NMR STUDIES OF 3 HE IN SOLID 4 HE SUNG SU KIM

Transcription

1 NMR STUDIES OF 3 HE IN SOLID 4 HE By SUNG SU KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 c 2011 Sung Su Kim 2

3 To my parents 3

4 ACKNOWLEDGMENTS I wish to thank all those who helped me. Without their help I could not have completed this project. First and foremost, I would like to thank my advisor, Dr. Neil. S. Sullivan for his extremely patient guidance, advice and support throughout all this work. I can not thank enough him for giving me a chance to work in his lab and a great experience I have had during my Ph. D. Many thanks should go to Larry, who had mounted transistor, resistors and capacitors on the low temperature preamplifier and to Mark who had made a great NMR cell. I cannot thank more to Greg and John for supplying liquid helium during all measurements even on Christmas day. This work would not be completed without a special thanks to Dr. Candela, Dr. Xia, Dr. Huan and Dr. Yin. Especially I would like to thank to Dr. Candela for helping me with developing the Mathematica program that is used to analyze the small NMR echo signal data from very dilute samples. I learned so many valuable low temperature technique from Dr. Xia. Taking NMR data might be a little painful, not only because the measurements need very long measuring time but also because of that reason we had to work literally day and night during measuring to save time. I would like to express my deepest gratitude to my committee members, Yoonseok Lee, Pradeep Kumar, Yasumasa Takano, and Clifford R. Bowers who are very supportive. I would also like to thank all lab members, Yu Ji, Yibing Tang and Dr. Jaha Hamida. During my Ph. D, everything has been grateful and I ve been blessed to have many friends here and in Korea. I am truly fortunate to have been able to enjoy and benefit from such a friendship with them. Hwang and TJ, thank you for coming here from Korea to meet me and for all good time we spent together. I thank to all Korean students in department and I really enjoyed our regular but more irregular meeting. Jinmyung, Byunghee, Koo, Jeen, Sohyun, Minjun and Inhea, thanks for having good time with me. Without friendship with them, my time would have been harsher than it would be. 4

5 Special thanks should go to all my family members for their constant supports and belief in my decision. I truly thank to my parents and my grandmother. Without their love and support I even wouldn t have started this long journey and they have been always in my side. I want to thank my sisters, Sookyung and Mikyung and my brother Choiwhan. I would not forget to thank to all my brother-in-laws and sister-in-law. I am truly and deeply indebted to so many people that there is no way to acknowledge them all, or even any of them properly. I offer my sincerest apologies to anyone ungratefully omitted. 5

6 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Overview of Nuclear Spin Relaxations in 3 He in Solid 4 He Theoretical Background on Dilute 3 He- 4 He Mixture Pure 3 He System Phonons Vacancy waves Tunneling excitations Mass Fluctuation Waves (MFW) for Dilute 3 He- 4 He Mixtures Nuclear Spin Relaxation Times in Dilute 3 He- 4 He Mixture Region 1-A Region 1-B Region Region Landesman Model: Strain Field Huang Model: MFW-MFW Interaction Potential EXPERIMENTAL DETAILS Overview Dilution Refrigerator NMR Coils Low Temperature Preamplifier Simple Source Follower Circuit Performance Sample Probe and the Signal to Noise Ratio Summary Pulsed NMR Method T 1 Measurements T 2 Measurements Sample Growth Thermometry He Melting Pressure Thermometry Resistance Thermometry

7 2.8 Pressure Measurements MICROSCOPIC DYNAMICS OF 3 HE IMPURITIES IN SOLID 4 HE IN THE PROPOSED SUPERSOLID PHASE Overview Concepts Experimental Detail Result and Discussion Anomalies in T 1 and T Annealing Effect Applying the NMR Theory to the Peaks in T 1 and T Possible Explanation for T 1 and T 2 Anomalies Summary CONCENTRATION DEPENDENCE OF T 1 AND T Overview General Concept Concentration Dependence of T Concentration Dependence of T Re-examination of Landesman s Model Summary PHASE SEPARATION OF VERY DILUTE 3 HE- 4 HE MIXTURE Overview Background Experimental Detail Experimental Result Discussion Summary NUCLEAR SPIN-SPIN RELAXATION TIMES IN HIGH TEMPERATURE REGION Overview Result and Discussion Summary CONCLUSION Summary First Microscopic Data in the Proposed Supersolid Phase Coherent and Incoherent Quantum Motion in Dilute 3 He in Solid He Mixture Nuclear Spin Relaxation Mechanism in the Phase Separated Region 104 7

8 7.1.4 New Consideration of Quantum Tunneling Motion of the 3 He Atoms in the Nuclear Spin Relaxations Future Work Frequency Dependent NMR Measurements Further Studies for the Nuclear Spin Relaxation Dependence on the Crystal Quality More Data from Different 3 He Concentration Samples APPENDIX A CALCULATION OF NMR SIGNAL AND NOISE B DATA ANALYSIS C RELAXATION TOPOLOGIES REFERENCES BIOGRAPHICAL SKETCH

9 Table LIST OF TABLES page 4-1 Table of T 1 and T 2 from Other Groups and Calculated J 34, M 2 and J eff Comparison of T A-1 Symbols and Units C-1 Topological Relaxation Time

10 Figure LIST OF FIGURES page 1-1 Schematic Drawing of the Vacancy Wave Excitation Simple Schematic Drawing of the Tunneling Excitation Schematic Representation of the Temperature Dependence of Nuclear Spin Relaxation Times Scattering Potential in the HCP Lattice Simple Schematic Drawing of Mass Fluctuation Wave (MFW) Motion High B/T Facility: Cryostat Phase Diagram of 3 He- 4 He Mixture Dilution Refrigerator Photo of Receiving Coil Photo of Transmitting Coil Schematic of NMR Crossed Coils Geometry of a Simple Saddle Coil Optimal Angle and Field - Uniformity Parameter for a Saddle Coil NMR Measurement Set-Up: NMR coils, Preamplifier and Tuning Capacitor Low Temperature Preamplifier Circuit Diagram Photo of Preamplifier Photo of a Low Noise Pseudomorphic High Electron Mobility Transistor (phemt) Two-Pulse NMR Hahn Echo Signal Free-Induction Decay (FID) from 500 ppm Sample after 90 o Pulse Schematic Drawing for Single Echo Pulse and Corresponding Spin Description Fit of Data Obtained by Averaged Single Echo Method for T Schematic Drawing for Carr-Purcell-Meiboom-Gill (CPMG): Multi-Echo Pulse Sequence Schematic Representation of the Blocked Capillary Method for Helium Crystal Growth

11 2-19 Solidification in the NMR Cell Calibration Data of Silicon Diode Thermometer Calibration Data of Carbon Resistance Thermometer Schematic Drawing of the Pressure Strain Gauge NMR Cell Design Determination of 3 He Concentration Temperature vs. T Temperature vs. T Normalized T 1 Peaks of 16 ppm and 24 ppm Samples Photo of the Tangles of Individual Dislocations in the Crystal of Helium Simple Schematic for Impurity Pinning to the Dislocation Network Probability and Binding Energy of 3 He Observed Concentration Dependence of the Nuclear Spin-Lattice Relaxation Times for Dilute 3 He in Solid 4 He Comparison of the Concentration Dependence of the Nuclear Spin-Spin Relaxation Times Reported in the Literature for Dilute 3 He in solid 4 He Variation of the Function F (x 3 ) = ( 4 3 )T 1T 2 x 2 3(M 2 /ω L ) 2 as a Function of 3 He Concentration Schematic of Phase Separated Liquid 3 He Inside Solid 4 He T 1 T 2 vs. x Temperature vs. T Temperature vs. T Comparison of Phase Separation Temperature Kinetics of the Growth of 3 He Nano-droplets for x 3 = 1000 ppm T 2 in the High Temperature Region for 250 ppm, 500 ppm and 1000 ppm B-1 Spin-Echo Data from a Sample with 16 ppm 3 He B-2 Spin-Echo Data from a Sample with 24 ppm 3 He B-3 Spin-Echo Data from a Sample with 500 ppm 3 He

12 B-4 Spin-Echo Data from a Sample with 900 ppm 3 He B-5 Spin-Echo Data from a Sample with 1870 ppm 3 He B-6 Spin Echo Data from a Sample with 2000 ppm 3 He

13 Chair: N. S. Sullivan Major: Physics Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NMR STUDIES OF 3 HE IN SOLID 4 HE By Sung Su Kim December 2011 Nuclear Magnetic Resonance (NMR) studies are performed on dilute concentrations of 3 He in hcp solid 4 He mixtures for a wide range of 3 He concentrations grown under constant volume method. The nuclear spin-lattice relaxation times, T 1 and the nuclear spin-spin relaxation times, T 2 have been measured for 3 He concentrations 16 x ppm using pulsed NMR techniques for a Larmor frequency of ω L = 2 MHz in the temperature region where significant non-classical rotational inertia fractions (NCRIFs) have been reported. Dramatic changes are observed in the T 1 and T 2 at both the proposed supersolid onset temperatures and the phase separation temperatures. The spin relaxation anomalies are explained by a quantum tunneling model and a phenomenological fit to the temperature dependence of the relaxation is provided in terms of a thermally activated process related to the anomalies in shear modulus measurements. The 3 He concentration dependence of T 1 and T 2 is discussed in terms of the impuriton model in the range of 10 5 x The crossover concentration between the coherent motion region and the incoherent motion region is estimated from the experimental data as occurring for 3 He concentrations x The temperature dependence of nuclear spin relaxation times are studied for 3 He concentrations, 500 x ppm for a molar volume V M = 20.7 cm 3. The formation of Fermi-liquid droplets of pure 3 He are observed after phase separation. The temperature dependence suggests that the interface 3 He atoms responsible for the nuclear 13

14 spin relaxations are degenerate due to the spherical symmetry and exchange narrowing, not solid-like. The temperature independent plateaus attributed to the quantum exchange motional narrowing are observed at the temperature of 0.25 T 1.3 K. The best fit for concentration dependence of the relaxation times shows deviations from the Landesman theory. The vacancy activation energy is determined to be 13.5 ± 0.3 K for a sample with x 3 = and molar volume V M = 20.9 cm 3. 14

15 CHAPTER 1 INTRODUCTION 1.1 Overview of Nuclear Spin Relaxations in 3 He in Solid 4 He NMR measurements of the nuclear spin-lattice relaxation times (T 1 ) and nuclear spin-spin relaxation times (T 2 ), give us important information about the properties of materials on the microscopic scale. For the case of dilute mixtures of 3 He in solid 4 He, which is a well characterized quantum system, large zero-point energy of 3 He and 4 He atom results in overlap of their wave functions and they can exchange their positions even at absolute zero temperature. Due to this large zero-point motion, 3 He atoms, which have one unpaired nuclear spin, move through the lattice by quantum tunneling and this motion of 3 He can be detected by the nuclear magnetic resonance (NMR) measurements. In this case, the measurements of nuclear spin relaxation times of 3 He atoms give us quantitative information about the mechanism of 3 He motion inside the 4 He lattice, and furthermore can give crucial information about the dynamics of the 4 He lattice. Nuclear spin relaxation times (T 1 and T 2 ) for very dilute 3 He in solid 4 He were measured in this thesis work and there are two reasons why this work is important. 1. There have been a large number of NMR experiments on pure 3 He 1,2 and non-dilute 3 He- 4 He mixtures 3 with theories that give good descriptions of the data but there have been no experiments on this system with very dilute 3 He in solid 4 He i.e., x ppm. To test and confirm the existing theory for nuclear spin relaxation in this quantum system, it is necessary to carry out NMR measurements on very dilute samples in the ultra low temperature region. 2. Since Kim et al. 4,5 have observed the anomalies for solid 4 He in the responses of torsional oscillators, i.e., the non-classical rotational inertia (NCRI) which could be the signature of the supersolid phase, the origin of the microscopic mechanism of the NCRI has become the focus of a large number of experiments designed to understand this unusual phase. NMR is an important tool to meet this demand because NMR is very 15

16 sensitive to the motion of the 3 He at the atomic level so by tracking the motion of the 3 He in a solid 4 He lattice, one can probe the microscopic dynamics of the solid 4 He. These properties are very difficult to study because very dilute 3 He in solid 4 He contains an extremely small number of nuclear spins and this results in very small NMR signal and long spin lattice relaxation times (T 1 ). 1.2 Theoretical Background on Dilute 3 He- 4 He Mixture Pure 3 He System To understand the properties of dilute 3 He in solid 4 He, we need to first understand pure 3 He excitations. It is believed that the 3 He motion in 3 He- 4 He mixture is very similar except for the added effect of the 4 He lattice. The NMR relaxation times are determined by the excitations of the system, so to analyze the NMR experiments, we need to determine the excited states of the system. There are three basic excitations in pure 3 He system: the phonons, vacancy waves and tunneling excitations Phonons The phonon excitations result from the displacements of the atoms with respect to their equilibrium lattice sites and usually these excitations are too small at low temperatures to generate considerable particle motion in the system. Therefore the NMR relaxation, which mostly depends on the motion of the 3 He atoms, is barely affected by this excitation. In other words, the contribution of this excitation to the nuclear spin relaxation times is not normally detectable in NMR measurements. It is therefore generally accepted that one may ignore this excitation when considering the nuclear spin-lattice relaxation times (T 1 ) and the nuclear spin-spin relaxation times (T 2 ). If we consider only two excitations (vacancy waves and tunneling excitations) in the system, the Hamiltonian for pure 3 He can be expressed as 6,7, H P M = ε(r)b + Rσ b Rσ + t(rr )b + Rσ b R σ + 1 φ 0 (R)b + Rσ 2 b+ Rσ b Rσ b Rσ (1 1) R,σ RR,σ R,σσ 16

17 where the operator b + Rσ creates a particle at R of spin σ in the ground state of a complete set of Wannier states. The first term is the Hartree energy of the system and the second term is the tunneling term, which enables particles to move from lattice site R to R and the third term represents the hard core repulsion part. Eqn. 1 1 describes well the system that has two simple excitations (vacancy waves and tunneling excitations) Vacancy waves Figure 1-1. Simple schematic drawing of the vacancy wave excitations. (a) The ground state of the lattice has one particle at each lattice site. (b) A vacancy state is created by moving a particle from R to R. (c) The vacancy can move around inside the lattice due to the tunneling term in the particle motion Hamiltonian (H P M ) in Eqn (d) The doubly occupied state also can propagate between the lattice sites. In the ground state of the quantum solid 3 He, normally one atom occupies one lattice site and this solid usually is called a commensurate solid. The vacancy is created by applying the vacancy creation operator C + V (RR ) given by, C + V (RR ) = b + Rσ b R σ(1 δ RR ) (1 2) Actually the vacancy creation process can create two lattice states when it is applied to the ground state of the system. One is an empty lattice site and the other is doubly occupied lattice site as shown schematically in Figure 1-1. In this figure, for simplicity the 17

18 spin states are not shown. These two states propagate continuously through the crystal lattice. These excitations are called vacancy waves Tunneling excitations The tunneling excitation was discussed by Guyer and Zane 7,8, who find out that the tunneling excitation is a process associated with the vacancy state resulting from the asymmetric wave function of a fermion. The tunneling motion associated with the virtual vacancy state is shown in Figure 1-2. Figure 1-2. Simple schematic drawing of the tunneling excitation. (a) The system is initially in the ground state. (b) The particle in the lattice R tunnels to R. (c) One of two particles in R returns to the lattice site R. The system goes back to the ground state. The first step of the tunneling excitation process starts from the commensurate ground state where one lattice is occupied by one atom as mentioned above. The next step starts when the atom in the lattice site R moves to the lattice site R. Now the lattice site R is empty. As a result this lattice site becomes a vacancy while the lattice site R becomes a doubly occupied site. Unlike the vacancy waves discussed in the previous section where the vacancy and the doubly occupied site propagate away from each other, in this case one of two particles in lattice site R returns to the lattice site R. As a result of this return process, the system goes back to the ground state (the commensurate state). The vacancy state which exists during the short time is called a virtual vacancy state. 18

19 Many theoretical and experimental works 9,10 show that the vacancy waves are strongly coupled with the phonons and the tunneling excitations. However, there are very few experimental studies of the interaction between the tunneling excitations and the phonons while some theoretical works have been published. 11, Mass Fluctuation Waves (MFW) for Dilute 3 He- 4 He Mixtures In dilute 3 He- 4 He mixtures, there are three pure crystal excitations as in pure 3 He: phonons, vacancy waves and the tunneling excitations. However there is also an additional excitation that is related to the motions of atoms. These motions have two aspects, one is the motion of the 4 He atom in the 3 He medium in the case of dilute 4 He inside 3 He, and the other is the motion of the 3 He atom in the 4 He medium in the case of dilute 3 He inside 4 He. These excitations resulting from the motion of the atoms is related to the mass fluctuation waves (MFWs). If we assume that there are N lattice sites in the ground state of this system then the wave function for the ground state is the sum of all commensurate lattice states which is given by N Φ 0 = φ R (x R ) (1 3) R=1 where φ R (x R ) is the wave function that describes a particle at x R and the excited state is given by Ψ RR = b + R b R Φ 0 (1 4) The tunneling term in the particle Hamiltonian for dilute 3 He- 4 He system is given by the elements of the matrix, Ψ RR H Φ 0 and the hard-core term is given from the elements of matrix, Ψ RR H Ψ RR. With the proper theoretical mode for the dilute 3 He- 4 He mixture, Guyer calculated the Hamiltonian which gives the tunneling excitation term in the dilute 3 He- 4 He system, 13 H T = [t λ (RR )t λ (RR )/( φ 0 )]b + Rλ b R λb + R λ b Rλ (1 5) RR λλ 19

20 where λ is the index for the different states. If we only consider the case λ λ (because λ = λ corresponds to the tunneling of particle to the neighboring site and then one particle returns to the original sites leaving the system in commensurate state: the tunneling excitation) then Eqn. 1 5 can be reduced below, H T = 2 RR [t 3 (RR )t 4 (RR )/φ 0 ]b + R4 b R 4b + R 3 b R3 (1 6) Guyer defined the operators as b + R4 b R 4b + R 3 b R3 = a + R a R (1 7) where a + R is the operator that creates a mass fluctuation at R and a R is the annihilation operator for mass fluctuation. Using simplified notation M(RR ) 34 = RR t 3 (RR )t 4 (RR )/φ 0 (1 8) he rewrote the tunneling Hamiltonian as H T = 2M(RR ) 34 a + R a R (1 9) The operator that makes a single 3 He atom tunnel through the 4 He lattice is given by a + k = R exp(ik R)a + R (1 10) The mass fluctuation wave (MFW) is defined by the excitation created by the operator a + k. The mass fluctuation waves in dilute 3 He- 4 He mixture interact strongly with phonons. On the other hand, the mass fluctuation waves do not interact strongly with tunneling excitations and the vacancy waves in dilute 3 He- 4 He mixtures because the interaction rates are proportional to the 3 He concentration x Nuclear Spin Relaxation Times in Dilute 3 He- 4 He Mixture The well-known temperature dependence of T 1 is shown in Figure 1-3. The nuclear spin relaxation times of both pure 3 He and dilute 3 He- 4 He mixture are similar in the high 20

21 temperature region (Region 1-A and Region 1-B). In the low temperature region, the dilute 3 He- 4 He mixture undergoes a well-known phase separation and its nuclear spin relaxation deviates from that of pure 3 He. Figure 1-3. Schematic representation of the temperature dependence of T 1. Three qualitative mechanisms for T 1 in different temperature regions characterize the relaxation topologies. In region 1, T 1 is determined by the direct coupling of the Zeeman energy to the particle motion excitation (the vacancy waves for 1-A and the tunneling excitation for 1-B). In region 2, T 1 is determined by the coupling between excitations in the system, e.g., tunneling -vacancy, and in this region an isotopic phase separation is observed for dilute 3 He- 4 He mixture labeled by PS. In region 3, T 1 is determined by the spatial diffusion of the excitations Region 1-A In region 1 where the system is tightly coupled to the thermal reservoir, the relaxation time measurement gives information about the vacancies in the system. In region 1-A, the energy from the nuclear spin system is transferred to the vacancy wave excitations. The frequency of the local field fluctuation associated with the vacancy 21

22 excitation is given by τ 1 v = x v zω 3 (V, 3) (1 11) where x v is the thermally activated vacancy concentration in the system, z is the number of near neighbors, and ω 3 (V, 3) is the tunneling frequency of 3 He atoms into the neighboring vacancy sites. In the high temperature region, the frequency of the local field fluctuation, that determines the nuclear spin relaxation, is the Larmor frequency (ω 0 = γh 0 ). ω 0 is much less than the local field fluctuation frequency (τv 1 ) because the number of thermally activated vacancies increase as the temperature increases. As a result in the high temperature region, ω 0 τ v 1, and the spin-lattice relaxation times are given by 1 T 1 = ( 10 3 )M 2τ v ω 2 dτ v (ω 0 τ v 1) (1 12) where M 2 is the Van Vleck second moment, ω d is the local field frequency with which the spin precesses in dipolar field of z neighbors atoms. As the temperature is lowered, the spin-lattice relaxation time (T 1 ) which is independent of the Larmor frequency (ω 0 ) becomes shorter and shorter. Because the number of a thermally activated vacancy decreases as the temperature is lowered and as a result the vacancy correlation time (τ v ) becomes longer and longer. From Eqn T 1 becomes shorter with decreasing temperature in this region. At the resonance temperature at which the local field fluctuation frequency (τv 1 ) is the same as the frequency of nuclear precession (Larmor frequency ω 0 ), i.e., ω 0 τ v 1, the spin can be flipped most effectively resulting in the shortest T 1 shown as a minimum in Figure 1-3. T 1 = minimum (ω 0 τ v 1) (1 13) As one lowers the temperatures further, i.e., ω 0 τ v 1, the spin-lattice relaxation time (T 1 ) is given by 1 T 1 ω2 d ω 2 0τ v (ω 0 τ v 1) (1 14) 22

23 As the temperature decreases the number of vacancy decreases resulting in weaker relaxation rate. This means that the correlation time τ v becomes longer. As a result the nuclear spin relaxation time becomes longer as it is clear from Eqn The spin-spin relaxation time (T 2 ) has much less bizarre temperature dependence than the spin-lattice relaxation time (T 1 ), because spin-spin relaxation does not involve energy transfer from the Zeeman system to other system (it involves energy transfer within the Zeeman system only) but instead phase changes for the nuclear magnetization. The spin-spin relaxation occurs in the transverse plane due to the fluctuation of the local field by the motion of the vacancy waves. The spin-spin relaxation times in region 1-A are given by 1 T 2 = 2 3 (M 2/ω 0 ){ 3 2 η [η/(1 + η2 )] + [η/(1 + 4η 2 )]} (1 15) where η = ω 0 τ v, M 2 is the second moment of the spectral density and ω 0 and τ v are defined above. For T 2, we have to consider both transverse local field (H d (ω 0 ) ) and z-direction local field (H d (ω = 0) z ) unlike the case of T 1 for which we only need to consider the transverse local field (H d (ω 0 ) ) fluctuation responsible for the spin flip. The first term in Eqn is due to the z component of the dipolar field. In high temperature region, ω 0 τ v 0 and the analysis is similar to the case for T 1. The spin-spin relaxation time, T 2 in this region is given by 1 T 2 ( 10 3 )M 2τ v ( 10 3 )ω2 dτ v (1 16) The temperature dependence of T 2 in this region is due to the temperature dependence of the vacancy correlation time, τ v. In this region, the energy flow chain can be expressed by the vacancy wave route of Zeeman vacancy wave phonon reservoir. The energy flow topology for this region is expressed by Z-VP for both pure 3 He system and dilute 3 He- 4 He mixture indicating that the weak link in the energy flow chain is the Zeeman-vacancy coupling. The detailed discussion about the relaxation topologies is added in the Appendix C. The energy transferred from the RF pulses to Zeeman system is delivered to vacancy waves 23

24 which are strongly connected to the phonons and the thermal reservoir while the coupling between Zeeman system and the vacancy waves is relatively weak. As the temperature decreases further the spin-spin relaxation times (T 2 ) become shorter until the particle motions are only due to the tunneling excitation and this is the starting point for region 1-B that will be discussed below Region 1-B In region 1-B, the energy transferred from the RF field to Zeeman system is delivered to the tunneling excitations. In this case the weak link in the energy flow chain is Zeeman-tunneling coupling. The topology of the energy flow for this region can be written by Z-TVP for pure 3 He while the energy flow topology for dilute 3 He- 4 He mixture is Z-MFP because in dilute 3 He in solid 4 He mixture, the mass fluctuation waves completely dominate the vacancy waves in region 1-B (at temperature as low as 600 mk). So the energy flow route of Zeeman mass fluctuation wave phonon reservoir will be a very good alternative to the vacancy route of Zeeman tunneling wave vacancy wave reservoir, if the mass fluctuation waves can get rid of the energy they received from Zeeman system. Region 1-B appears when the particle motion due to the vacancy waves is negligible. Because the number of vacancies is too small to be effective below a certain temperature. In this case the coupling of the Zeeman system to the tunneling excitation dominates the nuclear spin relaxation and T 1 is given by T 1 = 1 [J 1 (ω 0 /ω T ) + 4J 1 (2ω 0 /ω T )] (1 17) where J 1 and J 2 are given by Gaussian and Lorentzian approximations respectively. 1 J 1 (ω 0 /ω T ) = [ ] 1/2 exp( ω πm 2 /3ω 0/2ω 2 T 2 ) Gaussian (1 18) T J 1 (ω 0 /ω T ) = (πm 2 /6ω T )exp( ω 0 /ω T ) Lorentzian (1 19) where ω T is proportional to J and ω 0 is the Larmor frequency. When the nuclear precession frequency is slower than J, where J = ω T /b and b is a constant that depends on 24

25 the choice of the correlation functions of Eqn and Eqn For ω 0 /ω T 0, T 1 is given by T 1 ω T ω 2 d (1 20) which is its smallest value. If the Larmor frequency is larger than J and ω 0 /ω T, then T 1 is given by T 1 ω T ω 2 d ω T ω 2 d exp[ 1 2 (ω 0/ω T ) 2 ] In this region the fluctuation of the local field due to the tunneling excitation is (1 21) temperature independent, i.e., ω T is temperature independent so T 1 is also temperature independent. However ω T strongly depends on the molar volume. For T 2, the particle motions are also related to the tunneling excitation and follow the same discussion above for T 1 and is given by 1 T 2 = 2 3 (π 2 )1/2 (M 2 /ω T ){ exp[ 1 2 (ω 0/ω T ) 2 ] + exp[ 2(ω 0 /ω 2 T )]} (1 22) where the Gaussian approximation is used. In this region the Van Vleck second moment M 2 is related to the 3 He concentration, x 3 and the ω T is temperature independent. Consequently T 2 shows a temperature independent value and in this region T 2 only depends on the 3 He concentrations. The concentration dependence of nuclear spin relaxation times is discussed in Section and Section based on the theoretical formalism of Landesman and Huang Region 2 In the low temperature region, the Zeeman system and mass fluctuation waves are coupled, and T 1 and T 2 are given by T 1 1/{M 2 x 3 τ 3 [1/(1 + ω 2 τ 2 3 )] + [4/(1 + 4ω 2 0τ 2 3 )]} (1 23) T 2 1/ M 2 x 3 τ 3 { [1/(1 + ω2 τ 2 3 )] [4/(1 + 4ω2 0τ 2 3 )]} (1 24) 25

26 where τ 1 3 is the sum of two tunneling rates: a 3 He atom tunneling rate through 4 He and the tunneling rate for a 3 He atom into a vacant neighboring lattice site. In region 2, the Zeeman system is not tightly coupled to the thermal reservoir or lattice, and the energy flow after the RF pulse excitations is bottlenecked by the particle motion excitations. In region 1, the particle motion excitations are strongly coupled to the phonons and they are always at the same temperature with the lattice or thermal reservoir. When the particle motion excitation becomes uncoupled from the lattice or thermal reservoir, region 2 occurs. In this case the relaxation process involves three or more systems other than jut the two systems in the case of region 1. The intrinsic relaxation times are defined for the two system relaxation as in region 1, and the topological times are defined for the three or more systems as in region 2. The topological times are defined by 1 1 = (topological factor) (1 25) T topological T intrinsic The topological factor is defined as the energy constant of the systems which are involved in the relaxation process. In region 2, the small concentrations of 3 He are mobile through the 4 He lattice and constitute mass fluctuation waves which couple to the phonons in the system. As mentioned before, the mass fluctuation waves of 3 He atoms do not interact with the tunneling excitations. The energy flow topology is given by ZMF-P and the energy flow chain can be ordered: Zeeman mass fluctuation wave phonon reservoir. The weak link in the energy flow chain is mass fluctuation - phonon coupling. The topological relaxation times are given by = [k MF /(k z + k T MF )] (1 26) topological T MFP where T MFP is the intrinsic relaxation time when we only consider the weak coupling (long relaxation time) in the energy flow chain, i.e., relaxation from mass fluctuation waves to phonons. k MF and k z are topological factors for mass fluctuation wave system and Zeeman 26

27 system respectively and defined by k MF = de MF dβ k z = de z dβ (1 27) (1 28) where E MF and E z are the energy for mass fluctuation and Zeeman system respectively and β is inverse temperature Region 3 In region 3, the energy acquired from the RF pulses is transferred by the diffusion of the particle motion excitation while it is transferred by the coupling among the excitations in region 1 and 2. As the temperature decreases the weak connection in the energy flow chain is the link of the mass fluctuation waves to the phonons. It therefore takes a long time for the energy to reach the phonons. Once phonons receive the energy they transfer it to the reservoir and the relaxation is determined by the coupling efficiency of the two phonon systems which is the Kapitza resistance Landesman Model: Strain Field Landesman calculated the nuclear spin relaxation times of dilute 3 He impurities in solid 4 He in region 1-B. He assumed that the relaxation is based on the modulation of the dipolar interaction resulting from the tunneling of the 3 He atoms. He considered the constraints on the tunneling for dilute 3 He in solid 4 He due to the large elastic lattice interactions. For the hcp lattice there is an elastic deformation around each 3 He atom. The elastic interaction due to the lattice distortion around a 3 He atom is induced by the virtual phonon exchange of the 3 He at site i and j and given by V ij = K( a R ij ) 3 (1 3cos 2 θ) (1 29) a is the nearest neighbor distance and θ is the angle between R ij and the trigonal axis. Landesman assumed = 1 and K J 34. A schematic representation of the configuration of the strain field is shown in Figure 1-4. Landesman calculated the diffusion constant 27

28 Figure 1-4. The schematic of the elastic interaction. The distance between two 3 He atoms that are sitting in the lattice site i and j is R ij and the θ is the angle between R ij and the trigonal axis in the hcp lattice. using simple scattering due to the elastic interactions and found a diffusion constant of D x 4/3 3 instead of D x 1 3 as predicted by other theories He fitted the Sussex data 17 using his diffusion formula and derived the value for the tunneling rate of the isolated 3 He impurities from site to site, J 34 = 1.0 MHz and the elastic interaction between two 3 He atoms sitting on the neighboring lattice sites, K/2π = 1200 MHz. Up to numerical prefactors Landesman calculates the 3 He diffusion constant D L to be D L = 16 3 Γ(4 3 ) a 2 J 2 34 K(Λx 3 ) 4/3 = 0.36 a2 J 2 34 Kx 4/3 3 (1 30) 28

29 where Λ = Landesman expresses the diffusion in terms of an effective random-walk jump rate τ 1 r as D L = a2 6τ r (1 31) By comparing Eqn and Eqn. 1 31, one can deduce the expression for the effective random walk jumping rate τ 1 r as τ 1 r = 2.2J 2 34 Kx 4/3 3 When a 3 He jumps one lattice site, the typical change in elastic energy is (1 32) Ka3 d( ) r a 3 dr = 3Ka4 r 4 3Kx 4/3 3 (1 33) using the mean 3 He- 3 He separation of r a/x 1/3 3. In the absence of the elastic energy a 3 He atom would jump at the rate zj 34 in a band of states of width J 34. But with the elastic energy due to other 3 He atoms the bandwidth is increased to Kx 4/3 3. Hence the density of states and jump rate are reduced by a factor of J 34 /Kx 4/3 3, giving Eqn up to the prefactor. Landesman derived the spin-lattice relaxation time T 1 and spin- spin relaxation time T 2 using a simple scattering model by considering the scattering of the 3 He atoms by the elastic distortion field and found T 1 = 3 4 ω 2 0 BM 2 J eff x 2/3 3 = 0.03 ω 2 0K M 2 J 2 34x 2/3 3 (1 34) T 2 = BJ eff M 2 x 4/3 3 = 23.3 J 2 34 M 2 Kx 4/3 3 (1 35) where B = The effective tunneling rate J eff = J 2 34/K, and M 2 is the dipolar second moment for pure 3 He with the same lattice structure. He calculated the correlation time τ c through the Kubo formalism and found 18 τ 1 c = BJ eff x 1/3 3 = 23.3 J 2 34 Kx 1/3 3 (1 36) 29

30 As Landesman pointed out in his paper, there is a large difference between the NMR correlation frequency (the local field correlation frequency), τ 1 c in Eqn and the jumping frequency, τ 1 r in Eqn (i.e., τ 1 c τr 1 ). As shown in the two equations, the jumping frequency proportional to x 1/3 3 is much larger than the actual NMR correlation frequency which is proportional to x 4/ Huang Model: MFW-MFW Interaction Potential Huang discussed the effect of the long-range interaction induced by the mass fluctuations on the motion of the 3 He particle through the 4 He lattice. He derived the Hamiltonian for the dilute 3 He- 4 He mixture in the formalism of second quantization and averaged a single particle crystal states. The total Hamiltonian for the dilute 3 He system with 3 He concentration of x 3 is given by H(x 3 ) = E 0 (x 3 ) + H D + H w + H I (1 37) H I = H (1) I + H (2) I (1 38) where E 0 (x 3 ) is the ground state energy for the system with 3 He concentration of x 3. H D generates the displacement fluctuation and H w is responsible to the width fluctuation. The mass fluctuation mass fluctuation (MF-MF) interaction, H I is the sum of direct fluctuation, H (1) I and indirect fluctuation, H (2) I. H D and H w change the ground state energy slightly and describe the thermal excitations, and at low temperatures, the MF-MF interaction (H I ) can affect the motion of the 3 He in 4 He. Huang calculated the effect of this interaction especially on the dilute 3 He in 4 He. To discuss the intermediate concentration region he first discussed the very dilute region, i.e., MFW region, illustrated in Figure 1-5 In the MFW region, 3 He atoms are so rare that the mean separation is greater than the range of the distortion fields. One may assume that they form Bloch waves and propagate within the lattice but once 3 He atoms approach one another less than the distance r c then 3 He atoms can not propagate freely any more. As a result 3 He atoms are 30

31 Figure 1-5. The simple schematic of the mass fluctuation waves (MFW) motion. If a 3 He atom is close to another 3 He atom at distance less than r c, then 3 He atom is scattered due to the large interaction potential between the two 3 He atoms. scattered by each other because the energy difference due to the motion of one relative to the other is much larger than the tunneling rate J 34 in 3 He- 4 He. The MFW-MFW scattering interaction is given by Huang s calculation S I = V 0 ( R R ) (1 39) where V 0 is the interaction constant between two MFWs and V K. is the nearest neighbor distance. The cross-section of the MFW-MFW scattering is σ c = πr 2 = π 2 ( 2V 0 J 34 ) 1/2 (1 40) 31

32 The distance of nearest approach r is given by the following relation, J 34 = ( V 0 )( r ) r= r (1 41) Huang calculated the diffusion constant using these relations and found D I = J 34 λ = 1 π 2 J 34 x 3 ( J 34 3V 0 ) 1/2 (1 42) where λ is the MFW-MFW mean free path which is given by λ = 1 π x 3 ( J 34 3V 0 ) 1/2 (1 43) By requiring that the average distance between atoms be greater than the distance of the maximum approach r c, one can determine the upper limit for 3 He concentration for MFW region. Above this concentration 3 He will interact strongly with other 3 He atoms and they move incoherently through the lattice. He estimated the upper 3 He concentration, x Huang also calculated the nuclear spin-spin relaxation times (T 2 ) based on this model in the strongly interacting region and this is discussed in detail in Chapter 4 by analyzing the concentration dependence of our NMR data. 32

33 CHAPTER 2 EXPERIMENTAL DETAILS 2.1 Overview This chapter will discuss the experimental details that are used for the most of the NMR measurements carried out for this thesis work. All NMR measurements are carried out in an ultra low temperature environment using a dilution refrigerator. Section 2.2 gives a general background for the use of a dilution refrigerator. All NMR measurements are done on very dilute 3 He in solid 4 He and especially for x ppm, the number of nuclear spins in the sample volume of 0.15 cm 3 is of the oder of and this is a very small number. The magnitude of the NMR echo signal is therefore very small. A low temperature homemade preamplifier is consequently needed to improve the signal to noise ratio before the signal is transfered from the NMR coil to the room temperature amplifier. In Section 2.4, the low temperature preamplifier is discussed in detail. The pulsed NMR technique is discussed in Section 2.5. The NMR sample cell is constructed as a crossed coil arrangement as discussed in Section 2.3, and the general method for growing the helium crystals is discussed in Section 2.6. Section 2.7 and Section 2.8 describe the method of thermometry used at low temperatures and give a brief summary of pressure measurements respectively. 2.2 Dilution Refrigerator Most NMR measurements for this thesis work were carried out in the bay 3 high B/T facility in Micro-kelvin Laboratory, shown in Figure 2-1. The choice of refrigerator for specific measurements depends on the temperature range where the actual experiments will be carried out. For T 0.25 K, a 3 He evaporation cryostat is usually used. The temperature range of the dilution refrigerator is T 1.0 K and for T K, adiabatic demagnetization is usually used. Our measurements are done in the range of 0.01 T 0.6 K so a dilution refrigerator with high refrigerating capability is used. 33

34 Figure 2-1. The high B/T facility in Micro-kelvin laboratory in University of Florida. Photo courtesy of Micro-kelvin laboratory. The simple process of the dilution refrigerator relies on certain thermodynamic characteristics of 3 He and 4 He. The phase diagram of 3 He- 4 He mixture is shown in Figure 2-2. At temperatures below the triple point, the 3 He- 4 He mixture will separate into two liquid phases, a 3 He rich phase and a 4 He rich phase, divided by a phase separation boundary. The 3 He rich phase is mostly 3 He and the 4 He rich phase is a mixture of 4 He and 3 He which is always composed of at least 6 % 3 He, down to the lowest temperatures. This phase is shown in the diagram below and to the left of the triple point, along the equilibrium line. The two phases are maintained in liquid form. Since there is a boundary between both phases, extra energy is required for particles to go from one phase to another. If we pump on the vapor above the 4 He rich liquid phase then we remove mostly 3 He because of the larger vapor pressure of 3 He. To move 3 He across the boundary the dilute 34

35 Figure 2-2. Phase diagram of 3 He- 4 He. The lambda line is between the superfluid and normal liquid phase and the phase-separation line is between the superfluid and unstable state where the 3 He and 4 He are separated. liquid phase is connected to a still (Figure 2-3) which contains both liquid and vapor. As a result, 3 He will have to flow across the phase boundary from the 3 He rich side to the 4 He rich side to restore equilibrium. Crossing this phase boundary line produces the cooling due to the latent heat of mixing. In other words, the 3 He rich phase sends the 3 He to the dilute phase and absorbs energy in the form of heat from the phonons of the liquid in the mixing chamber and as a result the mixing chamber is cooled down. The samples are in thermal contact with the wall of the mixing chamber and they are cooled down also. The 3 He will cross the phase boundary and join the 4 He rich phase by restoring equilibrium, and the 3 He atoms lost 35

36 Figure 2-3. The schematic of dilution refrigerator. during this cycle are replenished by a constantly circulating flow of 3 He. A set of heat exchangers (Figure 2-3) is used to cool the incoming 3 He to the temperature of the liquid mixture in the dilute phase. In this way the dilution refrigerator cools the mixing chamber and thus the sample continuously. A simple schematic drawing of a dilution refrigerator is shown in Figure NMR Coils The NMR cell is designed to have a horizontal cylindrical shape with two coils: (i) a solenoidal NMR signal receiving coil wound around the cylindrical cell, and (ii) a thermally isolated saddle shaped radio frequency (RF) excitation coil for the RF pulses. 36

37 The receiving coil and the transmitting coil with the sample filling line attached are shown in Figure 2-4 and Figure 2-5 respectively. The solenoidal receiving coil has turns of high conductivity copper wire to achieve the desirable quality factor Q and NMR coil inductance Ls. Figure 2-4. Photo of receiving coil. The sample filling capillary line is attached with Stycast 2850GT. One of NMR signal output line goes to the preamplifier input and the other is connected to the ground. Photo courtesy of S. S. Kim. Figure 2-6 shows a schematic view of these two crossed coils. This crossed coil design offers two features. This design can reduce the unwanted pick-up of RF excitation in the receiving coil and provide the sample cell with a thermal isolation from the heat generated in the RF excitation coil. The cell was made from polycarbonate and one end of the cell is open to a pressure gauge and the other end is sealed by an insulating epoxy (Stycast 2850GT) cap through which a capillary line passes to provide a sample filling line. The transmission coil is a saddle-shaped coil and it can generate a uniform RF magnetic field perpendicular to the cylindrical axis of the NMR sample cell In our measurements, the signal is extremely small (typically < 1nV) and obtaining a uniform 37

38 Figure 2-5. The photo of transmitting coil. The receiving coil is placed inside the transmission coil and the filling line is from the receiving coil. Photo courtesy of S. S. Kim. Figure 2-6. The schematic of the NMR crossed coils. AC magnetic field B1 as illustrated in Figure 2-6 is very important. If B1 is not uniform then it reduces the magnitude of the NMR signal. 38

39 Figure 2-7. The geometry of a simple saddle coil. From the geometry shown in Figure 2-7, the field uniformity parameter Q and the optimal angle φ 0 is given as below Q = (15 3s) (2s 2 + 2s 3 ) sin 2 (φ 0 /2) = (15 + 3s + 6s2 + 12s 3 ) 2(5 + 3s + 4s 2 + 8s 3 ) (2 1) (2 2) where s = 1 + (h/d) 2 with h and D are the height and diameter of the saddle coil. Coincidentally, in the range 1 h/d 2, the curve showing Q as a function of h/d is almost identical in shape to that of φ 0 as a function of h/d, and therefore simply rescaling the vertical axis as shown in Figure 2-8 suffices to indicate how Q depends on h/d. For 39

40 our case, h/d 1.2 and the optimal angle φ 0 is selected about o. Figure 2-9 Figure 2-8. The graph of the angle φ 0 and the field-uniformity parameter Q as a function of saddle coil physical dimension. Research shows that the angle φ 0 and field uniformity factor Q has same vales in the range of 1 h/d 2. is a photo of the NMR set-up showing the configuration of preamplifier discussed in section 2.4 and NMR coils. With this set-up we reached down to 250 mk because of heating from the preamplifier. We moved the preamplifier away from the sample coil to obtain lower temperatures down to 10 mk. The circuit diagram of the preamplifier is shown in Figure

41 Figure 2-9. NMR measurement set-up: NMR coils, preamplifier and tuning capacitor. preamplifier and the tuning capacitor are located close to the NMR coils. Superconducting wires are used to connect the receiving coil to the preamplifier to provide high electrical conductivity but low thermal conductivity from the preamplifier circuit to the NMR sample. Photo courtesy of the Micro-kelvin laboratory. 2.4 Low Temperature Preamplifier All the NMR measurements were carried out for very dilute 3 He in solid 4 He samples where the number of nuclear spins is less than The NMR signal is therefore too small to be detected by the usual NMR spectrometer. For this kind of measurement a low temperature preamplifier is necessary to minimize the signal loss due to the mismatch of impedance between the signal and the coaxial cable which extends from low temperature to the room temperature. 41

42 2.4.1 Simple Source Follower Circuit Figure The circuit diagram of the low temperature preamplifier. The pseudomorphic high electron mobility transistor (phemt), Agilent type ATF-35143, is used for a simple source follower circuit. The operating point of the transistor is controlled by the bias voltage V B to match the high impedance of the NMR resonance coil (L s ) to that (50 Ω) of the RF coaxial cable which transfers the signal to the low noise amplifier at room temperature. The circuit design for the amplifier is a simple source follower using a pseudomorphic high electron mobility transistor (phemt) as shown in Figure Although a phemt generates appreciable heat ( mw), it has a high gain at low temperatures up to the UHF range and the planar geometry allows the phemt to be aligned parallel to the magnetic field, eliminating the Hall effects 22. The circuit is optimized to operate at 2 MHz 42

43 at which the nuclear spin-lattice relaxation time is 103 s at low temperature ( 10 mk). Figure The photo of the low temperature preamplifier. The preamplifier takes signal from NMR coil with Ω. The out-put impedance of the preamplifier (50 Ω) is matched with the room temperature amplier to improve the signal to noise ratio. Photo courtesy of the Micro-kelvin laboratory. The impedance mismatch is the main reason for the NMR signal loss at low temperatures. The impedance of the NMR signal is given by R = QωL. Q is the quality factor of the coil and ω is the Larmor frequency and L is the inductance of the NMR coil. For our case, Q 82, ω 2 MHz and L 100 µh and with these numbers the impedance of the NMR coil is Ω. As shown in Figure 2-10, the impedance of the NMR signal is 300 times larger than that of commercial coaxial cables, so without the preamplifier only a small fraction of the signal can be transferred to the cable. 43

44 Figure Photo of a low noise pseudomorphic high electron mobility transistor (phemt). The model of ATF is suitable for applications in cellular and personal communications service (PCS) base stations, a low earth orbit (LEO) satellite systems, multi-channel multi-point distribution service (MMDS), and other systems requiring very low noise figure with good intercept from 450 MHz to 10 GHz frequency range. Photo courtesy of the Agilent Technologies Inc. For the source follower circuit the gain and impedance are given by Voltage Gain : A V = V OUT V IN = g mr s g m R s + 1 1, for g mr s (2 3) Current Gain : A I = (2 4) Input Impedance : Z I = (2 5) Output Impedance : Z OUT = R s g m R s g m (2 6) The required transfer admittance g m is achieved by tuning the bias voltage V B to match the output impedance of the amplifier to the 50 Ω. This design has the advantage of minimizing the tendency of the circuit to oscillate because of the high gain of the devices used. The resistors were miniature metal-film resistors and the capacitors miniature ceramic chip capacitors tested at liquid nitrogen temperatures prior to assembly for integrity on thermal cycling. The value for these components are shown in the Figure

45 Figure 2-11 is a photo of the preamplifier and the incorporated Agilent type ATF pseudomorphic high electron mobility field effect transistor (phemt). The latter is shown separately in Figure This device has low power dissipation compared to others in the same class. The gate bias is supplied by an external supply to limit the total power dissipation to 0.5 mw and with the source resistance determined the operating point and thus the transconductance of the phemt. The circuit was operated at 2 MHz because at this frequency the nuclear spin-lattice relaxation time (T 1 ) is about 100 s. and this is the maximum time length for which we can obtain the tolerable signal-to-noise ratio with reasonable averaging time at low temperatures Performance A typical NMR spin echo signal for a sample containing 500 ppm of 3 He in solid 4 He at 400 mk is shown in Figure The observed signal/noise is 35 after signal averaging (10 pulse sequences) for a bandwidth of 17 khz. From the calibrated gain this corresponds to a noise temperature of approximately 1.1 K. A small cross-coupling from the transmission coil leads to a small uncertainty in the gain calibration. The quoted noise temperature is an upper limit Sample Probe and the Signal to Noise Ratio In NMR signal detection, the sample probe and the coupling between the sample probe and the preamplifier are the most critical parts of an NMR apparatus. For more sophisticated experiments, optimizing sample size and sample shape are also important but the NMR coils coupling to the preamplifier can be essential. In the pulsed NMR experiments, the sample material receives pulses from the transmitter that generates a radio frequency B 1 field perpendicular to the static DC B 0 field. The response to these pulses has to be converted into a signal voltage and fed to the preamplifier. The electrical noise is unavoidable and it originates in the electrical components themselves. Usually the current-carrying electrons have a random elements in their motions due to their thermal energy. Statistical thermodynamics shows that for a 45

46 Spin Echo (Arb. Unit) Time (ms) Figure NMR Hahn echo signal obtained after a two pulse sequence of 90 o τ 180 o. The red line and black line are signals from real and imaginary components of the spin echo. Theses signals are taken for a solid sample with x 3 = 500 ppm at 400 mk. system in the equilibrium at temperature T, the average energy associated with thermal excitations is k B T/2, where k B is Boltzmann s constant. For example, in a resistor there can be some electrons that slightly deviate from other electrons which remain in the equilibrium state. These deviations generate a potential difference as a fluctuating voltage. This is the origin of Johnson noise and the mean square magnitude of the voltage in a 46

47 frequency band f is given by v 2 = 4k B T R f (2 7) The Johnson noise is proportional to the temperature and the resistance. Another contribution to the noise is shot noise and this is due to the finite charge carried by electrons and it appears as a fluctuation in the current flowing and the frequency band f. i 2 = 2eI f (2 8) where I is the current and e is the electron charge. These two contributions to the noise are known as white noise. These contributions are fundamentally due to random processes, so the average of the noise voltage is zero. The mean square voltage has finite values thus when we mention the noise voltage of certain magnitude we refer to the root mean square (RMS) value. This property is important in data averaging to improve the signal noise ratio. The common way to reduce the noise in the NMR measurement is to repeat the measurements and then average them. For example, if we do the measurements N times and add them to calculate the average value of a noise voltage then the noise increases with the square root of N, i.e., So if we divide it by N the averaged noise value as below, i=n v i = Nv i (2 9) i=1 i=n i=1 v i N = 1 N v i (2 10) then the average noise actually decreases by the factor of 1/ N. In this way by averaging the NMR signal we can improve the signal to noise ratio. Typically for frequencies below 100 MHz, the noise introduced by the amplifier is ignored and the dominant noise is the Johnson noise from the tuned NMR coil which is given by Eqn. 2 7 In the parallel resonant configuration the coil resistance is R = QωL. The size of the signal after π/2 47

48 pulse is to be v s = NAωχ 0 B 0 (2 11) where N is the number of turns, A is the cross section of the coil and χ 0 is the magnetic susceptibility. The signal to noise ratio is given by the ratio of the signal voltage to the noise voltage. (S/N) = NAηωχ 0B 0 (4k B T R f) 1/2 (2 12) where η is the filling factor that is usually unity when the volume of the sample cell is equal to the volume of the NMR coil. NMR cell was designed to have η 1. The details of the calculations for the signal and noise voltage for our NMR measurements are give in Appendix A Summary A low power preamplifier employing a pseudomorphic transistor was used to realize low noise temperatures (T N ), T N 1.1 K, for pulsed NMR experiments at milli-kelvin temperatures by isolating the amplier and NMR coils from the sample cell itself. The circuit is easy to construct and in principle can operate up to UHF frequencies. The design allows the use of high power pulses for straightforward pulse sequences. The improvement over simple passive matching using a capacitive transformer to match a resonant coil to a 50 Ω transmission line is better than a factor of Pulsed NMR Method The pulsed NMR technique was used to measure spin-lattice relaxation times (T 1 ) and spin-spin relaxation times (T 2 ). In order to understand the pulsed NMR methods we need to consider the effect of a short radio frequency pulse on the equilibrium magnetization I z. The RF pulse generates an oscillating RF magnetic field B 1 cosω a t where ω a is very close to the Larmor frequency ω 0. The interaction with the nuclear spin is given by the Hamiltonian, H = γb 1 cosω a ti x + γb o I z (2 13) 48

49 where ω 0 = γb o in the applied static magnetic field and γ is the gyromagnetic ratio for 3 He. In a reference frame rotating at frequency ω a about the z-axis, we may write H = γb 1 I x + (ω 0 ω a )I z (2 14) The last form is negligible so the effect of B 1 applied for time t w is given by the unitary operator U = e ih t w = e i(γb 1t w)i x (2 15) which is equivalent to a rotation by angle θ = γb 1 t w about the x-axis. By choosing t w and/or B 1 one can therefore rotate the spin by any desired angle. Typical value of θ are 90 o and 180 o as described below T 1 Measurements In order to determine T 1, we measured either (i) the free-induction decay (FID) after a single 90 o pulse, or (ii) a single spin echo method for a 90 o 180 o pulse sequence. When we used the FID to measure T 1, we changed the delaying time t between pulses and applied 90 o pulses for each measurement. The typical FID data from a 500 ppm sample is shown in Figure It is important to note that immediately after the RF pulse there is a dead time due to the recovery of the receiving amplifier after the pulse. This dead time t d 10Q/ω µs. We increased the delay time gradually for each FID measurement. As we increased the delay time the FID signal size also increased exponentially. The fit of FID signal size with respect to the delay times gives the T 1 values. The other pulse sequence we used for T 1 measurements is the single echo pulsed method. The equilibrium magnetization I z is destroyed and then one applies a (90 o τ 180 o ) pulse sequence to observe an echo at 2τ. One then waits a time before repeating to observe the growth of the echo as t becomes comparable to the relaxation time. The time interval between 90 o and 180 o pulses is fixed as τ = 4 ms. The echo heights are given by h(t) = h( )[1 exp( t T 1 )]exp( 2 τ T 2 ) (2 16) 49

50 8000 Intensity (Arb. Unit) Time (ms) Figure The free- induction decay (FID) from 500 ppm sample after 90 o pulse. The black trace is the in-phase component of the FID and the red trace is the out of phase component. where h( ) is the echo height of the equilibrium state. We determined the h( ) size from the values of the echo height that are measured with a 4 5 hour waiting time. T 1 is sensitive to the variation in h( ), so more than one measurement at t = 5τ was made in the experiments. With fixed τ which is mostly chosen to be 4 ms, as we increase the waiting time t the echo height increases exponentially with time t and the best fit using Eqn gives the T 1 values. 50

51 Figure Schematic drawing for the formation of a single echo and corresponding spin evolution. A schematic drawing for the single echo pulse sequence and corresponding evolution of the spin are shown in Figure We used the FID sequence to measure T 1 for relatively high 3 He concentration samples. The single echo sequence was used for very dilute samples (16 ppm and 24 ppm samples). The difference between the values of spin-lattice relaxation times obtained from FID and single-echo method was negligible T 2 Measurements The spin-spin lattice relaxation times (T 2 ) are measured by using a multi spin echo method, the Carr-Purcell-Meiboom-Gill (CPMG) and also the averaged single spin echo method. The averaged single spin echo method needs longer measuring times because after applying 90 o τ 180 o pulse we have to wait a very long time (> T 1 ) until we start the next measurement with a different value of τ. We used this method as an alternative to check the accuracy of the T 2 values acquired by CPMG method. Figure 2-16 shows one of the fits of the echo heights obtained by the averaged single echo method to obtain 51

52 T 2 values. In averaged single echo method, as we increase the time interval τ with fixed waiting time nuclear spin echo heights decreases exponentially. Nuclear spin echo height (Arb. Unit) (ms) Figure Fit of data obtained by the averaged single echo method for T 2. The red line is a fit to single exponential decay. The multi-spin echo method can reduce the amount of data taking time and also minimize the diffusion effects on the echo heights which is incurred in the single 90 o τ 180 o sequence method. The Carr-Purcell-Meiboom-Gill sequence was used for multi-echo sequences, [90 o τ (180 o 2τ) n ] and the echo height at time τ is given by, h(τ) = h(0)exp( 2τ/T 2 2γ 2 G 2 D 2 τ 3 /3n 2 ) (2 17) 52

53 where G is the magnetic field gradient, D is the diffusion constant, and n is the number of 180 o pulses. Figure 2-17 shows the schematic drawing for CPMG sequence. Figure The schematic drawing for Carr-Purcell-Meiboom-Gill (CPMG) with four 180 o pulses. As τ becomes longer then the size of the nuclear spin-echo decreases following Eqn (We used eight 180 o pulses but for simplicity four pulses are shown in this figure.) Shortening the time interval between spin echoes reduces the diffusion-induced shortening of T 2 and improves the resolution of short T 2 components. Increasing the number of echoes increases signal-to-noise (SNR) and improves the resolution of long T 2 components. We need to optimize the number of echoes and the time interval between echoes to optimize data taking time and obtain data with good resolution. All pulses are applied along the positive x-axis and the initial preparation 90 o pulse tips the net magnetization in the z direction M 0 to be along the positive y-axis. For our measurements we mostly used eight 180 o pulses and averaged for 5 days for very dilute samples. 2.6 Sample Growth The blocked capillary method was used to grow a solid crystal of the 3 He- 4 He mixture and its density is kept constant The capillary is weakly thermally anchored inside the cryostat at the 1 K pot, the still, the heat exchanger, and the mixing chamber. During 53

54 Figure Schematic representation of the blocked capillary method for helium crystal growth. At room temperature, 3 He- 4 He gas mixture is pressurized to fill the NMR sample cell and a small external reservoir and the gas mixture is condensed in the cell at 4.2 K. After subsequent cooling the system, solidification starts at the coldest part of the filling line and crystallites grow from the filling line into the cell. During solidification the pressure drops because the helium sample solidifies with constant volume i.e. constant density. filling from a gas mixture of 3 He- 4 He at room temperature, the capillary line is kept at a higher temperature (around 4.2 K) to prevent solidification of helium inside the capillary line before the pressure of the NMR sample cell reaches the desirable pressure. 54

55 By increasing the pressure of the gas mixture of 3 He- 4 He we can condense the mixture into the NMR cell and during this condensation the gas mixture experiences both cooling and pressurizing until it reaches the values needed for crystal growth in the cell. After the pressure of the cell reaches the final pressure the system is cooled down until it hits the melting temperature. The capillary becomes blocked and the pressure follows the melting curve as observed by the decrease of the sample pressure measured by the in situ pressure gauge, as shown in Figure The maximum pressure is around 46 bar for all samples before condensing the gas mixture. With the pressure maintained higher than the melting pressure, the NMR cell is cooled down to the solid region of the phase diagram. Figure The schematic of the solidification of the helium in the NMR cell. When the system is cooled down the solidification of helium starts from the capillary line and the red arrow shows the direction of the crystallization in the cell. By running the dilution refrigerator at full capacity, the capillary line is cooled down and is blocked at some point which is the coldest part in the filling line. This blocking enables us to maintain the solid sample in the NMR cell and the direction of the 55

56 solidification is from the capillary line (filling line) into the cell, as shown schematically in Figure However, the 3 He has a much larger zero pint motion than 4 He, and as a result it has a higher binding interaction with walls. 3 He can therefore preferentially plate out on the capillary walls rather than in the cell. As a result the density of 3 He inside the 4 He usually changes by the order of 10 %. By cooling a liquid sample in the NMR sample cell from 4 K, the resultant solid sample pressure is typically bars and this is 40 % lower than the starting pressure. 2.7 Thermometry In our experiment in order to determine the correct temperature we used several thermometers, selecting the ones appropriate to the temperature range. When we cooled the system by running the dilution refrigerator from room temperature to condense the gas mixture into the NMR cell at temperature 4.2 K, we used a DT-670 silicon diode thermometer. Figure 2-20 shows the calibrated data for high temperature and the photo of the silicon diode thermometer. The calibration is carried out using a 3 He melting pressure thermometer. Below 1.2 K, a carbon resistance thermometer is typically used to read the temperature of the sample. The calibration data of carbon resistance thermometer is shown in Figure Most of our measurements were done in the range of 0.01 T 0.4 K, so almost all temperatures are read from the carbon resistance thermometer He Melting Pressure Thermometry The melting curve of 3 He exhibits a pronounced temperature dependence as reported in many papers This temperature dependence can be used as a thermometer particularly in the temperature range of 1 T 250 mk The temperature of the minimum of the melting curve, the superfluid transitions of liquid 3 He and the nuclear 56

57 Resistance (Ohm) Temperature (K) Figure Calibration data of silicon diode thermometer. Inset photo is the Lakeshore silicon diode DT-670 series. 28 anti-ferromagnetic ordering transition of solid 3 He on the melting curve provide well defined fixed points. The advantages of using a 3 He melting curve thermometer are the high resolution and reproducibility. The weak point of this thermometer is the large specific heat of liquid 3 He in the thermometer and also the need for a gas handling system with a capillary filling line. For our experiment we used a 3 He melting thermometer to calibrate our carbon resistance thermometer. Our sample temperatures are decided by the value of carbon resistance thermometer in the range of measurement temperature Resistance Thermometry Resistance thermometry exploits the temperature dependence of metals or semiconductors and it is the most widely used method in low temperature measurements even though the thermal conductivity, thermal contact and self-heating of the device might be a weak 57

58 7 Resistance (Kohm) Temperature (K) Figure Calibration data of carbon resistance thermometer. point. The carbon resistor is widely used for low temperature thermometer and this is not manufactured as specific thermometer. Therefore it is very cheap and has very stable R-T behavior. 2.8 Pressure Measurements A Straty-Adams strain gauge 34 was used to measure the pressure of the sample and is placed in one end of the NMR cell. In the construction of the capacitor the two movable plates are electrically insulated by a spacer, such as Kapton, mylar and a glass shim but those materials show a large temperature and pressure dependence at very low temperatures and because of this property, the strain gauge can also be used as a thermometer. 35 For this reason the capacitive strain gauge with a plastic spacer cannot work as an accurate pressure gauge. 58

59 In our experiment, we used a new self-contained capacitive pressure transducer without any spacer and the design of the compact strain gauge is shown in Figure The eddy current heating and the large heat capacity of conventional designs are avoided Figure Schematic drawing of the pressure gauge which is placed inside the NMR cell to measure an accurate sample pressure. This figure is reproduced with the permission from Ref. 36 by using coin silver for the diaphragms and high-purity titanium for the capacitive plates. Before carrying out the actual NMR measurements we tested the NMR cell with this pressure gauge inside the cell and this system worked very well up to very high pressures (P 46 bar). The final pressure of the solid samples and the molar volume V m of the samples were determined by the PVT data of Grilly and Mills 37 and the equation derived by Mullin, 38 V m (p, x3) = x 3 V 3 (p) + (1 x 3 )V 4 (p) 0.4x 3 (1 x 3 ) (2 18) using the measured pressure of the sample cell. Here V 3 and V 4 are the molar volumes of pure 3 He and the 4 He, respectively. 59

60 CHAPTER 3 MICROSCOPIC DYNAMICS OF 3 HE IMPURITIES IN SOLID 4 HE IN THE PROPOSED SUPERSOLID PHASE 3.1 Overview This chapter has focused on investigating the microscopic dynamics of 3 He atoms in solid 4 He by measuring the NMR relaxation times T 1 and T 2 in the region where significant non-classical rotational inertia fractions (NCRIFs) have been reported for solid 4 He. For 3 He concentrations x 3 = 16 ppm and 24 ppm, changes are observed for both the spin-lattice relaxation time T 1 and the spin-spin relaxation time T 2 at the temperatures corresponding to the onset of NCRIFs and, at lower temperatures, to the 3 He- 4 He phase separation. The magnitudes of T 1 and T 2 at temperatures above the phase separation agree roughly with existing theory based on the tunneling of 3 He impurities in the elastic strain field due to isotopic mismatch. However, a distinct peak in T 1 and a less well-resolved feature in T 2 are observed near the reported NCRIFs onset temperature, in contrast to the temperature-independent relaxation times predicted by the tunneling theory. 3.2 Concepts The discovery of a non-classical rotational inertia fraction (NCRIF) in solid 4 He by Kim and Chan 4,5 has generated enormous interest because the observed NCRIF could be the signature of a supersolid state. 39 Several independent experiments 40,41 have shown that the NCRIF magnitude and temperature dependence are strongly dependent on defects such as 3 He impurities and the quality of the crystals in terms of the density of dislocations. Furthermore recent studies of the elastic properties of solid 4 He by Beamish and colleagues 42 have revealed a significant frequency dependent change in the elastic shear modulus with an enhanced dissipation peak having a temperature dependence comparable to that observed for the NCRIF. These results for the shear modulus suggest that the dynamics of the 4 He lattice plays an important role in the low temperature bulk properties of solid 4 He and rather than observing a phase transition to a supersolid state 60

61 one may be observing a thermally excited dynamical response. It is therefore important to study the microscopic dynamics of 3 He impurities to probe the dynamics of the 4 He lattice and understand their relation to the NCRIF and shear-modulus phenomena. The measurement of the NMR relaxation times of dilute 3 He impurities in solid 4 He at low temperature is one of the most effective experiments to meet this need. The NMR relaxation rates are determined by quantum tunneling (via 3 He- 4 He atom exchange) and the scattering of the diffusing atoms by the crystal deformation field around the 3 He impurities and other lattice defects. The NMR relaxation rates are therefore very sensitive to the elastic properties of the solid 4 He and on any changes in the crystal ground state that would modify the tunneling rate. Although low-temperature NMR data for 3 He impurities in solid 4 He have been reported in Ref. 43,44 for higher 3 He concentrations x ppm, we obtained the first direct NMR data on isolated 3 He impurities in the region of x 3 and temperatures in which NCRIF has clearly been observed. Recently Toda et al. 43,44 reported simultaneous NCRIF and NMR data for a sample with x 3 = 10 ppm, but the NMR signal was only observable in their study from 3 He atoms in phase-separated clusters and not 3 He in solution in the 4 He lattice. 3.3 Experimental Detail As discussed in Chapter 2, the samples were prepared by mixing high purity gases and condensing the mixture at high pressure (46.2 bar) into a polycarbonate cell that contained a pressure gauge (Figure 3-1), and then solidifying the samples using the blocked capillary method. Thermal contact to the sample was provided by a solid silver cold finger extending from the dilution refrigerator. To obtain the lowest temperature 10 mk the preamplifier that was at first mounted close to the NMR cell (this first configuration is shown in Figure 2-9) was moved to the 4 K plate which is 1.8 m from the sample cell as shown in Figure 3-1. This move reduced the heat input generated by the preamplifier into the NMR probe. 61

62 Figure 3-1. Schematic representation of the low temperature NMR cell. The preamplifier and tuning capacitor are located on a 4 K cold plate located a distance of 1.8 m from the sample cell. The RF transmitting and receiving coils are simplified in this figure. [Figure reproduced with permission from Kim et al., Phys. Rev. Lett. 106, (2011). Copyright (2011) by the American Physical Society.] A first sample with x 3 = 16 ppm was annealed for 24 hours just below the melting point, while a second sample with x 3 = 24 ppm was annealed for only 30 minutes. For both samples the final pressure measured in situ at low temperature was ± 0.05 bar, corresponding to a molar volume V m 20.8 ± 0.1 cm 3. As described in Section 2.5, standard pulsed NMR techniques were used to measure the nuclear spin relaxation times: magnetization recovery following a spin echo to measure T 1, and a CPMG (Carr-Purcell-Meiboom-Gill 45 ) multiple-echo sequence to measure T 2. 62

63 The studies were carried out for a Larmor frequency of ω L /2π = 2 MHz as the expected relaxation times extrapolated from previous studies would be prohibitively long at higher frequencies. Nuclear spin Echo Amplitude (a.u.) ppm calibration 16 ppm cooling 16 ppm warming 24 ppm warming A Temperature (mk) A3 A2 Figure 3-2. Nuclear spin echo amplitudes for determination of the sample concentration. To determine the concentration of each sample we used a calibration method. A1 is the spin echo amplitude of the calibrated 1000 ppm, and A2 and A3 is the spin echo amplitudes for each sample. To determine the concentration of samples we calculated the ratio A1 A2 for the 16 ppm sample and for the 24 A3 A3 ppm sample. In order to determine the 3 He concentrations of samples we measured the amplitude of the NMR echoes at high temperatures (150 T 350 mk) which follows the Curie law and compared the signal to a standard reference sample with x 3 = 1000 ppm as shown in Figure

64 3.4 Result and Discussion Anomalies in T 1 and T 2 The observed temperature dependences of the nuclear spin-lattice relaxation time T 1 for samples with x 3 = 16 ppm and 24 ppm are compared with the dependence for a much higher concentration sample, x 3 = 500 ppm in Figure 3-3. A pronounced peak Spin-Lattice Relaxation time (s) 16 ppm cooling ppm warming 24 ppm warming 500 ppm L 16 ppm Cooling Warming L 24 ppm L 500 ppm Temperature (mk) Figure 3-3. Temperature dependence of the nuclear spin-lattice relaxation time T 1 for samples with x 3 = 16 ppm and 24 ppm compared to a high concentration sample (x 3 = 500 ppm, from Ref. 50 ). A peak in T 1 is observed for both samples at T 175 mk. In addition, for the x 3 = 16 ppm sample T 1 drops by a factor of 100 below 85 mk due to 3 He- 4 He phase separation. Solid (open) circles show data taken on warming (cooling). The dashed green lines represent T 1 values calculated for the impuriton model of Landesman. [Figure reproduced with permission from Kim et al., Phys. Rev. Lett. 106, (2011). Copyright (2011) by the American Physical Society.] at T 175 mk is observed for both the 16 ppm and 24 ppm samples in contrast with 64

65 the weak temperature dependence observed for the x 3 = 500 ppm sample. Samples with higher concentrations (500 ppm x ppm) were also studied 50,51 and had temperature dependences similar to that of the 500 ppm sample except for the expected shift in the phase separation temperature. The position of the peak in T 1 corresponds closely to the saturation temperature T 90 observed for NCRIF at this x 40 3 where T 90 is the temperature at which the normalized NCRIF is 90 % of the low temperature limiting value. By allowing for expected shifts with pressure and x 3, the T 1 peak we observe also correlates well with the shear dissipation peak observed by Syschenko et al. 42 Finally, the T 1 peak occurs at approximately the same temperature for this x 3 value as the sharp ultrasonic absorption anomaly reported by Ho et al. 52 In parallel with the observations for T 1, a less well resolved peak, which is followed by a minimum, in T 2 is observed for x 3 = 16 ppm at the same temperature for which the T 1 peak is seen (Figure 3-4). At lower temperatures large changes in T 1 and T 2 are observed due to 3 He- 4 He phase separation at T ps 85 mk. This value of T ps agrees well with the predictions of Edwards and Balibar. 53 The same features are observed for the sample with 24 ppm: a strong peak in T 1 at T 175 mk (Figure 3-3) and a weaker peak in T 2 (Figure 3-4). The T 1 and T 2 features at T 175 mk appear unrelated to the large changes in T 1 and T 2 associated with the phase separation at lower temperatures. To verify this fact, the 24 ppm sample was purposely never cooled to the phase-separation temperature. The thermal hysteresis around 85 mk in T 1 and T 2 reinforces the view that the changes at this temperature are due to 3 He- 4 He phase separation. 54 The NMR echo amplitudes (Figure 3-2) become temperature independent below the phase separation as expected for the formation of degenerate liquid 3 He droplets. 51 The results for the phase separation temperatures are in good agreement with those reported by Toda et al. for x ppm. 43 The samples with x 3 = 16 ppm and x 3 = 24 ppm have significantly different temperature dependences for T 2 at T 200 mk. As T 2 (unlike T 1 ) is sensitive 65

66 Spin-Spin Relaxation time (ms) Cooling Warming L 24 ppm L 16 ppm 16 ppm Cooling 16 ppm Warming 24 ppm Warming Landesman Model Temperature (mk) Figure 3-4. Temperature dependence of the nuclear spin-spin relaxation time T 2 for samples with x 3 = 16 ppm and 24 ppm. In addition to a strong, hysteretic drop in T 2 for x 3 = 16 ppm below 100 mk due to phase separation, both samples show poorly-resolved features near the temperature at which the T 1 peaks are observed, T 175 mk. As T 2 reflects the spectral density of 3 He motion near zero frequency it is sensitive to slow and static 3 He redistributions that do not affect T 1. This may also contribute to the temperature dependence of the T 2 values observed at T 200 mk for the x 3 = 24 ppm sample, which was cooled after less annealing than the x 3 = 16 ppm sample. [Figure reproduced with permission from Kim et al., Phys. Rev. Lett. 106, (2011). Copyright (2011) by the American Physical Society.] 66

67 to very slow and static changes in the positions of the 3 He impurities, it seems that the T 2 data cannot clearly resolve the effects leading to the T 1 peak from other effects. In particular, the higher-temperature data in Figure 3-4 suggest that T 2 is sensitive to the crystal quality Annealing Effect Normalized T 1 Peak % difference Annealing time: 24 h T 1 (s) Annealing time: 0.5 h ppm 24 ppm Temperature (mk) Temperature (mk) 16 ppm 24 ppm Figure 3-5. The normalized T 1 peaks of 16 ppm and 24 ppm sample. The solid blue line represents a fit for the 24 ppm sample and the solid red line is the fit for 16 the ppm sample. The absolute value for T 1 for the 16 ppm sample is much larger than that of 24 ppm sample because the spin-lattice relaxation time is proportional to the x 3 concentration as T 1 x 4/3 3 or T 1 x 4/3 3 in dilute 3 He- 4 He mixture. If the peaks are normalized removing the concentration dependence then only the annealing effect is seen in the peaks. 67

68 Annealing is a well known method to improve the quality of crystals. Annealing is performed by warming the sample close to the melting point (1.2 K) for some time, and cooling down slowly afterwards at 27 bar. The 16 ppm sample was annealed for 24 hours while the 24 ppm sample was annealed for 30 min. As shown in Figure 3-5 the normalized T 1 peak height of 16 ppm sample is 10 % smaller than the 24 ppm sample. The best fit using Lorentzian function shows that the peak position is mk for 16 ppm sample and mk for 24 ppm sample. Annealing decreases the peak in T 1 and this agrees well with other observations related to NCRI and shear modulus measurements. Rittner and Reppy 41 have found that annealing the helium sample reduces the period shift of their torsional oscillator in their experiment. Although Penzev et al. reported the observation of a 10 % increase of the NCRI after annealing, 55 and Kondo et al. found no effect of annealing, 56 it is now generally accepted that annealing reduces the disorder in the crystal and consequently it also decreases the magnitude of the NCRI. The especially careful analysis of annealing by Rittner and Reppy 57 and by Clark, West and Chan 58 have further confirmed this viewpoint. As shown clearly in the Figure 3-5, this features agree with the peaks in the nuclear spin relaxation times Applying the NMR Theory to the Peaks in T 1 and T 2 The nuclear spin dynamics of 3 He impurities in solid 4 He have been studied extensively for relatively high concentrations (x 3 90 ppm) and high temperatures (T 350 mk) The results have been described in terms of mobile 3 He impurities tunneling through the 4 He matrix by 3 He- 4 He exchange (J 34 ) with a mutual scattering due to the elastic deformation field surrounding each impurity (K(r) = K 0 r 3 ). 13,59 A reasonable fit to the T 1 and T 2 values observed for very low concentrations ( 20 ppm) at high temperature (T 200 mk) is obtained using the Landesman model 59 as shown by dashed green lines in Figure 3-3 and Figure

69 Landesman 59 shows that for high concentrations (x 3 (J 34 /K 0 ) 2, where 10 6 (J 34 /K 0 ) ), the correlation time is given by an effective exchange frequency J eff = J 2 34/K 0 which is determined from the elastic strain field K due to lattice deformation induced by the difference of occupied volume between 3 He and 4 He. Landesman s model is discussed in detail in Chapter 4. Applying Landesman s model the value of T 1 for relatively high temperatures (T 200 mk) is calculated using the parameters cited by Landesman in his paper. This impuriton model leads to T 1 = 5700 s and T 2 = 1.34 s for x 3 = 16 ppm in close agreement with the observed values shown by the dashed green lines in Figure 3-3. It is important to note that in this simple model T 1 J 1 eff and T 2 J eff. One would therefore expect a peak in T 1 to be accompanied by a dip in T 2 if the spectral density of the relaxation time (G(ω)) is specified by a single characteristic time. The experimental results show that this is not necessarily the case. This is not surprising as there are two characteristic times: the diffusion time for one 3 He atom to diffuse to the site of another 3 He atom (t D x 1/3 3 /J 34 ) and the scattering time for the interaction field K (t C < r K > 2 1/2 /J 2 34). 59 The results clearly show that an additional dynamical effect contributes to the NMR relaxation rates for T 175 mk. The 3 He atoms can also tunnel as weakly bound pairs. 60 The pair-tunneling model explains the resonant dips in T 1 observed 48 at ω L = 1.3 and 2.6 MHz since T 1 directly measures fluctuations in 3 He- 4 He separations at frequencies of ω L and 2ω L. However, none of these models predict temperature dependent relaxation times so they can not explain the observed T 1 peaks even though they can explain the T 1 and T 2 value roughly in the higher temperature region i.e. before the peaks appear Possible Explanation for T 1 and T 2 Anomalies The T 1 peak occurs at roughly the same temperature at which torsional oscillator and shear modulus anomalies are observed, 40,42 and it is tempting to infer a connection between all three phenomena. One possibility is that the sharp re-entrant peak in fluctuations near 175 mk signals a phase transition, possibly associated with supersolidity. 69

70 However, other explanations not involving a phase transition must be considered. Figure 3-6. Tangles of individual dislocations in the crystal of helium. This crystal picture is from a talk given by Beamish at the 25 th International Conference on Low Temperature Physics. Photo courtesy of John Beamish. Typically solid helium crystals contain defects such as isotopic impurities ( 3 He atoms), dislocations and grain boundaries. Several researchers suggest that at low temperature each dislocation line has nodes with other dislocation lines and this leads to a three dimensional network as shown in Figure 3-6. The typical dislocation density of 4 He crystal depends on the crystal growth method. The dislocation density N d in poor crystals typically grown by the constant volume method is about N d 10 9 cm 2. 61,62 The dislocation density in good crystals grown by the constant volume method 63 or by the constant temperature method 64 above 0.5 K 70

71 is N d 10 5 or 10 7 cm 2. The dislocation density of the best crystals that are grown by the constant temperature method below 0.2 K is N d 10 cm 2 or 10 2 cm This dislocation network could be pinned by intersection nodes of the network and Figure 3-7. Simple schematic for impurity pinning to the dislocation network. if the isotopic 3 He atoms exist in the crystal then they also can pin the dislocation network As shown in Figure 3-7, the distance between two nodes is L N, the length of the dislocation between two 3 He impurities is L IP and a is distance between nearest neighbors. If the temperature is lowered and the concentration of 3 He is increased then L IP is less than L N. In this case the impurity pinning is dominant. 70 As shown in Ref. 40 the x 3 -dependent onset temperature for the torsional-oscillator anomalies agrees well with the temperature, T IP (x 3 ), below which 3 He pinning of dislocation lines is expected to dominate over pinning by dislocation-network nodes 71

72 B T IP = ln( a x 3 L N ) where B 0.5 K is the binding energy of a 3 He impurity to a dislocation line, L N is (3 1) the average length of dislocation segments between nodes, and a = m is the nearest-neighbor distance. In this model there are roughly L N /a binding sites for a 3 He impurity on an inter-node segment of a dislocation line, and each binding site is occupied with probability P oc = x 3 x 3 + e B/T (3 2) For a total length of dislocation lines per unit volume Λ 0.2/L 2 N 1011 m 2 for 4 He crystals grown by the blocked-capillary method, 40 the concentration of 3 He binding sites on dislocations relative to the number of 4 He atoms in the sample is x d 10 8, much smaller than the concentrations of 3 He atoms x 3 present in our NMR experiments. It would be surprising if such a small concentration of binding sites x d could have a measurable effect on T 1. However, there are other indications that the density of dislocations or other defects must be much larger than the estimate quoted above, if the NCRIF observations are explained either by superflow along dislocations cores 58 or as a direct mechanical effect unconnected with supersolidity. 71 Figure 3-8 shows the temperature dependence of T 1 along with the occupation probability for 3 He binding sites and another model discussed below. With the expected 3 He-dislocation binding energy B 0.5 K it can be seen that the significant transition in P oc (T ) occurs at much lower temperatures than the peak in T 1 as shown in Figure 3-8 (b). If larger binding energies B = K are assumed, the drop in P oc (T ) occurs at temperatures close to those of the NMR anomaly. However, it is not clear how the partial occupation of 3 He binding sites would lead to reduced fluctuations of 3 He inter atomic vectors as inferred from the peak in T 1. Another phenomenological model we consider is to associate the T 1 anomalies with a thermally activated relaxation peak as has been used successfully to describe the 72

73 Spin-Lattice Relaxation time (s) ppm cooling 16 ppm warming 24 ppm warming (a) B = 1.81 K 16 ppm B = 1.88 K 24 ppm B = 0.5 K 16 ppm B = 0.5 K 24 ppm P oc 1 (b) Temperature (mk) B = 1.81 K 16 ppm B = 1.88 K 24 ppm B = 0.5 K 16 ppm B = 0.5 K 24 ppm 1 Figure 3-8. (a) Measured spin-lattice relaxation times for two low 3 He concentration samples (data points) along with fits to a phenomenological model based on thermally-activated relaxation of unknown degrees of freedom (smooth curves). (b) Temperature-dependent probability for a defect such as a dislocation to be occupied by a 3 He impurity. Here B is the 3 He thermal activation energy (a) or defect binding energy (b). [Figure reproduced with permission from Kim et al., Phys. Rev. Lett. 106, (2011). Copyright (2011) by the American Physical Society.] 73

74 shear-modulus shifts. 42 The smooth curves in Figure 3-8 (a) are fits to the form ωτ T 1 = [R 0 2R (ωτ) 2 ] 1, ωτ = ωτ 0 e B/T (3 3) Here R 0 and R 1 are fitting parameters giving the background relaxation rate and height of the peak in T 1, B is the activation energy, τ 0 is an attempt frequency, and ω is the frequency at which relaxation is being probed (e.g. ω L = s 1 ). For the dotted and dashed curves the activation energy was fixed at B = 0.5 K (the approximate binding energy of a 3 He impurity to a dislocation inferred in Ref. 40 while for the other curves B was allowed to increase to improve the fit to the data. Note that these fits determine only the combination ωτ 0 (not ω or τ 0 separately); for the best fit to the x 3 = 16 ppm data ωτ 0 = and B = 1.81 K. It must be emphasized that the fitting functions used in Figure 3-8 (a) are phenomenological and not derived from a simple microscopic model of the NMR relaxation. For example, a simple resonance between the thermally-activated motion of 3 He impurities and the Larmor frequency ω L would ordinarily lead to a minimum in T 1 (increased fluctuations at the Larmor frequency), not a peak as we observe. On the other hand, relaxation of 3 He impurities in solid 4 He is known to be controlled by a non-monotonic spectral density of fluctuations with a sharp feature at the frequency ( 3 MHz) at which nearest-neighbor 3 He pairs walk through the lattice via quantum tunneling. 60 Therefore a more complex mechanism such as the disruption of the quantum walking motion of 3 He pairs by resonant fluctuations of dislocation lines might be needed to explain the NMR data. 3.5 Summary The NMR measurements can only be understood in terms of a sharp change in the fluctuation spectrum and the same changes would also play a dominant role in the anomalies observed for the sound attenuation and the shear modulus. Specifically, if the fluctuation spectrum is associated with critical behavior at a phase transition, the attenuation and NMR relaxation rates would vary as T 1 T T 0 λ with 74

75 λ = 1/2 in agreement with the NMR and sound attenuation results. Alternatively, if a collective but non-critical change occurred in the lattice dynamics, the associated change in the elastic properties of the solid would result in changes of both the observed shear modulus and the NMR relaxation rates. The latter has been shown to depend on the elastic strain surrounding 3 He impurities. 59 Further studies for a wide frequency range and for samples grown at constant pressure to produce higher quality crystals are needed to distinguish between these interpretations. However, it is clear that a successful microscopic model of the lattice dynamics of solid 4 He must explain not only the NCRIF observations, but also the coincident anomalies that have been observed in ultrasound, 52 shear modulus, 42 and NMR relaxation as reported here. 75

76 CHAPTER 4 CONCENTRATION DEPENDENCE OF T 1 AND T Overview The concentration (x 3 ) dependence of the spin-lattice relaxation time, T 1, and spin-spin relaxation time, T 2, in the low temperature region where the temperature independent quantum tunneling determines the spin relaxation is discussed. In the intermediate concentration regime, impurities interact continuously with the strain field of high concentrations 59 or mass fluctuation- mass fluctuation (MF-MF) potential 15 and these interactions strongly attenuate the tunneling motions of the 3 He impurities. This damped tunneling motion of 3 He atoms modulates the nuclear spin relaxation times. In the very dilute regime the 3 He impurities behaves as individual particles or individual mass fluctuation waves (MFWs) that move through the lattice by tunneling and are scattered by the collisions between each other. There is a controversial cross over concentration above which the motion of 3 He atoms as determined by collisions no longer dominates the motion of 3 He atoms and instead 3 He atoms move under the influence of continuous interactions. Landesman predicts 59 this cross over concentration is much higher than (J 34 /K 0 ) or (J 34 /K 0 ) while Huang estimates it as (J 34 /3V 0 ) 3/ However the analysis of over a wide range of concentrations dependence of T 1 and T 2 measurements using the new data provided by this thesis work confirm that the cross over concentration is about 10 4 and the correlation times for T 1 and T 2 are different. 4.2 General Concept In quantum crystals the zero-point motion, resulting from small atomic mass and the weak attractive interaction between the atoms, is large enough to lead to a significant overlap of the wave functions of each atom and as a result of this overlap atoms can exchange their positions in the lattice

77 Dilute 3 He in solid 4 He mixture is an ideal system to test this quantum mobility of atoms. If a small amount of 3 He impurities are added to the perfect 4 He crystal, the 3 He may tunnel through the lattice by exchanging the position with a 4 He neighbor at a rate of J 34 or it may pair with another 3 He to form ( 3 He) 2 molecules 60 and tunnel while being assisted by the pair motion of 3 He 72. Andreev and Lifshitz first pointed out that it is possible to speak of a branch of defecton excitations whose energy ε can take on all possible values in a band of width ε which is proportional to the defect tunneling frequency, ε = zħj, where z is the number of nearest neighbors. When the tunneling frequency (rate) is large compared to the collision rate between impurities and the phonons, the coherent quantum tunneling is dominant. Vice versa, when the tunneling rate is small compared to the impurity scattering rate by the phonons, the coherent quantum tunneling loses the coherency and the impurity motion becomes a random walk through the lattice 46. Coherent tunneling occurs at low temperatures and incoherent tunneling dominates at higher temperatures because of the increasing number of thermal phonons. However at a fixed temperature where the temperature independent quantum tunneling motion is dominant, there are two 3 He concentration regions in the impuriton model. The intermediate concentration region is where the 3 He impurities can continuously interact with each other through the strain field resulting from the distortion of the lattice due to the difference of volume occupied by 3 He, 4 He, and vacancies. The strain field affects the motion of impurities and controls the tunneling rate of 3 He impurities. An alternative explanation for the motion of 3 He in this region considers the interaction between the well defined excitations: mass fluctuation wave (MFW). In this model one formulates the interaction between the MFWs in the averaged crystal instead of the real crystal. Another region is the very dilute region where the mean separation of 3 He impurities are outside the effective range of any strain field or mass fluctuation interaction potential 77

78 for most of the time. Therefore they may be considered as a dilute gas, each 3 He traveling freely (exchanging coherently) over several lattice spacings until encountering another 3 He impurity or other defects such as vacancies, where it is scattered. Several theorists 13,15 calculated the concentration limit as 10 3 above which the impuriton model is invalid. As the 3 He impurity concentration x 3 increases, the 3 He atom s motion must change from propagating as individual ballistic particles between collisions to one of continuous interaction in the strong field environment. The precise value of the crossover concentration is very important for understanding the property of the quantum motion of 3 He impurities. However the crossover concentration of 3 He for the transition from the very dilute gas-like region to the strong interaction region has not been clearly demonstrated by the experimental data prior to the studies reported here. In this section, the concentration dependence of T 1 and T 2 will be discussed based on the impuriton model and the crossover 3 He concentration will be estimated from the experimental data. 4.3 Concentration Dependence of T 1 Fig. 4-1 shows all T 1 data and the data reported by other group 49,54,73 76 with the concentration ranging from 10 5 to 10 2, and specific values are shown in Table 4-1. For intermediate concentration region, Eqn is used with the values of M 2 and J eff, that Landesman obtained for his elastic interaction model given below M 2 = s 1 (4 1) J eff /2π = 1.8 khz (4 2) Eqn gives the correlation values as τ c = x 1/3 3 s and from the calculation using Eqn and all numbers introduced by Landesman, the spin-lattice relaxation time (T 1 ) is calculated as T 1 = x 2/3 3 s. The green line in Figure 4-1 shows this values calculated by using Landesman s model. 78

79 10000 T dil 3 4/3 2.8 x T 1 (second) T L 3 2/ x x 3 (ppm) Figure 4-1. The observed concentration dependence of the nuclear spin-lattice relaxation times for dilute 3 He in solid 4 He. The red line represents the best fit using the collision model in the dilute region. The green line is the fit obtained by applying the Landesman s model in the intermediate region. The best fit using Landesman s numbers gives the correlation time for the intermediate region τ c = x 1/3 3 s. The correlation time for the dilute region is obtained as τ ch = x 1/3 3 s. (Experimental data: Kim et al., 73 Schratter et al., 49 Schuster et al., 74 Allen et al., 75 Hirayoshi et al., 76 Greenberg et al. 54 ) When the impurity concentration x 3 decreases this continual interacting system becomes invalid because the 3 He- 3 He collisions are not effective, and the only characteristic 79

80 time is given by 3 He- 4 He hopping between sites. The characteristic time is given by τ ch λ sep ν g = 1 x 1/3 zj (4 3) where the group velocity is ν g = azj 34 and the separation distance between two 3 He atoms is λ sep = a/x 1/3. z is the number of the nearest neighbors and in hcp structure z = 12. The nearest-neighbor distance is a = m. If 3 He atoms occupy a band of width J 34 then the maximum (mid-band) impuriton velocity is ν g azj 34 = (m/s). Landesman s values yield τ ch = x 1/3 3 s. (4 4) From Eqn. 1 34, Eqn and Eqn. 4 4 the T 1 can be given in dilute region, T 1 = 3 4 ω 2 L M 2 τ 1 c = 3 4 ω 2 L M 2 τ 1 ch = x 4/3 s. (4 5) These values are shown in Figure 4-1 as a solid red line which well matches with the data. The simplest model for scattering due to the elastic interaction would be that 3 He atoms propagate ballistically until they are within a distance R JK of each other, where R JK is the distance r between two 3 He atoms at which the elastic interaction energy K(a/r) 3 equals the tunneling energy J 34. R JK = (K/J 34 ) 1/3 a = 11a = (m) = 4 (nm). This gives a scattering cross-section σ Rij, 2 a mean free path λ a 3 σx 3 and a diffusion coefficient D dil ν g λ, D dil = ν ga 3 R 2 ij x 3 = J 34a 2 (J 34 /K) 2/3 x 3 = x 3 (m 2 /s) (4 6) Comparing Eqn to the Eqn. 4 6 and assuming whichever diffusion coefficient is lowest dominates, there will be a very mild crossover from D dil x 1 3 for x 3 < J 34 /K, to D AL x 4/3 3 for x 3 > J 34 /K. Landesman had estimated that the crossover concentration above which the continuously strong elastic field interaction is dominant is about 10 5 or However, the additional lower concentration data labeled as Kim et al. in the Figure 4-1, shows 80

81 that the crossover concentration is about 100 ppm ( 10 4 ) which is a little higher than that of Landesman s prediction. Table 4-1. T 1 and T 2 from other groups. J 34, M 2 and J eff are calculated using the Landesman s formula. J 34 = KJ eff, where K = 2π 1200 Hz 59. M 2 = 4 T 1 T 2 x T, J 3 ωl 2 eff = 1 (23.3) 2 3 T 2. All valuses are normalized to 2 MHz and ωl 2 x2/3 3 the red values are showing the normalized values. x 3 (ppm) f = ω L 2π (MHz) T 1 (s) T 2 (ms) V m (cm 3 /mol) J 34 (MHz) M 2 (sec 2 ) J eff (sec 1 ) E E E E E E E E E E E E E E E E E E E E E (2) 62(27.6) E Concentration Dependence of T 2 Figure 4-2 shows the T 2 data with respect to the 3 He concentration. All data points are selected from the same group as T 1 shown in the Figure 4-1. The green line results from the fit to the elastic strain field model (Eqn. 1 35) of Landesman using the same correlation time used for T 1, τ c = x 1/3 3 s. This fit is very poor 81

82 for the T 2 data and this is not sunrising because Landesman already had mentioned in his paper 59 that his model is not consistent with T 2 data. Huang did a detailed T L 4 4/3 5 x T 2 (second) T2 dil 1. 68s 0.01 T H x x 3 (ppm) Figure 4-2. Comparison of the concentration dependence of the nuclear spin-spin relaxation times reported in the literature for dilute 3 He in solid 4 He. The green line shows Landesman s elastic field interaction model using his values. The fit that worked for T 1 very well does not work for T 2. The red lines are from Huang s model 15 in the intermediate concentration region. The dotted red line shows x 3 independent T 2 induced from the individual ballistic collision model. (Experimental data: Kim et al., 73 Schratter et al., 49 Schuster et al., 74 Allen et al., 75 Hirayoshi et al. 76 ) calculation for the diffusion constant and the spin-spin relaxation time T 2 applying mass-fluctuation-mass-fluctuation interaction to the very low 3 He concentration mixture. 82

83 In his mass fluctuation wave (MFW) scattering model, the interaction between two MFWs prohibits the 3 He tunneling inside 4 He and disturbs the coherent motion of the 3 He impurities. The interaction between two MFWs is similar to the elastic strain field in Landesman s model. In his paper Huang derived the spin- spin relaxation time T 2 at zero Larmor frequency as T 2 (0) = W 2(x 3 ) RR x 1 3 (4 7) M 2 (1) where M 2 (1) is the second moment for x 3 = 1 and W 2 (x 3 ) RR is the rate of transition of a 3 He at R to the site R by interchange with 4 He while a spectator 3 He is at R. Huang calculated M 2 (1) 3.6ω d (RR ) 2, where ω d (RR ) is the frequency of the particle (MWF) motion due to the dipolar field. For the rigid lattice, Huang used ω d (RR ) = γµ/ R R rad/s, where γ = 10 4 rad/sg, µ = erg/g, and R R = m. With all these values he simplified Eqn. 4 7 as given below, T 2 (0) W 2(x 3 ) RR 3.6ω d (RR )x 3 W 2(x 3 ) RR x 1 3 (4 8) Huang obtained J 34 /V 0 = by fitting the diffusion data and calculated W 2 (x 3 ) RR = Inserting this value into Eqn. 4 8 results in T 2 = x 1 3 and this function is presented with black solid line in Figure 4-2. The red line T 2 = x 1 3 for J 34 /V 0 = which is used for the T 1 data in Landesman s model. The blue line stands for T 2 = x 1 3 resulted from the algebraic expression for W 2 (x 3 ) RR of Huang s model. Huang suggested that the concentration range for incoherent motion induced by this mass-fluctuation mass-fluctuation interaction is 10 3 x Below 10 3, the very dilute region called the MFW region, the 3 He atoms are very far from each other and Huang assumed the 3 He atoms then propagate with the dispersion relation and when the two 3 He atoms are closer to each other than the critical distance r c they can not propagate any more by the quantum tunneling. Because the energy difference induced by 83

84 this scattering is much larger than J 34. This scattering disturbs the tunneling motion and in this way MFW-MFW scattering mostly determines the relaxation times of 3 He atoms. Huang calculated the mean free path for MFW-MFW scattering as λ mfp = 1 π x 3 ( J 34 3V 0 ) 1/2 (4 9) where is the distance between the 3 He atoms and V 0 is the MFW-MFW interaction constant and J 34 is the exchange rate of the pair of 3 He- 4 He in a mixture. The velocity of the 3 He particle is given by v = J 34 (4 10) The characteristic time that then determines the spin relaxation time of 3 He in the MFW region is τ = λ mfp v (4 11) Inserting Eqn. 4 9 and Eqn into Eqn gives the characteristic time as τ = 1 πx 3 (3V 0 J 34 ) 1/2 (4 12) If one uses Huang s value for J 34 /3V and use J 34 = , then one obtains V 0 = and Eqn becomes τ = 1/x ( ) 1/2 = x 3 (s) (4 13) In our case, the Larmor frequency is ω L = 2 MHz and 1/ω L < τ. In this case the spin-spin relaxation time T 2 is given by T 2 = 1 x 3 M 2 τ (4 14) If one puts Eqn and Eqn. 4 1 into Eqn then the spin spin relaxation time in the MFW region is T 2 = 1 x 3 M 2 τ = x 3 = 1.2 (s) (4 15) x

85 In Figure 4-2 the dotted red line shows this T 2 in the MFW region (In Figure 4-2 it is labeled as dil to be consistent with the T 1 analysis.) The value of T 2 from the Huang s MFW model agrees quite well with our data in the very dilute region. Unfortunately Huang did not work on the T 1 because of the complexity of interaction among frequencies: J 34, J 33 and W 2 (x 3 ) RR. Huang suggested that the crossover concentration from MFW region (in our case very dilute region) to mass-fluctuation mass-fluctuation interaction region (in our case it is called intermediate region ) is about x However from our analysis using additional new data we obtained in the very dilute region it is clear that the crossover concentration is about x 3 < 10 4 and this is in a good agreement with the analysis for T 1 in section Re-examination of Landesman s Model Fig. 4-3 shows the product of T 1 and T 2 for each concentration of x 3 for several independent experiments. Multiplication of Eqn with Eqn gives 4 3 T 1T 2 x 2 3( M 2 ω L ) 2 = 1 (4 16) T 1 and T 2 are from Figure 4-1 and Figure 4-2 and inserting Eqn. 4 1 and using 2 MHz for the Larmor frequency one can calculate 4 3 T 1T 2 x 2 3( M 2 ω L ) 2 for each concentration, and from Landesman s model this should be unity if T 1 and T 2 have the same correlation time. However as shown in Figure 4-3, this is not unity for the concentration x Summary The new data obtained from this thesis work make it possible to investigate further the two theoretical models 15,59 that have been used to explain the concentration dependence of the nuclear spin relaxations in very dilute 3 He in solid 4 He. A more reliable determination of the cross-over concentration between coherent and incoherent motion of 3 He impurities in the 4 He lattice was made. Even though these two models use similar concepts the cross-over concentrations are different. Landesman found the upper limit 85

86 1 F ( x 3 ) x 3 (ppm) Figure 4-3. Variation of the function F (x 3 ) = ( 4 3 )T 1T 2 x 2 3(M 2 /ω L ) 2 as a function of 3 He concentration. For a unique correlation time F (x 3 ) = 1.0. (Experimental data: Kim et al., 73 Schratter et al., 49 Schuster et al., 74 Allen et al., 75 Hirayoshi et al. 76 ) concentration for the coherent 3 He motion to be x while Huang estimates the cross-over occurring at x The data set enhanced by the experiments reported here confirms that the cross-over concentration is x and this value is consistent with the analysis for both T 1 and T 2. Huang deduced this value from the diffusion constant and T 2 in the very dilute region (MFW region, borrowing his term used in his literature) but there were no experimental 86

87 data for him to apply his model at very dilute values of x 3. The new data from this thesis work are able to check his model for incoherent motion and his model agrees well with the data. 87

88 CHAPTER 5 PHASE SEPARATION OF VERY DILUTE 3 HE- 4 HE MIXTURE 5.1 Overview The temperature dependence of the nuclear spin-lattice relaxation times (T 1 ) and nuclear spin-spin relaxation times (T 2 ) are measured for a wide range of 3 He concentrations in dilute mixtures of 3 He in solid 4 He. Dramatic changes in T 1 and T 2 occur when phase separation occurs because the separated phase consists of essentially pure 3 He and pure 4 He. These isotopic phase separations are observed for all samples with 3 He concentrations in the range, 500 x ppm, for a molar volume V m = 20.7 cm 3. The temperature dependence of the amplitudes of the NMR signals after phase separation confirms that the phase-separated 3 He atoms form Fermi-liquid droplets below the phase separation temperatures and suggests that the interfaces between 3 He droplets and 4 He atoms are responsible for the nuclear spin relaxation. The spin-spin relaxation times (T 2 ) after phase separation increased abruptly and became temperature independent. This feature is similar to that observed for the higher x 3 concentration sample reported before Background Studies of solid and liquid 3 He clusters and films have revealed a rich array of new phenomena for both the magnetic properties and the dynamics at low temperatures, but a complete understanding of the interface interactions and kinetics has been elusive. The properties of these systems are very different from bulk properties at very low temperatures because the DeBroglie wavelength becomes larger than the range of atomic interactions and this results in interesting new quantum properties. As noted by Collins et al. 77 early NMR studies of the low temperature properties of several systems led to a wide belief that a 2 to 3 atom layer of a 2D-like solid forms at the interface between the liquid 3 He and the confining surface. 78,79 However recent studies of phase separated 3 He- 4 He solid solutions by Kingsley et al. 80 did not observe the temperature dependence expected in 88

89 this case but instead a temperature independent relaxation at the lowest temperatures. This observation is also consistent with the results reported by Hebral et al. 81 and by Mikhin et al. 82 In addition, the observed dependence of the relaxation on magnetic field varies considerably with experimental conditions from negligible dependence to linear dependence for constrained geometries in contrast to the classical quadratic dependence at high magnetic fields 83 for bulk systems and this variation needs to be understood. In order to resolve these questions we have performed NMR studies of the relaxation of the nuclear spins in 3 He clusters or droplets formed by phase separation for low 3 He concentrations from carefully annealed bulk samples of 3 He- 4 He solid solutions. The nuclear spin-lattice (T 1 ) relaxation times were measured for each concentration studied over a wide temperature range (15 T 300 mk). The values expected for T 1 depend on the relative strengths of the fluctuation spectrum of the 3 He motion at the Larmor frequency. The results of these studies are also important for developing an understanding of the NMR properties of very dilute 3 He (x 3 30 ppm) in the region where supersolid phenomena are observed Experimental Detail For these studies of the phase separation it is most important to have reliable values of the 3 He concentrations. A reference gas mixture was prepared with a 1000 ppm of 3 He concentration by mixing known volumes of high purity 3 He and 4 He gas. The NMR echo amplitudes of a standard RF pulse sequence for this reference mixture were measured at a given temperature which is well above the phase separation temperature. This measurement provided a reference point for comparison with lower concentration samples using the same method as described in Chapter 3 and shown in Figure 3-2 to calculate the correct concentration for each sample. Lower concentration samples were prepared by diluting the reference gas sample with ultra pure 4 He. Because some 3 He atoms can be lost by preferential condensation on the walls of the long capillary lines, NMR amplitudes 89

90 at a fixed temperature of 300 mk were used to determine the true concentration of the samples, as discussed in Section 3.3. The solid samples were formed using the blocked capillary method, and all had pressures at low temperature of 29.0 ± 0.03 bar as measured by an in-situ pressure gauge. At this pressure at low temperatures bulk 4 He is solid and bulk 3 He is liquid. Thus the simplest model of the phase-separated system would be droplets of liquid 3 He dispersed in a matrix of solid 4 He as shown in Figure 5-1. As described earlier the capillary was Figure 5-1. Schematic of phase separated 3 He- 4 He mixture. Under the pressure and temperature where 3 He is liquid and 4 He is solid, the mixture phase separates into the 3 He rich phase (liquid droplet of 3 He atoms) and 4 He rich phase ( 4 He lattice). heat sunk at the 1 K pot of a dilution refrigerator but otherwise thermally isolated. The samples were annealed for 24 hours just below the melting curve before cooling to low temperatures for data taking. The details of the experimental cell have been reported elsewhere. 50 Standard NMR pulse techniques were used to measure the nuclear spin-lattice (T 1 ) relaxation times using 90 x x RF pulses at 2 MHz. As discussed in Chapter 2 the 90