Anisotropic spin fluctuations

4. Elemental metals

4.1. Anisotropic spin fluctuations

Of particular interest is the situation in the paramagnetic regime, where no axis of magnetization exists on a macroscopic scale and the crystalline axes (in non-cubic systems) provide preferred orientations. The basic observables of muon relaxation above Tc or TN in anisotropic systems like the hexagonal lanthanide metals have been discussed by Karlsson (1990, 1995)). A recent discussion of correlation functions and longitudinal relaxation rates can also be found in Dalmas de R6otier and Yaouanc (1997). As outlined earlier (and to be discussed below), the muon spin relaxation rate in the fast fluctuating limit is given by 3` = y2 ( ~ ) . r~ where rc is the characteristic time for the fluctuations B 2 of the local field Bg. The temperature dependence of 3` (or r~) can then be used, for example, to make comparisons with predictions from dynamic scaling theories, as will be discussed for the case of Gd. Single-crystal measurements that combine fixed orientations (with respect to crystalline axes) o f the muon spin (at t = 0) with definite orientations of the local fields pick up different parts of the local field correlations

(B~(t),B~(O))

and

(B~(t),

B~(0)> at the muon site. In the present case we take the hexagonal c-axis as the main symmetry axis (parallel direction).

The muon spin relaxation rates 3`11 and 3,]_, observed in ZF measurements with muons implanted with initial spin polarization parallel and perpendicular to the c-axis, are given by

f0 G

3`11 = Y~

dt[(B~(t),BX~(O)> + (BY~(t),B~(O)>] ,

/o

3`]- = 7~ 2 dt[(B;(t),

B;(O)) +

(B~t(t),

B;(O))],

(56)

(57)

where z is the direction of the symmetry axis and x and y are the directions perpendicular to it. These are identical in case of uniaxial symmetry, which has been assumed in all the FtSR work. There is some anisotropy within the basal planes of the hexagonal rare- earth metals (it is involved in the spin-slip structures mentioned below), but it is much smaller than the anisotropy between the c-axis and the basal plane, and unlikely to be

~SR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 125

detectable in the relaxation rates of the equations above. By combining measurements of relaxation rates for muon spin alignments parallel and perpendicular to the c-axis a separation of fluctuations of the local field components is possible. But as eqs. (56) and (57) demonstrate, the rate )~ll only contains the perpendicular field correlations, while )~± contains both the perpendicular and parallel field correlation functions. Hence the evaluation of the directional dependence of field fluctuations or fluctuation times is not necessarily trivial.

The situation is more complex in transverse field experiments, where the orientation of the applied field with respect to the symmetry axis is an additional parameter. Two cases need to be considered for uniaxial symmetry:

(a) Sg points parallel to c and Bapp perpendicular to c. The correlation functions appearing in equations of the type (56) or (57) are (Karlsson 1995)

2 (8 t),B 0)) +

(b) S~ points perpendicular to c and Bapp parallel to c. Then one considers the correlation functions

= +

For uniaxial symmetry, this reduces to two distinct TF relaxation rates to be measured, as in ZF.

It is important to stress once more that p~SR measures the fluctuations of the localfields at the muon site and not directly the fluctuations of the spins creating those fields. This means in essence that for a quantitative analysis (for TF as well as for ZF measurements) the exact relations between the individual components of lanthanide spin correlations (Si(t)SJ(O)) and the muon field correlations (B~(t)U~(O)) must be worked out for each particular muon spin +-~ lanthanide crystal geometry. This step is often overlooked. The general tensorial relation between spin and field correlation has been given by Dalmas de R6otier et al. (1996).

Barsov et al. (1986c) directly use the short-time approximation to evaluate the field correlations. This leads to an exponential time dependence of the different components of the field correlations

i 2

(B~(t'),Bi~(t)) = ( ( , ~ ) ) exp ( - I t - / [ / U ) , (58) with orientationally dependent characteristic fluctuation times r~. Here the v{ are the lifetimes of the magnetic field components B~, which originate from the spin fluctuations determined by the correlation functions

(si(t)sJ(o)).

^{T h i s} means that the r~ do not directly

126 G.M. KALVIUS et al.

reflect the spin fluctuation times in different directions, since each may contain more than one directional component. Using the approximation given in eq. (58) the following expressions for the relaxation rates in uniaxial symmetry

Xll = 2G±

A± = GII + G±

with G ~ = y ~

2((

B~ c ,

2 TII,

with GII = y~ B " c

are obtained for the two orthogonal muon implantation geometries. Modifications of these formulae for intermediate situations, i.e., textured polycrystals, are straightforward:

Xll = 2 G ± r + (all + G ± ) (1 - r), Z± = G±(1 - Y) + g (Grl + G±) (1 + Y), 1

where Y = (cos 2 0) and 0 is the angle between SL~ and the symmetry axis (c-axis) in a particular crystallite. In this approximation it is also easy to write down the corresponding expressions for the TF conditions.

One basic problem (which has already been pointed out in sect. 3.2.2) o f the fast fluctuation limit is that a separation o f (]Bp~l 2} and rc is not possible. Furthermore, since the expression (58) for the different directional components i contain different correlation times T/, it is incorrect to express the total relaxation simply as

<IB I2>

in a strongly anisotropic case. Therefore, from a measurement of )~IF and 2± it is not trivial (among other considerations, one must know the muon interstitial site) to determine how much of the anisotropy comes from an anisotropic field distribution and how much relates to anisotropic fluctuation rates. One may hope that not too close to the magnetic transition (i.e., well inside the paramagnetic regime) (IBa 12) shows little anisotropy and that one senses the anisotropy of rc in ZE But an exact treatment of a btSR measurement on these terms has not been given to our knowledge.

4.2. Gadolinium

Gadolinium is probably the most extensively studied elemental lanthanide metal not only in the context of bulk magnetic measurements, but equally for ~tSR spectroscopy. This is not true, however, for neutron scattering, because of the excessively high absorption cross section of natural Gd. In part, this has been overcome by selecting a neutron wavelength where absorption is much less (Cable and Wollan 1968). Also, a single crystal of the weakly absorbing isotope 16°Gd has been produced. A review of the band structure can be found in Norman and Koelling (1993).

From a magnetic point of view, Gd is the simplest of the lanthanide metals. Exchange favors FM, while all other heavy lanthanide metals initially show AFM order. The Curie temperature (Tc = 293 K) is in a convenient temperature range. The Gd 3+ ion has a half- filled 4f shell. It is thus an S state ion featuring pure (but very strong) spin magnetism.

~tSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 127 CEF interactions vanish because of the S character (see sect. 5.1.1). The small residual anisotropy causes the moments to point along the c-axis just below T o About 100 K below Tc, the easy axis begins to move toward the basal plane, reaching quickly a maximum tilt angle of O ~ 60 ° at 180 K. On further lowering the temperature to ~50 K, O is reduced slowly to 30 ° , where it then stays to the lowest temperatures. It is thought that this spin turning arises from the action of a competing anisotropy having its origin in spin-orbit coupling.

The earlier ~SR studies were concerned with the peculiar temperature dependence of the spontaneous precession frequency caused by the interplay of the isotropic contact field and the anisotropic dipolar field at the muon site. The latter is clearly dependent on the spatial alignment of the Gd spins. Also, the muon stopping site needed to be determined.

These data will be discussed in sect. 4.2.2 below. Later studies included more detailed investigations of the dynamical critical behavior in the paramagnetic state (see following section).

4.2.1. P a r a m a g n e t i c r e g i o n a n d c r i t i c a l b e h a v i o r

Relaxation measurements in the paramagnetic state of Gd have been numerous. A typical example of a TF%tSR spectrum of Gd metal above Tc has been presented in fig. 9.

Such type of measurements by W~ickelg~rd et al. (1986, 1989) show the existence of spin correlations well above 2 T¢. The relaxation data can be represented by a power law of the type )~ oc ( T - T c ) -w. As can be seen from fig. 29, a crossover from w = 0.56 (high-temperature value) to w = 0.15 takes place at T - Tc ~ 10K. At about the same temperature the Knight shift (measured on a spherical polycrystalline sample where Lorentz and demagnetization fields cancel) changes from Curie-Weiss behavior (exponent 1) to a temperature dependence with exponent 1.25. The reason for this change of slope has to be ascribed to the onset of dipolar interactions between the Gd spins (see also discussion in the next paragraph). As described by Karlsson (1990), one consequence is a break-up of the longitudinal spin waves resulting in a reduced temperature dependence of spin fluctuation rates. Karlsson et al. (1990) discuss the influence of dynamic paramagnetic clusters on neutron and ~tSR parameters. W~ickelgfird et al. (1986) also attempted an estimate of correlation lengths from their relaxation data and found ~(1.2Tc) = 5 A and ~(1.5 Tc) = 3,~. The exponent w can be related (Hohenemser et al. 1982, 1989) to the correlation length exponent v and the dynamic scaling exponent z v i a w = v . ( z - 1.06). The value of w is consistent with the mean field values v -- 0.5 and z = 2.

The paramagnetic critical region was studied later in more detail with zero field ~tSR on single crystals (Hartmann et al. 1990a). Anisotropy of the relaxation rate for muons with their spins either parallel or perpendicular to the c-axis was found (see fig. 30, left). Such orientation-dependent muon relaxation will be discussed in more detail in connection with Er and Ho. The directional differences are comparatively small in Gd because the strong single-ion anisotropy is absent. Further extensive measurements by Henneberger et al.

(1997) have been compared to elaborate theoretical calculations (Dalmas de R6otier and

128 G . M . K A L V I U S et al.

0.1 a o

>,

0.01 (/)

>.. 1-

0 z U.l 0 LU rr LL

0,1

",. i

[] ,, !

0

^{k b a r}

I I H I i [ l i i r H i ] i I I I H I

6 k b a r

i . . . i'0 . . . i d0

T - T c [ K ]

Gadolinium metal

LM , ~ 0.1 rr Z

0 1 :

I-- ,<

X <, 12S UA

0.1

inlnl ~ n IllEFE I I I Ennlnll n

1 1 0 1 0 0

T - T c [ K ]

Fig. 29. Temperature dependence o f the frequency shift (left) and TF relaxation rate 0"ight) o f Gd metal at ambient (top) and 6 kbar applied pressure (bottom). The measurements under ambient conditions used a polycrystalline specimen (W/ickelghrd et al. 1986), the high-pressure data a single crystal (Sctneier et al.

1997). The lines are power law fits as discussed in the text. The various symbols in the top-left panel refer to different values o f the transverse field.

Yaouanc 1994, Dalmas de R~otier et al. 1994a, Frey et al. 1997). Henneberger et al.

(1997) shows that close to Tc, not only the influence of dipolar interactions, but also the uniaxial anisotropies are of importance to reproduce precisely the critical behavior of the

~tSR relaxation rates.

A theoretical summary analysis in terms of Gd spin dynamics above Tc combining the single-crystal ~tSR results with paramagnetic fluctuation data on Gd from perturbed angular correlation and M6ssbauer spectroscopies has been given by Frey et al. (1997).

In applying mode-coupling theory to hexagonal lattices, and ascribing the uniaxial anisotropy to dipolar couplings only, they can indeed reproduce the major features of the observations from the three different methods. Figure 30 (right) depicts the case of perpendicular FSR relaxation rates from their work. The first change in the slope o f m u o n spin relaxation rate vs. temperature occurs at the temperature marked TD. This effect is also visible in the data of Wfickelghrd et al. (1986) shown in fig. 29. A second change, but less pronounced, takes place at ira. It is found that Gd behaves for T > TD like an isotropic Heisenberg FM. Dipolar anisotropy comes into play at T < TD. Finally, for T < TA one must consider uniaxial anisotropy as well. The major result of the treatment by Frey et al.

(1997) is that the spin dynamics o f a ferromagnet with an hcp lattice (such as Gd) belongs to a new dynamic universality class (see Henneberger et al. 1999 for details). The fit of the muon data to the expected theoretical variation of)~(T) presented in fig. 30 raises the

gSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 129

.== .4 1 : • :

o,8_

r r 0 . 6 - ¢ ~ c) ~ g i i A

O

~x 0.4- [ fir ' ' "

(~ • S perp. c •

"~ o S parallel c

rrE 0 . 2 - ~ ~ r y , octahedralsites I \

"~- "~E . Data 1994 I \

~, ° 0 . 1 • Data 1989/92 ]

0 . 0 . . . ~ . . . ' . . . ]

293 294 2 9 5 2 9 6 n 0.01 0.1 1 10 100 1 0 0 0

T e m p e r a t u r e [ K ] T - T c [ K ]

Fig. 30. Critical spin fluctuations in Gd from ZF%tSR measurements. Left: Anisotropy of the relaxation rate )~

just above the Curie point. After Hartmann et al. (1990a). Right: Comparison of the temperature dependence of the relaxation rate measured in perpendicular geometry with the prediction of mode-coupling theory for two muon stopping sites. The rates of 1989 were re-analyzed, changing slightly their absolute values (compared to those plotted in the left-hand panel). T D and T a mark the points where dipolar and uniaxial anisotropies start

to play a significant role. After Henneberger et al. (1997).

question whether the assignment o f an octahedral muon site, which had been established in the ferromagnetic regime first by Graf et al. (1977) (see also Denison et al. 1979), should be changed to a tetrahedral site for temperatures above Tc.

4.2.2. Ferromagnetic region

Studies o f the spontaneous muon spin precession frequency in ferromagnetic gadolinium attracted much attention due to its peculiar behavior as function o f temperature. First data by Gurevich et al. (1975) were followed by measurements o f Graf et al. (1977), Nishida et al. (1978) and Hartmann et al. (1990a). The results o f the various groups are quite alike.

Minor differences are probably sample dependent. A characteristic ZF%tSR spectrum has been presented in fig. 26. The observed temperature dependence o f the precession frequency (see the curve labeled p = 0 G P a in fig. 31) shows initially the Brillouin- type rise usually seen after a second-order transition into the magnetically ordered state.

Further down in temperature, marked deviations from this monotonic rise occur, which can be explained fully by the change o f the dipolar field contribution to B~ as the Gd spins change their tilt angle with respect to the c direction. This effect is particularly noticeable in the temperature dependence o f the local field in Gd because Bdip and Boon are o f comparable magnitude. Using a formalism (Meier et al. 1978, Meier 1984) based on the condition o f a smooth temperature dependence o f the sum vector

X = Beon + Blor = B~ - B~ip, (59)

130 G.M. KALVIUS et al.

9

UJ ,'7 .J o O J

2250

1 I

_

l

1250- A ~" ' Ji'" \ _ / \, :', 1 o o o - " " / ~ ~ _

/\Le,

750. / \ i~.#,, 3

500-

- -o-.---4.5 kbar 250- ~ 6 kbar

0 ~

50 100 150 200 250 300 TEMPERATURE [ K]

17.5-

0.00-

, , . , , , • , . , .

[ ] Gd metal

U

215 220 225 230 235 240 245 TEMPERATURE [ K]

Fig. 31. Left: Temperature dependence of local magnetic field at the muon site in ferromagnetic Gd at ambient and applied pressure (Schreier et al. 1997). Right: Results of the detailed analysis of spectra in the spin turning regime. Two signals are present (open and solid symbols). Their meanings are described in the text (Hartmann

et al. 1994b). The lines are guides to the eye.

allows safe separation o f the contact and the dipolar contributions to the local field at the muon site, even for a polycrystalline sample (Graf et al. 1977, Denison et al.

1979). The result not only identified the muon stopping site as the octahedral interstitial position, but also allowed the extraction o f the temperature dependences o f the contact field contribution and the spin tiring angle O. The latter is found to be in excellent agreement with other determinations (CaNe and Wollan 1968, Corner and Tanner 1976, Graham 1962) as demonstrated in fig. 32 (top). ~tSR gives a steeper initial rise o f the tilting angle. The results shown in this figure are based on a recent evaluation o f single- crystalline data (Hartmann et al. 1994b). The differences with earlier results are minor.

The temperature dependence o f the contact field deviates slightly from that o f the bulk magnetization (Denison et al. 1979). More pronounced such deviations are seen in the 3d FM Fe and Ni. They are discussed in terms o f disturbances o f the local band structure by the muon (see, for example, Kanamori et al. 1981). The sign and magnitude o f the contact field o f - 0 . 7 0 T for T ~ 0 are in good agreement with the Knight shift measurements o f W/~ckelggtrd et al. (1986, 1989) mentioned earlier.

The single-crystal measurements by Hartmann et al. (1994b) add more details about the spin turning process. The study showed that the turning o f spins does not occur uniformly and simultaneously over the whole sample. The spectra between 230 K and 220 K are best described by the sum o f two subspectra. One exhibits an increase o f frequency (which indicates spin turning) with decreasing temperature while the other roughly shows

gSR STUDIES OF RARE-EARTH AND ACT1NIDE MAGNETIC MATERIALS 131

90 60 30 0 90 60 3O 0 t~ 90

" a O O 60

c -

30

e- 0

.~_

~_ 9o 60 30 0 90 60 30

G a d o l i n i u m

o.o o r \

j 6 . O k 0

0 50 100 1-50 200 250 300

T e m p e r a t u r e [K]

- - p S R [] n e u t r o n s

t o r q u e : '~ 1 9 7 6 - - - 1 9 6 2

Fig. 32. Spin turning angle (with respect to the c-axis) in ferromagnetic Gd metal as a function of temperature for ambient (top) and applied pressures derived from the temperature variation of the internal field as shown in fig. 31. Results from other measurements at ambient pressure are shown for comparison. The panels depicting the results at applied pressures also contain the data for ambient pressure (dotted line) to make the differences

more apparent. From Schreier (1999).

constant frequency, meaning that it reflects a portion o f the sample where spins have not yet started to turn. The amplitude o f the former sub-spectrum rises continuously while the amplitude o f the other decreases concomitantly until it vanishes at ~220 K, indicating that now all spins in the sample have turned (fig. 31, right).

4.2.3. High-pressure measurements

High-pressure g S R studies (Mutzbaner et al. 1993, Kratzer et al. 1994a, Schreier et al.

1997) on G d (a typical spectrum has been shown earlier in fig. 12) revealed that the

132 G.M. KALVIUS et al.

temperature dependence of the precession frequency (the field at the muon site) is highly sensitive to applied pressures even in the fairly low pressure range up to 0.6 GPa (6 kbar).

The dominant reason is the dependence of the spin turning process on pressure. The findings are summarized in fig. 32. Under hydrostatic pressure the onset of spin-turning gradually shifts from 230K to lower temperatures (while Tc increases slightly). In addition, the turn angles get larger with pressure below 170 K. At p = 0.6 GPa it has reached 90 ° over the full temperature interval between 170K and 50K. It is quite remarkable that the angle then decreases again rather sharply with reduced temperature, approaching again the low-temperature limit of 30 ° as under ambient pressure. The change of tilting angle has such strong influence on the measured precession frequency that the pressure dependence of the contact field can only be evaluated above the spin- turning temperature (i.e., T ~> 250K). There the values of B~ are found to be lower than one would expect if only the pressure-induced shift of the ordering temperature (dTc/dp = - 1 4 K/GPa according to Bartholin and Bloch 1968) are taken into account. The results suggest an increase of the size of the (negative) contact field with applied pressure, likely to be caused by an increase of conduction electron spin polarization as volume is reduced. The application of high pressure in the paramagnetic state (Schreier et al. 1997) has no measurable effect on the behavior of relaxation rates and frequency shifts. As fig. 29 (left) shows, the crossover behavior around Tc + 10K is still clearly present for both quantities. It can be inferred from these results that small volume reductions have no marked effect on the ~SR magnetic parameters of Gd in the critical regime just above Tc.

4.3. Holmium, dysprosium and erbium 4.3.1. Ambient pressure

4.3.1.1. Holmium. This R metal is a most impressive case demonstrating how the interplay of various magnetic and electric (CEF) interactions produces a rich variety of spin structures which are often almost continuously changing with temperature. Just below TN = 132 K, a helical AFM spin structure is stabilized. In lowering the temperature, the helix becomes distorted, as can be seen from the appearance of higher harmonics in the neutron diffraction pattern. The magnetic order runs through a series of spin- slip structures (Jensen 1996) until a second-order transition at Tc = 2 0 K leads into a shallow-cone-shaped FM spin arrangement. The (fairly weak) magnetization points along the c axis. The helical component is commensurate, but the moments are not uniformly arranged along the helix, being bunched around the b axis. The cone angle decreases continuously towards 80 ° as T --+ 0.

The original ~SR measurements on Ho (Gurevich et al. 1976, Barsov et al. 1986b) were unable to observe spontaneous spin precession. This has, however, recently been achieved in both polycrystalline (Krivosheev et al. 1997b) and single-crystal material (Schreier et al. 2000a, Schreier 1999). The data of the two studies are quite similar in the AFM regime, but near Tc the results of Schreier et al. (2000a) are more detailed. Figure 33 shows that the overall temperature dependence of the internal field is rather smooth and Brillouin-like (as for the case of Dy). Just below TN the data can be fitted to a critical