Magnetic Susceptibility Tensor Research Articles

Characterizing the magnetic susceptibility tensor of lanthanide-containing polymethylated-DOTA complexes.

Lanthanide complexes based on the DOTA (1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid) cage are commonly used as phase contrast agents in magnetic resonance imaging, but can also be utilized in structural NMR applications due to their ability to induce either paramagnetic relaxation enhancement or a pseudocontact shift (PCS) depending on the choice of the lanthanide. The size and sign of the PCS for any given atom is determined by its coordinates relative to the metal center, and the characteristics of the lanthanide's magnetic susceptibility tensor. Using a polymethylated DOTA tag (Ln-M8-SPy) conjugated to ubiquitin, we calculated the position of the metal center and characterized the susceptibility tensor for a number of lanthanides (dysprosium, thulium, and ytterbium) under a range of pH and temperature conditions. We found that there was a difference in temperature sensitivity for each of the complexes studied, which depended on the size of the lanthanide ion as well as the isomeric state of the cage. Using 17O-NMR, we confirmed that the temperature sensitivity of the compounds was enhanced by the presence of an apically bound water molecule. Since amide-containing lanthanide complexes are known to be pH sensitive and can be used as probes of physiological pH, we also investigated the effect of pH on the Ln-M8-SPy susceptibility tensor, but we found that the changes in this pH range (5.0-7.4) were not significant.

Application of orthorhombic standardization in magnetic susceptibility studies of localized spin models with S=1, 3/2, 2, 5/2

The standardization idea is nowadays tacitly accepted in EMR area, however, its usefulness in magnetism studies has not been fully recognized as yet. This idea arises due to intrinsic features of orthorhombic Hamiltonians of any physical nature, including the crystal (ligand) field (CF/LF) Hamiltonians or the zero-field splitting (ZFS) ones. Standardization limits the ratio of the orthorhombic parameter to the axial one to a fixed range between 0 and a specific value that depends on the notation used. For the ZFS parameters expressed in the conventional spin Hamiltonian (SH) notation the ratio λ=E/D can always be limited to the range (0, ±1/3) by appropriate choice of coordinate system. Implications of standardization of orthorhombic spin Hamiltonians for interpretation of experimental magnetic susceptibility data are considered. Using a numerical example, we show the existence of alternative solutions for ZFS parameters potentially obtainable from fitting experimental magnetic data and discuss their importance. For the first time algebraic applications of the standardization to the expressions for magnetic susceptibility tensor derived earlier for localized spin models with S=1, 3/2, 2, 5/2 and with rhombic anisotropy are explored. The numerical and algebraic results allow us to formulate an 'invariance principle'. These considerations facilitate interpretation of experimental magnetic data and provide an additional check of correctness of analytical magnetic susceptibility expressions.

Magnetic Anisotropy in Functionalized Bipyridyl Cryptates.

The magnetic properties of molecular lanthanoid complexes are very important for a variety of scientific and technological applications, with the unique magnetic anisotropy being one of the most important features. In this context, a very rigid tris(bipyridine) cryptand was synthesized with a primary amine functionality for future bioconjugation. The magnetic anisotropy was investigated for the corresponding paramagnetic ytterbium cryptate. With the use of a combination of density functional theory calculations and lanthanoid-induced NMR shift analysis, the magnetic susceptibility tensor was determined and compared to the unfunctionalized cryptate analogue. The size and orientation of the axial and rhombic tensor components show remarkably great resilience toward the decrease of local symmetry around the metal and anion exchange in the inner coordination sphere. In addition, the functionalized ytterbium cryptate also exhibits efficient near-IR luminescence.

Susceptibility tensor imaging (STI) of the brain.

Susceptibility tensor imaging (STI) is a recently developed MRI technique that allows quantitative determination of orientation-independent magnetic susceptibility parameters from the dependence of gradient echo signal phase on the orientation of biological tissues with respect to the main magnetic field. By modeling the magnetic susceptibility of each voxel as a symmetric rank-2 tensor, individual magnetic susceptibility tensor elements as well as the mean magnetic susceptibility and magnetic susceptibility anisotropy can be determined for brain tissues that would still show orientation dependence after conventional scalar-based quantitative susceptibility mapping to remove such dependence. Similar to diffusion tensor imaging, STI allows mapping of brain white matter fiber orientations and reconstruction of 3D white matter pathways using the principal eigenvectors of the susceptibility tensor. In contrast to diffusion anisotropy, the main determinant factor of the susceptibility anisotropy in brain white matter is myelin. Another unique feature of the susceptibility anisotropy of white matter is its sensitivity to gadolinium-based contrast agents. Mechanistically, MRI-observed susceptibility anisotropy is mainly attributed to the highly ordered lipid molecules in the myelin sheath. STI provides a consistent interpretation of the dependence of phase and susceptibility on orientation at multiple scales. This article reviews the key experimental findings and physical theories that led to the development of STI, its practical implementations, and its applications for brain research. Copyright © 2016 John Wiley & Sons, Ltd.

Canonical Quantization of Electromagnetic Field in the Presence of Nonlinear Anisotropic Magnetodielectric Medium with Spatial-Temporal Dispersion

Modeling a nonlinear anisotropic magnetodielectric medium with spatial-temporal dispersion by two continuum collections of three dimensional harmonic oscillators, a fully canonical quantization of the electromagnetic field is demonstrated in the presence of such a medium. Some coupling tensors of various ranks are introduced that couple the magnetodielectric medium with the electromagnetic field. The polarization and magnetization fields of the medium are defined in terms of the coupling tensors and the oscillators modeling the medium. The electric and magnetic susceptibility tensors of the medium are obtained in terms of the coupling tensors. It is shown that the electric field satisfy an integral equation in frequency domain. The integral equation is solved by an iteration method and the electric field is found up to an arbitrary accuracy.

Magnetic Anisotropies in Rhombic Lanthanide(III) Complexes Do Not Conform to Bleaney's Theory.

We report a complete set of magnetic susceptibilities of lanthanide complexes with a macrocyclic ligand based on a 3,6,10,13-tetraaza-1,8(2,6)-dipyridinacyclotetradecaphane platform containing four hydroxyethyl pendant arms (L(1)). The [LnL(1)](3+) complexes are isostructural along the lanthanide series from Ce(3+) to Yb(3+), with the only structural change observed along the series being the monotonous shortening of the Ln-donor distances due to lanthanide contraction. The (1)H NMR spectra point to a D2 symmetry of the [LnL(1)](3+) complexes in aqueous solution, which provides a unique opportunity for analysis of the rhombic magnetic anisotropies with an unequivocal location of the magnetic axes. The contact contributions for the observed paramagnetic shifts have been estimated with density functional theory calculations on the [GdL(1)](3+) complex. Subsequently, the pseudocontact shifts could be factored out, thereby giving access to the axial and rhombic contributions of the magnetic susceptibility tensor. Our results show that the calculated magnetic anisotropies do not follow the trends predicted by Bleaney's theory, particularly in the case of Ho(3+) and Er(3+) complexes.

Polarized Neutron Diffraction to Probe Local Magnetic Anisotropy of a Low-Spin Fe(III) Complex.

We have determined by polarized neutron diffraction (PND) the low-temperature molecular magnetic susceptibility tensor of the anisotropic low-spin complex PPh4 [Fe(III) (Tp)(CN)3]⋅H2O. We found the existence of a pronounced molecular easy magnetization axis, almost parallel to the C3 pseudo-axis of the molecule, which also corresponds to a trigonal elongation direction of the octahedral coordination sphere of the Fe(III) ion. The PND results are coherent with electron paramagnetic resonance (EPR) spectroscopy, magnetometry, and ab initio investigations. Through this particular example, we demonstrate the capabilities of PND to provide a unique, direct, and straightforward picture of the magnetic anisotropy and susceptibility tensors, offering a clear-cut way to establish magneto-structural correlations in paramagnetic molecular complexes.

Polarized Neutron Diffraction to Probe Local Magnetic Anisotropy of a Low‐Spin Fe(III) Complex

AbstractWe have determined by polarized neutron diffraction (PND) the low‐temperature molecular magnetic susceptibility tensor of the anisotropic low‐spin complex PPh4[FeIII(Tp)(CN)3]⋅H2O. We found the existence of a pronounced molecular easy magnetization axis, almost parallel to the C3 pseudo‐axis of the molecule, which also corresponds to a trigonal elongation direction of the octahedral coordination sphere of the FeIII ion. The PND results are coherent with electron paramagnetic resonance (EPR) spectroscopy, magnetometry, and ab initio investigations. Through this particular example, we demonstrate the capabilities of PND to provide a unique, direct, and straightforward picture of the magnetic anisotropy and susceptibility tensors, offering a clear‐cut way to establish magneto‐structural correlations in paramagnetic molecular complexes.

The influence of molecular order and microstructure on the R2* and the magnetic susceptibility tensor

In this work, we demonstrate that in the presence of ordered sub-voxel structure such as tubular organization, biomaterials with molecular isotropy exhibits only apparent R2* anisotropy, while biomaterials with molecular anisotropy exhibit both apparent R2* and susceptibility anisotropy by means of susceptibility tensor imaging (STI). To this end, R2* and STI from gradient echo magnitude and phase data were examined in phantoms made from carbon fiber and Gadolinium (Gd) solutions with and without intrinsic molecular order and sub-voxel structure as well as in the in vivo brain. Confidence in the tensor reconstructions was evaluated with a wild bootstrap analysis. Carbon fiber showed both apparent anisotropy in R2* and anisotropy in STI, while the Gd filled capillary tubes only showed apparent anisotropy on R2*. Similarly, white matter showed anisotropic R2* and magnetic susceptibility with higher confidence, while the cerebral veins displayed only strong apparent R2* tensor anisotropy. Ordered sub-voxel tissue microstructure leads to apparent R2* anisotropy, which can be found in both white matter tracts and cerebral veins. However, additional molecular anisotropy is required for magnetic susceptibility anisotropy, which can be found in white matter tracts but not in cerebral veins.

Polarized Neutron Diffraction as a Tool for Mapping Molecular Magnetic Anisotropy: Local Susceptibility Tensors in Co(II) Complexes.

Polarized neutron diffraction (PND) experiments were carried out at low temperature to characterize with high precision the local magnetic anisotropy in two paramagnetic high-spin cobalt(II) complexes, namely [Co(II) (dmf)6 ](BPh4 )2 (1) and [Co(II) 2 (sym-hmp)2 ](BPh4 )2 (2), in which dmf=N,N-dimethylformamide; sym-hmp=2,6-bis[(2-hydroxyethyl)methylaminomethyl]-4-methylphenolate, and BPh4 (-) =tetraphenylborate. This allowed a unique and direct determination of the local magnetic susceptibility tensor on each individual Co(II) site. In compound 1, this approach reveals the correlation between the single-ion easy magnetization direction and a trigonal elongation axis of the Co(II) coordination octahedron. In exchange-coupled dimer 2, the determination of the individual Co(II) magnetic susceptibility tensors provides a clear outlook of how the local magnetic properties on both Co(II) sites deviate from the single-ion behavior because of antiferromagnetic exchange coupling.

Electrodynamic boundary conditions for planar arrays of thin magnetic elements

Approximate electrodynamic boundary conditions are derived for an array of dipolarly coupled magnetic elements. It is assumed that the elements' thickness is small compared to the wavelength of an electromagnetic wave in a free space. The boundary conditions relate electric and magnetic fields existing at the top and bottom sides of the array through the averaged uniform dynamic magnetization of the array. This dynamic magnetization is determined by the collective dynamic eigen-excitations (spin wave modes) of the array and is found using the external magnetic susceptibility tensor. The problem of oblique scattering of a plane electromagnetic wave on the array is considered to illustrate the use of the derived boundary conditions.

Additive properties of molecular magnetic susceptibility of chlorosubstituted naphthalenes

A new method was proposed for the calculation of the molecular magnetic susceptibility tensor of aromatic systems. The method is based on additive schemes. The possibilities of the method were analyzed for a series of chlorosubstiuted naphthalenes. It is shown that the data for substituted naphthalenes can be obtained from the molecular magnetic susceptibility tensor of naphthalene and increments describing the substitution of the hydrogen atom by chlorine. The results of application of the additive schemes are well consistent with the results of the direct quantum chemical calculation.

Transverse susceptibility and the T3/2 law in the dynamic spin-fluctuation theory

We obtain explicit expressions for elements of the magnetic susceptibility tensor in the dynamic spin-fluctuation theory. Using an analytic continuation of the Green’s functions, we show that the transverse susceptibility has spin-wave poles at low temperatures, yielding the asymptotic T3/2 law for magnetization. We derive an explicit expression for the coefficient in the T3/2 law based on the multiband Hubbard Hamiltonian and real band structure. We demonstrate the correct low-temperature behavior of magnetization in the example of iron.

Polarized Neutron Diffraction study of the molecular magnetic anisotropy

The magnetic anisotropy is a prerequisite for a metal complex to behave as a single-molecule magnet (SMM). Unfortunately, today we do not fully understand the relationships between the local structural parameters and the magnetic anisotropy that results at the molecular level. This is an issue that has become recursive in this area. Out of the synthesis work which is still important, but generates a multiplication of SMMs with frustrating properties, there are various studies to understand these relationships among which most are theoretical studies. In this context, we believe that polarized neutron diffraction (PND) can provide an experimental and complementary point of view to these theoretical studies. PND is indeed well known to allow an accurate determination of the spin density in magnetic compounds and in the field of molecular magnetism it has provided unique information on the pathways and the nature of intra- or intermolecular magnetic coupling [1]. In the case of highly anisotropic paramagnetic materials, where local magnetic moments cannot be aligned by an external magnetic field, that is more tricky, but a method based on local magnetic susceptibility tensor, has been recently developed that allows now analysing the data in this case and obtaining the magnetization distribution [2]. This approach was first used for inorganic compounds. Our idea has been to use this approach to go beyond the reconstruction of spin density to study the magnetic anisotropy in molecular systems. In this paper, we present the results of such an approach applied for the first time to metal complexes that are simple mono and dinuclear cobalt(II) complexes.

On the Question of the Magnetic Susceptibility of Fractal Ferromagnetic Wires

With the help of a well-grounded measure, we have constructed an algorithm for calculating the longitudinal component of the magnetic susceptibility tensor of a fractal ferromagnetic wire. The calculations are based on the semiclassical kinetic equation for the magnon distribution function. The temperature and frequency dependences of the longitudinal magnetic susceptibility of a fractal wire is found.

THEORETICAL STUDY OF MAGNETIC SUSCEPTIBILITY AND OPTICAL ACTIVITY OF SMALL MOLECULES CONTAINING ONE CHIRAL CENTER

In the first part of this work, correlation between optical activity and elements of magnetic susceptibility tensor (MST) for five classes of model small molecules containing a single chiral center has been studied using quantum computational techniques at DFT-B3LYP level of theory with 6-311G basis set. Several molecular properties are used to reduce the MST elements prior to the examination of the correlation. This study showed a close correlation between optical activity and the off-diagonal MST elements reduced by the electronic spatial extent . In the second part of this work, effects of the type of substitutions and their distances from the chiral center on the optical activity of these molecules have been investigated. Results of this work show that optical activity is reduced effectively upon distancing the substitution groups from the chiral center. Furthermore, the set of substitutions with higher electron densities and larger differences in their electronic structures result in higher optical activity.

Magnetic Susceptibility Tensor of Anisotropic Ferromagnetic Magnetized Media

Using a method analogous to the Smit–Suhl method, we have calculated the magnetic susceptibility tensor of ferromagnetic media with allowance for the magnetocrystalline anisotropy and the anisotropy of the magnetomechanical ratio. An analysis has been carried out of the obtained formulas in the particular cases of an isotropic medium with anisotropic magnetomechanical ratio, crystals with uniaxial magnetocrystalline anisotropy, and composite materials with randomly distributed ferromagnetic inclusions of ellipsoidal shape.

Splitting of spin-wave modes in thin films with arrays of periodic perturbations: theory and experiment

A joint theoretical–experimental study focusing on the description of the ferromagnetic resonance response of thin films in the presence of periodic perturbations introduced on the upper film surface is presented. From the viewpoint of theory, these perturbations may exist in the form of any kind of one- or two-dimensional rectangular defect arrays patterned onto one surface of the magnetic film. Indeed, the defects may be pits or bumps, or ion-implanted regions with a lower saturation magnetization. The complete set of response functions, given by the components of the frequency and wave-vector dependent dynamic magnetic susceptibility tensor of the film exposed to microwave excitation, are provided and are used to explain the experimental data. This allows us to obtain the response of the system due to microwave absorption, from which the zero wave-vector spin-wave modes in the field-frequency spectra, including their intensity, are calculated. Explicit calculations for periodic defects featuring the shape of stripes, dots and rectangles are given in detail, as well as experimental results for stripe-like defects prepared either by topographical depressions or by ion implantation of thin magnetic films. The excellent agreement of the theoretical and experimental results manifests the validity of the presented model.

Comparison of Magnetic Susceptibility Tensor and Diffusion Tensor of the Brain.

Susceptibility tensor imaging (STI) provides a novel approach for noninvasive assessment of the white matter pathways of the brain. Using mouse brain ex vivo, we compared STI with diffusion tensor imaging (DTI), in terms of tensor values, principal tensor values, anisotropy values, and tensor orientations. Despite the completely different biophysical underpinnings, magnetic susceptibility tensors and diffusion tensors show many similarities in the tensor and principal tensor images, for example, the tensors perpendicular to the fiber direction have the highest gray-white matter contrast, and the largest principal tensor is along the fiber direction. Comparison to DTI fractional anisotropy, the susceptibility anisotropy provides much higher sensitivity to the chemical composition of the white matter, especially myelin. The high sensitivity can be further enhanced with the perfusion of ProHance, a gadolinium-based contrast agent. Regarding the tensor orientations, the direction of the largest principal susceptibility tensor agrees with that of diffusion tensors in major white matter fiber bundles. The STI fiber tractography can reconstruct the fiber pathways for the whole corpus callosum and for white matter fiber bundles that are in close contact but in different orientations. There are some differences between susceptibility and diffusion tensor orientations, which are likely due to the limitations in the current STI reconstruction. With the development of more accurate reconstruction methods, STI holds the promise for probing the white matter micro-architectures with more anatomical details and higher chemical sensitivity.

Combined NMR Analysis of Huge Residual Dipolar Couplings and Pseudocontact Shifts in Terbium(III)-Phthalocyaninato Single Molecule Magnets

Several small paramagnetic complexes combine large hyperfine NMR shifts with large magnetic anisotropies. The latter are a prerequisite for single molecule magnet (SMM) behavior. We choose the SMM tris(octabutoxyphthalocyaninato) diterbium (1) for a high resolution NMR study where we combined for the first time a comprehensive (1)H and (13)C chemical shift analysis of a SMM with the evaluation of large residual dipolar couplings (RDCs). The latter are a consequence of partial alignment of SMM 1 in the strong magnetic field of the NMR spectrometer. To the best of our knowledge RDCs in SMMs have never been reported before. We measured RDCs between -78 and +99 Hz for the (13)C-(1)H vectors of CH bonds and up to -109 Hz for (1)H-(1)H vectors of geminal hydrogen atoms (magnetic field of 14.09 T, temperature 295 K). Considerable negative Fermi contact shifts (up to -60 ppm) were determined for (13)C atoms at the phthalocyaninato core. Paramagnetic (13)C NMR shifts of the butoxy chains as well as all (1)H NMR chemical shifts are a result of pseudocontact shifts (pcs), and therefore it is easily possible to determine the positions of the respective nuclei in solution. Measurements of CH and HH vectors by RDC analysis are in accordance with the geometry as determined by the pseudocontact shifts, but in addition to that, RDCs give information about internal mobility. The axial component of the magnetic susceptibility tensor has been determined independently by pcs and by RDC.