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Low temperature thermal expansion of soda-borate glasses
Solid State Communications, Printed in Great Britain.
Vol. 48, No. 2, pp. 143-145,
LOW TEMPERATURE
0038-1098/83 $3.00 + .OO Pergamon Press Ltd.
1983.
THERMAL EXPANSION OF SODA-BORATE
GLASSES
Ester S. Piriango, S. Vieira and R. Villar Departamento
de Fisica Fundamental,
Universidad
Autdnoma
de Madrid, Canto Blanco, Madrid-34, Spain
(Received 17 April 1983 by M. Cardona) The thermal expansion of glassy (BzOs)l_x(NazO)x, for x = 0.06,0.16 and 0.25 has been measured in the temperature range 4 K < T < 20 K. The results are analysed in terms of a polynomial (II= aT + bT3 + CT’ + dT’ and the values of the coefficients are discussed. The linear term a is small and positive in the three glasses. This yields a small and positive Griineisen parameter for the two level systems. The cubic term is negative and is not affected by change in coordination, phonon dispersion being responsible for the fast increase in thermal expansion on increasing the temperature. thermal expansion
TETRAHEDRAL COORDINATION in solids plays an important role in the behaviour of the thermal expansion coefficient OLat low temperatures, as has been stressed by several authors [ 1, 21. This coefficient becomes negative below a certain temperature for nearly all amorphous as well as crystalline materials, whose atoms display a tetrahedral coordination [3,4]. For example, vitreous silica has a negative expansivity up to 1.50 K. Bz03 has a triangular coordination and following experiments of Krause and Kurkjian [2] and Villar [5] shows a positive and relatively high expansivity for T > 4 K. It is well known that addition of NazO to boron oxide alters the coordination, which gradually changes from triangular to tetrahedral [6]. If x is the sodium oxide molar concentration, the percent fraction of boron atoms tetrahedrally coordinated to oxygen atoms is given by the relation [6]: N=X 1 -x’
coefficient
(Ywould follow a law:
(Y = aT3+ bT5. This is the usual behaviour in a dielectric crystal at low temperatures where the only contribution is the anharmonicity of acoustic phonons. It can be observed in the figures that in our case the temperature dependence is more complex. Several features should be pointed out. First of all, the expansivity becomes negative within the range of our measurements. Second, for the samples with x = 0.16 and x = 0.25 another contribution with a lower power law than T3 is visible at the lowest temperatures. We have analysed our results with least squares tits to several expressions suggested by the theory, with a Minuit-CERN minimum search program. The smoothest fit of our results, following the criteria of the minimization program, is a polynomial of the form:
(1)
We have measured the thermal expansion of a series of borate glasses to analyse the effect of coordination on this property. Here we present the results for x = 0.06, 0.16 and 0.25. The samples were prepared from raw products Merck pro-analysis Bz03 and C03Na2. The experimental method and the analysis of results have been previously described [7] and are based in the precise measurement of capacitance changes first developed by White [8]. The samples were cut as rectangular prisms about 35 mm long in the direction of measurement with polished bases flat and parallel. A germanium calibrated thermometer to measure temperature and a manganin heater and carbon sensor for temperature control were assembled on each sample. The results are shown in Fig. 1 where we present a/T3 vs T2. This representation would produce a straight line if the
(Y = aT+bT3+cT5+dT7. There are other facets which support this fit in comparison with other trials, for example with an Einstein term instead of the T5 and T’ terms, and others. First, with this fit there is a correlation in the coefficients for the three samples measured. This correlation shows up in the following features: The linear term is small and positive for the three samples; the cubic term, which is negative, decreases in absolute value with increasing sodium oxide concentration; the same decreasing trend is presented by the TS term and even by the T’ term though we do not consider the latter very significant. This correlation between the coeffcients of the three glasses disappears with other fits. Second, the value of the coefficients in this fit does not change substantially when the temperature range of experimental points taken for the tit is changed.
143
144
LOW TEMPERATURE
THERMAL EXPANSION OF SODA-BORATE
GLASSES
Vol. 48. No. 2
3-100
0
50
ZOO
150
100 TEMPERATURE
TEMPERATURE
T2 [K21
Fig. 1. Thermal expansion of (Bz03)r_,(Naz0), plotted as (u/T3 vs T2, but with different ordinate scales to allow for very different expansivities at low temperatures. (*) (BzOa)w(Na20).5, (+) (B,O&(Na2O)i6 and (0) (B203),s(Naz0),,. in this plot a departure from a straight line means a departure from a bT3 + CT’ temperature dependence of the expansivity. The results of these fits are presented in Table 1, and we present also, in Fig. 2, the experimental data with the fitted curve up to 7 K. The existence of a linear temperature dependence in the specific heat and thermal expansion of insulating glasses is associated to other peculiarties in the propagation of heat, sound and light. The most important model was introduced by Anderson, Halperin and Varma [lo] and Phillips [ 111 and it allows for a wide explanation of experiments. Its central assumption is the existence of atoms or atom groups which may tunnel to neighbouring positions (two level systems). It is also necessary to assume an adequate distribution of barrier heights and of asymmetries between energy minima in order to explain the experiments, though some inconsistences remain which require supplementary assumptions [ 11, 121. Lyon et al. [ 131 have analysed in terms of the tunneling model the low temperature thermal expansion of Si02 and PMMA (polymethylmethacrylate) and suggest that the coefficient of the linear term may be negative for all glasses, giving rise to a high and negative Griineisen parameter for the two level systems. Recently Anderson and Ackerman [14] reported measurements of thermal
T2 lK21
Fig. 2. Experimental data of same three glasses plotted as a/T vs T2 for T < 7 K. The curves are the fitted curves to the data with coefficients of Table 1. Note ordinary scale to the right for (t) (B203)&Na20)r6, (*) i6 and (0) (B203),5(Na20)2s. In this plot (BzO&(NazO) a straight line means a aT + bT temperature dependence and the T = 0 extrapolation of the curve yields the linear term. expansion of several glasses for T < 1 K (Si02, AsZS3, PMMA and SC5) where the linear term was either high or low, positive or negative for different systems, that is, this parameter does not show a uniform behaviour in glasses. Our results support the above conclusion. The evidence of a positive linear term is clear in the samples x = 0.16 and x = 0.25. The specific heat of these glasses at low temperatures has not been measured and therefore we cannot calculate the Griineisen parameter 7~~s of the two level systems, which can be obtained as 3aV YTLS
=
kc'
where V is the molar volume, a is the coefficient of the T term in the expansion of the linear thermal expansion, k is the adiabatic compressibility and c is the coefficient of the linear contribution to the specific heat at constant pressure. Though the lowest temperature in our experiments is 4 K, in a temperature scale relative to the Debye T/0, temperature, it is not so high, because with the sound velocity data of Krause and Kurkjian [B] we may compute the following Debye temperatures,
Table 1. a, b, c and d are the coefficients in the fit CY= aT + bT3 + CT’ + dT’for the thermal expansion. n is the percent fraction of tetrahedrally coordinated boron atoms, after relation (1) and BDis the Debye temperature calculated using the sound velocity measurements of Krause and Kurkjian [9]
(B2WdNa20)6 (B2O&(Na20)16 (B203)&a20)25
a
b
IO-‘OK-2
lo-11
4.8 7.1 3.2
-12.5 -7.6 -1.7
C K-4
d
10-13K+
lo-‘4K-8
40.1 7.5 2.1
-1.5 -0.2 -0.04
6 19 33
326 414 472
Vol. 48, No. 2
LOW TEMPERATURE
THERMAL EXPANSION OF SODA-BORATE
increasing with x : &, (x = 0.06)
= 326 K,
&,(x=0.16)
=414K,
Bu (x = 0.25)
= 472 K.
A T3 contribution to the specific heat in SiOz and other glasses in excess to the T3 term calculated from elastic data has been proved. Loponen et al. [ 151 have measured the specific heat in the microsecond time scale and conclude that this excess T3 term may be of a different origin than the linear term. A remarkable discrepancy is also observed between the cubic term coefficient measured in the thermal expansion of SiOz and the one calculated from the pressure dependence of sound velocities. In Table 1 the T3 term is negative and decreases in absolute value for increasing x. The T5 term is responsible for the fast growing of thermal expansion in the sample for x = 0.06 and decreases with increasing x. This T5 term is associated to dispersion in the acoustic phonon branches in a similar way as for the specific heat. These results suggest that the fast change in expansivity reported by Krause and Kurkjian [2] and Villar [S] for Bz03 is due to phonon dispersion rather than to a high T3 term. In conclusion, for the low temperature thermal expansion of (B,03)r_,(Na0), (x = 0.6,0.16 and 0.25) a positive linear term is observed at the lowest temperatures not strongly dependent on x, supporting the results of Ackermann and Anderson that the Griineisen parameter of the two level systems is not high and negative for all glasses. The T3 and T5 terms seem to
GLASSES
145
show a strong dependence on x, T3 being always negative though no relation seems to exist between this sign and tetrahedral coordination. Acknowledgements - This work was partially supported by the Spain-USA Joint Research Program. REFERENCES 1. 2. 3. 4. 5. 6.
7.
::
10. 11. 12. 13. 14. 15.
G.K. White,Phys Rev. Lett. 34,204 (1975). J.T. Krause & C.R. Kurkiian, J. Amer. Ceram. Sot. 51,226(1968). a ’ P.W. Sparks & C.A. Sweson, Phys. Rev. 163,779 (1967j. T.F. Smith & G.K. White, J. Phys. C&203 1 (1975). R. Villar, PhD Thesis, Universidad Autonoma de Madrid (unpublished, 1978). D.L. Griscom, Borate Glasses, (Edited by L.D. Pye, V.D. Frechette & N.J. Kreidl), Vol. 12, p. 38. Plenum, New York (1978). R. Villar, M. Hortal & S. Vieira, Rev. Sci. Znstr. 51, 27 (1980). G.K. White, 0yogenics III, 151 (1961). J.T. Kranse & C.R. Kurkjian, Borate Glasses, (Edited by L.D. Pye, V.D. Frechette & N. J. Kreidl), Vol. 12, p. 577. Plenum, New York (1978). P.W. Anderson, B.I. Halperin & C.M. Varma, Phil. Msg. 25,1 (1972). W.A. Phillips, J. Low Temp. Phys. 7,351 (1972). J.L. Black,Phys Rev. Sly, 2740 (1978). . K.G. Lyon, G.L. Salinger & C.A. Swenson, Phys. Rev. B19,4231 (1979). D.A. Ackermann & A.C. Anderson, Phys Rev. Lett. 49, 1176 (1982). M.T. Loponen, R.C. Dynes, V. Narayananiurti & J.P. Garno, Phys Rev. B25,1161 (1982).
Vol. 48, No. 2, pp. 143-145,
LOW TEMPERATURE
0038-1098/83 $3.00 + .OO Pergamon Press Ltd.
1983.
THERMAL EXPANSION OF SODA-BORATE
GLASSES
Ester S. Piriango, S. Vieira and R. Villar Departamento
de Fisica Fundamental,
Universidad
Autdnoma
de Madrid, Canto Blanco, Madrid-34, Spain
(Received 17 April 1983 by M. Cardona) The thermal expansion of glassy (BzOs)l_x(NazO)x, for x = 0.06,0.16 and 0.25 has been measured in the temperature range 4 K < T < 20 K. The results are analysed in terms of a polynomial (II= aT + bT3 + CT’ + dT’ and the values of the coefficients are discussed. The linear term a is small and positive in the three glasses. This yields a small and positive Griineisen parameter for the two level systems. The cubic term is negative and is not affected by change in coordination, phonon dispersion being responsible for the fast increase in thermal expansion on increasing the temperature. thermal expansion
TETRAHEDRAL COORDINATION in solids plays an important role in the behaviour of the thermal expansion coefficient OLat low temperatures, as has been stressed by several authors [ 1, 21. This coefficient becomes negative below a certain temperature for nearly all amorphous as well as crystalline materials, whose atoms display a tetrahedral coordination [3,4]. For example, vitreous silica has a negative expansivity up to 1.50 K. Bz03 has a triangular coordination and following experiments of Krause and Kurkjian [2] and Villar [5] shows a positive and relatively high expansivity for T > 4 K. It is well known that addition of NazO to boron oxide alters the coordination, which gradually changes from triangular to tetrahedral [6]. If x is the sodium oxide molar concentration, the percent fraction of boron atoms tetrahedrally coordinated to oxygen atoms is given by the relation [6]: N=X 1 -x’
coefficient
(Ywould follow a law:
(Y = aT3+ bT5. This is the usual behaviour in a dielectric crystal at low temperatures where the only contribution is the anharmonicity of acoustic phonons. It can be observed in the figures that in our case the temperature dependence is more complex. Several features should be pointed out. First of all, the expansivity becomes negative within the range of our measurements. Second, for the samples with x = 0.16 and x = 0.25 another contribution with a lower power law than T3 is visible at the lowest temperatures. We have analysed our results with least squares tits to several expressions suggested by the theory, with a Minuit-CERN minimum search program. The smoothest fit of our results, following the criteria of the minimization program, is a polynomial of the form:
(1)
We have measured the thermal expansion of a series of borate glasses to analyse the effect of coordination on this property. Here we present the results for x = 0.06, 0.16 and 0.25. The samples were prepared from raw products Merck pro-analysis Bz03 and C03Na2. The experimental method and the analysis of results have been previously described [7] and are based in the precise measurement of capacitance changes first developed by White [8]. The samples were cut as rectangular prisms about 35 mm long in the direction of measurement with polished bases flat and parallel. A germanium calibrated thermometer to measure temperature and a manganin heater and carbon sensor for temperature control were assembled on each sample. The results are shown in Fig. 1 where we present a/T3 vs T2. This representation would produce a straight line if the
(Y = aT+bT3+cT5+dT7. There are other facets which support this fit in comparison with other trials, for example with an Einstein term instead of the T5 and T’ terms, and others. First, with this fit there is a correlation in the coefficients for the three samples measured. This correlation shows up in the following features: The linear term is small and positive for the three samples; the cubic term, which is negative, decreases in absolute value with increasing sodium oxide concentration; the same decreasing trend is presented by the TS term and even by the T’ term though we do not consider the latter very significant. This correlation between the coeffcients of the three glasses disappears with other fits. Second, the value of the coefficients in this fit does not change substantially when the temperature range of experimental points taken for the tit is changed.
143
144
LOW TEMPERATURE
THERMAL EXPANSION OF SODA-BORATE
GLASSES
Vol. 48. No. 2
3-100
0
50
ZOO
150
100 TEMPERATURE
TEMPERATURE
T2 [K21
Fig. 1. Thermal expansion of (Bz03)r_,(Naz0), plotted as (u/T3 vs T2, but with different ordinate scales to allow for very different expansivities at low temperatures. (*) (BzOa)w(Na20).5, (+) (B,O&(Na2O)i6 and (0) (B203),s(Naz0),,. in this plot a departure from a straight line means a departure from a bT3 + CT’ temperature dependence of the expansivity. The results of these fits are presented in Table 1, and we present also, in Fig. 2, the experimental data with the fitted curve up to 7 K. The existence of a linear temperature dependence in the specific heat and thermal expansion of insulating glasses is associated to other peculiarties in the propagation of heat, sound and light. The most important model was introduced by Anderson, Halperin and Varma [lo] and Phillips [ 111 and it allows for a wide explanation of experiments. Its central assumption is the existence of atoms or atom groups which may tunnel to neighbouring positions (two level systems). It is also necessary to assume an adequate distribution of barrier heights and of asymmetries between energy minima in order to explain the experiments, though some inconsistences remain which require supplementary assumptions [ 11, 121. Lyon et al. [ 131 have analysed in terms of the tunneling model the low temperature thermal expansion of Si02 and PMMA (polymethylmethacrylate) and suggest that the coefficient of the linear term may be negative for all glasses, giving rise to a high and negative Griineisen parameter for the two level systems. Recently Anderson and Ackerman [14] reported measurements of thermal
T2 lK21
Fig. 2. Experimental data of same three glasses plotted as a/T vs T2 for T < 7 K. The curves are the fitted curves to the data with coefficients of Table 1. Note ordinary scale to the right for (t) (B203)&Na20)r6, (*) i6 and (0) (B203),5(Na20)2s. In this plot (BzO&(NazO) a straight line means a aT + bT temperature dependence and the T = 0 extrapolation of the curve yields the linear term. expansion of several glasses for T < 1 K (Si02, AsZS3, PMMA and SC5) where the linear term was either high or low, positive or negative for different systems, that is, this parameter does not show a uniform behaviour in glasses. Our results support the above conclusion. The evidence of a positive linear term is clear in the samples x = 0.16 and x = 0.25. The specific heat of these glasses at low temperatures has not been measured and therefore we cannot calculate the Griineisen parameter 7~~s of the two level systems, which can be obtained as 3aV YTLS
=
kc'
where V is the molar volume, a is the coefficient of the T term in the expansion of the linear thermal expansion, k is the adiabatic compressibility and c is the coefficient of the linear contribution to the specific heat at constant pressure. Though the lowest temperature in our experiments is 4 K, in a temperature scale relative to the Debye T/0, temperature, it is not so high, because with the sound velocity data of Krause and Kurkjian [B] we may compute the following Debye temperatures,
Table 1. a, b, c and d are the coefficients in the fit CY= aT + bT3 + CT’ + dT’for the thermal expansion. n is the percent fraction of tetrahedrally coordinated boron atoms, after relation (1) and BDis the Debye temperature calculated using the sound velocity measurements of Krause and Kurkjian [9]
(B2WdNa20)6 (B2O&(Na20)16 (B203)&a20)25
a
b
IO-‘OK-2
lo-11
4.8 7.1 3.2
-12.5 -7.6 -1.7
C K-4
d
10-13K+
lo-‘4K-8
40.1 7.5 2.1
-1.5 -0.2 -0.04
6 19 33
326 414 472
Vol. 48, No. 2
LOW TEMPERATURE
THERMAL EXPANSION OF SODA-BORATE
increasing with x : &, (x = 0.06)
= 326 K,
&,(x=0.16)
=414K,
Bu (x = 0.25)
= 472 K.
A T3 contribution to the specific heat in SiOz and other glasses in excess to the T3 term calculated from elastic data has been proved. Loponen et al. [ 151 have measured the specific heat in the microsecond time scale and conclude that this excess T3 term may be of a different origin than the linear term. A remarkable discrepancy is also observed between the cubic term coefficient measured in the thermal expansion of SiOz and the one calculated from the pressure dependence of sound velocities. In Table 1 the T3 term is negative and decreases in absolute value for increasing x. The T5 term is responsible for the fast growing of thermal expansion in the sample for x = 0.06 and decreases with increasing x. This T5 term is associated to dispersion in the acoustic phonon branches in a similar way as for the specific heat. These results suggest that the fast change in expansivity reported by Krause and Kurkjian [2] and Villar [S] for Bz03 is due to phonon dispersion rather than to a high T3 term. In conclusion, for the low temperature thermal expansion of (B,03)r_,(Na0), (x = 0.6,0.16 and 0.25) a positive linear term is observed at the lowest temperatures not strongly dependent on x, supporting the results of Ackermann and Anderson that the Griineisen parameter of the two level systems is not high and negative for all glasses. The T3 and T5 terms seem to
GLASSES
145
show a strong dependence on x, T3 being always negative though no relation seems to exist between this sign and tetrahedral coordination. Acknowledgements - This work was partially supported by the Spain-USA Joint Research Program. REFERENCES 1. 2. 3. 4. 5. 6.
7.
::
10. 11. 12. 13. 14. 15.
G.K. White,Phys Rev. Lett. 34,204 (1975). J.T. Krause & C.R. Kurkiian, J. Amer. Ceram. Sot. 51,226(1968). a ’ P.W. Sparks & C.A. Sweson, Phys. Rev. 163,779 (1967j. T.F. Smith & G.K. White, J. Phys. C&203 1 (1975). R. Villar, PhD Thesis, Universidad Autonoma de Madrid (unpublished, 1978). D.L. Griscom, Borate Glasses, (Edited by L.D. Pye, V.D. Frechette & N.J. Kreidl), Vol. 12, p. 38. Plenum, New York (1978). R. Villar, M. Hortal & S. Vieira, Rev. Sci. Znstr. 51, 27 (1980). G.K. White, 0yogenics III, 151 (1961). J.T. Kranse & C.R. Kurkjian, Borate Glasses, (Edited by L.D. Pye, V.D. Frechette & N. J. Kreidl), Vol. 12, p. 577. Plenum, New York (1978). P.W. Anderson, B.I. Halperin & C.M. Varma, Phil. Msg. 25,1 (1972). W.A. Phillips, J. Low Temp. Phys. 7,351 (1972). J.L. Black,Phys Rev. Sly, 2740 (1978). . K.G. Lyon, G.L. Salinger & C.A. Swenson, Phys. Rev. B19,4231 (1979). D.A. Ackermann & A.C. Anderson, Phys Rev. Lett. 49, 1176 (1982). M.T. Loponen, R.C. Dynes, V. Narayananiurti & J.P. Garno, Phys Rev. B25,1161 (1982).