Interplay of rare earth and iron magnetism in RFeAsO with R = La, Ce, Pr, and Sm: A muon spin relaxation study and symmetry analysis

Interplay of rare earth and iron magnetism in RFeAsO with R = La, Ce, Pr, and Sm: A muon spin relaxation study and symmetry analysis

H. Maeter Institut für Festkörperphysik, TU Dresden, D–01069 Dresden, Germany    H. Luetkens Laboratory for Muon–Spin Spectroscopy, Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland    Yu. G. Pashkevich A. A. Galkin Donetsk Phystech NASU, 83114 Donetsk, Ukraine    A. Kwadrin Institut für Festkörperphysik, TU Dresden, D–01069 Dresden, Germany    R. Khasanov Laboratory for Muon–Spin Spectroscopy, Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland    A. Amato Laboratory for Muon–Spin Spectroscopy, Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland    A. A. Gusev A. A. Galkin Donetsk Phystech NASU, 83114 Donetsk, Ukraine    K. V. Lamonova A. A. Galkin Donetsk Phystech NASU, 83114 Donetsk, Ukraine    D. A. Chervinskii A. A. Galkin Donetsk Phystech NASU, 83114 Donetsk, Ukraine    R. Klingeler Leibniz–Institut für Festkörper– und Werkstoffforschung (IFW) Dresden, D–01171 Dresden, Germany    C. Hess Leibniz–Institut für Festkörper– und Werkstoffforschung (IFW) Dresden, D–01171 Dresden, Germany    G. Behr Leibniz–Institut für Festkörper– und Werkstoffforschung (IFW) Dresden, D–01171 Dresden, Germany    B. Büchner Leibniz–Institut für Festkörper– und Werkstoffforschung (IFW) Dresden, D–01171 Dresden, Germany    H.–H. Klauss h.klauss@physik.tu-dresden.de Institut für Festkörperphysik, TU Dresden, D–01069 Dresden, Germany
July 15, 2019
Abstract

We report zero field muon spin relaxation (SR) measurements on RFeAsO with R = La, Ce, Pr, and Sm. We study the interaction of the FeAs and R (rare earth) electronic systems in the non superconducting magnetically ordered parent compounds of RFeAsOF superconductors via a detailed comparison of the local hyperfine fields at the muon site with available Mössbauer spectroscopy and neutron scattering data. These studies provide microscopic evidence of long range commensurate magnetic Fe order with the Fe moments not varying by more than 15 % within the series RFeAsO with R = La, Ce, Pr, and Sm. At low temperatures, long range R magnetic order is also observed. Different combined Fe and R magnetic structures are proposed for all compounds using the muon site in the crystal structure obtained by electronic potential calculations. Our data point to a strong effect of R order on the iron subsystem in the case of different symmetry of Fe and R order parameters resulting in a Fe spin reorientation in the R ordered phase in PrFeAsO. Our symmetry analysis proves the absence of collinear Fe–R Heisenberg interactions in RFeAsO. A strong Fe–Ce coupling due to non–Heisenberg anisotropic exchange is found in CeFeAsO which results in a large staggered Ce magnetization induced by the magnetically ordered Fe sublattice far above . Finally, we argue that the magnetic R–Fe interaction is probably not crucial for the observed enhanced superconductivity in RFeAsOF with a magnetic R ion.

pacs:
75.30.Fv, 74.70.-b, 76.75.+i, 76.80.+y

I Introduction

The recent discovery of high temperature superconductivity in LaFeAsOF by Kamihara and coworkers has triggered intense research in the Fe–pnictides.Kamihara08 () In its wake, superconductivity with transition temperatures that exceed 50 K have been found in the oxopnictide materials in which La is substituted by R = Sm, Ce, Nd, Pr, and Gd, respectively. Chen08_XH-arXiv (); Chen08_GFb-arXiv (); Ren08a-arXiv (); Ren08b-arXiv (); Cheng08-arXiv () Besides the high critical temperatures, striking similarities to the properties of the high–T cuprates have been pointed out. As with the cuprates, the Fe–pnictides have a layered crystal structure with alternating FeAs and RO sheets, where the Fe ions are arranged on a simple square lattice.Kamihara08 () Superconductivity emerges in the pnictides when doping the antiferromagnetic parent compound either with electrons or holes which suppresses the magnetic order.Luetkens08a (); Luetkens08b-arXiv () These similarities raised the hope that cuprates and pnictides share a common mechanism for superconductivity, and that after 20 year of research on high–T cuprates the Fe–pnictides may provide new insight into the superconducting coupling mechanism and verify existing theories about high temperature superconductivity.

In contrast to the cuprates, the non superconducting magnetic parent compound is not a Mott–Hubbard insulator but a metal. Theoretical studies reveal a two–dimensional electronic structure with all Fe 3d bands contributing to the density of states at the Fermi level.Singh08-arXiv () Neutron diffractionCruz08 () and local probe techniques like Mössbauer spectroscopy and SRKlauss08 (); Carlo09 (); Bernhard09 () prove that the RFeAsO parent compounds order in a commensurate spin density wave (SDW) magnetic order with a strongly reduced ordered Fe moment. Neutron studies suggest a columnar magnetic structure with a Fe magnetic moment between 0.25  and 0.8  below  K that depends on R.Lynn08 () Due to the small size of the ordered SDW moment (compared to metallic iron with a moment of approx. 2.2  per Fe) and the lack of large single crystals the temperature dependence of the magnetic order parameter can be determined by local probe techniques with a much higher accuracy than with neutron scattering.Klauss08 () Note that the magnetic transition in the RFeAsO system is always preceeded by an orthorhombic structural distortion which appears at , which is about 10–20 K above .

It is still an open question why the rare earth containing systems in the series RFeAsOF with a localized R magnetic moment have a higher T than LaFeAsOF. One suggestion is based on a purely geometrical argument. It is argued that the different ionic radii of the rare earth elements change the Fe–As–Fe bond angles in the Fe–As plane and the FeAs–FeAs interplane distance. As a consequence, the planar anisotropy of the electronic properties may be better for superconductivity in case of R = Ce, Pr, Nd, etc.. On the other hand, the electronic interaction of the R 4f electrons with the Fe conduction band states may be crucial to enhance the density of states at the Fermi energy.

In this work we examine the interaction of the FeAs electronic bands with the rare earth subsystem by a detailed comparison of the local hyperfine fields at the muon site and Fe nucleus with neutron scattering results. These studies were performed on the undoped magnetically ordered parent compounds of the RFeAsOF superconductors. We report zero field muon spin relaxation measurements on powder samples of RFeAsO with R = La, Ce, Pr, and Sm. We provide microscopic evidence of static commensurate magnetic order of Fe moments. The Néel temperatures and the temperature dependence of the Fe sublattice magnetization were determined with high precision and are compared with available Mössbauer spectroscopy and neutron scattering results from the literature. In contrast to neutron studies our measurements prove that the size of the ordered Fe moment is independent of the rare earth ion. For a quantitative analysis of the SR spectra the muon site in the RFeAsO crystal structure has been determined by electronic potential calculations using a modified Thomas–Fermi approach. Spontaneous magnetic ordering of the rare earth magnetic moment is observed by SR below  4.4(3), 11(1), and 4.66(5) K for R = Ce, Pr, and Sm, respectively. Iron and R magnetic structures are proposed for all compounds on the basis of a detailed symmetry analysis and magnetic dipole field calculations at the muon site on the one hand and SR, Mössbauer spectroscopy and neutron scattering data on the other. Non–collinear magnetic order of Ce and Sm is found in the corresponding compounds, which we explain by a weak coupling of adjacent R planes in the R–O–R layer. In PrFeAsO the SR data suggest an Fe spin reorientation developing below , while the Fe order is unaltered below in the Sm and Ce compounds.

In CeFeAsO we find a sizable staggered magnetization of the Ce ions induced by the Fe subsystem already far above which amounts to approximately   near to . We argue that the neglect of the Ce magnetization may have caused the overestimation of the ordered Fe moment in recent neutron diffraction studies. Our symmetry analysis proves the absence of collinear Fe–R Heisenberg interaction in RFeAsO compounds. Using classical and quantum mechanical approaches we deduce Fe–Ce and Ce–Ce exchange coupling constants and show that the Fe–Ce non–Heisenberg exchange interaction is exceptionally strong and of the same order as the Ce–Ce exchange interaction. In the Sm and Pr compounds the observed coupling between the R and the Fe subsystems is found to be much weaker than in CeFeAsO. Therefore, we conclude that the magnetic R–Fe interaction is probably not crucial for the enhanced superconducting transition temperatures in RFeAsOF with magnetic 4f ions compared to LaFeAsOF, since only in CeFeAsO a strong R–Fe coupling is observed.

Ii Experimental

Polycrystalline RFeAsO have been prepared by using a two–step solid state reaction method, similar to that described by Zhu et al., and annealed in vacuum.Zhu_X08-arXiv () The crystal structure and the composition were investigated by powder X–ray diffraction. From the X–ray diffraction data no impurity phases are inferred.

In a SR experiment nearly 100% spin–polarized muons are implanted into the sample one at a time. The positively charged thermalize at interstitial lattice sites, where they act as magnetic micro probes. In a magnetic material the muon spin precesses about the local magnetic field at the muon site with the Larmor frequency (muon gyromagnetic ratio MHz T). With a lifetime of  s the muon decays into two neutrinos and a positron, the latter being predominantly emitted along the direction of the muon spin at the moment of the decay. Measurement of both the direction of positron emission as well as the time between muon implantation and positron detection for an ensemble of several millions of muons provides the time evolution of the muon spin polarization along the initial muon spin direction. Magnetic materials with commensurate order possess a well–defined local field at the muon site. Therefore, a coherent muon spin precession can be observed, which for powder samples has the following functional form (see e.g. Ref. Reotier97, ):

(1)

The occurrence of 2/3 oscillating and 1/3 non–oscillating SR signal fractions originates from the spatial averaging in powder samples where 2/3 of the magnetic field components are perpendicular to the spin and cause a precession, while the 1/3 longitudinal field components do not. The relaxation of the oscillation, , is a measure of the width of the static field distribution . Dynamical effects are also present in while the relaxation of the second term, , is due to dynamic magnetic fluctuations only.

If magnetically inequivalent muon sites exist in the crystallographic or magnetic structure, each of the sites contribute to the SR signal with its weight . In 100% magnetically ordered specimens . Therefore SR not only provides a highly sensitive measure of the magnetic order parameter via internal magnetic fields , but also allows to independently determine the magnetic volume fraction. This is not possible with non–local probes such as e.g. neutron diffraction.

Iii Muon spin relaxation results

In Fig. 1, zero field (ZF) SR time spectra are shown for RFeAsO with R = La, Ce, Pr, and Sm. At high temperatures above 150 K no muon spin precession and only a very weak depolarization of the polarization is observed. This weak depolarization and its Gaussian shape are typical for a paramagnetic material and reflect the occurrence of a small Gaussian–Kubo–Toyabe depolarization originating from the interaction of the spin with randomly oriented nuclear magnetic moments.Hayano79 () At temperatures below a well–defined spontaneous muon spin precession is observed in all compounds indicating a well–defined magnetic field at the muon sites. Therefore, long range static magnetic order with a commensurate magnetic structure is realized in all investigated compounds of the RFeAsO series. Accordingly, incommensurate order or spin glass magnetism can be excluded. Only in LaFeAsO, a second SR frequency with lower amplitude is observed in addition to the main precession signal . As described above, this indicates that two magnetically inequivalent muon stopping sites are present in the crystal lattice/magnetic structure.

Figure 1: Zero field SR spectra for RFeAsO with R = La, Ce, Pr, and Sm for three different temperatures: , , and .

The (static) magnetic order develops in the full sample volume below as evidenced by the magnetic SR signal fraction shown in the inset of Fig. 2. The observed 5–10% residual non–magnetic signal fraction observed in our measurements are due to muons that do not hit the sample but stop in the sample holder or cryostat walls.

Figure 2: Muon spin precession frequency as a function of reduced temperature for RFeAsO with R = La, Ce, Pr, and Sm. Inset: Magnetic signal fraction for RFeAsO with R = La, Ce, Pr, and Sm. Lines are guides to the eye.

The Néel temperatures obtained from the SR measurements (superscript ) are summarized in table 1. As we have shown previously for LaFeAsO, the Néel temperature of the iron subsystem and the structural transition temperature can also be determined from anomalies in the temperature dependence of the resistivity.Klauss08 () For all RFeAsO compounds investigated here, pronounced anomalies have been observed.Klauss08 (); Kimber08 (); Hess08-arXiv (); Hamann-Borrero-unpublished () The corresponding transition temperatures and are listed also in table 1. In addition magnetic, of the iron and the rare earth (superscript ) subsystems, and structural, , transition temperatures deduced from neutron (superscript ) and X–ray scattering (superscript ) experiments are given for comparison. Note that for SmFeAsO no neutron data are available due to the high neutron absorption of natural Sm.

R
La 139(1) 137(3)Cruz08 () 138.0 158(3) 156.0
Ce 4.4(3) 5Zhao08a () 137(2) 139(5)Zhao08a () 134.8 152.2 151.5
Pr 11(1) 11 123(2) 127.0Zhao08b () 127.0 136.0 136.0
Sm 4.66(5) 138(2) 136.5 158.5 160.0
Table 1: Magnetic and structural transition temperatures of RFeAsO with R = La, Pr, Sm, and Ce. In the cases where no reference is given the transition temperatures were measured exactly on the samples used in this study.Klauss08 (); Luetkens08b-arXiv (); Hess08-arXiv (); Kimber08 (); Hamann-Borrero-unpublished () All temperatures are given in Kelvin. See text for the abbreviations in the superscripts.

Within the experimental error, the Néel temperatures determined from SR, resistivity and neutron scattering agree very well. The structural transition is found to be clearly separated by 10–20 K from for all investigated compounds. Our highly sensitive SR investigations, prove the absence of static long range magnetic order between and . Also no static magnetic short range correlations or disordered magnetism, which would have been easily detected by SR, has been observed. However, our SR data do not rule out a dynamic nematic magnetic phase with broken Ising symmetry that has recently been proposed to develop below ,Xu08 (); Fang08 () provided that the fluctuations are faster than approx. 10 GHz.

iii.1 Iron magnetic order

Now we turn our discussion to the temperature dependence of the Fe sublattice magnetization. We will first concentrate on the temperature regime above the static order of the rare earth moment. The temperature dependence of the muon spin precession frequency for RFeAsO with R = La, Ce, Pr, and Sm is shown in Fig. 2 for comparison. All compounds display a very similar temperature dependence and absolute value of the SR frequency. Only the R = Ce compound shows a higher frequency and a stronger temperature dependence below . In Fig. 3 the SR frequency is shown together with the average magnetic hyperfine field at the Fe site from Fe Mössbauer spectroscopy McGuire08b-arXiv () and the square root of the magnetic Bragg peak intensity of available neutron scattering data.Huang08 (); Zhao08a (); Zhao08b () The scale for and is the same for all diagrams, and the scales for have been adjusted so that the SR and neutron data coincide as much as possible. In the following, we discuss the data for each compound separately.

Figure 3: Main SR frequency as a function of reduced temperature together with the average magnetic hyperfine field at the Fe site from Mössbauer spectroscopy McGuire08b-arXiv () and the square root of the magnetic Bragg peak intensity of available neutron scattering Huang08 (); Zhao08a (); Zhao08b () data. Note that the scale of and of is the same in all graphs. Some axes’ labels have been omitted for clarity. Typically, error bars of the SR frequency are smaller than the data points.

iii.1.1 LaFeAsO

In LaFeAsO all three observables , , and scale with each other since they all measure the size of the ordered Fe moment. A gradual increase of the ordered Fe moment observed below indicates a second order phase transition. The steep increase of the order parameter deviates from the mean–field temperature dependence. As we have shown previously, this can be understood qualitatively in the framework of a four band spin density model.Klauss08 () The onset of the magnetic order is accompanied by a broad static magnetic field distribution at the muon site (see Sec. II) that narrows rapidly with decreasing temperature, as can be seen in Fig. 4(a). The relative width of the field distribution is proportional to , i.e. diverges with as the temperature approaches the Néel temperature of the Fe sublattice from below for all systems presented here (not shown). Except for SmFeAsO no dynamic magnetic fluctuations are detected () for temperatures below .

iii.1.2 SmFeAsO

For SmFeAsO no neutron and Mössbauer data are available. Qualitatively and quantitatively the temperature dependence of the observed SR frequency is very similar to that of LaFeAsO and consistent with previously reported SR data.DrewNature () A sharp increase of the ordered Fe moment is observed below . Also the saturation value of  MHz indicates the same size of the ordered Fe moment in LaFeAsO and SmFeAsO assuming the same hyperfine coupling constants in these isostructural compounds. In contrast to the other systems, magnetic fluctuations in the time window of SR are detected in this system that cause the dynamic relaxation rate to increase gradually and to saturate between 10 K and 30 K. As reported by Khasanov et al., this can be associated with fluctuating Sm magnetic moments due to a thermally activated population of Sm crystal electric field levels.Khasanov08 () The temperature dependence of the dynamic relaxation rate is well described by

(2)

with a saturation value at low temperatures and an activation energy that is related to low lying Sm crystal electric field levels. The rough agreement of the activation energy of the Sm magnetic moment fluctuations with the Sm crystal electric field splitting has been confirmed by specific heat measurements reported by Baker and coworkers.Baker08-arxiv () A fit of Eq. (2) to the longitudinal relaxation rate is shown in Fig. 4 and yields s, s, and  meV. This value for is approximately two times higher than in the oxygen deficient SmFeAsO studied by Khasanov et al..Khasanov08 () However, one has to take into account that the oxygen deficiency in SmFeAsO causes a higher defect density and changes of the lattice parameters.Ren08 () Therefore, a different activation energy for the undoped SmFeAsO compared to SmFeAsO is possible.

Figure 4: top: Transverse relaxation rate for RFeAsO with R = La, Ce, Pr, and Sm. Lines are guides to the eye. bottom: Longitudinal relaxation rate for RFeAsO with R = Sm only. The solid line is a fit of equation 2 to the data. The dashed line is a guide to the eye

iii.1.3 PrFeAsO

Apart from a slightly reduced Néel temperature, PrFeAsO shows the same temperature dependence and saturation value of the Fe sublattice magnetization, i.e. as SmFeAsO. Note that again the saturation value for and are very similar to that of LaFeAsO indicating the same size of the ordered Fe moment. A close comparison of and reveals that is systematically reduced by a small amount compared to . This phenomenon can be explained by the same mechanism found in the Ce system (see below and Section VI). In the case of PrFeAsO the muon spin polarization function (1) did not approximate the data very well and a non–zero phase of the oscillation and a generalized exponential relaxation function had to be used to describe the data.

iii.1.4 CeFeAsO

A qualitatively different behavior is observed for the CeFeAsO compound. Neutron diffraction as well as SR data do not scale with the observed hyperfine field at the Fe site. The magnetic Bragg intensity as well as the internal magnetic field at the muon site measured by the muon precession frequency continuously increase below . Only the Mössbauer hyperfine field displays the same rapid saturation below with the same ordered moment as observed for the La and Pr compounds.McGuire08b-arXiv () Therefore we conclude that the SR as well as the neutron data do not solely measure the Fe sublattice magnetization, but also contain a significant contribution from the Ce sublattice. The Fe Mössbauer spectroscopy provides the most accurate measurement of the on–site Fe sublattice magnetization without sizable contribution from the rare earth moments due to a weak transferred hyperfine coupling. In contrast, a magnetization on the rare earth site induced by the Fe subsystem has the same symmetry as the Fe order and therefore contributes to the same Bragg peaks. In principle, neutron scattering can distinguish between the different contributions from the Fe and the Ce sublattice by fitting the different magnetic form-factors. This has not been done for in the neutron studies Huang08 (); Zhao08a (); Zhao08b () shown in Fig. 3, where the whole magnetic intensity has been attributed to originate from Fe moments alone. Similar to the neutron data, the local field at the muon site also contains a contribution from the rare earth magnetic sublattice. In the following, we model the above mentioned contribution of the induced Ce moment to the field at the muon site by an additional Curie–Weiss contribution. This can be interpreted as a local magnetization of paramagnetic moments of the Ce sublattice induced by the molecular field generated by the Fe sublattice. In turn, the induced Ce sublattice magnetization creates a dipole field at the muon site. The temperature dependence of the Fe molecular field has to be proportional to the Fe sublattice magnetization, i.e. the muon spin precession frequency observed in the other RFeAsO systems. This is plausible because we have shown that above the rare earth ordering temperature is almost independent of the rare earth ion. Thus we chose the following function to describe our data in a first approximation:

(3)

The first term in the last square brackets is used to describe the contribution of the Fe sublattice, the second term is the additional Curie–Weiss contribution of the Ce sublattice to the magnetic field at the muon site. Here, describes the hyperfine coupling constant of the Ce moments with the muon spin. A fit of this function to the SR frequency obtained for CeFeAsO is shown in Fig. 3 and the two contributions are highlighted as described in the legend. This simple model describes the data reasonably well for temperatures between 10 K and up to  K, yielding  MHz, , . The constants  K, and  K were obtained by restricting the fit range to the most relevant temperature region between 10 K and 50 K. The exponents and are close to the results obtained for the other systems without the Curie–Weiss contribution. The enhanced could be due to a slightly different muon site compared to other RFeAsO compounds. The deviation from this behavior below 10 K is attributed to the growth of antiferromagnetic correlations in the vicinity of .

Figure 5: Structure and electrostatic potential energy map of the (110)–plane for a muon in the Cmma orthorhombic phase of CeFeAsO. The A and B muon positions are marked by corresponding letters. The potential energy is given in atomic units. The origin of the unit cell has been moved for clarity.

The exceptionally strong coupling of the Ce is already reasonable considering the ground state properties of the R ion in the crystal electric field (CEF). The ground state of the Kramers ion Ce is the J-multiplet . The Sm Kramers ion has as lowest -multiplet with . The non–Kramers ion Pr stays in the ground multiplet with . The magnetic behavior of the R ions can be understood qualitatively from the susceptibility of free R ions:

(4)

Thus, because of the different g-factors (, , and ) one expects at least one order of magnitude less induced Sm magnetic moment compared to Ce (for equal magnitudes of Fe–Sm and Fe–Ce coupling constants, see section V.2). Probably, due to this feature, we observe very similar SR spectra in the Sm, Pr, and La compounds in the temperature range . since the exact sequence of CEF levels in PrFeAsO is not known up to now, it is more difficult to predict its behavior.

The above qualitative discussion neglects higher CEF levels and the Ce–Ce interactions. A more detailed and quantitative discussion of the coupling of the Ce sublattice to the Fe sublattice using classical and quantum approaches will be given in Sec. V and VI.

iii.2 Rare earth magnetic order

For R = Pr and Ce the onset of the rare earth magnetic order  K and 4.4(3) K, respectively causes a second order, i.e. smooth decrease of the muon precession frequency by approx. 2.2 MHz at 2 K in both cases, as can be seen in Fig. 3. However, in PrFeAsO the Pr magnetic order is accompanied by a maximum in the width of the magnetic field distribution, i.e. the relaxation rate (see inset of Fig. 4) that rapidly decreases at lower temperatures, whereas in CeFeAsO the Ce magnetic order causes the field distribution to broaden monotonically. Note that the magnetic field at the muon site caused by the static order of the Ce sublattice leads to a decrease of the observed SR frequency while the Fe order induced magnetization of the Ce sublattice above causes an increase of the frequency . Therefore, it is evident that the Ce moments order in a different structure than induced by the Fe magnetic system.

SmFeAsO shows first order Sm magnetic order, i.e. the Sm magnetic order parameter is discontinuous at  K. Contrary to the Ce and Pr magnetic order the Sm magnetic order causes the appearance of two satellite muon precession frequencies  MHz in addition to the main frequency observed above . The two satellites and the main frequency have signal fractions of 1.0(3):1.0(3):4.3(6). This shows that the Sm magnetic order has different symmetry compared to the Fe magnetic order, i.e. it causes a change of the magnetic unit cell. A detailed discussion of R magnetic structures and its interplay with the Fe sublattice will be given in Sec. VI, VII and VIII.

Iv Determination of the muon site

To determine the contributions from Fe and R magnetic order to the local magnetic field at the muon site it is necessary to determine the muon site in the lattice. For this purpose we used a modified Thomas Fermi approachReznik08 () and available structural data. The goal is to determine a self consistent distribution of the valence electron density from which the electrostatic potential can be deduced. The local minima of this potential at interstitial positions are regarded as possible stopping sites for muons. We verified the applicability of our approach by comparison of numerical results with experimentally determined muon sites in RFeOHolzschuh83 () and by a successful interpretation of SR spectra of the complex magnetic structures in layered cobaltites RBaCoO.Luetkens-Co ()

A potential map of CeFeAsO in the orthorhombic phase was calculated using structural dataZhao08a () at 1.4 K and is shown in Fig. 5. The calculations have been done without taking into account the host lattice relaxation around the muon. We observe two types of possible muon positions which are labeled A and B. The A type position is located on the line connecting the Ce and As ions along the –direction. Note that a similar location of the point with deepest potential was calculated for LaOeAs in the tetragonal phase using the general potential linearized augmented plane-wave method and local density approximation.Takenaka78 () The A type position with coordinates (0,1/4,z) has 4g local point symmetry mm2, i.e. the same as the R sites. The B type position has a general 16o local point symmetry with (x,y,z) coordinates. For CeFeAsO the coordinates of A are (0,1/4,0.41) and (0.44,0.19,0.04) for B. Note that the points B are located at oxygen ions which are typical points for muons in many oxides. The z and z coordinates vary in the third decimal if Ce is replaced by La, Pr or Sm.

A second important result is the comparison of the calculated muon precession frequency, i.e. the local magnetic dipole field at the muon site with the experimentally determined muon precession frequency for LaFeAsO. With the experimentally determined Fe magnetic moment Cruz08 () of 0.36  we calculate f MHz for site A and  MHz for site B. As shown by Klauss et al. in Ref. Klauss08, and Section III of this work, in LaFeAsO two muon frequencies are observed. One frequency with 23 MHz originating from the mayor volume fraction () and one lower frequency with 3 MHz which develops from a strongly damped signal below approximately 70 K. In RFeAsO with R = Sm, Pr, and Ce only the high frequency is present in the SR spectra. We conclude that site A is the main muon site since this gives the correct order of magnitude for . Probably the site B is also partially populated at low temperatures in the LaFeAsO. The fact that we obtain a 46 % smaller value than the experimental result is reasonable since our calculation only takes into account local dipole fields and neglects contact hyperfine contributions. Similar differences are found in LaCuO, the antiferromagnetic parent compound of the 214–cuprate superconductors.Klauss04 () As will be shown below, the magnetic field caused by the Fe magnetic order is directed along the crystallographic –axis. This is in agreement with recent SR experiments on Fe pnictide single crystals.luetkensInprep () In the following we use only muon site A and a renormalization factor of 1.86 for the local field caused by Fe magnetic ordering at this muon site to account for the contact hyperfine field for all R. This assumption is justified from our calculation of the electronic charge density distribution that shows very similar results for all R. This manifests itself in the almost identical position of muon site A in compounds with different R. In its turn this means that the renormalization due to the contact hyperfine interaction, which depends only on the electron density at the muon site, should be nearly the same within this series.

V Symmetry analysis of Fe and R magnetic order parameters in RFeAsO

v.1 Translational symmetry and magnetic modes in RFeAsO compounds

Iron magnetic order in RFeAsO sets in about 10–20 K below the tetragonal to orthorhombic structural phase transition. Cruz08 () The space group of the paramagnetic phase is with Fe and R ions occupying 4b and and 4g position respectively. Neutron studies of RFeAsO compounds revealed numerous magnetic Bragg peaks all of which can be reduced just to three types of magnetic propagation vectors.Cruz08 (); Kimber08 (); Zhao08a (); Zhao08b (); Qiu08 (); Huang08 (); Lynn08 () These vectors in an orthorhombic setting are:

(5)

Here we use an orthorhombic primitive cell with the unit cell vectors , , and where , , and are the distances between nearest neighbor Fe ions along a, b and c directions, denoted by the unit vectors , respectively. Note that in this setting nuclear Bragg peaks of the spacegroup are either or . Also, a ’magnetic setting’ with , , and is often used.

The primitive cells of the magnetic structures induced by each of the are not orthorhombic. Each of the magnetic primitive cells contains four magnetically inequivalent R–, Fe–, and muon–sites. To analyze the symmetry of the magnetic structures in a unified way we will label these four positions with numbers , with corresponding ordered magnetic moments for the R– and Fe–positions. To identify the different positions in the magnetic cell with positions in the nuclear cell (doubled along the –direction) we use the mapping where is one of the four possible positions in the magnetic primitive cell and a position in the nuclear cell as shown in Fig. 6.

Figure 6: Enumeration of positions used to describe different types of Fe and R magnetic order. (a) R and Fe positions and b) muon site A.

The propagation vector induces a magnetic lattice of type in which the primitive cell is built on the translations , , and the anti–translation along the z–axis .X1 () Thus, in –type magnetic structures the following positions are equivalent and mapped as denoted by ’’: , , , and .

The primitive cell of the magnetic lattice induced by the propagation vector is constructed from the translations , , and the anti translation .X1 () Then we obtain the identity of the positions: , , , and .

Finally, for the propagation vector the magnetic lattice is and the magnetic primitive cell can be built from the translations: , , .X1 () Therefore we have the mappings: , , , and .

In order to investigate the magnetic interactions in different R and Fe magnetic structures with the translational symmetry of the propagation vector () it is necessary to calculate the magnetic modes, i.e. basis functions of the irreducible representations (IR) of the propagation vector little groups . These groups are the same for all propagation vectors (5) and they are also identical with the groups for ferromagnetic order, i.e. propagation vector .X2 () Following a method described by Bertout, and Izyumov and Naish we introduce possible magnetic modes and , =1,2,3 as linear combinations of Fe (or R) sublattice magnetic moments , where denotes a particular atom (see above) in the respective magnetic primitive cell:X3 (); X4 ()

(6)

The components of these linear combinations are the basis functions of the eight one-dimensional IR with for the group (see table 32 in Ref. X2, ). The results of the symmetry analysis for the magnetic moments located at the 4b positions (Fe ions) and 4g positions (R ions and muon sites) are shown in Table 2.

symmetry elements (index 1 denotes an improper translation along the y–axis) (0,0,0) (,0,) (,0,0) (0,0,)
IR 2 2 2 Fe R & Fe R & Fe R & Fe R &
Table 2: Magnetic modes of the magnetic propagation vectors , i.e. basis functions of the irreducible representations (see text). The indexes indicate the non–zero Cartesian components of the vectors (6), i.e. for example is the –component of .

In accordance with neutron data the iron magnetic subsystem orders antiferromagnetically along the –direction and ferromagnetically along the –direction with doubling of the unit cell along the –axis for the R = La compound (magnetic propagation vector ) and without doubling for R = Ce and Pr (magnetic propagation vector ).Lynn08 () In all these structures Fe magnetic moments are directed along the –axis. These magnetic structures are described by and non–zero magnetic order parameters shown in Table 2.

The onset of a given type of magnetic order lowers the symmetry of the paramagnetic phase and creates a magnetic symmetry. The orientation of the magnetic moments and the distribution of local magnetic fields in the magnetic cell have the same symmetry. To determine the orientation of the magnetic field at a particular muon site we assign an artificial magnetic degree of freedom to this site. The set of magnetic degrees of freedom for different points (Wyckoff positions) forms a magnetic representation. A standard decomposition of this magnetic representation into irreducible representations of the space group allows the analysis of the symmetry of magnetic field distributions at the muon sites. This symmetry must be compatible with the given space group and the same as the symmetry of the magnetic order parameter. In other words, the symmetry of the magnetic field distribution at the muon site must belong to the same IR as the magnetic order parameter.

From Table 2 it becomes self–evident that the –type of magnetic structure of iron moments creates –type molecular fields at the rare earth and A–type muon sites. Therefore, a magnetic field distribution of any symmetry that induces a staggered-type ordering of the R magnetic moments leads to a magnetic field distribution at the A–type muon site of same symmetry. Due to symmetry reasons, some types of Fe magnetic order, for instance , do not create magnetic fields at the 4g positions. Thus, the presence of SR signals in the Fe antiferromagnetically (AFM) ordered phase of R = La, Pr, Ce and Sm compounds excludes the possibility for AFM magnetic order along the –axis and ferromagnetic (FM) order along –axis with iron magnetic moments directed along –axis.

The –type of Fe order and the corresponding direction of the magnetic field at the A–type muon site are shown in Fig. 7. The dipole fields at the A–type muon position for all types of Fe and R magnetic order are given in the Appendix A.

Figure 7: (a) L(K)–type Fe magnetic order realized in LaFeAsO and probably in SmFeAsO and its magnetic dipole field direction at the muon site A. (b) L(K) order observed in CeFeAsO and PrFeAsO.

v.2 Symmetry of the Fe–R magnetic interactions

To start with the strongest exchange interaction we analyze the symmetry of permutation modes (5) or exchange multiplets for Fe and R magnetic subsystems.X3 () The permutation modes for the different translational symmetries are listed together with the corresponding irreducible representations in table 3.

(0,0,0) (,0,) (,0,0) (0,0,)
IR Fe R Fe R Fe R Fe R
Table 3: Permutation modes, i.e. exchange multiplets of the four magnetic propagation vectors.

Since there are no permutation modes of the Fe– and R–subsystems which belong to the same irreducible representation there are no Heisenberg exchange interactions between the R– and Fe–subsystems for the case of or translational symmetry of the Fe magnetic order. This interaction exists only for magnetic structures with and translational symmetry. The respective Fe–R magnetic exchange Hamiltonian has the form:

(7)

However, for the cases of and translational symmetry the Fe and R subsystems can interact by the non–Heisenberg exchange (see Table 3). The part of these anisotropic Fe–R interactions relevant for the following considerations is given below:

(8)

Consequently, the onset of the type of Fe magnetic order creates an effective staggered magnetic field at the R site along the z–direction. The magnitude of this field is proportional to the value of the Fe–R coupling constant and the iron subsystem order parameter determines its temperature dependence. In the following we will show that this field exceeds a respective dipole field at the R sites by at least one order of magnitude. The mutual orientation of the and vectors depends on the sign of the Fe–R coupling constant . The orientation of the induced magnetic moment on the R site is shown in Fig. 8.

Figure 8: Direction of the magnetic moment on the R site induced by the L(K) Fe order in a) CeFeAsO and b) PrFeAsO. The sign of the coupling constant is different in the Ce and the Pr compound resulting in opposite orientation of the induced moments.

The magnitude of the –component of the rare earth magnetic moments is determined by the exchange field and the R ion crystal electric field (CEF).

Vi Determination of the magnetic structures and the Fe–R coupling constants from the temperature dependence of the SR response

Below the Néel temperature the SR response of RFeAsO is mainly determined by magnetic dipole fields created by the Fe and R subsystems on the A-type muon site. It can be calculated by:

(9)

If the are given in units of (see equation (6)) the magnetic dipole field is obtained in units of 0.1 T. These equations demonstrate a surprisingly close agreement of dipole fields for Fe magnetic order with () and without () doubling along z–axis. Note that the constants in equations (9) only vary in the third decimal if Ce is replaced by La, Pr or Sm. This explains the similar SR spectra of LaFeAsO with –, and CeFeAsO and PrFeAsO both with –type of Fe antiferromagnetic order. The high temperature fit of the SR frequency gives roughly the same saturation value  MHz. This corresponds to a value of 0.36  for the Fe saturation moment if one takes into account contact hyperfine fields by applying the renormalization factor of 1.86 as in LaFeAsO. Because of the similar order parameters of the Fe magnetic order found in SmFeAsO, LaFeAsO, and PrFeAsO we conclude that a –type of Fe antiferromagnetic order is also realized in SmFeAsO. However, SR studies alone cannot distinguish between the possible translational symmetries, i.e. and of the order parameter.

vi.0.1 CeFeAsO

The SR response drastically changes below the Ce ordering temperature . Contrary to the naive expectation that the Ce magnetic order would increase the local magnetic field at the muon site and therefore the SR frequency our experimental results show a decrease of the SR frequency below . According to our analysis of the magnetic field distribution at the A–type muon site this behavior indicates a breaking of the symmetry of the induced Ce order. Neutron diffraction studies of CeFeAsO revealed the appearance of –type magnetic Bragg peaks below in addition to –type peaks also present above .Zhao08a () The Ce magnetic structure proposed by Zhao et al. is a non–collinear arrangement of Ce moments in the -plane.Zhao08a () This structure is shown in Fig.9 a) and can be described as a linear combination of two order parameters: . This new type of R order produces the following magnetic field at the A-type muon site for temperatures below :

(10)
Figure 9: (a) Collinear Fe magnetic order () and non–collinear Ce order in CeFeAsO as proposed by Zhao et al. Zhao08a () which is in accordance with the SR measurements. (b) Magnetic structure of the Pr and Fe subsystems with translation symmetry which satisfactorily describes SR and Mössbauer data. Non–zero are the of Fe and of Pr order parameters. The Fe and Pr subsystems remain in the and type of exchange magnetic order, respectively. (c) The proposed Sm structure which explains the observed SR spectra. Fe magnetic ordering has translational symmetry with a doubling along the –axis, while the Sm magnetic order possesses translational symmetry without a doubling along the –axis).

Here =25.7 MHz accounts for the contribution of the iron subsystem, which, we suppose, preserves its magnetic structure found above . Despite different translational symmetry of the Ce order parameters, the magnetic fields at all eight muon sites have the same modulus (10), therefore no additional frequencies appear at temperatures below . With the Ce magnetic moment of 0.83  at 1.7 K reported by Zhao et al. Zhao08a () and supposing =0 we then obtain a SR frequency of  MHz instead of the experimentally observed 29.8 MHz at 1.9 K. Also the experiment reveals a remarkable decrease of the observed SR frequency compared to maximal value of 32 MHz at 4 K. The decrease of the SR frequency is not as rapid as expected, it develops only gradually below . As follows from equation (10), the magnetic dipole fields at the muon sites created by the Ce magnetic moments lie in the –plane and are rather weak compared to the dipole field created by the Fe moments. The slow decrease of the SR frequency below therefore reflects the gradual disappearance of the Fe induced type magnetization of the Ce moments in the AFM Ce phase below . One can conclude that the Ce–Ce coupling constant which creates the AFM Ce order parameter cannot immediately suppress the induced magnetization. As will be shown below the Ce–Ce coupling constant is of the same order as the Fe–Ce coupling constant . Using equation (10) we estimate the remanent order parameter =0.14 at 1.9 K.

Due to the significant contribution of Ce dipole fields, the SR frequency is unusually high already far above in CeFeAsO. This has been discussed phenomenologically in Sec. III. Here we carry out a more elaborate analysis of this observation. The lowest crystal electric field (CEF) excitations of CeFeAsO and SmFeAsO consist of three Kramers doublets. With the onset of antiferromagnetic iron order these doublets split. Inelastic neutron scattering studies of CeFeAsO revealed that in the paramagnetic phase the energies of the first and the second excited doublets are 216 K and 785 K respectively.Chi08 () The splitting at  K amounts to 10.8 K for the ground state doublet and 37.1 K and 66.1 K for the first excited levels. Using these experimentally deduced parameters and the wave functions of the corresponding CEF levels allows us to determine the Fe–Ce and Ce–Ce exchange interaction coupling constants. In the Fe antiferromagnetic phase above , they are obtained by fitting the theoretical temperature dependence of the Ce magnetic moment to the experimental data. In Fig. 10 the experimental value of the Ce single ion magnetic moment is shown. It is calculated from the SR frequency with the help of equation (9) after subtracting the Fe contribution (see inset of Fig. 10):

(11)

In this effective field approximation the Fe–Ce coupling constant and the Ce–Ce coupling constant both create a staggered effective field on the Ce sites. Its interaction with the Ce magnetic moments lifts the degeneracy of the Kramers doublets. The corresponding effective field on the Ce site is given by:

(12)

where and with and in units of T/. We neglect contributions from Fe–Ce and Ce–Ce dipole fields to the splitting of Kramers doublets under the assumption that the Fe–Ce and Ce–Ce (non–)Heisenberg exchange fields (see equation 12) are much stronger. The Fe–Ce staggered field has to have the same temperature dependence as , i.e. the averaged value of the iron order parameter . Therefore, it can be modeled as with and as determined in Sec. III.

According to the analysis of inelastic neutron scattering data Chi08 () the lowest Kramers doublet possesses the orbital momentum and does not contain any admixtures of the states. One can estimate the magnitude of the effective field from the splitting of the ground Kramers doublet at  K by using the relation:

(13)

where is the g-factor of the free ion and  K the splitting of the ground state doublet. If we account for just the Fe–Ce interaction it yields  T. This large Fe–Ce effective field indicates that dipole–dipole interactions (see Appendix B) do not play a significant role for the magnetization of the Ce sublattice by the ordered Fe sublattice above .

The temperature dependence of the Ce moment in the effective field has been calculated in a quantum mechanical approach according to equation (10) in Ref. Mesot97, by calculating the thermal population of all experimentally determined CEF levels.Chi08 () The result is shown in Fig. 10 as a green line. From our calculation it is clear that there is a deviation from Curie-Weiss behavior due to a sizable contribution to from higher Kramers doublets for temperatures above 50 K. In Sec. III we did a Curie-Weiss approximation of by using Eq. (3). A Curie-Weiss description neglects higher crystal field levels. This is the reason why we restricted the fit to the temperature range between 10 and 50 K in the first approximation done in Sec. III.

On the other hand, a systematic deviation between obtained from the quantum approach and the experimental data is observed in the low temperature region between 10 and 50 K. This indicates that the Ce–Ce interaction, which is neglected by this approach is significant in this temperature region and has to be taken into account for a proper description of the data. Therefore, we fit the Ce magnetic moment with contributions from the Fe–Ce and Ce–Ce exchange interactions by using the usual Brillouin function. In the case of S=1/2 this leads to the following equation:

(14)

where is given in Eq. (12) which should fulfill the boundary condition given by Eq. (13). This approach neglects the higher crystal field levels and therefore can only be applied at temperatures below 50 K. From a fit in the low temperature region we obtain the microscopic parameters  T and  T/. The fit is shown as a magenta line in the Fig. 10. Using 0.41  for iron saturated moment one can estimate a rather large Fe–Ce coupling constant  T/. The negative sign of the Ce–Ce coupling constant  T/ indicates that the type of Ce exchange magnetic order is energy favorable. From =4.4 K one can estimate the value of the coupling constant. It is responsible for the appearance of the in-plane order parameter.Zhao08a () This rough estimate gives  T/. Note, that the big difference between Ce–Ce exchange coupling constants along z- and y–directions points to a strong easy-plane anisotropy in the Ce subsystem.

Figure 10: Temperature dependence of the induced Ce magnetic moment for . has been extracted from the SR frequency by subtracting the contribution from the Fe subsystem (inset). The data are fitted by a quantum and a Brillouin approach valid in different temperature regions, see text.

In conclusion, for CeFeAsO we found a sizable staggered magnetization of the Ce ions induced by the Fe subsystem already far above which amounts to approximately   near to . With the help of a symmetry analysis as well as classical and quantum mechanical approaches we were able to deduce Fe–Ce and Ce–Ce exchange coupling constants. We demonstrated that in the Ce compound the Fe–Ce non-Heisenberg exchange interaction is exceptionally strong and of the same order as the Ce–Ce exchange interaction. The observed SR frequency in the Ce ordered phase is in agreement with the magnetic structure proposed by Zhao et al.,Zhao08a () but a small component from the type Ce magnetic order induced by the Fe sublattice has to be included to fully explain our data below .

vi.0.2 PrFeAsO

The ground state multiplet () of the non–Kramers ion Pr in the low symmetry crystal field splits into nine singlets. Since the exact structure of the CEF levels is not known, the Pr magnetic susceptibility cannot be estimated. The temperature dependence of the SR frequency in PrFeAsO is similar to the one observed in LaFeAsO. This suggests that the induced Pr moment is not as strong as in CeFeAsO. However, the comparison of the high temperature fit of the SR frequency for La, Ce, Sm, and Pr compounds yields the lowest value of  MHz for the Pr compound. Additionally, there is a small but systematic deviation between SR and Mössbauer data for PrFeAsO in the Fe AFM phase. This demonstrates that also in PrFeAsO a Fe induced ordered Pr moment is present in the Fe AFM phase. One can conclude that, in contrast to the Ce compound, the Fe–Pr coupling constant is small and negative. According to equation (9), the fields created by the Fe order parameter and the induced Pr order parameter then have to point in opposite directions at the muon site. Therefore the SR frequency in PrFeAsO in the Fe AFM phase is smaller compared to the one in the other compounds.

Recent neutron diffraction studies reveal that in the Pr ordered phase the Pr moments are directed along the –axis, which creates a non–zero magnetic order parameter, i.e. the same basis function of the irreducible representation as the induced Pr moments in the Fe ordered phase.Zhao08b () Therefore, it is reasonable to assume that also in the Pr–ordered state, the local field at the muon site created by the Fe and the Pr magnetic order have opposite signs. Respective magnetic structures are shown in Fig. 8(b). The magnetic structure shown can therefore qualitatively explain the 2 MHz drop of the SR frequency in the Pr ordered phase. However, the quantitative evaluation of the SR frequency using (9) and the Fe and Pr moment of 0.48 /Fe and 0.84 /Pr at 5 K that were deduced from neutron measurementsZhao08b () reveal a strong discrepancy. If the suggested magnetic structure would be realized a SR frequency near to zero should have been observed. Even if one includes a strong increase of the local Fe moment (neglecting all transferred hyperfine field contributions) below the Pr ordering temperature which might be suggested by the Mössbauer data,McGuire08b-arXiv () the calculated SR frequency is much too small. To remove the discrepancy between neutron and SR observations one has to conclude that in the PrFeAsO the Pr magnetic structure is either easy–plane non–collinear or collinear. In this case the situation would be similar to the Ce and Nd compounds where in addition to the order parameter also other –plane components like or exist. Similar conclusions have been drawn from recent neutron diffraction studies.Kimber08 () The consequences of non–collinear order parameters will be discussed in Sec. VII.

vi.0.3 SmFeAsO

Compared to La and Pr, the Sm compound demonstrates a very similar SR response in the Fe ordered AFM phase which implies a or magnetic Fe order parameter. In spite of the similarity of the lowest CEF levels in Sm and Ce compounds the low value of the Sm g-factor reduces the induced Sm magnetic moment and therefore its contribution to the SR frequency in the Fe AFM phase. While in the Sm ordered phase, in contrast to the Pr and Ce compounds, the SR spectra change drastically and several well resolved frequencies are observed. At the lowest temperatures three SR frequencies are observed: 15 MHz, 23 MHz, 31 MHz for which the approximate ratio of amplitudes is 1:4:1. This behavior indicates that for the given magnetic symmetry of the Sm order, at least three muon sites are inequivalent. This inequivalence can be caused by different translational symmetry of Sm and Fe magnetic order parameters or a complex non–collinear magnetic order of the Sm subsystem. A minimal model which can explain the SR spectra involves a order parameter for the Fe subsystem with and and order parameters for the Sm subsystem with translational symmetries. This magnetic structure is shown in Fig. 9(c). It results in three inequivalent muon sites. The first with a low local field at the muon positions (15 MHz at A5 and A7), the second where the field from the R system cancels (23 MHz at A2, A4, A6, and A8) and a third with a high local field (31 MHz at A1 an A3). The experimentally observed amplitudes are also reasonably well reproduced by the model. Within this magnetic model the size of the Sm magnetic moment can be estimated to be 0.4 /Sm at 1.9 K. Note that it is possible to fit the low temperature data with even more frequencies. The present statistical accuracy is not sufficient to determine the correctness of such a fit, but in a recent SR study the low temperature data have been fitted with five frequencies.DrewNature () One has to note that our symmetry analysis also allows more complicated and even incommensurate Sm magnetic structures.Lamonova ()

Vii Rare earth non–collinear order in RFeAsO

The above proposed R magnetic structures for the Ce, Sm and probably also in the Pr compound involve the well known exchange non–collinearity, i.e. a perpendicular orientation of neighboring R moments. As shown above, such a magnetic structure can be described by a composition of the Cartesian components of two different permutation modes like in the case of the Ce compound or like in the case of SmFeAsO. From a pure symmetry point of view, commensurate and non–collinear magnetic order is characteristic for crystallographic structures with higher than orthorhombic symmetry that implies existence of two– or three–dimensional IR for their space groups at k=0. From a pure energy point of view, the exchange non–collinearity results in competition of two or three different type of exchange multiplets (permutation modes) which belong to the same IR and therefore possess the same Heisenberg energy. In high symmetry magnets this competition is resolved by accounting for fourth or higher order magnetic interactions in the system’s free energy.Sobolev93 (); Pashkevich95 () Due to its crystallographic structure the condition for accidental energy degeneracy of and rare earth permutation modes is fulfilled in the RFeAsO compounds. The part of the Hamiltonian containing the RR exchange interactions for translational symmetry can be written in the form:

(15)

where

(16)

and

(17)

For simplicity we omit the notation . The symbol denotes an exchange interaction coupling constant between nearest neighbor rare earth ions on sites 1 and (see Fig. 11. To deduce equation (15) we used a permutation symmetry in which the interaction between ion 2 and 4 is equal to the interaction between 1 and 3 and so on. If the exchange constant