- Email: [email protected]
μSR and neutron scattering studies of spin dynamics
ARTICLE IN PRESS
Physica B 350 (2004) 17–25
mSR and neutron scattering studies of spin dynamics R. Cywinski* School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
Abstract Aspects of the complementary roles of muon and neutron beam techniques in the investigation of magnetic systems are presented and discussed, particularly in the context of studies of spin fluctuations and spin glass-like behaviour in itinerant electron antiferromagnets (e.g. YMn2, b-MnAl) and ferromagnets (e.g. Ni3Al). r 2004 Published by Elsevier B.V. PACS: 76.75.+i; 78.70.Nx; 75.40.Gb; 75.50.Lk Keywords: Muon spin relaxation; mSR; Neutron scattering; Spin dynamics; Itinerant electron magnetism; Spin glasses
1. Introduction The over-riding strength of mSR, a generic acronym for Muon Spin Relaxation, Rotation, and Resonance, is that it is a uniquely sensitive probe of extremely small microscopic magnetic fields and the distribution and dynamics of such fields within a sample [1]. Indeed the sensitivity of mSR is such that even the fields generated by nuclear moments are easily measured. Moreover, the dynamical response of mSR, typically covering the range from 104 to 1012 Hz, serves to bridge the gap that separates, for example, bulk measurements of magnetisation and susceptibility from neutron scattering methods. The field dynamics measured by mSR can be related either to intrinsic internal field fluctuations, or to the apparent field fluctuations caused by a *Tel.: +44-113-343-3841; fax: +44-113-343-3846. E-mail address: [email protected] (R. Cywinski). 0921-4526/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physb.2004.04.049
muon diffusing through a lattice. Whilst the former is of key importance in the study of magnetic phenomena, the latter is often the focus of interest as the muon itself behaves like a light isotope of hydrogen ðMm B19 amuÞ and therefore can be used to probe hydrogen-like diffusion processes. The unique sensitivity of mSR has secured its role as an increasingly important tool in the armory of the condensed matter scientist. In particular, over the past two decades, there have been extremely successful and often pioneering investigations of phenomena as diverse as spin glass dynamics, spin fluctuations in itinerant electron magnets, moment localization in ultra-small moment and heavy fermion systems, flux distributions in superconductors, hydrogen mobility, passivation in semiconductors, muonium chemistry, and more recently, surface and near surface effects, see for example Ref. [1].
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These and many other studies are a testimony to the wide applicability of mSR and to the disciplinary breadth of the muon beam community. In this respect it is tempting to consider mSR as a competitive technique with which to address many of the problems in condensed matter science that are traditionally the domain of neutron scattering. However, it is generally found that mSR and neutron scattering are entirely complementary, and when the two techniques are used together a considerably greater understanding is achieved. This paper will present and discuss some examples taken from the many investigations of spin dynamics in which such complementarity has been successfully exploited, but it will also highlight other situations in which there remain unresolved discrepancies between the results provided by neutron scattering and mSR studies of the same magnetic materials.
2. The lSR technique The leading muon beam facilities for condensed matter research are at PSI (Switzerland), ISIS (UK), KEK (Japan) and TRIUMF (Canada). All except those at TRIUMF are located alongside proton accelerator-driven spallation neutron scattering facilities. This is, of course, more than just coincidence: Such facilities are a rich source of pions and hence of muons. Pions are produced by high-energy (>500 MeV) proton bombardment of a graphite target. Positive, low-energy or ‘‘surface’’ muons are emitted isotropically by the decay of those positive pions which are at rest at the surface of the target: within its rest frame the pion decays with a half-life of 26.03 ns emitting a muon with energy of 4.119 MeV. As a consequence of maximal parity violation in the decay process the resulting muon is perfectly spin polarized with its spin antiparallel to the direction of travel. Muon spin relaxation, rotation and resonance all depend upon the implantation of such spin polarized positively charged muons within a sample. Fortunately, implantation and thermalisation of surface muons within a sample, at depths
of the order of millimeters, is extremely rapid (B1 ns) and depolarisation during the implantation and thermalisation processes is negligible. The subsequent time evolution of the spin vectors of the implanted muons is determined by each muon’s local magnetic environment. If an ensemble of muons experience a unique, internal or applied, off-axis magnetic field, then the muon spins precess coherently at the appropriate Larmor frequency. Conversely, spatial or temporal fluctuations in the magnetic field at the muon sites result in a dephasing or depolarisation of the muon spin ensemble. Muons decay with a half-life of 2.197 ms emitting positrons preferentially in the direction of the muon spin vector at the time of decay according to the probability function W ðyÞ ¼ 1 þ a0 cos y;
ð1Þ
where y is the angle between the muon spin and the direction of positron emission. The asymmetry parameter, a0 ; depends upon the emitted positron energy, but for most muon experiments a0 typically takes a value close to 0.25. By observing the temporal and angular distribution of emitted positrons it is thus possible to monitor directly precessional motion or the dephasing of the implanted muon spins. Muon spin relaxation is generally used to describe the time-dependent loss of polarisation of the muon spins by internal fields either in zero applied field or with a field applied parallel to the initial muon spin (z-) direction. In contrast rotation describes the dephasing of muon spins by local internal fields when a magnetic field is applied transverse to the initial muon spin direction (i.e. along the x-direction). Taking the simplest experimental geometry in which positron detectors are placed up- and down-stream of the sample (or forward, F ; and backward, B, with respect to the initial muon spin direction), the time dependence of the polarisation of the muon spin ensemble, PðtÞ is given by PðtÞ ¼
F ðtÞ aBðtÞ ; F ðtÞ þ aBðtÞ
ð2Þ
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where F ðtÞ and BðtÞ are the positron count rates in the forward and backward detectors and a is a calibration parameter associated with the relative efficiencies of the F and B detectors. For muon spin relaxation PðtÞ ¼ a0 G z ðtÞ
ð3Þ
whilst for muon spin rotation PðtÞ ¼ a0 G x ðtÞ cosðoL tÞ;
ð4Þ
where oL(=gm B, with gm =2p 1.355342 108 s1 T1) is the Larmor precession frequency of the muon in the applied transverse field, B: It is Gz ðtÞ and G x ðtÞ; the longitudinal relaxation and transverse depolarisation functions, respectively, that are of principal interest in mSR experiments. However, it is important to note that it is sometimes necessary to choose carefully the muon facility at which to perform a particular mSR experiment [1]. PSI and TRIUMF are both continuous muon sources, at which coincidence techniques are used to ensure that only one muon is present in the sample at any time, thereby providing an intrinsic time resolution which is determined only by that of the detection system. ISIS and KEK, on the other hand, are pulsed muon sources, producing well-defined periodic bunches of muons of some tens of nanoseconds half-width, thus defining a corresponding time resolution for the measurement of G z ðtÞ and G x ðtÞ: Experiments which attempt to measure the coherent precession of muon spins either in static internal or external transverse magnetic fields exceeding approximately 50 mT are thus far better suited to PSI and TRIUMF: at ISIS or KEK the convolution of the intrinsic muon pulse width with rapidly and coherently precessing signal (for example given by Eq. (4)) in fields greater than this leads to an effective loss of asymmetry, and hence of sensitivity. In this paper, we shall confine our discussion of mSR entirely to muon spin relaxation measurements of Gz ðtÞ and, moreover, to examples of those experiments that can be performed equally well at either pulsed or continuous muon facilities.
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3. Muon spin relaxation functions The local internal field responsible for the muon spin precession at each muon site originates from a dipolar interaction with surrounding nuclear or electronic spins, and from contact hyperfine fields associated with the spin density at the muon site. For any muon at a site at which the resultant field jBj makes an angle y with respect to the z-axis, the z-component of the muons spin vector sz ðtÞ varies with time according to sz ðtÞ ¼ cos2 y þ sin2 y cosðgm jBjtÞ:
ð5Þ
The relaxation function, G z ðtÞ; can then be calculated by taking the statistical average of all sz ðtÞ; i.e. ZZZ G z ðtÞ ¼ sz ðtÞPðBx ÞPðBy ÞPðBz Þ dBx dBy dBz ;
ð6Þ
where PðBi Þ are the probability distributions representing the i ¼ x-, y- and z-components of the resultant internal fields. For a concentrated system of randomly oriented static magnetic (nuclear or atomic) dipoles PðBi Þ are Gaussian distributed such that 1 PG ðBi Þ ¼ pffiffiffiffiffiffi expðB2i =2D2 Þ ð7Þ 2pD with D the second moment of the field distribution. For such a distribution Eq. (6) gives the wellknown Gaussian Kubo–Toyabe relaxation function z GKT ðtÞ ¼ 13 þ 23 ð1 s2 t2 Þ expð12 s2 t2 Þ;
ð8Þ
where s ¼ gm D [2]. In the limit of dilute dipolar spins, the internal field distributions are Lorentzian in form and the resulting Kubo–Toyabe function is [3] z ðtÞ ¼ 13 þ 23 ð1 atÞ expðatÞ GKT
ð9Þ
with a ¼ gm L; where 2L is the full-width at halfmaximum of the Lorentzian field distribution. At intermediate dipole concentrations the Voigtian Kubo–Toyabe provides an appropriate G z ðtÞ [4]. The Kubo–Toyabe functions given by Eqs. (8) and (9) are calculated in the static limit. Internal field dynamics, resulting from the muon hopping
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from site to site or from the internal fields themselves fluctuating, or both, are taken into account by the strong collision approximation, which assumes that the local field changes its direction at a time t according to a probability distribution PðtÞ ¼ expðntÞ; and that the field after such a ‘‘collision’’, chosen randomly from the distribution PðBi Þ; is entirely uncorrelated with the field before the collision. The resulting dynamic Kubo–Toyabe function must be evaluated numerically [2]. However, over much of the dynamic range analytical approximations suffice [1]. For example, in the limit of rapid field fluctuations ðR ¼ n=s > 10Þ the motionally narrowed dynamic Gaussian Kubo–Toyabe function is well represented by a simple exponential z GDKT ðtÞ ¼ expðltÞ
ð10Þ
2
with l ¼ 2s =n; while at intermediate fluctuation rates ð1oRo10Þ the Abragam function is suitable. z ðtÞ GDKT
¼ expflðexpðntÞ 1 þ ntÞg:
ð11Þ
For very slow fluctuations ðRo1Þ only the 13 tail of the Kubo–Toyabe function is affected and the form of the tail becomes z Gtail ðtÞ ¼ 13 expð23ntÞ:
ð12Þ
These analytic approximations are compared with numerical calculations of the dynamic Gaussian Kubo–Toyabe in Fig. 1. It should be noted that in studies of atomic spin fluctuations the motionally narrowed muon spin relaxation rate, l, can be related directly to the associated fluctuating local magnetic fields through the relation XZ N l¼G /Ba ð0ÞBa ðtÞS dt; ð13Þ a¼x;y
0
where G is a coupling constant. It is straightforward [5] to show that kB T X wðqÞ ð14Þ lðTÞ ¼ G N q GðqÞ in which wðqÞ is the wave vector-dependent susceptibility, GðqÞ is the line width of the spin fluctuations and the sum is taken over the Brillouin zone. As inelastic neutron scattering measures wðqÞ and GðqÞ directly it is therefore
Fig. 1. The dynamic Gaussian Kubo–Toyabe function (solid lines) compared with the exponential and Abragam approximations of Eqs. (10) and (11) (broken lines) for R>1, where R ¼ s=n; and time T is expressed as the dimensionless parameter st:
possible to compare mSR and neutron measurements of the temperature-dependent spin fluctuation spectrum. Such a comparison was made for the first time in studies of longitudinal spin fluctuations in the frustrated itinerant electron antiferromagnet YMn2 [5–7], as discussed in the next section. It should be noted, with reference to Eq. (14), that in the particular case of paramagnetic spin fluctuations, such as those addressed in this paper, the muon spin relaxation rate, l, is determined by a sum over the entire Brillouin zone, thereby accounting for both short- and long-range dynamic correlations of the neighbouring atomic spins. It is therefore often not necessary to determine the precise location of the muon within the lattice in order to fully interpret the muon spectra (although a determination of the muon site is generally straightforward). However, in other situations, for example in the presence of longrange magnetic order and associated coherent internal fields, the muon site or sites may be crucial for interpretation of the spectra [1].
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Although Eqs. (8–12) have proved widely applicable in mSR studies of both magnetic and nonmagnetic materials, they do not adequately describe muon spin relaxation in the class of highly disordered magnets known as spin glasses. For example, whilst it might be expected that mSR in a dilute spin glass would follow the dynamic Lorentzian Kubo–Toyabe form, it is found experimentally that, in the fast fluctuation limit above the spin glass temperature, Tg ; the muon spin relaxation is better described by [8] qffiffiffiffiffiffiffiffiffiffiffi z ðt; a; nÞ ¼ exp 4a2 t=n : ð15Þ GDSG This ‘‘root-exponential’’ time dependence has been derived explicitly using a model in which the dynamic variable range of the time modulated field at each muon site, determined by the muon’s proximity to neighbouring atomic spins, is approximated by a Gaussian distribution, whilst the probability distribution for a muon experiencing a particular variable range is chosen such that the overall internal field distribution is Lorentzian, precisely as expected for a dilute system [8]. The complementarity between neutron scattering and mSR in studies of spin dynamics above Tg is evident: Only the neutron spin echo technique can approach similar time regimes (B106 s), but because this technique is often count-rate limited, studies of the more dilute spin glasses such as CuMn and AuFe, can be difficult and time consuming. However, for such dilute spin glasses, mSR is able map the evolution of spin dynamics, often over several decades in frequency, as the glass transition is approached. Moreover, estimates of the spin relaxation rate extracted from the temperature-dependent root-exponential mSR spectra are found to be in close agreement with the rather sparse data obtained using neutron spin echo, as can be seen in Fig. 2 where the combined data indicate a common power-law dependence of the critical slowing down of the spin correlation times [8]. Whilst the root-exponential muon spin relaxation function (Eq. (15)) is also found to be appropriate for dilute superparamagnetic systems [9], recent extensive studies of more concentrated spin glasses and similar frustrated magnets have
Fig. 2. Spin correlation times extracted from mSR studies of dilute CuMn and AuFe spin glasses as a function of reduced temperature. The correlation times obtained from neutron spin echo studies of CuMn are also shown (from Ref. [8]).
shown that, generally, the relaxation function takes the form of a stretched exponential z GCSG ðtÞ ¼ expððltÞb Þ
ð16Þ
for which l is found to diverge as Tg is approached, but significantly b is also temperature dependent. Several explanations for this behaviour have been suggested, the simplest of which invokes a broadening distribution of the fluctuation frequencies n, and hence of l, with decreasing temperature. Interestingly the temperature dependence of b is often found to reflect almost precisely that found in Monte Carlo calculations of the realtime stretched exponential, or Kohlrausch, relaxation of the spin autocorrelation function of Ising spin glasses [11], i.e. /Sð0Þ SðtÞSpexpððt=tÞb
ð17Þ
as can be seen in Fig. 3, with b decreasing from unity at temperatures of approximately 4Tg to a limiting, and almost universal, value of 13 at Tg. The phenomenon of Kohlrausch, or stretched exponential, relaxation in the real-time correlations in structural glasses above the glass temperature is well known: whilst strong structural glasses
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for Mn, respectively) resulting in a spin liquid-like state which persists to the lowest temperatures [5– 7]. On the other hand, substitution of Al for Mn expands the lattice and rapidly stabilizes the Mn moments on the geometrically frustrated tetrahedrally coordinated transition metal sites, leading to a disordered spin glass-like state. In this context it should be noted that the mSR spectra for Y(MnAl)2 are of the stretched exponential form with b approaching 13 close to Tg [6]. For YMn2 and its related alloys neutron measurements show GðqÞ to be only weakly dependent upon q. Eq. (14) can therefore be written Fig. 3. Temperature dependence of the exponent b (a) estimated from the time-dependent spin autocorrelation function found in Ogielski’s Monte Carlo simulations of an Ising spin glass [11] and (b) measured by mSR for AuMn [10].
often exhibit stretched exponential relaxation with a temperature independent b of approximately 0.5, for fragile glasses b decreases from 1 at high temperatures to close to 13 at the glass transition. Similarities with the behaviour of muon spin relaxation in dilute and concentrated spin glasses, respectively, are evident, and suggest that studies of the relatively simple spin glass problem may shed light on the more complex dynamical processes in structural glasses. Neutron spin echo and mSR will have a key role to play in such studies.
4. Spin fluctuations in itinerant magnets 4.1. YMn2 and related compounds The C15 Laves Phase compound, YMn2, undergoes a discontinuous phase transition from a spin fluctuating Pauli paramagnetic state at high temperatures to a helical antiferromagnetically ordered state below 100 K. This transition is accompanied by a dramatic 5% increase in cell volume. The transition to the ordered state can be suppressed by weak external or chemical pressure (e.g. by 0.3 Gpa or substitution of only 2.5 at% Fe
lðTÞ ¼ G
kB TwL ; G
where wL is the local susceptibility given by 1 X wðqÞ: wL ¼ N q
ð18Þ
ð19Þ
The neutron scattering spectra of Y(Mn0.9Al0.1)2 show GðTÞ to be Arrhenius-like with an activation energy of 283 K, whilst wL ðTÞ is Curie–Weiss like with y ¼ 93 K: lðTÞ can thus be evaluated using Eq. (15) and then scaled to the measured lðTÞ with the coupling constant G as the only free parameter [6]. As can be seen in Fig. 4, the correspondence is excellent. The inset to Fig. 4 shows the temperature dependence of the line-width parameter, G; for YMn2 and its dilute alloys with Sc, Fe and Al substitution for Mn, as determined directly by neutron scattering and from mSR data scaled according to Eq. (18). It is clear that G narrows considerably on addition of Al as the YMn2 lattice expands and the Mn moments tend to localize. Moreover the approximately linear dependence of G with T for these dilute alloys is in full accord with predictions of SCR theory of spin fluctuations [12]. The neutron spectra from which wðqÞ and GðqÞ are extracted contain more information than the corresponding muon spectra, as the latter provide only the dynamical response averaged over the Brillouin zone and hence contain no spatial information. However, whilst each individual neutron spectrum required at least one day of
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Fig. 4. The measured mSR relaxation rate for Y(Mn0.9Al0.1)2 compared to l(T) calculated from scaled neutron data using Eq. (18) (solid line) [6]. The dashed line indicates the spin glass transition. Inset: Line-width parameters for YMn2 and its dilute alloys from neutron scattering (full symbols) and scaled mSR relaxation rates (open symbols) [5]. The lines in the inset are guide to the eye.
beam time to collect, all the mSR measurements shown in Fig. 4 were obtained in just a few hours. mSR is therefore an ideal technique for extensive and systematic studies of spin fluctuations. Furthermore, mSR substantially extends to lower frequencies the dynamical range made available by neutron scattering. 4.2. b-Mn and b-Mn1xAlx Combined mSR and neutron scattering studies have also provided some detailed insights into the nature of longitudinal quantum spin fluctuations and moment localisation in elemental b-Mn and its alloys with Al [13,14]. b-Mn is the only allotropic form of elemental manganese that does not possess a magnetic moment at any temperature. It crystallises with
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the cubic A13 structure with eight and 12 Mn atoms at the inequivalent site I and site II positions, respectively. NMR and NQR data suggest that whilst site I Mn atoms are very weakly magnetic, those at site II are associated with strong, slow antiferromagnetically correlated spin fluctuations [15]. Similarities with YMn2 are evident, particularly as the topology of the site II atoms, described as a network of corner sharing triangles, also leads geometric frustration. Inelastic polarized neutron scattering studies of the quantum fluctuations in b-Mn showed that the spectral width of the excitations decreased markedly from approximately 20 meV to only a few meV with increasing Al substitution at site II, reflecting the combined effects of lattice expansion and a lifting of the spin configurational degeneracy associated with the geometric frustration [15]. mSR has helped us develop a more complete picture of the evolution of the spin fluctuation spectra in b-Mn1xAlx [13]. For b-Mn the exponential mSR spectra exhibit extreme motional narrowing (lB0.02 ms1) at all temperatures, consistent with the broad and weakly temperature-dependent inelastic neuron line width. With the addition of Al, for xo0:09; the mSR spectra remain exponential, with l diverging towards low temperatures (3–5 K), indicating a critical slowing down of the Mn spin fluctuations with an exponent, g, very close to 1 (Fig. 5). However, as the Al concentration increases beyond x ¼ 0:09 the mSR spectra abruptly adopt the stretched exponential form, as described by Eq. (16), characteristic of concentrated spin glass behaviour. At these concentrations l again critically diverges (Fig. 5) but at much higher temperatures (30–40 K). The exponent b decreases from unity at high temperatures to 13 at the critical temperature. It is suggested that these features, summarized in Fig. 5, are evidence of a quantum spin–liquid to spin–glass transition in b-Mn1xAlx [13]. These mSR measurements have, in turn, prompted a more extensive time-of-flight inelastic neutron scattering study spin fluctuations in b-Mn and its alloys with Al, and the results of these studies provide clear evidence for non-Fermi liquid scaling behaviour in unalloyed b-Mn over a relatively wide temperature range [14]. Significantly, this is
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Fig. 5. Critical scaling of the muon spin relaxation rates in bMn1xAlx alloys. The inset shows the corresponding magnetic phase diagram [13]. Note the marked difference in behaviour for xp9% and x>9%.
below TC which collapse into diffusive modes above TC [18]. In addition, the results of magnetisation, NMR and resistivity measurements on Ni3Al are all in close agreement with the predictions of the self-consistent renormalisation (SCR) theory of spin fluctuations for weak ferromagnets. The atomic spin contribution to the mSR spectra of stoichiometric Ni3Al remains fully motionally narrowed down to 40 K, whilst below 40 K mSR, somewhat controversially, provides no evidence for long-range ferromagnetism [17]. Instead, a static but disordered magnetic ground state, represented in the mSR spectra by a static Voigtian Kubo–Toyabe function, begins to emerge and grow, accounting at the lowest temperatures for almost two-thirds of the mSR signal (Fig. 6). Although the Ni3Al mSR sample is found to be crystallographically well ordered, the low-temperature magnetic state revealed by mSR is intrinsically inhomogeneous.
the first reported example of such scaling in a pure elemental paramagnetic metal. 4.3. Itinerant electron ferromagnets: Ni3Al Interestingly, one of the earliest and most successful examples of the application of the zero field mSR technique was the detailed study of longitudinal spin fluctuations in the itinerant electron ferromagnet MnSi [2,16], a study which provided considerable support for SCR theory [12]. It is therefore surprising that there have since been very few comparable studies of other itinerant ferromagnets. However, a recent mSR investigation of the archetypal weak itinerant electron ferromagnet Ni3Al has highlighted potential problems associated with such studies [17]. Stoichiometric Ni3Al, has a reported Curie point of 40 K and a nickel moment of approximately 0.076 mB. Small-angle neutron scattering shows a sharp critical scattering peak at the reported Curie temperature, indicating a divergence of the spin correlation range, whilst inelastic neutron scattering reveals well-defined spin waves
Fig. 6. Zero field mSR spectra from Ni3Al. The solid lines represent a fit to a magnetically inhomogeneous model, with one component fully motionally narrowed and the second represented by a static Voigtian Kubo–Toyabe function. These two components are themselves multiplied by a static Gaussian Kubo–Toyabe function associated with the nuclear dipole fields at the muon site.
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mSR and neutron scattering have thus provided quite disparate views of the magnetic ground state of Ni3Al. Similar discrepancies have also been observed for another itinerant electron ferromagnet, PdNi, for which mSR provides no evidence of magnetic order below the reported Curie point [19]. It remains uncertain whether this is a consequence of the fundamental differences in the way that neutrons and muons probe spin correlations in such systems, or whether the muon itself, rather than acting as a passive probe, is directly responsible for locally perturbing the weak ferromagnetic order. This is clearly an area in need of further detailed theoretical and experimental investigation.
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is profound, and highlights the need for further work.
Acknowledgements The author wishes to thank his many mSR collaborators, including P.M. Bentley, R.I. Bewley, I.A. Campbell, M.R. Crook, J.A. Dann, J.L. Garcia-Munoz, A.D. Hillier, S.H. Kilcoyne, P. Manuel, C. Pappas, M.J. Pottinger, J.M. Preston, B.D. Rainford, C.A. Scott, J.R. Stewart and M.T.F. Telling, and the muon support scientists at both ISIS and PSI.
References 5. Conclusions In this paper, the complementary roles of mSR and neutron scattering in studies of both fast spin fluctuations and slow spin dynamics in some itinerant electron magnets have been considered. Whilst neutron scattering provides spatial and temporal information on spin correlations, mSR provides information only on the dynamics. Nevertheless, in the fast fluctuation limit mSR facilitates a rapid and systematic characterisation of the evolution of the spin dynamics as a function of, for example temperature or atomic substitution. In the slow fluctuation, spin glass-like limit, mSR extends considerably the dynamic range offered by neutron spin echo, whilst also providing some remarkable insights into the non-exponential nature of the dynamic spin correlations above Tg. Indeed, mSR has provided arguably the best information available to date on spin dynamics in the paramagnetic phase of spin glasses. In most cases the information obtained using neutron and muon techniques, in both the fast and slow spin fluctuation limits, is in excellent agreement and in accord with theory. Where there is disagreement, for example in the case of Ni3Al, it
[1] S.L. Lee, S.H. Kilcoyne, R. Cywinski (Eds.), Muon Science (Muons in Physics, Chemistry and Materials), Scottish Universities Summer School in Physics, IOP Publishing, Bristol, 1999. [2] R.S. Hayano, et al., Phys. Rev. B 20 (1979) 850. [3] Y.J. Uemura, et al., Phys. Rev. Lett. 45 (1980) 583. [4] M.R. Crook, R. Cywinski, J. Phys.: Condens. Matter 9 (1997) 1149. [5] B.D. Rainford, in: S.L. Lee, S.H. Kilcoyne, R. Cywinski (Eds.), Muon Science, Scottish Universities Summer School in Physics, IOP Publishing, Bristol, 1999, p. 115. [6] R. Cywinski, B.D. Rainford, Hyperfine Interaction 85 (1994) 215. [7] B.D. Rainford, et al., in: D. Wagner, W. Brauneck, A. Solontsov (Eds.), Itinerant Electron Magnetism: Fluctuation Effects, Kluwer Academic Publishers, Netherlands, 1998. [8] Y.J. Uemura, et al., Phys. Rev. B 31 (1985) 546. [9] R.I. Bewley, R. Cywinski, Phys. Rev. B 58 (1998) 11544. [10] I.A. Campbell, et al., Phys. Rev. Lett. 72 (1994) 1291. [11] A.T. Ogielski, Phys. Rev. B 32 (1985) 7384. [12] T. Moriya, Spin Fluctuations in Itinerant Electron Magnetis, Springer, Berlin, 1985. [13] J.R. Stewart, R. Cywinski, Phys. Rev. B 59 (1999) 4305. [14] J.R. Stewart, et al., Phys. Rev. Lett. 89 (18) (2002) 6403. [15] H. Nakamura, et al., J. Phys.: Condens. Matter 9 (1997) 4701. [16] R.S. Hayano, et al., Phys. Rev. Lett. 41 (1978) 1743. [17] S.H. Kilcoyne, R. Cywinski, Physica B 326 (2003) 171. [18] N. Bernhoeft, et al., Phys. Rev. B 28 (1983) 422. [19] B.D. Rainford, et al., Hyperfine Interactions 85 (1994) 323.
Physica B 350 (2004) 17–25
mSR and neutron scattering studies of spin dynamics R. Cywinski* School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
Abstract Aspects of the complementary roles of muon and neutron beam techniques in the investigation of magnetic systems are presented and discussed, particularly in the context of studies of spin fluctuations and spin glass-like behaviour in itinerant electron antiferromagnets (e.g. YMn2, b-MnAl) and ferromagnets (e.g. Ni3Al). r 2004 Published by Elsevier B.V. PACS: 76.75.+i; 78.70.Nx; 75.40.Gb; 75.50.Lk Keywords: Muon spin relaxation; mSR; Neutron scattering; Spin dynamics; Itinerant electron magnetism; Spin glasses
1. Introduction The over-riding strength of mSR, a generic acronym for Muon Spin Relaxation, Rotation, and Resonance, is that it is a uniquely sensitive probe of extremely small microscopic magnetic fields and the distribution and dynamics of such fields within a sample [1]. Indeed the sensitivity of mSR is such that even the fields generated by nuclear moments are easily measured. Moreover, the dynamical response of mSR, typically covering the range from 104 to 1012 Hz, serves to bridge the gap that separates, for example, bulk measurements of magnetisation and susceptibility from neutron scattering methods. The field dynamics measured by mSR can be related either to intrinsic internal field fluctuations, or to the apparent field fluctuations caused by a *Tel.: +44-113-343-3841; fax: +44-113-343-3846. E-mail address: [email protected] (R. Cywinski). 0921-4526/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physb.2004.04.049
muon diffusing through a lattice. Whilst the former is of key importance in the study of magnetic phenomena, the latter is often the focus of interest as the muon itself behaves like a light isotope of hydrogen ðMm B19 amuÞ and therefore can be used to probe hydrogen-like diffusion processes. The unique sensitivity of mSR has secured its role as an increasingly important tool in the armory of the condensed matter scientist. In particular, over the past two decades, there have been extremely successful and often pioneering investigations of phenomena as diverse as spin glass dynamics, spin fluctuations in itinerant electron magnets, moment localization in ultra-small moment and heavy fermion systems, flux distributions in superconductors, hydrogen mobility, passivation in semiconductors, muonium chemistry, and more recently, surface and near surface effects, see for example Ref. [1].
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These and many other studies are a testimony to the wide applicability of mSR and to the disciplinary breadth of the muon beam community. In this respect it is tempting to consider mSR as a competitive technique with which to address many of the problems in condensed matter science that are traditionally the domain of neutron scattering. However, it is generally found that mSR and neutron scattering are entirely complementary, and when the two techniques are used together a considerably greater understanding is achieved. This paper will present and discuss some examples taken from the many investigations of spin dynamics in which such complementarity has been successfully exploited, but it will also highlight other situations in which there remain unresolved discrepancies between the results provided by neutron scattering and mSR studies of the same magnetic materials.
2. The lSR technique The leading muon beam facilities for condensed matter research are at PSI (Switzerland), ISIS (UK), KEK (Japan) and TRIUMF (Canada). All except those at TRIUMF are located alongside proton accelerator-driven spallation neutron scattering facilities. This is, of course, more than just coincidence: Such facilities are a rich source of pions and hence of muons. Pions are produced by high-energy (>500 MeV) proton bombardment of a graphite target. Positive, low-energy or ‘‘surface’’ muons are emitted isotropically by the decay of those positive pions which are at rest at the surface of the target: within its rest frame the pion decays with a half-life of 26.03 ns emitting a muon with energy of 4.119 MeV. As a consequence of maximal parity violation in the decay process the resulting muon is perfectly spin polarized with its spin antiparallel to the direction of travel. Muon spin relaxation, rotation and resonance all depend upon the implantation of such spin polarized positively charged muons within a sample. Fortunately, implantation and thermalisation of surface muons within a sample, at depths
of the order of millimeters, is extremely rapid (B1 ns) and depolarisation during the implantation and thermalisation processes is negligible. The subsequent time evolution of the spin vectors of the implanted muons is determined by each muon’s local magnetic environment. If an ensemble of muons experience a unique, internal or applied, off-axis magnetic field, then the muon spins precess coherently at the appropriate Larmor frequency. Conversely, spatial or temporal fluctuations in the magnetic field at the muon sites result in a dephasing or depolarisation of the muon spin ensemble. Muons decay with a half-life of 2.197 ms emitting positrons preferentially in the direction of the muon spin vector at the time of decay according to the probability function W ðyÞ ¼ 1 þ a0 cos y;
ð1Þ
where y is the angle between the muon spin and the direction of positron emission. The asymmetry parameter, a0 ; depends upon the emitted positron energy, but for most muon experiments a0 typically takes a value close to 0.25. By observing the temporal and angular distribution of emitted positrons it is thus possible to monitor directly precessional motion or the dephasing of the implanted muon spins. Muon spin relaxation is generally used to describe the time-dependent loss of polarisation of the muon spins by internal fields either in zero applied field or with a field applied parallel to the initial muon spin (z-) direction. In contrast rotation describes the dephasing of muon spins by local internal fields when a magnetic field is applied transverse to the initial muon spin direction (i.e. along the x-direction). Taking the simplest experimental geometry in which positron detectors are placed up- and down-stream of the sample (or forward, F ; and backward, B, with respect to the initial muon spin direction), the time dependence of the polarisation of the muon spin ensemble, PðtÞ is given by PðtÞ ¼
F ðtÞ aBðtÞ ; F ðtÞ þ aBðtÞ
ð2Þ
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where F ðtÞ and BðtÞ are the positron count rates in the forward and backward detectors and a is a calibration parameter associated with the relative efficiencies of the F and B detectors. For muon spin relaxation PðtÞ ¼ a0 G z ðtÞ
ð3Þ
whilst for muon spin rotation PðtÞ ¼ a0 G x ðtÞ cosðoL tÞ;
ð4Þ
where oL(=gm B, with gm =2p 1.355342 108 s1 T1) is the Larmor precession frequency of the muon in the applied transverse field, B: It is Gz ðtÞ and G x ðtÞ; the longitudinal relaxation and transverse depolarisation functions, respectively, that are of principal interest in mSR experiments. However, it is important to note that it is sometimes necessary to choose carefully the muon facility at which to perform a particular mSR experiment [1]. PSI and TRIUMF are both continuous muon sources, at which coincidence techniques are used to ensure that only one muon is present in the sample at any time, thereby providing an intrinsic time resolution which is determined only by that of the detection system. ISIS and KEK, on the other hand, are pulsed muon sources, producing well-defined periodic bunches of muons of some tens of nanoseconds half-width, thus defining a corresponding time resolution for the measurement of G z ðtÞ and G x ðtÞ: Experiments which attempt to measure the coherent precession of muon spins either in static internal or external transverse magnetic fields exceeding approximately 50 mT are thus far better suited to PSI and TRIUMF: at ISIS or KEK the convolution of the intrinsic muon pulse width with rapidly and coherently precessing signal (for example given by Eq. (4)) in fields greater than this leads to an effective loss of asymmetry, and hence of sensitivity. In this paper, we shall confine our discussion of mSR entirely to muon spin relaxation measurements of Gz ðtÞ and, moreover, to examples of those experiments that can be performed equally well at either pulsed or continuous muon facilities.
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3. Muon spin relaxation functions The local internal field responsible for the muon spin precession at each muon site originates from a dipolar interaction with surrounding nuclear or electronic spins, and from contact hyperfine fields associated with the spin density at the muon site. For any muon at a site at which the resultant field jBj makes an angle y with respect to the z-axis, the z-component of the muons spin vector sz ðtÞ varies with time according to sz ðtÞ ¼ cos2 y þ sin2 y cosðgm jBjtÞ:
ð5Þ
The relaxation function, G z ðtÞ; can then be calculated by taking the statistical average of all sz ðtÞ; i.e. ZZZ G z ðtÞ ¼ sz ðtÞPðBx ÞPðBy ÞPðBz Þ dBx dBy dBz ;
ð6Þ
where PðBi Þ are the probability distributions representing the i ¼ x-, y- and z-components of the resultant internal fields. For a concentrated system of randomly oriented static magnetic (nuclear or atomic) dipoles PðBi Þ are Gaussian distributed such that 1 PG ðBi Þ ¼ pffiffiffiffiffiffi expðB2i =2D2 Þ ð7Þ 2pD with D the second moment of the field distribution. For such a distribution Eq. (6) gives the wellknown Gaussian Kubo–Toyabe relaxation function z GKT ðtÞ ¼ 13 þ 23 ð1 s2 t2 Þ expð12 s2 t2 Þ;
ð8Þ
where s ¼ gm D [2]. In the limit of dilute dipolar spins, the internal field distributions are Lorentzian in form and the resulting Kubo–Toyabe function is [3] z ðtÞ ¼ 13 þ 23 ð1 atÞ expðatÞ GKT
ð9Þ
with a ¼ gm L; where 2L is the full-width at halfmaximum of the Lorentzian field distribution. At intermediate dipole concentrations the Voigtian Kubo–Toyabe provides an appropriate G z ðtÞ [4]. The Kubo–Toyabe functions given by Eqs. (8) and (9) are calculated in the static limit. Internal field dynamics, resulting from the muon hopping
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from site to site or from the internal fields themselves fluctuating, or both, are taken into account by the strong collision approximation, which assumes that the local field changes its direction at a time t according to a probability distribution PðtÞ ¼ expðntÞ; and that the field after such a ‘‘collision’’, chosen randomly from the distribution PðBi Þ; is entirely uncorrelated with the field before the collision. The resulting dynamic Kubo–Toyabe function must be evaluated numerically [2]. However, over much of the dynamic range analytical approximations suffice [1]. For example, in the limit of rapid field fluctuations ðR ¼ n=s > 10Þ the motionally narrowed dynamic Gaussian Kubo–Toyabe function is well represented by a simple exponential z GDKT ðtÞ ¼ expðltÞ
ð10Þ
2
with l ¼ 2s =n; while at intermediate fluctuation rates ð1oRo10Þ the Abragam function is suitable. z ðtÞ GDKT
¼ expflðexpðntÞ 1 þ ntÞg:
ð11Þ
For very slow fluctuations ðRo1Þ only the 13 tail of the Kubo–Toyabe function is affected and the form of the tail becomes z Gtail ðtÞ ¼ 13 expð23ntÞ:
ð12Þ
These analytic approximations are compared with numerical calculations of the dynamic Gaussian Kubo–Toyabe in Fig. 1. It should be noted that in studies of atomic spin fluctuations the motionally narrowed muon spin relaxation rate, l, can be related directly to the associated fluctuating local magnetic fields through the relation XZ N l¼G /Ba ð0ÞBa ðtÞS dt; ð13Þ a¼x;y
0
where G is a coupling constant. It is straightforward [5] to show that kB T X wðqÞ ð14Þ lðTÞ ¼ G N q GðqÞ in which wðqÞ is the wave vector-dependent susceptibility, GðqÞ is the line width of the spin fluctuations and the sum is taken over the Brillouin zone. As inelastic neutron scattering measures wðqÞ and GðqÞ directly it is therefore
Fig. 1. The dynamic Gaussian Kubo–Toyabe function (solid lines) compared with the exponential and Abragam approximations of Eqs. (10) and (11) (broken lines) for R>1, where R ¼ s=n; and time T is expressed as the dimensionless parameter st:
possible to compare mSR and neutron measurements of the temperature-dependent spin fluctuation spectrum. Such a comparison was made for the first time in studies of longitudinal spin fluctuations in the frustrated itinerant electron antiferromagnet YMn2 [5–7], as discussed in the next section. It should be noted, with reference to Eq. (14), that in the particular case of paramagnetic spin fluctuations, such as those addressed in this paper, the muon spin relaxation rate, l, is determined by a sum over the entire Brillouin zone, thereby accounting for both short- and long-range dynamic correlations of the neighbouring atomic spins. It is therefore often not necessary to determine the precise location of the muon within the lattice in order to fully interpret the muon spectra (although a determination of the muon site is generally straightforward). However, in other situations, for example in the presence of longrange magnetic order and associated coherent internal fields, the muon site or sites may be crucial for interpretation of the spectra [1].
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Although Eqs. (8–12) have proved widely applicable in mSR studies of both magnetic and nonmagnetic materials, they do not adequately describe muon spin relaxation in the class of highly disordered magnets known as spin glasses. For example, whilst it might be expected that mSR in a dilute spin glass would follow the dynamic Lorentzian Kubo–Toyabe form, it is found experimentally that, in the fast fluctuation limit above the spin glass temperature, Tg ; the muon spin relaxation is better described by [8] qffiffiffiffiffiffiffiffiffiffiffi z ðt; a; nÞ ¼ exp 4a2 t=n : ð15Þ GDSG This ‘‘root-exponential’’ time dependence has been derived explicitly using a model in which the dynamic variable range of the time modulated field at each muon site, determined by the muon’s proximity to neighbouring atomic spins, is approximated by a Gaussian distribution, whilst the probability distribution for a muon experiencing a particular variable range is chosen such that the overall internal field distribution is Lorentzian, precisely as expected for a dilute system [8]. The complementarity between neutron scattering and mSR in studies of spin dynamics above Tg is evident: Only the neutron spin echo technique can approach similar time regimes (B106 s), but because this technique is often count-rate limited, studies of the more dilute spin glasses such as CuMn and AuFe, can be difficult and time consuming. However, for such dilute spin glasses, mSR is able map the evolution of spin dynamics, often over several decades in frequency, as the glass transition is approached. Moreover, estimates of the spin relaxation rate extracted from the temperature-dependent root-exponential mSR spectra are found to be in close agreement with the rather sparse data obtained using neutron spin echo, as can be seen in Fig. 2 where the combined data indicate a common power-law dependence of the critical slowing down of the spin correlation times [8]. Whilst the root-exponential muon spin relaxation function (Eq. (15)) is also found to be appropriate for dilute superparamagnetic systems [9], recent extensive studies of more concentrated spin glasses and similar frustrated magnets have
Fig. 2. Spin correlation times extracted from mSR studies of dilute CuMn and AuFe spin glasses as a function of reduced temperature. The correlation times obtained from neutron spin echo studies of CuMn are also shown (from Ref. [8]).
shown that, generally, the relaxation function takes the form of a stretched exponential z GCSG ðtÞ ¼ expððltÞb Þ
ð16Þ
for which l is found to diverge as Tg is approached, but significantly b is also temperature dependent. Several explanations for this behaviour have been suggested, the simplest of which invokes a broadening distribution of the fluctuation frequencies n, and hence of l, with decreasing temperature. Interestingly the temperature dependence of b is often found to reflect almost precisely that found in Monte Carlo calculations of the realtime stretched exponential, or Kohlrausch, relaxation of the spin autocorrelation function of Ising spin glasses [11], i.e. /Sð0Þ SðtÞSpexpððt=tÞb
ð17Þ
as can be seen in Fig. 3, with b decreasing from unity at temperatures of approximately 4Tg to a limiting, and almost universal, value of 13 at Tg. The phenomenon of Kohlrausch, or stretched exponential, relaxation in the real-time correlations in structural glasses above the glass temperature is well known: whilst strong structural glasses
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R. Cywinski / Physica B 350 (2004) 17–25
for Mn, respectively) resulting in a spin liquid-like state which persists to the lowest temperatures [5– 7]. On the other hand, substitution of Al for Mn expands the lattice and rapidly stabilizes the Mn moments on the geometrically frustrated tetrahedrally coordinated transition metal sites, leading to a disordered spin glass-like state. In this context it should be noted that the mSR spectra for Y(MnAl)2 are of the stretched exponential form with b approaching 13 close to Tg [6]. For YMn2 and its related alloys neutron measurements show GðqÞ to be only weakly dependent upon q. Eq. (14) can therefore be written Fig. 3. Temperature dependence of the exponent b (a) estimated from the time-dependent spin autocorrelation function found in Ogielski’s Monte Carlo simulations of an Ising spin glass [11] and (b) measured by mSR for AuMn [10].
often exhibit stretched exponential relaxation with a temperature independent b of approximately 0.5, for fragile glasses b decreases from 1 at high temperatures to close to 13 at the glass transition. Similarities with the behaviour of muon spin relaxation in dilute and concentrated spin glasses, respectively, are evident, and suggest that studies of the relatively simple spin glass problem may shed light on the more complex dynamical processes in structural glasses. Neutron spin echo and mSR will have a key role to play in such studies.
4. Spin fluctuations in itinerant magnets 4.1. YMn2 and related compounds The C15 Laves Phase compound, YMn2, undergoes a discontinuous phase transition from a spin fluctuating Pauli paramagnetic state at high temperatures to a helical antiferromagnetically ordered state below 100 K. This transition is accompanied by a dramatic 5% increase in cell volume. The transition to the ordered state can be suppressed by weak external or chemical pressure (e.g. by 0.3 Gpa or substitution of only 2.5 at% Fe
lðTÞ ¼ G
kB TwL ; G
where wL is the local susceptibility given by 1 X wðqÞ: wL ¼ N q
ð18Þ
ð19Þ
The neutron scattering spectra of Y(Mn0.9Al0.1)2 show GðTÞ to be Arrhenius-like with an activation energy of 283 K, whilst wL ðTÞ is Curie–Weiss like with y ¼ 93 K: lðTÞ can thus be evaluated using Eq. (15) and then scaled to the measured lðTÞ with the coupling constant G as the only free parameter [6]. As can be seen in Fig. 4, the correspondence is excellent. The inset to Fig. 4 shows the temperature dependence of the line-width parameter, G; for YMn2 and its dilute alloys with Sc, Fe and Al substitution for Mn, as determined directly by neutron scattering and from mSR data scaled according to Eq. (18). It is clear that G narrows considerably on addition of Al as the YMn2 lattice expands and the Mn moments tend to localize. Moreover the approximately linear dependence of G with T for these dilute alloys is in full accord with predictions of SCR theory of spin fluctuations [12]. The neutron spectra from which wðqÞ and GðqÞ are extracted contain more information than the corresponding muon spectra, as the latter provide only the dynamical response averaged over the Brillouin zone and hence contain no spatial information. However, whilst each individual neutron spectrum required at least one day of
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Fig. 4. The measured mSR relaxation rate for Y(Mn0.9Al0.1)2 compared to l(T) calculated from scaled neutron data using Eq. (18) (solid line) [6]. The dashed line indicates the spin glass transition. Inset: Line-width parameters for YMn2 and its dilute alloys from neutron scattering (full symbols) and scaled mSR relaxation rates (open symbols) [5]. The lines in the inset are guide to the eye.
beam time to collect, all the mSR measurements shown in Fig. 4 were obtained in just a few hours. mSR is therefore an ideal technique for extensive and systematic studies of spin fluctuations. Furthermore, mSR substantially extends to lower frequencies the dynamical range made available by neutron scattering. 4.2. b-Mn and b-Mn1xAlx Combined mSR and neutron scattering studies have also provided some detailed insights into the nature of longitudinal quantum spin fluctuations and moment localisation in elemental b-Mn and its alloys with Al [13,14]. b-Mn is the only allotropic form of elemental manganese that does not possess a magnetic moment at any temperature. It crystallises with
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the cubic A13 structure with eight and 12 Mn atoms at the inequivalent site I and site II positions, respectively. NMR and NQR data suggest that whilst site I Mn atoms are very weakly magnetic, those at site II are associated with strong, slow antiferromagnetically correlated spin fluctuations [15]. Similarities with YMn2 are evident, particularly as the topology of the site II atoms, described as a network of corner sharing triangles, also leads geometric frustration. Inelastic polarized neutron scattering studies of the quantum fluctuations in b-Mn showed that the spectral width of the excitations decreased markedly from approximately 20 meV to only a few meV with increasing Al substitution at site II, reflecting the combined effects of lattice expansion and a lifting of the spin configurational degeneracy associated with the geometric frustration [15]. mSR has helped us develop a more complete picture of the evolution of the spin fluctuation spectra in b-Mn1xAlx [13]. For b-Mn the exponential mSR spectra exhibit extreme motional narrowing (lB0.02 ms1) at all temperatures, consistent with the broad and weakly temperature-dependent inelastic neuron line width. With the addition of Al, for xo0:09; the mSR spectra remain exponential, with l diverging towards low temperatures (3–5 K), indicating a critical slowing down of the Mn spin fluctuations with an exponent, g, very close to 1 (Fig. 5). However, as the Al concentration increases beyond x ¼ 0:09 the mSR spectra abruptly adopt the stretched exponential form, as described by Eq. (16), characteristic of concentrated spin glass behaviour. At these concentrations l again critically diverges (Fig. 5) but at much higher temperatures (30–40 K). The exponent b decreases from unity at high temperatures to 13 at the critical temperature. It is suggested that these features, summarized in Fig. 5, are evidence of a quantum spin–liquid to spin–glass transition in b-Mn1xAlx [13]. These mSR measurements have, in turn, prompted a more extensive time-of-flight inelastic neutron scattering study spin fluctuations in b-Mn and its alloys with Al, and the results of these studies provide clear evidence for non-Fermi liquid scaling behaviour in unalloyed b-Mn over a relatively wide temperature range [14]. Significantly, this is
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Fig. 5. Critical scaling of the muon spin relaxation rates in bMn1xAlx alloys. The inset shows the corresponding magnetic phase diagram [13]. Note the marked difference in behaviour for xp9% and x>9%.
below TC which collapse into diffusive modes above TC [18]. In addition, the results of magnetisation, NMR and resistivity measurements on Ni3Al are all in close agreement with the predictions of the self-consistent renormalisation (SCR) theory of spin fluctuations for weak ferromagnets. The atomic spin contribution to the mSR spectra of stoichiometric Ni3Al remains fully motionally narrowed down to 40 K, whilst below 40 K mSR, somewhat controversially, provides no evidence for long-range ferromagnetism [17]. Instead, a static but disordered magnetic ground state, represented in the mSR spectra by a static Voigtian Kubo–Toyabe function, begins to emerge and grow, accounting at the lowest temperatures for almost two-thirds of the mSR signal (Fig. 6). Although the Ni3Al mSR sample is found to be crystallographically well ordered, the low-temperature magnetic state revealed by mSR is intrinsically inhomogeneous.
the first reported example of such scaling in a pure elemental paramagnetic metal. 4.3. Itinerant electron ferromagnets: Ni3Al Interestingly, one of the earliest and most successful examples of the application of the zero field mSR technique was the detailed study of longitudinal spin fluctuations in the itinerant electron ferromagnet MnSi [2,16], a study which provided considerable support for SCR theory [12]. It is therefore surprising that there have since been very few comparable studies of other itinerant ferromagnets. However, a recent mSR investigation of the archetypal weak itinerant electron ferromagnet Ni3Al has highlighted potential problems associated with such studies [17]. Stoichiometric Ni3Al, has a reported Curie point of 40 K and a nickel moment of approximately 0.076 mB. Small-angle neutron scattering shows a sharp critical scattering peak at the reported Curie temperature, indicating a divergence of the spin correlation range, whilst inelastic neutron scattering reveals well-defined spin waves
Fig. 6. Zero field mSR spectra from Ni3Al. The solid lines represent a fit to a magnetically inhomogeneous model, with one component fully motionally narrowed and the second represented by a static Voigtian Kubo–Toyabe function. These two components are themselves multiplied by a static Gaussian Kubo–Toyabe function associated with the nuclear dipole fields at the muon site.
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mSR and neutron scattering have thus provided quite disparate views of the magnetic ground state of Ni3Al. Similar discrepancies have also been observed for another itinerant electron ferromagnet, PdNi, for which mSR provides no evidence of magnetic order below the reported Curie point [19]. It remains uncertain whether this is a consequence of the fundamental differences in the way that neutrons and muons probe spin correlations in such systems, or whether the muon itself, rather than acting as a passive probe, is directly responsible for locally perturbing the weak ferromagnetic order. This is clearly an area in need of further detailed theoretical and experimental investigation.
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is profound, and highlights the need for further work.
Acknowledgements The author wishes to thank his many mSR collaborators, including P.M. Bentley, R.I. Bewley, I.A. Campbell, M.R. Crook, J.A. Dann, J.L. Garcia-Munoz, A.D. Hillier, S.H. Kilcoyne, P. Manuel, C. Pappas, M.J. Pottinger, J.M. Preston, B.D. Rainford, C.A. Scott, J.R. Stewart and M.T.F. Telling, and the muon support scientists at both ISIS and PSI.
References 5. Conclusions In this paper, the complementary roles of mSR and neutron scattering in studies of both fast spin fluctuations and slow spin dynamics in some itinerant electron magnets have been considered. Whilst neutron scattering provides spatial and temporal information on spin correlations, mSR provides information only on the dynamics. Nevertheless, in the fast fluctuation limit mSR facilitates a rapid and systematic characterisation of the evolution of the spin dynamics as a function of, for example temperature or atomic substitution. In the slow fluctuation, spin glass-like limit, mSR extends considerably the dynamic range offered by neutron spin echo, whilst also providing some remarkable insights into the non-exponential nature of the dynamic spin correlations above Tg. Indeed, mSR has provided arguably the best information available to date on spin dynamics in the paramagnetic phase of spin glasses. In most cases the information obtained using neutron and muon techniques, in both the fast and slow spin fluctuation limits, is in excellent agreement and in accord with theory. Where there is disagreement, for example in the case of Ni3Al, it
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