Radiation induced capacitance in barium aluminoborate glasses

Journal of Non-Crystalline Solids 321 (2003) 29–36 www.elsevier.com/locate/jnoncrysol

Radiation induced capacitance in barium aluminoborate glasses M.S.F. da Rocha *, W.M. Pontuschka, A.R. Blak Institute of Physics, University of S~ ao Paulo, C.P. 66318, 05315-970 S~ ao Paulo, SP, Brazil Received 9 April 2002; received in revised form 20 December 2002

Abstract Impedance analysis (IA) was applied to study the effect of electrically charged radiation-induced point defects on the dielectric properties of barium aluminoborate glasses. Changes observed in the real component e0 of the dielectric constant after X-ray irradiation have been investigated by comparing the dipolar contributions from the induced electrical charges, with those already present in the glass before irradiation. In this paper the results of impedance spectroscopy are presented and analyzed for the glass systems (1  x)((1/7)Al2 O3  (6/7)B2 O3 )  xBaO(Ax) and (1  xÞ((2/ 7)Al2 O3  (5/7)B2 O3 )  xBaO(Bx), where x represents the molar fraction of the alkaline earth oxide in the sample for x ¼ 0:3ðAxÞ and x ¼ 0:2, 0.3, and 0:4ðBxÞ. It is suggested that the most important contributions in the observed increase of e0 at room temperature are provided by the dipoles formed by a boron–oxygen hole center and a non-paramagnetic boron electron center (BEC2 , BOHCþ ), those composed by a BOHC and a neighboring Fe3þ impurity cation which has trapped an electron ([Fe2þ ] , BOHCþ ), and those composed by a BOHC and an alkaline earth electron center (AEEC , BOHCþ ). For glasses of high alkaline earth content it was observed a decrease of e0 and a model is suggested for the mechanism of this behavior. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 61.43.Fs; 61.82.)d; 77.22.)d; 78.90.+t

1. Introduction Radiation induced capacitance is a property bearing new potential information about the distribution of the charged local point defects produced by the trapping of electrons and holes in irradiated dielectric materials. Since the pioneering

*

Corresponding author. Tel.: +55-11 3091 6850; fax: +55-11 3031 2742/3813-4334. E-mail address: [email protected] (M.S.F. da Rocha).

work of Yasaitis and Smaller in 1953 [1], local induced defects have been well studied with EPR spectroscopy [2–11] in a set of alkali borate glasses. The local structure of the host material, prior to irradiation, has also been well characterized with NMR [12], Raman and M€ ossbauer [13,14] spectroscopy, and X-ray diffraction [15]. Previous work on the recombination mechanisms of the charge carriers located in their sites of metastable states has also been reported [13,14], for barium and calcium aluminoborate glasses showing similar defect states induced by irradiation.

0022-3093/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0022-3093(03)00087-5

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M.S.F. da Rocha et al. / Journal of Non-Crystalline Solids 321 (2003) 29–36

The aim of this work is to investigate how the microscopic point defects boron–oxygen hole center (BOHCþ ), boron electron center (BEC and BEC2 ), Fe2þ ion which trapped one electron ([Fe2þ ] ) and alkaline earth electron center (AEEC ), induced by X-ray in barium aluminoborate glasses are correlated to their macroscopic dielectric properties. Impedance analysis (IA) was selected to measure the resultant dipolar contribution of the induced point defect center charge distribution to the capacitance of parallel plate capacitors containing the glass samples of composition (1  x)((1/7)Al2 O3  (6/7)B2 O3 )  xBaO(Ax) and (1x)((2/7)Al2 O3 (5/7)B2 O3 )xBaO(Bx), where x represents the molar fraction of the alkaline earth oxide in the sample for x ¼ 0:3ðAxÞ and x ¼ 0:2, 0.3, and 0.4 ðBxÞ. Experimental evaluation of the dielectric constant was performed for the samples, before and after irradiation, and the results were compared.

2. Background The defect centers induced by ionizing radiation in borate glasses are in general described by microscopic models constructed on the basis of local information obtained from studies using several spectroscopic techniques, such as optical absorption, infrared and Raman spectroscopy, Xray diffraction, electron paramagnetic resonance (EPR), etc. The comparison between the structures of glasses and crystals of the same chemical composition is a common practice in the modeling processes. The specific models for the recombination mechanisms of the charge carriers are also very sensitive to the distribution and nature of the local chemical bonds where the free radicals appear after irradiation. Although the EPR spectra of borate glasses are already well understood, some controversy still remains [2–4]. The EPR spectra of irradiated borate glasses show several features of the hyperfine structure attributed to the isotopes B11 (80.22%, with I ¼ 3=2) and B10 (19.78%, with I ¼ 3). Krogh-Moe [16–21] identified several superstructural units in crystals containing boron, that later become the base of the more complete

classification of the borate glass structural units [22,23]. The immediate effect of the ionizing radiation on the borate glasses is the formation of e hþ pairs. The boron electron centers (BEC) in alkaline borate glasses was first identified by Griscom [5]. Electrons released during irradiation are trapped at metastable states of oxygen vacancy levels, continuously distributed around an average value of energy of about 0.2 eV [6] below the mobility edge of the conduction band, where the states are localized. As the photoelectrons are trapped at these vacancies, the formation of BEC takes place. The holes left in the valence band are self-trapped at the oxygen p–p levels bridging planar BO3 to tetrahedral BO4 structural units [7], giving rise to the so-called boron–oxygen hole centers (BOHC), whose average energy is about 1.0 eV [6] above the mobility edge of the valence band. It is meant here that the trapped charge-carrier (electron or hole), after overcoming the activation energy, reaches the extended states of the respective band acquiring mobility. Low amount of impurities such as iron and hydrogen atoms are often found in most of the samples, so that specific fractions of radiation induced electrons and holes are scavenged by the impurity ions, under conditions that are dependent on the chemical affinity and on the oxidation state of the glass. Since the BEC electrons are trapped at shallower levels than the holes of the BOHC, the former are released more quickly from their traps or recombine directly with the BOHCs at temperatures ranging from 80 to 320 K [6] until the complete elimination of the BECs. However, a considerable concentration of BOHCs remains, so that an equivalent number of electrons is still present somewhere else, e.g. at a non-paramagnetic boron electron center (BEC2 ), some trapping impurity sites such as [Fe2þ ] (an electron trapped by a Fe3þ ion) [11], and alkaline earth ions (AEEC or AEEC2 ) which are expected to behave as reported for alkali ions [24]. The BEC2 center is an ordinary BEC which received an additional electron and is supposed to be stable at room temperature. As the borate glasses are good electrical insulators, it is expected that most of the (BEC , BOHCþ ) or (BEC2 , BOHCþ ) pairs form local

M.S.F. da Rocha et al. / Journal of Non-Crystalline Solids 321 (2003) 29–36

metastable electrical dipoles occupying the first or second nearest neighboring sites in the glassy matrix. The same hypothesis is valid for the (AEEC , BOHCþ ), (AEEC2 , BOHCþ ) and the ([Fe2þ ] , BOHCþ ) pairs.

3. Experimental The glass samples were produced from a mixture of weighed amounts of the reagent grade raw materials B2 O3 (boric oxide), Al2 O3 (aluminum oxide) and Ba(OH)2  8H2 O (barium hydroxide), where the iron impurity content for the boric oxide and the barium hydroxide is 0.001 (wt%). The glasses were prepared in an alumina crucible at 1150 °C for 2 h. The melt was poured into an aluminum mould, and a sample of laminar shape was annealed inside an oven at 400 °C during 1 h. The oven was then turned off and slowly reached room temperature. The dielectric constants of the glasses were determined by measuring the capacitance in function of the frequency of the applied electric field to the capacitors, prepared following the procedure described below. The glass pieces were first ground and polished in order to plane down their surfaces. The plane surfaces of the Bx samples were sputter coated with gold and those of the Ax sample were covered with silvered glue. The latter were heat treated to evaporate the resin, avoiding undesirable Maxwell–Wagner effects [26]. The metallized samples are capacitors of parallel plates with areas of irregular shapes. The areas were determined from digitized photographs and were accurate to 1.5%. The thickness of each sample was determined as the average of several measurements at different positions. The capacitance of the empty cell corresponding to the sample was calculated with the equation C0 ¼ f e0 A=d, where f is a correction factor that accounts for edge effects and deviations of the metallized surfaces from two parallel plates, A is the area of the capacitor, and d the distance between the plates. In order to evaluate the edge effects, capacitance measurements were performed on parallel capacitor plates for several values of d. A graph of C=C0 d=A was plotted,

31

Table 1 Parameter formulas for impedance measurement Parameter jZj jY j h L C Q D

Equivalent circuit Series pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 þ X 2 1

tan ðX =RÞ X =x 1=ðxX Þ jX j=R R=jX j

Parallel pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 þ B2 tan1 ðB=GÞ 1=ðxBÞ B=x jBj=G G=jBj

and a polynomial pðd=AÞ of degree 6 was fit to the points in a least squares sense. The correction factors were then obtained directly, since they are independent of the particular sample shape. The correction DC in the capacitance C due to the deviation d from parallel plates is given by DC d2 =ð3d2 Þ. The impedance parameter measurements were performed with an impedance analyzer controlled with a software codified in visual basic language. The built-in frequency synthesizer was set for frequency measurements within the range of 100 Hz to 13 MHz (1 mHz maximum resolution), the OSC level was set to 1 V rms (5 mV resolution), and the internal dc bias voltage source, to 0 V. The analyzer measures the impedance R þ jX , when the circuit mode is set to an equivalent series circuit, and the admittance G þ jB, when the circuit mode is set to an equivalent parallel circuit. Other impedance parameters are calculated from these measured values using the relations given in Table 1.

4. Results In Table 2 are presented the geometric parameters, the capacitances, and the dielectric constants for the samples Ax and Bx. In the calculation of the capacitances C0theor of the empty cells, the correction factors due to edge effects and deviation from parallelism are included. Cexp and C0exp are the measured capacitances of the sample and respective empty cell. e0theor and e0exp are the

32

M.S.F. da Rocha et al. / Journal of Non-Crystalline Solids 321 (2003) 29–36

Table 2 Geometric parameters, capacitances, and dielectric constants for the samples Ax and Bx Sample

Geometric parameters

Capacitance (pF)

A.3

Area (cm2 ) Thickness (mm) d=A (100/cm) Edge effects dd correction

2.70 (3) 0.87 (4) 3.22 (15) 1.76 (2) 1.0007

C0

Area (cm2 ) Thickness (mm) d=A (100/cm) Edge effects dd correction

6.12 (6) 0.75 (2) 1.23 (3) 1.30 (3) 1.0002

Area (cm2 ) Thickness (mm) d=A (100/cm) Edge effects dd correction

3.98 (2) 0.75 (2) 1.88 (5) 1.483 (11) 1.0002

Area (cm2 ) Thickness (mm) d=A (100/cm) Edge effects dd correction

3.82 (5) 0.55 (7) 1.44 (18) 1.42 (18) 1.0054

Area (cm2 ) Thickness (mm) d=A (100/cm) Edge effects dd correction

2.69 (4) 0.72 (6) 2.68 (23) 1.66 (13) 1.0023

B.2a

B.2b

B.3

B.4

Dieletric constant 4.8 (2)

theor

Cexp

20.77 (5)

C0

theor

9.4 (3)

C0

exp

10.5 (5)

Cexp

45.95 (5)

C0

7.0 (2)

theor

Cexp

31.32 (5)

C0

theor

8.8 (16)

C0

exp

8.2 (5)

Cexp

50.13 (5)

C0

theor

5.5 (6)

C0

exp

5.6 (5)

e0theor

4.3 (2)

e0theor e0exp

4.89 (16) 4.4 (2)

e0theor

4.47 (13)

e0theor e0exp

6 (1) 6.1 (4)

e0theor e0exp

5.5 (6) 5.4 (5)

30.11 (5)

Cexp

The dd correction is a factor that accounts for the change in the capacitance C0 due to the deviation from parallelism of the capacitor plates.

B.3

6.2 6.0 5.8 5.6 B.4

5.4

ε'

theoretical and experimental values of e0 , given by the rates Cexp =C0theor and Cexp =C0exp , respectively. The graph in Fig. 1 shows the dielectric constant spectra of the glasses A.3 ðx ¼ 0:3Þ and Bx (x ¼ 0:2, 0.3, and 0.4), before irradiation and at room temperature, as a function of the frequency of the applied electric field. The samples B.2a and B.2b were made from the same glass piece, but they differ in area and shape. On the other hand, we observe that the curve of the sample B.2a is very close to the curve of the sample B.2b showing the reproducibility of the method. The X-ray effect on the dielectric properties of the glasses A.3 and B.2 is an increase in the observed capacitance of the sample. Figs. 2 and 3 show the effect of the irradiation on the dielectric constant e0 of the samples B.2b and A.3,

5.2 5.0 4.8 4.6

B.2a B.2b

A.3

4.4 2

3

4 5 6 Log[frequency(Hz)]

7

8

Fig. 1. Dependence of e0 with the alkaline earth concentration for the Bx glasses, and comparison with the curve of the A.3 glass, where the relative concentrations of B2 O3 and Al2 O3 were changed. All measurements were made at room temperature (24.0  0.2) °C, before irradiation.

5

5.7 5.6 5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 4.6 4.5 4.4 4.3

2.0x10

B.2b sample non irradiated irradiated

non irradiated irradiated

5

-2.0x10

5

-4.0x10

5

-6.0x10

5

-8.0x10

6

-1.0x10

6

-1.2x10

6

-1.4x10

0.0

2

3

4 5 6 Log[frequency(Hz)]

7

4

4

4

4

4

Z' (Ohm)

Fig. 4. Cole–Cole diagram showing the X-ray effect on the dielectric constant of the B.2b glass sample. It was irradiated with 150 kV–5 mA for 2.5 h. The measurements were made at room temperature.

5.50

A.3 sample

4.65 4.60 4.55 4.50 4.45 4.40 4.35 4.30 4.25 4.20 4.15

3

5.0x10 1.0x10 1.5x10 2.0x10 2.5x10 3.0x10

8

Fig. 2. X-ray effect on the dielectric constant e0 of the B.2b glass sample. It was irradiated with 150 kV–5 mA for 2.5 h. The measurements were made at room temperature.

non irradiated irradiated

B.4 sample non irradiated irradiated

5.45 5.40 5.35 5.30

ε

ε

33

B.2b sample

0.0

Z'' (Ohm)

ε'

M.S.F. da Rocha et al. / Journal of Non-Crystalline Solids 321 (2003) 29–36

5.25 5.20 5.15

2

3 4 5 Log[frequency(Hz)]

6

5.10 2

3

4 5 6 Log[Frequency(Hz)]

7

8

Fig. 3. X-ray effect on the dielectric constant of the A.3 glass sample. It was irradiated with 150 kV–5 mA for 2.5 h. The measurements were made at room temperature.

Fig. 5. X-ray effect on the dielectric constant of the B.4 glass sample. It was irradiated with 150 kV–5 mA for 2.5 h. The measurements were made at room temperature.

respectively. In Fig. 4 the Cole–Cole [27] diagram is presented for the sample B.2b, where an increase in the conductivity can be observed in the glass after irradiation. The graph in Fig. 5 shows a decrease in the dielectric constant e0 of the sample B.4 from X-ray irradiation. This behavior indicates that electrical dipoles already present in the glass before irradiation can be neutralized by the ionizing radiation, producing a decrease in the measured capacitance.

5. Discussion Most of the spectra show the same statistical fluctuation observed in Fig. 1 at frequencies below 1 kHz. This phenomenon is due in part to the thermal agitation of free electric charges in the leads and capacitor plates, since it was also observed for the empty cells. This effect was first measured by Johnson [28] using a vacuum tube amplifier and thermocouple, and was expressed as

34

hI 2 i ¼ ð2kT =pÞ

M.S.F. da Rocha et al. / Journal of Non-Crystalline Solids 321 (2003) 29–36

Z

1

2

RðxÞjY ðxÞj dx;

ð1Þ

0

where T is the absolute temperature of the input element, k the Boltzmann gas constant, R the real resistance of the element, and Y the complex transfer admittance. This expression was later developed by H . Nyquist [29] on a wholly theoretical basis. Hydrogen bonds and hydroxyls are normally found in borate glass systems [30]. Their presence in our glasses was confirmed by optical absorption measurements in the infrared. All the spectra show the characteristic bands of those radicals between 2.5 and 4.0 lm. The presence of bubbles in the glass could also explain the water absorption band at 3.5 lm. As a result, low frequencies would lead to cell polarization which could introduce in the spectra some additional deviations from the expected behavior. A spurious feature of the curves was observed as a slight increase at frequencies near 13 MHz. This effect was identified as the tale of a resonance peak, produced by the parasitic impedance of the measurement fixture, plus the impedance of the circuit being measured. The increase observed in e0 as x changes from 0.2 to 0.3 could be ascribed to the conversion of BO3 triangles of threefold coordinated borons to fourfold co-ordinated borons in tetrahedral BO 4 units [24,25]. Near x ¼ 0:2 the glass Bx undergoes a visible phase separation and a devitrification process that increases with decreasing x, as has already been observed for alkali-deficient glasses [31]. The result is a decrease that is observed in the curves of the samples B.2a and B.2b at the frequencies ranging from 100 Hz to 10 kHz, indicating the presence of dipoles with high relaxation times (sluggish dipoles) in the glass. For higher x, it is expected that some of the alkaline earth cations are compensated by nonbridging oxygens (NBOs) at the BO 3 units, by analogy with glasses of high alkali content [24,25]. This effect could be responsible for the observed decrease in e0 as x changes from 0.3 to 0.4. The response of the electric dipoles induced by the applied electric field seems to be higher at lower frequencies. This behavior suggests that the

effect of the ionizing radiation on the electrical distribution of charges tends to form sluggish electrical dipoles, as expected. The pairs (BOHCþ , BEC ) survive only at cryogenic temperatures at which the dipoles are practically frozen and cannot be detected with IA. By heating the sample to room temperature, the BEC and BOHC recombination eliminates 90% [9] of the BECs produced during irradiation. On the other hand, the [Fe2þ ] is responsible for the red luminescence observed in the same range of temperatures where the BOHCs disappear [9,10]. Before irradiation, the [Fe3þ ] concentration in aluminoborate glasses is (1.13  0.08) 1018 spins/ cm3 [10]. It decreases 40%, approximately, after irradiation, so that the [Fe2þ ] concentration is (0.60  0.04) 1017 spins/cm3 . Meanwhile, the BOHC resonance reaches an intensity 14.5 times greater than that of the [Fe2þ ] . This relation indicates that the [Fe2þ ] content is small and does not account to the changes observed in the capacitance, since a great number of BOHCs have to be charge compensated by some non-paramagnetic negative centers. A real possibility is the nonparamagnetic center [BEC2 ], not detected with EPR and supposed to be more stable than the [BEC ] center. Therefore, the contribution of the ionizing radiation to the dielectric constant of the glasses B.2 and A.3 must be due to the presence of dipoles formed by a BOHC and a cation that has previously trapped an electron, where the dipoles ([Fe2þ ] , BOHCþ ), (BEC2 , BOHCþ ), and (AEEC , BOHCþ ) are the most probable examples. An example of dipole neutralization in borate glasses is shown in Fig. 6, for oxide glasses of high content of alkali or alkaline earth cations. As the NBOs contained in non-irradiated oxide glasses of high concentration of alkali or alkaline earth cations have an extra negative charge, electrical dipoles are composed of a pair of a NBO and a positive charge unit of the modifier ion. The extra electron of a NBO can be removed from its p-orbital under the action of a X-ray photon. If the ejected electron is trapped by the glass modifier ion, a dipole (NBO , Aþ ) or (NBO , A2þ ) is neutralized, respectively for an alkali or an alkaline earth ion.

M.S.F. da Rocha et al. / Journal of Non-Crystalline Solids 321 (2003) 29–36

35

hole

NBO

-

dipole

A+

+

before irradiation

o

photo electron

o

+ AEC

(a)

A

NBO

++

A2+

-

NBO

before irradiation

o

NBO

OHC photo ++ quadrupole electron AEEC

-

(b)

(neutral)

after irradiation

hole

-

(neutral)

NBO

OHC

+

(neutral) (one charge neutralized)

A

NBO

dipole

after irradiation

Fig. 6. Example of dipole neutralization due to the X-ray action in (a) alkali or (b) alkaline earth oxide glasses.

(a) one electron is removed from the NBO of the (NBO , Aþ ) dipole and is trapped by the cation Aþ . The ionized oxygen becomes electrically neutral, and so it happens with the Aþ ion which has trapped the photoelectron. (b) One of the NBOs of the quadrupole is ionized, and the ejected electron is trapped by the cation A2þ , which is reduced to Aþ , leaving a remaining dipole composed of the second NBO, that may be also neutralized by a subsequent similar process.

6. Conclusions It was observed that the non-irradiated B.2 glass shows a smaller dielectric constant e0 as

compared with the other samples. It was demonstrated, also with IA, the appearance of additional dipoles induced with X-ray irradiation. The experimental results show that the X-ray induced intrinsic centers in the samples A.3 and B.2 form electric dipoles of high relaxation times. Their response to the applied electric field is more sensitive at lower frequencies and disappears at frequencies higher than 100 kHz. At room temperature, the contribution of the ionizing radiation to the dielectric constant of the glasses B.2 and A.3 is ascribed to the presence of dipoles formed by a BOHC and a cation which has previously trapped an electron, where the dipoles ([Fe2þ ] , BOHCþ ), (BEC2 , BOHCþ ), and (AEEC , BOHCþ ) are the most probable examples.

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M.S.F. da Rocha et al. / Journal of Non-Crystalline Solids 321 (2003) 29–36

The decrease of e0 observed in Fig. 5, for a high content alkaline earth glass sample, could be described by a mechanism in which one quadrupole (see Fig. 6(b)) formed by a couple of NBOÕs charge-compensated by a A2þ alkaline earth ion can be partially or even completely neutralized by the X-ray action in the glass. We suggest that a similar behavior is expected also for an alkali glass (see Fig. 6(a)), where a dipole of a NBO chargecompensated by a Aþ alkali ion can be neutralized by an electron–hole pair created by X-irradiation. The BEC contribution to the dielectric properties of our glasses could not be detected with IA, since, at cryogenic temperatures, the dipoles (BEC , BOHCþ ) are frozen. The alternative method of field induced thermally stimulated currents (FITSC) [26] will be considered in future studies. Acknowledgements The authors thank the Instituto de Pesquisas Energeticas e Nucleares, and the Laborat orio de Microscopia Eletr^ onica do Instituto de Fısica da Universidade de S~ ao Paulo for the experimental facilities support. This work was performed with financial support from Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico. References [1] E.L. Yasaitis, B. Smaller, Phys. Rev. 92 (1953) 1068. [2] M.C.R. Symons, J. Chem. Phys. 53 (1970) 468. [3] D.L. Griscom, P.C. Taylor, P.J. Bray, J. Chem. Phys. 53 (1970) 469. [4] P.C. Taylor, D.L. Griscom, J. Chem. Phys. 55 (7) (1971) 3610. [5] D.L. Griscom, J. Chem. Phys. 55 (3) (1971) 1113.

[6] W.M. Pontuschka, M.I.T. Oliveira, S.M. Del Nery, in: A.C. Wright, S.A. Feller, A.C. Hannon (Eds.), Borate Glasses, Crystals and Melts, Alden, Oxford, UK, 1997, p. 392. [7] D.L. Griscom, P.C. Taylor, D.A. Ware, P.J. Bray, J. Chem. Phys. 48 (11) (1968) 5158. [8] W.M. Pontuschka, S. Isotani, A. Piccini, N.V. Vugman, J. Am. Ceram. Soc. 65 (10) (1982) 519. [9] W.M. Pontuschka, S. Isotani, A. Piccini, J. Am. Ceram. Soc. 70 (1) (1987) 59. [10] S.M. Del Nery, W.M. Pontuschka, S. Isotani, C.G. Rouse, Phy. Rev. B 49 (6) (1994) 3760. [11] W.M. Pontuschka, L.S. Kanashiro, L.C. Courrol, Fiz. Khim. Stekla 27 (1) (2001) 54; Glass Phys. Chem. 27 (1) (2001) 37. [12] P.J. Bray, J. Non-Cryst. Solids 95 (1987) 45. [13] H.B. Pascoal, W.M. Pontuschka, H.R. Rechenberg, J. Non-Cryst. Solids 258 (1999) 92. [14] H.B. Pascoal, W.M. Pontuschka, H.R. Rechenberg, Appl. Phys. A 70 (2000) 211. [15] A.C. Wright, in: P.A. Fleury, B. Golding (Eds.), Coherence and Energy Transfer in Glasses, 9, Plenum, New York, 1984, p. 1. € R KEMI Band 14 (39) (1959) [16] J. Krogh-Moe, ARKIV FO 439. [17] J. Krogh-Moe, Acta Cryst. 13 (1960) 889. [18] J. Krogh-Moe, Acta Cryst. 15 (1962) 190. [19] J. Krogh-Moe, Phys. Chem. Glass. 3 (4) (1962) 101. [20] J. Krogh-Moe, Acta Chem. Scand. 18 (1964) 2055. [21] J. Krogh-Moe, Acta Cryst. 18 (1965) 77. [22] J. Krogh-Moe, Phys. Chem. Glass. 6 (2) (1965) 46. [23] D.L. Griscom, in: L.D. Pye, V.D. Frechette, N.J. Kreidl (Eds.), Borate Glass Structure, U.S. Naval Research Laboratory, 1978. [24] I.A. Shkrob, B.M. Tadjikov, A.D. Trifunac, J. Non-Cryst. Solids 262 (2000) 6. [25] P.J. Bray, J. Non-Cryst. Solids 95 (1987) 45. [26] J. Vanderschueren, J. Gasiot, in: P. Br€aunlich (Ed.), Thermally Stimulated Relaxation in Solids, Springer, Berlin, 1979, p. 135 (Chapter 4). [27] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics, London, 1983. [28] J.B. Johnson, Phys. Rev. 32 (1928) 97. [29] H. Nyquist, Phys. Rev. 32 (1928) 110. [30] M.E. Milberg, F. Meller, J. Chem. Phys. 31 (1) (1959) 126. [31] I.A. Shkrob, B.M. Tadjikov, A.D. Trifunac, J. Non-Cryst. Solids 262 (2000) 35.