Spin Values Research Articles

Magnetic materials as an environmentally friendly cooling and heating systems: Tuning magnetocaloric properties in the magnetic nanotubes

The magnetocaloric properties - which is important for designing energy efficient and environment friendly heating and cooling systems- of the magnetic nanotube are constituted by arbitrary core spin values S_C and the shell spin values S_(S ) have been investigated by mean field approximation. Several quantities have been calculated during this investigation, such as isothermal magnetic entropy change, full width at half maximum value and the refrigerant capacity, in order to suggest more efficient heating and cooling. The variation of these quantities with the values of the spins and exchange interaction between the core and shell is determined. Besides, recently experimentally observed double peak behavior in the variation of the isothermal magnetic entropy change with the temperature is obtained for the nanotube.

Measuring black hole spins through X-ray reflection spectroscopy and the relativistic precession model: the case of XTE J1859+226

ABSTRACT The development of techniques to measure accurately black hole spins is crucial to study the physics and astrophysics of these objects. X-ray reflection spectroscopy is currently the most popular method to estimate the spins of accreting black holes; so far it has provided a spin measurement of about 40 stellar-mass black holes in X-ray binaries and 40 supermassive black holes in active galactic nuclei. The relativistic precession model (RPM) is another method to measure the spins of stellar-mass black holes: it requires the measurement of the frequencies of three simultaneous quasi-periodic oscillations and can potentially provide precise estimates of the black hole mass and spin. However, the two methods do not seem to provide consistent results when applied to the same sources, which questions the reliability and accuracy of these measurements. Recently, the RPM has been applied to infer the spin of the black hole in XTE J1859+226. The authors found a* = 0.149 ± 0.005 (68 per cent CL). There are no other spin measurements of this source. We looked for archived RXTE observations of XTE J1859+226 with blurred reflection features and found 23 spectra suitable for measuring the spin. We employed two different models with relxill and relxillD and obtained a higher spin value from all these fits. From simultaneous fitting of seven spectra of higher quality, we found $a_* = 0.986^{+0.001}_{-0.004}$ and a* = 0.987 ± 0.003 (90 per cent CL, statistical) with relxill and relxillD, respectively. Our results confirm the discrepancy between the spin measurements inferred from the two techniques.

Translation-invariant Gibbs measures for Potts model with competing interactions with a countable set of spin values on Cayley tree

Translation-invariant Gibbs measures for Potts model with competing interactions with a countable set of spin values on Cayley tree

Generating images of the M87* black hole using GANs

ABSTRACT In this paper, we introduce a novel data augmentation methodology based on Conditional Progressive Generative Adversarial Networks (CPGAN) to generate diverse black hole (BH) images, accounting for variations in spin and electron temperature prescriptions. These generated images are valuable resources for training deep learning algorithms to accurately estimate black hole parameters from observational data. Our model can generate BH images for any spin value within the range of [−1, 1], given an electron temperature distribution. To validate the effectiveness of our approach, we employ a convolutional neural network to predict the BH spin using both the GRMHD images and the images generated by our proposed model. Our results demonstrate a significant performance improvement when training is conducted with the augmented data set while testing is performed using GRMHD simulated data, as indicated by the high R2 score. Consequently, we propose that GANs can be employed as cost-effective models for black hole image generation and reliably augment training data sets for other parametrization algorithms.

Gradient Gibbs measures with periodic boundary laws of a generalized SOS model on a Cayley tree

We consider gradient Gibbs measures corresponding to a periodic boundary law for a generalized solid-on-solid (SOS) model with spin values from a countable set on a Cayley tree. On the Cayley tree, detailed information on gradient Gibbs measures for models of SOS type is given in Botirov and Haydarov (2020 J. Stat. Mech. 093102), Henning et al (2019 Electron. J. Probab. 24 104), Haydarov and Rozikov (2022 Rep. Math. Phys. 90 81–101) and Kulske and Schriever (2017 Markov Process. Relat. Fields 23 553–90). We continue this work for the generalized SOS model. Namely, in this paper, the problem of finding gradient Gibbs measures which correspond to periodic boundary laws is reduced to a functional equation and, by solving this equation, all gradient Gibbs measures with four periodic boundary laws are found.

Clocking mechanism from a minimal spinning particle model

The clock hypothesis plays an important role in the theory of relativity. To test this hypothesis, a mechanical model of an ideal clock is needed. Such a model should have the phase of its intrinsic periodic motion increasing linearly with the affine parameter of the clock’s center of mass worldline. A class of relativistic rotators introduced by Staruszkiewicz in the context of an ideal clock is studied. A singularity in the inverse Legendre transform leading from the Hamiltonian to the Lagrangian leads to new possible Lagrangians characterized by fixed values of mass and spin. In free motion the rotators exhibit intrinsic motion with the speed of light and fixed frequency.

Testing theories of accretion and gravity with super-extremal Kerr discs

ABSTRACT Fitting the thermal continuum emission of accreting black holes observed across X-ray bands represents one of the principle means of constraining the properties (mass and spin) of astrophysical black holes. Recent ‘continuum fitting’ studies of Galactic X-ray binaries in the soft state have found best-fitting dimensionless spin values which run into the prior bounds placed on traditional models (a⋆ = 0.9999). It is of critical importance that these results are robust, and not a result solely of the presence of these prior bounds and deficiencies in conventional models of accretion. Motivated by these results we derive and present superkerr, an xspec model comprising of a thin accretion disc solution valid in the Kerr geometry for arbitrary spin parameter a⋆, extending previous models valid only for black holes (|a⋆| < 1). This extension into ‘super-extremal’ space–times with |a⋆| > 1 includes solutions which describe discs evolving around naked singularities, not black holes. While being valid solutions of Einstein’s field equations these naked singularities are not expected to be present in nature. We discuss how the ‘measurement’ of a Kerr spin parameter 1 < a⋆ < 5/3 would present compelling evidence for the requirement of a rethink in either standard accretion theory, or our theories of gravity.

Effects of electron–phonon coupling and Rashba spin–orbit interaction on thermodynamic and magnetic properties of quantum dots

The thermodynamic and magnetic properties of quantum-dot structures subjected to an applied magnetic field were studied, including the longitudinal optical–phonon interaction and the Rashba spin–orbit effect. The Schrödinger equation was solved to determine the energy levels. The partition function was evaluated by summing the accessible energy levels and was then utilized to calculate the thermomagnetic functions. In this paper, we present magnetic properties by considering three interacting polarons. Our results indicated that at B=0T, the susceptibility exhibits diamagnetic behavior for all values of the Rashba spin–orbit parameter. However, the magnetic susceptibility increases with an applied magnetic field, and the system exhibits paramagnetic behavior under moderate magnetic fields. However, in situations with and without the polaron effect, the susceptibility is saturated at 0(meV/T2) under large magnetic fields. In this study, we showed that the Rashba spin–orbit interaction (SOI) strengthens the cutoff magnetic field Bc (the B value at which the magnetic nature of the dot changes from diamagnetic to paramagnetic). Rashba SOIs reduce the mean energy of the system, including polaronic interactions. Under the polaron effect, the heat capacity curve shifts to lower temperatures. A quantitative description of the magnetocaloric effect (ΔS) as a function of the Wigner and Rashba spin–orbit parameters is presented.

An infinite-dimensional nonlinear equation related to Gibbs measures of a SOS model

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation.

Origin of Multiferroism of Ion Doped at Different Sites: Bi4Ti3O12 Bulk and Nanoparticles

Based on a microscopic model and the Green's function theory, the temperature, size, magnetic field, and ion doping dependence of the magnetic, electric, and dielectric properties in Fe‐and Nd‐doped ferroelectric Bi4Ti3O12 bulk and nanoparticles is investigated. The multiferroism can be achieved in two ways. At once the magnetism appears by substitution of the nonmagnetic Ti ion with Fe or other transition metal ions in bulk Bi4Ti3O12. Otherwise, the simultaneous existence of Fe and Fe gives rise to a superexchange mechanism responsible for ferromagnetic interaction. Moreover, multiferroism is possible also in pure Bi4Ti3O12 nanoparticles. Due to surface effects and oxygen vacancies leading to different valence states on the surface of Ti or Ti with spin value , there appears weak magnetism, which can be improved by Fe ion doping. It is demonstrated that doping with rare earth ions, for example Nd, at the Bi site is able to improve the ferroelectric, multiferroic, and dielectric properties.

Probing mean values and correlations of high-spin systems on a quantum computer

We consider simulation of the high spins on a quantum computer. The protocols which allow one to measure the mean value of spin and correlations between spins are proposed. As a result, we determine the time dependence of the mean values of spin-1 in the magnetic field prepared on the ibmq-santiago quantum computer. In addition, we study the evolution of two interacting spins on the ibmq-lima quantum computer. The time-dependencies of the mean value of spin-1 and correlations between these spins are detected. Finally, we generalize these protocols for the spins of arbitrary values.

Charged spinning and magnetized test particles orbiting quantum improved charged black holes

In the present work, we aimed to investigate the dynamics of spinning charged and magnetized test particles around both electrically and magnetically charged quantum-improved black holes. We derive the equations of motion for charged spinning test particles using the Mathisson-Papapetrou-Dixon ***equations with the Lorentz coupling term. The radius of innermost stable circular orbits (ISCOs), specific angular momentum, and energy for charged spinless, uncharged spinning, and charged spinning test particles around the charged and non-charged quantum-improved black holes are analyzed separately. We found that the quantum parameter increases the maximum spin value, smax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_\ extrm{max}$$\\end{document}, which leads to the nonphysical motion (superluminal motion) of the charged spinning test particle. In contrast, the black hole charge decreases its value. We also found that, in contrast to the Reissner Nordström black hole, spinning charged test particles in the quantum-improved charged black hole have higher smax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_\ extrm{max}$$\\end{document}; moreover, positively charged spinning particles can have higher values of smax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_\ extrm{max}$$\\end{document} near the extreme black hole cases when compared with uncharged spinning particles. Finally, we investigate the magnetized test particle’s dynamics in the spacetime of a quantum-improved magnetically charged black hole in Quantum Einstein Gravity using the Hamilton–Jacobi equation. We show that the presence of ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} increases the maximum value of the effective potential and decreases the minimum energy and angular momentum of magnetized particles at their circular orbits. We found an upper constraint in the black hole charge at the ISCO.

Singular points of the isotropic Non-Heisenberg magnet

The symmetry properties of an isotropic non-Heisenberg magnet with the values of the magnetic ion spin S = 1 or S = 3/2 are studied in this work. It is shown that the systems are invariant under the transformation group SO(3) or SU(2). The conditions under which magnets with spin S = 1 or S = 3/2 are invariant with respect to the transformation groups SU(2S + 1) are determined. The study of the dynamic properties of the magnets under consideration has shown that at SU(2S + 1) points the systems become two-level, similarly to systems with magnetic ion spin S = 1/2.

New black hole spin values for Sagittarius A* obtained with the outflow method

ABSTRACT Six archival Chandra observations are matched with eight sets of radio data and studied in the context of the outflow method to measure and study the spin properties of $\rm {Sgr ~A^{*}}$. Three radio and X-ray data sets obtained simultaneously, or partially simultaneously, are identified as preferred for the purpose of measuring the spin properties of $\rm {Sgr ~A^{*}}$. Similar results are obtained with other data sets. Results obtained with the preferred data sets are combined and indicate weighted mean values of the spin function of $F = 0.62 \pm 0.10$ and dimensionless spin angular momentum of $a_* = 0.90 \pm 0.06$. The spin function translates into measurements of the black hole rotational mass, Mrot, irreducible mass, Mirr, and spin mass–energy available for extraction, Mspin, relative to the total black hole dynamical mass, Mdyn. Weighted mean values of (Mrot/Mdyn) = (0.53 ± 0.06), (Mirr/Mdyn) = (0.85 ± 0.04), (Mspin/Mdyn) = (0.15 ± 0.04), Mrot = (2.2 ± 0.3) × 106 M⊙, Mirr = (3.5 ± 0.2) × 106 M⊙, and Mspin = (6.2 ± 1.6) × 105 M⊙ are obtained; of course (Mrot/Mirr) = (0.62 ± 0.10) since (Mrot/Mirr) = F. Values obtained for $\rm {Sgr ~A^{*}}$ are compared with those obtained for M87 based on the published spin function, which indicate that M87 carries substantially more rotational energy and spin mass–energy relative to the total (i.e. dynamical) black hole mass, the irreducible black hole mass, and in absolute terms.

New class of Gibbs measures for two-state hard-core model on a Cayley tree

In this paper, we consider a Hard-Core (HC) model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant. In addition, we study free energy of the model.

Spin state and magnetic coupling in polynuclear Ni(II) complexes from density functional theory: is there an optimal amount of Fock exchange?

Reliable prediction of the ground-state spin and magnetic coupling constants in transition-metal complexes is a well-known challenge for density functional theory (DFT). One popular strategy for addressing this long-standing issue involves the modification of the fraction of Fock exchange in a hybrid functional. Here we explore the viability of this approach using three polynuclear metal-organic complexes based on a Ni4O4 cubane motif, having different ground state spin values (S = 0, 2, 4) owing to the use of different ligands. We systematically search for an optimum fraction of Fock exchange, across various global, range-separated, and double hybrid functionals. We find that for all functionals tested, at best there only exists a very narrow range of Fock exchange fractions which results in a correct prediction of the ground-state spin for all three complexes. The useful range is functional dependent, but general trends can be identified. Typically, at least two similar systems must be used in order to determine both an upper and lower limit of the optimal range. This is likely owing to conflicting demands of minimizing delocalization errors, which typically requires a higher percentage of Fock exchange, and addressing static correlation, which typically requires a lower one. Furthermore, we find that within the optimal range of Fock exchange, the signand relative magnitude of Ni-Ni magnetic coupling constants are reasonably well reproduced, but there is still room for quantitative improvement in the prediction. Thus, the prediction of spin state and magnetic coupling in polynuclear complexes remains an ongoing challenge for DFT.

The spin measurement of MAXI J1348-630 using the Insight-HXMT data

ABSTRACT We report the results of fitting Insight-HXMT data to the black hole X-ray binary MAXI J1348-430, which was discovered on January 26th, 2019, We also inferred the ionization state and iron abundancewith the Gas Slit Camera on-board MAXI. Several observations at the beginning of the first burst were selected, with a total of 10 spectra. From the residuals of fits using disc plus power-law models, X-ray reflection signatures were clearly visible in some of these observations. We use the state-of-the-art $\tt {relxill}$ series reflection model to fit six spectra with distinct reflection signatures and a joint fit to these spectra. In particular, we focus on the results for the black hole spin values. Assuming Rin = RISCO, the spin parameter is constrained to be $0.82^{+0.04}_{-0.03}$ with 90 per cent confidence level (statistical only).

The influence of a static planetary atmosphere on spin transfer during pebble accretion

Context. Pebble accretion has been used to explain the small size of Mars, the heavy element contents of the gas giants, and the size distribution of asteroids. More recently, pebble accretion has been proposed as a means to explain not only the growth but also the prograde spin preference of most larger bodies in the Solar System. Pebble accretion could induce planetary and asteroid spin equal to or exceeding the spins currently measured. However, as these planetesimals grow, they start condensing the gas of the disc around them, forming an atmosphere within their Bondi radius. Aims. We study the effect an atmosphere has on the pebble orbits and spin build-up on the planet’s surface during pebble accretion in the extreme case of a static atmosphere. Pebble feedback to the gas is not taken into account. Methods. The equations of motion for pebbles in a planar, global frame with a planet and a central star were integrated using the AIS15 integrator of REBOUND. An adiabatic atmosphere was then placed around the planet, and the spin deposited on the planet’s surface was measured. These simulations were evaluated for different distances to the star, Stokes numbers, and planet masses. Results. In general, an atmosphere dampens the spin the planet’s surface receives by absorbing part of the angular momentum of the pebbles and circularising their orbits. This could prevent the excessive spin values predicted in some 3D pebble accretion simulations without an atmosphere. For planets larger than 0.5 M⊕, a stationary atmosphere absorbs all angular momentum, leaving no spin for the surface. Significant quantities of angular momentum are stored in the inner and intermediate atmosphere (<0.3 Bondi radii). Depending on the atmospheric and disc model, this spin could be transported either to the disc through atmospheric recycling or to the planet through drag between the surface and the atmosphere. Further research is required to quantify the spin transfer within the atmosphere.

Unveiling the accretion scenario of BH-ULXs using XMM–Newton observations

ABSTRACT We present a comprehensive spectrotemporal analysis of five ultraluminous X-ray sources (ULXs) with central object likely being a black hole, using archival XMM–Newton observations. These sources, namely NGC1313 X−1, NGC5408 X-1, NGC6946 X−1, M82 X−1, and IC342 X−1, reveal short-term variability with fractional variance of 1.42$-27.28~{{\ \rm per\ cent}}\, $ and exhibit Quasi-periodic Oscillations (QPOs) with frequency νQPO ∼ 8−667 mHz. Long-term evolution of ULXs energy spectra (0.3–10 keV; excluding M82 X−1) are described satisfactorily with a model combination that comprises a thermal Comptonization component (nthComp, yielding Γnth ∼ 1.48–2.65, kTe ∼ 1.62–3.76 keV, τ ∼ 8–20, y-par∼1.16–6.24) along with a standard disc component (diskbb, kTin ∼ 0.16–0.54 keV). We find that these ULXs generally demonstrate anticorrelation between disc luminosity and inner disc temperature as $L_{\rm disc} \propto T_{\rm in}^\alpha$, where α = −3.58 ± 0.04 for NGC1313 X−1 and IC342 X−1, α = −8.93 ± 0.11 for NGC6946 X−1, and α = −10.31 ± 0.10 for NGC5408 X−1. We also obtain a linear correlation between bolometric luminosity Lbol and Γnth that indicates spectral softening of the sources when Lbol increases. We observe that in presence of QPO, Comptonized seed photon fraction varies in between $\sim 5-20~{{\ \rm per\ cent}}\, $, while the Comptonized flux contribution (50$-90~{{\ \rm per\ cent}}\, $) dominates over disc flux. Utilizing νQPO and Lbol, we constrain ULXs mass by varying their spin (ak) and accretion rate ($\dot{m}$). We find that NGC6946 X−1 and NGC5408 X−1 seem to accrete at sub-Eddington accretion rate provided their central sources are rapidly rotating, whereas IC342 X−1 and NGC1313 X−1 can accrete in sub/super-Eddington limit irrespective to their spin values.

Analytic study of the Maxwell electromagnetic invariant in spinning and charged Kerr-Newman black-hole spacetimes

The Maxwell invariant plays a fundamental role in the mathematical description of electromagnetic fields in charged spacetimes. In particular, it has recently been proved that spatially regular scalar fields which are non-minimally coupled to the Maxwell electromagnetic invariant can be supported by spinning and charged Kerr-Newman black holes. Motivated by this physically intriguing property of asymptotically flat black holes in composed Einstein-Maxwell-scalar field theories, we present a detailed analytical study of the physical and mathematical properties of the Maxwell electromagnetic invariant FKNrθMaQ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{F}}_{\ extrm{KN}}\\left(r,\ heta; M,a,Q\\right) $$\\end{document} which characterizes the Kerr-Newman black-hole spacetime [here {r, θ} are respectively the radial and polar coordinates of the curved spacetime and {M, J = M a, Q} are respectively the mass, angular momentum, and electric charge parameters of the black hole]. It is proved that, for all Kerr-Newman black-hole spacetimes, the spin and charge dependent minimum value of the Maxwell electromagnetic invariant is attained on the equator of the black-hole surface. Interestingly, we reveal the physically important fact that Kerr-Newman spacetimes are characterized by two critical values of the dimensionless rotation parameter â≡a/r+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{a}\\equiv a/{r}_{+} $$\\end{document} [here r+ (M, a, Q) is the black-hole horizon radius], âcrit−=3−22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}_{\ extrm{crit}}^{-}=\\sqrt{3-2\\sqrt{2}} $$\\end{document} and âcrit+=5−25\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}_{\ extrm{crit}}^{+}=\\sqrt{5-2\\sqrt{5}} $$\\end{document}, which mark the boundaries between three qualitatively different spatial functional behaviors of the Maxwell electromagnetic invariant: (i) Kerr-Newman black holes in the slow-rotation â<âcrit−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{a}<{\\hat{a}}_{\ extrm{crit}}^{-} $$\\end{document} regime are characterized by negative definite Maxwell electromagnetic invariants that increase monotonically towards spatial infinity, (ii) for black holes in the intermediate spin regime âcrit−≤â≤âcrit+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}_{\ extrm{crit}}^{-}\\le \\hat{a}\\le {\\hat{a}}_{\ extrm{crit}}^{+} $$\\end{document}, the positive global maximum of the Kerr-Newman Maxwell electromagnetic invariant is located at the black-hole poles, and (iii) Kerr-Newman black holes in the super-critical regime â<âcrit+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\hat{a}<{\\hat{a}}_{\ extrm{crit}}^{+} $$\\end{document} are characterized by a non-monotonic spatial behavior of the Maxwell electromagnetic invariant FKNr=r+θMaQ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{F}}_{\ extrm{KN}}\\left(r={r}_{+},\ heta; M,a,Q\\right) $$\\end{document} along the black-hole horizon with a spin and charge dependent global maximum whose polar angular location is characterized by the dimensionless functional relation â2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\hat{a}}^2 $$\\end{document} · (cos2θ)max = 5 – 25\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 2\\sqrt{5} $$\\end{document}.