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We reconsider the recently proposed mimetic gravity, focusing in particular on whether the theory is able to reproduce the inferred flat rotation curves of galaxies. We extend the theory by adding a non-minimal coupling between matter and mimetic field. Such coupling leads to the appearance of an extra force which renders the motion of test particles non-geodesic. By studying the weak field limit of the resulting equations of motion, we demonstrate that in the Newtonian limit the acceleration law induced by the non-minimal coupling reduces to a Modified Newtonian Dynamics (MOND)-like one. In this way, it is possible to reproduce the successes of MOND, namely the explanation for the flat galactic rotation curves and the Tully-Fisher relation, within the framework of mimetic gravity, without the need for particle dark matter. The scale-dependence of the recovered acceleration scale opens up the possibility of addressing the missing mass problem not only on galactic but also on cluster scales: we defer a full study of this issue, together with a complete analysis of fits to spiral galaxy rotation curves, to an upcoming companion paper., 11 pages, accepted for publication in Class. Quant. Grav

Although, as part of a general phenomenon, the piezoelectric response of $\mathrm{Ba}(\mathrm{T}{\mathrm{i}}_{1\ensuremath{-}y}{M}_{y}){\mathrm{O}}_{3}$ ($M=\mathrm{Zr}$, Sn, Hf) increases in the vicinity of the orthorhombic ($Amm2$)-tetragonal ($P4mm$) and orthorhombic ($Amm2$)-rhombohedral ($R3m$) polymorphic phase boundaries, experiments in the last few years have shown that the same phase boundaries show significantly enhanced weak-field piezoproperties in the Ca-modified variants of these ferroelectric alloys, i.e., $(\mathrm{Ba},\mathrm{Ca})(\mathrm{Ti},M){\mathrm{O}}_{3}$. So far there is a lack of clarity with regard to the unique feature(s) which Ca modification brings about that enables this significant enhancement. Here, we examine this issue from a structural standpoint with $M=\mathrm{Sn}$ as a case study. We carried out a comprehensive comparative structural, ferroelectric, and piezoelectric analysis of the $Amm2$ phase in the immediate vicinity of the $P4mm\text{\ensuremath{-}}Amm2$ phase boundaries of (i) Ca-modified $\mathrm{Ba}(\mathrm{Ti},\mathrm{Sn}){\mathrm{O}}_{3}$, as per the nominal formula $(1\ensuremath{-}x)\mathrm{BaT}{\mathrm{i}}_{0.88}\mathrm{S}{\mathrm{n}}_{0.12}{\mathrm{O}}_{3}\text{\ensuremath{-}}(x)\mathrm{B}{\mathrm{a}}_{0.7}\mathrm{C}{\mathrm{a}}_{0.3}\mathrm{Ti}{\mathrm{O}}_{3}$ and (ii) without Ca modification, i.e., $\mathrm{Ba}(\mathrm{T}{\mathrm{i}}_{1\ensuremath{-}y}\mathrm{S}{\mathrm{n}}_{y}){\mathrm{O}}_{3}$. We found that the spontaneous lattice strain of the $Amm2$ phase is noticeably smaller in the Ca-modified counterpart. Interestingly, this happens along with an improved spontaneous polarization by enhancing the covalent character of the Ti-O bond. Our study suggests that the unique role of Ca modification lies in its ability to induce these seemingly contrasting features (reduction in spontaneous lattice strain but increase in polarization).

International audience; Post-Newtonian (PN) theory enables us to predict the waveform and the phaseevolution of the gravitational waves emitted by a system of two compact objectscoalescing in its inspiral phase. State-of-the-art works provide the phase ofthe expected signal up to 3.5PN 5 (i.e. up to 1/c 7). Comparison with numericalrelativity, as well as the promising evolution of gravitational wave detectorsand the new discoveries incite us to pursue this computation to a higher order.In our current program, we are aiming at reaching the phase of the signal at the4.5PN order (i.e. 1/c 9). For this purpose, the energy flux emitted by such asystem for circular orbits has to be known at 4.5PN. One difficulty of thiscomputation is the high-order interactions-commonly called tails-between themass of the system and its mass and current multipole moments during thepropagation of the signal. In Marchand et al. 2016 1 , we have computed thecontribution of all the tails entering the flux at the 4.5PN order, and inparticular the third-order mass-quadrupole interactions known as " tails-of-tails-of-tails ". Our work enables to fully determine the 4.5PN coefficient ofthe flux for circular orbits (while the 4PN coefficient is still unknown).

It was recently proposed that the effects usually attributed to particle dark matter on galaxy scales are due to the displacement of dark energy by baryonic matter, a paradigm known as emergent gravity. This formalism leads to predictions similar to Modified Newtonian Dynamics (MOND) in spherical symmetry, but not quite identical. In particular, it leads to a well defined transition between the Newtonian and the modified gravitational regimes, a transition depending on both the Newtonian acceleration and its first derivative with respect to radius. Under the hypothesis of the applicability of this transition to aspherical systems, we investigate whether it can reproduce observed galaxy rotation curves. We conclude that the formula leads to marginally acceptable fits with strikingly low best-fit distances, low stellar mass-to-light ratios, and a low Hubble constant. In particular, some unobserved wiggles are produced in rotation curves because of the dependence of the transition on the derivative of the Newtonian acceleration, leading, even in the most favorable case, to systematically less good fits than MOND. Then, applying the predicted transition from emergent gravity in a regime where it should a priori be applicable, i.e. in spherical symmetry and outside of the bulk of matter, we show that the predictions for the secular advances of Solar System planets' perihelia are discrepant with the data by seven orders of magnitude, ruling out the present emergent gravity weak-field formula with high confidence., 10 pages, 2 figures, accepted for publication in Phys. Rev. D

The electronic structures of nicotine, nicotinic acid and nicotinamide have been studied by valence photoemission spectroscopy (PES), core X-ray photoelectron spectroscopy (XPS), and near-edge X-ray absorp- tion fine structure (NEXAFS) and interpreted with the aid of quantum chemical calculations. Nicotinamide and nicotinic acid are closely related and show correspondingly similar spectral features, while nicotine is both struc- turally and spectroscopically diverse

We investigate the effects of the anomalous magnetic moment (AMM) in the EoS of a fermion system in the presence of a magnetic field. In the region of strong magnetic fields ($B>m^2$) the AMM is found from the one-loop fermion self-energy. In contrast to the weak-field AMM found by Schwinger, in the strong magnetic field case, the AMM depends on the Landau level (LL) and decreases with it. The effects of the AMM in the EoS at intermediate-to-large fields can be found introducing the one-loop, LL-dependent AMM in the effective Lagrangian that is then used to find the thermodynamic potential of the system. We compare the plots of the parallel and perpendicular pressures versus the magnetic field in the strong field region considering the LL-dependent AMM, the Schwinger AMM, and no AMM at all. The results clearly show a separation between the physical magnitudes found using the Schwinger AMM and the LL-dependent AMM. This is an indication of the inconsistency of considering the Schwinger AMM beyond the weak field region $B< m^2$ where it was originally found. The curves for the EoS, pressures and magnetization at different fields give rise to the well-known de Haas van Alphen oscillations, associated to the change in the number of LL contributing at different fields.

Electromagnetically induced transparency (EIT) in the five-level cascade scheme of 85Rb atoms is investigated by analytical method. In the weak field limit of the probe light, the linear susceptibility, absorption, and dispersion coefficients have been derived in analytical forms. Unusual EIT signatures of the system are studied in details.

We investigate the absorptive–dispersive properties of a weak field in a coupled semiconductor double-quantum-dot nanostructure. It is found that the enhanced index of refraction without absorption can be easily controlled via adjusting properly the corresponding parameters of the system. Our scheme opens the possibility to control index of refraction without absorption in a coupled semiconductor double-quantum-dot nanostructure.

The article substantiates the necessity of developing a theoretical model to study the influence of non-stationary processes on the shape of the recorded NMR signal in a weak magnetic field. Therefore, comparison of different ways of solving Bloch’s equations to describe the shape of the recorded NMR signals using a modulation technique is carried out. Moreover, mathematical models have been developed to describe the shape of the NMR signal, considering the peculiarities of its registration using modulation techniques in a weak field. Finally, the results of calculations, obtained by proposed methods, are compared with experimental data.

In this paper, for the first time in the context of time orbiting potential (TOP) trap, the necessary and sufficient conditions for the adiabatic evolution of weak field seeking states have been quantitatively examined. It has been well accepted since decades that adiabaticity has to be obeyed by the atoms for successful magnetic trapping. However, we show, on the contrary, that atoms can also be confined beyond the adiabatic limit. For the demonstration, we have considered a toy model of a single weak field seeking atom in its ground state and have calculated its survival probability inside a TOP trap. Our findings open new possibilities to relax the restrictions of atom trapping in laboratories.

We report the discovery of a new magnetic DA white dwarf (WD), WD 0011 − 721, which is located within the very important 20 pc volume-limited sample of the closest WDs to the Sun. This star has a mean field modulus ⟨|B|⟩ of 343 kG, and from the polarisation signal we deduce a line-of-sight field component of 75 kG. The magnetic field is sufficiently weak to have escaped detection in classification spectra. We then present a preliminary exploration of the data concerning the frequency of such fields among WDs with hydrogen-rich atmospheres (DA stars). We find that 20 ± 5% of the DA WDs in this volume have magnetic fields, mostly weaker than 1 MG. Unlike the slow field decay found among the magnetic Bp stars of the upper main sequence, the WDs in this sample show no evidence of magnetic field or flux changes over several Gyr.

It is shown that polynomial gravity theories with more than four derivatives in each scalar and tensor sectors have a regular weak-field limit, without curvature singularities. This is achieved by proving that in these models the effect of the higher derivatives can be regarded as a complete regularization of the delta-source. We also show how this result implies that a wide class of non-local ghost-free gravities have a regular Newtonian limit too, and discuss the applicability of this approach to the case of weakly non-local models., 18 pages. Matches the version published in JCAP

The paper discusses the need to develop a new approximation of the nuclear magnetic resonance signals. This approximation allows enhancing functional capabilities of express control devices. The universal mathematical model is developed to describe the shape of the NMR signals taking into account the peculiarities of their registration using a modulation technique in a weak field. The modeling results obtained are compared with experimental data.

We performed three-dimensional detection of magnetic nanoparticle (MNP) samples using the combination of a field-free line (FFL) and multiple pickup coils. In this method, two-dimensional maps of the signal fields generated by the MNP samples were used to reconstruct a three-dimensional image of the MNP concentration. In the experiment, a weak field gradient of the FFL (0.06–0.13 T m−1) was used. The degradation of the imaging quality owing to the weak field gradient was compensated using ten field maps. We successfully demonstrated the three-dimensional detection of two MNP samples. The Fe content of each sample was 10 and 20 μg and the spacing for both samples was 16 mm. The results confirm the effectiveness of the present method for MNP detection using a weak field gradient. We also quantitatively discuss how the imaging quality degrades when the field gradient is decreased.

The magnetoconductivity, which is the correction to the diagonal conductivity proportional to the second power of magnetic-field strength, is calculated in bilayer graphene in the presence of scatt...

Gravitational waves (GWs) in Modified Gravity are discussed and compared with General Relativity in view of possible detection of new modes. After a summary of some classes of modified theories, we recall GWs properties in General Relativity as the basic standard by which to compare any modified theory. Then, we discuss GWs in Extended Theories like [Formula: see text] gravity, scalar–tensor gravity, Gauss–Bonnet gravity and other gauge theories, as well as Teleparallel Equivalent General Relativity (TEGR), taking into account also torsion instead of curvature, to describe the gravitational field. As a general remark, all these theories lead to new polarizations in addition to the two standard modes of General Relativity. It is possible to show that, in 4 dimensions, the polarizations are at most 6. Theoretical and experimental implications of new modes are briefly discussed.

The magnetoconductivity proportional to B2 with B the strength of applied magnetic field is calculated in monolayer graphene in a self-consistent Born approximation for scatterers with long-range p...

A weak active region (NOAA 11158) appeared on the solar disk near the eastern limb. This region increased rapidly and, having reached the magnetic flux higher than 1022 Mx, produced an X-class flare. Only weak field variations at individual points were observed during the flare. An analysis of data with a resolution of 45 s did not indicate any characteristic features in the photospheric field dynamics during the flare. When the flux became higher than 3 × 1022 Mx, active region NOAA 10720 produced six X-class flares. The field remained quiet during these flares. An increase in the magnetic flux above ∼1022 Mx is a necessary, but not sufficient, condition for the appearance of powerful flares. Simple active regions do not produce flares. A flare originates only when the field distribution in an active region is complex and lines of polarity inversion have a complex shape. Singular lines of the magnetic field can exist only above such active regions. The current sheets, in the magnetic field of which the solar flare energy is accumulated, originate in the vicinity of these lines.

Many quantum optical and optoelectronic achievements have been obtained by using quantum coherence effects occurred in four-level quantum systems. Here we report a systemic study on the 1st, 2nd and 3rd order subschemes of the four-level scheme with the aid of Boolean algebra, and find 7 1st order subschemes and 53 2nd order subschemes in total. Further it is found that 472 3rd-order subschemes may appear in the realistic optical quantum coherence experiments when the applied lasers are distinguished only by the strong and weak field intensities.

We investigated the absorptive–dispersive property of a weak field in a four-level inverted-Y quantum well system. We find that the cycling field and relative phase of the applied fields can affect the absorptive–dispersive property of a weak field dramatically, which can be used to manipulate efficiently the ultrahigh index of refraction without absorption. Our study is much more practical than its atomic counterpart due to its flexible design and controllable quantum-interference effect. Thus, it may provide some new possibilities for technological applications in optoelectronics and solid-state optical communication

Laser assisted Coulomb scattering by relativistic electron and heavy electron (muon) is studied by using Salamin waves (Salamin 1993) in the Weak Field Approximation (WFA). Both electron and muon are described by the Dirac equation, with the anomalous magnetic moment effects fully included. The generalization of this paper to heavy electron (muon) gives interesting insights as to how the mass affects the magnitude of the differential cross sections. No significant difference in the muon's DCS with and without AMM effects was detected., Comment: I want to change the text

In this paper, we show that the global monopole spacetime is one of the exact solutions of Einstein equations by means of the method treating the matter field as a non-linear sigma model, without the weak field approximation applied in the original derivation by Barriola and Vilenkin in 1989. Furthermore, we find the physical origin of the topological charge in the global monopole spacetime. Finally, we generalize the proposal which generates spacetime from thermodynamical laws to the case that spacetime with global monopole charge.

We develop the foundations of an effective-one-body (EOB) model for eccentric binary coalescences that includes the conservative dynamics, radiation reaction, and gravitational waveform modes from the inspiral and the merger-ringdown signals. We use the same approach as is commonly employed in black-hole perturbation theory by introducing a relativistic parameterization of the dynamics that is defined by the orbital geometry and consists of a set of phase variables and quantities that evolve only due to gravitational radiation reaction. Specializing to nonspinning binaries, we derive the EOB evolution equations and compute the binary's radiative multipole moments that determine the gravitational waves through a decomposition into the fundamental frequencies of the motion. The major differences between our treatment and the quasi-Keplerian approach often used in post-Newtonian (PN) calculations are that the orbital parameters describe strong-field dynamics, and that expressing the multipole moments in terms of the frequencies simplifies the calculations and also results in an unambiguous orbit-averaging operation. While our description of the conservative dynamics is fully relativistic, we limit explicit derivations in the radiative sector to 1.5PN order for simplicity. This already enables us to establish methods for computing both instantaneous and hereditary contributions to the gravitational radiation in EOB coordinates that have straightforward extensions to higher PN order. The weak-field, small eccentricity limit of our results for the orbit-averaged fluxes of energy and angular momentum agrees with known PN results when expressed in terms of gauge-invariant quantities. We further address considerations for the numerical implementation of the model and the completion of the waveforms to include the merger and ringdown signals, and provide illustrative results., Comment: 25 pages, 8 figures, small changes to match published version

Some of the dark matter in the Universe is made up of massive neutrinos. Their impact on the formation of large scale structure can be used to determine their absolute mass scale from cosmology, but to this end accurate numerical simulations have to be developed. Due to their relativistic nature, neutrinos pose additional challenges when one tries to include them in N-body simulations that are traditionally based on Newtonian physics. Here we present the first numerical study of massive neutrinos that uses a fully relativistic approach. Our N-body code, gevolution, is based on a weak-field formulation of general relativity that naturally provides a self-consistent framework for relativistic particle species. This allows us to model neutrinos from first principles, without invoking any ad-hoc recipes. Our simulation suite comprises some of the largest neutrino simulations performed to date. We study the effect of massive neutrinos on the nonlinear power spectra and the halo mass function, focusing on the interesting mass range between 0.06 eV and 0.3 eV and including a case for an inverted mass hierarchy., Comment: 26 pages + appendix, 17 figures; v2: minor revision, published version; code available at https://github.com/gevolution-code/gevolution-1.1

This work relates to the famous experiments, performed in 1975 and 1979 by Werner et al., measuring neutron interference and neutron Sagnac effects in the earth's gravitational field. Employing the method of Stodolsky in its weak field approximation, explicit expressions are derived for the two phase shifts, which turn out to be in agreement with the experiments and with the previously obtained expressions derived from semi-classical arguments: these expressions are simply modified by relativistic correction factors., 7 pages, 1 figure

We study the tunneling of virtual matter-antimatter pairs from the quantum vacuum in the presence of a spatially uniform temporal electric background composed of of a strong slow field superimposed with a weak rapid field. After analytic continuation to Euclidean spacetime we obtain from the instanton equations two critical points. While one of them is the closing point of the instanton path, the other serves as an Euclidean mirror which reflects and squeezes the instanton. It is this reflection and shrinking which is responsible for an enormous enhancement of the vacuum pair production rate. We discuss how important features of this mechanism can be analysed and understood via such a rotation in the complex plane. Consistent with previous studies, we consider certain examples where we apply weak fields with a distinct pole structure in order to show that the reflection takes place exactly at the poles. We also discuss the effect of possible sub-cycle structures. We extend this reflection picture to fields which have no poles present and illustrate the effective reflections with explicit examples. An additional field strength dependence for the rate occurs in such cases. We analytically compute the characteristic threshold for this mechanism given by the critical combined Keldysh parameter. We discuss significant differences between these two types of fields. For various backgrounds, we present the contributing instantons and perform analytical computations for the corresponding rates treating both fields nonperturbatively. The validity of the results is confirmed by numerical computations. Considering different profiles for the strong field, we also discuss its impact on the critical combined Keldysh parameter.

The paper aims to apply the complex octonion to explore the influence of the energy gradient on the Eotvos experiment, impacting the gravitational mass in the ultra-strong magnetic fields. Until now the Eotvos experiment has never been validated under the ultra-strong magnetic field. It is aggravating the existing serious qualms about the Eotvos experiment. According to the electromagnetic and gravitational theory described with the complex octonions, the ultra-strong magnetic field must result in a tiny variation of the gravitational mass. The magnetic field with the gradient distribution will generate the energy gradient. These influencing factors will exert an influence on the state of equilibrium in the Eotvos experiment. That is, the gravitational mass will depart from the inertial mass to a certain extent, in the ultra-strong magnetic fields. Only under exceptional circumstances, especially in the case of the weak field strength, the gravitational mass may be equal to the inertial mass approximately. The paper appeals intensely to validate the Eotvos experiment in the ultra-strong electromagnetic strengths. It is predicted that the physical property of gravitational mass will be distinct from that of inertial mass.

Scalar field self-force effects on a scalar charge orbiting a Reissner-Nordstr\"om black hole are investigated. The scalar wave equation is solved analytically in a post-Newtonian framework, and the solution is used to compute the self-field as well as the components of the self-force at the particle's location up to 7.5 post-Newtonian order. The energy fluxes radiated to infinity and down the hole are also evaluated. Comparison with previous numerical results in the Schwarzschild case shows a good agreement in both strong-field and weak-field regimes., Comment: 16 pages, 3 eps figures, revtex macros

International audience; Although the perturbation series in quantum electrodynamics has been studied for eighty years concerning its high-order behavior, our present understanding is still poorer than for many other field theories. An interesting case is Schwinger pair creation in a constant electric field, which may possibly provide a window to high loop orders, simple non-perturbative closed-form expressions have been conjectured for the pair creation rate in the weak field limit, for scalar QED in 1982 by Affleck, Alvarez, and Manton, and for spinor QED by Lebedev and Ritus in 1984. Using Borel analysis, these can be used to obtain non-perturbative information on the on-shell renormalized N-photon amplitudes at large N and low energy. This line of reasoning also leads to a number of nontrivial predictions for the effective QED Lagrangian in either four or two dimensions at any loop order, and preliminary results of a calculation of the three-loop Euler–Heisenberg Lagrangian in two dimensions are presented.

The identification of scalable processes that transfer random mixtures of single-walled carbon nanotubes (SWCNTs) into fractions featuring a high content of semiconducting species is crucial for future application of SWCNTs in high-performance electronics. Herein we demonstrate a highly efficient and simple separation method that relies on selective interactions between tailor-made amphiphilic polymers and semiconducting SWCNTs in the presence of low viscosity separation media. High purity individualized semiconducting SWCNTs or even self-organized semiconducting sheets are separated from an as-produced SWCNT dispersion via a single weak field centrifugation run. Absorption and Raman spectroscopy are applied to verify the high purity of the obtained SWCNTs. Furthermore SWCNT - network field-effect transistors were fabricated, which exhibit high ON/OFF ratios (105) and field-effect mobilities (17 cm2/Vs). In addition to demonstrating the feasibility of high purity separation by a novel low complexity process, our method can be readily transferred to large scale production.

The effective-one-body theory (EOB) describes the conservative dynamics of compact binary systems in terms of an effective Hamiltonian approach. The Hamiltonian for moderately eccentric motion of two nonspinning compact objects in the extreme mass-ratio limit is given in terms of three potentials: $a(v)$, $\overline{d}(v)$, $q(v)$. By generalizing the first law of mechanics for (nonspinning) black hole binaries to eccentric orbits, [A. Le Tiec, Phys. Rev. D 92, 084021 (2015).] recently obtained new expressions for $\overline{d}(v)$ and $q(v)$ in terms of quantities that can be readily computed using the gravitational self-force approach. Using these expressions we present a new computation of the EOB potential $q(v)$ by combining results from two independent numerical self-force codes. We determine $q(v)$ for inverse binary separations in the range $1/1200\ensuremath{\le}v\ensuremath{\lesssim}1/6$. Our computation thus provides the first-ever strong-field results for $q(v)$. We also obtain $\overline{d}(v)$ in our entire domain to a fractional accuracy of $\ensuremath{\gtrsim}{10}^{\ensuremath{-}8}$. We find that our results are compatible with the known post-Newtonian expansions for $\overline{d}(v)$ and $q(v)$ in the weak field, and agree with previous (less accurate) numerical results for $\overline{d}(v)$ in the strong field.

Polarity reversals of the geomagnetic field are a major feature of the Earth's dynamo. Questions remain regarding the dynamical processes that give rise to reversals and the properties of the geomagnetic field during a polarity transition. A large number of paleomagnetic reversal records have been acquired during the past 50 years in order to better constrain the structure and geometry of the transitional field. In addition, over the past two decades, numerical dynamo simulations have also provided insights into the reversal mechanism. Yet despite the large paleomagnetic database, controversial interpretations of records of the transitional field persist; they result from two characteristics inherent to all reversals, both of which are detrimental to an ambiguous analysis. On the one hand, the reversal process is rapid and requires adequate temporal resolution. On the other hand, weak field intensities during a reversal can affect the fidelity of magnetic recording in sedimentary records. This paper is aimed at reviewing critically the main reversal features derived from paleomagnetic records and at analyzing some of these features in light of numerical simulations. We discuss in detail the fidelity of the signal extracted from paleomagnetic records and pay special attention to their resolution with respect to the timing and mechanisms involved in the magnetization process. Records from marine sediments dominate the database. They give rise to transitional field models that often lead to overinterpret the data. Consequently, we attempt to separate robust results (and their subsequent interpretations) from those that do not stand on a strong observational footing. Finally, we discuss new avenues that should favor progress to better characterize and understand transitional field behavior., Key Points We examine and assess an updated database of geomagnetic reversal recordsWe discuss robust and controversial features of the transitional fieldFuture work should involve millimeter‐size specimens and physics‐based models

In 2012 Coey proposed a theory on the mechanism of magnetic water treatment based on the gradient of the applied field rather than its absolute strength. We tested this theory by measuring the effect of very weak field magnets (≤ 10 G) containing strong magnetic inhomogeneities (ΔB = 770 G·m−1 (WCM 62081992) and 740 G·m−1 (WCM 62083545)) on tap water samples by the use of electric impedance spectroscopy (EIS) and laser scattering. Our results show an increased formation of nm-sized prenucleation clusters (dynamically ordered liquid like oxyanion polymers or “DOLLOPs”) due to the exposure to the magnetic field and thus are consistent with Coey’s theory which is therefore also applicable to very weak magnetic fields as long as they contain strong gradients.