μ+SR-spectroscopy in intermetallic compounds containing heavy-electrons

Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 455-461 North-Holland, Amsterdam

455

INVITED PAPER p, + S R - S P E C T R O S C O P Y IN I N T E R M E T A L L I C C O M P O U N D S C O N T A I N I N G HEAVY-ELECTRONS S. B A R T H , H.R. O T T Laboratorium f f r Festki~rperphysik, ETH-H6nggerberg, 8093 Ziirich, Switzerland

F.N. G Y G A X , B. HITTI, E. LIPPELT, A. S C H E N C K lnstitut ]'fir Mittelenergiephysik, ETH-Ziirich, c / o PSI, CH-5234 Villigen, Switzerland #+SR-investigations of various heavy-electron systems with different electronic ground states are presented. These experiments have revealed the presence of hitherto unknown weak magnetic correlations in some of these materials. The correlations partly coexist with superconductivity and can even develop without a distinct phase transition. They are quasistatic on the muon timescale in CeA13, CeCu2.1Si2, U2Zna7 , UCus, URuzSi2, UPt 3 and Uo.977Tho.033Be13 , but not in the case of UCdl~. The magnitude of the local magnetic fields at the muon sites points to small effective electronic f-moments ( < 0.5#B ) a n d / o r small coherence lengths. This weak magnetism seems to be an important feature of the electronic ground state of heavy-electron materials and deserves further investigations both from the experimental and theoretical point of view.

1. Introduction

The determining factor for the formation of a particular electronic groundstate in heavy-electron compounds is still a mystery [1-3]. Experiments revealing that small changes of external parameters or chemical composition can drastically affect the type and the properties of this groundstate [4 6] made it evident that there is a complicated interplay between different interactions and possible order parameters associated with different ground states. Heavy-electron compounds can be regarded as single-ion K o n d o compounds at elevated temperatures [7], whereas at low temperatures some interaction between individual K o n d o scattering centers develops [7] which is generally referred to as onset of coherence. Accordingly, the magnetic properties of these systems are determined by the competition of a renormalized Kondo-interaction and the Ruderman-Kittel- Kasuya-Yosida ( R K K Y ) interaction [7]. The occurrence of superconductivity per se in heavy-electron materials is surprising because it derives from an instability of the heavy f-electrons [1] which to a large extent are also responsible for the magnetic properties. As a logical consequence, superconductivity in these materials exhibits deviations from the conventional features of superconducting simple metals [8-10]. In this article we review the appli-

cation of the ~+SR-technique [11] to the investigation of static and dynamic magnetic properties of heavy-electron materials with different electronic ground states as identified by macroscopic experimental methods. The positive muon is an interstitial probe which monitors local magnetic fields and their distributions in solids. An advantage of the muon technique is its enormous sensitivity which alows to detect static magnetic correlations among moments of the order of 10 3/~B, also in absence of a perturbing external field. With I,+SR one can detect dynamic fluctuations (Trprocesses) with relaxation rates between 0.05 and 50 ~ts t [11]. In analogy to NMR, a muon Knight-shift K~t in the paramagnetic regime is defined by [11] I Bdip + Bhf ]

K~ -

iBex ' ]

(1)

where Bhr is the Fermi-contact hyperfine field, Baip the dipolar field of localized moments, and Be×t the external field. While in the NMR-definition [12] often only contributions from itinerant electrons are taken into account, Bj~p is included here because the degree of f-localization is not clear in heavy-electron systems. In ideal polycrystalline samples the contribution of Bdip in eq. (1) is averaged to zero, of course. K~ is a measure of the local electron susceptibility X~o~ which is

0304-8853/88/$03.50 © Elsevier Science Publishers B.V.

S. Barth et al. / Intermetallic compound9 containing heavv-electrons

456

eral conclusions a n d m a k e s o m e suggestions for further activities in this new field which m a y be c h a r a c t e r i z e d as small m o m e n t magnetism.

350

3oo

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2 5 O -o ¢)

>,3 O

200

o

150 b +

CeAI 3

o E 100 _

Hext = 0

5O

I

I

I

I

I

L

~

I

01

0.2

0.3

0.4

0.5

0.6

0.7

0.8

o

0

T(K)

Fig. 1. Temperature dependence of the spontaneous Larmor precession frequency vp or average local magnetic field at the muon site in CeAI 3. No external field was applied.

d o m i n a t e d by f-electrons in heavy-electron systems [1 3]. In the following we review the m a j o r i t y of p + S R - e x p e r i m e n t s p e r f o r m e d on heavy-electron systems so far. E m p h a s i s is put on the g r o u n d s t a t e b e h a v i o u r at very low temperatures, whereas results for the p a r a m a g n e t i c regime are only briefly m e n t i o n e d . Section 2 is solely d e v o t e d to CeAI3. Section 3 c o m p a r e s similar results in U2Zn]7, U R u 2 S i 2 a n d U C u s. The first o b s e r v a t i o n of dynamical r e l a x a t i o n in a heavy-electron c o m p o u n d , n a m e l y U C d u, is discussed in section 4. In section 5 some aspects of s u p e r c o n d u c t i v i t y a n d p o s s i b l e interference of m a g n e t i s m in C e C u 2 ] S i 2, UPt3 a n d Ul_ffh~Be13 for x = 0 a n d x = 0 . 0 3 3 are reviewed. In section 6 we finally d r a w some gen-

i

o.75

i

CeAI 5 transverse field

I

F

2000 G

g 0.50 +~~

o

Aco

0.25

015

,~.0 ,15 2'.0 T(K) Fig. 2. Temperature dependence of the asymmetries A1 and A 2 of the transvers-field p+-signal in CeAI3. An external field of 2000 G was applied. Acu is the asymmetry of a constant background signal of the Cu-target holder.

2. Frustrated magnetism in CeAI 3

In CeA13, p + S R - e x p e r i m e n t s in zero external field p r o v i d e d the very first evidence for the g r a d ual onset of static m a g n e t i c correlations a m o n g the C e - m o m e n t s below a b o u t 2 K [13]. A b r o a d d i s t r i b u t i o n of local magnetic fields with zero average starts to develop below 2 K at a p a r t of the m u o n sites. Below 0.7 K a s p o n t a n e o u s L a r m o r precession frequency v~ in a b s e n c e of an external field occurs, i n d i c a t i n g the onset of some coherent o r d e r in the vicinity of a p a r t of the m u o n sites. T h e weak t e m p e r a t u r e d e p e n d e n c e of % is ill u s t r a t e d in fig. 1. A t 30 m K , the frequency corres p o n d s to an average internal field of 220 G. This value decreases only by a b o u t 20% when the temp e r a t u r e is raised to 0.7 K. T h e s i m u l t a n e o u s o b s e r v a t i o n differently relaxing a n d evolving p ' p o l a r i z a t i o n c o m p o n e n t whose relative fractions exhibit a p r o n o u n c e d t e m p e r a t u r e d e p e n d e n c e led to the conclusion that these correlations develop in a spatially i n h o m o g e n e o u s way. This gradual onset of m a g n e t i c correlations between 2 a n d 0.7 K resembles the spin freezing in a spin glass-like C u M n where similar results were o b t a i n e d ~t+SR [14]. Fig. 2 d i s p l a y s the t e m p e r a t u r e d e p e n d e n c e of the relative fractions A~ a n d A 2 of the 2000 G transverse field p+-signal in CeAI 3. A(,o is a constant b a c k g r o u n d fraction originating from the C u - c o l d f i n g e r of the cryostat. A~ represents the fraction of m u o n s being subject to magnetically c o r r e l a t e d surroundings. It decreases from 65% at 30 m K to effectivley zero at 2 K. A 2, which represents the fraction of m u o n s located in a p a r a m a g n e t i c e n v i r o n m e n t , increases by the same a m o u n t . T h e p a r t i a l onset of coherent o r d e r at 0.7 K c a n n o t b e directly observed in a transverse field of 2000 G because it only leads to a l i n e b r o a d e n ing which is equal to the c o n s t a n t linewidth ~ of the c o m p o n e n t A l [15]. By m e a s u r e m e n t s in sufficiently strong longitudinal fields it was shown that the l i n e b r o a d e n i n g is exclusively of quasistatic origin. D i p o l a r field calculations for different o r i e n t a t i o n s of the nearest C e - n e i g h b o u r s at the most p r o b a b l e m u o n site which is an o c t a h e d r a l site s u r r o u n d e d by six

S. Barth et al. / Intermetallic compounds containing heavy-electrons

Al-atoms revealed that for an arbitrary f-moment of 1/% fields between 0 and 1800 G can be obtained. From this a lower limit of 0.1#B per Ce-atom may be estimated. It can be demonstrated that frustrated magnetic order is possible in CeA13 if one assumes antiferromagnetic interplane interactions among the Ce-moments [16]. Accordingly, the broad field distribution with zero average which starts to develop below 2 K may be attributed to such frustrated correlations. At 0.7 K most likely a partial change of the local order among neighbouring Ce-moments occurs [15]. Apparently, the smallness of the effective static f-moments or a very short coherence length rendered direct observation of these correlations by other techniques impossible up to now [1-3]. It should be empasized that no evidence for a macroscopic phase transition was found by ~t+SR or by any other experimental technique. In absence of any obvious trivial reasons for the observed phenomena it was speculated that the gradual onset of magnetism in CeAI 3 is driven by the temperature dependence of an effective R K K Y interaction as was recently also proposed for U2Znl7 [17]. 3.

Magnetic

ordering

in U 2 Z n t 7 , U C u s

and

URu2Si 2

A recently reported complicated anisotropy of the ~+-Knight-shift and relaxation rate in a U 2 Z n t 7 single crystal [18] turned out to be an artifact of a superposition of two components in the transverse field spectra of this substance [15]. These two components most probably correspond to two types of interstitial muon sites. The angular dependence of the two ~t+-frequencies in a transverse-field of 2000 G is depicted in fig. 3. v,(1) is temperature- and angle-independent and therefore drawn as a dashed line. The second frequency shows a marked angular and temperature dependence which above T N = 9.7 K can be exactly derived from the locally induced dipolar fields according to the magnetic susceptibility at the interstitial site (1/3, 2 / 2 , 5 / 6 ) of the hexagonal cry_stal lattice. This is illustrated in fig. 4 for external fields of 2000 and 4000 G. The relative fraction of the ~t+-signal which represents muons occupying this particular site exhibits a steplike decrease of about 40% at T N. This is caused by a

457

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,

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45

90 155 180 angle 8 (°) Fig. 3. Angular dependence of the two F+-frequencies in a transverse field of 2000 G in an U2Znl7 single crystal at 4.25, 12, 75 and 153 K. O is the polar angle between c-axis and the external field, ,~ the azimuthal angle between a-axis and the external field. ~(1) drawn as dashed line amounts to 27.09 MHz and is constant. The solid lines are fits of the angular dependence due to induced dipolar fields at the interstitital site (1/3, 2/3, 5 / 6 ) of the hexagonal crystal lattice.

sudden occurrence of a very broad distribution of local magnetic fields ( A B - - 1 0 0 0 G) [19] at the corresponding muon sites. Dipolar-field calculations for the simple antiferromagnetic structure deduced from neutron diffraction [20], however, revealed that the field should be zero at this site. Because it is unlikely that domain walls could produce such a large fraction of disturbed sites, these results could be taken as a hint that the magnetic structure in U 2 Z n ] 7 is more complicated 1500 ~ I000

.E

f U2Znl7

,

,

r

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single crystal

~¢~

~

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15

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20 20

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bulk susceptibility (10 -5 emu) mole Fig. 4. Scaling factor for induced dipolar fields at the interstitial site (1/3, 2/2, 5 / 6 ) in transverse fields of 2000 and 4000 G in a U2Zn17 single crystal as a function of the isotropic susceptibility X0 for different angles of q~= 0 and @= 90 ° The solid line represents the scaling above TN, the dashed line shows the expected behaviour below T N. Note that the scale has been reflected at T N.

S. Barth et al. / lntermetallic compounds containing heat!v-electrons

458

than assumed so far. Below T n, the induced dipolar fields acting on the remaining 60% of this c o m p o n e n t keep the value attained at T N. This means that the local susceptibility of the surrounding U-atoms becomes temperature independent below T N. This can also be seen in fig. 4. The expected scaling below T N as deduced from the behaviour above T N is drawn as dashed lines. It clearly does not coincide with the observed behaviour. The extremely weak dipolar coupling between muons occupying the second type of site and the U - m o m e n t s made similar observations for them impossible. So from the viewpoint of bt+SR, antiferromagnetic correlations in U2Zn]v seem to be non-uniform over the sample. It is unclear so far, if the temperature-independent local Pauli-type susceptibility of a part of the U-atoms below T N is a consequence of a transition from a 5f local state into a delocalized state, or a result of K o n d o screening, or even an indication that part of the U - a t o m s are not magnetically correlated. This fact deserves a some more detailed study because it might be related to the reduced ordered magnetic m o m e n t s observed by neutron experiments [20]. U C u 5 was reported to exhibit two spontaneous ix~ Larmor precession frequencies in zero-field below the antiferromagnetic phase transition at 15 K [19]. Their temperature dependence is displayed in fig. 5. In addition, a weakly-damped K u b o - T o y a b e c o m p o n e n t was detected below 15 K in more recent measurements. These different fractions of the bL+-signal show that there must be ~+-sites which are not symmetric with respect to the magnetic sublattices, and other ones which

I

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I

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2

4

6

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I

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14

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Fig. 5. T e m p e r a t u r e d e p e n d e n c e of the two s p o n t a n e o u s L a r m o r precession frequencies or local m a g n e t i c fields in UCu 5 below T N = 15 K. N o external field was applied.

are. Dipolar field calculations at the most probable m u o n sites make the occurence of one of the spontaneous frequencies and the K u b o - T o y a b e c o m p o n e n t plausible. N o explanation was found for the magnitude of the second spontaneous frequency. Again, this could be a hint of a more complicated magnetic structure than the one deduced from neutron scattering [21]. N o information about a second phase transition in UCu 5 at 1 K, the nature of which is u n k n o w n so far [22], could be obtained in these experiments because this phase transition was suppressed in our sampie, presumably by small a m o u n t s of impurities or imperfections [15]. MacLaughlin et al. [23] reported that the onset of magnetic order in U R u 2 S i 2 is accompanied by a weak increase of the zero-field linewidth by about I G. Dipolar field calculations revealed that either the ~ ~-site is approximately symmetric with respect to the magnetic sublattices or the m u o n suppresses magnetic order in its vicinity. In none of this three c o m p o u n d s any dynamical relaxation of the m u o n polarization was found.

4. Spin fluctuations in UCdll Transverse-field ~x+SR-measurements in U C d ~ confirmed the magnetic character of the phase transition at T N = 5 K in U C d l l [24]. At T n the asymmetry of the transverse-field bt t-signal of our U C d ~ - t a r g e t decreases abruptly by about 80%. This was interpreted as being due to a large dipolar field spread at crystallographically equivalent but magnetically non-equivalent ~t+-sites in U C d ~ . The remaining fraction of 20% is probably related to foreign phases, disturbed sites or sample-independent background. If the phase transition is approached from above T N, the Lorentzian transverse-field linewidth drastically increases. Interestingly, also in relatively strong longitudinal fields(Btong > 1000 G), a relaxation of the bt+-sig nal was observable in U C d ~ . Clearly, this is the first observation of dynamical (spin-lattice) relaxation in a heavy-electron system by ~t +SR. It is probably caused by the critical slowing down of the fluctuations of the U-moments. In our sample only 60% of the muons, however, were subject to these fluctuations. By combination of these results with results of transverse-field measurements it was found that another 20% of the muons in our

S. Barth et al. / Intermetallic compounds containing heavy-electrons

459

I

,i t .....

UCdll longitudinal

field

,~ I O O 0 G

~3

o 2000 G

I

10 -

,

p.+ in C e C u 2 j S i 2

' ° 4000 G o

g5 _o a) 0 | I

I 2

I

I

I

3

4

5

I 6

I 7

T~

T/T N

1

Fig. 6. Inverse spin-lattice relaxation time T1 i as a function of the normalized temperature (T N = 5 K) in UCdal in longitudinal fields of 1000, 2000 and 4000 G. The solid line is a fit according to eq. (2).

U C d ] l - s a m p l e are a f f e c t e d b y a f o r e i g n m a g n e t i c phase, presumably UO2, which orders antiferrom a g n e t i c a l l y at 30.8 K. S e e m i n g l y , this f o r e i g n p h a s e acts as a t r a p for the m u o n s , b e c a u s e suceptibility measurements did not show any a n o m a l y in the v i c i n i t y o f 30.8 K. Fig. 6 d i s p l a y s the i n v e r s e s p i n - l a t t i c e r e l a x a t i o n t i m e T]- ] ext r a c t e d f r o m the e x p o n e n t i a l d e c a y o f the ~t+p o l a r i z a t i o n as a f u n c t i o n o f t e m p e r a t u r e . T h e d a t a are well r e p r e s e n t e d b y a p o w e r law o f the f o r m [15] T1-1 -- (T--

TN)

-a

(2)

w i t h a = 0.4 (0.1).

5. Superconductivity and magnetism in CeCu 2.1Si 2, U P t 3 and U l _ x T h Bel3 with x = 0 and 0.033 S i m i l a r results as t h o s e in CeA13 w e r e r e c e n t l y o b t a i n e d in C e C u 2 n S i 2 [25]. B e l o w 0.8 K, static m a g n e t i c c o r r e l a t i o n s b e g i n to d e v e l o p . T h e y c o e x i s t w i t h s u p e r c o n d u c t i v i t y w h i c h sets in at Tc = 0.7 K. T h e y c a u s e a field d i s t r i b u t i o n w i t h z e r o a v e r a g e l e a d i n g to a G a u s s i a n line b r o a d e n i n g in z e r o - a n d t r a n s v e r s e - f i e l d s . I n fig. 7 we s h o w the z e r o - f i e l d r e l a x a t i o n r a t e in C e C u 2.1Si2 [25]. T h e a u t h o r s suggest s p i n glass o r i n c o m mensurate spin-density wave ordering with a small a v e r a g e d static m o m e n t o f the o r d e r o f 0 . 1 / x B / C e at T---, 0. T h e s h a p e o f the z e r o - f i e l d s p e c t r a also r e v e a l e d t h e p o s s i b l e e x i s t e n c e o f v e r y s l o w dyn a m i c s w h i c h where, h o w e v e r , n o t q u a n t i t a t i v e l y a n a l y s e d so far.

O

0.5

1 T(K)

Fig. 7. Zero-field ~t+ relaxation rate observed in CeCu/jSi 2. The onset of magnetic order happens of around 0.8 K, Tc is at 0.7 K [25].

Another heavy-electron superconductor, UPt3, was r e p o r t e d to o r d e r m a g n e t i c a l l y b e l o w 5 K w i t h an e f f e c t i v e static m o m e n t o f 0 . 0 0 1 ~ B / U as d e d u c e d f r o m a n e n h a n c e d z e r o - f i e l d ~t+SR-line w i d t h [26]. N e u t r o n m e a s u r e m e n t s c o n f i r m e d this claim, g i v i n g a slightly l a r g e r m o m e n t o f (0.02 + 0 . 0 1 ) / % [271. N o m a g n e t i c o r d e r i n g has b e e n o b s e r v e d in s u p e r c o n d u c t i n g UBe~3 so far 11]. H e f f n e r et al. [28] s t u d i e d t h e ~t÷ K n i g h t - s h i f t in U ~ _ x T h x B e ] 3 w i t h x = 0 a n d x = 0.033 in an e x t e r n a l field o f 5 k G . T h e y f o u n d t h a t K~t is n e g a t i v e a n d scales w i t h the b u l k s u s c e p t i b i l i t y b e t w e e n Tc a n d 100 K for b o t h x = 0 a n d x = 0.033. B e l o w T~, the absolute v a l u e of K~t d e c r e a s e s for x = 0, b u t rem a i n s c o n s t a n t for x = 0.033. T h e t e m p e r a t u r e d e p e n d e n c e o f K~t is s h o w n in fig. 8. T h e s t r o n g r e d u c t i o n in the c a s e o f x = 0 is c o n s i s t e n t w i t h

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Fig. 8. Temperature dependence of the p,+ Knight-shift Kp, in the normal and superconducting states of U]_ xTh x Bet3, x = 0 and x = 0.033 in an applied field of 5 kOe [28].

S. Barth et al. / lntermetallic compounds containing heat:v-electrons

460

0.3

~

i

~ ~ ~ ~ ~ll

]~1 T

~

~ l

U~_~Yh,Be,3

(It t :::L <~ 0.2

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0.1

0.2

/

Tel /

t

~ t,l',l

0.5

Tc / ,

1

2

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,

5

T(K)

Fig. 9. Temperature dependence of the zero-field relaxation rate AKT in superconducting Ul_~,ThxBe13, x = 0 and x 0.033. The superconducting transition temperature Tc for x = 0, and T~I and T¢2 for x = 0.033 are indicated by arrows [29].

b o t h even-parity p ai r in g or o d d - p a r i t y pairing with strong p i n n i n g of the o r d e r p a r a m e t e r to the lattice. T h e difference in the b e h a v i o u r of x = 0 and x = 0.033 b el o w Tc could either be due to strong s p i n - o r b i t c o u p l i n g for x = 0.033 or to different orde r parameters. M o s t interestingly, the zero-field ~t+SR linewidth as illustrated in fig. 9, shows a weak increase below the second phase transition (To2) in the T h - d o p e d material which m i g h t indicate the onset of m a g n e t i c correlations [29]. This would c o n f i r m the conclusions d r a w n from ultrasonic m e a s u r e m e n t s which gave evidence for itinerant a n t i f e r r o m a g n e t i s m in this material below T¢2 [30]. M o r e careful investigations, however, are needed to clarify this i m p o r t a n t point. N o d y n a m i c a l relaxation was found in U P t 3 and U 1 xThxBel3 with x = 0 and x = 0.033.

6. Conclusions and future aspects T h e presented e x p e r i m e n t a l results d e m o n s t r a t e the u n i q u e sensitivity of ~ + S R to detect static m a g n e t i c correlations with small involved effective m o m e n t s or small c o h e r e n c e lengths. O f course, with p.+ S R one c a n n o t m e a s u r e wave vectors, but it certainly is well suited as fast d e t e c t i o n m e t h o d for such weak m a g n e t i c effects. M o s t i m p o r t a n t i n f o r m a t i o n can be o b t a i n e d f r o m zero-field measurements, whereas in external fields u n w a n t e d shifts or b r o a d e n i n g s of the ~+-signals can occur because of i n h o m o g e n e i t y or a n i s o t r o p y effects.

~t+SR has cast new light on the physics of heavy-electron systems by the d et ect i o n of quasistatic m a g n e t i c correlations which can o ccu r in a spatially i n h o m o g e n e o u s way and even w i t h o u t a m a c r o s c o p i c phase transition. U p to n o w there are four heavy-electron systems which show superconductivity coexisting with m a g n e t i c correlations, n a m e l y CeCu2.1Si 2, U R u 2 S i :, U P t 3 and possibly U 1 xTh~Bel3 with x = 0.033. Often, no consistency b e t w e e n m a g n e t i c structures d e d u c e d f r o m n e u t r o n d i f f r a c t i o n an d the local m a g n e t i c fields derived f r o m ~ + S R is found. O n e possible e x p l a n a t i o n is that the m u o n itself locally changes the m a g n e t i c o r d e r or even produces it. It seems also m a n d a t o r y to reinvestigate some of these systems very carefully with n e u t r o n diffraction in view of the possible existence of c o m p l i c a t e d super-structures. Tl-processes can also be o b s e r v e d by ~t+SR when the fluctuation rate m at ch es the t i m e w i n d o w of the m u o n as ev i d en ced by the findings in U C d u and CeCu2ASi 2. A s s u m i n g that the small m o m e n t m a g n e t i s m is an intrinsic p r o p e r t y of heavy electron systems it should now be investigated theoretically w h e t h e r it can be explained within the f r a m e w o r k of a F e r m i - l i q u i d theory. In summary, it has turned out that ~t + S R provides an excellent tool for the investigation of weak m a g n e t i c co r r el at i o n effects. T h e studies perf o r m e d so far have raised a lot of new q u est i o n s which deserve intensive theoretical and experimental work in future.

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