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Vibrational spectra of rings in silicate glasses
] C ' U R N A L OF
ELSEVIER
Journal of Non-Crystalline Solids 168 (1994) 265-274
Vibrational spectra of rings in silicate glasses B. M i h a i l o v a *, N . Z o t o v
1, M . M a r i n o v , J. N i k o l o v , L. K o n s t a n t i n o v
Institute of Applied Mineralogy, Bulgarian Academy of Sciences, Rakovski Str. 92, 1000 Sofia, Bulgariu
(Received 19 May 1993; revised manuscript received 30 September 1993)
Abstract
lsotropic, HH and HV Raman as well as infrared spectra of isolated n-membered rings (n = 3, 4, 5, 6, 7) of ideal silicon-oxygen tetrahedra are calculated. The dependences of the spectral line frequencies and intensities on the number of tetrahedra in the ring as well as on the type and the degree of structural disorder are studied. The results show that the ring order influences the mode localization, thus changing the spectral line positions and intensities, while the structural disorder (ring puckering and topology) affects mainly the vibrations of the bridging oxygen atoms changing the frequencies and intensities of the peaks corresponding to the modes localized in this type of atom. The effect of the surrounding matrix upon vibrational properties of the rings is briefly discussed and a better, but by no means full, agreement with experiment is obtained by taking into account this effect.
1. Introduction
Among all methods for investigating the structure of disordered materials, R a m a n and infrared (IR) spectroscopies are widely employed [1]. The main approach in treating such spectra, however, usually consists in comparing them with data on corresponding crystalline analogues for which the theory of lattice dynamics is well developed [2,3]. This approach, although effective in some cases, cannot avoid the principal difficulties arising when studying amorphous materials in which long-range order is lacking [4]. For such materials, the only reasonable theoretical approach is to present their
Present address: Institute of Crystallography, University of Munich, Theresienstr. 41, 8000 Munich 2, Germany. * Corresponding author. Tel. +359-2 872 450. Telefax: + 359-2 834 979.
structure as being built of relatively simple structural units and to analyze the contribution of the modes of vibrations in these units to the spectrum of the entire system. This theoretical approach has been used by numerous authors in various modifications which can be classified in several groups: (i) molecular approximation [5-11] in which the coupling between the structural units is entirely neglected; (ii) computer modelling of the vibrational spectra of large clusters of atoms [12-15] which requires complicated and computer-lime consuming calculations; (iii) analytical and numerical calculations for small clusters with Bethe lattice boundary conditions [16,17] which give quite satisfactory results for the vibrational density of states of network glasses such as SiO 2, but which can hardly be extended to various systems since it is not
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B. Mihailova et aL /Journal of Non-CrystaUine Solids 168 (1994) 265-274
266
straightforward to construct appropriate average Bethe lattices and to solve the corresponding matrix equations for the Green's function; (iv) the nearest-neighbour central force model [18,19] which predicts with considerable accuracy the frequency limits of the vibrational bands and the number of states in each band for A X 2 and AX 3 glasses, but ignores the non-central forces and the disorder in the bridging and dihedral angles. The simplest structural units in tetrahedrally bonded network glasses are rings of T O 4 tetrahedra (T = Si, A1, P, . . . ) deformed a n d / o r oriented differently to each other. However, there are relatively few theoretical studies of rings in framework glasses [3,20-27], treating only specific aspects of the effect of ring structure on the vibrational spectra, e.g., the interpretation of the defect D 1 and D 2 lines in the HH-polarized Raman spectra of a-SiO 2 [28-32]. Recently a computational model for calculating Raman and IR spectra of isolated rings of SiO 4 tetrahedra with different values of the S i - O - S i bridging angle and topological disorder has been reported [331. The aim of the present paper is to discuss in detail the theoretically calculated polarized and isotropic Raman as well as IR spectra of rings of various order, puckering and topological disorder. The effect of the surrounding matrix is also considered.
Cartesian coordinate indices, m; is the mass of the ith atom, ric, is the a t h component of the position vector of the ith atom and U is the potential energy of the system. Using a Keatingtype potential, the dynamical matrix, D, can be presented as a sum of terms corresponding to stretching and bending forces between the nearest neighbours [10]. The elements of the stretching dynamical matrix O str, are defined as [7]
Fij,a~ = E
Kin rin'a rin'¢~,
n~i m i
rij,a rij,13 x 1/2 ' ~, m i m j ) rij rij
~
O bend, a r e
D ij,a~ bend : ~ij( F ij,ctfl ' " h -[- (1 -- 6ij ) + F ij,ctfll ×
, Fij,a[3 =
2.1. Modes of vibration
1
Dij,,t3
Fin
where K;j is the stretching force constant between atoms i and j: rij = I r# I, rij = ri-rj; 6ij is the Kroneker symbol. The elements of the bending dynamical matrix,
Fi~,~t3 =
The normal mode calculations are based on the well known method for determining the eigenfrequencies and eigenvectors of the equations of motion through direct diagonalization of the dynamical matrix [34] defined as
rin Kij
Sij,afl -
t
Sitj,afl =
f!
ill
( Sij,c~l3 "}- Sij,a ~ q- Sij,a[3),
(3)
/3piq hqi,a) E 2 m i (hpi,a qp~i,qvsi p4=q ×
2. Modelling
(2)
o sij,otl3 t r = ~ijFij,a B q_ (1 -- ~ij)Sij,oqs,
+
/3ipq
y"
--hip,~hip,t3(ipCqp,
p4=i,q4:i m i p4:q
E /3isj his ahjs fl¢isfjs, sv=i,j ( m i m j ) 1/2 ' ' --
/3ijs
-
Si~'~ = ~*i j
(mimj) 1/2 hij'"(hij't3 + hsi't3)fiY~sY'
siT, o, = s~i,j E
( m i m/3s; j ) 1/2 (hsi'a +
02U
(mimj)l/2 Ori~Orjt3 D str + D bend,
(1)
where i, j indicate the atom positions (i, j = 1,..,N, N is the number of atoms), a, /3 are the
hyi'")hji'ogsiCji"
Here /3isj is the bending force constant between atoms i, j, s, when the sth atom is on the top of
B. Mihailova
e t aL / J o u r n a l
of Non-Crystalline
the angle, 4) between ris and rjs, hi~,,, is determined by =
'
ri s cos2~b
ris
cos & -
rjs
I
(4)
~ij ~-
Here the index 1 denotes the vibrational modes, cot is the frequency of the lth mode and Tl and G t are invariants of the polarizability tensor Pt defined as T, --- ½Sp(P,),
and the definition of ~'~j is 1, O,
(8a)
G f = 3[(P.,, 1 -
K~j v~ 0 Kij = 0.
(5)
P 22,l) ,
-~- 3 [ ( P 1 2 / ) 2
The calculation of the intensities of the individual Raman modes follows the general principles given by Bell [14]. Knowing the eigenfrequencies and eigenvectors of the dynamical matrix the polarizability tensor, P/, can be found 1
pm
P,13,, c ~ -u 2 - - R[ m m m
_
m
m
m
m
,
1,t3J+PZ~,t3J]
(6)
m
P I , ~ , t - a h q~ q~ bl • q,,, pot,, m . , ~ j = ~ ( 1a mb + a m - a t m )(b,~,tq ~m + b , mj q , rn _ 2qmq~b,~ . q m ) .
IR(CO,) CXTl 2,
(7a)
IHH (Oil)
(7b)
O~
7 G ~ + 45T~ 2,
1HV(cot) cz 6G 2.
(7c)
p
-]- ( 1 3 , / )
2
-~-(
p
2 233)]
.
(8b)
The infrared intensity for the lth mode with frequency wl is directly determined from the atom vector displacements by [10] I r a ( w , ) cx ~ (
zm'u7 '' + zm"u'~ '') .
(9)
where z m' and z " " are the charges of the atoms connected by the mth bond. Finally, to calculate the spectrum profile, the lines were broadened by Lorentz functions of a half-width, F(cot), given according to Refs. [8,14] by
r( ,,,,)
Here u is the volume of the structural unit, the upper index, m, denotes the valence bonds, R m is the length of the mth bond, b~n is the vector describing the change of the mth bond due to the atom vibrations with eigenfrequency w l (e.g., b 7' m p m" t . = u t - u t , w h e r e u 7' and u 7' are the displacement vectors of the atoms connected by the mth bond), qm is the unit vector along the mth bond, m ab, anm are the parallel polarizabilities of the mth bond produced by bonding and non-bonding electrons, respectively, and a t is the transverse polarizability of the ruth bond. The summation in Eq. (6) is over all the bonds in the structural unit. m am The bond polarizabilities, a6, n and a t , have been calculated using the method of Lippincott and Stutman [35]. The isotropic and polarized Raman intensities, I R, IHH and IHV, are given by [34]
+ (P22j-P33,l)
+ ( P33,/- Plnl):]
2.2. Vibrational intensities
1
267
Solids 168 (1994) 265-274
( rl( co,) /co,),
(lO)
where II(co t) is the participation ratio for the eigenfrequency w t. On the basis of the method described above, a computer program for an I B M / P C was developed [36], which calculates the vibrational spectra of a particular structural unit and displays the atom vibrations in a given mode.
3. Results
3.1. Basic structural units The theoretical model described in the previous section was applied to calculate the isotropic and polarized Raman as well as IR spectra of planar and puckered rings of different number, n, of regular S i O 4 tetrahedra (n is referred to hereafter as ring order). For such rings, all O - S i O angles, &, are equal to the tetrahedral value &V = 109"47° and all S i - O bond lengths to the average S i - O distance in amorphous SiO2,
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
268
reduces the bridging angle values with respect to those for planar rings. The topological configurations of rings can be specified by the mutual orientation and the sign of ~ of the individual tetrahedra. The symmetry o f s u c h r i n g s is d e t e r m i n e d b y t h e n u m b e r , m , o f t e t r a h e d r a w i t h i n v e r s e o r i e n t a t i o n [33]. A b i n i t i o m o l e c u l a r o r b i t a l c a l c u l a t i o n s [38] h a v e i n d i c a t e d that such configurations can substantially decrease the stabilization energy of the rings. All normal mode calculations were carried out with the following values of the force constants: K'= 400 N/m for the Si-Obr bond stretching, K"= 500 N/m for the Si-Ono n bond stretching ( O b r a n d Ono n d e n o t e b r i d g i n g a n d n o n - b r i d g i n g o x y g e n s , r e s p e c t i v e l y ) , 1 3 ' = 50 N / m f o r t h e O Si-O bond bending and /3"= 5 N/m for the
( R s i _ o ) = 1.62 A [37]. F o r p l a n a r r i n g s o f o r d e r n , all S i - O - S i bridging angles, 7 n= 360 ° (1( 1 / n ) ) - ~ b T, a n d t h e r i n g s a r e o f p o i n t g r o u p s y m m e t r y Dnh. By tilting the tetrahedra along their bridging o x y g e n e d g e s a t a n a n g l e o f p u c k e r i n g , r/, o n e c a n f o r m v a r i o u s p u c k e r e d r i n g s . W h e n all t e t r a h e d r a are tilted in one and the same direction at the s a m e a n g l e ~7 ( t h e s o - c a l l e d r e g u l a r l y p u c k e r e d r i n g s ) , t h e y a r e o f p o i n t g r o u p s y m m e t r y Cnv a n d have bridging angles 7n(r/) = 2 arcsin(~l
-cos2(~br/2
) sin2(r/) }
× sin(yl~(-q)/2),
(11)
where 7~(r/) = 360°(1-1/n)-2 arctg(tg ( ~ b T / 2 ) / c o s r/). A s s e e n , t h e p u c k e r i n g a l w a y s
Table 1 Calculated eigenfrequencies and type of the vibrational modes of isolated n-membered planar rings Activity a
Frequency (cm- 1) b n=3
n=4
Type of the vibrational modes c n=5
n=6
n=7
R1 R2 R3 R4
930 s 625 vs 485 s 297 s
932 s 569 m 483 s 267 s
938 s 583 s 459 s 227 m
943 s 600 s 446 s 192 m
948 s 615 s 440 s 164 m
HI H2 H3 H4 H5 H6 H7 H8
1057 s 1008 s 876 vw 726 vw 492 vw 428 vw 301 w 317 m
1076 s 1010 s 805 w 772 w 495 vw 425 vw 178 w 305 m
1078 w 1011 s 839 m 700 w 497 w 434 vw 329 vw 298 s
1078 vw 1012 vs 869 s 640 vw 498 w 467 vw 386 vw 294 s
1077 m 1013 vs 890 s 598 vw 499 w 508 vw 399 w 291 s
H9 H10
245 m
235 m 111 vw
189 m 82 vw
181 m 59 vw
172 W 44 vw
IR1 IR2 IR3 IR4 IR5 IR6 IR7 IR8
1057 s 1014 s 876 m 726 m 501 s 428 m 301 vw 245 w
1098 s 1014 s 923 s 516 s 501 s 563 s 406 s 177 vw
1096 s 1014 s 933 s 448 s 501 s 589 s 406 s 159 vw
1087 s 1014 s 900 m 626 s 501 s 465 s 389 w 223 w
1097 s 1014 s 909 m 559 s 501 s 523 s 404 m 198 w
sym S i - O n o n bond stretching 'breathing' of Obr atoms O - S i - O bond bending ring 'breathing' S i - O b r bond stretching
asym Si-Ono n bond stretching Si-O bond stretching ring 'elongation' Si-Obr bond rocking ring 'elongation' ring 'elongation' Si-Ononbond stretching and bending S i - O n o n bond rocking Si-Ono n bond rocking asym Si-Obr bond stretching asym Si-Ono n bond stretching Si-O bond stretching Si-O-Si bond bending Si-Obr bond rocking Si-O-Si and O - S i - O bond bending ~rv-asym O vibrations Si-Ono n bond rocking
a R, modes active in the isotropic and polarized Raman spectrum; H, modes active only in the polarized Raman spectra; IR, infrared-active modes. b w, weak; s, strong; m, medium; v, very. c sym, symmetrical; asym, antisymmetrical.
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
S i - O - S i bond bending, which are the same as those used by Furukawa et al. [10]. 3.2. The effect o f ring order, n
To study the effect of ring order, n, on the isotropic Raman, H H Raman, H V Raman and IR spectra of rings, one can restrict the consideration only to the case of planar rings since the general behaviour of this effect is preserved for puckered rings. The symmetry of the Raman and IR active modes of vibration of n-membered planar rings obtained by the theory of irreducible representations [33] indicates that the total number of modes of a given symmetry does not depend on n. Most generally, the ring order, n, influences only the localization of the vibrational modes in different types of atom since the angle determining the bending force field depends on n. This influence appears in the spectra as a change in the peak position, intensity and width on varying n. The results of the calculations for planar rings with n = 3, 4, 5, 6 and 7 are given in Table 1, where the eigenfrequencies of the modes are listed together with the corresponding type of vibrations and their activity (R indicates modes active in the isotropic and polarized Raman spectrum, H such active only in the polarized Raman spectra, and IR the infrared-active modes). As seen from these data, in accordance with the predictions of the group theory, there are four modes active in the isotropic Raman spectrum which are not active in the IR spectrum since they preserve the symmetry of the structural unit. In addition, there are ten modes active only in the polarized Raman spectra. It should be noted that, except for the modes H2, H5 and H8 which are ~rh-antisymmetric, the other vibrational modes are non-degenerate for n = 4, while they are doubly degenerate for n = 3, 5, 6, 7. This fact, as well as the absence of the mode H10 for n = 3, is in good agrement with the results of the group theory analysis [33]. In the IR spectra, there are eight active modes, two of which are non-degenerate O-h-antisymmetric. The group theory predicts that, in the vibrational spectra of planar rings, there have to be four doubly degenerate ~rh-antisymmetric polar-
269
ized Raman modes and three non-degenerate Crh-antisymmetric IR active modes [33], while we have obtained, respectively, three and two such modes. The eigenfrequencies of the two modes, missing in the case of isolated planar rings, are approximately zero. It is worth noting that the ring order affects most strongly the frequency of the modes IR4 and IR6 corresponding to S i - O - S i bond bending. 3.3. The effects o f disorder
A comparison of calculated modes for planar rings with experimental data shows that none of the calculated spectra of planar rings are observed in the spectra of actual silicate glasses, i.e., the structure of such glasses cannot be modelled in terms of planar rings only. In fact, the results reported by others [20,21,38-42] have shown that rings with differing deformation can exist in silicate glasses. This fact makes it necessary to study in detail the effects of the non-planarity of the rings on their vibrational spectra. This type of disorder can be introduced in two ways. On the one hand, one can consider regularly puckered rings (see Fig. 1) for which the degree of disorder is given by the angle of puckering, r / ( t h e disorder in the S i - O - S i bridging angles in silicate glasses is then associated with random puckering of the rings). On the other hand, all the tetrahedra can be tilted at the same 77, but in different directions, i.e., the disorder in this case can be specified by the sign of 77, the number of inversely oriented tetrahedra and their relative position in the ring. This latter type of configuration repre-
Fig. 1. Geometry of a regularlypuckered four-membered ring, the angle of puckering "O= 320.
270
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
sents a simple but convenient model for studying the topological disorder in silicate glasses. 3.3.1. The effect o f ring puckering To study the effect of puckering, we calculated the polarized and isotropic R a m a n and I R spectra for n - m e m b e r e d regularly puckered rings (n = 3,4 . . . . . 7) by varying ~7 in the interval from 0 to 35 ° . This interval was found to demonstrate the effect of r/. Since the results for the polarized R a m a n spectra do not differ substantially from those for the isotropic R a m a n spectrum and, in addition, the change in n does not affect the character of the corresponding spectra, we consider below only the results obtained for the isotropic R a m a n and I R spectra of four-membered rings. As a whole, the puckering does not change the type of the modes existing in the planar rings and the modes given in Table 1 are present in puckered rings as well. In addition, the intensities of several modes which are zero for planar rings (not listed in Table 1) become, in accordance with the group theory [33], non-zero in the Raman a n d / o r I R spectra of puckered rings due to a decrease of symmetry from Dnh to Cnv. For the
a)
same reason, the modes R1, R2, R3 and R4, isotropic Raman-active for planar rings, become also IR-active for puckered rings, while the tr nantisymmetric IR-active modes IR2 and IR5 become also isotropic Raman-active for puckered rings. Except for the modes H2, H5, H8, IR2 and IR5 whose frequencies become increasingly dependent on n with increasing r/, the dependence of w on n for all other modes in the case of puckered rings is similar to that for planar rings. The calculated isotropic R a m a n spectra of puckered four-membered rings are shown in Fig. 2(a) for four different values of ft. As seen, the effect of puckering is most pronounced in the mid-frequency range (from about 400 to 700 c m - 1 ) on account of the intense Raman-active bond rocking mode IR5. For 7; 4:0 °, the intensity of this m o d e increases with increasing r/ due to an increase in the in-plane component of the atomic displacement vectors, while the intensity of the mode R2 decreases owing to the decrease of this component. This behaviour results in a resonant interaction of these two modes (Fig. 2(b)) at ~7 = 20 ° where the mode frequencies coincide (Fig. 2(c)). These modes exhibit a similar behaviour in the I R spectra as well.
b) •
•
-
•
i
-4--3 .,--t
0 l0 20 30 Angle of puckering (deg)
-4J C
c)
H
.
.
.
.
.
.
.
.
i
.
0 500 1000 Frequency (cm
5oo1-,
J
0 10 20 30 Ang]e of puckering (deg)
Fig. 2. (a) Calculated isotropic Raman spectrum of a regularly puckered four-membered ring for four different values of the angle of puckering -q. Dependence of the mid-frequency peak intensity, I (b), and of the peak position, to (c), on r/. e, mode R2; G, mode IR5. For details, see the text.
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
3.3.2. The effect of topology For the sake of brevity, the different topological ring configurations are specified below by 0 if for a particular tetrahedron 7/< 0, and 1 if r; > 0. For a four-membered ring, the case considered in detail below, there are consequently four possible topological configurations, namely 1111, 1110, 1100 and 1010 (e.g., the configuration 1100 contains two neighbouring tetrahedra oriented upwards and the next two downwards). The configuration 1111 corresponds to a puckered four-membered ring and will be used hereafter as a reference structure. The isotropic Raman and IR spectra calculated for these four configurations are shown in Fig. 3. As can be seen, the topological disorder changes the ring symmetry leading to a splitting of the mode frequencies. Since the mutual orientation of tetrahedra affects predominantly the vibration of the bridging oxygen atoms, the peaks corresponding to modes localized in such atoms are affected stronger than the other modes. For example, such strongly affected modes are IR1 at 1080 cm -1 resulting from Si-Obr bond stretching, or the mid-frequency modes corresponding to S i - O - S i bond bending. In general, the number of peaks in the vibrational spectra for a given topological configuration increases with decreasing point group order. Thus, for the 1110 configuration which has the lowest symmetry, in the case of four-membered rings the number of peaks
o
6oiobo
Frequency (cm-~)
obo Frequency (cm-~)
Fig. 3. Calculated isotropic Raman (a) and IR (b) spectra of the four possible topological configurations of a four-membered ring: 1111, 1110, 1100, 1010.
271
i
5
L
z
~0
56o
obo Frequency (cm-I)
Fig. 4. Measured IR absorption spectrum of fused silica.
both in the Raman and IR spectra is greatest. The results for rings of other order are similar to these. In fact, both the magnitude and the sign of r/ can change from one tetrahedron to the other, but this case does not provide information which may be easily and unequivocally interpreted and is not discussed.
4. Discussion
The effects of puckering and topological disorder of isolated rings, considered in Section 3.3., represent an idealized situation and can hardly be expected to describe, adequately, the case of real glasses. For example, the comparison between the calculated IR spectra (Fig. 3(b)) and experimentally measured spectra of fused silica (Fig. 4) shows that, while in the spectral ranges around 1100 and 500 cm ~ the agreement is satisfactory, the peak observed at about 800 cm 1 is not reproduced in the calculated spectrum. The same holds for the Raman spectra, namely the frequency of the peaks in the calculated spectra as well as the depolarization ratios in the midfrequency region obtained by us are close to those observed experimentally ([28]; Fig. 3), but as a whole the profiles of both spectra differ substantially from each other. This general behaviour does not change with the ring order, degree of puckering or topology.
272
B. Mihailova et aL /Journal of Non-Crystalline Solids 168 (1994) 265-274
a)
b)
-r--1
c~
v
~0 500
1000
0
500
1000
Frequency (cm-~)
Frequency (cm-~) c)
d)
==i ~0
~-'t 0
500 1000 Frequency (cm-I)
,
0
500
,
,
1000
Frequency (cm-~)
Fig. 5. Isotropic Raman (a), HH Raman (b), HV Raman (c) and IR (5d) spectra of a four-membered planar ring in which all non-bridging oxygen atoms are linked with hypothetical particles of an arbitrary mass.
A n e x t s t e p to i m p r o v e t h e m o d e l is to s p e c i f y boundary conditions for the peripheral atoms of t h e s t r u c t u r a l u n i t by c o r r e c t i n g t h e d y n a m i c a l
matrix elements. This correction can be realized m o s t g e n e r a l l y by c h a n g i n g e i t h e r t h e f o r c e c o n stants, t h e e f f e c t i v e m a s s o f a t o m s o r t h e i r local
a)
0
b)
500
1000
0
Frequency (cm-a)
500
1000
Frequency (cm-~)
c)
d)
2g
~
~0
~0
0
500
1000
Frequency (cm-I)
0
500
I000
Frequency (cm-a)
Fig. 6. Isotropic Raman (a), HH Raman (b), HV Raman (c) and IR (d) spectra of a four-membered planar ring with imposied boundary conditions as specified in the text (the ratio K/j8 is four times less than that used for isolated rings; four of the non-bridging O atoms are linked to the surrounding).
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
surrounding. In the case of isolated rings, the bridging and non-bridging oxygen atoms are distinguished from one another by the stretching force constants K ' and K", respectively. In a medium built of interconnected silicon-oxygen rings, this difference disappears but the calculations carried out with the same stretching force constant even for all the S i - O bonds do not change the results. The variation in the effective mass of the non-bridging O atoms does not provide reasonable data because it only shifts the peak positions as a whole and does not give rise to additional modes arising from the decrease in the degree of freedom of the system. A new weak low-frequency mode, in which the atoms of the ring vibrate normal to the plane of the contour of covalent bonds, appears when the dynamical matrix is corrected by changing the surrounding of the bonded non-bridging O atoms. For the purposes of this study, these atoms were assumed linked by the same stretching and bending force constants as previously specified to hypothetical particles (HP) of an arbitrary mass located along the Si-O,o n bond. The Raman and IR spectra of a four-membered planar ring calculated by imposing such boundary conditions are shown in Fig. 5. As seen, although this procedure improves the agreement between the calculated and the experimental positions of the high-frequency peaks [28], the main differences in the spectra remain. At the same time, variations in the mass of the HP as well as in their distance to the ring atoms do not change the spectral profiles. The afore-mentioned arguments show that the effect of the surrounding matrix on the vibrational density of states in glasses cannot be correctly described by the bonding of the non-bridging oxygen atoms, but a complete change of the dynamical matrix has to be introduced. For example, the correction of the dynamical matrix can be performed by changing the ratio K/~ for all the atoms in tetrahedra containing oxygen atoms bonded to the amorphous matrix. Fig. 6 shows the vibrational spectra of a four-membered planar ring in which four of the non-bridging oxygen atoms are linked and the ratios K/~s is four times smaller than that used in the case of iso-
273
lated rings. It is evident that the shape of the IR spectrum becomes similar to the spectra observed in silicate glasses (Fig. 4). The reduction of the ratio K/[3 leads also to an increase in the intensity ratio of the H H and HV modes in the midfrequency range, thus improving the agreement with experiment [28]. In addition, the intensity of the weak Raman peak at about 700 cm -1 increases, while the intensity ratio of the highfrequency and mid-frequency peaks in the spectrum decreases. Thus, the spectra, calculated with the reduced ratio K/B, approach those measured in silicate glasses, although differences remain. For example, the intensity ratio of the highfrequency and mid-frequency Raman peaks in the experimental spectra is smaller than that in the calculated spectra. One can further improve the fit to experiment or can model vibrational spectra of various glasses by specitying the imposed random boundary conditions. This trend lies, however, beyond the goal of the present paper and will be the subject of further investigations.
5. Conclusions
On the basis of a combined theoretical method for calculating the vibrational spectra of isolated rings of SiO 4 tetrahedra, the effects of ring order, puckering and topology are: (i) the ring order affects only the mode localization, thus changing the spectral line intensities and positions, (ii) the degree of puckering determines the value of the Si-O-Si angle and thus affects the Raman and IR peaks in the mid-frequency spectral range; and (iii) the ring topology determines the total number of Raman and IR active vibrations mainly affecting the modes localized in the bridging oxygen atoms and leading to a frequency splitting of these modes. By comparing the calculated spectra of rings and the experimental data on silicate glasses, various means for setting boundary conditions are proposed to account for the effect of the amorphous matrix and are shown to improve the
274
B. Mihailoua et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
agreement between calculated and experimental s p e c t r a . S u c h s t u d i e s a r e n o w in p r o g r e s s .
T h e f i n a n c i a l s u p p o r t o f this w o r k by t h e B u l garian Ministery of Science and Education under c o n t r a c t n u m b e r S R F - 2 1 3 is g r a t e f u l l y a c k n o w l edged.
6. References [1] D.L. Weaire, in: Topics in Current Physics, Vol. 24, Amorphous Solids, Low Temperature Properties, ed. W.A. Phillips (Springer, Berlin, 1981) p. 13. [2] E. Dowty, Phys. Chem. Miner. 14 (1987) 80 [3] S.K. Sharma, J.F. Mammone and M.F. Nicol, Nature 292 (1981) 140. [4] J.M. Ziman, Models of Disorder (Cambridge University, London 1979). [5] P.H. Gaskell, Phys. Chem. Glasses 8 (1967) 69. [6] J. Bock and G. Su, J. Am. Ceram. Soc. 53 (1970) 69. [7] J.B. Bates, J. Chem. Phys. 56 (1972) 1910. [8] S. Brawer, Phys. Rev. Bll (1975) 3173. [9] G. Lucovsky and R.M. Martin, J. Non-Cryst. Solids 8 (1978) 185. [10] T. Furukawa, K.E. Fox and W.B. White, J. Chem. Phys. 75 (1981) 3226. [11] E. Dowty, Phys. Chem. Miner. 14 (1987) 67. [12] R.J. Bell and P. Dean, Nature 212 (1966) 1354. [13] R.J. Bell, Rep. Prog. Phys. 35 (1972) 1315. [14] R.J. Bell, Meth. Comp. Phys. 15 (1976) 215. [15] R.A. Murray and W.Y. Ching, Phys. Rev. B39 (1989) 1320. [16] R.B. Langhlin and J.D. Joanmoponlos, Phys. Rev. B16 (1977) 2942. [17] R.A. Bario, F.L. Galeener and E. Martinez, Phys. Rev. B31 (1985) 7779. [18] P.N. Sen and M.F. Thorpe, Phys. Rev. B15 (1977) 4030. [19] M.F. Thorpe and F.L. Galeener, Phys. Rev. B22 (1980) 3078.
[20] F.A. Seifert, B.O. Mysen and D. Virgo, Am. Mineral. 67 (1982) 696. [21] S.K. Sharma, J.A. Phillpotts and D.W. Matson, J. NonCryst. Solids 71 (1985) 403. [22] A.M. Prima, Opt. Spectrosc. 9 (1960) 236. [23] B.D. Saksena, K.C. Agarwal and G.S. Jauhri, Trans. Faraday Soc. (1963) 276. [24] J.D. Kubicki and D. Sykes, Phys. Chem. Miner. 19 (1993) 381. [25] R.N. Nucho and A. Madhukar, Phys. Rev. B21 (1980) 1576. [26] P. Dean, Nature 210 (1966) 257. [27] S.V. King, Nature 213 (1967) 1112. [28] F.L. Galeener, Solid State-Commun. 44 (1982) 1037. [29] F.L. Galeener, J. Non-Cryst. Solids 49 (1982) 53. [30] F.L. Galeener and M.F. Thorpe. Phys. Rev. B28 (1983) 5802. [31] F.L. Galeener, R.A. Barrio, E. Martinez and R.J. Elliott, Rhys. Rev. Lett. (1984) 2430. [32] S.W. de Leeuw, H. He and M.F. Thorpe, Solid State Commun. 56 (1985) 343. [33] N. Zotov, B. Mihailova, M. Marinov and L. Konstantinov, in: Proc. Int. Workshop on the Dynamics of Disordered Materials II ILL, Grenoble, Mar. 1993, Physica A, in press. [34] E.B. Wilson, J.C. Decius and P.C. Cross, Molecular Vibrations (McGraw-Hill, New York, 1962). [35] E.R. Lippincott and J.M. Stutman, J. Phys. Chem. 68 (1964) 2926. [36] M. Marinov, B. Mihailova, J. Nikolov, N. Zotov and L. Konstantinov, in: Proc. 7th Int. School on Condensed Matter Physics, Varna, Bulgaria, Sept. 1992 (World Scientific, Singapore) p. 601. [37] R.L. Mozzi and B.E. Warren, J. Appl. Crystallogr. 2 (1969) 164. [38] B.C. Chakoumakos, R.J. Hill and G.V. Gibbs, Am. Mineral. 66 (1981) 1237. [39] A.C. Wright and J.A. Erwin Desa, Phys. Chem. Glasses 19 (1978) 140. [40] R.J. Bell and P. Dean, Philos. Mag. 25 (1972) 1381. [41] H.-X. He, J. Non-Cryst. Solids 89 (1987) 403. [42] M. Wada, K. Kamiya, H. Nasu, J. Matsuoka, T. Yoko, T. Fukunaga and M. Misawa, J. Non-Cryst. Solids 149 (1992) 203.
ELSEVIER
Journal of Non-Crystalline Solids 168 (1994) 265-274
Vibrational spectra of rings in silicate glasses B. M i h a i l o v a *, N . Z o t o v
1, M . M a r i n o v , J. N i k o l o v , L. K o n s t a n t i n o v
Institute of Applied Mineralogy, Bulgarian Academy of Sciences, Rakovski Str. 92, 1000 Sofia, Bulgariu
(Received 19 May 1993; revised manuscript received 30 September 1993)
Abstract
lsotropic, HH and HV Raman as well as infrared spectra of isolated n-membered rings (n = 3, 4, 5, 6, 7) of ideal silicon-oxygen tetrahedra are calculated. The dependences of the spectral line frequencies and intensities on the number of tetrahedra in the ring as well as on the type and the degree of structural disorder are studied. The results show that the ring order influences the mode localization, thus changing the spectral line positions and intensities, while the structural disorder (ring puckering and topology) affects mainly the vibrations of the bridging oxygen atoms changing the frequencies and intensities of the peaks corresponding to the modes localized in this type of atom. The effect of the surrounding matrix upon vibrational properties of the rings is briefly discussed and a better, but by no means full, agreement with experiment is obtained by taking into account this effect.
1. Introduction
Among all methods for investigating the structure of disordered materials, R a m a n and infrared (IR) spectroscopies are widely employed [1]. The main approach in treating such spectra, however, usually consists in comparing them with data on corresponding crystalline analogues for which the theory of lattice dynamics is well developed [2,3]. This approach, although effective in some cases, cannot avoid the principal difficulties arising when studying amorphous materials in which long-range order is lacking [4]. For such materials, the only reasonable theoretical approach is to present their
Present address: Institute of Crystallography, University of Munich, Theresienstr. 41, 8000 Munich 2, Germany. * Corresponding author. Tel. +359-2 872 450. Telefax: + 359-2 834 979.
structure as being built of relatively simple structural units and to analyze the contribution of the modes of vibrations in these units to the spectrum of the entire system. This theoretical approach has been used by numerous authors in various modifications which can be classified in several groups: (i) molecular approximation [5-11] in which the coupling between the structural units is entirely neglected; (ii) computer modelling of the vibrational spectra of large clusters of atoms [12-15] which requires complicated and computer-lime consuming calculations; (iii) analytical and numerical calculations for small clusters with Bethe lattice boundary conditions [16,17] which give quite satisfactory results for the vibrational density of states of network glasses such as SiO 2, but which can hardly be extended to various systems since it is not
0022-3093/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0022-3093(93)E0279-H
B. Mihailova et aL /Journal of Non-CrystaUine Solids 168 (1994) 265-274
266
straightforward to construct appropriate average Bethe lattices and to solve the corresponding matrix equations for the Green's function; (iv) the nearest-neighbour central force model [18,19] which predicts with considerable accuracy the frequency limits of the vibrational bands and the number of states in each band for A X 2 and AX 3 glasses, but ignores the non-central forces and the disorder in the bridging and dihedral angles. The simplest structural units in tetrahedrally bonded network glasses are rings of T O 4 tetrahedra (T = Si, A1, P, . . . ) deformed a n d / o r oriented differently to each other. However, there are relatively few theoretical studies of rings in framework glasses [3,20-27], treating only specific aspects of the effect of ring structure on the vibrational spectra, e.g., the interpretation of the defect D 1 and D 2 lines in the HH-polarized Raman spectra of a-SiO 2 [28-32]. Recently a computational model for calculating Raman and IR spectra of isolated rings of SiO 4 tetrahedra with different values of the S i - O - S i bridging angle and topological disorder has been reported [331. The aim of the present paper is to discuss in detail the theoretically calculated polarized and isotropic Raman as well as IR spectra of rings of various order, puckering and topological disorder. The effect of the surrounding matrix is also considered.
Cartesian coordinate indices, m; is the mass of the ith atom, ric, is the a t h component of the position vector of the ith atom and U is the potential energy of the system. Using a Keatingtype potential, the dynamical matrix, D, can be presented as a sum of terms corresponding to stretching and bending forces between the nearest neighbours [10]. The elements of the stretching dynamical matrix O str, are defined as [7]
Fij,a~ = E
Kin rin'a rin'¢~,
n~i m i
rij,a rij,13 x 1/2 ' ~, m i m j ) rij rij
~
O bend, a r e
D ij,a~ bend : ~ij( F ij,ctfl ' " h -[- (1 -- 6ij ) + F ij,ctfll ×
, Fij,a[3 =
2.1. Modes of vibration
1
Dij,,t3
Fin
where K;j is the stretching force constant between atoms i and j: rij = I r# I, rij = ri-rj; 6ij is the Kroneker symbol. The elements of the bending dynamical matrix,
Fi~,~t3 =
The normal mode calculations are based on the well known method for determining the eigenfrequencies and eigenvectors of the equations of motion through direct diagonalization of the dynamical matrix [34] defined as
rin Kij
Sij,afl -
t
Sitj,afl =
f!
ill
( Sij,c~l3 "}- Sij,a ~ q- Sij,a[3),
(3)
/3piq hqi,a) E 2 m i (hpi,a qp~i,qvsi p4=q ×
2. Modelling
(2)
o sij,otl3 t r = ~ijFij,a B q_ (1 -- ~ij)Sij,oqs,
+
/3ipq
y"
--hip,~hip,t3(ipCqp,
p4=i,q4:i m i p4:q
E /3isj his ahjs fl¢isfjs, sv=i,j ( m i m j ) 1/2 ' ' --
/3ijs
-
Si~'~ = ~*i j
(mimj) 1/2 hij'"(hij't3 + hsi't3)fiY~sY'
siT, o, = s~i,j E
( m i m/3s; j ) 1/2 (hsi'a +
02U
(mimj)l/2 Ori~Orjt3 D str + D bend,
(1)
where i, j indicate the atom positions (i, j = 1,..,N, N is the number of atoms), a, /3 are the
hyi'")hji'ogsiCji"
Here /3isj is the bending force constant between atoms i, j, s, when the sth atom is on the top of
B. Mihailova
e t aL / J o u r n a l
of Non-Crystalline
the angle, 4) between ris and rjs, hi~,,, is determined by =
'
ri s cos2~b
ris
cos & -
rjs
I
(4)
~ij ~-
Here the index 1 denotes the vibrational modes, cot is the frequency of the lth mode and Tl and G t are invariants of the polarizability tensor Pt defined as T, --- ½Sp(P,),
and the definition of ~'~j is 1, O,
(8a)
G f = 3[(P.,, 1 -
K~j v~ 0 Kij = 0.
(5)
P 22,l) ,
-~- 3 [ ( P 1 2 / ) 2
The calculation of the intensities of the individual Raman modes follows the general principles given by Bell [14]. Knowing the eigenfrequencies and eigenvectors of the dynamical matrix the polarizability tensor, P/, can be found 1
pm
P,13,, c ~ -u 2 - - R[ m m m
_
m
m
m
m
,
1,t3J+PZ~,t3J]
(6)
m
P I , ~ , t - a h q~ q~ bl • q,,, pot,, m . , ~ j = ~ ( 1a mb + a m - a t m )(b,~,tq ~m + b , mj q , rn _ 2qmq~b,~ . q m ) .
IR(CO,) CXTl 2,
(7a)
IHH (Oil)
(7b)
O~
7 G ~ + 45T~ 2,
1HV(cot) cz 6G 2.
(7c)
p
-]- ( 1 3 , / )
2
-~-(
p
2 233)]
.
(8b)
The infrared intensity for the lth mode with frequency wl is directly determined from the atom vector displacements by [10] I r a ( w , ) cx ~ (
zm'u7 '' + zm"u'~ '') .
(9)
where z m' and z " " are the charges of the atoms connected by the mth bond. Finally, to calculate the spectrum profile, the lines were broadened by Lorentz functions of a half-width, F(cot), given according to Refs. [8,14] by
r( ,,,,)
Here u is the volume of the structural unit, the upper index, m, denotes the valence bonds, R m is the length of the mth bond, b~n is the vector describing the change of the mth bond due to the atom vibrations with eigenfrequency w l (e.g., b 7' m p m" t . = u t - u t , w h e r e u 7' and u 7' are the displacement vectors of the atoms connected by the mth bond), qm is the unit vector along the mth bond, m ab, anm are the parallel polarizabilities of the mth bond produced by bonding and non-bonding electrons, respectively, and a t is the transverse polarizability of the ruth bond. The summation in Eq. (6) is over all the bonds in the structural unit. m am The bond polarizabilities, a6, n and a t , have been calculated using the method of Lippincott and Stutman [35]. The isotropic and polarized Raman intensities, I R, IHH and IHV, are given by [34]
+ (P22j-P33,l)
+ ( P33,/- Plnl):]
2.2. Vibrational intensities
1
267
Solids 168 (1994) 265-274
( rl( co,) /co,),
(lO)
where II(co t) is the participation ratio for the eigenfrequency w t. On the basis of the method described above, a computer program for an I B M / P C was developed [36], which calculates the vibrational spectra of a particular structural unit and displays the atom vibrations in a given mode.
3. Results
3.1. Basic structural units The theoretical model described in the previous section was applied to calculate the isotropic and polarized Raman as well as IR spectra of planar and puckered rings of different number, n, of regular S i O 4 tetrahedra (n is referred to hereafter as ring order). For such rings, all O - S i O angles, &, are equal to the tetrahedral value &V = 109"47° and all S i - O bond lengths to the average S i - O distance in amorphous SiO2,
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
268
reduces the bridging angle values with respect to those for planar rings. The topological configurations of rings can be specified by the mutual orientation and the sign of ~ of the individual tetrahedra. The symmetry o f s u c h r i n g s is d e t e r m i n e d b y t h e n u m b e r , m , o f t e t r a h e d r a w i t h i n v e r s e o r i e n t a t i o n [33]. A b i n i t i o m o l e c u l a r o r b i t a l c a l c u l a t i o n s [38] h a v e i n d i c a t e d that such configurations can substantially decrease the stabilization energy of the rings. All normal mode calculations were carried out with the following values of the force constants: K'= 400 N/m for the Si-Obr bond stretching, K"= 500 N/m for the Si-Ono n bond stretching ( O b r a n d Ono n d e n o t e b r i d g i n g a n d n o n - b r i d g i n g o x y g e n s , r e s p e c t i v e l y ) , 1 3 ' = 50 N / m f o r t h e O Si-O bond bending and /3"= 5 N/m for the
( R s i _ o ) = 1.62 A [37]. F o r p l a n a r r i n g s o f o r d e r n , all S i - O - S i bridging angles, 7 n= 360 ° (1( 1 / n ) ) - ~ b T, a n d t h e r i n g s a r e o f p o i n t g r o u p s y m m e t r y Dnh. By tilting the tetrahedra along their bridging o x y g e n e d g e s a t a n a n g l e o f p u c k e r i n g , r/, o n e c a n f o r m v a r i o u s p u c k e r e d r i n g s . W h e n all t e t r a h e d r a are tilted in one and the same direction at the s a m e a n g l e ~7 ( t h e s o - c a l l e d r e g u l a r l y p u c k e r e d r i n g s ) , t h e y a r e o f p o i n t g r o u p s y m m e t r y Cnv a n d have bridging angles 7n(r/) = 2 arcsin(~l
-cos2(~br/2
) sin2(r/) }
× sin(yl~(-q)/2),
(11)
where 7~(r/) = 360°(1-1/n)-2 arctg(tg ( ~ b T / 2 ) / c o s r/). A s s e e n , t h e p u c k e r i n g a l w a y s
Table 1 Calculated eigenfrequencies and type of the vibrational modes of isolated n-membered planar rings Activity a
Frequency (cm- 1) b n=3
n=4
Type of the vibrational modes c n=5
n=6
n=7
R1 R2 R3 R4
930 s 625 vs 485 s 297 s
932 s 569 m 483 s 267 s
938 s 583 s 459 s 227 m
943 s 600 s 446 s 192 m
948 s 615 s 440 s 164 m
HI H2 H3 H4 H5 H6 H7 H8
1057 s 1008 s 876 vw 726 vw 492 vw 428 vw 301 w 317 m
1076 s 1010 s 805 w 772 w 495 vw 425 vw 178 w 305 m
1078 w 1011 s 839 m 700 w 497 w 434 vw 329 vw 298 s
1078 vw 1012 vs 869 s 640 vw 498 w 467 vw 386 vw 294 s
1077 m 1013 vs 890 s 598 vw 499 w 508 vw 399 w 291 s
H9 H10
245 m
235 m 111 vw
189 m 82 vw
181 m 59 vw
172 W 44 vw
IR1 IR2 IR3 IR4 IR5 IR6 IR7 IR8
1057 s 1014 s 876 m 726 m 501 s 428 m 301 vw 245 w
1098 s 1014 s 923 s 516 s 501 s 563 s 406 s 177 vw
1096 s 1014 s 933 s 448 s 501 s 589 s 406 s 159 vw
1087 s 1014 s 900 m 626 s 501 s 465 s 389 w 223 w
1097 s 1014 s 909 m 559 s 501 s 523 s 404 m 198 w
sym S i - O n o n bond stretching 'breathing' of Obr atoms O - S i - O bond bending ring 'breathing' S i - O b r bond stretching
asym Si-Ono n bond stretching Si-O bond stretching ring 'elongation' Si-Obr bond rocking ring 'elongation' ring 'elongation' Si-Ononbond stretching and bending S i - O n o n bond rocking Si-Ono n bond rocking asym Si-Obr bond stretching asym Si-Ono n bond stretching Si-O bond stretching Si-O-Si bond bending Si-Obr bond rocking Si-O-Si and O - S i - O bond bending ~rv-asym O vibrations Si-Ono n bond rocking
a R, modes active in the isotropic and polarized Raman spectrum; H, modes active only in the polarized Raman spectra; IR, infrared-active modes. b w, weak; s, strong; m, medium; v, very. c sym, symmetrical; asym, antisymmetrical.
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
S i - O - S i bond bending, which are the same as those used by Furukawa et al. [10]. 3.2. The effect o f ring order, n
To study the effect of ring order, n, on the isotropic Raman, H H Raman, H V Raman and IR spectra of rings, one can restrict the consideration only to the case of planar rings since the general behaviour of this effect is preserved for puckered rings. The symmetry of the Raman and IR active modes of vibration of n-membered planar rings obtained by the theory of irreducible representations [33] indicates that the total number of modes of a given symmetry does not depend on n. Most generally, the ring order, n, influences only the localization of the vibrational modes in different types of atom since the angle determining the bending force field depends on n. This influence appears in the spectra as a change in the peak position, intensity and width on varying n. The results of the calculations for planar rings with n = 3, 4, 5, 6 and 7 are given in Table 1, where the eigenfrequencies of the modes are listed together with the corresponding type of vibrations and their activity (R indicates modes active in the isotropic and polarized Raman spectrum, H such active only in the polarized Raman spectra, and IR the infrared-active modes). As seen from these data, in accordance with the predictions of the group theory, there are four modes active in the isotropic Raman spectrum which are not active in the IR spectrum since they preserve the symmetry of the structural unit. In addition, there are ten modes active only in the polarized Raman spectra. It should be noted that, except for the modes H2, H5 and H8 which are ~rh-antisymmetric, the other vibrational modes are non-degenerate for n = 4, while they are doubly degenerate for n = 3, 5, 6, 7. This fact, as well as the absence of the mode H10 for n = 3, is in good agrement with the results of the group theory analysis [33]. In the IR spectra, there are eight active modes, two of which are non-degenerate O-h-antisymmetric. The group theory predicts that, in the vibrational spectra of planar rings, there have to be four doubly degenerate ~rh-antisymmetric polar-
269
ized Raman modes and three non-degenerate Crh-antisymmetric IR active modes [33], while we have obtained, respectively, three and two such modes. The eigenfrequencies of the two modes, missing in the case of isolated planar rings, are approximately zero. It is worth noting that the ring order affects most strongly the frequency of the modes IR4 and IR6 corresponding to S i - O - S i bond bending. 3.3. The effects o f disorder
A comparison of calculated modes for planar rings with experimental data shows that none of the calculated spectra of planar rings are observed in the spectra of actual silicate glasses, i.e., the structure of such glasses cannot be modelled in terms of planar rings only. In fact, the results reported by others [20,21,38-42] have shown that rings with differing deformation can exist in silicate glasses. This fact makes it necessary to study in detail the effects of the non-planarity of the rings on their vibrational spectra. This type of disorder can be introduced in two ways. On the one hand, one can consider regularly puckered rings (see Fig. 1) for which the degree of disorder is given by the angle of puckering, r / ( t h e disorder in the S i - O - S i bridging angles in silicate glasses is then associated with random puckering of the rings). On the other hand, all the tetrahedra can be tilted at the same 77, but in different directions, i.e., the disorder in this case can be specified by the sign of 77, the number of inversely oriented tetrahedra and their relative position in the ring. This latter type of configuration repre-
Fig. 1. Geometry of a regularlypuckered four-membered ring, the angle of puckering "O= 320.
270
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
sents a simple but convenient model for studying the topological disorder in silicate glasses. 3.3.1. The effect o f ring puckering To study the effect of puckering, we calculated the polarized and isotropic R a m a n and I R spectra for n - m e m b e r e d regularly puckered rings (n = 3,4 . . . . . 7) by varying ~7 in the interval from 0 to 35 ° . This interval was found to demonstrate the effect of r/. Since the results for the polarized R a m a n spectra do not differ substantially from those for the isotropic R a m a n spectrum and, in addition, the change in n does not affect the character of the corresponding spectra, we consider below only the results obtained for the isotropic R a m a n and I R spectra of four-membered rings. As a whole, the puckering does not change the type of the modes existing in the planar rings and the modes given in Table 1 are present in puckered rings as well. In addition, the intensities of several modes which are zero for planar rings (not listed in Table 1) become, in accordance with the group theory [33], non-zero in the Raman a n d / o r I R spectra of puckered rings due to a decrease of symmetry from Dnh to Cnv. For the
a)
same reason, the modes R1, R2, R3 and R4, isotropic Raman-active for planar rings, become also IR-active for puckered rings, while the tr nantisymmetric IR-active modes IR2 and IR5 become also isotropic Raman-active for puckered rings. Except for the modes H2, H5, H8, IR2 and IR5 whose frequencies become increasingly dependent on n with increasing r/, the dependence of w on n for all other modes in the case of puckered rings is similar to that for planar rings. The calculated isotropic R a m a n spectra of puckered four-membered rings are shown in Fig. 2(a) for four different values of ft. As seen, the effect of puckering is most pronounced in the mid-frequency range (from about 400 to 700 c m - 1 ) on account of the intense Raman-active bond rocking mode IR5. For 7; 4:0 °, the intensity of this m o d e increases with increasing r/ due to an increase in the in-plane component of the atomic displacement vectors, while the intensity of the mode R2 decreases owing to the decrease of this component. This behaviour results in a resonant interaction of these two modes (Fig. 2(b)) at ~7 = 20 ° where the mode frequencies coincide (Fig. 2(c)). These modes exhibit a similar behaviour in the I R spectra as well.
b) •
•
-
•
i
-4--3 .,--t
0 l0 20 30 Angle of puckering (deg)
-4J C
c)
H
.
.
.
.
.
.
.
.
i
.
0 500 1000 Frequency (cm
5oo1-,
J
0 10 20 30 Ang]e of puckering (deg)
Fig. 2. (a) Calculated isotropic Raman spectrum of a regularly puckered four-membered ring for four different values of the angle of puckering -q. Dependence of the mid-frequency peak intensity, I (b), and of the peak position, to (c), on r/. e, mode R2; G, mode IR5. For details, see the text.
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
3.3.2. The effect of topology For the sake of brevity, the different topological ring configurations are specified below by 0 if for a particular tetrahedron 7/< 0, and 1 if r; > 0. For a four-membered ring, the case considered in detail below, there are consequently four possible topological configurations, namely 1111, 1110, 1100 and 1010 (e.g., the configuration 1100 contains two neighbouring tetrahedra oriented upwards and the next two downwards). The configuration 1111 corresponds to a puckered four-membered ring and will be used hereafter as a reference structure. The isotropic Raman and IR spectra calculated for these four configurations are shown in Fig. 3. As can be seen, the topological disorder changes the ring symmetry leading to a splitting of the mode frequencies. Since the mutual orientation of tetrahedra affects predominantly the vibration of the bridging oxygen atoms, the peaks corresponding to modes localized in such atoms are affected stronger than the other modes. For example, such strongly affected modes are IR1 at 1080 cm -1 resulting from Si-Obr bond stretching, or the mid-frequency modes corresponding to S i - O - S i bond bending. In general, the number of peaks in the vibrational spectra for a given topological configuration increases with decreasing point group order. Thus, for the 1110 configuration which has the lowest symmetry, in the case of four-membered rings the number of peaks
o
6oiobo
Frequency (cm-~)
obo Frequency (cm-~)
Fig. 3. Calculated isotropic Raman (a) and IR (b) spectra of the four possible topological configurations of a four-membered ring: 1111, 1110, 1100, 1010.
271
i
5
L
z
~0
56o
obo Frequency (cm-I)
Fig. 4. Measured IR absorption spectrum of fused silica.
both in the Raman and IR spectra is greatest. The results for rings of other order are similar to these. In fact, both the magnitude and the sign of r/ can change from one tetrahedron to the other, but this case does not provide information which may be easily and unequivocally interpreted and is not discussed.
4. Discussion
The effects of puckering and topological disorder of isolated rings, considered in Section 3.3., represent an idealized situation and can hardly be expected to describe, adequately, the case of real glasses. For example, the comparison between the calculated IR spectra (Fig. 3(b)) and experimentally measured spectra of fused silica (Fig. 4) shows that, while in the spectral ranges around 1100 and 500 cm ~ the agreement is satisfactory, the peak observed at about 800 cm 1 is not reproduced in the calculated spectrum. The same holds for the Raman spectra, namely the frequency of the peaks in the calculated spectra as well as the depolarization ratios in the midfrequency region obtained by us are close to those observed experimentally ([28]; Fig. 3), but as a whole the profiles of both spectra differ substantially from each other. This general behaviour does not change with the ring order, degree of puckering or topology.
272
B. Mihailova et aL /Journal of Non-Crystalline Solids 168 (1994) 265-274
a)
b)
-r--1
c~
v
~0 500
1000
0
500
1000
Frequency (cm-~)
Frequency (cm-~) c)
d)
==i ~0
~-'t 0
500 1000 Frequency (cm-I)
,
0
500
,
,
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Fig. 5. Isotropic Raman (a), HH Raman (b), HV Raman (c) and IR (5d) spectra of a four-membered planar ring in which all non-bridging oxygen atoms are linked with hypothetical particles of an arbitrary mass.
A n e x t s t e p to i m p r o v e t h e m o d e l is to s p e c i f y boundary conditions for the peripheral atoms of t h e s t r u c t u r a l u n i t by c o r r e c t i n g t h e d y n a m i c a l
matrix elements. This correction can be realized m o s t g e n e r a l l y by c h a n g i n g e i t h e r t h e f o r c e c o n stants, t h e e f f e c t i v e m a s s o f a t o m s o r t h e i r local
a)
0
b)
500
1000
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Frequency (cm-a)
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1000
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c)
d)
2g
~
~0
~0
0
500
1000
Frequency (cm-I)
0
500
I000
Frequency (cm-a)
Fig. 6. Isotropic Raman (a), HH Raman (b), HV Raman (c) and IR (d) spectra of a four-membered planar ring with imposied boundary conditions as specified in the text (the ratio K/j8 is four times less than that used for isolated rings; four of the non-bridging O atoms are linked to the surrounding).
B. Mihailova et al. /Journal of Non-Crystalline Solids 168 (1994) 265-274
surrounding. In the case of isolated rings, the bridging and non-bridging oxygen atoms are distinguished from one another by the stretching force constants K ' and K", respectively. In a medium built of interconnected silicon-oxygen rings, this difference disappears but the calculations carried out with the same stretching force constant even for all the S i - O bonds do not change the results. The variation in the effective mass of the non-bridging O atoms does not provide reasonable data because it only shifts the peak positions as a whole and does not give rise to additional modes arising from the decrease in the degree of freedom of the system. A new weak low-frequency mode, in which the atoms of the ring vibrate normal to the plane of the contour of covalent bonds, appears when the dynamical matrix is corrected by changing the surrounding of the bonded non-bridging O atoms. For the purposes of this study, these atoms were assumed linked by the same stretching and bending force constants as previously specified to hypothetical particles (HP) of an arbitrary mass located along the Si-O,o n bond. The Raman and IR spectra of a four-membered planar ring calculated by imposing such boundary conditions are shown in Fig. 5. As seen, although this procedure improves the agreement between the calculated and the experimental positions of the high-frequency peaks [28], the main differences in the spectra remain. At the same time, variations in the mass of the HP as well as in their distance to the ring atoms do not change the spectral profiles. The afore-mentioned arguments show that the effect of the surrounding matrix on the vibrational density of states in glasses cannot be correctly described by the bonding of the non-bridging oxygen atoms, but a complete change of the dynamical matrix has to be introduced. For example, the correction of the dynamical matrix can be performed by changing the ratio K/~ for all the atoms in tetrahedra containing oxygen atoms bonded to the amorphous matrix. Fig. 6 shows the vibrational spectra of a four-membered planar ring in which four of the non-bridging oxygen atoms are linked and the ratios K/~s is four times smaller than that used in the case of iso-
273
lated rings. It is evident that the shape of the IR spectrum becomes similar to the spectra observed in silicate glasses (Fig. 4). The reduction of the ratio K/[3 leads also to an increase in the intensity ratio of the H H and HV modes in the midfrequency range, thus improving the agreement with experiment [28]. In addition, the intensity of the weak Raman peak at about 700 cm -1 increases, while the intensity ratio of the highfrequency and mid-frequency peaks in the spectrum decreases. Thus, the spectra, calculated with the reduced ratio K/B, approach those measured in silicate glasses, although differences remain. For example, the intensity ratio of the highfrequency and mid-frequency Raman peaks in the experimental spectra is smaller than that in the calculated spectra. One can further improve the fit to experiment or can model vibrational spectra of various glasses by specitying the imposed random boundary conditions. This trend lies, however, beyond the goal of the present paper and will be the subject of further investigations.
5. Conclusions
On the basis of a combined theoretical method for calculating the vibrational spectra of isolated rings of SiO 4 tetrahedra, the effects of ring order, puckering and topology are: (i) the ring order affects only the mode localization, thus changing the spectral line intensities and positions, (ii) the degree of puckering determines the value of the Si-O-Si angle and thus affects the Raman and IR peaks in the mid-frequency spectral range; and (iii) the ring topology determines the total number of Raman and IR active vibrations mainly affecting the modes localized in the bridging oxygen atoms and leading to a frequency splitting of these modes. By comparing the calculated spectra of rings and the experimental data on silicate glasses, various means for setting boundary conditions are proposed to account for the effect of the amorphous matrix and are shown to improve the
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agreement between calculated and experimental s p e c t r a . S u c h s t u d i e s a r e n o w in p r o g r e s s .
T h e f i n a n c i a l s u p p o r t o f this w o r k by t h e B u l garian Ministery of Science and Education under c o n t r a c t n u m b e r S R F - 2 1 3 is g r a t e f u l l y a c k n o w l edged.
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