Dynamics of βQuartz Structures of Vitreous SiO2 and BeF2John B. Bates Citation: J. Chem. Phys. 56, 1910 (1972); doi: 10.1063/1.1677474 View online: http://dx.doi.org/10.1063/1.1677474 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v56/i5 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 56, NUMBER 5 1 MARCH 1972

Dynamics of {3-Quartz Structures of Vitreous SiO 2 and BeF 2 * JOHN B. BATES

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

(Received 30 June 1971)

The optically active vibrational modes of vitreous SiO. and Bel'. were calculated using the j3-quartz models determined for these materials from recent x-ray diffraction experiments. The assumption of regular j3-quartz lattice models with the appropriate unit cell parameters resulted in good agreement between ob­served and calculated frequencies. The crystal potential functions of SiO. and BeF. were derived from a modified valence force field, and several force fields which differed in the type of interaction terms allowed were studied. Calculated mean amplitudes for Si-O and Be-F stretching motion and for 0·· ·0, Si·· ·Si, F· •• F, and Be· •• Be nonbonded displacements were in good agreement with the respective mean amplitudes derived from the x-ray data.

I. INTRODUCTION

One of the major difficulties encountered in interpret­ing the vibrational spectra of glassy materials is the lack of structural models from which selection rules can be derived and which provide the framework for dy­namical calculations. Such has been the case with vitreous Si02 and BeF2. Previous calculations of the vibrational spectra for these materials were based pri­marily on two methods: Bell et aU computed the spec­tra of glassy Si02, BeF2, and Ge02 using a statistical technique for obtaining the distribution of frequencies of a random network modeJ,2 while Bock and Su3 used the GF matrix method and a valence force field to cal­culate a set of vibrational frequencies for Si02 based on the a-quartz structure. The GF matrix method was also used by Gaskell4 to calculate the vibrational fre­quencies of a {3-cristobalite lattice model of silica. Apparently there have been no previous attempts to quantitatively correlate the observed spectra of Si02 or BeF2 with a model based on the structure of {3-quartz.

The structure of crystalline {3-quartz was shown by Narten5 to be the best model of all the known struc­tural polymorphs of Si02 for interpreting the observed x-ray diffraction pattern of vitreous Si02 and BeF2. Assuming these structures to be perfect lattices (i.e., not disordered), we have calculated their optically active (k=O) modes using the GF matrix method as developed for crystal vibrations.6 The Gop matrices for Si02 and BeF2 were derived from the respective unit cell parameters reported in Ref. S. A modified valence force field (MVFF) was determined for Si02 and BeF2 by least squares refinement of an initial set of the force constants to give the best fit of the calculated frequen­cies to sets of observed frequencies. The discussion of these calculations given below will also include results of a normal coordinate analysis of the optically active modes of crystalline {3-quartz. The MVFF determined for this material was used as a starting point for the Si02 calculations. The bonded and non bonded atom­atom mean amplitudes of Si02 and BeF2 were also com­puted and will be compared with the mean amplitudes derived from the diffraction data.5

II. EXPERIMENTAL RESULTS AND SELECTION RULES

A large number of papers which discuss measure­ments of the vibra.tional spectra of the crystalline modi­fications of quartz and of vitreous Si02 have been pub­lished. Much of this work reported prior to 1970 has been reviewed by Wong and AngelJ.1 More recently, Hass8 measured the Raman spectrum of Si02 over a temperature range 12-300oK and determined the polar­ization properties of the observed bands. The polariza­tion characteristics of the Raman bands of vitreous silica have also been measured in this Laboratory.9 A more detailed study of Raman scattering and infrared absorption from low-frequency modes in Si02 was dis­cussed in a paper by Stolen.lO The infraredll ,12 and Raman spectral2 of BeF2 and the spectra observed from inelastic neutron scattering experiments on crystalline and vitreous BeF213 have also been reported. However, the vibrational spectrum of BeF2 is not as well charac­terized as that of Si02 because of the lack of low-fre­quency optical data and the lack of spectroscopic data on the crystalline modifications of BeF2• Assignments of the vibrational modes of {3-quartz, Si02, and BeF2 will be discussed later.

The (3-quartz lattice has the space group symmetry given by DS4.14 The SieBe) atoms occupy D2 sites and the O(F) atoms occupy sites of C2 symmetry. There are three Si02(BeF2) formula units in the primitive cell. Using standard group-theoretical procedures, the ir­reducible representation (rep) for the 24 optically active (k=O) modes of Si02 and BeF2 is given by

rOP(Ds) =al+2112+3bl+2b2+4el+4e2.

This can be compared to the rep for a-quartz which is given by PP(Da) =4al+4112+8e. Thus, the (3-quartz selection rules determined by rop(Ds) predict only one totally symmetric mode, l'l(al), in contrast to the four al modes predicted by the selection rules for a-quartz [rop(Da)]. The observation of only one strong, polar­ized band in the Raman spectra of vitreous Si02 and BeF2 is supporting evidence in favor of the {3-quartz model for these glasses.

1910

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DYNAMICS OF VITREOUS Si0 2 AND BeF 2 1911

The vibrational frequencies observed from infrared and Raman spectral studies on {3-quartz and vitreous Si02 and BeF2 are collected in Table 1. The frequencies observed in crystalline a-quartz which can be corre­lated with those of the {3 structure are also given in Table 1. The frequencies and assignments for a-quartz were determined from polarized Raman studies,15 and those for {3-quartz were taken from recent experiments performed in this Laboratory.16

The vibrational frequencies for vitreous Si02 and BeF2listed in Table I were obtained from the references cited earlier. The assignments proposed for Si02 were based on near-frequency coincidence with the {3-quartz modes. For purposes of calculation, the e2 mode fre­quencies of Si02 listed at 700 and 250 cm-1 in Table I were estimated from the frequencies of the correspond­ing modes observed in {3-quartz. Bands in the 200-300-and 600-700-cm-1 regions have been observed in Raman scattering experiments from Si02 (see Table I, Ref. 3), although single peak frequencies at 700 and 250 cm-1

have not been identified. The assignments for BeF2 were made by comparison

of relative band intensities and by assuming a similar ordering of frequencies as that for Si02: The strongest

TABLE I. Observed frequencies of crystalline a- and t/-quartz and vitreous Si~ and BeF2• Frequencies in cm-t.

a Quartz· (Da)

464 (al)

1080 (~) 364 (~)

1085 (al) 356 (al)

778 (~) 495 (~)

1162 (e) 795 (e) 394 (e) 128 (e)

1072 (e) 697 (e) 450 (e) 265 (e)

• Reference 15. b Reference 16. c References 8 and 9.

t/ Quartzb

(D.)

462 (al)

(~)

(~)

Inactive (bl) Inactive (bl)

Inactive (b2) Inactive (b2)

1067 (el) 788 (el) 428 (el)

99 (el)

1173 (e2) 688 (e2) 409 (e2) 245 (e2)

Si~c.d

432 pf 280 p

1065 dp 910 800 dp 770 443 dp 410 60 dp 105 g

1200 dp 805 dp [700JI

380 dp [250Jl

d The freQ uencies reported from the optical measurements on SiO% and BeF, are accurate to ±IO em-I. The band reported at "'105 em-I from neutron scattering experiments on BeF2 is believed to be accurate to within ±20 em-I.

e References II and 12. f Estimated frequencies, see text. P. polarized; dpt depolarized. c Reference 13.

band in the Raman spectrum was at 280 cm-1 and was polarized,12 The assignment of this band to the single vl(al) mode is consistent with these observations. The bands at 910, 770, and 410 cm-1 in the infrared spec­trum of BeF21l,12 were assigned to el modes, which is consistent with the assignments given for Si02 and (3-quartz. The peak observed at ",,105 cm-1 in the den­sity-of-states function [g(w) ] of BeF2 determined from inelastic neutron scattering experiments was assigned to an el mode corresponding to the low-frequency mode observed in Si02 at ",,60 cm-1 (Table I). Although this mode should be optically active, its appearance has not been reported in either infraredll ,12 or Raman12 spec­tral studies with BeF2. The remaining bands in the Raman spectrum at ",,80S and 380 cm-1 were assigned to e2 modes. As pointed out above, the vibrational spec­tra of vitreous Si02 and BeF2 give supportive evidence for the {3-quartz lattice model proposed for these glasses from the x-ray diffraction experiments. The polariza­tion characteristics of the Si02 and the BeF2 Raman bands (Table I) exhibit the trend expected for struc­tures having D6 symmetry, since only the single al mode should be polarized.

Two additional Raman bands of Si02 at 490 and 600 cm-1 were also observed to be polarized.8,9 These bands may be associated with localized modes arising from vibrations about defects in fused quartz. Analysis of the x-ray diffraction data5 based on the {3-quartz model required 12% of the Si02 positions in the lattice to be vacant. With this concentration of defects in Si02, localized modes could be observed in the Raman spec­trum. These modes, which arise from vibrational motion of nonbridged 0 and Si atoms, would be effectively de­coupled from the remaining modes of the Si02 lattice.8 The fact that the bands at ",,490 and 600 cm-1 appear to be sharper in comparison to the remaining bands in the Raman spectra of Si02

8,9 is consistent with these assignments. Additional support for the assignment of these two bands as due to vibrations localized about defects is found in the work of Gaskel1.4 As was pointed out in Ref. 4, an increase in frequency of the bands in the S00-700-cm-1 region is observed as the number of nonbridging oxygen atoms increases in the series from Si02 to the orthosilicate glasses. The 490- and 600-cm-1

bands may be associated with modes localized at more than one type of defect in which the defect having the larger number of non bridged oxygen atoms would give rise to the higher frequency.

III. METHOD OF COMPUTATION

The vibrational frequencies of Si02 and BeF2 were calculated at k=O (assuming ordered lattice structures) by the GF matrix method applied to optically active lattice vibrations.6 Criteria for choosing a set of internal valence coordinates for describing the vibrational mo­tions of crystals were discussed previously.17 ,18 The

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1912 JOHN B.

TABLE II. Internal coordinate definitions for !3-quartz and vitreous SiD. and BeF2 calculations.

Atoms defining" Coordinate coordinate

(Sn) n j k

Si-Q (BeF) 2 stretch 2 3

3 5 4 10 5 4 5 6 4 6 7 4 8 8 4 13 9 7 8

10 7 9 11 7 11 12 7 12

o-Si-Q 13 2 3 (F-Be-F) 14 5 10 bend 15 2 5

16 2 10 17 3 5 18 3 1 10

o-Si-Q 19 5 4 6 (F-Be-F) 20 8 4 13 bend 21 5 4 8

22 5 4 13 23 6 4 8 24 6 4 13 25 8 7 9 26 11 7 12 27 8 7 11 28 8 7 12 29 9 7 11 30 9 7 12

Si-Q-Si 31 2 14 (Be-F-Be) 32 3 15 bend 33 1 5 4

34 4 8 7 35 4 6 16 36 7 9 17

a i-j-k indicates bonding of atoms in which the central atom j is bonded to i and k.

atoms involved in defining the internal coordinate set for the ,B-quartz lattice are shown in Fig. 1. Atoms numbered 1-9 are contained in the origin cell, while atoms numbered 10-17 are contained in unit cells ad­jacent to the origin cell. Three types of internal coordi­nates were employed: Si-O stretching, O-Si-O angle bending, and Si-O-Si angle bending. The choice of these coordinates required that a total of 36 internal coordinates must be employed on using the full D6 lattice symmetry to factor the secular determinant.

The internal coordinate set is given in Table II. The

BATES

the atoms (Fig. 1) involved in each coordinate. For bending coordinates, the numbers in the order i-j-k indicate that bending occurs about the central atom j. The 36 coordinates in Table II can be grouped into five symmetrically equivalent sets. These are given by

R = SI, "', SI2,

(h = SI3, S14, SI9, S20, S25, S26,

82 = SI5, SIS, S22, S2a, S2S, S29,

83= SI6, SI7, S21, S24, S27, sao,

<1>= S31, "', S36.

The 12 Si-O(Be-F) stretching coordinates all belong to the same set, R, while the 18 O-Si-O(F-Be-F) bending coordinates are composed of three sets consist-ing of six coordinates each. The six Si-Q-Si(Be-F-Be) bending coordinates all belong to the same set, <1>.

The Cartesian coordinates of the atoms in the Si02

and BeF2 lattices were computed from the respective unit cell parameters determined by Narten.5 The trans-formation from Cartesian displacement coordinates, fi,

to the internal displacement coordinates, St, defined in Table II is given by

an

St= L L Bt/fi l, t=1, ",,36, n=17,

I i=l

where the sum over 1 includes the origin cell (l=0) and all unit cells adjacent to the origin cell (l>0) which contain atoms that are used in defining S t (Table II). The B matrix for the optically active vibrations Bop (k=O) is computed by adding the displacements of atoms in cells 1>0 to the displacements of atoms in the origin cell (1=0) to which they are equivalent by primitive translations:

Btiop= L Btil. I

coordinates are defined by designating the numbers of FIG. 1. Atoms used in defining internal coordinates of Table II.

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DYNAMICS OF VITREOUS Si0 2 AND BeF 2 1913

Modified verSIOns of the Schachtschneider computer programsl9 were employed to compute Bop, Gop, and to solve the vibrational secular equations given by I GopFop- XE 1=0.

The five symmetrically equivalent sets of internal coordinates described above require five valence force constants (diagonal terms) in the MVFF for Si02 and BeF2• Various interaction constants (off-diagonal terms) were studied for each structure in which the goodness of fit to the observed frequencies as well as the values of the force constants were used as criteria in judging the acceptability of a particular force field.

Prior to solving for the eigenvalues and eigenvectors, the dynamical matrix was factored by a similarity trans­formation with the matrix U, which relates the internal coordinates s to a set of symmetry coordinates S, S = Us. The set S is determined by the rep for vibra­tional motion using the internal coordinates rather than Cartesian coordinates as basis functions. For the inter­nal coordinate set of Table II,

rint (D6) = Sal+ 21l2+ 2bl+3~+Sel+ 7 e2.

The symmetry coordinates derived from the internal

z Lx

z Lx

z Lx

z Lx

0, MODE 465cm-'

e, MODE 1063cm-'

e, MODE 62 cm-'

e2 MODE 398 cm-'

y

Lx

y

Lx

y Lx

FIG. 2. Cartesian displacement vectors for some normal modes of Si~. Calculated displacements multiplied by 6 for illustration.

z Lx

z Lx

z Lx

z Lx e2 MODE

300 cm-'

y

Lx

FIG. 3. Cartesian displacement vectors for some normal modes of BeF2• Calculated displacements multiplied by 6 for illustration.

coordinates of Table II are given in Appendix A. Sub­tracting r op (D6) from r iDt (D6), it is seen that the inter­nal coordinate set of Table II contains 13 redundant coordinates given by rred=4al+b2+el+3e2. Also, on comparing riDt(Da) with r OP (D6), it will be noted that the internal coordinate set of Table II does not gener­ate the third bl mode required by rOP(D6)' Thus, r int(D6) does not completely span the symmetry space defined by rOP(D6)' However, the bl modes are optically in­active, and, since the modes of one symmetry species do not mix with those which have a different symmetry, the internal coordinates chosen for these calculations (Table II) completely define the infrared and Raman­active normal coordinates for the {j-quartz models of Si02 and BeF2 •

The methods used to calculate the bonded and non­bonded vibrational mean amplitudes are described in Appendix B. The eigenvectors in terms of Cartesian displacement vectors for the normal modes of Si02 and BeF2 were also calculated, and a few are illustrated in Figs. 2 and 3, respectively. The figures were prepared by using the ORTEP program by Johnson.2o

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1914 JOHN B. BATES

TABLE III. Calculated frequencies for crystalline {3-quartz and vitreous Si02 and BeF2.

Calculated frequencies Assignmenta •b

Symmetry type {3 Quartz Si02 BeF2 Si02 BeF2

al 473 465 329 82,83 <I>

~ 1274 1317 770 R R 333 351 417 82 82

hi 1067 1066 689 R R 306 330 209 (82, 83) (82, 83)

b2 728 713 967 R R,<I> 443 401 315 (82, 83) <I>,R

el 1069 1063 914 R R 770 779 751 R R 427 413 368 (82, 83) <1>, (82,83)

96 62 90 <1>, (82,83) (82,8s)

eo, 1157 1184 821 R R 724 735 532 R R 389 398 300 81, (02, Os) <I>

262 240 269 (02, Os), 01 01> (02,03)

• Determined from the potential·energy distribution. b R. 01. 0" 8,. and <I> refer to the symmetrically equivalent sets of internal coordinates.

IV. RESULTS AND DISCUSSION

The analysis of the normal mode vibrations of vitre­ous Si02 assuming the .a-quartz lattice structure was initiated by refining the parameters of an MVFF to fit the frequencies recently observed from polarized Raman spectra of .a-quartz.16 The calculated frequencies are given in Table III, and the final set of force constants and mean amplitudes calculated at 25°C are given in Table IV. As shown in Tables I and III, a good fit be­tween the observed and calculated frequencies for .a-quartz was obtained with a seven-parameter MVFF. Initial trial force fields contained separate valence force constants for all three O-Si-O bending modes. The re­sulting sets of refined force constants were not accept­able since Ks1 for the 2-1-3-type O-Si-O bend (Table II) was calculated to be about 3 times larger than K9a for the 2-1-10-type O-Si-O bend. Since this result did not seem physically reasonable, an additional constraint was imposed on the vibrational potential function by setting Ks2 (2-1-S) =K9s(2-1-1O). Rapidly converging solutions were obtained with this potential field, and, as shown in Table IV, a reasonable set of force constants was obtained. It should be noted that in previous calcu­lations,3 the valence force constants for all three types of O-Si-O bending coordinates were equated to a single value: Kh (2-1-3) = K8z(2-1-S) =K9a(2-1-10).

As mentioned above, the assignment of the bands observed ilJ. infrared and Raman spectra of vitreous Si02

8•9 was based mainly on near-frequency coincidence with the fundamental frequencies observed for .a-quartz.

The final set of .a-quartz force constants (Table IV) was used as initial input for the Si02 calculations. The calculated frequencies from the final set of Si02 force constants are given in Table III, and the force constants and mean amplitudes calculated at 25°C are given in Table IV. The only significant difference between the calculated force constants of .a-quartz and those of Si02

occurred in the values for the Si-0-Si bending force constants. The decrease in frequency associated with Si-0-Si bending from 97 cm-1 in .a-quartz to 60 cm-I

in Si02 (Table I) resulted in a lower calculated value of the Si-0-Si bending force constant (Table IV) com­pared to the value calculated for .a-quartz. The assign­ment of the 60-cm-1 band in Si02 to a fundamental vibration of the Si02 lattice was discussed by Hasss and by Stolen.lo The observed and calculated temperature dependence for this band assuming first-order Raman scattering was good.s The assignment of the 6O-cm-1

band to an el vibration is supported by the conclusion of Stolenlo that both Raman scattering and infrared ab­sorption in the low-frequency region of silica arise from the same normal modes and by the fact that only the el modes of the .a-quartz structure are both Raman and infrared active. The reason for the ",,37 -cm-I difference between the low-frequency mode in vitreous Si02 and the corresponding mode in .a-quartz is not apparent from the dynamics of these materials based on a perfect lattice model for Si02• As shown in Table III and Fig. 1, the el mode calculated at 62 cm-I in Si02 consists primarily of the bending motion of the Si-0-Si coordi­nates. The defects in Si02 may be responsible for lower-

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DYNAMICS OF VITREOUS Si0 2 AND BeF 2 1915

ing the frequency of the Si-O-Si bend by reducing the net resistance to motion for this particular mode. Addi­tional work is needed to determine the influence of the defects in Si02 on the normal mode vibrations.

The assignments of the calculated frequencies of {3-quartz and Si02 (Table III) according to internal co­ordinate designations was based on the potential-energy distribution (ped). The notation of Table III indicates which internal coordinates are the primary contributors to a particular mode. Thus, for the lowest-frequency el mode, 'P, (82, 83) indicates that the (symmetrically equivalent) Si-0-Si coordinates are the principal con­tributors with minor contributions from the O-Si-O coordinates, 82 and 83• The separate contributions made by the sets 82 and 83 could not be made on the basis of the ped, since the force constants for these bending coordinates were equated to a single value (see above). It should be emphasized that the assignments given in Table III for {3-quartz, Si02 , as well as for BeF 2 (dis­cussed below), are only approximate descriptions. More accurate descriptions of each mode must be based on the calculated eigenvectors expressed in the Cartesian form as illustrated in Fig. 2.

The Si02 force constants for Si-O stretch and O-Si-O and Si-O-Si bends given in Table IV can be compared with those of Bock and Su.3 The latter report values of 4.8, 0.35, and 0.05 mdyn/ A, respectively, for Si-O stretch, o-Si-O bend, and Si-O-Si bend. It is not clear how the low value for the Si-O-Si bending constant

TABLE IV. Calculated force constants and mean vibrational amplitudes for p-quartz and vitreous Si~.

Coordinate or interaction

Si-D o-Si-D (2-1-3) o-Si-D (2-1-5) and (2-1-10) Si--D-Si Si-O (1-2), Si-O (1-3) Si-D (1-2), o-Si-D (2-1-3) Si-D (1-2), O-Si-D (2-HO)

Force constanta (mdyn/A)

p Quartz Si02

4.905 4.765 0.428 0.550 0.304 0.324 0.103 0.022 0.791 0.713

-0.592 -0.697 0.330 0.236

Mean amplitudes: (I!J.r;,J)1/2 A

Si-D 0···0 Si· .. Si

Calculated

pQuartz Si~

0.045 0.089 0.054

0.045 0.089 0.069

X-ray diffraction b

Si02

0.042 0.092 0.100

a The number of significant figures to which the force constants are reported is not necessarily commensurate with the accuracy of the observed freq uencies.

b Reference 5.

TABLE V. Calculated force constants and mean vibrational ampli­tudes for vitreous BeF2•

Coordinate or interaction

Be-F F-Be-F (2-1-3) F-Be-F (2-1-5) and (2-1-10) Be-F-Be Be-F, Be-F Be-F, F-Be-F (2-1-3) Be-F, F-Be-F (2-1-10)

Force constanta (mdyn/ A)

2.260 0.222 0.132 0.145 0.578 0.176 0.254

Mean amplitudes: (I!J.r;l)1I2 A

Be-F F .. ·F Be···Be

Calculated X-ray diffractionb

0.064 0.073 0.074

0.083 0.103 0.132

& The number of significant figures to which the force constants are reported is not necessarily commensurate with the accuracy of the observed freq uencies.

b Reference 5.

was determined in Ref. 3 since the lowest-frequency mode calculated (using the a-quartz model) was an al bending mode at 273 cm-l • It is stated that K(Si-O-Si) is the primary contributor to modes below 400 cm-l •

From our results (Table III), the value of K~ is fixed by the el mode at 62 cm-l and, to a much lesser extent, by the e2 mode at 269 cm-l .

The effect of estimating two el mode frequencies at 700 and 250 cm-1 on the force constants of Si02 cannot be well established. Without using these two frequencies in the refinement procedure, converging solutions could not be obtained without imposing unreasonable con­straints on the variation of the force constants. The acceptability of these estimated frequencies was briefly discussed earlier. The good agreement between the cal­culated frequencies of {3 quartz and those of Si02 in the ~, bl , and b2 blocks (Table III) also supports the plausibility of these estimates since the calculated fre­quencies in these symmetry blocks are fixed by the force constants Rand (82,83) whose values are, in turn, partly determined by the estimated frequencies.

The calculated frequencies for BeF2 based on the assignments proposed in Table I are reported in Table III. The assignments were determined from the ped as described above for Si02• For several modes, the primary and secondary contributors to the ped were different for Si02 and BeF2 as shown in the table. The calculated force constants and mean vibrational ampli­tudes are given in Table V. The same restraint applied to (3-quartz and Si02, i.e., KB2 (2-1-5) =KBa(2-1-1O), was applied to the BeF2 potential field. The larger value of K~ relative to KBI and K(Blo B2) in BeF2 (Table V) also resul ted in changes in the form of the normal

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1916 JOHN B. BATES

coordinates compared with those of Si02, as demon­strated by comparing the assignments based on the ped for BeF2 and Si02 given in Table III and by com­paring the Cartesian displacements for the modes illus­trated in Figs. 2 and 3.

The calculated Be-F mean amplitude (Table V) does not agree closely with the value determined from the x-ray diffraction experiments.5 This was somewhat sur­prising in view of the close agreement between the spectroscopic and diffraction values of (dR2)1/2 for Si02 (Table IV). Indeed, no force field of BeF2 studied in this work yielded Be-F mean amplitudes larger than 0.065 A. Since the mean amplitudes derived from diffraction data are determined from the net effect of all thermal motion at a given temperature, it may happen that low-frequency acoustic-type motions make significantly larger contributions to the Be-F mean amplitude than in the case of the Si-0 mean amplitude in fused quartz. This would be expected because of the much lighter mass of Be.

v. CONCLUSIONS

The vibrational analysis of the optical modes of the ~-quartz lattice model of vitreous Si02 and BeF2 was undertaken with the intention of providing additional support for the validity of these models as derived from x-ray diffraction experiments.5 The results of the above study showed that the observed spectra of these glassy materials appear to obey the optical selection rules de­rived from the lattice models in terms of observed in­tensities and polarization characteristics. The calcu­lated frequencies of the ~-quartz model of Si02 and BeF2 were in good agreement with the observed values, and the force constants which determined these fre­quencies were physically reasonable. Furthermore, the calculated mean amplitudes agreed closely with the corresponding values derived from diffraction experi­ments.5 It can thus be stated that the proposed model for vitreous Si02 and BeF2 permits a correlation and interpretation of a large amount of data relating to the static and dynamic properties of these materials.

ACKNOWLEDGMENT

The author wishes to thank Dr. C. K. Johnson of the ORNL Chemistry Division for help with the mean amplitude calculations and with the application of his ORTEP program for illustrating normal mode vibrations.

APPENDIX A: SYMMETRY COORDINATES

The solution for the eigenvalues and eigenvectors of the secular determinant, I GF - AE 1=0 is facilitated by first transforming the G and F matrices from the internal coordinate space to the space defined by a set of symmetry coordinates. The transformations are given by G= UGUt and F= UFUt where the matrix U relates the internal coordinates s to the symmetry coordinates S as given by

S=Us.

The symmetry coordinates employed in the calcula­tions discussed above are given in Table VI.

APPENDIX B: CALCULATION OF VIBRATIONAL MEAN AMPLITUDES

The vibrational mean amplitudes for Si-0 and Be-F bonded displacements were calculated19 from the equa­tion, 1;8= L1;QV. 1;Q is a diagonal matrix with elements };kkQ= (Qk2) = (h/87r2Pk) coth(hpk/2kT), where Pk is the frequency of the kth normal mode, and (Qk2)1/2 is the mean amplitude of the kth normal coordinate. The matrix L is determined by the transformation from normal coordinates, Q, to the internal coordinates, s, s= LQ. The 1;' matrix thus contains the mean-square displacements and interaction terms for the internal

TABLE VI. Transformation from internal to symmetry coordi­nates. Internal coordinate designation as given in Table II.

SI = (2vJ)-I(SI +S2+S3+S.+S5+S.+S7+SS+S9+SIO+Sl1 +SI2) S2= (1/61 /2 ) (SI3+S14+S19+S20+S25+S2.)

S3= (1/6112) (SI5+SIS+S22+S23+S28+S29) S.= (1/6112) (S16+S17+S21+S .. +S27+S30)

S.= (1/6112) (S31+S32+S33+S,.+S35+S3.)

S.= (2vJ)-I(SI +S2-S3 -S.+S5+S.-S7-SS+S9+SIO-Sl1-S12)

S7= (1/61/2) (S13-S14+S19-S20+S25-S2.)

SS= (2vJ)-I(SI-S2-S3+S,+S5-S.+S7-SS-S9+SIO-S11 +SI2)

S9= (1/6112) (SI.-S17+ S21-S .. - S27+S30)

SIO= (2vJ)-I(SI-S2+S3-S,+S5-S.-S7+SS-S9+SIO+Sl1-S12) Sl1 = (1/6112) (SI5-SIS+S22-S23-S2S+S29) S12= (1/6112) (S31-S32+S33-S,.-S35+S3.)

S13= (2vJ)-1(2s1-2s2-S5+S0+S9-SIO) S14= (2vJ)-1(2s3-2s,+S7-SS-S11+S12) SI5 = (2vJ) -I (2s15 - 2SIS-S22+S23+S28- S29)

SIO= (2vJ)-1(2s!6-2s17-S21 +S .. +S27- S30)

S17= (2vJ)-1(2s31-2s32-S33+S,,+S35-S30) SIS = !(S5-S0+S9-SIO)

S19= HS7-SS+S11-S12) S20 = HS22-S23+S28-S21!)

S21 = !(S21-S,,+S27-S30) S22= !(S33+S,.-S35-S36)

S23= (2vJ)-1(2sl+2s2-S5-S0-S.-slO) S .. = (2vJ) -I (2s,+ 2s. - S7 - Ss - S11 - S12) S2.= (1/6112) (2S13-S,.-S25)

S20= (1/61 /2 ) (2S14-S20-S2II) S27= (2vJ)-1(2sl.+2sls-S22-S23-S28-S21!) S28 = (2vJ)-1(2sI0+2s17-S21-S .. -S27-S30) S2. = (2vJ) -I (2s31 + 2s32 - S33 - S,. - S35 - 530) SOo= !(S.+S6-5.-SI0) S31 = HS7+SS-S11-S12) S32= (1/v!) (SI9-S2.) S .. = (1/v!) (S20-S2II) S,. =! (S22+S23-S28-S29)

S3.= HS21 +S24- S27-S30) S30= HS33-S,,+S35-S30)

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DYNAMICS OF VITREOUS Si0 2 AND BeF 2 1917

coordinates of Table II. The values of (.MSi-02)1/2 and (MBe-F2)1/2 are obtained from the diagonal elements of the respective 1:' matrices: (tlrinl/2=(1:kk,)1/2, k:::;12 (1:11'= 1:22'= 1:12,12').

The mean amplitudes for the nonbonded displace­ments, 0·· ·0, Si·· ·Si, F·· .F, and Be·· ·Be, were computed21 from the Cartesian mean-square amplitude matrix, 1:" (the equation relating 1:" to 1:Q is given in Refs. 19 and 21). If 1 denotes a unit vector between any two atoms, i and j, the value of (tlri/)1/2 can be calculated from the equation,

(Minl /2= {111:ii"l+ 1I1:jj"l- 2111:i/l\I/2.

1: ii", 1: jj", and 1: i/ are 3X3 submatrices of 1:" and are given by

[6X''''';) (tlXitlyj) (""''''';) ]

1:i/= (tlYitlXj) (tlyitlYj) (tlyitlz j) .

(tlZitlXj) (tlZitlYj) (tlzitlzj)

1: ii" and 1: j / are symmetric matrices whereas 1:i/ is (in general) not symmetric. The diagonal elements of 1:ii" and 1: j / are the mean-square displacements of the Cartesian coordinates, x, y, and z, for atoms i and j, respectivel y .19 ,21

THE JOURNAL OF CHEMICAL PHYSICS

• Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corp.

1 R. J. Bell, N. F. Bird, and P. Dean, J. Phys. C 1, 299 (1968). 2 R. J. Bell and P. Dean, Nature 212,1354 (1966). 3 J. Bock and G. J. Su, J. Am. Ceram. Soc. 53, 69 (1970). 4P. H. Gaskell, Phys. Chern. Glasses 8, 69 (1967). o A. H. Narten, J. Chern. Phys. 56, 1905 (1972), preceding

article. 6 T. Shimanouchi, M. Tsuboi, and T. Miyazawa, J. Chern.

Phys.35, 1597 (1961). 7 J. Wong and C. A. Angell, App!. Spectry. Rev. 4,155 (1971). 8 M. Hass, J. Phys. Chern. Solids 31,415 (1970). 9 J. B. Bates (uripublished results, 1971). lOR. H. Stolen, Phys. Chern. Glasses 11, 83 (1970). 11 L. R. Batsanova, G. S. Yur'ev, and V. P. Doronia, J. Struct.

Chern. (USSR) 9, 63 (1968). 12 A. S. Quist, J. B. Bates, and G. E. Boyd, ORNL Rept. No.

ORNL-4581, p. 166,20 May 1970; "Raman Spectra of Vitreous, Polycrystalline, and Molten BeF2 to 630°C and the Infrared Spectrum of Vitreous BeF2 at 25°C," Spectrochim. Acta (to be published) .

13 A. J. Leadbetter and A. C. Wright, J. Non-Crystalline Solids 3,239 (1970).

14 R. W. G. Wyckoff, Crystal Structures (Interscience, New York, 1960), Vo!. 1, p. 313.

10 J. F. Scott and S. P. S. Porto, Phys. Rev. 161,903 (1967). 16 J. B. Bates and A. S. Quist, J. Chern. Phys. 56,1528 (1972). 17 H. L. McMurry, A. W. Solbrig, Jr., J. K. Boyter, and C.

Noble, J. Phys. Chern. Solids 28,2359 (1967). 18 J. B. Bates, E. R. Lippincott, Y. Mikawa, and R. J. Jakobsen,

J. Chern. Phys., 52, 3731 (1970). 19 J. H. Schachtschneider, Shell Development Co., Tech. Rept.

Nos. 57-65 and 231-64, 1964. 20 C. K. Johnson, ORNL Report No. ORNL-3794, 1965. 21 S. J. Cyvin, Molecular V ibmtions and Mean Square Amplitudes

(Elsevier, Amsterdam, 1968), Chap. 13, p. 261.

VOLUME 56, NUMBER 5 1 MARCH 1972

Three-Body Dipole and Quadrupole Interactions in Neon and Argon

RONALD FOWLER

Oak Ridge National Laboratory,· Oak Ridge, Tennessee 37830

AND

H. W. GRABEN

Clemson University, Clemson, South Carolina 29631

(Received 16 September 1971)

Contributions from three-body dipole and quadrupole interactions to the third virial coefficients of neon and argon and to the cohesive energy of solid argon are evaluated. In both cases it is found that the triple-dipole term dominates the nonadditive effect while the dipole-dipole--quadrupole, dipole-quadrupole­quadrupole, and triple-quadrupole terms are of successively decreasing importance.

I. INTRODUCTION

The long-range nonadditive dispersion energy of a system of N atoms may be written in terms of two expansions, the Rayleigh-Schrodinger perturbative expansion and the multipolar expansion} The foremost term in the double expansion is the third-order triple­dipole term which has received considerable attention in recent years.2 It is of interest to know the importance of the fourth- and higher-order terms in the perturbative expansion3 and the higher multipole terms in the third order of perturbation theory. In this paper we focus

our attention on the latter area of interest. Ayres and Tredgold4 have derived an expression for the second term of the multipolar expansion in third order of perturbation theory, and have computed its influence on the cohesive energy of solid argon. Recently BellS has calculated the full mUltipolar expansion for the nonadditive third-order interaction energy. In addition to the well-known dipole-dipole-dipole (ddd) formula, he gives explicit expressions for the dipole-dipole­quadrupole (ddq) , dipole-quadrupole-quadrupole (dqq), and quadrupole-quadrupole-quadrupole (qqq) terms. His (ddq) term does not agree with that of

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