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ABSTRACT

This thesis reports neutron diffraction and nuclear quadrupole resonance

(NQR) studies of the influence of substitutional impurities, uniaxial stress and

hydrostatic pressure on the characteristics of the cubic to tetragonal phase

transition in the potassium hexachloroosmate antifluorite, K2OsCI6.

The studied structural phase transition, which occurs at about 45K, is

associated with an anomalous temperature dependence of the Brillouin zone

centre lattice mode. This mode, which involves rigid body librations of the 0sC16

octahedra, is predicted t o be a soft mode in the vicinities of the transition

temperature.

Previous research in this system has shown a clear dependency of the

phase transition characteristics on sample preparation, which is known to

determine the strain configuration of the cornpound. As an approach to

understand the subtleties of this phenornenon, the static and dynamic

parameters associated with the transition are studied as a function of applied

pressure and by incorporating controlled amounts of impurities.

A novel uniaxial pressure and cryogenic temperature controller device was

incorporated into the NQR spectrometer in order to measure axial pressure and

temperature dependencies of the resonance frequency, spin-lattice relaxation

tirne, spin-spin relaxation time and the lineshape of Fourier transform spectra.

The effect of kC162- and ReCI62- impurity ions substituted for 0sC162- ions

in K20sCI6 is studied using neutron powder diffraction. The introduction of the

impurity ions causes measurable changes in the lattice parameten but not in the

O s 4 bond length. The dynamical parameters of the model structure used to fit

the data are well described by a Debye model of the lattice vibrations. The

structural phase transition temperature is affected by the addition of the

impurity ions. Evidence is provided for suppression, in the presence of the

impurities, of the development of precursor dynamical clusten above the phase

transition temperature.

35CI NQR measurernents were obtained in a series of powder samples

containing a known concentration of impurities and in two single crystals

through the temperature range of the cubic to tetragonal structural phase

transition. For the powder samples the data are reported for a series of applied

hydrostatic pressures. In addition to the Fast Fourier Transform analysis, the

data are subjected to a Maximum Entropy Method appraisal in regions of

overlapping spectral components. Coexistence of the cubic phase and the

tetragonal phase over a range of temperatures is clearly depicted for the first

time through 3D-presentations of stacked FT (Fourier Transform) spectra for a

series of temperatures. The presence of the impurities causes the transition

temperature to shift and also results in the appearance of a satellite line shifted

with respect to that of the cubic phase resonance. The effect of an increasing

hydrostatic pressure is to shift the transition temperatures to lower values, and

to change certain quantitative aspects of the spectra.

For the single crystal NQR experiment, the effect of the uniaxial pressure

applied along the (1 00) and (1 1 1 ) directions are reported at 78.0 K and over

the temperature range 30 to 57 K. The free induction decay times decrease

with pressure due to the additional strains. The corresponding frequency domain

spectra indicate that the inhomogeneous broadening of the NQR signal is

dominated by point defects, but that the nature, number and/or distribution of

these defects is different in the two crystals. The spin-spin relaxation times are

dependent on the crystal orientation and are independent of pressure. The

measured T2 results agree with predicted values based on a second moment

calculation. Spin-lattice relaxation data further illustrate the difference between

the two crystals. It is described by a single exponential for the (1 1 1 ) crystal,

but by a double exponential for the (1 00) crystal. The latter behaviour indicates

the presence of dynamic clusters some 30 K above the temperature of the

phase transition. The application of pressure is seen to hinder their formation.

The additional stress causes a shift in the central frequency of the main

resonance line in the cubic phase. From the analysis of the results it is concluded

that the crystal dynamics are unchanged by the application of a small uniaxial

stress; the observed changes are due entirely to variations in the static

components of the electric field gradient. The application of pressure does not

produce an observable change in the frequencies of the lines in the tetragonal

phase. Alternatively, in the region of coexistence of the cubic and tetragonal

phases, it enhances the fraction of tetragonal phase present at a particular

temperature.

First and foremost I want to express deep gratitude to Prof. Robin L.

Armstrong for his constant encouragement throughout the course of my

research. It has been a rich experience to carry out this project under his

guidance.

This project has been accomplished thanks to the help of several people. I

would like to gratefully acknowledge al1 those who have contributed to the

scientific aspect of this thesis and have created a nice atmosphere for my

studies in Fredericton.

I am deeply indebted to Prof. Carlos Martin, Prof. Allan Sharp and Prof.

Bruce Balcom for their continuous support and to Prof. Abdelhaq Hamza for

useful discussions in the late stage of my program. My gratitude also goes to Mr.

Rodney MacGregor for his technical assistance with the NQR spectrometer, to

Mr. Todd Kelly, Mr. David Tree and Mr. Allan Edwards from the electronic shop,

Mr. Donald Hornibrook and Mr. James Merrill from the machine shop and Mr.

Attila Sandor from the glass shop for their valuable help, and also to al1 the staff

members and secretaries of the Physics Department who made my studies

possible and enjoyable, especially Prof. Ronald Lees, Prof. Colan Linton, Dr. Ker-

Ping Lee, Prof. Stephen Ross, Miss Joyce McBride, and Prof. Reinhold Kaiser.

I am grateful to Dr. Brian Powell, Dr. tan Swainson, Mr. Ronald Donaberger

and al1 the staff members of the Neutron and Solid State Division at the Chalk

River Nuclear Laboratories of AECL for their assistance with the neutron

diffraction experiments.

I would like to mention the memory of my father, Prof. José J Prado, who

freely supported me to pursue a scientific career.

Finally, I would like to extend my thanks to my mother, Delia, to Carlota,

Ines, Asun, Ignacio, Javier, Sofia, Amanda, Matt, Alfonso, Patrik, Gustavo,

Cristina, Patxi, Scott and Monika, and to the group of graduate students with

whom I have shared these gratibing years. In particular to my friends Charles

and Albert.

TABLE OF CONTENTS

ABSTRACT

ACKNOWLEDGEMENT

TABLE OF CONTENTS

LlST OF TABLES

LlST OF FIGURES

CHAPTER 1.

CHAPTER 2.

CHAPTER 3.

INTRODUCTION

References

ANTIFLUORITES STRUCTURE

Structure

K20sCf6

References

NUCLEAR QUADRUPOLE RESONANCE AND

RELAXATION PROCESSES

Introduction

Nuclear Quadrupole Interaction

Temperature and Pressure Effects

Spin-Lattice Relaxation

Spin-Spin Relaxation

page i

page iv

page vi

page xi

page xii

CHAPTER 4.

CHAPTER 5.

CHAPTER 6.

References

NEUTRON DIFFRACTION

Introduction

Nuclear Bragg Scattering

Debye-Waller Factors

References

EXPERIMENTAL DETAILS

Sample Preparation

Neutron Powder Diffractometer

NQR Spectrometer

References

PUBLICATION A

Influence of Substitutional lmpurities on the Static and Dynamical

Behaviour of KtOsCls in the Vicinity of the Structural Phase Transition: A

Neutron Diffraction Study.

6.0 Abstract page 6.2

6.1 Introduction page 6.3

6.2 Experimental page 6.4

6.3 Cubic Fit to the Powder Profiles page 6.8

6.4 The Tetragonal Phase page 6.1 1

6.5 Conclusions page 6.1 6

1 8 6.6 References

CHAPTER 7. PUBLICATION B

NQR Study of the Hydrostatic Pressure and lmpurity Effects in the Cubic-

Tetragonal Phase Transition of K20sCls.

Abstract

Introduction

Experimental

Fourier Transfomi Analysis

Maximum Entropy Method

Presentation of Results

Discussion and Conclusions

References

CHAPTER 8. PUBLICATION C

Uniaxial Pressure Effects in the Cubic Phase of K20sC16.

Abstract page 8.2

Introduction page 8.3

Apparatus, Sarnples, and Techniques page 8.4

Free Induction Decay Measurements page 8.7

Spin-Spin Relaxation Time

Measurernents

Spin-Lattice Relaxation Time

Measurements page 8.1 9

8.6 Conclusions page 8.24

8.7 References page 8.26

CHAPTER 9. PUBLICATION D

Nuclear Quadrupole Resonance Study of the Cubic to Tetragonal Phase

Transition in K20sC16: Uniaxial Pressure Effects.

9.0 Abstract

9.1 Introduction

9.2 Experimental

9.3 Results

9.3.1 Linesh

apes page 9.6

9.3.2 Spin-Lattice Relaxation page 9.1 5

9.4 Discussion and Conclusions

9.4.1 Lineshapes page 9.1 9

9.4.2 Spin-Lattice Relaxation page 9.24

9.5 References page 9.28

CHAPTER 10. SUMMARY AND CONCLUDING REMARKS

1 0.1 Introduction page 1 0.1

10.2 Influence of lrnpurities page 10.2

1 0.3 Influence of Pressure page 10.3

10.4 lnherent Factors page 10.4

APPENDIX

Maximum Entropy Method

and FFT Programs page A. 1

LIST OF TABLES

Table 2.1: Structural information for high temperature page 2.2

antifluorite compounds (from the International Tables for x-

ray Crystallography).

Table 2.2: Phase transition temperatures for four K2MCls page 2.4

compounds.

Table 7.1 : Transition temperatures for the six samples. page 7.24

Table 7.2: Pressure derivatives of Tc. page 7.25

Table 8.1: Relative frequency shifts A between the main page 8.1 5

resonance and the satellite Iine.

Table 9.1 : Tr* values and resonance frequencies a t 53 K. page 9.8

LIST OF FIGURES

Figure 2.1 : R z W 6 unit cell in the cubic phase.

Figure 2.2: Depiction of the ferro-rotation of the MX6

octahedra in the tetragonal cells; O indicates the rotation

angle.

Figure 2.3: Scanning electron microscope images of a

Kz[OsC16].98[ReCî6].02 crystal. (a): grain from the powder

sample. (b): magnified section of the crystal presented in

Figure 3.1 : Reference frame for the spin-spin coupling and rf

Figure 4.1 : Geometry of the scattering system.

Figure 4.2: Reciprocal lattice and Bragg scattering condition.

Figure 5.1 : C2 Dualspec powder diffractometer.

Figure 5.2: Block diagram of the NQR pulse spectrometer.

Figure 5.3: Resonance circuits. a) Probe for the uniaxial

pressure experiment; b) Probe for the hydrostatic pressure

sxperiment; c) The LC receiver filter.

Figure 5.4: Quadrature detector block diagram.

Figure 6.1 : The observed and calculated powder-diffraction

profiles and their difference for Kz[OsCl6Jo.ge[ReCls]o.or at

121 K. The structural model is for space group Fm3m.

Figure 6.2: Temperature dependencies of the lattice

constants a and the bond lengths r for the KzOsC16,

Kz[OsCls 10.98 [IrCk ]o.oz, and Kz[OsCl6lo.s~[ReCl~lo.oz samples

as extracted from diffraction profile-refinement analysis

using a model with space group Fm3m.

Figure 6.3 a: Temperature dependencies of the chlorine and

potassium Debye-Waller factors as given by profile-

refinement analysis using a model with space group Frn3m.

The solid and dashed lines are theoretical predictions for the

potassium and chlorine Debye-Waller factors, respectively,

with no adjustable parameten. KzOsCl6 sample.

Figure 6.3 b: Temperature dependencies of the chlorine and

refinement analysis using a model with space group Fm3m.

with no adjustable parameters. Kt [OsCl6]o.g8[IrCls]o.or

sample.

Figure 6.3 c: Temperature dependencies of the chlorine and

refinement analysis using a model with space group Fm3m.

with no adjustable parameters. Kt[OsCl6]0.98[ReCl6]0.02

sample.

Figure 6.4: Temperature dependencies of ~2 for the K20sC16, page 6.1 4

K t [OsCl6]0.98 [lrClsIo.oz, and Kt[OsClsIo.98[ReCl~lo.oz samples

as obtained from profile-refinement analysis using a model

with space group Fm3m. The dashed lines are "guides to the

Figure 7.1: 3sCI response in the KzOsC16 sampie at 25.OK page 7.8

and atmospheric pressure: (a) FID signal; (b) magnitude

spectrum as obtained by FFT.

Figure 7.2: Stacked plot of FT magnitude spectra showing page 7.9

the gross features of the evolution of the spectrum

spanning the temperature range of the structural phase

transition in KzOsCIs.

Figure 7.3: Details of the line-shape evolution of the K20sC16

spectrum shown in Figure. 7.2 covering the region of a few

degrees on either side of Tc.

Figure 7.4: Temperature evolution of the 3sCl spectrum of

K2[0sCls].se[PtCls].oz indicating the presence of a satellite

peak due to the Pt impurities.

Figure 7.5 a,b: Spectra of K2[0sCls].98[PtCl6 1.02 sarnple

taken a t 40.2K and atmospheric pressure: (a) FID signal; (b)

FT amplitude spectrum.

Figure 7.5 c,d: Spectra of K2 [OsCls ].98[PtC16~.oz sample

taken at 40.2K and atmospheric pressure: (c) FT magnitude

spectrum; (d) MEM output.

Figure 7.6 a,b: FT magnitude spectra for two samples a t page 7.1 8

atrnospheric pressure and temperatures significantly greater

than Tc: (a) kOsCk;(b) K2[O~Cl6].ga[lrCls].or. The satellite

peaks resulting from the addition of impurities are indicated

by arrows.

Figure 7.6 c,d: FT magnitude spectra for two samples a t page 7.1 9

atmospheric pressure and temperatures significantly greater

t h a n T c : ( C ) K2[OsC16].98[PtC16]. 0 2 ; (d )

Kz[OsC16].96[ReC16].04. The satellite peaks resulting from the

addition of impurities are indicated by arrows.

Figure 7.7: Temperature variation of the JsCl NQR resonance

lines for the Kz[OsCls].s8[lrCls].oz sample subjected to an

applied hydrostatic pressure of 69 MPa. Note the region of

coexistence of the cubic and tetragonal phases.

Figure 7.8 a: Maximum amplitude of the FT amplitude

spectrum in the cubic phase obtained by fitting Lorentzian

line-shapes t o the data. K2 [OsC16].98[ReCI6]. O 2 a t

atmospheric pressure; an exponential decay function is

Figure 7.8 b: Maximum amplitude of the FT amplitude

spectrum in the cubic phase obtained by fitting Lorentzian

line-shapes to the data. K2[0sCls].ss[irCls].or under 1 38 MPa

hydrostatic pressure; an exponential plus a linear term is

used to represent the decay.

Figure 7.9: Transition temperature, Tc, plotted as a function page 7.23

of applied hydrostatic pressure, P, for the six samples

studied.

Figure 7.10: Temperature variation of the 3sCl NQR page 7.26

resonance lines for the Kz [OS C l6].99[ITCl6].0 1 sample

subjected to an applied hydrostatic pressure of 234 MPa.

Note the region of coexistence of the cubic and tetragonal

phases.

Figure 7.1 1 : Tz* values as a function of temperature in the

cubic phase of KzOsCk and Kz[OsCls].ss[ReCl6].0~ samples

at atmospheric pressure.

Figure 7.1 2 a: Maximum amplitudes, as a function of

temperature, of the resonance lines in both the cubic and

tetragonal phases of the K~osC16 sample a t atmospheric

pressure.

Figure 7.1 2 b: Maximum amplitudes, as a function of

temperature, of the resonance lines in both the cubic and

tetragonal phases of the Kz[OsCI6].se[ReCl6].02 sample a t

atmospheric pressure.

Figure 7.13: FFT spectra of the Kz[OsCl6].ss[ReCls].oz

sample at atmospheric pressure. By cornparison with Fig.

3(b), it can be seen that the presence of the Re impurities

cause the xy-component (solid line) to grow faster as the

temperature decreases, once it appears.

Figure 8.1 : Uniaxial pressure device.

Figure 8.2: Typical 3sCI free induction decay signal. The solid

line through the data is a least square fit.

Figure 8.3: Tz* values obtained from fits to the free

induction decay signals.

Figure 8.4: Typical example of an inhomogeneously

broadened 3sCI resonance line as obtained by Fourier

transformation of a free induction decay signal. The dashed

lines are the symrnetric Lorentzian components; the solid

line their vector sum.

Figure 8.5 a: Peak frequencies of the main and satellite lines page 8.1 1

of the 35Cl spectra as a function of applied uniaxial pressure.

The solid lines are least squares fits. Results for the (100)

crystal.

Figure 8.5 b: Peak frequencies of the main and satellite lines page 8.1 2

of the 3sCI spectra as a function of applied uniaxial pressure.

The solid lines are least squares fits. Results for the (1 1 1)

crystal.

Figure 8.6: Change in the relative intensities, A d A l , of the page 8.1 6

satellite and main lines as a function of the applied uniaxial

pressure. The solid lines are least squares fits.

Figure 8.7: A representative 3sCI decay signal as obtained page 8.17

following a spin-echo sequence. The Gaussian fit to the data

yields the T2 value.

Figure 8.8: Tz values as a function of applied uniaxial page 8.1 8

pressure. The solid lines are the average values.

Figure 8.9: Typical semi-logarithrnic decay plots obtained for page 8.20

the (1 1 1 ) crystal following an inversion recovery sequence.

The least squares lines yield values of TI.

Figure 8.10: Typical semi-logarithmic decay plots obtained page 8.21

for the (100) crystal following an inversion-recovery

sequence. These data are f i t by a double exponential

function and yield two time constants, Tis and TI L.

Figure 8.1 1: TSI and TIL values for the (100) crystal as a page 8.22

function of the applied uniaxial pressure. The solid lines are

guides to the eye.

Figure 8.1 2: Plot of the amplitude ratio, As/AL, of the two page 8.23

components of the double exponential fit to the spin-lattice

relaxation data for the (1 00) ciystal. The solid line is a least

squares fit to these data.

Figure 9.1 : Cubic phase Fourier transform amplitude spectra page 9.7

at 53 K. The dashed lines are Lorentzian components which

sum to yield the solid lines through the data.

Figure 9.2: Temperature variation of the peak frequencies,

VQ, of the main components of the cubic phase spectra, with

and without pressure; vo = 16.884 MHz.

Figure 9.3: Temperature variation of the peak frequencies of page 9.1 1

the z- and xy-lines for the samples in the tetragonal phase,

with and without pressure. The solid Iines are fits to the

function A(B - T) 112.

Figure 9.4: Temperature dependence of Tz* values for the page 9.1 3

z- and xy-lines for the samples in the tetragonal phase, with

and without pressure. The solid lines are guides to the eye.

Figure 9.5: Spectra for the (100) crystal a t two page 9.1 4

temperatures in the coexistence region, with and without

pressure.

Figure 9.6: Variation of the maximum of the FT signal as a page 9.1 6

function of the delay time for the sample at 49.0 K and

atmospheric pressure.

Figure 9.7 a): Plot of TILTZ as a function of T for the (100) page 9.1 7

crystal, with and without pressure.

Figure 9.7 b): Plot of TlsTz as a function of T for the (1 00) page 9.1 7

crystal, with and without pressure

Figure 9.8: Temperature dependence of the ratio As/& as a page 9.18

function of T for the (100) crystal, with and without

pressure.

Figure 10.1: a) Fourier transform and b) MEM anaiysis page 10.2

corresponding to a tetragonal phase three component 3sCI

resonance spectrum.

Introduction

CHAPTER 1

INTRODUCTION

Chapter 7 - Page 7

The study of structural phase transitions is an extensive field in

condensed matter physics. It has been the focus of many theories such as the

early formulation by Landau and Lifshitz in the 1930's (Landau and Lifshitz,

1970). By using rnicroscopic models (Wilson and Kogut, 1974; Cowley, 1964

and Cochran, 1959), lattice instabilities are related to changes in the phonon

frequencies in the phase transition temperature vicinity.

There are intrinsic and extrinsic mechanisms triggering structural

distortions in the crystals. Externally applied stress and the presence of

impurities produce additional crystalline strains that result in noticeable

variations in the intra- and inter-molecular forces. This is directly reflected in the

phonon spectra. Dependence of dynamic parameten on sample preparation has

been observed (Armstrong et al., 1986). A qualitative understanding of the

influence of these factors on the characteristics of phase transitions has been

pursued; these discussions have involved such factors as the critical

temperature, precunor effects and anomalies in the lattice vibration spectrum.

Antifluorite (R2MX6) and perovskite (RMXs) families of compounds

represent relatively simple systems in which phase transition processes are well

studied. These compounds play a crucial role in linking theoretical models to the

measurable parameters associated with the transitions. As an example, the rigid-

sphere description (Krupski, 1989a and Brown, 1964) is based on geometrical

introduction Chapter 1 - Page 2

arguments, considering the sites of the ions and the lattice constants.

In an attempt to undentand the different features driving the phase

transitions, Nucfear Quadrupole Resonance (NQR), neutron scattering and optical

spectroscopy experiments have been undertaken in single crystal and

polycrystalline antifluorite cornpounds. These studies are summarized in various

publications (Armstrong, 1989, 1988, 1980, 1975b and Armstrong et al.,

197Sa).

A characteristic of the antifluorite structures is the strong bond among

the M-X atoms. This coupling causes the MX6 octahedra t o behave as a

molecular unit. The equilibrium position of these octahedra can rotate about the

cell axes (Armstrong, 1988 and O'Leary, 1969) and thereby cause a structural

phase transition. For example, the occurrence of cubic-tetragonal phase

transitions in antifluorite crystals involving both, a rotation of the octahedra and

a distortion of the cubic cell have been reported (McElroy et al., 1980, for

KzOsCls, KzReCls and (NH4)zPtls).

The present thesis describes experiments on a particular antifluorite,

potassium hexachloroosmate (KzOsCl6). A t high temperature this crystal

displays cubic symmetry. The structure can be viewed as a face centred cubic

lattice of OsCl6 octahedral units, whose cell size is a n 9.7 A. The K+ ions lie on a

a a a a simple cubic lattice of dimension 2, displaced by ( I 7 , i. 4 , * 7) from the

origin of the face centred cubic lattice. Details of this structure are presented in

Chapter 2.

The high temperature phase transition of K2OsCl6 is from a cubic (Oh5) to

a tetragonai structure (C4hS). It occurs at a criticai temperature of about 45K

(Armstrong, 1970). This transition was determined t o be second order

ln troduc tion Chapter 1 - Page 3

displacive using specific heat and thermal expansion data (Novotny et al., 1977;

Willemsen et al., 1977 and Martin, 1975). The order parameter is the time

averaged rotation angle of the OsCl6 octahedral unit around a lattice axis. At

high temperature, the rnean direction of the Os41 bond is along a cell edge.

Information on the lattice vibration modes and structural parameters is

provided by neutron scattering measurements. This can be achieved because

the thermal neutron energy is of the order of the lattice excitation energy and

the neutron wavelengths are comparable to the unit cell dimensions (Bacon,

1 975). Inelastic neutron scattering results contributed to the identification of

the rotary lattice mode behaviour in K20sC16 and K2ReCI6. (Sutton et al., 1983;

Mintz and Armstrong, 1979; Lynn et al., 1978 and Shirane, 1971). The

experiments have shown that the r-point rotary mode softens as the transition

temperature is approached (Pelzl et al., 1 977 and Winter et al., 1 976). This

frequency evolution of the modes is responsible for the development of an

instability and finally, for the structural change to a tetragonal symmetry

(O'Leary and Wheeler, 1 970 and O'Leary, 1 969).

Neutron powder diffraction results in the tetragonal phase in K20sC16

(Armstrong et al., 1987 and 1978) were used to estimate the angles of rotation

of the 0sC16 unit. For K2ReCls, NMR (Brown et al., 1973) and x-ray (O'Leary and

Wheeler, 1970) measurements provided this information. In al1 cases a maximum

rotation angle of approxirnately 3O is reported. The structural transition is also

accompanied by a small distortion of the cell of about 0.2% (Wruk et al., 1990).

Changes in the electric field gradients (EFG) a t the nuclei sites can be

stlidied by NQR. This technique has proved to be a sensitive tool to monitor

alterations to the structurai parameters, providing insight into the mechanisms

that trigger the phase transitions. Typically, changes in the resonance frequency

Introduction Chapter 7 - Page 4

by one part in IO5 are detected. 3sCI NQR Fourier transform (FT) spectra for

K20sC16 present a relatively narrow, thus high amplitude resonant line. It is

therefore a good candidate for the present study.

In the cubic phase, the rotary lattice mode is, because of its syrnmetiy,

neither Raman nor infrared active (Van Driel et al., 1972). However, NQR

spectroscopy can reveal information on the softening of the mode.

The motional average of the EFG depends on the temperature (Bayer,

1951 and Kushida et al., 1956) and the extemal stress applied. The defect

distribution in the crystal produces asymmetries in the NQR spectral lineshapes

(Stoneham, 1969). Then, the strains within the sample can be characterized by

a detailed analysis of the resonant components. To help elucidate the static and

dynamic contribution to the NQR lineshape and the anisotropic responses of the

crystalline parameters, hydrostatic (Krupski and Armstrong, 1 989b and

Armstrong and Krupski, 1990) and unidirectional pressure experiments (Zamar

and Brunetti, 1991 and 1988 and Brunetti, 1980) have been performed.

NQR spin-lattice relaxation times also reflect the evolution of the

frequency of the rotary lattice mode (Dimitropoulos et al., 1992 and Bonera et

al., 1970). By selecting different directions for the application of compression

stresses one can observe the influence on the relaxation rates of externally

modified crystalline forces. Directional effects are also reflected in the spin-spin

relaxation time, as described by a second moment calculation. The present

thesis contains the first NQR study of the effects of uniaxial stress in an

antifluorite compound.

Evidence of dynamic clusters presenting the symmetry of the low

temperature phase a few degrees above the critical temperature has been

In traduction Chapter 1 - Page 5

reported (Ryan et al., 1986, by x-ray scattering in RbCaF3 and Armstrong and

Martin, 1975c, by 35CI NQR in KtOsCls). Neutron powder diffraction profiles are

also sensitive to the development of these precursor effects. Profile refinement

is used to detect alterations in the lattice parameters and precursor dynamical

clusten, when impurities are present. The dynamical parameten of the model

structure, used to fit the data, are corroborated by using a Debye model of the

lattice vibrations to calculate translational Debye-Waller factors for the atoms.

A combined study using 35CI NQR and neutron diffraction techniques,

monitoring the development of precursor cluster effects in the high temperature

regime, is presented in this thesis. The focus is on deriving conclusions on the

dependence of the phase transition parameten on the pressure and impurity

concentration, considering the characteristics of the clusters, rotary-lattice

modes and critical temperatures (Krupski, 1 990).

Two low temperature pressure chambers for the NQR spectrometer were

used. A novel uniaxial stress arrn and heat exchanger was designed and build as

part of the project. For the hydrostatic pressure experiment (up to 270 MPa),

the chamber and temperature comptroller were improved with the guidance of

Professor Marcin Krupski (Institute of Molecular Physics, Polish Academy of

Science, Poznan, Poland) during his one year research leave at the University of

New Brunswick (1 992). A number of changes were made to the spectrometer in

order to increase the signal to noise ratio. Hardware features are presented in

the experimental chapter. For the NQR spectral analysis, resolution was

improved using a combined FT and Maximum Entropy Method (Mackowiak, 1994;

Eguchi et al., 1989; Stephenson, 1988 and Laue et al., 1985).

Neutron diffraction profiles were taken on the C2 (DUALSPEC) powder

diffractometer at the NRU reactor at Chalk River Laboratories of AECL, with the

In traduction Chapter 1 - Page 6

assistance of Dr. Brian M. Powell. The profiles were refined using the Rietveld

General Structure Analysis Systern code (GSAS, Larson and Von Dreele, 1990).

INTRODUCTION REFERENCES

Armstrong R L, Kmpski M and Su S 1990 Can. J. Phys. 68 88

Armstrong R L 1989 Progress in NMR Spectroscopy 21 15 1

Armstrong R L 1988 Magnetic Resonance and Related Phenornena, 24 th

Ampere Congress, Poznan, Elsevier 54

Armstrong R L, Morra R, Svare I and Powell B M 1987 Can. J. Phys. 65 386

Armstrong R L, Ramia M E and Morra R M 1986 J. Phys. C 19 4563

Armstrong R L 1980 Physics Reports 57 343

Armstrong R L, Mintz D, Powell B M and Buyen W J 1978 Phys. Rev. B 17 1260

Armstrong R L and van Driel H M 197Sa Advances in Nuclear Quadrupole

Resonance Vol. 2, Heyden and Son Ltd. 179

Armstrong R L 1975b J. Magn. Reson. 20 21 4

Armstrong R L and Martin C A 197Sc Phys. Rev. Lett. 35 294

Armstrong R L and Baker G L 1970 Can. J. Phys. 48 241 1

Bacon 1 975 Neutron Diffraction, Clarendon Press, Oxford

Bayer H 1951 2. Phys. 130 227

Bonera G, Bona F and Rigamonti A 1970 Phys. Rev. B 2 2784

Brown A, Jeffrey K R and Armstrong R 1 1973 Phys. Rev. B 8 121

Brown I D 1964 Can. J. Chem. 42 2758

Brunetti A H 1 980 J. Molec. Struct. 58 5 1 3

Cochran W 1959 Phys. Rev. Lett. 3 41 2

Cowley R A 1 964 Phys. Rev. 1 34 981

Dimitropoulos C, Bona F and Pelzl J 1992 2. Naturforsch. 47a 261

Eguchi T, Mano K and Nakamura N 1989 2. Naturfonch 44a 1 S

Krupski M 1990 High Pressure Research 4 466

Krupski M 1 989a Phys. Stat. Sol. a 78 1 751

Krupski M and Armstrong R L 1989b Can. J. Phys. 67 566

Kushida T, Benedek G B and Bloembergen N 1956 Phys. Rev. 104 1364

Landau L D and Lifshitz E M 1970 Statistical Physics, Addison-Wesley Publishing

Co., Don Mills, Ontario

Larson A C and Von Dreele R 8 1990 Preprint No. LANSCE, MS-H805. Los

Alamos National Laboratories

Laue E Dl Skilling J, Staunton J, Sibisi S and Brereton R G 1985 J. Magn. Reson.

62 437

Lynn J W, Patterson H Hl Shirane G and Wheeler R G 1978 Sol. St. Comm. 27

Mackowiak M 1994 Molec. Phys. Rep. 6 188

Martin C A 1975, Ph0 thesis, University of Toronto

McElroy R G, Hwang J Y and Armstrong R L 1980 J. Mag. Reson. 37 49

Mintz J D and Armstrong R L 1979 Can. J. Phys. 58 657

Novotny V, Martin C A, Armstrong R L and Meinke P P 1977 Phys. Rev. B 1 5

OILeary G P and Wheeler 1970 Phys. Rev. B 1 4409

OILeary G P 1969 Phys. Rev. Lett. 23 782

Pelzl J, Engels P and Florian R 1977 Phys. Status Solidi b 82 145

Ryan T W, Nelsen R J, Cowley R A and Gibaud A 1986 Phys. Rev. Lett. 56 2704

Shirane G 1971 Structural Phase Transitions and Soft Modes, Samuelson E J,

Andersen E and Feder J, Universiteforlaget, Oslo, Nomay

Stephenson O S 1988 Progress in Nuclear Magnetic Resonance Spectroscopy,

Emsley J W, Feeney J and Sutcliffe L Hl Pergamon Press, Oxford 51 2

Stoneham A M 1969 Rev. Mod. Phys. 41 82

Sutton M, Armstrong R L, Powell B M and Buyers W 1983 Phys. Rev. B 27 380

van Driel H M, Wiszniewska Ml Moores B M and Armstrong R L 1972 Phys. Rev. B

6 1596.

Willemsen H W, Martin C A, Meincke P P and Armstrong R L 1977 Phys. Rev. B

Wilson K G and Kogut J 1974 J. Phys. Rep. Cl 2 75

Winter J, Rossler K, Bolz J and Pelzl J 1976 Phys. Status Solidi b 74 194

Wruk N, Pelzl J, Hock K H and Saunden G A 1990 Philos. Mag. B 61 67

Zamar R C and B ~ n e t t i A H 1991 J. Phys.: Cond. Matt. 3 2401

Zamar R C and Brunetti A H 1988 Phys. Stat. Sol. b 150 245

Antifluotite Structure. K20sCI6 Chapter 2 - Page 1

CHAPTER 2

ANTIFLUORiTE STRUCTURE

K20sCl 6

2.1 STRUCTURE

The high temperature phase of antifhorite compounds (R2MX6) is cubic,

with space-group Oh5 (Fm3m). This structure is analogous to that of CaF2. The

calcium ions are replaced by (MXs)-2 molecular units and the fluorine ions by R+

cations. A diagram of the unit cell is displayed in Figure 2.1 and information

about the symmetry is presented in Table 2.1. Many of these compounds

undergo phase trmsitions to structures of lower symmetry as the temperature

is lowered. Using information obtained from a published compilation in

hexahalometallates (Rossler and Winter, 1977), the phase transition

temperatures for those compounds relevant to this thesis are presented in Table

For the chemical formula R2MX6 (hexahalometallates), M represents heavy

metallic ions, for exarnple Pt, Ir, Os or Re. At high temperature these ions define

a face centred cubic lattice (a sites following Wyckoff notation). They are in the

centre of an octahedron whose corners are occupied by the X ions (e sites). The

X halogen ions may be CI, Br or 1. The R ions may be K, Rb, Cs or NH4; they are

located a t c sites, forming a simple cubic sublattice of half-size with respect to

the face centred cubic lattice. It is worth mentioning that the e points are not

symmetry sites but lie on lines of symmetry, so there is no direct relation

between the lattice constant and the M-X bond length.

Antinuorite Structure. KzOsC16 Chapter 2 - Page 2

Table 2.7: Structural information for antifluorite compounds (from the

International Tables for x-ray Crystallography).

Chemical formula: R2MX6.

Symmetry: Fm3m (No. 225) - Oh

The lattice parameter is a, = 10 A, with the interatomic M-X distance

a0 slightly less than 4. Structural data for some RzMX6 crystais were reported by

Takazawa et al. (1 990) and Rossler and Winter (1 976). The structural stability

of the compounds has been related to geometrical parameten through the so-

called rigid-sphere model (Krupski, 1 990 and 1 989 and Brown, 1964).

Of particular interest in this thesis are the structural phase transitions

characterized by a rotation of the equilibrium position of the MX6 octahedral unit

(McElroy et al., 1980; McElroy and Armstrong, 1978; Winter et al., 1976 and

Grundy and Brown, 1970). The rotations can be ferro or antiferrorotations.

Antifluorite Structure. K20sC16 Chapter 2 - Page 3

Figure 2.2 represents a ferrorotation, displaying the orientation of the MX6

octahedra in a tetragonal phase.

Fig. 2.1 R2Mx6 unit ce//

in the cubic phase.

Rossler and Winter (1 977) studied the structural phase transitions in antifluonte

compounds. They reported that the transitions show systematic trends as a

function of periodic properties of the lattice constituents. In particular they

found that a shift to lower values of the transition temperature with increasing

size of the cations and decreasing size of the halogen can be noticed.

An tifluorite Structure. K20sC16 Chapter 2 - Page 4

Table 2.2: Phase transition temperatures for foor K2MC16 compounds.

The present study uses single crystal and polycrystalline KrOsC16 samples.

This crystal is cubic, Oh5 (F m3m), at room temperature and undergoes a

structural transition to a tetragonal phase, C4h5 (1 4/m), at around 45K.

Compound r

K20sC16

K2ReC16

K2kC16

K2PtC16

The metal-halogen bond is strong for KzOsCls. The Os-CI distance remains

unchanged even after the crystal undergoes the structural phase transition

(Armstrong et al., 1987). Therefore, the 0sC16 octahedra complex is considered

a rigid molecular unit. Figure 2.1 displays the osmium and chlorine atoms

defining the octahedra unit.

Figure 2.3 is a scanning electron microscope image of a grain from a

powder sample. Figure 2.3(b) provides clear evidence of the lattice planes. The

appearances of the single crystals were similar to those shown in the image of

the grain.

Tc (KI

111, 103, 76, 11

none reported

nature

structural

antiferromagnetic

An tif h o rite Structure. K&kCI6 Chapter 2 - Page 5

Details about the structural parameters are presented in chapter 6, where

the refinement of neutron powder diffraction profiles is reported.

Fig. 2.2: Depiction of the ferro-rotation of the MX6 octahedra in the tetragonal

cells; 8 indicates the rotation angle.

An tifluonte Structure. K20sC/6 Chapter 2 - Page 6

Fig. 2.3: Scanning electron microscope images of a K2[OsC16].g8[ReC16~.02

crystal. (a): grain from the powder sample. (b): magnified section of the crystal

presented in (a).

Antifluonte Structure. K20sCI6 Chapter 2 - Page 7

CHAPTER 2 REFERENCES

Armstrong R L, Morra R M, Svare I and Powell B M 1987 Can. J. Phys. 65 386

Brown I D 1964 Can. J. Chem. 42 2758

Grundy H D and Brown I D 1970 Can. J. Chem. 48 1 1 SI

Krupski M 1990 High Pressure Research 4 466

Kmpski M 1989 Phys. Stat. Sol. a 78 751

McElroy R G C, Hwang J Y and Armstrong R L 1980 J. Mag. Reson. 37 49

McElroy R G C and Armstrong R L 1978 Phys. Rev. 18 1352

Rossler K and Winter J 1977 Chem. Phys. Lett. 46 566

Takazawa H, Ohba S, Saito Y and Sano M 1990 Acta Cryst. 846 166

Winter J, Rossler K, Bolz J and Pelzl J 1976 Phys. Stat. Sol. 74b 193

NQR and Relaxation Processes Chapter 3 - Page 1

CHAPTER 3

NUCLEAR QUADRUWLE RESONANCE AND

RELAXATION PROCESSES

3.1 INTRODUCTION

In solids, nuclei with an electric quadrup ole moment experience an

interaction with the crystalline electric field gradient (EFG). The lattice vibrations

cause a motional averaging of the EFG that is pressure and temperature

dependent. When this time average is non-zero, a set of stationary nuclear

energy levels results. Nuclear Quadrupole Resonance (NQR), is the study of

magnetic dipole transitions between these levels. Typically, the resonance

frequency lies in the radio-frequency (rf), region of the electromagnetic

spectrum. The first observation of pure NQR was reported by Dehmelt and

Kruger (1 950). The measurements of the resonance frequencies contain

information about the EFG so that NQR is a powerful tool to detect anomalous

behaviour in the lattice. Because of its intrinsic sensitivity, NQR makes an ideal

probe to study the subtleties associated with structural phase transitions.

By proposing a model for the interaction mechanism, one is able to relate

the temperature and pressure evolution of the NQR frequency and line-shape to

structural and motional characteristics of the lattice. Moreover, an energy

exchange process between the spin system and the lattice (the so-called spin-

lattice relaxation process) can be studied, providing an undentanding of the

behaviour of the vibrational modes. Hindered rotations often provide powerful

mechanisms for such nuclear relaxation.

In the subsequent sections of this chapter, the principles underlying the

effects of pressure and temperature on the NQR frequency and relaxation

processes are presented.

3.2 ELECTRIC QUADRUPOLE INTERACTION: RESONANCE FREQUENCY

Abragam (1 983) described the electrostatic interaction between the

nucleus and its neighboring electrons by the Hamiltonian:

where A; is the mth component of the tensor operator corresponding to the Ith

nuclear multipole moment, and B;" is the mth component of the electronic

tensor operator. A: = O for odd i-values because the stationary nuclear States

have well defined parity.

The case 1 = O corresponds to the Coulomb interaction between point

charges; 1 = 2 represents the interaction between a nuclear electric quadrupole

moment and the crystalline EFG. This latter multipole component can be

expressed as the product of two second rank tenson (Slichter, 1990; Das and

Hahn, 1 958 and Cohen and Reif, 1 957):

QO = A (3 1: - 12)

and Tm, the components of the EFG, are:

with l r=r nucleus

I represents the nuclear spin operator, with axis of quantization along the z-

direction. A = eQ where Q is the quadrupole moment of the nucleus, 2 1(21 - 1)

defined as

with p., the charge density at the position r. = (x,. y,,, z,,) of a nucleus with

magnetic quantum number tn (-1 -< m 5 1) and e the electronic charge.

A transformation is made to a system of axes whose z-direction is parallel

to the principal component of the EFG. The axes are labelled according to the

magnitude of the eigenvalues, ~v,i 2 iV,i 2 IV,~. lnvoking Laplace's equation, i.e.

v2v=0, only two parameten are necessary to describe the EFG tensor, namely

the field gradient parameter q and the asymmetry parameter q:

The quadrupole Hamiltonian may be expressed as

The components of an alternating magnetic field with a non-vanishing

projection perpendicular to the EFG direction can induce magnetic dipole

transitions between the quadrupole energy levels. In the present work, the case

3 I = y is studied. Two degenerate energy levels are present, producing a unique

NQR frequency, w, given by

Transitions other than those corresponding to Am= I 1 are expected to

be weak, except for large values of q and are forbidden for axial symmetry, i.e.

= O, where

These results show the relation between the resonance frequency, which

can be measured, and the EFG at the nuclear site.

3.3 TEMPERATURE AND PRESSURE EFFECTS

The EFG fluctuates as a result of the lattice vibrations. But the frequency

of the lattice modes (=1013 Hz) is much higher than the quadrupole resonance

frequency (4 O7 HZ), so the latter only reflects an EFG time average value. The

lattice vibration spectra and amplitudes depend on the temperature, and so

does the EFG average. This determines the temperature evolution of the NQR

frequency.

It can be shown that the lowest order non-zero effect on the NQR

frequency is proportional to the mean square displacement of the ions in the

lattice (Wang, 1955). As a result, the frequency varies with the temperature

according to a weighted function of the lattice normal mode CO-ordinates.

Inspection of contributions from individual modes has to be complemented with

results from other techniques such as Raman or IR spectroscopy to obtain useful

information.

Bayer (1 951) proposed a simple mode1 where the lattice presents a

unique torsional mode. Two atoms are bound and the one containing the

resonant nucleus moves perpendicular to the bond direction. For a small angle of

rotation 8, about the y axis the quadrupole Hamiltonian becomes:

The first term is responsible for the stationary quadrupole energy levels, while

the others give rise to spin-lattice relaxation. Because of the high frequency of

the lattice vibrations, the rotation angle can be replaced by its mean square

value. In the case of a small tonional oscillation about the y-axis and the bond

equilibrium direction taken as the z-axis

where V, is the static lattice value of the resonance frequency and < 8;> is

determined by the lattice vibration characteristics.

< 8 :> can be evaluated by cornparhg the energy of the torsional

oscillator, ha,, to the mean energy of a Planck oscillator:

where I is the moment of inertia of the oscillator. This leads to the expression

In Bayer's simple model, only one vibration mode has been considered.

This theory was generalized to include al1 normal modes of vibration by Kushida

NQR and Relaxation Processes

(1 955). The result is

where A, is

placed in a

Chapter 3 - Page 7

a weighting factor associated with the i-mode. When the nucleus is

rigid molecule or ion and the torsional and translational modes are

not strongly coupled, Ai can be interpreted as the inverse of the moment of

inertia of the ith mode.

The volume of the sample changes with temperature. As a result extra

considerations have to be made in a constant pressure experiment. Variations in

the equilibrium position of the ions produce changes in the EFG. Furthermore,

the normal mode frequencies may be volume dependent. Assuming vQ is a

function of temperature and volume, it can be shown (Kushida et al., 1956 and

Wallace, 1972) that

where P is the pressure, v the volume, a , = ~ l [av]p is the isobaric volume

thermal expansion coefficient and PT=-V l [av]T ap is the isofhermal

compressibility of the solid.

A procedure in thermodynamics known as correction to fixed volume is

used to reduce results acquired at varying volume to data a t constant volume

(Wallace, 1972). Considering that torsional oscillations are in general

asymmetric, but equally possible in the x and y directions, and combining

equations (3.3.2) with (3.3.6) (Martin and Armstrong, 1 975) it follows that

with v, the volume of the sample at T = T,. The mean squared angular

displacement < 0 > can be detennined by the experimental values of v&T)

and then related to the lattice vibrational modes. Since the contribution from

different modes can not be uniquely identified by this method, only an overall

behaviour can be obtained. Using a mode expansion given by O'Leary and

Wheeler (1 969 and 1970) and assuming the interna1 mode frequencies of the

MX, complexes are reasonably independent of the crystalline environment

(Debeau and Poulet, 1 970), Martin and Armstrong (1 975) were able to account

for the observed decrease in the average lattice vibrational frequency in K2OsCI6

in the cubic phase as the transition temperature is approached.

Effects of sample strains in the EFG and the vibrational modes can be

studied by monitoring the temperature evolution of the NQR frequency and the

relaxation processes when additional pressure is applied to the sample. Zamar

(1 991 and 1 988) and Brunetti (1 980) have formulated an equation to estimate

the dynamic and static contributions to the EFG by means of the resonance

frequency differences corresponding to nuclei in chemically non-equivalent sites.

The main contribution to the EFG is given by intramolecular effects,

leaving only 5% of the total contribution to the intermolecular components.

Nevertheless, as seen from the Kushida et al. (1956) analysis, the

intermolecular contribution dispbys a marked temperature dependence.

Structural parameten obtained by neutron diffraction profile refinements

in K,OsCI, data (Armstrong et al., 1987), have shown no significant change in

the Os-CI bond distance over a large temperature range. This encourages one to

follow the analysis reported by Brunetti (1980). presenting a qualitative

discussion linking the resonance frequency to strain effects in the crystal.

Equation (3.3.5) can be approximated by a simpler expression in the high

hoi temperature regime, Le. -a 1. The contributions of the intramolecular kT

vibrations, the crystalline field and the low frequency rotational modes

(-20 cm-' ) to the NQR frequency can be separated. Using

the resonance frequency becomes,

where F(T) and v, are the intramolecular and the crystalline field terms,

respectively. The summation is over N rotational modes. This approximation may

not hold in the tetragonal phase of K,OsCI, because of the low temperature and

the increase of the lattice mode frequencies.

When moderate pressure is applied to the ciystal, changes in the NQR

frequency are expected to arise from the crystalline EFG and the lattice mode

frequency contributions. No significant change is expected in the intramolecular

vibration term (see equation 3.3.9). Brown (1960) proposed a linear

dependence of the lattice frequencies with temperature leading to the following

NQR and Relaxation Processes Chapter 3 - Page 1 O

behaviour of the NQR frequency difference (Brunetti, 1980):

Av, = Av, + a T + b~~

In this expression a is related to the variation of the harrnonic terms of the

intermolecular potential and b to changes in the anharmonic terms of the

tonional frequency.

Zamar and Brunetti (1991) developed a model to identifi static and

dynamic contributions to changes in the EFG when uniaxial isothermal stress is

applied to the sample. The EFG variation arises from alterations in the rigid

lattice and in the thermal average from perturbations on the force constants.

Equation 3.3.5 can be rewritten as

where e, is the kth component of the strain tensor.

For small external stress in a given pressure configuration, y, a linear

dependence of voon the strain tensor can be assumed (Zamar and Brunetti,

1 988), giving

where Q: and Qi are the dynamic and static cornponents of the frequency

derivatives, [2] , respectively. €=O

NQR and Relaxation Processes Chapter 3 - Page 7 7

Assuming the crystalline configuration is slightly distorted, temperature

evolution of the pressure derivative (in a small temperature range around the

reference state) is expected to arise mainly from the explicit dependence on the

dynamic contribution. Then

where DY is the temperature independent static term and

where S, =

By measuring the pressure and temperature influence on the NQR

frequency, this formulation allows one to make an estimate of the static and

dynarnic contributions to the EFG a t the sites of the resonant nuclei.

When approaching the phase transition, additional considerations must be

made. An extra distortion of the unit cell may appear as dynamic clusten with

the symmetry of the low temperature phase are triggered. This phenornenon

affects the distribution of strains in the crystal and increases the contribution of

the static component. Moreover, anomalies in the phonon spectra charactenstics

are expected in the vicinity of the critical temperature (Bonera et al., 1970),

resulting in a more complex dynamic tem.

3.4 SPIN-LATTICE RELAXATION

The energy exchange of the spin system with the lattice is characterized

by the spin-lattice relaxation time Tl. Different mechanisms can drive the

approach to equilibriurn and, when the sources of the relaxation are undentood,

a quantitative measure of the lattice vibration spectra is achieved. Nevertheless,

except when one vibration mode is the predominant mechanism, NQR results

provide only an average over al1 modes.

Once the relation between the relaxation rates and the mode frequencies

is established (Armstrong and Jeffrey, 1971 and Bonera et al., 1970), effects of

structural anomalies as the phase transition temperature is approached may be

predicted from the Tl-values (Dimitropoulos et al., 1992; Jain et al. , 1992 and

Armstrong et al., 1 986).

In antifluonte compounds the rotary lattice modes present an efficient

spin-lattice relaxation mechanism. These modes correspond to rigid-body

rotations of the octahedra units. The interna1 modes of these units do not

provide an important contribution to the relaxation process.

The transitions between different energy levels are generated by the non-

secular elements of the Hamiltonian, Hm,,,. (given by equation 3.2.4). Using f i n t

order perturbation theory, the transition probability from state m to m' at time t

is (Merzbacher, 1970):

where h O,,, m . is the energy difference between the two States.

Following a similar procedure as the one reported by Bloembergen et al.

(1 948) for magnetic relaxation, a correlation time rc can be defined by the

relation

where rc is the time for which G(T,) a G(0). Furthermore, Abragam (1 983)

showed that for t u r, the transition probability per unit time is

indicating that the transition probability induced by the lattice is the spectral

density of the autocorrelation of the non-secular components of the interaction

Hamiltonian. In order to produce spin-lattice relaxation, the lattice vibrational

spectrum has to contain components at the frequency determined by the

separation of the quadrupole energy levels.

The time dependence of the population of the levels is described by a set

3 of differential equations. In the case of 1 = - two probabilities are present, W, 2 '

(Am = il) and w, (Am = u), leading to

Bonera et al. (1 970) darived expressions for the transition rates near the

phase transition temperature. They considered the anornalous temperature

dependence of unstable lattice modes. This analysis, based on a Raman second-

order two-phonon process (Van Kranendonk, 1954), relates the relaxation

parameters to the rnicroscopic rnechanisms responsible for the transition. A

successful description of the spin-lattice relaxation parameters using this model

has been reported for the antifhorite family of compounds (see Armstrong et

al., 1986 for K,OsCI, and Dimitropoulos et al., 1992 for (NH,),TeCI,). For

K20sCI, the relaxation mechanism corresponds to a weakly damped phonon

process. Only this case is presented here.

In the above mentioned model, the transition probability is divided into

two terms. Fint, there is a contribution from a temperature independent phonon

spectrum, which is proportional to TL for the high temperature regime. Second,

there is a term depending on the structure of the crystal and on the different

causes of the fluctuating EFG, associated with the lattice vibrations. Then

with W" depending on the spectral density and containing information about

any anomalies present.

A particular dispersion relation produces a distinctive variation of the

transition rate with the temperature. Using a central peak frequency variation of

the form oi a (T -Tc) (Cochran, 1960), combined with a quadratic dispersion

relation, gives

wviba T~ -' Cr-T,)

for T > Tc (3.4.6)

A peak in the transition rate is present even with the assumption of a

NQR and Relaxation Processes Chapter 3 - Page IS

linear dispersion, but the particular shape of the peak is sensitive to the phonon

spectrum assumed.

3.5 SPIN-SPIN RELAXATION

The large nuclear concentrations and the small distance separating the

nuclei in bulk matter leads to relatively strong spin-spin interactions. In liquids

the coupling efficiency is considerably reduced by the rapid motion of the nuclei.

In a pure quadrupole resonance experirnent the magnetic dipole

interactions and the EFG fluctuations due to strains within the crystal are the

main causes of line broadening. The phenomena is characterized by the so-called

spin-spin relaxation time T,. This process does not involve an energy exchange

between the spin system and the lattice, but rather a rapid transfer of energy

among the spins. Thermal equilibrium is established inside the nuclear spin

system in a time shorter than T,. The calculation of the resulting lineshape is an

important problem of nuclear magnetism.

A general expressions for the second moment of the lineshape was

presented by Van Vleck (1 948)

H is the Hamiltonian of the spin system, including the dipole-dipole interaction

and 1,, is the component of the spin parallel to the applied rf field.

Calculation of the second moment in the case of pure quadrupolar

resonance, assurning identical resonant nuclei and common directions and

magnitudes of axially symmetric EFG, has been presented by Pratt (1 980) and in

the case of powder samples by Abragam and Kambe (1 953). Kano (1 958)

extended the analysis for the case in which the EFG at the sites of the resonant

nuclei display the same magnitude but different directions.

The second moment in the case of K,OsCI, is calculated considering

different contributions. The chlorine resonant spins are in sites of the same EFG

magnitude. The couplings are divided in two groups, namely interactions among

nuclei with parallel ('like' spins) and non-parallel ('semi-like' spins). Additionally,

there is coupiing between the resonant CI nuclei and the potassium ions, in sites

of non-parallel EFG. There is no dipolar coupling to the Osmium nuclei since their

nuclear spin is zero.

Structural parameters are obtained using the neutron diffraction

technique. Then the second moment for the K,OsCI, powder sample can be

calculated using the above

mentioned models. A new

expression is derived in this

thesis for the single crystal case,

where a dependence on the

orientation of the sample is

predicted.

Figure 3.1 displays the

reference frame: (x, y. Z) where Ij

is parallel to z and rj, is the

position of the spin I',, whose symmetry axis is along z', perpendicular to z. It

should be noticed that for the crystal studied the semi-like spins are always

perpendicular to each other.

The total Hamiltonian can be written as H = Ho + H' where Ho is the

unperturbed component and H' represents the magnetic interactions. Then:

Ho = HI + H2 with

Sis a scalar representing the quadrupole strength; it determines the NQR

28 frequency through vo=

3 for 1=- H', and H', are the interaction terms

between parallel spins, as discussed by Abragam and Kambe (1953). The

coupling for a general relative orientation of spins is given by (Slichter, 1990)

NQR and Relaxation Processes

Po In this expression A = (=) Y 2h ', a= (1 - 3y2),

Chapter 3 - Page 18

3 3 bi=-Yq' and ~ * - - ( q * ) ~ 2 - 4

with \r and q defined according to the reference angles shown in Figure 3.1.

Finally, y=cos8 and q*=sine efio, y is the magnetogyric ratio of the nuclei and p,,

7 Wb is the permeability of free space (h=41~ x1 O- -). Am

For the coupling between non-parallef spins, in the present case: G., = h, thereby

reducing greatly the number of terms involved in the interaction component.

The eigenvectors of H,, fm,,m;s, are represented by Im,,m;>, and the

eigenvalues are

2 E,= -S<m,,m;lI, + I'~,lm,.m;>

leading to

E,=-6(m: + mf)

1 5 9 This gives rise to three energy levels: Eo= - 2 6. -y 6 and - - 6 . 2

Resonance energy gaps, hv,, are found between the first and second and the

second and third levels.

In order to discard the matrix elements whkh contribute only to satellite

lines, the interaction Hamiltonian has to be truncated suitably, keeping the

components that commute with H, Then,

where A' is the part of H' which commutes with Ho and Wjk represents the

interaction between a patticular couple of spins j and k. The coupling between

the rf field and the spin system is also truncated appropriately, defining the spin

operator, I,,. Then. the second moment of the line is

As a result of the truncation, the diagonal elements < mj,mr~,,hj,mk >, of

the 1,, representation matrix are null. The off-diagonal components are defined

by the projection in the xy plane of 1,,, represented by the operator I,,, where

I,, = Ixjcos@ + 1, sin @ + Ixkcos@ + I,, sin Q (3.5.1 1 )

This can be rearranged, to give

where @ is the angle between the projection of the coi1 axis in the XY plane and

the x direction (Figure 3.1 ).

Now that the form of Il, and R' are known, the second moment of the

lineshape can be calculated by using equation 3.5.10. The procedure is

straightforward but lengthy; it required the aid of a cornputer code for the

rnatrix manipulation and the evaluation of the interaction terms belonging t o

different nuclei sites. Results for two K,OsCI, single crystals are shown in

chapter 8.

In chapters 7 to 9, 35CI NQR studies of the hydrostatic pressure effects

on K20sCI, samples with different concentrations of impurities and the uniaxial

pressure effects in two single crystals are reported. Particular emphasis is placed

on the discussion of the presence of dynamic tetragonal clusters and changes

on the transition temperature.

Abragam A 1983 The Principles of Nuclear Magnetisrn, Oxford University Press,

London

Abragam A and Kambe K 1953 Phys. Rev. 91 894

Armstrong R L, Morra R M. Svare I and Powell B M 1987 Can. J. Phys. 65 386

Armstrong R L and Jeffrey K R 1971 Can. J. Phys. 49 49

Bayer H 1951 2. Physik 130 227

Bloembergen N, Pourcell E M and Pound R V 1948 Phys. Rev. 73 879

Bonera G, Bona G and Rigamonti A 1970 Phys. Rev. B 2 2784

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Cochran W 1960 Advan. Phys. 9 387

Cohen M H and Reif 1957 Solid State phys. 5 321

Das T P and Hahn E L 1958 Solid State Physics 1, Seitz F and Turnbull D,

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Debeau M and Poulet H 1 969 Spectrochim. Acta 25 A 1 553

Dehmelt H G and Kruger H 1950 Natumrissenschaften 37 1 1 1

Dimitropoulos C, Borsa F and Pelzl J 1992 Z. Naturforsch. 47a 261

Jain S K, Goyal R P and Gupta B R 1992 lnfrared Phys. 33 589

Kano K 1958 J. Phys. Soc. Japan 13 975

Kushida J, Benedek G 6 and Bloembergen 1 956 Phys. Rev. 1 04 1 3 64

Kushida J 1 95 5 J. Science Hiroshima University A1 9 327

Martin C A and Armstrong R L 1 975 J. Mag. Reson. 20 41 1

Menbacher E 1970 Quantum Mechanics, Wiley, New York

O'Leary G P and Wheeler R G 1970 Phys. Rev. BI 4409

O'leary G P 1969 Phys. Rev. Lett. 23 782

Pratt J C 1 980 J. Mag. Reson. 38 31 9

Slichter C P 1 990 Principles of Magnetic Resonance, 3rd edition, Springer-Verlag,

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Van Kranendonk J 1954 Physica 20 781

Van Vleck J H 1948 Phys. Rev. 74 11 68

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Zamar R C and Brunetti A H 1988 Phys. Stat. Sol. b 150 245

Zamar R C and Brunetti A H 1991 J. Phys.: Cond. Matter 3 2401

Neutron Diffraction Chapter 4 - Page 1

CHAPTER 4

NEUTRON DIFFRAnION

4.1 INTRODUCTION

The scattering of thermal neutrons is a powerful technique for the study

of structural and dynamic properties of matter. Neutrons penetrate solids and

are not sensitive to the exact conditions of the surfaces. They are scattered by

the nuclei within the material, becoming a probe of bulk matter.

Thermal neutron wavelengths which lie in the range 0.5 to 5 A, are

comparable to the separation of nuclei in crystals. Consequently, when a neutron

is scattered by a material, a measurable change in the neutron momentum will

occur. In addition, the energy of thermal neutrons lie in the range -4 to 300

meV which is matched to typical energies of the low lying excitations in solids.

This simultaneous matching of both the wavelength and the energy of the

neutrons to the characteristic interatomic distances and excitation energies of

the target system provides a powerful combination to help understand

properties of materials.

In the present thesis, K,OsCI, neutron powder diffraction rneasurements

are presented. The effects of adding impurities on the lattice dimensions, the

dynamical parameten and the development of precursor clusters are reported.

Previous results provide complementary information; the soft-mode dynamics

Neutron Ditfraction Chapter 4 - Page 2

has been studied by inelastic-neutron-scattering (Sutton et al., 1983 and Mintz

et al., 1 979) and the longitudinal rotary-lattice mode has been identify as the

soft-mode.

This chapter concentrates on the basic concepts of powder neutron

diffraction and the measurernent of dynamic parameten that can be obtained

by refinement of the profiles.

4.2 NUCLEAR BRAGG SCATTERING

The wavelength li of thermal neutrons is given by the de Broglie relation

where k and E are the neutron wavevector and energy, respectively. A t 300K I

u 1.7 A. Neutrons display a pronounced interference effect when scattered from

condensed matter systems.

In the presence of a series of nuclei rigidly bound at the sites R, and using

Born's approximation (see for example Landau, 1965)' a scattering cross

section can be written based on the number of neutrons scattered per unit time

da into the element of solid angle d a = sine d e d$ (see Figure 4.1 ): N (- ) dR,

where N is the flux of incident neutrons defined as the number psr unit area per

unit time. Cross-sections have the dimensions of area, and are given in barns (1

barn = 1 m2). The differential cross-section can be written as (Bacon, 1975

and Marshall and Lovesey, 1 97 1 )

Neutron Diffraction

where the coherent cross section is

and the incoherent cross section is

Chapter 4 - Page 3

(4.2.2)

with b the scattering length given by the strength of the Fermi pseudo-potential,

characteristic of the target nucleus, and r = k - k' is the difference between the

incident and scattered neutron wavevectors.

Detector

It is clear from eq. 4.2.4

that the nature of the coherent

and incoherent contribution to

the cross-section is different.

The coherent term rises from

interference between the waves

scattered from different nuclei. In

the incoherent scattering there is

no interference and the cross-

section is isotropie.

Fig. 4.7: Geometry of the scattering system.

The position of an atorn in a crystal can be wntten as

where I is the position of the unit cell and d the position of the atom within the

A reciprocal lattice vector s can be defined such that

= 1 for al1 i (4.2.6)

For a large crystal eq. 4.2.3 becomes (Marshall and Lovesey, 1971 )

where 6 is the Dirac S function, (2- is the reciprocal lattice volume and F(r) is "O

the nuclear unit-cell structure factor,

ird - F , O = C e b d

Sirnilarly eq. 4.2.4 becomes

da 2 2

(an i n = d { d - l b d l 1

As can be seen from eq. 4.2.7, no coherent elastic scattering is produced

from a perfect crystal unless K coincides with a reciprocal lattice vector (see

Figure 4.2).

If 0 is the angle between the incident and scattered neutron beam, Bragg

scattering occurs when

Fig. 4.2: Reciprocal la ttice and Bragg

scat t e h g condition.

This refers to a determined orientation

of the crystal. The total cross-section is

obtained by integrating 4.2.7 over dQ,

i.e. over al1 the orientations of k', gMng

for elastic scattering

In the present thesis, powder neutron diffraction measurements are

presented. For this technique an incident monochromatic beam is used. An

average over al1 orientations of r is made (Marshall and Lovesey, 1971 )

The azimuthal angles dependence is eliminated after the averaging. Then,

the scattering direction is presented in cones, called Debye-Scherrer cones,

which f o m an angle 0 with respect to k given by the condition

which is the same condition as eq. 4.2.1 0.

The total cross-section can then be evaluated

IF,l2 iç the mean value of F(T)I~ of contributions frorn al1 reciprocal vectors with

magnitude T.

lntensities in an experiment with powder samples are lower than with a

single crystal but polycrystalline samples are in general easier to obtain.

4.3 DEBYE-WALLER FACTORS

The cross-section for the elastic coherent scattering of nuclei undergoing

harrnonic vibrations can be calculated by considering the instantaneous position

of the nuclei:

R,, = I +d + u,, (4.3.1 )

where u,, represents the position of the dth nucleus within the Ph cell with

respect to the average position.

In the general case of a lattice consisting of interpenetrating Bravais

lattices, Marshall and Lovesey (1 971 ) formulated

where the nuclear unit-cell factor is given by

itd - F,(T)= E e bde-W(K)

1 W(K) is the Debye-Waller factor: W ( r ) = p { K . Q } ~ > , and <...> denotes thermal

average. Then, the structure factor is not a simple scaling component.

In the case of a cubic lattice and considering that u is a random thermal

displacement uncorrelated with the direction of u, the thermal average is

In a neutron powder diffraction experiment a t a given temperature the

Debye-Waller factor can be calculated through measuring the dependence of the

Bragg peak intensity on the scattering angle.

In the Debye model of lattice vibration the cubic lattice is assumed to

behave as an isotropic continuous solid in which al1 waves propagate with the

same speed, whatever their length and direction, although the speeds of

longitudinal and transverse modes differ (James, 1 958).

A report on K20sCI, (Armstrong et al., 1987) provides temperature-

dependent mean square displacement of the atoms. The refined dynamic

parameters behave as predicted from the elastic properties of a harmonic cubic

Neutron Difiaction Chapter 4 - Page 8

crystal. These results encourage one to proceed with a Debye model analysis of

the samples studied in the present thesis.

An approximate expression for the Debye-Waller factor is given by

(James, 1 958)

where rn is the atornic mass, x = - and e, the Debye temperature. The T

function N x ) is defined as

and is tabulated (International x-Ray Tables).

The Debye temperature is calculated from the relation (Launay, 1956)

with V, the atomic volume, p the crystal density and f(s,t) a function of the elastic

constants, s = '1 '12 0'44 and t= tabulated by de Launay (1 956). The '12+'44 C 44

elastic constants can be calculated from acoustic mode dispersion curves

deduced from inelastic neutron scattering measurements.

Substitution of the constants in eq. 4.3.7 yields the Debye temperature

and then the mean square displacement can be calculated using eq 4.3.5.

In a neutron diffraction experiment, a correction to the Debye-Waller

factor due to absorption should be made. In the case of a spherical or cylindrical

sample the correction is reduced to a simple adjustment of the parameters a t

the end of the refinement process (Hewat, 1979 and James, 1958),

with the correction factor

AB can be expressed in a simple formula as presented by Hewat (1 979):

where the parameten b, and b, are determined by the sample symmetry (Rouse

et al., 1970 and Hewat, 1979), r is the container radius and q is the linear

absorption coefficient, determined by a neutron transmission experiment.

It should be noticed that the addition of impurities in the samples may

produce variation in the elastic constants and hence in the Debye temperature.

-P Chapter 6 presents a powder neutron diffraction study op'K,OsCI,. The

structural parameters resulting from the profile refinement analyses are

reported. A cornparison of the predicted Debye-Waller factors values and fitted

parameters for the case of a pure powder and two other samples containing

impurities is shown.

Neutron Diffraction Chapter 4 - Page 1 1

Armstrong R L, Morra R, Svare I and Powell B 1987 Can J. Phys. 65 386

Bacon G E 1975 Neutron Diffraction, 3d edn., Oxford. Clarendon Press

Hewat A W 1979 Acta Cryst. A35 248

International Tables for x-Ray Crystallography 1952 The Kynoch Press,

Birmingham, England

James R W 1958 The optical principles of the diffraction of x-rays. Vol II. Bragg

W L. Bell G and Sons Ltd. London, England 21 9

Landau L D and Lifshitz E M 1965 Quantum Mechanics, Znd edn., Pergamon Press

de Launay J 1956 J. Chem. Phys. 24 1071

Marshall W and Lovesey S W 1971 Theory of Thermal Neutron Scattering,

Oxford, Clarendon Press

Mintz D, Armstrong R L, Powell B M and Buyen W J 1979 Phys. Rev. B 19 448

Rouse K D, Cooper, York E J and Chakera A 1970 Acta Cryst. A26 682

Sutton M, Armstrong R L, Powell B M and Buyers W J 1983 Phys. Rev. B 27 380

Experimen ta1 De tails Chapter 5 - Page 1

CHAPTER 5

EXPERIMENTAL DETAILS

5.1 SAMPLE PREPARATION

A polycrystalline sample of K20sCI6 and five other powder samples

containing different concentrations o f substitutional impurities were

recrystallized for the neutron powder diffraction and hydrostatic pressure 35CI

NQR experirnents. Supplies of K,OsCI,, K,ReCI,, K21rCk and K2PtC16 were

obtained from Johnson Matthey Chemicals Limited, Wayne, PN. The stated

purity was 99.9 %. Two K20sC16 single crystals (grown by S. Mroczkowski at

Yale University) were cut and polished to perform the uniaxial pressure NQR

measurements.

All powder samples were prepared starting with the sanie batch of

K,OsCI,. lsostructural substitutional impurities were selected and added in a

controlled rnanner. The pure K20sC16 compound and measured amounts of

K,ReCI,, K21rC16 or K2PtC16 (Krupski and Armstrong, 1994) were dissolved in an

aqueous solution of 8 N hydrochlonc acid and then recrystallized by evaporation

in vacuum at room temperature. About 3 g of each sample were prepared. The

concentration of HCI was selected so to obtain a high S/N in the NQR spectra

(Ishikawa et al., 1 986). The size of the crystallites was measured t o be a few

micrometers using electmn microscope images.

Experimental Details Chapter 5 - Page 2

A microprobe x-ray study confirmed that the impurities were located in

substitutional positions.

The symmetry presented by the single crystals allowed us to identify the

lattice planes by inspection of the angles between the edges (see for example a

grain of the powder sample in Figure 2.2). When the desired faces were

recognized, the crystals were cut so to have (100) and (1 11) planes,

respectively, top and bottom. These faces were polished with fine sandpaper

and then with a slightly wet cloth. The sides of the crystals were cut to be

normal to the (1 00) and (1 1 1 ) planes in order to avoid extra strains. Crystal

volumes of around 40 mm3 were achieved.

5.2 NEUTRON POWDER DIFFRACTOMETER

Neutron diffraction experiments require intense beams as provided by

reactors and spallation sources. The powder diffraction measurements presented

in this thesis were made a t the NRU reactor at Chalk River on the Dualspec

powder diffractometer, which has an 800-element arc-shaped detector covering

an 80° range of scattering angles (20). Rotation of the detector bank produced

a step size of 0.05~ for the profiles. A diagram of the diffractometer is

presented in Figure 5.1.

In the reactor neutron scattering experirnent the neutrons emerge from

the core of the reactor with Maxwellian energy distribution. Low-sensitivity

fission counters are used to monitor the white radiation at the gate of the

reactor and the monochromatic beam (Figure 5.1). The neutron beams are

horizontally collirnated by Soller slits, consisting of several steel sheets plated

Experîmen ta1 De tails Chapter 5 - Page 3

with cadmium. The white beam is Bragg-reflected with an angle 29, by a silicon

single crystal monochromator.

800-wire Detector -7

Monitor /

Seller Slits

Monitor

1 Collimator

Fig. S. 1 : CZ Dualspec powder diffractometer.

Powder specimens were positioned in a thin-walled vanadium can of about

5 mm diameter and 5 cm long and then mounted in a top loading helium

cryostat. The sample was rotated continuously about a vertical axis to minimize

Experimental Details Chapter 5 - Page 4

the effects of preferred orientations. A programmable temperature comptroller

sets the temperature with an uncertainty smaller than 0.1 K. A calibrated

thermocoupfe placed on top of the sample is used to determine the

temperature.

After the diffraction profiles are acquired, the data were fitted using

Rietveld (1 969) refinement techniques (Powley, 1972), in which the count a t

each position is weighted by the factor l/oi, with q the standard deviation of

the count. General Structure Analysis System package (GSAS, Larron and Von

Dreele, 1990) was used for the fitting process. This computer code was initially

run remotely and then installed on a VAX system at MTC, Mechanical

Engineering, UN6 Fredericton.

The parameten to be refined fall into two groups (Bacon, 1975), those

describing the characteristics of the diffractometer and those which depend on

the crystal structure. The first group consists of the neutron wavelength, the

zero position of the counter, and three peak shape parameten U, v and W. which

describe the variation with 20 of the angular width h of the Gaussian peaks

according to the equation (Thomas, 1977; Cooper and Sayer 1975 and Rietveld,

There is also a correction for asymmetry of the reflection curves, which affects

the angular width at low angles. The structural parameters are scaling factors:

the dimensions and angles of the unit cell, and the coordinates and thermal

parameten (see Chapter 4) of the individual atorns. Rigid body constraints are

also introduced and are implemented as an option in the GSAS code (Swainson,

Experjmental De tails Chapter 5 - Page 5

1994). The preferred orientation of the crystallites in the powder (Prince, 1983

and Howard, 1982) is an additional correction that can be taken into account.

As a refinement strategy in the process to obtain least-squares fits to

the profiles with the GSAS code, the background, histogram scale factor,

diffractometer zero position and lattice parameters were initially varied.

Refinement of the isotropie temperature factors, interatomic distances in the

octahedra complexes, preferred orientation parameters and peak shape

parameters U, v and w followed. Output error for the fitted parameters are

known to be underestimated by a factor of three (Torrie et al., 1994 and Powell

et al., 1982)

More details about the experimental setup and profile analysis for the

neutron powder diffraction experiment on K,OsCI, are presented in Chapter 6.

5.3 NQR SPECTROMETER

The NQR experirnents were carried out using a Fourier transform pulse

spectrometer system. Spectra were acquired using a Tecmag unit and a

Macintosh cornputer with commercial software, MacNMR. The analysis was

performed with the MacNMR software and with a series of C (ThinkC for

Macintosh) and Quick Basic language programs written for Fourier transform and

Maximum Entropy calculation, as presented in this thesis appendix.

Two different setups were used for the measurements. A helium gas

hydrostatic pressure chamber (up to 270 MPa) for powder samples was

developed by Prof. Marcin Krupski. The original instrument (Krupski and

Armstrong, 1989; Krupski et al., 1987 and Krupski, 1983) was improved and a

new temperature comptroller was added by Krupski (1 994). A new chamber and

uniaxial pressure device (for pressures up to 10 MPa) were designed and

constructed as part of this thesis project. A description of the uniaxial pressure

apparatus and accompanying helium flow temperature comptroller system are

presented in Chapter 8.

Figure 5.2 displays a block diagram of the NQR spectrometer as used for

the hydrostatic and uniaxial pressure experiments. All components were placed

in a closed metal rack to improve the RF shielding and ground conditions of the

spectrometer.

PULSE GENERATION AND SIGNAL AVERAGING (TecMag )

! 'O synthesizer ' 0 ) Fluke 61 608 quadrature

- 1 detector 1 synthesizer

- 1 Fluke 6039A Vi

Fig. 5.2: Block diagram of the NQR pulse spectrometer.

Experimental De tails Chapter 5 - Page 7

The pulse generation and signal averaging section controls the vanous

parts of the spectrometer and averages the free induction and echo signals in

order to improve the signal to noise ratio.

In the transmitter section, the spectrometer produces the high voltage rf

pulses that excite the nuclear spin system. The rf pulse is generated by a Fluke

61 608 frequency synthesizer, a rf gate circuit and the Amplifier Research 2001

power amplifier. A Fluke 61 39A synthesizer generates a reference signal for the

quadrature detection as described below. The pulse sequence is directed

through the MacNMR software and driven by the Tecmag pulse kit. Pulses of

about 10 ps and a continuous rf signal are input to the phase shift and gating

unit and then amplified up to 500 W by the power amplifier. Decoupling

capaciton to ground were added to the Tecmag unit, and phase shift and gating

component outputs to filter a low frequency noise present in the previous setup

of the spectrometer.

For the sample probe, a

series-parallel resonant circuit

was used for the uniaxial

pressure experiments (Figure 5.3

a) and a parallel resonant circuit

for the hydrostatic pressure

measurements (Figure 5.3 b).

These two different schemes

were selected as a result of

considerations of the impedance

characteristics of the coaxial line

transmitting the signal to the

pressure chambers. An HP

Ekpethen ta1 De tails Chapter 5 - Page 8

481 SA vector impedance meter and an HP 5244L Electronic Counter were used

to tune the circuits with an impedance of 50 a. To prevent high frequency noise

signals from entering the receiver amplifier circuit, a low-pass LC filter is placed

before the pre-amplifier (Figure 5.2 and 5.3 c).

The receiver components amplify the signal generated in the probe circuit

which appears as free induction decay or echo response of the order of

microvolts. Two amplification stages are used; a pre-amplification and then a

final gain, achieved by an Anzac AM 108 amplifier. The input signal for the

Anzac amplifier is gated by a Relcom switch as driven by the Pulse kit unit.

During the time the r f pulse is on the transmitter gate is open t o allow

the high voltage pulse t o pass. The receiver gate is closed during this period.

After the large transient voltage is damped into the probe, the transmitter gate

is closed and the receiver gate is opened to acquire the nuclear spin induced

signal. The closing of the transmitter gate reduces the leakage reaching the

receiver from the transmitter.

After the receiver stage, the signal is processed by a quadrature detector

(Abragam et al., 1992 and Fukushima and Roeder, 1981). The introduction of

quadrature detection (recording two signal channels whose reference signals

differ by 90°) introduces sorne advantages. An early spectrometer using this

approach is described by Redfield and Gupta (1 971 ). A benefit of this detection

setup is that linear combinations of the two signals can be made which

effectively change the phase of the reference signal. Thus, the proper phase can

be determined after the data is recorded. A second advantage is that it permits

one to carry out a complex Fourier transfom.

A block diagram describing the quadrature detection process is displayed

in Figure 5.4. At this stage the amplified probe signal is mixed with a high

frequency sine signal, with resulting frequencies: v,+lOMHzIv,. The additive

component is filtered at the input of the video amplifier so it can be ignored.

The low frequency signal, v,+lOMHz-v,, is split and mixed separately with two

1 OMHz sine signals, with phases O and n/2, and fed into the video amplifier. A

digital signal averaging unit integrates the two-channel output, which is then

displayed on the Macintosh computer. The digital averaging of transient signals

enables one t o achieve effectively long averaging times, a big advantage over

early boxcar integrators where the storage time was determined by the time

constant of the capacitor in which the voltage was stored.

For the receiver as a whole, the noise figure is that of the pre-amplifier,

and the bandwidth is primarily determined by the output filter of the phase

sensitive detector.

Chapters 7 to 9 describe details about the spectrometer setups for three

different NQR experiments. The same spectrometer was used for the powder

(Chapter 7) and single crystal (Chapten 8 and 9) experiments, with different

probe tuning circuits, as illustrated in Figure 5.3 a and b.

Experimental De tails Chapter 5 - Page 1 O

Fluke 61 608

Fluke 6039A

probe amp Y V r vo + F D

shifter

divider

vo+ 1 OMHz-V, QUADRATURE

O R / 2

mixer mixer 4 I

video amplifier

Fig. 5.4: Quadrature de tector block diagram.

1 OMHz ) mixer , *

vO + 1OMHz 4

filter

Experimental Details Chapter 5 - Page I 1

Abragam et al. 1992 Pulsed Magnetic Resonance, Bagguley D M S, Clarendon

Press, Oxford.

Bacon G E 1975 Neutron Diffraction, 3rd edn., Oxford, Clarendon Press

Cooper M J and Sayer 1975 J. Appl. Cryst. 8 61 5

Fukushima E and Roeder S B 1981 Experimental Pulse NMR, Addison-Wesley Pub.

Comp..

Howard C J 1 982 J. Appl. Ctyst. 1 5 61 5

lshikawa A, Sasane A, Mori Y and Nakamura D 1986 2. Naturoforsch. A 41 326

Krupski M and Armstrong R L 1994 Molec. Phys. Rep. 6 169

Krupski M and Armstrong R L 1989 Can. J. Phys. 67 566

Krupski M, Armstrong R L Mackowiak M and Zdanowska-Fraczek M 1987 Can. J.

Phys. 65 134

Krupski M 1 983 Phys. Status Solidi A 78 75 1

Larson A C and R B Von Dreele 1990 Preprint # LANSCE, MSwH805. Los Alamos

National Laboratories.

Powell B M Dolling G and Tome B H 1982 Acta Crystallogr. B 281 28

Powley H M 1972 Adv. Struct. Res. Diffr. Methods 4 32

Prince E 1983 J. Appl. Cryst. 16 508

Redfield A G and Gupta R K 1971 Adv. Mag. Res. 5 81

Rietveld H M 1969 J. Appl. Cryst. 2 65

Swainson I P 1994 Znd Workshop on Neutron Powder Diffraction, Chalk River

La bora tories

Thomas 1977 J. Appl. Cryst. 10 1 2

Torrie B H O'Donovan C and Powell B M 1994 Molec. Phys. 82 643

Influence of Substitutional lmpurities ... Chapter 6 - Page 1

CHAPTER 6

INFLUENCE OF SUBSTITUTIONAL IMPURITIES ON THE STATIC AND

DYNAMICAL BEHAVIOUR OF

K,OsCI, IN THE VlClNlTY OF THE STRUCTURAL PHASE TRANSITION:

A NEUTRON DIFFRACTION STüDY

Pablo Prado and Robin L. Armstrong

Department of Physics

University of New Brunswick

New Brunswick, Canada

Brian M. Powell

AECL, Chalk River Nuclear Laboratories, Chalk River, ON,

CANADA KOJ 1 JO

Reference: Prado P J, Armstrong R L and Powell B 1995 Can. J. Phys. 73 626

This article is included in the present thesis under permission of the Canadian

Journal of Physics publisher.

hfluence of Substitutional Impurifies. .. Chapter 6 - Page 2

ABSTRACT

The effect of IrC162- and R~CI,~- impurity ions substituted for 0sCls2- ions in

K,0sC16 is studied using neutron powder diffraction. The introduction of the

impurity ions causes measurable changes in the lattice parameten but not in the

Os-CI bond length. The dynamical parameters of the model structure used to fit

the data are well described by a Debye model of the lattice vibrations. The

structural phase transition temperature is affected by the addition of the

impurity ions. Evidence is provided for suppression, in the presence of the

impurities, of the development of precursor dynamical clusters above the phase

transition temperature.

Influence of Substitutional Impurities ... Chapter 6 - Page 3

6.1 INTRODUCTION

The static and dynamic properties of antifluorite crystals have been the

subject of considerable study over the past thirty years [l-71. Of particular

interest has been the investigation of structural phase transitions in these

materials and the softening of the rotary lattice modes associated with the

correlated motions of the octahedral ions as a precursor to the onset of a phase

transition from a high temperature cubic phase to a low temperature tetragonal

phase. A rigid-ion model has been developed using input data from infrared and

Raman spectroscopy and elastic neutron scattering [5,6].

The materials are never pure and the residual impurities are believed to

play a role in the detailed dynamics of the ciystals. It has been pointed out [ Z ]

that a complete understanding of structural phase transitions awaits

experiments on samples in which impurities are deliberately added and their

concentrations controlled. As a fint step, it was decided to prepare a series of

K20sCI, samples from the same batch and with small concentrations of ~ e C b ~ -

and ions replacing the OS CI,^- ions. The choice of K,OsCI, was dictated by

the significant previous literature available and the sensitivity of this compound

to precursor behaviour [7-91. [kCi6I2- and [ReCI6I2- were selected as

substitutional impurities because K,lrC16 and K2ReC16 are isostructural at room

temperature with K20sC16 and they too are well studied [ I l . It is the dynamics

of the MX6 group that dominate the behaviour of the phase transitions in R,MX6

compounds [5,6]. Previous studies of K,[ReC16],,[ReBr,]x have been reported

Influence of Substitutional lmpunties. .. Chapter 6 - Page 4

[IO]. The present study is the first to be reported in which the M atom is

introduced as the impurity.

The first study reported using these samples [8] used 3 s ~ I nuclear

quadru pole resonance frequency determinations to document the influence of

hydrostatic pressure and concentration of substitutional impurities on the phase

transition temperature Tc. The results were interpreted using the rigid sphere

model [Il].

The present neutron diffraction investigation on this set of samples was

undertaken to observe whether or not the development of precunor clusters

above Tc is affected by the addition of impurities. The approach taken parallels

that reported in an earlier study on "pure" K20sCI, [7]. In the course of the

experiment the following additional information was obtained: (a) structural

parameten, (b) dynamical parameters, and (c) confirmation that the impurity

atoms have been incorporated in substitutional positions.

Section 2 contains the experimental details. The fit of the profiles to a

cubic structural model is described in Section 3. At the lowest temperatures

studied the samples have undergone a transition to a tetragonal structure as

discussed in Section 4. The final section contains the conclusions.

A series of powdered crystalline samples of K2[OsCI,],,[MC16], with

concentrations x = 1.4% of substitutional impurities with M = Ir and Re were

prepared. Supplies of KZOsC16, K,lrC16 and K,ReC16 were obtained from Johnson

Matthey Chemicals Limited with stated purity of 99.9%. The new samples were

Influence of Substitutional Impurities. .. Chapter 6 - Page 5

prepared by the recrystallization of measured amounts of the pure compounds

dissolved in aqueous solutions of 8 M hydrochloric acid [8]. The concentration of

hydrochloric acid was selected to provide high signal-to-noise ratio NQR spectra

[12]. Recrystallization was performed by evaporation in vacuum at room

temperature. The size of the crystallites is of the order of a few Pm. About 3

gm of each sample were prepared.

As a confirmation that the impurities were located in substitutional

positions we carried out an x-ray study. The results from the microprobe showed

an impurity concentration distribution which seemed to correspond to the size

of the crystallite selected for observation. The larger the crystallite, the lower

the concentration of impurities. Since the neutron experiment is not sensitive to

scattering from individual crystallites it yields an average over the distribution.

For the neutron diffraction experirnents, four individual samples of about

2.7 g were placed in thin walled vanadium cans 5mm in diameter. The cans

contained K20sC16 , K2 [OsC1610.9e [lrC1610. 2 , K2 [ O S C ~ ~ ~ ~ . ~ ~ [ReC1610, 2 , and

K2[OsCI,]o~,,[ReCI,],,,. These were mounted in a helium flow ctyostat in which

the temperature can be controlled to better than 0.1 K. The samples were

cooled to the lowest temperature, 40K, in about six houn. Higher temperatures

were obtained in sequence, with about two houn taken to raise the temperature

each step, followed by a two hour waiting period before data were collected.

Voltages were read from a calibrated thermocouple that sits on top of the

sample. During the experiments the cryostat was oscillated to minimize errors

due to polycrystalline quality.

The experiments were performed on the C2 (DUALSPEC) powder

diffractometer at the NRU reactor at the Chalk River Laboratories of AECL. A

silicon (53 1 ) monochromator was used giving a calibrated wavelength of

Influence of Substitutibnal Impurities. .. Chapter 6 - Page 6

1.50452(6) A. The 800-wire detector covered 80° in scattering angle. Soller

slits provided collirnation of 0.4 deg between the source and monochromator.

The profiles were obtained using four positions of the detector, two at low

scattering angle (5-8S0) and two at high angle (40-120°). The resultant

separation between data points is O.OSO. The acquisition tirne per profile was

about 3 hrs.

Neutron diffraction profiles for the K,OsCI,, K,[0~Cl,J,,~,[lrCl,b ,,, and

K2[O~C16]o.sa[ReC16]o~02 samples were taken at 40.0, 44.5, 50.0, 84.0 and

1 2 1 .O K, while a single profile for the K, [OsCI,],,,[ReCI,], sample was

recorded a t 50.0 K.

The data for each sample were merged to yield intensity profiles of

scattered neutrons as a function of scattering angle from 5 to 120 deg. The

diffraction profiles were refined using the Rietveld General Structure Analysis

System (GSAS) code [13]. The background function was fitted to the flat

regions between the peaks. Three width parameters and an asymmetry were

used to describe the angle dependent width of the Debye-Scherrer peaks [14-

171. These parameters were obtained for each sample by fitting to the 121 K

data and the values obtained were held fixed for the other temperatures for the

three cases where several temperatures were measured. The fitted structural

parameters were derived by minimiring the quantity

where 1, and 1, are the observed and calculated counts, respectively, for each

channel, w are the weights assigned by the GSAS program, and the summation is

over al1 channels. The X Z parameter defined as

Influence of Substitutional Imp~rities~.. Chapter 6 - Page 7

with N the number of data points and n the number of adjustable parameters, is

used as a measure of the quality of the fit.

Lattice parameters, fractional coordinates and thermal factors were

obtained from al1 profiles. For the samples containing impurities, the relative

concentration of the two components was initially fixed for the refinement.

When, as a final step, this quantity was allowed to Vary no significant variation

occurred.

A correction to the translational Debye-Waller factors corresponding to

motions of the potassium and chlorine atorns due to absorption was made. The

transmission was calculated using the work of Armstrong et al. [7] yielding q r =

0.27(3) where r is the container radius. The absorption correction <~2>,, to the

mean square displacernent from equilibrium <u2> is given by 11 8,191

Substitution of

AB = 7i2p,q r + b2(q r12]

with the neutron wavelength and b, = -0.0368, b, = -0.3750 yields

= -5.3 x 10-4 AZ.

Influence of Substitutional Impurities ... Chapter 6 - Page 8

6.3 CUBlC FIT TO THE POWDER PROFILES

All the profiles were fint fitted assuming a cubic (Fmfm) unit cell with a

single lattice parameter a. As an example, the K,[OsCI,],,,[ReCI,],,, profile a t

121 K is shown in Figure 1. The data points are indicated by crosses and the

theoretical profile by the solid line. The difference spectrurn is defined by

The l$,mfi,e = F$, value is defined as

where 1 is the sum over al1 channels. For this fit ï$ is 0.057. This compares with

an expected value %,,, defined as

of 0.048, based solely on the statistical errors.

Figure 2 shows the temperature variation of the lattice parameter a for

the K,OsCI,, K,[OSC~,],~, [IrCI,],,,, and K2 [ O S C ~ & ~ ~ [ReCI,],,, samples. The

values for the K20sCI, sample are in reasonable agreement with previous

determinations [7]. A t each temperature the lattice parameter for the

Influence of Substitutional lmpuntes... Chapter 6 - Page 9

K2[OsC16]o~,,[lrC16]o~02 sample is smaller than for the K,0sC16 sample, and for

the K2[OsC16],,,[ReCI,]o~02 sample is greater than for the K20sC16 sample. That

is, the presence of ReCIG2- ions as a substitutional irnpurity causes the lattice to

expand slightly and the presence of IrCl,2- ions causes the lattice to contract.

This behaviour is as expected from a consideration of the lattice parameters of

K21rCI, and K,ReCI, [20,21]. In fact, if we add 2% of the difference between a

for pure K,OsCI, and pure K2ReCI, [21] at 121 K we obtain a = 9.71 8 A which

agrees well with the measured value for the K,[OsCI,],,,[ReCI,],, sample.

Fig. 1. The observed and calcula ted po wder-diffraction profiles and their

difference for K,[OsCI JO ,[ReCI JO ,, at 12 1 K. The structural mode1 is for

space group Fm3m

Influence of Substitutional lmpurities... Chapter 6 - Page 10

In each case the lattice parameter varies linearly with the temperature within

experimental error; the slopes provide linear thermal expansion coefficients. The

value for the pure compound, 3 .4~1 K-', agrees with previous determinations

17,221. The values for the K, [OsCl,],,,[ IrCi , ] ,.,, sample and the

K,[OSCI,~,,[R~C~~]~.~~ sample are slightly less and slightly greater,

respectively, than for the K,OsCI, sample.

Fig. 2. Temperature

dependencies of

t h e lat tice

constants a and the

bond lengths r as

extracted from

diffraction profile

refinemen t analyss

using a mode1 with

space group Frn3m.

Figure 2 also shows the temperature variation of the potassium-halogen

interatomic distance r for the three samples. Although there appean to be some

variation, the average values of r for the three sarnples over the temperature

range studied are identical to within experimental error, namely r = 2.324(2) A. This value agrees with, and is more accurate than, the previous neutron

diffraction determination [7 ] .

Influence of Substitutional /mpurities. .. Chapter 6 - Page 7 7

The diffraction profile for the K,[OSC~~]~,,[R~CI,]~~~~ sample taken at 50

K yielded results for a and r consistent with those above. That is to Say, the a

value is slightly larger than, and the r value is the same as that for the

K, OS CI^]^^,, [ReC16],,, sample within experimental error.

Translational Debye-Waller facton for the potassium and chlorine atoms

were also obtained from the fits and corrected for absorption as indicated

above. These are shown in Figure 3(a), 3(b) and 3(c) for the K20sCI,,

ICI ] K2[0~C161098 6 0.02, and K2 [ 0 s C 161a98[ReC1610.02 samples. respectively.

Debye-Waller facton for the M atoms were zero to within the erron of the fits

and were therefore set to

zero for the final fits. For

Potassiun -

0.020, . , . , . , . 1 ' 1 '

container used in the

present experiments, the

absorption correction was

always less than 10%. In

cont rast, corrections

comparable t o the

magnitude of the Debye

Waller factors themselves

Chlorine

0.000 . , . , , , , , 1 were required in the

20 40 60 80 100 120 140 experiment of Armstrong T(K)

et al. [7].

the small diameter sample

Fig. 3 a

Influence of Substitutional lmpunties ... Chapter 6 - Page 12

0.020 I . m I . 1 "

Chlorine Potassium . i

0.01 5 - Tc /'

0.01 O 0

0.005 - K2C 6198[ 6102

0.000 20 40 60 80 100 120 140

Fig. 3 c

The solid and dashed lines on the plots in Figure 3 are theoretical curves

with no adjustable parameten calculated assuming a Debye spectrum of lattice

Influence o f Substitutional Impurities.. . Chapter 6 - Page 1 3

vibrations and a Debye temperature of 190 K as calculated by Armstrong et al.

[7] . Accordingly

where x = efl, m is the atomic mass and the function MX) is tabulated [23].

The solid lines give the potassium Debye Waller factors; the dashed lines

the chlorine Debye-Waller factors. There is good agreement for both Debye-

Waller factors for the K,OsCI, sample except for the chlorine Debye-Waller

factor at the two lowest temperatures. Note that this was not the case for the

previous experiment [7] for the potassium atoms, probably because of errors

resulting from the very large absorption factor corrections that were required.

ici ] For the K,[OsC~610,g,[ , 0.0 , and K, [OsC~,Io,,, [ReCI,l,.,, samples, the

potassium Debye-Waller factors agree with the theoretical curves while the

chiorine data lie somewhat above them. It might be noted, however, that a

Debye temperature of 180 K gives good agreement in each case, except for the

chlorine Debye Waller factor for the K,[OsC16],g,[lrCI,]o~02 sample for the two

lowest temperatures. Since it is to be expected that the elastic constants and

therefore the Debye temperatures will be somewhat different in the samples

containing the impurities than in the pure sample, this may be the cause of the

discrepancy observed.

Figure 4 is a plot of ~2 versus temperature for the cubic structure fits to

the powder profiles for the K, OsCI,, K,[OsC16]o,g,[lrC16]o~,,, and

K, OSC CI,]^,,, [ReC16]o.02 samples. Except at the lowest temperatures for the

Influence of Substitutional lrnpunties ... Chapter 6 - Page 14

K,OsCI, and K,[OsC16]o~ss[ReCI,]o~02 samples the value of $ for al1 profiles is

2.7 f 0.5. At the lowest temperatures the value of ~2 increases.

Fig. 4. Temperature

dependencies of x* for the

K,&C16,

K2[OsC1 J 0 . 9 d l ~ I J a o ~

K ~ ~ O ~ C I , I , . ,,[R~CI,I,. ,, samples as obtained from

profile-refinement analysis

using a mode1 with space

group Fm3m. The dashed

lines are "guides to the

eye". The vertical arrows

indicate the transition

temperatures.

6.4 THE TETRAGONAL PHASE

It has been shown that K,0sC16 undergoes a structural phase transition

[24] to a tetragonal structure (1 4/m) as the temperature is lowered. Krupski et

al. [8] have determined the phase transition temperatures for the K,0sC16,

K, [OsCl,],,, [lrCI,]o.02, and K,[OSCI,]~~,,[R~CI,]~,~~ samples. They are 45.9 K,

40.5 K and 47.0 K, respectively with an uncertainty of 0.5 K in each case. This

result is a clear indication of the presence of the impurities in the samples.

Influence of Substitutional lmpurities.. . Chapter 6 - Page 7 5

These temperatures are indicated on Figures 3 and 4. It was previously observed

by Armstrong et al. [7] that for cubic structure fits to K,OsCI, profiles the value

of ~2 began to increase below 80 K and well above the phase transition

temperature. This behaviour was interpreted as evidence for the occurrence of

precunor clusters signifjting the onset of the phase transition to a tetragonal

structure at about 45 K. The present data in Figure 4 for pure sample are

entirely consistent; the dashed line provides a guide to the eye. The quantitative

differences between the present and the previous (7) X* values is due to the

difference in the statistics of the two experiments. What is interesting about

Figure 4 is that there is little or no evidence for the formation of precursor

clusters above Tc in either the K2[OsC16],,,[ ICI , ] ,,, or the

K,[OSC~,~~~,[R~CI,]~~~~ sample. For the latter sample with a Tc = 47.0 K it is

only the profile a t 40 K for which ~2 is significantly greater than 2.7 and at this

temperature the sample is already in the tetragonal phase. For the

K2[OsCl,],g,[lrC16]o~oz sample ~2 is essentially constant; the phase transition

has not yet occurred. The conclusion is that the presence of the impurities

suppresses the development of precunor clusters.

For the profiles taken at 40 K and 44.5 K a tetragonal (1 4/m) structural

model was fitted to the data. The structural parameters are two lattice

constants, a and c; an angle, 0, corresponding to the ferro-rotation of the

equilibriurn orientation of the OsCI, octahedra, and two Debye-Waller factors.

The bond length, r, obtained from the cubic fits was used. The rigid body rnodel

(optional with GSAS) was implemented thereby fixing the OsCI, octahedral shape

and atlowing the ion to rotate around the c axis. As expected, this mode1 made

no improvement for the K,[~SC~,]~~~~[~~CI,]~~~~ sample. The values of ~2 for the

fits for the K20sCI, and Kz [ O S C ~ & ~ ~ [ReCI,],,z samples were reduced to the

Influence of Substitutionûl lmpurities... Chapter 6 - Page 16

values at the higher temperatures. Values of e of approximately Z0 were

obtained.

The deviation of the chlorine Debye-Waller factors as noted in Figure 3

for the K20sC16 and K2[0sC16]o~,,[ReC16]o~02 samples a t 40 K and 44.5 K from

the predicted behaviour can now be explained as a consequence of the

occurrence of structural phase transitions. At these temperatures for these

samples we expect a static component in the chlorine atom displacements

relative to the cubic phase; we expect no such component for the potassium

atoms. Indeed, the values from the tetragonal fit are significantly closer to the

predicted values.

For the 50 K profiles a combined tetragonal-cubic model was fitted to the

profiles for the K,0sC16, K,[OsC16],,,[lrC16],,, and K2[0sC16]o~,,[ReC16]o~02

samples. The tetragonal phase fraction was an adjustable parameter. The quality

of the fits was better than for the pure cubic phase fits; the tetragonal phase

fraction for the K,0sC16 sample was 0.36, for the K , [ O S C ~ ~ ] ~ ~ , , [ ~ ~ C I ~ ] ~ ~ ~ ~ sample

was 0.08 and for the K,[OsCI,], ,,[ReC16],02 sample was 0.22. This result tends

to support the conclusion that the presence of the impurities tends to suppress

the formation of tetragonal phase clusten above Tc.

6.5 CONCLUSIONS

This experiment has provided a verification that the I ~ C I , ~ ~ and R~cI,~-

impurity ions in the samples studied are indeed in substitutional positions for the

OS CI,^- ions. Since the model used to fit the neutron scattering data assumed

influence of Substitutionai lmpurities.. . Chapter 6 - Page 1 7

that the impurity ions were in substitutional positions, if this were not so, the

model would have failed.

The effect of the introduction of the impurity ions on the static

parameten of the model structure, namely the lattice parameter and the bond

length, are deduced. The addition of IrCI, tends to reduce the lattice constant

whereas the addition of ReCl, tends to increase it. The bond length is unchanged

within experimental error.

Smaller diameter sample tubes were used for the present study than in

the previous study on KzOsCk [7] and consequently the absorption correction is

much smalter. It follows that the Debye-Waller factors for both the potassium

and chlorine ions are much better defined. The temperature dependencies of

these parameters are now well described by a Debye model of the lattice

vibrations. The addition of impurities does not significantly change these

dynamical parameters.

The temperature dependence of the goodness of fit parameter ~2

provides evidence of the development of precursor dynamical clusten. The

present experiment indicates that the presence of the impurities significantly

suppresses the development of these clusters. This conclusion is supported by

fitting the profiles to tetragonal and combined cubic-tetragonal structural

rnodels.

Influence o f Substitutional lmputfties... Chapter 6 - Page 18

1 . R.L. Armstrong. Physics Reports 57, 343 (1 980).

2. R.L. Armstrong. Prog. Nucl. Magn. Reson. Spectrosc. 20,1 51 (1 989).

3. R.L. Armstrong. 24th Ampere Congress, Magn. Reson. and Related

Phenornena, Poznan 53, (1 988).

4. D. Nakamura. J. Mol. Struct. 1 1 1 , 341 (1 983).

S. G.P. O'Leary and R.G. Wheeler. Phys. Rev. B I , 4409 (1 970).

6. M. Sutton, R.L. Armstrong, B.M. Powell and W.J.L. Buyen. Phys. Rev. B27,

380 (1 983).

7. R.L. Armstrong, R.M. Morra, 1. Svare and B.M. Powell. Can J. Phys. 65, 386

(1 987).

8. M. Krupski and R.L. Armstrong. Mol. Phys. Rep. 6, 169 (1 994).

9. R.L. Armstrong and M.E. Ramia. J. Phys. C 18, 2977 (1 985).

10. C. Dimitropoulos, J. Pelzl, H. Lerchner, M. Regelsberger, K. Rossler and A.

Weiss. J. Magn. Reson. 30, 41 5 (1 978).

11. M. Krupski. Phys. Stat. Sol. (a) 78, 751 (1 983).

12. A. Ishikawa, A. Sasane, Y. Mori and D Nakamura. 2. Naturofrsch. 41 a. 326

(1 986).

13. A.C. Larson and R.B. Von Dreele; LANSCE, MS-H805. Los Alamos National

Laboratories (1 990).

14. C.J. Howard. J. Appl. Cryst. 15, 61 5 (1 982).

1 5. M.W. Thomas. J. Appl. Ctyst. 10, 1 2 (1 976).

1 6. M.J. Cooper and J.P. Sayer. J. Appl. Cryst. 8, 61 5 (1 975).

17. H.M. Rietveld. J. Appl. Cryst., 2, 65 (1 968).

18. A.W. Hewat. Acta Cryst. A35, 248 (1 979).

NQR Study of Hydrostatic Pressure ... Chapter 6 - Page 1 9

19. R.W. James. The optical principles of the diffraction of X-ray. The crystalline

state. Vol II. Edited by W.L. Bragg. G.Bell and Sons Ltd. London, Efigland.

219, (1958).

20. V.J. Minkiewicz, Ga Shirane, B.C. Frazer, R.G. Wheeler, and P.B. Dorain. J.

Phys. Chern. Solids. Pergamon Press 29,881 (1 968).

21. H. Takazawa, S. Ohba, Y. Saito and M. Sana. Acta Cryst. 846, 166 (1 990).

22. H. Willemsen, C.A. Martin, P.P.M. Meincke and R.L. Armstrong. Phys. Rev.

B I 6 ,2283 (1 977).

23. International X-ray Tables. Kynoch Press, Birmingham, England 2, 264

(1 959).

24. R.L. Armstrong, D. Mintz, B.M. Powell and W.J.L. Buyers. Phys. Rev. 61 7,

1260 (1 978).

NQR Study of Hydrostatic Pressure ... Chapter 7 - Page 1

CHAPTER 7

NUCLEAR QUADRUPOLE RESONANCE STUDY OF HYDROSTATIC

PRESSURE AND SUBSTITUTIONAL IMPURITY EFFECTS

IN THE CUBlC TO TETRAGONAL PHASE TRANSITION IN

K20sCI 6

Pablo Prado and Robin L. Armstrong

New Brunswick, Canada

Marcin Krupski

lnstitute of Molecular Physics

Polish Acaderny of Sciences

Poznan, Polarid

Reference: Accepted for publication. Prado P J, Armstrong R L and Krupski M

September 1996 Can. J. Phys.

Journal of Physics publishers.

ABSTRACT

Samples of K2[OsC16]~-xx[MC16]x with M = Pt, Ir or Re and X = 0.01, 0.02 or 0.04

were prepared from a single batch of K2ûsC16. 3 5CI Nuclear Quadrupole

Resonance measurements of Free Induction Decay signals were obtained. Data

are reported through the temperature range of the cubic t o tetragonal

structural phase transition, and for a series of applied hydrostatic pressures. The

data are subjected t o Fast Fourier Transforrn analysis, complemented by a

Maximum Entropy Method appraisal in regions of overlapping spectral

cornponents. Coexistence of the cubic phase and the tetragonal phase over a

range of temperatures is clearly depicted for the fint time through 3D-

presentations of stacked FFT spectra for a series of temperatures. The presence

of the impurities results in the appearance of a satellite line shifted t o a higher

frequency than that of the cubic phase resonance for P t and Ir, and to a lower

frequency for Re. The impurities also cause the transition temperature to shift

to higher values for Re, and to lower values for Pt and Ir. The effect of an

increasing hydrostatic pressure is to shift the transition temperatures t o lower

values, and t o change certain quantitative aspects of the spectra.

Over the past three decades, extensive studies of antifluorite compounds

have been reported. Of particular interest is the mechanism responsible for the

structural phase transitions that occur in various of these solids. For example, it

has been shown that the softening of the rotary lattice mode triggers the cubic

(Oh5) to tetragonal (C&) phase transitions that occur in KzOsCls and KzReCls.

The mode softening is accompanied by the occurrence of precursor dynamic

tetragonal-phase clusters while the sample is still in the cubic phase. The details

of the cluster dynamics are dependent on the impurity content of the sample,

and on the magnitude of any externally applied stress.

Nuclear Quadrupole Resonance (NQR), neutron diffraction, neutron

inelastic scattering, and optical spectroscopy studies of the structural details

and dynamic processes in antifluorite crystals have been carried out. The results

are summarized in several publications [l -61.

The phase transitions in K20sCI6 and KzReCI6 have been studied by

chlorine NQR. The investigations have included monitoring the temperature

evolution of the line-shape through the region of the cubic to tetragonal phase

transition. KtOsC16 is the optimal antifluorite compound for NQR experiments

because of the sharpness of its resonance lines, and hence its intrinsic high

resolution.

Because of the extreme sensitivity of the NQR technique to perturbations

of the local electric field gradients, it is the ideal tool for the study of the

subtleties associated with structural phase transitions in the presence of

impurities, and when pressure is applied. This has been illustrated previously 17-

81 in experiments to observe the effect of applied hydrostatic pressure on

crystaîs of K20sC16.

The structural instability in antifluorite crystals has been accounted for by

a rigid-sphere model [7,9]. Information from NQR experiments concerning the

phase transition temperature, Tc, and its pressure derivative, dTc/dP, in K20sCk

has been discussed [7,10] in t e n s of this model. These parameten are

compared with values deduced in the present work.

It has been shown that substitutional impurities in K20sCI6, as well as the

application of hydrostatic pressure, produce changes in the lattice constant

[Il ,121. Further, it has been shown by neutron diffraction measurements [12]

that substitutional impurities influence the temperature range over which the

dynamical processes associated with the phase transition occur. In the present

work the evolution of the 35Cl NQR line-shape is followed through the

temperature range of the structural phase transition. The conclusions drawn are

compared with those deduced from the neutron diffraction data. These data

were obtained from experiments carried out on three of the powder samples

used for the present work.

A t those temperatures where overlapping lines are present, the Maximum

Entropy Method (MEM) for spectral analysis [13,14] was used in combination

with Fast Fourier Transform (FFT) analysis to determine the 3sCl NQR resonance

frequencies. This procedure aids in the observation of the evolution of the phase

transition and thereby in the determination of meaningful values for Tc and

dTc/dP.

Section 2 gives a discussion of the sample preparation technique and of

the NQR apparatus used to acquire the data. Section 3 provides comments on

the FFT method of analysis; Section 4 on the MEM. The two procedures are

shown to be complementary; they are used in combination for the determination

of the resonance frequencies in regiom of strong spectral overlap. In Section 5,

the evolution of the NQR line-shapes through the temperature range (20K < Tc

< 60K) is presented. Data are reported for K20sC16 and five samples of

Kz[OsCi6]1.x[MC16]x with M = Pt, Ir or Re, and X = 0.01, 0.02 or 0.04, and for

pressures ranging from atmospheric to 270 MPa. A discussion of the results and

conclusions drawn is given in Section 6.

7.2 EXPERIMENTAL

Six powder samples were prepared from the same batch of K20sCls

powder obtained from Johnson and Matthey. The batch was divided into six,

more or less equal portions. Each portion was dissolved in an aqueous, 8M

hydrochloric acid solution. In five of the portions some measured additional

amount, from 1% to 4% of one of K2ReCI6, K21rCI6, or KzPtCI6, was dissolved in

the solution (Krupski, 1994). Six powder samples, one of K2OsCI6 and five of

Kz[OsC16]i.x[MC16]x with M = Re, Ir, or Pt, were produced by recrystallization

from these solutions in vacuum a t room temperature, yielding approximately 2.7

g of each compound. Use of an 8M hydrochloric acid solution has been shown to

optimally enhance the signal to noise ratio for NQR spectra obtained from the

resulting powders [15].

As a confirmation that the impurities were located in substitutional

positions we carried out an x-ray study. The results from the microprobe showed

an irnpurity concentration distribution which seemed to correspond to the site

NQU Study of Hydrostatic Pressure ... Chapter 7 - Page 6

of the crystallite selected for observation. The larger the crystallite, the lower

the concentration of impurities. Since the neutron expriment is not sensitive to

scattering from individual crystallites it yields an average over the distribution.

A neutron scattering expriment [12] carried out using the same samples

provided further verification that the IrC16z- and ReC16z- impurity ions in the

samples studied are indeed in substitutional positions for the 0sC162- ions. Since

the model used to fit the neutron scattering data assurned that the impurity

ions were in substitutional positions, if this were not so, the model would have

failed. This experiment also showed that the phase transition temperature

changed substantially in the sarnples containing the impurities. Such a change

would not have been observed if the impurities had not been incorporated into

the lattice.

A Tecmag Fourier transfon pulse spectrometer was used to measure the

35CI NQR free induction decay (FID) responses from the six powden. Signals

following a single n/2 pulse were averaged for typically 200 acquisitions in a

1024 channel window. Spectra were recorded as a function of ternperature for a

series of values of applied hydrostatic pressure, spanning the range of the cubic

to tetragonal structural phase transition occurring in the vicinity of 40K.

A high pressure device, previously described by Krupski and Armstrong

1161, was used to apply hydrostatic pressures up to 270 MPa to the samples. A

heat exchanger was added around the high pressure chamber. Temperature

stability of 0.1 K was achieved by using a regulated flow of liquid helium and a

system of electric heaten coupled to a Lake Shore Temperature Controller.

Details as to how temperatures and pressures were detenined and calibrated

are given in reference [16]. Spectra were obtained at atmospheric pressure and

two or three elevated pressures for each sample. Twenty to thirty ternperature

values between 20K and 60K were monitored in each case. In total, more than

500 spectra were recorded in a series of experiments on the six samples.

When the spectral lines were clearly resolved, determination of the

resonance frequencies was carried out using a FFT algorithm and a least squares

fitting routine with one or two Lorentzian line-shapes for the cubic and

tetragonal phases, respectively. However, in many instances three or even four

lines were present and two or more of them strongly overlapped. In such cases

it was useful to employ a combination of FFT and MEM analyses.

7.3 FOURIER TRANSFORM ANALYSE

Figure 1 (a) shows a tetragonal phase 3sCI FID signal taken a t 30K for the

K2OsCI6 sampie at atmospheric pressure. Aithough 1024 points were acquired

with a 1 ps dwell time, only the first 500 are displayed. Following FFT analysis,

the spectrum shown in Figure 1 (b) is obtained. It consists of two separated

resonance lines which are fitted individually by Lorentzian line-shapes. The peak

frequencies and widths can then be readily determined. The same is, of course,

true for the single resonance line observed a t temperatures significantly greater

than Tc.

A mp. 1.2 I I 1 1

0.8 - a

0.4 - -

0.0 - 1

-0.4 - -

- 0 . 8 - 1 " ' ' " " " " " ' m ' m m " ' - -

60 40 20 O -20 -40 -60 v- v,(k Hz)

Fig. 1. 35Cl respunse in the K20sC16 sample at 25.OK and atmospheric pressure:

(a) FI0 signal; (b) magnitude spectnrm as ubtained by FFT.

The gross features of the line-shape evolution in the K20sCls sample

through the phase transition are vividly portrayed in Figure 2. This 3D

presentation of the data contains the FT signal magnitude as a function of

frequency for a series of temperatures.

Fig. 2. Stacked plot of F i magnitude spectra showing the gross features of the

evolution of the spectrum spanning the temperature range of the structural

phase transition in K20sC16.

For temperatures in the vicinity of Tc, a third component is present in the

spectrum, and it overlaps with one of the lines noted above. The details of the

line-shape evolution in the region covering a few degrees on either side of Tc are

depicted in figure 3. It can be noticed that there is a decrease in the intensity of

the cubic phase component of the spectrum, and an increase in the intensity of

the z-component of the tetragonal phase as the temperature is lowered. The

solid lines indicate the growth of the tetragonal phase xy- and z-components.

Figure 3 shows the evolution of the 35CI NQR spectrum in K20sC16 in the vicinity

of the phase transition with hitherto unattainable clarity.

In the samples containing impurities an extra line appean in the spectrum.

This is illustrated in Figure 4 where an additional satellite peak appears in the

K20so.9sPto.02Ck sample. Notice that this small peak occurs at a higher

frequency than the main resonance. The solid line provides a guide to the eye; it

indicates that the satellite peak remains essentially unchanged through the

phase transition, but in the tetragonal phase overlaps with, and broadens, the

high frequency xy-component.

Note that in the region of overlapping lines, the FFT analysis does not

permit accurate values of the central frequencies of the constituent

components. To overcome this limitation the FFT analysis was combined with

MEM as discussed in the next section.

Fig. 3. Details of the line-shape evolution of the K20sC16 spectrum shown in

figure 2 covering the region of a few degrees on either side of Tc.

Fig. 4. Tempera ture e volution of the 3sCl spectrurn of Kf l~Cls l . .~~[PtCI~ jI02

indicating the presence of a satellite peak due to the Pt imputities.

NQR Study of Hydrostatic Pressure.. . Chapter 7 - Page 1 3

7.4 MAXIMUM ENTROPY METHOD

The MEM for spectral analysis is a non-linear fitting procedure that

maximizes the output entropy information [17-191. This technique has been

used successfully in NQR spectroscopy [13,14], and in I D and 20 NMR

experiments [20-251. There is a substantial literature containing discussions

about resolution and sensitivity improvement, and the loss of intensity

information in different situations [26-331. The aim of the present work is not

to discuss the general applicability of the MEM. The objective is to use the

technique to assist in the resolution of overlapped spectral lines, without

concern for recovering relative intensity values.

The MEM is used to fit the time domain data. Although spurious peaks can

be created in the MEM output [14], this is of no concern to us because we use

the technique in conjunction with FFT analysis. Therefore, the number of peaks

is known even though their frequencies can not always be accurately deduced

from the FFT spectra. Unambiguous resolution of overlapping peaks is possible

with the MEM since it emphasizes weak features over strong ones [28].

However, only poor information conceming relative intensities and line-shapes of

the component peaks is recovered.

For the present purposes a C language version of the algorîthm presented

by Mackowiak [13] was written. A characteristic parameter of the algorithm is

m, the "prediction filter error length". This parameter is chosen by trial and error

to optimize the output profile. A wrong choice of m value can result in the

production of extra peaks, or in the failure to resolve al1 of the peaks present.

The C program has reduced the processing time, allowing the m parameter to be

changed in an interactive and rapid fashion.

For spectra where more than one resonance is present, the spectrometer

frequency is usually set between the frequencies of the spectral lines in order

that the various resonances are irradiated with equal power. The subsequent FFT

analysis reproduces the true relative intensities of the components. However,

selection of the spectrometer frequency in this manner causes a complication

for the MEM analysis. Lines that are more or less equidistant from the irradiation

frequency now overlap, and a considerable distortion is produced in the output

amplitude. This distortion depends not only on the choice of the parameter ml

but also on the irradiation frequency selected. When using only the MEM

reconstruction the problem can be circumvented by applying a spectrometer

offset frequency [14].

As a consequence of using insufficiently hard rf pulses and short delay

times before data acquisition, the amplitude spectra resulting from FFT analysis

exhibit phase shifts of first and higher order in the frequency. Phase shifts of

other than first order are extremely hard to correct in a consistent and

systematic manner. Therefore, magnitude spectra were usually created and

interpreted for the present experiments. These spectra are characterised by

somewhat wider lines than amplitude spectra, and therefore the difficulties with

overlapping lines is increased. However, the MEM permits the resolution of peaks

not differentiated by FFT analysis, extracting them from a low signal to noise

spectrum and a highly overlapped profile.

As an example, consider the case of the K2[0~C16].g8[PtC16].02 sample a t

40.2K and atmospheric pressure. Under these conditions the compound is in the

tetragonal phase. The FID is displayed in Figure 5(a). The resulting FFT deduced

amplitude spectrum is shown in Figure 5(b). Notice the large uncorrected phase

shift. Figure 5(c) is the corresponding magnitude spectrum. Only two peaks are

readily discernible, one centred at -4.9 kHz, and the other at about 10 kHz from

the reference frequency. The latter resonance is broader than the former,

suggesting the presence of a third weaker component. Indeed, we anticipate

such a spectral feature from a consideration of spectra obtained in this sample

at higher temperatures and depicted in Figure 4. The third line is due t o the

presence of the Pt impurities in the sample.

a mp. 10

Fig. 5 a,b. Spectra of K2[O~C16].98[PtC/6].02 sample taken at 40.2K and

atmospheric pressure: (a) FI0 s&nal; (b) FT amplitude spectrum.

Use of the MEM routine, with m = 100, results in Figure 5(d). From this

plot we can readily identify not only the peak centred at 4.9 kHz, but two other

peaks at 10.3 kHz and 16.1 kHz. The latter is the weak peak due t o the

presence of the Pt irnpurities in the sample.

MEM am p.

'O - Fig. 5 c,d. Spectra of KZ[O~C/6~98[ptC/6].02 sample taken at 40.ZK and

atmospheric pressure: (c) FT magnitude spectrum; (d) MEM output.

The significance of combining the FFT and MEM techniques should be

noted. The FFT analysis has provided the direction of the frequency shift for

each of the components identified by the MEM with respect to the reference

frequency. The MEM has yielded accurate measures of the resonance

frequencies of each component; the FFT analysis can then provide the line-

shapes and relative intensities.

NQR Study of Hydrostatic Pressure-.. Chapter 7 - Page 17

The MEM has also been applied for temperatures and pressures of

samples where both the cubic and tetragonal phases coexist. It is diffkult, using

only FFT analysis, to resolve the tetragonal phase components as they first

begin to appear as the temperature is reduced. The lines are weak and the

spectra are noisy. However, these tetragonal phase resonance lines can be

readily resolved using the MEM.

7.5 PRESENTATION OF RESULTS

The 35CI NQR spectrum of a R2MX6 compound in the cubic phase is

expected to exhibit a single resonance line, and in the tetragonal phase to

consist of two lines. Figure 6 shows the 3sCI NQR response for four of the

samples in the cubic phase at temperatures well above Tc. For the KzOsCls

sample at T = 64.8K and atmospheric pressure, a single resonance line is

observed; the FT magnitude spectrum is presented in Figure 6(a). For the other

three samples, each containing impurities and each at temperatures well above

Tc, an additional peak is observed. Figures 6(b) and 6(c) are FT magnitude

spectra for Kz[O~C16].s8[lrC16].02 and Kz[OsC16].g8[PtC16].oz, respectively, a t

atrnospheric pressure. In each instance the additional peak, as indicated by the

arrow, occun a t a higher frequency than that of the main resonance. Figure 6(d)

is the FT magnitude spectrum for K2[0sC16].g6[ReC16],04; the additional peak, as

indicated by the arrow, is observed at a lower frequency than that of the main

resonance. These additional peaks are not a surprise. They are a consequence of

the presence of the impurities in the samples [34]; they have been previously

reported in the cubic phase of K2OsCI6 [35].

Fig. 6 a,b. FT magnitude spectra for two samples at atmospheric pressure and

tempera tures significan tly grea ter than Tc: (a) K20sCl6;(b) K2[OsC/61 98[/c/6]. The satellite peaks resulting from the addition of impurities are indicated by

arro ws.

Sets of resonance frequency vs temperature data were obtained for each

of the 6 samples at every value of the applied hydrostatic pressure. The central

frequencies of al1 of the resonances were determined, in general by combining

the FFT analysis with the MEM technique. As an exarnple, we show in Figure 7 a

NQR Study of Hydrostatic Pressure-.. Chapter 7 - Page 19

frequency vs temperature plot for the K2[OsC16].g8[lrC16],02 sample subjected to

a hydrostatic pressure of 69 MPa. The high and low frequency lines for the

tetragonal phase are labelled xy and z, respectively. For simplicity. the satellite

line due to the Ir impurities is not displayed.

Fig. 6 c,d. FT magnitude spectra for two samples at atmospheric pressure and

tempera tures sig nifican tly grea ter than Tc: (c) Kt [Os Cl6]. CJ~[&C/~]. 02; (d)

K2[0sC/6j94[ReC/6]. 04. The satellite peaks resulting from the addition of

impurities are indicated by arrows.

t= Tet ragonal

9 : Cubic phase

Fig. 7. Temperature variation of the 3 SC1 NQR resonance lines for the

Kz[OsC~].g8[l~16].02 sample subjected to an applied hydrostatic pressure of 69

MPa. Note the region o f coexistence of the cubic and tetragonal phases.

For each sample a t every pressure studied, a temperature range exists

over which the cubic and tetragonal phases coexist. This is seen explicitly in the

data contained in both Figure 3 and Figure 7. This is a real phenomenon and not

due to a temperature gradient; the temperature difference across the sample is

less than 0.01 K, whereas the coexistence range is typically 5K. The definition of

the phase transition temperature, Tc, is therefore somewhat arbitrary. We shall

define Tc as the temperature at which the cubic line vanishes as the

temperature is lowered. To determine Tc we plot the peak intensity of the cubic

phase line as a function of temperature, and extrapolate to zero intensity. Figure

NQR Study of Hydrostatic Pressure... Chapter 7 - Page 2 1

8 illustrates the procedure. For low pressure spectra the data are well

represented by an exponential decay function. Figure 8(a) shows the variation of

the maximum amplitude, Am,,, of the cubic phase line as a function of

temperature for the K2[0sC16].98[ReC16].02 sample at atmospheric pressure. A

value of Tc = 46.OK is deduced by extrapolation. For high pressure spectra the

decay is faster. To represent the data over a similar temperature range it was

necessary t o add a linear component to the exponential decay. Figure 8(b) gives

the maximum amplitude as a function o f temperature for the

Kz[OsC16].g8[lrC16]~02 sample subjected to a hydrostatic pressure of 1 38 MPa. A

value of Tc = 34.2K is obtained.

50 I - Fig. Ba. Maximum - amplitude of the FT

amplitude spectrum in the

cubic phase obtained by - fit ting Loren tzian line-

shapes to the data. - Kz[OsCl6J98tReC~61.0 2 a t

atrnospheric pressure; an - exponential decay function

O i i i i ' i i ' " i i i i l i i i i is used.

45 50 55 60 65

Values of Tc deduced from the analysis are listed in Table 1 and plotted

as a function of the hydrostatic pressure in Figure 9. The solid lines are linear

regressions. As the pressure is increased, the frequency shift between the cubic

phase line and the z component of the tetragonal phase decreases, rnaking Tc

determinations more difficult. The extreme cases are the 234 MPa pressure data

for the K2[0sCls]~ss[lrC16]~ol and K2[0~C16].ss[PtC16]nz samples. In each instance

the presence of the zsomponent is observed only as a broadening of the cubic

phase line; its frequency is obtained through MEM analysis. The positions of the

resonance lines for the K2 [OsC16]~gg[lrC16]~oi sample under 234 MPa of

hydrostatic pressure are plotted in Figure 10. The solid line through the three

data points for the xy-component of the tetragonal phase is extrapolated to the

temperature at which the z-cornponent vanishes; it is assumed that the xy-

component exists, but is too weak to be detected, over this range.

Fig. 8b. Maximum

amplitude of the FT

amplitude spectrum in the

cubic phase obtained by

fit ting Lorentzian line-

shapes to the data.

KdosCl61 9 9 [ ~ ~ / 6 1 . O 1 under

138 MPa hydrostatic

pressure; an exponential

plus a linear term is used

L I I I I 1 I

30 35 40 45 50 55 60 65 to represent the decay.

œ - - - 9

30 - II

20 - - - rn .

10 - - 9

O p i ' _ i ' l i m a ' l ' l m m ' l " i l i m " l i r

I mpuri t y concentrati on

r 4%Re V 2%Re

Fig. 9. Transition temperature, Tc, plotted as a function of applied hydrostatic

pressure, P, for the six samples studied.

Although it is possible to identiw both the cubic phase resonance and the

two tetragonal phase lines for the K 2 [ O ~ C l ~ ] . ~ ~ [ l h l ~ ] . ~ ~ and

K2[0~C16].9e[PtC16],op samples, there is insufficient data to obtain a reliable value

for Tc. Therefore, in Figure 9, Tc values appear on the plot at only two values of

hydrostatic pressure even though measurements were done a t three pressures.

Figure 9 indicated that the Tc values are sarnple dependent. The addition

of Re as an impurity causes Tc to increase, whereas the addition of Ir or P t

causes Tc to decrease. Figure 9 also shows that as the hydrostatic pressure is

increased, Tc decreases for al1 samples. This latter observation is as predicted

for a phase

derivatives,

transition driven by the softening o f a ï'-point mode [36]. Pressure

dTc/dP, were calculated from the linear fits. They are recorded in

Table 2. As a consequence o f the small number o f pressures measured for each

sample, the error associated with the derivatives arising from the Tc

determinations is estimated t o be 10%. Therefore, the dTc/dP values should be

taken as

Table 7:

being sample independent within experimenta! error.

Transition temperatures for the six samples.

atm 43.1 atm 39.7 atm 44.0

138 3 6.2 69 3 6.8 138 37.0

138 34.2

P (MPa)

Tc (K)

P (MPa)

Tc (K)

P (MPa)

f c (K)

NQR Study of Hydrostatic Pressure... Chapter 7 - Page 25

Table 2: Pressure derivatives o f Tc

The frequency vs temperature curves, as shown in Figure 7, Vary with the

composition of the sarnple and with the applied hydrostatic pressure. These

modifications are undoubtedly related to some structural characteristic of the

tetragonal distortion since the frequency splitting, Av, between the xy- and z-

lines of the tetragonal phase is proportional to the square of the rotation angle

of the MX6 octahedra [37]. Empiflcai curves were fit to each of the lines for al1

six samples and for each pressure at which measurements were taken. Three

functions were found adequate to represent the three lines in al1 of the

experiments. The temperature dependence of the frequency of the cubic phase

line was described by a linear equation. The temperature dependencies of the

frequencies of the tetragonal phase lines were represented by two different

expressions. The Aine was described by a quadratic curve; the xy-line by the

function A(B - T)1/2 + C. A typical result obtained by applying these equations in

a least squares fitting procedure is shown by the solid lines in Figure 7.

T(K) Fig. 10. Temperature variation of the JsCl NQR resonance lines for tne

K2[OsC/6~gg[lK/6].o sample subjected to an applied hydrostatic pressure of 234

MPa. Note the region of coexistence of the cubic and tetragonal phases.

The addition of impurities is responsible not only for the extra

cornponents in the cubic phase, but also for a broadening of the main high

temperature lines. This latter phenornenon is detected as a reduction of the T2+

decay tirnes in al1 of the samples containing impurities as compared to the

"pure" sample. This is illustrated in Figure 11 where T2* values are plotted for

the K20sC16 and Kz [ 0 ~ C l ~ ] , ~ ~ [ R e C l ~ ] . ~ ~ samples a t atmospheric pressure. For

these two samples the Tc values differ by only 0.7K.

4 ' Tet ragonal phase

- \ '\ - z \

\ - \ \

1-‘" Cubi c phase a : - ° a

24 25 26 27 28 29 30 31 32

pure v 2%Re -

~ I i I ~ l i I r I i l a

45 50 55 60 65 70 75 80

Fig. 1 1. Tt* values as a

function of temperature in

the cubic phase of K20sC16

and K~[osCl6hdReC/61.02

samples at atmospheric

pressure.

The maximum amplitudes of the peaks have been determined for lines in

regions of well defined phase for al1 of the samples. We display the results for

the cubic and xy-resonances for the K20sC16 and K2[0sC16].ss[ReC16].oz samples

at atmospheric pressure in Figure 12. The changes in amplitude as Tc is

approached from either side are described by exponential functions. A

comparison of Figure 12(a) with Figure 12(b) suggests that the range over

which coexistence of the cubic and tetragonal phases occun is reduced in the

sample containing the impurities.

The tetragonal distortion causes the six identical chlorine nuclei in each

MC16 complex in the cubic phase to be divided into two groups of identical nuclei

with four in the one and two in the other. The former group gives rise to the xy-

line; the lat ter to the z-line. As a result the intensity of the xy-line is predicted

to be twice that of the z-line. But the intensity of each resonance is proportional

NQR Study of Hydrostatic Pressure,.. Chapter 7 - Page 28

to the maximum amplitude divided by the corresponding Tz' value, not simply to

the maximum amplitude. This then is the explanation of why the ratios

amp,/(ampz + ampw) at 25K in Figures 1 2(a) and (b) are about 0.5 rather than

0.67. Since the xy-line is approximately twice as wide as the z-line [see Figure

1 (a)], it follows that the intensity of the xy-line a t 25K is, within experimental

error, twice that of the z-line as should be the case.

Fig. V a . Maximum amplitudes, as a function of temperature, of the resonance

lines in both the cubic and tetragonal phases of the K20sCI6 sample at

a tmospheric pressure.

Fig. 12b. Maximum amplitudes, as a function of temperature, of the resonance

lines in both the cubic and tetragonal phases of the K2[O~C16].g8[ReC~02

sample at atmospheric pressure.

7.6 DISCUSSION AND CONCLUSIONS

The present experirnents have shown that the replacement of Os ions in

K2OsCI6 by Pt, Ir or Re ions produces a satellite line in the 3sCi NQR spectrum.

This line is easily observed in the cubic phase, shifted from the main resonance

by 10 to 20 kHz. It remains essentially unchanged as the compounds undergo

the structural transition to the tetragonal phase but overlaps with one of the

tetragonal components. For Pt and Ir impurities, the satellite line appears at a

higher frequency than the main resonance; for Re impurities it is shifted to a

NQR Study of Hydostatic Pressure ... Chapter 7 - Page 30

lower frequency. It is interesting to note that the 3sCl resonance frequencies of

both K2PtCl6 and K21rC16 are higher than that of K20sCk, whereas that of

K2ReC16 is lower.

The electronic structures of K2PtCI6, KzlrC16, K20sC16, and K2ReCls are

similar [38]. The anti-bonding n* orbital is full for K2PtC16, has one hole for

K21rC16, two holes for K20sCi6, and three holes for K2ReC16. When holes are

present, n- bonding occun. The local change in bonding due to the presence of

impurities can be expected to affect the behaviour of the samples. It is

therefore not surprising that the shifts of the satellite peaks relative to that of

the main resonance are in the directions that are observed.

The presence of impurities has a significant effect on the temperature of

the structural phase transition, shifting it systematically to lower temperatures

for P t and Ir impurities, and to higher temperatures for Re impurities. These

changes have been accounted for [11,39] using the "rigid sphere" rnodel.

According to this model the shifts are caused by differences in the ionic radii

altering the size of the lattice.

Co-existence of the cubic phase and tetragonal phase over a range of

temperatures has been clearly depicted for the fint tirne through displays of the

overall temperature behaviour of the FFT spectra using a 3D-presentation. These

plots have also demonstrated the need for employing the MEM to resolve the

positions of the peaks through this range.

It is not unprecedented for there to be regions of coexistence in

transitions which exhibit first-order character, even in the absence of

appreciable thermal gradients, as is the case here [40]. For transitions, such as

the one in KrOsC16, in which rotations are coupled to strain, it is possible to get

a distribution of micro-strains within different grains due to defects, so that

different portions o f the sample are in a different state of order at a given

temperature. Similar behaviour has been reported [41] for the fint order cubic-

orthorhombic transition in c-AIP04 where coexistence is observed over a

temperature interval of about 20 deg. This substance is a framework structure

in which A104 and PO4 tetrahedra rotate and induce a substantial strain. The

coexistence is attributed t o the occurrence of a range of transition

temperatures.

Because of the intrinsic lack of perfection of real crystals no structural

phase transition is purely second order. That is, there is some first order

character associated with the transition in K2OsCI6. Although it is difficult to

obtain an accurate measure of the first order discontinuity in the order

parameter from the NQR data, it is clear that the discontinuity is of the order of

10 per cent of the saturation value of the order parameter.

The presence of the impurities alters the range over which CO-existence

of the cubic and tetragonal phases occurs. We saw evidence of this in figure 12.

The presence of irnpurities also affects the sharpness of the phase transition.

For example, consider Figs. 3(b) and 13; the FFT spectra are plotted over the

same temperature range in both cases. The former shows the transition region

for KzOsCls; the latter for K2[0sC16].ss[ReC16].02. Bath experiments were carried

out at atmospheric pressure. Note that the xy-component, once it appears,

grows faster as the temperature is further decreased for the sample containing

the Re impurities than for the sample that does not.

lncreasing the hydrostatic pressure applied t o the samples, causes the

temperature to decrease as predicted for a phase transition driven by the

softening of a r-point mode. lncreasing pressure also alters the frequency

NQR Study of Hydrostatic Pressure. .. Chapter 7 - Page 32

Fig. 1 3. FFT spec tra of the K2 [ O S C / ~ ~ ~ ~ [ R ~ C I ~ ] . 2 sample a t a tmospheric

pressure. By comparison with figure 3, it can be seen that the presence of the

Re impurities cause the xy-component (solid line) to grow faster as the

temperature decreases, once it appears.

splitting of the spectral components causing the Aine of the tetragonal phase

to merge with the cubic resonance. However, the use of the MEM provided a

means to resolve these lines, as seen in figure 10 for the ~2 [O~C~6]~gs [~~~~6] .o l

under 234 MPa of hydrostatic pressure.

Softening of the rotary lattice mode causes the intensity of the cubic

phase resonance to decrease as the temperature approaches Tc from above.

This decay competes with the linear increase in intensity predicted by the Debye

model for a stable lattice as temperature is lowered. This is illustrated in figure

8(b). The development of precunor dynamic clusters exhibiting the symmetry

of the tetragonal phase accompanies the softening of the rotary mode. The

occurrence of these clusters is reflected in modifications t o the line shape as

reported for the pure sample [SI. The addition of irnpurities produces a

broadening of the resonance lines (figure 11) which obscures the subtle

contribution to the line shape due to precunor dynamic clusteo.

Neutron diffraction experiments [12] carried out on three of the samples

used for the present experiments yielded transition temperatures consistent

with, but less accurate than, those determined by NQR. An analysis of the

neutron diffraction data revealed that the goodness of fit parameter increased

as the temperature was lowered towards Tc. This effect was attributed to the

occurrence of precursor dynamic clusters. A sirnitar result would occur in a

region of CO-existence of tetragonal and cubic phases since the analysis was

carried out using a cubic phase model. However, the growth of the goodness of

fit parameter began well outside the region of CO-existence and therefore can

not be accounted for by this phenomenon.

NQR Study of Hydrostatic Pressure... Chapter 7 - Page 34

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Uniaxial Pressure Effects in the Cubic ... Chapter 8 - Page 7

CHAPTER 8

UNIAXIAL PRESSURE EFFECTS IN THE CUBlC PHASE OF

KzOsCl 6

Pablo J. Prado and Robin L. Armstrong

Fredericton, New Brunswick

Canada E3B SA3

Reference: Prado P J and Armstrong R L 1996 J. Phys.: Condens. Matter 8 5621

This article is included in the present thesis under permission of the Journal of

Physics: Condensed Matter publishen.

ABSTRACT

Chlorine nuclear quadrupole resonance (NQR) measurements of the effect of the

application of uniaxial pressure applied along the (1 00) and (1 1 1 ) directions in

single crystals of K20sCls are reported at 78.0 K. The free induction decay times

decrease with pressure due to the additional strains introduced. The

corresponding frequency domain spectra indicate that the inhomogeneous

broadening of the NQR signals is dominated by point defects, but that the

nature, nurnber and/or distribution of these defects is different in the two

crystals. The spin-spin relaxation tirnes are dependent on the crystal orientation

and are independent of pressure. The measured T2 results agree with calculated

values. Spin-lattice relaxation data further illustrate the difference between the

two crystals. It is described by a single exponential for the (1 1 1 ) crystal, but by

a double exponential for the (1 00) crystal. The latter behaviour indicates the

presence of dynamic clusten some 30 K above the temperature of the phase

transition. The application of pressure is seen to hinder their formation.

Uniaxial Pressure Effects in the Cubic.. ... Chapter 8 - Page 3

8.1 INTRODUCTION

Extensive nuclear quadrupole resonance, neutron scattering, and optical

spectroscopic studies of the structural and dynamic characteristics of

antifluorite crystals have been reported. The work has been summarized in

several review articles (Armstrong. 1975, 1980, 1989, 1989; Armstrong and

van Driel, 1975).

These experiments have provided an understanding of the rnechanism

responsible for the cubic (OhS) to tetragonal (C4$) phase transition occurring in

these compounds, specifically in K2ReC16 and K20sCI6. The structural phase

transitions occurring in these compounds have, by inelastic neutron scattering

measurements, been shown to be driven by the softening of the rotary lattice

mode. A considerable number of chlorine nuclear quadrupole resonance (NQR)

studies of K20sCI6 have been undertaken; it is the ideal antifluorite compound

for NQR investigation because it exhibits a narrow line and thus a strong signal.

This work has shown that the softening of the rotary mode is accompanied by

the development of precursor dynamic tetragonal phase clusters in the cubic

phase. NQR, because it is a local probe, is particularly sensitive to the

establishment of local order.

The high sensitivity of the NQR technique to perturbations in the local

electric field gradients caused by the presence of point defects or dislocations,

or by the application of pressure, makes it an ideal probe for the study of the

subtleties associated with structural phase transitions. It has been used to study

the effect of applied hydrostatic pressure on the phase transition in K20sC16

Uniaxial Pressure Effects in the Cubic. .. Chapter 8 - Page 4

(Krupski and Armstrong, 1989; Armstrong, Krupski and Su, 1990). It is

therefore an appropriate technique to choose to investigate the effect of

uniaxial pressure on the same phase transition.

Section 2 provides a discussion of the apparatus, sampks and techniques

used to study two single crystals of K2OsCI6 a t 78.0 K. A novel uniaxial pressure

device is described. 3sCI free induction decay measurements are presented and

discussed in Section 3, spin-spin measurements in Section 4, and spin-lattice

relaxation time measurements in Section S. The conclusions are surnmarized in

Section 6.

8.2. APPARATUS, SAMPLES AND TECHNIQUES

A Tecmag Fourier transform pulse spectrometer was used to measure the

35CI NQR signals in the KzOsC16 crystals. The experiments were carried out at

78.0 K because of the ease of maintaining a stable temperature using a liquid

nitrogen bath.

A new uniaxial pressure device was developed based in part on ideas of

Zamar et al. (1983) and Berlinger and Müller (1977). The design, shown in

Figure 1, incorporates an easy to calibrate force arm and an easy to access

sample space. The brass chamber which contains the rf coi1 mounted on a 1 cm

long glass tube is immened in a liquid nitrogen dewar. The rf voltage is fed to

the coi1 by a 50 coaxial line, in this application at a frequency of 16.87790

MHz. Two copper-constantan thermocouples were positioned at the top and

bottom interior of the vesse1 to measure the temperature gradient. Measured

temperature differences were in al1 cases less than 0.1 K. A Lake Shore

temperature sensor focated about 2 mm from the sample was used to monitor

the temperature stability in the vicinity of the crystal. Variations were less than

0.1 K over the course of each experiment.

Pressure is exerted on the crystal surfaces by means of two polished

glass cylinden. The top cylinder is attached to the pressure rod with a teflon

separator to provide thermal and electric isolation. The teflon washen used to

align the pressure rod in the stainless steel tube are a loose fit to minimize

friction. The bottorn cylinder seats on a pivot through teflon and brass supports.

The use of this pivot prevents strains from being exerted on the crystal which

could cause it to breaks. The force applied by the horizontal bar is easy to

calibrate because of the lever mechanism.

Helium gas is pumped into the system in order to avoid air condensation

around the rnoving parts. The helium pressure is kept slightly above

atmospheric; a balloon is used as a gauge.

Two single crystals of K20sCI6, grown by S. Mroczkowski a t Yale

University, were used for the experiments. The crystals were cut so as to have

(100) and (1 1 1) planes, top and bottom. These faces were polished with fine

sandpaper and finally with a slightly wet cloth. The sides of the crystals were cut

to be normal to (1 00) and (1 11) faces. The final volumes of the crystals with

the polished (100) and (1 11) faces were 0.045 and 0.041 cm3, respectively.

Henceforth, the two samples will be referred to as the (1 00) and the (1 11)

crystals.

STA INLESS SEEL TUBE

=ON WASHER

BRASS KESSEL

Fig 7. Uniaxial pressure de vice.

For each experiment the sample was allowed to equilibrate for 24 hrs at

Uniaxial Pressure Effects in the Cubk. .. Chapter 8 - Page 7

liquid nitrogen temperature before measurements were begun. The temperature

was 78.0 I 0.1 K. Measurements were taken for seven pressures in the range O

to 9 MPa. The upper limit for the pressure was dictated by the strength of the

antifluorite crystals. For pressures significantly greater than 9 MPA the crystals

fractured. The pressure was applied in steps, starting without load, adding load

up to the maximum value, and then removing it back to the initial situation. By

this procedure it was confirmed that the application of the pressure was a

reversible process, and that the crystals were not damaged in the course of the

experiments. Free induction decay (FID) signals following a single n/2 pulse were

averaged over 200 acquisitions. Spin-spin relaxation (T2) times were deduced

from maximum echo amplitudes following a n/Z-r/Z-TI sequence for a series of

T-values; 80 scans were taken for each value of T. Spin-lattice relaxation (Tl)

times were obtained from a n-r-n/Z inversion recovery sequence. A series of r

values was carefully chosen to cover the time scales of both the expected fast

and slow decays (Martin and Armstrong, 1975: Armstrong, Rsmia and Morra,

1986); 80 scans were taken for each value of T.

8.3. FREE INDUCTION DECAY MEASUREMENTS

Figure 2 shows a typical 3sCI free induction decay signal. The actual data

presented are for the (1 1 1) crystal at a pressure of 2.9 MPA. The line through

the data is a least squares fit to the function

The values of Tz* obtained from fits of this type to al1 of the data are shown in

Figure 3. The lines through the data are linear least squares fits. We see from

the graph that the value of T2* at zero applied pressure is different for the two

crystals. This illustrates that the two crystals are not identical; they have

different intemal strains because of different defects or dislocations. We also

see that in each case Tz* decreases in a linear fashion as uniaxial pressure is

applied over this low pressure range. A decrease is to be expected as the

application of the pressure will introduce additional strains.

O 400 800 1200 1600

t (P ) Fig 2. T y p i ~ a l 3 ~ ~ 1 Free induction decay signal. The solid line through the data is

a least square fit.

Fig.3. Tf values obtained from fits to the free induction decay signals.

The free induction decays were Fourier transformed to obtain amplitude

spectra. These spectra in each case consist of an asymmetric, inhomogeneous

line that can be decomposed into two symmetric components. Figure 4 provides

a typical example; it was obtained from the (100) sample under 2.3 MPa

pressure. This behaviour has been reported previously for both powder samples

(Rarnia and Armstrong, 1985) and single crystal samples (Armstrong and Ramia,

1985) of K2OsCI6. The behaviour has been explained in terms of the strains

produced in the samples by the presence of point defects and dislocations

(Stoneham, 1969).

Uniaxial Pressure Effects in the Cubic ... Chapter 8 - Page 1 0

Mag 10 - I I I I I I 1

1 1 1 1 l 1 1 1 1 1 1 1 1 1 ~ 1 1 1 1 l l ~ l l l l l ~ l l l l l l

-20 -15 -10 -5 O 5 10 15 20 v - v, (kHz)

Fig 4. Typical example of an inhomogeneously broadened 3 5 ~ 1 resonance line as

obtained by Fourier transformation of a free induction decay signal. The dashed

lines are the symmetric Lorentzian components; the solid line their vector sum.

The decomposition of the asymmetric lines required five independent

parameters, namely the central frequency and the line width of each cornponent,

and their relative intensity. Both pure Lorentzian line shapes, and combined

Lorentzian-Gaussian line shapes were tried. In no case was the fit improved, or

the parameters changed significantly, by adding a Gaussian component. It was

concluded that the observed line shapes could be decomposed into two

Lorentzian components consisting of a main line and a satellite shifted to a

lower frequency. The dashed lines in figure 4 are the two symmetric

Uniaxial Pressure Effects in the Cubic. .. Chapter 8 - Page I 7

components and the solid line is their vector sum.

The fact that the component lines can be represented by Lorentzian

functions indicates that the inhomogeneous broadening in the present case is

dominated by the presence of point defects in the crystals (Stoneham, 1969).

This result is in contrast with a previous experiment (Armstrong and Ramia,

1985). In that case, the two component lines were of predominantly Gaussian

character, indicating that the broadening was dominated by dislocations in the

crystal. This difference illustrates that for a given crystal either point defects or

dislocations can provide the dominant broadening mechanism. NQR is a sensitive

probe of crystal imperfections.

Fig. Sa

0.5 I I I I I -

m O O A 9

0.0 - -

(1 00) 0 Mainpeak

Figure 5 is a plot of

the peak frequencies of

the main and satellite lines

of the 35CI NQR spectra as

a function o f applied

uniaxial pressure for both

crystals. Figure 5 gives

the results for the (1 00)

and (1 1 1 ) crystals. The

lines through the data are

least squares fits. We see

that in each case there is

a small, but measurable,

shift of frequency with

pressure. The shift is to

lower frequencies for the

(1 1 1 ) crystal, and to higher frequencies for the (1 00) crystal. Note the high

intrinsic accuracy of NQR frequency determinations. The maximum observed

- Satellite Iine - m

9 - O f T - 9 a &

O 9 O L - - m

1 1 1 b 1 I I I 1 1

O 2 4 6 8 10

changes are about 400 Hz in 16.9 MHz, which is less than 1 part in 40,000.

Previous measurements of the effect of hydrostatic pressure on the 3sCI

NQR frequency in a powder sample of KzOsCls (Armstrong and Baker, 1970)

revealed a decrease in frequency with increasing pressure. A value of ( a ~ / d P ) ~ =

-0.7(5) kHz MPa-1 was reported. The explanation presented was that the

application of pressure tends to destroy the n-bonding within the CI-Os bonds.

On the basis of this result, a frequency variation for the 0-9 MPa range of about

600 Hz is predicted. In the present case changes of about 400 Hz were

observed for the (1 11) crystal and of about 100 Hz for the (1 00) crystal.

The difference in frequency shifts for the two crystals can be explained

by taking account of the effects of a uniaxial pressure (Zamar and Bninetti,

Fig. Sb

0.5 I I I I I '

0.0 [ -- -

O 2 4 6 8 10 P (MPa)

- (111) - . C

- l w o ~ ~ I Satellite Iine ' ' * I 1 a I 1 I 1

1988) which produce a perturbation in the electric field gradients which is a

combination of contractions and dilations in different crystalline directions. For

the (1 00) crystal, the 3W.I nuclei associated with the CI-Os bonds lying along the

direction of the applied pressure are not observable in the NQR experiment.

These bonds would experience the maximum effect of the pressure with the

consequence that the n-bonding and associated averaged electric field gradient

at the 3sCI sites would be expected to decrease. On the other hand, the CLOS

bonds perpendicular to the direction of the applied pressure would be

significantly less affected by the pressure. Further, it is possibk to imagine that

in this case the sign of the effect would be revened, resulting in a small increase

in frequency with pressure.

Stoneham (1 969) has presented a review of the calculation of the shapes

of inhomogeneously broadened resonance lines in solids. These calculations

permit a cornparison of predicted line shapes and experimental measurements

for each of the various broadening mechanisms. In the continuum approximation,

and for a random distribution of defects, three different cases are identified. For

strain broadening by straight-edge or screw dislocations the line shape is given

where E is a norrnalized frequency, and a and b are parameters which describe

the dislocation characteristics. If b-O, the line shape is a Gaussian; othemise the

line shape decrease faster than a Gaussian in the wings. For broadening by

random electric fields the line shape is given by

Uniaxial Pressure Effects in the Cubic.. . Chapter 8 - Page 1 4

This is the Holtsmark distribution which is intermediate between a Lorentzian

and a Gaussian. For broadening due to point defects by random strains and

electric field gradients the line shape is given by the Lorentzian

I(E) = ( 2 7 ~ ) ~ ' $ dx exp(i&x) exp( - alxl)

with a the half width.

In order to explain the presence of a satellite line (Stoneham, 1969) it is

necessary to consider an inner region surrounding each of the centres

responsible for the line broadening and in which the lattice is treated as discrete,

as well as an outer region in which the continuum approximation is valid. When

this is done, the final expression shows that I(E) consists of a main line on which

is superimposed one or more satellite lines of the same shape but with different

intensities as determined by details of the defects present.

Since the component lines of the observed profiles were Lorentzians, it

follows that the cause of line broadening for the crystak studied is the presence

of point defects. The results further show that the distribution of defects is

altered by the application of the applied stress. The effect can be quantified by

introducing the relative frequency shift

where v l is the frequency of the line centre of the main resonance and vz is the

frequency of the line centre of the satellite. Values of A for no external stress

and 8.8 MPA are presented in Table 1. The difference in the values of A for no

external stress is a result of differences in the two crystals. However, it is also

seen that the effect of applied stress is much larger for the (1 11) crystal than

for the (1 00) crystal. Since depends not only on the positions of the defects,

but also on the set of relevant interna1 variables that characterize it, any

attempt to relate this parameter to a particular variable is unlikely t o be

profitable.

Table 1: Relative frequency shifts A between the main resonance and the

satellite line.

Figure 6 gives plots of the change in the relative intensity, A2/Ai, of the

satellite and the main peaks with applied pressure for the (1 00) and the (1 1 1 )

crystals. When comparing the two intensities a correction of about 5% was

performed due to the small differences in line widths or T2' values (Mintz and

Armstrong, 1980). The correction applied is

where the subscripts 1 and 2 refer to the high and low frequency peaks,

respectively, and b = 36 us is the delay time after the n/2 pulse. The solid lines

are least squares fits.

We see from Figure 6 that A21Al for the (1 11) crystal decreases

significantly as the applied pressure is increased, whereas the ratio is essentially

independent of pressure for the (100) crystal. All other things being equal, we

Uniaxial Pressure Effects in the Cubic. .. Chapter 8 - Page 7 6

expect A21AI for the (1 1 1 ) crystal to change more rapidly with pressure than for

the (1 00) crystal. The reason is probably the same one that explains the larger

change in frequency for the (1 11) crystal as compared to the (100) crystal.

These changes are intimately connected to the alteration of the defect

distribution.

0.7 8 a 1 1 1 - Fig 6. Change in the I

relative in tensities, A2/A i , - I of the satellite and main

lines as a function of the . 0.2 - appiied uniaxial pressure.

The solid lines are least r

squares fits.

0.01 ' i I l 1 i I i 1 i

O 2 4 6 8 10 P(M Pa)

But al1 other things are not equal for these two crystals as can be seen

from a comparison of the A21AI values for zero applied pressure. The values are

dramatically different. This fact indicates that the point defects are quite

different for the two crystals. This is somewhat surprising since the crystals

were produced in the same laboratory using the same procedure. The result

therefore suggests that the relative intensities of the two symmetric

components of the asymmetric line is very sensitive to the exact nature,

number, and/or distribution of the point defects.

8.4. SPIN-SPIN RELAXATION TlME MEASUREMENTS

Spin-spin relaxation times, Tz, were deduced from the data acquired using

the spin echo sequence. The decay of the spin echo amplitude as a function of

the delay time .r yielded a Gaussian from which the value of T2 was obtained. A

representative decay is shown in Figure 7. It was obtained for an applied

pressure of 8.8 MPa. The spin echo amplitude is plotted as a function of T from

which a T2 value of (940 f 20) ~s is deduced from the expression

Fig 7. A representative 3sCl decay signal as obtained following a spin-echo

sequence. The Gaussian fit to the data yields the T2 value.

Uniaxial Pressure Effects in the Cubic. .. Chapter 8 - Page 7 8

Fig. 8

Figure 8 is a plot of

T2 as a function of the

applied pressure. The solid

lines are the average

values. We see that the

values o f T2 are

independent of pressure

for the two samples.

Because the 35Cl NQR

frequency changes with

pressure, one might

expect that T2 would also

change with pressure.

However, the change of

frequency is very small

and is only observable because the high intrinsic NQR sensitivity. In comparison,

the sensitivity t o changes in Tz is low. Therefore. there is no inconsistency

presented by the apparent constancy of the T2 results. The averaged values of

Tt obtained for the two crystals are as follows: (937 f 20) ps for the (100)

crystal and (890 f 20) us for the (1 11) crystal.

Martin and Armstrong (1 975) reported T2 values in a powder sample of

K2OsCI6 in the temperature range 45 to 300 K. No temperature dependence was

apparent and an average Tz value of 1017 ps deduced. This result was

compared to an estimated ngid lattice second moment based on Van Vleck's

(1 948) formula and using equations given by Abragam and Kambe (1 953) and

Kano (1 958) assuming a dipole-dipole interaction with the surrounding nuclei.

Uniaxial Pressure Effects in the Cubic... Chapter 8 - Page 19

The Os-CI distance was taken to be 1/4 of the lattice parameter and the

calculation included up to third nearest neighbours. The resultant value for T2 is

1060 us.

We have repeated the calculation using recently reported structural

parameten (Prado, Armstrong and Powell, 1995) and employing a computer

program to include enough neighbouring nuclei to ensure four significant digits in

the relaxation time output. A value of TZ = 899.1 ps is obtained.

Second moment calculations were carried out for single crystal samples

following the procedure described by Kano (1 958), based on the symmetry of

the K20sCI6 cubic antifluorite structure. These calculations yielded Tt values of

885.6 ps and 930.3 ps for the (1 11) and (100) orientations, respectively.

These results are in excellent agreement with the experimentally measured

values.

8.5. SPIN-LATTICE RELAXATION TlME MEASUREMENTS

Spin-lattice relaxation t h e (Tl) data were obtained from the data

acquired using an inversion recovery sequence. TypEal semi-logarithmic plots of

the magnetization as a function of delay tirne r are given in Figures 9 and 10.

Data taken for O and 7.0 MPa applied pressure for the (1 11) crystal are

shown in Figure 9. These data indicate a pure exponential decay described by

time constants Tl = (1 53 f 3) ms and (1 51 + 2) ms, respectively. This result is

in agreement with previous measurements (Martin and Armstrong, 1975).

Similar behaviour is obsewed at al1 other pressures. The Tl data are independent

of applied pressure; the average value is calculated to be Tl = (1 52 I 3) ms.

i n[(M- B)/A]

Fig 9. Typical semi-logarithmic decay plots obtained for the ( 1 7 1 ) ctystal

fol10 wing an inversion recovery sequence. The least squares lines yield values of

Data taken for O and 7.8 MPa applied pressure for the (100) crystal are

shown in Figure 10. These data are reminiscent of those obtained previously

from powder and single crystal samples of KzOsC16 (Martin and Armstrong,

1975; Armstrong, Ramia and Morra, 1986). They may be characterized by a

double exponential fit of the form

Uniaxial Pressure Effects in the Cubic ... Chapter 8 - Page 2 1

where B includes both M(z =-) and background contributions, and subscripts L

and S refer to the long and short components of the decay, respectively. The

parameters AL and A~ are measures of the number of nuclei relaxing with time

constants T 1 ~ and Tl S. respectively. The Tl L values were fitted first and then

fixed to fit the Tls values. A visual inspection of Figure 10 suggests that the TiL

and Tis values at the two pressures are quite sirnilar, but that the ratio As/AL is

different.

l n[(M- B)/A, ] I 1 I 1 I I 1

9 - . - -

-3 - - 9

O 100 200 300 400 500 600 (ms)

Fig 10. Typical semi-logarithmic decay plots obtained for the ( 1 00) crystal

following an inversion-recovery sequence. These data are fit by a double

exponen tial function and yield two time constants, T l s and Tl L.

P (MPa)

Figure 11 is a plot

of the T 1 ~ and Tl s values

as a function of the

applied pressure as

obtained for the (100)

crystal. These data

indicate that the Tl values

are independent o f

pressure; the lines

represent the average

values, namely Tl L =

(141 f 3) ms and T1s =

(5.2 + 0.5) ms.

Figure 12 is a plot of the ratio As/AL as a function of the applied pressure.

The data indicate a significant decrease in the ratio over the pressure range

studied. They can be represented by a straight line with slope ~(2 .4 f 0.l)xlO-2

MPa-1. The solid line is the least squares fit to the data.

Previous work has shown that chlorine relaxation in K20sC16 is dominated

by fluctuations of the electric field gradient having the symmetry of the rotary

lattice mode, and that these fluctuations occur on two very different tirne scales

in localized regions of the sample (Armstrong, Ramia and Morra, 1986). The TiL

component is the response to fluctuations on the time scale of the lifetime of

normal-mode excitations, whereas the Tl s component is the response to

Uniaxa/ Pressure Effects in the Cubic.. . Chapter 8 - Page 23

fluctuations on the time scale of the lifetime of the dynamic clusters that are

the precursor to the structural phase transition that occurs at a lower

temperature. The ratio As/AL is a measure of the relative number of nuclei

associated with dynamic

clusters at 78.0 K. The

fact that this ratio

decreases with increasing

pressure suggests that

pressure tends t o inhibit

the formation o f the

dynamic clusters much as

does the addition of

impurities (Prado,

O - 1 Armstrong and Powell,

1995). The fact that the

0.0 data for the (1 1 1 ) crystal O 2 4 6 8 10

can be represented by a P (MPa)

single exponential, which is

Fig. 12 equivalent to saying that

As is zero, indicates that

there is a significant difference in the point defects in the two crystals studied,

and that in the case of the (1 11) crystal dynamic clusters have been totally

suppressed by the defects at 78.0 K.

The observation that the various Ti values measured are al1 independent

of applied pressure suggests that the changes which must occur are too small to

be detected with the sensitivity characteristic of this type of measurement.

Uniaxial Pressure EfKects in the Cubic.. . Chapter 8 - Page 24

8.6. CONCLUSIONS

A new design of uniaxial pressure cell has been presented and applied to

the measurement of the 3sCI NQR in single crystal samples of K20sC16 at liquid

nitrogen temperature.

Two single crystals were cut so that a uniaxial pressure could be applied

perpendicular to (1 00) and (1 1 1 ) faces, respectively. Free induction decay

signals, and spin-spin and spin-lattice relaxation measurements were carried out

at seven different applied pressures in the range from O to 9 MPa.

The free induction decay times, T2*, decrease linearly with pressure due

to the additional strains introduced by the application of the uniaxial pressure.

The free induction decay signals were Fourier transformed to generate spectra in

the frequency domain. In each case the spectra consisted of a single asymmetric

line which was successfully decomposed into two Lorentzian lines. This result

implies that the inhomogeneous broadening of the NQR signals is dominated by

the presence of point defects. The central frequency of the main component is

affected slightly by the pressure and by an amount that is consistent with the

results of previous measurements of K20sC16 samples under hydrostatic

pressure. The ratio of intensities of the two component peaks is different for the

two samples a t zero applied pressure. This is strong evidence that the nature,

number, and/or distribution of the point defects is different in the two crystals.

The different pressure dependence of this ratio is qualitatively undentandable.

The spin-spin relaxation behaves as expected for a K20sCI6 rigid lattice,

and the relaxation times Tz, are independent of pressure. This is not surprising

given the magnitude of the applied pressures and their small affect on the

Uniaxial Pressure Effects h the Cubic. .. Chapter 8 - Page 25

resonance frequencies. Second moment cakulations yield Tz values in excellent

agreement with the measured results for the (1 1 1 ) and (1 00) orientations.

The spin-lattice relaxation behaves dramatically different for the two

crystals; it is describable by a single exponential for the (1 1 1 ) crystal and by a

double exponential for the (100) crystal. However, in previous experiments on

single crystal samples of K2OsCI6 both types of behaviour have been reported.

The conclusion is that the two crystals used for the present experiments are

significantly different. This agrees with the conclusion reached from a

consideration of the line shape data. AI1 Tl values measured are independent of

pressure. This is as expected and is for the same reason that the T2 values are

independent of pressure. However, for the (100) crystal the amplitude ratio of

the signals decaying with the short and long relaxation times, Tl s and TI^,

respectively, decreases with pressure. R is believed that this is evidence that the

application of uniaxial pressure hinden the formation of dynamic precursor

clusters much as does the addition of impurities to a sample.

Abragam A and Kambe K 1953 Phys. Rev. 91 894

Armstrong R L 1 975 J. Magn. Reson. 20 21 4

Armstrong R L 1980 Physics Reports 57 343

Armstrong R L 1989 Magnetic Resonance and Related Phenornena, 24th Ampere

Congress Poznan 1988 Elsevier 54

Armstrong R L 1989 Progress in NMR Spectroscopy 21 151

Armstrong R L and Ramia M E 1 985 J. Phys. C 1 8 2977

Armstrong R L and Baker G L 1970 Can. J. Phys. 48 241 1

Armstrong R L and van Driel H M 1975 Advances in Nuclear Quadrupole

Resonance Vol. 2, Heyden and Son Ltd., New York. 179

Armstrong R L, Krupski M and Su S 1990 Can. J. Phys. 68 88

Berlinger W and Müller K A 1977 Rev. Sci. Instrum. 48 1 161

Kano K 1958 J. Phys. Soc. Japan 13 975

Krupski M and Armstrong R L 1989 Can. J. Phys. 67 566

Martin C A and Armstrong R L 1 975 J. Magn. Reson. 20 41 1

Mintz J D and Armstrong R L 1980 Can. J. Phys. 58 657-663

Prado P, Armstrong R L and Powell B 1995 Can. J. Phys. 73 626

Ramia M E and Armstrong R L 1985 Can. J. Phys. 63 350

Stoneham A M 1969 Rev. Mod. Phys. 41 82

Van Vleck J H 1948 Phys. Rev. 74 1 1 68

Zamar R C and Brunetti A H 1988 Phys. Stat. Sol. B 150 245

Zamar R C, Brunetti A H and Pusiol D J 1983 J. Mol. Struct. 1 1 1 171

NQR Study of the Cubic to Tetragonal-..Chapter 9 - Page 7

CHAPTER 9

NUCLEAR QUADRUPOLE RESONANCE STUDY OF THE CUBlC TO

TETRAGONAL PHASE TRANSiTlON IN

K~ osa6: UNIAXIAL PRESSURE EFFECTS

Pablo J. Prado and Robin L. Armstrong

Fredericton, NB

Canada E3B SA3

Reference: Accepted for publication. Prado P J and Armstrong R L October 1996

Cm. J. Phys.

Journal of Physics publishers.

NQR Study of the Cubic to Tetragonal. ..Chapter 9 - Page 2

3sCl Nuclear Quadrupole Resonance measurements are reported on two single

crystal samples of KzOsC16 subjected to uniaxial pressure. The crystals are cut

to allow the stress to be applied in the (1 00) and (1 1 1) directions. Lineshape

and spin-lattice relaxation data are obtained over the temperature range 30 to

57 K spanning the transition temperature for the cubic to tetragonal structural

phase transition. Since the Iineshapes are well represented by Lorentzians it can

be concluded that the source of line broadening is the presence of point

defects. The application of pressure causes a shift in the central frequency of

the main resonance line in the cubic phase. From the analysis of the results it is

concluded that the crystal dynamics are unchanged by the application of a srnall

uniaxial stress; the observed changes are due entirely to variations in the static

components of the electric field gradient. The application of pressure does not

affect the frequencies of the lines in the tetragonal phase. The application of

pressure in the region of coexistence of the cubic and tetragonal phases

enhances the fraction of tetragonal phase present at a particular temperature.

The spin lattice relaxation results are discussed in terms of a Raman two phonon

model.

9.1 . INTRODUCTION

Antifluorite compounds have been the subject of a large number of

nuclear quadrupole resonance (NQR), neutron scattering, and optical

spectroscopy experiments [l-51. An important aspect of this work has been the

study of the dynamic processes which drive the cubic (O$) to tetragonal (C&)

structural phase transition. For K20sC16 and K2ReCI6 the relevant dynamics

relates to the softening of the r-point rotary-lattice mode accompanied by the

development of tetragonal phase precunor clusten. The resultant structural

alteration consists of a ferro-rotation of the equilibrium positions of the MX6

octahedral units, and a small tetragonal distortion of the unit cell [6-81.

The r-point rotary-lattice mode in RzMX6 crystals is neither Raman nor

infrared active in the cubic phase 191. However, X- nuclear relaxation

measurements using the NQR technique have provided evidence for the

softening of this mode, as well as for the occurrence of the precunor clusten,

as the transition temperature, Tc, is approached from above.

NQR is a remarkably sensitive technique for the study of structural

changes in crystals through modifications to local electric field gradients (EFG's).

The antifluorite K20sC16 is the compound of choice for chlorine NQR

investigations because of its relatively namw linewidth and associated high S/N

ratio. The primary interest in studies of K20sC16 is not in the compound itself,

but rather is in the information that may be gleaned on the mechanism

responsible for structural phase transitions in this class of matenals.

NQR Study of the Cubk to Tetragonal. ..Chapter 9 - Page 4

In the past few years several experiments have been canied out to try to

understand the influence of intrinsic and extrinsic factors involved in driving the

structural phase transition in K20sC16. In order to study the influence of

impurities, neutron diffraction [IO] and 35CI NQR measurements [l 11 in

polycrystalline samples of K2OsCI6 doped with PtCI6, kCI6 and ReC16 were

undertaken. These studies have shown that the structural parameters, the

critical temperature for the phase transition, and tetragonal phase precursor

cluster formation are altered in the presence of a controlled concentration of

substitutional ions. Because the values of the parameten are dependent on the

sample preparation process and on the purity of the starting components, the

measurements were taken on samples prepared in the same manner, from the

same batch of K20sC16.

A JsCI NQR study of the effect of uniaxial pressure on single crystals of

K20sCI6 has been reported [12]. This work provided evidence of directional

dependent responses which complements the information obtained from

experiments in which hydrostatic pressure was applied to KzOsCI6 samples [13,

In the present work lineshapes and Tl relaxation times are measured in two

single crystals of KzOsC16 as a function of both uniaxial stress, and temperature,

in the region of the structural phase transition. The crystals were cut to allow

the stress to be applied in the (1 00) and (1 11 ) directions, respectively. We

believe this to be the first NQR study of the effect of uniaxial stress on an

antifluorite crystal in the vicinity of a structural phase transition.

NQR Study of the Cubic to Tetragonal..Chapter 9 - Page 5

9.2. EXPERIMENTAL

A cryostat constructed for conducting NQR experiments under uniaxial

pressure was described in a previous work [12]. Temperatures below 78 K are

achieved by blowing cold helium gas through a brass spiral channel surrounding

the vesse1 [13]. The helium pressure is controlled; the dewar pressure is

increased by pumping helium gas into it through a set of valves. A resistor which

is wound around the brass spiral, and through which an electrical current is

driven, is used to set the temperature a t the sample site. A Lake Shore

controller is employed to establish and regulate the temperature. The heat

exchange unit is located in a liquid nitrogen dewar.

Uniaxial stress is applied to the crystals [12] by means of two polished

glass cylinders. The magnitude of the force exerted is determined directiy from

the mass of weights used.

The coi1 is positioned at the centre of the chamber and the rf voltage is

fed to the coi1 by a 50 f2 coaxial line. The temperature is monitored to within

0.1 K by means of a copper-constantan thermocouple located about 2 mm from

the sample. Copper-constantan thermocouples placed at the top and bottom of

the chamber were used to detect temperature gradients across the chamber;

these were always less than 0.1 K over 10 cm.

Two crystals, grown from HCI aqueous solution by S. Mroczkowski at Yale

University, were cut to exhibit parallel faces perpendicular to the (1 00) and

(1 11) directions, respectively; they will subsequently be referred to as the

(100) and (1 11) crystals. The crystal volumes were 45 and 41 mm3,

respectively.

NQR Study of the Cubic to Tetragonal...Chapter 9 - Page 6

35CI NQR spectra of the KZOsCl6 crystals were taken using a Tecmag

Fourier transform pulse spectrometer. Measurements were carried out at

atmospheric pressure and at 5 MPa uniaxial applied pressure for a series of

temperatures between 28 and 55 K. Temperature stability was better than 0.2

K. Each time that the temperature was changed, the system was left to

equilibrate for more than one hour before measurements were taken. By doing

so, previously noted non-equilibriurn phenornenon [15] were eliminated.

Free induction decay (FID) signals following n/Z pulses were averaged

over 300 acquisitions to obtain adequate S/N ratios. Spin- lattice relaxation

spectra recorded from n-~-n/Z inversion-recovery sequences were averaged

over 100 acquisitions. Tl values were deduced from fits to the maximum FT

signal amplitude as a function of delay time. When more than one relaxation

component was present, special care was taken in the choice of the delay tirnes

in order that the different tirnes scales were adequately covered.

9.3. RESULTS

9.3.1 LINESHAPES

At temperatures above Tc a single resonance line is expected since in the

cubic phase al1 chlorine nuclei occupy equivalent positions in the lattice. The

peak width is determined by spin-spin relaxation processes and the intrinsic

characteristics of the sample, including the presence of impurities and crystal

strains that arise because of point defects and dislocations in the lattice. The

presence of crystal imperfections can result in the appearance of satellite lines

11 61.

Fig. 7. Cubic phase Fourier transform amplitude spectra at 53 K. The dashed

lines are Lorentzian components which sum to yield the solid lines through the

da ta.

Figure 1 shows cubic phase Fourier transform amplitude spectra acauired

at 53 K for the two crystals under atmospheric pressure. It is seen that the

resonance lines exhibit an asymmetry; this feature remains after appfying

external stress. The spectra are in each case well represented by two

Lorentzians with the low frequency cornponent shifted by 1.5 kHz from the main

resonance and having an amplitude approximately 1 /5 of that resonance. Table

1 lists the FID time constant, T2*, and central frequencies, VQ, o f the main

resonances for both samples and at both atmospheric pressure and 5 MPa

applied uniaxial pressure. The results show that T2* is higher for the (100)

crystal than for the (1 1 1 ) crystal, and that Tz* decreases for both crystals as

stress is applied. The VQ value is higher for the (1 00) crystal than for the (1 1 1 )

crystal. As pressure is applied VQ decreases for the (1 1 1 ) crystal and increases

for the (1 00) crystal. Sirnilar behaviour was previously observed at 78 K for the

same two crystals [12].

Table 7 . T2* values and resonance frequencies at 53 K.

Evolution of the peak frequencies, VQ, with temperature in the cubic

phase is presented in Figure 2 for both samples; v, is the operating frequency of

the spectrometer. The data for the (1 00) crystal are shown in the main body of

the figure; six temperatures are measured, both a t atmospheric pressure and at

an appiied pressure of 5 MPa. The application of stress at each temperature

causes the frequency to increase, but the temperature derivative of the

frequency, ( dv~ /dT )~ is independent of pressure. The data for the (1 1 1 ) crystal

are shown in the insert; only two temperatures were measured. In this case the

application of stress causes the frequency to decrease, but again, the

temperature derivative of the frequency appean to be pressure independent.

At temperatures below Tc two resonance lines are expected. In the

tetragonal phase those nuclei that lie along the axis of distortion experience a

different EFG than those in a plane perpendicular to the axis of distortion.

NQR Study of the Cubic to Tetragonal...Chapter 9 - Page 7 0

O P = OMPa

P = 5 MPa

Fig. 2. Temperature variation o f the peak frequencies, VQ, of the main

components of the cubic phase spectra, with and without pressure; vo = 7 6.884

MHz. Solid lines represent least square fits to the data in the (700) case. Data

points for the ( 1 1 1 ) crystal are represented &y straight lines of equal slope.

The observed spectra conoist of two lines, namely the z- and the xy-lines;

they are affected by a second order phase shift. Phase corrections were

performed over the individual components, after which lineshape parameters

were deduced from a least squares Lorentzian fitting procedure.

NQR Study of the Cubic to TetragonaL..Chapter 9 - Page 7 1

Evolution of the peaks frequencies, VQ with temperature for both samples

in the tetragonal phase is shown on Figure 3. The data suggest that the

frequencies of the tetragonal phase cornponents are, within experimental error,

unaffected by the application of the uniaxial pressure. However, it is important

to note that the experimental errors associated with these frequency

determinations are larger than the differences in frequency observed with and

without pressure, for the samples in the cubic phase.

v, (MHz)

Fi@ 3. Temperature

variation of the

peak frequencies of

the z- and xy-lines

for the samples in

the tetragonal

phase, with and

without pressure.

The solid lines are

fits to the function

A(B - r ) 1 4

NQR Study of the Cubic to Tetragonal. Xhapter 9 - Page 72

T2* values for the tetragonal phase are deduced from the Iinewidths

obtained by fitting the observed profiles. The results are displayed in Figure 4.

For both sarnples the xy-components are approximately twice as broad as the z-

lines; hence, the T2* values are about one-half. This behaviour has been

observed before [17]. The data suggest that as the ternperature is decreased

below Tc, the Tz* values at first decrease and then approach a temperature

independent value. Uniaxial pressure applied to the (1 00) crystal causes a

systematic reduction in the T2* values for both components. No definitive

pressure effect can be deduced for the (1 1 1 ) crystal.

There exists a region of about 3 K about Tc where the cubic and

tetragonal phases coexist and where small changes in either temperature or

applied pressure have a large effect. In this region the lines overlap and it is not

possible to determine the resonance frequencies of al1 of the peaks from a

multiple line fitting of the FT spectra. However, the frequencies can be

determined by using the Maximum Entropy Method as previously described [Il].

Spectra for the (100) crystal at two temperatures in the coexistence

range, 46.5 and 47.7 K, are shown in Figure 5. The spectra a t each temperature

are norrnalized so that the total area under the spectra, with and without

applied pressure, are the same. The spectra are very different at the two

temperatures, and the application of uniaxial pressure is seen to have a dramatic

effect at both temperatures. It is very apparent that the relative amplitudes of

the cubic phase component and the z-component of the tetragonal phase are

dramatically affected by the pressure. It is also seen that the frequency of the

xy-component of the tetragonal phase shifts to a higher value and the

resonance line is broadened with the application of pressure in this small

temperature range; this is particularly evident in the spectra at 46.5 K.

NQR Study of the Cubic tu Tetragona l... Chapter 9 - Page 13

120 P = S MPa

Fig. 4. Temperature dependence of Tt* values for the z- and xy-lines for the

samples in the tetragonal phase, with and without pressure. The solid lines are

guides to the eye.

In order to check if the application of uniaxial pressure is a revenible

process, the spectra shown in Figure 5 were chosen because of their sensitivity

to pressure. At each temperature, the pressure was applied and released four

NQR Study of the Cubic to Tetragonal. ..Chapter 9 - Page 7 4

times. No noticeable changes were observed in the atmospheric pressure

spectrum or the spectrum taken for an applied pressure of 5 MPa. It was

concluded that the priicess was reversible.

Fig. 5. Spectra for the ( 7 00) crystal at two temperatures in the coexistence

region, with and without pressure.

9.3.2 SPIN-LATTICE RELAXATION

In an earlier study of the (1 00) crystal at 78 K 1121 it was found that the

spin-lattice relaxation contained two components. In addition to the usual long

cornponent due to the coupling of the fluctuating electric field gradients with

the chlorine nuclear quadrupole moments, a short component was alto present.

This component was associated with the formation of tetragonal phase dynamic

clusters.

Here we report measurements of the spin-lattice relaxation for the same

(1 00) crystal in the vicinity of Tc as a function of temperature, with and without

applied uniaxial pressure. The data were obtained using an inversion-recovery

sequence. The relaxation parameten were determined by fitting the maximum

FT signal, M(T) as a function of the delay time, :

where the subscripts L and S refer to the long and short cornponents of the

decay and B is the signal a t T >> TiL. Delay time tables were constructed for

the inversion recovery sequence which covered the time scales of both the long

and the short components.

An example of the variation of the maximum of the FT signal as a function

of the delay time is shown in Figure 6 for the crystal at 49.0 K and under

atmospheric pressure. The data for the long T values were fitted first. Then,

after fixing the resultant parameters, the contribution to the short delay values

NQR Study of the Cubic to Tetragonal. Xhapter 9 - Page 1 6

was obtained. The results of the least square process are shown by the solid

l n ((M- B)/AL)

o.5 5 7 Fig. 6. Variation of the

The temperature dependencies of both TIL and T 1 ~ for P = O and P = 5

MPa are presented in figure 7. The results for P = O are consistent with those

reported previously [18]. A clear dependence on pressure is seen. Figure 7(a)

indicates that both the slope and the intercept of the linear fit to the TILT2 vs T

data are altered; figure 7(b) shows the same tendency for the TisT2 vs T data.

The relative contribution of the short component to the long component

(As/AL) exhibits, at most, a weak dependence on temperature at both P = O and

-3 .O -3 -5

P = 5 MPa. The application of stress significantly reduced the contribution of the

short component to the overall response, giving As/AL pSo~pa = 0.34 f 0.02 and

As/AL P-SMPa = 0.20 f 0.03.

O 50 100 150 200 4ms)

- - - - - - - - - - - O -

i I i i f i i 1 i i i ~ ~ f i ~ i ~ ~ ~ ~ ~

maximum of the FT signal

as a function of the delay

time for the sample at

49.0 K and atmospheric

pressure.

NQR Study of the Cubic to Tetragonal Lhapter 9 - Page 1 7

- O P = OMPa

- P = 5 MPa - LI

P = OMPa 1 P = 5 MPa

Fig. 7. a) Plot of TlLT2 as

a function of T for the

(100) crystal, with and

without pressure.

Fig. 7. b) Plot of Tl sT2 as

a function of T for the

(100) crystal, with and

without pressure.

NQR Study of the Cubic to Tetragonal. ..Chapter 9 - Page 7 8

Below Tc, in the tetragonal phase, the recovery curves are pure

exponentials. Because of the overlap of the xy- and Aines the parameters

deduced from the fitting have a large associated error. At least partly for this

reason no definitive pressure dependence was observed. The data obtained for

the z- and xy-lines of the (1 00) crystal for an applied pressure P = 5 MPa are

plotted in Figure 8; they are not distinctly different from one another. The T 1 ~

data taken for the cubic phase at the same pressure are also shown. The Tl

data taken together exhibit an obvious minimum at the temperature of the cubic

to tetragonal structural phase transition.

Fîg. 8. Plot of Ti as a fonction of T for all of the lines in the cubic and tetragonal

phases for a uniaxial pressure of 5 MPa.

9.4. DISCUSSION AND CONCLUSIONS

The lines in the cubic phase can be adequately represented by Lorentzian

line shapes (Figure 1 ). This was also the case for the experiment carried out at

78K on the same crystals [12]. It was concluded that the source of line

broadening is the presence of point defects [16]. The effect of the applied

stress can be quantified by using the relative frequency shift: AV/VQ = 7.1 x 1 0-6

and -8.3 x 10-6 for the (100) and (1 11) crystals, respectively. Since these

changes depend not only on the positions of the defects, but ako on a set of

interna1 variables that characterize the defects, it is not possible to relate this

parameter to a particular variable.

Lattice vibrations produce a fluctuating EFG at the chlorine nuclear sites.

The frequencies of these vibrations are much higher than the NQR resonance

frequencies. Therefore, the nuclei sense a temperature dependent average of

the fluctuating EFG. In the cubic phase the resonance frequency is observed to

increase in a linear fashion as the temperature decreases (Figure 2). This

behaviour has been reported before [19, 201 and explained using a theory

proposed by Kushida 1211. In the present experiment it is seen that the

parameters which describe the variation of the frequency with temperature

depend on the magnitude and the direction of the applied uniaxial stress.

For the (1 00) crystal the temperature derivative is essentially the same

at both pressures that were studied: ( d ~ ~ / d T ) ~ , ~ = -0.1 74(2) and

( ~ v ~ / Q T ) ~ ~ ~ ~ ~ ~ = -0.1 75(4) kHz/K. The frequency shift with pressure is

NQR Study of the Cubic tu TetragonaLChapter 9 - Page 20

(Av/AP)T = 0.024(4) kHz/MPa. In the case of the (1 1 1 ) sample, the frequency

shift with pressure is in the opposite direction: ( A v ~ A P ) ~ = -0.028 kHz/MPa.

This results agree with those found previously a t 78 K [12]. Linear changes in

frequency with applied pressui. lave been reported for NaC103 [22] and for

paradichlorobenzene [2 31.

Martin and Armstrong [20] analyzed the variation of frequency with

temperature in K20sCI6 using an approach proposed by Kushida [21] and

assuming the predominance of the rotary-lattice mode in the normal mode

expansion. They obtained the following expression for the temperature and

volume dependence of the frequency:

with a = (~lp/h) (d~/dP)~. In this equation, vs is the static value of V, <@(T)> is

the mean angular displacement of the Os-CI bonds from their equilibrium

position, ap is the isobaric volume thermal expansion, and & is the isothermal

compressibility. A relation between <@(T)> and the mode frequency proposed

by O'Leary (241, permits an estimation of the frequency of the mode

responsible for the temperature evolution. This calculation yields a reduction of

17% in a certain average of the rotary-lattice mode frequency over a 7 0 K range

above Tc and thereby provides evidence of the condensation of the rotary-

lattice mode phonons.

Additional considerations are required to account for the pressure

effects. Brunetti [25] has described a simple model to account for the

difference in the NQR frequency for different molecular arrangements. The

equation derived could apply to results obtained in different phases, or results

with and without applied pressure. The expression, based on the quasi-harmonic

NQR Study o f the Cubic to Tetragonal. ..Chapter 9 - Page 2 1

approximation, and assuming a high temperature approximation is:

with Av, the change in the crystalline EFG, A deterrnined by the change in

the hamonic terni of the interrnolecubr potential, and B a measure of the

change due to the anharrnonicities of the molecular tonional frequencies.

The present experirnent has shown that (dvdoT), is unchanged for an

applied stress of 5 MPa from its value at atmospheric pressure. This result

irnplies that tne application of an external stress has produced a significant

change only in the static component v,; the coefficients A and B are zero to

within experimental error. It can be concluded that the crystal dynamics are

unchanged by the application of a small uniaxial stress.

Zamar and Brunetti [22] have derived the following expression which

explicitly includes the different pressure configurations, y, in the NQR frequency

analy sis:

with Ay = dpd/6T, and where & and $yd are the static and dynamic

contributions, respectively. Since the experiments have indicated that the

pressure derivative of frequency is independent of temperature, it can be

concluded that Ar = O; only the static contribution ByS is significant to the

change resulting from the application of a srnall uniaxial stress. Evidence of this

type of behaviour has been reported in other cubic crystals such as NaCIO, [26]

and NaBrO, [27].

The parameter is related to the stress tensor a and the strain tensor e,

through the expression $ys = 4, &Ski (daJaP)Y with Ski = (OE JaaJT. The

parameter Q, = [~~(T) /vJ(dv~/&~) reflects the variation in the rigid lattice

frequency, vs, in the presence of a particular stress. Depending on the

anisotropic nature of the tensors and the weights of the various components, grS

can be quite different for one direction of applied stress as compared to another

and indeed can even change sign, as in the present case.

In the tetragonal phase, due to the presence of chlorine nuclei in two non

equivalent sites, two resonance lines occur; they are the xy- and z-lines shown in

Figure 3. Based on previous NQR [28] and neutron scattering [29] experiments,

it is expected that the frequency splitting Av of the two lines should Vary as

(Tc - T)'IZ for (Tc - T)/Tc 5 0.1. The solid curves in the figure are fits of the

function A(B - T)l12 + C to the data for both the xy- and z- lines; they provide

an adequate description of the data. This behaviour reflects the temperature

dependence of the rotary-lattice mode. In principle, a detailed analysis of this

type of data can provide information on the Gruneisen tensor [22]; in the

present case there is insufficient data for such an analysis. Nonetheless, we can

conclude that the tonional frequencies are higher in the tetragonal phase and

that they exhibit a larger temperature dependence [25].

The application of uniaxial stress has produced no clear change in the

frequency response of either sample. It should be noted that the experimental

uncertainty in the frequency determinations is greater than for the cubic phase.

As a result of the splitting of the high temperature line the intensities of the

individual components are reduced, the peaks overlap and phase distortion

occurs. As seen from Figure 4 the tetragonal phase lines are wider than the

NQR Study of the Cubk to TetragonaLChapter 9 - Page 23

cubic phase lines, and with the addition of applied stress become still wider.

The lineshape in a small temperature range about Tc exhibits dramatic

changes with pressure for both crystals. Coexistence of the cubic and tetragonal

phases occurs over this range. The results may also reflect critical behaviour.

Figure 5 shows results obtained a t 46.5 and 47.7 K for the (100) crystal. The

application of uniaxial pressure is seen to increase the fraction of the tetragonal

phase that is present. The pressure defines a unique direction and therefore

promotes the development of tetragonal phase cells, in the present case with

their c-axes parallel to the coi1 axis. That this should happen for even a small

extemal stress in the temperature regirne where the two phases coexist, is not

surprising. However, the explanation of the shift in the frequency and the

broadening of the xy-line with pressure is not so obvious.

Measurements of definitive changes in the 3sCI NQR frequency and line

shape under the application of a modest uniaxial pressure to an antifluorite

crystal have been presented. The observed changes can be understood in a

qualitative manner. In principle, such measurements offer the potential of

providing quantitative information on the Gruneisen tensor.

9.4.2 SPIN-LATICE RELAXATION

The spin-lattice relaxation process was studied as a means to obtain

information on the uniaxial pressure dependence of the lattice vibrational modes.

It is known that the relaxation is driven by the anharmonic Raman process [30]

NQR Study of the Cubic to Tetragonal...Chapter 9 - Page 24

T , T ~ = A

with Tl a value of the rotary-lattice mode phonons averaged over the Brillouin

zone. The minimum at Tc in the plot of Tl vs T shown in Figure 8 is clear

evidence of the softening of the rotary lattice mode.

Figure 7(a) indicates that Tl, increases when a uniaxial stress is applied.

This is almost certainly a direct consequence of the increase in the torsional

oscillation frequency due t o the small reduction in the distance between the

OsCI, groups and the neighbouring K+ ions. For a Raman two-phonon model [3 1 ]

in which the soft mode is weakly damped

The parameter T, is not necessarily equal to the transition temperature Tc.

The Tl, data in Figure 7(a) are well represented by this equation. The

parameters deduced are A = (21 I 2) ms and T, = (41.6 f 0.8) K. The

application of a uniaxial stress of 5 MPa caused the parameters to change to A =

(30 I 2) ms and T, = (43.9 f0 .6 ) K. The fits are shown by the solid lines. The

large change in A (almost 50%) reflects the extreme sensitivity of the present

experiment to alterations of the derivatives of the EFG as a result of changes in

the lattice vibrational spectrum.

Previously reported data [18, 201 taken at atmospheric pressure on

single crystal and powder samples of KzOsCls yielded comparable but not

identical results. For the single crystal T,, data, A = 23 ms and T, = 37K; for

NQR Study of the Cubic to TetragonalLChapter 9 - Page 25

the powder data A = 22 ms and T, = 42K. This is further evidence that small

differences in the number and spatial distribution of impunties between samples

can affect the measured parameten.

Figure 7(b) is a plot of the Tl, data as a function of the temperature.

Because of the large errors present in the determination of the Tl, values, it is

not possible to extract accurate parameten. Nonetheless, it is clear that Tl,

increases when the stress is applied. Since both Tl, and Tl, are govenled by the

same basic rnechanism [18], it is t o be expected that they will both be affected

in a sirnilar manner by the application of a stress. Although the Tl, data can be

represented by the same form of linear equation as used for the case of the Tl,

data, the values of the parameter To are very different. Such was not the case

with the earlier data, and the reason for the discrepancy is not understood.

Finally, it should be noted that measurernents on the same (100) crystal

taken at 78 K gave no indication of a pressure dependence for either Tl, or Tl,.

This suggests that the mathematical forrn use to represent the present data

near Tc holds only over a limited temperature range above Tc.

The explanation of the reduction of the ratio AdA, with the application of

uniaxial pressure is that the stress acts to inhibit the formation of dynarnical

clusten much as does the addition of impurities [ l O]. Our previous study of this

same crystal at 78 K [12] showed that AdAL decreases in a linear fashion with

applied pressure. A comparison of the actual numerical values indicates that the

ratio is remarkably temperature insensitive from 49 to 78 K.

The Tl data taken in the tetragonal phase are much closer near Tc than

NQR Study of the Cubic to TetragonaLChapter 9 - Page 26

any previously reported. The fact that the present data (Figure 8) indicate that

Tl, Tl, near Tc is not inconsistent with the earlier result [18] that Tl, > Tl,

for T < 30 K. The former result was interpreted as evidence for a two-

dimensional character to the critical fluctuations. The same conclusion can be

extracted from the present results, but with the qualification that the two-

dimensional character is not as mrked near T,.

Spin-lattice relaxation measurernents are far less precise than frequency

measurements and therefore a less sensitive probe of subtle changes in the EFG

due tij either impurities or pressure. Nonetheless, the present Tl data have

indicated that the application of even a modest uniaxial pressure applied to an

antifluorite crystal produces measurable effects which in turn contain

information on the changes to the phonon spectrum. The link between Tl values

and the details of the phonon spectrum poses, however, an intractable problem.

NQR Study of the Cubic to Tetragona /... Chapter 9 - Page 27

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18. R.L. Armstrong, M.E. Ramia and R.M. Morra. J. Phys. C 19, 4363 (1 986).

19. R.L. Armstrong and G.L. Baker. Can. J. Phys. 48, 241 1 (1 970).

20. C.A. Martin and R.L. Armstrong. J. Magn. Reson. 20, 41 1 (1 975).

21. T.Kushida, G.B. Benedek and N. Bloembergen. Phys. Rev. 104, 1364

(1 956).

22. Zamar R.C. and A.H. Brunetti. J. Phys.: Cond. Matter 3, 2401 (1 991).

23. Zamar R.C. and A.H. Brunetti. Phys. Stat. Sol. b 150, 245 (1988).

24. G.P. O'Leary. Phys. Rev. Lett. 23,782 (1969).

25. A.H. Bmnetti J. Molec. Struct. 58, 5 1 3 (1 980).

26. D.D. Early, C.I. Stutz, S.F. Harley, D.C. Dening and R.F. Tipsword. J. Chem.

Phys. 62, 301 (1 975).

27. C.J. Whidden, C.D. Whilliams and R.F. Tipsword. J. Chem. Phis. 5 0, 507

(1 969).

28. M. Wiszniewska and R.L. Armstrong. Can. J. Phys. 5 1 , 781 (1 973).

29. J.D. Mintz and R.L. Armstrong. Can. J. Phys. 58 , 657 (1 979).

30. H.M. van Driel, M.M. McEnnan and R.L. Armstrong. J. Mag. Reson. 18, 485

(1 975).

31. G. Bonera, F. Bona and A. Rigamonti. Phys. Rev. B 2, 2784 (1 970)

Conclusions Chapter 1 O - Page 1

CHAPTER 10

SUMMARY AND CONCLUDING REMARKS

1 0.1 INTRODUCTION

The present work has reported the influence of impurities and pressure on

the cubic to tetragonal structural phase transition in K,OsCI,, which occurs a t

approximately 45K. Special emphasis is taken in relation to precursor effects and

in the softening of the rotary mode driving the displacive structural transition.

The study involved a series of neutron diffraction and NQR experiments. The

results are compiled in chapten 6 to 9 in the body of this thesis.

The importance of the topic is a result of the numerous systems in which

a sof? phonon mode provides the mechanism driving a phase transition.

Antifluorite compounds have been selected because they give rich information

about the details of the process. The static and dynarnic nature of the transition

is affected by intrinsic properties of the samples, in particular dislocations and

point defects which determine the configuration of strains within the sample.

Then, this project aims to explain the characteristic changes in the transition

after adding known concentrations of impurities and systematically applying

hydrostatic and uniaxial pressure to the samples.

A neutron diffraction experiment (chapter 6), an NQR hydrostatic

Conclusions Chapter 10 - Page 2

pressure experiment (chapter 7) on powder samples containing controlled

concentrations of substitutional impurities and two NQR axial stress studies on

single crystals (chapter 8 and 9) were carried out. Indidual sections present

the data and corresponding conclusion; they are not fully repeated here. A

summary of the main conclusions is presented as follows.

10.2 INFLUENCE OF IMPURITIES

The introduction of substitutional impurities in K,OsCI, produces changes

in the lattice parameter as seen by a neutron diffraction experiment (chapter 6).

Measured variations are consistent with the relative ionic sizes of OsCI, and the

added MCI, impurities. The Os-CI bond length is unchanged within experimental

error. In the cubic phase, the temperature dependence of the Debye-Waller

factors is described by a Debye model of the lattice vibrations and the addition

of impurities does not significantly change these dynamical parameters.

Nevertheless, the impurities do suppress the development of precursor

tetragonal clusters, which are present in the high temperature cubic phase.

As seen from the neutron diffraction profiles and NQR experiments, the

presence of impurities has a significant effect on the transition temperature and

in the range over which coexistence of the two phases occun. Tc is shifted

systematically to lower temperatures after adding P t and Ir impurities, and to

higher temperatures for Re impurities, in accord with differences in the ionic radii

that cause an alteration in the size of the lattice, as described by the rigid

sphere model (chapter 7). Furthermore, the addition of impurities produces a

broadening of the resonance cornponents which obscures the contribution to

the lineshape due to precursor dynamic clusters.

1 0.3 INFLUENCE OF STRESS

lncreasing the hydrostatic pressure applied to the samples causes Tc to

decrease as expected when the transition is dnven by the softening of a r-point

mode. Pressure also alters the splitting of the spectral components causing the

z-line of the tetragonal phase to merge with the cubic resonance line in the

region of coexistence (chapter 7).

The spin-lattice relaxation time increases when the axial pressure is

applied, indicating an increase in the frequency of the rotary lattice mode due to

the small reduction in the distance between the OsCI, groups and the

neighbouring K+ ions. The analysis of the relaxation parameten shows that the

stress acts to inhibit the formation of dynamical clusters much as does the

addition of impurities (chapter 9).

The study of the evolution of the resonance frequency with temperature

in the cubic phase allows us to conclude that the dynamic contributions are

unchanged by the application of a small uniaxial stress (chapter 9), while a

measurable change is produced in the static contribution to the resonance line.

The free induction decay times decrease linearly with pressure due to the

additional strain introduced in the lattice. lnhomogeneous broadening of the NQR

signals is observed and shown to be dominated by the presence of point

defects. For the two single crystals studied, differences in the intensity ratios of

the two components of the resonance line, even when pressure is not applied, is

reported (chapter 8); this indicates a sample dependence of the strain

configuration.

In the tetragonal phase, a larger temperature dependence of the

resonance frequency is noticed and the application of axial stress produces no

observable change in the frequency response. As a result of the splitting of the

high temperature cubic line the intensities of the individual components are

reduced, while phase distortion occurs. The tetragonal phase linewidths are

larger than the cubic phase ones, and with the addition of applied stress they

become even wider, making impossible the detection of small changes in the

frequency values.

In the ïegion of coexistence of the cubic and tetragonal phases, the

additional stress produces a dramatic change of the relative magnitude of the

three components. An increase in the fraction of tetragonal phase present is

observed.

10.4 INHERENT FACTORS

The spin-lattice relaxation time behaves differently for the two studied

single crystals studied (chapter a), presenting a unique relaxation component in

one case and a double exponential decay in the other. The additional short

relaxation component is associated with the dynamical tetragonal clusten. It is

concluded that the dynamics of the response is driven by a non-controlled strain

configuration determined during the sample preparation process. This is

supported by the lineshape analysis mentioned above.

Since the contribution of the short relaxation tirne vs the long one

decreases with the axial stress, it is believed that the application of uniaxial

pressure hinden the formation of dynamic precunor clusters much as does the

addition of impurities to a sarnple.

An orientation dependence of the spin-spin relaxation time was found. A

second moment calculation is presented (chapter 8) in order to estimate the

relaxation parameten. The predicted values are in excellent agreement with the

results, indicating that the differences found between the two samples are

based solely on the orientation of the single crystal with respect to the coi1 axis.

As a conclusion it should be remarked that the set of NQR parameters

measured in the vicinity of the transition temperature (Le., resonance frequency,

relaxation times, and lineshape of Fourier transform spectra) as a function of the

hydrostatic and axial pressure, in conjunction with the structural and dynamic

parameters obtained by refinement of the powder neutron diffraction profiles in

a set of samples with controlled concentrations of substitutional impurities, has

proven to be a dariQing approach towards understanding the anomalies that

characterire a displacive structural phase transition. Of particular interest are

the systems where precursor effects are present.

MEM and FT Appendix - Page 1

MAXIMUM ENTROPY AND FOURIER TRANSFORM

The computer programs for spectral analyses by Maximum Entropy

Methods are based on the algorithm presented by Mackowiak (1994, Mol. Phys.

Rep. 6 188).

In order to analyze the NQR hydrostatic and uniaxial pressure spectra

presented in this thesis, the code was first written in Microsoft QuickBasic and

then using the C language (Think C 5.0 for Macintosh) ta improve the tirne

performance which is mainly determined by the various loops of the algorithrn. A

few code lines have been added to Mackowiak's QuickBasic prograrn to display

the input and output spectra.

Given a real vector datq of length n, and given the filter error parameter

m, the routine returns a vector cof of length m with components cof(j) = ai,

and a scalar pm = a,, which are the coefficients of the MEM spectral estimation.

The algorithm is recursive, building up the answer for larger and larger values of

m until the desired value is reached. Using the parameters returned by the

algorithm, evalua calculates the power spectrum estimate as a function of fdt.

As an example of the application, figures 7.5 c and d illustrate the Fourier

transform and MEM outputs corresponding to a tetragonal phase three

component 35CI resonance spectrum of one of the samples presented in chapter

7. Clear evidence of the third weak component is shown in the MEM output

spectrum (figure 7.5 d). This feature is not resolved by the magnitude of the FT

(figure 7.5 c).

The Fourier transfomi spectra were obtained using the FFT option of the

MacNMR software (versions 4.5.9 and 5.4). In order to compare the MEM

outputs to the Fourier transform results, a computer codc in QuickBasic

language was written to achieve an efficient display of the transform data in an

adequate format.

Using an IDL (Interactive Data Language) code, the Fourier transform

outputs were displayed in a stack format, as presented in chapter 7. In the

exarnple shown, the code inputs files with MacNMR format, and then proceeds to

extract the resonance frequency and point separation from each file. The

magnitude values are extrapolated to get the final display. This allows one to

see the temperature evolution of the resonance lines.

The computer programs for the MEM analysis, FFT calculation, the

overlapped display and a format change program that transfer the binary

MacNMR files to ASCII lines are included below.

MAXIMUM ENTROPY METHOD PROGRAMS

List of variables:

MEM/out

evalua

amp[ 1

input file name (QB routine)

input file name (ThinkC routine)

input number of data points

input vector (n dimensions)

prediction filter error parameter

output file name

output MEM spectrum number of points

output channel (frequency x dwell tirne)

output coefficients vector (m dimensions)

output scalar parameter

generates spectral estimate (QB routine)

spectral estimate vector (ThinkC routine)

MEM MlCROSOFT QUlCK BASIC PROGRAM:

DEFSNG evalua, fdt, n, m , pm DIM datq!(256), wk1!(25S),wk2!(25S),wkm!(3),cof!(3)

INPUT "# channels=", n INPUT "filter errer=", m INPUT "# MEM spectrum points =", nfdt ' INPUT "File name", nm$ nm$ = "MEM/inU

OPEN nm$ FOR INPUT AS #1 FOR i = 1 TO n

INPUT # I I datq(i)

datq(i) = datq(i)/20000! NEXT i CLOSE #1

PSET (0,290) LINE -(500,290), 33 PSET (0,35) LINE -(500,35), 33 FOR i=O TO 10

PSET (i*50,290) LfNE -(50*i,285), 33 PSET (i*50,35) LINE -(50*i,40), 33 PRlNT USlNG "#.## "; i * .O5

NEXT i

p = O! FORj= 1 TOn

p = p + datq(j)A2 N W j p m = p / n PRlNT pm wkl(1) = datq(1) wk2(n-1) = datq(n) FOR j-2 TO n-1

wkl (j) = datqü) wk2Ü-1) = datq(j)

NEXT j FOR k=l TO m

pneum = O! denom = O! FOR j= 1 TO n-k

pneum = pneum + wkl (j) * wk2(j) denom = denom + wkl ( j )A2 + wk2(j)A2

NEXT j cof(k) = 2! * pneurn / denom prn = prn * (1 ! - cof(k)A2) FOR i=l TO k-1

cof(i) = wkm(i) - cof(k) * wkm(k-i) NUCT i IF k = m THEN GOTO 10 FORi= 1 TOk

wkm(i) = cof(i)

NUCT i FORj= l T o n - k - 1

wkl (j) = wkl (j) - wkm(k) * wkZ(j) wkZ(j) = wkZ(j+l) - wkm(k) * wkl ü+l)

NEXTj NEXT k PRlNT "never get here"

10 OPEN "MEM/outM FOR OUTPlJT AS #1 PSET (0,280) FORj= 1 T o n

fdt = .5 * j / nfdt theta# = 6.2831 853071 7959# * fdt wpr# = COS(theta#) wpi# = SIN(theta#) wr# = 1# wi# = O# sumr = 1 ! sumi = O! FOR i = 1 TO m

wtemp# = wr# wr# = wr# * wpr# - wi# * wpi# wi# = wi# * wpr# + wtemp# * wpi# sumr = sumr-cof(i) * (WH) sumi = sumi-cof(i) * (wi#)

NUCT i evalua = pm / (surnrA2 + sumiA2) PRlNT #1, evalua * 100 IF j = l THEN LINE -(fdt * 1000, 280 - evalua * 5000), 30 LlNE -(fdt * 1000, 280 - evalua * 5000), 33

NEXT j CLOSE #1

SOUND 100,3,50 END

MEM THlNK C 5.0 PROGRAM:

/* Think C 5.0 source program by Pablo Prado

Based on Burg algorithm / see QuickBasic program wntten by M. Mackowiak

MEM creates the Maximum Entropy Spectrum */

Mefine WindlD 1 1 28 /* ID for windows created byResEdit */ Mefine WindlD2 129 #define WindlD3 130

Mefine m 5 /* Filter */ #define n 5 1 2 /* Number of channels (input) */ #define nfdt 1 0 2 4 /* Output */ #define WindLength .5

FILE *in, *out;

int i, j, k, WindHigh = 130; float datq[ n + 1 1, cof[ m + 1 1, w k l [ n + 1 1, wk2[ n + 1 1, wkm[ m + 1 1; float fdt, pm, pneum, denom, sumr, sumi, p, amp[ nfdt + 1 1, amax, drnax; double pi = 3.1 41 592653589793, wpr, wpi, wr , wi , ang, wtemp, nothing;

void MEM( void ); void ToolBoxlnit( void ); void Windowlnit 1 ( void ); void Windowlnit2( void ); void Windowlnit3( void );

void main( void )

M W 1; ToolBoxlnit( ); Windowlnit 1 ( ); Windowlnit2( );

while ( !Button( ) ); 1

void f

ToolBoxlnit( void )

void Windowlnitl ( void ) /* Window for MEM Spectrum */ i

WindowPtr window; window = GetNewWindow( WindlD1, nil, O ); if( window == ni1 )

1 SysBeep( 1 O ); ExitToShell( ); 1

amax = amp[ 1 1; for( i=l; i <= n; i += 1 )

f if ( amp[ i ] >= amax ) amax = amp[ i 1; 1

ShowWindow( window ); SetPort( window ); MoveTo( WindLength * n / 2, 20 ); DrawString( "\p MEM Spectmm" ); MoveTo( WindLength * n + 2, WindHigh + 12 ); DrawString( "\p fdt" ); MoveTo( 5, 5 ); LineTo( 5, WindHigh + 1 0 ); LineTo( WindLength * n, WindHigh + 1 O ); MoveTo( 10, WindHigh - amp[ 1 ] * WindHigh / amax ); f o r ( i = l ; i < = n - l ; i + = 1 )

LineTo( ( i - 1 ) * Windlength + 10, WindHigh - amp[ i ] *

MEM and F 7 Appendix - Page 8

WindHigh / amax ); for( i = O; i <= 4; i += 1 )

i MoveTo ( 10 + Windlength * i * n / 5, WindHigh + 10); LineTo( 10 + WindLength * i * n / 5, WindHigh + 7); 1

void Windowlnit2 ( void ) /* Window for Comments i

SysBeep( 1 O ); ExitToShell( ); 1

ShowWindow( window ); SetPort( window ); MoveTo( 6, 20 ); DrawString( "\p COMMENTS" ); MoveTo( 6,40 ); DrawString( "\p Try new rn" );

void Windowlnit3( void ) /* Window for FI0 i

i SysBeep( 10 ); ExitToShell( ); 1

dmax = datq[ l 1; for( i = 1; i <= n; i += 1 )

i if ( datq[ i ] >= dmax ) dmax = datq[ i 1; 1

ShowWindow( window ); SetPort( window ); MoveTo( WindLength * n / 2 ,20 );

DrawString( "\p FID" ); MoveTo( WindLength * n + 2, WindHigh + 12 ); DrawString( "\p ch#" ); MoveTo( 5, 5 ); LineTo( 5, WindHigh + 1 O ); LineTo( WindLength * n, WindHigh + 10 ); MoveTo( 1 O, WindHigh - datq[ 1 ] * WindHigh / dmax); for ( i= l ; i < = n - 1 ; i + = 1 )

LineTo( ( i - 1 ) * WindLength + 10, WindHigh - datq[ i ] * WindHigh / drnax);

for( i = O; i <= 4; i += 1 ) i MoveTo ( 1 0 + WindLength * i * nfdt / 5, WindHigh + 1 0); LineTo( 10 + WindLength * i * nfdt / 5, WindHigh + 7); 1

1 /* MEM calculation */ void MEM( ) {

in = fopen ( "MEM/in","rt" ); f o r ( i = l ; i < = n ; i + = l )

1 fscanf( in, "%fin", &datq[ i ] ); fscanf( in, "%finu, nothing ); 1

fclose( in ); p = O; for( j = 1; j <= n ; j += 1 )

P = P + (daW j 1 * datq[ j 1); p m = p / n ; wkl [ 1 ] = datq[ 1 1; wk2[ n - 1 ] = datq[ n 1; f o r ( j = 2 ; j < = n - 1 ; j + = 1 )

{ wkl [ j ] = datq[ j 1; wk2[j-l ] = datqu]; 1

f o r ( k = l ; k < = m ; k + = l ) { pneum = 0;

MEM and fT Appendjx - Page 1 O

pneum = pneum + (wkl [ j ] * wk2[ j 1); denom = denom + (wkl [ j ] * wkl [ j 1) + (wk2[ j ] * wk2[ j 1); 1

cof[ k ] = 2 * pneum / denom; pm = pm* ( 1 -cof[ k ] *cof[ k]); f o r ( i = l ; i < = k - 1 ;i+= 1 )

cof [ i ] = w k m [ i ] - c o f [ k ] * w k m [ k - i l ; i f ( k < m )

f o r ( i = l ; i < = k ; i + = l ) wkm[ i ] = cof[ i 1;

f o r ( j = l ; j < = n - k - I ; j + = l ) i w k l [ j ] = w k l [ j ] -wkm[k] *wk2[ j 1; w k 2 [ j ] = w k Z [ j + l 1 - w k m [ k ] * w k l [ j + l 1; 1

out = fopen ( "MEM/outt8, "wt" ); f o r ( i = l ; i < = n ; i + = 1 )

fdt = .S * i / nfdt; ang = 2 * pi + fdt; wpr = COS( ang ); wpi = sin( ang ); wr = 1; wi = 0; sumr = 1 ; sumi = 0; f o r ( j = l ; j < = m ; j + = l )

i wtemp = wr; wr = wr * wpr - wi * wpi; wi = wi * wpr + wternp * wpi; sumr = sumr - cof[ j ] * wr; sumi = sumi - cof[ j ] * wi; 1

amp[ i ] = pm / (sumr * sumr + sumi * sumi); fprintf( out, "%f %Rn", fdt, amp[ i ] ); 1

fclose( out );

MEM and FT Appendix - Page 1 1

FAST FOURIER TRANSFORM MICROSOFT QUlCK BASIC PROGRAM:

DIM Yrea1(1030), Yimag(1030) DEFSNG n OPEN "FID.reW FOR INPUT AS #f OPEN "FID.irnm FOR INPüT AS #2 OPEN "FFT.out" FOR OUTPUT AS #3 Npts=l024 pi-3.1 41 59265357# PRlNT

FOR C l TO Npts INPüT #1, Yreal(i) INPUT #2, Yimag(i) PRIM i, Yreal(i), Yimag(i)

NUCT i

CALL FFT(Yreal(),Yimag (), N pts)

PRlNT FOR i= l TO Npts

PRlNT i, Yreal(i), Yimag (i) PRlNT #3, i, Yreal(i), Yimag(i)

NEXT i CLOSE #1 CLOSE #2 CLOSE #3 SOUND 300, 5,50 END

SUB FFT(Yreal(),Yimag(), Npts) STATIC n = lNT(Npts) m = LOG(n)/LOG(2) nv2 = n/2 nrnl = n-1 j = l FOR i = 1 TO nml

IF i < j THEN treal = Yrealu) timag = Yimagü) Yrealü) = Yreal(i)

Yimagü) = Yimag(i) Yreal(i) = treat Yimag(i) = timag

END IF k = nv2 WHlLE k < j

j = j - k k = k/2

WEND j = j + k

NUCT i FORL= 1 Tom

le = 2AL le1 = le/2 ureal = 1 ! uimag = O! wreal = COS(3.l416/le 1 ) wimag = -SIN(3.1416/lel) FORj = 1 TOlel

FOR i = j TO n STEP le ip = i + le1 treal = Yreal (ip) * ureal - Yimag(ip) * uimag timag = Yreal (ip) * uimag + Yimag(ip) * ureal Yreal(ip) = Yreal(i) - treal Yimag(ip) = Yimag(i) - timag Yreal(i) = Yreal(i) + treal Yimag(i) = Yimag(i) + timag

NUCf i treal = ureal * wreal - uimag * wimag uimag = ureal * wimag + uimag * wreal ureal = treal

NEXT j NEXT L FOR i = 1 1 0 n/2

tempr = Yreal(i) tempi = Yimag(i) Yreal(i) = Yreal(n/2 + i) Yimag(i) = Yimag(n/Z + i) Yreal(n/Z + i) = tempr Yimag(n/Z + i) = tempi

NEXT i END SUB

IDL DISPLAY PROGRAM:

F = fltarr(22, 1024) mag = fltarr(22J024) T = fltarr(22) magin = fltarr(22,1230) filenarne=pickfile(/read,GET-PATH = pathname)

For J = 1,22 DO BEGIN ext = STRTRIM(STRING(J), 1 ) filename = pathname + ext print,filename get-lun, unit openr,unit, filename readflunitl in T(J- 1 ) = in

for i = 0, 1023 DO BEGIN readf,unit,in 1 ,inZ,in3 F((J-1 ), i) = in1 mag((J-l ), i) = sqrt((inZ)A2 +(in3)AZ)

freq=findgen( 1 2 30) freq = (freq * .0009765625)/2 + 16.6 magin((J-1 ), *)= interpol (mag((J- 1 ), *),F((J-1 ), *),freq)

close, unit free-lun, unit

END SAVE, magin, Tl freq, FILENAME = 'Out.datl window, 1

surface , xtitle= 'T(K)', ytitle='V(MHz)', zstyle=4, magin(0:28,450:800), T(O:28),freq(450:800), BACKGROUND = 255, COLOR = O, /upper-only, Xrange=[20,60], AZ=50, AX =6O, charsize=2, /horizontal

MEM and R Appendix - Page 14

REFORMAT THINK C 5.0 PROGRAM:

* Reads a MacNMR file (binary) and converts it in ascii with first column the actual frequency, second column the real part of the signal and third the imaginary.

parameters: dwell time = dt # channels = block-size

char spectrum-name[32], *data, *skip; int df, sa, sk, skip-amount, freq-size, freq-pos; long int byte-size, data-size, blocksize, output-size; short double freq; main ()

long int n, b; int k, tot, f, fr; char *v; FILE *header-file, *spectrum, *workfile,*spectra; float real, imag, of, Temp, fch, dt, x;

printf("Number of files:"); scanf("%dU, &tot); spectra = fopen("names","r"); for(f= 1 ; f<=tot ; f += 1 )

i fscanf(spectra, "%f %sibs", &Temp, spectrum-name); dt=l ; block-size = 2048; fch= 1 /(dt*blocksize); skip-amount = 4096; freq-sire = 8;

freq-pos = 1 62; data-size = blocksize*2*4;

data = (char *)malloc(data,size); skip = (char *)malloc(skip-amount); spectrum = fopen(spectrum-namell'rb"); sk= fread(skip, 1, freq-pos, spectrum);

printf("Skipped first %d bytes \nu ,sk); fr = fread(&freq , freq-site, 1, spectrurn); sa = fread(skip, 1, (skip-amount - freq-pos - freq-size), spectrum);

printf(5kipped last %d bytes \n'',sa); df = fread(data , 1, data-size, spectrum);

printf("\nDone a file\nU); fclose(spectrurn); sprintf(v,"%dU,f); workfile = fopen(v, "wt"); fprintf(workfile, N%fWl Temp); for(k = 1 ; k <= blocksize*2 - 1 ; k = k + 2)

I real = 0.0; n = k-1; memcpy(&real,data+(4*n),4); memcpy(&imag, data+4*(n+ 1 ), 4); real = rea1/10000; imag = imag/l0000; x = ((blocksize/2)-(k/2+1 ))*fch+freq; fprintf(workfile,"%f %f %finN, x, real, imag ); 1

fclose(workfile); free(skip); free(data); 1

fclose(spect ra); 1

ABSTRACTnlc-bnc.ca/obj/s4/f2/dsk3/ftp04/nq23872.pdf · 2004. 9. 1. · Kz[OsC16].98[ReCî6].02...

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THE OF MANITOBA OPERATION BASED COSTING MODEL FOR ...collectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/NQ57506.pdf · Operation Based Costing, a cost measurement technique specifically

collectionscanada.cacollectionscanada.ca/obj/s4/f2/dsk3/ftp04/mq30577.pdf · Abstract A major objective in knowledge discovery in Internet database research is to sup port exploration

ARTHROSCOPIC TRANSFER OF OSTEOCHONDRAL ...collectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/nq31898.pdfstandard errors (percentage of bone in the three fields of interest for each anatomic

Hemodialysis Prescription: Dose Adequacy by Continuous …collectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/mq60844.pdf · 2004. 9. 1. · dialysis in the rend unit at Thunder Bay Regional

1*1 - Library and Archives Canadacollectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/nq25051.pdf · INTRODUCTION 1.1 Motivation for the Research ..... -1 1.2 Background ... being Dr. Clark

abundance DNA Aspetgillus flavipes ulomumcollectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/nq23602.pdf · iii sequenced. Primers Ranking the simple sequenœ repeats were synthesized and

INVESTIGATION OF THE LOSS OF CREOSOTE COMPONENTScollectionscanada.gc.ca/obj/s4/f2/dsk3/ftp04/MQ58738.pdf · different compounds in different ties at the beginning of experîments