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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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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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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
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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