- Email: [email protected]
Dielectric behaviour of Nd ions in the lead borate glass
JOURNA L O F
ELSEVIER
Journal of Non-Crystalline Solids 172-174 (1994) 1324-1327
Dielectric behaviour of Nd ions in the lead borate glass Luis Cadillon Costa*, Sushil Kumar Mendiratta Departamento de Fisica, Universidadede Aveiro, 3800 Aveiro, Portugal
Abstract Accurate measurements of the relaxation in the time domain show that the low frequency relaxation processes in glasses are well described by the Williams-Watts (WW) relation. Measurements of the complex permittivity in the frequency domain have been often claimed, on the other hand, to satisfy the Cole-Davidson (CD) relation, e*= eo~ + [(Co- e~o)/(1 +jo)z)'], 0 < ~. In order to check with what accuracy the systems whose time domain behaviour is given by WW relation also satisfy the CD equation in the frequency domain, frequency measurements of the glasses in the frequency region 1 Hz-10 GHz are taken using different techniques. Absorption was observed in two different frequency regions and they are well described by the two relations. The temperature dependence of relaxation time parameters was also studied in order to characterize the physics of the relaxation process. The dependence of the parameters on the concentration of Nd sheds some light on the state of aggregation of the Nd ions in the glass.
1. Introduction Dielectric properties of disordered materials have been investigated for a long period either from the point of view of characterizing the material for applications or from the point of view of characterizing the dominant polar species which relaxes. However, there is another interesting aspect of the problem that is not explored so often. The parameters that characterize a measured time response or frequency response contain information not only about the relaxing species but also about the structure around the microscopic relaxing unit in which the actual relaxation takes place. In our previous
* Corresponding author. Tel: + 351-34 370 200. Telefax: + 35134 24 965.
studies of time response [ 1] on a series of Gd-based lead borate glasses, we had shown how the variation of the parameters in the stretched exponential response, also referred as K o h l r a u s c h Williams-Watts (KWW), q~ = ¢o exp( - t/z) p, with the concentration of the modifier ion can be correlated with the structural changes inferred from other independent techniques. In this paper, we show how the high frequency response measurements (1 H z - 1 0 G H z ) fit the Cole-Davidson (CD) formula e * = e ~ + [ ( e o -e~)/(l+jogz)~]. The same samples were submitted to time response and we try to find correlation between the high frequency parameters, especially z and c
0022-3093/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0 0 2 2 - 3 0 9 3 ( 9 4 ) 0 0 0 8 2 - X
L.C. Costa, S.K. Mendiratta / Journal of Non-Crystalline Solids 172 174 (1994) 1324 1327
2. Experimental procedures Time domain measurements in the seconds range were performed in a fashion similar to already reported [1]. Samples were of thickness 1 mm and diameter 10mm and the polarizing field was 100 kV/m. Measurements were performed at various temperatures between 80 and 300 K. In the frequency domain, different techniques were used in different regions. For frequencies from 1 Hz to 100 kHz, we used a sensitive lock-in amplifier to measure in-phase and out-of-phase current to give real and imaginary parts of complex dielectric constant. The absolute calibration in this range was done by 1 kHz measurements on a six-digit precision capacitance bridge. Measurements up to 1 MHz were done with the help of a precision H P capacitance bridge. The range 10 M H z - 1 G H z was covered by measurements with an rf impedance meter in which the amplitude of the incident and reflected wave is measured and a 50 f~ short is used for calibration purposes. Between 1 and 10 GHz, the resonant cavity method [3] was employed and here the samples were in the form of cylinders.
1325
Table 1 Stretched exponential and Cole Davidson (CD) parameters for various glasses. KWW
CD
Composition
z (s)
13
4o
r (ps)
:t
0.00Nd 0.01Nd 0.03Nd 0.07Nd 0.10Nd 0.20Nd
22 -22 25 29 31
0.45 0.50 0.47 0.48 0.48
40
908 948 986 1033 1073 1167
0.88 0.87 0.85 0.73 0.78 0.78
79 97 96 96
1.2 '*
~
0.9
_.10
1.1" ,
C(,,+) 1.0
0 +~
......
O +" ~"
2 .....
.
0.7
O 0.91
3. Results The discharge current, i(t), can be calculated from the polarization, ~(t), as i(t)= d~/dt. The procedure to calculate the parameters fl, ~o and z from i(t) was the same as has been described before [1]. In Table l and Fig. I(A) we show these parameters for a series of Nd glasses. From the data, we see that the relaxation time, z, is not very sensitive to concentration but at concentration of about 0.07 it starts increasing while the polarization, ~o, although increasing initially, becomes almost fiat starting from precisely this concentration. In Table 2 we show the variation of the K W W parameters with temperature for two compositions for the sole purpose of showing how the parameters vary with the temperature. In Fig. 2 is shown a Cole-Cole plot of e" against d, for a glass sample of the composition 0.03Nd2Oa • PbO" 2B203 at T = 300 K. It is easily seen that the points do not lie on a semicircle, indicating a non-Debye behaviour or a distribution
X
•
/O . . . . . O . . . . . . . . . . . . . . . .
100.
O
/
,° l
o ;/
70-
,30 ~ ( d
.~ .................
20 4ot
i A
i
~o5
o~2
o.1 x
~
Fig. 1. (A) z and 4o parameters of the KWW relation versus concentration of neodymium oxide. (B) z and ct parameters of CD equation versus concentration of neodymium oxide.
of relaxation times. We fit, therefore, the data by the skewed arc or the Cole-Davidson function, ,~* = ~:+ + [(,~0 - , % ) / ( !
+ jo~)~'].
1326
L.C. Costa, S.K. Mendiratta / Journal of Non-Crystalline Solids 172-174 (1994) 1324-1327
Table 2 Stretched exponential parameters at different temperatures for two compositions of neodymium oxide x = 0.03
x = 0.07
T
z (s)
fl
$o
r (s)
fl
~o
150 200 250 300
35 30 25 22
0.30 0.36 0.42 0.50
17 38 54 79
35 32 25 22
0.30 0.36 0.40 0.47
22 44 68 97
t h e t i m e r e s p o n s e . W e first c a l c u l a t e t h e a p p r o x i m a t e v a l u e s o f these p a r a m e t e r s f r o m the a s y m p t o t i c p a r t o f the d a t a a n d t h e n use t h e m as s t a r t i n g v a l u e s in a s e a r c h for the best v a l u e s t h r o u g h a n o n - l i n e a r c u r v e fitting a l g o r i t h m . F o r (oor)>> l, we c a n e x p a n d [4] the C o l e - D a v i d s o n r e p r e s e n t a t i o n in the p o w e r s o f (~oz)-1:
(e' - e~)/e" = cotg(~rr/2) + [1 + cotg2(0~x/2)]~(toz) -1 .
4. Discussion For determining the two parameters that characterize the relaxation in this formula i.e., r and ~, we follow a strategy similar to that adopted for fitting
Thus we can obtain a first estimate of = from the high frequency part. On the other hand, the low frequency part of the data yields a first estimate oft. The low frequency expansion in the powers of (toz) is given by ~' = es - 2g[(1 + cO/2-1d'(~oz) + 0(~o~)4.
10
x=O.03
t
i
• 1
0 5
6
7
8
9
10
11
12
13
14
15
Fig. 2. Cole-Cole plot of e" against e', for a glass sample of the composition 0.03Nd203 • PbO" 2B203 at T = 300 K.
L.C. Costa, S.K. Mendiratta / Journal o/'Non-Crystalline Solids 172-174 (1994) 1324 1327
x= 0.00
20
40
60
~f
Fig. 3. ~:' versus e"f for the lead borate glass matrix. Thus we see that, in the low frequency region, the plot of e' versus (E'og) is a straight line with slope proportional to 3; as an example we show in Fig. 3 such a plot for x = 0.00. From the starting values of c~ and z thus obtained it is easy to proceed with the elaborate curve fitting procedure. It is worth mentioning that, in the high frequency (low frequency) approximation, we included only those datapoints starting from the highest (lowest) frequency measured which fit the straight line of the respective formula within a preassigned small e r r o r (X2 < 10-4"). In Table 1 and Fig. I(B) we summarize the best fit parameters for each glass. We cannot help but notice that the parameter ct has a sort of minimum value at the critical concentration x = 0.07. Besides, the relaxation time monotonically increases with concentration for this high frequency relaxation process following a trend similar to that of the low frequency relaxation time of the K W W relation.
5. Conclusion We have analyzed the low frequency as well as high frequency relaxation processes in the same
1327
samples of the glass system and find that low frequency response measured in time domain is well described by the stretched exponential while the high frequency complex dielectric constant data fits the empirical Cole-Davidson (or skewed arc) function. The variation of the stretched exponential parameters with the composition shows a break in the trend at a critical concentration of modifier ions of about 7 mol %. Beyond this concentration, we have argued before [5], the majority of the modifier ions are coupled to each other through bridging oxygen ions. On the other hand, although no clear physical meaning has been so far associated with the Cole-Davidson parameter ct [6], we find it difficult to believe that the minimum in the ct versus concentration curve at the same critical concentration is merely a coincidence. It is worth noting that has been taken as a measure of the skewness of the arc and the asymmetry of the distribution of relaxation times in the modified Debye model. It would therefore seem plausible that the structural changes which imply alterations in the relaxation processes should also change the parameter values [7]. We are currently undertaking the dielectric study of other glass systems in which the structural changes have been inferred by magnetic and EPR measurements. The authors thank Jfllio Gonqalves for preparing the samples and for general technical help and J N I C T for financial support.
References [1] S.K. Mendiratta and L.C. Costa, J. Non-Cryst. Solids, 131 133 (1991) 990. [2] M.A. Valente, PhD thesis, University of Aveiro (1993). I-3] F. Henry, th~sede Doctorat d'Etat en Sciences, Paris (1982). 1-4] J. Berberian and R. Cole, J. Chem. Phys. 84 (1986) 12. [5] S.K. Mendiratta, L.C. Costa and E.G. Sousa, J. Mater. Sci. Lett. 9 (1990) 301. [6] A.K. Jonscher, Dielectric Relaxation in Solids (Chelsea Dielectric, London, 1983). [7] K.L. Ngai, R.W. Rendell and A,F. Yee, Macromolecules23 (1990) 648.
ELSEVIER
Journal of Non-Crystalline Solids 172-174 (1994) 1324-1327
Dielectric behaviour of Nd ions in the lead borate glass Luis Cadillon Costa*, Sushil Kumar Mendiratta Departamento de Fisica, Universidadede Aveiro, 3800 Aveiro, Portugal
Abstract Accurate measurements of the relaxation in the time domain show that the low frequency relaxation processes in glasses are well described by the Williams-Watts (WW) relation. Measurements of the complex permittivity in the frequency domain have been often claimed, on the other hand, to satisfy the Cole-Davidson (CD) relation, e*= eo~ + [(Co- e~o)/(1 +jo)z)'], 0 < ~. In order to check with what accuracy the systems whose time domain behaviour is given by WW relation also satisfy the CD equation in the frequency domain, frequency measurements of the glasses in the frequency region 1 Hz-10 GHz are taken using different techniques. Absorption was observed in two different frequency regions and they are well described by the two relations. The temperature dependence of relaxation time parameters was also studied in order to characterize the physics of the relaxation process. The dependence of the parameters on the concentration of Nd sheds some light on the state of aggregation of the Nd ions in the glass.
1. Introduction Dielectric properties of disordered materials have been investigated for a long period either from the point of view of characterizing the material for applications or from the point of view of characterizing the dominant polar species which relaxes. However, there is another interesting aspect of the problem that is not explored so often. The parameters that characterize a measured time response or frequency response contain information not only about the relaxing species but also about the structure around the microscopic relaxing unit in which the actual relaxation takes place. In our previous
* Corresponding author. Tel: + 351-34 370 200. Telefax: + 35134 24 965.
studies of time response [ 1] on a series of Gd-based lead borate glasses, we had shown how the variation of the parameters in the stretched exponential response, also referred as K o h l r a u s c h Williams-Watts (KWW), q~ = ¢o exp( - t/z) p, with the concentration of the modifier ion can be correlated with the structural changes inferred from other independent techniques. In this paper, we show how the high frequency response measurements (1 H z - 1 0 G H z ) fit the Cole-Davidson (CD) formula e * = e ~ + [ ( e o -e~)/(l+jogz)~]. The same samples were submitted to time response and we try to find correlation between the high frequency parameters, especially z and c
0022-3093/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0 0 2 2 - 3 0 9 3 ( 9 4 ) 0 0 0 8 2 - X
L.C. Costa, S.K. Mendiratta / Journal of Non-Crystalline Solids 172 174 (1994) 1324 1327
2. Experimental procedures Time domain measurements in the seconds range were performed in a fashion similar to already reported [1]. Samples were of thickness 1 mm and diameter 10mm and the polarizing field was 100 kV/m. Measurements were performed at various temperatures between 80 and 300 K. In the frequency domain, different techniques were used in different regions. For frequencies from 1 Hz to 100 kHz, we used a sensitive lock-in amplifier to measure in-phase and out-of-phase current to give real and imaginary parts of complex dielectric constant. The absolute calibration in this range was done by 1 kHz measurements on a six-digit precision capacitance bridge. Measurements up to 1 MHz were done with the help of a precision H P capacitance bridge. The range 10 M H z - 1 G H z was covered by measurements with an rf impedance meter in which the amplitude of the incident and reflected wave is measured and a 50 f~ short is used for calibration purposes. Between 1 and 10 GHz, the resonant cavity method [3] was employed and here the samples were in the form of cylinders.
1325
Table 1 Stretched exponential and Cole Davidson (CD) parameters for various glasses. KWW
CD
Composition
z (s)
13
4o
r (ps)
:t
0.00Nd 0.01Nd 0.03Nd 0.07Nd 0.10Nd 0.20Nd
22 -22 25 29 31
0.45 0.50 0.47 0.48 0.48
40
908 948 986 1033 1073 1167
0.88 0.87 0.85 0.73 0.78 0.78
79 97 96 96
1.2 '*
~
0.9
_.10
1.1" ,
C(,,+) 1.0
0 +~
......
O +" ~"
2 .....
.
0.7
O 0.91
3. Results The discharge current, i(t), can be calculated from the polarization, ~(t), as i(t)= d~/dt. The procedure to calculate the parameters fl, ~o and z from i(t) was the same as has been described before [1]. In Table l and Fig. I(A) we show these parameters for a series of Nd glasses. From the data, we see that the relaxation time, z, is not very sensitive to concentration but at concentration of about 0.07 it starts increasing while the polarization, ~o, although increasing initially, becomes almost fiat starting from precisely this concentration. In Table 2 we show the variation of the K W W parameters with temperature for two compositions for the sole purpose of showing how the parameters vary with the temperature. In Fig. 2 is shown a Cole-Cole plot of e" against d, for a glass sample of the composition 0.03Nd2Oa • PbO" 2B203 at T = 300 K. It is easily seen that the points do not lie on a semicircle, indicating a non-Debye behaviour or a distribution
X
•
/O . . . . . O . . . . . . . . . . . . . . . .
100.
O
/
,° l
o ;/
70-
,30 ~ ( d
.~ .................
20 4ot
i A
i
~o5
o~2
o.1 x
~
Fig. 1. (A) z and 4o parameters of the KWW relation versus concentration of neodymium oxide. (B) z and ct parameters of CD equation versus concentration of neodymium oxide.
of relaxation times. We fit, therefore, the data by the skewed arc or the Cole-Davidson function, ,~* = ~:+ + [(,~0 - , % ) / ( !
+ jo~)~'].
1326
L.C. Costa, S.K. Mendiratta / Journal of Non-Crystalline Solids 172-174 (1994) 1324-1327
Table 2 Stretched exponential parameters at different temperatures for two compositions of neodymium oxide x = 0.03
x = 0.07
T
z (s)
fl
$o
r (s)
fl
~o
150 200 250 300
35 30 25 22
0.30 0.36 0.42 0.50
17 38 54 79
35 32 25 22
0.30 0.36 0.40 0.47
22 44 68 97
t h e t i m e r e s p o n s e . W e first c a l c u l a t e t h e a p p r o x i m a t e v a l u e s o f these p a r a m e t e r s f r o m the a s y m p t o t i c p a r t o f the d a t a a n d t h e n use t h e m as s t a r t i n g v a l u e s in a s e a r c h for the best v a l u e s t h r o u g h a n o n - l i n e a r c u r v e fitting a l g o r i t h m . F o r (oor)>> l, we c a n e x p a n d [4] the C o l e - D a v i d s o n r e p r e s e n t a t i o n in the p o w e r s o f (~oz)-1:
(e' - e~)/e" = cotg(~rr/2) + [1 + cotg2(0~x/2)]~(toz) -1 .
4. Discussion For determining the two parameters that characterize the relaxation in this formula i.e., r and ~, we follow a strategy similar to that adopted for fitting
Thus we can obtain a first estimate of = from the high frequency part. On the other hand, the low frequency part of the data yields a first estimate oft. The low frequency expansion in the powers of (toz) is given by ~' = es - 2g[(1 + cO/2-1d'(~oz) + 0(~o~)4.
10
x=O.03
t
i
• 1
0 5
6
7
8
9
10
11
12
13
14
15
Fig. 2. Cole-Cole plot of e" against e', for a glass sample of the composition 0.03Nd203 • PbO" 2B203 at T = 300 K.
L.C. Costa, S.K. Mendiratta / Journal o/'Non-Crystalline Solids 172-174 (1994) 1324 1327
x= 0.00
20
40
60
~f
Fig. 3. ~:' versus e"f for the lead borate glass matrix. Thus we see that, in the low frequency region, the plot of e' versus (E'og) is a straight line with slope proportional to 3; as an example we show in Fig. 3 such a plot for x = 0.00. From the starting values of c~ and z thus obtained it is easy to proceed with the elaborate curve fitting procedure. It is worth mentioning that, in the high frequency (low frequency) approximation, we included only those datapoints starting from the highest (lowest) frequency measured which fit the straight line of the respective formula within a preassigned small e r r o r (X2 < 10-4"). In Table 1 and Fig. I(B) we summarize the best fit parameters for each glass. We cannot help but notice that the parameter ct has a sort of minimum value at the critical concentration x = 0.07. Besides, the relaxation time monotonically increases with concentration for this high frequency relaxation process following a trend similar to that of the low frequency relaxation time of the K W W relation.
5. Conclusion We have analyzed the low frequency as well as high frequency relaxation processes in the same
1327
samples of the glass system and find that low frequency response measured in time domain is well described by the stretched exponential while the high frequency complex dielectric constant data fits the empirical Cole-Davidson (or skewed arc) function. The variation of the stretched exponential parameters with the composition shows a break in the trend at a critical concentration of modifier ions of about 7 mol %. Beyond this concentration, we have argued before [5], the majority of the modifier ions are coupled to each other through bridging oxygen ions. On the other hand, although no clear physical meaning has been so far associated with the Cole-Davidson parameter ct [6], we find it difficult to believe that the minimum in the ct versus concentration curve at the same critical concentration is merely a coincidence. It is worth noting that has been taken as a measure of the skewness of the arc and the asymmetry of the distribution of relaxation times in the modified Debye model. It would therefore seem plausible that the structural changes which imply alterations in the relaxation processes should also change the parameter values [7]. We are currently undertaking the dielectric study of other glass systems in which the structural changes have been inferred by magnetic and EPR measurements. The authors thank Jfllio Gonqalves for preparing the samples and for general technical help and J N I C T for financial support.
References [1] S.K. Mendiratta and L.C. Costa, J. Non-Cryst. Solids, 131 133 (1991) 990. [2] M.A. Valente, PhD thesis, University of Aveiro (1993). I-3] F. Henry, th~sede Doctorat d'Etat en Sciences, Paris (1982). 1-4] J. Berberian and R. Cole, J. Chem. Phys. 84 (1986) 12. [5] S.K. Mendiratta, L.C. Costa and E.G. Sousa, J. Mater. Sci. Lett. 9 (1990) 301. [6] A.K. Jonscher, Dielectric Relaxation in Solids (Chelsea Dielectric, London, 1983). [7] K.L. Ngai, R.W. Rendell and A,F. Yee, Macromolecules23 (1990) 648.