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Influence of the neutron dose on the coupling between the tunnelling states and the phonons in neutron irradiated quartz
Volume 126, number 5,6
PHYSICS LETI’ERS A
11 January 1988
INFLUENCE OF THE NEUTRON DOSE ON THE COUPLING BETWEEN THE TUNNELLING STATES AND THE PHONONS IN NEUTRON IRRADIATED QUARTZ C. LAERMANS and V. ESTEVES Department of Physics, Catholic University of Leuven VS.H.D., Celestijnenlaan 200 D, 3030 Leuven, Belgium Received 24 February 1987; revised manuscript received 31 July 1987 accepted for publication 15 November 1987 Communicated by J.I. Budnick
Low temperature ultrasonic attenuation measurements of neutron irradiated quartz are reported. The results of a systematic study for four neutron doses and five frequencies have been analysed in terms of the tunnelling model originally proposed for glasses. It is found that the coupling ofthe tunnelling states with the phonons is smaller than in vitreous silica and increases with neutron dose,
1. Introduction
diated quartz also explained by a strong coupling between phonons and localized low-energy excitations
At low temperatures amorphous solids exhibit anomalous dynamic properties which can be explained by the existence of configurational tunnelling states [1]. These centres couple strongly to phonons and dominate the behaviour of specific heat, thermal conductivity ultrasonic attenuation and other properties at low In spite of the fact that this behaviour is nearly universally found in the many glasses so far studied, virtually nothing is known about the microscopic origin of these states which in a more simplified way can be described as two-level systems (TLS) [3]. Dynamic studies in
formally described within the framework of the tunnelling model [1]. These excitations or two-level systems (TLS) are supposed to be single atoms or groups of atoms which undergo quantum mechanical tunnelling between two configurational equilibrium states. The model does not specify the microscopic nature of the tunnelling particle. In order to search for additional information on structural aspects of the TLS, intrinsically related to disorder, neutron damaged quartz might be cornpared with vitreous silica, which is another polymorphic form of Si02 and the glass for which low
crystalline quartz locally disordered by high energy neutrons were carried out in order to obtain a better understanding of this “anomalous” behaviour of glasses and showed a non-linear acoustic attenuation as in glasses [4]. The study of an irradiated crystal can thus be a tool for the investigated ofthe low ternperature anomalies of amorphous solids. Other dynarnic studies have supported this evidence [5—7]as have studies in electron irradiated quartz ~2• The evidence from these experiments is in irra-
temperature properties have been most studied to date. This approach should lead to an easier interpretation than would the study of excitations in glass where the atomic environment is less ordered and therefore more difficult to describe. We report new measurements of the ultrasonic attenuation in neutron irradiated quartz at lower neutron doses, than was previously the case [5—9].The
~
vi S2
A review of the low temperature properties ofamorphous solids is given in ref. [2]. For a recent review see ref. [7].
main purpose however is to report on a detailed analysis of a systematic study of the ultrasonic attenuation in neutron irradiated quartz for four neutron doses and five frequencies. The neutron doses cover more than a decade and the frequencies the
0375-960l/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
341
Volume 126, number 5,6
PHYSICS LETTERS A
range 0.1 to 0.7 0Hz. The typical parameters of the tunnelling model )~and Py~are directly deduced from the data. Evidence is presented that the coupling between phonons and two-level systems is smaller than in vitreous silica. In addition the increase in coupling with the neutron dose is also demonstrated. This is an interesting observation in view of the microscopic description of the two-level systems. 2. Theoretical considerations At low temperatures the interaction between the two-level systems and the ultrasonic wave in an amorphous solid is described by two processes: the reSonant and the relaxation process. Under the preseat experimental conditions the resonant absorption is saturated and we have only to deal with the relaxational attenuation. A sound wave travelling through an ensemble of two-level systems, disturbs their thermal equilibrium so that the TLS tend to relax into a new equilibrium. Two temperature regimes have to be considered depending on whether WTm>~ 1 or WTm << l~ in which rm is the minimum relaxation time. The corresponding expressions for the ultrasonic attenuation are [10]: (1) Low temperature regime: W~m>> l,
~
&~T
—
Q~7+-_)_~ 2y~\T3
(l)
a’-T
_~ —
_.~-~c~-?ii~ liv ü~• 3~2
(2) High temperature regimç: =
~.
3. Sample characterization and experimental technique X-cut single crystals of very high purity synthetic quartz into 8 mm rods with 3 were mm shaped diameter. The end long facescylindrical were polished optically flat and parallel for ultrasonic studies. Neutron activation analysis on a sample of the same origin yielded as impurity content in ~.Lg/g: Na 2.3; Fe 0.7; Ga 0.06; Cr 0.0016; Cl 9.8; Zn 0.4; Sb 0.06; Co 0.0022.
anm ~<
(2)
neutrons (E~0.3MeV) at doses indicated in table 1. The third columm refers to calibration of fast neutron flux for E~0.1 MeV. The samples N 3, N4 and N5 were irradiated in the
(3)
same nuclear reactor cycle. This is specific to these samples which therefore constitute a reliable set of samples as regards the ever present question of the reliability of dose units. The sample K 10, although not irradiated in the same nuclear reactor cycle was
l~
~it CQ W/V~
The constant C5 and the average coupling constant ~ are related to the fundamental parameters y~and y, by: =
P~~
(4)
and y~/v~ + 2y~/v,~ l/v~+ 2/vt The symbols have the following meaning. -2
gitudinal relaxation absorption for the low temperature regime. c4’: longitudinal relaxation absorption for the high temperature regime (plateau). f3~ mean value of the ultrasonic velocity. fi: spectral density of two-level systems TLS. yii: coupling parameter between the longitudinal and transverse phonons respectively and the two-level systems TLS. The other symbols have their usual meaning. The quantities c4~T and c4’ are the measured values for the w°T3regime and the high temperature attenuation plateau. Combining eqs. (2) and (3) we are able to deduce directly the parameters of the tunnelling model: Py~and The spectral density of states P and the coupling parameters can be deduced if we assume that ~‘ y~and y~ 2y~as is the case for vitreous silica [3].
Al content is below the detectable limit of 10 j.tg/g. Four crystals were irradiated at SCK, Mol with fast
or
cv
11 January 1988
—
—
342
(5) c4~T: lon-
irradiated in similar conditions. The irradiations were carried out at 60°C.An unirradiated sample, labelled V, was measured for comparison at the different frequencies. The experiments were performed using the standard pulse—echo technique for rftransmitting and receiving. Longitudinal waves in quartz were generated by surface wave excitation.
Volume 126, number 5,6
PHYSICS LETTERS A
11 January 1988
Table 1 Set ofultrasonic attenuation measurements. Sample
Dose (n/cm 2) E~0.3MeV
V
unirradiated
K 10
l.68x i0’~
2) Dose (n/cm E~0.lMeV
2.7x l0’~
Frequency
(MHz)
Temperature range(K)
340 400 480 540 640
4.2—300 4.2—400 1.3—300 4.2—300 1.3—300
480
1.3—300
640
1.3—300
N 6.6x lOIS
1.1
x l0’~
340 480 640
1.3—300 1.3—300 1.3—300
N4
l.2x l0’~
2.Ox iO’~
340 480 640
1.3—300 1.3—300 1.3—300
N5
2.6x l0’~
4.1
155
1.3—120
3
x l9’~
340
1.3—300
400 480 540 640
1.3—300 1.3—300 1.3—300 1.3—300
4. Experimental results and discussion ilo Figs. 1 and 2 give typical curves obtained for the ultrasonic attenuation as a function of temperature in quartz. 2 a typical forneutron-irradiated an unirradiated sample ofIn thefig. same origin iscurve also given, for comparison. The low temperature region, below about 30 K, where the two-level systems dommate the ultrasonic attenuation, is of interest for this work. As can be seen in fig. 2 at the lowest temperatures the neutron irradiation causes a strong in-
crease ofthe ultrasonic attenuationcompared to that for the unirradiated crystal. The temperature dependence can be described by a T3 law, a behaviour attributed to the relaxational interaction of the
1:
.640 MHz .480 I*lz 2 1 68~i&8 n/cm
01
ultrasonic waves with the two-level systems in the
regime served of frequency that (flm*~ the inultrasonic 1this (see temperature eq. attenuation (1)). Itrange, hasisalso in independent a~eement been obwith the TLS model (see also ref~.[11,22]). At about 2.5 K a departure from the T3 law occurs and the attenuation increases less steeply until about 10 K when it starts to level off. In this temperature region the transition between the regimes corm ~ 1 and
001 1
10
i- (K)
102
Fig. 1. Ultrasonic attenuation as a function of temperature in
neutron irradiated quartz; dose: 1.68 x l0’~n/cm2, E> 0.3 MeV.
343
Volume 126, number 5,6
(C
PHYSICS LETTERS A
11 January 1988
I
~
I
I
/ 4
-~
-v
“°
C °
—
/t
~1.U it o C
20
-
8 S
/
-
:
0 01
~///~
640MHz
200
400 600 800 Frequency (MHz)
Fig. 3. Ultrasonic attenuation at the “plateau” as a function of 0 0 frequency in neutron irradiated quartz: sample N n/cm2, E~O.3MeV). 5 (2.6x 1019
unirradiated 5ri/cm2 • 6.6xlO
frequency for the sample N ___________________________________ ib 1b2
io~
Temperature (K)
Fig. 2. Ultrasonic attenuation as a function of2,temperature E~O.3MeV) in neutron and for unirradiated irradiated quartz quartz. (dose: 6.6x 1Q18 n/cm
1 occurs (see eqs. (1) and (3)). Above 10 K a clear plateau is observed in fig. 1 which extends to about 30 K. The plateau is predicted by the tunnelling model but is seldom observed so clearly at these frequencies. It is attributed to the WTm << 1 re gime of the relaxation attenuation for which an w T behaviour is expected (see eq. (3)) Above 30 K the attenuation starts to rise again. This increase is due to other mechanisms such as the anhannonic three phonon processes related with the crystalline phase, present in the sample, and the structural relaxation [91 as observed in vitreous silica. For the highest doses these processes become observable at lower temperatures as in K 10 and as a consequence the plateau is not so clearly seen (see fig. 2). Fig. 3 gives the attenuation at the plateau as a function of 0) Tm <<
~.
5. It is seen that the frequency dependence predicted by the tunnellingmodel explains the measurements very well. To our knowledge this behaviour has never been verified before forEqs. neutron (2) and irradiated (3) lead quartz directly to the parameter However, we note that these equations have been ~
~.
derived for amorphous solids which are isotropic. Irradiated quartz, at least for the doses used here is still mainly crystalline and therefore anisotropic. The expression between brackets in eq. (1) represents a sum over the phonon modes and over the different crystal directions. The phonon velocities depend on the crystal direction and the coupling constants y~ and Vt may also be anisotropic. We take this into account by considering ~ as an average over phonon modes and over crystal directions and by using a calculated mean value for the sound velocity as previously determined for quartz by Doulat et al. [141: 47 x lO~m/s. The derived values for ii as a function of the frequency and fast neutron dose are given in table 2. For the purpose of comparison a value of -
~ in vitreous silica was calculated in the same way °“
83
During the course of the preparation of this manuscript, vibrating reed experiments have shown the existence of a plateau in the internal friction in neutron irradiated quartz over almost three decades in temperature (frequency about 1 kHz) [121. The observations demonstrate the validity of the tunnelling model for much longer time scales than in previous experiments. They show the validity of the standard distribution used in the tunnelling model also for neutron irradiated quartz.
344
Data about the hypersonic attenuation at 35 GHz support these findings [1 3]. A comparison has been madebetween the ultrasonic attenuation we find in N 5 at 350 MHz, at 10 K in the plateau region, and the hypersonic attenuation at 35 GHz, also at 10 K. The hypersonic data were on a brother sample of N5, one of our samples irradiated simultaneously with N5. ~ .08 wasfound in this broad frequency range. Although the 35 GHz data do not show a plateau it seems that this behaviour, as is statedby the authors, can be explained by the tunnelling model and therefore supports our findings.
Volume 126, number 5,6
PHYSICS LETTERS A
11 January 1988
Table 2 Fundamental parameters ofthe TLS theory. Sample
Frequency (MHz)
~ (eV) (from eq. (2))
~ (cv) average
y~(cv)
y~(eV) (average)
1.68
480 640
0.38 0.35
0.37
0.53 0.49
0.51
6.6
340 480 640
0.43 0.37 0.35
0.40
0.61 0.52 0.49
0.54
8n/cm2) Dose (10 E~O.3 MeV
K 10 N 3
N4
12
340 480 640
0.42 0.41 0.40
0.41
0.60 0.58 0.56
0.58
N5
26
340 400 480 540 640
0.49 0.50 0.45 0.44 0.43
0.46
0.69 0.71 0.64 0.62 0.61
0.65
105
0.54°~
a-SiO, ‘~
0.76°~
Calculated in a similar way from recent results in a-Si02 (ref. [151).
from recent data [15]. A striking conclusion can be drawn from these findings: the coupling parameter increases with neutron dose. In addition, with increasing neutron dose it evolves to a value which is closer and closer to that in vitreous silica although remaining slightly smaller. This observation seems to be in disagreement with previous findings [5], but the disagreement is only apparent. Indeed in a phonon echo experiment in one ofitour N1 2, E>~0.3 MeV) wassamples, found that (dose 8 x 1018 n/cm ~ in neutron irradiated quartz is “similar” to that in vitreous silica. Since for vitreous silica y~ 2y~a similarity between the ~ should also imply a similarity between the Our data agree with this “similarity” since the ~ in neutron irradiated quartz are only slightly smaller than in vitreous silica. The coupling parameters ~ and y 5 cannot be directly derived from these data. However using 2y~and ~ ~ an estimate can be given (see table 1). The parameters derived in this way are an average over the different crystal directions, and therefore it is not certain that they can be compared to the value deduced from phonon echo experiments in sample N, which was 1.2 eV deduced for the xdirection [5]. In addition it has been observed that the values for the coupling parameters derived from different types of experiments are different and that ~.
phonon echo experiments give mostly a higher value. (For vitreous silica values ranging from 0.4 to 2 eV have been reported [16].) The origin of these discrepancies is however not yet clear. In order to be able to compare to vitreous silica a value for ~ has been derived from recent ultrasonic attenuation data in a similar way as used for the determination of our y~.It is seen that also y~is systematically smaller than in vitreous silica, it increases dose and approaches more andthat more the valuewith of vitreous silica. Using eq. (3) the parameters CQ and Py~can be derived directly. This was again done for the different frequencies and neutron doses. For the longitudinal sound velocity in the x-direction a value VQ = 5.75 x 1 0~rn/s was used [5]. The results are given in table 3. For all the doses studied Py~is smaller than in vitreous silica and it increases with dose. Since this increase is much more rapid than that of v~it has to be attributed mainly to the increase of the density of states of the two-level systems with neutron dose. This has been explained before in view of the increasing amorphization of the quartz upon increasing neutron irradiation [17]. We will come back to this subject furtheron. The observation of an increasing coupling constant with increasing neutron dose is an interesting observation in view of the search for a microscopic 345
Volume 126, number 5,6
PHYSICS LETTERS A
Table 3 Numerical results for Ca and
11 January 1988
Py~.
Sample
Dose
KlO
E~0.3 MeV 1.68
2)
(1018 n/cm
Frequency
C l0—~ 1.2 1.4
Py~ 3) (lo~erg cm 1.1 1.2
Py~(average) (10~erg cm3)
(MHZ) 480 640 340 480 640
2.0 2.8 3.1
1.8 2.4 2.7
2.3
8 x
1.2
N 3
N4
12
340 480 640
4.1 4.4 4.7
3.6 3.9 4.1
3.8
N5
26
340 400 480 540 640
7.7 7.3 8.9 9.4 9.4
6.7 6.4 7.7 8.2 8.6
7.5
a-Si02 ~l
6.6
105
481
Calculated in a similar way from recent results in a-Si02 (ref. [15]).
description of the TLS. Quartz is known to amorphize gradually upon increasing irradiation with fast neutrons. A review of the effects of irradiation on quartz has been given in a previous paper [7]. For the doses used in the experiments in question the sample consist basically of two phases: a crystalline matrix and amorphous regions embedded in the matrix [17,18]. From mass density measurements performed on our samples [19] it is known that the crystalline matrix is still the major part of the samples: the mass density for the highest dose used has only decreased upon irradiation by 0.1% to be cornpared2)with about quartz 14% atbecomes saturating dose (2 x 1020 for which fully amorphous. n/cm For the doses used the amorphous islands are homogeneously spread over the crystal and the matrix has mainly the properties of the unirradiated crystal. Previously it has been put forward that the amorphous islands may be the location of the two-level systems [171, although more recent experiments suggested that this is not necessarily the case [20,21]. Our new data cannot give a decisive answer to this question, but add new elements to the discussion. Since the coupling between the phonons and the twolevel systems is only slightly different from that in vitreous silica we may state that the TLS have a similar nature in neutron irradiated quartz as in vitreous 346
silica. This has been suggested before [51. However since the coupling is not “the same” this suggests that the TLS are not “the same” and they change with neutron dose. It has been observed in X-ray studies that in the defected regions the configuration of the atoms changes with neutron dose [181. The different local arrangements of the atoms in the vicinity of the TLS could explain the change of the coupling of the TLS to the phonons with dose. In addition the amorphous islands in the crystal are highly cornpressed [181. Our findings therefore suggest that TLS in a dense environment couple less to phonons than if they are less dense environment. Since samples are in stilla mainly crystalline it can also notthe be ruled out that the coupling is anisotropic: the TLS may couple more in one crystal direction than in another. Our estimated values for y~and y, are mean values. It is known that the disordering process is different for the different crystal directions [17,18]. If evidence could be found of a directionality of the TLS this would help to come closer to a microscopic description of the two-level systems in neutron irradiated quartz and possibly in vitreous silica since the basic building units (the Si0 4 tetrehedra) in both forms of SiO2 are the same.
Volume 126, number 5,6
PHYSICS LETTERS A
5. Conclusion In this paper new data of the ultrasonic attenuation as a function of temperature are reported for neutron irradiated quartz. The main purpose of the paper however is to report the results of a detailed analysis of the data obtained in a systematic study of the ultrasonic attenuation as a function of temperature in neutron irradiated quartz for four neutron doses and five frequencies The neutron doses cover more than a decade. The data are analysed in term of the tunnelling model, put forward to explain the low temperature dynamic anomalies in glasses. The main result is the finding that the coupling parameterwhich describes the coupling ofthe TLS with the phonons, is smaller than in vitreous silica and increases with neutron dose approaching more and more the value for vitreous silica, ~.
Acknowledgement The authors thank SCKJCEN, Mol, Belgium for the irradiation and in particular J. Cornelis. They are also grateful to the Belgian 11KW for financial support. Interesting discussions with Professor S. Hunklinger are gratefullyacknowledged. One of us (V.E.) thanks INIC, Instituto National de Investigaçao Cientifica, Lisboa (Portugal) for a grant. They also thank Oak Ridge National Laboratory and especially R. Weeks for the neutron activation analysis. ~ The results and the conclusionsgiven in this paper are part of the Ph.D. thesis of one ofus [22].
References [I]P.W. Anderson, B.I. Halperin and C. Varma, PhIlos. Mag. 25 (1972) 1; W.A. Philips, J. Low Temp. Phys. 7 (1972) 351.
Il January 1988
[2] W.A. Philips, ed., Amorphous solids: low-temperature properties (Springer, Berlin, 1981). [3] S. Hunklinger and W. Arnold, in: Physical acoustics, Vol. 12, eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1976) p. 155—215. [4] C. Laermans, Phys. Rev. Lett. 42 (1979) 250. [5] B. Golding, J.E. Graebner, W. Haemmerle and C. Laermans, Bull. Am. Phys. Soc. 24 (1979) 495; B. Golding and J.E. Graebner, in: Phonon scattering in condensed matter, ed. H.J. Marts (Plenum, New York, 1980) ~. 1-20. [6] W. Gardner and A.C. Anderson, Phys. Rev. B 23 (1981) [7] C. Laermans, in: Structure and bonding in noncrystalline solids, eds. 0. Walrafen and A. Revesz (Plenum, New York, 1986) p. 329—367. [8] C. Laermans and V. Esteves, in: Phonon scattering in condensed matter, eds. W. Eisenmenger, K. Lassmann and S. Dottinger (Springer, Berlin, 1983) pp. 407—409. [9] V. Esteves, A. Vanelstraete and C. Laermans, in: Phonon physics, eds. J. Kollar, N. Kroo, N. Menyhard and T. Sikios (World Scientific, Singapore, 1985) p. 45. [10]J. JackIe, Z. Phys. 257 (1972) 212. [I 1] C. Laermans, V. Esteves and A. Vanelstraete, Radiat. Eff. 97 (1986) 175. [12] A. Venelstraete, C. Laermans, L. Lejan’aga, M. von Schickfus and J. AppI. Suppl. 26 (1987) 733; S. Z.Hunklinger, Phys. (Cond.Japan. Matter), to bePhys. published. [13] R. Vacher, J. Pelous, N. Elkhayati Elidnssi and M. Boissier, in: Phonon scattering in condensed matter, eds. A.C. Anderson and J.P. Wolfe (Springer, Berlin, 1987) p. 61. [14] J. Doulat, M. Locatelli and R. Rivallin, in: Phonon scattering in solids, eds. L.J. Challis et al. (Plenum, New York, 1976) ~,. 383. [15] S. Hunklinger, private communication. [16] J. Graebner and B. Golding, Phys. Rev. B 19 (1979) 964. [17] D. Grasse, 0. Kocar, H. Peisl, S.C. Moss and B. Golding, Phys. Rev. Lett. 46(1981)261. [18]R. Comes, M. Lambert and H. Guinier, in: Interaction of radiation in solids, ed. A. Bishay (Plenum, New York, 1967) p.319. [19]A. Vanelstraete and C. Laermans, internal report (1985). [20]A.C. Anderson, J.A. Mc. Millan andF.J. Walker, Phys. Rev. B24 (1981) 1124. [21]C. Laermans and A. Vanelstraete, Phys. Rev. B 35 (1987) 6399, (22] V. Esteves, Ph.D. Thesis, Leuven (6 October 1986), unpublished.
347
PHYSICS LETI’ERS A
11 January 1988
INFLUENCE OF THE NEUTRON DOSE ON THE COUPLING BETWEEN THE TUNNELLING STATES AND THE PHONONS IN NEUTRON IRRADIATED QUARTZ C. LAERMANS and V. ESTEVES Department of Physics, Catholic University of Leuven VS.H.D., Celestijnenlaan 200 D, 3030 Leuven, Belgium Received 24 February 1987; revised manuscript received 31 July 1987 accepted for publication 15 November 1987 Communicated by J.I. Budnick
Low temperature ultrasonic attenuation measurements of neutron irradiated quartz are reported. The results of a systematic study for four neutron doses and five frequencies have been analysed in terms of the tunnelling model originally proposed for glasses. It is found that the coupling ofthe tunnelling states with the phonons is smaller than in vitreous silica and increases with neutron dose,
1. Introduction
diated quartz also explained by a strong coupling between phonons and localized low-energy excitations
At low temperatures amorphous solids exhibit anomalous dynamic properties which can be explained by the existence of configurational tunnelling states [1]. These centres couple strongly to phonons and dominate the behaviour of specific heat, thermal conductivity ultrasonic attenuation and other properties at low In spite of the fact that this behaviour is nearly universally found in the many glasses so far studied, virtually nothing is known about the microscopic origin of these states which in a more simplified way can be described as two-level systems (TLS) [3]. Dynamic studies in
formally described within the framework of the tunnelling model [1]. These excitations or two-level systems (TLS) are supposed to be single atoms or groups of atoms which undergo quantum mechanical tunnelling between two configurational equilibrium states. The model does not specify the microscopic nature of the tunnelling particle. In order to search for additional information on structural aspects of the TLS, intrinsically related to disorder, neutron damaged quartz might be cornpared with vitreous silica, which is another polymorphic form of Si02 and the glass for which low
crystalline quartz locally disordered by high energy neutrons were carried out in order to obtain a better understanding of this “anomalous” behaviour of glasses and showed a non-linear acoustic attenuation as in glasses [4]. The study of an irradiated crystal can thus be a tool for the investigated ofthe low ternperature anomalies of amorphous solids. Other dynarnic studies have supported this evidence [5—7]as have studies in electron irradiated quartz ~2• The evidence from these experiments is in irra-
temperature properties have been most studied to date. This approach should lead to an easier interpretation than would the study of excitations in glass where the atomic environment is less ordered and therefore more difficult to describe. We report new measurements of the ultrasonic attenuation in neutron irradiated quartz at lower neutron doses, than was previously the case [5—9].The
~
vi S2
A review of the low temperature properties ofamorphous solids is given in ref. [2]. For a recent review see ref. [7].
main purpose however is to report on a detailed analysis of a systematic study of the ultrasonic attenuation in neutron irradiated quartz for four neutron doses and five frequencies. The neutron doses cover more than a decade and the frequencies the
0375-960l/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
341
Volume 126, number 5,6
PHYSICS LETTERS A
range 0.1 to 0.7 0Hz. The typical parameters of the tunnelling model )~and Py~are directly deduced from the data. Evidence is presented that the coupling between phonons and two-level systems is smaller than in vitreous silica. In addition the increase in coupling with the neutron dose is also demonstrated. This is an interesting observation in view of the microscopic description of the two-level systems. 2. Theoretical considerations At low temperatures the interaction between the two-level systems and the ultrasonic wave in an amorphous solid is described by two processes: the reSonant and the relaxation process. Under the preseat experimental conditions the resonant absorption is saturated and we have only to deal with the relaxational attenuation. A sound wave travelling through an ensemble of two-level systems, disturbs their thermal equilibrium so that the TLS tend to relax into a new equilibrium. Two temperature regimes have to be considered depending on whether WTm>~ 1 or WTm << l~ in which rm is the minimum relaxation time. The corresponding expressions for the ultrasonic attenuation are [10]: (1) Low temperature regime: W~m>> l,
~
&~T
—
Q~7+-_)_~ 2y~\T3
(l)
a’-T
_~ —
_.~-~c~-?ii~ liv ü~• 3~2
(2) High temperature regimç: =
~.
3. Sample characterization and experimental technique X-cut single crystals of very high purity synthetic quartz into 8 mm rods with 3 were mm shaped diameter. The end long facescylindrical were polished optically flat and parallel for ultrasonic studies. Neutron activation analysis on a sample of the same origin yielded as impurity content in ~.Lg/g: Na 2.3; Fe 0.7; Ga 0.06; Cr 0.0016; Cl 9.8; Zn 0.4; Sb 0.06; Co 0.0022.
anm ~<
(2)
neutrons (E~0.3MeV) at doses indicated in table 1. The third columm refers to calibration of fast neutron flux for E~0.1 MeV. The samples N 3, N4 and N5 were irradiated in the
(3)
same nuclear reactor cycle. This is specific to these samples which therefore constitute a reliable set of samples as regards the ever present question of the reliability of dose units. The sample K 10, although not irradiated in the same nuclear reactor cycle was
l~
~it CQ W/V~
The constant C5 and the average coupling constant ~ are related to the fundamental parameters y~and y, by: =
P~~
(4)
and y~/v~ + 2y~/v,~ l/v~+ 2/vt The symbols have the following meaning. -2
gitudinal relaxation absorption for the low temperature regime. c4’: longitudinal relaxation absorption for the high temperature regime (plateau). f3~ mean value of the ultrasonic velocity. fi: spectral density of two-level systems TLS. yii: coupling parameter between the longitudinal and transverse phonons respectively and the two-level systems TLS. The other symbols have their usual meaning. The quantities c4~T and c4’ are the measured values for the w°T3regime and the high temperature attenuation plateau. Combining eqs. (2) and (3) we are able to deduce directly the parameters of the tunnelling model: Py~and The spectral density of states P and the coupling parameters can be deduced if we assume that ~‘ y~and y~ 2y~as is the case for vitreous silica [3].
Al content is below the detectable limit of 10 j.tg/g. Four crystals were irradiated at SCK, Mol with fast
or
cv
11 January 1988
—
—
342
(5) c4~T: lon-
irradiated in similar conditions. The irradiations were carried out at 60°C.An unirradiated sample, labelled V, was measured for comparison at the different frequencies. The experiments were performed using the standard pulse—echo technique for rftransmitting and receiving. Longitudinal waves in quartz were generated by surface wave excitation.
Volume 126, number 5,6
PHYSICS LETTERS A
11 January 1988
Table 1 Set ofultrasonic attenuation measurements. Sample
Dose (n/cm 2) E~0.3MeV
V
unirradiated
K 10
l.68x i0’~
2) Dose (n/cm E~0.lMeV
2.7x l0’~
Frequency
(MHz)
Temperature range(K)
340 400 480 540 640
4.2—300 4.2—400 1.3—300 4.2—300 1.3—300
480
1.3—300
640
1.3—300
N 6.6x lOIS
1.1
x l0’~
340 480 640
1.3—300 1.3—300 1.3—300
N4
l.2x l0’~
2.Ox iO’~
340 480 640
1.3—300 1.3—300 1.3—300
N5
2.6x l0’~
4.1
155
1.3—120
3
x l9’~
340
1.3—300
400 480 540 640
1.3—300 1.3—300 1.3—300 1.3—300
4. Experimental results and discussion ilo Figs. 1 and 2 give typical curves obtained for the ultrasonic attenuation as a function of temperature in quartz. 2 a typical forneutron-irradiated an unirradiated sample ofIn thefig. same origin iscurve also given, for comparison. The low temperature region, below about 30 K, where the two-level systems dommate the ultrasonic attenuation, is of interest for this work. As can be seen in fig. 2 at the lowest temperatures the neutron irradiation causes a strong in-
crease ofthe ultrasonic attenuationcompared to that for the unirradiated crystal. The temperature dependence can be described by a T3 law, a behaviour attributed to the relaxational interaction of the
1:
.640 MHz .480 I*lz 2 1 68~i&8 n/cm
01
ultrasonic waves with the two-level systems in the
regime served of frequency that (flm*~ the inultrasonic 1this (see temperature eq. attenuation (1)). Itrange, hasisalso in independent a~eement been obwith the TLS model (see also ref~.[11,22]). At about 2.5 K a departure from the T3 law occurs and the attenuation increases less steeply until about 10 K when it starts to level off. In this temperature region the transition between the regimes corm ~ 1 and
001 1
10
i- (K)
102
Fig. 1. Ultrasonic attenuation as a function of temperature in
neutron irradiated quartz; dose: 1.68 x l0’~n/cm2, E> 0.3 MeV.
343
Volume 126, number 5,6
(C
PHYSICS LETTERS A
11 January 1988
I
~
I
I
/ 4
-~
-v
“°
C °
—
/t
~1.U it o C
20
-
8 S
/
-
:
0 01
~///~
640MHz
200
400 600 800 Frequency (MHz)
Fig. 3. Ultrasonic attenuation at the “plateau” as a function of 0 0 frequency in neutron irradiated quartz: sample N n/cm2, E~O.3MeV). 5 (2.6x 1019
unirradiated 5ri/cm2 • 6.6xlO
frequency for the sample N ___________________________________ ib 1b2
io~
Temperature (K)
Fig. 2. Ultrasonic attenuation as a function of2,temperature E~O.3MeV) in neutron and for unirradiated irradiated quartz quartz. (dose: 6.6x 1Q18 n/cm
1 occurs (see eqs. (1) and (3)). Above 10 K a clear plateau is observed in fig. 1 which extends to about 30 K. The plateau is predicted by the tunnelling model but is seldom observed so clearly at these frequencies. It is attributed to the WTm << 1 re gime of the relaxation attenuation for which an w T behaviour is expected (see eq. (3)) Above 30 K the attenuation starts to rise again. This increase is due to other mechanisms such as the anhannonic three phonon processes related with the crystalline phase, present in the sample, and the structural relaxation [91 as observed in vitreous silica. For the highest doses these processes become observable at lower temperatures as in K 10 and as a consequence the plateau is not so clearly seen (see fig. 2). Fig. 3 gives the attenuation at the plateau as a function of 0) Tm <<
~.
5. It is seen that the frequency dependence predicted by the tunnellingmodel explains the measurements very well. To our knowledge this behaviour has never been verified before forEqs. neutron (2) and irradiated (3) lead quartz directly to the parameter However, we note that these equations have been ~
~.
derived for amorphous solids which are isotropic. Irradiated quartz, at least for the doses used here is still mainly crystalline and therefore anisotropic. The expression between brackets in eq. (1) represents a sum over the phonon modes and over the different crystal directions. The phonon velocities depend on the crystal direction and the coupling constants y~ and Vt may also be anisotropic. We take this into account by considering ~ as an average over phonon modes and over crystal directions and by using a calculated mean value for the sound velocity as previously determined for quartz by Doulat et al. [141: 47 x lO~m/s. The derived values for ii as a function of the frequency and fast neutron dose are given in table 2. For the purpose of comparison a value of -
~ in vitreous silica was calculated in the same way °“
83
During the course of the preparation of this manuscript, vibrating reed experiments have shown the existence of a plateau in the internal friction in neutron irradiated quartz over almost three decades in temperature (frequency about 1 kHz) [121. The observations demonstrate the validity of the tunnelling model for much longer time scales than in previous experiments. They show the validity of the standard distribution used in the tunnelling model also for neutron irradiated quartz.
344
Data about the hypersonic attenuation at 35 GHz support these findings [1 3]. A comparison has been madebetween the ultrasonic attenuation we find in N 5 at 350 MHz, at 10 K in the plateau region, and the hypersonic attenuation at 35 GHz, also at 10 K. The hypersonic data were on a brother sample of N5, one of our samples irradiated simultaneously with N5. ~ .08 wasfound in this broad frequency range. Although the 35 GHz data do not show a plateau it seems that this behaviour, as is statedby the authors, can be explained by the tunnelling model and therefore supports our findings.
Volume 126, number 5,6
PHYSICS LETTERS A
11 January 1988
Table 2 Fundamental parameters ofthe TLS theory. Sample
Frequency (MHz)
~ (eV) (from eq. (2))
~ (cv) average
y~(cv)
y~(eV) (average)
1.68
480 640
0.38 0.35
0.37
0.53 0.49
0.51
6.6
340 480 640
0.43 0.37 0.35
0.40
0.61 0.52 0.49
0.54
8n/cm2) Dose (10 E~O.3 MeV
K 10 N 3
N4
12
340 480 640
0.42 0.41 0.40
0.41
0.60 0.58 0.56
0.58
N5
26
340 400 480 540 640
0.49 0.50 0.45 0.44 0.43
0.46
0.69 0.71 0.64 0.62 0.61
0.65
105
0.54°~
a-SiO, ‘~
0.76°~
Calculated in a similar way from recent results in a-Si02 (ref. [151).
from recent data [15]. A striking conclusion can be drawn from these findings: the coupling parameter increases with neutron dose. In addition, with increasing neutron dose it evolves to a value which is closer and closer to that in vitreous silica although remaining slightly smaller. This observation seems to be in disagreement with previous findings [5], but the disagreement is only apparent. Indeed in a phonon echo experiment in one ofitour N1 2, E>~0.3 MeV) wassamples, found that (dose 8 x 1018 n/cm ~ in neutron irradiated quartz is “similar” to that in vitreous silica. Since for vitreous silica y~ 2y~a similarity between the ~ should also imply a similarity between the Our data agree with this “similarity” since the ~ in neutron irradiated quartz are only slightly smaller than in vitreous silica. The coupling parameters ~ and y 5 cannot be directly derived from these data. However using 2y~and ~ ~ an estimate can be given (see table 1). The parameters derived in this way are an average over the different crystal directions, and therefore it is not certain that they can be compared to the value deduced from phonon echo experiments in sample N, which was 1.2 eV deduced for the xdirection [5]. In addition it has been observed that the values for the coupling parameters derived from different types of experiments are different and that ~.
phonon echo experiments give mostly a higher value. (For vitreous silica values ranging from 0.4 to 2 eV have been reported [16].) The origin of these discrepancies is however not yet clear. In order to be able to compare to vitreous silica a value for ~ has been derived from recent ultrasonic attenuation data in a similar way as used for the determination of our y~.It is seen that also y~is systematically smaller than in vitreous silica, it increases dose and approaches more andthat more the valuewith of vitreous silica. Using eq. (3) the parameters CQ and Py~can be derived directly. This was again done for the different frequencies and neutron doses. For the longitudinal sound velocity in the x-direction a value VQ = 5.75 x 1 0~rn/s was used [5]. The results are given in table 3. For all the doses studied Py~is smaller than in vitreous silica and it increases with dose. Since this increase is much more rapid than that of v~it has to be attributed mainly to the increase of the density of states of the two-level systems with neutron dose. This has been explained before in view of the increasing amorphization of the quartz upon increasing neutron irradiation [17]. We will come back to this subject furtheron. The observation of an increasing coupling constant with increasing neutron dose is an interesting observation in view of the search for a microscopic 345
Volume 126, number 5,6
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Table 3 Numerical results for Ca and
11 January 1988
Py~.
Sample
Dose
KlO
E~0.3 MeV 1.68
2)
(1018 n/cm
Frequency
C l0—~ 1.2 1.4
Py~ 3) (lo~erg cm 1.1 1.2
Py~(average) (10~erg cm3)
(MHZ) 480 640 340 480 640
2.0 2.8 3.1
1.8 2.4 2.7
2.3
8 x
1.2
N 3
N4
12
340 480 640
4.1 4.4 4.7
3.6 3.9 4.1
3.8
N5
26
340 400 480 540 640
7.7 7.3 8.9 9.4 9.4
6.7 6.4 7.7 8.2 8.6
7.5
a-Si02 ~l
6.6
105
481
Calculated in a similar way from recent results in a-Si02 (ref. [15]).
description of the TLS. Quartz is known to amorphize gradually upon increasing irradiation with fast neutrons. A review of the effects of irradiation on quartz has been given in a previous paper [7]. For the doses used in the experiments in question the sample consist basically of two phases: a crystalline matrix and amorphous regions embedded in the matrix [17,18]. From mass density measurements performed on our samples [19] it is known that the crystalline matrix is still the major part of the samples: the mass density for the highest dose used has only decreased upon irradiation by 0.1% to be cornpared2)with about quartz 14% atbecomes saturating dose (2 x 1020 for which fully amorphous. n/cm For the doses used the amorphous islands are homogeneously spread over the crystal and the matrix has mainly the properties of the unirradiated crystal. Previously it has been put forward that the amorphous islands may be the location of the two-level systems [171, although more recent experiments suggested that this is not necessarily the case [20,21]. Our new data cannot give a decisive answer to this question, but add new elements to the discussion. Since the coupling between the phonons and the twolevel systems is only slightly different from that in vitreous silica we may state that the TLS have a similar nature in neutron irradiated quartz as in vitreous 346
silica. This has been suggested before [51. However since the coupling is not “the same” this suggests that the TLS are not “the same” and they change with neutron dose. It has been observed in X-ray studies that in the defected regions the configuration of the atoms changes with neutron dose [181. The different local arrangements of the atoms in the vicinity of the TLS could explain the change of the coupling of the TLS to the phonons with dose. In addition the amorphous islands in the crystal are highly cornpressed [181. Our findings therefore suggest that TLS in a dense environment couple less to phonons than if they are less dense environment. Since samples are in stilla mainly crystalline it can also notthe be ruled out that the coupling is anisotropic: the TLS may couple more in one crystal direction than in another. Our estimated values for y~and y, are mean values. It is known that the disordering process is different for the different crystal directions [17,18]. If evidence could be found of a directionality of the TLS this would help to come closer to a microscopic description of the two-level systems in neutron irradiated quartz and possibly in vitreous silica since the basic building units (the Si0 4 tetrehedra) in both forms of SiO2 are the same.
Volume 126, number 5,6
PHYSICS LETTERS A
5. Conclusion In this paper new data of the ultrasonic attenuation as a function of temperature are reported for neutron irradiated quartz. The main purpose of the paper however is to report the results of a detailed analysis of the data obtained in a systematic study of the ultrasonic attenuation as a function of temperature in neutron irradiated quartz for four neutron doses and five frequencies The neutron doses cover more than a decade. The data are analysed in term of the tunnelling model, put forward to explain the low temperature dynamic anomalies in glasses. The main result is the finding that the coupling parameterwhich describes the coupling ofthe TLS with the phonons, is smaller than in vitreous silica and increases with neutron dose approaching more and more the value for vitreous silica, ~.
Acknowledgement The authors thank SCKJCEN, Mol, Belgium for the irradiation and in particular J. Cornelis. They are also grateful to the Belgian 11KW for financial support. Interesting discussions with Professor S. Hunklinger are gratefullyacknowledged. One of us (V.E.) thanks INIC, Instituto National de Investigaçao Cientifica, Lisboa (Portugal) for a grant. They also thank Oak Ridge National Laboratory and especially R. Weeks for the neutron activation analysis. ~ The results and the conclusionsgiven in this paper are part of the Ph.D. thesis of one ofus [22].
References [I]P.W. Anderson, B.I. Halperin and C. Varma, PhIlos. Mag. 25 (1972) 1; W.A. Philips, J. Low Temp. Phys. 7 (1972) 351.
Il January 1988
[2] W.A. Philips, ed., Amorphous solids: low-temperature properties (Springer, Berlin, 1981). [3] S. Hunklinger and W. Arnold, in: Physical acoustics, Vol. 12, eds. W.P. Mason and R.N. Thurston (Academic Press, New York, 1976) p. 155—215. [4] C. Laermans, Phys. Rev. Lett. 42 (1979) 250. [5] B. Golding, J.E. Graebner, W. Haemmerle and C. Laermans, Bull. Am. Phys. Soc. 24 (1979) 495; B. Golding and J.E. Graebner, in: Phonon scattering in condensed matter, ed. H.J. Marts (Plenum, New York, 1980) ~. 1-20. [6] W. Gardner and A.C. Anderson, Phys. Rev. B 23 (1981) [7] C. Laermans, in: Structure and bonding in noncrystalline solids, eds. 0. Walrafen and A. Revesz (Plenum, New York, 1986) p. 329—367. [8] C. Laermans and V. Esteves, in: Phonon scattering in condensed matter, eds. W. Eisenmenger, K. Lassmann and S. Dottinger (Springer, Berlin, 1983) pp. 407—409. [9] V. Esteves, A. Vanelstraete and C. Laermans, in: Phonon physics, eds. J. Kollar, N. Kroo, N. Menyhard and T. Sikios (World Scientific, Singapore, 1985) p. 45. [10]J. JackIe, Z. Phys. 257 (1972) 212. [I 1] C. Laermans, V. Esteves and A. Vanelstraete, Radiat. Eff. 97 (1986) 175. [12] A. Venelstraete, C. Laermans, L. Lejan’aga, M. von Schickfus and J. AppI. Suppl. 26 (1987) 733; S. Z.Hunklinger, Phys. (Cond.Japan. Matter), to bePhys. published. [13] R. Vacher, J. Pelous, N. Elkhayati Elidnssi and M. Boissier, in: Phonon scattering in condensed matter, eds. A.C. Anderson and J.P. Wolfe (Springer, Berlin, 1987) p. 61. [14] J. Doulat, M. Locatelli and R. Rivallin, in: Phonon scattering in solids, eds. L.J. Challis et al. (Plenum, New York, 1976) ~,. 383. [15] S. Hunklinger, private communication. [16] J. Graebner and B. Golding, Phys. Rev. B 19 (1979) 964. [17] D. Grasse, 0. Kocar, H. Peisl, S.C. Moss and B. Golding, Phys. Rev. Lett. 46(1981)261. [18]R. Comes, M. Lambert and H. Guinier, in: Interaction of radiation in solids, ed. A. Bishay (Plenum, New York, 1967) p.319. [19]A. Vanelstraete and C. Laermans, internal report (1985). [20]A.C. Anderson, J.A. Mc. Millan andF.J. Walker, Phys. Rev. B24 (1981) 1124. [21]C. Laermans and A. Vanelstraete, Phys. Rev. B 35 (1987) 6399, (22] V. Esteves, Ph.D. Thesis, Leuven (6 October 1986), unpublished.
347