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Low-frequency acoustic properties of neutron-irradiated quartz
Physica B 263—264 (1999) 139—142
Low-frequency acoustic properties of neutron-irradiated quartz J. Classen *, I. Rohr , C. Enss , S. Hunklinger , C. Laermans Institut fu( r Angewandte Physik, Universita( t Heidelberg, Albert-Ueberle-Str. 3-5, 69120 Heidelberg, Germany Department of Physics, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
Abstract We present results of low-frequency acoustic (vibrating reed) measurements on quartz crystals irradiated with six different neutron doses up to 2;10 n/cm in the temperature range from 7 mK to 1 K. From the temperature dependence of the sound velocity and of the internal friction information on the density of states of tunneling states and their coupling to phonons is obtained. Comparisons are made with recent ultrasonic measurements. 1999 Elsevier Science B.V. All rights reserved. Keywords: Tunneling states; Neutron irradiation; Elastic properties
The low-temperature properties of almost all amorphous solids are governed by the influence of tunneling states, see for example Ref. [1]. Quartz crystals irradiated by fast neutrons are an interesting model system to investigate the density, the dynamics, and perhaps the nature of tunneling defects in ordinary glasses like vitreous silica. By variation of the irradiation dose it is possible to study the transition from an almost perfectly crystalline to an entirely amorphous state. Over the past years the elastic properties of neutron-irradiated quartz have been investigated in detail above 0.3 K by means of ultrasonic experiments [2—5]. However, no elastic measurements have been performed yet at temperatures below 0.3 K, and only very little information is available on the
* Corresponding author. Fax: #49-6221-549262; e-mail: [email protected].
low-frequency elastic properties of neutron-irradiated quartz even at higher temperatures [6]. We have performed vibrating reed experiments on six quartz samples irradiated with different neutron doses in the temperature range from 7 mK to room temperature. The radiation dose varied between 4.6;10 and 200;10 n/cm (neutron energies '300 keV). In addition, an unirradiated quartz crystal was investigated. In this short communication we will discuss only the low-temperature data (¹(1 K). A more comprehensive discussion will be given elsewhere [7]. All measurements have been performed on synthetic quartz samples irradiated at SCK Mol, Belgium. Typical sample dimensions were 8;2; 0.2 mm. The long side (8 mm) of the reeds was parallel to the z-direction of the crystals and the 2 mm side parallel to the x-direction. Fig. 1 shows the temperature dependence of the internal friction (upper panel) and of the relative
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 3 0 - 1
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J. Classen et al. / Physica B 263—264 (1999) 139—142
the low-temperature properties of amorphous solids [1], the slopes of the sound velocity below and above the maximum as well as the height of the internal friction in the plateau region are a direct measure of the parameter C"PM c/ov where PM is the density of states of tunneling systems (assumed to be constant), c the coupling constant between phonons and tunneling systems (“deformation potential”), o the mass density, and v the sound velocity. The tunneling model predicts dv/v"C ln(¹/¹ ) at very low temperatures and
(1)
(2) dv/v"!C ln(¹/¹ ) above a few 100 mK when relaxation sets in. In the latter temperature range the internal friction is predicted to have the value Q\"pC/2 Fig. 1. Temperature dependence of the internal friction (upper panel) and of the relative change of sound velocity (lower panel) below 1 K of an unirradiated quartz crystal ( f"3.1 kHz) and of quartz samples irradiated with neutron doses 4.6;10 n/cm (5.8 kHz), 47;10 n/cm (3.1 kHz), and 200;10 n/cm (2.9 kHz). Sound velocity data for different samples were offset for clarity.
change of sound velocity (lower panel) below 1 K of an unirradiated quartz crystal and of quartz samples irradiated with neutron doses 4.6, 47, and 200; 10 n/cm, respectively). All samples show a fairly similar qualitative behavior but drastic quantitative differences. At very low temperatures the sound velocity increases as the logarithm of temperature, passes a maximum, and decreases towards higher temperatures, again nearly logarithmically. The internal friction increases strongly with rising temperature and becomes almost independent of temperature above a few 100 mK (“plateau region”). This behavior of dv/v and Q\ is typical of solids containing two-level tunneling states with a broad distribution of energy splittings and relaxation times [8]. According to the phenomenological tunneling model, which is quite successfully used to describe
(3)
independent of temperature and frequency. From the above equations it becomes clear that C can be understood as a “macroscopic coupling constant” which determines the integral influence of the tunneling states on the measured quantities. Using Eqs. (1)—(3) we have derived the parameters C from our experimental data. The results are shown in Fig. 2 as a function of the neutronirradiation dose. Rather than discussing the differences of the absolute values of C derived from different methods (for a possible explanation of this effect see Ref. [9]) we would like to stress that the relative change of the parameter C as a function of neutron dose is almost the same for all three methods of determination: The macroscopic coupling constant increases approximately linearly with increasing neutron dose for small irradiation intensities and saturates for doses above 100;10 n/cm. The saturation value is about 40% smaller than in vitreous silica. It is interesting to compare these results with recent data derived from experiments where longitudinal ultrasound was applied to neutron-irradiated quartz samples cut along different crystal axes. Fig. 3 shows the parameter C derived from ultrasound attenuation measurements for crystals cut in x- (open squares) [3,5] and in z-direction (full circles) [4] in comparison to our
J. Classen et al. / Physica B 263—264 (1999) 139—142
141
Fig. 2. “Macroscopic coupling constants” C as a function of neutron dose for all samples investigated. C was derived from the plateau value of Q\ (Eq. (3), open circles), from the slope of the sound velocity above the maximum of sound velocity (Eq. (2), open squares) and from the slope of dv/v below the maximum (Eq. (1), full circles). Solid lines are guides to the eye only. Due to experimental problems the low-temperature slope of the sound velocity could not be measured reliably for the sample irradiated with 100;10 n/cm.
Fig. 3. “Macroscopic coupling constants” C as a function of neutron dose. Open circles denote data derived from vibrating reed measurements of the plateau height of Q\ around 1 K using Eq. (3) (same data as in Fig. 2). Also shown are values of C derived from ultrasound experiments for crystals cut in x(open squares) [3,5] and in z-direction (full circles) [4]. Solid lines are only guides to the eye.
data derived from the plateau value of the internal friction (open circles). The results obtained from ultrasonic experiments on the z-cut samples are fairly close to the vibrating reed data while the x-cut samples exhibit a pronounced maximum of C at a dose of 47;10 n/cm. The strong anisotropy observed in the ultrasonic measurements has been interpreted to arise from the anisotropy of the tunneling motion of two-level-systems in the crystalline regions of the sample [4]. For a neutron dose of 47;10 n/cm the fraction of these only slightly distorted crystalline regions is known to be still about of the total sample volume [2] but these regions consist almost entirely of the so-called a -a - or Dauphine´-twins, i.e., groups of SiO -tet rahedra which are rotated 180° with respect to each other around the optical axis (i.e. the z-axis). Due to the spatial arrangement of the microtwins it is more difficult to perform coupled motions of SiO -tet rahedra, which have been proposed to be the origin of the tunneling states in SiO , along the optical axis than perpendicular to the z-axis [4]. As a consequence, the coupling parameter c between
phonons and tunneling systems and thus the macroscopic coupling constant C exhibits a strong anisotropy. The monotonic dose dependence of C observed in our vibrating reed measurements is consistent with this picture. In these experiments the dominant strain component is longitudinal and parallel to the z-axis. Hence we may expect that the values of C derived from the low-frequency experiments are similar to those for the ultrasound experiments on z-cut samples. Despite the very large difference (factor 10) in experimental frequency — and thus in relaxation times of the relevant tunneling systems — the results do in fact agree even quantitatively within &30%. This demonstrates that the distribution of relaxation times must be similarly broad as for amorphous solids. Even though the parameter C does not show any peculiarity for the reed irradiated with 47;10 n/cm we find a pronounced behavior of this sample when comparing the positions of the sound velocity maxima for the different specimen. As can be seen in the lower panel of Fig. 1 the maximum
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occurs for the highly twinned sample at a lower temperature than both for the sample with higher and with lower irradiation dose. According to the tunneling model the maximum temperature should vary as ¹ Jc\ . Since thermal phonons of all
polarizations and directions can contribute to the relaxation processes, the deformation potential relevant here is an average over all directions. We interpret the minimum in ¹ for the highly twin ned sample to occur from a maximum of this averaged value of c. The maximum in turn may arise from the particularly strong coupling of phonon modes causing particle displacements perpendicular to the z-axis. This picture seems to be fully consistent with the microscopic model for the origin of the tunneling systems mentioned above. To further investigate the strong anisotropy of the phonon coupling observed in high-frequency experiments we are planning to do vibrating reed
measurements on neutron-irradiated quartz samples cut along different crystal axes.
References [1] W.A. Phillips (Ed.), Amorphous Solids — Low Temperature Properties, Springer, Berlin, 1981. [2] A. Vanelstraete, C. Laermans, Phys. Rev. B 42 (1990) 5842. [3] V. Keppens, C. Laermans, Nucl. Instr. and Meth. B 91 (1994) 346. [4] C. Laermans, V. Keppens, Phys. Rev. B 51 (1995) 8158. [5] E. Peeters, C. Laermans, D.A. Parshin, M. Coeck, Nucl. Instr. and Meth. B 141 (1998) 634. [6] A. Vanelstraete, C. Laermans, L. Lejarraga, M.v. Schickfus, S. Hunklinger, Z. Phys. B 70 (1988) 19. [7] J. Classen, I. Rohr, C. Enss, S. Hunklinger, C. Laermans, Eur. Phys. J. B, to appear. [8] J. Classen, C. Bechinger, C. Enss, G. Weiss, S. Hunklinger, Ann. Phys. (Leipzig) 3 (1994) 315. [9] C. Enss, S. Hunklinger, Phys. Rev. Lett. 79 (1997) 2831.
Low-frequency acoustic properties of neutron-irradiated quartz J. Classen *, I. Rohr , C. Enss , S. Hunklinger , C. Laermans Institut fu( r Angewandte Physik, Universita( t Heidelberg, Albert-Ueberle-Str. 3-5, 69120 Heidelberg, Germany Department of Physics, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
Abstract We present results of low-frequency acoustic (vibrating reed) measurements on quartz crystals irradiated with six different neutron doses up to 2;10 n/cm in the temperature range from 7 mK to 1 K. From the temperature dependence of the sound velocity and of the internal friction information on the density of states of tunneling states and their coupling to phonons is obtained. Comparisons are made with recent ultrasonic measurements. 1999 Elsevier Science B.V. All rights reserved. Keywords: Tunneling states; Neutron irradiation; Elastic properties
The low-temperature properties of almost all amorphous solids are governed by the influence of tunneling states, see for example Ref. [1]. Quartz crystals irradiated by fast neutrons are an interesting model system to investigate the density, the dynamics, and perhaps the nature of tunneling defects in ordinary glasses like vitreous silica. By variation of the irradiation dose it is possible to study the transition from an almost perfectly crystalline to an entirely amorphous state. Over the past years the elastic properties of neutron-irradiated quartz have been investigated in detail above 0.3 K by means of ultrasonic experiments [2—5]. However, no elastic measurements have been performed yet at temperatures below 0.3 K, and only very little information is available on the
* Corresponding author. Fax: #49-6221-549262; e-mail: [email protected].
low-frequency elastic properties of neutron-irradiated quartz even at higher temperatures [6]. We have performed vibrating reed experiments on six quartz samples irradiated with different neutron doses in the temperature range from 7 mK to room temperature. The radiation dose varied between 4.6;10 and 200;10 n/cm (neutron energies '300 keV). In addition, an unirradiated quartz crystal was investigated. In this short communication we will discuss only the low-temperature data (¹(1 K). A more comprehensive discussion will be given elsewhere [7]. All measurements have been performed on synthetic quartz samples irradiated at SCK Mol, Belgium. Typical sample dimensions were 8;2; 0.2 mm. The long side (8 mm) of the reeds was parallel to the z-direction of the crystals and the 2 mm side parallel to the x-direction. Fig. 1 shows the temperature dependence of the internal friction (upper panel) and of the relative
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 3 0 - 1
140
J. Classen et al. / Physica B 263—264 (1999) 139—142
the low-temperature properties of amorphous solids [1], the slopes of the sound velocity below and above the maximum as well as the height of the internal friction in the plateau region are a direct measure of the parameter C"PM c/ov where PM is the density of states of tunneling systems (assumed to be constant), c the coupling constant between phonons and tunneling systems (“deformation potential”), o the mass density, and v the sound velocity. The tunneling model predicts dv/v"C ln(¹/¹ ) at very low temperatures and
(1)
(2) dv/v"!C ln(¹/¹ ) above a few 100 mK when relaxation sets in. In the latter temperature range the internal friction is predicted to have the value Q\"pC/2 Fig. 1. Temperature dependence of the internal friction (upper panel) and of the relative change of sound velocity (lower panel) below 1 K of an unirradiated quartz crystal ( f"3.1 kHz) and of quartz samples irradiated with neutron doses 4.6;10 n/cm (5.8 kHz), 47;10 n/cm (3.1 kHz), and 200;10 n/cm (2.9 kHz). Sound velocity data for different samples were offset for clarity.
change of sound velocity (lower panel) below 1 K of an unirradiated quartz crystal and of quartz samples irradiated with neutron doses 4.6, 47, and 200; 10 n/cm, respectively). All samples show a fairly similar qualitative behavior but drastic quantitative differences. At very low temperatures the sound velocity increases as the logarithm of temperature, passes a maximum, and decreases towards higher temperatures, again nearly logarithmically. The internal friction increases strongly with rising temperature and becomes almost independent of temperature above a few 100 mK (“plateau region”). This behavior of dv/v and Q\ is typical of solids containing two-level tunneling states with a broad distribution of energy splittings and relaxation times [8]. According to the phenomenological tunneling model, which is quite successfully used to describe
(3)
independent of temperature and frequency. From the above equations it becomes clear that C can be understood as a “macroscopic coupling constant” which determines the integral influence of the tunneling states on the measured quantities. Using Eqs. (1)—(3) we have derived the parameters C from our experimental data. The results are shown in Fig. 2 as a function of the neutronirradiation dose. Rather than discussing the differences of the absolute values of C derived from different methods (for a possible explanation of this effect see Ref. [9]) we would like to stress that the relative change of the parameter C as a function of neutron dose is almost the same for all three methods of determination: The macroscopic coupling constant increases approximately linearly with increasing neutron dose for small irradiation intensities and saturates for doses above 100;10 n/cm. The saturation value is about 40% smaller than in vitreous silica. It is interesting to compare these results with recent data derived from experiments where longitudinal ultrasound was applied to neutron-irradiated quartz samples cut along different crystal axes. Fig. 3 shows the parameter C derived from ultrasound attenuation measurements for crystals cut in x- (open squares) [3,5] and in z-direction (full circles) [4] in comparison to our
J. Classen et al. / Physica B 263—264 (1999) 139—142
141
Fig. 2. “Macroscopic coupling constants” C as a function of neutron dose for all samples investigated. C was derived from the plateau value of Q\ (Eq. (3), open circles), from the slope of the sound velocity above the maximum of sound velocity (Eq. (2), open squares) and from the slope of dv/v below the maximum (Eq. (1), full circles). Solid lines are guides to the eye only. Due to experimental problems the low-temperature slope of the sound velocity could not be measured reliably for the sample irradiated with 100;10 n/cm.
Fig. 3. “Macroscopic coupling constants” C as a function of neutron dose. Open circles denote data derived from vibrating reed measurements of the plateau height of Q\ around 1 K using Eq. (3) (same data as in Fig. 2). Also shown are values of C derived from ultrasound experiments for crystals cut in x(open squares) [3,5] and in z-direction (full circles) [4]. Solid lines are only guides to the eye.
data derived from the plateau value of the internal friction (open circles). The results obtained from ultrasonic experiments on the z-cut samples are fairly close to the vibrating reed data while the x-cut samples exhibit a pronounced maximum of C at a dose of 47;10 n/cm. The strong anisotropy observed in the ultrasonic measurements has been interpreted to arise from the anisotropy of the tunneling motion of two-level-systems in the crystalline regions of the sample [4]. For a neutron dose of 47;10 n/cm the fraction of these only slightly distorted crystalline regions is known to be still about of the total sample volume [2] but these regions consist almost entirely of the so-called a -a - or Dauphine´-twins, i.e., groups of SiO -tet rahedra which are rotated 180° with respect to each other around the optical axis (i.e. the z-axis). Due to the spatial arrangement of the microtwins it is more difficult to perform coupled motions of SiO -tet rahedra, which have been proposed to be the origin of the tunneling states in SiO , along the optical axis than perpendicular to the z-axis [4]. As a consequence, the coupling parameter c between
phonons and tunneling systems and thus the macroscopic coupling constant C exhibits a strong anisotropy. The monotonic dose dependence of C observed in our vibrating reed measurements is consistent with this picture. In these experiments the dominant strain component is longitudinal and parallel to the z-axis. Hence we may expect that the values of C derived from the low-frequency experiments are similar to those for the ultrasound experiments on z-cut samples. Despite the very large difference (factor 10) in experimental frequency — and thus in relaxation times of the relevant tunneling systems — the results do in fact agree even quantitatively within &30%. This demonstrates that the distribution of relaxation times must be similarly broad as for amorphous solids. Even though the parameter C does not show any peculiarity for the reed irradiated with 47;10 n/cm we find a pronounced behavior of this sample when comparing the positions of the sound velocity maxima for the different specimen. As can be seen in the lower panel of Fig. 1 the maximum
142
J. Classen et al. / Physica B 263—264 (1999) 139—142
occurs for the highly twinned sample at a lower temperature than both for the sample with higher and with lower irradiation dose. According to the tunneling model the maximum temperature should vary as ¹ Jc\ . Since thermal phonons of all
polarizations and directions can contribute to the relaxation processes, the deformation potential relevant here is an average over all directions. We interpret the minimum in ¹ for the highly twin ned sample to occur from a maximum of this averaged value of c. The maximum in turn may arise from the particularly strong coupling of phonon modes causing particle displacements perpendicular to the z-axis. This picture seems to be fully consistent with the microscopic model for the origin of the tunneling systems mentioned above. To further investigate the strong anisotropy of the phonon coupling observed in high-frequency experiments we are planning to do vibrating reed
measurements on neutron-irradiated quartz samples cut along different crystal axes.
References [1] W.A. Phillips (Ed.), Amorphous Solids — Low Temperature Properties, Springer, Berlin, 1981. [2] A. Vanelstraete, C. Laermans, Phys. Rev. B 42 (1990) 5842. [3] V. Keppens, C. Laermans, Nucl. Instr. and Meth. B 91 (1994) 346. [4] C. Laermans, V. Keppens, Phys. Rev. B 51 (1995) 8158. [5] E. Peeters, C. Laermans, D.A. Parshin, M. Coeck, Nucl. Instr. and Meth. B 141 (1998) 634. [6] A. Vanelstraete, C. Laermans, L. Lejarraga, M.v. Schickfus, S. Hunklinger, Z. Phys. B 70 (1988) 19. [7] J. Classen, I. Rohr, C. Enss, S. Hunklinger, C. Laermans, Eur. Phys. J. B, to appear. [8] J. Classen, C. Bechinger, C. Enss, G. Weiss, S. Hunklinger, Ann. Phys. (Leipzig) 3 (1994) 315. [9] C. Enss, S. Hunklinger, Phys. Rev. Lett. 79 (1997) 2831.