United States

The synthesis of single-crystalline and single-domain ferromagnetic nanowire materials has been an area of intense research,&lt;sup&gt;1-3&lt;/sup&gt; however the control of geometric parameters (diameter, lengths, tip morphology and uniformity) in nanowire syntheses is mostly empirical. Here, we demonstrate a simple method for precise control of geometric parameters in nanowire synthesis by establishing both steady-state growth conditions and a uniform rate of particle growth through the continuous and controlled additions of precursors. To prepare Co nanowires, we used both the &#39;LaMer&#39; and &#39;extended LaMer&#39; processes, which allows us to fine control over diameter, aspect ratio, and morphology uniformity of the nanowires with reproducibility. Meanwhile, the structural property analysis of a series of nanowires revealed both qualitative and quantitative information on the growth kinetics of the nanowires. It has been demonstrated how the precursor flow rate, metal precursor mole ratio and the reducing ability of long-chain amine affect the nanowire morphology. We then show how such fine geometric control affects the magnetic coercivity of nanowire assemblies. Based on the shape anisotropy and orientation of the nanowire assemblies, a record high room-temperature coercivity exceeding 10 kOe has been measured in Co nanowires with a diameter of 8 nm.

![](https://assets.underline.io/uploads/markdown_image/1/image/30b2d410cd16b38f3094e1caeb5b2bd1.jpg)

Fig. 1 Schematic illustration of the effect of (a) flow rate of Co-precursor, (b) mole ratio of RuCl3 to Co-precursor and (c) long-chain amine content on the growth kinetic of Co nanowires. (d-f) The TEM images of Co nanowires prepared through the continuous flow of Co-precursor at various decreasing flow rates.

2022 Joint MMM-INTERMAG

Effects of growth kinetics on structural and magnetic properties of Co nanowires

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The synthesis of single-crystalline and single-domain ferromagnetic nanowire materials has been an area of intense research,1-3 however the control of geometric parameters (diameter, lengths, tip morphology and uniformity) in nanowire syntheses is mostly empirical. Here, we demonstrate a simple method for precise control of geometric parameters in nanowire synthesis by establishing both steady-state growth conditions and a uniform rate of particle growth through the continuous and controlled additions of precursors. To prepare Co nanowires, we used both the 'LaMer' and 'extended LaMer' processes, which allows us to fine control over diameter, aspect ratio, and morphology uniformity of the nanowires with reproducibility. Meanwhile, the structural property analysis of a series of nanowires revealed both qualitative and quantitative information on the growth kinetics of the nanowires. It has been demonstrated how the precursor flow rate, metal precursor mole ratio and the reducing ability of long-chain amine affect the nanowire morphology. We then show how such fine geometric control affects the magnetic coercivity of nanowire assemblies. Based on the shape anisotropy and orientation of the nanowire assemblies, a record high room-temperature coercivity exceeding 10 kOe has been measured in Co nanowires with a diameter of 8 nm.

![](https://assets.underline.io/uploads/markdown_image/1/image/30b2d410cd16b38f3094e1caeb5b2bd1.jpg)

Fig. 1 Schematic illustration of the effect of (a) flow rate of Co-precursor, (b) mole ratio of RuCl3 to Co-precursor and (c) long-chain amine content on the growth kinetic of Co nanowires. (d-f) The TEM images of Co nanowires prepared through the continuous flow of Co-precursor at various decreasing flow rates.

technical paper

Ferrites are the most used permanent magnets (PMs) due to their availability and low cost, being an alternative to PMs based on critical rare-earths in new applications such as electromobility [1]. In this work we will show the possibility of 3D-printing objects using strontium ferrite powder obtained by successful recycling of the residue generated from manufacturing commercial ferrite magnets. Tuning the morphology and microstructure of the Sr-ferrite residue by post-processing made possible to end with Sr-ferrite powder with PM properties superior, not only to the residue precursor, but to those of the brand new commercial powder (3.5 fold increase in coercivity and a 25% increase in remanence) [2]. The recycled ferrite was used as precursor for synthesizing composites by the solution casting method [3], and extruding filaments for advanced manufacturing of magnets (with a particle content up to 65 wt%) by Fused Filament Fabrication. Magnetic characterization of composites, filaments and printed objects was performed by Vibrating Sample Magnetometer (VSM), showing that the magnetization scales accordingly to the content of ferrite particles and that there is no deterioration of the PM properties of the starting particles along processing (Fig. 1). Scanning Electron Microscopy (SEM) analysis showed a homogeneous distribution of the particles in the polymer matrix (Fig. 2) [4]. This work has shown an efficient path for developing new and alternative PMs by the combination of recycling and 3D-printing. Acknowledgements. Authors acknowledge financial support from MICINN by NEXUS project (PID2020-115215RB-C21), industrial collaboration PLASMAG with IMA S.L.U. and from the Regional Government of Madrid by NanoMagCOST project (P2018/NMT-4321).

![](https://assets.underline.io/uploads/markdown_image/1/image/ac2f3c775797ab289c0ec1e59559f011.jpg)

Fig. 1. (a) First and second quadrants of the hysteresis loops of starting powder and composites with different content of ferrite. (b) Normalized hysteresis loops of the starting powder, composite and 3D-printed object. Insets show a 3D-printed piece based on ferrite particles.

![](https://assets.underline.io/uploads/markdown_image/1/image/ce076f52b81f87799a6418acd5c64cfb.jpg)

Fig. 2. SEM images of: (a) cross section and (b) closer view of a magnetic filament with a 50 wt% of ferrite; and (c) and (d) inner morphology of a 3D-printed object with a particle content of 65 wt%.

A Sustainable Route for Permanent Magnets Fabrication: Additive Manufacturing Applied to Recycled Ferrite Residues

Additive manufacturing (AM) of composites is attracting much interest in high-tech sectors for fabricating complex high-performance objects with tailored properties [1]. For permanent magnets (PMs), the challenge is to develop magnets by AM with no geometrical restrictions, high filling factor (FF), and non-deteriorated PM properties [2], together with finding alternatives (e.g., improved ferrites and promising MnAlC-based alloys) to rare earth-based magnets [3]. Composites (PM particles/polymer) were synthesized by solution casting, followed by extrusion of filaments for 3D-printing, being reported for the very first time (Fig. 1). Several alternative PM materials were studied (gas-atomized τ-MnAlC, Sr-ferrite and hybrids —Sr-ferrite/NdFeB). Particle size and fine-to-coarse particles ratio (FP/CP) play a key role on the flexibility and powder loading of MnAlC filaments (length&gt;10 m), reaching FF&gt;80% and non-deteriorated PMs properties (Fig. 1) [4]. These results will be compared to the obtained for filaments based on Sr-ferrite (Hc=239 kA/m, FF=92%), NdFeB (Hc=812 kA/m, FF=93%) and hybrids (Hc=629 kA/m, FF=90%) (Fig. 2). MnAlC-based objects were 3D-printed under controlled temperature, proving that alternative PM materials can be efficiently synthesized and processed to develop novel PMs by AM (Fig. 1) [4]. Acknowledgements. Authors acknowledge collaborations with B. Skårman, H. Vidarsson and P.-O. Larsson (Höganäs, Sweden) by the industrial contract GAMMA, and A. Nieto and R. Altimira (IMA, Spain), and financial support from: EU M-ERA.NET by COSMAG project (PCI2020-112143); MICINN by NEXUS project (PID2020-115215RB-C21); CM by NanoMagCOST project (P2018/NMT-4321).

![](https://assets.underline.io/uploads/markdown_image/1/image/3da8e2e0919c7ea3ed6b01fc3af543f4.jpg)

Fig. 1. (a) Gas-atomized MnAlC, polymer, and composite; (b) SEM image of MnAlC particles and size distribution; (c), (d) image and SEM cross section of a MnAlC/ABS filament; (e) MnAlC-based 3D-printed object; (f) normalized magnetic response of MnAlC-based samples; and (g) magnetic flux density of a 3D-printed MnAlC-based disc.


![](https://assets.underline.io/uploads/markdown_image/1/image/b00d3d1a8d0386d3ac76cfe5a8bfe75e.jpg)

Fig. 2. (a) SEM and EDX of a hybrid composite; (b) second quadrant of the hysteresis loops for the hybrid, Sr ferrite- and NdFeB-based materials. Insets show composite pellets and SEM cross section of a Sr ferrite-based filament.

Optimizing the Particle Size Distribution in Permanent Magnet Composites to Extrude Flexible Filaments for Additive Manufacturing

3D printing of permanent magnets has recently received considerable attention [1,2]. In previous work the realization of isotropic magnets was demonstrated using fused filament fabrication, which results in the fabrication of polymer bonded magnets, with properties very similar to state-of-the-art magnets fabricated by injection molding [1]. In Ref [3] the easy axis of the printed magnet was aligned by printing the magnetic filament on magnetized hard magnets. Within this talk we will present a significant step forward by producing magnets where the easy axis can be locally aligned in-situ during the printing process. This is realized by redesigning the printer nozzle of a fused filament fabrication printer (FFF) so that permanent magnets can be placed next to the nozzle. The magnetic strayfield of the permanent magnets is optimized so that it aligns the easy axis of anisotropic ferromagnetic particles inside a paste-like compound material when it is in the melted state. Two different magnetic materials are investigated: (i) Strontium hexaferrite inside a PA6 matrix (Sprox® 10/20p), fill grade: 49 vol% and (ii) Sm2Fe17N3 inside a PA12 matrix, fill grade: 44 vol%. In the presence of the external alignment field, the Sr ferrite particles inside the PA6 matrix can be well aligned as shown in Fig. 1, where the hard axis hysteresis loop (blue) and easy axis hysteresis loop (orange) is shown. The ratio of the remanence of the hard axis loop and the easy axis hysteresis loop is 2.2, which shows a significant alignment. No significant alignment for Sm2Fe17N3 could be achieved, where the ratio of the easy and hard axis remanence is just 1.15. The results show the feasibility to fabricate magnets with arbitrary and locally defined easy axis, since the permanent magnets (or alternatively electromagnet) can be mounted on a rotatable platform. Hence, highly optimized magnets can be produced such as Hallbach arrays which can not be produced by any other method.

![](https://assets.underline.io/uploads/markdown_image/1/image/738ec0c61c56217e9864e877356c108d.jpg)

Hysteresis loops in hard and easy axes of a hard magnetic ferrite (Sprox) measured with the VSM. The 3d printed magnetic material is aligned in-situ during the print process by permanent magnets next to the printer nozzle.

In-situ alignment of anisotropic hard magnets of 3D printed magnets

Additive manufacturing (AM) via 3D printing technologies have become a frontier in materials research, including its application in the development and recycling of permanent magnets1,2. The limited availability of the rare earth elements (REEs) for magnets present a demanding challenge and necessitates reduction in the losses of REEs from conventional or subtractive manufacturing. AM allows for a minimal waste production and re-utilization of recovered rare earths in the magnet processing. In-situ alignment during 3D printing of magnetic materials has opened new horizons for manufacturing of complex permanent magnets3,4. The introduction of functionalized magnetic 3D printing as an AM process allows rapid prototyping of multi-directionally aligned magnetic systems, and reduced energy requirement due to the elimination of post-production alignment3. In this work, we have developed a robust multiphysics model to simulate the alignment of magnetic particles in a polymer matrix, during 3D printing. Magnetization vs. field measurements of extruded and printed Sm-Co bonded in PLA (15 vol.%) and Nd-Fe-B+Sm-Fe-N bonded in Nylon 12 (65 vol.%), with and without alignment, confirmed the alignment of magnetic particles. A parametric analysis was performed to understand the effect of process variables, like alignment field, loading fraction, printing speed and particle size, on the degree of alignment of the samples (Fig.1)5. The model is upgraded to consider variable particle size distributions, non-homogenous particle spacing and different printing temperature. The model predicts a competing behavior between particle-fluid and particle-particle interactions under magnetic field. The model provides a framework to efficiently predict the DoA in tandem with a functionalized-magnetic 3D printer and allows the user to adjust the operating parameters according to the desired DoA. This work is supported by the Critical Materials Institute (CMI), an Energy Innovation Hub funded by the U.S. Department of Energy (DOE), Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office. Ames Laboratory is operated for the U.S. DOE by Iowa State University of Science and Technology under Contract No. DE-AC02-07CH11358.

![](https://assets.underline.io/uploads/markdown_image/1/image/900c7448b55f774cce490496f838c293.jpg)

Modeling in-situ magnetic alignment during magnetic 3D printing

Strontium ferrite/polyamide12 (PA12) magnetic composite fused deposition modeling filaments made using a twin-screw extruder are being studied to be used for Magnetic Field Assisted Additive Manufacturing (MFAAM) of permanent magnets. A previous study shows that the magnetic viscosity (MV) varies with field and field angle. The Mx MV is maximum in the easy plane (1.1%) and decreases with a factor of two along the filament’s macroscopic hard axis [1]. The My MV is nearly zero along the hard axis and in the easy plane but maximum for intermediate angles. Here we study the time dependence as a function of temperature using a MicroSense EZ9 biaxial VSM to determine the melt viscosity. Samples were made by loading ~14 mg of SrO(Fe2O3)6/PA12 (26 or 40 wt%) cylindrical-shaped filament parts in pyrex glass tubes. The filament is compacted with PEEK stoppers on both sides to secure its position and avoid direct exposure to the atmosphere while heating. Our aim is to study the rotation of the magnetic particles in the molten PA12 matrix. We first heat the sample to 200 oC to orient all the particles in the y-direction using a field of 22 kOe. Then the sample temperature is lowered to the target temperature after which the field is lowered to zero. Finally, a field step is applied in the x-direction and Mx and My are monitored as a function of the time. The rotation of the moment vector with time is calculated from the data. Measurements are done for the fields of 500 and 2000 Oe, and at the temperatures of 120, 124, 128, 132, and 136°C. Up to 128°C, no strong time dependence was observed. Between 128 and 132°C, the magnetic particles orientations are no longer fixed in the matrix, but they can rotate and line up along the field direction. The time dependence of the rotation of the magnetic particles in the PA12 matrix is calculated from Mx and My (Fig. 1 and Fig. 2). The time constant decreases with temperature and field. The melt viscosity of the molten PA12 matrix at 136 oC is determined from the data is 1.4E5 Pa.s at 500 Oe field.

![](https://assets.underline.io/uploads/markdown_image/1/image/252cfa512dda08ad988cba344352f796.jpg)


![](https://assets.underline.io/uploads/markdown_image/1/image/2e86b69ac993ac1e74be8a59fc537023.jpg)

Time Dependence Analysis as a Function of the Temperature of Magnetic Polymer Composite Filaments

Soft materials embedded with hard magnetic particles have recently gained widespread recognition as small-scale actuators due to their capacity to be a rapid and precise shape-shifting material[1, 2]. Strontium ferrite (SrFe12O19) particles have been shown to be a great candidate for such applications, since it is an inert hard magnetic material [3]. The study of magnetic properties of hard-magnetic soft materials is a critical step to better design these upcoming actuators. The preparation of such material is done by mixing the particles into the uncured elastomer (PDMS), in liquid form, and then pouring the mixture in a mold for curing. If the samples are subjected to a magnetic field during the curing process, chains of particles are formed in the direction of the applied field (see Fig 1). We expect that the individual particles rotate in the fluid such that their easy axes becomes parallel to the curing field, thus creating an easy axis in the composite in this same direction. The magnetic properties of such types of composites, or for any other soft matrix, cannot yet be found in literature. In this study, we analyzed three concentrations of strontium ferrite particles in PDMS under three field configurations, resulting in 9 different samples. The concentrations used were 1:1, 2:1 and 4:1 ratios of PDMS to strontium ferrite per weight. All three samples of each concentration were cured either in a zero magnetic field, or over the north pole of a permanent magnet, or over the side of said magnet (see Fig 2). A biaxial vibrating sample magnetometer (VSM) was used to measure torque curves and the hysteresis curves parallel and perpendicular to the curing field. The samples cured in a field show a squareness ratio of up to 0.94 while the samples cured in zero field, only close to 0.5. The samples cured in a field show a magnetic anisotropy with an easy axis parallel to the curing field. Anisotropy values are compared to the crystal anisotropy of strontium ferrite to determine the shape anisotropy contribution due to the observed chaining.

![](https://assets.underline.io/uploads/markdown_image/1/image/862eb027f9f4f8e973ec3a62a4ee66d8.jpg)

Fig. 1 PDMS/strontium ferrite composite showing alignment of particles when cured under magnetic field.

![](https://assets.underline.io/uploads/markdown_image/1/image/cd5f70171405db2e6e00e095c1d763be.jpg)

Fig. 2 Curing process of samples

Magnetic Properties of PDMS Embedded with Strotium Ferrite Particles Cured Under Different Magnetic Field Conditions

Networks of interconnected magnetic nanowires offer an exciting platform to explore 3-dimensional (3D) nanomagnetism, where their structure, topology and frustration may be used as additional degrees of freedom to tailor the materials properties. We have achieved quasi-ordered metallic networks over cm-scale areas. Using a cost-effective multiple azimuthal angle ion-tracking and etching method with suitable fluence, interconnected nanoporous polycarbonate membranes were first fabricated. A subsequent electrochemical deposition process led to interconnected metallic nanowire networks, with density as low as 40 mg/cm3 (Fig. 1) [1]. New magnetization reversal mechanisms in cobalt networks (Fig. 1d) are captured by the first-order reversal curve (FORC) method, which demonstrate the evolution from strong demagnetizing dipolar interactions to intersections-mediated domain wall pinning and propagation, and eventually to shape-anisotropy dominated magnetization reversal (Fig. 2). By varying the applied magnetic field strength, orientation, and sequence, it is in principle possible to selectively address a certain subset of the nanowire networks, encode digital information into their magnetic state, and propagate it through the networks for potential 3D magnetic memory and logic applications. Such network of magnetic nanowires also has the potential to implement repeatable multi-state memristors in neuromorphic circuits, due to the expected discrete nature of domain wall motion through the intersecting networks. Interestingly, another random configuration of metallic nanowire networks has found applications in deep-submicron particulate filtration, relevant to combatting COVID-19 and air pollution, due to the extremely large surface areas and the excellent mechanical strength [2]. This work has been supported by the NSF (ECCS-1611424 and ECCS-1933527), SRC/NIST nCORE SMART center, DTRA, U.C. Davis and Georgetown University.

![](https://assets.underline.io/uploads/markdown_image/1/image/235e34862100e61fc1216607d02eed81.jpg)

Figure 1: (a) Side-view and (b) top-view SEM images of an interconnected copper network, tracked at five angles as illustrated by the schematics. Inset in (a) shows that at an intersection the wires are physically connected. Top-view SEM images of (c) a copper network and (d) a cobalt network, tracked at three angles. This produces intersections in planes that contain the three irradiated angles (which are highlighted with white dashed lines).

![](https://assets.underline.io/uploads/markdown_image/1/image/46695a346d2d61a65f1ffb4d1ef4c0ad.jpg)

Figure 2. FORC diagrams shown in 2D contour plots for (a-c) a single-angle irradiated sample, and (d-i) three-angle irradiated sample. The schematic diagrams in the lower right corner of panels (a-i) indicate the orientation of the wires (black lines) and membrane (blue box) relative to the applied field shown by the large black arrow.

3D Nanomagnetism in Interconnected Magnetic Nanowire Networks

The fabrication of micrometric magnetic materials and their integration into portable devices reveal great interests for telecommunications, automotive, biomedical and space applications, but still remain highly challenging. The bottom-up approach for nanostructured magnets is an interesting alternative route to the classical rare earth metallurgy or thin film deposition. Indeed, the recent progresses in the magnetic nanoparticle synthesis allow a good control of the particle size, shape and chemical composition. Single-crystalline Co nanorods (NRs), which combine a high magnetization and a large magnetic anisotropy, can be prepared by the polyol process [1]. These anisotropic particles constitute building blocks of primary choice for permanent magnet applications. [2] We reported recently a versatile approach for the fabrication and direct integration of nanostructured magnetic materials of controlled shaped at specific locations onto silicon substrates [3]. The magnetophoresis-assisted capillary assembly of magnetic nanoparticles leads to the fabrication of high-performance Co-based permanent magnets (Figure 1). Integrated submillimeter magnets as well as millimetric self-standing magnets exhibiting magnetic properties competing with NdFeB-based composited were obtained through this cost- and time-efficient process. The proof-of-concept of electromagnetic actuation of a MEMS cantilever by means of these supercrystals highlights their potentiality as efficient integrated magnetic materials within nomadic devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/ea95a49ae8b5e560ab72881a728b9d7b.jpg)

Nanostructured magnets prepared by magnetophoresis-assisted capillary assembly of Co nanorods

Nanostructured magnets prepared by magnetophoresis-assisted capillary assembly of Co nanorods

We study the effect of thermal fluctuations on dipole coupled mesospins, the building blocks of artificial spin ice (ASI), in a micromagnetic framework. Taking into account the internal magnetic texture, we focus specifically on the fluctuations at the edges, which have previously shown to exhibit a temperature dependent switching between two states that are essentially half of a C- or S-state [1]. We study two different systems, the first being a system of two coupled magnets and the second being a fourfold coordinated ASI vertex. For the two mesospin system with parallel magnetization, we find magnonic mode splitting at the edges, which is relatively unaffected by temperature. When the magnetization in the two mesospins is opposite, we find the appearance of an asymmetric potential, which can be quantified through the magnetic texture correlations. The Arrhenius law is used to find the attempt frequency and the energy barriers associated with switching, the latter are found to be inversely proportional to the spacing. We find a logarithmic dependence of the attempt frequency on the energy barrier, which is indicative of entropy-energy compensation at small spacings (large barriers). The temperature dependent edge dynamics for the ASI vertex is evaluated for the four different types (Type I-IV), each displaying their own characteristic dynamics. An example of such dynamics is the collective switching of the handedness associated with the perpendicular edge magnetization in a type IV vertex configuration. These results indicate the presence of rich texture dynamics, even at the level of the individual building blocks of the ASI.

Magnetic Texture Fluctuations in Coupled Mesospin Systems

Artificial spin-ice systems (ASI) have been studied for 15 years, allowing detailed studies to examine the frustration, ordering and monopole transport in a variety of two-dimensional geometries [1]. The nanostructuring of magnetic materials in three-dimensions presents a new paradigm in condensed matter physics and allows access to a wide range of new physical phenomena [2]. With respect to ASI systems, there are several reasons why 3D geometries are interesting. This includes the realisation of a degenerate ice-rule, mimicking of bulk spin-ice geometries and studying the influence of surfaces upon ordering. In this talk, I will outline the realisation of a 3D artificial spin-ice system [3] which takes the geometry of a diamond-bond lattice (Fig. 1), capturing the arrangement of spins in bulk pyrochlore systems. By using magnetic force microscopy (Fig. 2), I will show that the vertex states upon the surface can be determined, and the global switching of the lattice can be measured. We find very different magnetic charge dynamics along two principal directions upon the lattice. For a field applied along the surface termination, local energetics force magnetic charges to nucleate over a larger characteristic distance, reducing their magnetic Coulomb interaction and producing lone, uncorrelated monopoles. In contrast, applying a field transverse to the surface termination yields large numbers of highly correlated monopole-antimonopole pairs. A combination of micromagnetic and Monte-Carlo simulations are used to understand the results. These suggest it is the difference in effective chemical potential that yields the striking differences in monopole transport. We anticipate that our study will inspire a new generation in artificial spin-ice research whereby the ground states in complex 3D frustrated lattices are explored.

![](https://assets.underline.io/uploads/markdown_image/1/image/68e61fe5f53c1aa5f59122b57dafa546.jpg)

Fig 1. Atomic force microscopy image of the 3DASI system. Distinct layers within the system are labelled L1 - L3.

![](https://assets.underline.io/uploads/markdown_image/1/image/6da2b1e596bc9e183b4530846549ee08.jpg)

Fig 2. Magnetic force microscopy of the 3DASI surface, showing a number of ice-rule and monopole states.

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