# Magnetic Neutron Scattering of Thermally Quenched K-Co-Fe Prussian Blue
Analogue Photomagnet

Daniel M. Pajerowski  NIST Center for Neutron Research, Gaithersburg, MD
20899-6012, USA  Department of Physics and National High Magnetic Field
Laboratory, University of Florida, Gainesville, FL 32611-8440, USA  V. Ovidiu
Garlea  Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831-6393, USA  Elisabeth S. Knowles  Department of Physics and
National High Magnetic Field Laboratory, University of Florida, Gainesville,
FL 32611-8440, USA  Matthew J. Andrus  Department of Chemistry, University of
Florida, Gainesville, FL 32611-7200, USA  Matthieu F. Dumont  Department of
Physics and National High Magnetic Field Laboratory, University of Florida,
Gainesville, FL 32611-8440, USA  Department of Chemistry, University of
Florida, Gainesville, FL 32611-7200, USA  Yitzi M. Calm  Department of Physics
and National High Magnetic Field Laboratory, University of Florida,
Gainesville, FL 32611-8440, USA  Stephen E. Nagler  Quantum Condensed Matter
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, USA  Xin
Tong  Instrument and Source Design Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831-6393, USA  Daniel R. Talham  Department of Chemistry,
University of Florida, Gainesville, FL 32611-7200, USA  Mark W. Meisel
Department of Physics and National High Magnetic Field Laboratory, University
of Florida, Gainesville, FL 32611-8440, USA

###### Abstract

Magnetic order in the thermally quenched photomagnetic Prussian blue analogue
coordination polymer K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot$1.42D2O has
been studied down to 4 K with unpolarized and polarized neutron powder
diffraction as a function of applied magnetic field. Analysis of the data
allows the onsite coherent magnetization of the Co and Fe spins to be
established. Specifically, magnetic fields of 1 T and 4 T induce moments
parallel to the applied field, and the sample behaves as a ferromagnet with a
wandering axis.

###### pacs:

75.50.Xx, 75.25.-j, 75.30.Gw, 75.50.LK

##  I Introduction

Manipulating magnetization with photons is now a major research focus because
it may yield materials capable of dense information storage. An epitomic
example of a photomagnetic coordination polymer is potassium cobalt
hexacyanoferrate, KαCo[Fe(CN)6]${}_{\beta}\cdot$nH2O (from now on referred to
as Co-Fe, with the crystal structure shown in Fig. 1), which displays magnetic
order and an optical charge transfer induced spin transition (CTIST).Sato1996
The details of the magnetism in Co-Fe have been investigated with bulk probes
such as magnetization,Sato1999  and AC-susceptibility,Pejakovic2002  as well
as atomic level probes such as X-ray magnetic circular dichroism
(XMCD),Champion2001  and muon spin relaxation ($\mu$-SR).Salman2006  However,
we utilize neutron scattering because it is capable of extracting the magnetic
structure, including the length and direction of the magnetic moments
associated with different crystallographic positions.

Neutron scattering research has been important in understanding the structure
of materials similar to Co-Fe. For example, neutron diffraction has been used
to elucidate the location of water molecules, to identify the long-range
magnetic order, and to explore the spin delocalizetion in Prussian
blue.Buser1977 ; Herren1980 ; Day1980  Later work used similar techniques to
investigate hydrogen adsorption in Cu3[Co(CN)6]2, along with vibrational
spectroscopy,Hartman2006  and neutron vibrational spectroscopy was also
measured in Zn3[Fe(CN)6]2.Adak2010  Likewise, magnetic structure determination
with neutrons was used to explore negative magnetization in
Cu0.73Mn0.77[Fe(CN)${}_{6}]\cdot$nH2O,Kumar2008  and to extract on-site
moments in Berlin greenKumar2005 ; Kumar2004  and in (NixMn1-x)3[Cr(CN)6]2
molecule-based magnets.Mihalik2010

Figure 1: (Color online) Co-Fe unit cell. Crystallographic positions of atoms
within the unit cell are illustrated with cubes for K (cyan), Co (red), Fe
(blue), C (white), N (black), and coordinated D2O, O positions (yellow), while
the interstitial D2O density is displayed using contoured isosurfaces (green)
and Fe-C bonds are displayed as tubes (white). Details of structure
determination are presented in Section III.

To this end, we have performed neutron powder diffraction (NPD) on deuterated
Co-Fe samples in magnetic states resulting from thermal quenching. Briefly, at
room temperature, the photomagnetic Co-Fe with optimal iron vacancies is
paramagnetic, with a transition to the diamagnetic low-spin state when cooling
below nominally 200 K. It is below 100 K that applied light may convert
molecules from diamagnetic to paramagnetic and back, and below around 20 K
where this effect is most striking due to the large susceptibility of the
magnetically ordered state. However, a magnetically ordered state may also be
achieved at low temperatures by thermally quenching,Park2007 ; Chong2011
where the paramagnetic 300 K state is cooled so quickly to below 100 K that it
does not relax to the diamagnetic ground state. It is this magnetic, thermally
quenched state that we study with NPD as a function of magnetic field, while
complementary magnetization, transmission electron microscopy (TEM), Fourier
transform infrared spectroscopy (FT-IR), and elemental analysis have also been
performed on the sample. We find that Co-Fe possesses a correlated spin glass
ground state that is driven via magnetic field to behave as a ferromagnet with
a wandering axis.

##  II Experimental DetailsNISTdisclaimer

###  II.1 Synthesis

To begin preparation of KαCo[Fe(CN)6]${}_{\beta}\cdot$nD2O powder, a 75 mL
solution of 0.1 mol/L KNO3 in D2O was added to a 75 mL solution of
20$\times$10-3 mol/L K3[Fe(CN)6] in D2O, and stirred for ten minutes. While
continuing to stir, a 300 mL solution of 5$\times$10-3 mol/L CoCl2 in D2O was
added drop-wise over the course of two hours. Stirring of the final solution
was allowed to continue two additional hours subsequent to complete mixing.
Next, the precipitate was collected by centrifugation at 2000 rpm (210 rad/s)
for 10 min (600 s) and dried under vacuum. This procedure was repeated 14
times until 4.37 g of powder was collected. Potassium ferricyanide, anhydrous
cobalt chloride, and potassium nitrate were all purchased from Sigma-Aldrich.
To remove water, the potassium nitrate was heated in an oven to 110 ∘C (383 K)
for 4 h before use. All other reagents and chemicals were used without further
purification. Deuterium oxide was purchased from Cambridge Isotope
Laboratories, Inc.

###  II.2 Instrumentation

Neutron powder-diffraction experiments were conducted using the HB2A
diffractometer at the High Flux Isotope Reactor,Garlea2010  using the Ge[113]
monochromator with $\lambda$ = 2.41 $\mathrm{\AA}$ (0.241 nm). Sample
environment on HB2A utilized an Oxford 5 T, vertical-field magnet with helium
cryogenics. Neutron polarization was achieved with a 3He cell that produced
79$\%$ polarization at the beginning of the experiment and decayed to 63$\%$
polarization after 20 hours at the end of the experiment, to give an average
polarization of 71$\%$ for both up and down polarization measurements, and we
did not perform polarization analysis after the sample but instead followed
established methods for powder diffraction with polarized
neutrons.Lelievre2010 ; Wills2005  The flipping difference spectra were
obtained by subtracting the diffraction data, measured with the incident
neutron polarization parallel to the applied field and magnetization, from the
data recorded with the incident polarization antiparallel to the field.
Magnetic measurements were performed using a Quantum Design MPMS XL
superconducting quantum interference device (SQUID) magnetometer. Infrared
spectra were recorded on a Thermo Scientific Nicolet 6700 spectrometer. Energy
dispersive X-ray spectroscopy (EDS) and TEM were conducted on a JEOL 2010F
super probe by the Major Analytical Instrumentation Center at the University
of Florida (UF). The UF Spectroscopic Services Laboratory performed combustion
analysis.

###  II.3 Analysis Preparations

For NPD, 4.37 g of powder were mounted in a cylindrical aluminum can. Thermal
quenching to trap the magnetic state was achieved by filling the cryostat bath
with liquid helium and directly inserting a sample stick from ambient
temperature. To avoid hydrogen impurities, the powder was wetted with
deuterium oxide, and to avoid sample movement in magnetic fields, an aluminum
plug was inserted above the sample. To measure magnetization, samples heavier
than 10 mg were mounted in gelcaps and held in plastic straws. Thermal
quenching in the SQUID was achieved by equilibrating the cryostat to 100 K,
and directly inserting the sample stick from ambient temperature. For
measurements in 10 mT, samples are cooled through the ordering temperature in
10 mT, and for 1 T and 4 T measurements, there is no observed thermal
hysteresis. For FT-IR, less than 1 mg amounts of sample were suspended in an
acetone solution and deposited on KBr salt plates and allowed to dry. For EDS
and TEM, acetone suspensions of the powder were deposited onto 400 mesh copper
grids with an ultrathin carbon film on a holey carbon support obtained from
Ted Pella, Inc.

###  II.4 Diffraction analysis scheme

Intensities were fit to the standard powder diffraction equation with a
correction for absorption,

$\displaystyle I(\theta)$ $\displaystyle=$ $\displaystyle
A_{0}\frac{m_{hkl}|F(hkl)|^{2}}{\sin\theta\,\sin
2\theta}\;\eta(\theta)~{}~{}~{},$ (1)

with

$\displaystyle\eta(\theta)$ $\displaystyle=$ $\displaystyle
e^{-(1.713-0.037\sin^{2}\theta)\mu
R+(0.093+0.375\sin^{2}\theta)\mu^{2}R^{2}},$ (2)

where $A_{0}$ is an overall scale factor, $m_{hkl}$ is the multiplicity of the
scattering vector, $F$ is the structure factor, $\theta$ is the scattering
angle, $\mu$ is the linear attenuation coefficient, and $R$ is the radius of
the sample cylinder. For our sample and experimental arrangement, $\mu
R=0.17$, which has little effect on the observed intensities aside from scale.
The structure factor has nuclear ($F_{N}$) and magnetic ($F_{M}$)
contributions, and for unpolarized neutrons

$|F|^{2}~{}=~{}|F_{N}+F_{M}|^{2}~{}=~{}|F_{N}|^{2}+|F_{M}|^{2}~{}~{}~{}.$ (3)

On the other hand, $F_{N}$ and $F_{M}$ can coherently interfere for polarized
neutrons such that, for moments co-linear with $P$,

$|F|^{2}~{}=~{}|F_{N}+F_{M}|^{2}~{}=~{}|F_{N}|^{2}+|F_{M}|^{2}\pm
2PF_{N}F_{M}~{}~{}~{},$ (4)

where $P$ is the neutron polarization fraction and the sign of the final term
depends upon up or down neutron polarization.Schweizer2006  For nuclear
scattering,

$F_{N}(hkl)~{}=~{}\sum_{j}{n_{j}b_{j}e^{iG\cdot d_{j}}e^{-W_{j}}}~{}~{}~{},$
(5)

where the sum is over all atoms in the unit cell, $n$ is related to the
average occupancy, $b$ is the coherent nuclear scattering length, $G$ is the
$hkl$ dependent reciprocal lattice vector, $d$ is the direct space atomic
position, and $W=BQ^{2}/16\pi^{2}$ is the Debye-Waller factor. For magnetic
scattering, all coherent scattering is modeled to be along the applied field,
which is perpendicular to the scattering plane, so that

$F_{M}(hkl)~{}=~{}\frac{\gamma r_{0}}{2}\sum_{j}{m_{j}(Q)e^{iG\cdot
d_{j}}e^{-W_{j}}}~{}~{}~{},$ (6)

where $\frac{\gamma r_{0}}{2}~{}=~{}2.695$ fm, and the magnetization can be
written as

$\displaystyle m_{j}(Q)$ $\displaystyle=$ $\displaystyle\langle
L_{z}\rangle_{j}f_{L,j}(Q)~{}+~{}2\langle S_{z}\rangle_{j}f_{S,j}(Q)$ (7)
$\displaystyle=$ $\displaystyle g_{J,j}\langle J_{z}\rangle_{j}f_{J,j}(Q)$
$\displaystyle=$ $\displaystyle\langle
J_{z}\rangle_{j}(g_{L,j}f_{L,j}(Q)~{}+~{}g_{S,j}f_{S,j}(Q))~{}~{}~{},$

where $\langle J_{z}\rangle$ is the average total angular momentum, $\langle
L_{z}\rangle$ is the average orbital angular momentum, $\langle S_{z}\rangle$
is the average spin angular momentum, $f_{J}(Q)$ is the magnetic form factor
for the total angular momentum, $f_{L}(Q)$ is the magnetic form factor for the
orbital angular momentum, $f_{S}(Q)$ is the magnetic form factor for the spin
angular momentum, and $g_{J}$, $g_{L}$ and $g_{S}$ may be determined by
Wigner’s formula.Sakurai1994  The tabulated form factor values within the
dipole approximation are used for the spin and orbital form
factors.Clementi1974  Squared differences between observed and calculated
intensities were minimized using a Nelder-Mead simplex algorithm. Open-source
Python 2.7 libraries were utilized to aid in plotting routines, matplotlib
1.0.1 and Mayavi2, and computation, NumPy 1.6.1 and SciPy 0.7.2. Reported
uncertainties of fit parameters are the square root of the diagonal terms in
the covariance matrix multiplied by the standard deviation of the residuals.

Figure 2: (Color online) Neutron powder diffraction of Co-Fe at $T$ = 40 K. Observed scattering is shown as open circles (obs), a full fit including sample mount contributions is shown as a black line (calc’d), the diffuse background is illustrated with a magenta line (bgr), and the residuals of the fit are shown below the zero line with a green line (resid). The signal due to Co-Fe is emphasized with a red filling. Experimental uncertainties derived from counting statistics are smaller than the plotting symbols. Table 1: Atomic coordinates and occupancies for Co-Fe at $T$ = 40 K. atom | position | $n$ | x | y | z
---|---|---|---|---|---
Co | 4a | 1 | 0.5 | 0.5 | 0.5
Fe | 4b | 0.73 | 0 | 0 | 0
C | 24e | 0.73 | 0.212 | 0 | 0
N | 24e | 0.73 | 0.313 | 0 | 0
K | 8c | 0.135 | 0.25 | 0.25 | 0.25
O | 24e | 0.27 | 0.243 | 0 | 0
D | 96k | 0.135 | 0.303 | 0.060 | 0.060
Figure 3: (Color online) Magnetic neutron powder diffraction of Co-Fe at $T$ =
4 K as a function of applied magnetic field. The difference between the $T$ =
40 K diffractogram and the $T$ = 4 K diffractogram (open circles) along with
profile fits to intensities (red line from model #1 for 4 T and 1 T data in
Table II) are shown for (a) 4 T, (b) 1 T, and (c) 10 mT. Additionally, the
difference between the $T$ = 4 K up-neutron-polarization diffractogram and the
$T$ = 4 K down-neutron-polarization diffractogram (open circles) along with
profile fits to intensity (red line from model #5 for polarized data in Table
II) are shown for (d) 1 T. Uncertainty bars on experimental data points are
statistical in nature representing one standard deviation from the mean, using
counting statistics.

##  III Results and Analyses

The nuclear crystal structure of Co-Fe can be modeled with space group
$Fm\overline{3}m$ (No. 225), where ferricyanide molecules and cobalt ions are
alternately centered on the high symmetry points of the unit cell, with heavy-
water bound to cobalt when ferricyanide is absent, and potassium ions and
heavy-water molecules filling in voids.Herren1980 ; Buser1977 ; Hanawa2003
This structure is used as a starting point to fit the $T$ = 40 K thermally
quenched Co-Fe contribution to the measured intensity profile, Fig. 2, which
also has sample mount contributions due to $P63/mmc$ (No. 194) D2O and
$Fm\overline{3}m$ (No. 225) aluminum.Dowell1960 ; Hull1917  Incomplete
trapping of the high-temperature state in Co-Fe gives rise to a highly
microstrained nuclear structure,Hanawa2003  and to account for this effect
during refinement, we use an asymmetric double sigmoidal peak shape, namely

$y_{a2s}~{}=~{}\frac{I}{2.49w}\left(1-\frac{1}{1+e^{-\frac{\theta-\theta_{c}}{3.43w}}}\right)\left(\frac{1}{1+e^{-\frac{\theta-\theta_{c}}{w}}}\right)~{}~{}~{},$
(8)

where $I$ is the intensity, $w$ is the width, and $\theta_{c}$ is the center
of the reflection, and these fits yield an effective lattice constant of 10.23
$\mathrm{\AA}$. Observed Co-Fe reflections that can be clearly separated from
sample holder reflections are used to extract structure factors. In modeling
the unit cell, the cobalt to iron ratio was determined with EDS, while the
room temperature oxidation states with FT-IR. The carbon and nitrogen content
were established with combustion analysis, and the potassium ions provide
charge balance. Finally, the heavy-water concentration and positions were
refined along with the scale factor to fit the structure factors.

Table 2: Comparison of the eight magnetic models, as described in the text, numbered (#) $1-8$ for Co-Fe at $T$ = 4 K in different magnetic fields tabulated as “cond.”, which is shorthand for experimental condition, where “$P$” designates the data acquired with polarized neutrons. Here, “align.” is short for “moment alignment,” where + denotes parallel alignment of moments and - denotes antiparallel alignment of moments. The units of $m_{z,Co}$ and $m_{z,Fe}$ are $\mu_{B}$, and the units of M are $\mu_{B}$ mol-1. The sum of the residuals are normalized to model 1 for each experimental condition. # | cond. | align. | $g_{S,Co}$ | $g_{L,Co}$ | $g_{S,Fe}$ | $g_{L,Fe}$ | $J_{z,Co}$ | $J_{z,Fe}$ | $m_{z,Co}$ | $m_{z,Fe}$ | M | $\sum_{j}{residual^{2}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 4 T | + | 10/3 | 1 | 2/3 | 4/3 | 0.63 $\pm$ 0.02 | 0.13 $\pm$ 0.03 | 2.7 | 0.3 | 2.6 | 1.000
2 | 4 T | - | 10/3 | 1 | 2/3 | 4/3 | 0.62 $\pm$ 0.02 | 0.00 $\pm$ 0.02 | 2.7 | 0.0 | 2.4 | 1.019
3 | 4 T | + | 10/3 | 1 | 2 | 0 | 0.64 $\pm$ 0.02 | 0.12 $\pm$ 0.04 | 2.8 | 0.2 | 2.6 | 1.000
4 | 4 T | - | 10/3 | 1 | 2 | 0 | 0.63 $\pm$ 0.02 | 0.00 $\pm$ 0.11 | 2.7 | 0.0 | 2.5 | 1.020
5 | 4 T | + | 2 | 0 | 2/3 | 4/3 | 1.08 $\pm$ 0.03 | 0.16 $\pm$ 0.03 | 2.2 | 0.3 | 2.1 | 1.028
6 | 4 T | - | 2 | 0 | 2/3 | 4/3 | 1.04 $\pm$ 0.03 | 0.00 $\pm$ 0.03 | 2.1 | 0.0 | 1.9 | 1.049
7 | 4 T | + | 2 | 0 | 2 | 0 | 1.09 $\pm$ 0.03 | 0.12 $\pm$ 0.04 | 2.2 | 0.2 | 2.1 | 1.030
8 | 4 T | - | 2 | 0 | 2 | 0 | 1.05 $\pm$ 0.03 | 0.00 $\pm$ 0.03 | 2.1 | 0.0 | 1.9 | 1.048
1 | 1 T | + | 10/3 | 1 | 2/3 | 4/3 | 0.37 $\pm$ 0.03 | 0.20 $\pm$ 0.09 | 1.6 | 0.4 | 1.7 | 1.000
2 | 1 T | - | 10/3 | 1 | 2/3 | 4/3 | 0.37 $\pm$ 0.02 | 0.00 $\pm$ 0.04 | 1.6 | 0.0 | 1.4 | 1.020
3 | 1 T | + | 10/3 | 1 | 2 | 0 | 0.38 $\pm$ 0.03 | 0.12 $\pm$ 0.07 | 1.6 | 0.2 | 1.6 | 1.004
4 | 1 T | - | 10/3 | 1 | 2 | 0 | 0.38 $\pm$ 0.05 | 0.00 $\pm$ 0.11 | 1.6 | 0.0 | 1.5 | 1.019
5 | 1 T | + | 2 | 0 | 2/3 | 4/3 | 0.64 $\pm$ 0.04 | 0.20 $\pm$ 0.09 | 1.3 | 0.4 | 1.4 | 1.001
6 | 1 T | - | 2 | 0 | 2/3 | 4/3 | 0.64 $\pm$ 0.10 | 0.00 $\pm$ 0.21 | 1.3 | 0.0 | 1.2 | 1.022
7 | 1 T | + | 2 | 0 | 2 | 0 | 0.65 $\pm$ 0.04 | 0.13 $\pm$ 0.07 | 1.3 | 0.3 | 1.3 | 1.006
8 | 1 T | - | 2 | 0 | 2 | 0 | 0.63 $\pm$ 0.09 | 0.00 $\pm$ 0.08 | 1.3 | 0.0 | 1.1 | 1.021
1 | $P$ | + | 10/3 | 1 | 2/3 | 4/3 | 0.38 $\pm$ 0.01 | 0.06 $\pm$ 0.02 | 1.6 | 0.1 | 1.5 | 1.000
2 | $P$ | - | 10/3 | 1 | 2/3 | 4/3 | 0.40 $\pm$ 0.02 | 0.00 $\pm$ 0.01 | 1.7 | 0.0 | 1.5 | 1.058
3 | $P$ | + | 10/3 | 1 | 2 | 0 | 0.36 $\pm$ 0.02 | 0.18 $\pm$ 0.05 | 1.6 | 0.4 | 1.6 | 1.004
4 | $P$ | - | 10/3 | 1 | 2 | 0 | 0.40 $\pm$ 0.02 | 0.00 $\pm$ 0.02 | 1.7 | 0.0 | 1.5 | 1.065
5 | $P$ | + | 2 | 0 | 2/3 | 4/3 | 0.68 $\pm$ 0.02 | 0.06 $\pm$ 0.02 | 1.4 | 0.1 | 1.3 | 0.995
6 | $P$ | - | 2 | 0 | 2/3 | 4/3 | 0.71 $\pm$ 0.02 | 0.00 $\pm$ 0.02 | 1.4 | 0.0 | 1.3 | 1.058
7 | $P$ | + | 2 | 0 | 2 | 0 | 0.66 $\pm$ 0.03 | 0.16 $\pm$ 0.06 | 1.3 | 0.3 | 1.3 | 1.005
8 | $P$ | - | 2 | 0 | 2 | 0 | 0.74 $\pm$ 0.03 | 0.00 $\pm$ 0.05 | 1.5 | 0.0 | 1.3 | 1.079

To begin, refinement yielded interstitial heavy-water pseudo-atoms at the 8c
position ($n$ = 0.618, $B$ = 5) and the 32f position (x = 0.3064, $n$ = 0.333,
$B$ = 20), after which all other parameters were fixed (Table 1) and Fourier
components of the heavy-water were further refined to give the interstitial
distribution shown in Fig. 1. These refinements give a chemical formula of
K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot$1.42D2O. Moreover, at room
temperature, the more complete chemical formula with oxidation states of the
metal ions included is
K0.27Co${}^{2+}_{0.94}$Co${}^{3+}_{0.06}$[Fe3+(CN)6]0.58
[Fe2+(CN)6]0.15[D2O6]${}_{0.27}\cdot$1.42D2O, or more compactly represented by
Co${}^{2+}_{0.94}$Co${}^{3+}_{0.06}$Fe${}^{3+}_{0.58}$Fe${}^{2+}_{0.15}$.

Having highly ionic wavefunctions, the magnetic ground states of Co and Fe in
Co-Fe are well described with ligand field theory.Figgis2000  As displayed in
Fig. 1, the iron atoms are octahedrally coordinated by carbon atoms that
introduce a ligand field splitting parameter ($\Delta_{\mathrm{Fe}}$) of
approximately 0.70 aJ (35,000 cm-1 or 4.3 eV), and typical Fe Racah parameters
put $d^{5}-$Fe3+ into a ${}^{2}T_{2g}$ ground state, and $d^{6}-$Fe2+ into a
diamagnetic ${}^{1}A_{1g}$ ground state. Similarly, cobalt atoms are
octahedrally coordinated with oxygen and nitrogen atoms to give
$\Delta_{\mathrm{Co}^{3+}}~{}\approx~{}$0.46 aJ (23,000 cm-1 or 2.9 eV) for
$d^{6}-$Co3+ that has a diamagnetic ${}^{1}A_{1g}$ ground state, and
$\Delta_{\mathrm{Co}^{2+}}~{}\approx~{}$0.20 aJ (10,000 cm-1 or 1.2 eV) for
$d^{7}-$Co2+ that has a ${}^{4}T_{1g}$ ground state, using typical Co Racah
parameters. At temperatures much less than the spin-orbit coupling energy,
only the lowest energy total angular momentum levels are appreciably occupied,
so that the relevant states are Fe3+[$J=1/2$, $g_{J}=(2+4k)/3$, $g_{L}=4k/3$,
$g_{S}=2/3$] and Co2+[$J=1/2$, $g_{J}=(10+2Ak)/3$, $g_{L}=2Ak/3$,
$g_{S}=10/3$], where $A$ is expected to be nearly 1.5 due to the weak ligand
field, and $k$ is the orbital reduction parameter. It is worth noting that
analogous orbitally degenerate terms have been observed for
$d^{5}-$Fe3+(${}^{2}T_{2g}$) in K3Fe(CN)6,Figgis1969  and
$d^{7}-$Co2+(${}^{4}T_{1g}$) in K1.88Co[Fe(CN)6]${}_{0.97}\cdot$3.8H2O and
Na1.52K0.04Co[Fe(CN)6]${}_{0.89}\cdot$3.9H2O.Matsuda2010  Alternatively, if
interaction with the lattice drastically quenches the orbital moment, spin-
orbit coupling no longer splits the ground states and the magnetic parameters
become Fe3+[$J=1/2$, $g_{J}=2$, $g_{L}=0$, $g_{S}=2$] and Co2+[$J=3/2$,
$g_{J}=2$, $g_{L}=0$, $g_{S}=2$]. To estimate the relative proportion of the
different oxidation states in the thermally quenched state, the effective
paramagnetic moment is linearized as a function of temperature for the 300 K
and 100 K states,PajerowskiPHD  and the measured lattice constant is compared
to a weighted average of quenched and ground-state lattice constants,Chong2011
to self-consistently give
Co${}^{2+}_{0.90}$Co${}^{3+}_{0.10}$Fe${}^{3+}_{0.54}$Fe${}^{2+}_{0.19}$ for
the magnetic quenched state at 100 K and below that is analyzed in detail
herein.

Cooling the sample further, subsequent to quenching, the bulk magnetization
measured in 10 mT showed the well-documented upturn at around 15 K
corresponding to the onset of magnetic order. Therefore, additional NPD was
performed at 4 K in applied fields of 10 mT, 1 T, and 4 T, Fig. 3, to compare
to the scattering in the paramagnetic state. Furthermore, polarized NPD was
performed at 4 K in an applied field of 1 T, Fig. 3 (d), where the difference
between diffractograms for up and down incident neutron polarizations
increases signal to noise of the measured magnetic structure at reflections
with large nuclear contributions. For each of the three experimental
conditions where magnetic scattering is observed, we compare the results of
eight plausible but different models that all have moments along the applied
field. Specifically, each possible case considers various combinations of the
parallel or antiparallel alignment of Co and Fe moments when each ion
possesses either spin-only or orbitally degenerate magnetic states, Table 2.
The analyses indicate that most magnetism resides on the Co 4a site for all
models, with a parallel alignment of Fe and Co moments giving the best fits
and $\chi^{2}$ surfaces suggesting a reduced but present orbital moment on
both ions. No magnetic scattering is observed in 10 mT, and increased coherent
magnetic scattering appears with increasing field, which is consistent with
the presence of significant random anisotropy, where a correlated spin glass
(CSG) is the ground state and sufficiently large fields cause entrance into a
ferromagnetic phase with wandering axis (FWA) state or at even larger fields a
nearly collinear (NC) state.Chudnovsky1986

Analytical expressions for the magnetization process for magnets with random
anisotropy are availableChudnovsky1986  for the Hamiltonian

$\displaystyle\mathcal{H}~{}=~{}-\mathcal{J}\sum_{i,j}{S_{i}\cdot
S_{j}}-D_{r}\sum_{i}{(\hat{n_{i}}\cdot S_{i})^{2}}$ $\displaystyle-
D_{c}\sum_{i}{(S_{i}^{z})^{2}}-g\mu_{B}\sum_{i}{H\cdot S_{i}}~{}~{}~{},$ (9)

where $\mathcal{J}$ is the superexchange constant, $S$ is the spin operator,
$D_{r}$ is strength of the random anisotropy, $\hat{n}$ is the direction of
the random on-site anisotropy, $D_{c}$ is the strength of the coherent
anisotropy, $g$ is the Land$\acute{\mathrm{e}}$ factor, and $H$ is the applied
field. In the FWA regime where the applied field energy is larger than the
random anisotropy energy but much less than the exchange field, the low
temperature magnetization is

$\displaystyle M_{FWA}$ $\displaystyle=$ $\displaystyle
M_{S}-\frac{6\sqrt{2}D_{r}^{2}\Omega
M_{S}}{5\pi^{2}a^{3}(z\mathcal{J})^{3/2}(H+H_{C})^{1/2}}$ (10)
$\displaystyle=$ $\displaystyle
M_{S}\left(1-\frac{D_{FWA}^{1/2}}{(H+H_{C})^{1/2}}\right)~{}~{}~{},$

where $M_{S}$ is the saturation magnetization, $\Omega$ is the integrated
local anisotropy correlation function, $z$ is the number of magnetic
neighbors, $a$ is the mean distance between neighboring spin sites, $H_{C}$ is
the coherent anisotropy field, and $D_{FWA}$ is a measure of the random
anisotropy to superexchange strengths. For the NC phase that is reached when
the applied field and coherent anisotropy field are much larger than the
exchange field, the low temperature magnetization is

$\displaystyle M_{NC}$ $\displaystyle=$ $\displaystyle
M_{S}-\frac{4D_{r}^{2}M_{S}}{15a^{6}(H+H_{C})^{2}}$ (11) $\displaystyle=$
$\displaystyle M_{S}\left(\frac{1-D_{NC}^{2}}{(H+H_{C})^{2}}\right)~{}~{}~{},$

where $D_{NC}$ is a measure of the random anisotropy. Based upon the magnetic
ordering temperature, Co-Fe is expected to be in an FWA-like phase, and
although both FWA and NC expressions may be fit to the low temperature
magnetization data, Fig. 4, the parameters extracted by the NC fit are not
consistent with the derivation limit for Eq. 11, further suggesting an FWA-
like state.

Figure 4: (Color online) SQUID magnetization of Co-Fe at $T$ = 2 K.
Experimental magnetization is shown as open circles (SQUID magnetization) and
model fits as a red line (model fit), where FWA and NC are visually
indistinguishable; $M(1~{}\mathrm{T})=1.1\pm 0.1~{}\mu_{B}$ mol-1 and
$M(4~{}\mathrm{T})=1.6\pm 0.2~{}\mu_{B}$ mol-1. The units of $M_{S}$ are
$\mu_{B}$ mol-1, and $H_{C}$, $D_{FWA}$, and $D_{NC}$ are all units of Tesla.
Uncertainty bars represent one standard deviation from the mean, where
statistics are generated by measuring the magnetization of the 14 synthesis
batches required to generate 4.37 g for NPD.  Figure 5: (Color online) An
illustration of magnetic structure for different magnetic field regimes. Here,
the magnetic field points towards the top of the page, short arrows represent
iron moments, and long arrows represent cobalt moments. (a) The Co-Fe sample
cooled in zero field has a CSG-like state with no average on-site moment, as
shown for the measurement of magnetic scattering in 10 mT, Fig. 3 (c). (b) The
application of magnetic field cants the moments towards the field (Fig. 3 (b)
and (d) ), and (c) larger fields induce larger average moments (Fig. 3 (a) ).
(d) More complicated mesoscopic states that contain texture are also possible,
but are not unambiguously determined with our data.

##  IV Discussion

We have presented neutron diffraction and bulk magnetization measurements of
K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot$1.42D2O that suggest a CSG ground
state that enters a FWA-like state in applied magnetic fields of the order 1 T
and larger (Fig. 5). This conclusion is based upon the field dependence of the
magnetization, and particularly the diffraction experiments that show an
absence of long range order in the 10 mT data and ordered moments that are
induced by the applied field, ruling out a high-field domain magnetization
process. This random-anisotropy-based magnetization process explains the
appreciable slope observed for Co-Fe even in fields of 7 T at 2 K (Fig. 4), in
a way similar to the magnetization process at low temperature and high
magnetic fields for the vanadium tetracyanoethylene molecular magnet prepared
using solvent based methods.Zhou1993  This magnetization process is different
than for other reported cubic complex cyanide systems that have magnetically
ordered ground states and saturate magnetization at 2 K and 7 T.Mihalik2010 ;
Kumar2004 ; Kumar2005  The CSG ground state is consistent with previous AC-
susceptibility measurements of K1-2xCo1+x[Fe(CN)6]$\cdot y$H2O (0.2
$\leq~{}x~{}\leq$ 0.4, $y~{}\approx$ 5) that showed glassy
behavior,Pejakovic2002  although the relative orientation we find for Co and
Fe at 1 T is contrary to the XMCD experiment that reported antiparallel Co and
Fe at 1 T in Rb1.8Co4[Fe(CN)6]${}_{3.3}\cdot$13H2O and
K0.1Co4[Fe(CN)6]${}_{2.7}\cdot$18H2O.Champion2001  For fields of 1 T and 4 T,
we find a parallel alignment of Co and Fe moments minimizes the residuals
between model and data, and this alignment is clearly seen for the low-angle 4
T peaks, Fig. 6. However, the lack of coherent scattering for CSG means we
cannot strictly discuss the nature of the superexchange interaction in the
ground state, although we infer some significant ferromagnetic character based
upon the high-field regime. One must also be careful about applying
qualitative Goodenough-Kanamori rules to this system,Goodenough1955 ;
Goodenough1958 ; Kanamori1959  as the single-electron states are not only
mixed from electrostatic interactions, but also due to the aforementioned
presence of spin-orbit coupling. A band structure calculation using a full
potential linearized augmented plane wave method resulted in an
antiferromagnetic ground state for Co-Fe, with +0.296 $\mu_{B}$ on the Co site
and -0.280 $\mu_{B}$ on the Fe site,takegahara2002  although such values do
not agree with experimental findings.

For the Co-Fe presented, care was taken to ensure that the average particle
size was greater than $\approx$100 nm (as measured by TEM) to avoid finite
size effects,Pajerowski2007  and the FWA-like phase can explain the previously
reported changes in low temperature high field magnetization with particle
size as a tuning of the local random anisotropy with size, larger particles
requiring higher fields to saturate at base-temperature as they are deeper in
a glassy phase. The saturation value for bulk
K0.27Co[Fe(CN)6]0.73[D2O6]${}_{0.27}\cdot 1.42$D2O, $M_{S}=2.7\pm
0.3~{}\mu_{B}$ mol-1, is comparable to a variety of similar states with
different degrees of orbital reduction on Co and Fe sites; $exempli~{}gratia$,
considering complete orbital moments on both Co and Fe gives $M_{S}\approx
2.5~{}\mu_{B}$ mol-1 and spin-only moments gives $M_{S}\approx 3.2~{}\mu_{B}$
mol-1.

The NPD experiments show a model-independent increase in coherent magnetic
scattering as a function of field, and a slightly form-factor dependent ratio
of Co to Fe moments, Table 2. A parallel alignment of Co and Fe is found for
both 1 T and 4 T, and when an antiparallel alignment is forced, Fe moments go
to zero to achieve best fits. The moment ratio is heavily dictated by the
low-Q peaks where the form-factor has little effect, while the scale of the
moments is different depending upon the presumed shape of the scatterer.
Previous neutron diffraction measurements have shown covalency effects, due to
$\sigma$-bonding and $\pi$-back-bonding with CN, to be important in the
chemically similar CsK2[Fe(CN)6].Figgis1990  Covalency can increase the
direct-space size of the moments, thereby decreasing the reciprocal-space size
even in the presence of orbital moments. For Co-Fe, the smaller reciprocal-
space form-factors give magnetizations most similar to those determined by
SQUID, although covalency makes assignment of orbital and spin magnetism based
upon spatial distribution inconclusive. We do not refine the form-factor in
this manuscript because high parameter covariance is introduced. Finally,
small quantitative differences between SQUID and NPD moment values may also be
due to sample inhomogeneity overestimating the NPD moments, and unitemized
experimental uncertainties due to the complicated and highly un-stoichiometric
formulation of the Co-Fe material, but our conclusions remain robust with
respect to such perturbations. The line-widths for magnetic and nuclear NPD
are similar, suggesting comparable domain sizes for the scattering objects,
but it is possible that the induced moments have a texture over some other
length scale, Fig. 5 (d), a possibility suggested by cluster-glass behavior in
AC susceptibility studies.Pejakovic2002  As shown in the Appendix, regions of
coherent magnetization at an angle $\xi$ from the applied field would rescale
the measured NPD longitudinal moment by
$\cos\xi\sqrt{\frac{2}{\cos^{2}\xi+1}}$, but the similarity between the
polarized and unpolarized magnetic diffraction further suggests that such an
effect is small in our samples.

Figure 6: (Color online) Visual comparison of magnetic configurations for Co-
Fe. Here, the low angle H = 4 T data (open circles) are set side by side with
model #1 for 4 T from Table II (thick-solid, red line), model #2 for 4 T from
Table II (dotted, black line), a ferrimagnetic structure with a 3:1 Co:Fe spin
ratio (dashed, blue line), and a ferromagnetic structure with a 3:1 Co:Fe spin
ratio (thin-solid, black line). Uncertainty bars are representative of one
standard deviation from the mean, using counting statistics.

##  V Conclusions

The neutron diffraction and bulk magnetization measurements of Co-Fe suggest a
magnetization process that evolves from a correlated spin glass to a quasi-
ferromagnetic state with increasing magnetic field, where average Co and Fe
moments are induced to lie along the applied field. When considering memory
storage applications of molecule based magnetic materials, structure-property
relationships that may give rise to coherent and random anisotropy will be
important to consider.

###### Acknowledgements.

DMP acknowledges support from the NRC/NIST post-doctoral associateship
program. Research at High Flux Isotope Reactor at ORNL was sponsored by the
Scientific User Facilities Division, Office of Basic Energy Sciences, U. S.
Department of Energy. This work was supported, in part, by NSERC, CFI, and NSF
through Grants No. DMR-1005581 (DRT) and No. DMR-0701400 (MWM).

*

##  Appendix A Effect of Canting on Intensity

A powder sample consisting of domains canted at an angle $\xi$ away from the
applied field, with random rotational distribution, may give the same
unpolarized NPD signal as domains along the field, but with a different
magnetic moment. With the magnetic field along the z-axis, and the scattering
vector along the x-axis, the uncanted magnetic moment is simply

$\mathbf{M_{u}}~{}=~{}(0,0,M)~{}~{}~{},$ (12)

and the canted moment can be expressed as

$\mathbf{M_{canted}}~{}=~{}M(\sin\xi\cos\phi,\sin\xi\sin\phi,\cos\xi)~{}~{}~{},$
(13)

where $\xi$ is the canting angle, $\phi$ is the rotation angle about the
field, and $M$ is the magnitude of the magnetic moment. The interaction
vectorSchweizer2006

$\left|\mathbf{M_{\bot}}\right|^{2}~{}=~{}\sum_{\alpha,\beta}{\left(\delta_{\alpha\beta}-\widehat{Q}_{\alpha}\widehat{Q}_{\beta}\right)M^{*}_{\alpha}M_{\beta}}~{}~{}~{},$
(14)

then gives a dependence of the intensity on the canting angle such that

$\left|\mathbf{M_{u,\bot}}\right|^{2}~{}=~{}M^{2}~{}~{}~{},$ (15)

and

$\left|\mathbf{M_{canted,\bot}}\right|^{2}~{}=~{}M^{2}{\frac{\cos^{2}\xi+1}{2}}~{}~{}~{},$
(16)

where random $\phi$-angles have been averaged over. Therefore, an uncanted
model with a z-component (relevant to compare with longitudinal magnetization)
of $M$ can give the same unpolarized NPD intensity as a canted model with a
z-component of $M\cos\xi\sqrt{\frac{2}{\cos^{2}\xi+1}}$.

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