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The effect of heat treatment on the heat capacity of vitreous silica between 0.3 and 4.2 K
Journal of Non-Crystalline Solids 28 (1978) 67-76 © North-Holland Publishing Company
THE EFFECT OF HEAT TREATMENT ON THE HEAT CAPACITY OF VITREOUS SILICA BETWEEN 0.3 AND 4.2 K R.L. FAGALY * and R.H. BOHN Department of Physics and Astronomy, The University of Toledo, Toledo, Ohio 43606, USA Received 8 January 1977 Revised manuscript received 26 September 1977
The heat capacity of five samples of vitreous silica have been measured, using a modified heat-pulse method between 0.3 and 4.2 K. Each sample was subjected to a different heat treatment and the total heat capacity shows a small dependence on thermal history. Generally, the temperature dependence of the heat capacity can be described by an equation of the form: e = e 1T + c3 T3 + c5 TS. These results are also compared to earlier measurements.
1. Introduction Heat-capacity measurements [ 1 - 4 ] on vitreous silica below 20 K have shown heat capacities in excess of that predicted by the Debye model. Measurements between 2 and 20 K are often described by using a combination of Debye and Einstein models. Thus, the heat capacity can be written as
m
, [O'E)
C = CD T3 + ~ aiCl.z~T , i=1
where cD = 8.03 erg/g • K 4 (0o = 494 K) [5] and the summation involves a series of Einstein heat capacities with suitably chosen Einstein temperatures (0~). Usually three such terms can describe the data (m = 3). Below 2 K the data can be described by using
c :~CnV",
(1)
n
where el ~ 10 erg/g • K 2, c3 ~ 20 erg/g • K 4 and the other terms are small or zero. The linear term appears to be intrinsic to glasses and the model that appears to provide the best explanation is the tunneling model of Anderson et al. [6] and Phillips [7], hereafter referred to as AHV-P. The cubic term is larger than expected from the Debye model and might appear especially enhanced if there were higher* Present address: Physics Department, Illinois Institute of Technology, Chicago, Illinois 60616, USA. 67
68
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica
order terms in eq. (1) that were not included in the fit to the data. In particular, the effect of dispersion might have to be included. Leadbetter [8] has mentioned that there is very strong dispersion of the transverse acoustic mode (TA) branches in several simple glasses. This could lead to an apparent enhancement of the cubic term in the heat capacity. With this impetus, we have investigated the heat capacity of several samples of vitreous silica in the range 0 . 3 - 4 . 2 K in an effort simultaneously to determine values for cl, c3, and dispersive terms for a given sample. Each of the samples also had a different thermal history. The results for different samples were then compared in an effort to detect any differences that might be due to thermal history.
2. Experiment~
2.1. Sample preparation The heat capacities of five samples of Infrasil * glass were found over the temperature range 0 . 3 - 4 . 2 K. Infrasil is a type I glass produced from quartz by fusion under an inert atmosphere. According to Brtickner's classification [9], a type I glass has very low OH impurities but relatively higher metallic impurities. All five samples were given 1 h of heat treatment at 1300°C to remove any previous thermal history. Four of the samples were given additional heat treatment. All samples were air-quenched after the 1300°C heat treatment and after the final heat treatment. It might be noted that the densities for samples 1 - 4 are the same, to within the experimental uncertainty. Normally, one might expect differences in the density for samples stabilized at different temperatures. However, when one considers the shallow minimum in the volume versus temperature graph for the liquid at 1550°C that is characteristic of a type I silica glass [9], and the size of our uncertainties, it should not be too surprising that the densities are similar. Table 1 Heat treatment Sample number
Heat treatment
1 2 3 4 5a
1300°C 1300°C 1300°C 1300°C 1300°C
1h 1h 1h 1h 1h
Density (gm/cm 3) IO00°C 70 h llO0°C 22 h 1200°C 7 h
2.201 (+-0.007) 2.198 2.206
1400°C 5 min
2.213
a Sample five was given a short HF acid bath to remove any surface crystallinity that might have formed during the heat treatment at 1400°C. The higher density may indicate that some devitrification occurred. * Trademark of Amersil Quartz Division, Engelhard Industries, Inc.
R.L. Fagaly, R.G. Bohn / Heat capacity o]" vitreous silica
69
2.2. Cryostat
The samples rested on nylon pins mounted on an M-shaped sample holder (fig. 1). The sample holder, made of 0.08 cm thick copper, allowed two samples to be measured during a single experiment. Thermal contact between the sample holder and the mixing chamber of a dilution refrigerator was provided by seven 4-40 screws threaded into the mixing-chamber body. Apiezon "N" grease was used to aid thermal contact, both to the 4-40 threads and the mixing-chamber bottom. A sapphire wafer, indium soldered to the back of the sample holder, provided a ther-
VACUUM ISOLATION CAN
-0 CM
GRAPHITE SUPPORT ROD ~
SAMPLE !
~YLONP,Ng
HEAT ANGER
1
1
.,.
SAMPLE 2 u
SAIMPLE HOL'DER
Fig. 1. Experimental apparatus. The electrical leads are left out for clarity.
R.L. Fagaly,R.G. Bohn /Heat capacity of vitreous silica
70
mal sink for electrical leads. The thermal links that connect the sample and the mixing chamber are also attached to the sapphire wafer. A calibrated germanium resistor was mounted directly on to the sample holder. The sample holder and sample(s) were enclosed in an aluminium foil radiation shield prior to an experiment.
2.3. Experimental technique The technique used to measure the heat capacity has been discussed in detail elsewhere [10]. In brief, this is a modified heat-pulse technique in which the sample was thermally connected to a heat sink (a dilution refrigerator) by a small gold wire. The time constant of the thermal link ranged from 4 to 40 s. This technique allowed the heat capacity to be determined in the presence of an external (vibrational) heat leak. Uncertainties (oc) in the measured heat capacity are estimated as follows: T < I K , oc/C<0.04; 1 K < T < 0 . 4 K , oc/C<0.06; 0 . 4 K < T ,
oc/C<0.12.
3. Results After the heat capacities were found, the data were fit to a power series in T. Four forms: (I) clT+c3 T3, (II) clT+c2 T2 +c3 T3, (IlI) c IT+caT 3 +csT 5, (IV) Cl T + c2 T 2 + c3 T 3 + c5 T 5, were used to fit the data. An F-value [11] was used to test the "goodness of fit", and form III was preferred over form IV. As can be seen from fig. 2, there is very little difference between forms III and IV in describing the trend of the data. For these reasons, the entries listed in table 2 are the values associated with form III. Fig. 3 shows the general trend of the data for samples 1, 2, 4, and 5 with the data points being omitted for clarity. As can be seen in figs. 2 and 4 a fifth-order term is necessary properly to describe the data. Let us compare our results with those of other authors. If form I is used to describe our samples below 2K, cl ~ 10 erg/g" K 2 and c3 ~ 20 erg/g" K 4 which is in agreement with Zeller and Pohl [2]. Neither Zeller and Pohl nor Stephens [3] gave any information on the thermal history of their samples. The measurements of Flubacher et al. [1] on glass annealed at 1100°C showed similar behavior to our samples (fig. 4), however, no information on annealing times was given. As mentioned earlier, the coefficient of the T 3 term using the Debye model is 8.03 erg/g" K 4 based on elastic constants [5] and Brillouin scattering [1 ]. This im-
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica 100
io
I
i
I
i
'1
i
I
i
71
I
7O
\ 4a
\ (..)
20
t 0.4
I
1
I 0.7
i
I
[ 1
I 2
I
| 4
|
I
Fig. 2. Heat capacity of 1100°C heat treated Infrasil (sample 2) plotted as C/T 3 versus T. Here, I, II, III and IV are plots of the different possible fits to the data points. For clarity some data points which duplicate others are not shown.
plies a Debye temperature of 494 K. Actually, the temperature dependence of the form used to fit the experimental data will influence how much excess cubic (c3 - CD) heat capacity exists. For example, Flubacher et al. [1] were able to fit their data (taken between 2.3 and 19 K) by considering the excess heat capacity due to three optical (Einstein) modes and a small amount of dispersion (c s = 0.01 erg/g. K 6 and c7 = 7 × 10 - s erg/g-K8). This fit places essentially all the excess heat capacity in the three optical modes. If their data from 2.3 to 5 K is re-exam-
Table 2 Heat capacities. Sample number
cI (erg/g • K 2)
c3 (erg/g • K 4)
c5 (erg/g • K 6)
Temperature range
1 2 3a 4 5
10.2 13.8 10.4 9.2 16.8
13.2 12.3 21.5 13.4 11.5
0.90 0.89 0.55 0.80
0.37-4.2 K 0.40-4.2 K 0.65-1.66 K 0.33-3.3 K 0.46-4.2 K
(0.6) b (0.7) (0.6) (0.7) (1.6)
(0.4) (0.3) (0.6) (0.6) (0.8)
(0.03) (0.02) (0.06) (0.05)
a It should be noted that the data for sample 3 cover a temperature range from 0.65 to 1.66 K. Because of this, a value for c 5 could not be determined. Subsequently it has been ignored in the discussion. b The values in parentheses are the uncertainties of the coefficients derived from the fitting program [12].
72
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica
~1\ '.... s '\ •\\ \ " ".. ~ 2',.\\\ ' . ."..
, t ,,i ,.N
J
, t s 'i
t
"... \\
"'.
\ . -\: \ '~..N,. :::..'%.,
/~..'" S .<.<-.y
-,~. ,, '~,.._.'~,.
I 0,4
I
t
I 0.7
I
< ..,,..'./
I
I
I 2
1
I
I 4
I
Fig. 3. Heat capacity of Infrasil subjected to different heat treatments plotted as C/T 3 versus T. The curves are plots of the best (form III) fit to the data over the temperature range in which data was taken. Numbers refer to the samples listed in table 1.
1000
i
i
i
i
i Iii
I
i
i
i
i
i
iii
I
I L~lJl
I
100
\ o W1400'C 10 803
_ _
--Co
I
I
t
I t IIIJ
I 1
i
10
T (K) Fig. 4. The heat capacity of vitreous silica plotted as C/T 3 versus T. ZP are the results of ZeUer and Pohl [2]; S is the result of Stephens [3] ; W l 0 0 0 and W1400 are the results of White and Birch [4] for 1000°C and 1400°C respectively; and F the results of Flubacher et al. [1]. Here, c D is the Debye heat capacity (8.03 erg/g • K4).
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica
73
Table 3 Heat capacities. Reference Flubacher [ 1 ] Zeller [2] Stephens [3] White [4] 1000°C 1400°C
Form used
C1
C3
c5
(erg/g • K2)
(erg/g • K4)
(erg/g • K6)
Temperature range
III a I I Ill a III a
16 10 12 14 8.2
11.5 18 17.5 11.5 12.7
0.65 1.00 0.76
2.3-4.6 K 0.1-1 K 0.5-2 K 1.2-4.2 K 1.3-4.2 K
a The coefficients listed are the result of fits [ 12] to data in the cited reference.
ined, the heat capacity can also be expressed as C = C l T + c a T 3 + c s T s without resorting to optical modes. White and Birch [4] measured the heat capacity of IR Vitreosil *, a type I glass [9] that had been annealed for several hundred hours, one at 1000°C and the other at 1400°C. In comparing the results of White and Birch to the heat capacities of samples 1 and 5, there is good agreement near 4 K. Below 2 K, the data of White and Birch do not agree well with our data. However, it is possible that their data are not low enough to distinguish differences in the linear contribution to the heat capacity. Some difference in the 1400°C samples might be expected from the (slightly) different thermal histories. It can be seen from figs. 3 and 4 that the heat capacities of various samples of vitreous silica are similar. An average heat capacity could be used to describe the trend of the data and is given as C(erg/g. K) = [12 + 3] T + [12.5 -+ 1] T 3 + [0.8 + 0.15] T s .
(2)
As the effects of heat treatment appear to be small (<25%), chi-square (X2) tests [11] were performed to determine if there was any real change in the heat capacity from sample to sample. The ×2 test is based on the assumption that the best description of a set of data is one which minimizes the weighted sum of squares of deviations of the data from the fitting function, when the ×2 statistic is divided by the number of degrees of freedom, which is the difference between the number of data points and the number of fitting parameters, one obtains the reduced chisquare X2. It can be shown [ 11 ] that if the fitting function is a good approximation to the actual function that describes the data, then the reduced ×3 should be approximately 1. This indicates that the dispersion of the data about the fit is not due to inaccuracies of the fit. If ×2 is larger than 1, then tables can be used that indicate the probability of exceeding such a value of X2. In the following, the subscripts x a n d y in ×~y refer respectively to the data used * Trademark of Thermal Syndicate Limited, UK.
74
R.L. Fagaly, R.G. Bohn /Heat capacity of vitreous silica
Table 4 X2,y 2 x1,1 2 Xl,l+2 2 x2,2 2 ×2,1+2 2 x1+2,1+2
v
X2
X2/v
Confidence level
48
50.3
1.05
35%
48
62.1
1.29
5%
36
41.2
1.15
15%
36
74.5
2.07
<0.1%
87
136.6
1.57
0.1%
in the X2 test, and the least-squares fit to the particular set of data denoted by y. For example, X~, 1+2 means that the test was performed on the data of sample 1, using the best (least-squares) fit found by combining the data of samples 1 and 2. Here, X2 values are based on the experimental error estimates mentioned above, and are consistent from run to run. The confidence level is the probability that X2 , calculated for a random selection of data that are correctly described by a particular fit, will exceed the observed value of X2. If samples 1 and 2 are both collections of data that result from two measurements of identical samples, one would expect that a larger collection of data would give a smaller reduced chi-square X2/u, with u degrees o f freedom. We see that data from samples 1 and 2 taken separately can be fit by polynomials of the form c l T + c a T 3 + c s T 5 with reasonable residual errors as measured by a ×2 test. On the other hand, neither sample is well described by a fit based on the combined data for both samples, and in fact the fit based on the combined data when tested against a single sample would be rejected by a X2 test at the 1% confidence level. As a further check, the data of the various samples were tested against fits to various combinations of sample data. Table 5 shows the trend for samples 1 and 2. Similar results occur with the other samples. From the above results, it can be argued that the heat capacity does differ from sample to sample and that the trends seen in fig. 3, although small, do occur.
Table 5 2y Xx,
v
X2
2 X1,1+2+5 2 X1,1+2+3+4+5 2 ×2,1+2+5
48
48
2 X2,1+2+3+4+S
×2Iv
Confidence level
84.6
1.76
0.5%
131.1
2.73
<<0.1%
36
56.6
1.57
1.5%
36
92.1
2.56
<<0.1%
75
R.L. Fagaly, R.G. Bohn / Heat capacity o f vitreous silica IOC
1
I
\\
i
1 i i [
I
i
i
I
i
5
7¢
c~ 4c
\
N
N
'
~
~. 20
w14oo'c .....
[
I 0.4
T
I
I 0.7
I
I
//f"
?.:'~.'.':.'.'"
I
I
1
2
I
I
I
I
4
Fig. 5. Heat cal~acity of vitreous silica plotted as C/T 3 versus T. . . . . Infrasil 1000°C heat treatment- sample 1; - - I n f r a s i l 1400°C heat treatment- sample 5; . - . - . IR Vitreosil 1000°C heat treatment [4]; . . . . . . IR Vitreosil 1400°C heat treatment [41.
Additional support for the belief that heat treatment is the cause of the change in the heat capacity from sample to sample comes from the data of White and Birch [4]. Despite differences in the time spent at the treatment temperatures, their results show ordering similar to our results (fig. 5), although the temperatures achieved by White and Birch do not allow for the effect of the linear term to be dominant. We also see that in the region above 2 K, the data of White and Birch matches our data for the respective treatment temperatures.
4. Conclusion Summarizing, over the temperature range 0.1-4.2 K the heat capacity of vitreous silica can be described by eq. (2). The linear term is felt to be due to a tunneling process, as described by AHV-P. The magnitude of Cs would indicate that dispersion is very important in vitreous silica. Traditional methods of estimating the effect of dispersion [13] yield values of cs two to three orders of magnitude too small. Bilir and Phillips [14] have shown that crystoballite, a crystalline form of silica, shows large dispersion, and evidence to support the belief that vitreous silica has a similar dispersive nature has been given by Leadbetter [ 15]. The effect of heat treatment is to produce small changes (<25%) in the heat capacity. Increasing the fictive temperature tends to enhance the heat capacity below 2 K and to decrease the heat capacity above 2 K. Until more accurate data
76
R.L. Fagaly, R.G. Bohn / Heat capacity o f vitreous silica
are available, one cannot make specific comments on the effect of heat treatment on the individual terms in eq. (2).
Acknowledgement We would like to thank Steven Barber and the Owens-Illinois Technical Center for supplying and characterizing the samples.
References [1] P. Flubacher, A.J. Leadbetter, J.A. Morrison and B.P. Stoicheff, J. Phys. Chem. Solids 12 (1959) 53. [2] R.C. Zeller and R.O. Pohl, Phys. Rev. B4 (1971) 2029. [31 R.B. Stephens, Phys. Rev. B8 (1973) 2896; B13 (1976) 852. [4] G.K. White and J.A. Birch, Phys. Chem. Glasses 6 (1965) 85. [5] O.L. Anderson, J. Phys. Chem. Solids 12 (1959) 41. [6] P.W. Anderson, B.I. Halperin and C.M. Varma, Phil. Mag. 25 (1972) 1. [7] W.A. Phillips, J. LOW Temp. Phys. 7 (1972) 351. [8] A.J. Leadbetter, Proc. Int. Conf. on Phonon Scattering in Solids, Paris, 1972, ed. H.J. Albany (CEN, Saclay, 1972). [9] R. Briickner, J. Non-Crystalline Solids 5 (1970) 123. [10] R.L. Fagaly and R.G. Bohn, Rev. Sci. Instr. 48 (1977) 1502. [11] P:R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGrawHill, New York, 1969). [12] Hewlett-Packard HP 67/97 Users Library Program No. 00674D, Hewlett-Packard Corvallis Division, Corvallis, Oregon 97330 (1977). [13] A.J. Dekker, Solid State Physics (Prentice-Hall, New Jersey, 1957). [14] N. Bilir and W.A. Phillips, Phil. Mag. 32 (1975) 113. [15] A.J. Leadbetter, J. Chem. Phys. 51 (1969) 779.
THE EFFECT OF HEAT TREATMENT ON THE HEAT CAPACITY OF VITREOUS SILICA BETWEEN 0.3 AND 4.2 K R.L. FAGALY * and R.H. BOHN Department of Physics and Astronomy, The University of Toledo, Toledo, Ohio 43606, USA Received 8 January 1977 Revised manuscript received 26 September 1977
The heat capacity of five samples of vitreous silica have been measured, using a modified heat-pulse method between 0.3 and 4.2 K. Each sample was subjected to a different heat treatment and the total heat capacity shows a small dependence on thermal history. Generally, the temperature dependence of the heat capacity can be described by an equation of the form: e = e 1T + c3 T3 + c5 TS. These results are also compared to earlier measurements.
1. Introduction Heat-capacity measurements [ 1 - 4 ] on vitreous silica below 20 K have shown heat capacities in excess of that predicted by the Debye model. Measurements between 2 and 20 K are often described by using a combination of Debye and Einstein models. Thus, the heat capacity can be written as
m
, [O'E)
C = CD T3 + ~ aiCl.z~T , i=1
where cD = 8.03 erg/g • K 4 (0o = 494 K) [5] and the summation involves a series of Einstein heat capacities with suitably chosen Einstein temperatures (0~). Usually three such terms can describe the data (m = 3). Below 2 K the data can be described by using
c :~CnV",
(1)
n
where el ~ 10 erg/g • K 2, c3 ~ 20 erg/g • K 4 and the other terms are small or zero. The linear term appears to be intrinsic to glasses and the model that appears to provide the best explanation is the tunneling model of Anderson et al. [6] and Phillips [7], hereafter referred to as AHV-P. The cubic term is larger than expected from the Debye model and might appear especially enhanced if there were higher* Present address: Physics Department, Illinois Institute of Technology, Chicago, Illinois 60616, USA. 67
68
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica
order terms in eq. (1) that were not included in the fit to the data. In particular, the effect of dispersion might have to be included. Leadbetter [8] has mentioned that there is very strong dispersion of the transverse acoustic mode (TA) branches in several simple glasses. This could lead to an apparent enhancement of the cubic term in the heat capacity. With this impetus, we have investigated the heat capacity of several samples of vitreous silica in the range 0 . 3 - 4 . 2 K in an effort simultaneously to determine values for cl, c3, and dispersive terms for a given sample. Each of the samples also had a different thermal history. The results for different samples were then compared in an effort to detect any differences that might be due to thermal history.
2. Experiment~
2.1. Sample preparation The heat capacities of five samples of Infrasil * glass were found over the temperature range 0 . 3 - 4 . 2 K. Infrasil is a type I glass produced from quartz by fusion under an inert atmosphere. According to Brtickner's classification [9], a type I glass has very low OH impurities but relatively higher metallic impurities. All five samples were given 1 h of heat treatment at 1300°C to remove any previous thermal history. Four of the samples were given additional heat treatment. All samples were air-quenched after the 1300°C heat treatment and after the final heat treatment. It might be noted that the densities for samples 1 - 4 are the same, to within the experimental uncertainty. Normally, one might expect differences in the density for samples stabilized at different temperatures. However, when one considers the shallow minimum in the volume versus temperature graph for the liquid at 1550°C that is characteristic of a type I silica glass [9], and the size of our uncertainties, it should not be too surprising that the densities are similar. Table 1 Heat treatment Sample number
Heat treatment
1 2 3 4 5a
1300°C 1300°C 1300°C 1300°C 1300°C
1h 1h 1h 1h 1h
Density (gm/cm 3) IO00°C 70 h llO0°C 22 h 1200°C 7 h
2.201 (+-0.007) 2.198 2.206
1400°C 5 min
2.213
a Sample five was given a short HF acid bath to remove any surface crystallinity that might have formed during the heat treatment at 1400°C. The higher density may indicate that some devitrification occurred. * Trademark of Amersil Quartz Division, Engelhard Industries, Inc.
R.L. Fagaly, R.G. Bohn / Heat capacity o]" vitreous silica
69
2.2. Cryostat
The samples rested on nylon pins mounted on an M-shaped sample holder (fig. 1). The sample holder, made of 0.08 cm thick copper, allowed two samples to be measured during a single experiment. Thermal contact between the sample holder and the mixing chamber of a dilution refrigerator was provided by seven 4-40 screws threaded into the mixing-chamber body. Apiezon "N" grease was used to aid thermal contact, both to the 4-40 threads and the mixing-chamber bottom. A sapphire wafer, indium soldered to the back of the sample holder, provided a ther-
VACUUM ISOLATION CAN
-0 CM
GRAPHITE SUPPORT ROD ~
SAMPLE !
~YLONP,Ng
HEAT ANGER
1
1
.,.
SAMPLE 2 u
SAIMPLE HOL'DER
Fig. 1. Experimental apparatus. The electrical leads are left out for clarity.
R.L. Fagaly,R.G. Bohn /Heat capacity of vitreous silica
70
mal sink for electrical leads. The thermal links that connect the sample and the mixing chamber are also attached to the sapphire wafer. A calibrated germanium resistor was mounted directly on to the sample holder. The sample holder and sample(s) were enclosed in an aluminium foil radiation shield prior to an experiment.
2.3. Experimental technique The technique used to measure the heat capacity has been discussed in detail elsewhere [10]. In brief, this is a modified heat-pulse technique in which the sample was thermally connected to a heat sink (a dilution refrigerator) by a small gold wire. The time constant of the thermal link ranged from 4 to 40 s. This technique allowed the heat capacity to be determined in the presence of an external (vibrational) heat leak. Uncertainties (oc) in the measured heat capacity are estimated as follows: T < I K , oc/C<0.04; 1 K < T < 0 . 4 K , oc/C<0.06; 0 . 4 K < T ,
oc/C<0.12.
3. Results After the heat capacities were found, the data were fit to a power series in T. Four forms: (I) clT+c3 T3, (II) clT+c2 T2 +c3 T3, (IlI) c IT+caT 3 +csT 5, (IV) Cl T + c2 T 2 + c3 T 3 + c5 T 5, were used to fit the data. An F-value [11] was used to test the "goodness of fit", and form III was preferred over form IV. As can be seen from fig. 2, there is very little difference between forms III and IV in describing the trend of the data. For these reasons, the entries listed in table 2 are the values associated with form III. Fig. 3 shows the general trend of the data for samples 1, 2, 4, and 5 with the data points being omitted for clarity. As can be seen in figs. 2 and 4 a fifth-order term is necessary properly to describe the data. Let us compare our results with those of other authors. If form I is used to describe our samples below 2K, cl ~ 10 erg/g" K 2 and c3 ~ 20 erg/g" K 4 which is in agreement with Zeller and Pohl [2]. Neither Zeller and Pohl nor Stephens [3] gave any information on the thermal history of their samples. The measurements of Flubacher et al. [1] on glass annealed at 1100°C showed similar behavior to our samples (fig. 4), however, no information on annealing times was given. As mentioned earlier, the coefficient of the T 3 term using the Debye model is 8.03 erg/g" K 4 based on elastic constants [5] and Brillouin scattering [1 ]. This im-
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica 100
io
I
i
I
i
'1
i
I
i
71
I
7O
\ 4a
\ (..)
20
t 0.4
I
1
I 0.7
i
I
[ 1
I 2
I
| 4
|
I
Fig. 2. Heat capacity of 1100°C heat treated Infrasil (sample 2) plotted as C/T 3 versus T. Here, I, II, III and IV are plots of the different possible fits to the data points. For clarity some data points which duplicate others are not shown.
plies a Debye temperature of 494 K. Actually, the temperature dependence of the form used to fit the experimental data will influence how much excess cubic (c3 - CD) heat capacity exists. For example, Flubacher et al. [1] were able to fit their data (taken between 2.3 and 19 K) by considering the excess heat capacity due to three optical (Einstein) modes and a small amount of dispersion (c s = 0.01 erg/g. K 6 and c7 = 7 × 10 - s erg/g-K8). This fit places essentially all the excess heat capacity in the three optical modes. If their data from 2.3 to 5 K is re-exam-
Table 2 Heat capacities. Sample number
cI (erg/g • K 2)
c3 (erg/g • K 4)
c5 (erg/g • K 6)
Temperature range
1 2 3a 4 5
10.2 13.8 10.4 9.2 16.8
13.2 12.3 21.5 13.4 11.5
0.90 0.89 0.55 0.80
0.37-4.2 K 0.40-4.2 K 0.65-1.66 K 0.33-3.3 K 0.46-4.2 K
(0.6) b (0.7) (0.6) (0.7) (1.6)
(0.4) (0.3) (0.6) (0.6) (0.8)
(0.03) (0.02) (0.06) (0.05)
a It should be noted that the data for sample 3 cover a temperature range from 0.65 to 1.66 K. Because of this, a value for c 5 could not be determined. Subsequently it has been ignored in the discussion. b The values in parentheses are the uncertainties of the coefficients derived from the fitting program [12].
72
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica
~1\ '.... s '\ •\\ \ " ".. ~ 2',.\\\ ' . ."..
, t ,,i ,.N
J
, t s 'i
t
"... \\
"'.
\ . -\: \ '~..N,. :::..'%.,
/~..'" S .<.<-.y
-,~. ,, '~,.._.'~,.
I 0,4
I
t
I 0.7
I
< ..,,..'./
I
I
I 2
1
I
I 4
I
Fig. 3. Heat capacity of Infrasil subjected to different heat treatments plotted as C/T 3 versus T. The curves are plots of the best (form III) fit to the data over the temperature range in which data was taken. Numbers refer to the samples listed in table 1.
1000
i
i
i
i
i Iii
I
i
i
i
i
i
iii
I
I L~lJl
I
100
\ o W1400'C 10 803
_ _
--Co
I
I
t
I t IIIJ
I 1
i
10
T (K) Fig. 4. The heat capacity of vitreous silica plotted as C/T 3 versus T. ZP are the results of ZeUer and Pohl [2]; S is the result of Stephens [3] ; W l 0 0 0 and W1400 are the results of White and Birch [4] for 1000°C and 1400°C respectively; and F the results of Flubacher et al. [1]. Here, c D is the Debye heat capacity (8.03 erg/g • K4).
R.L. Fagaly, R.G. Bohn / Heat capacity of vitreous silica
73
Table 3 Heat capacities. Reference Flubacher [ 1 ] Zeller [2] Stephens [3] White [4] 1000°C 1400°C
Form used
C1
C3
c5
(erg/g • K2)
(erg/g • K4)
(erg/g • K6)
Temperature range
III a I I Ill a III a
16 10 12 14 8.2
11.5 18 17.5 11.5 12.7
0.65 1.00 0.76
2.3-4.6 K 0.1-1 K 0.5-2 K 1.2-4.2 K 1.3-4.2 K
a The coefficients listed are the result of fits [ 12] to data in the cited reference.
ined, the heat capacity can also be expressed as C = C l T + c a T 3 + c s T s without resorting to optical modes. White and Birch [4] measured the heat capacity of IR Vitreosil *, a type I glass [9] that had been annealed for several hundred hours, one at 1000°C and the other at 1400°C. In comparing the results of White and Birch to the heat capacities of samples 1 and 5, there is good agreement near 4 K. Below 2 K, the data of White and Birch do not agree well with our data. However, it is possible that their data are not low enough to distinguish differences in the linear contribution to the heat capacity. Some difference in the 1400°C samples might be expected from the (slightly) different thermal histories. It can be seen from figs. 3 and 4 that the heat capacities of various samples of vitreous silica are similar. An average heat capacity could be used to describe the trend of the data and is given as C(erg/g. K) = [12 + 3] T + [12.5 -+ 1] T 3 + [0.8 + 0.15] T s .
(2)
As the effects of heat treatment appear to be small (<25%), chi-square (X2) tests [11] were performed to determine if there was any real change in the heat capacity from sample to sample. The ×2 test is based on the assumption that the best description of a set of data is one which minimizes the weighted sum of squares of deviations of the data from the fitting function, when the ×2 statistic is divided by the number of degrees of freedom, which is the difference between the number of data points and the number of fitting parameters, one obtains the reduced chisquare X2. It can be shown [ 11 ] that if the fitting function is a good approximation to the actual function that describes the data, then the reduced ×3 should be approximately 1. This indicates that the dispersion of the data about the fit is not due to inaccuracies of the fit. If ×2 is larger than 1, then tables can be used that indicate the probability of exceeding such a value of X2. In the following, the subscripts x a n d y in ×~y refer respectively to the data used * Trademark of Thermal Syndicate Limited, UK.
74
R.L. Fagaly, R.G. Bohn /Heat capacity of vitreous silica
Table 4 X2,y 2 x1,1 2 Xl,l+2 2 x2,2 2 ×2,1+2 2 x1+2,1+2
v
X2
X2/v
Confidence level
48
50.3
1.05
35%
48
62.1
1.29
5%
36
41.2
1.15
15%
36
74.5
2.07
<0.1%
87
136.6
1.57
0.1%
in the X2 test, and the least-squares fit to the particular set of data denoted by y. For example, X~, 1+2 means that the test was performed on the data of sample 1, using the best (least-squares) fit found by combining the data of samples 1 and 2. Here, X2 values are based on the experimental error estimates mentioned above, and are consistent from run to run. The confidence level is the probability that X2 , calculated for a random selection of data that are correctly described by a particular fit, will exceed the observed value of X2. If samples 1 and 2 are both collections of data that result from two measurements of identical samples, one would expect that a larger collection of data would give a smaller reduced chi-square X2/u, with u degrees o f freedom. We see that data from samples 1 and 2 taken separately can be fit by polynomials of the form c l T + c a T 3 + c s T 5 with reasonable residual errors as measured by a ×2 test. On the other hand, neither sample is well described by a fit based on the combined data for both samples, and in fact the fit based on the combined data when tested against a single sample would be rejected by a X2 test at the 1% confidence level. As a further check, the data of the various samples were tested against fits to various combinations of sample data. Table 5 shows the trend for samples 1 and 2. Similar results occur with the other samples. From the above results, it can be argued that the heat capacity does differ from sample to sample and that the trends seen in fig. 3, although small, do occur.
Table 5 2y Xx,
v
X2
2 X1,1+2+5 2 X1,1+2+3+4+5 2 ×2,1+2+5
48
48
2 X2,1+2+3+4+S
×2Iv
Confidence level
84.6
1.76
0.5%
131.1
2.73
<<0.1%
36
56.6
1.57
1.5%
36
92.1
2.56
<<0.1%
75
R.L. Fagaly, R.G. Bohn / Heat capacity o f vitreous silica IOC
1
I
\\
i
1 i i [
I
i
i
I
i
5
7¢
c~ 4c
\
N
N
'
~
~. 20
w14oo'c .....
[
I 0.4
T
I
I 0.7
I
I
//f"
?.:'~.'.':.'.'"
I
I
1
2
I
I
I
I
4
Fig. 5. Heat cal~acity of vitreous silica plotted as C/T 3 versus T. . . . . Infrasil 1000°C heat treatment- sample 1; - - I n f r a s i l 1400°C heat treatment- sample 5; . - . - . IR Vitreosil 1000°C heat treatment [4]; . . . . . . IR Vitreosil 1400°C heat treatment [41.
Additional support for the belief that heat treatment is the cause of the change in the heat capacity from sample to sample comes from the data of White and Birch [4]. Despite differences in the time spent at the treatment temperatures, their results show ordering similar to our results (fig. 5), although the temperatures achieved by White and Birch do not allow for the effect of the linear term to be dominant. We also see that in the region above 2 K, the data of White and Birch matches our data for the respective treatment temperatures.
4. Conclusion Summarizing, over the temperature range 0.1-4.2 K the heat capacity of vitreous silica can be described by eq. (2). The linear term is felt to be due to a tunneling process, as described by AHV-P. The magnitude of Cs would indicate that dispersion is very important in vitreous silica. Traditional methods of estimating the effect of dispersion [13] yield values of cs two to three orders of magnitude too small. Bilir and Phillips [14] have shown that crystoballite, a crystalline form of silica, shows large dispersion, and evidence to support the belief that vitreous silica has a similar dispersive nature has been given by Leadbetter [ 15]. The effect of heat treatment is to produce small changes (<25%) in the heat capacity. Increasing the fictive temperature tends to enhance the heat capacity below 2 K and to decrease the heat capacity above 2 K. Until more accurate data
76
R.L. Fagaly, R.G. Bohn / Heat capacity o f vitreous silica
are available, one cannot make specific comments on the effect of heat treatment on the individual terms in eq. (2).
Acknowledgement We would like to thank Steven Barber and the Owens-Illinois Technical Center for supplying and characterizing the samples.
References [1] P. Flubacher, A.J. Leadbetter, J.A. Morrison and B.P. Stoicheff, J. Phys. Chem. Solids 12 (1959) 53. [2] R.C. Zeller and R.O. Pohl, Phys. Rev. B4 (1971) 2029. [31 R.B. Stephens, Phys. Rev. B8 (1973) 2896; B13 (1976) 852. [4] G.K. White and J.A. Birch, Phys. Chem. Glasses 6 (1965) 85. [5] O.L. Anderson, J. Phys. Chem. Solids 12 (1959) 41. [6] P.W. Anderson, B.I. Halperin and C.M. Varma, Phil. Mag. 25 (1972) 1. [7] W.A. Phillips, J. LOW Temp. Phys. 7 (1972) 351. [8] A.J. Leadbetter, Proc. Int. Conf. on Phonon Scattering in Solids, Paris, 1972, ed. H.J. Albany (CEN, Saclay, 1972). [9] R. Briickner, J. Non-Crystalline Solids 5 (1970) 123. [10] R.L. Fagaly and R.G. Bohn, Rev. Sci. Instr. 48 (1977) 1502. [11] P:R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGrawHill, New York, 1969). [12] Hewlett-Packard HP 67/97 Users Library Program No. 00674D, Hewlett-Packard Corvallis Division, Corvallis, Oregon 97330 (1977). [13] A.J. Dekker, Solid State Physics (Prentice-Hall, New Jersey, 1957). [14] N. Bilir and W.A. Phillips, Phil. Mag. 32 (1975) 113. [15] A.J. Leadbetter, J. Chem. Phys. 51 (1969) 779.