90 G.M. KALVIUS et al.
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S
~tSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 91 One finds Blor = g~/0Mdom 1 where n d o m is the magnetization of the domain in which the Lorentz sphere is situated. Strictly speaking this result is true only for non-conducting compounds since in metallic ferromagnets an additional (but usually small) contribution to Maom arises from conduction electron polarization which is of course not part of Bdip. The demagnetizing field can be expressed as Bdem =
-#oHMs
where A/" is the demagnetization tensor (a quantity depending on the geometrical shape of the sample) and Ms is the total sample magnetization. For a spherical sample A/" = ½. Using such a shape and enforcing Maom =Ms by fully saturating the material, magnetically reduces eq. (15) toB~ = B~i p +Bco n + Bapp, (16)
which represents a significant simplification.
In general Bdem is a troublesome quantity (especially for a polycrystalline sample consisting of oddly shaped grains) since it produces non-homogeneous fields which substantially increase (B~) and hence static damping. Of course, a contribution by Bdern can be avoided altogether by measuring in zero field. This is the standard procedure in case of ordered magnetism (see sect. 3.3). If ferro- or ferrimagnetism is present one must be sure that the sample shows no remanent magnetization. In paramagnets the magnetization is usually small for the fields used in vSR. It can be a problem, however, for Knight shift measurements in strong paramagnets such as the heavy rare earths.
The crucial point in the Lorentz construction is how to select the size of the Lorentz sphere. Its radius must be large compared to interatomic distances, otherwise the dipolar sum strongly depends on the sphere's volume. On the other hand it must be less than domain dimensions in order to have a unique definition of Mdom. These conditions can, however, be fulfilled quite well. Finally, we point out that the Lorentz construction has originally been developed for ferromagnets. No macroscopic or domain magnetization exists in antiferromagnets under ZF conditions and both the Lorentz and the demagnetization fields are not present. An antiferromagnet, however, has a well- defined susceptibility. In consequence a magnetization develops in the presence of an external field (spin canting). Then Blot and Bdem must be taken into account.
Bdip and Boon differ in their directional dependences. While Boon will point into (or opposite to) the direction of Mdom, this is in general not the case for Bdip. Symmetry arguments show that B~i p always vanishes if cubic symmetry exists for the location of surrounding dipoles with respect to muon site. Furthermore, any isotropic distribution of dipoles (i.e., a powder sample) leads to B~i p = 0 when one averages over the whole crystal.
One must not forget, however, that Bg =0 does not imply (B~)=0! In addition, due to their difference in spatial dependence, the vanishing o f B~i p does not necessarily mean that Boon also vanishes. We return to this problem in sect. 3.3.
One can imagine influences of the muon also o n Bdip. Firstly, the muon distorts the lattice around its position (self-trapping, small polaron). This can change the dipolar sum over the first neighbor shell. Secondly, the muon may not be that well localized at r -- 0, even if it does not diffuse from site to site. The muon could have an extended wave
92 G.M. KALV1US et al.
fimction ranging over the whole interstitial site or it may carry out jump motions within the interstitial hole (caged diffusion). A certain averaging of B~ip(r ) is the most obvious consequence (see Meier 1984). Whether an overlap of the muon wave function with the electronic wave functions of neighboring atoms can also influence the magnitude of their magnetic moment is an open question.
3.2. #SR spectroscopy of dia- and paramagnets
The purpose of this and the next section is to familiarize the reader with the basic features of gSR spectra of magnetic materials and the parameters which determine their shape.
We shall stay on very general grounds. In actual magnetic studies the situation may be considerably more complex. The discussion primarily follows the original development of ~tSR spectral theory which, as already stated, closely followed the NMR analogue.
It is thus geared to interaction with nuclear moments. This is in fact the situation that prevails in diamagnets. The extension to electronic moments is not always straightforward.
Paramagnets are a simple case, because in general the electronic moments can be viewed as approximately point-like, independent dipoles. One important difference is that nuclear dipoles can in most situations considered to be static, but electronic moments are usually dynamic. Nuclear spin relaxation times are seldom shorter than 10 .4 s, so their correlation time is much longer than the muon life time. In contrast, electronic spin fluctuation frequencies in free paramagnets are beyond the GHz regime and therefore reside in the fast fluctuation limit, which will be discussed below. The extension to ordered moments is more complex and not fully solved in the more exotic cases. We shall treat the basics for ferro- and antiferromagnets in the following section and spin glass behavior in sect. 8.
Special cases will be discussed as the need arises.
In a material containing dipolar moments (either electronic or nuclear) the muon at its resting place (to be discussed in sect. 3.6) experiences an effective magnetic field B~ (either static or fluctuating in time), as discussed in the previous section.
The individual members of the muon ensemble that generates the ~tSR spectrum (see sect. 2.6) do not see exactly the same field Bg, even if all muons come to rest at one type of internal site. This is due to small changes in the local surroundings, in particular with respect to the field generated by nearby dipoles with different spatial orientation. By B~ we denote in the following the mean field sensed by the muon ensemble (for clarity we no longer use the notation (B~)). We will introduce, where necessary, an appropriate parameter which describes the distribution of fields around its mean B a. In addition to getting information on the magnitude of Bg and its distribution, a major objective of ~tSR in magnetic materials is the measurement of the muon spin relaxation function which then can be related to the dynamics of the spin ensemble in the sample material. We turn now in succession to ~tSR spectra obtained in the three fundamental experimental geometries, excluding muon spin resonance and level crossing data. A review on "gSR relaxation functions in magnetic materials" appeared recently (Uemura 1999).
IxSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 93
3.2.1. Transverse field measurements (Knight shift)
"Transverse field" (TF) means that the applied field is oriented perpendicular to the initial muon spin polarization. As mentioned, this does not necessarily mean that the field is oriented perpendicular to the muon beam. With a surface beam, it may be oriented along the beam momentum when the muon spin has previously been turned by a spin rotator (see sect. 2.5). We restrict this discussion (except for some short remarks) to the strong-field limit, that is to say, we assume that the local quantization axis for muon spin and its surroundings is determined by the externally applied field alone. Then only the secular term in the Zeeman interaction of the local moments with
Bapp
need to be considered (details can be found, for example in Schenck 1985, chapter 2.3.1). Sensing a transverse field, the muon spin will precess in the plane perpendicular to the field axis, which generates the asymmetry spectrum~4( t) = aoGx( t) cos(2arf~t). (17)
The asymmetry A(t) has been defined by eq. (9). The spin precession frequency f~ is directly proportional to the magnitude of B~:
f~ = (y~/2zg)Bg. (18)
We shall discuss the information contained infg below. Gx(t) is the transverse muon spin relaxation function as indicated by the index x (the z-axis is commonly fixed parallel to the muon spin).
In the case of a dense system of randomly oriented moments, the field distribution can be assumed to have Gaussian shape. Truly random orientation is certainly fulfilled for a nuclear moment system (except at extremely low temperatures, which are out of the reach of btSR). For electronic moments it is strictly true only for a free paramagnet. This field distribution is added to Bapp and in summary B~ is distributed. The width of this field distribution can be characterized by its second moment, the so called polycrystalline Van
Vleck moment, originally derived for nuclear moments:
2 4 2 2 2 6 4 2 V - " 2r-6
°v2v= ( 4 ~ ) ~o E l ( I + 1)7('7ih ri ~ ( 4 ~ ) 2 ~ yu 2 - ~ f ~ i i ' (19)
i i
where I is the nuclear angular momentum, D the corresponding gyromagnetic ratio and ri is the distance to the ith dipole from the muon site. An analogous expression could be used for electronic moments. In general one can write
0.2= ]]2 ( bt}, B 2 (20)
with (B 2) being the width of the (Gaussian) field distribution. This leads to the transverse field relaxation fimcfion
G x ( t , r ) = e x p { - a 2 r 2 I ( t ) - l + e x p ( - t ) ] } , (21) which is commonly referred to as Abragam relaxation. 1/r is the fluctuation rate o r b s . Inherent assumptions as to the relaxation process are hidden in eq. (21). The derivation
94
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0.6
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G.M. KALVIUS et al.
i i
4 6 8
t ~
Fig. 16. Transverse field muon spin re- laxation function (Abragam relaxation).
The various curves are labeled by the value o f r - a (in rad). They are the envelopes of the oscillatory spin precession patterns.
by Abragam (1961) is based on a Gaussian-Markovian process meaning that the field correlation function takes the form
(72
As seen from eq. (17), the relaxation function Gx(t) is the envelope of the spin precession pattern cos(27cf~t). Its form is shown in fig. 16 for r as the parameter. In the static limit (r ---+ oc) we have pure Gaussian relaxation
Gx(t) = exp [-½azt2] . (23)
In the fast fluctuation limit ( r ~ 0), the decay of muon spin polarization becomes exponential:
Gx(t) = exp[-a2Tt] = exp[-,~t]. (24)
According to eq. (20) one finds
= O'2T = 7~
(B2) r. (25)
One often denotes a as the static and )~ as the dynamic relaxation rate. They are usually given in p~s -l (corresponding to 106 rad/s), One notices, especially from fig. 16, that the damping becomes weaker with rising fluctuation rate. This effect is known in NMR as motional narrowing. Static (Gaussian) relaxation of muon spin precession is also sometimes called inhomogeneous broadening, again borrowing from NMR. We illustrate
gSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 95
A(t)
o o ~
(a) Time
A(t)
~ I i (b) me
Fig. 17. Transverse field p.SR spectra showing the limiting cases of (a) static (Gaussian) and (b) full dynamic (exponential) depolarization. From Karlsson (1995).
the limiting situations of static and dynamic damping with the two spectra shown in fig. 17.
According to eq. (21), transverse field measurements allow, in principle, separation of the static field width (oc a) from the fluctuation rate ( l / r ) in an intermediate case.
In practice, this is rather difficult and the combination of zero and longitudinal field measurements are more powerful in this respect, as will be shown further below. In the static limit (v --+ oc) we can extract the second moment of the field distribution (see eq. 20). In the fast fluctuation limit only the product a 2 T appears (eq. 24) and independent information on one of the two quantities is needed.
One basic assumption leading to eq. (21) was the Gaussian shape of the field distribution. We had stated that this requires a dense system of moments. In dilute systems (if less than ~10% of the surrounding atoms have a moment - see Kittel and Abrahams 1953) the field distribution is close to a Lorentzian shape (Walstedt and Walker 1974).
A Lorentzian has no second moment and the distribution width must be characterized by its half width at half maximum. It will not show motional narrowing: whether the local field is static or fluctuating, the experimenter will observe exponential relaxation, whose relaxation rate is basically insensitive to the field fluctuation rate 1/T. In this case not much reliable information can be extracted from TF data. Dilute moment systems are therefore usually studied with ZF- and LF gSR.
We now discuss the information obtainable from the muon spin precession fre- quencyfg. The strong-field limit means that the external field is large compared to internal field contributions andf~ is mainly determined by Bapp. Furthermore, for a truly random distribution of internal fields we expect their mean to be zero. The applied field, however, causes (a perhaps very weak, but still finite) magnetization of the sample and thus destroys the full randomness of moment orientation. As a result, a small internal field adds to the applied field. The resulting spin precession frequency V~ bs = 7~B~ is slightly shifted from the value v ° = ~/aBapp expected if the external field alone were present. This is known as the muonic Knight shift. One defines the Knight shift constant
K - B~ . (26)
Bapp
96
0"5 j ,, 300 mT