0.05 Gauss within ±0.5 cm from the center: 12 Helmholtz Coil Magnetic Field Variation along Center Axis 11.9 Magnetic Field (G) 11.8 11.7 11.6 11.5 11.4 11.3 11.2 -1.5 -1 -0.5 0 0.5 Vertical Distance from Center (cm) 1 1.5 Figure 4.14 Helmholtz Coil Axial Magnetic Field Variation The radial variation along the plane is seen to be uniform over approximately 2/3 of the radius using a similar calculation. Therefore, the modulation field at the sample is uniform to better than 1% of the peak magnetic field. 67 4.8 Voltage-Controlled Oscillator and Phase Noise The voltage-controlled oscillator used in the prototype was Z-Communications, Inc. model CLV1100E. The supply and varactor node must be carefully filtered to achieve optimal phase noise performance. A measurement of the CLV1100E phase noise is shown in Figure 4.7.1, which can be compared to the phase noise output of a Hewlett-Packard Model 8665B Synthesized Signal Generator. The phase noise at 1 kHz offset from carrier is 80dBc/Hz in both cases. Z-Comm CLV1100E Phase Noise @ 950 MHz, use LT1761 Regulator w/ Noise Bypass -20 -40 dBc/Hz -60 -80 HP8665B -100 -120 -140 -160 0 10 1 10 2 10 3 4 10 10 Carrier Offset (Hz) 5 10 Figure 4.15 Measured Voltage-Controlled Oscillator Phase Noise 6 10 68 4.9 Grounding and Shielding An important consideration in the RF system design is to prevent leakage of the RF signal to avoid unwanted modulation with the driver circuit. This is one reason why the electroplated resonator design is essential, because by using an electroplated shield, the audio modulation field penetrates the sample without attenuation, and the radio-frequency field is completely contained, since the metal thickness is several times the RF skin-depth. Ground is also an important consideration. Isolating the large currents from the modulation driver is absolutely essential. This can be achieved by using floating differential signal inputs to the driver amplifier, which is standard practice in audio design. A good reference on grounding and shielding is the book by H. Ott [13.] 4.10 Sample Preparation and Heat Treatment The vanadium doped magnesium oxide samples used in the experiment were signal crystal magnesium oxide grown using an arc furnace, obtained from Escete Technology (The Netherlands.) Single crystal samples are not essential, and ceramic samples can be fabricated by sintering high purity powders. The heat treatments described below were performed at Caltech using a tube furnace equipped with a gas flow controller. In the magnesium oxide samples as grown, most of the vanadium is incorporated in the trivalent state. Heat treatment in a reducing atmosphere is essential to convert a significant fraction of the vanadium ions to the divalent state [14,15,16]. (The paramagnetic resonance spectrum of the trivalent state cannot be measured at room temperature because of the nonzero orbital angular momentum, a therefore large spin-lattice coupling.) Annealing at 1100°C in an atmosphere 4 parts argon and 1 part hydrogen is one effective recipe for 69 reducing the vanadium ions to the divalent state. Excessive heat treatment results in formation of a black precipitate in the magnesium oxide samples. The X-band (i.e., high-field) spectrum of a sample of magnesium oxide as grown is shown below in Figure 4.16, where the six larger lines are from the manganese ions (I = 5/2), and the eight smaller resonances are from divalent vanadium (I = 7/2): 4 1 V++ / Mn++ MgO ESR Spectrum x 10 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 3200 3300 3400 3500 3600 Gauss 3700 3800 3900 Figure. 4.16 V++/Mn++/MgO Spectrum in Samples as Grown The spectra shown in Figures 4.16 and 4.17 were measured at Caltech using a commercial X-band ESR spectrometer. The effect of annealing the sample is shown very clearly in the next figure, where the eight lines of the vanadium spectrum are now much more intense than the manganese spectrum (Fig. 4.17). The concentration of divalent manganese ions can be taken as constant in comparing the two figures. 70 4 1.5 V++ MgO spectrum H2 anneal x 10 1 0.5 0 -0.5 -1 -1.5 3200 3300 3400 3500 3600 Gauss 3700 3800 3900 Figure 4.17 Spectrum after Annealing 6 hours @ 1100 °C in H2/Argon Atmosphere Given that the concentration of ions is proportional to the area under the resonance curve, we can see that the divalent vanadium concentration is increased by at least a factor of ten following heat treatment. Other possible heat treatments are discussed in the article by Wertz et. al. [14]. The chemistry of magnesium oxide, and the effects of various heat treatments and doping combinations can involve many factors [17,18,19,20]. Precipitates may be formed while annealing [19,20], as was observed for samples annealed for 12 hours at 1100°C, where a black precipitate is formed in the absence of oxygen. Therefore, developing a “recipe” for growing high-purity magnesium oxide ceramic containing divalent vanadium is a subject for further study. 71 BIBLIOGRAPHY, CHAPTER 4 [1] Christopher D. Delfs and Richard Bramley, “Zero-field Electron Magnetic Resonance Spectra of Copper Carboxylates,” J. Chem. Phys. 107, 8840, 1997. [2] Christopher D. Delfs, Richard Bramely, “The Zero-Field ESR Spectrum of a Copper Dimmer,” Chemical Physics Letters, 264, 333, 1997. [3] Richard Bramley and Steven J. Strach, “Zero-Field EPR of the Vanadyl Ion in Ammonium Sulfate,” J. of Magnetic Resonance, 61, 245, 1985. [4] Richard Bramley and Steven J. Strach, “Electron Paramagnetic Resonance Spectroscopy at Zero Magnetic Field,” Chem. Rev., 83, 49-82, 1983. [5] Pilbrow, J.R., Troup, G. J., Tirkel, A. Z., Hutton, D. R., Gruner, L., and McLaren, N. R., United States Patent 4,803,624, “Electron Spin Resonance Spectrometer,” February 7, 1989. [6] Eugene N. Ivanov, M. E. Tobar, R. A. Woods, “Microwave Interferometry: Application to Precision Measurements and Noise Reduction Techniques,” IEEE Trans. on Ultrasonics, Ferroelectrics, and Frequency Control, 45, 1526, 1998. [7] Eugene N. Ivanov, Micheal E. Tobar, Richard A. Woode, “Applications of Interferometric Signal Processing to Phase-Noise Reduction in Microwave Oscillators,” IEEE Trans. on Microwave Theory and Techniques, 46, 1537, 1998. [8] W. Piasecki, W. Froncisz, and James S. Hyde, “Bimodal Loop-gap Resonator,” Rev. Sci. Instrum., 67, 1896, 1996. [9] S. Pfenninger, W. Froncisz, J. Forrer, J. Luglia, and James S. Hyde, “General Method for Adjusting the Quality Factor of EPR Resonators,” Rev. Sci. Instrum., 66, 4857, 1995. [10] T. Christides, W. Froncisz, T. Oles, James S. Hyde, “Probehead with Interchangeable Loop-gap Resonators and RF Coils for Multifrequency EPR/ENDOR,” Rev. Sci. Instrum., 65, 63, 1994. [11] W. Froncisz and James S. Hyde, “The Loop-gap Resonator: A New Microwave Lumped Circuit ESR Sample Structure,” J. Magnetic Resonance, 47, 515, 1982. [12] W. N. Hardy and L. A. Whitehead, “Split-ring Resonator for use in Magnetic Resonance from 200-2000 MHz,” Rev. Sci. Instrum., 52, 213, 1981. 72 [13] Henry W. Ott, Noise reduction techniques in electronic systems, New York, Wiley 1988. [14] John E. Wertz, John W. Orton, and P. Auzins, “Spin Resonance of Point Defects in Magnesium Oxide,” Journal of Applied Physics, Supplement to vol. 33, no. 1, pp. 322328, January 1962. [15] D. H. Dickey and J. E. Drumheller, “Forbidden Hyperfine Transitions in the Electron Paramagnetic Resonance of V++ in MgO,” Physical Review, vol. 161, no. 2, pp. 279282, September 1967. [16] M. D. Sturge, “Optical Spectrum of Divalent Vanadium in Octahedral Coordination,” Physical Review, vol. 130, no. 2, pp. 639-646, April 1963. [17] W. D. Kingery ed. (MIT), Structure and Properties of MgO and Al203 Ceramics, Advances in Ceramics, vol. 10, The American Ceramic Society, Inc., 1984. [18] A.F. Henriksen, “Precipitation in MgO,” Ph.D. Thesis, M.I.T., 1978. [19] K.N. Woods and M.E. Fine, “Nucleation and Growth of Magnesioferrite in MgO Containing 0.9 % Fe3+,” J. Am. Ceram. Soc., vol. 52, no. 4, pp. 186-8, 1969. [20] W. H. Gourdin, W. D. Kingery, and J. Driear, “The Defect Structure of MgO Containing Trivalent Cation Solutes: The Oxidation-Reduction Behavior of Iron,” J. Mater. Sci., vol. 14, pp. 2074-82, 1979. 73 APPENDICIES, CHAPTER 4 74 Appendix I: Manufacturing Drawings for Loop-Gap Resonator 75 Appendix II: Contact Information for Manufacturers and Suppliers: Loop-Gap Resonator Components: Precision Ground Fused Silica Rods: Rayotek Scientific, Inc. 11494 Sorrento Valley Rd., Suite J San Diego, CA 92121 Ph: (858) 558-3671 Fax: (858) 558-6213 Custom Electroplating: Hudson Plating Works 8740 Remmet Avenue Canoga Park, CA 91304 Ph: (818) 709-5409 Fax: (818) 709-1045 www.hudsonplatingworks.com Silvering Solutions: Peacock Laboratories, Inc. 54th & Paschall Ave., Philadelphia, PA 19143 Ph: (215) 729-4400 Fax (215) 729-1380 www.peacocklabs.com HE-300 A, B, and C 77-cleanser 93-sensitizer Rouge (for preparing MACOR surface): Mc-Master Carr www.mcmaster.com Jeweler’s Rouge, Red, P/N 4783A3 MACOR® (Corning Inc., Corning, New York): Astro Met, Inc. 9974 Springfield Pike Cincinnati, Ohio, 45215-1425 Ph: 513-772-1242 Fax: 513-772-9080 www.astromet.com Rexolite®: C-LEC Plastics, Inc. Philadelphia, Pennsylvania Ph: (215) 708 7731 Fax: (215) 708 7728 www.rexolite.com 76 SMA Connectors: Pasternack Enterprises, Inc. PO Box 16759 Irvine, CA USA 92623-6759 Ph: (949) 261-1920 Fax (949) 261-7451 e-mail: sales@pasernack.com www.pasternack.com PE4122, SMA female panel mount, 4-hole, solder attachment for RG405 Flexure Stage and Piezo Driver: Thorlabs, Inc. 435 Route 206 North Newton, NJ 07860 Ph: 1-973-579-7227 Fax: 1-973-300-3600 E-mail: sales@thorlabs.com www.thorlabs.com 3-Axis Flexure Stage Single Channel Piezo Driver Model MDT694 Piezo Adjuster: Elliot Scientific, Ltd. 3 Allied Business Center, Coldharbour Lane, Harpenden, Herts. AL5 4UT Ph: +44(0) 1582 766300 Fax: +44(0) 1582 766340 E-mail: sales@elliotscientific.com MDE218 Piezo Adjuster Frequency Discriminator RF Components: Phaseshifter: Advanded Technical Materials, Inc. 49 Rider Avenue, Patchoque, NY 11772 Ph: 631-289-0363 Fax: 631-289-0358 E-mail: atm@atmmicrowave.com www.atmmicrowave.com P/N P1603D, 90°/GHz Phaseshifter Quadrature Hybrid: Krytar, Inc. 1292 Anvilwood Court, Sunnyvale, CA 94089 Ph: (408) 734-5999 Fax (408) 734-3017 Toll Free: 1 (877) 734-5999 sales@krytar.com www.krytar.com Model 3005070 90° Hybrid Coupler 77 Voltage Controlled Oscillator: Z-Communications, Inc. 9939 Via Pasar, San Diego, CA 92126 Ph: (858) 621-2700 Fax: (858) 621-2722 CLV1100E Voltage Controlled Oscillator, MINI-14S – Style Package Frequency Mixer: Mini-Circuit, Inc. P.O. Box 350166, Brooklyn, New York 11235-0003 Ph: (718) 934-4500 Fax: (718) 332-4661 Distribution 800-654-7949 417-335-5935 Fax 417-335-5945 ZFM-150 Frequency Mixer, 10dBm, 2 GHz Magnesium Oxide: Single Crystal Vanadium Doped Magnesium Oxide: eSCeTe Single Crystal Technology B. V. Ir. Schiffstraat 220 NL-7500 DW Enschede The Netherlands Ph: +31 53 435 6146 Fax: +31 53 435 2134 escete@escete.com Software and Equipment: Leveled RF Power: Hewlett Packard 8347A RF Amplifier 100 kHz – 3 GHz Frequency Counter: Hewlett Packard 5350A Microwave Frequency Counter Power Meters: Hewlett Packard 437B Power Meter Hewlett Packard E4418B EPM Series Power Meter Power Supply for DC Modulation Coils: Agilent E3644A DC Power Supply Data Acquisition Card: DaqBoard/500TM www.iotech.com Software: National Instruments LabVIEW 7 Express, GPIB Interface  78 Chapter 5 SENSITIVITY 5.1 Introduction This chapter analyses the noise sources in the zero-field electron spin resonance spectrometer. These considerations are very important because, as will be discussed in Chapter 6, the width of the atomic resonance and the sensitivity of the atomic resonance measurement determine the short-term stability of a passive atomic frequency standard. The three basic components of the spectrometer which contribute to random instabilities are (Figure 5.1): (1) frequency fluctuations and tuning voltage noise of the loop-gap resonator, (2) phase noise of the voltage-controlled oscillator, and (3) the noise floor of the mixer or receiver. In addition, technical sources of background interference include acoustic pickup by the loop-gap resonator, magnetic pickup of the audio modulation field, and spurs on the RF carrier from 60 Hz pickup. The following three sections will analyze the random and technical noise sources of each of the basic components, along with possible improvements to the prototype design. 79 Resonator Tuning Voltage Oscillator Phase-Noise Loop-Gap Resonator Phase Adjuster Receiver Noise Floor Voltage-Controlled Oscillator Figure 5.1: Noise Sources in the Frequency Discriminator 5.2 Resonator Noise The loop-gap resonator is tuned using a piezo adjuster to control the capacitive gap. Noise from the piezo driver modulates the resonant frequency. Similarly, common-mode pick-up of the magnetic field modulation causes a spurious background at the spectrometer output. In the ideal case, the two pieces of the loop-gap resonator are parallel, and therefore, the capacitive gap has a uniform spacing, as shown in Figure 5.2: 80 Silica Beam Piezo Adjust Resonator Holder Adjustable Capacitive Gap Flexure Stage Figure 5.2: Tunable Loop-Gap Resonator Side View The resonator is tuned using the piezo adjuster, which controls the approach of the capacitive gap. In the prototype, this gap is between 10 microns and 25 microns, and consequently the alignment of the resonator critical. In practice, the two resonator pieces are not perfectly aligned and at some lower frequency the resonator will short. A 150 V piezo driver is used to adjust the resonator gap and therefore the tuning. Noise from the driver causes a random modulation of the resonator gap, and therefore random frequency modulation of the resonator, which adds to the spectrometer noise floor. One method to analyze the contribution is to measure the resonator frequency as a function of piezo voltage. The prototype loop-gap resonator tunes from ~ 1050 MHz @ 150 V to 1250 MHz @ 0V, with a mean slope of approximately 1.3 MHz / V. The signal which results at the output of the frequency discriminator from noise on the piezo driver is as follows. 81 First, analyze the result of a small sinusoidal voltage on the piezo electrodes: ∆ V = v ⋅ sin( ω m t ) (5.1) The loop-gap resonator is frequency modulated ∆f = K ⋅ v ⋅ sin(ω mt ) (5.2) In the equation, K is the slope of the frequency tuning curve in MHz/V. The resulting phase modulation of the signal coupled through the resonator is: ∆θ ≅ 2QL ⋅ K ⋅ v ⋅ sin(ω m t ) f0 (5.3) In the equation, f0 is the center frequency of the resonator (MHz) and QL is the loaded-Q of the coupled resonator. In Figure 5.1, the mixer inputs are adjusted in quadrature to implement a phase detector, and assuming 6dB conversion loss: ∆VM = Q 1 1 ⋅ VRF ⋅ sin θ ≅ ⋅ VRF ⋅ ∆θ = VRF ⋅ L ⋅ K ⋅ v ⋅ sin(ω m t ) 2 f0 2 (5.4) Therefore, noise with a white noise spectrum from the piezo driver results in a white noise floor at the frequency discriminator output, with a transfer function: Vout Q K = V RF L V piezo f0 (5.5) For example, assuming the carrier power at the RF port of the mixer is 0dBm, the resonator loaded-Q is ~400, the measurement frequency is 1100 MHz, and the sensitivity of the resonator tuning voltage is of 1.3 MHz/V, the transfer function Equation 5.5 is 0.15. Noise on the piezo driver equal to 6nV Hz results in 0.9nV 82 Hz at baseband, i.e., is equivalent to the thermal noise of a 50 Ohm resistor at 300K. A large electrolytic capacitor on the output of the piezo driver is an effective noise filter in combination with the 50 Ohm output impedance of the driver. This capacitor limits the bandwidth of the automatic frequency control loop used to maintain tracking between the local oscillator frequency and resonator center frequency. Noise and spurs from the piezo driver pose a problem if a faster loop is required. A second consideration, which is very significant in the prototype design, is the sensitivity of the tunable resonator to vibrations and acoustics. Referring to Figure 5.2, vibrations of the flexure stage platform cause motion of the cantilevered beam holding one piece of the loop gap resonator. Sound waves also excite the beam. The combination of high-Q and a small capacitive gap means the resonator is very sensitive to these mechanical sources of noise. Vibrations and Acoustics The mechanically tuned loop-gap resonator used in the prototype is sensitive to vibrations and acoustics because of the very close spacing of the capacitor electrodes (~ 10 microns) and a mechanical design which uses a cantilevered beam from the flexure stage. The resonant frequency is given by: ( f 0 )2 = 1 g = 2 2 (2π ) LC (2π ) Lε 0 A (5.6) and variations in the resonator frequency with the capacitance gap g are: ∆f 0 1 ∆g = ⋅ f0 2 g (5.7) The frequency deviation cause by mechanical vibration results in a phase variation of the carrier and therefore a voltage at the frequency-discriminator output: 83 ∆VM ≅ Q 1 ∆g 1 1 ⋅ VRF ⋅ ∆θ = VRF ⋅ L ⋅ f0 2 2 f0 2 g (5.8) In practice, the largest vibration interference is below 500 Hz, which depends on the mechanical resonances of the loaded beam used to position the resonator (Figure 5.2). Choosing a magnetic field modulation frequency above 500 Hz is essential to avoid this interference. Any excessive vibration saturates the receiver chain. Modulation Coil Acoustics The Lorentz force between the modulation coils and windings generates sound and vibrations: Magnetic Field Direction Axial Current Direction Radial Current Direction Figure 5.3 Lorentz Force Between Helmholtz Coils Depending on the angle of fringing of the magnetic field at the modulation coils, there is a net force directed along the axis of the coil pair. There is no net force in the radial direction for circular symmetry. The Lorentz force is proportional to the square of the current, and therefore the vibrations or acoustics of the modulation coils generate acoustics at the second harmonic of the bidirectional modulation waveform. This signal is not rejected by the modulation waveform scheme discussed in Chapter 4 (Figure 4.3). A typical example of the measured background signal is shown below: Volts at Spectrometer Output (Modulation ON, Modulation OFF) 84 1 0 -1 -2 -3 -4 -5 -6 -7 1 1000 1 05 1050 11 1100 1 15 1150 12 1200 1 25 1250 Frequency (MHz) Figure 5.4 Background Pickup Signal During Spectrometer Sweep Notice that the sensitivity to pickup increases at lower frequencies because the capacitive gap g is smaller, as in Equation 5.8. An improved tunable resonator design would take into account the problem with acoustic pickup. The current design, which uses a cantilevered beam for positioning the resonator, is especially sensitive to acoustics. 5.3 Receiver Noise Figure Thermal noise is the only fundamental noise source in the frequency discriminator (Figure 5.1), provided the amplitude and phase noise of the carrier can be adequately reduced. The prototype design is far from optimal, in that no low-noise amplifier is used ahead of the mixer, and a 6 dB attenuator is used to match the mixer input port, with an equivalent loss 85 of power. In a refined design, however, thermal noise would be the limiting noise source. Carrier-suppression techniques may also be used to reduce noise from the source to below the receiver noise floor [1,2,3]. Another consideration is 1/f noise of the low-noise amplifier and mixer at high carrier powers, which has been studied for precision frequency discriminators [4]. 5.4 Oscillator Phase Noise This section analyzes the contributions of phase and amplitude noise of the oscillator to the measurement noise floor. While amplitude noise is shown to be a second-order effect when a frequency discriminator is used for measuring magnetic resonance, the phase noise of the oscillator is often the dominant random noise source in the spectrometer or frequency standard. Techniques for reducing the close-in phase noise of a voltage-controlled oscillator are well known [3]. In particular, a second frequency discriminator may be used to measure and suppress (using feedback) the oscillator phase noise. The prototype oscillator uses a simple voltage-controlled LC oscillator. A coaxial resonator oscillator could have up to 20 dB better phase noise. Phase Noise Consider a phase modulated carrier: V (t ) = V cos[2πf c t + β sin( 2πf m t )] (5.9) Assuming β << 1 this phase modulated signal is approximately equal to: V (t ) ≅ V cos(2πf c t ) − βV sin( 2πf m t ) sin( 2πf c t ) (5.10) V (t ) ≅ V cos(2πf c t ) − βV 2 86 [cos(2πf c t − 2πf m t ) − cos(2πf c t + 2πf m t )] (5.11) The resonator dispersion introduces delay to the modulation sidebands equal to: ∆θ m = ± 2QL f m f0 (5.12) and therefore the signal couple coupled through the resonator is βV [cos(2πf c t − 2πf m t − ∆θ m ) − cos(2πf c t + 2πf m t + ∆θ m )] (5.13) βV ⎡ [cos(2πfct − 2πfmt − ∆θm ) − cos(2πfmt + 2πfmt + ∆θm )]⎤⎥ ⋅ [V sin(2πfct) + βV sin(2πfmt ) cos(2πfct )] V (t ) ≅ ⎢V cos(2πfct ) − 2 ⎣ ⎦ (5.14) V (t ) ≅ V cos(2πf c t ) − 2 The mixer output is: V (t) ≅ V 2 [β sin(2πf mt) ⋅ cos2 (2πf c t) − β 2 [cos(2πf ct − 2πf mt − ∆θm ) − cos(2πf mt + 2πf mt + ∆θm )] ⋅ sin(2πf ct)] 1 1 V (t ) ≅ V 2 β [sin(2πf m t ) ⋅ − [sin(2πf m t +∆θ ) + sin(2πf m t + ∆θ )]] 2 4 V (t ) ≅ V 2 β 2 [sin(2πf mt ) − sin(2πf mt +∆θ )] ≅ V 2 β 2 ∆θ cos(2πf mt) ≅ V 2 β 2QL f m 2 ⋅ f0 cos(2πf mt ) (5.15) (5.16) (5.17) The single-sideband power of the phase modulation is: PFM , NB ,SSB = 1 2 β 2 which results in a modulated signal power at the frequency discriminator output: (5.18) 87 PPN , SSB = 1 2 ⎛ QL f m ⎞ ⎟ β ⎜ 2 ⎜⎝ f 0 ⎟⎠ 2 (5.19) Hence, the transfer function between the phase-noise sideband power and the baseband noise floor is: ⎛Q f ⎞ = ⎜⎜ L m ⎟⎟ PPhaseNoise ⎝ f 0 ⎠ Pout 2 (5.20) In the above, fm is the offset from carrier. Equation 5.20 shows that the 1/f2 phase noise region of the source generates a white noise power spectrum at baseband, and the 1/f3 phase noise region is demodulated to 1/f noise. If the RF carrier power is 0dBm and the phase noise is -80dBc/Hz at 3.3 kHz offset (see Chapter 4.7), then the noise power at baseband is: 2 ⎛ 400 * 3.3kHz ⎞ P=⎜ ⎟ − 80dBm = −138dBm ⎝ 1100MHz ⎠ The noise power is equivalent to 27nV (5.21) Hz at 50 Ohms. The phase noise may easily dominate the noise floor of the spectrometer. Amplitude Noise The amplitude noise of the oscillator is a second-order effect, as will be shown below. This “nulling” of the carrier amplitude noise is one reason for using the frequency discriminator design. Consider a sinusoidally amplitude modulated carrier: 1 V (t ) = V cosω mt cosω ct = V [cos(ω m − ω c )t +cos(ω m +ω c)t ] 2 The response of the loop-gap resonator is: (5.22) 88 Vo 2 s RC = 2 Vi s + 2 s RC + 1 LC (5.23) Vo 2 jω RC = 2 Vi ω0 − ω 2 + 2 jω RC (5.24) Vo 2 j (ω0 + ω m ) RC 2 jω0 RC j RC = 2 ≅ = 2 Vi ω0 − (ω0 + ω m ) + 2 j (ω0 + ω m ) RC − 2ω0ω m + 2 jω0 RC − ω m + j RC (5.25) The approximation is valid to the order of the ratio of the modulation frequency to the carrier frequency. For example, if the modulation frequency is 10 kHz and the carrier frequency is 1 GHz, the ratio is 10-5. The approximation is extremely accurate up to modulation frequencies of 1 MHz (~ 0.1%). The phase shift of the AM modulation sidebands is: θ= π − tan −1 (1 ω m RC ) (5.26) − tan −1 ( f 0 2QL ∆f ) (5.27) 2 or in terms of loaded-Q: θ= π 2 The dispersion at the resonator center frequency is: ∂θ 2Q = L ∂∆f ∆f = 0 f0 (5.28) and the approximate phase shift of the modulation sidebands is: ∆θ m = ± 2QL f m f0 (5.29) 89 If the mixer is driven in quadrature, and assuming 6dB conversion loss, the mixer output is: 1 VB (t ) = V [cos((ω c − ω m )t − ∆θ m ) +cos((ω c +ω m )t + ∆θ m )]sin(ω c t ) 2 (5.30) 1 VB (t ) ≅ V [sin (ω m t + ∆θ m ) − sin (ω m t + ∆θ m )] 4 (5.31) which is zero provided that the resonator has flat dispersion. The amplitude noise of the carrier is therefore a second order effect. If the frequency discriminator phase is badly detuned from the resonator center frequency, then amplitude noise may be important. 60Hz Spurs A final technical noise source is magnetic pick-up of 60 Hz, which results in 60 Hz harmonics at the mixer output and may dominate the noise floor depending on the lock-in amplifier bandwidth. 90 5.5 Prototype Sensitivity The chart summarizes the contributions of the various noise mechanisms to the system noise floor. The most important are the phase noise of the RF source and acoustic pickup from the modulation coils. Noise Source Oscillator Phase Noise Transfer Function ⎛Q f ⎞ Pout = ⎜⎜ L m ⎟⎟ PPhaseNoise ⎝ f 0 ⎠ 2 Techniques Reduce close-in phase noise or use carrier suppression design Piezo Driver Noise Vout Q K = V RF L f0 V piezo Driver output filter Receiver Noise Figure -171 dBm/Hz @ 3dB NF Fundamental Acoustics Q 1 ∆g 1 ∆VM ≅ VRF ⋅ L ⋅ f0 2 f0 2 g Re-Design Resonator 91 BIBLIOGRAPHY, CHAPTER 5 [1] F. L. Walls, “Suppressed Carrier Based PM and AM Noise Measurement Techniques,” Proc. IEEE International Frequency Control Symposium, pp. 485-492, 1997. [2] E. S. Ferre-Pikal, M. C. Delgado Aramburo, F. L. Walls, K. M. Lakin, “1/f Frequency Noise of 2-GHz High-Q Thin-Film Sapphire Resonators,” IEEE Trans. on Ultrasoonics, Ferroelectrics, and Frequency Control, vol. 48, no. 2, pp. 506-510, March 2001. [3] E. N. Ivanov, M. E. Tobar and R. A Woode, “Advanced Phase Noise Suppression Technique for Next Generation of Ultra-Low Noise Microwave Oscillators,” Proc. IEEE International Frequency Control Symposium, pp. 314320, 1995. [4] D. Halford, A. E. Wainwright, and J. A. Barnes, “Flicker Noise of Phase in RF Amplifiers and Frequency Multipliers: Characterization, Cause and Cure,” Proc. IEEE Frequency Control Symposium, pp. 340-341, 1968.  92 Chapter 6 FREQUENCY STABILITY 6.1 Frequency Stability of a Passive Atomic Frequency Standard This section calculates the frequency stability of a passive atomic frequency standard, which depends on the width of the atomic resonance signal, the sensitivity of the atomic resonance measurement, the free-running phase noise and the bandwidth of the frequency control loop. Many atomic clocks use a simple integrator to lock the voltage-controlled oscillator to the atomic resonance signal (Figure 6.1): Voltage-Controlled Oscillator Atomic Resonance Noise Floor Integrator Oscillator Phase-Noise Figure 6.1: Passive Atomic Frequency Standard 93 Assume the slope of the voltage-controlled oscillator frequency-voltage tuning curve is KV, and the slope of the atomic resonance error signal is: ⎛ ∂E ⎞ ⎜⎜ ⎟⎟ ⎝ ∂f ⎠ (6.1) The difference in the frequency of the voltage-controlled oscillator and the center of the atomic resonance is: ∆f = f O − f A (6.2) The transfer function of the integrator is modeled using: H (s) = − 1 RCs (6.3) In the figure above, the loop gain is given by the product of these three quantities: A( s ) = − ⎛ ∂E ⎞ G 1 K V ⎜⎜ ⎟⎟ = − RCs s ⎝ ∂f ⎠ (6.4) When the system is open loop, a noise voltage vn at the integrator input generates a signal VE proportional to A(s) at the output of the atomic resonance measurement. The equation which describes the closed-loop system is: (v n + V E ) A( s ) = V E (6.5) and therefore H 1 ( s) = VE A( s ) −G = = v n 1 − A( s) s + G The frequency deviation of the voltage-controlled oscillator is: (6.6) 94 ∆f = VE ∂E ∂f (6.7) The transfer function between injected noise vn and the oscillator frequency deviation is: 1 ∆f = H 1 (s) v n ⎛ ∂E ⎞ ⎜⎜ ⎟⎟ ⎝ ∂f ⎠ (6.8) For atomic clocks, the power spectrum of the normalized frequency deviation is conventional [1,2]: y (t ) ≡ f O− f A ∆f = fA fA (6.9) The transfer function between the normalized frequency deviation and the receiver noise is: y = vn 1 H 1 ( s) ⎛ ∂E ⎞ ⎟⎟ f A ⎜⎜ ⎝ ∂f ⎠ (6.10) and, therefore, the power spectral density of the voltage noise Sn is related to the power spectral density of the frequency deviation Sy according to: Sy = Sn 2 ⎛ ∂E ⎞ f A ⎜⎜ ⎟⎟ ⎝ ∂f ⎠ 2 H 1 (s) 2 (6.11) For frequency offsets within the loop bandwidth, the frequency stability is: S y, A = Sn ⎛ ∂E ⎞ f A ⎜⎜ ⎟⎟ ⎝ ∂f ⎠ 2 2 (6.12) 95 The frequency stability improves by decreasing the noise floor or increasing the slope of the error signal. For example, if the slope of the error signal is 1µV / MHz, the receiver noise floor is 1nV / Hz , and the resonance frequency is 1100 MHz, then the power spectrum of the normalized frequency deviations is approximately 10-12. To take into account the phase noise of the voltage-controlled oscillator, we can show that the overall frequency stability is [3,4]: 2 S y = S y ,O H 1 ( s ) + S y , A H 2 ( s ) 2 (6.13) where Sy,O is the (open-loop) frequency spectrum of the voltage-controlled oscillator, and Sy,A is the “intrinsic” stability of the atomic resonance defined in Equation 6.12. The relationship between the power spectral density of phase and the equivalent power spectral density of the normalized frequency is [2]: 2 ⎛ f ⎞ ⎟⎟ Sθ ( f ) S y ( f ) = ⎜⎜ f ⎝ O⎠ (6.14) where f is the offset from carrier. The function H1(s) is the low-pass characteristic defined in Equation 6.6. The “filtering” of the oscillator phase noise is a high-pass function H2(s): H 2 (s) = 1 − G s = s+G s+G (6.15) Therefore, well outside the loop-bandwidth, the phase-noise is the same as the open-loop noise of the oscillator; inside the loop the frequency stability is determined by the atomic resonance stability (6.12). For offset frequencies in the vicinity of the loop-bandwidth, the overall frequency noise can be calculated from (6.13). We can also see intuitively that the feedback loop is designed to adjust the voltage at the integrator input to zero. Hence, within the loop bandwidth, noise at the integrator input is 96 compensated by a deviation in the voltage-controlled oscillator frequency from the center of the atomic resonance (Figure 6.2): Voltage Instantaneous Frequency Nominal Frequency Frequency Figure 6.2: Atomic Resonance Signal showing Instantaneous and Mean Frequency This frequency deviation is determined by the slope of the atomic resonance signal, and therefore, within the loop bandwidth: ∆f = vn ⎛ ∂E ⎞ ⎜⎜ ⎟⎟ ⎝ ∂f ⎠ (6.16) This equation leads directly to (6.12). Since, as described in Equation 6.13, the best possible frequency stability inside the loop bandwidth is for a noiseless voltage-controlled 97 oscillator, (6.12) can be considered as the “intrinsic” stability of the atomic resonance, which is seen to depend on the measurement sensitivity and the atomic resonance width. 6.2 Measurement of Frequency Stability The long-term stability of a frequency source is usually characterized in the time domain. A simple technique for measuring the frequency stability is shown below: Frequency Counter 123456789 Log Frequency Data Reference Oscillator DUT Oscillator Figure 6.3 Direct Frequency Counter Measurement of Allan Variance In Figure 6.2, a frequency counter samples the frequency of the device under test and the frequency measurement data is logged to a computer. The variations between successive frequency measurements characterize the frequency stability of the oscillator. The usual metric for the time-domain frequency stability is the two-sample variance, or Allan variance [1,2]: 98 ⎤ 1 ( y k +1 − y k ) 2 ⎥ ∑ ⎦ ⎣ 2( M − 1) k =1 ⎡ M −1 σ y 2 (τ ) ≅ ⎢ (6.17) where y is the normalized frequency deviation (Equation 6.9) averaged over the sampling interval: t +τ y= 1k τ t∫ y (t )dt (6.18) k The direct frequency counter technique assumes the reference oscillator internal to the frequency counter is more stable than the DUT oscillator. The estimate of the Allan variance should also account for the dead time between measurements, as described in [5]. The article contains formulas to correct the error introduced by dead time, depending on the power-law characteristic of the noise processes. Additional measurement techniques are described in [6]. The Allan variance, defined in (6.17), and the power spectrum of the normalized frequency deviation Sy(f) are related by the transformation [1]: ∞ σ y2 (τ ) = 2∫ S y ( f ) 0 sin 4 (πfτ ) df (πfτ ) 2 (6.19) The transformation between the Allan variance and spectral density of the frequency fluctuations is simple when the spectral density is a power law: S y ( f ) = hα f α (6.20) White noise frequency fluctuations result in an Allan variance: ∞ σ y2 (τ ) = 2 ∫ h0 0 h sin 4 (πfτ ) df = 0 2 2τ (πfτ ) (6.21) 99 and flicker frequency noise results in a variance [2]: h−1 sin 4 (πfτ ) df = 2 log e 2h−1 0 f (πfτ ) 2 σ y2 (τ ) = 2∫ ∞ (6.22) White noise results in τ-1 scaling of the Allan variance, and 1/f noise sources result in a constant “flicker floor” in the Allan variance plot. For example, assuming the short-term stability of the passive atomic frequency standard is due to a white noise process, using (6.12) and (6.21) shows that the Allan variance is given by: σ y2, A (τ ) = 1 2τ Sn ⎛ ∂E ⎞ f A ⎜⎜ ⎟⎟ ⎝ ∂f ⎠ 2 2 (6.23) The short-term stability of a frequency standard is often quoted as the Allan deviation at 1 second: σ y , A (τ = 1) = 1 Sn 2 ⎛ ∂E ⎞ f A ⎜⎜ ⎟⎟ ⎝ ∂f ⎠ (6.24) For example, if the slope of the error signal is 1µV / MHz, the receiver noise floor is 1nV / Hz , and the resonance frequency is 1100 MHz, the Allan deviation is approximately 0.64 × 10-7.at 1 second integration. 100 6.3 Solid-State Atomic Frequency Standard The prototype solid-state atomic frequency standard is the same as the zero-field electron spin resonance spectrometer described in Chapter 4, with the addition of a feedback loop to lock the voltage-controlled oscillator to the center of the magnetic resonance signal (Figure 6.4): Helmholtz Coils 90O Hybrid Modulation Driver Electrical Resonator Voltage-Controlled Oscillator Lock-In Amplifier Phase Adjuster Noise Filter Phase Detector Integrator Figure 6.4 Solid-State Atomic Frequency Reference In the prototype measurements, the bandwidth of the frequency-locking loop was approximately 1 Hz. The proof-of-concept experiment needs refinement to reduce the technical noise sources to below the fundamental random noise processes; however, frequency-locking has been demonstrated as shown on the next page in Figure 6.5: 101 9 Frequency Lock - x 10 1.119 1.1185 Frequency 1.118 1.1175 1.117 1.1165 1.116 1.1155 0 200 400 600 800 1000 1200 Time (seconds) ~ 30 mins 1400 1600 1800 Figure 6.5 Frequency-Locking of Solid-State Atomic Frequency Standard The solid-state atomic clock generates a reference frequency at 1116.5 MHz. The oscillator is locked to the magnetic resonance for more than 20 minutes, but shows considerable frequency drift. The frequency drift is due thermal drift of the alignment of the loop-gap resonator, which results in drift in the background interference signal from acoustic pickup. At 1600 seconds, the drift is such that the zeroing is lost and the loop comes unlocked. The short-term frequency variations are due to 60 Hz spurs close to the lock-in modulation frequency, not the random noise floor. With improvements in the design, Allan deviations at 1 second on the order of 10-8 or better appear feasible, as will be discussed in the conclusion. 102 6.4 Frequency Stability – Thermal Noise Limit The theoretical short-term frequency stability can be calculated assuming thermal noise of the receiver is the limiting noise source. In practice, other noise sources may limit system performance; however, with proper design the magnetic resonance measurement will approach the thermal limit as discussed in Chapter 5. For example, in the prototype design, thermal drift in the alignment of the capacitive gap results in a relatively large background signal. The variations in the gap result in changes in sensitivity to the acoustics and common mode pickup. A second source of noise is spurs from 60 Hz harmonic pickup. In the current prototype these systematic noise sources mask the thermal noise floor. The frequency stability assuming thermal noise as a limiting source is calculated using Equation 6.24. An example measurement of the vanadium magnetic resonance is shown in Figure 2.3, Chapter 2. The lock-in amplifier gain was 4 × 10-6 and the peak-to-peak resonance signal is 1.2 µV, with a width of 6 MHz. The slope of the error signal from the atomic resonance is 0.2 µV / MHz. The measured spectrum shown at the end of Chapter 2 (Figure 2.5) used a mixer with a 6 dB pad on the input to improve matching. The conversion gain of the mixer and attenuator loss account for 13 dB signal loss which subtracts directly from the sensitivity. If a lownoise amplifier were used ahead of the mixer, the conversion loss would not limit the sensitivity. With these assumptions, the signal may be increased by 13 dB (factor of 4.5 in voltage) to estimate the thermal limit on sensitivity. As a result, we use 0.9 µV / MHz as the error signal slope in the estimate below. The noise of a 50 Ohm resistor at 300K is 0.9nV / Hz . Using Equation 6.24, with a reference frequency of 1115 MHz, the thermal limit on the Allan variance at 1 second is seen to be 0.63 × 10-6. This calculation is not a “fundamental limit” on the frequency stability, in that many additional factors, such as the concentration of paramagnetic ions, the carrier power, and the resonator design, may be optimized to improve the short-term stability. 103 BIBLIOGRAPHY, CHAPTER 6 [1] IEEE Std 1139-1999, IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology – Random Instabilities, March 1999. [2] J. A. Barnes et al., “Characterization of Frequency Stability,” IEEE Transactions on Instrumentation and Measurement, vol. 20, no. 105, 1971. [3] J. Vanier and L. Bernier, “On the Signal-to-Noise and Short-Term Stability of Passive Rubidium Frequency Standards,” IEEE Transactions on Instrumentation and Measurements, IM-30, 277, 1981. [4] J. Vanier, M. Tetu and L. Bernier, “Transfer of Frequency Stability from an Atomic Reference to a Quartz-Crystal Oscillator,” IEEE Transactions on Instrumentation and Measurements, (1979), IM-28, 188 [5] J. A. Barnes and D. W. Allan, “Variances Based on Data with Dead Time between the Measurements,” NIST Technical Note 1318, 1990. [6] J. Vanier and M. Tetu, “Time Domain Measurement of Frequency Stability” Proc. 10th PTTI Meeting, pp. 247-291, Nov. 1978. 104 Chapter 7 CONCLUSION The dissertation has developed the original concept of a solid-state atomic clock, and presented, in detail, a prototype system to realize the concept. The prototype system uses only low-cost radio-frequency components which may easily be integrated into a compact atomic frequency standard. The “atomic” resonance signal was derived from the zero-field electron spin resonance of divalent vanadium in a magnesium oxide lattice. The thesis also showed an original measurement of the zero-field magnetic resonance of vanadium in magnesium oxide. As was discussed in Chapter 3, for the mF = 0 to mF = 0 magnetic field independent transitions, there is an optimal concentration of paramagnetic ions which increases the paramagnetic resonance signal without significantly increasing the resonance width. The maximum concentration is limited by exchange coupling or second-order dipolar broadening. For magnetic resonance lines with a first-order Zeeman shift, the short-term stability is independent of concentration, because the dipolar resonance width increases in proportional to concentration. With increased doping, the slope of the magnetic resonance signal is unchanged, and therefore the frequency stability is constant (Chapter 6, Equation 6.24). This is why the existence of a second-order magnetic field independent transition is so significant for the performance of a small solid-state atomic clock, because it implies the possibility of increasing the signal intensity without first-order broadening of the resonance width. A second topic for further study is the effect of higher order terms proportional to S3 on the apparent resonance width at zero-field, considered at the beginning of Chapter 2. These additional terms in the vanadium spin Hamiltonian may account for the asymmetry of the resonance signal measured in Figure 2.3. 105 The temperature dependence of the zero-field magnetic resonance was not investigated in this thesis. However, previous high-field measurements (Chapter 2, [12]) indicate the temperature coefficient of divalent vanadium in magnesium oxide is on the order of a few parts-per-million per degree. The hyperfine resonance frequency would decrease with increasing temperature. However, very careful measurements on the higher order terms in the isoelectronic Cr3+/MgO system showed that the factors proportional to S3 have strong positive temperature dependence (Chapter 2, [16]). Hence, the combined effect of the hyperfine splitting and the higher order terms may reduce the overall temperature dependence at zero-field. Another possibility is to look at different materials systems where the Hamiltonian is a combination of hyperfine and crystal field terms, or to look at multiple dopants in a signal crystal, to minimize the measured temperature coefficient. Therefore, several research questions remain, as far as the possibilities for exploiting narrow magnetic field independent transitions to optimize the short-term stability, and the possibilities of developing a materials system with reduced temperature dependence or even a turn-over temperature. The best technique for measuring the magnetic field independent transitions is also an open question, where we have suggested the possibility of a “rotating polarization” measurement as one solution. The short-term stability is always a trade-off with the volume of the resonator, because for a larger resonator, the RF magnetic field density is lower at a given carrier power, and generally, the unloaded quality factor Q of the resonator increases. For example, the Q of a loop-gap resonator is approximately 500. This trade-off with sample volume is similarly the case for miniaturized rubidium atomic clocks. There is no “one answer” for the frequency stability of the solid-state atomic frequency standard – the theoretical stability will depend largely on the acceptable power consumption. In particular, for a large resonator volume, the power to drive the audio magnetic field is considerable. Similarly, regulating the temperature of a large resonator uses considerable power. The trade-offs involving power consumption and the volume (i.e., unloaded-Q) of the resonator are characteristic of oscillator design in general, provided 106 thermal noise is dominant and not excess device noise, or other noise sources which scale with carrier power. Several improvements in the prototype design were also suggested in Chapters 4, 5, and 6, which are summarized below: • adding carrier suppression or using a second frequency discriminator to demonstrate a thermal noise limited measurement at high carrier power • miniaturization of the electrical resonator, and reduction in sensitivity to acoustics • ceramic manufacturing of high purity vanadium doped magnesium oxide and developing an optimal heat treatment recipe • finding the optimal doping concentration for best short-term stability by analysis of the second-order dipolar broadening of the magnetic field independent transitions • studies of the long-term stability of solid-state atomic frequency standard The goal is to demonstrate a spectrometer with noise performance limited by the receiver noise figure. In the current demonstration, technical noise sources remain the limiting factors. We anticipate a significant improvement in the measured short-term stability with the use of narrow magnetic-field independent resonances and better spectrometer sensitivity. In conclusion, this work opens the possibility for developing a commodity frequency standard with short-term stability and accuracy in the range of 10-7 to 10-9. The new frequency standard is entirely solid-state, works at room temperature, and without large magnetic fields or hermetic packaging. The possibilities for long-term stability better than crystal oscillators make the solid-state atomic frequency standard an interesting technology to address the needs of a number of communications and navigation systems, for both military and next-generation consumer products.