On the structure of phosphosilicate glasses

Journal of Non-Crystalline Solids 306 (2002) 209–226 www.elsevier.com/locate/jnoncrysol

On the structure of phosphosilicate glasses V.G. Plotnichenko *, V.O. Sokolov, V.V. Koltashev, E.M. Dianov Fiber Optics Research Center, General Physics Institute, Russian Academy of Sciences, 38 Vavilov Street, Moscow 119991, Russia Received 21 June 2001; received in revised form 8 February 2002

Abstract Vibrational spectra of phosphosilicate glasses with P2 O5 concentrations up to 15 mol% are investigated by the methods of Raman spectroscopy and quantum-chemical modeling. We have found that the Raman band at 1320 cm
1 characteristic for such glasses is not simple and may be decomposed into two components with frequencies at 1317 and 1330 cm
1 caused in our opinion by single phosphorus centers (O@PO3 tetrahedra surrounded by SiO4 ones) and by double phosphorus centers (pairs of O@PO3 tetrahedra bonded by a common oxygen atom). In the investigated phosphosilicate glasses manufactured by MCVD and SPCVD methods the ratio of concentrations of single and double centers varies from 1:5 to 1:2. A novel interpretation of the Raman bands distinct from the traditional one is suggested. The approach to the Raman spectra analysis developed in this article can be applied for control and optimization of manufacturing process of phosphosilicate and similar glasses as well as optical fibers.
2002 Elsevier Science B.V. All rights reserved. PACS: 61.43.Fs; 63.50.þx; 78.20.Bh; 78.30.
j

1. Introduction Phosphorus is one of the main dopants in highpurity silica glass (v-SiO2 ) used in fiber optics technology to form an optimal refractive index profile in a fiber and to modify the viscosity of its core and cladding [1]. Phosphosilicate glasses (P2 O5 )x (SiO2 )1
x with P2 O5 concentration x K 15 mol% are used in developing stimulated Raman fiber lasers and amplifiers [2,3]. Phosphosilicate glasses are also sensitive to UV radiation near 190 nm [4] which allows one to form the refractive

*

Corresponding author. Tel.: +7-095 135 8093; fax: +7-095 135 8139. E-mail address: [email protected] (V.G. Plotnichenko).

index gratings in phosphosilicate-core fibers [2]. Finally, phosphosilicate glasses doped with rareearth elements are considered to be a potential material for optical amplifiers, converters and sources of visible and near IR radiation [5,6]. However, despite of wide applications of phosphosilicate glasses, their structure and optical properties are yet to be investigated sufficiently. Phosphosilicate glass is thought currently to consist of silicon–oxygen, SiO4 , and phosphorus– oxygen, O@PO3 , tetrahedra bonded randomly in a three-dimensional network where each silicon atom is bonded with four silicon or phosphorus atoms by oxygen linkages, and each phosphorus atom has only three such bridging bonds. In the fourth vertex of the O@PO3 tetrahedron there is a non-bridging oxygen atom bound with the central

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phosphorus atom with a double bond, O@P. Owing to a high polarizability of the double bond an intensive band with a maximum at 1320 cm
1 caused by the stretching vibrations of the O@P bonds arises in the Raman spectra [7–9]. Similar band with a frequency at 1390 cm
1 [10,11] is characteristic as well for vitreous P2 O5 (v-P2 O5 ) where the atomic network is formed by O@PO3 tetrahedra [12]. Our analysis of the Raman spectra of phosphosilicate glass samples manufactured by modified chemical vapor deposition (MCVD) and surface plasma chemical vapor deposition (SPCVD) methods in different laboratories has shown that the intensity of the O@P band grows with molar concentration of P2 O5 in (P2 O5 )x (SiO2 )1
x glass [9]. The frequency of maximum and shape of the band do not depend on P2 O5 concentration in glasses with x K 15 mol% manufactured under the same conditions. We have found as well that this band is composite and may be represented by two components with frequencies at about 1317 and 1330 cm
1 , whose intensity ratio is determined by the features of the production process, and, probably, by the subsequent thermal history of the samples. We assume that the low-frequency component corresponds to O@PO3 tetrahedra bonded only with the SiO4 tetrahedra (single phosphorus centers) and the high-frequency one is caused by pairs of the O@PO3 tetrahedra bound together by PAOAP linkages (double phosphorus centers). The random structure of the phosphosilicate glass network is insufficient to explain such an interpretation, since for low P2 O5 concentration ( K 15 mol%) in absence of correlations in arrangement of phosphorus atoms, mainly single (bonded with silicon atoms only) O@PO3 tetrahedra occur in the glass and the number of any groups of O@PO3 tetrahedra should not exceed 5% of the total amount of phosphorus tetrahedra. Thus the intensity of the low-frequency component should be at least 20 times as high as the intensity of the high-frequency component. However, we observe in the Raman spectra just an inverse relation: the intensity of the low-frequency component is 5–10 times lower than that of the high-frequency component. This may be explained by predominant formation of double phosphorus

centers in the investigated glasses. Since in manufacturing phosphosilicate glasses by chemical vapor deposition (CVD) methods, being most frequently used in fiber optics, the oxidation of phosphorus oxychloride occurs, and molecules of the most stable phosphorus oxide, P2 O5 , arise, the formation of double phosphorus centers seems to be quite natural. The main goal of this work was to verify this hypothesis both theoretically and experimentally and to interpret the vibrational spectrum of phosphosilicate glass and its dependence on phosphorus concentration. For this purpose the quantum-chemical modeling of the structure and vibrational properties (vibrational frequencies, IR absorption and Raman scattering intensities) of phosphorus centers in phosphosilicate glass was performed and Raman spectra of phosphosilicatecore optical fibers made in different laboratories by MCVD and SPCVD methods were measured and analyzed.

2. Quantum-chemical calculations All calculations were performed with the help of the GAMESS (US) program [13] in Hartree–Fock approximation using basis sets and effective core potentials (ECP) developed in Ref. [14] for oxygen and fluorine and in Ref. [15] for phosphorus, silicon, chlorine, bromine and iodine. One extra d-type polarization function was included in ECP basis for each atom (f ¼ 0:80, 0.55, 0.80, 0.395, 0.75, 0.389, 0.266 for O, P, F, Si, Cl, Br and I atoms, respectively). 3-21G standard basis set was used for hydrogen. As shown in Ref. [16,17], such a choice of the basis provided a good description of properties of the systems under consideration. To verify our approach we have calculated several molecules of phosphorus oxihalogenides, O@PA3
x Bx , for A, B ¼ H, F, Cl, Br and I. Results for some molecules are collected in Table 1. In this table and everywhere in what follows the IR ab
2 , sorption intensities are given in Debye2 /amu/A 4
and the Raman intensities in A /amu. The comparison of the calculated geometrical parameters with the experimental data available

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211

Table 1 Results of the quantum-chemical calculations of phosphorus oxihalogenides molecules, O@PA3
x Bx Molecule
) O@P bond length (A
) PAX bond length (A O@PAX angle (deg) XAPAX angles (deg) O@P vibration frequency (cm
1 ) Corrected frequency Experimental frequency [18]
2 ) IR absorption (Debye2 /amu/A
4 /amu) Raman intensity (A

O@PH3

O@PF3

O@PCl3

O@PBr3

O@PI3

1.467 1.395 116.808 101.238 1376.33 1273 – 8.0 7.8

1.428 1.520 117.135 100.834 1512.68 1400 1416.8 7.7 4.7

1.440 1.985 114.614 103.873 1429.16 1322 1321.5 6.0 10.5

1.445 2.191 114.024 104.559 1400.22 1295 1277 5.4 13.5

1.451 2.430 113.110 105.600 1371.19 1268 – 5.0 16.2

[18] confirms the conclusion made in Ref. [16,17] about the basis set choice. The Hartree–Fock approximation is known to overestimate systematically the vibrational frequencies. Comparing the calculated and experimental [18] frequencies of the stretching vibration of O@P bonds in O@PA3
x Bx molecules we have found that the scaling factor for this vibration in these molecules is 0.925 and the estimated average accuracy of calculated frequency of this vibration is 15 cm
1 (better than 2%). All calculated vibrational frequencies are given further with the appropriate scaling factors taken into account. One of the crystalline polymorphs of P2 O5 (hexagonal [19,20]) is formed by P4 O10 molecules. According to certain models such molecules may occur both in v-P2 O5 and in (ultra) phosphate glasses. Therefore we have calculated the P4 O10 molecule using our approach. The results of calculations are as follows (the experimental values from the review [20] are given in brackets): O@P
(1:40  0:03 A
); OAP bond lengths – 1.423 A

bond lengths – 1.600 A (1:60  0:01 A); O@PAO angles – 117.57 (117); OAPAO angles – 100.29 (101); PAOAP angles – 125.67 (124.5); frequencies of the stretching O@P bond vibrations – 1434 (A1 ) and 1401 (F2 ) cm
1 (1430 and 1405 cm
1 , respectively); IR absorption intensities – 0.0 and 13.0, Raman intensities – 15.5 and 7.7, respectively. The calculations of molecules prove sufficient reliability of our approach for quantum-chemical modeling of the vibrational properties of silica glass, phosphorus centers in phosphosilicate glass

and v-P2 O5 , which has been performed in a molecular cluster model. The main results of the modeling are given in Table 2.

2.1. Silica glass Vibrational properties of the silica glass network were simulated by the cluster ðH3 SiAOÞ3 B SiAOASiBðOASiH3 Þ3 containing two SiO4 tetrahedra bonded together by common bridging O atom and each connected to three Si atoms with dangling bonds saturated by H atoms. According to calculations, in the equilibrium configuration the SiAO bond lengths were equal to 1.612 and
for central SiAOASi linkage and to 1.608 1.610 A
for other linkages (there are two short and 1.607 A and two long SiAO bonds in each SiO4 tetrahedron). The SiAOASi angles were equal to 144.2 both in the central linkage SiAOASi and in other linkages, and all the OASiAO angles in SiO4 tetrahedra were 109.4. Hence the calculation reproduces the mean geometrical parameters of silica glass [21]. Our approach also allows one to describe well the most typical vibrational properties of the silica glass network. For further consideration the antisymmetric stretching vibrations of the OASi bonds in the SiAOASi linkages are of the most interest. There are two types of such vibrations with frequencies 1194 and 1091 cm
1 . According to Ref. [22], the first-type vibrations corresponds to LO phonons, and the second-type ones to TO phonons. IR absorption intensities for these

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Table 2 Results of the quantum-chemical modeling Model

Geometry

Silica glass

SiAO bond lengths

Single phosphorus center

Double phosphorus center

P2 O5 glass

Vibrations

Frequency

IR

Raman

SiAOASi angles OASiAO angles

1.612 1.610 144.2 109.4

SiAO bonds antisymmetric stretching in the SiAOASi linkages

1194 1091

8.8 16.1

0.1 0.1

O@P bond length PAO bond lengths SiAO bond lengths O@PAO angles OAPAO angles PAOASi angles

1.458 1.563 1.669 114.7 103.8 134.4

O@P bond stretching

O@P bond lengths PAO bond lengths in the PAOAP linkage PAO bond lengths in the PAOASi linkages SiAO bond lengths in the PAOASi linkages O@PAO angle in the PAOAP linkage O@PAO angles in the PAOASi linkages OAPAO angle in the PAOAP linkage OAPAO angles in the PAOASi linkages PAOASi angles

1.455 1.601

138.0

O@P bond length PAO bond lengths O@PAO angles OAPAO angles PAOAP angles

1.449 1.571 112.3 103.0 130.9

1316

6.2

1.9

PAO and SiAO bonds antisymmetric stretching in the PAOASi linkages

1068 1067

24.5

0.6

1.550

O@P bond stretching

1331

8.8

2.1

1.660

PAO bonds antisymmetric stretching in the PAOAP linkage

1120

6.5

0.3

PAO and SiAO bonds antisymmetric stretching in the PAOASi linkages

1102

24.5

0.7

1100

23.7

0.3

1391 1386

7.8 8.0

1.9 2.5

1020

25.4

1.7

111.8 116.4 103.2 104.0

O@P bond stretching PAO bonds antisymmetric stretching in the PAOAP linkages


, angles in degrees, frequencies in cm
1 , IR absorption intensities in Debye2 /amu/A
2 , Raman intensities A
4 /amu. Bond lengths in A

vibrations are 8.8 and 16.1, respectively, and the Raman intensities are 0.1 for the vibrations of the both types. Notice that to compare the results of the calculations directly with the experiment, in which the polarizations of the incident and scattered light are not fixed, the calculated Raman intensities are given hereafter averaged over the polarization. 2.2. Single phosphorus center Single phosphorus center in phosphosilicate glass designated as O@PðOASiÞ3 was simulated by O@PðAOASiH3 Þ3 cluster containing a single

O@PO3 tetrahedron bonded with three Si atoms, the dangling bonds of those saturated by H atoms. Calculated configuration of the single phosphorus center is shown in Fig. 1. The O@P double bond
, PAO bond lengths were 1.563 length was 1.458 A
, SiAO bond lengths were 1.659 A
, O@PAO A angles were 114.7, OAPAO angles were 103.8, PAOASi angles were 134.4. Calculated vibrational properties of the O@ PðOASiÞ3 center were as follows: frequency of the O@P bond stretching vibration was 1316 cm
1 , IR absorption and Raman intensities were 6.2 and 1.9, respectively; frequencies of antisymmetric stretching vibrations of the OAP and OASi bonds

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213

Fig. 1. Calculated configuration of the single phosphorus center. Fig. 2. Calculated configuration of the double phosphorus center.

in the PAOASi linkages were 1068 and 1067 cm
1 , for both vibrations IR absorption and Raman intensities being 24.5 and 0.6, respectively. 2.3. Double phosphorus center The calculated frequency of the double bond vibration in the single phosphorus center appears to be somewhat lower than the value 1320 cm
1 experimentally observed for the maximum of the corresponding Raman band in phosphosilicate glasses. This suggests that there are relatively few such centers in phosphosilicate glasses, i.e. single phosphorus atoms surrounded by SiO2 network, since phosphorus mainly forms centers with more complex structure. For low P2 O5 concentration these centers are primarily two-atom ones with two O@PO3 tetrahedra bonded together by common bridging oxygen atom. In what follows these centers are designated as O@PAOAP@O. The double phosphorus center in phosphosilicate glass was simulated by the cluster containing two O@PO3 tetrahedra bonded to each other by common bridging O atom. Each of the tetrahedra was bonded to two Si atoms with dangling bonds saturated by H atoms. Calculated configuration of the double phosphorus center is shown in Fig. 2. According to the calculations, length of the O@P
, lengths of PAO bonds double bonds was 1.455 A


, lengths of in the PAOAP linkage was 1.601 A PAO and SiAO bonds in the PAOASi linkages
, respectively, O@PAO were 1.55 and 1.66 A angles were 111.8 in the PAOAP linkage and 116.4 in the PAOASi linkages, OAPAO angles between PAOAP and PAOASi linkages and between two PAOASi linkages were 103.2 and 104.0, respectively, PAOASi angles were 138.0. In general, the stretching vibrations of the O@P double bonds in the O@PAOAP@O double center interact with each other. As a result, two combined vibrational modes, co-phase and opposite-phase, arise. However, this interaction turns out to be weak and the frequencies of these combined modes differ slightly (less than by 2 cm
1 ). Calculated frequencies of these stretching vibrations are 1331 cm
1 , IR absorption and Raman intensities for each O@P bond in the double center are 8.8 and 2.1, respectively. Frequency of antisymmetric stretching vibration of the OAP bonds in the PAOAP linkage is 1120 cm
1 , IR absorption and Raman intensities are 6.5 and 0.3, respectively. Frequencies of antisymmetric stretching vibrations of the OAP and OASi bonds in the PAOASi linkage are 1102 and 1100 cm
1 , IR absorption intensities are 20.1 and 23.7, Raman intensities are 0.7 and 0.3, respectively.

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2.4. Vitreous P2 O5 The vibrational properties of v-P2 O5 network were simulated by calculating clusters of two types similar to the clusters used for single and double phosphorus centers in phosphosilicate glass. The first cluster, O@PðAOAH2 P@OÞ3 , contains one O@PO3 tetrahedron bonded with three O@P groups with dangling bonds of the P atom saturated by H atoms. The second cluster, ðO@PH2 A OÞ2 @ðO@PÞAOAðP@OÞ@ðOAH2 P@OÞ2 , contains two O@PO3 tetrahedra bonded to each other by common bridging O atom, each tetrahedron bonded with two O@P groups with dangling bonds of the P atom saturated by H atoms. In the calculated equilibrium configuration of the first cluster the O@P double bond length was
, the PAO bond lengths were 1.574 A
, 1.446 A O@PAO angles were 115.7, OAPAO angles were 102.6, PAOAP angles were 130.8. In the calculated equilibrium configuration of the second cluster the O@P double bond lengths were 1.449
, the PAO bond lengths were 1.571 A
, O@PAO A angles were 112.3, OAPAO angles were 103.0, PAOAP angles were 130.9. So the calculation reproduces the mean geometrical parameters of v-P2 O5 and II and III orthorhombic crystalline polymorphs of P2 O5 [12,19]. The calculation resulted in the following vibrational properties of the first cluster: the O@P bond stretching vibration frequency was 1357 cm
1 , IR absorption and Raman intensities were 6.1 and 2.1, respectively, antisymmetric stretching vibration frequency of OAP bonds in the PAOAP linkage was 1024 cm
1 , IR absorption and Raman intensities were 25.5 and 1.2, respectively. In the second cluster, similar to the O@PAOAP@O double center, the O@P double bond stretching vibrations interact with each other giving rise to co-phase and opposite-phase combined vibrational modes. This interaction is considerably stronger than that in the O@PAOAP@O double center, the calculated frequencies of these modes being 1391 and 1386 cm
1 , respectively. IR absorption intensities are 7.8 and 8.0, and Raman intensities are 1.9 and 2.5, respectively. Frequency of antisymmetric stretching vibration of OAP bonds in the PAOAP linkage is 1020 cm
1 , IR absorption

and Raman intensities are 25.4 and 1.7, respectively. Thus, the calculations suggest that there is a general trend of change of vibrational frequencies and Raman intensities in the process of association of O@PO3 tetrahedra: frequency of the O@P bond stretching vibrations and the corresponding Raman intensity increase achieving their maxima in v-P2 O5 , the frequency of antisymmetric stretching vibrations of OAP bonds in the PAOAP linkages practically does not change, the corresponding Raman intensity increasing. These changes of frequencies and intensities are most pronounced for the association of two single O@PO3 tetrahedra and become considerably lower with the number of the tetrahedra increasing.

3. Experiment We have measured the Raman spectra of two sets of fibers with a core made of phosphosilicate glass, (P2 O5 )x (SiO2 )1
x , manufactured by MCVD method in different laboratories [9,23]. According to measurements of the refractive index profile in the fiber preforms, the maximal P2 O5 concentration in the fiber cores was 4.6, 8.5, 14.2, 15.0 mol% in the set 1 and 6.6, 8.8, 11.0, 13.2, 14.7 mol% in the set 2 [9,23]. 1 Besides, we have also measured the Raman spectrum of a phosphosilicate-core fiber manufactured using the SPCVD method [24]. And finally, the Raman spectra of a fiber with 9 mol% of P2 O5 in the core manufactured in the same laboratory and by the same technology, as the set 1 fibers, were measured before and after the 244 nm UV irradiation by KrF laser with the dose density of about 1 kJ cm
2 (see Ref. [25]). The Stokes–Raman scattering spectra excited by the 514.5 nm light of an Ar laser 2 were measured using a triple spectrograph 3 with a spectral resolution of about 1 cm
1 . The spectra were measured without regard for the polarization of

1

The refractive index measurements are described in Ref.

[23]. 2 3

Spectra Physics Stabilite 2000. Jobin Yvon T64000.

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215

both incident and scattered light. The laser beam was focused by a microscope 4 upon a spot of 1–3 lm in diameter into the core at the input end of the fiber under investigation. The core diameter of the fibers was about 7 lm. The length of the fibers was about 5 m. A backward geometry was used for collecting of the laser light scattered from the fiber core. The experimental approach is described in Ref. [26] as well. To analyze the vibrational properties and to compare them with the results of quantum-chemical calculations, it is convenient to use the reduced Raman spectra defined as follows [27]: Ire ðxÞ ¼ I e ðxÞ

x 4

ðX0
xÞ ðnðxÞ þ 1Þ

ð1Þ

where Ire ðxÞ is the reduced Raman intensity, I e ðxÞ is the measured Raman intensity, x is the Stokes shift, X0 is the frequency of the incident light, nðxÞ is the Bose distribution function. The reduced experimental Raman spectra for the fibers sets 1 and 2 are given in Figs. 3 and 4, respectively. Fig. 5 shows the influence of UV irradiation on the reduced spectrum of the phosphosilicate-core fiber.

Fig. 3. Reduced Raman spectra of the set 1 fibers with the maximal P2 O5 concentration in the core: (a) 4.6 mol%, (b) 8.5 mol%, (c) 14.2 mol%, (d) 15.0 mol%.

3.1. Decomposition of experimental Raman spectra For our purposes the points of the greatest interest are the Raman band with the frequency of the maximum at about 1320 cm
1 caused by stretching vibrations of O@P double bonds and an adjacent frequency range J 860 cm
1 with the Raman bands caused by antisymmetric stretching vibrations of the SiAO and PAO bonds in SiAOASi, PAOASi and PAOAP linkages (bridging O atoms moving parallel to the SiASi or SiAP lines) [8,25]. This spectral range is separated from the lower-frequency part and hence can be analyzed separately. To interpret Raman bands and to understand their relation to the glass structure we have decomposed the reduced spectra in the 860–1460 cm
1 frequency range in components described by

4

Olympus BH2-UMA.

Fig. 4. Reduced Raman spectra of the set 2 fibers with the maximal P2 O5 concentration in the core: (a) 6.6 mol%, (b) 8.8 mol%, (c) 11.0 mol%, (d) 13.2 mol%, (e) 14.7 mol%.

Voigt functions. Various decompositions of each spectrum were analyzed using the v2 criterion. We

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V.G. Plotnichenko et al. / Journal of Non-Crystalline Solids 306 (2002) 209–226

Fig. 5. Reduced Raman spectra of the fiber with 9 mol% of P2 O5 in the core: (a) before UV irradiation, (b) after UV irradiation (KrF laser at 244 nm, dose density about 1 kJ cm
2 ).

have found that no more than five components can be determined uniquely from each of the experimental spectra to ensure both the required accuracy of decomposition relevant to experimental accuracy of the measured spectra and unambiguity of the decomposition. According to the quantumchemical calculations cited above, two components describe the O@P vibrational band (1320 cm
1 ) and other three components describe the bands of SiAOASi, PAOASi and PAOAP linkage vibrations. For greater number of components either the errors of the Voigt function parameters turned

out to be unsuitably high or the decomposition became ambiguous, in other words, there were at least two different sets of components providing similar (in the sense of the v2 criterion) approximation of the experimental spectrum. The main results of the decomposition are given in Tables 3 and 4. As an example of the decomposition, Fig. 6 shows the experimental spectrum and the fitting curves for one of the set 2 fibers (13.2 mol% of P2 O5 in the core). Relative intensities of the decomposition components are shown in Figs. 7 and 8 versus P2 O5 concentration in glass for the fibers of the sets 1 and 2. Notice that for each fiber the intensities of all the components are normalized to the intensity of the 1330 cm
1 component (O@P stretching bonds in the double phosphorus center). To interpret the decompositions obtained and, in particular, to study the concentration dependencies (relative) Raman intensities of vibrational modes in phosphosilicate glass network are required. We shall use the Raman intensities obtained in the above-described quantum-chemical modeling.

4. Discussion The frequencies of two decomposition components of the experimental Raman band at 1320 cm
1 given in Tables 3 and 4 agree well with the calculated frequencies of the O@P bond stretching vibrations in the single and double phosphorus centers, respectively. Therefore it is reasonable to suppose that the components of this band are mainly caused just by the single and double

Table 3 Results of decomposition of Raman spectra of the set 1 fibers in the 880–1460 cm
1 frequency range 4.6 mol%

8.5 mol%

14.2 mol%

15.0 mol%

Frequency

Raman intensity

Frequency

Raman intensity

Frequency

Raman intensity

Frequency

Raman intensity

1 2 3 4 5

1327.8  1.4 1314.2  1.1 1179.5  9.2 1137.6  2.3 1027.7  2.6

1.00  0.03 0.15  0.04 0.48  0.15 0.13  0.09 0.16  0.01

1331.5  1.2 1317.3  1.1 1179.1  3.8 1145.7  2.2 1022.3  1.3

1.00  0.01 0.15  0.02 0.39  0.02 0.06  0.01 0.16  0.01

1329.7  1.2 1313.7  1.1 1188.0  4.4 1149.3  2.3 1017.4  1.4

1.00  0.01 0.15  0.02 0.30  0.02 0.10  0.02 0.15  0.03

1330.5  1.2 1314.9  1.1 1192.0  8.9 1149.0  2.3 1018.8  1.7

1.00  0.02 0.15  0.01 0.29  0.05 0.12  0.03 0.14  0.05

v2

0:20 10
3
1

0:30 10
3

Frequencies in cm , Raman intensities are normalized to the 1330 cm

1:50 10
3
1

component for each fiber.

2:00 10
3

0:35 10
4

1331.5  1.1 1317.4  1.1 1180.5  2.0 1147.7  1.3 1020.4  1.3

1.00  0.01 0.12  0.01 0.27  0.01 0.18  0.02 0.14  0.01

217

0:65 10
4 0:40 10
4 v2

Frequencies in cm
1 , Raman intensities are normalized to the 1330 cm
1 component for each fiber.

1:10 10
4 0:92 10
4

1329.6  1.1 1315.4  1.1 1188.4  3.0 1144.9  1.2 1018.1  1.5 1.00  0.01 0.13  0.01 0.39  0.01 0.11  0.01 0.16  0.01 1.00  0.01 0.12  0.02 0.50  0.01 0.07  0.01 0.17  0.01 1330.0  1.1 1316.8  1.1 1171.4  1.5 1142.0  1.1 1019.5  1.3 1 2 3 4 5

1330.6  1.1 1316.2  1.1 1183.1  1.9 1144.8  1.1 1017.8  1.3

1.00  0.01 0.12  0.01 0.44  0.01 0.10  0.01 0.16  0.01

1328.2  1.1 1314.5  1.1 1180.0  1.9 1141.5  1.1 1016.8  1.3

1.00  0.01 0.12  0.01 0.31  0.02 0.15  0.01 0.14  0.01

Frequency Frequency

13.2 mol%

Raman intensity 11.0 mol%

Frequency Frequency

Raman intensity 8.8 mol%

Raman intensity 6.6 mol%

Table 4 Results of decomposition of Raman spectra of the set 2 fibers in the 880–1460 cm
1 frequency range

Frequency

Raman intensity

14.7 mol%

Raman intensity

V.G. Plotnichenko et al. / Journal of Non-Crystalline Solids 306 (2002) 209–226

Fig. 6. Decomposition of the reduced Raman spectrum of the set 2 fiber with 13.2 mol% of P2 O5 in the core. Open circles are the experimental points, thin lines are the components of the decomposition, thick line is the result of the fitting.

phosphorus centers. Using the intensities of the decomposition components of this band and the calculated Raman intensities for the O@P bond stretching vibrations in the single and double phosphorus centers, one may estimate roughly the ratio of concentrations of such centers. It turns out that in the investigated phosphosilicate glasses there are more double centers than single ones. On the average, the approximate ratios between the concentration of the double centers and that of the single ones are 3 in the set 1 fibers, 4 in the set 2 fibers, and 2 in the SPCVD-manufactured fiber. More precise estimations are given in what follows. On the strength of the results of our calculations and the analysis of concentration dependencies of the intensities we have attributed other components of the experimental Raman spectra (Figs. 3–5) to SiAOASi linkages (1185 cm
1 ), PAOASi and SiAOASi linkages (1025 cm
1 ), and PAOAP linkages (1150 cm
1 ), as shown in Fig. 9. Our interpretation differs from the only original interpretation of the Raman spectra of phosphosilicate glasses [8] we know. In Ref. [8] the bands with the maxima at frequencies about 1200

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Fig. 7. Experimental dependence of relative intensities of the decomposition components on P2 O5 concentration in the set 1 fibers cores (for each concentration the intensities of all the components are normalized to the intensity of the  1330 cm
1 component): (j) 1330 cm
1 component; ( ) 1317 cm
1 component; (N) 1185 cm
1 component; (.) 1150 cm
1 component; (r) 1025 cm
1 component.

Fig. 8. Experimental dependence of relative intensities of the decomposition components on P2 O5 concentration in the set 2 fibers cores (for each concentration the intensities of all the components are normalized to the intensity of the 1330 cm
1 component): (j) 1330 cm
1 component; ( ) 1317 cm
1 component; (N) 1185 cm
1 component; (.) 1150 cm
1 component; (r) 1025 cm
1 component.

and 1020 cm
1 are attributed to PAOAP linkages, the band at 1145 cm
1 ––to PAOASi linkages, and nothing is said about the SiAOASi linkages at all. Most likely, the interpretation given in Ref. [8] is based on the data for (ultra) phosphate glasses and v-P2 O5 (notice that our calculations give for the frequency of antisymmetric stretching of OAP bonds in PAOAP linkages in v-P2 O5 just 1020 cm
1 ). However, as is shown below, the assignments made in Ref. [8] contradict to the dependencies of the Raman band intensities on P2 O5 concentration in glass. In general, using all the data on the intensities of decomposition components of the experimental Raman spectra and the calculated Raman intensities for the phosphorus centers and v-SiO2 one is able to estimate not only the ratio between concentrations of the single and double phosphorus centers but also the relative concentrations of O@P bonds and of SiAOASi, PAOASi and PAOAP linkages, and even the P2 O5 molar concentration in phosphosilicate glass, and hence, the absolute

Fig. 9. An interpretation of the components of the experimental Raman spectrum of a phosphosilicate-core fiber (shown as measured, i.e. not reduced).





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concentrations of all these bonds and linkages. However, this is possible only on the basis of certain additional assumptions concerning the structure of the glass network. 4.1. Model of phosphosilicate glass network A model of phosphosilicate glass network based on the assumption of a complete absence of correlations in arrangement of phosphorus atoms is developed in Ref. [7,28]. The model allows one to find relative concentrations of O@P bonds and of SiAOASi, PAOASi and PAOAP linkages in this glass. We have extended this model to the case when there are no other phosphorus centers in glass except for double and single centers (both the single and the double centers may either be isolated or form any groups). In this model the structure of phosphosilicate glasses expressed by the formula (P2 O5 )x (SiO2 )1
x is conveniently presented as ððIÞn ðIIÞ1
n Þx (SiO2 )1
x where (I) and (II) are single and double phosphorus centers, respectively, and n is a parameter of relative concentration of these centers. The above-mentioned ratios between double and single phosphorus centers allow one to estimate roughly the n parameter to be 0.25 and 0.20 on the average for set 1 and set 2 fibers, respectively, and 0.30 for the SPCVD-manufactured fiber. In phosphosilicate glass (with P2 O5 molar concentration equal to x) described by such model, the relative concentrations, fi , of single (I) and double (II) phosphorus centers, O@P double bonds and SiAOASi, PAOASi and PAOAP linkages in respect to total number, N, of bridging linkages in glass are as follows: fðIÞ ¼ 12nxð1
xÞN
1 ; i h fðIIÞ ¼ 8ð1
nÞxð1
xÞ þ ð2 þ nÞ2 x2 N
1 ; fO@P ¼ ½8ð1
nÞxð1
xÞ þ 2ð2 þ nÞxð2 þ nxÞ N
1 ; fSiAOASi ¼ 4ð1
xÞ2 N
1 ; fPAOASi ¼ 4ð2 þ nÞxð1
xÞN
1 ; i h fPAOAP ¼ 8ð1
nÞxð1
xÞ þ 2ð2 þ nÞ2 x2 N
1 ; ð2Þ

219

where N ¼ 4ð1
xÞ2 þ 4ð2 þ nÞxð1
xÞ þ 8ð1
nÞ xð1
xÞ þ 2ð2 þ nÞ2 x2 . These expressions for relative concentrations, fi , are derived without considering obvious dependence of phosphosilicate glass density on P2 O5 content. In our model this dependence should be derived from experimental data and it may be readily taken into account. Unfortunately no such data have come to our notice. However, since the density of v-P2 O5 [20] is not too different from that of v-SiO2 , one would expect that the phosphosilicate glass density depends only slightly on P2 O5 content, at least for P2 O5 concentrations up to 15 mol%. The relative concentrations, fi (2), and corresponding Raman intensities derived using fi and the quantum-chemically calculated intensities for each phosphorus center and linkage (see Table 2) are shown versus molar P2 O5 concentration in Figs. 10 and 11, respectively, for the parameter of relative concentration of single and double phosphorus centers, n, equal to 0.2. To make it possible to compare the calculated and experimental results, the relative concentrations are normalized to the concentration of O@P bonds in the double centers, and the relative intensities, similar to the experimental Raman spectra, are normalized to the total intensity of O@P bond stretching vibrations in the double center. To illustrate the behavior of the relative concentrations, fi , they are shown for the complete range of P2 O5 molar concentration, from 0 to 100%, while corresponding Raman intensities are given for the range from 1 to 15 mol% actual for our measurements. The comparison of Figs. 10 and 11 with the results of decomposition of experimental Raman spectra proves qualitatively the validity of assignments of the components in the 860–1200 cm
1 range to antisymmetric stretching vibrations of certain linkages, made in foregoing calculations, since the intensities of the components do depend on P2 O5 molar concentration in accordance with our model. On the contrary, the assignments made in Ref. [8] contradict this model since the intensity of the component attributed in Ref. [8] to the PAOAP linkages decreases with growth of P2 O5 concentration.

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Fig. 10. The relative concentrations in the phosphosilicate glass model for the parameter of relative concentration of the single and double phosphorus centers n ¼ 0:2: (a) O@P bonds in the single phosphorus centers (fðIÞ ), (b) O@P bonds in the double phosphorus centers (2fðIIÞ ), (c) total concentration of O@P bonds (fðIÞ þ 2fðIIÞ ), (d) SiAOASi linkages (fSiAOASi ), (e) PAOASi linkages (fPAOASi ), (f) PAOAP linkages (fPAOAP ).

Fig. 11. Calculated Raman intensities in the phosphosilicate glass model for the parameter of relative concentration of the single and double phosphorus centers n ¼ 0:2: (a) O@P bonds in the single phosphorus centers, (b) O@P bonds in the double phosphorus centers, (c) total intensity of Raman scattering on O@P bonds, (d) SiAOASi linkages (LO-type vibrations), (e) PAOASi linkages and SiAOASi linkages (TO-type vibrations of the latters), (f) PAOAP linkages.

4.2. Estimations of concentrations Now we are to estimate the relation between the single and double phosphorus centers as well as P2 O5 molar concentration from our Raman measurements. The model considered in the previous section together with the calculated Raman intensities for each phosphorus center and linkage makes it possible to obtain the dependencies between the n parameter and P2 O5 molar concentration for experimental relative Raman intensities of O@P bond stretching vibrations in single phosphorus centers and of antisymmetric stretching vibrations of SiAO bonds in SiAO linkages. These dependencies make it possible to find both the relative concentration parameter of the single and double phosphorus centers, n, and P2 O5 molar concentration by solving a system of two nonlinear equations. Such dependencies are shown in the Figs. 12 and 13 for MCVD-manufactured fibers of the set 1 and 2, respectively, and in Fig. 14 for the SPCVD-

manufactured fiber. These figures illustrate graphically the solution: the points where the respective lines cross give the P2 O5 molar concentration and the n parameter for each fiber (look for further explanations in the figures captions). Notice that only the close vicinity of the cross point are shown in the figures. For the set 1 fibers with the P2 O5 molar concentration in the core, according to the refractive index measurements, being 4.6, 8.5, 14.2 and 15.0 mol% we have obtained the following estimations of P2 O5 molar concentration: 6:1  3:0, 7:7  0:5, 10:1  0:7 and 10:5  2:5 mol%, respectively. For the set 2 fibers with P2 O5 molar concentration in the core, according to the refractive index measurements, being 6.6, 8.8, 11.0, 13.2 and 14.7 mol%, the estimations 5:6  0:3, 6:5  0:4, 7:3  0:4, 9:3  0:6 and 10:7  0:7 mol%, respectively, are obtained. For the SPCVD-manufactured fiber, the P2 O5 concentration is estimated to be 1:9  0:7 mol%.

V.G. Plotnichenko et al. / Journal of Non-Crystalline Solids 306 (2002) 209–226

221

Fig. 12. Determining P2 O5 molar concentration and the relative concentration parameter of the single and double phosphorus centers, n, in the set 1 fibers on the basis of experimental data on the relative Raman intensities of the O@P bonds stretching vibrations in the single phosphorus centers (nearly vertical lines) and the relative Raman intensities of the antisymmetric stretching vibrations of the SiAO bonds in the SiAOASi linkages (nearly horizontal lines). Bold lines are the mean values, thin lines are the confidence intervals for the fibers with assumed P2 O5 molar concentrations: solid lines – 4.6%, dash line – 8.5%, dot lines – 14.2%, dash dot lines – 15.0%. The point of respective lines crossing give P2 O5 concentration and the n parameter for each fiber.

Fig. 13. Determining P2 O5 molar concentration and the relative concentration parameter of the single and double phosphorus centers, n, in the set 2 fibers on the basis of experimental data on the relative Raman intensities of the O@P bonds stretching vibrations in the single phosphorus centers (nearly vertical lines) and the relative Raman intensities of the antisymmetric stretching vibrations of the SiAO bonds in the SiAOASi linkages (nearly horizontal lines). Bold lines are the mean values, thin lines are the confidence intervals for the fibers with assumed P2 O5 molar concentrations: solid lines – 6.6%, dash line – 8.8%, dot lines – 11.0%, dash dot lines – 13.2%, dash double dot lines – 14.7%. The point of respective lines crossing give P2 O5 concentration and the n parameter for each fiber.

Within the limits of the accuracy of our analysis, the n parameter turn out to be one and the same for all the set 1 fibers (0:20  0:03) and for all the set 2 fibers (0:16  0:02). Notice that these values are close enough to each other (again within our accuracy). For the SPCVD-manufactured fiber the n parameter is 0:28  0:07. So it seems reasonably safe to suggest that the relation between the single and double phosphorus centers is determined mainly by the manufacturing process and does not depend practically on the P2 O5 concentration. Evidently the values of P2 O5 molar concentration in the fiber cores obtained on the basis of our approach turn out to be systematically lower than the results based on the refractive index measurements. This is explained by a non-uniform radial

distribution of P2 O5 in the fiber cores [23]. The measurements of the Raman spectra in fibers provide the Raman intensity averaged over the core. It is reasonable to suggest that for homogeneous samples the approach based on the Raman spectra analysis and other methods of measurement of P2 O5 concentration should give close results. 4.3. Influence of UV irradiation In our previous article [25] the experimental frequencies were obtained for the most intensive Raman bands in phosphosilicate glass caused by phosphorus centers with the O@P double bond, intensities of those decreasing considerably after UV irradiation. In the frequency range

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are connected to fourfold-coordinated phosphorus atoms, and hence, in decrease of the Raman intensity at about 1150 cm
1 , since the frequencies and Raman intensities for vibrations of PAOAP and PAOASi linkages connected to fivefold-coordinated phosphorus atoms are lower than those of the same linkages connected to fourfold-coordinated atoms [25]. The analysis of experimental Raman spectra of the fibers before and after UV irradiation allows us to conclude that

Fig. 14. Determining P2 O5 molar concentration and the relative concentration parameter of the single and double phosphorus centers, n, in the SPCVD-manufactured fiber on the basis of experimental data on the relative Raman intensities of the O@P bonds stretching vibrations in the single phosphorus centers (nearly vertical lines) and the relative Raman intensities of the antisymmetric stretching vibrations of the SiAO bonds in the SiAOASi linkages (nearly horizontal lines). Bold lines are the mean values, thin lines are the confidence intervals. The point of respective lines crossing give P2 O5 concentration and the n parameter.

860–1460 cm
1 there are two of these bands, 1320 and 1150 cm
1 . From the aforesaid it follows that the first band is caused by the stretching vibrations of O@P double bonds of different types while the second one is mainly contributed by an antisymmetric stretching vibration of PAO bonds in the PAOAP linkages and, to a lesser degree, by the same vibrations of PAO and SiAO bonds in the PAOASi linkages. According to Ref. [25], the reduction of the 1320 cm
1 band intensity after UV irradiation is caused by a decrease in the O@P double bond concentration owing to the transition of a part of phosphorus atoms from the fourfold-coordinated form (O@PO3 ) into the fivefold-coordinated form (PO5 ) in a photoinduced reaction of non-bridging oxygen atoms with SiAOASi linkages. Clearly, such changes result in decrease of concentration of those PAOAP and PAOASi linkages, which

• Raman intensity of the 1320 cm
1 band, and hence the total concentration of O@P double bonds decreases by three times after UV irradiation; • the ratio between intensities of the 1320 cm
1 band components changes considerably: the low-frequency component intensity decreases only slightly, and practically all the reduction of the Raman band total intensity is caused by a decrease in the high-frequency component intensity; • the frequencies of the Raman band components do not change. It is reasonable to assume that under UV irradiation the O@P double bonds disappear both in single phosphorus centers and in double ones. In the double phosphorus center either one O@P bond or both bonds may disappear. In the first case a new phosphorus center arises, with one O@P double bond and a PAOAP linkage between fourfold- and fivefold-coordinated phosphorus atoms. Our calculation shows that in this center the frequency and the Raman scattering intensity of the O@P bond stretching vibration are close to those in the single phosphorus center and the frequency of the Raman scattering intensity of the antisymmetric stretching vibrations of PAO bonds in the PAOAP linkage are close to those in the double phosphorus center. Hence the new phosphorus centers do contribute to the low-frequency component of the 1320 cm
1 band. So reactions of three types may occur under UV irradiation, with the O@P double bonds disappearing in each of them. Unfortunately, basing only on the decomposition of the 1320 cm
1 band before and after irra-

V.G. Plotnichenko et al. / Journal of Non-Crystalline Solids 306 (2002) 209–226

diation, it is impossible to estimate the fractions of the phosphorus centers participating in these reactions. However one can find the limiting fractions of the phosphorus centers participating in reactions of disappearing of one and two O@P bonds. If all the single centers disappear under UV irradiation then one double bond turns out to disappear in 50% of the double centers and both double bonds turn out to disappear in 6% of the centers. If no single centers disappear at all under UV irradiation then one double bond disappears in 35% of the double centers and both double bonds disappear in 25% of them. To estimate the fraction of the single centers disappearing in the real situation one should use the data on changes in intensities of components of the Raman spectrum in the range 860–1200 cm
1 caused by the linkages of different types. However the PAOAP and PAOASi linkages connected with both fourfold-coordinated and fivefold-coordinated phosphorus atoms must be taken into account to decompose the Raman spectra in this range in the UV-irradiates glasses. In other words, at least two extra components must be taken into account. The experimental accuracy available gives no way to perform such a decomposition. Nevertheless, the decrease of Raman scattering near 1150 cm
1 due mainly to a reduction of concentration of the PAOAP linkages connected at least with one fourfold-coordinated phosphorus atom suggests the second limiting case to be more close to the reality. 4.4. On the estimations of Raman cross section in phosphosilicate glass Analyzing the operation of a stimulated Raman laser based on phosphosilicate-core fiber the authors of Ref. [3] make three assumptions being of interest for our discussion, namely 1. the laser generation occurs independently on P2 O5 and SiO2 fiber core glass constituents; 2. the Raman gain coefficient for the 1320 cm
1 band is proportional to P2 O5 molar concentration and that for the 440 cm
1 band––to SiO2 molar concentration;

223

3. the maximum value of the Raman gain coefficient for the 1320 cm
1 band corresponds to the value measured for v-P2 O5 in Ref. [10]. Under these assumptions Raman gain coefficients were estimated for the wavelengths 1.24 and 1.31 lm and the conclusion was made that the Raman cross section ratio for the 1320 cm
1 band in v-P2 O5 and for the 440 cm
1 band in v-SiO2 was considerably higher than that obtained in Ref. [10]. The assumptions made in Ref. [3] seem to be incorrect. Firstly, the laser generation cannot occur independently on P2 O5 and SiO2 glass constituents. The matter is that, besides the vibrational modes, which actually are classified as vibrations of P2 O5 and SiO2 constituents, there are the combined modes caused by interaction of vibrations of the glass constituents. The simplest of them are the vibrations of PAOASi linkages with frequency varying approximately in the limits 440–480 cm
1 with P2 O5 concentration growth [25]. The corresponding Raman band overlaps with the 440 cm
1 band of the v-SiO2 network caused mainly by vibrations of SiAOASi linkages. Secondly, as follows from the foregoing, the Raman gain coefficient for the 1320 cm
1 band is not proportional to P2 O5 molar concentration in the glass since the gain is determined by the concentration of O@P bonds at least of two types causing this band. And finally, strictly speaking, it is not correct to use the Raman cross section for v-P2 O5 since, as it follows from the results of above quantum-chemical modeling, the Raman cross section on a O@P bond increases with associating O@PO3 tetrahedra together: the Raman intensity is minimum in single phosphorus center of phosphosilicate glass and it is maximum for v-P2 O5 (see Table 2). Since the fiber used in Ref. [3] was made in the same laboratory and by the same technology as the set 2 fibers, it is safe to assume that n  0:2 for this fiber. For P2 O5 concentration x  13 mol% [3] the relative concentrations of various bonds in the fiber core turn out to be fI  0:06, fII  0:16, fSiAOASi  0:62 and fPAOASi  0:20. Using the calculated ratio 2.3 between Raman intensities for O@P bonds in double and single phosphorus centers and assuming the Raman intensity for the

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O@P bond in v-P2 O5 to be equal to the corresponding intensity for the double phosphorus center, we conclude that the ratio between complete Raman cross sections for the O@P bond stretching vibrations in phosphosilicate glass under investigation and in v-P2 O5 is 2.8. It is somewhat higher than the value 2.6 found in Ref. [3]. This seems natural since the Raman intensity for the O@P bond in v-P2 O5 is higher than in the double phosphorus center. So, as a matter of fact, there is no contradictions with the results of Ref. [10]. On the other hand, the relative amount of SiAOASi linkages is well below the unity (0.62) so it seems too low to explain the Raman gain in the 440 cm
1 band observed in Ref. [3] (the experimental value of the Raman gain coefficient corresponds to the concentration fSiAOASi  0:95). This implies that more than 30% of the gain in this band is caused by other contributions, namely, as follows from aforesaid, by PAOASi linkages. Unfortunately, the clusters used for the quantumchemical modeling are insufficiently large to calculate exactly the ratio between the Raman intensities for PAOASi and SiAOASi linkage vibrations contributing to this band. Nevertheless the calculations allow us to estimate roughly the ratio between these intensities as 1.5–3. With fSiAOASi and fPAOASi taken into account this just results in the contribution of PAOASi linkages of the order of 30%.

5. Summary In this work the vibrational properties of phosphosilicate glasses with P2 O5 concentration up to 15 mol% manufactured by the MCVD and SPCVD methods were investigated using the Raman spectroscopy and quantum-chemical modeling. The main results and conclusions consist in the following: • the experimental Raman spectra can be decomposed in the frequency range 860–1460 cm
1 in five components; • the Raman band at 1320 cm
1 typical for phosphosilicate glasses is not simple but contains at least two components, their frequencies being

1317 and 1330 cm
1 in the investigated glasses and the relative intensity ratio depending on manufacturing techniques; • the low-frequency component of this Raman band is caused by single phosphorus centers (O@PO3 tetrahedra) and the high-frequency one––by double phosphorus centers (pairs of O@PO3 tetrahedra bonded together by common oxygen atom). The investigated phosphosilicate glasses contain 2–4 times as much double centers as single ones. Other three components of the Raman spectra are caused by Si–O–Si (1185 cm
1 ), PAOASi and SiAOASi (1025 cm
1 ) and PAOAP (1150 cm
1 ) linkages; • up to 60% of the double phosphorus centers are destroyed under 244 nm UV irradiation. In this work we have proposed a model of the network of phosphosilicate glass allowing one to calculate the concentrations of phosphorus and silicon sites, O@P double bonds and SiAOASi, PAOASi and PAOAP linkages for any P2 O5 concentration in the glass. We have shown that using this model together with the results of decomposition of experimental Raman spectra allows one to find both the P2 O5 content and the concentration of all these species in the glass. The main object of research in this work has been the optical fibers drawn from preforms manufactured by CVD methods for which, as is noted above, the formation of double phosphorus centers seems natural. In the framework of our approach the analysis of the Raman spectra of phosphosilicate glasses manufactured by other methods is of obvious interest. In our opinion, such approach to the analysis of Raman spectra may be used to optimize and control any process of manufacturing either phosphosilicate glasses, or other similar glasses and optical fibers on their basis.

Acknowledgements The authors are grateful to M.M. Bubnov, K.M. Golant, A.N. Guryanov, G.A. Ivanov and V.F. Khopin for the phosphosilicate-core fibers and for valuable discussions.

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Appendix A. Model of phosphosilicate glass network A model of phosphosilicate glass network developed in Refs. [7,28] allows one to find relative concentrations of O@P bonds and of SiAOASi, PAOASi and PAOAP linkages in such glass. The model is based on the assumption of a complete absence of correlations in arrangement of phosphorus atoms in the glass network. The main assumption of the present work is that there are two types of phosphorus centers in the phosphosilicate glass, single phosphorus centers (O@PO3 tetrahedra bonded only with the SiO4 tetrahedra) and double phosphorus centers (pairs of the O@PO3 tetrahedra bound together by PAOAP linkages). We have extended the model of Refs. [7,28] to such a case. The main points of our model are as follows. (1) There are no other phosphorus centers in glass except for double and single centers. The structure of phosphosilicate glass with composition (P2 O5 )x (SiO2 )1
x is presented as ððIÞn ðIIÞ1
n Þx (SiO2 )1
x where (I) and (II) are single and double phosphorus centers, respectively, and n is a parameter of relative concentration of these centers. (2) Both the single and the double centers may either be isolated or form any groups. Two single centers are joined together by a PAOAP linkage forms a double center. A single center joined by a PAOAP linkage with a double one forms a pair of double centers, and so on. So there are three structural units in the phosphosilicate glass: (a) SiO2 unit with four SiAO bonds; (b) P2 O5 single-center unit with six ðIÞAO bonds; (c) P2 O5 double-center unit with four ðIIÞAO bonds. Probabilities of formation of these bonds are denoted as p0 , p1 , and p2 , respectively. (3) There are six types of oxygen linkages in the glass: (a) SiAOASi, (b) SiAOAðIÞ, (c) SiAOAðIIÞ,

225

(d) ðIÞAOAðIÞ, (e) ðIÞAOAðIIÞ, (f) ðIIÞAOAðIIÞ. Probabilities of formation of these linkages are denoted as p1 , p2 , p3 , p4 , p5 , and p6 , respectively. The probabilities pi and pi are related by obvious formulae: p1 ¼ p20 N
1 , p2 ¼ 2p0 p1 N
1 , p3 ¼ 2p0 p2 N
1 , p4 ¼ p21 N
1 , p5 ¼ 2p1 p2 N
1 , p6 ¼ p22 N
1 . The normalizing factor, N
1 , is nothing but a complete number of bridging linkages of any type in phosphosilicate glass. In phosphosilicate glass with composition ((I)n (II)1
n )x (SiO2 )1
x the bond formation probabilities are readily shown to be p0 ¼

2ð 1
x Þ 2 þ nx

p1 ¼

3nx 2 þ nx

p2 ¼

2ð1
nÞx : 2 þ nx

The relative concentrations, fi , of single (I) and double (II) phosphorus centers, O@P double bonds and SiAOASi, PAOASi and PAOAP linkages in respect to total number, N, of bridging linkages in the phosphosilicate glass are expressed in terms of the pi probabilities as follows: fðIÞ ¼ p2

fðIIÞ ¼ p3 þ p4 þ p5 þ p6

fO@P ¼ p2 þ 2ðp3 þ p4 þ p5 þ p6 ÞfSiAOASi ¼ p1 fPAOASi ¼ p2 þ p3 fPAOAP ¼ p3 þ 2ðp4 þ p5 þ p6 Þ and the total number of linkages N ¼ p1 þ p2 þ 2ðp3 þ p4 þ p5 þ p6 Þ: The explicit formulae for the relative concentrations, fi , of single (I) and double (II) phosphorus centers, O@P double bonds and SiAOASi, PAOASi and PAOAP linkages and the total number, N, of bridging linkages in the phosphosilicate glass derived from these expressions are given in Section 4 (see (2) and what follows).

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