Anisotropy in RFe 2 Intermetallics : An NMR Study

ةـصلاخ : تانويأ صاوخ نيابت انيصقتسإ دقل ضرلأا ةردانلا . + 3 oH Tb + 3 mT . نينرـلا فـيط داـمتعا مادختـساب ىونل لاجملا ةدش ىلع يوونلا يسيطانغملا mT 169 , bT 159 , Ho 165 تابكرملا يف ةدوجوملا 2 TmFe 2 TbF 2 HoFe ةلصلا تاذ تابكرملا و . لا ىغطي ةدـش ىطعي يذلا و يوونلا يسيطانغملا نينرلا فيط ىلع تانويلأا هذهل قيقدلا لعافت نم ددرتلا زيح يف باطقتسلاا 2 ىلا 7 زتريهاجيج . ةيرظنلا تاباسحلاب تاسايقلا ةنراقمب انمق امك . ABSTRACT: The anisotropy of the Ho, Tb and Tm rare-earth ions in RFe2 compounds is investigated using the field dependence of the NMR spectra of Tm, Tb and Ho in polycrystalline TmFe2, TbFe2, HoFe2 and related pseudo-binary compounds. The very large hyperfine interaction in these ions dominates the NMR spectrum; the NMR frequencies in the 2 to 7 GHz range provide therefore a measure of the localized 4 f moment and its orientation in the crystal field. We compare the NMR measurements to single ion computations that include magnetostriction.


Introduction
he cubic Laves phases series RFe 2 where R is a rare-earth element is the simplest of the several inter- metallic compounds in which rare-earth and iron magnetism coexist.For more than 30 years the RFe 2 compounds and their pseudo-binary (RR') Fe 2 and R(FeM) 2 alloys where M is a transition metal have been the subject of intense investigations.These compounds have provided and continue to provide (Sawickki et al, 2000) a fertile ground to test theoretical models.More recently epitaxially grown films (Lee et al, 2000, de la Fuente et al, 2001, Gordeev et al, 2001, Oster et al, 2005), X-ray magnetic circular dichroism investigations (Fujiwara et al, 2005, Dumesnil et al, 2002, Parlebas et al, 2006) and the hydrated RFe 2 materials (Paul-Boncour, 2004) have attracted the attention of several researchers.However, there are some basic questions in the RFe 2 that are not yet completely resolved.There is still uncertainty about the anisotropy of the iron sub-lattice, about the T contribution of magneto-elastic effects to the measured anisotropy (Mougin et al, 2000), about the crystal-field interaction at the lanthanide site and about the role of the ordered super-lattice of vacancies in the C15 Laves phases (Gratz et al, 1996).Also, the distribution of the total magnetization between the iron and lanthanide sublattices and the relative contribution of localized and itinerant moments is still an open question (Fujiwara et al 2005).
In the present paper the anisotropy of the rare-earth ion is investigated using the field dependence of the NMR spectra of 169 Tm(I = 1/2), 159 Tb(I = 3/2) and 165 Ho(I = 7/2) in polycrystalline TmFe 2 , TbFe 2 , HoFe 2 and some pseudo-binary compounds.In the Ho 3+ , Tb 3+ and Tm 3+ ions the very large hyperfine interaction dominates the NMR spectrum; the NMR frequencies in the 2 to 7 GHz range provide therefore a measure of the localized 4 f moment and its orientation in the crystal field.We compare our NMR measurements to single-ion computations that include the distortion of the cubic environment of the lanthanide ion due to the magnetostriction.The results from the computation suggest that the magnetostriction does not have a significant effect on the NMR spectrum.

Theory
In this section we describe the model used for the computation of the magnitude and the orientation of the moments in RFe 2 compounds subjected to a magnetic field applied in any arbitrary direction but before doing so, we need to clarify a possible confusion between two senses of the term "anisotropy".
There are two distinct aspects to the anisotropy of a magnetic material, the energy anisotropy and the magnetization anisotropy.The former, conventionally specified by classical "anisotropy constants" denotes the angular variation of the free energy, whereas the latter denotes the angular variation of the magnitude of the ionic moment.For the RFe 2 compounds in which the exchange interaction ex H may, at least to a first approximation, be considered isotropic both types of anisotropy are manifestations of the crystal-field interaction cf H at the lanthanide site.The relationship between energy and magnetization anisotropy is not straightforward.For example, it is not necessarily the case that the hardest direction of magnetization (i.e. the direction for which the free energy is greatest) is the direction in which the ionic moment is most quenched.(1979).

The effective electronic Hamiltonian
The orientation of the lanthanide and the iron moments in a crystal of RFe 2 subjected to a magnetic field applied at an arbitrary angle is a compromise between several competing interactions.There is the ferrimagnetic coupling between the lanthanide and the iron, the crystal-field interaction, the interaction with the applied field and indirectly the magnetostriction which affects the crystal field.If one assumes that the iron sub-lattice is isotropic in both senses, the magnitude of the iron moment is constant and its direction always collinear to the total field Fe tot B at the iron site, the equilibrium orientation of the two moments in the presence of a field applied at an arbitrary direction is as shown on Figure 1 where the n n B O denote the cubic crystal-field terms in the notation of Lea, Leask and Wolf (1962) where K is a unit vector parallel to the total angular momentum J of the 4 f electrons of the ion and me K is the magneto elastic constant.The indices 1, 2 and 3 denote respectively the directions of the three four-fold axis of the cubic unit cell.The crystal-field parameters are phenomenological parameters determined experimentally.In transparent compounds these parameters can be accurately obtained from the optical absorption spectrum.In metals the determination of the crystal field parameters is not straightforward; they are obtained by combinations of different experimental techniques which can produce conflicting results (Mcmorrow et al, 1989).For the RFe 2 series there is to our knowledge no detailed comprehensive analysis for the crystal field parameters.To date the most widely accepted sets of crystal field parameters for the RFe 2 are the ones obtained by Germano and Butera (1981).The parameters for the Tb, Ho and Tm compounds are given in Table 1 together with the exchange and the magnetoelastic constant.Germano and Butera obtained the crystal field parameters from specific heat measurements.In their analysis the magenostriction is not taken into account, the site symmetry is considered exactly cubic; this plus the fact that the contribution from the crystal field to the specific heat is relatively small may raise questions concerning the accuracy of the parameters.Nevertheless most of the predictions based on these parameters are in fairly good agreement with experiments; these parameters are used faute de mieu for the present work.The computed direction of spontaneous magnetization in HoFe 2 , TmFe 2 and TbFe 2 at zero temperatures is respectively the <100>, <110> and <111> direction.
Table 1.The exchange fields, the crystal-field parameters and the magnetoelestic constants obtained by Germano and Butera (1981)  In TmFe 2 the computation reveals a low-lying first excited state that strongly prefers the <111> direction.The existence of this excited state is responsible for the reorientation of the magnetization at about 55 K reported by Germano and Butera.Above 55 K the <111> direction is the preferred orientation for the magnetization.The experiments reported in the present work were carried out at liquid-helium temperatures where only the ground state is significantly populated therefore the <110> direction is strongly preferred in TmFe 2 .

Computation
In zero applied field and in the absence of magnetostriction the ground state and ground state energy of the lanthanide ion can be readily obtained form the Hamiltonian 1.The expectation value of z J for the ground state is then obtained to calculate the intra-ionic contribution to the hyperfine splitting and the rare-earth's moment.In the presence of an applied field at an arbitrary orientation one must first determine the orientation of ln tot B .
To determine the equilibrium orientation of ln tot B in an applied field we have minimized with respect to the polar angles of the iron moment, Fe θ and Fe φ the free energy per formula unit at absolute zero including the magnetostriction.At zero temperature the free energy is simply . µ is the contribution from the iron sub-lattice and the expectation values are calculated using the ground state of the lanthanide ion.In writing equation (3) it is implicitly assumed that the iron sub-lattice is isotropic in both senses.
The minimization of the free energy (3) is not straightforward.The term 2 V depends on the orientation of the lanthanide moment that is itself the object of the computation.The computation was done by iterations.An educated guess of the orientation of the lanthanide moment was used for the first iteration.The computation was repeated until self-consistency and minimum energy was achieved.It was observed that the term 2 V does not affect significantly the outcome of the computation.
In order to investigate the behavior of polycrystalline samples in applied fields we have computed the magnitude and orientation of the lanthanide moment in HoFe 2 , TbFe 2 and TmFe 2 at various fields applied in 30 different orientations equally spaced over the portion of the unit sphere defined by the <100>, <111> and <110> directions.The results for the 8 T field are summarized in Table 2 and Figure 2. Figure 2 is a planar projection of the portion of the unit sphere under consideration.The orientation of the moment is represented on the Figure by a point corresponding to the intersection of the direction of the moment and the surface of the unit sphere.The shaded areas around the <100>, <110> and <111> axis represent the loci of the direction of the holmium, the thulium and the terbium moments respectively, when the direction of the applied field is scanned across the entire area of the unit sphere under consideration.One can see that an 8 T field applied in any direction hardly moves the thulium moment away from its <110> preferred orientation, 5 degrees at most when the field is along the <111> direction.Holmium, on the other hand, can be more readily moved away from the <100> preferred orientation, 25 degrees when the field is along the <110> axis.
Table 2.The computed orientation of the iron's magnetic moment m, the lanthanide magnetic moment µ and the total field ln tot B at the lanthanide site for an 8 tesla field applied along each of the principal crystallographic directions.The column before last gives the computed intra-ionic contribution to the dipolar hyperfine splitting and the last column gives the total field-dependent contribution (intra-ionic plus the contribution from applied field).ψ is the angle between the applied field and the lanthanide moment.Figure 2. The planar projection of the portion of the unit sphere under consideration.The angles along both axes are in degrees.The shaded areas around the <100>, <110> and <111> axis represent the loci of the direction of the holmium, the thulium and the terbium moments respectively, when the direction of an applied 8 T field is scanned across the entire area of the unit sphere under consideration.

B
Table 2 gives the computed polar angles of the iron moment m , the lanthanide moment µ and the total field at the lanthanide site ln tot

B
when the 8 tesla field a B is applied along each of the principal crystallographic orientations.The last two columns give computed field dependent contributions to the dipolar hyperfine splitting which will be discussed in the next section.

The hyperfine splitting
The theory of the hyperfine splitting in rare-earth compounds is described in detail in McCausland and Makenzie (1979), Bunbury et al (1989) and Li et al (1996) whose notation we shall follow.The hyperfine splitting of the electronic states is described to first order by the effective nuclear spin Hamiltonian { } where I is the nuclear spin, t a a t and t P are respectively the dipolar and the quadrupolar hyperfine parameters.
The z-axis is chosen to be parallel to J .Preliminary computations have shown that in the exchange dominated RFe 2 compounds second order contributions to the hyperfine parameters are at most 0.05%.Therefore, the second order terms (Bunbury et al 1989) have been neglected in the present work.Because of the strong hyperfine coupling in Ho 3+ , Tb 3+ and Tm 3+ the total dipolar hyperfine constant is dominated by the dipolar intraionic term (5) ' z a a J = where a, the dipolar hyperfine coupling constant, is 6497 MHz for Ho 3+ , 3180 MHz for Tb 3+ and 2333 MHz for Tm 3+ , the intra-ionic contribution a' is therefore very sensitive to small variations of the 4 f moment and provides an accurate measure of the moment.
The extra-ionic contribution to the dipolar splitting is where z B′′ is the component along J of B′′ the extra-ionic field at the lanthanide nucleus.′′ B is the vector sum of various terms where a B is the applied field, dip B is the dipolar field, p B is the field due to conduction electrons polarized by the spin of the parent ion and n B is the transferred hyperfine field from neighboring rare-earth and iron ions.
To a good approximation ( ) + + B B B can be regarded as independent of the applied field.Since we are interested in the field dependence of the hyperfine splitting, we will consider the field independent part of the extra-ionic contribution to the dipolar hyperfine splitting as a free parameter to be determined from the zero-field NMR spectrum; thf a is equal to the difference between the computed intraionic contribution and the measured total dipolar hyperfine parameter in zero fields.The total dipolar hyperfine splitting is therefore where ψ is the angle between the applied field and the lanthanide moment and γ is the gyromagnetic factor.
The field dependent contribution ( ) a Bcos γ ′ + ψ computed in a field of 8 T applied along each of the three principal crystallographic axis is given in Table 2.

Experiment.
The compounds were prepared by melting the constituents in an arc furnace under an atmosphere of purified argon.For the thulium compounds, the starting materials contained a 3% excess of Tm to compensate for the anticipated loss by evaporation.The resulting ingots were re-melted three times, annealed for two weeks at 900 0 C and finally crushed to form a fine powder sample.X-ray examination revealed less than 5% of non-Laves phases.Approximately 5 mg of the powder, embedded in epoxy resin, was located close to the current antinode of the central conductor of a tunable coaxial resonator of the type described by McCausland and Makenzie (1979).The NMR measurements were carried out at 1.3 K and 4.2 K in magnetic fields up to 8 T using the microwave spin-echo spectrometer described by Carboni et al (1989).The spectra were obtained by varying the frequency at fixed values of the applied field.For TmFe 2 and TbFe 2 the measured relaxation times are of the order of several micro seconds which is an order of magnitude larger than the pulse separation (200 ns) used to take the spectra so the NMR lines are negligibly distorted by relaxation.In pure Ho Fe 2 the transverse relaxation time is of the order of 200 ns, too short to obtain a reliable spectrum.The seven lines of the 165 Ho (I = 7/2) spectrum could not be unequivocally identified.The 165 Ho NMR data presented here were obtained mainly from specimens with a few percent of holmium substituted in TmFe 2 , LuFe 2 and YFe 2 where the relaxation times are less prohibitive and the spectra in low fields are resolved.The spectrum of 159 Tb(I = 3/2) consists of three lines; because of the strong quadrupolar hyperfine coupling the lines are very well resolved; the separation between the lines is of the order of 350 MHz.The spectrum of 169 Tm(I = 1/2) consists of a single line.
A summary of the hyperfine parameters measured in the various compounds investigated at liquid helium temperatures and in zero fields is given in Table 3.For comparison previously published data from other R': RFe 2 materials have also been included in the table.

Discussion.
The NMR line of 169 Tm:TmFe 2 , in zero fields and in an applied field of 7.5 tesla are shown in Figure 3.The field dependence of the measured total hyperfine parameter t a of 169 Tm:TmFe 2 , is shown in Figure 4-a; the variation is linear with a slope of (3.42 + 0.05) MHz/T.
The slope is less than the value of 3.52 MHz/T expected for the nuclear spin parallel to the applied field.We also observe that the applied field causes only a small broadening of the NMR line (see Figure 5).
shows that there is a significant quenching of <J z > if the moment is rotated away from the <110> orientation with a concomitant reduction of the hyperfine splitting.The reduction is 28% if the thulium moment is forced along the <100> direction corresponding to a shift of more than 350 MHz in the NMR frequency.Therefore, if the rare-earth moments rotate to become close to the direction of the applied field one expects a line broadening of the order of 200 to 300 MHz.This is not observed.
When the thulium ion is placed in a non magnetic and isotropic host like 1% 169 Tm (Tm:LuFe 2 ) we observe that a t decreases with applied field with a slope of (-13 + 2) MHz/T, much larger than the -3.52 MHz/T expected for the nuclear spin anti-parallel to the applied field.The line broadens rapidly with the applied field; the line width is 10 MHz in a field of 0.7 T and the line is too broad for reliable measurements in a field of 1 T. Assuming that the lutetium sub-lattice does not contribute significantly to the anisotropy these observations can be interpreted as follows: the applied fields causes a significant rotation of the iron moments towards a direction parallel to the applied field.Because of the strong ferimagnetic coupling between the thulium ion and the iron sublattice, the thulium moments are gradually rotated away from the strongly preferred <110> orientation hence the sharp reduction in the intraionic contribution to a t and the line broadening.Finally we note from table 3 that when the thulium ion is placed in the HoFe 2 host there is an increase of about 10 MHz in the dipolar hyperfine parameter and not the expected reduction of about 350 MHz if the thulium moment was forced along the <100> orientation preferred by the HoFe 2 host.One can conclude that thulium moment has remained close to the <110> direction and that therefore the iron together with the holmium sublattice are not along the <100> direction.The increase in the dipolar hyperfine parameter is due to the larger transferred hyperfine field in the holmium host due the fact that Ho 3+ has a larger projected spin (σ Ho = 2.0) than Tm 3+ (σ Tm = 1.0) (McCausland and Makenzie, 1979).A refined analysis of the data presented in this section is still in progress and will be presented in a separate publication (Carboni et al 2009, to be published).The field dependence of the measured total dipolar hyperfine parameter a t of 159 Tb:TbFe 2 , is shown on Figure 4b; the variation is linear with a slope of (9.5 + 0.1) MHz/T.The field dependence of the line-width is given in Figure 5.The data shows that like for TmFe 2 the anisotropy is, as predicted by the computation, very large and the terbium ion remains close to the <111> direction even at high fields.A detailed investigation of the zero field hyperfine splitting of Tb 3+ in various RFe 2 hosts has been done by W M de Azevedo (de Azevedo, 1986 and de Azevedo et al 1985).
Except in the TmFe 2 host, the NMR spectra from 165 Ho in the RFe 2 hosts are poorly resolved because the transverse relaxation time is comparable to the minimum pulse separation available on the spectrometer.Moreover, the NMR lines broaden rapidly with the applied field hence the large uncertainty on the hyperfine parameters for the holmium given in table 3.In the HoFe 2 compound it was not possible to obtain the complete 7-lines spectrum therefore it was not possible to obtain a value for the dipolar hyperfine parameter.The quadrupolar parameter P t obtained from the spacing between adjacent lines for this compound is (55 + 5) MHz.For holmium in the TmFe 2 host the spectrum is well resolved in fields up to 5 T.However, as can be seen in Figure 5, the line-width increases sharply with the applied field.If one assumes that the line broadening is mainly due to a spread of the orientations of the Ho 3+ moments with respect to the crystallites axis then, since it has been established that the Tm 3+ moment remains along the <110> direction, one can conclude that the applied field does, as predicted by the computation, achieve some rotation of holmium moment away from the <100> direction.The slope of a t versus field is (8.3 + 0.2) MHz/T, significantly less than the expected 8.9 MHz/T; this is almost certainly due to a gradual quenching of the holmium moment as it is rotated away from its preferred direction.

B
Figure 1.The orientation with respect to the direction of the applied field B a of the iron moment m, the lanthanide moment µ, the total field ln tot B at the lanhanide site and the total field Fe tot B at the iron site.Fe ex B and

Figure 3 .
Figure 3.The 169 Tm NMR line of thulium in TmFe 2 in zero fields and in an applied field of 7.5 T.

Figure 5 .
Figure 5.The behavior of the line width of the central line of the NMR spectra of 169 Tm:TmFe 2, 159 Tb:TbFe 2 and 165 Ho: Ho(TmFe 2 ).

B a m µ ln ex B ln tot B Fe ex B Fe tot B Direction of the applied field [ ]
for HoFe 2 , TbFe 2 and TmFe 2 .