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ABSTRACTSOLID HELIUM IN VYCOR GLASSJonathan Maloney, M.S. Department of Physics Northern Illinois University, 2011 Laurence Lurio, DirectorIn 2004, Kim and Chan (Kim & Chan, 2004) found evidence that solid helium could exhibit superfluid behavior. This odd state of matter was later dubbed “supersolidity”. Experiments have shown that the supersolid behavior is strongly dependent upon the quality of the helium crystals. More specifically, crystals of a very poor quality show the strongest supe

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ABSTRACT

SOLID HELIUM IN VYCOR GLASS

Jonathan Maloney, M.S. Department of Physics Northern Illinois University, 2011 Laurence Lurio, Director

In 2004, Kim and Chan (Kim & Chan, 2004) found evidence that solid helium could exhibit superfluid behavior. This odd state of matter was later dubbed “supersolidity”. Experiments have shown that the supersolid behavior is strongly dependent upon the quality of the helium crystals. More specifically, crystals of a very poor quality show the strongest supersolid effects. Vycor glass provides an excellent medium to produce very poor quality crystals. In fact, it was inside Vycor that supersolidity was first observed by Kim and Chan. However, the exact nature of solid helium as well as the crystal structure are unknown. Using x-ray synchrotron radiation, solid helium in Vycor glass was studied in order to identify the structure of the crystal. A single peak is observed up to 114 bar in the diffraction pattern. Above 114 bar, three peaks are observed. The low pressure phase is identified as body-centered cubic and the high pressure phase as a mixture of body-centered cubic and hexagonal close-packed. Unfortunately, higher order peaks could not be observed, preventing definitive confirmation of these symmetries. Stacking faults in the crystal lattice are believed to be the reason behind the suppression of higher order peaks.

NORTHERN ILLINOIS UNIVERSITY DE KALB, ILLINOIS DECEMBER 2011

SOLID HELIUM IN VYCOR GLASS

BY JONATHAN MALONEY c 2011 Jonathan Maloney

A THESIS SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE MASTER OF SCIENCE

DEPARTMENT OF PHYSICS

Thesis Director: Laurence Lurio

ACKNOWLEDGEMENTS
First, I would like to thank my Mother for always telling to me to follow my heart; and my Father for always encouraging me to pursue my passion and to “reach for the stars.” Also, I would like to thank my other Mother, Jennie, for the endless barrage of encouragement (“You’re one terrific kid!!”); and my other Father, David, for teaching me the meaning of both discipline and hard work. To all of you, I am deeply grateful; your words and life lessons never fell upon deaf ears. Next, I would like to thank Dr. Carpenter, my graduate advisor in Geophysics, who encouraged me to follow my passion and pursue a graduate degree in Physics. I would also like to thank my current advisor, Dr. Lurio, as well Dr. Bera for all the help and guidance with this challenging project.

DEDICATION

Perseverance- steady persistence in a course of action, a purpose, a state, etc., especially in spite of difficulties, obstacles, or discouragement. I would like to dedicate this thesis to anyone who has ever persevered in the pursuit of their dreams.

TABLE OF CONTENTS

Page LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2 1.3 First sighting of supersolidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Importance of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moving Towards a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 4 7 8 13 21 21 22 23 28 31 35 37 v

2 EXPERIMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Crystallographic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 3.1.2 3.1.3 3.2 3.3 Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Debye-Waller Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Stacking Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES

Figure 1.1 1.2 1.3 1.4 2.1 2.2 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Bulk 4 He Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic view of a torsional oscillator . . . . . . . . . . . . . . . . . . . . . Resonant period as a function of temperature for solid 4 He in Vycor glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Period Vs. Temperature for annealed samples of solid 4 He . . . . . . . Schematic Diagram of Dilution Refrigerator. . . . . . . . . . . . . . . . . . Diagram of experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . Picture of x-ray scattering captured by the CCD camera . . . . . . . . Diffraction patterns of Vycor, Helium plus Vycor, and Helium at a temperature of 0.8K and 98 bar . . . . . . . . . . . . . . . . . . . . . . . . . . Diffraction Pattern for 4 He at 22.5bar and 0.5K . . . . . . . . . . . . . . Diffraction Pattern for 4 He at 79bar and 2.23K . . . . . . . . . . . . . . . Diffraction Pattern for 4 He at 56.5bar and 2K . . . . . . . . . . . . . . . . Diffraction Pattern for 4 He at 56.6bar and 2K . . . . . . . . . . . . . . . . Diffraction Pattern for 4 He at 98bar and 0.8K . . . . . . . . . . . . . . . . Diffraction Pattern for 4 He at 114bar and 0.7K . . . . . . . . . . . . . . . Diffraction Pattern for 4 He at 130bar and 0.7K . . . . . . . . . . . . . . .

Page 2 3 5 6 9 10 14 14 16 16 17 17 18 18 19 19 20 25

3.10 Diffraction Pattern for 4 He at 162bar and 0.5K . . . . . . . . . . . . . . . 3.11 Liquid and Solid6 Diffraction Pattern . . . . . . . . . . . . . . . . . . . . . . 3.12 Debye-temperature (ΘM ) Vs. T(K) . . . . . . . . . . . . . . . . . . . . . . .

vi Figure 3.13 Diffraction Pattern of 4 He at 98bar and 0.8K with BCC & HCP . . . 3.14 Diffraction Pattern of 4 He at 114bar and 0.7K with BCC & HCP . . 3.15 Diffraction Pattern of 4 He at 130bar and 0.7K with BCC & HCP . . 3.16 Diffraction Pattern of 4 He at 162bar and 0.5K with BCC & HCP . . 3.17 Solid8 Without Stacking Faults . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 Gaussian Fit for 4 He at 98bar and 0.8K . . . . . . . . . . . . . . . . . . . . 3.19 Gaussian Fit for 4 He at 114bar and 0.7K. . . . . . . . . . . . . . . . . . . . 3.20 Gaussian Fit for 4 He at 130bar and 0.7K. . . . . . . . . . . . . . . . . . . . 3.21 Gaussian Fit for 4 He at 162bar and 0.5K. . . . . . . . . . . . . . . . . . . . 3.22 Density Vs. Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23 Intensity Vs. Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Postulated Phase Diagram for 4 He in Vycor . . . . . . . . . . . . . . . . . Page 26 26 27 27 28 32 32 33 33 34 34 36

CHAPTER 1 INTRODUCTION

In 2004, Kim and Chan (Kim & Chan, 2004) discovered, experimentally, an exotic state of matter that was considered a theoretical possibility as early as 1969. A “supersolid” is a quantum solid in which a fraction of the mass is superfluid. Since the discovery of this unusual state of matter, intense experimental and theoretical efforts have been made to explain the origins of supersolidity. Currently, we still do not fully understand this exotic state of matter. First, it is important to understand what makes a fluid superfluid before addressing the notion of a supersolid. In 1937, Allen and Misener discovered that below a temperature of 2 K, liquid 4 He undergoes a phase transition (Fig. 1.1) and then moves without friction (i.e. zero viscosity)(Allen & Misener, 1938). Kapitza made a similar observation around the same time in Moscow and called this newly found state “superfluid”(Kaptiza, 1938). The motion could not be explained using the classical laws of hydrodynamics and thermodynamics. The new “tools” of the time, quantum mechanics, had to be used in order to properly understand this bizarre state. London and Tisza proposed that superfluid 4 He had to be a quantum fluid in which atoms are indistinguishable from one another (London, 1938; Tisza, 1938). Additionally, the atoms all move together coherently and accumulate in the same state, which is known as BoseEinstein condensation (BEC), which pertains to bosons (i.e. integer spin particles) only. Today, thanks to quantum mechanics, the underlying physics of this quantum state of matter is well understood.

2

Figure 1.1: Bulk 4 He Phase Diagram Now, a solid is rigid because its atoms or molecules are “localized” (i.e. occupy particular positions in space). Also, each atom is “distinguishable”, which means that an atom at one location can be distinguished from another at some other location. On the other hand, a superfluid moves without friction because the atoms are delocalized and indistinguishable from one another. Presently, it is not clear how a system can be solid and superfluid at the same time (i.e. supersolid). In a supersolid, part of the mass flows without friction through the rest, which remains solid. However, there is no agreed-upon framework for how this occurs. This unusual state of matter was potentially discovered in 2004 by Kim and Chan. Torsional oscillator experiments (Fig. 1.2) performed on solid 4 He samples revealed the first experimental evidence. Since the initial discovery, several groups have confirmed that anomalies exist in the rotational properties known as non-classical rotational inertia (NCRI), elastic properties, and the specific heat of solid 4 He (Lin & Clark, 2007). The experimental evidence strongly supports the

3 hypothesis that solid 4 He is a supersolid, but the interpretation of the data is not that simple.

Figure 1.2: Schematic view of a torsional oscillator (Kim & Chan, 2004) There are currently two sides to the supersolid debate. One side believes that supersolidity may only exist in crystals with defects such as dislocations in single crystals, grain boundaries in polycrystals, or glassy regions (i.e., extrinsic defects). While the other side believes that supersolidity can be an intrinsic property of the ground state in ideal crystals and that disorder only enhances the effect. The consensus among scientists is undecided at this point.

1.1

First sighting of supersolidity

This new state of matter was of great interest to a number of research groups, but it wasn’t until 2004 that Kim and Chan found evidence for its existence. The experimental setup that they used consisted of a torsional oscillator containing a cylindrical cell with an annular space filled with a sample material suspended from a torsion rod (Fig 1.2).

4 The period,τ , of the oscillator is related to the rotational inertia, I, by the following relationship: τ = 2π I/K (K=elastic constant). A shift in the resonance

period of the oscillator is the conventional indicator of the superfluid transition. The inertia decreases when the material becomes superfluid because the superfluid remains motionless while the walls oscillate. Thus, the amplitude of the period shift varies with temperature T. In 2004, Kim and Chan observed a similar shift in the oscillator period in solid, not liquid, 4 He. Around 200mK, the oscillator period decreased (Fig. 1.3). This behavior was absent in control experiments with either an empty cell or a cell containing 3 He (3 He atoms are fermions and cannot undergo BEC consistent with 4 He, whose atoms are bosons). This was interpreted as a strong indication for the presence of supersolidity.

1.2

The Importance of Disorder

The initial discovery by Kim and Chan sparked a period of intense activity among groups of experimentalists as well as theorists. One of the imperative issues was the question of whether supersolidity could be an intrinsic property of 4 He crystals, or a phenomenon completely dependent upon the presence of extrinsic defects (grain boundaries, dislocation lines, etc.). Thus, the acute issue was determining whether there were vacancies (i.e., defects) in the ground state of a perfect 4 He crystal and whether their presence could explain the observed aberration. Several theorists have looked at the energy cost and gain associated with creating vacancies in a crystal and came up “empty-handed”(Ceperley & Bernu, 2004; Prokof’ev, 2007). They found that the probability of vacancies existing under ex-

5 perimental conditions should be negligible, and that they could not be responsible for supersolidity. Thus, it was then proposed that supersolidity may instead be due to extrinsic defects within the crystal.

Figure 1.3: Resonant period as a function of temperature for solid 4 He in Vycor glass (Kim & Chan, 2004) There were two types of defects that were suggested, dislocation cores and grain boundaries. These are regions of the crystal where local stresses give rise to vacancies naturally, allowing for the exchange of atoms as well as mass flow. Now, two types of experiments were investigated and they both seemed to confirm that disorder is crucial in supersolidity. One of these experiments explored alternating mass flow (a.c.) using torsional oscillators. Rittner et al. (Rittner & Reppy, 2006) found that in annealed samples (reduces the defect density in crystals) the NCRI decreased below the noise level in the measurements (Fig 1.4). Chan et al. later confirmed that annealing reduces the NCRI (Clark et al., 2007). On the other hand, when the same type of measurement

6 is performed on a quench-frozen sample (produces a large amount of disorder), the NCRI was found to be as large as 20% of the total helium inertia.

Figure 1.4: Period Vs. Temperature for annealed samples of solid 4 He (Rittner & Reppy, 2006) The other type of experiment was conducted by Sasaki et al. (Sasaki et al., 2006). Instead of investigating a.c. mass flow, Sasaki et. al decided to look for direct (d.c.) mass flow in torsional oscillator experiments. They preceded to build a two-part sample cell containing various types of 4 He crystals in equilibrium with liquid 4 He. They monitored the respective levels of the liquid-solid interface using an optical cryostat. In the end, these experiments did indeed provide evidence for supersolid d.c. mass flow. However, this phenomenon was only observed in polycrystals containing grain boundaries (i.e., contained many defects). Single crystals did not exhibit any flow (i.e., contained minimal defects). Considering the experiments conducted, it is apparent from these two experiments that defects play a crucial role in supersolidity.

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1.3

Moving Towards a Solution

There is well-documented science supporting the argument that defects plays a pivotal role in supersolidity. Furthermore, research groups have shown that the probability of vacancies existing under experimental conditions could not be responsible for supersolidity (Ceperley & Bernu, 2004; Prokof’ev, 2007). Thus, the last question to be answered through experimentation is whether or not defects are absolutely necessary for supersolidity. One way to introduce a large number of defects into the crystal is to grow it inside of a porous media such as Vycor glass. As of yet, the exact nature of solid helium in Vycor is not known, which includes crystal symmetry. Mulders et al. has shown that solid 4 He grown in aerogel is highly polycrystalline (Mulders et al., 2008). Through x-ray diffraction experiments, they were able to show that bulk 4 He expressed an HCP crystal structure and a crystallite size of approximately 100nm. Additionally, Wallacher et al. performed a neutron diffraction study of solid 4 He in mesoporous glass (similar to Vycor, but different) to study the crystal structure. They concluded that the crystal structure is BCC (Wallacher et al., 2005). In the present work, transmission x-ray diffraction experiments on solid helium in Vycor glass were carried out in order to identify the crystal structure for the first time. From the crystal structure, we can then infer how this unusual state of matter is behaving under various temperatures and pressures. The energies, types, and densities of defects depends strongly on the background crystal structure. Thus, studies of crystal structures of Helium in different porous media can be correlated with changes in defect properties and supersolid behavior in order to shed more light on what is occurring in this transition.

CHAPTER 2 EXPERIMENT

The experiment was conducted on beam line 33 ID-I of the Advanced Photon Source at Argonne National Laboratory in December 2010. A photon energy of 24 keV was used as well as a 100µ x 100µ beam size and Pilatus x-ray detector. A dilution refrigerator (Fig. 2.1) was used throughout the study in order to achieve the desired mK range. The refrigerator is a custom designed, homemade unit (Mulders, 2010). The cooling power of the refrigeration unit is 80mW at 100mK with a base temperature of 40mK. The unit is a total of 44in long. The top 9.5in is occupied by the pulse tube motor which sits on top of a 12in x 12in mounting plate. The bottom 34.5in is composed of the vacuum can and the sample cell, which contains two Beryllium (Be) windows. The porous glass chosen for this study is a controlled pore glass , “Vycor.” Vycor is a thermal shock, and high temperature resistant glass. It is composed of 96% silica and can be readily manufactured into a variety of shapes. In general, porous glasses, such as Vycor, are permeated by an interconnected network of tubular channels (Huber & Knorr, 1999). Views from a transmission electron microscope reveals these tubular channels to be about 30 nm in length and 7 nm in diameter (Levitz et al., 1991). A capacious variety of materials have been condensed into the pores such as organic chain molecules (Jackson & McKenna, 1993), semiconductors (Hendershot et al., 1993), metals (Unruh et al., 1993), and the rare gases (Molz et al., 1993). Throughout this study, helium was the only substance investigated.

9

Figure 2.1: Schematic Diagram of Dilution Fridge

10 The experimental setup allows for the collection of data from x-ray scattering (Fig. 2.2). A tablet of the Vycor glass is mounted in the sample cell. The two Be windows in the sample cell permit the incoming and outgoing x-ray beam passage. Temperatures above and below the solid Helium transition temperature (∼ 200mK) were investigated. Regardless, the discussion of this report will focus on the <200mK range.

Figure 2.2: Diagram of experimental setup Last, after gathering all the necessary information, the raw data was filtered through two programs to arrive at a diffraction pattern. The first program (program 1) is responsible for separating the “live” and “dark” data, the “live” data being information that was gathered when the beam was on and the ‘dark’ data being information that was gathered when the beam was off. The “dark” data allowed for a proper subtraction of the background noise (i.e., cosmic rays, etc.) to see scattering from the Vycor glass and 4 He only. Last, the program is also responsible for generating an output file for later use.

11 Next, the output file is then analyzed by the second program (program 2). This program is responsible for creating a mask, normalizing the data to the background, correcting for geometrical factors, and creating a circular average of the difference. The mask is responsible for removing the area behind the beamstop because no scattering occurs here. The average scattering intensity would be distorted otherwise. Also, the mask removes the stray scattering from x-rays reflecting off of the sample cell windows and scattering behind the scatter guard slit. Last, some helium crystals grow in the region between the edges of the sample cell and the Vycor glass. As a result, Bragg spots occur within the image due to these helium crystals and must be removed since we are interested in the helium within Vycor and not bulk helium. The “normalization” is responsible for removing any scattering that is not from the helium, since this is the only portion of interest to this study. In order to do this, an image is taken when only solid helium is present and this image is then subtracted from an image taken when only the Vycor glass is present (i.e., an empty run). However, this subtraction is not correct because the incident intensities between two different measurements will not be exactly the same. Thus, this must also be accounted for in the analysis. To correct this, the intensity in some region of the image where the subtraction is expected to produce zero intensity is measured. Then, one image is normalized to the other so that the region produces an average of zero intensity after subtraction. The geometrical factors that were corrected for include a number of physical phenomena. First, the polarization factor is included. This factor is responsible for correcting for x-rays dissipated due to scattering from the target perpendicular to the incident beam. Next, a correction for the attenuation due to the sapphire crystal (i.e., sample cell window) is made. Afterwards, a “ r12 ” correction is conducted.

12 This corrects for the solid angle subtended by each pixel element. The solid angle decreases the farther the pixel is situated away from the source, which varies as
1 . r2

Last, a correction is made for the camera efficiency due to the fact that the phosphor (i.e., a substance that exhibits luminescence) in the camera is not thick enough to stop 100% of the incident x-rays. Lastly, the sample is isotropic (i.e. the same in every direction) and the scattering should only depend on the total scattering angle, not the particular direction of the scattering. Therefore, the average intensity at a particular scattering wave vector (Q) is of interest, and not the azimuthal angle (φ). Thus, the circular average groups all the data together which has the same Q-value and outputs the average value. All of the above corrections are important because without them the correct intensity due to scattering would be unobtainable. In turn, proper deduction of the underlying physics would be impossible. The final program used to better understand the crystallographic structures for each run (i.e., under various temperatures and pressures) was a Gaussian fitting program. This program was responsible for fitting Bragg peaks on theoretical models of body-centered cubic (BCC), hexagonal close-packed (HCP), and a hybrid BCC/HCP state as well as the actual diffraction patterns. These fits were then used to distinguish which crystal structure was most likely being expressed. The program works by applying a linear-regression technique to a non-linear function: in this case, a Gaussian function (f (x) = e−x ) (Bevington, 1969).
2

CHAPTER 3 RESULTS AND DISCUSSION

To properly interpret the results of the data gathered from the December 2010 beam run, the raw data had to be analyzed first (Fig. 3.1). Then, theoretical models of both HCP and BCC were enhanced to include a number of crystallographic phenomena in order to properly model what was being observed. Octave, similar to MatLab, was used for all relevant programming and subsequent data analysis. First, the raw data was processed by programs 1 & 2. The crux of the entire data analysis process relied heavily on the accuracy of these first two programs. The scattering produced from the helium was minuscule compared to that produced by the Vycor glass. Fig. 3.2 is a plot that displays the scattering due to Vycor glass only, helium and Vycor glass, as well as the final diffraction pattern of Solid7 after all corrections have been made. It is apparent that the scattering from the Vycor glass (red) is essentially the same (the two plots lie on top of one another) as the scattering produced by helium and Vycor glass (black). Thus, it was exigent that every factor be accounted for in order to properly extract the scattering due to the helium in Vycor only. Next, an output file is created which is then plotted as ‘Intensity Vs. Q,’ where Q is defined as 2π (s − so ) λ

Q=

(3.1)

14 λ= wavelength of incident radiation s= scattered radiation so = incident radiation

Figure 3.1: Picture of x-ray scattering captured by the CCD camera

Figure 3.2: Diffraction patterns of Vycor, Helium plus Vycor, and Helium at a temperature of 0.8K and 98 bar

15 Q is a mathematical “map” from the raw data to an object that can be interpreted. This “object” is recognized as the diffraction pattern. In the end, the file names for each set of data include: Liquid22bar (Fig.3.3), Solid3 (Fig. 3.4), Solid4 (Fig. 3.5), Solid5 (Fig. 3.6), Solid6 (Fig. 3.7), Solid7 (Fig. 3.8), Solid8 (Fig. 3.9), and Solid9 (Fig.3.10). After processing the raw data, in which a number of samples were grown, the below graphs were produced. After careful evaluation, Solid6, Solid7, Solid8, and Solid9 were selected for further analysis. However, these diffraction patterns are exhibiting a few odd features that need to be explained. For example, Fig. 3.11 is a plot displaying files Liquid22bar and Solid6. The Liquid22bar file is known to be a liquid due to the broad peak and the fact that the width of the peak is proportional to the inverse size of the ordered region. In a liquid, local order only persists over a few atom lengths so the broad peak is consistent with theory. On the other hand, the one peak present at 98 bar and 0.8 K may or may not be a Bragg peak. Thus, a few features that need to be explained are: one, the peak is very broad. Two, there’s only one peak visible when in a typical diffraction pattern there are many. Last, there is an abundance of background at high Q-values.

16

Figure 3.3: Diffraction Pattern for 4 He at 22.5bar and 0.5K

Figure 3.4: Diffraction Pattern for 4 He at 79bar and 2.23K

17

Figure 3.5: Diffraction Pattern for 4 He at 56.5bar and 2K

Figure 3.6: Diffraction Pattern for 4 He at 56.6bar and 2K

18

Figure 3.7: Diffraction Pattern for 4 He at 98bar and 0.8K

Figure 3.8: Diffraction Pattern for 4 He at 114bar and 0.7K

19

Figure 3.9: Diffraction Pattern for 4 He at 130bar and 0.7K

Figure 3.10: Diffraction Pattern for 4 He at 162bar and 0.5K

20

Figure 3.11: Diffraction Pattern for Liquid and Solid6 Comparison One can be explained by the relationship between crystallite size (D) and the peak width by the following formula (Mulders et al., 2008): Sλ WF W HM cos θ

D=

(3.2)

S= shape factor of order one λ= x-ray wavelength WF W HM = width of the peak θ= diffraction angle Thus, the crystallite size is inversely related to the peak width. The Gaussian fitting program revealed a peak size around 7nm which is consistent with a large

21 peak width. Two can be explained by stacking faults as well as the Debye-Waller factor which is addressed in sections 3.1.3 & 3.2. Last, the abundance of background at high Q can be explained by diffuse scattering due to zero-point motion.

3.1

Crystallographic Phenomena

After the proper diffraction pattern is arrived at, the next step is to decide what crystal structure is being expressed. Thus, theoretical models of both the HCP and BCC (FCC was also formulated, but ended up not being relevant to any of the diffraction patterns analyzed) crystal structures were created for comparison. After the two theoretical models for HCP and BCC were formulated, they were both modified to include a number of phenomena. These phenomena include the structure factor, multiplicity, and the Debye-Waller factor. The logic for including these factors is that there were peaks predicted by the two theoretical models that were missing. After including the factors, the theoretical models properly illustrated the observed diffraction patterns.

3.1.1

Structure Factor

The structure factor (F) is a mathematical description of how a material scatters incident radiation. In turn, F is dependent upon the type (i.e., x-rays, neutrons, etc.) of incident radiation. The mathematical formulation is represented by

F =
n

fn eQ·rn .

(3.3)

22 f= atomic scattering factor For crystal structure determinations, the structure factor plays a very important role, since the atomic positions (rn ) only appear within this factor (Warren, 1969). Overall, it is imperative that this factor be included in order to properly deduce the underlying physics.

3.1.2

Multiplicity

In general, multiplicity (mhkl ) arises from the fact that there will be several sets of hkl -planes (miller indices), having different orientations in the crystal, but equivalent in that they have the same d (planar spacing) and F 2 (F= structure factor) values. The value depends on the crystal symmetry and hkl. Example For cubic symmetry, a few equivalent sets of planes are the following: 100,100,010,01,0,001,001 m100 =6,

110,100,110,110,101,101,101,101,011,011,011,011 m110 =12, 111,111,111,111,111,111,111,111 m111 =8.

The cubic system has a multiplicity of 48(highest degree of symmetry), in the tetragonal system it is 16, and in the orthorhombic system it is 8(lowest degree of symmetry) (Warren, 1969).

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3.1.3

Debye-Waller Factor

The Debye-Waller factor (D-W) is an attenuation of coherent scattering, which is caused by displacements of the atoms from their average positions within the crystal lattice. These displacements are due to both the thermal energy in the crystal and zero-point energy. The thermal energy decreases with temperature. In turn, the attenuation decreases as well. At zero temperature (0 K) in helium, only zero-point motion remains. The Debye-Waller factor has the form

I(Q, T ) ∝ e−2MQ ,

(3.4)

where MQ is the Debye-Waller factor along the direction of Q. Next, The Debye-Waller factor can be expanded as 1 2 )( u4 − 3 u2 )Q4 + O(Q6 ), Q Q 12

2MQ = u2 Q2 − ( Q

(3.5)

where u2 is the mean square atomic deviation along the direction of Q. If the Q deviations of the atoms from their average positions form a Gaussian shape (i.e., a characteristic “bell curve” shape), the Debye-Waller factor can be simplified to:

2MQ = u2 Q2 . Q

(3.6)

Solid helium has been shown to have a Gaussian momentum distribution n(p) using neutron scattering within small experimental uncertainties (Blasdell et al., 1993). But, this does not mean that solid helium also has a Gaussian position

24 distribution. However, as of now, there has been no measurement in crystal helium that shows otherwise within experimental uncertainty (Azuah et al., 1995). With an assumed Gaussian position distribution, combining I(Q, T ) ∝ e−2MQ and 2MQ = u2 Q2 result in Q ln(I(Q, T )) = − u2 Q2 + const, Q

(3.7)

showing that if a series of integrated intensity measurements are taken for many elastic peaks (Q measured for each) and the temperature is fixed, u2 can be Q measured directly. If the natural logarithm of the integrated intensities versus Q2 is plotted, then the negative of the resulting slope is the u2 value for that particular Q temperature. It is also important to note that u2 ( u2 = u2 ) values are very temperature Q dependent. A representation that is relatively insensitive to temperature is far more desirable considering that it will be easier to scale. Traditionally, u2 values have been converted into their equivalent Debye-temperatures ΘM according to the following formula u
2

3¯ 2 T 2 h = mkB Θ3 M

Θ2 M + 4T 2

ΘM T

0

d e −1

(3.8)

which depends on the temperature T, u2 , and the atomic mass m (where m is 4.0026032 amu for 4 He) (Arms et al., 2003). Fig. 3.12 produced by Arms et al., represents the Debye temperature measurements for HCP 4 He.

25

Figure 3.12: Debye-temperature (ΘM ) Vs. T(K) (Arms et al., 2003) Next, the Debye-temperature was extrapolated from this graph to temperatures relevant to this study (<200mK). After the desired Debye-temperature was reached, the Debye-Waller factor was then obtained through eqn. 3.8. After properly modifying the HCP and BCC code, the below graphs were produced (Fig. 3.13-3.16). From these graphs, the picture is less opaque on what crystal structure each data file (i.e., 4 He under various temperatures and pressures) is exhibiting. The Solid6 and Solid7 appears to have a BCC structure while Solid8 and Solid 9 appear to manifest a hybrid state between both BCC and HCP.

26

Figure 3.13: Diffraction Pattern of 4 He at 98bar and 0.8K with BCC & HCP

Figure 3.14: Diffraction Pattern of 4 He at 114bar and 0.7K with BCC & HCP

27

Figure 3.15: Diffraction Pattern of 4 He at 130bar and 0.7K with BCC & HCP

Figure 3.16: Diffraction Pattern of 4 He at 162bar and 0.5K with BCC & HCP

28

3.2

Stacking Faults

The last major obstacle to overcome was the inclusion of stacking faults. Up until this point, there was no clear explanation for why the 102 peak was missing (Fig. 3.17). Wallacher et al. conducted a study on solid helium in mesoporous glass and ruled out the possibility of random stacking (i.e., stacking faults) (Wallacher et al., 2005). However, Huber et al. found that random stacking was the key to the disappearance of the [102] peak for solidified nitrogen in porous glass (Huber et al., 1998). Hence, the suppression of the [102] peak due to stacking faults deserved a closer look.

Figure 3.17: Solid 8 Without Stacking Faults

29 After stacking faults were included, it became clear why the [102] peak was suppressed beyond visibility. The explanation starts by looking at the structure factor (F =
n

fn eiQ·rn ).

First, define a set of lattice vectors a, ˆ c and their corresponding reciprocal ˆ b, ˆ ˆ ˆ l. lattice vectors h, k, ˆ These vectors satisfy the following relationships:   ˆ ˆ b ˆ ˆ  h · a = 2π h · ˆ = 0 ˆ h·c=0    ˆ ˆ ˆ b ˆ ˆ k · a = 0 k · ˆ = 2π k·c=0     ˆ  l·a=0 ˆ · ˆ = 0 ˆ · c = 2π. ˆ l b l ˆ

Now, recall that the hexagonal close-packed arrangement is ABAB. . . (the other arrangements BCBC. . . and CACA. . . are structurally identical)(Guinier, 1963). The three atom positions (A,B,C) are formulated in the following way:    ra = na a + nbˆ ˆ b    b 2 a b) r = na a + nbˆ + 3 (ˆ + ˆ + ˆ  b     rc = na a + nbˆ + 1 (ˆ + ˆ + ˆ b 3 a b) where na and nb are some integer value.

c ˆ 2 c ˆ 2

30 Next, consider the [102] peak (h=1,k=0,l=2) without stacking faults (ABAB...). The structure factor (F) is given by   F = iQ·rn   n fn e     iQ·ra,b  =  n fa,b e       = f [eiQ·ra + eiQ·rb ]    ˆ ˆ 2 a ˆ c ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = f [ei(1h+0k+2l)·(na a+nb b) + ei(1h+0k+2l)(na a+nb b+ 3 (ˆ+b)+ 2 ) ]      = f [e2πi + ei(2π+ 2 (2π)+2π) ] 3       = f [1 + e 4πi ]  3       =∅ Thus, the structure factor is going to be non-zero and is therefore going to make a contribution (i.e., produce a Bragg peak) to the diffraction pattern. If random stacking (i.e., stacking faults) is now included (ABABCAB...), a different outcome is produced. In this case, the structure factor is given by   F = iQ·rn   n fn e      = fa,b,c eiQ·ra,b,c        = f [eiQ·ra + eiQ·rb + eiQ·rc ]   
4πi

ˆ a = f [1 + e 3 + ei(1h+0k+2l)·(na a+nb b+ 3 (ˆ+b)+ 2 ) ]      = f [1 + e 4πi + ei(2π+ 1 (2π)+2π) ] 3 3       = f [1 + e 4πi + e4πi e 2πi ]  3 3       =0

ˆ

ˆ

ˆ

ˆ

1

ˆ

c ˆ

Here the structure factor disappears. Thus, there will be no Bragg peak created along this direction, which is why the [102] peak is not found.

31 Beyond the [102] peak, the random-stacking model should not have any effect on the [100] and [002] peaks. The [002] is oriented along the c-axis and stacking faults have no impact on c-axis spacing. The [100] is perpendicular to the c-axis and will produce scattering of different phases depending on the lattice arrangement. However, these phases do not depend on the position along the c-axis. Thus, the stacking order will have the same [100] intensity as long as the total number of atoms remains fixed. Last, stacking faults should have an effect on the [101] peak. Since we are considering this as a hybrid state (HCP and BCC), the peak produced by the BCC structure will “overshadow” any attenuation in the [101] peak produced by the stacking faults. Thus, there will be no noticeable attenuation in the [101] intensity.

3.3

Gaussian Fitting

Focusing on the files Solid6, Solid7, Solid8, and Solid 9, the following graphs (Fig. 3.18-3.21) were produced by the Gaussian fitting program. These graphs further enforce that Solid6 and Solid7 are a BCC structure and that Solid8 and Solid9 are hybrid states displaying both characteristics of BCC and HCP.

32

Figure 3.18: Gaussian Fit for 4 He at 98bar and 0.8K

Figure 3.19: Gaussian Fit for 4 He at 114bar and 0.7K

33

Figure 3.20: Gaussian Fit for 4 He at 130bar and 0.7K

Figure 3.21: Gaussian Fit for 4 He at 162bar and 0.5K

34 The Gaussian fitting program allowed for the density and intensity variation of the crystal phases with pressure to be calculated (Figs. 3.16 & 3.17) as well as crystallite size. The density is obtained from the position of the Bragg peaks, while the intensity is obtained by comparing the heights of the Bragg peaks. It was found that the density of BCC crystals increases initially with pressure and then plateaus slightly at higher pressure. On the other hand, the density of HCP crystals increases until highest experimental pressure. For intensity, BCC drops intensity initially and then increases with pressure up to 130 bar. At this point, HCP begins to grow. For pressures above 130 bar, BCC peak intensity decreases and HCP peak intensity increases. Last, the crystallite size decreases from 7nm to 4nm with increasing pressure. These sizes are commensurate with the average pore size of 7nm.

Figure 3.22: Density Vs. Pressure

Figure 3.23: Intensity Vs. Pressure

CHAPTER 4 CONCLUSIONS

I carried out transmission x-ray diffraction experiments on solid helium in Vycor glass in order to identify the crystal structure. Theoretical models were used to help understand what crystal structure was present at each run. It appears that for pressures up to 114 bar a single peak is observed in the diffraction pattern. This low pressure phase, after acknowledging all information (i.e., D-W factor, stacking faults, etc.), is identified as a BCC structure. Above 114 bar, three peaks are observed in the diffraction pattern. This high pressure phase is identified as a hybrid state between both BCC and HCP. A a proposed crystallographic phase diagram for 4 He is presented below (Fig. 4.1). This phase diagram was modeled after all the information gathered throughout this study. Note, the BCC/HCP phase on the diagram is not a phase in which a true hybrid state exists. Rather, this state is a coexistence of BCC and HCP throughout the porous networks where the two phases remain mutually exclusive from pore to pore within the Vycor. This conclusion is reached from the knowledge that crystallographic phase transitions are discontinuous occurrences and not continuous, as illustrated on the diagram. The observed crystallographic phase transition has the possiblity of representing a supersolid phase transition as well. It would behoove future research scientists to perform torsional oscillator experiments around these temperatures and pressures. Only then can it be definitively known if the observed crystallographic phase transition corresponds to a supersolid phase transition.

36

Figure 4.1: Postulated Phase Diagram for 4 He in Vycor

REFERENCES

Allen, J., & Misener, A. (1938). Flow of Liquid Helium II. Nature, 141 . Arms, D., Shah, R., & Simmons, R. (2003). X-ray Debye-Waller Factor Measurements of Solid 3 He and 4 He. Physical Review B , 67 . Azuah, R., Stirling, W., Glyde, H., Sokol, P., & Bennington, S. (1995). Momentum Distributions in Quantum and Nearly Classical Liquids. Physical Review B , 51 . Bevington, P. (1969). Data Reduction and Error Analysis for the Physical Sciences. New York, New York: McGraw-Hill Book Company. Blasdell, R., Ceperley, D., & Simmons, R. (1993). Neutron and PIMC Determination of the Longitudinal Momentum Distribution of HCP, BCC, and Normal Liquid
4

He. Zeitschrift Fur Naturforschung Section A-A Journal of Physical Sciences,

48 . Ceperley, D., & Bernu, B. (2004). Ring Exchanges and the Supersolid Phase of 4 He. Physical Review Letters, 93 . Clark, S., West, J., & Chan, M. (2007). Nonclassical Rotational Inertia in Helium Crystals. Physical Review Letters, 99 . Guinier, A. (1963). X-ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies. San Francisco, California: W. H. Freeman and Company.

38 Hendershot, D., Gaskill, D., Justus, B., Fatemi, M., & Berry, A. (1993). Organometallic Chemical-Vapor-Deposition and Characterization of IndiumPhosphide Nanocrystals in Vycor Porous-Glass. Applied Physics Letters, 63 . Huber, P., & Knorr, K. (1999). Adsorption-Desorption Isotherms and X-ray Diffraction of Ar Condensed into a Porous Glass Matrix. Physical Review B , 60 . Huber, P., Wallacher, D., & Knorr, K. (1998). Solid Nitrogen Confined in Porous Glass. Journal of Low Temperature Physics, 111 . Jackson, C., & McKenna, G. (1993). The Melting Behavior of Organic Materials Confined in Porous Solids. Journal of Chemical Physics, 93 . Kaptiza, P. (1938). Viscosity of Liquid Helium Below the Lambda Point. Nature, 141 . Kim, E., & Chan, M. (2004). Probably Observation of a Supersolid Helium Phase. Nature, 427 . Levitz, P., Ehret, G., Sinha, S., & Drake, J. (1991). Porous Vycor Glass: The Microstructure as Probed By Electron Microscopy, Direct Energy Transfer, SmallAngle scattering, and Molecular Adsorption. Journal of Chemical Physics, 95 . Lin, X., & Clark, A. (2007). Heat Capacity Signature of the Supersolid Transition. Nature, 449 . London, F. (1938). The Lambda Phenomenon of Liquid Helium and the BoseEinstein Degeneracy. Nature, 141 . Molz, E., Wong, A., Chan, M., & Beamish, J. (1993). Freezing and Melting of Fluids in Porous Glasses. Physical Review B , 48 .

39 Mulders, N. (2010). Dilution Refrigerator. Mulders, N., West, J., Chan, M., Kodituwakku, C., Burn, C., & Lurio, L. (2008). Torsional Oscillator and Synchrotron X-ray Experiments on Solid 4 He in Aerogel. Physical Review Letters, 101 . Prokof’ev, N. (2007). What Makes a Crystal Supersolid? Advances in Physics, 56 . Rittner, S., & Reppy, J. (2006). Observation of Classical Rotational Inertia and Nonclassical Supersolid Signals in Solid 4 He. Physical Review Letters, 97 . Sasaki, S., Ishiguro, R., Caupin, F., Maris, H., & Balibar, S. (2006). Superfluidity of Grain Boundaries and Supersolid Behaviour. Science, 313 . Tisza, L. (1938). Transport Phenomena in Helium ii. Nature, 141 . Unruh, K., Huber, T., & Huber, C. (1993). Melting and Freezing Behavior of Indium Metal in Porous Glasses. Physical Review B , 48 . Wallacher, D., Rheinstaedter, M., Hansen, T., & Knorr, K. (2005). Neutron Diffraction Study of He Solidified in a Mesoporous Glass. Journal of Low Temperature Physics, 138 . Warren, B. (1969). X-ray Diffraction. Reading, Massachusetts: Addison-Wesley Publishing Company.

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