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Precision ultrasonic velocity measurements for the study of the low temperature acoustic properties in defective materials
Materials Science and Engineering, A 122 (1989) 77-81
77
Precision Ultrasonic Velocity Measurements for the Study of the Low Temperature Acoustic Properties in Defective Materials A. VANELSTRAETE and C. LAERMANS
Kathofieke Universiteit Leuven, Department of Physk's, Celestifnenlaan 2(X)l), 3030 Leuven (Belgium) (Received May 30, 1989)
Abstract
A very precise method for measurements of the variation in the velocity of sound in solids" in the 500 MHz frequency range is presented. The method is based on a pulse interference method and allows the observation of relative velocity changes of the order of 10 7 or better. The availability of this method made it possible to detect the "glassy" behaviour of the low temperature velocity of sound in electron-irradiated quartz and in neutron-irradiated quartz for low doses and to fit the results to the tunnelling model. The fittings give more convincing evidence that the tunnelling states in electron-irradiated quartz are of similar nature to those in neutron-irradiated quartz. The density of states, however, is very much smaller in electron-irradiated quartz than in neutron-irradiated quartz for a similar dose.
1. Introduction At low temperatures, amorphous solids exhibit dynamical properties, which are totally different from those in crystals (for a review see ref. 1). They can phenomenologically be described by the presence of low energy tunnelling excitations [2, 3], which seem to be intrinsic to the disordered state. Thermal and acoustic experiments showed that these tunnelling states (TSs) have a wide distribution of energies and relaxation times. Their microscopic nature, however, is still unknown. In search of a model substance, other workers as well as ourselves have shown similar excitations to exist in partly disordered crystals. The key idea in studying defective crystals is that defects in a still mainly crystalline environment are easier to study than in the random network of the glass. Hence, the study of the dynamic properties of defective crystalline solids allows a care0921-5093/89/$3.50
ful comparison between the type and/or the amount of disorder present and the TSs. Of the various materials which are investigated as models of the glassy state, neutron-irradiated quartz is one of the most attractive, as it allows continuous structural variations from the perfectly ordered crystal to the amorphous network disorder. High energy neutron irradiation causes so-called displacement cascades in quartz, resulting in highly disordered regions embedded in a mainly crystalline environment. For sufficiently high doses, these regions overlap and long-range order gradually decreases, resulting in an amorphous SiO 2 phase. Studies of the acoustic saturation, ultrasonic absorption and velocity, thermal conductivity and specific heat in neutron-irradiated quartz revealed similar anomalies to those in amorphous solids [4-6] (for a recent review see ref. 7). They were explained by the presence of TSs which are similar in nature to those in vitreous silica and with a density of states which is smaller than in vitreous silica and which increases with increasing neutron dose. Thermal conductivity [8] and, recently, ultrasonic measurements [9, 10] have given evidence for the existence of TSs in electron-irradiated quartz. Electrons, because of their small mass, transfer relatively little energy to the atoms which they hit. This excludes the production of highly disordered regions caused by cascade processes, as is the case for neutron irradiation. In this paper we report a very precise method for the measurement of the variation in the velocity of sound in solids in the 500 MHz frequency range. It is a modified pulse interference method, which leads to a precision of 10-7. The method made it possible to measure the very small temperature dependence of the ultrasonic velocity at low temperatures in electron-irradiated quartz and in neutron-irradiated quartz for © Elsevier Sequoia/Printed in The Netherlands
78 smaller doses. The data will be reported as well as the fittings to the tunnelling model and the deduced tunnelling parameters.
2. Experimental set-up and sample characterization The measurements were performed using a single-ended pulse echo technique for r.f. generation and detection of the ultrasound. The ultrasonic waves were generated by surface wave excitation. All experiments were done at a frequency of 648 MHz. The velocity measurements were performed, using a pulse interference method [11]. This method is schematically shown in Fig. 1. Two r.f. pulses originating from the same continuous r.f. source are applied to one end of the sample. Each of the pulses gives rise to a number of echoes on the scoop, originating from multiple reflections of the signal at the other end of the specimen (see Fig. l(a)). The time interval t' between the two pulses is taken to be equal to the time between two subsequent echoes (3/~s in our case). Hence, on the scoop the (n + 1)th echo of the first pulse interferes with the nth echo of the second pulse (see Fig. l(c)), leading to positive or negative interference depending on the phase difference between the two echoes. Since both signals originate from the same signal source and since they follow the same path to the sample, a change in interference signal can only arise from a change in phase difference within the specimen itself, which is induced by a change in the velocity of sound in our case. It can be shown that the relative velocity change corresponds to the relative change in frequency, for which the same minimum in the interference signal is
obtained:
v( T)-v( To)_f( T)-f( To) v(To) f(To)
With the use of a very stable r.f. signal source (model SMG from Rohde and Schwarz in our case) and fast microwave switches for r.f. pulse generation, relative velocity changes of 10-7 can be detected for the used frequency. One natural and three synthetic quartz samples, X cut and of high purity were shaped into a cylindrical rod with a 3 mm diameter and with 16 mm and 8 mm lengths respectively. Two of the synthetic quartz samples, labelled K9 and N4, were neutron irradiated in the same reactor at the Studiecentrum voor Kernenergie, Mol, Belgium, up to a dose of 0.85 x 10 L8 neutrons cm -2 and 1.2 × 1019 neutrons cm -2 (E>~0.3 MeV) respectively. The natural quartz specimen, E3, was irradiated with electrons (E = 3 MeV; flux, 2.6 /zA cm -2) in a Van de Graaff generator at Centre d'Etudes Nucl6aires, Grenoble, up to a dose of 1.0 × 1 0 20 electrons cm -2.
3. Theoretical considerations In the tunnelling model [2, 3] the TSs arise from the tunnelling of atoms or groups of atoms in double-well potentials. The energy difference between the two eigenstates is given by E = (A2+A02)l/2 where A is the asymmetry of the double-well potential. A 0 is the tunnel splitting, given by A 0 = hQ exp( - 2) where it describes the overlap of the wavefunctions, h~/2 is the ground state energy of the particle in an isolated well. One of the basic assumptions of the tunnelling model is that A and it are independent of each other and are uniformly distributed: P(A, it) dA dit = t5 dA dit
(b)
t' Fig. 1. The pulse interference method: (a) first pulse with echoes; (b) second pulse with echoes; (c) double pulse with echoes for positive interference.
(1)
(2)
with P a constant, the density of states of the TSs. For a sound wave travelling through an ensemble of TSs, two different mechanisms are of importance. Firstly, it will be resonantly absorbed by those TSs having an energy splitting E corresponding to the phonon energy (E=hw). Because of the wide distribution of energy splittings, this process occurs at all frequencies and leads to a frequency-independent but temperature-dependent change in the velocity of sound given by [2, 3]
79
v(
\ I",,2 pv ,
(3)
\ T,,]
provided that hw "~ kT. T~, is an arbitrary reference temperature and 71 represents the coupling of the TSs with the longitudinal phonons. The second process, the relaxation absorption, is the result of the modulation of energy splittings of the double-well potentials by the sound wave. In this way the equilibrium thermal distribution of the TSs is disturbed. In dielectric amorphous materials, relaxation occurs via interaction with thermal phonons. Below 1 K the simple one-phonon or direct process is dominant, i.e. one single thermal phonon is absorbed or emitted by the relaxing system. In this case, the relaxation rate is given by [1]
[&'2[E'3
4k3 [yl 2 2yt2/ K3 = p--~7~15 + v s]
(4)
where y~ and 7, represent the coupling of the TSs with the longitudinal and transverse phonons respectively. The variation in the velocity of sound due to the resonant and the relaxation process is then given by [ 12]
+(< res
(5)
\ V /rel
with
Av
PYl~-
V'~ Fel
/OVl- /*"mi.2 ~
u4 + 7 ~
1 sech2
AV ,,,= _(72 ln(T]\T0]
dE
du
l/rain
with u =Ao/E=(rm/r//2. Analytic solutions can be found for the limiting cases Wrm'>l and wr~ "~ 1, with rm, the smallest relaxation time of the TSs with energy E. At low temperatures, where O)rm">1 holds, the relaxation process gives
(6)
The expressions given so far are valid provided that the TSs relax via absorption or emission of a single thermal phonon. At higher temperatures, above a few kelvins, multiphonon processes should be taken into account. For two-phonon processes, the Raman process is most important. A modified relaxation time r has to be introduced in(5): r - l = r d - l + rr -l
(E)
with
V
a negligible contribution to the change in the velocity of sound [12]. Hence the total variation in the velocity of sound in this low temperature range is that of the resonant process (see (3)). For the ~orm'~ 1 regime (higher temperatures), the variation in the velocity of sound depends logarithmically on temperature and is frequency independent [ 12]:
(7)
in which r~ is the relaxation time for one-phonon processes (see (4)) and gr the relaxation time for the Raman process [12]. This additional process causes a stronger temperature dependence of the velocity change in the transition region from the wr m,> 1 to the o)Tm~ l regime than that predicted for the direct process. 4. Experimental results and discussion
Figures 2 and 3 give the variation Av/t, = {v(T) - v ( To)}/v( To) as a function of the temperature at 648 MHz for the neutron-irradiated specimens N4 and K9, for the electron-irradiated specimen E3 and for an unirradiated quartz sample. It can be seen that the smallest changes which can be detected are about 2 x 10 -7. It is clear that, in order to detect the temperature dependence in those samples, such a precision is necessary. The observed temperature dependence of the irradiated samples is qualitatively similar to that in amorphous solids [13] but very much smaller (a factor of about 102). As can be seen in Fig. 2, the velocity curves of the neutronirradiated specimen and the unirradiated sample are quite different. In N4 the velocity is temperature dependent whereas, below 15 K, no velocity changes were observed in the unirradiated specimen. The behaviour of Av/v in N4 is typical for the presence of TSs. Indeed, at the lowest temperatures {from 0.3 to 2 K) the velocity increases
8O
5O
4O
3O
2O
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= 0 o
O 0 n 0 0
f = 648 Hllz
-10
grn
-20
O
n-irrad.
£;
unirrad
O
N4
0
0 0
-30 l
-40 I 0.1
i
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i~
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~
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TEMPERATURE (1()
Fig. 2. Velocity change as a function of temperature for neutron-irradiated quartz N4 (dose, 1.2×10 j9 neutrons cm- 2; E >/0.3 MeV ) and for unirradiated quartz (frequency, 648 MHz; T~)=0.33 K). 80~ 70
6O 50 i 4O
3O
s
< =
20 10
.-:: >
0 -10
f = 648 MHz
-20 -30
O
: e - - i r r a d , . K3
.~. : n - i r r a d ~ 1(9
-40 -50
i
0.1
i
, ,,,,,i
,
,
.,,i,I
1
0
1 '11~MP~RAq'I] R ~
•
,
i
,ill,
100
(W)
Fig. 3. Measurements of the velocity change as a function of temperature and fittings of the data to the tunnelling model (frequency, 648 MHz; T,=0.45 K): o, electron-irradiated quartz, sample E3; +, neutron-irradiated quartz, sample K9.
logarithmically with increasing temperature. It is the result of the resonant interaction between the TSs and the acoustic phonons (see (3)). The condition hto~. kT of (3) was fulfilled (at the lowest temperatures hto-~0.1kT). Above 2 K , the change in the velocity of sound deviates from the
logarithmic law and a maximum is seen at about 4 K. In the tunnelling model, this behaviour is the consequence of the relaxational process, which causes a decrease in the velocity with increasing temperature (see (5)). When this contribution compensates that of the resonant process, a maximum in Av/v is observed. At higher temperatures, for which ~0rm~ 1, a logarithmic decrease in the velocity is expected with a slope half of that of the resonant part (see (6)). This logarithmic decrease was not observed in our case, nor was it detected in high frequency measurements in glasses [13]. The stronger temperature decrease in Av/v can be attributed to higher order phonon relaxational processes, e.g. the Raman process. The velocity curves for K9 and E3 show a similar behaviour to those for N4. However, the velocity changes are much smaller in K9 and in E3. This indicates that the density P of states is much smaller. We also note that, to obtain comparable velocity changes, a much higher dose is needed in the case of electron irradiation than in neutron-irradiated quartz. Although we reported before the logarithmic increase in Av/v at the lowest temperatures in electron-irradiated [10] and neutron-irradiated quartz [5], the measured typical "glassy" behaviour of Av/v at higher temperatures allows careful study of the effect of the relaxation process and makes it possible to determine both the density P of states of the TSs and the coupling parameter 7~. Therefore a quantitative analysis of the data was performed in the framework of the tunnelling model. The velocity change given in (5) was calculated numerically for each temperature. Raman processes were also taken into account in order to describe the drastic decrease in Av/v above 5 K. The best-suited theoretical curves for K9 and E3 are given in Fig. 3. As can be seen, there is a good agreement between the theoretical curves and the data. From the fits, the parameters C ('~-"/~12), K 3 and K7 (the Raman parameter [12]) can be determined independently. They are given in Table 1, as well as the values for the density P of states and the coupling parameter y~. These parameters were deduced from C and K3, making use of the expression yl2/Vl 2= ~/t2/Vt 2, which was found to be valid for most of the amorphous solids [3]. As can be seen from Table 1, the density of states increases with increasing neutron dose; P is about a factor of 5 larger in K9 than in N4. The observed increase in the density of states with increasing neutron dose is in agreement with
81
TABLE 1 Tunnel state parameters for neutron- and electron-irradiated quartz, obtained from fittings of the ultrasonic velocity data to the tunnelling model
Parameter
Sample E3
.Sample K9
Sample N4
Dose (cm 2) E (MeV) C ( × 1 0 ") K ~ ( × 1 0 7 K ~s i) KT(K 7 s I) PFle(l(Y'gcm Is e) 7J (eV) P ( x 10-~'~erg i cm 3)
1.0 × 102(telectrons 3 1.6 29 5×10 ~ 1.4 1.0 0.6
0.85 × 10 I~ neutrons i> 0.3 3.7 39 2×10.~ 3.2 1.1 1.0
1.2 x 10 ~ neutrons >/0.3 21 42 3×1().~ 18 1.1 5.4
previous results [7]. A more striking result that follows from Table 1 is that, for similar doses, electrons induce fewer TSs in quartz than neutrons do, whereas the parameter 7~ which describes the coupling between the TSs and the longitudinal phonons is found to be similar for both irradiations, within the accuracy of 15% for 7~. Hence the tunnelling states induced by electrons are of the same nature as those in neutronirradiated quartz, in spite of the very different disordering processes, but the induced density of states is very different for the same dose. We can conclude that the availability of the precision method for measurements of the variation in the velocity of sound in the 500 MHz range made it possible to detect the "glassy" behaviour of the low temperature velocity of sound in electron-irradiated quartz and to fit the results to the tunnelling model. This leads to rather precise values for 71 and to conclusive evidence that the tunnelling states are of a similar nature to those in neutron-irradiated quartz in spite of the very different disordering processes. The density of states, however, is very much smaller in electron-irradiated quartz than in neutron-irradiated quartz for the same dose. Acknowledgments The authors thank the Centre d'Etudes Nucldaires, Grenoble, for the electron irradiation and the Studiecentrum voor Kemenergie, Mol (and in particular J. Cornelis), for the neutron irradiations.
They are also grateful to the Belgian Interuniversitair Instituut voor Kernwetenschappen and the Katholieke Universiteit Leuven for financial support. References 1 W.A. Phillips (ed.), Amorphous Solids: Low Temperature Properties, Springer, Berlin, 1981. 2 P. W. Anderson, B. 1. Halperin and C. M. Varma, Philos. Mag., 25 (1972) 1. W. A. Phillips, J. Low Temp. Phys., 7 (1972) 351. 3 S. Hunklinger and W. Arnold, in W. P. Mason and R. N Thurston (eds.), Physical Acoustics, Vol. 12, Academic Press, New York, 1976, p. 155. 4 C. Laermans, Phys. Rev. Lett., 42 (1979) 250. 5 B. Golding, J. E. Graebner, W. H. Haemmerle and C. Laermans, Bull. Am. Phys. Soc,, 24 (1979) 495. B. Golding and J. E. Graebner, in H. J. Marls (ed.), Phonon Scattering in Condensed Matter, Plenum, New York, 1980, p. 11. 6 J. W. Gardner and A. C. Anderson, Phys. Rev. B, 32 (1981)474. 7 C. Laermans, in G. Walrafen and A. Revesz (eds.), Structure and Bonding in Non-Crystalline Solids, Plenum, New York, 1986, p. 325. 8 C. Laermans, A. M. de Go6r and M. Locatelli, Phys. Lett. A, 80(1980)331. A. M. de Go6r, M. Locatelli and C. Laermans, J. Phys. (Paris), Colloq. C6, 42 ( 1981 ) 78. 9 C. Laermans and A. Vanelstraete, Phys. Rev. B, 34 (1986) 1405. 10 A. Vanelstraete and C. Laermans, Phys. Rev. B, 39 (1989) 3905. 11 R. Truell, C. Elbaum and B. B. Chick, Ultrasonic" Methods in Solid State Physics, Academic Press, New York, 1969. 12 P. Doussineau, A. Levelut, M. Matecki, W. Sch6n and W. D. Wallace, J. Phys. (Paris), 46 ( 1985 ) 979. 13 L. Pich6, R. Maynard, S. Hunklinger and J. Jfickle, Phys. Rev. Lett., 32 (1974) 1426.
77
Precision Ultrasonic Velocity Measurements for the Study of the Low Temperature Acoustic Properties in Defective Materials A. VANELSTRAETE and C. LAERMANS
Kathofieke Universiteit Leuven, Department of Physk's, Celestifnenlaan 2(X)l), 3030 Leuven (Belgium) (Received May 30, 1989)
Abstract
A very precise method for measurements of the variation in the velocity of sound in solids" in the 500 MHz frequency range is presented. The method is based on a pulse interference method and allows the observation of relative velocity changes of the order of 10 7 or better. The availability of this method made it possible to detect the "glassy" behaviour of the low temperature velocity of sound in electron-irradiated quartz and in neutron-irradiated quartz for low doses and to fit the results to the tunnelling model. The fittings give more convincing evidence that the tunnelling states in electron-irradiated quartz are of similar nature to those in neutron-irradiated quartz. The density of states, however, is very much smaller in electron-irradiated quartz than in neutron-irradiated quartz for a similar dose.
1. Introduction At low temperatures, amorphous solids exhibit dynamical properties, which are totally different from those in crystals (for a review see ref. 1). They can phenomenologically be described by the presence of low energy tunnelling excitations [2, 3], which seem to be intrinsic to the disordered state. Thermal and acoustic experiments showed that these tunnelling states (TSs) have a wide distribution of energies and relaxation times. Their microscopic nature, however, is still unknown. In search of a model substance, other workers as well as ourselves have shown similar excitations to exist in partly disordered crystals. The key idea in studying defective crystals is that defects in a still mainly crystalline environment are easier to study than in the random network of the glass. Hence, the study of the dynamic properties of defective crystalline solids allows a care0921-5093/89/$3.50
ful comparison between the type and/or the amount of disorder present and the TSs. Of the various materials which are investigated as models of the glassy state, neutron-irradiated quartz is one of the most attractive, as it allows continuous structural variations from the perfectly ordered crystal to the amorphous network disorder. High energy neutron irradiation causes so-called displacement cascades in quartz, resulting in highly disordered regions embedded in a mainly crystalline environment. For sufficiently high doses, these regions overlap and long-range order gradually decreases, resulting in an amorphous SiO 2 phase. Studies of the acoustic saturation, ultrasonic absorption and velocity, thermal conductivity and specific heat in neutron-irradiated quartz revealed similar anomalies to those in amorphous solids [4-6] (for a recent review see ref. 7). They were explained by the presence of TSs which are similar in nature to those in vitreous silica and with a density of states which is smaller than in vitreous silica and which increases with increasing neutron dose. Thermal conductivity [8] and, recently, ultrasonic measurements [9, 10] have given evidence for the existence of TSs in electron-irradiated quartz. Electrons, because of their small mass, transfer relatively little energy to the atoms which they hit. This excludes the production of highly disordered regions caused by cascade processes, as is the case for neutron irradiation. In this paper we report a very precise method for the measurement of the variation in the velocity of sound in solids in the 500 MHz frequency range. It is a modified pulse interference method, which leads to a precision of 10-7. The method made it possible to measure the very small temperature dependence of the ultrasonic velocity at low temperatures in electron-irradiated quartz and in neutron-irradiated quartz for © Elsevier Sequoia/Printed in The Netherlands
78 smaller doses. The data will be reported as well as the fittings to the tunnelling model and the deduced tunnelling parameters.
2. Experimental set-up and sample characterization The measurements were performed using a single-ended pulse echo technique for r.f. generation and detection of the ultrasound. The ultrasonic waves were generated by surface wave excitation. All experiments were done at a frequency of 648 MHz. The velocity measurements were performed, using a pulse interference method [11]. This method is schematically shown in Fig. 1. Two r.f. pulses originating from the same continuous r.f. source are applied to one end of the sample. Each of the pulses gives rise to a number of echoes on the scoop, originating from multiple reflections of the signal at the other end of the specimen (see Fig. l(a)). The time interval t' between the two pulses is taken to be equal to the time between two subsequent echoes (3/~s in our case). Hence, on the scoop the (n + 1)th echo of the first pulse interferes with the nth echo of the second pulse (see Fig. l(c)), leading to positive or negative interference depending on the phase difference between the two echoes. Since both signals originate from the same signal source and since they follow the same path to the sample, a change in interference signal can only arise from a change in phase difference within the specimen itself, which is induced by a change in the velocity of sound in our case. It can be shown that the relative velocity change corresponds to the relative change in frequency, for which the same minimum in the interference signal is
obtained:
v( T)-v( To)_f( T)-f( To) v(To) f(To)
With the use of a very stable r.f. signal source (model SMG from Rohde and Schwarz in our case) and fast microwave switches for r.f. pulse generation, relative velocity changes of 10-7 can be detected for the used frequency. One natural and three synthetic quartz samples, X cut and of high purity were shaped into a cylindrical rod with a 3 mm diameter and with 16 mm and 8 mm lengths respectively. Two of the synthetic quartz samples, labelled K9 and N4, were neutron irradiated in the same reactor at the Studiecentrum voor Kernenergie, Mol, Belgium, up to a dose of 0.85 x 10 L8 neutrons cm -2 and 1.2 × 1019 neutrons cm -2 (E>~0.3 MeV) respectively. The natural quartz specimen, E3, was irradiated with electrons (E = 3 MeV; flux, 2.6 /zA cm -2) in a Van de Graaff generator at Centre d'Etudes Nucl6aires, Grenoble, up to a dose of 1.0 × 1 0 20 electrons cm -2.
3. Theoretical considerations In the tunnelling model [2, 3] the TSs arise from the tunnelling of atoms or groups of atoms in double-well potentials. The energy difference between the two eigenstates is given by E = (A2+A02)l/2 where A is the asymmetry of the double-well potential. A 0 is the tunnel splitting, given by A 0 = hQ exp( - 2) where it describes the overlap of the wavefunctions, h~/2 is the ground state energy of the particle in an isolated well. One of the basic assumptions of the tunnelling model is that A and it are independent of each other and are uniformly distributed: P(A, it) dA dit = t5 dA dit
(b)
t' Fig. 1. The pulse interference method: (a) first pulse with echoes; (b) second pulse with echoes; (c) double pulse with echoes for positive interference.
(1)
(2)
with P a constant, the density of states of the TSs. For a sound wave travelling through an ensemble of TSs, two different mechanisms are of importance. Firstly, it will be resonantly absorbed by those TSs having an energy splitting E corresponding to the phonon energy (E=hw). Because of the wide distribution of energy splittings, this process occurs at all frequencies and leads to a frequency-independent but temperature-dependent change in the velocity of sound given by [2, 3]
79
v(
\ I",,2 pv ,
(3)
\ T,,]
provided that hw "~ kT. T~, is an arbitrary reference temperature and 71 represents the coupling of the TSs with the longitudinal phonons. The second process, the relaxation absorption, is the result of the modulation of energy splittings of the double-well potentials by the sound wave. In this way the equilibrium thermal distribution of the TSs is disturbed. In dielectric amorphous materials, relaxation occurs via interaction with thermal phonons. Below 1 K the simple one-phonon or direct process is dominant, i.e. one single thermal phonon is absorbed or emitted by the relaxing system. In this case, the relaxation rate is given by [1]
[&'2[E'3
4k3 [yl 2 2yt2/ K3 = p--~7~15 + v s]
(4)
where y~ and 7, represent the coupling of the TSs with the longitudinal and transverse phonons respectively. The variation in the velocity of sound due to the resonant and the relaxation process is then given by [ 12]
+(< res
(5)
\ V /rel
with
Av
PYl~-
V'~ Fel
/OVl- /*"mi.2 ~
u4 + 7 ~
1 sech2
AV ,,,= _(72 ln(T]\T0]
dE
du
l/rain
with u =Ao/E=(rm/r//2. Analytic solutions can be found for the limiting cases Wrm'>l and wr~ "~ 1, with rm, the smallest relaxation time of the TSs with energy E. At low temperatures, where O)rm">1 holds, the relaxation process gives
(6)
The expressions given so far are valid provided that the TSs relax via absorption or emission of a single thermal phonon. At higher temperatures, above a few kelvins, multiphonon processes should be taken into account. For two-phonon processes, the Raman process is most important. A modified relaxation time r has to be introduced in(5): r - l = r d - l + rr -l
(E)
with
V
a negligible contribution to the change in the velocity of sound [12]. Hence the total variation in the velocity of sound in this low temperature range is that of the resonant process (see (3)). For the ~orm'~ 1 regime (higher temperatures), the variation in the velocity of sound depends logarithmically on temperature and is frequency independent [ 12]:
(7)
in which r~ is the relaxation time for one-phonon processes (see (4)) and gr the relaxation time for the Raman process [12]. This additional process causes a stronger temperature dependence of the velocity change in the transition region from the wr m,> 1 to the o)Tm~ l regime than that predicted for the direct process. 4. Experimental results and discussion
Figures 2 and 3 give the variation Av/t, = {v(T) - v ( To)}/v( To) as a function of the temperature at 648 MHz for the neutron-irradiated specimens N4 and K9, for the electron-irradiated specimen E3 and for an unirradiated quartz sample. It can be seen that the smallest changes which can be detected are about 2 x 10 -7. It is clear that, in order to detect the temperature dependence in those samples, such a precision is necessary. The observed temperature dependence of the irradiated samples is qualitatively similar to that in amorphous solids [13] but very much smaller (a factor of about 102). As can be seen in Fig. 2, the velocity curves of the neutronirradiated specimen and the unirradiated sample are quite different. In N4 the velocity is temperature dependent whereas, below 15 K, no velocity changes were observed in the unirradiated specimen. The behaviour of Av/v in N4 is typical for the presence of TSs. Indeed, at the lowest temperatures {from 0.3 to 2 K) the velocity increases
8O
5O
4O
3O
2O
'~0 <
= 0 o
O 0 n 0 0
f = 648 Hllz
-10
grn
-20
O
n-irrad.
£;
unirrad
O
N4
0
0 0
-30 l
-40 I 0.1
i
l i i
i~
j
~
, , ,i,,110
'
~
'
'''"100
TEMPERATURE (1()
Fig. 2. Velocity change as a function of temperature for neutron-irradiated quartz N4 (dose, 1.2×10 j9 neutrons cm- 2; E >/0.3 MeV ) and for unirradiated quartz (frequency, 648 MHz; T~)=0.33 K). 80~ 70
6O 50 i 4O
3O
s
< =
20 10
.-:: >
0 -10
f = 648 MHz
-20 -30
O
: e - - i r r a d , . K3
.~. : n - i r r a d ~ 1(9
-40 -50
i
0.1
i
, ,,,,,i
,
,
.,,i,I
1
0
1 '11~MP~RAq'I] R ~
•
,
i
,ill,
100
(W)
Fig. 3. Measurements of the velocity change as a function of temperature and fittings of the data to the tunnelling model (frequency, 648 MHz; T,=0.45 K): o, electron-irradiated quartz, sample E3; +, neutron-irradiated quartz, sample K9.
logarithmically with increasing temperature. It is the result of the resonant interaction between the TSs and the acoustic phonons (see (3)). The condition hto~. kT of (3) was fulfilled (at the lowest temperatures hto-~0.1kT). Above 2 K , the change in the velocity of sound deviates from the
logarithmic law and a maximum is seen at about 4 K. In the tunnelling model, this behaviour is the consequence of the relaxational process, which causes a decrease in the velocity with increasing temperature (see (5)). When this contribution compensates that of the resonant process, a maximum in Av/v is observed. At higher temperatures, for which ~0rm~ 1, a logarithmic decrease in the velocity is expected with a slope half of that of the resonant part (see (6)). This logarithmic decrease was not observed in our case, nor was it detected in high frequency measurements in glasses [13]. The stronger temperature decrease in Av/v can be attributed to higher order phonon relaxational processes, e.g. the Raman process. The velocity curves for K9 and E3 show a similar behaviour to those for N4. However, the velocity changes are much smaller in K9 and in E3. This indicates that the density P of states is much smaller. We also note that, to obtain comparable velocity changes, a much higher dose is needed in the case of electron irradiation than in neutron-irradiated quartz. Although we reported before the logarithmic increase in Av/v at the lowest temperatures in electron-irradiated [10] and neutron-irradiated quartz [5], the measured typical "glassy" behaviour of Av/v at higher temperatures allows careful study of the effect of the relaxation process and makes it possible to determine both the density P of states of the TSs and the coupling parameter 7~. Therefore a quantitative analysis of the data was performed in the framework of the tunnelling model. The velocity change given in (5) was calculated numerically for each temperature. Raman processes were also taken into account in order to describe the drastic decrease in Av/v above 5 K. The best-suited theoretical curves for K9 and E3 are given in Fig. 3. As can be seen, there is a good agreement between the theoretical curves and the data. From the fits, the parameters C ('~-"/~12), K 3 and K7 (the Raman parameter [12]) can be determined independently. They are given in Table 1, as well as the values for the density P of states and the coupling parameter y~. These parameters were deduced from C and K3, making use of the expression yl2/Vl 2= ~/t2/Vt 2, which was found to be valid for most of the amorphous solids [3]. As can be seen from Table 1, the density of states increases with increasing neutron dose; P is about a factor of 5 larger in K9 than in N4. The observed increase in the density of states with increasing neutron dose is in agreement with
81
TABLE 1 Tunnel state parameters for neutron- and electron-irradiated quartz, obtained from fittings of the ultrasonic velocity data to the tunnelling model
Parameter
Sample E3
.Sample K9
Sample N4
Dose (cm 2) E (MeV) C ( × 1 0 ") K ~ ( × 1 0 7 K ~s i) KT(K 7 s I) PFle(l(Y'gcm Is e) 7J (eV) P ( x 10-~'~erg i cm 3)
1.0 × 102(telectrons 3 1.6 29 5×10 ~ 1.4 1.0 0.6
0.85 × 10 I~ neutrons i> 0.3 3.7 39 2×10.~ 3.2 1.1 1.0
1.2 x 10 ~ neutrons >/0.3 21 42 3×1().~ 18 1.1 5.4
previous results [7]. A more striking result that follows from Table 1 is that, for similar doses, electrons induce fewer TSs in quartz than neutrons do, whereas the parameter 7~ which describes the coupling between the TSs and the longitudinal phonons is found to be similar for both irradiations, within the accuracy of 15% for 7~. Hence the tunnelling states induced by electrons are of the same nature as those in neutronirradiated quartz, in spite of the very different disordering processes, but the induced density of states is very different for the same dose. We can conclude that the availability of the precision method for measurements of the variation in the velocity of sound in the 500 MHz range made it possible to detect the "glassy" behaviour of the low temperature velocity of sound in electron-irradiated quartz and to fit the results to the tunnelling model. This leads to rather precise values for 71 and to conclusive evidence that the tunnelling states are of a similar nature to those in neutron-irradiated quartz in spite of the very different disordering processes. The density of states, however, is very much smaller in electron-irradiated quartz than in neutron-irradiated quartz for the same dose. Acknowledgments The authors thank the Centre d'Etudes Nucldaires, Grenoble, for the electron irradiation and the Studiecentrum voor Kemenergie, Mol (and in particular J. Cornelis), for the neutron irradiations.
They are also grateful to the Belgian Interuniversitair Instituut voor Kernwetenschappen and the Katholieke Universiteit Leuven for financial support. References 1 W.A. Phillips (ed.), Amorphous Solids: Low Temperature Properties, Springer, Berlin, 1981. 2 P. W. Anderson, B. 1. Halperin and C. M. Varma, Philos. Mag., 25 (1972) 1. W. A. Phillips, J. Low Temp. Phys., 7 (1972) 351. 3 S. Hunklinger and W. Arnold, in W. P. Mason and R. N Thurston (eds.), Physical Acoustics, Vol. 12, Academic Press, New York, 1976, p. 155. 4 C. Laermans, Phys. Rev. Lett., 42 (1979) 250. 5 B. Golding, J. E. Graebner, W. H. Haemmerle and C. Laermans, Bull. Am. Phys. Soc,, 24 (1979) 495. B. Golding and J. E. Graebner, in H. J. Marls (ed.), Phonon Scattering in Condensed Matter, Plenum, New York, 1980, p. 11. 6 J. W. Gardner and A. C. Anderson, Phys. Rev. B, 32 (1981)474. 7 C. Laermans, in G. Walrafen and A. Revesz (eds.), Structure and Bonding in Non-Crystalline Solids, Plenum, New York, 1986, p. 325. 8 C. Laermans, A. M. de Go6r and M. Locatelli, Phys. Lett. A, 80(1980)331. A. M. de Go6r, M. Locatelli and C. Laermans, J. Phys. (Paris), Colloq. C6, 42 ( 1981 ) 78. 9 C. Laermans and A. Vanelstraete, Phys. Rev. B, 34 (1986) 1405. 10 A. Vanelstraete and C. Laermans, Phys. Rev. B, 39 (1989) 3905. 11 R. Truell, C. Elbaum and B. B. Chick, Ultrasonic" Methods in Solid State Physics, Academic Press, New York, 1969. 12 P. Doussineau, A. Levelut, M. Matecki, W. Sch6n and W. D. Wallace, J. Phys. (Paris), 46 ( 1985 ) 979. 13 L. Pich6, R. Maynard, S. Hunklinger and J. Jfickle, Phys. Rev. Lett., 32 (1974) 1426.