Piezo-optic behavior of forsterite, Mg2SiO4

J. Phys.Chem.Solids, 1972,Vol. 33, pp. 1251-1255. PergamonPress. Printedin Great Britain

PIEZO-OPTIC

BEHAVIOR

OF FORSTERITE,

MgzSiO4*

J. L. KIRK and K. VEDAMt Materials Research Laboratory, T h e Pennsylvania State University, University Park, Penna 16802, U.S.A.

(Received 2 April 1971 ; in revised form 16 September 1971) Abstract--The piezo-optic behavior of forsterite, crystallizing in the orthorhombic system, has been investigated up to a maximum pressure of 7 kbars. It is found that the variation of the refractive indices with pressure, dn/dP, are 0-035 x 10-S/kbar, 0-04e x 10-8/kbar, and 0.063 x 10-S/kbar for the ha, ha, and n~ respectively. T h e s e values are the lowest on record. T h e corresponding values for the variation of the refractive indices with volume strain are 0-044, 0.059 and 0-080 respectively. These results are interpreted in terms of the bonding and coordination number of the oxygen ions. 1, INTRODUCTION

2. EXPERIMENTAL DETAILS

THE PIEZO-OPTICbehavior of inorganic oxides is extremely fascinating and perplexing, for in some oxides such as a-quartz[l], vitreous silica[2], etc., the refractive indices increase with hydrostatic pressure, while in some others such as MgO[3], etc. the refractive index was found to decrease with hydrostatic pressure even though in all these substances their optical properties in the visible region of the spectrum are determined mainly by the oxygen ions. However, in these materials the structural arrangement of the oxygen ions is quite different: SiO2 can be considered as an open structure compared to the close packing in MgO. Further, the coordination number of oxygen in MgO is 6 whereas it is only 2 in SiO2. Forsterite (Mg2SiO4) crystallizing in the orthorhombic system can be considered as a compound of 2MgO and SiO2 with the stacking of oxygen ions in a distorted hexagonal close packing array. This article describes the results of the measurements on the pressure variation of the refractive indices of this geophysically interesting material, forsterite.

The experimental method and the computations involved in this type of measurement have already been discussed at length in regard to similar investigations with uniaxial crystals[l, 4]. In brief, the changes in the refractive indices were measured by observing the shift of the localized interference fringes across a fiducial mark on the crystal for h5893 appropriately polarized. The change in the thickness of the sample was considered by using the elastic constants data of forsterite as determined by Graham and Barsch[5] of this laboratory. The change in the refractive index, An, was evaluted from the formula An = ( p k -- 2nAt ) [2to,

(1)

where p is the number of fringes shifted, to is the initial thickness of the crystal, At is the change in thickness of the crystal under pressure, and h is the wavelength of light employed. Since forsterite is biaxial, it is necessary to determine the values of d n / d P for the three principal refractive indices. Hence, three crystal plates perpendicular respectively to the a, b, c axes were cut from the same single crystal boule on which the elastic constants measurements[5] were made. The source of the synthetic boule and the results of the wet

*Work supported by a grant from the National Science Foundation. tAIso affiliated with the Department of Physics. 1251

1252

J.L. KIRK and K. VEDAM

chemical as well as spectroscopic analysis have already been described elsewhere[5]. The final dimensions of the experimental specimens were about 0-8 x 0.5 x 0.3 cm with the edges parallel to the crystallographic axes. 3. RESULTS AND DISCUSSION

Volume Strain (%1 0.2 0,3 0.4

0.1

Volume Strain 0.2 0.3

0.1 I

I

~%1

0.4

0.5

I

L

I

FORSTERITE n ~ = 1.651

]

3

"4"

The observed pressure dependence of the three principal refractive indices of forsterite are shown in Figs. 1-3. It is evident that measurements with each specimen (say, e.g. a plate) using appropriately polarized light will yield data on dn/dP for two of the three refractive indices (rib = n,~ = nz, a n d nc = na = nz for the a plate)[6]. H e n c e measurements with three plates of different orientations provide enough data to cross check the results for all three refractive indices. Further, two specimens of different thicknesses were also used for a and b plates. In each of the Figs. 1-3, results obtained on two different orientations are plotted and it is seen that the mutual agreement between them is good within the experimental error, which is also indicated in the figures. The data from b cut samples in Figs. 2 and 3 appear to be systematically higher than the corresponding results from a and c cut samples. However, since this deviation is well within the experimental error, no particular significance should be attached to this. All three refractive indices increase with pressure at a very small rate; in fact, the dn/dP 0

4~

0.5

* 3 xlO"5

x ee93 A 0

I

I

I

I

I

I

i

2

3

4

5

6

7

Pressure (k bar)

Fig. 2. Variation of the refractive index na of forsterite with pressure and volume strain T = 22~ [] Sample a-l, 9 Samplea-2, 0 Sample b-l, 9 Sample b-2. 50

0.1

Volume Strain (%1 0.2 0.3 0.4

i

I

I

0.5

I

I 6 0

I FORSTERI T E I

9 e~

9 9O 9

9

94 0

O9

O

O ~

3x10-5

9

tX

I

~, 5893

Pressure ( k bar)

Fig. 3. Variation of the ref~cdve index n~ of forsterite with pressure and volume strain T = 22~ O Sample

b-l, 9 Sample b-2, A Sample c.

FORSTERITE nct" = 1.635

!

0

I

2

, 3

,

4 S Pressure (k bar)

6

, 7

Fig. 1. Variation of the refractive index r~ of forsterite

with pressure and volume strain T= 22~ [] Sample a-I, 9 Samplea-2, A Samplec.

slopes observed in forsterite are the smallest observed in over forty different materials investigated thus far. These include glasses, polymers and numerous ionic and covalent crystals. In general, the values of d n l d P observed in these solids are one to two orders of magnitude larger than those in forsterite. Assuming that the refractive index increases linearly over the entire pressure range of the present measurements, the slopes of d n / d P were calculated using a least squares fit to a straight line for all the data points and were

F O R S T E R I T E , Mg2SiO4

found to be 0.035 • 10-3/kbar, 0.046 • 10-s/ kbar and 0.063 • 10-a/kbar for the n~, no, and nv, respectively. Since the stress-strain relation is essentially linear for the entire stress level employed in the study, both the stress and the corresponding volume strain are included in the Figs. 1-3. The variation of the refractive indices with volume strain, again as evaluated using a least squares fit, were found to be 0-044,0"059 and 0.080 for the n~, na, and nv, respectively. Since - An/(AV/Vo) = p (dn/dp),

where p is the density, these values correspond to the p(dn/cha) values for the three principal refractive indices of forsterite as well. Table 1 lists these values along with the values of the refi-active indices, the volume strain at 7 kbar, etc. The corresponding values for MgO and SiO2 are also included in this table. It is seen that the refractive index of MgO decreases with pressure while that of SiOz increases irrespective of the fact that it is in crystalline phase or amorphous form. Since Mg2SiO4 is one of the stable crystalline phases in the binary system MgO-SiO2, one may be tempted to predict the behavior of Mg2SiO4 from stoichiometry and a knowledge of the behavior of the end members. Such an approach would obviously lead to erroneous

1253

results; in fact, in this case even the sign of d n / d P would be wrong. As mentioned earlier, in all these materials the optical behavior in the visible region of the spectrum is mainly dependent on the polarizability of the oxygen ions and the contribution of the different cations Mg 2+ and SP + is negligible compared to that of the oxygen ions. H e n c e the available volume per 0 2- ion was evaluated using the same procedure as followed by Tessman et al. [7] and the values obtained are 17.5A 3 in MgO, 17.8A a in forsterite, 18.7A ~ in a-SiO2 and 22.6A a in vitreous silica. The volume per O z- in MgO and forsterite are almost equal since the stacking of the oxygen ions in the former is c.c.p. and in the latter it is h.c.p. However, it is seen that consideration of the volume of 0 2ions alone also does not lead to a correct prediction of dn/dP of forsterite. On the other hand, consideration of the structural details of these materials provides a clue to the understanding of their optical behavior. In MgO each (doubly ionized) oxygen ion is bonded to six nearest neighbor magnesium ions in octahedral coordination. On the other hand, in a-quartz as well as in vitreous silica, each oxygen is bonded to two silicon atoms which are bonded tetrahedrally to tour oxygen atoms. The structure[8] of forsterite can be considered as made up of

Table 1. Piezo-optic properties o f forsterite and other related materials

Material

Forsterite

Ref. index h5893

dn/dP (10 -3 kbar -I)

n~=nb=nv=l'635 n~=n~=nz=l'651

0"03s

n~ = n . = n x = l ' 6 7 0

/dn'~ P~d-pp]

Vol. strain at 7 kbar (%)

10~x An for 7 kbar

102 • An at 0-25% vol. strain

0"0245 0"032 0.044

0"0111 0"014~ 0"0198

--0"55

0"06s

0"044 0-05~ 0"080

-- 0" 17~

--0"302

--0"41

--0-123

--0"075

1"78

0"711 0-747

0"098 0"095

-- 1"96

0"650

0"081

O'04e

MgO

n = 1 "7379

a-Si02

no= 1"544 h e = 1-553

1.065 1"11o

0"39a 0"421

Vit-SiO2

n = 1-4584

0"915

0"335

- -

1254

J.L.

K I R K and K. V E D A M

2. II

~.ll Mg

Mg

JO 2.11 -Mg 2.11

Mg Mognesiurn Oxide

si,x ~

j,

si

L61~oL61

cz - Ouortz

cr b Fig. 4. Plan of the structure Mg2SiO4. Small black circles represent Si, shaded circles Mg, and open circles O atoms. Light and heavy lines are used to distinguish between SiO4 tetrahedra at different heights.

Fig. 5, Coordination around oxygen ions in MgO, o~-Si02" and forsterite.

discrete (SiO4) 4- tetrahedra linked by magnesium ions, each of which has six nearest neighbor oxygen ions. The plan of this structure perpendicular to [100] direction is shown in Fig. 4. From this it is seen that each oxygen is bonded to one silicon and at the same time bonded to three magnesium ions. Figure 5 shows the atomic arrangement around the oxygen in each of these materials, along with the bond distances [9], etc. From this we may, as a first approximation, infer that the immediate surroundings of the oxygen ions in forster-

ite are made up of the combination of the atomic arrangements of MgO and a-SiO~. Table 2 gives the values of the mean refractive indices of MgO, ct-SiO2 and forsterite. It is seen that the mean refractive index of forsterite is almost halfway between those of MgO and a-SiO2. Further, the number of oxygens per unit volume, N, in these materials is almost equal (see Table 2). The polarizability of the oxygen ions in these materials was evaluated using the same procedure as Tessman et al. [7] and are also given in Table 2.

Forsterite

Table 2.

Polarizability of oxygen ions in A 3 No. of oxygen ions per unit volume • 10-22 Mean refractive index, no Mean An for 0.25% vol. strain x 102

MgO A

ct-SiO2 B

Mg2SiO4

89 + B)

1 "70

1 "42

1 "56

1 "56

5-35

5.31

5"43

5"33

1-74

1-55

1-65

1 "645

0.097

0-015

0"011

-- 0-075

F O R S T E R I T E , MgzSiO4

Again it is seen that the polarizability of the oxygen ions in forsterite is almost equal to one-half the sum of the polafizabilities of oxygen in MgO and o~-SiO2. If now these materials are subjected to the same volume strain (say, 0-25 per cent), it-Is natural to expect the average change in the refractive indices of forsterite to be about one-half the sum of what is observed in MgO and tx-SiO2 under identical circumstances. As seen from Tables 1 and 2, this expectation is borne out to be true; but it must be emphasized that the close agreement between the expected and calculated values must be considered as fortuitous in view of the approximations and averaging procedure adopted at various stages. However, it clearly brings out the fact that in inorganic oxide materials, the refractive index and its variation with volume and possibly temperature as well are mainly dependent on the type of bonding and the coordination number of the oxygen ions. Recent quantum mechanical calculations of Yamashita et al. [11] have clearly shown that the 0 2- ion, which is normally unstable in free space (dissociating to O- ion and an electron), may be stabilized in a crystal such as MgO by the local crystal field. This is the reason why Tessman, Kahn and Shockley[7] could not arrive at a unique value of the polarizability for the 0 2- ions in crystals. In fact, their analysis showed that its polarizability can have any value in the range 0-5-3.2(,~) a whereas they were able to assign specific

1255

values to the polarizabilities of the numerous other ions to two or even three significant figures. The present results which depend on the variation of the electronic polarizability with interatomic distance bring out the importance of the type of bonding in the determination of the polarizability and its variation. Acknowledgements-The authors would like to express their sincere thanks to Prof. R. E. N e w n h a m of The Pennsylvania State University and to the referee of this journal for useful suggestions. REFERENCES 1. V E D A M K. and D A V I S T. A., J. Opt. Soc. Am. 57, 11401"1967). 2. V E D A M K., S C H M I D T E. D. D. and R O Y R., J. Am. Ceram. Soc. 49, 531 (1966). 3. V E D A M K. and S C H M I D T E. D. D., Phys. Rev. 146, 548 (1966). 4. D A V I S T. A. and V E D A M K.,J. Opt. Soc. Am. 58, 1446 (1968). 5. G R A H A M E. K., JR. and B A R S C H G. R., J. Geophys. Res. 74, 5949 (1969). 6. There appears to be confusion in the literature about the nomenclature. In this work the choice of the axes were such that a = 4.78 ~ , b = 10.21 ]~, c = 5.78 ,~. 7. T E S S M A N J. R., K A H N A. A. and S H O C K L E Y W., Phys. Rev. 92, 890 (1953). 8. B I R L E J. D., G I B B S G. V., M O O R E P. B. and S M I T H J. V.,Amer. Mineral. 53, 807 (1968). 9. The two non-equivalent Mg ions in forsterite are each bonded to six oxygen ions in octahedral coordination with the mean M-O distances of 2'103 and 2.135 ,~., respectively. Hence Fig. 5 indicates these values and their average. The Si-O distances are from a recent compilation [ 10]. 10. B R O W N G. E. and G I B B S G. V., Am. Mineral. 54, 1528 (1969). 11. Y A M A S H I T A J. and A S A N O S., J. phys. Soc. Japan 28, 1143 (1970); Y A M A S H I T A J. and K O J I M A , M., J. Phys. Soc. Japan 7, 261 (1952).