- Email: [email protected]
Ultrasonic attenuation measurements in neutron-irradiated quartz: a microscopic model for the tunneling states
. iii!!
CQs
Nuclear Instruments and Methods iu Physics Research B 116 (1996) 511415
me
__
NOMB
Beam Interactions with Materials&Atoms
ELSEVIER
Ultrasonic attenuation measurements in neutron-irradiated quartz: a microscopic model for the tunneling states V. Keppens, C. Laermans
*,M. Coeck
Katholieke Universiteit Leuven, Dept. Physics, Celestijnenlaan
2000, B-3001 L.euven, Belgium
Abstract Ultrasonic attenuation measurements are carried out in neutron-irradiated z-cut quartz for three different doses, in a frequency range from 70 to 320 MHz. The data are analyzed using the tunneling model, and the typical TS-parameters are derived. A comparison with the results obtained from similar x-cut samples shows that the coupling of the tunneling states with the longitudinal phonons is direction-dependent. This confirms the anisotropic behaviour of the tunneling states and gives support to the microscopic picture of the TS as a rotation of coupled SiO, tetrahedra.
1. Introduction At low temperatures, amorphous solids show properties that are quite different from the crystalline behaviour [l]. An adequate description of these “anomalies” can be obtained by the tunneling model [2,3]. This model assumes the existence of low-temperature excitations, the so-called tunneling states (TS), with a wide distribution of energies and relaxation times. But in spite of its success, this model is purely phenomenological and gives no indication about the microscopic nature of the TS. In search of the TS-origin, special attention is paid to crystalline solids with defects, such as irradiated crystals. Neutron-irradiated quartz in particular turned out to be very interesting as a model for the glassy state. By varying the neutron dose, different degrees of disorder can be induced and a gradual transition from a perfect crystal to a completely amorphous material can be obtained. A systematic study of neutron-irradiated quartz therefore allows a careful comparison between the induced disorder and the dynamic properties. Studies of the acoustic saturation, absorption and velocity change, as well as thermal conductivity and specific heat measurements on neutron-irradiated quartz showed “glassy anomalies” which can be explained by the presence of TS, similar in nature to those in vitreous silica and with a density of states increasing with the dose [4,5]. In our laboratory, a systematic study of the TS in neutron-irradiated x-cut quartz is being carried out, by means of ultrasonic measurements [6]. From these mea-
* Corresponding author. Tel.: + 32/16 32 71 28; fax: + 32/ 16 32 79 87.
surements, we could show that the TS do not only reside in the “amorphous” ’ regions; an important part of the TS has to be attributed to the remaining “crystalline” * part of the sample [7,8]. In order to obtain more information about the microscopic origin of these “crystalline” TS, we started recently ultrasonic measurements in neutron-irradiated z-cut quartz [9]. This study revealed a remarkable anisotropy in the behaviour of the TS, which has to be attributed to the tunneling mechanism itself. These results gave evidence for the 01i-o 2 Dauphin6 twins as microscopic origin of the “crystalline” TS, leading to a description of the tunneling mechanism as a rotation of coupled SiO, tetrahedra. In this paper, we report the results of further attenuation measurements in neutron-irradiated z-cut quartz, carried out for three different doses, at several frequencies (in a range from 70 to 320 MHz). From numerical fits of these data, the typical TS-parameters are derived and will be compared with those found in comparable x-cut samples (irradiated under the same conditions and with similar doses).
2. Theoretical considerations According to the tunneling model, the TS arise from the tunneling of atoms or groups of atoms in a double well.
’ By “amorphous” we mean similar to vitreous silica but still somewhat governed by the symmetry of the crystalline matrix. 2 The word “crystalhue” is used for the slightly damaged parts that are similar to u-quartz, but with a disturbed perk&city, mainly due to point defects.
0168-583X/%/$15.00 Copyright 0 1996 Ekevier Science B.V. All rights reserved PII SO168-583X(96)00099-7
512
V. Keppens et al./Nucl.
The energy difference
between the eigenstates
Instr. and Meth. in Phys. Res. B I16 (1996) 511-515
is given by
with A the asymmetry of the double well. A, is the tunnel splitting and can be written as A, = fi D exp (- A) where A describes the overlap of the wavefunctions. One of the basic assumptions of the tunneling model is that A and h are independent of each other and have a uniform distribution: P( A, h)d Ad A = Fd Ad A, with p a constant, the density of states of the TS. To describe the effect of the TS on the low temperature ultrasonic properties, two different mechanisms have to be considered. Firstly, there is the resonant absorption of the sound wave by those TS having an energy splitting E corresponding to the phonon energy (E = fi 0). Because of the wide distribution of energy splittings, this process occurs at all frequencies. However, for our absorption rireasurements, this process need not to be considered, since its contribution in this high-energy experiment is negligible compared to the second mechanism, the relaxation process. The latter results from the modulation of the energy splittings A by the sound wave. The TS are brought out of thermal equilibrium and relaxation occurs via interaction with the thermal phonons. At low temperatures, the most likely process is the one-phonon process, giving rise to a relaxation rate [ 11:
E=/a
(1) where yr and yt represent the coupling of the TS with respectively the longitudinal and transverse phonons. The general expressions for the attenuation due to this process are given in Ref. [lo] and can be calculated numerically. Analytic solutions can be derived for the limiting cases or,,, > 1 (with TV the smallest relaxation time of the TS) and wr,,, +z 1 and are briefly discussed here. At the lowest temperatures, where the condition wr,,, z-=-1 holds, the relaxation attenuation is expected to be frequency independent and proportional to T3 [lo]:
dence in the transition wrm -=z 1 regime.
3. Experiments
region from the wrm > 1 to the
and discussion
Three z-cut specimens of quartz, labeled Z-N4, Z-N5 and Z-N6, have been irradiated up to a neutron dose of respectively 9.9 X 10” n/cm2, 3.0 X 1019 n/cm* and 4.7 X 1019 n/cm* (E 2 0.3 MeV). On these specimens, measurements of the ultrasonic attenuation have been carried out as a function of temperature using a pulse-echo technique. For converting the applied electromagnetic signal into an elastic wave, a LiNbO,-transducer was attached to the samples. Figs. 1 to 3 show the attenuation as a function of temperature for respectively Z-N4, Z-N5 and Z-N6. The observed behavior is qualitatively similar to the absorption found before in irradiated x-cut samples (see e.g. Ref. 161) and unambiguously shows the presence of tunneling states: the T3 dependence at the lowest temperatures is typical of the wrm >> 1 regime of the relaxation attenuation (see Eq. (2)). At higher temperatures, the absorption gradually levels off to a temperature-independent plateau, which corresponds to the wrm (< 1 regime (Eq. (2)). The T3 behavior is frequency-independent for all doses, whereas the plateau varies linearly with the frequency, as predicted by Eqs. (1) and (2). The figures also show that - for a comparable frequency - the absorption increases with increasing dose, as observed before for x-cut quartz [6].
1
0 310MHz
The wr,,, e 1 condition is satisfied at higher temperatures and leads to a temperature-independent attenuation that varies linearly with frequency [lo]: cr=Y+ 2Vl
ITCoPy: 2pv:’
(3)
The expressions given so far are valid provided that the TS relax via absorption or emission of a single thermal phonon. At higher temperatures, above a few kelvin, a two-phonon Raman-process has to be taken into account [IO]. This additional process causes a stronger temperature depen-
E 0.001 0.1
,
, , , ,(,,, ( 1 TEMPERATURE
10
, , , ,,,,
J 100
(K)
Fig. 1. Ultrasonic attenuation as a function of temperature in Z-N4 (dose: 9.9X lOI n/cm’) for two frequencies. -: fitcurve.
V. Keppens et al./Nucl. 10
Instr. and Meth. in Phys. Res. B 116 (1996) 511-515 10
t
t 0 280 MHz
0 320MHz
0 70MHz
0 75 MHz
0.001 0.1
1
10
TEMPERATURE
’
’
0.1
100
8
‘**sl’l
’
’
““3”
1 TEMPERATURE
(K)
3
10
’
I’“’
100
(K)
Fig. 2. Ultrasonic attenuation as a function of temperature in Z-N5 (dose: 3.0 X 10” n/cm? for three frequencies. -: fitcurve.
Fig. 3. Ultrasonic attenuation as a function of temperature in Z-N6 (dose: 4.7X 10” n/cm’) for three frequencies. -: fitcurve.
Since the measurements follow the predictions of the tunneling model, they have been quantitatively analyzed using this model and taking into account both one-phonon and Raman-processes. The obtained fitcurves are shown in Figs. l-3 and am in good agreement with the experimental data. Table 1 gives the values of the TS-parameters C (-&f> and K, and the Raman parameter K,, derived from these fits. As can be seen, the values obtained for
different frequencies are consistent with each other, although they are independently determined. Table 2 contains the parameters obtained from attenuation measurements in x-cut quartz, irradiated with similar doses [6,11,12]. A first comparison shows the same tendency for x-cut and z-cut samples: an increase of the neutron dose mainly affects C and thus Fyf. For the parameter K,, similar values are found in the x-cut and z-cut samples, as
Table 1 TS-parameters derived from attenuation measurements in neutron-irradiated z-cut quartz Dose [n/cm’] Z-N4 9.9 x 10’8
ZN5 3.0 x 10’9
Z-N6 4.7 x 10’9
c (10-6)
30 (145 MHz) 32 (3 10 MHz) a
K, (10’ K-3s-‘)
12 (145 MHz) ll(31OMHz) a
&:
31045 MHz) 33 (310 MHz) a
52 (75 MHz) 52 (145 MHz) 58 (320 MHz) a 11 (75 MHz) 11 (145 MHz) 12 (320 MHz) a 54 (75 MHz) 54 (145 MHz) 60 (320 MHz) a 3 X 10’ (75 MHz) 4 X 10’ (145 MHz) 4 X lo* (320 MHz) a
110 (70 MHz) lOO(130 MHz) llO(280MH~)~ 17 (70 MHz) 16 (130 MHz) 18 (280 MHz) a 110 (70 MHz) 105 (130 MHz) 110 (280 MHz) a 1 X lo3 (70 MHz) 1 X lo3 (130 MHz) 2 X lo4 (280 MHz) a
(106gcm-‘s-2)
K, (K-’ s- ‘)
3 X 10’ (145 MHz) 5 X lo* (310 MHz) a
The accuracy for C and F-y: is lo%, for K, 20%. For K, only an order of magnitude can be detcrmmed. a These data have been reported before in Ref. [9] but ate added here for the sake of completeness.
514
V. Keppens et al./Nucl.
Instr. and Meth. in Phys. Res. B 116 (1996) 511-515
Table 2 TS-parameters derived from attenuation measurements in neutron-irradiated x-cut quartz Dose [n/cm’ I
c (10-6) (lo7 K-3~-‘) py: (IO6 gem-‘s-‘1 K, (K-’ s- ‘) K,
N4 [61 1.2x 10’9 648 MHz
N5 [ll] 2.6X lOI
648 MHz
4.7X lOI 340 MHz
21 42 18 3x
49 42 42
160 46 130
104
2x
104
N6 1121
8X lo3
The accuracy for C and 77: is lo%, for K, 20%. For K, only an order of magnitude can be determined.
expected, since it represents an average coupling over all directions and polarisations. The slight increase with the dose, which was found earlier in x-cut quartz 1131, is confirmed in the z-cut samples. However, a quantitative comparison of the obtained.values for &F reveals for all frequencies the same striking difference: when comparing two similar doses, we find p-y: to be significantly smaller in the z-cut samples than in the x-cut samples. As mentioned above, we reported this anisotropic behavior already before, for one frequency [9]. The data reported here confirm the anisotropic behavior in a wide frequency range. Since the irradiations took place in the same rector, under the same conditions, the damage in the x- and z-cut samples should be very similar for similar doses. Therefore the density of states of the TS must be the same in the samples irradiated up to the same dose. This means that (&f ), 4 (&f), can be reduced to a difference in y,: the coupling of the TS with ultrasonic phonons traveling along the crystallographic z-direction is significantly smaller than the coupling with phonons in the x-direction. The difference as observed through our measurements is much too high to be explained by the anisotropy of the crystal and must therefore be attributed to the tunneling mechanism itself. As explained in Ref. [9], the observed anisotropy gives evidence for the OL1-~z twin domains as microscopic origin of the “crystalline” tunneling states. These twins are induced in quartz by irradiation and consist of groups of SiO, tetrahedra which are related to each other by a rotation over 180” around the threefold axis (c- or z-axis). They are essentially the same as the twins observed at the ol-p transition (which takes place in quartz at 573°C) [ 141. According to Comes et al., the presence of radiation-induced twins can explain the disorder in the less damaged “crystalline” parts of neutron-irradiated quartz 1151. Both (Y1- and cr.,-configutitions have a trigonal structure, but the presence of both structures in equal amounts would explain the apparent hexagonal symmetry observed for higher doses [15]. Since both the hexagonal symmetry and
the density of states saturate at a similar dose 171, it has been suggested that the TS in the “crystalline” fraction of the irradiated samples might be related to these domains. The observation of TS in electron-irradiated quartz is also in agreement with this suggestion, since twinning as a consequence of electron irradiation has been clearly observed [16]. The movements involved in the twinning motion can be described as small rotations of SiO, tetrahedra, which involve movements of the oxygen atoms up to about 0.4 A. These movements take place mainly in the plane perpendicular to the crystallographic z-axis. The SiO, tetrahedra stay rigid but are slightly tilted along the binary axis [ 171;the Si atoms remain mainly fixed, whereas the oxygen atoms move essentially in the x-direction. The anisotropic behaviour of the TS gives support to the suggestion that part of the TS is related to the twins: if configuration changes inherent to the twins are involved in the interaction with the ultrasonic phonons, it can indeed be expected that a longitudinal wave traveling in the x-direction - the direction of the moving oxygen atoms can more easily induce a transition of a tetrahedron (or a chain of tetrahedra) than a wave in any other direction, leading to a higher coupling. The anisotropy as observed from our measurements gives therefore experimental evidence for the twins as microscopic origin for at least part of the TS, confirming previous suggestions. Considering the movements involved in the twinning, the tunneling mechanism can be described as small rotations or tilts of coupled tetrahedra through distances of about 0.4 A [9]. Scrutinizing Tables 1 and 2 again it is striking that the parameter C has a dose dependence which is different for both orientations. In fact for the x-cut sample it is strongly non-linear while for the z-cut sample C is more closely proportional to the dose. For the dose N6 it is known that the crystalline part of the sample is not trigonal anymore but hexagonal. It consists of the OL 1 and CL 2 modification in similar amounts, which makes the ultrasonic attenuation strongly anisotropic. At low doses the “crystalline” TS seem to play a minor role (no anisotropy) while at higher dose the crystalline part is full of twins and the “crystalline” TS play a more dominant role. The behavior of C with dose indicates that the relative contribution of the crystalline TS increases with dose, with as a consequence a stronger than linear increase of the TS DOS with dose in the x-direction in which those TS participate most. It is interesting to note that the results of attenuation measurements carried out after heat treatment of neutronirradiated x-cut quartz also fit within the microscopic picture of the twins as origin of part of the TS. Recently, we could indeed show that the modifications observed after annealing can be interpreted in terms of modifications of the twin domains, induced by heat treatments above the ct-p transition temperature [lS]. The consistency of the whole gives therefore additional support to the twinning model and the resulting description of the TS in neutronirradiated quartz as a rotation of coupled SiO, tetrahedra.
V. Keppens er al./Nucl.
Instr. and M&r. in Phys. Res. B 116 (1996) 511-515
Acknowledgements The authors thank the SCK (Mel, Belgium), in particular J. Comelis, for the neutron irradiation. They are also grateful to the IIKW (Belgium) for financial support.
References [I] W.A. Phillips [2] [3] [4] [5]
[6] [7]
(ed.), Amorphous solids: Low temperature properties (Springer-Verlag, Berlin, 1981). P.W. Anderson, B.I. Halperin and C.M. Varma, Philos. Mag. 25 (1972) 1. W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351. C. Laermans, Phys. Rev. Lett. 42 (1979) 50. For a review, see C. Laermans, in: Structure and Bonding in Non-crystalline Solids, eds. G. Walrafen and A. Revesz (Plenum, New York, 1986) p. 325. A. Vanelstraete and C. Laermans, Phys. Rev. B 42 (1990) 5842 and references herein. A. Vanelstraete and C. Laermans, Phys. Rev. B 38 (1988) 6312.
515
[8] V. Keppens and C. Laermans, Nucl. Instr. and Meth. B 91 (19941346. [9] C. Laermans and V. Keppens, Phys. Rev. B 51 (1995) 8215. [lo] P. Doussineau, C. Fr&ois, R.G. Leisure, A. Levelut and J.Y. Prieur, J. Phys. (Paris) 41 (1980) 1193. [II] V. Keppens and C. Laermaus, Nucl. Instr. and Meth. B 65 (1992) 223. [12] V. Keppens, Ph.D. thesis, K.U. Leuven, unpublished (1995). [13] C. Laermans and V. Esteves, Phys. Lett. 126 (1988) 341. [14] D. Grasse, J. Peisl and B. Domer, Nucl. Instr. and Meth. B 1 (1984) 183. [15] R. Comes, M. Lambert and A. Guinier, in Interaction of Radiation with Solids, ed. A. Bishay (Plenum, New York, 1967) p. 319. [16] J.J. Comer, J. Cryst. Growth 15 (1972) 179. [17] H. Grimm and B. Domer, J. Phys. Chem. Solids 36 (1975) [18] ?Keppens and C. Laermans, Phys. Rev. B 51 (1995) %02; V. Keppens and C. Laermans, in: Microstructure of Irradiated Materials, eds. I.M. Robertson, S.J. Zmkle, L.E. Rehn and W.J. Phytian, MRS hoc. 373 (1995) p. 323.
CQs
Nuclear Instruments and Methods iu Physics Research B 116 (1996) 511415
me
__
NOMB
Beam Interactions with Materials&Atoms
ELSEVIER
Ultrasonic attenuation measurements in neutron-irradiated quartz: a microscopic model for the tunneling states V. Keppens, C. Laermans
*,M. Coeck
Katholieke Universiteit Leuven, Dept. Physics, Celestijnenlaan
2000, B-3001 L.euven, Belgium
Abstract Ultrasonic attenuation measurements are carried out in neutron-irradiated z-cut quartz for three different doses, in a frequency range from 70 to 320 MHz. The data are analyzed using the tunneling model, and the typical TS-parameters are derived. A comparison with the results obtained from similar x-cut samples shows that the coupling of the tunneling states with the longitudinal phonons is direction-dependent. This confirms the anisotropic behaviour of the tunneling states and gives support to the microscopic picture of the TS as a rotation of coupled SiO, tetrahedra.
1. Introduction At low temperatures, amorphous solids show properties that are quite different from the crystalline behaviour [l]. An adequate description of these “anomalies” can be obtained by the tunneling model [2,3]. This model assumes the existence of low-temperature excitations, the so-called tunneling states (TS), with a wide distribution of energies and relaxation times. But in spite of its success, this model is purely phenomenological and gives no indication about the microscopic nature of the TS. In search of the TS-origin, special attention is paid to crystalline solids with defects, such as irradiated crystals. Neutron-irradiated quartz in particular turned out to be very interesting as a model for the glassy state. By varying the neutron dose, different degrees of disorder can be induced and a gradual transition from a perfect crystal to a completely amorphous material can be obtained. A systematic study of neutron-irradiated quartz therefore allows a careful comparison between the induced disorder and the dynamic properties. Studies of the acoustic saturation, absorption and velocity change, as well as thermal conductivity and specific heat measurements on neutron-irradiated quartz showed “glassy anomalies” which can be explained by the presence of TS, similar in nature to those in vitreous silica and with a density of states increasing with the dose [4,5]. In our laboratory, a systematic study of the TS in neutron-irradiated x-cut quartz is being carried out, by means of ultrasonic measurements [6]. From these mea-
* Corresponding author. Tel.: + 32/16 32 71 28; fax: + 32/ 16 32 79 87.
surements, we could show that the TS do not only reside in the “amorphous” ’ regions; an important part of the TS has to be attributed to the remaining “crystalline” * part of the sample [7,8]. In order to obtain more information about the microscopic origin of these “crystalline” TS, we started recently ultrasonic measurements in neutron-irradiated z-cut quartz [9]. This study revealed a remarkable anisotropy in the behaviour of the TS, which has to be attributed to the tunneling mechanism itself. These results gave evidence for the 01i-o 2 Dauphin6 twins as microscopic origin of the “crystalline” TS, leading to a description of the tunneling mechanism as a rotation of coupled SiO, tetrahedra. In this paper, we report the results of further attenuation measurements in neutron-irradiated z-cut quartz, carried out for three different doses, at several frequencies (in a range from 70 to 320 MHz). From numerical fits of these data, the typical TS-parameters are derived and will be compared with those found in comparable x-cut samples (irradiated under the same conditions and with similar doses).
2. Theoretical considerations According to the tunneling model, the TS arise from the tunneling of atoms or groups of atoms in a double well.
’ By “amorphous” we mean similar to vitreous silica but still somewhat governed by the symmetry of the crystalline matrix. 2 The word “crystalhue” is used for the slightly damaged parts that are similar to u-quartz, but with a disturbed perk&city, mainly due to point defects.
0168-583X/%/$15.00 Copyright 0 1996 Ekevier Science B.V. All rights reserved PII SO168-583X(96)00099-7
512
V. Keppens et al./Nucl.
The energy difference
between the eigenstates
Instr. and Meth. in Phys. Res. B I16 (1996) 511-515
is given by
with A the asymmetry of the double well. A, is the tunnel splitting and can be written as A, = fi D exp (- A) where A describes the overlap of the wavefunctions. One of the basic assumptions of the tunneling model is that A and h are independent of each other and have a uniform distribution: P( A, h)d Ad A = Fd Ad A, with p a constant, the density of states of the TS. To describe the effect of the TS on the low temperature ultrasonic properties, two different mechanisms have to be considered. Firstly, there is the resonant absorption of the sound wave by those TS having an energy splitting E corresponding to the phonon energy (E = fi 0). Because of the wide distribution of energy splittings, this process occurs at all frequencies. However, for our absorption rireasurements, this process need not to be considered, since its contribution in this high-energy experiment is negligible compared to the second mechanism, the relaxation process. The latter results from the modulation of the energy splittings A by the sound wave. The TS are brought out of thermal equilibrium and relaxation occurs via interaction with the thermal phonons. At low temperatures, the most likely process is the one-phonon process, giving rise to a relaxation rate [ 11:
E=/a
(1) where yr and yt represent the coupling of the TS with respectively the longitudinal and transverse phonons. The general expressions for the attenuation due to this process are given in Ref. [lo] and can be calculated numerically. Analytic solutions can be derived for the limiting cases or,,, > 1 (with TV the smallest relaxation time of the TS) and wr,,, +z 1 and are briefly discussed here. At the lowest temperatures, where the condition wr,,, z-=-1 holds, the relaxation attenuation is expected to be frequency independent and proportional to T3 [lo]:
dence in the transition wrm -=z 1 regime.
3. Experiments
region from the wrm > 1 to the
and discussion
Three z-cut specimens of quartz, labeled Z-N4, Z-N5 and Z-N6, have been irradiated up to a neutron dose of respectively 9.9 X 10” n/cm2, 3.0 X 1019 n/cm* and 4.7 X 1019 n/cm* (E 2 0.3 MeV). On these specimens, measurements of the ultrasonic attenuation have been carried out as a function of temperature using a pulse-echo technique. For converting the applied electromagnetic signal into an elastic wave, a LiNbO,-transducer was attached to the samples. Figs. 1 to 3 show the attenuation as a function of temperature for respectively Z-N4, Z-N5 and Z-N6. The observed behavior is qualitatively similar to the absorption found before in irradiated x-cut samples (see e.g. Ref. 161) and unambiguously shows the presence of tunneling states: the T3 dependence at the lowest temperatures is typical of the wrm >> 1 regime of the relaxation attenuation (see Eq. (2)). At higher temperatures, the absorption gradually levels off to a temperature-independent plateau, which corresponds to the wrm (< 1 regime (Eq. (2)). The T3 behavior is frequency-independent for all doses, whereas the plateau varies linearly with the frequency, as predicted by Eqs. (1) and (2). The figures also show that - for a comparable frequency - the absorption increases with increasing dose, as observed before for x-cut quartz [6].
1
0 310MHz
The wr,,, e 1 condition is satisfied at higher temperatures and leads to a temperature-independent attenuation that varies linearly with frequency [lo]: cr=Y+ 2Vl
ITCoPy: 2pv:’
(3)
The expressions given so far are valid provided that the TS relax via absorption or emission of a single thermal phonon. At higher temperatures, above a few kelvin, a two-phonon Raman-process has to be taken into account [IO]. This additional process causes a stronger temperature depen-
E 0.001 0.1
,
, , , ,(,,, ( 1 TEMPERATURE
10
, , , ,,,,
J 100
(K)
Fig. 1. Ultrasonic attenuation as a function of temperature in Z-N4 (dose: 9.9X lOI n/cm’) for two frequencies. -: fitcurve.
V. Keppens et al./Nucl. 10
Instr. and Meth. in Phys. Res. B 116 (1996) 511-515 10
t
t 0 280 MHz
0 320MHz
0 70MHz
0 75 MHz
0.001 0.1
1
10
TEMPERATURE
’
’
0.1
100
8
‘**sl’l
’
’
““3”
1 TEMPERATURE
(K)
3
10
’
I’“’
100
(K)
Fig. 2. Ultrasonic attenuation as a function of temperature in Z-N5 (dose: 3.0 X 10” n/cm? for three frequencies. -: fitcurve.
Fig. 3. Ultrasonic attenuation as a function of temperature in Z-N6 (dose: 4.7X 10” n/cm’) for three frequencies. -: fitcurve.
Since the measurements follow the predictions of the tunneling model, they have been quantitatively analyzed using this model and taking into account both one-phonon and Raman-processes. The obtained fitcurves are shown in Figs. l-3 and am in good agreement with the experimental data. Table 1 gives the values of the TS-parameters C (-&f> and K, and the Raman parameter K,, derived from these fits. As can be seen, the values obtained for
different frequencies are consistent with each other, although they are independently determined. Table 2 contains the parameters obtained from attenuation measurements in x-cut quartz, irradiated with similar doses [6,11,12]. A first comparison shows the same tendency for x-cut and z-cut samples: an increase of the neutron dose mainly affects C and thus Fyf. For the parameter K,, similar values are found in the x-cut and z-cut samples, as
Table 1 TS-parameters derived from attenuation measurements in neutron-irradiated z-cut quartz Dose [n/cm’] Z-N4 9.9 x 10’8
ZN5 3.0 x 10’9
Z-N6 4.7 x 10’9
c (10-6)
30 (145 MHz) 32 (3 10 MHz) a
K, (10’ K-3s-‘)
12 (145 MHz) ll(31OMHz) a
&:
31045 MHz) 33 (310 MHz) a
52 (75 MHz) 52 (145 MHz) 58 (320 MHz) a 11 (75 MHz) 11 (145 MHz) 12 (320 MHz) a 54 (75 MHz) 54 (145 MHz) 60 (320 MHz) a 3 X 10’ (75 MHz) 4 X 10’ (145 MHz) 4 X lo* (320 MHz) a
110 (70 MHz) lOO(130 MHz) llO(280MH~)~ 17 (70 MHz) 16 (130 MHz) 18 (280 MHz) a 110 (70 MHz) 105 (130 MHz) 110 (280 MHz) a 1 X lo3 (70 MHz) 1 X lo3 (130 MHz) 2 X lo4 (280 MHz) a
(106gcm-‘s-2)
K, (K-’ s- ‘)
3 X 10’ (145 MHz) 5 X lo* (310 MHz) a
The accuracy for C and F-y: is lo%, for K, 20%. For K, only an order of magnitude can be detcrmmed. a These data have been reported before in Ref. [9] but ate added here for the sake of completeness.
514
V. Keppens et al./Nucl.
Instr. and Meth. in Phys. Res. B 116 (1996) 511-515
Table 2 TS-parameters derived from attenuation measurements in neutron-irradiated x-cut quartz Dose [n/cm’ I
c (10-6) (lo7 K-3~-‘) py: (IO6 gem-‘s-‘1 K, (K-’ s- ‘) K,
N4 [61 1.2x 10’9 648 MHz
N5 [ll] 2.6X lOI
648 MHz
4.7X lOI 340 MHz
21 42 18 3x
49 42 42
160 46 130
104
2x
104
N6 1121
8X lo3
The accuracy for C and 77: is lo%, for K, 20%. For K, only an order of magnitude can be determined.
expected, since it represents an average coupling over all directions and polarisations. The slight increase with the dose, which was found earlier in x-cut quartz 1131, is confirmed in the z-cut samples. However, a quantitative comparison of the obtained.values for &F reveals for all frequencies the same striking difference: when comparing two similar doses, we find p-y: to be significantly smaller in the z-cut samples than in the x-cut samples. As mentioned above, we reported this anisotropic behavior already before, for one frequency [9]. The data reported here confirm the anisotropic behavior in a wide frequency range. Since the irradiations took place in the same rector, under the same conditions, the damage in the x- and z-cut samples should be very similar for similar doses. Therefore the density of states of the TS must be the same in the samples irradiated up to the same dose. This means that (&f ), 4 (&f), can be reduced to a difference in y,: the coupling of the TS with ultrasonic phonons traveling along the crystallographic z-direction is significantly smaller than the coupling with phonons in the x-direction. The difference as observed through our measurements is much too high to be explained by the anisotropy of the crystal and must therefore be attributed to the tunneling mechanism itself. As explained in Ref. [9], the observed anisotropy gives evidence for the OL1-~z twin domains as microscopic origin of the “crystalline” tunneling states. These twins are induced in quartz by irradiation and consist of groups of SiO, tetrahedra which are related to each other by a rotation over 180” around the threefold axis (c- or z-axis). They are essentially the same as the twins observed at the ol-p transition (which takes place in quartz at 573°C) [ 141. According to Comes et al., the presence of radiation-induced twins can explain the disorder in the less damaged “crystalline” parts of neutron-irradiated quartz 1151. Both (Y1- and cr.,-configutitions have a trigonal structure, but the presence of both structures in equal amounts would explain the apparent hexagonal symmetry observed for higher doses [15]. Since both the hexagonal symmetry and
the density of states saturate at a similar dose 171, it has been suggested that the TS in the “crystalline” fraction of the irradiated samples might be related to these domains. The observation of TS in electron-irradiated quartz is also in agreement with this suggestion, since twinning as a consequence of electron irradiation has been clearly observed [16]. The movements involved in the twinning motion can be described as small rotations of SiO, tetrahedra, which involve movements of the oxygen atoms up to about 0.4 A. These movements take place mainly in the plane perpendicular to the crystallographic z-axis. The SiO, tetrahedra stay rigid but are slightly tilted along the binary axis [ 171;the Si atoms remain mainly fixed, whereas the oxygen atoms move essentially in the x-direction. The anisotropic behaviour of the TS gives support to the suggestion that part of the TS is related to the twins: if configuration changes inherent to the twins are involved in the interaction with the ultrasonic phonons, it can indeed be expected that a longitudinal wave traveling in the x-direction - the direction of the moving oxygen atoms can more easily induce a transition of a tetrahedron (or a chain of tetrahedra) than a wave in any other direction, leading to a higher coupling. The anisotropy as observed from our measurements gives therefore experimental evidence for the twins as microscopic origin for at least part of the TS, confirming previous suggestions. Considering the movements involved in the twinning, the tunneling mechanism can be described as small rotations or tilts of coupled tetrahedra through distances of about 0.4 A [9]. Scrutinizing Tables 1 and 2 again it is striking that the parameter C has a dose dependence which is different for both orientations. In fact for the x-cut sample it is strongly non-linear while for the z-cut sample C is more closely proportional to the dose. For the dose N6 it is known that the crystalline part of the sample is not trigonal anymore but hexagonal. It consists of the OL 1 and CL 2 modification in similar amounts, which makes the ultrasonic attenuation strongly anisotropic. At low doses the “crystalline” TS seem to play a minor role (no anisotropy) while at higher dose the crystalline part is full of twins and the “crystalline” TS play a more dominant role. The behavior of C with dose indicates that the relative contribution of the crystalline TS increases with dose, with as a consequence a stronger than linear increase of the TS DOS with dose in the x-direction in which those TS participate most. It is interesting to note that the results of attenuation measurements carried out after heat treatment of neutronirradiated x-cut quartz also fit within the microscopic picture of the twins as origin of part of the TS. Recently, we could indeed show that the modifications observed after annealing can be interpreted in terms of modifications of the twin domains, induced by heat treatments above the ct-p transition temperature [lS]. The consistency of the whole gives therefore additional support to the twinning model and the resulting description of the TS in neutronirradiated quartz as a rotation of coupled SiO, tetrahedra.
V. Keppens er al./Nucl.
Instr. and M&r. in Phys. Res. B 116 (1996) 511-515
Acknowledgements The authors thank the SCK (Mel, Belgium), in particular J. Comelis, for the neutron irradiation. They are also grateful to the IIKW (Belgium) for financial support.
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