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Diffusion of positive muons in single crystals of tantalum
Volume 67A, number 3
PHYSICS LETTERS
7 August 1978
DIFFUSION OF POSITWE MUONS IN SINGLE CRYSTALS OF TANTALUM H. SCHILLING, M. CAMANI, F.N. GYGAX, W. RÜEGG and A. SCHENCK Laborato.‘y for High Energy Physics, Swiss Federal Institute of Technology (ETH), Zurich, do SIN, CH-5234 Villigen, Switzerland Received 11 April 1978
The diffusion of positive muons in Ta has been measured between 29 K and 91 K. The results are well reproduced by an 9 s~1. Arrhenius relationship with activation energy Fa = (41.6 ±3.0) meV and preexponential factor ~o (3.0~~)X i0 The muon seems to be locally in a tunneling state. In addition, the neighboring Ta nuclei show a very strong quadrupole interaction.
Recent studies of the interstitial diffusion of positive muons in various metals [1] have revealed the muon to be a very sensitive probe of the involved mechanisms which also determine hydrogen diffusion. We report here on results obtained from pSR measurements in single crystals of tantalum [21 in the ternperature range between 29 K and 91 K and for applied fields from 150 G to 5500G. The diffusion of hydrogen has been studied down to temperatures of 95 K [31. Tantalum presents thus the first case in which the investigated temperature ranges for muon and hydrogen diffusion are so close that a comparison be. comes meaningful. Hydrogen is supposed to diffuse via tetrahedral interstitial sites [4] However, the lattice strain field induced by a hydrogen impurity has been found to display a cubic symmetry rather than a tetragonal one [5,6] It has been suggested, among other explanations, that the cubic symmetry may be due to the formation of a delocahzed state of hydrogen over a planar ring array of 6 tetrahedral interstitial sites around the (111>-axis [6]. From X-ray measure.
.
The cylinder axes of the different samples corresponded to the (100>, (110> and (lii) crystal axes, which were oriented parallel to the applied field. The samples, from Metals Research Corp., England, had a specified purity of better than 99.9%. They have been used as obtained without further purification. The pSR-technique consists in obsei-ving the muon spin rotation via the anisotropic distribution of positrons from the muon decay, by recording a positron rate versus elapsed muon life time histogram [8] The spin rotation signal shows a damping due to small field inhomogeneities originating from the dipole moments of the Ta nuclei (S = 7/2, IIJ = 2.36 nm). Such a damping will be reduced by diffusive motion of the muon; in NMR this leads to a motional line narrowing. The damping function is commonly given by the expression [9] .
~
—
( ~
(1) 2 2 =
A
0 exp ~—2u Te [exp(—t/Tc) 1 + t/T~1 2 is}.proporA0 is an asymmetry parameter, u tional toeffective the second moment of the Ta nuclear dipole field distribution and assumed to be constant over the considered temperature region. rc is the correlation time associated with the muon motion, which can be viewed as the average residence time of the j.z~at some particular site. In order to extract the asymmetry function (eq. —
ments of inferred the lattice constant ofhydrogen-charged Ta it can be that the local linear dilation of the Ta-lattice amounts to 5.2% [7]. The present experiment was carried out at the SIN superconducting muon channel with a standard pSR set-up. About 5—6 X 106 p+ decay events were recorded in one spectrum. The single crystal Ta targets were of cylindrical shape, 40 mm long and 8 mm wide.
231
Volume 67A, number 3
PHYSICS LETTERS
(1)), the full time histogram (8 lAs) was divided into 12 subsections and an average time independent asymmetry parameter A(t~)was fitted in each subsection i. The results were corrected for a time independent background but not for any background lASR signal, which was estimated to be small. This procedure facilitated a global fit to all data, as described further below, Fig. I shows the normalized results for 10 out of 28 histograms taken between 29 K and 91 K with an ap-
TANTALUM 1.20
1.10
.
90k <111>
1 00
+
f
.~
o~o 110 80K <111>
-
1
1 ~ 0go 1.10 59K <100> 1 00 ~ 090 110
.
7 August 1978
plied field of 250 G and for different crystal orientations with respect to the applied field. In the analysis it has been assumed that the diffusion rate r~‘could be expressed by an Arrhenius law: = ~ exp(—E~/kT), (2) with Ea the activation energy. Eq. (2) was then inserted into eq. (1) which was subsequently fitted simultaneously to all Ä(t1) for all 28 pSR spectra. The 1. ± results are: Ea = (41.6 ±3.0) meY, u0 =(3.0 X 10~s~,u=(0.1094 ±0.0011)ps Various consistency tests proved that the results were stable against selective use of input data; in particular,no effect of crystal orientation was found. The field dependence of the depolarization rate was measured for a fixed temperature of 30 K for all three crystal orientations. Since the diffusion rate is already negligibly small at 30 K the depolarization function can be approximated by exp(—a2t2). The resuits from corresponding fits are shown in fig. 2. The solid linethrough with u all =(0.l073 ±0.010)ps~represents a best fit the obtained u-values. No field and
55K <100>
1 00
>-
~ 110 100
~ ~
090 110
e::
no orientation dependence is vriible. The larger scattering at high fields is believed to originate from iahomogeneities in the applied field. The activation energy found in this experiment co-
~
1.00 <
0go 110 ~
~
100
~
~z
090 110
Z
100
o w
090 110
hydrogen diffusion in the Ta inactivation the temperature range for incides very well with energy found
TANTALUM 0.19
45K <111>
<111> 40K<100>
— ~ <100> <100>—~ ___________________________________
1.00 110
o L) w u
090 0.90 ~ 1.00
~0.10~ ‘0 015
0 80 0.70 050 0.40 0
0.05
1
2 3 4 5I 6I TIME [MICROSEC]
7J
8
*~ +
~
I
<111>—
30k
.1
0.2 0.5BEXT[KGAUSSI 1 _______________ 5 Fig.a function 2.0.02 The___________________ dipolar spread o,_________ obtained 30 K,results is shown as of thefield external field. The solidatcurve from a fit through the experimental points. The short solid lines at t
the left and right borders are calculated values for the proposed tunneling state (see text) in the low and high field limit respectively, assuming no lattice relaxation. t
Fig. 1. Asymmetry values 1(t
t
t
1) plotted for 10 out of 28 histograms, taken at 250 G. The solid lines represent the result from a simultaneous fit of eq. (1) to all 28 spectra (see text). 232
Volume 67A, number 3
PHYSICS LETI‘ERS
95 K—250 K: E~= 40 ± 6meV [3] . It is hard to imagine that this agreement could be fortuitous. It is interesting to note further in this respect, that the activation energy for deuterium in the range between 95 K and 250 K also agrees with the hydrogen activation energy, in contrast to the situation in Nb [3]. The preexponential factor found in this experiment is however smaller by an order of magnitude than the one obtained for hydrogen = (8.8 ±5.3) X 1010 s~).This difference can be understood phenomenologically if one postulates that both hydrogen and muon diffusion are limited by the same bottle-neck (e.g. traps). The lower preexponential factor found for muons follows then from the fact that only the slow step of the diffusion, to be associated with the escape rate from the bottle-neck, influences the damping of the muon precession signal, while fast diffusion outside the bottle-neck region leads to considerable motional narrowing, i.e. to no depolarization [10]. For hydrogen, on the other hand, one has measured the effective total diffusion rate. The field independence of u at 30 K can be explained by a very strong quadrupole interaction of the Ta nuclei (Q0 = 4.2 b) in an electric field gradient
(vs‘
(EFG) [11,12] ‚which must be of orderItV~~/e = q 3 at the nearest Ta-neighbors. is interesting 1.6 A— to note that the EFG due to the lA~in fcc Cu is only q 0.27 A—3 [12] . It seems unlikely that such a large EFG could arise solely from the screened muon potential. Rather it is probably of crystalline origin, as can be expected if the muon causes a lattice distortion, thereby destroying locally the cubic symmetry of the crystal. The independence of u on crystal orientation and its value at 30 K have been analyzed in terms of vanous models for the local state of the muon including possible lattice relaxation and electric field gradient effects. It appears that a model in which the muon is in an extended (tunneling) state involving the four tetrahedral and the four trigonal sites in the (100) plane, together with a linear lattice dilation of ‘-‘13% most consistently reproduces the general features of our data 1~l As shown by Birnbaum and Flynn [13], this model also has the merit that it leads to approxi. mate cubic symmetry of the strain field, in accordance -
*1
Details of this analysis will be reported in a future publica-
7 August 1978
with results for H in Ta [5,6] . The existence of the tunneling state proposed in ref. [5] can be ruled out, since it would predict a smaller u than observed. If the present hypothesis of a bottle-neck limited diffusion is indeed correct, it is tempting to speculate on some possible connection between bottle-neck and tunneling state. In Nb, for instance, one has found evidence that a tunneling state may be formed around a nitrogen impurity [14—16]. In summary, we have found evidence that in Ta the muon behaves very similarly to hydrogen, despite its considerably smaller mass. References [1] For a recent review see: A. Seeger, in: Hydrogen in metals, eds. G. Alefeld and J. Völld, to be published.
[21 by: Some ~iSR-studies in Ta have been O.preliminary Hartmann et al., Hyp. Int. 4 (1977) 824;reported R.H. Heffner et al., Hyp. mt. 4 (1977) 833; M. Camani et aI., 7th Intern. Conf. on High energy phys. nucl. struct. (Zürich, 1977), Abstract Vol. of Proc. (SIN, 1977); D. Herlach et al., Verhandl. DPG (VI) 13 (1978) 277. Völkl, H.C. Bauer, U. Freudenberg, M. Kokkinidis, G. Lang, K.-A. Steinhauser and G. Alefeld, Proc. Conf. Hydrogen in metals (Tokyo, 1977) (Univ. of Tokyo Press).
[31 J.
[41 H.D. Carstanjen, Proc. lind Intern. Conf.
on Ion beam surf. layer analysis (Karlsruhe, 1975). [5] J. Buchholz, J. VOlk! and G. Alefeld, Phys. Rev. Lett. 30(1973) 318. [61 T.H. Metzger and J. Peisl, Proc. 2nd Intern. Congr. on Hydrogen in metals (Paris, 1977). [7] H. Pfeiffer and H. Peisl, Phys. Lett. 60A (1977) 363. [8] See e.g.: A. Schenck, and particle physics intermediate energies,in: ed.Nuclear J.B. Warren (Plenum, New at York, 1976). [9] A. Abragam, The principles of nuclear magnetism (Clarendon, Oxford, 1970). [10] M. Borghini, O. Hartmann, E. Karlsson, K.W. Kehr,
[11] [121 [13] [14] [15] [16]
TO. Niinikoski, L.O. Norlin, K. Pernestal, D. Richter, J.C. Saulié and E. Walker, CERN-preprint 1978, to be published. 0. Hartmann, Phys. Rev. Lett. 39(1977) 832. M. Camani, F.N. Gygax, W. Rüegg, A. Schenck and H. Schilling, Phys. Rev. Lett. 39 (1977) 836. H.K. Birnbaum and C.P. Flynn, Phys. Rev. Lett. 37 (1976) 25. C.G. Chen and H.K. Birnbaum, Phys. Stat. Sol. (a) 36 (1976). H.K. Birnbaum et al., Phys. Lett. 65A (1978) 435. C. Morkel, W. Wipf and K. Neumaier, Phys. Rev. Lett. 40 (1978) 947.
tion.
233
PHYSICS LETTERS
7 August 1978
DIFFUSION OF POSITWE MUONS IN SINGLE CRYSTALS OF TANTALUM H. SCHILLING, M. CAMANI, F.N. GYGAX, W. RÜEGG and A. SCHENCK Laborato.‘y for High Energy Physics, Swiss Federal Institute of Technology (ETH), Zurich, do SIN, CH-5234 Villigen, Switzerland Received 11 April 1978
The diffusion of positive muons in Ta has been measured between 29 K and 91 K. The results are well reproduced by an 9 s~1. Arrhenius relationship with activation energy Fa = (41.6 ±3.0) meV and preexponential factor ~o (3.0~~)X i0 The muon seems to be locally in a tunneling state. In addition, the neighboring Ta nuclei show a very strong quadrupole interaction.
Recent studies of the interstitial diffusion of positive muons in various metals [1] have revealed the muon to be a very sensitive probe of the involved mechanisms which also determine hydrogen diffusion. We report here on results obtained from pSR measurements in single crystals of tantalum [21 in the ternperature range between 29 K and 91 K and for applied fields from 150 G to 5500G. The diffusion of hydrogen has been studied down to temperatures of 95 K [31. Tantalum presents thus the first case in which the investigated temperature ranges for muon and hydrogen diffusion are so close that a comparison be. comes meaningful. Hydrogen is supposed to diffuse via tetrahedral interstitial sites [4] However, the lattice strain field induced by a hydrogen impurity has been found to display a cubic symmetry rather than a tetragonal one [5,6] It has been suggested, among other explanations, that the cubic symmetry may be due to the formation of a delocahzed state of hydrogen over a planar ring array of 6 tetrahedral interstitial sites around the (111>-axis [6]. From X-ray measure.
.
The cylinder axes of the different samples corresponded to the (100>, (110> and (lii) crystal axes, which were oriented parallel to the applied field. The samples, from Metals Research Corp., England, had a specified purity of better than 99.9%. They have been used as obtained without further purification. The pSR-technique consists in obsei-ving the muon spin rotation via the anisotropic distribution of positrons from the muon decay, by recording a positron rate versus elapsed muon life time histogram [8] The spin rotation signal shows a damping due to small field inhomogeneities originating from the dipole moments of the Ta nuclei (S = 7/2, IIJ = 2.36 nm). Such a damping will be reduced by diffusive motion of the muon; in NMR this leads to a motional line narrowing. The damping function is commonly given by the expression [9] .
~
—
( ~
(1) 2 2 =
A
0 exp ~—2u Te [exp(—t/Tc) 1 + t/T~1 2 is}.proporA0 is an asymmetry parameter, u tional toeffective the second moment of the Ta nuclear dipole field distribution and assumed to be constant over the considered temperature region. rc is the correlation time associated with the muon motion, which can be viewed as the average residence time of the j.z~at some particular site. In order to extract the asymmetry function (eq. —
ments of inferred the lattice constant ofhydrogen-charged Ta it can be that the local linear dilation of the Ta-lattice amounts to 5.2% [7]. The present experiment was carried out at the SIN superconducting muon channel with a standard pSR set-up. About 5—6 X 106 p+ decay events were recorded in one spectrum. The single crystal Ta targets were of cylindrical shape, 40 mm long and 8 mm wide.
231
Volume 67A, number 3
PHYSICS LETTERS
(1)), the full time histogram (8 lAs) was divided into 12 subsections and an average time independent asymmetry parameter A(t~)was fitted in each subsection i. The results were corrected for a time independent background but not for any background lASR signal, which was estimated to be small. This procedure facilitated a global fit to all data, as described further below, Fig. I shows the normalized results for 10 out of 28 histograms taken between 29 K and 91 K with an ap-
TANTALUM 1.20
1.10
.
90k <111>
1 00
+
f
.~
o~o 110 80K <111>
-
1
1 ~ 0go 1.10 59K <100> 1 00 ~ 090 110
.
7 August 1978
plied field of 250 G and for different crystal orientations with respect to the applied field. In the analysis it has been assumed that the diffusion rate r~‘could be expressed by an Arrhenius law: = ~ exp(—E~/kT), (2) with Ea the activation energy. Eq. (2) was then inserted into eq. (1) which was subsequently fitted simultaneously to all Ä(t1) for all 28 pSR spectra. The 1. ± results are: Ea = (41.6 ±3.0) meY, u0 =(3.0 X 10~s~,u=(0.1094 ±0.0011)ps Various consistency tests proved that the results were stable against selective use of input data; in particular,no effect of crystal orientation was found. The field dependence of the depolarization rate was measured for a fixed temperature of 30 K for all three crystal orientations. Since the diffusion rate is already negligibly small at 30 K the depolarization function can be approximated by exp(—a2t2). The resuits from corresponding fits are shown in fig. 2. The solid linethrough with u all =(0.l073 ±0.010)ps~represents a best fit the obtained u-values. No field and
55K <100>
1 00
>-
~ 110 100
~ ~
090 110
e::
no orientation dependence is vriible. The larger scattering at high fields is believed to originate from iahomogeneities in the applied field. The activation energy found in this experiment co-
~
1.00 <
0go 110 ~
~
100
~
~z
090 110
Z
100
o w
090 110
hydrogen diffusion in the Ta inactivation the temperature range for incides very well with energy found
TANTALUM 0.19
45K <111>
<111> 40K<100>
—
1.00 110
o L) w u
090 0.90 ~ 1.00
~0.10~ ‘0 015
0 80 0.70 050 0.40 0
0.05
1
2 3 4 5I 6I TIME [MICROSEC]
7J
8
~
I
<111>—
30k
.1
0.2 0.5BEXT[KGAUSSI 1 _______________ 5 Fig.a function 2.0.02 The___________________ dipolar spread o,_________ obtained 30 K,results is shown as of thefield external field. The solidatcurve from a fit through the experimental points. The short solid lines at t
the left and right borders are calculated values for the proposed tunneling state (see text) in the low and high field limit respectively, assuming no lattice relaxation. t
Fig. 1. Asymmetry values 1(t
t
t
1) plotted for 10 out of 28 histograms, taken at 250 G. The solid lines represent the result from a simultaneous fit of eq. (1) to all 28 spectra (see text). 232
Volume 67A, number 3
PHYSICS LETI‘ERS
95 K—250 K: E~= 40 ± 6meV [3] . It is hard to imagine that this agreement could be fortuitous. It is interesting to note further in this respect, that the activation energy for deuterium in the range between 95 K and 250 K also agrees with the hydrogen activation energy, in contrast to the situation in Nb [3]. The preexponential factor found in this experiment is however smaller by an order of magnitude than the one obtained for hydrogen = (8.8 ±5.3) X 1010 s~).This difference can be understood phenomenologically if one postulates that both hydrogen and muon diffusion are limited by the same bottle-neck (e.g. traps). The lower preexponential factor found for muons follows then from the fact that only the slow step of the diffusion, to be associated with the escape rate from the bottle-neck, influences the damping of the muon precession signal, while fast diffusion outside the bottle-neck region leads to considerable motional narrowing, i.e. to no depolarization [10]. For hydrogen, on the other hand, one has measured the effective total diffusion rate. The field independence of u at 30 K can be explained by a very strong quadrupole interaction of the Ta nuclei (Q0 = 4.2 b) in an electric field gradient
(vs‘
(EFG) [11,12] ‚which must be of orderItV~~/e = q 3 at the nearest Ta-neighbors. is interesting 1.6 A— to note that the EFG due to the lA~in fcc Cu is only q 0.27 A—3 [12] . It seems unlikely that such a large EFG could arise solely from the screened muon potential. Rather it is probably of crystalline origin, as can be expected if the muon causes a lattice distortion, thereby destroying locally the cubic symmetry of the crystal. The independence of u on crystal orientation and its value at 30 K have been analyzed in terms of vanous models for the local state of the muon including possible lattice relaxation and electric field gradient effects. It appears that a model in which the muon is in an extended (tunneling) state involving the four tetrahedral and the four trigonal sites in the (100) plane, together with a linear lattice dilation of ‘-‘13% most consistently reproduces the general features of our data 1~l As shown by Birnbaum and Flynn [13], this model also has the merit that it leads to approxi. mate cubic symmetry of the strain field, in accordance -
*1
Details of this analysis will be reported in a future publica-
7 August 1978
with results for H in Ta [5,6] . The existence of the tunneling state proposed in ref. [5] can be ruled out, since it would predict a smaller u than observed. If the present hypothesis of a bottle-neck limited diffusion is indeed correct, it is tempting to speculate on some possible connection between bottle-neck and tunneling state. In Nb, for instance, one has found evidence that a tunneling state may be formed around a nitrogen impurity [14—16]. In summary, we have found evidence that in Ta the muon behaves very similarly to hydrogen, despite its considerably smaller mass. References [1] For a recent review see: A. Seeger, in: Hydrogen in metals, eds. G. Alefeld and J. Völld, to be published.
[21 by: Some ~iSR-studies in Ta have been O.preliminary Hartmann et al., Hyp. Int. 4 (1977) 824;reported R.H. Heffner et al., Hyp. mt. 4 (1977) 833; M. Camani et aI., 7th Intern. Conf. on High energy phys. nucl. struct. (Zürich, 1977), Abstract Vol. of Proc. (SIN, 1977); D. Herlach et al., Verhandl. DPG (VI) 13 (1978) 277. Völkl, H.C. Bauer, U. Freudenberg, M. Kokkinidis, G. Lang, K.-A. Steinhauser and G. Alefeld, Proc. Conf. Hydrogen in metals (Tokyo, 1977) (Univ. of Tokyo Press).
[31 J.
[41 H.D. Carstanjen, Proc. lind Intern. Conf.
on Ion beam surf. layer analysis (Karlsruhe, 1975). [5] J. Buchholz, J. VOlk! and G. Alefeld, Phys. Rev. Lett. 30(1973) 318. [61 T.H. Metzger and J. Peisl, Proc. 2nd Intern. Congr. on Hydrogen in metals (Paris, 1977). [7] H. Pfeiffer and H. Peisl, Phys. Lett. 60A (1977) 363. [8] See e.g.: A. Schenck, and particle physics intermediate energies,in: ed.Nuclear J.B. Warren (Plenum, New at York, 1976). [9] A. Abragam, The principles of nuclear magnetism (Clarendon, Oxford, 1970). [10] M. Borghini, O. Hartmann, E. Karlsson, K.W. Kehr,
[11] [121 [13] [14] [15] [16]
TO. Niinikoski, L.O. Norlin, K. Pernestal, D. Richter, J.C. Saulié and E. Walker, CERN-preprint 1978, to be published. 0. Hartmann, Phys. Rev. Lett. 39(1977) 832. M. Camani, F.N. Gygax, W. Rüegg, A. Schenck and H. Schilling, Phys. Rev. Lett. 39 (1977) 836. H.K. Birnbaum and C.P. Flynn, Phys. Rev. Lett. 37 (1976) 25. C.G. Chen and H.K. Birnbaum, Phys. Stat. Sol. (a) 36 (1976). H.K. Birnbaum et al., Phys. Lett. 65A (1978) 435. C. Morkel, W. Wipf and K. Neumaier, Phys. Rev. Lett. 40 (1978) 947.
tion.
233