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USING THE EASYSPIN TOOLBOX IN MATLAB TO ANALYZE THE PRESSURE DEPENDENT MAGNETISM OF AN S = 1, ONE-DIMENSIONAL SPIN CHAIN
By
Orlando Trejo
Department of Physics Submitted as an Undergraduate Thesis
April 8, 2019
UNIVERSITY OF FLORIDA
2019
© 2019 Orlando Trejo
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To my mother, Carla Trejo
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ACKNOWLEDGEMENTS
Throughout my undergraduate research career I have received plenty of
support and advice from many people. Here I would like to acknowledge some of the
individuals and institutions who have helped me in various ways.
First, I would like to thank my advisor, Mark W. Meisel. His guidance was
valuable in helping me understand and complete this work. He shared his knowledge of
scientific concepts and his experimental expertise all while challenging me to solve and
ponder on the most fundamental aspects of the project. I would also like to
acknowledge the graduate student in our laboratory, John M. Cain, who trained me on
using several lab instruments. John never hesitated to answer any of my questions and
often expanded on the science at work, so a thanks goes to him for all of his insight on
much of the phenomena I encountered in the lab and on this project.
I would also like to thank Jared Singleton and Dr. Engelhardt at Francis
Marion University in Florence, South Carolina, who provided an analysis of the
Quantum Monte Carlo simulations on S = 1, Heisenberg chains. A thanks also goes to
Erik Čižmár, our colleague at P.J. Šafárik University in Košice, Slovakia, who I did not
have the pleasure of meeting in person, but nonetheless helped me with several coding
issues with the EasySpin toolbox.
Finally, aspects of this work and research experience were made possible by
funding from the National Science Foundation (NSF) supporting the National High
Magnetic Field Laboratory (DMR-1644779) and the single-investigator research and
training activities of Meisel Group (DMR-1708410).
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TABLE OF CONTENTS page
ACKNOWLEDGEMENTS ................................................................................................ 4
LIST OF FIGURES ........................................................................................................... 6
ABSTRACT ...................................................................................................................... 7
CHAPTER
1 INTRODUCTION ....................................................................................................... 8
NBCT and the Quantum Critical Point ....................................................................... 8EasySpin and the interacting Hamiltonian ............................................................... 10 Test Systems ........................................................................................................... 12 Summary ................................................................................................................. 14
2 TESTING THE CAPABILITIES OF EASYSPIN ....................................................... 15
Chain length dependence of QMC simulations ....................................................... 15 The Padé Approximations ....................................................................................... 16 Impurities ................................................................................................................. 17 Summary ................................................................................................................. 19
3 RESULTS ................................................................................................................ 20
Simulation results of D and J ................................................................................... 20
4 Conclusions and Future ........................................................................................... 22
APPENDIX
A Analayzing the E term in the Zero-Field splitting interaction .................................... 23
B Comparing simulations of the new library functions with previous algorithms of the EasySpin toolbox ............................................................................................... 24
C Nearest-neighbor interaction dependence on EasySpin versus the Pade approximation .......................................................................................................... 25
REFERENCES ............................................................................................................... 26
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LIST OF FIGURES
Figure page 1-1 Nearest neighbor interactions for the NBCT complex ......................................... 10
1-2 NBCT magnetic suscpetibility as a function of temperature taken at various pressures. ............................................................................................................ 11
1-3 Microscopic comparison of spin exchange interaction of trimer system of the EasySpin simulations against the data. ............................................................... 14
1-4 Giant Spin analysis of the single-ion anisotropy and comparing the EasySpin simulations to the data. ....................................................................................... 15
2-1 Chain length dependence of QMC simulations ................................................... 16
2-2 Padé Approximations compared to the EasySpin simulations. ........................... 18
2-3 Evaluating Simulations and the NBCT susceptibilities at ambient pressure. ...... 19
3-1 Magnetic susceptibility sesults of the simulations compared to the NBCT complex at several pressures. ............................................................................. 21
A-1 Susceptibility dependence of E term ................................................................... 24
A-2 Comparing library functions with previous homemade algorithms ...................... 25
A-3 Difference plot between EasySpin simulation and the Padé Approximation with the parameters shown ................................................................................. 26
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USING THE EASYSPIN TOOLBOX IN MATLAB TO ANALYZE THE PRESSURE DEPENDENT MAGNETISM OF AN S = 1, ONE-DIMENSIONAL SPIN CHAIN
By
Orlando Trejo
Department of Physics Undergraduate Thesis
April 8, 2019
The EasySpin toolbox, which operates within MATLAB software, was originally
designed to analyze electron paramagnetic resonance spectra. Yet, previous versions
of the toolbox (<v4.5) have been used to generate algorithms which allowed the
isothermal magnetization and temperature-dependent magnetic susceptibility to be
modelled. Now, EasySpin (>v5.2) contains library functions which allow these magnetic
properties to be calculated. The first step was to compare the simulation results of the
new version against the previous algorithms. Next, the magnetism of the spin trimer,
[Mn3O(O2PPh2)3(mpko)3]ClO4, was simulated and these results are discussed.
Ultimately, the goal was to analyze the magnetic behavior of the S = 1, one-dimensional
spin chain [Ni(HF4)(3-Cpy)4]BF4 at several pressures. The results indicate a trend that
the nearest-neighbor interactions weaken and the single ion anisotropy remains at
D ≈ 4.5 K, as the pressure rises. The implications of these results are discussed.
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CHAPTER 1 INTRODUCTION
1.1 NBCT and the Quantum Critical Point
The magnetic properties of [Ni(HF4)(3-Cpy)4]BF4 (NBCT) have previously been
studied at ambient pressure [1] and were characterized with the Hamiltonian
ℋ = 𝐽 𝑆! ∙ 𝑆!!!"!
+ 𝐽´ 𝑆! ∙ 𝑆![!"]
+ 𝐷 (𝑆!!)!!
− 𝜇!𝐵 ∙ 𝑔 ∙ 𝑆 (1-0)
where the interactions and notation are discussed later. Manson et al. [1] report on the
nearest-neighbor coupling and single-ion anisotropy of NBCT which are interpreted in
section 1.2. When considering the two-dimensional array of chains with nearest
neighbor interactions, as seen in figure 1-1, the interchain coupling terms, J´, are
negligible, indicating that NBCT consists of nearly isolated S = 1, one-dimensional
magnetic chains. This observation allowed the susceptibility of the system to be fitted to
that of an S = 1 Heisenberg chain with J = 4.86 K. Furthermore, Manson et al. report1 a
single ion anisotropy of D ≈ 4.3 K. These results place the D/J ratio near unity, where a
quantum critical point (QCP) in energy space is found, separating the Haldane phase
from the Large-D phase [2].
1 The single-ion anisotropy (D) was studied using the Angle Overlap Model with UV-vis data [1].
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Figure 1-1: Cartoon of Ni chain array showing the intrachain interactions, J, and interchain interactions J´, used in Equation (1-0).
Cain et al. [3, 4] have used pressure in an attempt to drive NBCT through the
QCP, near D/J ~ 1. Using a SQUID magnetometer and a measurement probe equipped
with a pressure cell, the temperature-dependent magnetic susceptibility was measured
at various pressures (figure 1-2).
The experimental and theoretical aspects of the QCP and the associated
quantum phase transition go beyond the scope of this work. Here the goal is to use the
EasySpin toolbox in MATLAB software to simulate and predict the pressure effects on
the low-field magnetic interactions. In the following section the methods and interacting
Hamiltonian that are used in the onset of the analysis are described.
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Figure 1-2: Plot of susceptibility vs. temperature of NBCT (randomly oriented single crystals) measured at various pressures [3, 4].
1.2 EasySpin and the Interacting Hamiltonian
The EasySpin toolbox is a free to use program that operates within the
framework of the MATLAB software [5]. It was originally developed by electron
paramagnetic resonance (EPR) spectroscopists with the main function to simulate and
fit a wide range of EPR spectra. However, the newest versions of EasySpin (>v5.2.23)
now contain library functions which calculate a number of magnetometry parameters
such as the temperature-dependent magnetic susceptibility and the isothermal
magnetization of clusters and chains.
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EasySpin has the capability of incorporating various interactions and system
conditions2 into the computations, and therefore, the most dominant interactions that
describe NBCT should be considered. It is important to specify that in the following
analysis, the magnetism is characterized for randomly oriented crystals in the low-field
limit, as 𝐵⟶ 0, where the temperature is varied from 2 K to room temperature.
When a weak external magnetic field, 𝐵, is applied to a spin system the Zeeman
effect accounts for energy splitting according to
ℋ!""#$% = −𝜇!𝐵 ∙ 𝑔 ∙ 𝑆 , (1-1)
where 𝜇! is the Bohr magneton, 𝑔 is the Landé g-tensor and 𝑆 is the spin vector.
Unpaired electrons also give rise to unquenched angular momentums at low
temperatures, which create sufficiently strong crystal fields that may lift the degeneracy
and produce more energy states. Specifically, Ni(II) chains often show a splitting of the
S = 1 triplet state given by a gap D, the so called Single Ion Anisotropy (SIA) [6]. This
splitting can be characterized by the contribution to the Hamiltonian
ℋ!"# = 𝐷 (𝑆!!)!!
+ 𝐸 (𝑆!!)!!
− (𝑆!!)! .
(1-2)
The E term is left out of the simulations as its effects are small and most apparent at
temperatures below the region of applicability (shown in figure A-1). Setting E = 0,
ℋ!"# = 𝐷 (𝑆!!)!!
. (1-3)
2 EasySpin allows the user to apply system conditions such as indicating an open chain or closed chain with periodic boundary conditions (PBC). If the sample is a single-crystal, one can also set the crystal orientation of the crystal [5].
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Combining Equations (1-1) and (1-3) with the Heisenberg Hamiltonian which only
takes into account the intrachain interactions, J, we arrive at the Hamiltonian assumed
to describe the magnetic behavior of the NBCT system,
ℋ = 𝐽 𝑆! ∙ 𝑆!!!"!
+ 𝐷 (𝑆!!)!!
− 𝜇!𝐵 ∙ 𝑔 ∙ 𝑆 , (1-4)
and when interactions to a neighboring chain are included as J´, Equation (1-0) is
realized.
EasySpin can therefore be used to simulate and extract the nearest-neighbor
interaction term, J, and a single-ion anisotropy term, D. In the following section simple
test systems are considered in order to demonstrate the capabilities of the EasySpin
toolbox.
1.3 Test Systems
As a first approach, the library functions in EasySpin (v5.2.23) were compared
with previous “homemade” algorithms which were used to report on the magnetism of
Mn(II) and Fe(II) trimers [7]. The Murray group used these Easyspin (<v4.2) algorithms
to simulate the temperature-dependent magnetic susceptibility and isothermal
magnetization. This work was successfully reproduced when using the new library
functions in EasySpin (v5.2.23) (Appendix B, figure A-2).
Next, the magnetic susceptibility of the spin trimer, [Mn3O(O2PPh2)3(mpko)3]ClO4,
was simulated and compared to previous fitting results obtained by Olajuyigbe et al. [8].
When determining the nearest-neighbor coupling, an equilateral model was used where
all nearest-neighbor interactions were assumed equal and no single-ion anisotropy was
considered (D = 0). According to the report, an exchange term of J = 16.0(1) cm-1 and
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g = 1.94(1) are expected. The simulation, shown in figure 1-3, indicates a J term and
Landé g-factor in agreement with the literature.
Figure 1-3: Simulations results using the EasySpin toolbox (blue) and the Christou group fit compared to the manganese data.
Olajuyigbe et al. then used a Giant Spin model with a total ground state spin,
S = 6 and J = 0 in order to determine the single ion anisotropy. The reduced
magnetization (M/𝑁!!) was measured at several field strengths and the literature gives
D = -0.29(2) cm-1 and g = 1.94 [8]. Accordingly, the EasySpin simulations are consistent
with D = -0.29 cm-1 and g = 1.94, as shown in figure 1-4.
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Figure 1-4: The reduced magnetization, using the GSM with S = 6, plotted as a function of field per temperature unit.
1.4 Summary
The magnetic properties of NBCT at ambient pressure place the system near a
QCP separating the Haldane phase from the Large-D phase [2]. These results have
motivated the method of using pressure to drive NBCT through a quantum phase
transition [3, 4]. In this section, the spin chain Hamiltonian was introduced and the
dominant interactions were presented.
It was demonstrated that the new library functions are consistent with the
“homemade” algorithms implemented on previous versions of the toolbox to analyze
simple clusters. Furthermore, comparisons show that EasySpin reproduces the results
for the low-field, temperature-dependent, D = 0 analysis of the susceptibility and the
isothermal, high-field Giant Spin magnetization analysis of a spin trimer reported in the
literature. In the next section the precision in the simulations is established by
comparing them to results obtained from the Quantum Monte Carlo (QMC) method.
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CHAPTER 2 TESTING THE CAPABILITIES OF EASYSPIN
2.1 Chain length dependence of QMC simulations
One of the main restrictions in using the EasySpin toolbox is the limitation on the
number of spin sites. The EasySpin simulations were restricted to a maximum of eight
spins on a standard laboratory computer, as it was not feasible to include more sites
due to the excessive memory requirement. Yet, the Quantum Monte Carlo method has
the ability to simulate thousands of sites. The following analysis considers the scaling of
the thermodynamic limit with chain-length for QMC results of Heisenberg
antiferromagnetic chains, which will later be used to determine the precision of the
EasySpin simulations.
Figure 2-1: Plot showing the magnetic susceptibility per spin versus temperature with
varying chain length of the QMC simulations. Work by Jared Singleton and Larry Engelhardt, FMU, Florence, South Carolina.
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Jared Singleton and Dr. Engelhardt at Francis Marion University (FMU) in
Florence, South Carolina, produced QMC simulations of the magnetic susceptibility per
spin as a function of temperature for S = 1 chains of varying length, N. It was observed
that for a closed chain with periodic boundary conditions, the thermodynamic limit is
approached quickly, such that N = 60 is effectively the same as N = ∞. Comparatively,
figure 2-1 shows the results for an open chain. At low temperature the thermodynamic
limit is approached slowly and N = 60 is close to the N = ∞ limit for T/J > 0.5. Hence,
assuming J ≈ 5 K, as predicted by Manson at al. [1], the QMC simulations will suffice for
temperatures T > 2.5 K.
2.2 The Padé Approximations and the T/J ≥ 1 rule
The Quantum Monte Carlo (QMC) method can be used to fit the temperature-
dependent magnetic susceptibility of Heisenberg antiferromagnetic chains (HAFCs) to a
high precision. However, the QMC method is a computationally intensive task and in
2013, Law et al. decided to use the well-known Padé Approximations (PA) to
approximate the QMC results of HAFCs for spin susceptibilities of S = 1, 3/2, 2, 5/2 and
7/2 [9].1 In the published work all calculations were performed for N = 2000 spin sites
and the deviations between the PA and fitted data were reported to be of the same
order of magnitude as the QMC simulation error. Hence, the precision of the EasySpin
simulations can be analyzed by comparing them to the PA established by Law et al. [9].
As described in section 1.1, the nearly isolated magnetic chains of NBCT allow
the susceptibility to be modelled by an S = 1 HAFC with J ≈ 5.0 K, g ≈ 2.10 and D = 0.
Hence, a PA was constructed using these parameters and the magnetic susceptibility
1 The Padé approximations were completed on closed chains using periodic boundary conditions using D = 0 and an antiferromagnetic spin-exchange term J [9].
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was simulated in EasySpin. Figure 2-2 shows the comparison, where the inset indicates
the difference. Further analysis on the J dependence reveal that the simulations stay
within 1% of the PA for temperature values when T/J ≥ 1 (figure A-3). Therefore, the
N = 8 simulations scale as N = 2000 for T/J ≥ 1.
Figure 2-2: Padé approximation of QMC results vs. EasySpin simulations. Inset shows the difference plot which remains within 1% difference down to T = 5 K.
2.3 Impurities
In order to model the sharp rise at low temperature, an intrinsic contribution of
paramagnetic impurities was included so that the total susceptibility is given by
𝜒!"#$% = 1 − 𝑝 𝜒!"# + 𝑝𝜒!" (2-1)
where 𝑝 indicates the amount of impurities present in the sample. In the following
simulations an impurity level of 7% was assumed. The magnetic susceptibility data
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acquired at ambient pressure was then simulated using Equation (2-1) and with the
parameters given by Manson et al. The simulation stays within 1% of the data for
T/J ≥ 1 and the results are shown in figure 2-3.
Figure 2-3: Plot of the susceptibility vs Temperature. In red is the data at ambient pressure. In black is the simulation taken to a temperature of T = 4.86 K. The Landé g-factor is proportional to the magnetic moment and therefore scales
the high temperature saturation limit of the magnetic susceptibility, which remains fairly
constant as a function of pressure. Furthermore, the methods of synthesis indicate the
impurity factor should be the same in all data sets. Therefore, the 7% impurity
assumption and g = 2.10 are considered fixed for the rest of the analysis.
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2.4 Summary
The QMC results for HAFCs reaches the thermodynamic limit with a minimum of
60 spin sites for T/J > 0.5. In the case where QMC simulations of 2000 sites were used,
the PA constructed by Law et al. [9], effectively approximates the QMC results to within
the fitting error. Hence, the PA were implemented as a means to test the EasySpin
simulations to the QMC results.
The PA was then constructed with the parameters presented by Manson et al.
[1]. With D = 0 and J as the parameter, it was shown that EasySpin simulations
(8 spins) remain within 1% difference of the QMC results (2000 spin) for T/J ≥ 1. Finally,
the temperature-dependent magnetic susceptibility data acquired at ambient pressure
was simulated with D > 0. A small intrinsic paramagnetic contribution of 7% was
required to simulate the low temperature tail and achieve best agreement with the
experimental data.
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CHAPTER 3 RESULTS
3.1 Simulations results of D and J
The temperature-dependent magnetic susceptibility was simulated for several
pressures using Equation (1-4) with a 7% impurity contribution, as in Equation (2-1) and
g = 2.10. As the pressure was increased an overall trend of decreasing J terms was
observed. Subsequently, according to the T/J ≥ 1 rule, smaller J values correspond to
simulation curves with a larger temperature range. This observation allowed the
simulation program to give a better indication of the single-ion anisotropy, a low
temperature effect, for higher pressures. The curves were simulated invoking Equation
(1-4) and a summary of the extracted parameters is shown in table 3-1.
Figure 3-1: Plot of the susceptibility of the NBCT sample at several pressures. The black lines indicate EasySpin simulations using finite D and J. In blue are simulations with D = 0 and finite J.
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Table 3-1: Parameters extracted from the simulation in figure 3-1.
Pressure (GPa) J (K) D (K) g ~D/J
1.47 2.12(2) 4.5(8) 2.11 2.1 1.05 2.25(2) 4.6(7) 2.11 2.0 0.35 3.62(2) 0 – 5 2.11 ?? 0.00 4.83(2) 4.3
(Manson et al.) 2.11 0.9
For pressures up to 0.35 GPa, EasySpin gives no indication of the single-ion
anisotropy as D = 0 and D = 5 K remain within the same percent difference. At higher
pressure, the nearest-neighbor coupling term decreases. Imposing the T/J ≥ 1 rule, the
simulations at high pressure become valid through lower temperatures and as a result,
the D term was predicted at about D ≈ 4.5 K.
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CHAPTER 4 CONCLUSIONS AND FUTURE
The results predict an increasing D/J ratio with pressure that reaches a value of
D/J ≈ 2.1. This change may indicate that NBCT is indeed being driven through the QCP
at D/J ~ 1 [2]. NBCT is therefore a promising candidate as a complex that can be
pressure tuned through a quantum phase transition. At this point the parameters
calculated using EasySpin are only plausible due to the assumptions considered in
developing the model and more work remains to be done.
First, the paramagnetic contribution to the temperature-dependent magnetic
susceptibility was included as a method to account for the low-temperature tail, shown
figure 1-2. However, the actual content of impurities is not known and the effect seen at
low temperature may be quantum mechanically driven.
When analyzing the single ion anisotropy, it is apparent that the D term effect on
the susceptibility is more dominant at lower temperatures. Yet, the uncertainty in the
single ion anisotropy is still quite large even at 1.44 GPa. This may come as a result of
the low number of spin chains used in the simulations.
Single crystal synthesis is currently being discussed as a way to better analyze
the magnetism of NBCT. Additionally, integration of the EasySpin toolbox with the
HiPerGator2 computing system is forthcoming. The success of this endeavor will allow
NBCT to be simulated with more spin sites and open the possibilities for studying other
clusters and chains more effectively.
2 HiPerGator is a higher computing system part of The University of Florida.
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APPENDIX A: Analyzing the E term in the Zero-Field splitting interaction
Figure A-1: Susceptibility versus temperature plotted for several values of the E term
shown in equation (1-2).
From Equation (1-2), the E term effect on the simulation was studied. Figure A-1
shows the susceptibility plotted for several valued of E. It is clear that the E term affects
the characteristic upturn of the data. However, this effect occurs at low enough
temperatures that for the purposes of simplifying the simulations the E term is
subsequently left out, i.e. E = 0.
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APPENDIX B: Comparing the new library functions with previous algorithms of the EasySpin
toolbox
Figure A-2: Simulations with new EasySpin library function. Plot shows the
reproducibility compared to the work by Guillet et al. [7].
Previously, the Murray group studied the magnetism of Fe(II) and Mn(II)
complexes with EasySpin (<v4.2) [7]. The magnetism was simulated by appropriating
“homemade” algorithms with the EasySpin version at the time. A first analysis involved
comparing the new library function of EasySpin (v5.2.23) to the previous algorithms.
The simulation in the new version were constructed with the same parameters as
previously predicted. In the plot shown above, the trimer consisted of two equal nearest
neighbor interaction, J’ and a third value J.
There is a small deviation in the manganese simulations that seems to grow as
the temperature increases. The effect may be due to thermal fluctuation which increase
with temperature, however at this time the deviation has not been explored further.
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APPENDIX C: Nearest-neighbor interaction dependence on EasySpin versus the
Padé approximation
Figure A-3: Difference plot between EasySpin simulation and the Padé Approximation
at different J values.
The J dependence was analyzed for J values above and below the predicted nearest-
neighbor interacting term of J ≈ 5.0 K for NBCT. The difference between the EasySpin
simulations and the Padé approximation with the same parameters was then plotted.
The difference remains below 1% for T/J ≥ 1, for all J terms considered.
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REFERENCES
[1] J.L. Manson et al., Inorg. Chem. 51 (2012) 7520.
[2] S. Hu, B. Normand, X. Wang and L. Yu, Phys. Rev. B 84 (2011) 220402(R).
[3] J. M. Cain et al., to be submitted (2019).
[4] M.K. Peprah et al., Bull. Am. Phys. Soc. MAR19 (2019) L38.04, http://meetings.aps.org/Meeting/MAR19/Session/L39.4
[5] S. Stoll, EasySpin – EPR Spectrum Simulation, www.easyspin.org/.
[6] C.P. Landee and M.M. Turnbull, J. Coord. Chem. 67 (2014) 375.
[7] G.L. Guillet et al., Chem. Commun. 49 (2013) 6635.
[8] O.A. Adebayo, K.A. Abboud, and G. Christou, Inorg. Chem. 56 (2017) 11352.
[9] J.M Law, H. Benner, and R.K. Kremer, J. Phys. Condens. Matter 25 (2013) 65601.