- Email: [email protected]
Muonium atom in the Bloch state
Physb=5916=Binni=Venkatachala=BG
Physica B 289}290 (2000) 459}463
Muonium atom in the Bloch state R. Kadono *, W. Higemoto , K. Nagamine , F.L. Pratt Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan
Abstract We present a clear signature of muonium (Mu) occupying a Bloch state in KCl at a temperature ((10 mK) much lower than the Mu energy band width (K0.15 K), i.e., the resonant enhancement of muon spin relaxation at a "eld where the Mu Zeeman splitting crosses a van Hove singularity in the Mu density of states. This feature (similar to the muon-level crossing resonance) can be used to study the energy band structure of Mu in a crystalline host. 2000 Elsevier Science B.V. All rights reserved. Keywords: Coherent tunneling; Bloch state; Muonium; van Hove singularity
It is now established that the di!usion of interstitial muon(ium) at lower temperatures is described by the tunneling of a muon(ium)-polaron state (with an e!ective tunneling matrix D) associated with two-phonon exchange processes, where the hopping frequency is l(¹)KD/X(¹) with the damping factor X(¹)J¹? due to interaction with the phonon/electron bath. While the hopping rate levels o! at l(¹)KD as temperature is lowered to satisfy X(¹)(D, the above picture of localized atoms undergoing occasional tunneling is still valid as long as the temperature is high enough for Mu to occupy all the possible energy eigenstates in the Mu energy band D (i.e., ¹
* Corresponding author. Fax: #81-298-64-5623. E-mail address: [email protected] (R. Kadono).
a coherent Bloch state; the Mu begins to occupy a well de"ned eigenstate of energy (i.e., the bottom of the Mu energy band) which leads to the delocalization of the state vector due to Heisenberg's uncertainty principle. The issue has been theoretically scrutinized previously (for a recent review, see for example Ref. [4]), where the temperature dependence of the longitudinal spin relaxation rate (¹\) was mainly discussed as a possible clue for probing such a state. A more detailed numerical evaluation by Kondo predicted that the magnetic xeld dependence of ¹\ in the Bloch state would be strongly modulated by the van Hove singularities in the Mu density of state [5]. Very recently, we have observed a peak of relaxation rate in KCl cooled down below 10 mK at around 0.15 T, which is in excellent agreement with the above prediction [6]. This feature provides a new type of &robust' evidence that Mu is in the Bloch state. In the case of Mu, one measures the longitudinal muon spin relaxation rate induced by #uctuation of local magnetic "elds acting on the Mu orbital
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 3 5 - X
Physb=5916=Binni=VVC=BG
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R. Kadono et al. / Physica B 289}290 (2000) 459}463
electron, i.e., ¹\Kd a C(u ), GH GH G H
(1)
where d is the rms value of the random local "eld, C(u) is the spectral density determined by the time correlation function for the local "eld, and u is the GH relevant Mu Zeeman frequency with the respective amplitude a under an applied magnetic "eld [7,8]. GH When ¹
Fig. 1. (a) Tunneling matrix of the Mu-polaron state, D. When ¹
shown in Fig. 2b, we did not see any signi"cant change in the lSR spectrum with cooling down the specimen from 4 K to 20 mK (which seems to be common to the case in Ref. [9]). Based on the interpretation that this result was due to insu$cient thermal conductivity, we prepared a mixture of freshly ground single-crystal KCl and silver
Physb=5916=Binni=VVC=BG
R. Kadono et al. / Physica B 289}290 (2000) 459}463
Fig. 2. (a) A sectional view of the tail of the cryostat. The He}He dilution refrigerator insert has its own LHe shield attached directly to LHe reservoir. The bottom part of the LHe space is shielded with a Te#on ring to eliminate the liquid near the beam windows by He vapor generated by the LHe bubbler. (b) A bulk crystal KCl was mounted with ECO bond, which turned out to have poor thermal contact, and (c) KCl grains mixed with Ag powder and vacuum grease were moulded into the sample holder. The current result was obtained with method (c).
powder (99.9%, 2 lm diameter) and moulded into a sample holder with a small amount of grease (Apiezon) to secure good thermal contact with the mixing chamber (see Fig. 2c). The weight ratio among KCl, Ag powder, and grease was about 1 : 1 : 0.1 (the corresponding volume ratio being 1 : 0.23 : 0.25). Compared with using the bulk crystal, a crude estimation suggests that about 10 times improved e$ciency was expected for cooling those ground crystals due to their larger surface area. The temperature was monitored by a calibrated resistance thermometer at the mixing chamber. (Because of the experimental circumstances, we were not able to use nuclear orientation thermometer and therefore only an upper limit is given for the base temperature of the present experiment.) Conventional muon spin relaxation (lSR) measurements were performed under a longitudinal magnetic "eld (for KCl crystals in this condition the only muon spin relaxation comes from Mu). The lSR time spectra were analyzed by a single exponential decay, A(t)"A(0)exp(!t/¹ ). The results are shown in Fig. 3, where one can notice a clear di!erence between the data at 3.9
461
Fig. 3. Muon spin relaxation rate for Mu in KCl (a) at 3.9 K and (b) below 10 mK. For the solid curves see the text. The dot}dashed curve in (b) is the best "t to a Lorentzian spectrum, whereas the dashed curve is the Lorentzian spectral component "tted in conjunction with a Gaussian peak around 0.15 T to give the solid curve.
K (Fig. 3a) and below 10 mK (Fig. 3b). In particular, there is a broad peak in the relaxation rate in Fig. 3b, which is absent in Fig. 3a. The spectral density in Fig. 3a is reproduced by assuming a Lorentzian distribution plus a constant background relaxation (K2.5;10\ s\), as shown by the solid curve. The magnitude of local "eld d (&1 mT) is about one half of the value at higher temperature, which is similar to the situation in NaCl [2]. The same model, however, completely fails to "t the data in Fig. 3b because of the peak around 0.15 T. Since the peak structure seems to be more pronounced than that in Fig. 1, which was predicted for a speci"c case, we adopt a phenomenological model in which C(u) is represented by the sum of a Lorentzian spectrum and a Gaussian peak. A "tting analysis with this model yields a "tted peak position at 0.16(1) T, corresponding to an energy of 0.11(1) K. This is close to the value of D K0.15 K estimated from the leveling o! of the
Physb=5916=Binni=VVC=BG
462
R. Kadono et al. / Physica B 289}290 (2000) 459}463
jump frequency and thus provides strong evidence that the peak originates from the energy band structure of the Mu atom in KCl. In other words, this peak structure in the spectral density below 10 mK is a clear signature that Mu is in the Bloch state. The Lorentzian part is well reproduced by assuming the Bloch state with d &1.2 mT. The peak is associated with a Gaussian width of about 40 mK, which may provide the actual Mu temperature. As discussed previously, the possibility of attributing the observed peak to anisotropic hyper"ne structure is ruled out by the "eld dependence of the initial muon polarization P (t) [6]. The X quenching pattern of the initial decay positron asymmetry A(0) was in good agreement with the case of isotropic hyper"ne structure with the known value of the Mu hyper"ne parameter (u "2p;4.28;10 s\) for both temperatures except a slight complication due to the presence of a fast spin relaxation process terminated within 10\ s. The remaining issue is the reduction of d from the value seen at higher temperatures (K2 mT). This may be interpreted by a scenario similar to that for Mu in NaCl [2]. The important point here is that d re#ects the rms value of the nhf "elds. While this value remains unchanged for localized Mu, it must be reduced by N\ for a Mu simultaneously probing the local "elds at N sites due to the e!ect of statistical averaging [10,11]. In other words, the characteristic length scale in this regime is the particle wavelength j and therefore we would observe dM "d (a/j) as an e!ec tive value of the nuclear hyper"ne parameter [4]. This interpretation for the present result (i.e., dM K1.2 mT, d K2 mT) immediately leads to an estimation
d j K1.4. " dM a
(2)
This rather short wavelength is in line with the relatively high Mu tempearture inferred from the peak width in Fig. 3b, which might originate from the intrinsic time scale of Mu thermalization. While an estimation based on the presumption of X(¹)J¹ provides a thermal relaxation time of the order of 1 s [4,9], we stress that the empirical
law X(¹)J¹ leads to much shorter thermalization times although it may place the current limit for the base temperature of Mu itself (K30}40 mK). The present result demonstrates that one can potentially study the energy band structure of the `polaron banda for a hydrogen isotope in any crystalline solid where one "nds stable Mu upon muon implantation. We believe that this will add an important new dimension to the study of atomic centers and point defects in solids. We thank J. Kondo for illuminating discussions and for showing us his theoretical calculation prior to publication. Thanks are also given to the RIKEN sta! for their technical support. This work was partially supported by a Monbusho Grant-inAid for Scienti"c Research on Priority Areas.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
R.F. Kie# et al., Phys. Rev. Lett. 62 (1989) 792. R. Kadono et al., Phys. Rev. Lett. 64 (1990) 665. Yu. Kagan, N.V. Prokof'ev, Phys. Lett. A 150 (1990) 320. V.G. Storchak, N.V. Prokof'ev, Rev. Mod. Phys. 70 (1998) 929. J. Kondo, J. Phys. Soc. Japan 68 (1999) 3315. R. Kadono et al., Phys. Rev. Lett. 83 (1999) 987. M. Celio, Helv. Phys. Acta 60 (1987) 600. H.K. Yen., M. Sc. Thesis, University of British Columbia, 1988, unpublished. W.A. MacFarlane et al., Hyper"ne Interactions 85 (1994) 23. A.M. Stoneham, Phys. Lett. A 94 (1983) 353. K.W. Kehr, K. Kitahara, J. Phys. Soc. Japan 56 (1987) 889.
Comments V.G. Storchak: In case of a delocalized state the ,rst thing one has to get is a (¹-dependence of q\. How come you did not see these primary features and see the peak which is a secondary feature coming with the primary feature only? My second point is that the calculated time for Mu thermalization in matter at a temperature of about 10 mK is about 1 s which is much larger than the muon lifetime [Storchak and Prokof1ev, Rev. Mod. Phys. 70 (1998) 929]. No way it is thermalized in your measurements. As a consequence there will be no Boltzman distribution in the band and no enhanced population in the bottom of the band. ¹hen one cannot observe the resonance. So my point is that the peak observed may be explained in any other way but NO¹ because of Mu delocalization.
Physb=5916=Binni=VVC=BG
R. Kadono et al. / Physica B 289}290 (2000) 459}463 My third point: how come you see just a twofold reduction in d ? ¹his can only mean there is no delocalization. R. Kadono: 1. Although we have only two points in terms of temperature we can show you that the ¸orentzian part of the spectral density for the `(10 mKa data is smaller that that at 3.9 K (by &70%). Given that our estimate of Mu temperature (30}40 mK) is adequate, it is too premature to discuss 1/¹ at this point. In general, we just remind you that the ¹-dependence is often di+erent from what you ` theoreticallya expect (e.g. ¹\ dependence in KCl). 2. Prokof 1ev also suggested to be empirical, namely, to take the observed ¹ dependence of C(J¹). ¹hen, instead of 1 s, we obtain the thermalization time ranging from 0.25}16 ls depending on how one evaluates the normalization. ¹he suggested Mu temperature higher than the nominal temperature would mean that we are in this sort of marginal situation. However, you should be able to see the peak feature if the Mu `temperaturea (or `average energya) is of the order of 30 mK. 3. ¹his opinion is just a matter of de,nition of the word `delocalizationa. =e think it legitimate to call the state `delocalizeda as long as the wavelength exceeds the unit cell.
463
Generally speaking, your points are mostly based on theoretical `implausibilitya of the matter we are discussing. If you think our interpretation is wrong, you ought to come up with your model to explain the peak feature. =e will be happy to assess your model carefully in regard to our data. K.W. Kehr: >our result on the mean free path implies that the Bloch states of Mu in KCl do not extend very far over the crystal. =ould it be possible to study Mu in an insulating crystal where the nuclei do not possess magnetic moments? ¹he -uctuating magnetic ,elds caused by these nuclei should then be absent, leading to more extended Bloch states. R. Kadono: ¹here is a problem to apply the current technique to study the Mu di+usion in crystals without nuclear moment. =e rely on the -uctuation of the local ,eld (in this case the nuclear hyper,ne ,eld) to probe the translational motion of Mu. But even if we can come up with other methods, I presume that the local ,eld is not the factor to prevent the development of the Bloch state. ¹he energy scale of the nuclear hyper,ne coupling is 10 MHz, which is more than one order of magnitude less than that of the tunneling matrix element.
Physica B 289}290 (2000) 459}463
Muonium atom in the Bloch state R. Kadono *, W. Higemoto , K. Nagamine , F.L. Pratt Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan
Abstract We present a clear signature of muonium (Mu) occupying a Bloch state in KCl at a temperature ((10 mK) much lower than the Mu energy band width (K0.15 K), i.e., the resonant enhancement of muon spin relaxation at a "eld where the Mu Zeeman splitting crosses a van Hove singularity in the Mu density of states. This feature (similar to the muon-level crossing resonance) can be used to study the energy band structure of Mu in a crystalline host. 2000 Elsevier Science B.V. All rights reserved. Keywords: Coherent tunneling; Bloch state; Muonium; van Hove singularity
It is now established that the di!usion of interstitial muon(ium) at lower temperatures is described by the tunneling of a muon(ium)-polaron state (with an e!ective tunneling matrix D) associated with two-phonon exchange processes, where the hopping frequency is l(¹)KD/X(¹) with the damping factor X(¹)J¹? due to interaction with the phonon/electron bath. While the hopping rate levels o! at l(¹)KD as temperature is lowered to satisfy X(¹)(D, the above picture of localized atoms undergoing occasional tunneling is still valid as long as the temperature is high enough for Mu to occupy all the possible energy eigenstates in the Mu energy band D (i.e., ¹
* Corresponding author. Fax: #81-298-64-5623. E-mail address: [email protected] (R. Kadono).
a coherent Bloch state; the Mu begins to occupy a well de"ned eigenstate of energy (i.e., the bottom of the Mu energy band) which leads to the delocalization of the state vector due to Heisenberg's uncertainty principle. The issue has been theoretically scrutinized previously (for a recent review, see for example Ref. [4]), where the temperature dependence of the longitudinal spin relaxation rate (¹\) was mainly discussed as a possible clue for probing such a state. A more detailed numerical evaluation by Kondo predicted that the magnetic xeld dependence of ¹\ in the Bloch state would be strongly modulated by the van Hove singularities in the Mu density of state [5]. Very recently, we have observed a peak of relaxation rate in KCl cooled down below 10 mK at around 0.15 T, which is in excellent agreement with the above prediction [6]. This feature provides a new type of &robust' evidence that Mu is in the Bloch state. In the case of Mu, one measures the longitudinal muon spin relaxation rate induced by #uctuation of local magnetic "elds acting on the Mu orbital
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 3 5 - X
Physb=5916=Binni=VVC=BG
460
R. Kadono et al. / Physica B 289}290 (2000) 459}463
electron, i.e., ¹\Kd a C(u ), GH GH G H
(1)
where d is the rms value of the random local "eld, C(u) is the spectral density determined by the time correlation function for the local "eld, and u is the GH relevant Mu Zeeman frequency with the respective amplitude a under an applied magnetic "eld [7,8]. GH When ¹
Fig. 1. (a) Tunneling matrix of the Mu-polaron state, D. When ¹
shown in Fig. 2b, we did not see any signi"cant change in the lSR spectrum with cooling down the specimen from 4 K to 20 mK (which seems to be common to the case in Ref. [9]). Based on the interpretation that this result was due to insu$cient thermal conductivity, we prepared a mixture of freshly ground single-crystal KCl and silver
Physb=5916=Binni=VVC=BG
R. Kadono et al. / Physica B 289}290 (2000) 459}463
Fig. 2. (a) A sectional view of the tail of the cryostat. The He}He dilution refrigerator insert has its own LHe shield attached directly to LHe reservoir. The bottom part of the LHe space is shielded with a Te#on ring to eliminate the liquid near the beam windows by He vapor generated by the LHe bubbler. (b) A bulk crystal KCl was mounted with ECO bond, which turned out to have poor thermal contact, and (c) KCl grains mixed with Ag powder and vacuum grease were moulded into the sample holder. The current result was obtained with method (c).
powder (99.9%, 2 lm diameter) and moulded into a sample holder with a small amount of grease (Apiezon) to secure good thermal contact with the mixing chamber (see Fig. 2c). The weight ratio among KCl, Ag powder, and grease was about 1 : 1 : 0.1 (the corresponding volume ratio being 1 : 0.23 : 0.25). Compared with using the bulk crystal, a crude estimation suggests that about 10 times improved e$ciency was expected for cooling those ground crystals due to their larger surface area. The temperature was monitored by a calibrated resistance thermometer at the mixing chamber. (Because of the experimental circumstances, we were not able to use nuclear orientation thermometer and therefore only an upper limit is given for the base temperature of the present experiment.) Conventional muon spin relaxation (lSR) measurements were performed under a longitudinal magnetic "eld (for KCl crystals in this condition the only muon spin relaxation comes from Mu). The lSR time spectra were analyzed by a single exponential decay, A(t)"A(0)exp(!t/¹ ). The results are shown in Fig. 3, where one can notice a clear di!erence between the data at 3.9
461
Fig. 3. Muon spin relaxation rate for Mu in KCl (a) at 3.9 K and (b) below 10 mK. For the solid curves see the text. The dot}dashed curve in (b) is the best "t to a Lorentzian spectrum, whereas the dashed curve is the Lorentzian spectral component "tted in conjunction with a Gaussian peak around 0.15 T to give the solid curve.
K (Fig. 3a) and below 10 mK (Fig. 3b). In particular, there is a broad peak in the relaxation rate in Fig. 3b, which is absent in Fig. 3a. The spectral density in Fig. 3a is reproduced by assuming a Lorentzian distribution plus a constant background relaxation (K2.5;10\ s\), as shown by the solid curve. The magnitude of local "eld d (&1 mT) is about one half of the value at higher temperature, which is similar to the situation in NaCl [2]. The same model, however, completely fails to "t the data in Fig. 3b because of the peak around 0.15 T. Since the peak structure seems to be more pronounced than that in Fig. 1, which was predicted for a speci"c case, we adopt a phenomenological model in which C(u) is represented by the sum of a Lorentzian spectrum and a Gaussian peak. A "tting analysis with this model yields a "tted peak position at 0.16(1) T, corresponding to an energy of 0.11(1) K. This is close to the value of D K0.15 K estimated from the leveling o! of the
Physb=5916=Binni=VVC=BG
462
R. Kadono et al. / Physica B 289}290 (2000) 459}463
jump frequency and thus provides strong evidence that the peak originates from the energy band structure of the Mu atom in KCl. In other words, this peak structure in the spectral density below 10 mK is a clear signature that Mu is in the Bloch state. The Lorentzian part is well reproduced by assuming the Bloch state with d &1.2 mT. The peak is associated with a Gaussian width of about 40 mK, which may provide the actual Mu temperature. As discussed previously, the possibility of attributing the observed peak to anisotropic hyper"ne structure is ruled out by the "eld dependence of the initial muon polarization P (t) [6]. The X quenching pattern of the initial decay positron asymmetry A(0) was in good agreement with the case of isotropic hyper"ne structure with the known value of the Mu hyper"ne parameter (u "2p;4.28;10 s\) for both temperatures except a slight complication due to the presence of a fast spin relaxation process terminated within 10\ s. The remaining issue is the reduction of d from the value seen at higher temperatures (K2 mT). This may be interpreted by a scenario similar to that for Mu in NaCl [2]. The important point here is that d re#ects the rms value of the nhf "elds. While this value remains unchanged for localized Mu, it must be reduced by N\ for a Mu simultaneously probing the local "elds at N sites due to the e!ect of statistical averaging [10,11]. In other words, the characteristic length scale in this regime is the particle wavelength j and therefore we would observe dM "d (a/j) as an e!ec tive value of the nuclear hyper"ne parameter [4]. This interpretation for the present result (i.e., dM K1.2 mT, d K2 mT) immediately leads to an estimation
d j K1.4. " dM a
(2)
This rather short wavelength is in line with the relatively high Mu tempearture inferred from the peak width in Fig. 3b, which might originate from the intrinsic time scale of Mu thermalization. While an estimation based on the presumption of X(¹)J¹ provides a thermal relaxation time of the order of 1 s [4,9], we stress that the empirical
law X(¹)J¹ leads to much shorter thermalization times although it may place the current limit for the base temperature of Mu itself (K30}40 mK). The present result demonstrates that one can potentially study the energy band structure of the `polaron banda for a hydrogen isotope in any crystalline solid where one "nds stable Mu upon muon implantation. We believe that this will add an important new dimension to the study of atomic centers and point defects in solids. We thank J. Kondo for illuminating discussions and for showing us his theoretical calculation prior to publication. Thanks are also given to the RIKEN sta! for their technical support. This work was partially supported by a Monbusho Grant-inAid for Scienti"c Research on Priority Areas.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
R.F. Kie# et al., Phys. Rev. Lett. 62 (1989) 792. R. Kadono et al., Phys. Rev. Lett. 64 (1990) 665. Yu. Kagan, N.V. Prokof'ev, Phys. Lett. A 150 (1990) 320. V.G. Storchak, N.V. Prokof'ev, Rev. Mod. Phys. 70 (1998) 929. J. Kondo, J. Phys. Soc. Japan 68 (1999) 3315. R. Kadono et al., Phys. Rev. Lett. 83 (1999) 987. M. Celio, Helv. Phys. Acta 60 (1987) 600. H.K. Yen., M. Sc. Thesis, University of British Columbia, 1988, unpublished. W.A. MacFarlane et al., Hyper"ne Interactions 85 (1994) 23. A.M. Stoneham, Phys. Lett. A 94 (1983) 353. K.W. Kehr, K. Kitahara, J. Phys. Soc. Japan 56 (1987) 889.
Comments V.G. Storchak: In case of a delocalized state the ,rst thing one has to get is a (¹-dependence of q\. How come you did not see these primary features and see the peak which is a secondary feature coming with the primary feature only? My second point is that the calculated time for Mu thermalization in matter at a temperature of about 10 mK is about 1 s which is much larger than the muon lifetime [Storchak and Prokof1ev, Rev. Mod. Phys. 70 (1998) 929]. No way it is thermalized in your measurements. As a consequence there will be no Boltzman distribution in the band and no enhanced population in the bottom of the band. ¹hen one cannot observe the resonance. So my point is that the peak observed may be explained in any other way but NO¹ because of Mu delocalization.
Physb=5916=Binni=VVC=BG
R. Kadono et al. / Physica B 289}290 (2000) 459}463 My third point: how come you see just a twofold reduction in d ? ¹his can only mean there is no delocalization. R. Kadono: 1. Although we have only two points in terms of temperature we can show you that the ¸orentzian part of the spectral density for the `(10 mKa data is smaller that that at 3.9 K (by &70%). Given that our estimate of Mu temperature (30}40 mK) is adequate, it is too premature to discuss 1/¹ at this point. In general, we just remind you that the ¹-dependence is often di+erent from what you ` theoreticallya expect (e.g. ¹\ dependence in KCl). 2. Prokof 1ev also suggested to be empirical, namely, to take the observed ¹ dependence of C(J¹). ¹hen, instead of 1 s, we obtain the thermalization time ranging from 0.25}16 ls depending on how one evaluates the normalization. ¹he suggested Mu temperature higher than the nominal temperature would mean that we are in this sort of marginal situation. However, you should be able to see the peak feature if the Mu `temperaturea (or `average energya) is of the order of 30 mK. 3. ¹his opinion is just a matter of de,nition of the word `delocalizationa. =e think it legitimate to call the state `delocalizeda as long as the wavelength exceeds the unit cell.
463
Generally speaking, your points are mostly based on theoretical `implausibilitya of the matter we are discussing. If you think our interpretation is wrong, you ought to come up with your model to explain the peak feature. =e will be happy to assess your model carefully in regard to our data. K.W. Kehr: >our result on the mean free path implies that the Bloch states of Mu in KCl do not extend very far over the crystal. =ould it be possible to study Mu in an insulating crystal where the nuclei do not possess magnetic moments? ¹he -uctuating magnetic ,elds caused by these nuclei should then be absent, leading to more extended Bloch states. R. Kadono: ¹here is a problem to apply the current technique to study the Mu di+usion in crystals without nuclear moment. =e rely on the -uctuation of the local ,eld (in this case the nuclear hyper,ne ,eld) to probe the translational motion of Mu. But even if we can come up with other methods, I presume that the local ,eld is not the factor to prevent the development of the Bloch state. ¹he energy scale of the nuclear hyper,ne coupling is 10 MHz, which is more than one order of magnitude less than that of the tunneling matrix element.