Relevant crystal field interaction in the magnetically ordered state

Accepted Manuscript Relevant crystal field interaction in the magnetically ordered state U. Köbler, I. Radelytskyi, H. Szymczak PII: DOI: Reference:

S0304-8853(18)32266-2 https://doi.org/10.1016/j.jmmm.2018.11.028 MAGMA 64587

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

19 July 2018 5 October 2018 4 November 2018

Please cite this article as: U. Köbler, I. Radelytskyi, H. Szymczak, Relevant crystal field interaction in the magnetically ordered state, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/ j.jmmm.2018.11.028

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Relevant crystal field interaction in the magnetically ordered state U. Köblera,*, I. Radelytskyib,**, H. Szymczakb a b

Research Centre Jülich, Institute PGI, D-52425 Jülich, Germany Institute of Physics, PAS, PL-02-668, Warsaw, Poland

ABSTRACT

Experimental data are discussed showing that in contrast to the paramagnetic phase, in the magnetically ordered state the action of the crystal electric field on the spin dynamics is quantized. In the Curie-Weiss regime of the paramagnetic susceptibility the spin dynamics is determined by local exchange interactions between individual pairs of spins and by single particle anisotropies (crystal field interaction). As we know from Renormalization Group (RG) theory, these local interactions are of no importance on the spin dynamics in the long range ordered state. On the other hand, a sufficiently strong crystal electric field is known to decrease the saturation magnetic moment for T→0. In the critical paramagnetic range and for all lower temperatures the spin dynamics is controlled by a field of delocalized bosons instead by exchange interactions between spins. As we could show, the bosons are essentially magnetic dipole radiation emitted by the precessing spins. It is observed that the spontaneous generation of magnetic dipole radiation involves all N=2S+1 spin states, and is different in magnets with an integer and a half-integer spin. The dynamics of the boson field therefore is quantized and can be characterized by a limited number of universality classes. The effect of a relevant crystal field interaction is to reduce the number of thermodynamically relevant spin states per magnetic atom by ΔN=1 or multiples thereof. This happens as discrete crossover events and reduces the saturation magnetic moment for T→0 in discrete steps. The dynamics remains quantized. Each reduction by ΔN=1 changes the universality class. Since a crossover is a threshold induced event, we have to distinguish between a relevant and a non-relevant crystal field interaction. Only a sufficiently strong crystal field interaction can become relevant. The crossover from S to Seff=S−1/2 can occur in the critical paramagnetic range, and manifests as a functional change in the temperature dependence of either the longitudinal or the transverse susceptibility. A very particular observation is that in the insulating magnets a relevant crystal field interaction lets the magnetic heat capacity collapse to its absolute minimum. The magnetic entropy saturates at the lowest possible value of R·ln(2), irrespective of the value of Seff (R=gas constant). This does not mean that a crossover to atomistic Ising behavior has occurred. For the metallic magnets the action of a relevant crystal electric field is also to reduce the number of relevant spin states for T→0 but the magnetic entropy saturates only gradually below the expected value of R·ln(2S+1). Indications are discussed that each reduction of the spin by ΔS=1/2 generates an additional energy band in the magnon excitation spectrum. In the metals the gap energy of the lowest magnon band is lower than in the insulators. Due to the lower excitation gap, spin dynamics and magnetic heat capacity are less suppressed in the metals compared to the insulators. Keywords: Boson fields Universality Crossover events Relevant interactions *

Corresponding author. Tel.: +492461611689; fax: +492461612410. E-mail address: [email protected] (U. Köbler). **

Present address: Centre for Neutron Science (JCNS) at MLZ, D-85748 Garching, Germany.

1. Introduction As we know from RG theory [1,2], on approaching the magnetic ordering temperature from the paramagnetic side, all microscopic details such as spins and interactions between spins become unimportant for the critical dynamics (of the spins!). Instead, spin dynamics is controlled by a boson field. Independence of the boson controlled dynamics from interactions between the spins and therefore from the spin structure is called universality. Universality further means that the critical dynamics can be classified by a limited number of (rational) critical exponents. Here we consider the effect of the crystal electric field that just belongs to the local interactions. As we will show, the crystal electric field is not completely irrelevant in the ordered range. A sufficiently strong and therefore relevant crystal field interaction reduces the saturation magnetic moment in discrete steps, and induces a crossover to another universality class. The finite number of critical exponents (universality classes) results from the three possible dimensionalities of the boson field and from the quantization of the spin [3]. According to RG theory, the new dynamic symmetry in the critical range is invariance with respect to transformations of the length scale. This is the definition of a continuum. As a consequence, the critical dynamics is as for a continuous magnetic medium. All discrete microscopic details, such as spin structure, lattice structure and crystal field interaction are unimportant. In other words, in contrast to the high-temperature Curie-Weiss regime of the paramagnetic phase, in the boson controlled temperature range, that is, in the critical paramagnetic range and for all lower temperatures, the action of the crystal electric field is quantized. The excitations of a continuous medium are freely propagating bosons. We will call the bosons of the continuous magnetic solid Goldstone bosons [4]. Invariance with respect to transformations of the length scale implies that the momentum of the Goldstone bosons is a conserved quantity. In other words, the bosons propagate ballistic, independent of the lattice structure. This propagation mode is the origin of universality. Lattice parameter and zone boundary are of no importance for the bosons. In fact, universality, in the sense of field theories, is the typical thermodynamic behavior of a field of freely propagating bosons. Unimportance of the spins and of the interactions between the spins has two severe consequences. First, if spins are unimportant, Hamiltonians are no longer suitable concepts. Instead, field theories are adequate. Second, if the exchange interactions between the spins are unimportant, there is no longer thermal energy in the system of the interacting spins (magnons). Instead, thermal energy has passed via a crossover event to the Goldstone boson field. The boson field now is the thermodynamically relevant excitation system, and hosts all thermal energy. As a consequence, the observed magnetic heat capacity is the heat capacity of the Goldstone boson field. The heat capacity of the magnons is negligible. Since the bosons are the relevant excitations in the critical range above and below the ordering temperature it follows that the magnetic ordering transition is driven by the Goldstone boson field instead by exchange interactions between the spins. This particular ordering type is, however, not understood at all. In other words, the spins have lost the responsibility for their dynamics and receive their dynamics from the Goldstone boson field. The heat capacity of the Goldstone boson field determines the spin dynamics. Note that the heat capacity is the only thermodynamic observable of a boson field. In fact, it is observed that thermal decrease of the spontaneous magnetization with respect to T=0 scales with the heat capacity of the Goldstone

boson field [3]. Since the observed spin dynamics is the dynamics of the boson field, field theories of magnetism can restrict to the energy degrees of freedom of the boson field. Spins need not to be considered. Although the findings of RG theory refer to the magnetic degrees of freedom on approaching the critical temperature from the paramagnetic side, they are of much more general validity and apply to the elastic and to the electronic degrees of freedom on approaching T=0 as well [5]. Remarkably, the issues of RG theory precisely conform to the assumptions made already in 1912 by Peter Debye in his famous theory of the low temperature heat capacity of solids [6]. Debye´s theory is the historically first field theory of solid state physics and can serve as a model for future field theories of magnetism. On treating the low temperature heat capacity of solids Debye has replaced the dynamics of the interacting massive atoms by the dynamics of non-interacting and mass-less bosons. These are evidently physically different systems that exist in parallel. The bosons of the elastic continuum are the well-known sound waves. We will call them Debye bosons. Debye has made, implicitly, a second very important assumption. He has assumed that at sufficiently low temperatures all thermal energy is in the boson field and not in the system of the vibrating atoms. The heat capacity of the solid then is the heat capacity of the Debye boson field. The heat capacity of the vibrating atoms, that is, the heat capacity of the acoustic phonons is negligible. As a consequence, it suffices to consider the energy degrees of freedom of the boson field exclusively. This is an enormous analytical simplification. Exactly the same scenario was born out 62 years later by RG theory for the magnetic degrees of freedom on approaching the finite magnetic ordering temperature from the paramagnetic side [1,2]. In other words, in the elastic case the functionality of the magnetic ordering temperature has shifted to T→0. For the elastic degrees of freedom T=0 is a critical temperature completely equivalent to the finite magnetic ordering temperature for the magnetic degrees of freedom. The only difference of the elastic system is that there are no temperatures lower than T=0. The Debye boson field does not order. Sufficiently close to a critical temperature, either T=0 or a finite ordering temperature, it is necessary to replace the atomistic dynamics by the dynamics of a boson field. Characteristic for boson dynamics is a finite critical range and rational critical exponents [7]. It is evident that a symmetry selection principle must be operative when thermal energy is either in the boson field or in the system of the interacting spins. This symmetry selection principle has been called relevance by RG theory. As is well-known from all parts of physics, symmetries can be considered as the generators of particles. Equivalently, particles can be classified by symmetries. This applies to the two types of translational symmetry of solids as well. What we have to learn is that for all degrees of freedom (elastic, magnetic, electronic…) we have to distinguish between the excitations of the continuous translational symmetry (bosons) and the excitations of the discrete translational symmetry of the atomic structure (magnons, phonons, electronic band states…). The actual dynamics can and, in fact, has to be classified by one of the two translational symmetries. Due to the symmetry selection principle of relevance, a mix of symmetries does not occur. In other words, the two associated excitation spectra become activated alternately only. Relevance has the severe consequence that all available energy states of boson field and atomistic system are not populated according to the Boltzmann factor. Detailed balance holds for the relevant excitation system

only. The heat capacity of the non-relevant system is negligible. This means that the dispersion relation of the non-relevant excitation system is thermally not populated. At the time of Debye, the categorical necessity to distinguish between the two translational symmetries and their excitation spectra (phonons, sound waves), was not yet clear [8]. Crossover events were unknown. As a consequence, Debye did not know how to stop the exploding T3 function of the heat capacity of his boson field. In order to simulate the finite Dulong-Petit heat capacity limit of the atomic(!) lattice, Debye has rather arbitrarily cut the linear dispersion of the Debye bosons at an appropriate energy. Note that incorporation of atomistic parameters into a field theory such as the total number of atoms is strictly forbidden. This discontinuous manipulation of the dispersion of the Debye bosons is, of course, unphysical and has the nonsensical consequence that the solid will no longer have elastic properties for temperatures of larger than corresponds to the cut-off energy (kB·ΘD) [8]. This is in disagreement with the experimentally well established fact that all solids have elastic properties up to melting temperature and even beyond [9]. At Debye´s cut-off energy no anomaly at all is visible in the dispersion of the Debye bosons [8] or in the temperature dependence of either the elastic constants or the sound velocities [9]. As we now know, the T3 function ceases at the crossover of thermal energy from the Debye boson field to the lattice system. This crossover commonly occurs at a temperature of 10…30 K, and is most clearly visualized by a rather sharp maximum in the thermal conductivity [5]. Due to their large mean free path, bosons are the predominant carriers of heat transport -if they are the relevant excitations. In insulators thermal conductivity is exclusively due to the Debye bosons [5]. In the approximation of an infinite mean free path of the Debye bosons, thermal conductivity is proportional to the heat capacity of the Debye boson field. The observed maximum in the thermal conductivity results from the crossover of thermal energy from the Debye boson field to the atomistic system of lattice vibrations. For temperatures of larger than crossover temperature the Debye boson field is no longer the relevant excitation system and its heat capacity tends to zero. This lets thermal conductivity of the insulators decrease to zero. Lattice theories then become adequate [10]. For temperatures of larger than crossover temperature, the system of the lattice vibrations (phonons) is relevant, and the dispersion relation of the Debye bosons is no longer thermally populated. Conversely, when the Debye boson field is the relevant excitation system the dispersion of the acoustic phonons is no longer thermally populated. As the strongly decreasing thermal conductivity for temperatures of larger than crossover temperature shows, the heat transport by phonons is negligible. The two excitation systems are, however, not completely independent of each other but occasionally interact significantly. Lattice theories therefore can give correct description of the phonon dispersions only if interactions between phonons and Debye bosons are negligible [8]. In the same way, spin wave theory can give correct description of the magnon dispersions only when magnon-Goldstone boson interactions are negligible. This, however, is never perfectly the case. For strong Debye boson-phonon interaction the dispersion of the acoustic phonons can assume over a large q-range the linear dispersion of the Debye bosons [8]. Note that because of their different (translational) symmetries, the two dispersion relations can attract each other. Fortunately, also for finite Debye boson-phonon interactions the dispersion of the Debye bosons remains a very nearly linear function of wave vector, and the lowtemperature heat capacity of the Debye boson field is given to a good approximation by Debye´s T3 function. In contrast to the acoustic phonons that always have continuous (gap

less) excitation spectrum, one manifestation of the interactions between magnons and Goldstone bosons is the magnon gap [13,14]. In our context it is important to emphasize that the Goldstone boson field changes its properties fundamentally at the magnetic ordering temperature. Below the magnetic ordering temperature the Goldstone boson field is in an ordered state. Note that in a pure field theory of magnetism spins and interactions between spins do not occur. The dynamics of the interacting spins is replaced by the dynamics of the boson field. It therefore follows stringently that the order parameter and its dimensionality (its components) must be properties of the boson field. In contrast to the disordered Debye boson field, in the ordered state of the Goldstone boson field (for T
γ=4/3 for the paramagnetic susceptibility [7,12]. Two-dimensional and three-dimensional boson fields result by some dynamic coupling of the one-dimensional field components of the differently oriented magnetic domains [13,14]. In other words, the dimensionality of the boson field is a property on the length scale of the magnetic domains. The dimensionality of the exchange interactions is unimportant, and can be different from the dimensionality of the Goldstone boson field [17]. Note that domains are not considered in spin wave theory. It is evident that a field theory of magnetism will be rather complicated due to the vector character of the ordered Goldstone boson field. The mechanism that couples the one-dimensional basis fields to result in a two-dimensional or a three-dimensional global field is completely nonunderstood. When coupled, the one-dimensional basis fields within the domains have to be classified as non-relevant for the dynamics. It seems to be a general phenomenon of solid state physics that upon approaching a critical temperature, either T=0 or a finite ordering temperature, a crossover from atomistic dynamics to boson dynamics always occurs. In other words, order-disorder phase transitions seem generally to be driven by boson fields [18-20]. Ordered boson fields seem always associated with a domain or mosaic structure. The crossover from atomistic dynamics to boson dynamics happens at a finite distance from critical temperature. In the magnetic solids the crossover from exchange defined atomistic dynamics to boson defined continuum dynamics is at the passage from Curie-Weiss susceptibility to critical susceptibility. At this crossover the dynamic symmetry gets “broken”. The two susceptibilities have analytically different temperature dependencies. Such a functional change is typical for a crossover event. The distance of the crossover from the critical temperature defines the width of the critical range. However, for most order-disorder phase transitions the driving bosons are unexplored. Indirect evidence for boson dynamics is provided by the rational critical exponents and by the finite width of the critical range [7,18,19]. Note that in the atomistic models of magnetism the width of the critical range either at T=0 or at T=Tc is zero. Crossover events are unknown. For the elastic and the electronic degrees of freedom the crossover from atomistic dynamics to boson dynamics that occurs on approaching T→0, commonly is at a temperature of 10…30 K [5]. Here we will show that the action of the crystal electric field on the spin dynamics changes from continuous in the Curie-Weiss range of the paramagnetic phase to discrete (quantized) at the crossover to critical susceptibility. A finite crystal field interaction requires a finite orbital moment in the total moment of the magnetic atom. In the compounds of the 3d-elements, finite orbital moment contributions reveal by a g-factor of larger than g=2.00 [21]. On the other hand, examples of pure spin magnetism (and no crystal field interaction) are tetragonal MnF2 [22] and hcp gadolinium [23]. The g-factors of the two compounds are g=2.026 for MnF2 [22] and g=2.01 for hcp Gd [23]. As a consequence, the Curie-Weiss susceptibility is very isotropic (Θ// = Θ┴). A finite crystal field interaction gives rise to an anisotropic Curie-Weiss temperature, Θ. Among the 3d-elements the Co2+ ion is known to have the largest orbital moment contribution in its magnetic moment. This applies in particular to Co2+ ions on octahedral lattice sites. The paramagnetic susceptibility can be extremely anisotropic as in the case of K2CoF4 [24]. Below we will discuss CoF2 and LuCoGaO4 as typical examples of a crystal field interaction that becomes relevant at the crossover from Curie-Weiss susceptibility to critical susceptibility. This means that at this crossover the number of thermodynamically relevant spin states gets

reduced. Since the number of spin states left remains an integer it is possible to define an effective spin with, commonly, Seff=S-1/2. As a consequence, in the boson controlled temperature range the dynamics remains quantized, and can be characterized by universality classes. This general behavior is not changed by the crystal electric field. We therefore have to ask whether crystal field interaction is relevant or not. Only a sufficiently strong crystal field interaction will become relevant and will reduce the spin quantum number and thereby changes the universality class. Finite but non-relevant interactions are negligible. When the crystal field becomes relevant this gives rise to a threshold induced discrete crossover event. In other words, relevance assures quantized and perfectly stabilized universality classes [13,14]. Quite generally, a low magnetic ordering temperature enhances the weight of the crystal electric field relative to the magnetic interactions and increases the probability for the crystal field to become relevant. In other words, the lower the magnetic ordering temperature is, the higher is the probability for the crystal field to become relevant in the ordered state. We can assume that in homologous compound series the strength of the crystal electric field is nearly composition independent but the ordering temperature commonly depends sensitively on various parameters such as the distance between the magnetic moments (the lattice constant). For instance, among the monoxides MnO (TN=122 K), FeO (TN=198 K), CoO (TN=291 K) and NiO (TN=523 K), MnO has the lowest ordering temperature (and the largest lattice parameter) [25]. As the observed saturation magnetic moments indicate, crystal field interaction is relevant only in MnO. Relevant means that the spin of the Mn2+ ion is reduced from S=5/2 to Seff=2. The observed saturation moment of MnO of mS= 4.58±0.03 μB/Mn is considerably lower than ~5.5 μB/Mn expected for S=5/2 but agrees reasonably with Seff=2 [26]. For NiO and for CoO the reported saturation magnetic moments are 1.90±0.06 μB/Ni for NiO [26], 3.80±0.06 μB/Co for CoO [27]. These saturation moment values are reasonably consistent with S=1 for NiO and S=3/2 for CoO. For FeO the situation is less clear because it is very difficult to prepare stoichiometric material with 1:1 composition [28]. Powder material is chemically unstable and partly decomposes into Fe3O4 or into Fe2O3. The reported saturation magnetic moment of 3.32 μB/Fe seems to be too low [25]. For chemically pure FeO powder material the spontaneous sublattice magnetization decreases according to T3 function [30]. This is the universality class of the one-dimensional boson field in magnets with integer spin and therefore confirms S=2. Moreover, the observed magnon gap energy of ~3.1 meV is a typical value for a magnet with integer spin (S=2) [29]. For half-integer spin the magnon gap usually is lower than ~1 meV [30]. Since the g-factor in the magnetic 3d-element compounds is theoretically not precisely known and can change under the action of a relevant crystal field, definite conclusion on the value of Seff from the observed saturation magnetic moment is not unambiguously possible. Clear evidence for the quantized dynamics is provided by the exponent ε of the Tε power function observed either in the thermal decrease of the spontaneous magnetization or in the magnetic heat capacity [13,14,30]. Knowing ε it is possible to conclude on whether the effective spin quantum number is integer or half-integer. This commonly allows definite decision on the value of Seff. Since the axial lattice distortions are relevant for all monoxides [25] the boson fields are one-dimensional. The universal exponent ε therefore is ε=3 for integer spin (MnO, NiO, FeO) but ε=5/2 for CoO with S=3/2. These exponent values are consistently confirmed experimentally [30]. In particular ε=3 for MnO proves integer spin

(Seff=2) [30]. For half-integer spin (S=5/2) the exponent would be ε=5/2. This exponent is verified for CoO with S=3/2 (see Fig. 6 below). A similar scenario as for the monoxides holds for the bi-fluorides MnF2 (TN=67.7 K), FeF2 (TN=75.8 K), CoF2 (TN=39.1 K) and NiF2 (TN=74.1 K) [31]. Among these compounds CoF2 has the lowest ordering temperature. Only for CoF2 crystal field interaction is relevant and reduces the spin quantum number from S=3/2 to Seff=1 [30,31]. Consistent with Seff=1, the reported saturation magnetic moments are 2.213±0.017 μB/Co [31], 2.57±0.02 μB/Co [32] and 2.60±0.04 μB/Co [33]. For S=3/2 a saturation magnetic moment of ~3.5 μB/Co can be anticipated. For the other bi-fluorides the observed saturation moments conform to the expected value of the divalent ions: 5.04±0.06 μB/Mn for MnF2 (S=5/2), 3.93±0.04 μB/Fe for FeF2 (S=2) and 1.96±0.02 μB/Ni for NiF2 (S=1) [31]. For the compounds of the 3d-elements the common effect of a relevant crystal field is to reduce the number of states per magnetic atom by ΔN=1 only. However, for materials with a very low ordering temperature it is possible that the crystal field interaction reduces the number of states by ΔN=2. As an example of ΔN=2 we discuss RbFeCl3 (see Fig. 11 below) [34]. Due to the extremely low ordering temperature of TN=2.55 K the effective spin of the Fe2+ ion is Seff=1 in RbFeCl3 instead of S=2. On the other hand, a strong orbital moment can have the opposite effect and can increase the number of states by ΔN=1. This occurs also as a crossover event. The orbital moment contribution then adds one state to the 2S+1 spin states. This is observed for the two Co2+ compounds CoO and CoCl2·2H2O (see Figs. 7, 8 below). However, for CoO this effect is material dependent and is observed for powder material only. For CoO single crystal material (Fig. 6 below) the crystal field is not relevant (S=3/2) [27]. For the compounds or alloys of the Rare Earth elements with an explicit orbital moment, crystal field interaction is generally relevant for materials with ordering temperatures of lower than Tc~100 K [35,36]. For still lower ordering temperatures the total moment quantum number J can be reduced by several times of ΔJ=1/2 (Figs. 13,16,21 below).

2. Crystal field effects in insulators In the magnetic insulators a relevant crystal field reduces the magnetic heat capacity dramatically. On one hand, in most of the 3d-element compounds a relevant crystal field reduces the number of spin states per magnetic atom by ΔN=1 only. This means that one of the 2S+1 spin states is excluded from thermodynamics and needs not to be considered. Reduction by ΔN=1 is equivalent to a reduction of the spin quantum number from S to Seff=S−1/2. This changes the dynamic universality class. Reduction by ΔN=2 or ΔS=1 leaves the spin quantum number integer (or half-integer) and therefore does not change the universality class (see Fig. 11 below). However, the observed saturation magnetic moment is reduced accordingly. On the other hand, the relevant crystal field lets the magnetic heat capacity collapse to its absolute minimum. The magnetic heat capacity is as for a two-level system, i.e. for a system with spin S=1/2, irrespective of the value of Seff. The magnetic entropy saturates at R·ln(2) [37−41]. This does not mean that the system has undergone a crossover to the atomistic Ising model. Ising behavior can be excluded by the observation of ε the same universal T power functions in the thermal decrease of the spontaneous

magnetization as for magnets with a non-relevant crystal field interaction (see Fig. 12 below). Note that the Ising model is a typical atomistic model and will not be realized when the Goldstone boson field is the relevant excitation system [4,42−45]. On the other hand, condition for boson dynamics is that the spin has three-components and can precess. Since genuine Ising spins do not precess they are unable to generate Goldstone bosons (magnetic dipole radiation). The boson field therefore gets not populated with field quanta and the dynamics is in fact determined by exchange interactions [42,43]. For magnets with relevant crystal field interaction thermal decrease of the spontaneous magnetization is as for a symmetry-equivalent magnet with a non-relevant crystal field but with a spin of S=Seff. In spite of the asymptotic entropy limit of R·ln(2) the observed saturation magnetic moment and the exponent ε conform to Seff [4,13,14]. Among the 3d-elements the best candidates to demonstrate crystal field effects are compounds of the Co2+ ion. The magnetic moment of the Co2+ ion is known for strong orbital moment contributions. Crystal field interaction therefore is generally strong and mostly relevant. As a first example we consider tetragonal CoF2. Commonly it is assumed that in all transition metal bi-fluorides the spin structure is as for MnF2 [22,31,32]. The spin structure established for MnF2 can, however, not be correct for CoF2. MnF2 has a pure spin moment of S=5/2. There is no single particle anisotropy (Θ//=Θ┴) [22]. Nevertheless all Mn2+ moments are rigidly aligned parallel to the tetragonal c-axis [22,32,46]. Strong coupling of the Mn2+ moments to the c-axis therefore is beyond atomistic concepts. The stabilized collinear spin structure results from a one-dimensional Goldstone boson field. There is only one domain type [16]. Bulk MnF2 can be considered as one large magnetic domain. Typical for the single domain situation in MnF2 is that the longitudinal susceptibility (along tetragonal c-axis) tends to zero for T→0 [22]. This shows that all Mn2+ moments are coupled rigidly to the c-axis, that is, to the axis of the one-dimensional boson field. As a quantitative measure of how strong the Mn2+ moments are coupled to the one-dimensional boson field we have identified the magnon excitation gap [13,14] that amounts to 1.1 meV for MnF2 [16,47-49]. Due to this interaction, the one-dimensional boson field furnishes the spin system with an axial anisotropy that is not known in the atomistic models. Another measure of the axial anisotropy provided by the onedimensional boson field to the spin system is the spin-flop field that is as large as 120 kOe in MnF2 [50]. For comparison, the spin flop field of the cubic antiferromagnet EuTe with a pure spin moment of S=7/2 is 770±200 Oe only [51]. In cubic EuTe there are domains along all three space directions [51] and the resulting global boson field is isotropic. Moreover, the spontaneous sublattice magnetization of MnF2 decreases according to T5/2 function [11]. This universality class applies to the spontaneous magnetization of magnets with a onedimensional boson field and half-integer spin (S=5/2) [13,14]. The spontaneous sublattice magnetization of the cubic antiferromagnet EuTe decreases according to T2 function [52]. The T2 function applies to the heat capacity of the three-dimensional boson field in magnets with half-integer spin. In contrast to MnF2 the longitudinal susceptibility of CoF2 remains finite for T→0 [22]. This is indicative of a multi-domain spin configuration. In fact, the spontaneous sublattice magnetization of CoF2 decreases according to T9/2 function (see Fig. 6 below) [30]. The T9/2 function is the universality class of the isotropic boson field in magnets with integer spin (Seff=1). The isotropic boson field requires magnetic domains along all three space directions. The one-dimensional boson fields associated with the differently oriented domains are

coupled to a three-dimensional global field [13,14]. As a consequence, in CoF2 there must be domains oriented with nearly equal probability along all three space directions. Within each domain the spin structure is one-dimensional as for bulk MnF2.

800

CoF2 BO c-axis TN= 37.5 K

S=3/2 , g=2.48

3

(g/cm )

600

-1

(g )

 = -36.5 K

TCO= 85 K

400

TN 200

Seff= 1, g=2.70  =+8.4 K

0 0

50

100

150

200

250

T (K)

Fig. 1. Reciprocal paramagnetic susceptibility of CoF2 measured transverse to tetragonal c-axis (this work). The slope of the high-temperature linear section of χ-1(T) conforms to the full spin of S=3/2 of the Co2+ ion with a fitted Landé splitting factor of g=2.48. Below crossover temperature at TCO=85 K crystal field interaction is relevant and decreases the spin quantum number to S eff=1. The fitted Landé splitting factor now is g=2.70. Seff=1 and g=2.70 are in good agreement with the observed saturation magnetic moments of mS=2.57±0.02 μB/Co [32] and mS=2.60±0.04 [33].

Fig. 1 shows the reciprocal paramagnetic susceptibility of CoF2 perpendicular to the tetragonal c-axis as a function of absolute temperature. Our data of Fig.1 and Fig. 2 are in good agreement with those of [22]. Two Curie-Weiss lines with different slopes can be distinguished in Fig. 1 [7]. Crossover to a relevant crystal field is at TCO=85 K. We must assume that the crossover to the critical range coincides with the crossover in spin quantum number at TCO=85 K. Note that when two crossover events coincide the critical range can be unusually large. The different slopes of χ-1(T) are indicative of different magnetic moments. In other words, change of spin quantum number is a discrete event and gives rise to an analytical change in the temperature dependence of the susceptibility. From the hightemperature slope of χ-1(T) a reasonable g-factor of g=2.48 results assuming the full moment of S=3/2 for the Co2+ ion. From the larger slope for T
700 650

CoF2 BO // c-axis

550

= -107 K

( g )

-1

3

(g/cm )

600

500

g = 2.91 TN=37.5 K

450

S=3/2

400 350 300 250 0

50

100

150

200

250

T (K)

Fig. 2. Reciprocal paramagnetic susceptibility of CoF2 measured parallel to tetragonal c-axis. Assuming S=3/2 for the Co2+ ion, a Landé splitting factor of g=2.91 is obtained from the slope of the Curie-Weiss line. No definite anomaly appears at ~85 K (compare Fig. 1). Note the much different Curie-Weiss temperature compared to the perpendicular susceptibility (Fig. 1) [22].

The longitudinal susceptibility shows a completely different behavior (Fig. 2). In particular, the amplitude crossover at TCO=85 K is absent. This shows that the crystal field is anisotropic. Only in the basal plane of CoF2 the crystal field is strong enough to become relevant. In contrast to MnF2, the longitudinal susceptibility of CoF2 remains finite for T→0 [22]. As a consequence, in CoF2 not all spins are aligned parallel to c-axis. In contrast to MnF2, axial anisotropy is low in CoF2. There must be domains with spin orientations along a- and b-axis as well. Since there are no typical two-dimensional structural elements in the lattice structure of CoF2 [22], it follows plausibly that the global Goldstone boson field is three-dimensional (see Fig. 6 below). Quite generally, isotropic boson fields can occur also in non-cubic crystals. However, in non-cubic magnets the isotropic universality class occasionally is metastable and can depend on sample preparation. For CoF2 the Curie-Weiss temperature and gfactor of the longitudinal susceptibility are considerably different compared to the transverse susceptibility. The strong paramagnetic anisotropy proves that there are considerable orbital moment contributions in the magnetic moment of the Co2+ ion (g>2). As a summary, in contrast to MnF2, the paramagnetic susceptibility of CoF2 is strongly anisotropic. The anisotropy results from crystal field interaction (single particle anisotropy). However, in the ordered state of CoF2 the dynamic symmetry is isotropic (~T9/2 universality class). This shows that the atomistic single particle anisotropy is not relevant in the boson controlled ordered state. In the Curie-Weiss regime of the susceptibility the crystal electric field manifests as paramagnetic anisotropy but in the ordered state by a reduced spin quantum number. Isotropic dynamics in the ordered state of CoF2 is indicative of an isotropic domain configuration. When the boson fields of the different domains are dynamically coupled to result in an isotropic global field, the typical axial anisotropy within each domain still persists

but the associated boson field is not the relevant field for the global dynamics. In other words, we have to distinguish between non-relevant boson fields and the boson field that is relevant for the dynamics. In MnF2 the situation is opposite. The paramagnetic susceptibility is very isotropic [22] but in the ordered state MnF2 is very anisotropic because there is only one domain type along c-axis [16]. Since the one-dimensional boson field is relevant also for the spin dynamics, the order parameter decreases according to ~T5/2 universality class [11]. In the heat capacity of CoF2 contributions due to Debye bosons and due to Goldstone bosons can be identified [37,38]. As the phenomenon of magnetostriction shows, the two boson fields interact significantly. As a consequence, the heat capacities of the two boson fields do not superimpose but appear alternately as a function of temperature. Note that this is different in the low-temperature heat capacity of the metals [8]. Since the bosons of the electronic degrees of freedom and the bosons of the elastic degrees of freedom (the Debye bosons) do virtually not interact, the linear-in-T heat capacity of the electronic bosons and the T3 heat capacity of the Debye bosons superimpose [5]. As a consequence of the weak interaction, the electronic bosons must have a very large mean free path [5]. 18 10 15

30

35

TN= 37.6 K

14

Seff=1

12

-1

-1

T (K)

25

CoF2

16

heat capacity (JK mole )

20

Goldstone bosons

10

Debye bosons

~T

9/2

8

ZnF2

6 4

TCO= 24.3 K

2

SrF2 E. Catalano, J.W. Stout, J. Chem. Phys. 23 (1955)1803.

0 0

10000

20000

30000 3

40000

50000

3

T (K )

Fig. 3. Heat capacity of CoF2 (open points) as a function of temperature to the third power [37,38]. Crossover between low-temperature ~T3 function due to dominating Debye bosons and hightemperature ~T9/2 function due to dominating Goldstone bosons can be identified at TCO=24.3 K. Because of the finite interaction between Debye bosons and Goldstone bosons [3] the two heat capacity contributions do not superimpose but appear alternately as a function of temperature. The prefactor of the initial T3 function is larger than for the non-magnetic reference compounds ZrF2 and SrF2 [38,53]. The enhanced pre-factor of the T3 function must be attributed to the heat capacity of the Goldstone bosons. In the same way, the heat capacity contribution of the Debye boson must be included in the pre-factor of the T9/2 function.

As can be seen in Fig. 3, below crossover at TCO=24.3 K the heat capacity of CoF2 is determined by the T3 function of the Debye boson field. For T>TCO the T9/2 universality class of the Goldstone boson field prevails (compare Fig. 6 below). On one hand, appearance of the well-established power functions of temperature proves the stability of the universality classes

also when the associated boson fields interact. On the other hand, it becomes clear that an exact separation into magnetic and non-magnetic heat capacity contributions is not possible. We can assume that the heat capacities of the two boson fields are finite for all temperatures in Fig. 3. As can be seen, the pre-factor of the T3 function of CoF2 is much larger than for the non-magnetic reference compounds ZnF2 and SrF2 [38,53]. We have to assume that the heat capacity of the Goldstone bosons increases the pre-factor of the Debye T3 function in the heat capacity of CoF2. On the other hand, for T>TCO=24.3 K the heat capacity of the Debye bosons must be included in the pre-factor of the T9/2 function of the Goldstone bosons. This behavior conforms to a conclusion of RG theory that sufficiently weak but non-relevant energies do not change the universality class but enter the pre-factor of the universal power function only. The intrinsic temperature dependence of the non-relevant energy degrees of freedom then becomes not apparent, As a suitable quantitative measure of the non-magnetic heat capacity contribution of CoF2, Fig. 3 includes the heat capacities of the diamagnetic reference compounds ZnF2 and SrF2 [38,53]. In the temperature window of Fig. 3, the heat capacity of the two non-magnetic compounds is given by Debye´s T3 function. Moreover, the heat capacities of ZnF2, SrF2 and CaF2 [54] are nearly identical. The large validity range of Debye´s T3 function indicates that ZrF2 and SnF2 are fairly hard materials. In fact, the calorimetric Debye temperature of ZnF2 and SrF2 has a relatively large value of ΘD=355 K. Note, however, that the correct Debye temperature, the elastic Debye temperature, has to be calculated from the measured sound velocities [54]. On the other hand, in most cases elastic and calorimetric Debye temperatures agree rather well [54].

6

Rln(2) -1

(J K mole )

5

CoF2

4

entropy

-1

2+

Co 3

T

3

; S=3/2

TN=37.7 K

Seff= 1

2

1 E. Catalano, J.W. Stout, J. Chem. Phys. 23 (1955) 1803.

0 0

10

20

30

40

50

60

70

80

90

T (K)

Fig. 4. Magnetic entropy of CoF2 as a function of absolute temperature. Crystal field interaction is relevant and decreases the spin quantum number from S=3/2 to Seff=1. Consistent with Seff=1 is an observed saturation magnetic moment of ~2.6 μB/Co [31-33]. However, the magnetic entropy saturates at R·ln(2) instead at R·ln(3). This was noticed already in 1955 [37,38]. Below TN=37.7 K the entropy can approximately be described by T3 function.

The difference between the heat capacity of CoF2 on one hand and the heat capacities of ZnF2 and SrF2 on the other hand provides a qualitative measure of the magnetic part of the heat capacity of CoF2. In order to obtain a reasonable quantitative result for the magnetic entropy of CoF2 it is advisable to take the heat capacity of the non-magnetic reference compounds SrF2 or ZnF2 [38,53] as the non-magnetic heat capacity background of CoF2. The temperature dependence of the magnetic entropy evaluated in this way can be seen in Fig. 4. The experimental accuracy is well sufficient to exclude that the magnetic entropy saturates at R·ln(4)=11.53 JK-1mole-1, assuming S=3/2 or at R·ln(3)=9.14 JK-1mole-1, assuming Seff=1. Instead, the magnetic entropy saturates as for a system with only two states at R·ln(2)=5.76 JK-1mole-1. This has been noticed already in 1955 [37,38]. In other words, the magnetic heat capacity is much too small for a system with Seff=1. As the observed T9/2 function in the thermal decrease of the spontaneous magnetization shows (see Fig. 6 below), the effective spin is integer (Seff=1). The T9/2 universality class pertains to magnets with an isotropic boson field and integer spin.

16

MnF2 S=5/2

Rln(6) 14

Rln(5) FeF2 S=2

-1

entropy (JK mole )

12

10

-1

Rln(3) 8

NiF2 S=1 6

Rln(2) CoF2 Seff =1

4

2

J.W. Stout, E. Catalano, J. Chem. Phys. 23 (1955) 2013.

0 0

20

40

60

80

100

120

T (K)

Fig. 5. Temperature dependence of the magnetic entropies of the four bi-fluorides as a function of temperature [37,38,53]. For MnF2 (S=5/2), FeF2 (S=2) and NiF2 (S=1) crystal field interaction is not relevant and the entropy saturates at the expected value of R·ln(2S+1). For CoF2 with the lowest ordering temperature (TN=39 K) of all four bi-fluorides the crystal field is relevant and the magnetic entropy saturates at R·ln(2). If the crystal field would not be relevant the magnetic entropy of CoF2 would saturate at the two times larger value of R·ln(4).

For the other bi-fluorides with larger ordering temperatures and non-relevant crystal field interaction the magnetic entropies saturate reasonably at the expected values of R·ln(2S+1) (Fig. 5) [38]. This shows in fact, that in the spontaneous generation process of the Goldstone bosons all 2S+1 spin states are involved, and that the observed magnetic heat capacity is the heat capacity of the Goldstone boson field. Note that spin wave theory treats non-quantized

spins. As a consequence, discrete saturation values of the magnetic entropy are beyond spin wave theory. In order to illustrate the various aspects of the important symmetry selection principle of relevance, established by RG theory [1], Fig. 6 shows the normalized spontaneous magnetization of CoO and CoF2 single crystals as a function of the reduced temperature [27,32]. These data are obtained from transition energies between hyperfine-split nuclear spin states using inelastic neutron scattering with high energy resolution [56,57]. This experimental technique is equivalent to nuclear magnetic resonance (NMR). The hyperfine field at the nucleus of the magnetic atom can be assumed to be proportional to the spontaneous magnetization. For the CoO single crystal, crystal field interaction is not relevant. The spin of the Co2+ ion has its full value of S=3/2. On the other hand, the tetragonal lattice distortion is relevant [25,55] and decreases the dimensionality of the boson field from three-dimensional isotropic (~T2 function) to one-dimensional (~T5/2 function). For the CoF2 single crystal, crystal field interaction is relevant and decreases the spin of the Co2+ ion from S=3/2 to Seff=1 [31]. The lower than cubic (tetragonal) lattice symmetry is not relevant and the spontaneous magnetization decreases according to the heat capacity of the isotropic threedimensional boson field (~T9/2).

CoF2

1,0

TN=39.25 K

0,9

M(T) / M(T=0)

~T

9/2

Seff=1

0,8 5/2

0,7

~T

0,6

TN=289.9 K

CoO S=3/2 0,5 0,4 0,0

0,2

0,4

0,6

0,8

1,0

T/TN

Fig. 6. Normalized spontaneous magnetization of CoO and of CoF2 single crystals as a function of the reduced temperature [27,32]. These data are obtained from observation of transition energies between nuclear spin states of the Co ion using inelastic neutron scattering with a high energy resolution. For the CoO single crystal, crystal field interaction is not relevant and the spin of the Co2+ ion is S=3/2. Due to a relevant lattice distortion [25,55] the relevant boson field is one-dimensional (T5/2 universality class). In CoF2 crystal field interaction is relevant, and the effective spin is Seff=1. The tetragonal lattice symmetry is not relevant and the boson field is isotropic (T9/2 universality class).

A similar result as for CoF2 (Figs. 1-4) is obtained for CoCl2. As neutron scattering data obtained on powder material indicate, thermal decrease of the sublattice magnetization is according to T3 function [58]. The effective spin therefore is integer (Seff=1). T3 function means a one-dimensional boson field. As a consequence, in contrast to tetragonal CoF2, the

trigonal lattice symmetry of CoCl2 is relevant and decreases the dynamic symmetry to onedimensional. Assumption of Seff=1 is reasonably consistent with the observed saturation magnetic moment of 3.1±0.6 μB/Co [58]. Moreover, the relevant crystal field interaction is confirmed by heat capacity measurements [59]. The magnetic entropy of CoCl2 saturates at R·ln(2) instead of saturating at R·ln(4) [59]. However, whether crystal field interaction is relevant or not can depend on sample preparation and/or sample morphology (see discussion of Fig. 18 below) [60]. Note that the domain structure is decisive for the dimensionality of the global boson field. The domain structure can depend on strain in the sample. In other words, differently prepared chemically pure samples of the same material can belong to different universality classes. Moreover, it seems possible that a large orbital moment contribution in the magnetic moment can add one orbital state to the N=2S+1 spin states. This seems to happen in CoO. Measurements of the spontaneous magnetization on a CoO single crystal reveal T5/2 dependence (Fig. 6). This is the universality class of the one-dimensional boson field in magnets with half-integer spin. Onedimensional symmetry is consistent with the spontaneous tetragonal lattice distortion of CoO [25,55]. It therefore can be concluded that the crystal field is not relevant in the single crystal and that the spin of the Co2+ ion is S=3/2. The observed saturation magnetic moment of 3.80±0.06 μB/Co is consistent with S=3/2 [27]. However for CoO powder material the sublattice magnetization decreases according to T9/2 function [61]. This is the universality class of the isotropic boson field in magnets with integer spin. As a consequence, in the powder material crystal field interaction is relevant but the tetragonal lattice distortion is not relevant. According to an observed saturation magnetic moment of 3.98±0.06 μB/Co the spin cannot be Seff=1 but must be Seff=2 [55]. The increased number of states by ΔN=+1 must be attributed to the orbital degrees of freedom [31]. Typical for explicit orbital moment contributions is a g-factor of slightly lower than g=2.

Rln(2)

6

TN

CoCl22H2O

-1

(J K mole )

5

S=3/2 Seff= 2

entropy

-1

4

N=+1

3

TN=17.2 K 2 1 T. Shinoda, H. Chihara, S. Seki, J. Phys. Soc. Japan 19 (1964) 1637.

0 0

10

20

30

40

50

T

(K)

60

70

80

90

Fig. 7. Magnetic entropy of CoCl2·2H2O evaluated from heat capacity data as a function of temperature [62]. Crystal field interaction is relevant as is evidenced by an entropy limit of only R·ln(2). Since the heat capacity is precisely described by T9/2 function (Fig. 8), the spin of the Co2+ ion is integer. According to a saturation magnetic moment of 3.3±0.4 μB/Co [63] the spin is likely to be

Seff=2. The number of states therefore is increased by ΔN=+1 from N=4 (S=3/2) to N=5 (Seff=2) by the action of the relevant crystal field.

A similar effect seems to occur in CoCl2·2H2O [62]. As can be seen in Fig. 7, the magnetic entropy of CoCl2·2H2O saturates rather perfectly at R·ln(2). Crystal field interaction therefore is relevant. For non-relevant crystal field interaction the magnetic entropy would saturate at R·ln(4). From the observed T9/2 function in the low-temperature heat capacity of CoCl2·2H2O it follows that the effective spin is integer (Fig. 8). The estimated heat capacity of the Debye boson field (~T3) is distinctly smaller than the observed heat capacity and can be assumed to be included in the pre-factor of the T9/2 function. The T3 curve in Fig. 8 is calculated assuming ΘD=300 K. According to an observed saturation magnetic moment of 3.3 μB/Co [63] we must assume that the effective spin is not Seff=1 but Seff=2. In other words, the number of states per Co2+ ion is increased from N=4 for S=3/2 to N=5 for Seff=2. As a conclusion, any change of the number of the relevant states involving the crystal electric field lets the heat capacity drop to its absolute minimum. 8 10

5

12

14

T (K)

16

-1

-1

heat capacity (cal K mole )

T. Shinoda, H. Chihara, S. Seki, J. Phys. Soc. Japan 19 (1964) 1637.

4

CoCl22H2O 3

TN = 17.2 K fitted exponent: 4.70±0.15

2

Seff = 2

cp Debye (estimated)

1

0 0

100000

200000

T

9/2

300000

9/2

(K )

Fig. 8. Heat capacity of monoclinic CoCl2·2H2O as a function of T9/2 [62]. T9/2 function means integer spin and isotropic boson field. Integer spin is in disagreement with S=3/2 of the free Co 2+ ion and proves relevant crystal field interaction. For CoCl2·2H2O the action of the crystal field is to increase the number of the relevant states per Co2+ ion from N=4 for S=3/2 to N=5 for Seff=2. An increased number of states is attributed to an explicit orbital moment contribution. The estimated heat capacity of the Debye boson field is given by curved line.

Another example of a spin quantum number crossover occurring in the paramagnetic phase is hexagonal LuGaCoO4 (see also Fig. 15 below). In contrast to tetragonal CoF2 (Fig. 1), in LuGaCoO4 this crossover shows up in the longitudinal susceptibility (Fig. 9). As a possible explanation of the different behavior compared to CoF2 we can assume that the spin quantum number crossover occurs in the susceptibility along the crystallographic axis with the larger lattice parameter. For CoF2 c/a=0.677 and for LuGaCoO4 c/a=7.415. As can be seen in Fig. 9,

in LuGaCoO4 the crossover from S=3/2 to Seff=1 occurs at TCO=85.2 K. We can assume that the quantum state crossover at TCO=85.2 K coincides with the crossover to the critical range. Below and above TCO=85.2 K the g-factor stays very nearly constant at g~2.8. The crossover at TCO=85.2 K is absent in the transverse susceptibility (Fig. 10). As a comparison of Fig. 9 and Fig. 10 shows, the Curie-Weiss temperatures of the longitudinal and of the transverse susceptibility are much different. The large paramagnetic anisotropy is indicative of the large content on orbital magnetism in the magnetic moment of the Co2+ ion.

40000 35000

Seff=1 , g=2.82

25000

S=3/2 , g=2.86

TCO=85.2 K

20000 15000

LuGaCoO4

10000

H// c



-1

-1

(g Oe emu )

30000

 = -70 K 5000

2+

Co : S=3/2 0

0

50

100

150

200

250

300

T (K)

Fig. 9. Paramagnetic susceptibility of LuGaCoO4 measured parallel to hexagonal c-axis as a function of absolute temperature. The high-temperature slope conforms to the full spin of the Co2+ ion of S=3/2 with a fitted g-factor of g=2.86. Below crossover at TCO=85.2 K, the slope of χ-1(T) can be described by Seff=1 and g=2.82.

65000 60000

H

S=3/2 , g=2.48

c



-1

-1

(g Oe emu )

55000 50000

LuGaCoO4

45000

g=2.48 =-199 K

TN=20 K

40000

2+

Co

35000

: S=3/2

30000

0

50

100

150

200

250

300

T (K)

Fig. 10. Paramagnetic susceptibility of LuGaCoO4 measured transverse to hexagonal c-axis as a function of temperature. The slope of the Curie-Weiss line conforms to S=3/2 and g=2.48. The large

paramagnetic anisotropy of LuGaCoO4 reveals by the different Curie-Weiss temperatures in measurements parallel to c-axis (Fig. 9) and transverse to c-axis (Fig. 10).

For the compounds of the 3d elements with a very low ordering temperature it is possible that a relevant crystal field interaction decreases the number of thermodynamically relevant states per magnetic atom by ΔN=2. This is nothing unusual for the compounds of the Rare Earth elements with an explicit orbital moment (see Figs. 16,21 below). As an example of a 3d compound with ΔN=2 Fig. 11 shows neutron scattering data of the spontaneous magnetization of hexagonal RbFeCl3 [34]. For unclear reasons the ordering temperature of RbFeCl3 is TN=2.55±0.05 only. The low ordering temperature increases the weight of the crystal field interaction with respect to the magnetic interactions. As the observed saturation magnetic moment of ~2.1 μB/Fe2+ unambiguously shows, the effective spin of the Fe2+ ion is Seff=1 in RbFeCl3. Decrease from S=2 of the free Fe2+ion to Seff=1 means a loss of two spin states. Integer spin of Seff=1 is consistent with the thermal decrease of the spontaneous magnetization according to T9/2 function. In other words, the hexagonal lattice symmetry of RbFeCl3 is not relevant.

5

~T

2

F.F.Y. Wang, D.E. Cox, M. Kestigian, Phys. Rev. B 3 (1971) 3946.

m ( B/Fe)

4

RbFeF3 S=2 TN=100.5 K

3

2

RbFeCl3 Seff=1

1 G.R. Davidson, M. Eibschütz, D.E. Cox, V.J. Minkiewicz, AIP Conf. Proc. 5 (1971) 436.

0 0,0

0,1

0,2

0,3

TN=2.55 K 0,4

(T/TN)

0,5

0,6

0,7

9/2

Fig. 11. Spontaneous magnetization of hexagonal RbFeCl3 [34] and of cubic RbFeF3 [64] obtained using neutron scattering as a function of reduced temperature to a power of 9/2. According to the observed saturation magnetic moments the spin of the Fe2+ ion is S=2 in RbFeF4 but Seff=1 in RbFeCl3. Crystal field interaction is relevant only in RbFeCl3 and has decreased the number of relevant states per Fe2+ ion by ΔN=2. In nominally cubic RbFeF3 lattice distortions become relevant at low temperatures and induce a dimensionality crossover from 3D-isotropic (T9/2 function) to 3Danisotropic (T2 function) [66].

Comparison with cubic RbFeF3 illustrates the situation without relevant crystal field interaction [64]. Using neutron scattering the spontaneous magnetization of RbFeF3 has been determined for all temperatures below TN=100.5±0.5 K. The observed saturation magnetic moment of 4.6±0.2 μB/Fe2+ is consistent with the full spin moment of S=2 of the Fe2+ ion.

heat capacity / R

Moreover, observation of T9/2 function in the thermal decrease of the spontaneous magnetization confirms that the spin is integer. For the isotropic boson field in magnets with integer spin the critical exponent is β=1/3 [7]. The observed value for RbFeF3 of β=0.329±0.004 is in excellent agreement with β=1/3. Note, however, that β=1/3 has a broad abundance and applies equally to the magnets with a one-dimensional boson field, irrespective of the spin quantum number [65]. RbFeF3 undergoes a structural phase transition from very nearly cubic to possibly orthorhombic at about T~86 K [64,66]. This structural phase transition gives not rise to a sharp anomaly in the temperature dependence of the spontaneous magnetization. However, the observed crossover to low-temperature T2 function provides clear evidence of a lattice distortion. Only if the lattice distortion is sufficiently strong it becomes relevant [66]. This is a threshold induced crossover event. In other words, lattice distortions can be finite but if they are sufficiently weak they become not relevant. Note that the T2 universality class pertains to the 3D-anisotropic boson field in magnets with an integer spin (curve labeled by T2 in Fig. 11) [12,13]. We now consider the two RE compounds DyPO4 and DyVO4. The Dy3+ ion has the electronic configuration 6H15/2 [35]. This means, the spin quantum number is S=5/2 and the orbital quantum number is L=5. Both materials have very similar antiferromagnetic ordering structures with Néel temperatures of only TN=3.043K (DyVO4) [67] and TN=3.40±0.1K (DyPO4) [68]. The observed saturation magnetic moments of DyVO4 of 9.0μB [69] and of DyPO4 of 9.0±0.3μB [70] agree perfectly with each other. They are only slightly reduced with respect to the free ion value of 10μB. The slightly reduced saturation magnetic moment can be described by Jeff=7 instead of J=15/2 and shows that the effect of the crystal field is surprisingly weak but is relevant. As a consequence, only one of the N=2J+1=16 states is excluded from dynamics by the crystal electric field. Assuming Jeff=7 a saturation magnetic moment of 9.33μB/Dy results using the free-ion g-factor of g=4/3. For the next lower integer J-value of Jeff=6 the calculated magnetic moment is 8μB/Dy. In view of an observed magnetic moment of 9.0±0.3μB, Jeff=6 can be excluded.

4

P.J. Becker et al. Phys. Lett. 31A (1970) 499.

J.H. Colwell, et al. Phys. Rev. Lett. 23 (1969) 1245.

3

DyVO4

DyPO4

TN=3.04 K

TN=3.39 K

Jeff=7

Jeff=7

2 fitted exponent: 4.418±0.111

1

0 1

2

T (K)

3

4

Fig. 12. Heat capacities divided by gas constant of DyVO4 [67] and of DyPO4 [68] as a function of temperature. The T9/2 function fitted to the low-temperature data of DyVO4 proves integer effective spin (Jeff=7). For the free Dy3+ ion J=15/2. For the analysis of the DyPO4 heat capacity data see Fig. 13.

Integer total moment number of Jeff=7, estimated on account of the observed saturation magnetic moments, is confirmed by the T9/2 universality class in the low temperature tails of the heat capacities of DyVO4 [67] and DyPO4 (Figs. 12,13) [68]. As can be seen in Fig. 12 the heat capacities of the two materials are, practically, identical except for the slightly different ordering temperatures. Due to the extremely low ordering temperature the magnetic heat capacity dominates. The T3 function of the non-relevant Debye boson field is completely absent. Possible weak heat capacity contributions of the Debye bosons can be assumed to be included in the pre-factors of the T9/2 functions (see Fig. 13). Low-temperature heat capacity data of DyVO4 are well described by a single T9/2 function (solid curve in Fig. 12). Lowtemperature heat capacity data of DyPO4 consist of two T9/2 sections with crossover at TCO=2.32 K (Fig. 13). This type of crossover with only a change of the pre-factor of the universal power function indicates a significant change of a non-relevant parameter. Note that non-relevant energy degrees of freedom do not change the universality class but modify the pre-factor of the universal power function only. We can assume that the non-relevant energy degrees of freedom are the Debye bosons. In other words the different pre-factors of the two T9/2 functions in Fig. 13 are caused by a varying importance of the Debye bosons. It is reasonable to assume that the intrinsic heat capacity of the Debye boson field is given by T3 function for all temperatures of Fig. 13. Nevertheless, a discrete amplitude crossover event is induced in the observed heat capacity of DyPO4 if the heat capacity of the Debye bosons has exceeded a threshold. This further illustrates the stability of the universality classes against weak perturbations. T (K) 1.8 2.0 2.2 1,4

2.6

2.8

3.0

DyPO4 TN= 3.391 K

1,2

heat capacity / R

2.4

Jeff = 7

1,0 0,8 0,6 0,4

J.H. Colwell et al. Phys. Rev. Lett. 23 (1969) 1245.

TCO=2.32 K

0,2 0,0 0

20

40

60

80

T

9/2

100 9/2

(K )

120

140

160

180

Fig. 13. Heat capacity divided by gas-constant R of DyPO4 as a function of T9/2 [68]. Crossover between two sections with T9/2 dependence is at TCO=2.32 K. Change in slope is indicative of a changing importance of the non-relevant Debye bosons.

Fig. 14 shows the magnetic entropies of DyVO4 [67] and of DyPO4 [68] as a function of temperature. It can be seen that asymptotically the entropies approach the lowest possible value of R·ln (2) only very slowly. In other words the magnetic heat capacity is much too low in view of an effective moment value of Jeff=7. For Jeff=7 the entropy should saturate at R·ln(15) which is a factor of 3.9 larger than the observed value of R·ln(2). Saturation of the entropy at R·ln(2) does not mean that a crossover to atomistic Ising behavior has occurred [42,43]. Ising behavior can be excluded by the observed universal T9/2 power functions in the heat capacity. Quite generally, in the compounds of the RE elements with an explicit orbital moment, Ising behavior is unlikely to occur. This is because an orbital moment with L=3 can have 2L+1=7 possible orientations in the crystal electric field. Ising behavior requires only two orientations.

0,7

ln(2) 3+

0,6

entropy / R

0,5

Dy

T

9/2

6

; H15/2

S=5/2 , L=5 , J=15/2 Jeff = 7

0,4

DyVO4

0,3

P.J. Becker et al. Phys. Lett. 31A (1970) 499.

0,2

DyPO4 J.H. Colwell et al. Phys. Rev. Lett. 23 (1969) 1245.

0,1 0,0 0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

T (K)

Fig. 14. Magnetic entropy of DyVO4 [67] and of DyPO4 [68] as a function of temperature. Crystal field interaction is relevant and has decreased the total moment from J=15/2 to J eff=7. The moment reduction by ΔJ=1/2 is weak but lets the heat capacity collapse to its absolute minimum [37-41]. The entropy slowly approaches R·ln(2) only. For Jeff=7 the entropy should saturate at R·ln(15).

3. Crystal field effects in metals On discussing crystal field effects in metallic magnets we restrict on the nominally cubic inter-metallic Rare Earth (RE) alloys with the composition REAl2 [35,71-73]. These alloys order ferromagnetically [35]. For the RE elements with an explicit orbital moment, crystal field interaction is generally strong. Alloying the Rare Earth elements with a non-magnetic metal decreases the magnetic ordering temperature strongly, and thus increases the weight of the crystal field relative to the magnetic interactions. This favors relevance of the crystal field interaction. For the compounds or alloys of the RE elements a relevant crystal field interaction

can easily be identified by a comparison of the experimental saturation magnetic moment with the theoretically well-known saturation moment [35]. Since the total moment number J and the g-factor are precisely known, the relation between the experimental saturation moment and the theoretical one scales with the ratio of the effective total moment number Jeff number and the full moment number J. The only limitation is the experimental uncertainty in the observed saturation magnetic moment. We start discussion with NdAl2 [30,72,74] and HoAl2 [56]. Very remarkable with the ferromagnetic inter-metallic REAl2 alloys is that they exhibit no critical behavior above and below the Curie temperature. In other words, the Curie-Weiss temperature extrapolated from the high-temperature paramagnetic susceptibility agrees with the Curie temperature [56]. Fig. 15 illustrates this for the paramagnetic phase of NdAl2. However as for CoF2 (Fig. 1) and for LuGaCoO4 (Fig. 9) not a single Curie-Weiss line is observed in the temperature dependence of χ-1(T) [74]. At TCO=182 K a crossover between two linear sections can be identified. For T>TCO=182 K it can reasonably be assumed that the Nd3+ ion has its full moment of J=9/2. From the slope of the Curie-Weiss line for T>TCO a g-factor of g=0.695 follows which is lower by only 4.6% than the theoretical value of 8/11=0.727. The larger slope for T
2500

NdAl2 4

I9/2

2000

3

(g/cm )

g=8/11

TCO=182 K

Jeff=J=9/2

1500

= 53.5 K gexp = 0.695

1/

1000

TC 500

Jeff=8/2 gexp=0.703 0 50

100

150

200

250

300

T (K)

Fig. 15. Reciprocal paramagnetic susceptibility of a single crystal of the cubic ferromgnet NdAl2 as a function of temperature (this work). At the crossover temperature at TCO=182 K the moment quantum number changes from J=9/2 of the free Nd3+ ion for T>TCO to Jeff=8/2 for T
numbers. The fitted g-factors are slightly below g=8/11=0.727 only. No typical crossover to critical behavior occurs (see text). The Curie temperature is at TC=80 K.

No critical behavior is observed in the ordered phase of NdAl2 as well (Fig. 16). Using zero-field neutron scattering [76] a thermal decrease of the spontaneous magnetization according to T3 is observed. The T3 function means a one-dimensional boson field and integer spin [13,14]. The T3 function is indicative of an axial lattice distortion. From the observed saturation magnetic moment of ~2.35 μB/Nd [35] it follows that the effective moment is Jeff=3. Using the g-factor of the free Nd3+ ion of g=8/11 a saturation magnetic moment of 2.18 μB/Nd can be calculated for Jeff=3. The slightly larger experimental value might be due to polarization contributions of the conduction electrons. As a conclusion, in the ordered state the number of thermodynamically relevant states per Nd3+ ion is further reduced to N=7. The total number of “quenched” states therefore is ΔN=3. No indication of the two further crossover events can be identified in the temperature dependence of the spontaneous magnetization. Possibly, the two additional crossover events coincide with the Curie temperature. Surprisingly, the T3 function holds for all temperatures in the ordered range (compare Fig. 19). No crossover to critical behavior can be identified. For magnets with onedimensional boson field and integer spin (T3 universality class) the critical exponent of the spontaneous magnetization is β=1/3 [65]. T (K) 30 40

50

60

70

1,0

80

NdAl2 3+

Nd

M(T) / M(T=0)

0,8

4

; I9/2

S=3/2, L=6, J=9/2 Jeff=3

0,6

0,4

TC=80 K 0,2

0,0 0

100000

200000

300000 3

400000

500000

3

T (K )

Fig. 16. Normalized zero-field spontaneous magnetization of NdAl2 obtained for a single crystal using neutron scattering as a function of absolute temperature to a power of three. T3 universality class means integer spin and one-dimensional boson field. The low dimensionality indicates a relevant axial lattice distortion. In view of a saturation magnetic moment of ~2.35 μB/Nd [35], the moment quantum number can be assumed to be Jeff=3. With g=8/11 and Jeff=3 a saturation magnetic moment of mS=2.18 μB/Nd follows. No crossover to critical behavior can be identified.

20

Rln(10)

16

NdAl2

14

I9/2

4

S=3/2, L=6, J=9/2

12

-1

-1

entropy (JK mole )

18

TN=77.2 K

10 8 6

Rln(2)

4

C. Deenadas et al. J. Phys. Chem. Solids 32 (1971) 1853.

2 0 0

20

40

60

80

100

120

140

T (K)

Fig. 17. Magnetic entropy of NdAl2 as a function of temperature [72]. For this metallic material the magnetic entropy exceeds the value of R·ln(2) of the insulators but approaches the theoretical value of R·ln(10) only slowly.

Fig. 17 shows the magnetic entropy of NdAl2 as a function of temperature [72]. As nonmagnetic heat capacity background, the heat capacity of LaAl2 is taken [72]. As can be seen in Fig. 17, the magnetic entropy approaches the theoretical limit of R·ln(10) very slowly. The slow saturation correlates with the susceptibility result in Fig. 15. As can be seen in Fig. 15 the crossover to the full moment of J=9/2 occurs at TCO=182 K. This temperature is beyond the temperature window of Fig. 17. We should add one example of the occasionally dramatic difference between the magnetic properties of powder samples and of single crystals of chemically identical materials [60]. Although the absolute magnetization differences between powder and single crystal are small, the analytical behavior of the temperature dependence of the spontaneous magnetization can be completely different. This reveals by a comparison of the NdAl2 single crystal data of Fig. 16 with the powder data of Fig. 18. From the different Tε functions at low temperatures it follows that the action of the crystal electric field is different for the single crystal and for the powder sample. This can be a matter of the different domain structure. Note that the T3 function (Fig. 16) means one-dimensional boson field and integer spin, but the T5/2 function in Fig. 18 means one-dimensional boson field and half-integer spin. Moreover, the NdAl2 powder sample in Fig. 18 shows definite critical behavior of mean field type. For the T5/2 symmetry class the standard critical exponent is β=1/3 [12]. Mean field critical behavior pertains to magnets with half-integer spin and isotropic boson field [7]. Note the large width of the boson controlled critical range in Fig. 18. In the classical (atomistic) mean field model the critical power function is the first term of a power series in (Tc-T). The width of the critical range is, so to say, zero [3]. As a consequence, for the NdAl2 powder sample a dimensionality crossover from axial to isotropic coincides with the common crossover from Tε function to critical power function. In those cases it is observed that the critical range at Tc

is unusually large. In fact, the critical range of the NdAl2 powder sample of Fig. 18 is as large as 0.38·TC. A common value would be 0.18·TC.

30

40

50

60

T (K)

70

80

120 100 80

M (a.u.)

~T

fitted exponent: 2.578±0.139

5/2

60 40

 =1/2

NdAl2 powder TC= 79.6 K

20 0 0

10000

20000

30000

T

5/2

40000

50000

60000

5/2

(K )

Fig. 18. Spontaneous magnetization of NdAl2 powder material as a function of temperature to a power of 5/2. Instead of the T3 dependence observed at all temperatures for the single crystal (Fig. 16), crossover from low-temperature T5/2 dependence to critical behavior with mean-field critical exponent of β=1/2 is identified. T5/2 function is indicative of half-integer spin and axial lattice distortion. For this symmetry class the critical exponent should be β=1/3 [12]. However, for powder samples β=1/2 is frequently observed (see text) [60].

Mean field critical behavior observed for the NdAl2 powder sample can, however, have two reasons. Either lattice distortions are weak in the critical range and not relevant or mean field critical behavior results by an averaging process over all differently oriented powder grains [60]. Assuming that the Goldstone bosons are able to tunnel from grain to grain it is not unlikely that in this way dynamic isotropy can result. Note that the Goldstone bosons are essentially magnetic dipole radiation generated by the precessing spins [60]. Tunneling between adjacent grains therefore appears possible. This type of averaging is not observed in the vicinity of T=0 but it occurs in the critical range near Tc only [60,61]. As a further example of a metallic ferromagnet we now discuss cubic HoAl2 with the electronic configuration 5I8. This configuration means S=2, L=6 and J=8. As for NdAl2, above and below the Curie temperature no critical behavior is observed. The paramagnetic susceptibility follows a single Curie-Weiss law for all temperatures [56]. In particular, the Curie-Weiss behavior continues up to T→TC. In other words, no crossover to critical behavior occurs (Fig. 19). The Curie-Weiss temperature agrees with the Curie temperature. For cubic magnets with an isotropic boson field and with integer spin the critical exponent of the paramagnetic susceptibility commonly is γ=4/3 [7,75]. The experimental effective magnetic moment agrees well with J=8 and g=5/4. The fitted g-factor assuming J=8 is g=1.26. The spontaneous magnetization below TC has been evaluated from a transition energy between two nuclear spin states (=hyperfine interaction) using inelastic neutron scattering

with high energy resolution [56,57]. This method is equivalent to the NMR technique. The hyperfine energy (field) at the magnetic atom is to a good approximation proportional to the spontaneous magnetization. As can be seen in Fig. 19 all data of the ordered phase are excellently described by one T9/2 function. On approaching T→TC, no indication of a crossover to critical behavior can be noticed in the spontaneous magnetization (compare Fig. 16). The reported saturation magnetic moments for HoAl2 range from 8.1±0.2 μB/Ho [77] to 9.17 μB/Ho [35]. These values are clearly inferior to the theoretical value of 10μB/Ho. As the T9/2 power function shows, the effective moment number is integer. It is reasonable to assume Jeff=7. In other words two states per Ho atom are excluded from dynamics by the action of the crystal electric field. Using the free-ion g-factor of g=5/4 a saturation moment of 8.75 μB/Ho results for Jeff=7, in reasonable agreement with the observed values. Assuming, alternatively, Jeff=6 a too low saturation magnetic moment of 7.5 μB/Ho results. Surprisingly, the two crossover events from J=8 to Jeff=7 are not resolved by the data of Fig. 19. We must assume that the two crossover events coincide, as for NdAl2, with the Curie temperature. As a conclusion, a relevant crystal field completely destroys the critical dynamics. No indication of the spin-reorientation transition at TSR=20 K is obtained in the temperature dependence of the spontaneous magnetization of HoAl2 (Fig. 19) [78]. This is in accordance with the findings of RG theory that events on the microscopic scale of the inter-atomic distance are not relevant for the boson controlled global spin dynamics. Universality means a dynamic behavior that is independence of spin structure. However, according to the heat capacity data [78], the critical behavior at the Curie transition at TC=30 K is of mean field type.

25

5

HoAl2 9/2

4 3

energy ( eV)

~T

fitted exponent: 4.474±0.136

15

1/ (mole/cm )

Jeff= 7

20

J=8 g=5/4 gexp=1.26

10

3

2

TC=30 K

1

1

5 T. Chatterji et al. Solid State Commun. 161 (2013) 42.

0

0

20

40

60

80

100

T (K)

Fig. 19. Reciprocal paramagnetic susceptibility (open points) and spontaneous magnetization (filled points) of HoAl2 as a function of temperature [56]. On both sides of the Curie temperature (TC=30 K) no critical behavior occurs. All data in the ordered range follow T9/2 function (for explanation see text). The effective spin therefore is integer (Jeff=7). Two states are suppressed (non relevant) by the crystal electric field.

24

Rln(16)

22

-1

-1

magn. entropy (JK mole )

20

DyAl2

18

ErAl2

16

J=15/2 TN= 12 K

14

J=15/2 TN= 58 K

12 10 8 6

Rln(2)

4 T. Inoue et al. J. Phys. Chem. Solids 38 (1977) 487.

2 0 0

20

40

60

80

100

T (K)

Fig. 20. Magnetic entropies of ErAl2 and of DyAl2 as a function of temperature [73]. In contrast to the magnetic insulators (Figs. 4,5,7,14), the magnetic entropy of the metallic magnets saturates not perfectly but close to the expected value of R·ln(2J+1). Dashed horizontal line labeled by R·ln(2) indicates the magnetic entropy limit of the insulators.

Fig. 20 displays magnetic entropy data of ErAl2 and of DyAl2 as a function of temperature [73]. The electronic configurations of the two materials are 4I15/2 (ErAl2) and 6H15/2 (DyAl2). It is evident that the magnetic entropy of these metallic ferromagnets exceeds the entropy limit of R·ln(2) of the insulating magnets. For both materials, the magnetic entropy approaches the expected limit of R·ln(16) reasonably. According to an observed saturation magnetic moment of ~7.5 μB/Er3+, the effective moment number of ErAl2 is Jeff=6 for T→0 [4,35]. The theoretical saturation moment is 9 μB/Er3+. With Jeff=6 and g=6/5 the calculated saturation magnetic moment is 7.2 μB/Er3+. The number of non-relevant states therefore is ΔN=3. As usual, the observed saturation magnetic moment is slightly larger due to polarization contributions of the conduction band. For DyAl2 the observed saturation magnetic moment is ~9.7 μB/Dy3+ [35]. The theoretical saturation magnetic moment is 10 μB/Dy3+. The effective moment number therefore is Jeff=7. With Jeff=7 and g=4/3 the calculated saturation magnetic moment is 9.33 μB/Dy3+. The number of thermodynamically deactivated (non-relevant states) therefore is ΔN=1. The lower number of “quenched” states for DyAl2 (ΔN=1) compared to ErAl2 (ΔN=3) can be correlated with the higher ordering temperature of DyAl2. As we have already mentioned, for a homologous compound series the number of nonrelevant states, ΔN, should scale with the ratio of the crystal field energy to the magnetic interaction energy. For the heavy REAl2 alloys it can be assumed that the crystal field is approximately constant. The crystal field interaction energy then is given by the orbital quadrupole moment that can be approximated by L·(L+1). A measure of the magnetic interaction energy is the ordering temperature. The ordering temperature should scale approximately with the square of the spin quantum number, S·(S+1). The content of S·(S+1) in the total moment squared J·(J+1) is given by the de Gennes factor G=(g-1)2J·(J+1). The

content of orbital moment squared, L·(L+1), in J·(J+1) can be written as J·(J+1)-G. The number of quenched states, ΔN, therefore should scale approximately with [J·(J+1)-G]/G=[1(g-1)2]/(g-1)2. Fortunately, in this expression the quantum number J is canceled and we need not care about the difference between J and Jeff. Moreover, the systematic difference between the actual, the boson driven ordering temperature, and the estimated ordering temperature due to the exchange interactions can also be assumed to be constant.

REAl2

4

TmAl2 3

N

ErAl2

2

HoAl2

TbAl2

1

DyAl2 GdAl2

0 0

5

10

15

20 2

25 2

30

35

[1-(gJ-1) ]/(gJ-1)

Fig. 21. Number of states quenched by the crystal electric field, ΔN, for the heavy REAl2 alloys. The quantity on the abscissa is an approximate expression for the ratio of crystal field interaction energy to magnetic interaction energy (see text). The effective moment number is Jeff=J-ΔN/2 with J as full moment number of the RE3+ ion. All abscissa coordinate values are rational numbers.

Fig. 21 displays for the heavy REAl2 compounds the number of states quenched by the crystal electric field, ΔN, as a function of the quantity [1-(g-1)2]/(g-1)2. As can be seen, for the heavy REAl2 the just rationalized simple relation holds reasonably well. We should mention that the behavior of the REAl2 alloys of the first half of the RE elements is not as systematic.

4. Conclusions For all investigated magnets the dynamics in the vicinity of the critical point T=0 is quantized and can be classified by one of the six “critical” exponents ε. Dynamic quantization is well-known for the critical range at the magnetic ordering temperature [7] but it holds for the “critical” dynamics at T=0 as well. Quite generally, sufficiently close to a critical temperature, either T=0 or T=Tc the dynamics is determined by a boson field. Boson dynamics is easily recognized by the finite widths of the critical range and by the rational critical exponents [7]. The universal power functions at T=0 are power functions of absolute temperature (Tε). The two universal power functions at T=0 and at T=Tc overlap [12,60]. Dynamic quantization therefore holds for all temperatures in the ordered state, and for the critical range above ordering temperature. However, the dimensionality of the relevant boson

field and the effective spin quantum number can be different at T=0 and at T=Tc [12,60]. The associated crossover events, commonly, coincide with the intersection of the two universal power functions at T=0 and at T=Tc. Here we have shown that in the case of a relevant crystal field interaction the dynamics remains quantized and can be classified by the same six universality classes (i.e. by one of the six exponents ε) as for the magnets with a non-relevant crystal field. The effect of a relevant crystal field is to eliminate one or more of the N=2S+1 (or N=2J+1) spin states per magnetic atom from dynamics. Due to the symmetry selection principle of relevance, the number of (relevant) states remains an integer, and universality holds as for the magnets with non-relevant crystal field interaction. The integer number of states left allows definition of an effective spin as Seff=S-ΔN/2 with ΔN as the number of quenched states. The decreased number of relevant states decreases the saturation magnetic moment accordingly. As a consequence, the saturation magnetic moment is also quantized. However, in the case of the 3d-metal compounds the quantization of the saturation magnetic moment is less apparent because the g-factor stays not necessarily constant upon the action of the crystal electric field. The dynamic universality classes at T=0 are more clearly specified than the universality classes at T=Tc. The universality classes at T=0 are distinguished by the three dimensionalities of the Goldstone boson field and by whether the spin quantum number is integer or half-integer. In this way six empirically well established universality classes result [13,14]. The spin dependence of the dynamics follows from the fact that integer and halfinteger spins precess differently and generate different types of field quanta (magnetic dipole radiation). The quantized, that is, boson defined universal dynamics is in contrast to spin wave theory that treats non-quantized spins. As a consequence, the observed discrete saturation values of the magnetic entropy (Fig. 5) are beyond spin wave theory. Saturation of the magnetic entropy at R·ln(2S+1) proves that all 2S+1 spin states are involved in the generation process of the Goldstone bosons. Moreover, observation of dynamic universality (independence of spin structure) proves that magnons are not the relevant excitations. However, in the insulating magnets with a relevant crystal field interaction the magnetic entropy saturates at the lowest possible value of R·ln(2), irrespective of the value of S eff [3741]. In other words, in spite of Seff>1/2 the heat capacity is as for a two-level system. This does not mean that universality is lifted, and that a crossover to the atomistic Ising behavior has occurred. Ising behavior can be excluded owing to the observed universal power functions in the temperature dependence of the spontaneous magnetization (Fig. 6,8,16,19). Only two possible spin orientations can be explained by the perfect one-dimensional Goldstone boson field within each magnetic domain [13]. The one-dimensional boson field interacts with the spins (magnons) and furnishes the spin system with an axial anisotropy that is not known in the atomistic models. As we have argued earlier, the magnon gap energy commonly of the order of a few meV- is a measure of the boson-magnon interaction strength [13]. The gap indicates a stabilized collinear spin structure. In other words, the onedimensional boson field within each domain dominates over the local exchange anisotropy and the single particle anisotropy and warrants the perfect collinear spin structure. From the fact that for a relevant crystal field interaction in insulators only the two spin orientations ±Seff with respect to the axis of the boson field are possible, it can be concluded that the axial anisotropy provided by the one-dimensional boson field is strongly enhanced. This means that the magnon gap must be increased. A larger magnon gap means a

strengthened collinear (one-dimensional) spin structure. This then depresses spin dynamics and decreases the magnetic heat capacity. In fact, in a material with a relevant crystal field, such as CoF2, thermal equivalent of the magnon gap energy (Egap/h=1.15 THz or Egap/k= 55 K) is larger than the Néel temperature of TN=38 K [79,80]. For spin wave theory this is a paradox. In other words, in insulators the effect of a relevant crystal field is to increase the magnon gap energy. On the other hand, magnons (exchange interactions) are non-relevant excitations. This reveals impressively from the fact that also for magnets with a large magnon gap, thermal decrease of the spontaneous magnetization is according to an universal power function of temperature and not according to an exponential function. As a consequence, a large magnon gap is of no importance on the universality class but it depresses spin dynamics strongly. The absolute heat capacity values are strongly reduced. For CoF2 with a magnon gap of larger than conforms to ordering temperature, the spontaneous magnetization decreases by universal T9/2 power function (Fig. 6). This universality class applies to the heat capacity of the isotropic boson field in magnets with integer spin (Seff=1). Isotropic dynamic symmetry means that the one-dimensional boson fields of the differently oriented domains are dynamically coupled to result in an isotropic global field [13,14]. Explanation of this coupling mechanism constitutes one of the big challenges for future field theories of magnetism. As a conclusion, in the boson controlled ordered state, the action of the crystal electric field is quantized and can be classified by Seff or, equivalently, by the exponent ε. On the other hand, for sufficiently high temperatures, or for activation energies of larger than crystal field interaction energy, crossover to the full spin moment, S, occurs (Figs. 1,15). This crossover happens at a well defined thermal energy (or temperature). As a consequence, we can expect discrete energy states in the magnetic excitation spectrum. Note that the excitation spectrum of the bosons always is gap-less. The discrete excitation energies must occur in the magnon spectrum. In fact, in the magnon excitation spectrum of CoF2 a second, high-energy magnon band is observed in the energy range 5.2…6.1 THz (at T=4.2 K) [80]. We can attribute the full spin quantum number of S=3/2 to this high-energy magnon band. By the same argument we can attribute the effective spin of Seff=1 to the low-energy magnon band of the range 1.14…1.85 THz. In other words, it appears that there exist ΔN+1 different magnon bands with ΔN as the number of non-relevant states (Fig. 21). This systematic is reasonably confirmed also for TbAl2 [81], for DyAl2 [82], for HoAl2 [83] and for ErAl2 [84]. According to Fig. 21 for TbAl2 and for DyAl2 one of the 2J+1 states per RE ion is “quenched” by the crystal electric field. As a consequence, it can be expected that, as for CoF2, only one further high-energy magnon band occurs. This high-energy magnon band can be attributed the full Jvalue. A second, high-energy magnon band is, in fact, confirmed for TbAl2 [81] and for DyAl2 [82]. In contrast to the insulating CoF2, in the metallic RE alloys thermal equivalent of the energy of the lowest magnon band is smaller than the Curie temperature. As a consequence, the action of the crystal field is weaker in the metals compared to insulators. The collinear spin structure is less stabilized and the magnetic heat capacity therefore is not much suppressed. For HoAl2 two of the 2J+1=17 states of the Ho3+ ion are quenched by the crystal electric field (Fig. 21). As a consequence, ΔN+1=3 magnon bands should occur in the magnon excitation spectrum. This is, in fact, confirmed experimentally [83]. For ascending excitation energies we can label the three magnon bands of HoAl2 by Jeff=14/2, Jeff=15/2 and J=16/2. As

for all metallic magnets, thermal equivalent of the gap energy of the lowest magnon band of HoAl2 is lower than the ordering temperature. For HoAl2 the energy of the lowest magnon gap is ~1.4 meV [83,85]. This energy corresponds to T=16.2 K, and is lower than the Curie temperature of TC=30 K. The spin dynamics therefore is not much suppressed and it can be understood that the entropy saturates approximately at the expected value of R·ln(17) [72,73,86]. For ErAl2 three of the 2J+1=16 states are quenched (Fig. 21) [3]. As a consequence four magnon bands should be observed. Unfortunately the magnon spectrum of ErAl2 is not sufficiently elaborated experimentally to confirm this explicitly [84]. On the other hand, assumption of four magnon bands is not in disagreement with the experimental results. A nice example of the first half of the RE elements confirming the just sketched systematics is NdAl2. According to the electronic configuration 4I9/2 the total moment quantum number is J=9/2. The theoretical saturation magnetic moment is 3.28 μB/Nd3+. The observed saturation moment is 2.5±0.1 μB/Nd3+ [87]. The effective moment quantum number can be assumed to be Jeff=7/2. With Jeff=7/2 and g=8/11 a saturation moment of 2.55 μB/Nd3+ is calculated, in good agreement with the observed one. The number of non-relevant states therefore is ΔN=2. In fact, in the magnon excitation spectrum of NdAl2 two well resolved additional high-energy magnon bands are observed [88]. For each quantum number crossover event an anomaly should be observed either in the susceptibility or in the spontaneous magnetization. However, in the here investigated REAl2 alloys a clear crossover event is observed, if at all, only in the paramagnetic range (Fig. 15). No indication of the anticipated further crossover events at lower temperatures is obtained. We therefore must conclude that the missing crossover events coincide with the Curie temperature. Connected with this fact seems to be that neither the spontaneous magnetization nor the paramagnetic susceptibility exhibit critical behavior at TC (Fig. 15,16,19) [89]. The Curie-Weiss temperature agrees with the Curie temperature. This does not mean that classical mean field behavior is realized. Moreover, for the spontaneous magnetization mean field critical behavior with β=1/2 is not observed. In other words, we can assume that the relevant crystal field interaction suppresses the critical dynamics at the Curie temperature. Critical behavior is observed in the heat capacity only [72,73,78]. The observed heat capacity anomaly at TC [72,73,86] seems to be due to the spin quantum number crossover that is pinned to TC. In spite of a totally quenched critical dynamics, the magnetic entropy of the REAl2 alloys saturates only slightly below the expected value of R·ln(2J+1) [72,73]. If the quantum number crossover events coincide with the Curie temperature it can be expected that the ΔN high-energy magnon bands merge in the lowest magnon band just at the Curie temperature. This temperature dependence of the excited magnon bands remains to be shown experimentally. Note that magnons at zone boundary commonly persist in the paramagnetic range. As a final conclusion, in the insulating and in the metallic magnets a relevant crystal field acts differently [90]. Common to metals and to insulators is that the effect of a relevant crystal field is to exclude one or more of the 2S+1 spin states from dynamics. However, in the metallic magnets spin dynamics is less suppressed compared to the insulating magnets. This can be correlated with the lower magnon gap energies in the metallic magnets. In the insulators a relevant crystal field shifts the magnon energies to larger values than conforms to the critical temperature. This strongly strengthens the collinear spin structure and allows for only two spin orientations along the one-dimensional boson field. This then reduces the spin

dynamics dramatically. The magnetic entropy thereby saturates at the lowest possible value of R·ln(2). For a more detailed outline of the here discussed phenomena the reader is referred to Ref. [61].

References [1] K.G. Wilson, J. Kogut, Phys. Rep. 12C (1974) 75. [2] E. Brézin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. B 10 (1974) 892. [3] A. Hoser, U. Köbler, Acta Phys. Polon. A 127 (2015) 350. [4] J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127 (1962) 965. [5] U. Köbler, Int. J. Thermodyn. 20 (2017) 210. [6] P. Debye, Ann. Physik, 39 (1912) 789. [7] U. Köbler, J. Magn. Magn. Mater, 453 (2018) 17. [8] U. Köbler, Int. J. Thermodyn. 18 (2015) 277. [9] R.F.S. Hearmon in Landolt-Börnstein, vol. III/11, ed. by K.-H. Hellwege and A.M. Hellwege, (Springer, Berlin, 1984), p.1 [10] M. Born, K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1956). [11] U. Köbler, A. Hoser, J. Magn. Magn. Mater. 325 (2013) 87. [12] U. Köbler, Acta Phys. Polon. A 128 (2015) 398. [13] A. Hoser, U. Köbler, J. Phys.: Conf. Series 746 (2016) 012062. [14] U. Köbler, Acta Phys. Polon. A 127 (2015) 1694. [15] A. Hoser, U. Köbler, Physica B : Condens. Matter (2017) [16] M.P. Schulhof, R. Nathans, P. Heller, A. Linz, Phys. Rev. B 4 (1971) 2254. [17] U. Köbler, Acta Phys. Polon. A 127 (2015) 356. [18] P. Heller, Rep. Prog. Phys. 30 (1967) 731. [19] R.K. Pathria, Statistical Mechanics, 2nd Edit. (Butterworth-Heinemann, Oxford 1996). [20] U. Köbler, V. Yu. Bodryakov, Int. J. Thermodyn. 18 (2015) 200. [21] K. Katsumata, M. Matsuura, H.P.J. Wijn, in Landolt-Börnstein, vol. III/27j1, ed. by H.P.J. Wijn (Springer, Berlin, 1994) [22] J.W. Stout, L.M. Matarrese, Rev. Mod. Phys. 25 (1953) 338. [23] H.E. Nigh, S. Legvold, F.H. Spedding, Phys. Rev. 132 (1963) 1092 [24] D.J. Breed, K. Gilijamse, A.R. Miedema, Physica 45 (1969) 205. [25] W.L. Roth, Phys. Rev. 110 (1958) 1333. [26] A.K. Cheetham, D.A.O. Hope Phys. Rev. B 27 (1983) 6964. [27] T. Chatterji, G.J. Schneider, Phys. Rev. B 79 (2009) 212409. [28] F.B. Koch, M.E. Fine, J. Appl. Phys. 38 (1967) 1470. [29] G.E. Kugel, B. Hennion, C. Carabatos, Phys. Rev. B 18 (1978) 1317. [30] U. Köbler, A. Hoser, J.-U. Hoffmann, Physica B 382 (2006) 98. [31] J. Strempfer, U. Rütt, S.P. Bayrakci, Th. Brückel, W. Jauch, Phys. Rev. B 69 (2004) 014417. [32] T. Chatterji, B. Ouladdiaf, T.C. Hansen, J. Phys.: Condens. Matter 22 (2010) 096001. [33] W. Jauch, M. Reehuis, A.J. Schultz, Acta Crystallogr. A 60 (2004) 51. [34] G.R. Davidson, M. Eibschütz, D.E. Cox, V.J. Minkiewicz, AIP Conf. Proc. 5 (1971) 436. [35] K.H.J. Buschow, Rep. Prog. Phys. 42 (1979) 1373. [36] A. Herr, M.J. Besnus, A.J.P. Meyer, Coll. Int. CNRS No 242 (1975) 47. [37] E. Catalano, J.W. Stout, J. Chem. Phys. 23 (1955) 1803. [38] J.W. Stout, E. Catalano, J. Chem. Phys. 23 (1955) 2013. [39] A. De Combarieu, J. Mareschal, J.C. Michel, J. Sivardière, Solid State Commun. 6 (1968) 257.

[40] J.D. Cashion, M.R. Wells, J. Magn. Magn. Mater. 177-181 (1998) 781. [41] A. Berton, B. Sharon, J. Appl. Phys. 39 (1968) 1367. [42] H. Ikeda, K. Hirakawa, Solid State Commun. 14 (1974) 529. [43] E.J. Samuelsen, Phys. Rev. Lett. 31 (1973) 936. [44] L. Onsager, Phys. Rev. 65 (1944) 117. [45] C.N. Yang, Phys. Rev. 85 (1952) 808. [46] T. Chatterji in: Neutron Scattering from Magnetic Materials, Ed. T. Chatterji (Elsevier B.V. Amsterdam, 2006) pp. 25-91. [47] A. Okazaki, K.C. Turberfield, R.W.H. Stevenson, Phys. Lett. 8 (1964) 9. [48] O. Nikotin, P.A. Lindgard, O.W. Dietrich, J. Phys. C: Solid State Phys. 2 (1969) 1168. [49] M.G. Cottam, V.P. Gnezdilov, H.J. Labbé, D.L. Lockwood, J. Appl Phys. 76 (1994) 6883. [50] Y. Shapira, S. Foner, Phys. Rev. B 1 (1970) 3083. [51] J.W. Battles, G.E. Everett, Phys. Rev. B 1 (1970) 3021. [52] U. Köbler, A. Hoser, H.A. Graf, M.-T. Fernandez-Diaz, K. Fischer, Th. Brückel, Eur. Phys. J. B 8 (1999) 217. [53] Y.S. Touloukian, E.H. Buyco: Thermophysical Properties of Matter, vol. 5.: Specific Heat of Nonmetallic Solids (IFI/Plenum, New York-Washington, 1970). [54] G.A. Alers, in: Physical Acoustics, vol. III B, ed. by W.P. Mason, (Academic Press, New York, 1965), p.1. [55] W. Jauch, M. Reehuis, H.J. Bleif, F. Kubanek, P. Pattison, Phys. Rev. B 64 (2001) 052102-1. [56] T. Chatterji, N. Jalarvo, A. Szytula, Solid State Commun. 161 (2013) 42. [57] T. Chatterji, N. Jalarvo, J. Phys.: Condens. Matter 25 (2013) 156002. [58] M.K. Wilkinson, J.W. Cable, E.O. Wollan, W.C. Koehler, Phys. Rev. 113 (1959) 497. [59] R.C. Chisholm, J.W. Stout, J. Chem. Phys. 36 (1962) 972. [60] U. Köbler, A. Hoser, J. Magn. Magn. Mater. 349 (2014) 88. [61] U. Köbler, A. Hoser : Experimental Studies of Boson Fields in Solids, World Scientific Publishing Company, Singapore (2018). [62] T. Shinoda, H. Chihara, S. Seki, J. Phys. Soc. Japan 19 (1964) 1637. [63] D.E. Cox, G. Shirane, B.C. Frazer, A. Narath, J. Appl. Phys. 37 (1966) 1126. [64] F.F.Y. Wang, D.E. Cox, M. Kestigian, Phys. Rev. B 3 (1971) 3946. [65] M. Eibschütz, S. Shtrikman, D. Treves, Solid State Commun. 4 (1966) 141. [66] V. Scatturin, L. Corliss, N. Elliott, J. Hastings, Acta Cryst. 14 (1961) 19. [67] P.J. Becker, G. Dummer, H.G. Kahle, L. Klein, G. Müller-Vogt, H.C. Schopper, Phys. Lett. 31A (1970) 499. [68] J.H. Colwell, B.W. Mangum, D.D. Thornton, J. C. Wright, H.W. Moos, Phys. Rev. Lett. 23 (1969) 1245. [69] G. Will, W. Schäfer, J. Phys. C : Solid State Phys. 4 (1971) 811. [70] H. Fuess, A. Kallel, F. Tchéou, Solid State Commun. 9 (1971) 1949. [71] U. Köbler, A. Hoser, Eur. Phys. J. B 60 (2007) 151. Neff RE-Pt2-alloys. Mtheor/Mexp=No/Neff [72] C. Deenadas, A.W. Thompson, R.S. Craig, W.E. Wallace, J. Phys. Chem. Solids 32 (1971) 1853. [73] T. Inoue, S.G. Sankar, R.S. Craig, W.E. Wallace, K.A. Gschneidner, Jr., J. Phys. Chem. Solids 38 (1977) 487. [74] U. Köbler, A. Hoser, J. Magn. Magn. Mater. 299 (2006) 145. [75] G.H.J. Wantenaar, S.J. Campbell, D.H. Chaplin, T.J. McKenna, G.V.H. Wilson, Phys. Rev. B 29 (1984) 1419. [76] U. Köbler, A. Hoser, J.-U. Hoffmann, C. Thomas, Solid State Commun.137 (2006) 301. [77] A.H. Millhouse, H.-G. Purwins, E. Walker, Solid State Commun. 11 (1972) 707. [78] M. Kahn, K.A. Gschneidner, V.K. Pecharsky, J. Appl. Phys. 110 (2011) 103912.

[79] P. Martel, R.A. Cowley, R.W.H. Stevenson, Canad. J. Phys. 46 (1968) 1355. [80] R.A. Cowley, W.J.L. Buyers, P. Martel, R.W.H. Stevenson, J. Phys. C: Solid State Phys. 6 (1973) 2997. [81] W. Bührer, M. Godet, H.-G. Purwins, E. Walker, Solid State Commun. 13 (1973) 881. [82] T.M. Holden, W. J. L. Buyers, H.-G. Purwins, J. Phys. F : Met. Phys. 14 (1984) 2701. [83] J.J. Rhyne, N.C. Koon, J. Magn. Magn. Mater. 31-34 (1983) 608. [84] H.B. Stanley, B.D. Rainford, W.G. Stirling, J.S. Abell, Physica 130B (1985) 355. [85] W. Schelp, W. Drewes, H.-G. Purwins, G. Eckold, J. Magn. Magn. Mater. 50 (1958) 147. [86] T.W. Hill, W.E. Wallace, R.S. Craig, T. Inoue, J. Solid State Chem. 8 (1973) 364. [87] N. Nereson, C. Olsen, G. Arnold, J. Appl. Phys. 37 (1966) 4575. [88] A. Furrer, H.-G. Purwins, Phys. Rev. B 16 (1977) 2131. [89] N. Nereson, C. Olsen, G. Arnold, J. Appl. Phys. 39 (1968) 4605. [90] Crystal Field Effects in Metals and Alloys, ed. by A. Furrer, Plenum Press, New-York/London (1977).

Highlights: The spin dynamics in the long-range ordered state is proven to be quantized. A relevant crystal electric field removes one or more of the N=2S+1 spin states. Each reduction of the spin by ΔS=1/2 implies a change in universality class.