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ABSTRACT

Study of ZrSiO4 Phase Transition Using Perturbed Angular Correlation Spectroscopy

By Matthew P. Rambo

The mineral zirconium silicate, ZrSiO4, or zircon is of interest due to its low coefficient of

thermal expansion. Metamict zircon heated above 800o C undergoes a structural displacive

phase transition. These properties are favorable for applications in the foundry industry as formliners, wave guide materials, and actinide-bearing structures for nuclear waste containment.

Perturbed angular correlation (PAC) techniques are used to study the short range order of any

phase transition at a Zr-site. PAC measurements of the electric field gradient (EFG) were

obtained for two samples from room temperature to 1100o C. Samples for the PAC experiments

were prepared from commercial materials obtained fromAldrich andAlfa Aesar. The primary

EFG parameters for both samples were consistent with previously published data. The

quadrupole interaction frequency decreased linearly with increasing temperature and the

asymmetry values were near zero. Uncharacteristic observations in the anisotropy may be

resultant of an after effect condition. Evidence of a displacive phase transition in the temperature

range of 800o C could not be confirmed.


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Study of ZrSiO4 Phase Transition Using Perturbed Angular Correlation Spectroscopy

A Thesis

Submitted to the

Faculty of Miami University

in partial fulfillment of

the requirements for the degree of

Master of Science

Department of Physics

By

Matthew P. Rambo

Miami University

Oxford, Ohio

2005

Advisor

Dr. Herbert Jaeger

Reader

Dr. Michael J. Pechan

Reader

Dr. Joseph R. Priest


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Table of Contents

Paragraph Page

1 ZIRCON AN INTRODUCTION...................................................................................... 1

1.1 OCCURRENCE .......................................................................................................... .........1

1.2 STRUCTURE ............................................................................................................. .........2

1.3 APPLICATIONS.......................................................................................................... ........4

1.4 PURPOSE OF THIS STUDY ......................................................................................... ........4

2 HISTORICAL NOTES.........................................................................................................5

2.1 GAMMA GAMMA ANGULAR CORRELATIONS.................................................................5

2.1.1 Unperturbed -Angular Correlations ....................................................................72.1.2 Perturbed -Angular Correlations.........................................................................8

2.1.3 The Electric Quadrupole Interaction.....................................................................14

3 EXPERIMENTAL SETUP ................................................................................................18

3.1 PACSPECTROMETER ............................................................................................. ........18

3.1.1 Gamma Detectors..................................................................................................19

3.1.2 Coincidence Electronics........................................................................................21

3.1.3 Routing and MCA.................................................................................................22

3.1.4 PAC Furnace .........................................................................................................23

3.2 SPECTROMETER CALIBRATION................................................................................ .......24

3.2.1 Energy Calibration ................................................................................................25

3.2.2. Time Calibration ...................................................................................................26

3.3 SAMPLE PREPARATION............................................................................................ .......27

3.4 DATA REDUCTION................................................................................................... .......27

3.4.1 Background Subtraction........................................................................................29

3.4.2 Error Propagation..................................................................................................30

3.4.3 Nonlinear Regression............................................................................................31

3.4.4 Computer Model Fitting........................................................................................32

4 RESULTS AND DISCUSSION ............................................................................................. 33


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4.1 ZIRCON SAMPLES STUDIED ..................................................................................... .......33

4.2 RESULTS OF THE PACEXPERIMENTS .................................................................... .........34

4.2.1 Aldrich Zircon Results ..........................................................................................34

4.2.2 Alfa/Aesar Zircon Results.....................................................................................37

4.3 DISCUSSION............................................................................................................. .......40

4.3.1 Aldrich......................................................................................................... ..........41

4.3.2 Alfa/Aesar .............................................................................................................44

4.3.3 Aldrich and Alfa/Aesar Combined........................................................................47

5 SUMMARY AND CONCLUSION....................................................................................51

6 REFERENCES.......................................................................................................... ..........53

List of Tables

Table 3.1 PCA II memory sector assignments.............................................................................22

Table 3.2 Isotope energy spectra...................................................................................................25

Table 4-1 Impurity content determined by neutron activation analysis.......................................33

Table 4-2 Typical impurities found in zircon samples.................................................................34

List of FiguresFigure 1.1: Tetragonal crystal lattice structure of ZrSiO4 ...............................................................2

Figure 1.2: Chains of alternating edge-sharing SiO4 tetrahedra and ZrO8 triangular dodecahedra

extending parallel to c: Figures taken from Ref. 27................................................................3

Figure 1.3: Chains of edge-sharing ZrO8 dodecahedra: Figures taken from Ref. 27......................3

Figure 2.1: Classical picture of the nuclear reorientation that an extranuclear EFG produces.

This EFG arises from the electrons bound to the probe nucleus and from all of the other

electrons and nuclei in the lattice. In the classical picture, the nuclear spin vector I

precesses about the Vzz-axis of the EFG tensor with the quadrupole frequency Q(16). .........9

Figure 2.2: The reorientations perturb the angular correlation as a function of time. ..................10

Figure 2.3: (LEFT) Decay scheme of181Hf to 181Ta PAC Probe. The primary decay path

involves (1) -decay from the I = - ground state of181Hf to the I = + 615 keV level of


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181Ta, (2) -ray emission from the I = 1/2 level to the I = 5/2 intermediate level and -ray

emission to the I = 7/2 ground state of181Ta. I indicates the spin of the respective quantum

level. The symbols E2 (electric quadrupole) and MI (magnetic dipole) indicate the

multipolarities of the -transitions. (RIGHT) In the absence of an external EFG, all of the

intermediate quantum states are degenerate..........................................................................12

Figure 2.4: Schematic diagram of the m-states associated with a spin I = 5/2 intermediate

quantum level. The dependence of the energy eigenvalues on the asymmetry parameter

for the nuclear electric quadrupole interaction of a spin I = 5/2 nucleus with an extranuclear

EFG. Three frequencies, 1, 2, and 3 characterize transitions between m-states............12

Figure 3.1: PAC Spectrometer arrangement. The fast branch consists of the four constant

fraction discriminators (CFDs) and the time to amplitude converter (TAC). The slow

branch includes the four linear amplifiers (LAs) and single channel analyzers (SCAs) and

the routing logic. The fast branch measures the time between 1 and 2 and the slow

branch checks that the -rays are the correct energy(19,20,21)..................................................19

Figure 3.2: MCA counts versus time spectrum for one detector pair. ..........................................23

Figure 3.3: PAC aluminum furnace showing ceramic sample tube, water and electrical

connections................................................................................................................... .........24

Figure 4-1: Typical PAC spectra (left) and their Fourier transforms (right) for the Aldrich zircon

sample........................................................................................................................ ............35Figure 4-2: Temperature dependence of the fitted EFG parameters during the heating and cooling

cycle. Measurements for cooling cycle and heating cycle are represented as open and filled

symbols, respectively. ...........................................................................................................36

Figure 4-3: Anisotropy as a function of temperature for Aldrich. ................................................37

Figure 4-4: Typical PAC spectra (left) and their Fourier transforms (right) for the Alfa/Aesar

zircon sample.........................................................................................................................38

Figure 4-5: Temperature dependence of the fitted EFG for the Alfa/Aesar Zircon sample for

cooling cycle data..................................................................................................................39

Figure 4-6: Anisotropy as a function of temperature for the Alfa/Aesar data. .............................40

Figure 4-7: Regression fit for data above and below 700o C. .......................................................42


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Figure 4-8: Asymmetry calibration curve determined for the theoretical calculation of the ratio

2 /1 using equations 2.16 and 2.17. ..................................................................................43

Figure 4-9: Temperature dependence of the fitted EFG for both Aldrich and Alfa/Aesar Zircon

samples (C = cooling data; H = heating data).......................................................................46

Figure 4-10: Combined anisotropy plot of Aldrich and Alfa/Aesar as function of temperature (C

= cooling data; H = heating data)..........................................................................................47


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Dedication

It is my honor to dedicate this thesis to my beloved wife Lydia, my one and only true love.


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1 Zircon An Introduction

Zircon is a transparent, translucent, or opaque mineral, composed chiefly of zirconium

silicate, ZrSiO4, and crystallizes in the tetragonal system. It has a hardness of 7.5 and a

specific gravity of 4.2 to 4.86; it shines with an adamantine luster. Zircon may occur ascolorless crystals or in shades of green, gray, red, blue, yellow, or brown. The clear,

transparent yellow, orange, red, and brown varieties are often used as gemstones and are

known as hyacinth or jacinth; translucent or opaque varieties, and most of the colorless types,

are known as jargon or jargoon. When subjected to high temperatures, zircons either change

color or lose their color, and assume a greater brilliance. Colorless zircons are known as

Matura diamonds or white zircons. A blue variety, produced by heat treatment and known as

blue zircon, is also commonly used as a gemstone. It is also found, often together with gold, as

rounded grains in streams and along sandy beaches. (1)

1.1 OccurrenceThe mineral zirconium silicate, ZrSiO4, commonly known as zircon, Figure 1.1, occurs as

an accessory mineral in crustal igneous, metamorphic, and sedimentary rocks, as well as in lunar

materials, meteorites, and tektites(2).

The element zirconium is present at a level of 0.02 to 0.03% in the earths crust and is

more abundant than copper, nickel, lead, and zinc. The two main mineral sources are zircon and

baddeleyite. Zircon, ZrSiO4, is the most abundant and widely distributed zirconium mineral.

Deposits are mined in Australia, India, South Africa, former USSR, and the USA(3).


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Figure 1.1: Tetragonal crystal lattice structure of ZrSiO4

1.2 StructureCrystalline ZrSiO4 has a tetragonal unit cell with space group I41 /amd, and 4 formula

units per unit cell. The density is= 4.714 g/cm3 and the lattice parameters are a = 6.605 ,

c = 5.987 . All Zr atoms are 8-fold coordinated by oxygen atoms forming ZrO8 triangular

dodecahedra subunits. The Si atoms are surrounded by four oxygen atoms forming SiO4

tetrahedra subunits.

The principal structure is a combination of alternating edge-sharing polyhedra subunits

that form chains parallel to the c-axis. Unit cells in the alternating chain are either ZrO8

dodecahedra subunits of zircon (Zr) or SiO4 tetrahedra subunits of silicon (Si), Figure 1.2. The

chains are connected laterally by edge-sharing ZrO8 dodecahedra, Figure 1.3. See discussion

from Z. Mursic, T. Vogt, F. Frey for an alternative description of the lattice structure(4).


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Figure 1.2: Chains of alternating edge-sharing SiO4 tetrahedra

and ZrO8 triangular dodecahedra extending parallel to c:Figures taken from Ref. 27.

Figure 1.3: Chains of edge-sharing ZrO8 dodecahedra: Figures

taken from Ref. 27.


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1.3 ApplicationsPrimary application is in the foundry industry as a form liner because zircon has a low

coefficient of thermal expansion. Its structural properties make it favorable as a proposed wave

guide material and actinide-bearing structures for nuclear waste containment(5).

1.4 Purpose of This StudyThe research within this thesis focuses on applying Perturbed Angular Correlation

(PAC), techniques to the ceramic mineral zircon. As an analytical tool, it will provide

information about the microscopic structure and examine a phase transition first proposed by

Mursic et al. PAC, unlike diffraction studies, explores the short-range order of a few unit cells

from the probe nuclei. This work aims to confirm previous research and increase the

understanding of phase transitions in ZrSiO4. Providing new knowledge from this research willextend the usefulness of zirconium silicate materials for scientific and industrial purposes alike.


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2 Historical Notes

In 1940, D.R. Hamilton published a paper in which he treated the theory of the

directional correlation of a gamma-gamma (-) cascade(6). The first successful correlation

measurement was performed seven years later by Brady and Deutsch(7). By todays standards,the initial equipment was primitive, consisting of Geiger counters that combined low detection

efficiency with a very poor time resolution and no energy resolution. With the introduction of

scintillation detectors, -angular correlation experiments became the standard technique to

determine spins and parities of nuclear states.

It was soon realized that extranuclear fields may perturb, and sometimes completely

obscure, the angular correlation

(8)

. This property permitted the determination of the nuclearg-factor(9), the nuclear quadrupole moment(10), and offered a tool to investigate solid-state

properties(11). Systematically, improvements in theory and experimental methods by many

scientists, lead to the first report of a time-differential PAC measurement(12).

The first PAC experiments were intended to measure magnetic and electric hyperfine

fields. A suitable nuclear probe usually a chemical compound or a dilute alloy was introduced

into the sample. One of the aims of the experiments of that early period was to systematicallycollect data on magnetic hyperfine fields in ferromagnetic materials, like iron, cobalt, nickel

and gadolinium, and on electric field gradients in chemical compounds and non-cubic

metals(13).

2.1 Gamma Gamma Angular CorrelationsPAC spectroscopy is a nuclear technique used for characterization of material properties

utilizing hyperfine interactions between the probe nuclei and their environments. Utilizing thewell-established theory for describing the angular correlation involving the emission of two -

rays during nuclear decay, it is possible to determine the short-range structure of crystalline

materials near the probe site. Radioactive nuclei that decay with the emission of two

consecutive -rays are restricted by angular momentum conservation from emitting both -rays


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in random directions. Depending on the nuclear properties, the established direction of the first

-ray emission restricts the direction of the second -ray emission. If a large number of nuclei

with the same orientation are well isolated from any external fields, the angular correlation has

a well-defined radiation pattern, W(). Once the probe is placed into the sample, the

configuration of the lattice defines a characteristic extranuclear field that interacts with the

intermediate nuclear moment, perturbing the known distribution pattern, causing it to precess

with specific frequencies. This results in a perturbation of the correlation function

W() W(, t).

To measure the time distribution when the first -ray is ascertained at some angle 1 by

the first moveable detector, the second moveable detector must identify an event from the

second -ray within the proper time frame at some other angle 2. For the purpose of

experimental and mathematical simplicity, the detectors shall remain fixed. In a macroscopic

sample, the probe nuclei are randomly oriented resulting in an isotropic distribution of the

emitted -rays. By waiting for a 1 in the direction of detector 1, we experimentally only

consider nuclei that are oriented in this direction. This is equivalent to working with a set of

nuclei in a particular orientation. The direction of the z-axis is also defined to be in that

direction making 1 = 0 and 2 = were is the angle between the emission of1 and 2. This

way the experimental setup is less complicated, since the mathematical description of theprobability distribution function is only dependent on one angle instead of two.

Experimentally, the primary quantity measured is the angular correlation function,

W(, t), in which is the angle of separation for detectors 1 and 2. After the first -ray is

detected (t) is the time elapsed waiting for the detection of the second -ray in detector 2. Thus,

the angular correlation function, W(, t), records the coincidence counting rate as a function of

the intermediate quantum level lifetime at a particular detector angle . The measured angularcorrelation function for static quadrupole interactions with the probe is modulated by a

perturbation function. Extracting the precise modulation frequencies, i, from the perturbation

function provides the components of the electric field gradient (EFG) tensor in the principal

axis coordinate system.


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The objective of the PAC measurement is to determine the perturbation function with

sufficient precision that one can confidently fit it to the appropriate model and extract the

parameters characteristic of the probe site. In zircon, the only nuclear interaction of any

consequence is that between the nuclear electric quadrupole moment Q and the EFG at the

nucleus(14).

( )4 2 1zz

Q

eQV

I I =

h(2.1)

Where e is the electronic charge, h is Plank's constant divided by 2, and I is the

intermediate quantum spin level, and Vzz is the z-component of the EFG in the principal axis

system.

Because of the importance of the anisotropic angular distribution, the rest of this

discussion is presented in incremental sections starting with the unperturbed -angular

correlation of the nuclei in free space and working up through model fitting of the perturbation

function.

The rest of this section will focus on a general description for obtaining the desired

angular correlation function. For a more rigorous and mathematically complete treatment of

Perturbed Angular Correlations, consult the textbook of Schatz and Weidinger(15).

2.1.1 Unperturbed -Angular CorrelationsIn a -- cascade, an angular correlation exists between the emission directions of the

first and second -ray that depends on the spins of the nuclear levels and the multipolarity of

the emitted radiation.

The probability that 2 will be emitted at an angle with respect to the direction of1 is

given by the unperturbed angular correlation function, W(). The unperturbed correlationfunction for the angular correlation between 1 and 2 is,

( ) ( )max

W cosk

k k

k

A P = (2.2)


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The index k is a positive even integer including zero, (k=0,2,4kmax). The index kmax

is determined by the nature of the transition and the spin angular momentum I of the excited

and intermediate states. The Pk(cos) terms are Legendre polynomials and the constants Ak are

well-defined anisotropy parameters describing the anisotropy with respect to the z-axis.

2.1.2 Perturbed -Angular Correlations

When the probe nuclei are present in a well-defined crystal lattice site, there is an

extranuclear field present due to the surrounding charge distribution. The probes experience

hyperfine shifts separating the energy levels of all quantum levels, Figure 2.4.

The magnitude of the extranuclear field is directly proportional to the size of the

intermediate energy level splitting. Extranuclear fields of large magnitude induce a larger

separation of the energy levels. During the lifetime of the intermediate quantum state, the

hyperfine interaction alters the orientation of probe spin vectors. The reorientation can be

described as the precession of the nuclear spin vector about a principle axis coordinate system

parallel to the z-axis, Figures 2.1 & 2.2.

The nuclei may experience a significant reorientation due to interactions between the

nuclear multipole moments and the corresponding electromagnetic fields caused bysurrounding charges and spins(14). As a nucleus precesses during the lifetime of the

intermediate quantum level, the probability of emitting the second -ray in the direction of

detector 2 changes and becomes a function of the time between emissions. The nucleus is no

longer oriented in the initial direction from which the first -ray was emitted. The

reorientations perturb the angular correlation as a function of time, and the angular correlation

function becomes:

( ) ( )

max

W , ( ) cos

k

k k k

kt A G t P = (2.3)

The effect of the extranuclear perturbation is defined by the perturbation function

Gk(t)(14).


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For this work, the -cascade of the probe nuclei is from the decay of the 181Ta isotope

were I = 5/2 for the intermediate state. The index kmax = 4, but the value of the coefficient A4 is

small enough that only the index k = 2 of the summation need be retained (14), so that the

perturbed angular correlation function becomes:

( ) ( )2 2 2W , 1 ( ) cost A G t P = + (2.4)

Figure 2.1: Classical picture of the nuclear reorientation that

an extranuclear EFG produces. This EFG arises from the

electrons bound to the probe nucleus and from all of the other

electrons and nuclei in the lattice. In the classical picture, the

nuclear spin vector I precesses about the Vzz-axis of the EFG

tensor with the quadrupole frequency Q(16).


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Figure 2.2: The reorientations perturb the angular correlation

as a function of time.

For a static quadrupole interaction the perturbation function G2(t) is of the form:

( ) ( )3

23

coso i ii

G t S S t =

= + (2.6)

The Si are parameters only weakly dependent on asymmetry . The i terms are related

to the differences in energy between the individual substates m and for I = 5/2

3 2 1 21

m mE E

= =

=h

(2.7)

5 2 3 22

m mE E

= =

=h

(2.8)

5 2 1 2

3

m mE E

= =

= h (2.9)

2.1.2.1 Probe Nuclei and Decay SchemeIdeally, the environment of interest to be measured is unaltered when introducing probe

nuclei. For zirconium compounds, the difficulty of inserting probe nuclei without altering the

original environment is taken care of by natural process. On the order of 1 at % the Hf

impurity is chemically very similar to Zr allowing it to occupy a substantial number of Zr-sites

on the crystal lattice.

The actual PAC probe isotope, tantalum 181Ta, is created when the impurity, 180Hf, is

irradiated with a flux of thermal neutrons for a few hours in a nuclear fission reactor. The

probe nucleus enables the PAC technique to sample the EFG at zirconium sites due to the

electric quadrupole interaction with the probe nucleus.


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The radioactive probe must decay anisotropically through two consecutive transitions to

the ground state in a gamma-gamma (-) cascade, Figure 2.3(16).

After radioactive -decay unstable Hf isotopes decay with the emission of a 133 keV

-ray (1) to populate the intermediate quantum level, I=5/2+ & Q = 2.5 barn, of the probe

nucleus for 10.8 ns. While in the intermediate state, the nuclear moments will interact with the

extranuclear environment. The final transition to the nuclear ground state transpires with the

emission of a 482 keV -ray (2), Figure 2.3.

In the "Semi classical picture", the nucleus can have electric and magnetic multipole

moments depending on its geometry, angular momentum, and charge. The particular

characteristics of a nucleus depend on the interaction between the moments with an electric or

magnetic field. The strongest types of the interactions involve the magnetic dipole moment

with the magnetic field or the electric quadrupole moment, Q, with an electric field gradient(17).

During the decay process, a nuclear probe will release energy in the form of a gamma ray

consisting of definite parity with properties of a certain multipolarity l.

The value of the multipolarity constant l determined from Maxwell's equations

identifies which type of interaction is strongest and most pertinent for measurements. For the

Hf probe, the only significant transitions of interest are the electric quadrupole transitions were

l = 2. According to the selection rules, (m = -l, , l) in whole integer steps, only intermediate

substates m = +- 5/2, +- 3/2, and +- 1/2 are possible.


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Figure 2.3: (LEFT) Decay scheme of181

Hf to181

Ta PAC Probe. The primary decay path

involves (1) -decay from the I = - ground state of181Hf to the I = + 615 keV level of181

Ta, (2) -ray emission from the I = 1/2 level to the I = 5/2 intermediate level and -rayemission to the I = 7/2 ground state of

181Ta. I indicates the spin of the respective

quantum level. The symbols E2 (electric quadrupole) and MI (magnetic dipole) indicate

the multipolarities of the -transitions. (RIGHT) In the absence of an external EFG, all ofthe intermediate quantum states are degenerate.

Figure 2.4: Schematic diagram of the m-states associated with a spin I = 5/2 intermediatequantum level. The dependence of the energy eigenvalues on the asymmetry parameter for the nuclear electric quadrupole interaction of a spin I = 5/2 nucleus with anextranuclear EFG. Three frequencies, 1, 2, and 3 characterize transitions betweenm-states.


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2.1.2.2 Hyperfine SplittingWhen the probe is placed into the sample, Figure 2.1, the effect of the EFG on the probe

causes hyperfine splitting of the intermediate energy levels, Figure 2.4. The magnitude of the

hyperfine splitting, Vzz, is directly proportional to the strength of the interaction between

external electromagnetic fields and the probe.

Splitting of the intermediate state into m-substates is associated with the spin I = 5/2

intermediate quantum level. Following the quantum selection rules, possible values of the

degenerate substates are limited to, m=+- 5/2, +- 3/2, and +-1/2. Figure 2.4 is a schematic

diagram of the m-states associated with the nuclear spin-I intermediate quantum level and the

transitions between m-states.

Detection of the initially emitted -rays targets nuclei that have a specific orientation.

After the first -ray emission, a preferential population for one of the degenerate m-substates

associated with the spin-I intermediate quantum level is made. Depending on the substate,

populated certain probe orientations are made. Subsequent depopulation of these preferred

m-states by the emission of the second -ray causes the direction of this emission to be

correlated with the emission direction of the first -ray. When the hyperfine interaction of the

intermediate nucleus with the extranuclear environment occurs, this spatial correlation isdisrupted. In the parlance of quantum mechanics, the intermediate quantum level makes an

m-to-m' transition. That is, the nucleus "flips" from one orientation to another. As a result, the

second -ray is emitted in a different direction than where it would have been emitted if the

intermediate nucleus had not flipped: The hyperfine interaction perturbs the angular

correlation(18).


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2.1.3 The Electric Quadrupole Interaction

2.1.3.1 The Electric Field GradientThe surrounding charge distribution established by the lattice characteristics of the

sample creates an extranuclear force field or electric field gradient, EFG, of the probe

environment. Measuring the interactions of the EFG with the electric quadrupole moment of

the probe nuclei allows the unique characteristics of the lattice to be determined. The gradient

of the electric field, Eq 2.10, describes the change in the electric field established by the

surrounding atoms of the local environment.

2 2 22

2 2 2

V V VV

x y z

= + +

(2.10)

2xx yy zzV V V V = + + (2.11)

2

2ii

VV

i

=

(2.12)

The principal coordinate system is that in which the EFG tensor is diagonalized.

Mathematically the EFG is a second rank, traceless tensor. In the principal-axis system all

off-diagonal terms are zero. Since a requirement of Poisson's equation is that the trace vanish,

Vxx + Vyy + Vzz = 0, only two of the three components are needed to completely describe theEFG in the principal axis coordinate system.

0 0

0 0

0 0

xx

yy

zz

V

EFG V

V

=

(2.13)

The EFG may be characterized using the magnitude Vzz and asymmetry parameter .

By convention, the component largest in magnitude is chosen to be Vzz and the smallestcomponent in magnitude is chosen to be Vxx. Vxx and Vyy are used to define the asymmetry

parameter eta, :

xx yy

zz

V V

V

(2.14)


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Because of the choice of x, y, and z-axis the asymmetry parameter, has values

between zero and one, 0 1 , and gives the relative difference between the x and y

components.

If the EFG is of equal magnitude on x and y axes, it can be seen from Eq 2.14 that = 0

and the environment around the probe nucleus is said to be axially symmetric. On the other

hand, = 1 for a maximally non-symmetric probe environment.

To characterize the interaction between the EFG (Vzz, ) and the quadrupole moment, Q

it is convenient to define the quadrupole frequency Q.

( )4 2 1zzQ

eQV

I I = h (2.15)

For simplicity the quadrupole interaction frequency Q is redefined in terms ofq

during data analysis. The quadrupole interaction frequency can be written as 20Q q = by

making the substitutions into equation 2.1 for I = 5/2 and h/2 for where zzeQV hq = .

The quadrupole frequency, Q, is connected to the interaction frequencies i that

correspond to transitions of the spin I = 5/2 intermediate quantum levels. Each of the

frequencies is a multiple of the quadrupole frequency.

1

12 3 sin arccos

3Q

=

(2.16)

( )21

2 3 sin arccos3Q

= (2.17)

( )3 12 3 sin arccos3Q = +

(2.18)

Where and are defined as:


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The first term 1 is used to determine Q both defined above. Using the ratio

2 /1 determines the asymmetry parameter .

Only in perfect crystals is it possible for all the probes to experience the same

environment. It is more realistic that the sample will have flaws and lattice defects due to

impurities and random dislocations. An adequate model adjustment for such small variations is

to assume the relative width, =Vzz / Vzz, of a Lorentzian distribution characterized as the

magnitude strength of the z-component, Vzz, about the average frequency. The small variation

of the EFG from one probe site to the other in the crystalline sample showing up as damping in

the perturbation function. To describe the damping introduces the term

( )ig te (2.25)

Where gi is a number approximated such that g1 1, g2 2, and g3 3 for close to zero(15).

The corrected perturbation function for experimental samples now takes the form

( )3

21

G ( ) cos( ) ig to i ii

t S S t e

=

= + (2.26)


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3 Experimental Setup

The spectrometer used in this experiment uses four detectors and an Oxford

Instrument's PCA II PC plug-in card for the multi-channel analyzer (MCA). The PCA II card

plugs into an expansion slot of a PC and emulates an MCA with 8192 channels. Theaccumulated data is digitally stored in one of the channels. While the experiment is in

progress, it is possible to perform live data analysis in order to determine if the statistics of the

spectrum are sufficient for least-squares fitting(19).

3.1 PAC SpectrometerA time-differential PAC (TDPAC) spectrometer is an instrument that allows the

measurement of magnetic fields and electric-field gradients in the neighborhood of radioactive

probe nuclei(20)(21). The spectrometer is primarily responsible for detecting the distinct -rays,

1 and 2, emitted during the two level cascade of the probe nuclei. Once the proper -rays are

identified, the lifetime of the intermediate state is determined by measuring the time between

the detection of1 and 2. This technique records the time-dependent spectrum of events in

which 1 enters detector 0 and 2 enters detector 2 at angle () a time (t) later.

A typical spectrometer, Figure 3.1, consists of one or more pairs of scintillation

detectors, some sort of coincidence electronics, and a MCA to record the count versus time

spectrum for each pair of detectors.

For this experiment, the spectrometer consists of four detectors that are set to a fixed

angle of 90 from each other on a horizontal plain around the sample. The block diagram in

Figure 3.1 shows the four detectors, the coincidence electronics, and the MCA used to record

and store the spectrum.

Using a spectrometer arrangement consisting of two detector pairs allows four times the

data accumulation of a system with only two detectors.


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19

Figure 3.1: PAC Spectrometer arrangement. The fast

branch consists of the four constant fraction discriminators

(CFDs) and the time to amplitude converter (TAC). The

slow branch includes the four linear amplifiers (LAs) and

single channel analyzers (SCAs) and the routing logic. Thefast branch measures the time between 1 and 2 and the

slow branch checks that the -rays are the correct

energy(19,20,21)

.

3.1.1 Gamma DetectorsEach detector is composed of two parts, the scintillator crystal and the photomultiplier

tube. The cylindrical barium fluoride (BaF2) scintillator, with dimensions 2" x 2", is mounted

in front of a photomultiplier tube. Interaction of a -ray with the scintillator results in a

fluorescence for which the number of visible and ultraviolet (UV) photons created is

proportional to the energy of the deposited -ray. Light from the scintillation process enters the

photomultiplier and strikes the cathode. Freed electrons from the cathode are accelerated and

multiplied along the chain of dynodes. When a -ray enters the scintillator, two basic


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interaction processes with the material are useful in detection: the photoelectric and Compton

effect(15).

The scintillator crystal intercepts the -ray emitted from the decaying probe nuclei and

converts the energy of the -ray to lower energy photons. The scintillation photons enter the

photomultiplier tube to be converted to an electrical pulse.

The BaF2 scintillator has the advantage of a very short response time and a high density

of 4.88 g/cm3. The high density increases the absorption efficiency of-rays. This allows a

high level of time resolution of about 300 ps for -ray energies of 511 keV(15).

The scintillator is encapsulated in a layer of aluminum open at one end for attachment

to the photomultiplier. The aluminum housing acts as a reflector for the scintillator photons. A

thin layer of glycerol is used as a coupling medium between the scintillator to the phototube to

match the index of refraction of the BaF2 with the phototube glass. In order to take advantage

of the short response time, the coupling fluid must not be absorbent in the ultraviolet range.

The joint between the phototube and the scintillator is covered with black electrical tape. The

black tape and the aluminum housing increase output of photons produced from -rays by

keeping external light from the photomultiplier.

A photomultiplier tube (PMT) consists of a photo cathode, a number of dynodes, and

the anode. When light strikes the photo cathode electrons are ejected proportional to the energy

of the initial -ray. The gradually increasing potential difference between each dynode

accelerates electrons from one dynode to the next. The total potential difference of 2200 V

from the cathode to anode is supplied by a high voltage power supply. There are about 14

dynodes between the cathode and the anode were the potential difference between each dynodeis about (2200/14) V. When electrons are accelerated from the previous dynode and collide

with the next dynode, an increasing number of electrons are ejected to the next dynode. This

process continues up through to the anode were the number of electrons are no longer

proportional to the energy of the initial -ray.


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The saturated signal taken from the anode is used for the fast branch of the

spectrometer. The second output of the detector is taken off the eleventh dynode. The

amplitude of this dynode pulse is proportional to the energy of the initial -ray. This signal is

used to discriminate -ray energies in the slow branch of the spectrometer. However, due to the

short time response of the BaF2 scintillator, the anode signals of the detectors have a width of

only about 3 ns(19).

3.1.2 Coincidence ElectronicsStorage of a valid event requires the determination of the energies and the

measurement of the time between the two photons. The former is done by the so-called slow

coincidence while the latter is done by the fast coincidence, Figure 3.1. The fast branch

processes the timing signals from the anode and the slow branch processes the energy signals

off the eleventh dynode.

3.1.2.1 Fast CoincidenceThe fast branch is equipped with a Constant Fraction Discriminator (CFD) Canberra

2126 and the Time-to-Amplitude Converter (TAC) EG & G 467. From the anode of the

photomultiplier tube, the CFD forms a sharp pulse thats correlated to the time the photon was

absorbed in the detector. The two CFD signals from detectors (0 & 1) are designated for startinputs of the TAC and CFD signals from detectors (2 & 3) are designated for stop inputs.

The TAC creates an analog pulse with amplitude linearly proportional to time elapsed

between start and stop signals from the CFD. A delay cable between CFD (2 & 3) and the

TAC stop input is used to move the time-zero point about 200 channels. This pulse is then

digitized by the analog-to-digital converter, which is part of the MCA (Oxford Instrument

PCAII), an then stored as delayed coincidence events in the MCA's memory.

3.1.2.2 Slow CoincidenceThe slow branch originates with the energy pulse off the seventh dynode of each

detector. The energy pulse is amplified and shaped by the linear amplifier (LA) Canberra


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2022. The amplitude of the bipolar output signal is proportional to the energy of the photon

adsorbed in the detector. The LA bipolar output enhances the efficiency by decreasing circuit

dead time. The signal is then fed to the single channel analyzer (SCA) Canberra 2035A for

selection of specific photon energies. The SCA will produce a logic pulse (TTL Pulse) if the

amplitude of the LA pulse fits between the upper and lower discrimination level.

3.1.3 Routing and MCA

The routing logic is used to determine valid coincidences between 1 and 2 as well as

the MCA memory sector the event will be recorded in. The circuitry of the routing logic

registers a true event if the output signals of a 1 SCA and a 2 SCA occur within 1 s of each

other. The routing logic then generates a gate signal for the input of the MCA. This signal on

the MCA input gate allows the TAC output to be accepted by the MCA. Depending on which

set of SCA's supplied the correct pair of1 and 2 events determines which of the four memory

sectors the TAC output will be stored in.

Table 3.1: PCA II memory sector assignments

Standard MCA software is used to record the number of counts versus time of each

detector pair and display the counts per channel, Figure 3.2. The amplitude of the TAC output

determines which channel is to be incremented. The number of events are plotted along the

y-axis.

Detector pair Angle Between Sector Stored in Channels

0 & 2 180o 1 0-2047

0 & 3 90o 2 2048-4095

1 & 2 90o 3 4096-6143

1 & 3 180o 4 6144-8191


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Figure 3.2: MCA counts versus time spectrum for one

detector pair.

Using four fixed detectors, as opposed to two, is a more efficient configuration. The

detectors do not have to be moved between measurements to get correlations at 90o or 180o.

With four detectors and the current routing logic, it is possible to measure two correlations of

90o and two correlations of 180o simultaneously. Consequently, this configuration is four times

as efficient as the basic two-detector configuration(22).

3.1.4 PAC FurnaceThere are very specific requirements that must be met by the furnace. The walls of the

furnace must be such that they minimize interference with the detection of the -rays in the

detectors. It must also be small enough to allow the detectors to be spaced closely.

The furnace is made of aluminum, because it has a low -ray absorption, and it is heated

with a wire element. The element is encased in an evacuated chamber wrapped around an

alumina tube that is closed at one end. Around the chamber is a water jacket that keeps the rest

of the furnace from getting to hot for the detectors to be placed closely. During operation, thevacuum is maintained with a mechanical pump and cooled continuously with a circulation of

chiller water through the water jacket, Figure 3.3.


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Passing a current through the heating element causes the furnace temperature to rise. To

prevent the magnetic field from the current carrying wire from interfering with the PMT the

wire is doubled before it is wrapped around the ceramic tube. Producing a flow of current in

both directions cancels out the magnetic field to minimize any effects at the PMT. Evacuation

of the heating element chamber helps to minimize oxidation and heat loss. Oxidation is also

reduced with the aid of a refractory cement, Alundum, used to keep the wire in place around

the tube(23).

Figure 3.3: PAC aluminum furnace showing ceramic sample

tube, water and electrical connections.

3.2

Spectrometer Calibration

Before any measurements can be made, the spectrometer must be calibrated for both

energy and time measurements. Before a new sample is placed in the spectrometer and after a

few sets of collected spectra, the calibration must be checked and adjusted. Energy calibration

involves setting the upper and lower level discriminator on the four SCAs. This is most


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conveniently done with an MCA (PCA II card) and a calibration source. The time calibration

involves locating the time-zero, to, channels where t = 0 for the four memory sectors and

determining the time-per-channel units of the MCA (PCA II card) (23).

3.2.1 Energy Calibration

The energy calibration is done using a sample containing 181Hf isotopes. The first step

is to place the 181Hf isotope in the spectrometer and connect the input of the PCA II card to the

delayed amplifier signal output. Next, open the window of the SCA until the entire energy

spectrum of the isotope is displayed. There are four prominent peaks visible in the

accumulated spectrum. The largest low-energy peak is mostly due to 133 keV photons from

the 1/2 5/2 decay, while the two smaller high-energy peaks are from the 346 keV resulting

from decay of the intermediate state to the 9/2 state and the 482 keV from the intermediate to

ground state decay, respectively see Figure 2.3. The small low-energy peak near 70 keV is an

X-ray peak resulting from Hf fluorescence. The 136 keV peak from the 9/2 state to ground

state decay cannot be resolved from the large 133 keV peak, Table 3.2.

Table 3.2: Isotope energy spectra

Energy (keV) Transition States

70 X-ray Peak

133 1/2 5/2

136 9/2 to Ground State

346 5/2 Intermediate to 9/2

482 5/2 to Ground State 7/2

With the PCA II card being gated by the output signal of the SCA, the window of the SCA for

detectors 0 and 1 is narrowed so that only the 133 keV (1) is detected and the same is done for


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the SCA window for detectors 2 and 3 so that only the 482 keV (2) peak is detected2. Special

care is taken in setting the window of the 482 keV peak so that any contribution from the

346 keV peak is minimized. Such a contribution may cause a distortion of the PAC spectrum

near the t = 0 channel.

3.2.2. Time CalibrationThe time calibration used to be a two-step process to first determine the time-zero

channel (t0) for each of the memory sectors and second to determine the time scale of the TAC.

The determination of the time-zero channel t0 was done with the use of a22Na source. By

definition, t0 marks the first channel of the spectrum were the elapsed time between two valid

events is zero. The 22Na source decays with a half-life of 2.6 years to 22Ne with the emission of

a positron. The positron will eventually interact with an electron to be annihilated in two

511 keV photons that are simultaneously created and travel away from each other in opposite

directions. If the photon pair is absorbed by two detectors 180o apart, an event is recorded into

the corresponding memory sector. After several events are recorded, the resulting prompt

spectrum marks the channel to for each sector.

Ideally, the prompt spike has a width of one channel. In reality, the limited time

resolution of the spectrometer produces a prompt similar in shape to a Gaussian distributionabout to with the full width at half maximum (FWHM) being a measure for the coincidence

time resolution of the detector pairs. Typical FWHM for this spectrometer are around 1 s.

The new method of determining the to is to now use the data itself. By doing a

numerical derivative to the left of to, the steepest part of the ascend is determined. This allows

the to location to be determined with in 1/10 of a channel. The raw data is then shifted to line

up within 1/10 of a channel.

The second calibration step is determining the time scale of the TAC. A time calibrator

(EG & Ortec model 462) generates a START and STOP pulse separated by integer multiples of

a constant time period. The TAC input receives the generated pulse from the time calibrator


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and sends the output to the PCA II card. The spectrum of the time-calibrator consists of spikes

repeated at equal intervals. The length of each interval corresponds to the constant time period

of the time calibrator. The channel number of each spike is used in the evaluation ofthe time

calibration using a least square fit to determine the channel width in units of time(19). Location

of the channel to, and time calibration are stored in a calibration file on a PC to be recalled

when the data is analyzed.

3.3 Sample PreparationThe samples must be able to withstand high temperature of the furnace and small enough

to be considered point like in comparison to the size of the detectors. They must also withstand

a suitable environment for probe activation. Activation is carried out by irradiating the samplein a flux of thermal neutrons to promote neutron capture in the probe nuclei.

The powdered sample is sealed in an evacuated quartz ampoule to prevent exploding due

to a buildup of internal pressure at high temperatures. Size and shape of the tube is on the order

of a few millimeters so that elements, such as Zr, with a high atomic number greatly reduce the

attenuation of-rays from absorption or scattering.

Small size is also preferred because in a large sample each point inside the sample will

subtend angles with respect to the detectors that greatly differ from 90o or 180o.

3.4 Data ReductionThis section deals with how to obtain the perturbation function and separate it out of the

extraneous data collected in the coincidence count rate spectrum.

The TDPAC spectra are stored in the memory of the PCA II board. The spectra consist

of the number of counts per channel as a function of time, Ci,j(t), known as the coincidence

count rate. Data reduction is the process by which the desired perturbation function A2G2(t) is

experimentally determined. A Fourier transform of this function reveals the interaction


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frequencies, k, needed to calculate the asymmetry parameter, , and Vzz, the z-component of

the EFG.

The coincidence counts, Ci,j(t), from the detector pair (i, j) takes into account the time, t,elapsed between 1 entering the first detector (i) and 2 entering the second detector (j). The

natural-radioactive dependence of-radiation must also be taken into consideration as part of

the coincidence spectrum. Ci,j(t) depends exponentially on the lifetime of the intermediate

state.

For any channel of the spectrum the differential coincidence count Cij(t) at time t

( )ij o ij ij ij ijC (t) N P (t) (t) T t B= + (3.1)

Addition of the term Bij accounts for the random time-independent background

coincident count rate. The decay rate of the parent nuclei No is the number of decays per unit

time, the function Pij(t), shown below, is the probability that 1-2 are detected a time t apart,

and ij(t) represents the channel width for the accumulation time Tij(t) of the data.

( ) ( ) ( )t1ij i jP t e w ,t= (3.2)

( ) ( ) ( )2 2 2w ,t 1 A G t P cos= + (3.3)

is the lifetime of the intermediate state, i defines the efficiency of detector i, and w(, t)

represents the time-dependent angular correlation function ( is the angle between detectors

(i, j)). A2 is the anisotropy of the correlation, G2(t) the perturbation function, and P2(cos) a

Legendre polynomial.

With all of the above substitutions the coincidence rate can be written as

( )( )

( ) ( )toN

ij ij 2 2 2 ijC t E (t)e 1 A G t P cos B

= + + (3.4)


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( ) ( ) ( )ij i j ij ijE t T t t= (3.5)

3.4.1 Background SubtractionThe number of counts for real events without coincidences, C*ij(t), is found by

subtracting the background counts from the raw counts.

ij ij ijC (t) C (t) B = (3.6)

The background counts for each sector are determined by an average of the events in the last

100 channels of the raw counts.

100

ij ij

1B C (t)

100

last

= (3.7)

The PAC spectrometer used for this research is designed to record coincidence spectra.

Correlation measurements of 90o result from the detector pair (1, 2) and (0, 3) and two

correlations of 180o from the pair (1, 3) and (0, 2). The background corrected coincidence

count rates, C*ij(t), reduces to a proportionality for the angular correlation function W(ij, t).

( ) ( ) ( )1

2 2

* *1,2 0,3 2 2 2 2 2

P (0)

1C (t) C (t) w( 90 ,t) 1 A G t P cos90 1 A G t

2=

= = + = 14243

(3.8)

( ) ( ) ( )

2

* *1,3 0,2 2 2 2 2 2

P ( 1) 1

C (t) C (t) w( 180 ,t) 1 A G t P cos180 1 A G t =

= = + = +1442443

(3.9)

Substituting Eij(t) is made for the detector efficiencies, channel width and system dead

time(23). Eij(t) can be neglected since all of the detectors are of the same make and model thus

detector efficiencies are assumed to be the same for all 4 detectors. Thus, it is reasonable to

assume that this parameter will be the same for each detector and hence divided out in the

calculation of the spectrum ratio R(t), defined below. The object is to extract the perturbation

function A2G2(t) from the coincidence count rate description.

The geometric average of the amplitudes and integrals of the 90o, C*90, and 180o, C*180,

time distributions reflect the extent of he spatial correlation of the -rays, that is the anisotropy

of the -cascade.


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*90 0,3 1,2C C (90 ,t) C (t) C (t) = =o (3.10)

*180 0,2 1,3C C (180 ,t) C (t) C (t) = =o (3.11)

The spectrum ratio is a combination for the geometric averages in such a way that allows theextraction of the perturbation function.

( ) 180 90180 90

C CR t 2

C 2C

=

+(3.12)

Making the substitutions from above into the spectrum ratio gives

( )( ) ( )

( ) ( )

W 180 ,t W 90 ,tR t 2

W 180 ,t 2W 90 ,t

=

+

o o

o o(3.13)

Using the results from Equation (3.8) and Equation (3.9) Equation (3.13) becomes:

( )( ) ( )

( ) ( )

12 2 2 22

12 2 2 22

1 A G t 1 A G tR t 2

1 A G t 2 1 A G t

+ = + +

(3.14)

and reduces to:

( ) ( )2 2R t A G t= (3.15)

This relationship enables the determination of the perturbation function in terms of the

four measured angular correlations.

3.4.2 Error PropagationDue to the nature of the experimental measurements, there is a certain amount of error

that propagates through to the determination of the A2G2(t) function. Determination of the

error in the A2G2(t) function one must consider a Poisson distribution for both the coincidence

counts and background subtraction for each detector pair. According to Poisson statistics, the

error for the coincidence counts, background, and R (t) is

( ) ( )ij ijC t C t= (3.16)


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ij

ijB

B

N= (3.17)

( )

2 2

ij ijR t ij ij ij

R R

C BC B

= + (3.18)

performing the partial derivations and simplification results in

( ) ( )

( ) ( )2 2

102 13 03 12 02 13 03 12N

R t A G t * * * *02 13 03 12

C C C C B B B B

3 C C C C

+ + + + + + += = (3.19)

Were N is the number of channels averaged for the background calculation. The above

calculations for the standard deviation as given by the Poisson distribution define the error bars

in the perturbation function.

3.4.3 Nonlinear RegressionAfter the experimental perturbation function is obtained, the interaction frequencies as

well as information about line shape and probe-site population are obtained by nonlinear

regression. This procedure requires an initial set of input parameters. The computer program

takes the initial approximation for the parameters A2, i, , and site fractions fi and calculates

theoretical values from the appropriate model given in the next section Equation (3.22). These

calculated values are then compared to the experimental data using the 2 technique(24). This

technique determines the weighted difference between data and calculated values and adjusts

the free parameters until this difference is minimized, according to

( )222 1

i

N

ie c i

i

y y x

= (3.20)

where yie are the experimentally measured data, yc (xi) are the calculated values of the

perturbation function, N is the number of data points, and2(24) is the variance of the Poisson

probability distribution function as determined from the error bars of the data points.


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3.4.4 Computer Model FittingIf all of the Hf probe nuclei are in equivalent positions in the lattice, they all experience

the same interactions. Fitting the experimental A2G2(t) to the appropriate function, all of the

parameters needed to describe the EFG at the probe location can be uniquely determined. The

EFG Equation (2.13) is characterized by z-component, Vzz, which is short for 2 2V z , and the

asymmetry parameter, Equation (2.14).

Only in perfect crystals is it possible for all the probes to experience the same

environment. It is more realistic that the sample will have defects due to impurities and

random dislocations. An adequate model adjustment for small variations is to assume that

there is a distribution of EFGs about the mean Vzz that can be described by a Lorentzian line

slope function having a width of =Vzz / Vzz. With this, the perturbation function becomes

( ) ( ) ( )3

21

G cos ig to i ii

t S S t e

=

= + (3.21)

The relative width, =Vzz / Vzz, of a Lorentzian distribution characterized as the

magnitude strength of the z-component, Vzz, about the average frequency.

If there is more than one type of probe site in the crystal lattice, the experimental Gexp(t)

spectrum is a weighted sum of the G2i(t) functions characteristic of the different sites.

( ) ( )#

exp 2GMax of Sites

i i

i

t f G t = (3.22)

Where fi are the fractions of probe sites in distinct environments, and G2i(t) are the

perturbation functions as in Equation (3.21), the PAC time spectrum for the ith phase(25).

1 2 3 1Nf f f f+ + + =K (3.23)


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4 Results and Discussion

4.1 Zircon Samples StudiedCrystalline ZrSiO4 has a tetragonal unit cell with space group I41 /amd, and 4 formula

units per unit cell. All Zr atoms are 8-fold coordinated by oxygen atoms forming ZrO8

triangular dodecahedra subunits. The Si atoms are surrounded by four oxygen atoms forming

SiO4 tetrahedra subunits (Figures 1.2 & 1.3). The principal structure is a combination of

alternating edge-sharing polyhedra subunits forming chains parallel to the c-axis. Unit cells in

the alternating chain are either ZrO8 dodecahedra subunits or SiO4 tetrahedra subunits. The

chains are connected laterally by edge-sharing ZrO8 dodecahedra. Alternatively it is

advantages to view the ZrO8 dodecahedra as two interpenetrating tetrahedra ZrO4 with different

Zr-O distances. The smaller of the two tetrahedra is corner-linked with the SiO4 tetrahedra

were as the larger is edge-linked to the SiO4 tetrahedra(26).

Zirconium silicate samples used for this study were prepared with materials obtained

from two commercial suppliersAldrich andAlfa Aesar. Impurities of interest within these

samples include Hf, U, and Th. Typically, the Hf content is close to 1 at% and the U and Th

content in natural zircons can be as high as 4000 and 2000 ppm. Impurity content and quantity

was determined by neutron activation analysis is provided in Table 4-1. Most zircons also

contain Al, Ti, Fe, and P impurities at concentrations of the order 0.01 to 0.1 at% Table 4-2(27)

.Hf is chemically very similar to Zr so it can be expected that all Hf impurities are located in Zr

lattice sites. In conjunction with beta decay process of181Hf and the similarity to Zr, the Hf

impurity is a good candidate as a nuclear probe for hyperfine interaction studies on the Zr

lattice site.

Table 4-1: Impurity content determined by neutron activation analysis.

SourceHf

[at%]U

[ppm]Th

[ppm]

Aldrich 1.08(4) 208(7) 120(5)

Alfa/Aesar 0.90(4) 212(7) 160(5)


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Table 4-2: Typical impurities found in zircon samples.

Hf

[at%]

Al

[at%]

Ti

[at%]

Fe

[at%]

P

[ppm]

1 0.5 0.1 0.06 800

Ra

[pCi/g]

U

[ppm]

Th

[ppm]

Pb

[ppm]

70 200 100 50

4.2 Results of the PAC Experiments

4.2.1 Aldrich Zircon ResultsData for PAC studies are collected as a function of temperature. First, radiation damage

as a result of activating the probe nuclei in the neutron bath must be eliminated. The radiation

damage is removed during the annealing process while bringing the sample up to the initial

collection temperature of 1052o C. Spectra are then collected for incrementally decreasing

temperatures down to 600o C and will be referred to as cooling cycle data. A final set of

spectra were collected for incrementally increasing temperatures to verify reproducibility of the

data and is referred to as heating cycle data. Heating cycle spectra were recorded at 775o C,

874o C, 975o C, and 1030o C. Representative PAC spectra for the Aldrich Zircon sample are

presented inFigure 4-1. Control of the sample temperature was held to within + 1o C.


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Figure 4-1: Typical PAC spectra (left) and their Fourier transforms (right) for the

Aldrich zircon sample.

Annealing of damage due to neutron irradiation occurs during the initial heating cycle.It has been shown that annealing of neutron radiation damage is complete in the vicinity of

300o C (29). Without the annealing process, unwanted interference in the collected spectra

appears in the form of broader line widths and a small set of random background frequencies.

This is why it is standard practice to collect the initial data set at higher temperatures. The

Fourier transform of A2G2(t) at the annealing temperature provide good initial values i and

needed for the least squares fitting function.

Modulation characteristics for all of the recorded spectra are well fitted with a

single-site model for the electric quadrupole interaction Eq: 3.22. This is to be expected due to

the similarity of the Hf to Zr chemistry. It is reasonable to assume that all of the probe nuclei

are situated in well-defined environments throughout the lattice structure of the sample.

The EFG parameters as a function of temperature, q quadrupole interaction frequency,

asymmetry, and frequency distribution are show in Figure 4-2. q vs. T decreases linearly

with temperature.

0 10 20 30 40 50 60

t [ns]

A2

G2

(t)

1052 C

800 C

600 C

0 500 1000 1500 2000

[Mrad/sec]

1052 C

800 C

600 C


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Figure 4-2: Temperature dependence of the fitted EFG parameters

during the heating and cooling cycle. Measurements for cooling cycle

and heating cycle are represented as open and filled symbols,

respectively.

630

635

640

645

650

655

660

665

q

[MHz]

Cooling

Heating

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0

10

20

30

40

50

60

550 650 750 850 950 1050 1150

T[C]

[

MHz]


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37

The anisotropy A2 data (Figure 4-3) are fairly constant from 1050o C and below for the

initial cooling cycle data. Below 850o C there is a decrease in the value of the data to a final

value near 0.07. The anisotropy parameter A2 is typically between 0.11 and 0.07 for these

measurements. What seems unusual is that the A2 values steadily decay with increasing

temperature during the heating cycle as well. Similar behavior has been observed in other

zircon samples around the same temperature range(29).

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

550 650 750 850 950 1050 1150

T[C]

-A2

Cooling

Heating

Figure 4-3: Anisotropy as a function of temperature for Aldrich.

4.2.2 Alfa/Aesar Zircon ResultsThe Alfa/Aesar data was collected for incrementally decreasing temperatures down to

500o

C after the initial heating to 1100o

C. Representative PAC spectra for the Alfa/AesarZircon sample are shown in Figure 4-4. Control of the sample temperature was held to within

+ 1o C.


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Figure 4-4: Typical PAC spectra (left) and their Fourier

transforms (right) for the Alfa/Aesar zircon sample.

The EFG parameters as a function of temperature, q quadrupole interaction frequency,

asymmetry, and frequency distribution are show in Figure 4-5. q vs. T decreases linearly

with temperature.

0 500 1000 1500 2000

[Mrad/sec]

1100 C

850 C

500 C

0 10 20 30 40 50

t [ns]

A2

G2

(t)

1100 C

850 C

500 C


47/63


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The anisotropy A2 data in Figure 4-6 are fairly constant throughout the entire

temperature range, with an average value of 0.11.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

400 500 600 700 800 900 1000 1100

T[C]

-A2

Figure 4-6: Anisotropy as a function of temperature for

the Alfa/Aesar data.

4.3 Discussion

Using the PAC technique, experimental evidence of a phase transition as described by

Mursic et al.(4) is expected to be observed as a step like change ofq and or or a change in the

slope ofq vs. T. It is anticipated that the most likely indication of a displacive phase transition

in the vicinity of the Zr site near 800o C will be observed in the q vs. T and the vs. T plot.

Changes in the vs. T plot are also possible, but not as likely. If the phase transition takes

place about all probe sites, a vs. T plot will not indicate a step-like deviation in the slope

since indicates the distribution over all probe sites.

The anticipated structural change is described as a rearrangement of the Zr-O

coordination. This reordering process is proposed to be induced by the tilting of edge-linked


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silicate tetrahedra. The displacive structural change is mainly due to the steady increase in

volume of the dodecadeltahedra along the rigidly coupled edge sharing tetrahedra.

With increasing temperature, volume expansion of the SiO4 tetrahedra respond

differently depending on how they are linked to the ZrO4 tetrahedra. Corner-linked tetrahedra

are more loosely bound and able to rotate having little effect on the Zr-O coordination distance.

More rigidly bound edge-linked tetrahedra force the Zr-O distance to increase as the SiO4

tetrahedra tilt to compensate for the volume expansion at higher temperatures.

This structural change occurs in the temperature range where metamict zircon crystals

recrystalize. Details of the structural changes that occur during recrystallization are not fully

understood. It is the intent of this study to help with the understanding of the recrystallization

process and to support the evidence presented by Mursic et al.

4.3.1 AldrichTemperature dependence of the EFG parameters, quadrupole interaction frequency q,

asymmetry , and frequency distribution for the Aldrich sample show in Figure 4-2 are

described in more detail below.

The quadrupole interaction frequency plot q vs. T decreases linearly, as expected, with

temperature. A linear fit encompassing the entire data set for heating and cooling results in

(T) 689(4)MHz - 52(4) (krad/s K) Tq

= . (4.1)

There is some evidence for a slight change in the slope between 700o C & 750o C. To

asses the magnitude of the change in slope a linear regression on either side of the breaking

point was performed. In Figure 4-7, a linear regression fit for T < 700o C is

(T) 679(5)MHz - 36(7) (krad/s K) Tq = (4.2)

A linear regression fit for T > 750o C is

(T) 685(6)MHz - 47(7) (krad/s K) Tq =


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625

630

635

640

645

650

655

660

665

550 650 750 850 950 1050 1150

T[C]

q[MHz]

Figure 4-7: Regression fit for data above and below 700o

C.

Indication of a significant change in slope is indiscriminant with respect to the

magnitude of the error for T < 700o C. Not enough data points were available for T < 700o C to

significantly reduce the error and indicate an appreciable change in the linearity of the q vs. T

data.

The asymmetry data is very near zero with an average value = 0.0670.014

indicating the sample is almost axially symmetric over the measured temperature range.

Considering only the cooling cycle data there appears to be a downward step of about 40%

from the nominal values between 800o C and 850o C. These results are very similar to those

obtained in a previous study with Aldrich zircon samples, Jaeger, Rambo etal.(27).

Comparison of this small step to the one mentioned in the previous study could not be

substantiated due to the large overlapping error bars.

The asymmetry parameter is determined from the ratio of2 /1 defined in equations

2.16 and 2.17. Using the defined equations a calibration curve is created for the range of the

q > 700q


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For both the asymmetry and frequency distribution plots, the resemblance of a step-like

change is not quantifiable, but it should be noted that it is very near the temperature range

where Mursic et al. describes the subtle rearrangement of the Zr-O coordination and where

natural zircon undergoes the metamictization process to recrystalize.

To summarize, the expected observation for a change in the linearity of the q vs. T and

vs. T data was not present and the slight change in the linearity of the vs. T is not

significant. There is some evidence for a slight change in the slope between 700o C & 800o C

for q vs. T, but not enough low temperature data is available to verify.

4.3.2 Alfa/AesarTemperature dependence of the EFG parameters, quadrupole interaction frequency q,

asymmetry , and frequency distribution for the Alfa/Aesar sample show in Figure 4-5 are

described in more detail below. q vs. T decreases linearly with temperature. A linear fit

encompassing the entire data set results in

(T) 687(1)MHz - 53(2) (krad/s K) Tq

= . (4.3)

The asymmetry parameter is very near zero with an average value = 0.100.01indicating the sample is almost axially symmetric over the measured temperature range.

There is no clear step like change in the linearity of the q vs. T and vs. T data as

would be expected for a phase transition.

The frequency distribution is observed to increase linearly with decreasing

temperature below 750o

C. For temperatures above 750o

C levels off to an average of = 38(2) MHz. This type of behavior for may indicate that the lattice is transitioning or

shifting to an increasingly well ordered crystalline structure for increasing temperatures. The

transition point to where the values begin to level off are very near the temperature range where


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Mursic et al. describes the subtle rearrangement of the Zr-O coordination and where natural

zircon undergoes the metamictization process to recrystalize.

Plotting the Aldrich and Alfa/Aesar together, Figure 4-9, it is clear that the quadrupole

interactionq for both set of data are in close agreement. There are some differences in the

asymmetry and frequency distribution data primarily in the form of an amplitude shift. The

Alfa/Aesar data is shifted above the Aldrich with larger separation for decreasing temperature.

The previously described downward step in the asymmetry for the Aldrich data is not present in

the Alfa/Aesar data. It is clear that between 700o C and 850o C the asymmetry values differ

between the two samples. It is interesting to note that the error for the two data points at

750o C and 800o C for the Alfa/Aesar sample are significantly reduced.

Even more contrasting is the frequency distribution data. While both sets of data are

relatively flat within the error calculations for higher temperatures above 850o C there is a

contrasting difference for decreasing temperatures. Both sets of data seem to be experiencing a

transitional period within the same temperature range, but with two different final outcomes.

The Aldrich sample makes an arguably small downward step, but remains relatively flat. In

contrast the Alfa/Aesar data experiences a more drastic change for increasing values at lower

temperatures.

In both the asymmetry and frequency distribution data the heating cycle data taken

on the Aldrich sample seem to make a transition from the Aldrich data to the Alfa/Aesar data.

Comparison of the anisotropy A2 data, Figure 4-10, is consistent with the findings for

the asymmetry and frequency distribution data. From 750o C and above the anisotropy

behavior is consistent with only an amplitude shift between them. For decreasing temperatures

the amplitude difference shows an increasing trend. It is expected that with the addition of

another data point for the Aldrich data at 500o C the behavior would be similar to that of the

Alfa/Aesar sample and trend upward. The anisotropy average and standard deviation for

Aldrich is 0.096 0.012 and for Alfa/Aesar is 0.113 0.009.


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Figure 4-9: Temperature dependence of the fitted EFG for both

Aldrich and Alfa/Aesar Zircon samples (C = cooling data; H = heating

data).

620

625

630

635

640

645

650

655

660

665

670

q[MHz]

Alfa/Aesar-C

Aldrich-C

Aldrich-H

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0

10

20

30

40

50

60

70

80

400 500 600 700 800 900 1000 1100

T[C]

[

MHz]


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0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

400 500 600 700 800 900 1000 1100

T[C]

-A2

Alfa/Aesar-C

Aldrich-C

Aldrich-H

Figure 4-10: Combined anisotropy plot of Aldrich and Alfa/Aesar as

function of temperature (C = cooling data; H = heating data).

4.3.3 The Anisotropy Parameter

The anisotropy A2 is a nuclear constant related to the solid angle of the emitted gamma

photons during the decay process and only varies based on the type of probe nucleus.

Theoretically A2 = -0.295(5)(30) for this study, but in typical experimentation the value is

around -0.10 and -0.15. Factors that may influence the experimentally determined A2 value

includes sample size, detector solid angle, and scattering corrections due to the sample

container. For this study the A2 value was typically around 0.11 and 0.07, Figures 4-3 and 4-6.

The observed decrease in A2 at lower temperatures indicates something unrelated to thetheoretical determination of the parameter is influencing the experimental data. The concern

with the variation of A2 with temperature is because there should be no temperature

dependence with the anisotropy parameter.


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The after effect is well documented in cases of electron capture decay as in 111In to111Cd(31). Evidence of the after effect following the beta decay - process as in that of the 181Hf

probe nuclei to 181Ta is not so widely accepted(28). This is because of the long half-life (18s) of

the initial energy state (615 keV) before the first emission in the -cascade. There should be

plenty of time for the near field EFG to stabilize after the beta decay process. - transitions are

the result of the probe nuclei transforming one of its neutrons to a proton with the emission of an

electron and an antineutrino. Before decaying, the 181Hf atom has 4 valence electrons that are

engaged in the bonding of neighboring atoms. When the nucleus beta decays to 181Ta, the new

atoms equilibrium state should contain 5 valence electrons, but only 4 are present post decay.

So to achieve proper 181Ta-atomic equilibrium one more electron must be acquired. The after

effect is highly temperature dependent with reduced observable effects on the correlated

emissions at higher temperatures. For higher temperatures or in good conductors, free electrons

are quickly captured by the unstable state of the 181Ta nucleus before the emission of correlated

gamma photons. In high purity oxides the missing electron is acquired through thermal

excitation of electrons to the conduction band. At lower temperatures, the needed electron

cannot be acquired in time before the first gamma emission. If an electron is obtained after

gamma1 is emitted, then the intermediate state sees a different EFG due to its presence.

Interaction with this "extra" field gradient will cause a rapid decay of the anisotropy evident in

the apparent increased error bars of the A2G2 fit at lower temperatures (Figure 4-1). The errorbars have not changed, but rather the amplitude has decreased. The magnitude of the A2G2

modulation during the reconfiguration of the electronic shell is reduced due to the inhibited

probability of detecting correlated gamma emissions. The magnitude of the extra EFG is

localized to the probe nucleus and is unlikely to have a significant affect on the surrounding

lattice structure. Therefore observations of changing anisotropy values are an artifact of the

decreased A2G2 amplitude and do not indicate any physical changes in the lattice structure. It is

more likely a short-range observation for the environment directly around the probe nuclei for

which the PAC technique is sensitive to.

Explaining the temperature dependence of the anisotropy values for the Aldrich sample

as a consequence of the after affect is plausible though not widely accepted, but can be


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corroborated(29). As to why the same type of behavior is not observed in the Alfa/Aesar sample

is not known.


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5 Summary and Conclusion

In a paper presented by Mursicet al.(4),they report the discovery of a previously

unknown displacive structural change in zircon, ZrSiO4, in the vicinity of 1100 K or 826o C

where metamict zircon crystals recrystalize.

Evidence of the structural transition presented by Mursic et al. was obtained with

neutron diffraction that looks at the long range order of the crystalline lattice. PAC is more of a

short range nuclear technique for observing lattice structure only a few unit cells from the

probe nuclei. This study will also help in comparing the results obtained using a long range

and short range technique for studying crystalline lattice structures.

Evidence of a displacive structural transition using the PAC technique was expected to

show up as a distinct change in slope for q vs. T data indicating environmental variations about

the Zr lattice site. Unfortunately there was a lack of clear evidence to indicate a displacive

structural transition near 1100 K / 826o C as previously reported by Mursic et al.

At lower temperatures T < 700o C there were indications for a slight variation in the

slope. The slope and standard error obtained from the linear fit for T < 700o C & T >750o C

were compared with disappointing results. The standard error is greatly increased for the

T < 700o C due to only having three data points available for the linear regression. The

presence of a distinct change in the slope for the q vs. T data cannot be confirmed with the

limited number of data points below 700o C. More data collected in the lower temperature

region is needed to more conclusively indicate whether or not there is enough change in the

slope to indicate a distinctly different environment about the Zr lattice site.

Explaining the uncharacteristic linear decline of the anisotropy at lower temperatureswith the after effect phenomena seems plausible, but cannot be verified from the data contained

in this work. The decaying anisotropy for decreasing temperatures (as a result of the extra field

gradient) is likely due to a nuclear decay, but there needs to be more data collected in the lower

temperature ranges to be more conclusive. It was expected that physical changes in the lattice


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structure about the Zr lattice position would appear as a shift in the slope of the q vs. T data.

Even though the observation of a physical shift in the lattice structure was not apparent it is

interesting to note that it may be possible to further investigate the after effect phenomena with

the PAC technique. PAC is a short range order technique sensitive to the environment about

the probe nuclei. The immediate environment of the probe nuclei expected to be altered by a

lattice shift could have been easily altered as a consequence of the after effect phenomena.

Further investigation is required to substantiate as to weather or not the observa

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