... AND TUNNELLING. IN Sc. F.N. GYGAX, A. AMATO, A. SCHENCK ... Paul Scherrer Institut (PSI), CH-5232 Villigen PSI, Switzerland. The angular and magnetic ...
Hyperf'meInteractions 85 (I 994) 73-78
#+ L O C A L I Z A T I O N
73
AND
TUNNELLING
I N Sc
F.N. G Y G A X , A. A M A T O , A. S C H E N C K
Institut f~r Mittelenergiephysik der E T H Zarich, CH-5232 Villigen PSI, Switzerland
I.S. ANDERSON Institut Laue Langevin, B.P. 156, F-380~2 Grenoble Cddex 9, France J.J. RUSH
Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA G. S O L T
Paul Scherrer Institut (PSI), CH-5232 Villigen PSI, Switzerland The angular and magnetic field dependence of the transverse pSR relaxation rate in single crystal Sc shows unambiguously that, at low temperatures, the muon is localized along the ~-axis joining interstitial tetrahedral sites. The best agreement with the measured results, covering a complete range of high and low field data, is obtained on assuming a muon state with fast tunnelling between two adjacent sites at positions z/c = -t-0.09 above and below the a - b plane. Both contributions from the muon-induced electric field gradient and that inherent to the crystal were considered. W e have p e r f o r m e d /zSR m e a s u r e m e n t s on single c r y s t a l s a m p l e s of ScH~, x = 0, 0.05 a n d 0.25, a i m e d at d e t e r m i n i n g b o t h t h e m u o n site in t h e h c p l a t t i c e a n d t h e d y n a m i c s at t e m p e r a t u r e s b e l o w 300 K [1]. In w h a t follows we discuss t h e results o b t a i n e d for t h e p u r e Sc samples. 1.
Muon
Site
T h e / ~ S R r e l a x a t i o n r a t e s have b e e n m e a s u r e d as a f u n c t i o n of t h e applied m a g n e t i c field in a t r a n s v e r s e field g e o m e t r y at various t e m p e r a t u r e s b e t w e e n 10 a n d 300 K. Scans were p e r f o r m e d for different values of t h e angle /9 b e t w e e n t h e h e x a g o n a l ~-axis of t h e c r y s t a l a n d t h e direction of t h e e x t e r n a l l y a p p l i e d field. All m e a s u r e m e n t s show basically a levelling of t h e # S R r e l a x a t i o n [2] b e l o w 50 K. T y p i c a l r e s u l t s at low t e m p e r a t u r e are p r e s e n t e d in Fig. 1. Accordingly, b e t w e e n 10 a n d 50 K, w h e r e t h e 9 J.C. Baltzer AG, Science Publishers
74
F.N. Gygax et aL / #+ localization in Sc
# S R signals have clearly gaussian form, neither thermally activated nor other long range motion of the muon is indicated and we concentrate on the #+ site determination for that temperature domain. 0.35 ~ "~" -+ --+r~ ^
~ t +"'4-
,, ),~
o.2s v b
0.20 '%(,
0.15
~' "x
0.10 t
i
I
20
i
40 60 TEMPERATURE (K)
80
100
Fig. 1. Selection of transverse field relaxation rates ~ measured in a pure Sc single crystal as function of t e m p e r a t u r e (lower range). D a t a shown for two crystal orientations in different fields: + for 0 = 0 ~ and 296 m T , o for 8 = 0 ~ and 36 m T , • for 0 = 45 ~ and 36 mT. T h e lines are m e a n t to guide the eye.
The # S R relaxation rates measured as function of the applied field B for 8 = 0 ~ and 8 = 45 ~ (Fig. 2) reach saturation values for B >_ 0.3 T. This indicates that the field distribution at the #+ site, originating from the 0.40
.......
~
',',,e
= 900
o
e =0 ~
0.35 T
0.30
10 -~
-i..~_.~..~.~ . . - . r
9
*~-
r
10-2
10 -t
10o
FIELD B (T)
Fig. 2. Relaxation rate ~ measured in Sc single crystals as function of the transverse magnetic field. T h e t e m p e r a t u r e was kept below 50 K and for most of the d a t a was fixed at 12-I-1 K. D a t a sets taken for the three crystal orientations 0 = 0 ~ (diamonds), 6 = 45 ~ (open circles) and ~ = 90 ~ ( x ) . A simple calculation of the second m o m e n t of the lattice dipolar field at t h e / J + (at a shifted T-site, assuming a radial E F G a n d 5.4% lattice expansion in a and b around the muon) yields the dashed line for 8 = 0 ~ the solid line for 9 = 45 ~ and the d o t t e d line for 9 = 90 ~
magnetic nuclear Sc moments, has reached the Van Vleck limit. Prom the
75
F.N. Gygax et aL / #+ localization in Sc
measured a(B) dependences it is not obvious where the "low-field" limit is to be found exactly, however it is certainly below 10 mT. One sees t h a t the a(8) dependences measured at 503 m T (Fig. 3a) and 7.8 m T (Fig. 3b) show two drastically different patterns.
i
0.40
0.40
0.35
0.35
~. 0.30
0.30
b
b
0.25
0.25 0.20
0.20
7.8 mT , , ,
200 e
(~
,
,
,
,
250
,
,
.
.
.
300 e
.
.
.
.
.
.
.
350
.
.
.
.
.
.
400
450
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Fig. 3. Transverse relaxation rate ~ measured versus the crystal orientation 0, (a) in a field of 503 mT at 10 and 30 K, and (b) in a field of 7.8 mT at 30 K. The solid lines are calculated by assuming, like in Fig. 2, a shifted T-site for the/~+ and a lattice expansion (see text). We have performed calculations with increasing levels of sophistication to u n d e r s t a n d the measured second moment of the field distribution, M2, corresponding to the square of a(8, B), by assuming various sites for the #+ in the crystal lattice. Starting at the Van Vleck limit, i.e. with the simplest calculation, it is clear t h a t an octahedral ( 0 ) site calculation gives an inadequate description of the angular p a t t e r n obtained experimentally (Fig. 4a). It should be noticed that in the Sc lattice the crystallographically equivalent "tetra.hedral" interstitial sites split into two magnetically non-equivalent sub-sites. Thus the calculation of the effective second moment necessitates the proper average of M2 over the sub-sites. For the muon at the center of a basic Sc t e t r a h e d r o n (T), i.e. with z/c = 0.1185 in the undisturbed Sc lattice (c is the ~-axis lattice constant and z the distance between the #+ and the basal lattice plane), the calculated a(8) function (Fig. 4a) reflects already the m a i n trends in the data. As a next step, more features of the measured a(0) are reproduced when one assumes a muon shifted from the center of the T-site towards the a - b basal plane of the tetrahedron, i.e. towards the next neighbour tetrahedral site. In fact talcing z/c = 0.05 and a 5.4% expansion of a and b for the lattice around the #+ gives the more satisfactory solid lines in Fig. 3a. To fit the low field d a t a of Fig. 3b (see the resulting solid line),
F.N. Gygax et al. / #+ localization in Sc
76
0.40
0.35 t
"• r
/"'",
o.,7,/
:~
0.30
~176
b
0.25
0,25i-
\,--,/
/
, '.h-----~...' \ \
\ 0.20
t!
200
\
!
High Field
\
250
350
300 o
(o)
t
O,20P
!
400
450
\
/
/
,,
\
Low Field 200
250
300 o
(o)
350
400
450
Fig. 4. (a): calculated relaxation rate a(B) in the Van Vleck limit. For the three cases the Sc lattice is undistorted. Dashed line for muons at O-sites, dotted line for p+ at the center of T-sites, solid line for/z + at shifted T-sites (z/c = 4-0.05). (b): calculated relaxation rate tr(0) at low magnetic fields. The muon is assumed at a shifted T-site (z/c = 4-0.05), a and b of the lattice are expanded by 5.4% around the/~+. Dashed line for a purely axial EFG, parallel to the ~-axis. With a purely radial EFG (i) dotted line at the low-field limit, (ii) solid line for B = 7.8 mT and uO = 50 kttz.
we a s s u m e d in a d d i t i o n a radial electric field g r a d i e n t ( E F G ) a r o u n d t h e #+ i m p l y i n g a q u a d r u p o l a r f r e q u e n c y u o of a b o u t 50 kHz for t h e closest Sc nuclei. (A p u r e l y axial E F G along t h e ~-axis - Fig. 4b - is clearly not s u p p o r t e d by t h e d a t a at 7.8 roT.) At this point one m i g h t a l r e a d y b e satisfied w i t h this m u o n site d e t e r m i n a t i o n r e p r o d u c i n g quite well t h e d a t a over a large range in applied field. However, it was n e c e s s a r y to a s s u m e a significant shift in t h e T-site position a n d a n ad hoc lattice expansion. F u r t h e r m o r e , t h e calculations do not r e p r o d u c e well t h e a ( B ) d a t a at v e r y low field (Fig. 2, solid line for 8 = 45 ~ a n d d a s h e d line for 8 = 0~ T h e a g r e e m e n t can be greatly i m p r o v e d b y considering t h e fact t h a t t h e large shift o b t a i n e d for t h e a s s u m e d static m u o n position t o w a r d s t h e a - b plane can be t a k e n as a n i n d i c a t i o n for a s t a t e w i t h t h e #+ tunnelling b e t w e e n a d j a c e n t T-sites across t h e plane. T h e calculations in this m o d e l show t h a t t h e m u o n d i s p l a c e m e n t s f r o m t h e two s y m m e t r i c a l T - " c e n t e r s " is i n d e e d smaller a n d t h a t an a g r e e m e n t w i t h t h e e x p e r i m e n t can be o b t a i n e d in this case even w i t h an u n r e l a x e d lattice. T h i s c a n be qualitatively u n d e r s t o o d by simple c o n s i d e r a t i o n s a b o u t t h e effect of # - t u n n e l l i n g on t h e second m o m e n t of t h e field distribution. As a result t h e b e s t possible a s s u m p t i o n for t h e #+ d y n a m i c s is a r a p i d (u >> 1/T,, u0) tunnelling b e t w e e n a d j a c e n t T-sites, w h o s e equilibritmi positions are slightly shifted f r o m t h e ideal T - c e n t e r s t o w a r d s each o t h e r ,
F.N. Gygax et aL I
#+ localization
77
in Sc
with z/c = 4-0.09 (here r, is the #+ lifetime and v0 the reorientation frequency of the nuclei). Below the Van Vleck limit the effect of the predominantly radial EFG generated by the muon is prevalent. The strength of the quadrupolar coupling is found to be vQ ~ 67 kHz for the neighbouring Sc nuclei. It contains a small contribution from an inherent crystalline axial EFG parallel to the ~-axis (v~" ~ 9 kHz) which leads to the upturn of the calculated a(O~ B) for B decreasing below 10 mT, in agreement with the data - see Fig. 5a. As can be seen from Fig. 5b,c the tunnelling model with the above parameters describes well the a(e) data measured at high and low field. 0.40:" . . . . . . .
~'-'"
"".O
= 90~
0
o.35~ . S.o o. ~ ";5.-:'*-ft'--=o ~ :a.
0.30
b
0.25
--**~." 5~
0.20 ( 0 ) ~o -~
TUNNELLING
......
i ~ -~
......
i~-'
......
i6 ~
FIELD B (T) 0.40'
~176 o.o[ y
(b) 0.35 ,--,
o.,of_./
~l. 0.30
b
-% ,S/
b
0.25
0.25~: 0.20~
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[,
200
250
300 e
350
(o)
400
450
7.8 mT i
.
200
.
.
.
1
,
250
,
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,
i
.
500
.
.
.
e (0)
r
.
350
.
.
.
.
.
400
.
.
450
Fig. 5. Data compared to the calculated a(/9, B) assuming p+ tunnelling between adjacent T-sites ( z / c = +0.09), vQ = 67 kHz and u~~ = 9 kHz. The Sc lattice is undistorted. Lines and data points in (a) identified as in Fig. 2. (b) for B = 503 mT and (c) for B = 7.8 mT.
2.
Discussion
The field and orientation dependences of the muon relaxation rate at low temperature are well described by a model invoking a rapid tunnelling
78
F.N. Gygax et al. / #+ localization m Sc
between adjacent tetrahedral sites separated by 2z = 0.18c. This separation is smaller t h a n expected from the "ideal" position but is remarkably near to the value 2z = 0.194c found for D in Sc [3]. Furthermore, rapid local hopping (,,~101~ s -1) of H between neighbouring T sites has been observed for H in Sc [4]. In this case the t e m p e r a t u r e dependence of the hopping rate is well described by a model in which the tunnelling rate is d a m p e d by coupling to conduction electrons. Considering the rather close separation of the adjacent T sites and the apparent shallow potential between t h e m (as m e a s u r e d by inelastic n e u t r o n scattering [4]) the observation of tunnelling p h e n o m e n a is not surprising, particularly for the light #+. Further measurements, particularly to study the interplay between muon and H in a scandium host, are under way.
Acknowledgement We wish to thank R. Feyerherm ( E T H Zurich) as well as V.G. Olshevsky and V.Yu. Pomjakushin ( J I N R Dubna) for their help in d a t a taking.
References [1] F.N. Gygax, A. Amato, I.S. Anderson, J.J. Rush and A. Schenck, Z. Phys. Chem., (1993), in press. [2] A careful examination indicates a very light drop of a at the lowest temperature. This feature is not discussed here but will be treated in a future report. [3] C.K. Saw, B.J. Beaudry, and C. Stassis, Phys. Rev. B 27 (1983) 7013. [4] N.F. Berk, J.J. Rush, T.J. Udovic, and I.S. Anderson, J. Less. Comm. Met., 172174 (1991) 496.
