muon diffusion in pure bismuth: evidence for an extended muonic state

Volume 132, number 4 PHYSICS LETTERS A 3 October 1988

M U O N DIFFUSION IN PURE BISMUTH: EVIDENCE FOR AN EXTENDED M U O N I C STATE

R. 'KADONO 1, K. NISHIYAMA, K. NAGAMINE Meson Science Laboratory, Faculty of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan

and

T. MATSUZAKI Metal Physics Laboratory, Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako-shL Saitama 351-01, Japan

Received 6 June 1988; revised manuscript received 31 July 1988; accepted for publication 12 August 1988 Communicated by J.I. Budniek

The diffusion of positive muons (1~ ÷ ) in ultra-pure monocrystalline bismuth has been studied experimentally by the method of zero and low longitudinal field muon spin relaxation (ZF-, LLF-laSR). The results indicate that the conventional interpretation of the spin rotation spectrum by simple motional narrowing does not hold at low temperature in this system. Existence of an extended state (which is metastable) is proposed on the basis of an analysis using a two-state model of diffusion.

The diffusion of positive muons (~t ÷ ) in metals at low temperature has attracted considerable interest as a fundamental problem, because the quantum me- chanical tunnel :between interstitial sites plays an im- portant role in the motion of light particles such as the positive muon. The possible dramatic effect of quantum diffusion on the migrat ion of light interstitial particles in perfect crystals was first pointed out by Kagan and Klinger [ 1 ]. They predicted from the study of polaronic effects that the diffusion rate D, of the interstitial particle would show a divergent increase with decreasing T (e.g., D~oc T - 9 in fcc lat- tices) at sufficiently low temperature. Motivated by such a possibility, there have already been many experimental investigations of hydrogen and muon in various metals such as Fe, AI, Cu, Nb, V and Bi (for a recent review o f hydrogen in metals, see, e.g., ref. [2]).

In most cases the diffusivity of the positive muon has been investigated by the conventional muon spin rotation technique in which a magnetic field is ap-

Present address: TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A3.

plied transverse to the initial polarization direction (TF-~tSR). However, the interpretation of such spectra is often ambiguous since the observed transverse relaxation function Gx(t) is not sensitive to the mechanism causing the spin relaxation (depolarization). In pure diamagnetic metals the origin of the muon spin depolarization is twofold: the inhomo- geneity of the field at muonic sites due to the static random local field (/ '2) and /or the spin relaxation from dynamical fluctuations in the local field due to the self-diffusion of muons (Tl) . Since both result in dephasing of the muon spins, the TF-~tSR spectrum alone does not allow a unique interpretation.

Considerably more detailed information can be obtained by measuring the longitudinal spin relaxation in zero field (ZF-btSR) or low longitudinal field (LLF-~tSR). These LF-~tSR spectra have the following unique characteristics: (a) the time spectrum has a large asymptotic component which is damped only by the T~ mechanism [3], and (b) the line shape of the spectrum shows a behavior sensitive to the presence of trapping/detrapping processes [4]. This method was successfully applied to clarify the temperature (T) dependence of the diffusion (hopping)

0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

195


rate in pure copper [ 5-9 ] and in niobium [ 10 ]. In this paper we shall report on an experimental

investigation on the diffusion of positive muons in pure bismuth (Bi). The present study was motivated by the previous work on the T-dependence of the spin depolarization rate (a) in Bi, obtained by TF-IxSR and shown schematically in fig. 1 [ I l -13 ]. It suggests that the muon seems to be almost immobile below l0 K (region I). As the temperature increases, the muon seems to become mobile in region II (20 ~ 60 K, showing a reduction of tr interpreted to be due to the motional narrowing). Then, the mobility seems to be reduced again in region III (around 90 K) [11,12]. It has also been shown that a is weakly dependent on temperature in region II.

In the conventional picture, the transition from region I to II is interpreted to be due to the detrapping of muons from a shallow trap, while the transition from region II to III attributed to the trapping to a deep trap where the presence of an impurity-trapped state is implicitly assumed. It is also pointed out that the transition from region III to II is caused by a divergent increase of the mobility predicted from the onset of coherent diffusion [ 14 ].

A proper understanding of the diffusion requires that the location of the muon in a crystal lattice be known. A recent TF-IxSR experiment in monocrystalline Bi (rhombohedral) indicates that the muon's site is different in region I and I I I [ 13 ]. The two sites which have been proposed are the two distorted-oc-

0 . 2

°r, j

:=L

b

0.1 o~

Q)

' ' ' ' 1

0.0 =~ , , I ~ , , 5 10

Temperature

' ' ' ' ' ' ' 1

®

50 100

(K)

Fig. 1. A schematic plot of the temperature dependence of the spin relaxation rate a in bismuth measured by the TF-p.SR method.

tahedral interstitial sites which have different crys- talline symmetry.

In order to further explore the behaviour of muons in Bi from the viewpoint of diffusion, we performed a ZF- and LLF-~tSR experiment in high-purity monocrystalline Bi. The Bi sample was carefully pre- pared at the Cryogenic Centre, University of Tokyo [ 15]. The starting material (99.9999% Bi-ingot from Cominco American Co. Ltd.) was purified by zone- refining (238 times) in a quartz cell evacuated to 10-s Torr by a sputter ion pump. After purification the monocrystal was grown in a graphite crucible by the Czochralski method under high vacuum (2 × 10 -7 Torr) by using a sputter ion pump. The present sample is estimated to have an impurity level below 1 ppm. To avoid stress-induced defects in the crystal, the sample was shaped by spark-erosion. The target sample ( ~ 130 g) was mounted on a cold head (Oxford Institute model CF-104) by a sample holder made of pure aluminium (A1 99.99%).

The experiment was conducted at the Ix2 port of the superconducting muon channel at BOOM facility (Booster Meson Facility, placed in KEK) oper- ated by the Meson Science Laboratory of the University of Tokyo which provides a pulsed (about 103 m u o n s in 50 ns width) muon beam at a repe- tition rate of 20 Hz [ 16 ]. A highly polarized ( > 80%) muon beam of momentum 75 MeV/c was implanted into the Bi crystal whose trigonal (c-)axis was placed parallel with the initial muon polarization. A low longitudinal field was applied along the muon polarization for the 1 mT LLF data. The decay posi- tron detection system has been already reported elsewhere [ 6 ]. The ~t-e decay time spectrum was fit by the following function,

N ( t ) =No e x p ( - t / z , ) [ 1 +AGz( t ) ] + B ,

where No is a normalization factor, z, is the average muon decay life time, A is the experimental asym- metry, Gz(t) is the ZF and/or LLF relaxation function discussed below, and B is a constant background. The ZF and LLF spectra at each temperature were analyzed in a combined manner to determine the parameters in the model function uniquely.

It was revealed that the observed time spectra could not be reproduced by the simple Kubo-Toyabe function with hopping below 90 K. the chi-square (X 2) minimization fitting of the ZF-LLF combined spec-

196

tra by the Kubo-Toyabe function yielded poor X 2 values (x2/Nf~ 2-3, Nr is the degree of freedom) as well as unreasonable parameter values. This result strongly suggests that the diffusion of muons in Bi involves more than one site. Here we should note that the LLF=I~SR data were necessary to clarify the sit= uation. We introduce a model of diffusion involving the contribution from another (meta-stable) state, which was first proposed by Seeger [ 17 ]. The spin relaxation function can then be written as

G~(t)=G~(t)+GE(t) ,

G~(t) =Pro exp( - Vtt)gm(t), t

G2(t)=p~g~z(t)+pmVt f e-~'~gm(u)g~(t-u) du, o

1.0

where gin(t) and g~(t) are the Kubo-Toyabe relaxation functions at the metastable (m) or stationary (s) state with static local field widths Am, As; Pt is the transition rate from the metastable to the stationary state, and Pm, Ps are the initial probabilities for each state (Pm+P,-=-I was assumed). The dynamical modulation due to hopping was introduced only for the stationary state g~(t). This description con'e= sponds to the simplest case of the two=state model [18].

Examples of the observed time spectra (Gz(t) ) are shown in fig. 2 with the best fit curves by the above function. Probably because of the small distortion due to the reduction of efficiency at the initial part of the time spectra (0-2 bts), the absolute chi-square of the fitting analysis was still not always satisfac- tory. However, the fitting analysis yielded fairly im-

0.0

. ¢ , .

O.5 0

1.o

1.0

v o

0.5

0.0

, . I . -

O.5 o

15

I I

I I

50 K ~ I l l


1.oJ ' ' I . g t

~ o.5 ,

o.o

1.0"

~'~N 0.5

0.0

o g ~ 15 o g ,b Time (p.sec) Time (~sec)

Fig. 2. Time spectra obtained by ZF- and LLF-pSR measurement in high purity bismuth at typical temperatures (filled circles: ZF spectra, open circles: LLF ( = 10 G) spectra). The LLF spectra show little damping at 11-95 K.

197


proved x2/Nfree (,~, 1--2) compared with that by the Kubo-Toyabe function. We note that the constant background B, which is crucial for the reliability of the time spectrum at later part, was negligible up to 20 gs (B/no< 10 -4) for all the observed spectra.

Despite the drastic change of the ZF spectra, the LLF spectra at 1 mT ( 10 G) clearly indicate that the dynamical modulation is small all over the temperatures below 90 K, which could not be reproduced by the simple Kubo-Toyabe function (see the dif- ference between 11 K and 95 K or 65 K and 113 K). A preference to the gaussian-like shape of the ZF spectra also suggests that the "motional narrowing" is not a correct interpretation. On the other hand, the effective dipolar width, which is reflected in the oscillation amplitude of the LLF spectra, shows a considerable change with temperature. As shown in figs. 3 and 4, these features are more clearly dem- onstrated in the T-dependence of the model parameters.

Fig. 3a implies that the probability for the muon to be in the m-state (Pro) is close to unity below 60 K, but it drops to zero at higher temperatures. The

1.0

0.8

= 0 , 6

0.4

0.0 0~4

~ 0 . 3

"~ 0.2

• ~ 0.1

0.(1

Pin

" ' . . .

I I i i J i i i i i i i i j r r l

. . . . j . . . . . . . . j

(b) [] . . . . . . . . = . . = . . . . ~ . . . . . . . . . . . . . . . ~ , . ~ . ~ ~

. . . . , ,

tO 50 tO0 Temperature (K I

Fig. 3. (a) Fraction of the metastable state Pr. as a function of temperature. It shows that about 85% of the muons are in the metastable State in the region II, which rapidly quenches at 60- 90 K. (b) Static dipolar width of relaxation at metastable (,Jr.) and stationary state (,J~). zt,. shows a temperature dependence quite similar to that oftr in fig. 1.3~ is almost constant except for a slight shift at 60-70 K.

10 o

10_I ::L

,.~ t0-z

lo-3

10 I " 7

t0 -I

== • [ = l0 -z

O

-,- 10.3 5

(al • ..0....¢

• . . . . ~ " . . . . . ~ . " . . • . . . .

' " . . @ - ' :

. . . . . . . . . . ~J~

(b) /

• "-.. ~ /

. . /

Ii - I ' ' L I i * . ~ , ~

l0 50 I00

Temperature (KI

Fig. 4. (a) Transition rate ~t from metastable to stationary state, and (b) hopping rate of muons in bismuth as a function of temperature. One can see a sharp peak of vt around 60-70 K. The dashed line shows the best fit curve by the Arrhenius formula.

static dipolar widths/Ira and As are shown in fig. 3b which should be compared to the T-dependence of the TF relaxation rate tr shown in fig. 1. Thus, the T- dependence of tr observed by the TF-pSR method should be mainly attributed to a change in Pm and Am of the m-state and not to dynamical modulation• Our best fit fields As=0.2989(28) ~ts -~ below 60 K and As = 0.271 (16) ps - i above 60 K. Provided that the second moment observed in ref. [ 13 ] is related to As, the s-states are interpreted to be the two distorted- octahedral interstitial sites at respective temperature regions. Here we would like to mention that an analysis assuming the diffusion-limited trapping [ 10] below 90 K (which corresponds to the case Pro= 1 and gin(t)= 1 in Gz(t)) yielded a relatively large static dipolar width (i.e., As,,~0.45 p s - ' ) as well as poor chi-square values, which is less favorable for the case of muons in Bi [13 ].

The smallness of Am ( ~ 0.094 (2) pS- ~ in region I and ~0.04( 1 ) ~ts -1 in region I I ) strongly suggests that the state is quite extended (only weakly local- ized). I f we assume that the extended muon wave function is isotropically distributed around n sites, it is estimated to be n ~ 10 in region I from the simple statistical theorem A 2 =A2/n [ 19,20]. Since the

198


observed dm is close to zero in region II, it is also consistent with the interpretation that the muon is rapidly diffusing through the m-states in this temperature region (hopping rate v as large as 2J2m/v ~ 0). In this case the transition from region I to II might be due to the onset of fast incoherent diffusion between m-states with very small activation energy.

The sharp peak in the transition rate vt at around 65 K shown in fig. 4a could be due to a critical temperature for the transition from the m-state to the s- state. The spectra above 90 K (see the 113 K spectra in fig. 2) are consistent with the Kubo-Toyabe relaxation function with motional narrowing. The hopping rate, obtained by assumingpm is zero above 120 K, is shown in fig. 4b. The small bumps of the T-dependence seen in the hopping rate at the transition regions from II to III might be interpreted as an effect associated with the transition from the extended to the stationary state. An analysis of the hopping rate above 100 K assuming an Arrhenius formula of v= Vo e x p ( - E a / k T ) , yields parameter value E ,=0 .144(3 ) eV and Vo= 1.51(4) X 1012 s -1 (see fig. 4b). If we assume that phonon-assisted tunneling between equivalent s-states is predominant in this region [ 21 ], the tunneling matrix element J is estimated to be

J'~Jo exp( - 5E,/2kOv) ~.Jo × 10- ~ ,

where Jo ( ~ 6.6 (2) meV) is a bare tunneling matrix and OD ( = 119 K) is the Debye temperature of Bi. This implies that the effective tunneling matrix Jbe- tween two s-states is too small to realize the extended (coherent) state predicted by Kagan and Klinger. The only remaining possibility is that the m-states are realized as a mixed state made of the two different interstitial sites or the tetrahedral-like sites.

As a conjecture, the existence of such an extended state may be related with a small conduction electron density in Bi (~. 10 ppm at 4.2 K). Because of the infinitely small excitation energy of the conduction electrons at the Fermi surface, the Coulomb in- teraction between muons and the conduction electrons in metals causes a damping of the coher- ence of the muon-lattice small-polaron state [22- 24 ]. The damping results in a reduction of the muon tunneling matrix by a factor (T/D) x (where D is the Fermi energy, K is the muon-electron interac-

tion coupling constant), which was observed in the case of muons in pure copper with K ~ 0.16 [ 6,7 ]. In the case of Bi, the small electron density would lead to the reduction of muon-electron coupling (i.e., K ~ 0) to recover the muon tunneling matrix element.

In summary, the ZF- and LLF-I~SR spectra in high purity Bi indicate that a large fraction of muons oc- cupies a state extended over several lattice sites below 60 K, which is strongly suggested from the smallness of dm(<As) in this temperature region. The very small Am in region II is also consistent with the rapid muon diffusion through m-states.

We would like to thank Dr. Mitsuru Suzuki for the loan of the excellent bismuth sample, the staff of Me- son Science Laboratory for the support of the experiment. We acknowledge Professor T. Yamazaki, Dr. J. Kondo and Dr. R.F. Kiefl for valuable dis- cussions and comments. This work was partially supported by a Grant-in-Aid of the Special Project Research on Meson Science of the Ministry of Ed- ucation, Science and Culture.

References

[ 1 ] Yu. Kagan and M.I. Klinger, J. Phys. C 7 (1974) 2791. [2] Y. Fukai and H. Sugimoto, Adv. Phys. 34 (1985) 263;

A. Schenck, Muon spin rotation spectroscopy (Hilger, Bris- tol, 1985).

[3] R.S. Hayano, Y.J. Uemura, J. Imazato, N. Nishida, T. Yamazaki and R. Kubo, Phys. Rev. B 20 (1979) 850.

[4] K.G. Petzinger, Phys. Lett. A 75 (1980) 225. [5 ] C.W. Clawson, K.M. Crowe, S.S. Rosenblum, S.E. Kohn,

C.Y. Huang, J.L. Smith and J.H. Brewer, Phys. Rev. Lett. 51 (1983) 114.

[6] R. Kadono, J. Imazato, K. Nishiyama, K. Nagaminc, T. Yamazaki, D. Richter and J.-M. Welter, Phys. Lett. A 109 (1985) 61.

[7 ] R. Kadono, J. Imazato, T. Matsuzaki, K. Nishiyama, K. Nagamine, T. Yamazaki, D. Richter and J.-M. Welter, sub- mitted to Phys. Rev. B.

[8]J.H. Brewer, M. Cello, D.R. Harshman, R. Keitel, S.R. Kreitzman, G.M. Luke, D.R. Noakes, R.E. Turner, E.J. Ansaldo, C.W. Clawson, K.M. Crowe and C.Y. Huang, Hyp. Int. 31 (1986) 191.

[9] S.R. Kreitzman, J.H. Brewer, D.R. Harshman, R. Keitel, D.LI. Williams, K.M. Crowe and E.J. Ansaldo, Phys. Rev. Lett. 56 (1986) 181.

[10] C. Boekema, R.H. Heffner, R.L. Hutson, M. Leon, M.E. Schillaci, W.J. Kossler, M. Numan and S.A. Dodds, Phys. Rev. B 26 (1982) 2341.

199


[ 11 ] V.G. Grebinnik, I.I. Gurevich, A.I. Klimov, V.N. Mayorov, A.P. Manich, E.V. Melnikov, B.A. Nikolsky, A.V. Pirogov, V.I. Selivanov, V.A. Suetin and V.A. Zhukov, JETP Lett. 25 (1977) 322.

[12] S.G. Barsov, A.L. Getalov, V.G. Grebinnik, V.A. Gordeev, I.I. Gurevich, Yu.M. Kagan, A.I. Klimov, S.P. Kruglov, L.A. Kuzmin, A.B. Lazarev, S.M. Mikirtychyants, B.A. Nikol- sky, A.V. Pirogov, A.N. Ponomarev, V.I. Selivanov, G.V. Shcherbakov, V.A. Suetin and V.A. Zhukov, Hyp. Int. 17- 19 (1984) 145.

[ 13] D. Baumann, F.N. Gygax, E. Lippelt, A. Schenck,R. Schmid and S. Barth, Hyp. Int. 31 (1986) 53; F.N. Gygax, A. Schenck, A.J. van der Wall and S. Barth, Phys. Rev. Lett. 56 (1986) 2842.

[14 ] Yu. Kagan and N.. Prokof'ev, preprint, IEA-4461/9 (I.V. Kurchatov Atom. Ener. Inst. ).

[15]H. Bando, Ph.D. Thesis, University of Tokyo (1983), unpublished.

[ 16] K. Nagamine, Hyp. Int. 8 (1981) 787. [ 17] A. Seeger, Phys. Lett. A 93 (1982) 33;

• A. Seeger and L. Schimmele, Hyp, Int. 17-19 (1984) 133. [ 18] E. Sato, T. Hatano, Y. Suzuki, M. Imafuku, M. Sunaga, M.

Doyama, Y. Morozumi, T. Suzuki and K. Nagamine, Hyp. Int. 17-19 (1984) 203.

[ 19] A.M. Stoneham, Phys. Lett. A 94 (1983) 353. [ 20 ] K.W. Kehr and K. Kitahara, J. Phys. Soc. Japan 56 ( 1987 )

889. [21]C.P. Flynn and A.M. Stoneham, Phys. Rev. B 1 (1970)

3966. [ 22 ] J. Kondo, Physica B 125 ( 1984 ) 279; 126 ( 1984) 377. [23] K. Yamada, Prog. Theor. Phys. 72 (1984) 195;

K. Yamada, A. Sakurai and S. Miyazawa, Prog. Theor. Phys. 73 (1985) 1342.

[24] U. Weiss and H. Grabert, Phys. Lett. A 108 (1985) 63.

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muon diffusion in pure bismuth: evidence for an extended muonic state

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