Proc. of the Yamada Conf. VII, Muon Spin Rotation, Shimoda, 1983 Hyperfine Interactions 17-19 (1984) 3-16

POSITIVE PIONS IN SOLID STATE PHYSICS K. MAIER

Max-Planck-lnstitut far Metallforschung, lnstitut far Physik, Posrfach 800665, D-7000 Stuttgart 80, Germany Received 18 April 1983

Crystallographic sites of positive pions are determined using channelling effects of the decay muons. Different applications of this rather new method are illustrated. Experimental results obtained on high-perfection metal and semiconductor crystals demonstrate the relevance of the method and lead already to new solid state information. A muon detection system with high energy resolution was developed using a low pressure multiwire chamber and a Si-detector.

1. Introduction In the sequence of the " h y d r o g e n isotopes" extending from the positron (e § to the triton, the positive pion (~r+) stands closest to the muon. This simple particle a p p e a r e d in solid state physics only four years ago [1]. The purpose of this review is to outline the m e t h o d of d e t e r m i n a t i o n of :r + sites in crystals using channelling effects of the decay muons. In the second part, the applications of positive pions are described. A s a "light isotope" of hydrogen the :r + can tell us details about the crystallographic sites p r e f e r e d by particles of unit electrical charge. F u r t h e r m o r e the 1r+ can be used as a probe atom that may get t r a p p e d at crystal defects in much the same way as positrons and muons. The application of :r + in defect studies is closely related to the diffusion of the ~r § - a field that is of great interest in itself. With the mass lying between those of the muon and the p r o t o n , the ~r+ will help us to u n d e r s t a n d q u a n t u m mechanical processes in the diffusion of light positive particles [2]. T h e third part of the p a p e r outlines the experimental technique [3, 4] in particular the detecting system for the 4 M e V decay muons. A similar e x p e r i m e n t a l technique may also be applied to muons [5, 6] and to the heavier K+-mesons [7] (mk ~ 0.5 rap) notwithstanding the fact that in the later case successful experiments will have to wait for much stronger K + beams than those available at present. In contrast to /~SR and nuclear magnetic resonance ( N M R ) in the channelling e x p e r i m e n t s information in the signal is not accumulated during the total time the 9 J.C. Baltzer AG, Scientific Publishing Company, and Yamada Science Foundation

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K. Maier, Positive pions

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particle stays in the crystal. W e observe the :r + site in the crystal at the time of the decay. With pulsed :r § b e a m s , such as in the 17 M H z m o d e at SIN, we are able to study the fate of the pion in the crystal as a function of time. Kinetic effects like t h e r m a l i s a t i o n , diffusion, trapping, and d e t r a p p i n g at defects can be m e a s u r e d in a very direct way [8]. In addition to these special e x p e r i m e n t a l advantages the :r § e x p e r i m e n t s do not require special p r o p e r t i e s of the material, e.g. nuclear magnetic m o m e n t s or high scattering cross sections for neutrons. W h a t is necessary are crystals of g o o d quality, i.e. the mosaic s p r e a d should not exceed a tenth of a degree. T h e sensitivity to impurities is small c o m p a r e d to muons due to the shorter :r § lifetime (r, = 85 r=). F o r the m a j o r i t y of metals, semiconductors, and ionic crystals it is possible to p r e p a r e specimens where i m p l a n t e d :r + are not influenced by the presence of impurities that might act as trapping centers.

2. The method T h e general idea is rather simple. I m p l a n t e d :r § are slowed down in times short c o m p a r e d with the lifetime of the ~r+ (z,~ = 2.6.10 -8 s) [8]. T h e electrostatic repulsion by the nuclei tends to localize the :r + in interstitial sites. T r a p p i n g at defects (e.g. vacancies) leads to different :r + sites in the crystal (Fig. 1). M o r e than 99% of the :r + decay according to ~r§ ~ / ~ + + v, in m o n o e n e r g e t i c muons and muon neutrinos. T h e 4.12 M e V decay muons are excellent projectils to p e r f o r m channelling experim e n t s with internal sources ( s o m e t i m e s called blocking experiments). To get inform a t i o n on the :r + site in the crystal we may use the muons as signals in two different ways.

M u o n intensity profiles. With an energy and position sensitive d e t e c t o r the muon intensity profile a r o u n d a main crystallographic direction is m e a s u r e d (Fig. 2). T h e solid angle for which lattice-steering effects can be o b s e r v e d is roughly d e t e r m i n e d

(o

0000 0 0/0

0\0

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ce

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A

0 0 ,0,,

~0 <0 . ~0 0~ 1o70 6 ~0 1

--,7o,

Fig. 1. Different :r+ sites in a two dimensional lattice. The decay muons may leave the crystal with increased probability from positions B, C, D (channelling). The muons are blocked from position A and B in (10) direction (blocking). Position D is a trapped site (not in the channel center) and the profile shape is different.

v

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6

K. Maier, Positive pions

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by Lindhard's critical angle ~Pcr [9]. For muons withEkin = 4.12 MeV we obtain ~pcr = 0.15 ( Z / d ) 1t2 (~P~rin degrees, d i n / k ) , where Z is the atomic number and d the lattice spacing of the crystal used. Pion sites in a channel lead to an increase of the muon intensity (channelling profile), pions located in a chain of host atoms lead to a decrease of the measured decay muon intensity (blocking profile) [10, 11]. These simple geometrical arguments allow us to distinguish between tetrahedral and octahedral intersitial sites in cubic structures. For a more precise investigation the muon profile may be calculated by computer simulation. From comparison of the measured and simulated profiles we not only get the pion site but also the vibration amplitude of the pion. However, only muons with energies between about 3.5 MeV and 4.1 MeV show lattice steering effects, muons having lost more than about 0.6 MeV by small angle scattering from the electrons have lost their information on the pion site. According to the estimate by Van Vliet [12] the mean multiple-scattering angle zl~p after a penetration length of 10 ~ m at room temperature amounts to 0.18 ~ (Ta) for best channelled muons in a (100) direction and 0.58 ~ for muons with high transverse energy. However, with 150 MeV/c pion beams the range straggling and therefore the implantation depth is much larger. M u o n energy profiles. In the case of lattice steering (channelling or blocking) the muon energy spectrum contains additional information on the Jr+ site in the host lattice. Within the range of the 4 MeV decay muons (200 pm in Ta) the stopping density of the implanted ~r+ is constant. If the pions decay at interstitial sites close to the center of a channel formed by rows of atoms in a main crystallographic direction, the muons can leave the crystal along trajectories which are stabilized close to the channel center (best channelled muons). Those muons travel through regions with small electron density and hence small stopping power compared with the total stopping power in a random direction. Therefore we observe an increased muon intensity at high muon energies. At lower energies, e.g. 3 MeV, the intensity in a channelling and in a random direction is the same because channelled muons undergo dechannelling by small angle scattering with electrons (Fig. 3). Similar arguments are valid in the case of blocking, however the scattering problem is more serious. The energy spectrum contains information of the pion site and the pion state (vibrational amplitudes) just like the intensity profiles. In addition we can study the dechannelling mechanism over distances up to 50/zm.

a) PIONS AS LIGHT ISOTOPES OF HYDROGEN In a period of great scientific and technological interest in the behaviour of hydrogen in metals and alloys it has become possible to perform experiments on shortlived elementary particles of the same nuclear charge as hydrogen. The gap between the positron, the muon and the proton mass is partly bridged using positive pions. On the one hand, the pion is still much lighter than the ordinary hydrogen nuclides and should therefore exhibit quantum effects more clearly than these. On the other

1.01

K. Maier, Positive pions

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o

300 I I I I

200

9

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I 0,7 --

I

I I

I

00

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Energy

{MeV)

I

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50

0

Implantation Depth (iJm)

Fig. 3. Muon energy spectrum of decay muons from positive pions stopped in Ta at 200 K. Polycrystalline Ta, Ta ( 111 ) direction.

hand, the pion is still heavy compared with electrons, so that the theoretical treatment of the behaviour in condensed matter is similar to that of protons, deuterons tritons and positive muons. So far positive pions as light hydrogen isotopes have been used in the following ways.

Pion/"hydrogen" sites. Disregarding the isotope effect, we can determine hydrogen sites in crystals using positive pions. This can be done over large temperature intervals independent of the hydrogen solubility. The short lifetime of the pion guarantees that in materials of reasonable purity the measured site is not affected by impurities. In order to prove that the pion is a "good hydrogen" nucleus we determine the pion site in a tantalum single crystal since Ta is a material with high hydrogen solubility and the behaviour of hydrogen and deuterium is well understood [13] (Fig. 2). The muon intensity was measured along the (100) and the (111) direction. Heights and widths of the two profiles are compatible with tetrahedral interstices. This is in agreement with the determination of deuteron sites [14]. Similar experiments were performed on molybdenum (tetrahedral), gold (octahedral) and germanium (see pions as probe atoms). Since Mo, Au and Ge are materials with low hydrogen solubility direct determination of hydrogen sites are difficult if not impossible. Hence the first pion experiments have already produced new solid state information. A careful analysis of the muon intensity profiles in Ta (111) indicates that at the lowest temperatures the height decreases and the width increases. A similar behaviour was observed in Au (110). The next step will be experiments extending

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K. Maier, Positivepions

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down to very low temperatures (<1 K) in order to look for pions in stable and metastable interstitial sites, e.g. tetrahedral and octahedral sites in Au [15]. b) PIONS AS PROBE OSCILLATORS IN CRYSTAL POTENTIALS The muon flux distribution is sensitive to the vibrations (thermal or quantum mechanical) of the pions. This allows us to study the potential in the neighbourhood of the pion sites and thus to gain further information relevant for the understanding of hydrogen in condensed matter, in particular in materials with low hydrogen solubility. From the shape of the measured profiles we may conclude that in Ta the vibration amplitudes of the pion are rather small (--< 2-10 -11 m at 200 K). C) PION DIFFUSION - DEFECT SPECTROSCOPY WITH PIONS The diffusion of positive pions in single crystals is measured in an indirect way. We use a crystal with a fixed concentration of trapping centers and observe the diffusion of the pions to these sinks. As trapping centers we may use foreign atoms or intrinsic atomic defects, e.g. vacancies. The lattice steering enables us to distinguish between trapped and freely diffusing pions, provided the trapped and free positions are sufficiently different. A shift of a few tenths of an ,~ with respect to the free pion location is enough to change the lattice steering pattern significantly. In the simplest cases, the traps may be described by a capture radius r.. The characteristic time for the trapping of the pions (by an atomic concentration C) of traps is then given by r -

VA

(1)

4m'oCD~

where VA is the atomic volume and Dr the diffusion coefficient of the pion. If r, is of the order of magnitude of ~,~ we are able to observe the transition between free and trapped pions and to deduce D~, provided we may get a good estimate of r0. With a pulsed pion beam we can measure the trapping as a function of time between 3 and 60 ns after implantation [8]. Tantalum: Fig. 4 shows the temperature dependence of the FWHM of the lattice steering pattern in Ta (111). The pure Ta crystal shows the temperature dependence associated with the vibration of the sr§ in tetrahedral interstices and the possible occupancy of metastable octahedral sites at the lowest temperature. At room temperature the pions are unaffected by the oxygen atoms, since the binding enthalpy is Ioo small for the :r + to be bound to the O-atoms. The F W H M of the doped crystal is about the same as that of the undoped crystal. At 55 K, however, the majority of the :r + are trapped by oxygen and decay at displaced interstitial sites. We observe a significant broadening of the channelling peak. With the known defect concentration this leads to a lower limit for the pion diffusivity, D(55 K) --> 10 -tl m2s -l. Between 55 K and 80 K detrapping occurs, which is in qualitative agreement with /zSR experiments [16].

1.01

K. Maier, Positive pions

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II ~

(111) To (,I11) 1.7-10-~O normalized to equal FWHM at 273K Ta

o

\ k

ea

!

tt_

0.5

l

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100

2OO

300 T(KI

--

Fig. 4. Width of the muon intensity profile in Ta as a function of temperature. At low temperatures the Jr+ get trapped at O-impurities.

Germanium: Fig. 5 shows the (110) muon intensity profiles in ultra high purity germanium and in Ge with 10 ~7 Si atoms/cm 3. In ultra high-purity germanium the channelling peak broadens markedly over a quite narrow temperature interval below 80 K [16]. From a comparison of the measured profiles with computer simulations we may deduce that above 80 K the pions occupy tetrahedral interstitial sites and that below 60 K the pion decays in hexahedral and/or bond centred sites. (With the present measurements and computer simulations it has not been possible to distinguish between hexahedral and bond centred positions.) It is difficult to understand why the pion should change its interstitial site in such a narrow temperature region. From /~+SR measurements it is known that muons form muonium (/~+e-) in high purity germanium below about 90 K. If the behaviour of the pion is similar we may see the site of the pions above 80 K and the position of the pionium "nucleus" below 60 K. Lattice steering experiment in Ge with 1017Si atoms/cm 3 do not show the broadening of the muon intensity profile in the temperature region discussed, and we think we do not observe pionium in this crystal. At low temperatures (20 K), however, we observe the growth of off-center peaks in the channelling profile, indicating non-te-

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K. Maier, Positive pions

11

trahedral pion positions, they might be associated with trapping at the Si-atoms. We estimate a diffusivity for pionium D,~(20 K) -> 10 -4 cmEs -1 [17]. Recent #+SR measurements in the same Si-doped germanium show that the muonium (Mu) signal disappears with increasing Si-content. This behaviour is explained by trapping and ionization of the Mu at Si atoms. It is found that muonium diffuses rather fast (D r 10 -4 cmEs-1 at 20 K) in Ge [18]. For a further test of the suggestion "pionium formation in Ge" additional lattice steering experiments on crystals of different Si contents are necessary.

3. Experimental techniques The experimental set-up has to satisfy the following requirements (Fig. 6): (i) In order to resolve the muon intensity profiles the angular resolution has to be better than the critical angles (about 0.2~ At SIN, at which all experiments discussed in this paper have been carried out, the incident ~r§ flux is distributed over several cm 2 cross section. A crystal size of not less than 2 x 2 c m 2 is highly desirable for intensity reasons. Good angular resolution and large crystal area is only possible if the distance between crystal of high perfection and detector is at least 6 m. The growth of enough big crystals presents serious problems e.g. for the bcc-transition metals, but with the help of carefully aligning small crystals to a reasonable large "quasi-single crystal" (a technique which has been developed for neutron scattering experiment), ~r+/# + lattice steering measurements on nearly all crystalline solids are possible. (ii) In order to keep the measuring time reasonable (2h) a two-dimensional position sensitive detector with a good energy resolution is necessary. The maximum lr + stopping density is available at 150 MeV/c :r § momentum. However the width

L

Fig. 6. Schematic drawing ofthe experimentalsetups. 1 aligned crystal, 21aser for optical alignment of the crystal, 3 Pion degrader (beryllium), 4 vacuum chamber, 5 position and energy sensitive muon detector.

12

K. Maier, Positive pions

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of the range straggling distribution is too large for lattice steering. However only muons coming from depths of not more than 20 # may be accepted, this corresponds to about 1% of the stopped pions. As a consequence the detector should resolve the muon energy with the best resolution available. The muon profiles in this paper have been measured with a windowless positionsensitive scintillation detector (1.5 mm thick) with a counting area of 300 mm in diameter. A b o u t 70% of the scintillation light is collected in four photomultiplier tubes. The coordinates X , Y of an event are determined by X = x 1 / ( x I + x2) , Y =

Yl/(Y~ + Y2),

(2)

where xl, x2, y~, and Y2 denote the pulse heights of the four tubes. The division according to (2) is done on-line with analogue dividers. The spatial resolution of the detector is better than 2 cm, its energy resolution for 4 M e V muons corresponds to F W H M = 300 keV. In front of and behind the scintillators are anti-counter tubes in order to reduce the background due to high-energy positrons crossing the detector under small angles (thus having a prolonged path length and giving rise to scintillation light similar to that of 4 MeV muons). For a pion m o m e n t u m of 150 MeV/c and a sample area of 4 c m 2 a signal-to-background ratio of 15 : 1 has been obtained. In order to improve the muon energy resolution a new detection system was developed (Fig. 2). The muon position is measured with a low pressure multiwire proportional chamber (MWPC). A t 40 mb isobutane pressure the detection efficiency for 4 M e V muons is already close to unity, but the windows between the vacuum and the chamber do not have to withstand atmospheric pressure. With a diameter of 150 ram, 10/l thick mylar foils provide a reasonably large safety factor. The energy and angular straggling for the muons passing through the chamber is reasonable small. Behind the chamber, the muons are focused with a quadrupole lens on to a Si-surface barrier detector measuring the muon energy (diameter 70

:I

v

9

// 5 1

2

3

2

6

4

Fig. 7. Muon detector system with high energy resolution. 1-3 low pressure multiwire chamber, 4 quadrupole lens, 5 Si-surface barrier detector, 6 positrons anti-counter for puls pile-up reduction, 1 thin mylar foils, 2 cathode planes (evaporated Al-stripes on 5/~ mylar foils) 3 anode plane with 20/~m diameter wires.

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K. Maier, Positive pions

13

mm, thickness 1 mm). The energy resolution is 80 keV for 4 MeV muons and 50 keV for 5.3 MeV a particles from a 24tAm calibration source. At present the spatial resolution of our chamber is 0.5 cm. A much better resolution is possible but not necessary for this experiment.

4. Conclusions The behaviour of positive pions may be investigated in all crystalline solids using lattice steering effects of their decay muons. Like positive muons, positive pions may be employed both as light "hydrogen isotopes" and as probe particles. Though the method is rather new, a series of successful experiments have already been performed. Improved muon detection systems and stronger pion sources like the projected spallation neutron sources and kaon factories will increase the power and the range of applications of the method [19].

Acknowledgements The author gratefully acknowledges the great help of H.D. Carstanjen, G. Flik, D. Herlach, G. Jtinemann, M. Krenke, H. Rempp, A. Seeger, and W. Sigle, who collaborated on the :r § lattice steering experiments. The measurements on Ge were performed on crystals kindly supplied by E. Hailer (Berkeley). The metal crystals were grown and machined by P. Kepler, W. Maisch and R. Henes (Max-Planck-Institut fUr Metallforschung, Stuttgart). I am further indebted to M. Sch/ifer and G. Wiederoder for their thechnical assistance and to Miss H. Schweyer for preparing the typescript. The new muon-detector was built in an unbureaucratic collaboration with R. Abela, SIN (calculation of muon trajectories), D. Anderson, C E R N detector group (help in chamber design and production) and P. Glasow, Siemens Company (Si surface barrier detector). The experiments were made possible by the financial support of the Bundesministerium fUr Forschung und Technologie, Bonn through the programme "Mittelenergiephysik" and, last but not least, by excellent experimental conditions at SIN.

References [1] K. Maier, D. Herlach, A. Seeger, and H.-D. Carstanjen, Proposal for an Experiment at SIN, RA-79-02 (1979) [2] A. Seeger, This conference, p. 75 [3] K. Maier, Proc. Conf. Nuclear Physics Methods in Materials Research, Darmstadt, 1980, ed. K. Bethge, H. Baumann, H. Jex, and F. Rauch (Vieweg, Braunschweig, 1980) p. 264

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K. Maier, Positive pions

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[4] K. Maier, G. Flik, A. Seeger, D. Herlach, H. Rempp, G. Jrinemann, and H.-D. Carstanjen. Nucl. Instr. Meth. 194 (1982) 159 [5] B.D. Patterson, A. Bosshard, U. Straumann, P. Trufl, A. Wriest, and Th. Wichert, This conference, p. 965 [6] K. Maier, G. Flik, D. Herlach, G. Jrinemanrt, A. Seeger, and H.-D. Carstanjen, Phys. Lett. 86A (1981) 126 [7] A. Seeger, Proc. Intern. Conf. on Hypernuclear and Kaon Physics, Heidelberg, 1982, ed. B. Povh, p. 379 [8] G. Jilnemann, H.-D. Carstanjen, G. Flik, D. Herlach, K. Maier, A. Seeger, and W. Sigle. This conference, p. 959 [9] J, Lindhard, K. Dan. Vid. Selsk Mat.-Fys. Medd. 34 (1965) no. 14 [10] D.S. Gemell, Rev. Mod. Phys. 46 (1974) 129 [11] R. Sizmann and C. Varelas, FestkOrperprobleme XVII, ed. J. Treusch, (Vieweg, Braunschweig, 1977) p. 261 [12] D. Van Vliet, Harwell A E R E Report 6395 (1970) [13] E. Yagi, T. Kobayashi, S. Nakamura, Y. Fukai and K. Watanabe, to be submitted to J. Phys. Soc. Japan [14] H.-D. Carstanjen, Phys. Stat. Sol. a59 (1980) 11 [15] A. Seeger, Phys. Lett. 93A (1982) 33 [16] K. Maier and A. Seeger, Int. Symposium on the Electronic Structure and Properties of Hydrogen in Metals, Richmond, 1982, ed. P. Jena (Plenum Press, 1983, p. 601) [17] G. Flik, University of Stuttgart (FRG) PhD.-Thesis 1983 [18] K.-P. D6ring, K.-P. Arnold, M. Gladisch, N. Haas, E.E. Haller, D. Herlach, W. Jacobs, M. Krause, M. Kraut, H. Ort and A. Seeger, This conference, p. 629 [19] A. Seeger, Proc. 5th Meeting of Int. Coll. on Advanced Neutron Sources (ICANS-V) Jrilich 1981, ed. S. Bauer and D. Filges, Jill-Conf. 45 (1981) p. 113

Discussions

A.M. STONEHAM Question: Your interesting result of a site change in Ge has (as you said) a possible explanation of pionium formation in one case. Another possibility (also a form of "spur" reaction) is that bound excitons are involved, with temperature-dependent recombination. Two possible experiments are (a) optical generation of excitons and (b) looking at gray tin, where the zero gap should exclude such effects. K. MAIER Answer: Both experiments can be done. Experiments with a pulsed pion beam are planned to look to the kinetics of the observed site change, T. YAMAZAKI Question: What is the basis for your argument to ascribe the anomalous temperature dependence of the channeling width to pionium formation? I don't think the channeling experiment alone indicates the electronic structure. K. MAIER Answer: The channeling experiment does not indicate the electronic structure. The argument for pionium formation and the site change in a narrow temperature range is difficult to understand with a single pion. In addition #SR experiments show the change from a muon to a muonium in a similar region. If Si-impurities are present, no site change occurs and no muonium is observed in the #SR experiment.

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K. Maier, Positive pions

15

A. S E E G E R Comment to question by Yamazaki: At the present stage it is not possible to prove that pionium formation is responsible for the strong temperature dependence of the channeling profiles in Ge. However, it appears that it provides the simplest and most convincing explanation of the experimental findings. The argument is as follows. The alternative explanation could be that the pions remain in one and the same state (say, "naked" :r +) and that with increasing temperature they occupy metastable states of higher energy. At high enough temperature one could then have equipartition between the two type of sites. However, the high temperature channeling profiles do not look as if they could be composed easily of the low temperature profile and a second one. Furthermore, in the situation just described, the population of the high energy levels should take place gradually as the temperature increases. However, experimentally the transition is found to be exceedingly sharp. With successive measuring temperature about 10% apart, no intermediate curve has been found. Such an exceedingly narrow temperature range is difficult to understand by population of different sites by the same particle but could be understood in terms of an electronic transition. Future measurements with the time-differential method should throw further light on this question. A. WEIDINGER Question: Can you deduce from the height of the channeling peak in the Ge-experiment 1) which percentage of muons is showing the channeling effect at all and 2) how many participate in the transformation from the low to the high temperature results? If might be interesting to compare these results with the fractions of Mu, Mu + and/.t + in Ge. K. M A I E R Answer: With a quantitative analysis of measured pulse heights one has to be careful in the present state of the experimental technique. Nevertheless, the peak height in Ge along (110) direction indicates that the majority of the pions stay in an interstitial position in the channel. To understand the large change in the peak width it is necessary that at least half of the pions jumps in off-centered positions.