- Email: [email protected]
Ultrasonic study of the tunneling states in quartz amorphized by neutron irradiation
Nuclear Instruments and Methods in Physics Research B 91 (1994) 346-349 Noah-Homed
Beam Interactions with Materials&Atoms
Ultrasonic study of the tunneling states in quartz amorphized by neutron irradiation V. Keppens * and C. Laermans Katholiekz Universiteit Leuven, Dept. Physics, Celestijmmlaan ZOOD,B-3001 Leaven, Belgium
The ultrasonic absorption and velocity change is measured in neutron irradiated quartz, for a dose of 1 X 10” n/cm2. These experiments are carried out as a function of temperature at two different frequencies. The results are analysed using the tunneling model and the TS-parameters P and y, are derived from numerical fits. A comp~ison with previous me~urements reveals that the increase of P with the dose, observed so far, is no longer continued; instead a remarkable drop of the density of states is found in this highly irradiated sample. This confirms that a considerable part of the TS in samples that are only slightly damaged by irradiation must be attributed to the defective crystalline regions rather than to the radiation induced amorphous regions.
1. Introduction Since it became clear that amorphous solids behave completely different from their crystalline ~unterp~ts [l], different models have been proposed to describe these “anomalies”. However, up to now only the tunneling model [2,3], which assumes the existence of low temperature excitations, the so-called tunnel~g states (TS), can describe both thermal and acoustic “glassy” properties. But in spite of its success, this model is purely phenomenological and the microscopic nature of the TS is still unknown. In search of the TS origin, the study of neutron irradiated quartz is very appropriate. Its advantage lies in the possibility of introducing in a perfect crystal different degrees of disorder by varying the neutron dose. Studies of acoustic saturation, absorption and velocity change, as well as thermal conductivity and specific heat measurements on neutron irradiated quartz showed “glassy anomalies” [4,5] 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. In our laboratory, a systematic study of the TS in neutron irradiated quartz has been started [6]. Until now, ultrasonic absorption and velocity change measurements are performed in a dose range from 0.85 X 1018 up to 4.7 X 101’ n/cm’, leaving the samples mainly crystalline. In order to make a reliable comparison with related damage studies of other authors, accurate mass density measurements of our irradiated samples were carried out. The change of mass density is indeed
* Corresponding author. Tel. + 32 16 20 10 15, fax + 32 16 20 13 68.
a measure for the induced damage, since upon neutron
irradiation, quartz gradually evolves to an amorphous form of SiO,, with a mass density being about 15% smaller than that of the crystal, but still 3% higher than that of vitreous silica [7]. A similar compacted form evolves from irradiating a-SiOz and is called the “metamict phase”. In the work reported here, we extended our ultrasonic measurements to a dose of 1 X 10” n/cm’. After this irradiation the specimen turned out to be fully amorphous. The obtained data are numerically fitted and will be discussed in view of previous results for lower doses.
2. Theoretical ~nside~tions According to the tunneling model, the TS arise from the tunneling of atoms or groups of atoms in a double well. The energy difference between the eigenstates is given by E = (A2 + Az)1/2 with A the asymmetry of the double well. A, is the tunnel splitting and can be written as A, = M2 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, A) dA dh =F dA dh, 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 ~~es~nding to the phonon energy (E = hw). Because of the wide distribution of energy splittings, this process occurs at all frequencies. When
0148-583X/94/$07.~ 0 1994 - Elsevier Science B.V. Al1 rights reserved SSIlI 016%583X(93)El043-L
V? fippens,
C. Laermans/Nucl.
Imtr. and Meth. in Phys. Res. B 91 (1994) 346-349
347
&J a kT, this process leads to a frequency independent but temperature dependent change in the velocity of sound, given by IS]:
with To an arbitrary reference temperature and y, the coupling of the TS with the longitudinal phonons. For our abso~tion meas~ements, this process need not to be considered, since its contribution in this high-energy experiment is negligible compared to the second mechanism, the relaxation process. This process results from the modulation of the energy splittings A by the sound wave. The TS are brought out of thermal equilibrium and relaxation occurs via interaction with the thermal phonons. At low temperatures, the most likely process is the one-phonon process, giving rise to a relaxation rate [l]:
c’=K,(f)‘(-$-rcoth(
~
0
&)
with
(2) where yt and yr represent the coupling of the TS with respectively the lon~tudinal and transverse phonons. The general expressions for (Y and AU/U due to this process are given in ref. [9] and can be calculated numerically. Analytic solutions can be derived for the limiting cases WT, z=- 1 (with ~~ the smallest relaxation time of the TS) and wrm < 1 and are briefly discussed here. At the lowest temperatures, where the condition WT,,,B 1 holds, the contribution to the velocity change can be neglected, whereas the relaxational attenuation is expected to be frequency independent and proportional to T3 [9]:
The wad * 1 condition is satisfied at higher temperatures and leads to a temperature independent attenuation that varies linearly with frequency [9]: &=?rwC= 2%
IE-2
7&y: 2pu:’
The variation in the velocity of sound depends logarithmically on temperature in this regime and is frequency independent [9]:
The expressions given so far are vahd provided that the TS relax via absorption or emission of a single
Fig. 1. Ultrasonic absorption as a function of temperature for K17N2 (dose: 1 X 1020 n/cm’). 0: f = 80 M.k& + : f = 210 Mhz: -: fit curve.
thermal phonon. At higher temperatures, above a few Kelvin, a two-phonon Raman-process has to be taken into account [9]. This additional process causes a stronger temperature dependence in the transition region from the wan SE-1 to the ~7~ c 1 regime.
An x-cut single crystal of quartz, labeled KlP’N2, was exposed to a neutron dose of 1 X 102’ n/cm2 (Er 0.3 MeV). X-ray Laue photographs and mass density measurements indicated that the sample is completely amorphized by this irradiation and turned into the compacted metamict form of SiO,. On this specimen, the acoustic absorption and velocity change are measured as a function of temperature. These accurate Au/u measurements, carried out at 210 MHz, are performed by using a modified pulse-interference method; the absorption is measured at 80 and 210 MHz with a pulse-echo technique. For converting the applied electromagnetic signal into an elastic wave, a LiNbO, transducer was attached to the sample. Fig. 1 shows the absorption after subtracting a temperature independent residual attenuation CY~,due to geometrical factors. These measurements show undoubtedly the presence of TS: as predicted by expressions (3) and (41, a frequency independent T3 behaviour levels off to a plateau that is proportional to the frequency. The velocity change, given in Fig. 2, IV. OXIDES AND CERAMICS
V; Keppens, C. Laermans /Nucl. Instr. and Meth. in Phys. Rex B 91 (1994) 346-349
348
unirradiated
a-SiO,
I
K17N2
l 0
0.:
1
TEMPERATURE
0
I
60
/
80
I
100
1
3
DOSE (10” n/cm’)
for fit
shows the typical
“glassy” behaviour as well: Au/u logarithmically (Eq. (l)), reaches a maximum
between 2 and 3 K and then decreases with increasing temperature (Eq. (5)). Figs. 1 and 2 also show the best fitted curves for the absorption and velocity measurements respectively. They are obtained from numerical calculations to the tunneling model and allow to determine the TS parameters, given in Table 1. For comparison, the previously reported [6] parameters obtained for N6 are added. We note that the discrepancy in the absolute values of the parameters whether derived from absorption or velocity measurements is not specific for irradiated quartz but has been observed for most of the amorphous solids. It is possibly due to the fact that different kinds of experiments probe different TS.
Fig. 3. Density of states of the TS, derived from absorption measurements, as a function of the neutron dose.
Considering the relative changes between N6 and K17N2, we see that the results of absorption and velocity measurements agree with each other: both show a drastic decrease of the parameter C, resulting in a decrease of Pyf with about 40% when the dose is doubled. Since K,, and thus the derived coupling y,, is only very slightly affected by the dose, this drop can be attributed to a decrease of the density of states of the TS. This is surprising when we consider our previous results in a dose range from 0.85 x 1018 to 4.7 x 1019 n/cm2, where an increase of p with the dose was found [lo]. In Fig. 3, the values for p obtained from absorption measurements are plotted as a function of the dose. The density of states in N6 was found to be half of that in vitreous silica, and there were indications that the TS would be saturated at this dose [ll].
Table 1 TS parameters for K17N2 and N6 derived from absorption and velocity measurements. 10% is obtained; for K, 30%, yr 15% and for p 25%
For the values C and PyF an accuracy of
From AU/ u
From absorption
Dose [n/cm*] c [lo-s] K, [lo7 K-3 s-l] Py,2 ilO6 g/cm sq P [1030 cmW3 erg-l] yI [eVl
40
(K)
Fig. 2. Velocity change as a function of temperature Kl7N2 (dose: 1X 10” n/cm’), f = 210 MHz; -: curve.
increases
I
A
K17N2 210 MHz
N6 161 195 MHz
K17N2 210 MHz
N6 161 195 MHZ
1 x 1020 238 18 177 147 0.67
4.7 x 1019 300 15 300 260 0.69
1 x 1020 125 63 93 22 1.3
4.7 x 1019 155 54 130 31 1.3
V. Keppens, C. Laermans /Nucl.
In@. and Meth. in Phys. Rex. B 91 (1994) 346-349
In that point of view, the decrease of F found in R17N2 is unexpected. But when we compare N6 and K17N2, we should keep in mind that we are considering two quite different samples: K17N2 (with a mass density p = 2.255 g/cm3> is fully amorphized by irradiation and reached the compacted metamict phase, N6 (p = 2.571 g/cm3> instead contains so\me amorphous regions, but is still 65% “crystalline”. The high density of states found in this sample was understood by assuming that an im~rtant fraction of these TS find their origin in the (distorted) “crystalline” part of the sample. The observed decrease of p in K17N2 can be seen as a confirmation of these “crystalline TS” in N6: by increasing the dose, the crystalline regions - and the TS originating in them - will vanish, leaving a fully irradiated compacted sample with only “amorphous TS”. The relatively low density of states found in the amorph~ed sample could then indicate that compacted amorphous regions are less favourable to create TS than slightly defective regions. In order to obtain more evidence for these ideas, we started recently ultrasonic experiments in fully irradiated a&O,. Since upon irradiation, vitreous silica evolves to the same material as fully irradiated quartz, important information might result from these measurements.
4. Conclusion Ultrasonic absorption and velocity measurements in quartz amorphized by irradiation show a decrease in the density of states of the TS. This confirms that in samples that are only slightly damaged by irradiation, a considerable part of the TS must be attributed to the
349
defective crystalline regions rather than to the radiation induced amorphous regions.
Acknowledgements The authors thank G. King for performing the X-ray recording. They are also grateful to the SCK (Mel, Belgium) for the neutron irradiation and the IIKW ~Belg~urn) for financial support. One of us (V.K.) thanks Vlaamse Leergangen Leuven for additional support.
References [l] W.A. Phillips (ed.), Amorphous solids: Low temperature properties (Springer, Berlin, 1981). [2] P.W. Anderson, B.I. Haiperin and C.M. Varma, Phiios. Mag. 25 (1972) 1. [3] W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351. [4] C. Laermans, Phys. Rev. Lett. 42 (1979) 50. [5] For a review, see C. Laermans, in: Structure and Bonding in Non-Castle Solids, eds. G. Walrafen and A. Revesz (Plenum, New York, 1986) p. 325. [6] A. Vanelstraete and C. Laermans, Phys. Rev. B 42 (1990) 5842 and references herein. [7] W. Primak, Phys. Rev. 110 (1958) 1240. [8] S. Hu~inger and W. Arnold, Phys. Acoustics eds. W.P. Mason and R.N. Thurston, vol. 12 (Academic Press, New York, 1976) p. 155. [9] P. Doussineau, C. Frenois, R.G. Leisure, A. Levelut and J.Y. Prieur, J. Phys. (Paris) 41 (1980) 1193. [lo] V. Keppens and C. Laermans, Nucl. Instr. and Meth. B 65 (1992) 223. Ill] A. Vanelstraete and C. Laermans, Phys. Rev. B 38 (1988) 6311.
IV. OXIDES AND CERAMICS
Beam Interactions with Materials&Atoms
Ultrasonic study of the tunneling states in quartz amorphized by neutron irradiation V. Keppens * and C. Laermans Katholiekz Universiteit Leuven, Dept. Physics, Celestijmmlaan ZOOD,B-3001 Leaven, Belgium
The ultrasonic absorption and velocity change is measured in neutron irradiated quartz, for a dose of 1 X 10” n/cm2. These experiments are carried out as a function of temperature at two different frequencies. The results are analysed using the tunneling model and the TS-parameters P and y, are derived from numerical fits. A comp~ison with previous me~urements reveals that the increase of P with the dose, observed so far, is no longer continued; instead a remarkable drop of the density of states is found in this highly irradiated sample. This confirms that a considerable part of the TS in samples that are only slightly damaged by irradiation must be attributed to the defective crystalline regions rather than to the radiation induced amorphous regions.
1. Introduction Since it became clear that amorphous solids behave completely different from their crystalline ~unterp~ts [l], different models have been proposed to describe these “anomalies”. However, up to now only the tunneling model [2,3], which assumes the existence of low temperature excitations, the so-called tunnel~g states (TS), can describe both thermal and acoustic “glassy” properties. But in spite of its success, this model is purely phenomenological and the microscopic nature of the TS is still unknown. In search of the TS origin, the study of neutron irradiated quartz is very appropriate. Its advantage lies in the possibility of introducing in a perfect crystal different degrees of disorder by varying the neutron dose. Studies of acoustic saturation, absorption and velocity change, as well as thermal conductivity and specific heat measurements on neutron irradiated quartz showed “glassy anomalies” [4,5] 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. In our laboratory, a systematic study of the TS in neutron irradiated quartz has been started [6]. Until now, ultrasonic absorption and velocity change measurements are performed in a dose range from 0.85 X 1018 up to 4.7 X 101’ n/cm’, leaving the samples mainly crystalline. In order to make a reliable comparison with related damage studies of other authors, accurate mass density measurements of our irradiated samples were carried out. The change of mass density is indeed
* Corresponding author. Tel. + 32 16 20 10 15, fax + 32 16 20 13 68.
a measure for the induced damage, since upon neutron
irradiation, quartz gradually evolves to an amorphous form of SiO,, with a mass density being about 15% smaller than that of the crystal, but still 3% higher than that of vitreous silica [7]. A similar compacted form evolves from irradiating a-SiOz and is called the “metamict phase”. In the work reported here, we extended our ultrasonic measurements to a dose of 1 X 10” n/cm’. After this irradiation the specimen turned out to be fully amorphous. The obtained data are numerically fitted and will be discussed in view of previous results for lower doses.
2. Theoretical ~nside~tions According to the tunneling model, the TS arise from the tunneling of atoms or groups of atoms in a double well. The energy difference between the eigenstates is given by E = (A2 + Az)1/2 with A the asymmetry of the double well. A, is the tunnel splitting and can be written as A, = M2 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, A) dA dh =F dA dh, 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 ~~es~nding to the phonon energy (E = hw). Because of the wide distribution of energy splittings, this process occurs at all frequencies. When
0148-583X/94/$07.~ 0 1994 - Elsevier Science B.V. Al1 rights reserved SSIlI 016%583X(93)El043-L
V? fippens,
C. Laermans/Nucl.
Imtr. and Meth. in Phys. Res. B 91 (1994) 346-349
347
&J a kT, this process leads to a frequency independent but temperature dependent change in the velocity of sound, given by IS]:
with To an arbitrary reference temperature and y, the coupling of the TS with the longitudinal phonons. For our abso~tion meas~ements, this process need not to be considered, since its contribution in this high-energy experiment is negligible compared to the second mechanism, the relaxation process. This process results from the modulation of the energy splittings A by the sound wave. The TS are brought out of thermal equilibrium and relaxation occurs via interaction with the thermal phonons. At low temperatures, the most likely process is the one-phonon process, giving rise to a relaxation rate [l]:
c’=K,(f)‘(-$-rcoth(
~
0
&)
with
(2) where yt and yr represent the coupling of the TS with respectively the lon~tudinal and transverse phonons. The general expressions for (Y and AU/U due to this process are given in ref. [9] and can be calculated numerically. Analytic solutions can be derived for the limiting cases WT, z=- 1 (with ~~ the smallest relaxation time of the TS) and wrm < 1 and are briefly discussed here. At the lowest temperatures, where the condition WT,,,B 1 holds, the contribution to the velocity change can be neglected, whereas the relaxational attenuation is expected to be frequency independent and proportional to T3 [9]:
The wad * 1 condition is satisfied at higher temperatures and leads to a temperature independent attenuation that varies linearly with frequency [9]: &=?rwC= 2%
IE-2
7&y: 2pu:’
The variation in the velocity of sound depends logarithmically on temperature in this regime and is frequency independent [9]:
The expressions given so far are vahd provided that the TS relax via absorption or emission of a single
Fig. 1. Ultrasonic absorption as a function of temperature for K17N2 (dose: 1 X 1020 n/cm’). 0: f = 80 M.k& + : f = 210 Mhz: -: fit curve.
thermal phonon. At higher temperatures, above a few Kelvin, a two-phonon Raman-process has to be taken into account [9]. This additional process causes a stronger temperature dependence in the transition region from the wan SE-1 to the ~7~ c 1 regime.
An x-cut single crystal of quartz, labeled KlP’N2, was exposed to a neutron dose of 1 X 102’ n/cm2 (Er 0.3 MeV). X-ray Laue photographs and mass density measurements indicated that the sample is completely amorphized by this irradiation and turned into the compacted metamict form of SiO,. On this specimen, the acoustic absorption and velocity change are measured as a function of temperature. These accurate Au/u measurements, carried out at 210 MHz, are performed by using a modified pulse-interference method; the absorption is measured at 80 and 210 MHz with a pulse-echo technique. For converting the applied electromagnetic signal into an elastic wave, a LiNbO, transducer was attached to the sample. Fig. 1 shows the absorption after subtracting a temperature independent residual attenuation CY~,due to geometrical factors. These measurements show undoubtedly the presence of TS: as predicted by expressions (3) and (41, a frequency independent T3 behaviour levels off to a plateau that is proportional to the frequency. The velocity change, given in Fig. 2, IV. OXIDES AND CERAMICS
V; Keppens, C. Laermans /Nucl. Instr. and Meth. in Phys. Rex B 91 (1994) 346-349
348
unirradiated
a-SiO,
I
K17N2
l 0
0.:
1
TEMPERATURE
0
I
60
/
80
I
100
1
3
DOSE (10” n/cm’)
for fit
shows the typical
“glassy” behaviour as well: Au/u logarithmically (Eq. (l)), reaches a maximum
between 2 and 3 K and then decreases with increasing temperature (Eq. (5)). Figs. 1 and 2 also show the best fitted curves for the absorption and velocity measurements respectively. They are obtained from numerical calculations to the tunneling model and allow to determine the TS parameters, given in Table 1. For comparison, the previously reported [6] parameters obtained for N6 are added. We note that the discrepancy in the absolute values of the parameters whether derived from absorption or velocity measurements is not specific for irradiated quartz but has been observed for most of the amorphous solids. It is possibly due to the fact that different kinds of experiments probe different TS.
Fig. 3. Density of states of the TS, derived from absorption measurements, as a function of the neutron dose.
Considering the relative changes between N6 and K17N2, we see that the results of absorption and velocity measurements agree with each other: both show a drastic decrease of the parameter C, resulting in a decrease of Pyf with about 40% when the dose is doubled. Since K,, and thus the derived coupling y,, is only very slightly affected by the dose, this drop can be attributed to a decrease of the density of states of the TS. This is surprising when we consider our previous results in a dose range from 0.85 x 1018 to 4.7 x 1019 n/cm2, where an increase of p with the dose was found [lo]. In Fig. 3, the values for p obtained from absorption measurements are plotted as a function of the dose. The density of states in N6 was found to be half of that in vitreous silica, and there were indications that the TS would be saturated at this dose [ll].
Table 1 TS parameters for K17N2 and N6 derived from absorption and velocity measurements. 10% is obtained; for K, 30%, yr 15% and for p 25%
For the values C and PyF an accuracy of
From AU/ u
From absorption
Dose [n/cm*] c [lo-s] K, [lo7 K-3 s-l] Py,2 ilO6 g/cm sq P [1030 cmW3 erg-l] yI [eVl
40
(K)
Fig. 2. Velocity change as a function of temperature Kl7N2 (dose: 1X 10” n/cm’), f = 210 MHz; -: curve.
increases
I
A
K17N2 210 MHz
N6 161 195 MHz
K17N2 210 MHz
N6 161 195 MHZ
1 x 1020 238 18 177 147 0.67
4.7 x 1019 300 15 300 260 0.69
1 x 1020 125 63 93 22 1.3
4.7 x 1019 155 54 130 31 1.3
V. Keppens, C. Laermans /Nucl.
In@. and Meth. in Phys. Rex. B 91 (1994) 346-349
In that point of view, the decrease of F found in R17N2 is unexpected. But when we compare N6 and K17N2, we should keep in mind that we are considering two quite different samples: K17N2 (with a mass density p = 2.255 g/cm3> is fully amorphized by irradiation and reached the compacted metamict phase, N6 (p = 2.571 g/cm3> instead contains so\me amorphous regions, but is still 65% “crystalline”. The high density of states found in this sample was understood by assuming that an im~rtant fraction of these TS find their origin in the (distorted) “crystalline” part of the sample. The observed decrease of p in K17N2 can be seen as a confirmation of these “crystalline TS” in N6: by increasing the dose, the crystalline regions - and the TS originating in them - will vanish, leaving a fully irradiated compacted sample with only “amorphous TS”. The relatively low density of states found in the amorph~ed sample could then indicate that compacted amorphous regions are less favourable to create TS than slightly defective regions. In order to obtain more evidence for these ideas, we started recently ultrasonic experiments in fully irradiated a&O,. Since upon irradiation, vitreous silica evolves to the same material as fully irradiated quartz, important information might result from these measurements.
4. Conclusion Ultrasonic absorption and velocity measurements in quartz amorphized by irradiation show a decrease in the density of states of the TS. This confirms that in samples that are only slightly damaged by irradiation, a considerable part of the TS must be attributed to the
349
defective crystalline regions rather than to the radiation induced amorphous regions.
Acknowledgements The authors thank G. King for performing the X-ray recording. They are also grateful to the SCK (Mel, Belgium) for the neutron irradiation and the IIKW ~Belg~urn) for financial support. One of us (V.K.) thanks Vlaamse Leergangen Leuven for additional support.
References [l] W.A. Phillips (ed.), Amorphous solids: Low temperature properties (Springer, Berlin, 1981). [2] P.W. Anderson, B.I. Haiperin and C.M. Varma, Phiios. Mag. 25 (1972) 1. [3] W.A. Phillips, J. Low Temp. Phys. 7 (1972) 351. [4] C. Laermans, Phys. Rev. Lett. 42 (1979) 50. [5] For a review, see C. Laermans, in: Structure and Bonding in Non-Castle Solids, eds. G. Walrafen and A. Revesz (Plenum, New York, 1986) p. 325. [6] A. Vanelstraete and C. Laermans, Phys. Rev. B 42 (1990) 5842 and references herein. [7] W. Primak, Phys. Rev. 110 (1958) 1240. [8] S. Hu~inger and W. Arnold, Phys. Acoustics eds. W.P. Mason and R.N. Thurston, vol. 12 (Academic Press, New York, 1976) p. 155. [9] P. Doussineau, C. Frenois, R.G. Leisure, A. Levelut and J.Y. Prieur, J. Phys. (Paris) 41 (1980) 1193. [lo] V. Keppens and C. Laermans, Nucl. Instr. and Meth. B 65 (1992) 223. Ill] A. Vanelstraete and C. Laermans, Phys. Rev. B 38 (1988) 6311.
IV. OXIDES AND CERAMICS