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Dimensionality crossover upon magnetic saturation in Fe, Ni and Co
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 1202–1208
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Dimensionality crossover upon magnetic saturation in Fe, Ni and Co U. Ko¨bler a,, A. Hoser b, C. Thomas a a b
¨ r Festko ¨rperforschung, Ju ¨ lich Forschungszentrum, D-52425 Ju ¨ lich, Germany Institut fu Hahn-Meitner-Institut Berlin, D-14109 Berlin, Germany
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 February 2008 Received in revised form 25 August 2008 Available online 21 November 2008
The magnetic saturation process of iron, nickel and cobalt single-crystal spheres is studied using neutron scattering in a vertical magnetic field. It is observed that upon magnetic saturation, the scattering intensities decrease instead of increasing. This indicates a decreasing coherent scattering with field. The spin precession around the field axis therefore can be assumed to be incoherent along directions transverse to the field. Comparison of the temperature dependence of the spontaneous magnetization measured by zero field NMR on the one hand and by the macroscopic magnetization on the other hand shows that Fe, Ni and Co are three-dimensional (3D) in the zero field ground state but one dimensional (1D) in the magnetically saturated state. The observed decrease in neutron scattering intensity is consistent with this conclusion. The change in dimensionality is associated with a crossover. Our neutron scattering study shows that the crossover occurs at a field that is smaller than the demagnetization field. The dimensionality crossover, therefore, is driven not by the field but by the associated forced magnetostriction. & 2008 Elsevier B.V. All rights reserved.
PACS: 05.10.Cc 75.10.Lp 75.50.Cc Keywords: Low-dimensional ferromagnetism Crossover phenomena Universality
1. Introduction It was noticed long ago that the temperature dependence of the macroscopic spontaneous magnetization and the temperature dependence of the 57Fe hyperfine field of bcc iron are not identical [1,2]. This effect was attributed to a temperature-dependent hyperfine coupling constant [2]. In other words, the macroscopic magnetization was considered to reflect the true temperature dependence of the order parameter. As we will show this conclusion is not correct. The two methods give different results because they pertain to different symmetries. Quite generally it has to be considered that upon magnetic saturation the symmetry changes from 3D to axial. In the zero field 57Fe NMR, Mo¨ssbauer effect or neutron scattering measurements, the natural domain structure is left unchanged and the system is evidently 3D isotropic. Typical for the 3D state of cubic materials is an isotropic distribution of magnetic domains with an equal domain population along all three space directions. This can be assumed to hold perfectly for bcc iron with very small anisotropy and no remanent magnetization. On the other hand, in macroscopic measurements of the spontaneous magnetization the sample is magnetically saturated, i.e. the domain structure has changed from isotropic multi-domain state to axially symmetric single-domain state. In the single-domain state all spins are
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E-mail address: [email protected] (U. Ko¨bler). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.11.008
oriented parallel to the external field. It can be assumed that due to magnetostriction the lattice symmetry also is no longer cubic but axially distorted. Those distortions usually are weak and difficult to observe experimentally [3]. One must be aware that the field-induced change of symmetry can change the magnetic properties. Speaking in terms of the renormalization group (RG) theory the question is whether the weak symmetry change is relevant for the dynamics or not [4]. Fortunately, in practically all insulating ferromagnets the fieldinduced symmetry change is not relevant. This means that there is no discrepancy between the temperature dependence of hyperfine field and macroscopic spontaneous magnetization. A relevant field-induced symmetry change seems to be particular to the itinerant ferromagnets Fe, Ni and Co. Temperature dependence of the macroscopic magnetization shows that these ferromagnets become 1D in the magnetically saturated state. The dimensionality crossover from 3D to 1D is not noticed in the field parallel magnetization measurements. As is well known, the magnetization of Fe, Ni and Co shows normal saturation behaviour with no remanent magnetization [5]. Using neutron scattering it becomes apparent that there is only weak or no coherent scattering. This can be explained assuming that the spins along lines perpendicular to the field axis precess largely independently of each other around the field axis. This seems to be typical for 1D bulk ferromagnets. For a generic 1D ferromagnet such as NdAl2 no further fieldinduced symmetry reduction is possible. We have observed that the magnetic neutron scattering intensities of NdAl2 disappear
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completely upon magnetic saturation in a field [6]. Interestingly, NdAl2 is cubic above the Curie temperature. One-dimensional (1D) behaviour results due to a relevant axial lattice distortion. A well-investigated generic 1D antiferromagnet is MnF2. In pioneering neutron scattering studies it has been shown that the correlation length transverse to the tetragonal c-axis is finite at the Ne´el transition and stays constant down to T ¼ 0 [7]. Classification of the dynamic dimensionality of ordered magnets is now conveniently possible on account of empirically known universality classes [8,9]. As we know from the RG theory, universality is observed in the vicinity of stable-fixed points (SFPs) and is represented by power functions of the distance from the stable-fixed point SFP. For the SFP T ¼ 0 these are power functions of absolute temperature. Universal temperature dependence of the magnetic order parameter means that the deviations with respect to saturation at T ¼ 0 are given over a large temperature range by a power function of absolute temperature with exponent that depends on the dimensionality of the relevant interactions only. Observation of exact power functions is demonstrated by the zero field NMR data of bcc iron and fcc nickel shown in Fig. 1. Due to the cubic symmetry, the interactions can be assumed to be isotropic in the zero field ground state. Additionally, the universal exponent depends on whether the spin quantum number is integer or half integer [9]. This is equivalent to an odd or even number of relevant states per magnetic atom. T2 function in a cubic material means half-integer spin. Considering the different saturation moments of bcc Fe (2.217 mB/Fe) and fcc Ni (0.617 mB/Ni) it is suggestive to attribute an effective spin of Seff ¼ 32 to iron but Seff ¼ 12 to Ni [10]. As we will show below the effective spin of iron can change with magnetic field and with temperature as well. The observed universality cannot be explained on the basis of the material-specific magnon dispersions. This does mean that the magnons are not the relevant excitations of the dynamics. In order to understand that the magnons are not relevant, three important points have to be considered: First, as is well known, the correlation length increases on approaching the critical point from the paramagnetic side and
T (K) 100 200
300
400
500
550
1.00
νres (T) / νres (0)
0.98 57Fe
NMR bcc iron
M.A. Butler et al.(1972)
0.96
0.94 61Ni
NMR fcc nickel
J. Englich (2002)
0.92 0
50000 100000 150000 200000 250000 300000 T2 (K2)
Fig. 1. Normalized zero field 57Fe NMR frequencies of bcc iron after Ref. [1] and normalized zero field 61Ni NMR frequencies of fcc nickel after Ref. [16] as a function of absolute temperature squared. T2 universality class pertains to isotropic magnets with half-integer spin. It can be assumed that the effective spin of iron is Seff ¼ 32 (in the zero field state) but Seff ¼ 12 for nickel. This roughly conforms to the different saturation moments of 2.217 mB/Fe and 0.617 mB/Ni.
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finally becomes infinitely large at the critical point. It is evident that for an infinitely large correlation length all microscopic details such as the discrete atomistic lattice structure as well as the spin structure are no longer important for the dynamics. As a consequence continuum theories such as field theoretical approaches rather than atomistic models now are adequate concepts [11]. Continuous dynamic symmetry is the reason for the universality of the critical exponents. Universality means independence of atomistic details such as the spin structure. Second, continuous symmetry applies not only asymptotically for T-Tc but sets in at a well-defined crossover temperature above the critical point [12]. This is approximately at the change from Curie–Weiss susceptibility to critical susceptibility. In particular, continuous dynamic symmetry holds for all lower temperatures down to T ¼ 0. In other words, from a dynamical point of view the ordered magnet has to be treated as a continuum. A general dynamic continuum theory of magnetism is, however, not available up to now. Only for the critical range field theoretical approaches have successfully been applied in the calculation of critical exponents [11]. Third, the change from atomistic to continuous dynamic symmetry is a fundamental process and has important conceptual consequences. In a continuum there are no spins and no interactions between spins. As a consequence instead of atomistic interactions such as magnons, a new type of excitation that is particular to the continuum must become relevant for the dynamics. The excitations of the continuous magnet we call Goldstone–Salam–Weinberg (GSW) excitations [13]. The GSW quasi-particles are massless and non-interacting bosons. They are generated through the spontaneous symmetry break at the phase transition [13]. In superconductors spin-zero bosons are well known as Cooper pairs. Quite analogous, we can assume that the GSW bosons of the magnetic materials are spincompensated quasi-particles. As a result, the GSW bosons have no magnetic moment. It is evident that non-magnetic and massless bosons cannot be observed using inelastic neutron scattering. Because the GSW bosons are massless and do not interact, they have linear dispersion in all 3D magnets and can be described by plane waves. Using the wave picture it is clear that magnetic plane waves carry no magnetic moment. Note that these waves are not spin waves. Spin waves never have linear dispersion exactly. Moreover, unlike the magnon excitation spectra that can have a gap the excitations of the continuous magnet are continuous, i.e. gapless. Linear dispersion in all solids is well known from the Debye bosons that are responsible for the universal T3 function of the low-temperature heat capacity of the diamagnetic solids [14]. Universality results from the analytically identical dispersion and density of states in all solids. The GSW bosons also control the temperature dependence of the magnons [15]. From the fact that the GSW bosons are relevant, we can conclude that they have lower dispersion energy and/or higher density of states compared to the magnons. The functionality of the atomistic near neighbour exchange interactions is essentially to define non-universal, i.e. materialspecific properties such as spin structure and ordering temperature. The saturation magnetic moment also is material specific and is determined by intra-atomic interactions. In other words, the two ending points of the spontaneous magnetization curve are given by atomistic interactions. The temperature function(s) between these ending points are variable and are defined by the GSW bosons. As we will see very weak external perturbations are sufficient to change the detailed temperature dependence between the SFPs T ¼ Tc and T ¼ 0. In particular, the GSW bosons also define the critical behaviour and the dimensionality.
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2. Experimental results Fcc nickel is a good candidate to start with because in nickel the effective spin is a constant ðSeff ¼ 12Þ. Fig. 2 nicely visualizes the problem. The normalized hyperfine field values measured by 61Ni NMR vs. absolute temperature to a power of 52 are shown [16]. On T5/2 scale, the T2 dependence seen in Fig. 1 appears curved. Additionally, the normalized spontaneous magnetization after Ref. [17] is depicted. It can be seen that the spontaneous magnetization follows T5/2 function quite well. The fitted exponent is e ¼ 2.4170.03. The zero field NMR data are excellently described by T2 function (fitted exponent e ¼ 1.9770.029). We have identified T5/2 universality class as characteristic of 1D magnets with half-integer spin [9]. As a conclusion upon magnetic saturation a dimensionality crossover from isotropic 3D (T2 function) to 1D (T5/2 function) is induced. This crossover must be at a field value that is smaller than the demagnetizing field. Normalization of the two experimental data sets to unity for T-0 supposes that the saturation magnetic moment is left unchanged by application of the demagnetization field (0.22 T for the Ni sphere). This assumption seems to be reasonable but is, of course, not strictly approved. This is because the hyperfine coupling constant is not absolutely known for the two experimental conditions. Independent of this difficulty it can be seen in Fig. 2 that the discrepancy between the zero field NMR results and the spontaneous magnetization is largest at intermediate temperatures and that the two fit curves approach each other towards the ordering temperature. This can be expected since the transition temperature should be identical in the two measuring processes. In other words, the absolute difference between the two experimental results is only small but quantitative analyses of the two temperature functions shows that this small effect has dramatic consequence on the universal exponent and therefore on the dimensionality. This shows that the universality classes are meta-stable and can depend sensitively on weak external parameters such as axial magnetostriction forced by an external
T (K) 100
200 250
300
350
400 59
nickel TC = 630.2K
nickel
spontaneous magnetization ~T5/2 J. Crangle, G.M. Goodman (1971)
0.95
58 m (emu/g)
M (T) / M (0)
1.00
field [18]. This sensitivity is not to be expected if the magnons would define the temperature dependence simply because the applied field is negligibly small compared to the internal Weiss molecular field. This further demonstrates that the universality classes cannot be defined by the magnons. For completeness, we should mention that T5/2 function of the spontaneous magnetization does not hold perfectly down to T-0. Although the deviations from T5/2 function are smaller than 1%, they can be interpreted as dimensionality crossover to T3/2 function at TDC 120 K. Fig. 3 shows the low-temperature section of the spontaneous magnetization after Ref. [17] on linear temperature scale. Below 120 K these data can excellently be fitted by T3/2 function. The fitted exponent is e ¼ 1.5070.02. This is a very unusual result because T3/2 means 3D anisotropic universality class but T5/2 is the 1D universality class. In principle, the symmetry of T3/2 class is higher than the 1D universality class observed at higher temperature (T5/2 function). Increase of symmetry with decreasing temperature normally should not occur. We shall discuss this phenomenon together with a similar observation on iron (see Fig. 5). For bcc iron the situation is more complicated than for nickel. Iron is a so-called weak ferromagnet [19]. This means that due do particularities in the band structure the spontaneous magnetization shows tendency to instability and can be fundamentally manipulated by weak external perturbations. Fig. 4 shows normalized spontaneous magnetization data of bcc iron after Ref. [17] as a function of T3 together with normalized zero field 57Fe NMR data after Ref. [1]. It can be seen that for T4350 K the spontaneous magnetization is excellently described by T3 function (fitted exponent: e ¼ 2.9770.03). This is a very surprising result because T3 universality class pertains to 1D magnets with integer spin [9]. We therefore have to conclude that not only a dimensionality crossover but additionally a crossover in spin quantum number from Seff ¼ 32 to Seff ¼ 2 is induced upon magnetic saturation. This seems to be a signature of the magnetic instability. In Fig. 4, the spontaneous magnetization data of Ref. [17] with To350 are omitted. As for nickel crossover to a different asymptotic power function for T-0 can be identified. Data below 280 K are well described by T2 function (see Fig. 5). T2 dependence was observed already in 1929 by Weiss and Forrer
0.90
~T3/2
TDC = 120K
57
~T5/2
zero field 61Ni NMR ~T2 J. Englich et al. (2000)
56
0.85 0
1000000
2000000 T5/2 (K5/2)
3000000
4000000
Fig. 2. Normalized zero field 61Ni NMR frequencies after Ref. [16] and normalized spontaneous magnetization of fcc nickel after Ref. [17] as a function of T5/2. Zero field data are excellently described by T2 function [9], while for the magnetically saturated state T5/2 function holds. T2 universality class pertains to isotropic 3D magnets with half-integer spin, T5/2 universality class to 1D magnets with halfinteger spin [9]. Change of exponent upon magnetic saturation means that dimensionality crossover from 3D to 1D is induced.
J. Crangle, G.M. Goodman, Proc. Roy. Soc. London A 321 (1971) 477. 55 0
50
100
150 T (K)
200
250
300
Fig. 3. Enlarged low-temperature region of macroscopic spontaneous magnetization of nickel after Ref. [17] as a function of temperature. Crossover from low temperature T3/2 to high temperature T5/2 function at dimensionality crossover TDC120 K can be identified. This is unusual because T3/2 universality class pertains to higher symmetry (3D anisotropic) than T5/2 universality class (1D).
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T (K) 300 400
500
600
650
700
750
1.00 bcc iron
0.98
M (T) / M (0)
0.96
spontaneous magnetization ~T3 J. Crangle, G.M. Goodman (1971)
0.94 0.92 0.90
zero field 57Fe NMR~T2 M.A. Butler et al. (1972)
0.88 0.86 0.84 0
1x108
2x108 T3
3x108
4x108
5x108
(K3)
Fig. 4. Normalized zero field 57Fe NMR frequencies of bcc iron after Ref. [1] and normalized spontaneous magnetization after Ref. [17] as a function of T3. T2 universality class pertains to isotropic magnets with half-integer spin while T3 universality class pertains to 1D magnets with integer spin. Upon magnetic saturation crossover not only in dimensionality but also in effective spin is induced. Data of Ref. [17] for To350 K are not shown. These data are analyzed in Fig. 5.
T (K) 100
150
200
250
300
M (T) / M (T = 0)
1.000 J. Crangle, G.M. Goodman, Proc. Roy. Soc. London, A 321 (1971)477.
0.995 57FeNMR
0.990
bcc iron P. Weiss, R. Forrer Ann. Physique 12 (1929)20. M.A. Butler, G.K. Wertheim, D.N.E. Buchanan Phys. Rev. B5 (1972)990.
0.985
0.980 0
10
20
30
40 50 60 T2 / 1000 (K2)
70
80
90
Fig. 5. Normalized low-temperature spontaneous magnetization data of bcc iron from the three different literature sources [1,17,20]. The three data sets agree very well and confirm asymptotic T2 function for T-0. For T4T315 K crossover to T3 function follows. T2 fit function to the zero field 57Fe NMR data shown in Fig. 1 is indicated by dashed line.
[20] who measured the spontaneous magnetization of iron for the first time with high precision. In other words, there is a wellresolved crossover from low temperature T2 function to high temperature T3 function at crossover-temperature T315 K in the spontaneous magnetization of iron. For the T2 function the spontaneous magnetization data are larger with respect to the T3 function. Note that, nevertheless, we have normalized the T3 function in Fig. 4 to unity for T-0. In order to visualize the low temperature T2 dependence of the macroscopic spontaneous magnetization Fig. 5 compiles data from the three different literature sources [1,17,20]. It can be seen
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that all data sets agree very nicely and that T2 function is confirmed within the typical precision of macroscopic magnetization measurements. For comparison T2 function fitted to the zero field 57Fe NMR data (see Fig. 1) is indicated by dashed line [1]. We should note that the low-temperature T2 function is not confirmed by the data of Refs. [5,21]. The spontaneous magnetization data of R. Pauthenet and of B.E. Argyle et al. deviate towards larger values for T-0 with respect to T2 function. Since these deviations can reasonably be described by adding a T1 term to the T2 function and since they are not observed by the other authors we attribute the excess magnetization for T-0 to paramagnetic impurities. It is evident that in the presence of an additive T1 term fit of the spontaneous magnetization data by a single power function results in a smaller exponent than e ¼ 2. In Refs. [5,21], it was claimed that the spontaneous magnetization data are in agreement with Bloch’s T3/2 function calculated by the atomistic spin wave theory [22,23]. This is not in agreement with the conclusion to be drawn from Fig. 5. Interpretation of the asymptotic T2 function in the spontaneous magnetization of iron is not unique because neither the relevant symmetry nor the spin quantum number is clear. The T2 function pertains to 3D isotropic symmetry and half-integer spin or to 3D anisotropic symmetry and integer spin [9]. Since it appears unlikely that spin and symmetry change at one crossover event as a function of temperature the crossover from T3 to T2 function is, perhaps, as for nickel, from 1D to 3D anisotropic but in contrast to nickel for integer spin. This would mean that in the spontaneous magnetization the effective spin is Seff ¼ 2 for all temperatures and the observed saturation moment of 2.217 mB/Fe applies to Seff ¼ 2. As for nickel asymptotic T2 function means a higher symmetry than T3 function observed at higher temperatures. This could be a consequence of the complicated interplay of magnetostriction, i.e. lattice distortion and dimensionality of the relevant excitations. The spin quantum number of iron is not only meta-stable with respect to an applied magnetic field but also with respect to temperature in the zero field state. In the temperature dependence of zero field 57Fe NMR and Mo¨ssbauer data crossover from low temperature T2 to high temperature T9/2 function at crossover temperature T615 K is identified [24]. Since in the zero field state iron is cubic for all temperatures, this crossover can safely be interpreted as change from Seff ¼ 32 to Seff ¼ 2, i.e. as change from 4 to 5 relevant states per Fe atom. To summarize, in the macroscopic spontaneous magnetization of iron crossover from low temperature T2 to high temperature T3 function is observed while in the zero field measurements the crossover is from T2 to T9/2. The crossover temperatures are 315 and 615 K, respectively. We have performed neutron scattering measurements on single-crystalline spheres of Fe, Ni and Co with diameter between 10 and 12 mm in a vertical field in order to observe some anomaly that can be identified as crossover from 3D to 1D state upon magnetic saturation. The neutron wavelength was l ¼ 2.44 A˚. Fig. 6 shows normalized results of two independent measurements of the field dependence of (11 0) scattering intensity for the Fe sphere. In agreement with Co and Ni the expected intensity increase with increasing field is not observed. For horizontal scattering vector 2/3 of all moments should be detected in the zero field state with isotropic domain distribution. Isotropic domain configuration can be assumed because there is no remanent magnetization. On the other hand all moments should be detected for magnetic saturation with all moments aligned parallel to the field, i.e. perpendicular to the scattering vector. As a consequence a considerable intensity increase can be expected. Note that Fig. 6 displays normalized total scattering intensities, i.e. nuclear and magnetic contributions.
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BDC = 0.65 T
I110 (B0) / I110 (B0 = 0)
1.00
0.98
0.96
BD = 0.71 T
0.94
0.92
0.0
bcc iron sphere ( 11 0 ) 0.2
0.4
0.6
0.8
B0 (T) Fig. 6. Normalized (11 0) neutron-scattering intensities of a spherical bcc iron single crystal as a function of a vertical magnetic field. The shown intensities contain nuclear and magnetic contributions. The two data sets are for independent measurements on different instruments (E6 and V2 of HMI). In spite of quantitative differences the two results reveal consistently a sudden intensity drop at BDC0.65 T that we identify as 3D to 1D dimensionality crossover (DC) associated with spin quantum number crossover from Seff ¼ 32 to Seff ¼ 2 (see text). The smooth intensity decrease conforms to the observations made in Ref. [26] and further supports symmetry change to 1D.
In all neutron scattering experiments on larger single crystals we have to consider that the absolute value of the scattered intensities can be reduced due to extinction, i.e. due to repeated scattering processes of the neutron on its way across the sample. Our data in Fig. 6 are not corrected for those effects. In Ref. [25] extinction effects have been systematically studied in zero field investigations of iron. Extinction can become increasingly important if the size of the domains grows upon magnetic saturation whereby the magnetic mosaic structure, so to say, decreases. This might reduce the scattering intensities with applied field. We, therefore, have performed the same scattering experiment as for the sphere but on powder material in order to obtain information on the importance of extinction. For powder material extinction should be negligible. Also for the Fe powder sample scattering intensities slightly decrease as a function of field. This is in agreement with a detailed investigation of structurally imperfect iron samples with correspondingly reduced extinction [26]. In the experiment of Ref. [26], ideally imperfect iron samples with increased mosaic spread were fabricated through mechanical treatment such as rolling in order to diminish extinction. At the expected field-induced crossover, a rather sharp anomaly as is in fact seen in Fig. 6 should occur. A sharp anomaly will be seen only on a spherical sample with a homogeneous internal field and domain distribution. On the other hand, extinction will not produce sharp anomalies in the field dependence of scattering intensities. The fact that the scattering intensities do not increase as a function of field, we consider as consistent with the conclusion drawn from the spontaneous magnetization that iron becomes 1D in the magnetically saturated state. One-dimensional symmetry means a strongly reduced transverse magnetic coherence length. This reduces the coherent magnetic-scattering intensities. For our purpose, the sudden drop in intensity at an external field of BDC 0.65 T is the most important feature of Fig. 6. This anomaly is observed in two independent experiments on the same iron sphere (open and filled symbols in Fig. 6) though with somewhat different absolute values.
For a very isotropic material like iron, it is very unlikely that the sudden change in scattering intensity at BDC0.65 T is due to a sudden change in domain structure [27]. For instance, the macroscopic magnetization of a spherical iron sample shows perfect behaviour of an ideal isotropic ferromagnet: linear field dependence up to demagnetization field and absolutely no remanence. As a consequence, the sudden intensity drop in Fig. 6 can reasonably be identified as combined crossover in dimensionality and in spin quantum number. We should note that the iron sphere is doped with 4% silicon for bcc phase stabilization. The demagnetization field for this sphere is BD0.71 T only but larger than BDC0.65 T. Nominally, the internal field is zero for fields smaller than the demagnetization field. This shows that the crossover is not driven by field but by the small axial lattice distortion due to forced magnetostriction [18]. Also in hcp cobalt crossover to 1D symmetry is induced upon magnetic saturation. Consistent with the hexagonal lattice structure below T0 ¼ 703 K [28], the symmetry class in the zero field ground state is 3D anisotropic. For half-integer spin this is T3/2 [9,29]. Upon magnetic saturation crossover to the same 1D universality class as for nickel represented by T5/2 is induced (see Fig. 2). T5/2 is excellently confirmed by the magnetization data of Ref. [30]. This universality class belongs to half-integer spin. In agreement with nickel but in contrast to iron the spin quantum number therefore seems not to be changed at the dimensionality crossover. We should note that asymptotically for T-0 T5/2 universality class can be identified in the zero field 59Co NMR data [31]. Since T5/2 function applies to deviations from saturation of less than two parts in thousand these details are not resolved by Fig. 7. Neutron scattering investigations on single-crystalline Co sphere exhibit definite anomaly at BDC ¼ 0.437 T that can be identified as dimensionality crossover. The total scattering intensities, i.e. nuclear and magnetic intensity shown by Fig. 8 are for field and hexagonal c-axis vertical, i.e. perpendicular to scattering plane. Again, decreasing scattering intensities as a function of field are not consistent with alignment of all moments in the vertical field and point to reduced coherent scattering in the
T (K) 200 300 M (T) / M (T = 0); νres (T) / νres (T = 0)
1206
400
500
600
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1.00
hcp cobalt
0.99
spontaneous magnetization H.P. Myers, W. Sucksmith (1951)
0.98
~T5/2
0.97 T0 = 703 K 0.96 hcp fcc
0.95 ~T3/2 0.94
zero field 59Co NMR M. Kawakami, H. Enokiya (1986)
0.93 0
2
4
6
8
10
12
14
T5/2/106 (K5/2) Fig. 7. Normalized zero field 59Co NMR frequencies of hcp cobalt after Ref. [29] and normalized spontaneous magnetization after Ref. [30] as a function of T5/2. The different temperature functions are attributed to a dimensionality crossover upon magnetic saturation from 3D anisotropic universality class (T3/2 function) to 1D universality class (T5/2). Open circles are magnetization data after Ref. [5].
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2.4 hcp cobalt
intensity / 106 (a.u.)
2.3 ~Bo2
2.2
BD
BDC = 0.437 T
2.1
2.0
1.9 0.0
0.1
0.2
0.3
0.4 Bo (T)
0.5
0.6
0.7
0.8
Fig. 8. Absolute neutron-scattering intensity of the (1 0 0) Bragg reflection measured on a spherical cobalt single crystal as a function of field. Hexagonal caxis and magnetic field are vertical, i.e. perpendicular to scattering plane. For this geometry scattering intensity should have reached maximum at the demagnetization field of BD0.64 T. Instead, scattering intensities decrease indicating decreasing coherent scattering. Anomaly at BDC ¼ 0.437 T is interpreted as dimensionality crossover (DC) from 3D to 1D. For 1D symmetry scattering intensities are considerably reduced but since demagnetization field BD is clearly visible the observed scattering intensities must contain finite magnetic contributions.
1D state. Scattering intensities might additionally decrease due to extinction. Change of domain configuration as well as the associated change of extinction can be expected to be a monotonous and steady function of temperature. Fig. 8 shows sudden change from quadratic to linear field dependence at BDC ¼ 0.437 T. This rather sharp event can safely be interpreted as dimensionality crossover. The field value of BDC ¼ 0.437 T is distinctly lower than the demagnetization field of BD ¼ 0.64 T and shows that the crossover is at nominally zero internal field. The crossover therefore must be driven by the size of the domains and the associated magnetostrictive lattice deformations. Note that the fact that the demagnetization field is visible in Fig. 6 indicates that there are still magnetic contributions in the total scattering intensity. This is not so clear for iron (see Fig. 6). For the nickel sphere practically no intensity change with field is observed. This means that there is also no visible intensity anomaly at the demagnetization field.
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macroscopic magnetization concern the detailed temperature dependence between these two stable ending points. By the same argument the field-induced magnetic changes are much too strong to be explained by the atomistic magnons. As a conclusion the magnons are not the relevant excitations for the dynamics [24]. This is confirmed by the observed universal power functions in the zero field methods as well as in the macroscopic magnetization. Universality cannot be explained by the material-specific magnons. Note that the magnon dispersions are nearly linear in antiferromagnets but nearly quadratic in ferromagnets. This does not fit the spin structure independent universal power functions for T-0 and for T-Tc. Instead the observed power functions are defined by the excitations of the magnetic continuum, the Goldstone–Salam–Weinberg bosons [13]. These bosons have linear dispersion and analytically simple densities of states in all 3D solids and define the universal temperature functions at the SFPs T ¼ 0 and T ¼ Tc. As a consequence, they also define the critical behaviour and the dimensionality. GSW bosons and dimensionality are meta-stable in the itinerant ferromagnets iron, nickel and cobalt. We have shown that upon magnetic saturation crossover to 1D universality class occurs. One-dimensional symmetry is verified on account of empirically known universal exponents [9]. The presented field-dependent neutron scattering intensities are consistent with 1D symmetry established by the spontaneous magnetization [26]. Decreasing neutron-scattering intensities with increasing vertical field can be explained by reduced phase coherence in the spin precession for spins arranged transverse to the field axis. This seems to be typical for 1D ferromagnets [6]. In the spontaneous magnetization, the time-averaged constant expectation value along z-axis is measured. On the other hand, the observation time of thermal neutrons is of the order of 1013 s only. This is short compared to the Larmor precession of 108 s. Neutrons therefore do not measure the time-averaged precession and see disordered spins. In insulating ferromagnets commonly no dimensionality crossover occurs upon magnetic saturation. This means these ferromagnets remain 3D also in the magnetically saturated state and the same temperature dependence of the order parameter is obtained in microscopic zero field measurements as well as in the macroscopic spontaneous magnetization. We can assume that in the 3D state the transverse interactions give rise to phase coherence in the spin precession such that the full coherent neutron-scattering intensities are observed. In 1D magnets there are only weak or no transverse interactions.
3. Conclusions Acknowledgments We have shown that the difference between the temperature dependence of the spontaneous magnetization measured by zero field 57Fe, 59Co and 61Ni NMR and the temperature dependence of the macroscopic spontaneous magnetization has a maximum of a few percent in the itinerant ferromagnets iron, cobalt and nickel. The essential difference in the two experimental methods is that due to application of the demagnetization field the macroscopic spontaneous magnetization refers to the axially symmetric singledomain state, but the zero field methods to the isotropic multidomain state. We can assume that a small axial lattice distortion, i.e. a symmetry change is associated with magnetic saturation. An applied magnetic field of the order of the demagnetization field is extremely weak compared to the Weiss molecular field. As a consequence no dramatic changes either in the absolute value of ordering temperature or in the saturation magnetic moment will be induced upon magnetic saturation. The observed different temperature functions in the zero field methods and in the
We thank N. Stu¨sser and K. Habicht of HMI/Berlin for the excellent support received during the neutron scattering experiments. Thanks are also to R. Zeller (IFF, Ju¨lich Forschungszentrum) for enlightening discussions concerning the band structure particularities of the itinerant ferromagnets. References [1] [2] [3] [4] [5] [6]
M.A. Butler, G.K. Wertheim, D.N.E. Buchanan, Phys. Rev. B 5 (1972) 990. G.B. Benedek, J. Armstrong, J. Appl. Phys. Suppl. 32 (1961) 106S. B.E. Argyle, N. Miyata, T.D. Schultz, Phys. Rev. 160 (1967) 413. K.G. Wilson, J. Kogut, Phys. Rep. 12C (1974) 75. R. Pauthenet, J. Appl. Phys. 53 (1982) 8187. U. Ko¨bler, A. Hoser, J.-U. Hoffmann, C. Thomas, Solid State Commun. 137 (2006) 301. [7] M.P. Schulhof, R. Nathans, P. Heller, A. Linz, Phys. Rev. B 4 (1971) 2254. [8] U. Ko¨bler, A. Hoser, D. Hupfeld, Physica B 328 (2003) 276. [9] U. Ko¨bler, A. Hoser, Physica B 362 (2005) 295.
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¨bler et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 1202–1208 U. Ko
[10] U. Ko¨bler, J. Englich, O. Hupe, J. Hesse, Physica B 339 (2003) 156. [11] J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. B 21 (1980) 3976. [12] U. Ko¨bler, A. Hoser, R.M. Mueller, K. Fischer, J. Magn. Magn. Mater. 315 (2007) 12. [13] J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127 (1962) 965. [14] G.A. Alers, in: W.P. Mason (Ed.), Physical Acoustics, Vol. III-B, Academic Press, New York, London, 1965, p. 1. [15] U. Ko¨bler, A. Hoser, W. Scha¨fer, Physica B 364 (2005) 55. [16] J. Englich, Charles University Prague, private communication. [17] J. Crangle, G.M. Goodman, Proc. R. Soc. London A 321 (1971) 477. [18] B.E. Argyle, N. Miyata, Phys. Rev. 171 (1968) 555. [19] A.P. Malozemoff, A.R. Williams, V.L. Moruzzi, Phys. Rev. B 29 (1984) 1620. [20] P. Weiss, R. Forrer, Ann. Phys. 12 (1929) 20.
[21] B.E. Argyle, S.H. Charap, E.W. Pugh, Phys. Rev. 132 (1963) 2051. [22] F. Bloch, Z. Phys. 61 (1930) 206. [23] F. Keffer, in: S. Flu¨gge, H.P.J. Wijn (Eds.), Handbuch der Physik, vol. 18/2, Springer, Berlin, 1966, p. 1. [24] U. Ko¨bler, A. Hoser, Eur. Phys. J. B 60 (2007) 151. [25] R.M. Moon, C.G. Shull, Acta Crystallogr. 17 (1964) 805. [26] A.S. Arrott, T.L. Templeton, J. Appl. Phys. 57 (1985) 3763. [27] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, Heidelberg, 2000. [28] M. Braun, R. Kohlhaas, O. Vollmer, Z. Angew. Phys. 25 (1968) 365. [29] M. Kawakami, H. Enokiya, J. Phys. Soc. Japan 55 (1986) 4038. [30] H.P. Myers, W. Sucksmith, Proc. R. Soc. London A 207 (1951) 427. [31] U. Ko¨bler, A. Hoser, J. Englich, A. Snezhko, M. Kawakami, M. Beyss, K. Fischer, J. Phys. Soc. Japan 70 (2001) 3089.
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Dimensionality crossover upon magnetic saturation in Fe, Ni and Co U. Ko¨bler a,, A. Hoser b, C. Thomas a a b
¨ r Festko ¨rperforschung, Ju ¨ lich Forschungszentrum, D-52425 Ju ¨ lich, Germany Institut fu Hahn-Meitner-Institut Berlin, D-14109 Berlin, Germany
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 February 2008 Received in revised form 25 August 2008 Available online 21 November 2008
The magnetic saturation process of iron, nickel and cobalt single-crystal spheres is studied using neutron scattering in a vertical magnetic field. It is observed that upon magnetic saturation, the scattering intensities decrease instead of increasing. This indicates a decreasing coherent scattering with field. The spin precession around the field axis therefore can be assumed to be incoherent along directions transverse to the field. Comparison of the temperature dependence of the spontaneous magnetization measured by zero field NMR on the one hand and by the macroscopic magnetization on the other hand shows that Fe, Ni and Co are three-dimensional (3D) in the zero field ground state but one dimensional (1D) in the magnetically saturated state. The observed decrease in neutron scattering intensity is consistent with this conclusion. The change in dimensionality is associated with a crossover. Our neutron scattering study shows that the crossover occurs at a field that is smaller than the demagnetization field. The dimensionality crossover, therefore, is driven not by the field but by the associated forced magnetostriction. & 2008 Elsevier B.V. All rights reserved.
PACS: 05.10.Cc 75.10.Lp 75.50.Cc Keywords: Low-dimensional ferromagnetism Crossover phenomena Universality
1. Introduction It was noticed long ago that the temperature dependence of the macroscopic spontaneous magnetization and the temperature dependence of the 57Fe hyperfine field of bcc iron are not identical [1,2]. This effect was attributed to a temperature-dependent hyperfine coupling constant [2]. In other words, the macroscopic magnetization was considered to reflect the true temperature dependence of the order parameter. As we will show this conclusion is not correct. The two methods give different results because they pertain to different symmetries. Quite generally it has to be considered that upon magnetic saturation the symmetry changes from 3D to axial. In the zero field 57Fe NMR, Mo¨ssbauer effect or neutron scattering measurements, the natural domain structure is left unchanged and the system is evidently 3D isotropic. Typical for the 3D state of cubic materials is an isotropic distribution of magnetic domains with an equal domain population along all three space directions. This can be assumed to hold perfectly for bcc iron with very small anisotropy and no remanent magnetization. On the other hand, in macroscopic measurements of the spontaneous magnetization the sample is magnetically saturated, i.e. the domain structure has changed from isotropic multi-domain state to axially symmetric single-domain state. In the single-domain state all spins are
Corresponding author. Tel.: +49 2461 61 3324; fax: +49 2461 61 2610.
E-mail address: [email protected] (U. Ko¨bler). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.11.008
oriented parallel to the external field. It can be assumed that due to magnetostriction the lattice symmetry also is no longer cubic but axially distorted. Those distortions usually are weak and difficult to observe experimentally [3]. One must be aware that the field-induced change of symmetry can change the magnetic properties. Speaking in terms of the renormalization group (RG) theory the question is whether the weak symmetry change is relevant for the dynamics or not [4]. Fortunately, in practically all insulating ferromagnets the fieldinduced symmetry change is not relevant. This means that there is no discrepancy between the temperature dependence of hyperfine field and macroscopic spontaneous magnetization. A relevant field-induced symmetry change seems to be particular to the itinerant ferromagnets Fe, Ni and Co. Temperature dependence of the macroscopic magnetization shows that these ferromagnets become 1D in the magnetically saturated state. The dimensionality crossover from 3D to 1D is not noticed in the field parallel magnetization measurements. As is well known, the magnetization of Fe, Ni and Co shows normal saturation behaviour with no remanent magnetization [5]. Using neutron scattering it becomes apparent that there is only weak or no coherent scattering. This can be explained assuming that the spins along lines perpendicular to the field axis precess largely independently of each other around the field axis. This seems to be typical for 1D bulk ferromagnets. For a generic 1D ferromagnet such as NdAl2 no further fieldinduced symmetry reduction is possible. We have observed that the magnetic neutron scattering intensities of NdAl2 disappear
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completely upon magnetic saturation in a field [6]. Interestingly, NdAl2 is cubic above the Curie temperature. One-dimensional (1D) behaviour results due to a relevant axial lattice distortion. A well-investigated generic 1D antiferromagnet is MnF2. In pioneering neutron scattering studies it has been shown that the correlation length transverse to the tetragonal c-axis is finite at the Ne´el transition and stays constant down to T ¼ 0 [7]. Classification of the dynamic dimensionality of ordered magnets is now conveniently possible on account of empirically known universality classes [8,9]. As we know from the RG theory, universality is observed in the vicinity of stable-fixed points (SFPs) and is represented by power functions of the distance from the stable-fixed point SFP. For the SFP T ¼ 0 these are power functions of absolute temperature. Universal temperature dependence of the magnetic order parameter means that the deviations with respect to saturation at T ¼ 0 are given over a large temperature range by a power function of absolute temperature with exponent that depends on the dimensionality of the relevant interactions only. Observation of exact power functions is demonstrated by the zero field NMR data of bcc iron and fcc nickel shown in Fig. 1. Due to the cubic symmetry, the interactions can be assumed to be isotropic in the zero field ground state. Additionally, the universal exponent depends on whether the spin quantum number is integer or half integer [9]. This is equivalent to an odd or even number of relevant states per magnetic atom. T2 function in a cubic material means half-integer spin. Considering the different saturation moments of bcc Fe (2.217 mB/Fe) and fcc Ni (0.617 mB/Ni) it is suggestive to attribute an effective spin of Seff ¼ 32 to iron but Seff ¼ 12 to Ni [10]. As we will show below the effective spin of iron can change with magnetic field and with temperature as well. The observed universality cannot be explained on the basis of the material-specific magnon dispersions. This does mean that the magnons are not the relevant excitations of the dynamics. In order to understand that the magnons are not relevant, three important points have to be considered: First, as is well known, the correlation length increases on approaching the critical point from the paramagnetic side and
T (K) 100 200
300
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νres (T) / νres (0)
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M.A. Butler et al.(1972)
0.96
0.94 61Ni
NMR fcc nickel
J. Englich (2002)
0.92 0
50000 100000 150000 200000 250000 300000 T2 (K2)
Fig. 1. Normalized zero field 57Fe NMR frequencies of bcc iron after Ref. [1] and normalized zero field 61Ni NMR frequencies of fcc nickel after Ref. [16] as a function of absolute temperature squared. T2 universality class pertains to isotropic magnets with half-integer spin. It can be assumed that the effective spin of iron is Seff ¼ 32 (in the zero field state) but Seff ¼ 12 for nickel. This roughly conforms to the different saturation moments of 2.217 mB/Fe and 0.617 mB/Ni.
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finally becomes infinitely large at the critical point. It is evident that for an infinitely large correlation length all microscopic details such as the discrete atomistic lattice structure as well as the spin structure are no longer important for the dynamics. As a consequence continuum theories such as field theoretical approaches rather than atomistic models now are adequate concepts [11]. Continuous dynamic symmetry is the reason for the universality of the critical exponents. Universality means independence of atomistic details such as the spin structure. Second, continuous symmetry applies not only asymptotically for T-Tc but sets in at a well-defined crossover temperature above the critical point [12]. This is approximately at the change from Curie–Weiss susceptibility to critical susceptibility. In particular, continuous dynamic symmetry holds for all lower temperatures down to T ¼ 0. In other words, from a dynamical point of view the ordered magnet has to be treated as a continuum. A general dynamic continuum theory of magnetism is, however, not available up to now. Only for the critical range field theoretical approaches have successfully been applied in the calculation of critical exponents [11]. Third, the change from atomistic to continuous dynamic symmetry is a fundamental process and has important conceptual consequences. In a continuum there are no spins and no interactions between spins. As a consequence instead of atomistic interactions such as magnons, a new type of excitation that is particular to the continuum must become relevant for the dynamics. The excitations of the continuous magnet we call Goldstone–Salam–Weinberg (GSW) excitations [13]. The GSW quasi-particles are massless and non-interacting bosons. They are generated through the spontaneous symmetry break at the phase transition [13]. In superconductors spin-zero bosons are well known as Cooper pairs. Quite analogous, we can assume that the GSW bosons of the magnetic materials are spincompensated quasi-particles. As a result, the GSW bosons have no magnetic moment. It is evident that non-magnetic and massless bosons cannot be observed using inelastic neutron scattering. Because the GSW bosons are massless and do not interact, they have linear dispersion in all 3D magnets and can be described by plane waves. Using the wave picture it is clear that magnetic plane waves carry no magnetic moment. Note that these waves are not spin waves. Spin waves never have linear dispersion exactly. Moreover, unlike the magnon excitation spectra that can have a gap the excitations of the continuous magnet are continuous, i.e. gapless. Linear dispersion in all solids is well known from the Debye bosons that are responsible for the universal T3 function of the low-temperature heat capacity of the diamagnetic solids [14]. Universality results from the analytically identical dispersion and density of states in all solids. The GSW bosons also control the temperature dependence of the magnons [15]. From the fact that the GSW bosons are relevant, we can conclude that they have lower dispersion energy and/or higher density of states compared to the magnons. The functionality of the atomistic near neighbour exchange interactions is essentially to define non-universal, i.e. materialspecific properties such as spin structure and ordering temperature. The saturation magnetic moment also is material specific and is determined by intra-atomic interactions. In other words, the two ending points of the spontaneous magnetization curve are given by atomistic interactions. The temperature function(s) between these ending points are variable and are defined by the GSW bosons. As we will see very weak external perturbations are sufficient to change the detailed temperature dependence between the SFPs T ¼ Tc and T ¼ 0. In particular, the GSW bosons also define the critical behaviour and the dimensionality.
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2. Experimental results Fcc nickel is a good candidate to start with because in nickel the effective spin is a constant ðSeff ¼ 12Þ. Fig. 2 nicely visualizes the problem. The normalized hyperfine field values measured by 61Ni NMR vs. absolute temperature to a power of 52 are shown [16]. On T5/2 scale, the T2 dependence seen in Fig. 1 appears curved. Additionally, the normalized spontaneous magnetization after Ref. [17] is depicted. It can be seen that the spontaneous magnetization follows T5/2 function quite well. The fitted exponent is e ¼ 2.4170.03. The zero field NMR data are excellently described by T2 function (fitted exponent e ¼ 1.9770.029). We have identified T5/2 universality class as characteristic of 1D magnets with half-integer spin [9]. As a conclusion upon magnetic saturation a dimensionality crossover from isotropic 3D (T2 function) to 1D (T5/2 function) is induced. This crossover must be at a field value that is smaller than the demagnetizing field. Normalization of the two experimental data sets to unity for T-0 supposes that the saturation magnetic moment is left unchanged by application of the demagnetization field (0.22 T for the Ni sphere). This assumption seems to be reasonable but is, of course, not strictly approved. This is because the hyperfine coupling constant is not absolutely known for the two experimental conditions. Independent of this difficulty it can be seen in Fig. 2 that the discrepancy between the zero field NMR results and the spontaneous magnetization is largest at intermediate temperatures and that the two fit curves approach each other towards the ordering temperature. This can be expected since the transition temperature should be identical in the two measuring processes. In other words, the absolute difference between the two experimental results is only small but quantitative analyses of the two temperature functions shows that this small effect has dramatic consequence on the universal exponent and therefore on the dimensionality. This shows that the universality classes are meta-stable and can depend sensitively on weak external parameters such as axial magnetostriction forced by an external
T (K) 100
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nickel TC = 630.2K
nickel
spontaneous magnetization ~T5/2 J. Crangle, G.M. Goodman (1971)
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field [18]. This sensitivity is not to be expected if the magnons would define the temperature dependence simply because the applied field is negligibly small compared to the internal Weiss molecular field. This further demonstrates that the universality classes cannot be defined by the magnons. For completeness, we should mention that T5/2 function of the spontaneous magnetization does not hold perfectly down to T-0. Although the deviations from T5/2 function are smaller than 1%, they can be interpreted as dimensionality crossover to T3/2 function at TDC 120 K. Fig. 3 shows the low-temperature section of the spontaneous magnetization after Ref. [17] on linear temperature scale. Below 120 K these data can excellently be fitted by T3/2 function. The fitted exponent is e ¼ 1.5070.02. This is a very unusual result because T3/2 means 3D anisotropic universality class but T5/2 is the 1D universality class. In principle, the symmetry of T3/2 class is higher than the 1D universality class observed at higher temperature (T5/2 function). Increase of symmetry with decreasing temperature normally should not occur. We shall discuss this phenomenon together with a similar observation on iron (see Fig. 5). For bcc iron the situation is more complicated than for nickel. Iron is a so-called weak ferromagnet [19]. This means that due do particularities in the band structure the spontaneous magnetization shows tendency to instability and can be fundamentally manipulated by weak external perturbations. Fig. 4 shows normalized spontaneous magnetization data of bcc iron after Ref. [17] as a function of T3 together with normalized zero field 57Fe NMR data after Ref. [1]. It can be seen that for T4350 K the spontaneous magnetization is excellently described by T3 function (fitted exponent: e ¼ 2.9770.03). This is a very surprising result because T3 universality class pertains to 1D magnets with integer spin [9]. We therefore have to conclude that not only a dimensionality crossover but additionally a crossover in spin quantum number from Seff ¼ 32 to Seff ¼ 2 is induced upon magnetic saturation. This seems to be a signature of the magnetic instability. In Fig. 4, the spontaneous magnetization data of Ref. [17] with To350 are omitted. As for nickel crossover to a different asymptotic power function for T-0 can be identified. Data below 280 K are well described by T2 function (see Fig. 5). T2 dependence was observed already in 1929 by Weiss and Forrer
0.90
~T3/2
TDC = 120K
57
~T5/2
zero field 61Ni NMR ~T2 J. Englich et al. (2000)
56
0.85 0
1000000
2000000 T5/2 (K5/2)
3000000
4000000
Fig. 2. Normalized zero field 61Ni NMR frequencies after Ref. [16] and normalized spontaneous magnetization of fcc nickel after Ref. [17] as a function of T5/2. Zero field data are excellently described by T2 function [9], while for the magnetically saturated state T5/2 function holds. T2 universality class pertains to isotropic 3D magnets with half-integer spin, T5/2 universality class to 1D magnets with halfinteger spin [9]. Change of exponent upon magnetic saturation means that dimensionality crossover from 3D to 1D is induced.
J. Crangle, G.M. Goodman, Proc. Roy. Soc. London A 321 (1971) 477. 55 0
50
100
150 T (K)
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300
Fig. 3. Enlarged low-temperature region of macroscopic spontaneous magnetization of nickel after Ref. [17] as a function of temperature. Crossover from low temperature T3/2 to high temperature T5/2 function at dimensionality crossover TDC120 K can be identified. This is unusual because T3/2 universality class pertains to higher symmetry (3D anisotropic) than T5/2 universality class (1D).
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1x108
2x108 T3
3x108
4x108
5x108
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Fig. 4. Normalized zero field 57Fe NMR frequencies of bcc iron after Ref. [1] and normalized spontaneous magnetization after Ref. [17] as a function of T3. T2 universality class pertains to isotropic magnets with half-integer spin while T3 universality class pertains to 1D magnets with integer spin. Upon magnetic saturation crossover not only in dimensionality but also in effective spin is induced. Data of Ref. [17] for To350 K are not shown. These data are analyzed in Fig. 5.
T (K) 100
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bcc iron P. Weiss, R. Forrer Ann. Physique 12 (1929)20. M.A. Butler, G.K. Wertheim, D.N.E. Buchanan Phys. Rev. B5 (1972)990.
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Fig. 5. Normalized low-temperature spontaneous magnetization data of bcc iron from the three different literature sources [1,17,20]. The three data sets agree very well and confirm asymptotic T2 function for T-0. For T4T315 K crossover to T3 function follows. T2 fit function to the zero field 57Fe NMR data shown in Fig. 1 is indicated by dashed line.
[20] who measured the spontaneous magnetization of iron for the first time with high precision. In other words, there is a wellresolved crossover from low temperature T2 function to high temperature T3 function at crossover-temperature T315 K in the spontaneous magnetization of iron. For the T2 function the spontaneous magnetization data are larger with respect to the T3 function. Note that, nevertheless, we have normalized the T3 function in Fig. 4 to unity for T-0. In order to visualize the low temperature T2 dependence of the macroscopic spontaneous magnetization Fig. 5 compiles data from the three different literature sources [1,17,20]. It can be seen
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that all data sets agree very nicely and that T2 function is confirmed within the typical precision of macroscopic magnetization measurements. For comparison T2 function fitted to the zero field 57Fe NMR data (see Fig. 1) is indicated by dashed line [1]. We should note that the low-temperature T2 function is not confirmed by the data of Refs. [5,21]. The spontaneous magnetization data of R. Pauthenet and of B.E. Argyle et al. deviate towards larger values for T-0 with respect to T2 function. Since these deviations can reasonably be described by adding a T1 term to the T2 function and since they are not observed by the other authors we attribute the excess magnetization for T-0 to paramagnetic impurities. It is evident that in the presence of an additive T1 term fit of the spontaneous magnetization data by a single power function results in a smaller exponent than e ¼ 2. In Refs. [5,21], it was claimed that the spontaneous magnetization data are in agreement with Bloch’s T3/2 function calculated by the atomistic spin wave theory [22,23]. This is not in agreement with the conclusion to be drawn from Fig. 5. Interpretation of the asymptotic T2 function in the spontaneous magnetization of iron is not unique because neither the relevant symmetry nor the spin quantum number is clear. The T2 function pertains to 3D isotropic symmetry and half-integer spin or to 3D anisotropic symmetry and integer spin [9]. Since it appears unlikely that spin and symmetry change at one crossover event as a function of temperature the crossover from T3 to T2 function is, perhaps, as for nickel, from 1D to 3D anisotropic but in contrast to nickel for integer spin. This would mean that in the spontaneous magnetization the effective spin is Seff ¼ 2 for all temperatures and the observed saturation moment of 2.217 mB/Fe applies to Seff ¼ 2. As for nickel asymptotic T2 function means a higher symmetry than T3 function observed at higher temperatures. This could be a consequence of the complicated interplay of magnetostriction, i.e. lattice distortion and dimensionality of the relevant excitations. The spin quantum number of iron is not only meta-stable with respect to an applied magnetic field but also with respect to temperature in the zero field state. In the temperature dependence of zero field 57Fe NMR and Mo¨ssbauer data crossover from low temperature T2 to high temperature T9/2 function at crossover temperature T615 K is identified [24]. Since in the zero field state iron is cubic for all temperatures, this crossover can safely be interpreted as change from Seff ¼ 32 to Seff ¼ 2, i.e. as change from 4 to 5 relevant states per Fe atom. To summarize, in the macroscopic spontaneous magnetization of iron crossover from low temperature T2 to high temperature T3 function is observed while in the zero field measurements the crossover is from T2 to T9/2. The crossover temperatures are 315 and 615 K, respectively. We have performed neutron scattering measurements on single-crystalline spheres of Fe, Ni and Co with diameter between 10 and 12 mm in a vertical field in order to observe some anomaly that can be identified as crossover from 3D to 1D state upon magnetic saturation. The neutron wavelength was l ¼ 2.44 A˚. Fig. 6 shows normalized results of two independent measurements of the field dependence of (11 0) scattering intensity for the Fe sphere. In agreement with Co and Ni the expected intensity increase with increasing field is not observed. For horizontal scattering vector 2/3 of all moments should be detected in the zero field state with isotropic domain distribution. Isotropic domain configuration can be assumed because there is no remanent magnetization. On the other hand all moments should be detected for magnetic saturation with all moments aligned parallel to the field, i.e. perpendicular to the scattering vector. As a consequence a considerable intensity increase can be expected. Note that Fig. 6 displays normalized total scattering intensities, i.e. nuclear and magnetic contributions.
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BDC = 0.65 T
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0.96
BD = 0.71 T
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0.92
0.0
bcc iron sphere ( 11 0 ) 0.2
0.4
0.6
0.8
B0 (T) Fig. 6. Normalized (11 0) neutron-scattering intensities of a spherical bcc iron single crystal as a function of a vertical magnetic field. The shown intensities contain nuclear and magnetic contributions. The two data sets are for independent measurements on different instruments (E6 and V2 of HMI). In spite of quantitative differences the two results reveal consistently a sudden intensity drop at BDC0.65 T that we identify as 3D to 1D dimensionality crossover (DC) associated with spin quantum number crossover from Seff ¼ 32 to Seff ¼ 2 (see text). The smooth intensity decrease conforms to the observations made in Ref. [26] and further supports symmetry change to 1D.
In all neutron scattering experiments on larger single crystals we have to consider that the absolute value of the scattered intensities can be reduced due to extinction, i.e. due to repeated scattering processes of the neutron on its way across the sample. Our data in Fig. 6 are not corrected for those effects. In Ref. [25] extinction effects have been systematically studied in zero field investigations of iron. Extinction can become increasingly important if the size of the domains grows upon magnetic saturation whereby the magnetic mosaic structure, so to say, decreases. This might reduce the scattering intensities with applied field. We, therefore, have performed the same scattering experiment as for the sphere but on powder material in order to obtain information on the importance of extinction. For powder material extinction should be negligible. Also for the Fe powder sample scattering intensities slightly decrease as a function of field. This is in agreement with a detailed investigation of structurally imperfect iron samples with correspondingly reduced extinction [26]. In the experiment of Ref. [26], ideally imperfect iron samples with increased mosaic spread were fabricated through mechanical treatment such as rolling in order to diminish extinction. At the expected field-induced crossover, a rather sharp anomaly as is in fact seen in Fig. 6 should occur. A sharp anomaly will be seen only on a spherical sample with a homogeneous internal field and domain distribution. On the other hand, extinction will not produce sharp anomalies in the field dependence of scattering intensities. The fact that the scattering intensities do not increase as a function of field, we consider as consistent with the conclusion drawn from the spontaneous magnetization that iron becomes 1D in the magnetically saturated state. One-dimensional symmetry means a strongly reduced transverse magnetic coherence length. This reduces the coherent magnetic-scattering intensities. For our purpose, the sudden drop in intensity at an external field of BDC 0.65 T is the most important feature of Fig. 6. This anomaly is observed in two independent experiments on the same iron sphere (open and filled symbols in Fig. 6) though with somewhat different absolute values.
For a very isotropic material like iron, it is very unlikely that the sudden change in scattering intensity at BDC0.65 T is due to a sudden change in domain structure [27]. For instance, the macroscopic magnetization of a spherical iron sample shows perfect behaviour of an ideal isotropic ferromagnet: linear field dependence up to demagnetization field and absolutely no remanence. As a consequence, the sudden intensity drop in Fig. 6 can reasonably be identified as combined crossover in dimensionality and in spin quantum number. We should note that the iron sphere is doped with 4% silicon for bcc phase stabilization. The demagnetization field for this sphere is BD0.71 T only but larger than BDC0.65 T. Nominally, the internal field is zero for fields smaller than the demagnetization field. This shows that the crossover is not driven by field but by the small axial lattice distortion due to forced magnetostriction [18]. Also in hcp cobalt crossover to 1D symmetry is induced upon magnetic saturation. Consistent with the hexagonal lattice structure below T0 ¼ 703 K [28], the symmetry class in the zero field ground state is 3D anisotropic. For half-integer spin this is T3/2 [9,29]. Upon magnetic saturation crossover to the same 1D universality class as for nickel represented by T5/2 is induced (see Fig. 2). T5/2 is excellently confirmed by the magnetization data of Ref. [30]. This universality class belongs to half-integer spin. In agreement with nickel but in contrast to iron the spin quantum number therefore seems not to be changed at the dimensionality crossover. We should note that asymptotically for T-0 T5/2 universality class can be identified in the zero field 59Co NMR data [31]. Since T5/2 function applies to deviations from saturation of less than two parts in thousand these details are not resolved by Fig. 7. Neutron scattering investigations on single-crystalline Co sphere exhibit definite anomaly at BDC ¼ 0.437 T that can be identified as dimensionality crossover. The total scattering intensities, i.e. nuclear and magnetic intensity shown by Fig. 8 are for field and hexagonal c-axis vertical, i.e. perpendicular to scattering plane. Again, decreasing scattering intensities as a function of field are not consistent with alignment of all moments in the vertical field and point to reduced coherent scattering in the
T (K) 200 300 M (T) / M (T = 0); νres (T) / νres (T = 0)
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T5/2/106 (K5/2) Fig. 7. Normalized zero field 59Co NMR frequencies of hcp cobalt after Ref. [29] and normalized spontaneous magnetization after Ref. [30] as a function of T5/2. The different temperature functions are attributed to a dimensionality crossover upon magnetic saturation from 3D anisotropic universality class (T3/2 function) to 1D universality class (T5/2). Open circles are magnetization data after Ref. [5].
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Fig. 8. Absolute neutron-scattering intensity of the (1 0 0) Bragg reflection measured on a spherical cobalt single crystal as a function of field. Hexagonal caxis and magnetic field are vertical, i.e. perpendicular to scattering plane. For this geometry scattering intensity should have reached maximum at the demagnetization field of BD0.64 T. Instead, scattering intensities decrease indicating decreasing coherent scattering. Anomaly at BDC ¼ 0.437 T is interpreted as dimensionality crossover (DC) from 3D to 1D. For 1D symmetry scattering intensities are considerably reduced but since demagnetization field BD is clearly visible the observed scattering intensities must contain finite magnetic contributions.
1D state. Scattering intensities might additionally decrease due to extinction. Change of domain configuration as well as the associated change of extinction can be expected to be a monotonous and steady function of temperature. Fig. 8 shows sudden change from quadratic to linear field dependence at BDC ¼ 0.437 T. This rather sharp event can safely be interpreted as dimensionality crossover. The field value of BDC ¼ 0.437 T is distinctly lower than the demagnetization field of BD ¼ 0.64 T and shows that the crossover is at nominally zero internal field. The crossover therefore must be driven by the size of the domains and the associated magnetostrictive lattice deformations. Note that the fact that the demagnetization field is visible in Fig. 6 indicates that there are still magnetic contributions in the total scattering intensity. This is not so clear for iron (see Fig. 6). For the nickel sphere practically no intensity change with field is observed. This means that there is also no visible intensity anomaly at the demagnetization field.
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macroscopic magnetization concern the detailed temperature dependence between these two stable ending points. By the same argument the field-induced magnetic changes are much too strong to be explained by the atomistic magnons. As a conclusion the magnons are not the relevant excitations for the dynamics [24]. This is confirmed by the observed universal power functions in the zero field methods as well as in the macroscopic magnetization. Universality cannot be explained by the material-specific magnons. Note that the magnon dispersions are nearly linear in antiferromagnets but nearly quadratic in ferromagnets. This does not fit the spin structure independent universal power functions for T-0 and for T-Tc. Instead the observed power functions are defined by the excitations of the magnetic continuum, the Goldstone–Salam–Weinberg bosons [13]. These bosons have linear dispersion and analytically simple densities of states in all 3D solids and define the universal temperature functions at the SFPs T ¼ 0 and T ¼ Tc. As a consequence, they also define the critical behaviour and the dimensionality. GSW bosons and dimensionality are meta-stable in the itinerant ferromagnets iron, nickel and cobalt. We have shown that upon magnetic saturation crossover to 1D universality class occurs. One-dimensional symmetry is verified on account of empirically known universal exponents [9]. The presented field-dependent neutron scattering intensities are consistent with 1D symmetry established by the spontaneous magnetization [26]. Decreasing neutron-scattering intensities with increasing vertical field can be explained by reduced phase coherence in the spin precession for spins arranged transverse to the field axis. This seems to be typical for 1D ferromagnets [6]. In the spontaneous magnetization, the time-averaged constant expectation value along z-axis is measured. On the other hand, the observation time of thermal neutrons is of the order of 1013 s only. This is short compared to the Larmor precession of 108 s. Neutrons therefore do not measure the time-averaged precession and see disordered spins. In insulating ferromagnets commonly no dimensionality crossover occurs upon magnetic saturation. This means these ferromagnets remain 3D also in the magnetically saturated state and the same temperature dependence of the order parameter is obtained in microscopic zero field measurements as well as in the macroscopic spontaneous magnetization. We can assume that in the 3D state the transverse interactions give rise to phase coherence in the spin precession such that the full coherent neutron-scattering intensities are observed. In 1D magnets there are only weak or no transverse interactions.
3. Conclusions Acknowledgments We have shown that the difference between the temperature dependence of the spontaneous magnetization measured by zero field 57Fe, 59Co and 61Ni NMR and the temperature dependence of the macroscopic spontaneous magnetization has a maximum of a few percent in the itinerant ferromagnets iron, cobalt and nickel. The essential difference in the two experimental methods is that due to application of the demagnetization field the macroscopic spontaneous magnetization refers to the axially symmetric singledomain state, but the zero field methods to the isotropic multidomain state. We can assume that a small axial lattice distortion, i.e. a symmetry change is associated with magnetic saturation. An applied magnetic field of the order of the demagnetization field is extremely weak compared to the Weiss molecular field. As a consequence no dramatic changes either in the absolute value of ordering temperature or in the saturation magnetic moment will be induced upon magnetic saturation. The observed different temperature functions in the zero field methods and in the
We thank N. Stu¨sser and K. Habicht of HMI/Berlin for the excellent support received during the neutron scattering experiments. Thanks are also to R. Zeller (IFF, Ju¨lich Forschungszentrum) for enlightening discussions concerning the band structure particularities of the itinerant ferromagnets. References [1] [2] [3] [4] [5] [6]
M.A. Butler, G.K. Wertheim, D.N.E. Buchanan, Phys. Rev. B 5 (1972) 990. G.B. Benedek, J. Armstrong, J. Appl. Phys. Suppl. 32 (1961) 106S. B.E. Argyle, N. Miyata, T.D. Schultz, Phys. Rev. 160 (1967) 413. K.G. Wilson, J. Kogut, Phys. Rep. 12C (1974) 75. R. Pauthenet, J. Appl. Phys. 53 (1982) 8187. U. Ko¨bler, A. Hoser, J.-U. Hoffmann, C. Thomas, Solid State Commun. 137 (2006) 301. [7] M.P. Schulhof, R. Nathans, P. Heller, A. Linz, Phys. Rev. B 4 (1971) 2254. [8] U. Ko¨bler, A. Hoser, D. Hupfeld, Physica B 328 (2003) 276. [9] U. Ko¨bler, A. Hoser, Physica B 362 (2005) 295.
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[10] U. Ko¨bler, J. Englich, O. Hupe, J. Hesse, Physica B 339 (2003) 156. [11] J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. B 21 (1980) 3976. [12] U. Ko¨bler, A. Hoser, R.M. Mueller, K. Fischer, J. Magn. Magn. Mater. 315 (2007) 12. [13] J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127 (1962) 965. [14] G.A. Alers, in: W.P. Mason (Ed.), Physical Acoustics, Vol. III-B, Academic Press, New York, London, 1965, p. 1. [15] U. Ko¨bler, A. Hoser, W. Scha¨fer, Physica B 364 (2005) 55. [16] J. Englich, Charles University Prague, private communication. [17] J. Crangle, G.M. Goodman, Proc. R. Soc. London A 321 (1971) 477. [18] B.E. Argyle, N. Miyata, Phys. Rev. 171 (1968) 555. [19] A.P. Malozemoff, A.R. Williams, V.L. Moruzzi, Phys. Rev. B 29 (1984) 1620. [20] P. Weiss, R. Forrer, Ann. Phys. 12 (1929) 20.
[21] B.E. Argyle, S.H. Charap, E.W. Pugh, Phys. Rev. 132 (1963) 2051. [22] F. Bloch, Z. Phys. 61 (1930) 206. [23] F. Keffer, in: S. Flu¨gge, H.P.J. Wijn (Eds.), Handbuch der Physik, vol. 18/2, Springer, Berlin, 1966, p. 1. [24] U. Ko¨bler, A. Hoser, Eur. Phys. J. B 60 (2007) 151. [25] R.M. Moon, C.G. Shull, Acta Crystallogr. 17 (1964) 805. [26] A.S. Arrott, T.L. Templeton, J. Appl. Phys. 57 (1985) 3763. [27] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, Heidelberg, 2000. [28] M. Braun, R. Kohlhaas, O. Vollmer, Z. Angew. Phys. 25 (1968) 365. [29] M. Kawakami, H. Enokiya, J. Phys. Soc. Japan 55 (1986) 4038. [30] H.P. Myers, W. Sucksmith, Proc. R. Soc. London A 207 (1951) 427. [31] U. Ko¨bler, A. Hoser, J. Englich, A. Snezhko, M. Kawakami, M. Beyss, K. Fischer, J. Phys. Soc. Japan 70 (2001) 3089.