American Mineralogist, Volume 61, pages 1280-1293, 1976

Effects of temperature and pressure on the crystal structure of forsterite

Rosnnr M. HlzeNl

Haruard Uniuersity, Department of Geological SciencesC amb ridge, M as sachuse tts 02 I 3 8

Abstract

The crystal structure of synthetic forsterite has been studied as a function of temperatureand pressure. Crystal structures have been refined from three-dimensional diffraction datacollected at temperatures from -196'C to 1020'C and at pressures to 50 kbar. Volumecompressibility and thermal expansion of individual magnesium octahedra are comparable tothose of bulk foresterite. However, the silicon tetrahedron neither expands nor compresseswithin the range of conditions investigated. Thus, changes in Mg-O bond distances accountfor virtually all of forsterite's volume response to changes in temperature and pressure. ShortMg-O bonds vary significantly less with temperature and pressure than do long bonds,implying that shorter bonds are stronger. Octahedral distortions increase with temperature,but show no strong trends with increasing pressure.

IntroductionThe characterization of minerals at high pressures

and temperatures is an essential part of the earthsciences. Many physical properties of minerals in-cluding compressibility, thermal expansivity, andelectrical conductivity are functions of temperatureand pressure. Since a solid's physical properties are adirect consequence of its atomic arrangement, thecontinuous variation of atomic positions resultingfrom changes in ? and P may thus be used to under-stand more fully the variation of physical propertieswith these intensive parameters. The principal objec-tive of this study is to determine the atomic coordi-nates and thermal vibration pararneters of magnesianolivine at a variety of temperatures and pressures inorder better to define the equation of state of thisimportant mineral.

The crystal structure of olivine was first determinedby Bragg and Brown (1926); subsequent refinem€ntswere reported by Belov e, al. (1951), Hanke andZemann (1963), Birle et al. (1968), and Brown (1970).Lattice parameters of magnesian olivine have beenevaluated by many authors (e.g. Louisnathan andSmith, 1968). Thermal expansion of olivine has beenstudied by several investigators including Kozu et al.(1934), Rigby et al. (1946), Skinner (1962), and Soga

I Present address: GeophysicalN. W., Washington, D. C. 20008.

and Anderson (1967). More recently, variations ofcrystal structure with temperature have been reportedfor pure forsterite and FoseFaoz (Smyth and Hazen,1973), FoogFar' (Brown and Prewitt, 1973), and purefayalite (Smyth, 1975). Compressibility studies onolivine are numerous, and data of Bridgman (1948),Olinger and Duba (1971), and Schock et al. (1972)are frequently cited. Recent studies of Olinger andHalleck (1975) offer improved values for linear andvolume compressibilities of olivine.

Olivine has two crystallographrcally distinct oc-tahedral sites, one tetrahedral site, and three distinctoxygen positions. It displays significant deviationsfrom ideal close-packing of oxygens, and these devia-tions are most conveniently visualized in terms ofdistortions of the octahedral and tetrahedral cationcoordination polyhedra from their regular forms(compare Fig. la and Fig. lb). The M(l) site hasi symmetry, and its coordination polyhedron ap-proximates an octahedron flattened along a three-fold axis. The M(2) octahedron, which has mirrorsymmetry, has larger average M-O distances thanM(l), and does not approximate any simple dis-tortion model (Dollase, 1974). As noted by Brown(1970), the three olivine cation polyhedra have anumber of oxygen-oxygen edges in common, includ-ing two between adjacent M(l)'s, two between M(l)and M(2), two between M(l) and Si, and one be-tween M(2) and Si. Distortions of the hexagonal

Laboratory, 2801 Upton Street,

I 280

FORSTERITE l 2 E l

composition of 99.99 *'percent MgzSiOr, and th€indices of refraction are d = 1.635 + 0.002, P : 1.650+ 0.002, and 7 : 1.670 + 0.002, which correspond toreported values for pure magnesian olivine (Hein-riques, 1957). The crystals are optically free of frac-tures or other visible defects, and a good (010) cleav-age is present. An irregular crystal of approximately400 X 300 X 300 pm maximum dimensions was se-lected for preliminary X-ray study. A suite of a-, b',and c-axis precession photographs was obtained toconfirm previous unit-cell and symmetry determina-tions. A diffraction symbol of mmmPbn- wasobserved, which corresponds to the reported olivinespace group Pbnm.

Unit-cell dimensions of forsterite were determinedon the oriented single crystal from precision back-reflection Weissenberg photographs. Refinement of82 diffraction spot pairs yielded orthorhombic cellparameters of a : 4.7535 + 0.0004, b : 10.1943 +0.0005, and c : 5.9807 + 0.0004 A, with a resultingunit-cell volume of 289.80 + 0.05 A'. These valuesclosely agree with determinations by Yoder and Sa-hama (1957).

Dala collection and inslrumentation

Procedures for data collection from single crystalsat room temperature, liquid-nitrogen temperature,and high temperature are described by Hazen (1976),while high-pressure X-ray diffraction techniques weresimilar to those of Hazen and Burnham (1974,1975).High-pressure diffraction data have been correctedfor diamond pressure cell absorption as well as speci-men absorption (Hazen, 1976).

Results

Room temperalure

A reference room-temperature and pressure struc-ture refinement was made with 956 observed diffrac-

TI

A.

B.

l._ ! _{

Frc. l. Olivine crystal structure: A-ideal HCP model, B-actual structure, (after Brown, 1970).

close-packed oxygen array have been related to

cation-cation repulsions across these shared edgesin agreement with Pauling's third rule.

Experimental

Specimen description

Colorless and transparent single crystals of syn-thetic forsterite have been manufactured by the Crys-tal Products Division of Union Carbide Corporation,and were kindly provided by Professor T. J. Shank-land. This essentially pure forsterite has a reported

TABLE lA. Forsterite unit-cell parameters from 23" to I 105.C

T ( t c )o ^

v o l ( A J )o

b (A)o

a (A)_ o .

c (A)

23180305450

4 . 7 5 2 ( 3 ) x4 . 7 5 8 ( s )4 . 7 6 s ( 5 )4 . 7 6 7 ( 3 )

4 . 7 7 5 ( 5 )4 . 7 8 0 ( 3 )4 .785 (7 ' )4 . 7 e 7 ( 7 )4 . 8 0 s ( 7 )

1 0 . 1 9 3 ( 8 )1O.272(7O)r o . 2 3 1 ( 1 O )1o . 248 (8 )

1 0 . 2 7 3 ( 1 0 )ro .292 (8 )1 0 . 3 3 ( 1 )1 0 . 3 5 ( 2 )

5 , 9 7 7 ( 5 )5 . 9 9 r ( 6 )5 . 9 9 8 ( 6 )6 . oo9 (s)

6 . o 2 3 ( 6 )6 . o 3 2 ( 5 )6 .o47 (9 )6 . o58 (8 )6 . 069 (8 )

2 8 e . s ( 3 )2 9 r , r ( 5 )2 9 2 . 4 ( 5 )293 .6 (3 )

2e5 . 5 ( 5 )296 .8 (3 )298 .9 ( 8 )3 0 o . 7 ( 1 2 )

6 1 0750900

1020I105

* Parenthes ized f i sures re fe r to the esd o f leas t un i ts c i ted .

1282 ROBERT M. HAZEN

tion maxima representing all hkl's in one octant ofreciprocal space from 5o to 80o 2d (MoKa). Of 1096measured reflections, ll6 including all 90 space-group extinct reflections, were unobserved. In addi-tion, 24 of the strongest reflections, for which lF,u"l(( lF"rt"l, were rejected due to presumed secondaryextinction. A calculated linear absorption coefficientof 10.594 cm-r was applied, and transmission factorsvaried from 78 to 83 percent. Refinement of forsteriteincluded 40 variables, representing I I atomic posi-tional parameters, 28 anisotropic vibrational parame-ters, and a scale factor. A weighted R-factor of 4.8percent (4.9 percent unweighted) was obtained forthis refinement. Forsterite refinement conditions,positional parameters, and temperature factors arelisted in Tables 1,2, and 3. Room temperature andpressure values are in close agreement with previous23"C forsterite refinements by Birle et al. (1968) andBrown (1970).

Cell dimension and structural factor measuremew qr-196" to I 105"C

Forsterite unit-cell parameters and structure fac-tors were measured at several temperatures from-196"C to above 1000'C. Preliminary results of thehigh-temperature portion of this study were reportedby Smyth and Hazen (1973). The crystal chosen forhigh-temperature work by Smyth and Hazen wasmounted in air directly on the join of the PtlPtroRhro

thermocouple by natural adhesion to hot platinum.Unit-cell constants and intensity data were collectedfirst at 23"C and then at reported temperatures of300o, 600o, and 900' (see below).

In the present study, the crystal was mountedwithin a silica capillary, and unit-cell constants offorsterite were measured at temperatures to I105'C(see Table la). An attempt at a 1020'C data collec-tion was then made, but fewer than 150 diffractionswere measured before the heater burned out. Duringthe rapid cooling following heater failure, the silicacapillary fractured and the crystal was lost. Thus, itwas not possible to recheck the crystal at room tem-perature.

Another forsterite crystal of approximately rec-tangular shape, 200 x 180 X 120 pm, was mountedon a coppar pin; unit-cell and intensity data werecollected at liquid-nitrogen temperature using theCryo-Tip system. Unit-cell parameters were redeter-mined following the low-temperature study, and wereunchanged from previous room-temperature meas-urements. In each of the six structure refinements,made a t -196o,23o, 300o,600o,900o, and 1020"C,all Pbnm space group-extinct reflections were unob-served. From 6 to 24 of the strongest reflections wererejected from each refinement due to presumed sec-ondary extinction (i.e.,lF"6"l (lF""r"l). Results arepresented in Tables 1,2, and 3, in the same format asthat of Brown (1970) for ease of comparison.

TABLE 18. Forsterite unit-cell parameters and refinement conditions

T ^ - N o . o f N o . o f N o . o f("c) P condi t ions

" t i l r l i l c f i ) 'or ( i3) .8.(%1" .Bl%) ' o i "" i " .a * . : . " t .a"

- 3 5 0

- 6 7 5- 1 0 0 0

1 0 2 0

-I96

2 3

2 3

2 3

h

1 . 7 5 3 5 ( 4 r r 0 . 1 9 4 3 ( s ) 5 . 9 8 0 7 ( 4 ) 2 8 9 . 8 0 ( s ) 4 . 8 4 9z J ' ? t ' a o o ^ !

; : : ;T " ' " 'o t " " n . ru r 1 r1 r0 .24 ( r ) 5 . eee (6 )

, ' c 4 . 7 i a ( 5 \ 1 0 . 2 9 ( 1 ) 6 . 0 r 7 ( 6 )n c 4 , 7 9 5 ( 5 \ r o . 3 6 ( r ) 6 . 0 6 0 ( 6 )

2 9 2 . 6 ( 5 ) 5 . 4 5 . 6

2 9 6 . 0 ( s ) 5 . 7 s . 9

3 0 0 . 8 ( 5 ) 5 . s s . 9

d d

r l - t - 1 2 - 9 -

4 . 9 5 . 5

7 - o " 9 - 4 "

; . : d s . e d

9 . 8 - 1 2 . 3 -

1 0 9 6

6 7 6

6 8 6

1 3 5

7 4 2

969

964

962

L O 2 6

9 5 6

569 7

564 6

5 6 5 8

t 0 8 9, , S i l i c a c a p -" . , , 4 . 7 9 e ( 5 ) 1 0 . 3 6 ( r ) 6 . 0 6 S ( 6 ) 3 0 1 . 4 ( 5 )l t l a r y m o u n t

a r \ r ^ - f i ^^ 1 ^ ^ 1 5 ) 1 0 . r 8 ( 1 ) 5 . 9 ' 7 6 ( 6 ) 2 8 8 . 6 ( 5 )m o u n t

D F c < c , r r a

2 o k b " ; i i - * - -

4 . 7 4 3 ( 5 1 1 0 . 0 e ( r ) 5 . e 5 4 ( 6 ) 2 8 s . 0 ( 5 )

4 0 k b , 4 . 7 3 4 ( 5 ) r 0 . 0 2 ( r ) 5 . 9 4 0 ( 6 ) 2 8 r . 8 ( s )

5 0 k b " 4 . 7 r 2 ( 5 ) 9 . 9 7 ( 1 ) 5 . 9 5 5 ( 6 ) 2 7 9 . 1 ( 5 \

646

329

3 7 4

790

3

6

l 3

2S tanda rd

2 3 1 a t m m o u n t ( a f t e r 4 - 7 4 9 1 5 ) r 0 . 1 9 ( 1 ) 5 . 9 8 0 ( 6 ) 2 e 9 . 5 ( S ) 3 . 9 4 - 2h i q h p )

a )

b )

d )

l l " ,o . l - l r " " r " l l ' : .0P a r e n t h e s i z e d f i g u r e s r e f e r t o t h e e s d o f l e a s t u n i t s

D a t a o f S n y t h a n d H a z e n ( 1 9 7 3 ) , w i r h c o r r e c t e d E e h p e r a r u r e

I s o t r o p i c t e m p e r a t u r e f a c t o r s o n l y ,

c i t e d .

e s c l m a E e s .

FORSTERITE

Tnslr 2. Forsterite positional parameters, isotropic temperature factors, and r. m. s. equivalents

23' c - : so " "b - a r r " " b - r ooo "cb 1 0 2 0 ' c 20kb

l 283

Atom HCP

P a r a -

m e t e r 40 kb1 a t m a f -

t e r h i g h P50kb

M ( 1 )

zB

0t / 4| /4

3 / a0 . o 8 3 3r / 4

3 /40 . 0 8 3 3r /4

t / 4o . 4 \ 6 7t / 4

t / 4o . L667

0

000o . o g ( z ) "0 . 0 3 4

0 . 9 9 1 4 ( 3 )o . 2 ' 7 7 2 ( r ll / 40 . 0 8 ( 2 )0 . 0 3 1

o . 4 2 6 1 ( 2 )0 . 0 9 3 9 ( 1 )t / 4

- o . 0 2 ( 2 10 . 0 0

0 . 7 6 6 1 ( 6 )0 . 0 9 1 9 ( 3 )r / 40 . 1 2 ( 4 )o . 0 3 9

o . 2 2 0 2 \ 6 10 . 4 4 5 9 ( 3 )1- /40 . 0 6 ( 4 )0 . 0 2 6

o . 2 1 7 ' 7 1 4 )o . t 6 2 A 1 2 )

0 . 0 3 3 3 ( 3 )0 . 0 7 ( 3 )o . o 2 9

oo00 . 2 6 ( 1 )0 . 0 5 7

0 . 9 9 r 5 ( 2 )o - 2 7 7 4 l t \t / 4o . 2 2 ( r l0 . 0 5 3

o . 4 2 6 2 ( r )0 . 0 9 4 0 ( r )r / 40 . 0 8 ( 1 )0 . 0 3 1

0 . 7 6 s 7 ( 3 )0 . 0 9 1 3 ( 2 )t / 4o . 2 6 ( 2 10 . 0 s 7

0 . 2 2 1 5 ( 4 )o . 4 4 7 4 1 2 )l / 4o . 2 4 ( 2 )0 . 0 5 5

o . 2 7 1 7 \ 2 )0 . 1 6 2 8 ( 1 )

0 . 0 3 3 r ( 2 )o . 2 8 ( 2 10 . 0 6 0

o000 . 6 7 ( 3 )0 . 0 9 2

0 . 9 9 r s ( s )0 . 2 7 8 0 ( 2 )r / 40 . 6 6 ( 4 )0 . 0 9 1

o . 4 2 s 7 ( 4 )0 . 0 9 3 9 ( 2 )r /4o . 3 3 ( 3 )0 . 0 6 5

0 . 7 6 5 7 ( 1 0 )0 . 0 9 1 0 ( s )t / 40 . 7 0 ( 6 )0 . 0 9 4

0 . 2 1 7 8 ( 9 )o . 4 4 9 2 ( s )r /40 . 6 0 { 6 )0 . 0 8 7

o . 2 a 2 2 1 6 )o . 1 6 1 9 ( 3 )

0 . 0 3 4 7 ( 6 )o . 6 5 ( 4 )0 . 0 9 0

000r . 2 0 l 4 lo . I 2 2

0 . 9 9 1 9 ( 5 )o . 2 1 4 5 l 2 ll / 4r . 1 9 ( 4 )o . r 2 2

0 . 4 2 s ' 7 l 3 l0 . 0 9 4 r ( r )t / 40 . 6 6 ( 3 )0 . 0 9 1

o - ' 1 6 3 1 ( 9 )0 . o 9 0 6 ( 4 )),/4t . t 2 \ ' 7 )0 . 1 1 9

o . 2 1 7 8 ( 9 )o . 4 4 e 1 ( 4 )r / 4r - 0 2 ( 7 )0 . 1 1 3

o . 2 4 2 2 ( 6 10 . 1 6 r 9 ( 3 )

0 . 0 3 s 2 ( s )r . 1 s ( s )0 . 1 2 0

000r . 7 1 1 4 )0 . 1 5 0

o . 9 9 2 4 ( 4 )o . 2 7 7 2 ( r )t / 41 . 7 0 ( 4 )o - r 4 7

o . 4 2 6 3 l 3 )0 . 0 9 4 3 ( 1 )r / 40 . 9 7 ( 3 )0 . 1 1 1

0 . 7 6 3 1 ( 8 )0 . 0 9 1 4 ( 4 )t / 4r - 5 1 ( 6 )o . L 3 ' 7

0 . 2 r 7 8 ( e )0 . 4 4 9 7 ( 4 )r / 41 . 4 0 ( 6 )0 . 1 3 3

o . 2 4 4 3 ( 5 10 . 1 6 2 9 ( 3 )0 . 0 3 s 9 ( s )r . 6 7 ( s )o . L 4 4

0o0r . 8 ( 3 )0 . 1 5

0 . 9 9 r ( 3 )0 . 2 8 0 ( 1 )7 /41 - 8 ( 3 )0 . r 5

o . 4 2 ' 1 1 2 )0 . 0 9 4 ( r )L / 41 . 0 ( 2 )O . I I

o . 7 3 1 1 2 )0 . 0 8 8 ( 2 )| /41 . 0 ( s )0 . 1 1

o . 2 2 4 ( 4 )0 . 4 s 8 ( 3 )r /4r . 0 ( 5 )O . I I

0 . 1 6 0 ( 2 )

0 . 0 3 0 ( 3 )r . 8 ( 4 )o . r 5

0000 . 1 6 ( 9 )o . 0 4 6

0 . 9 9 2 ( r )o . 2 7 7 ( r )r / 40 . 3 7 ( r 0 )0 . 0 6 9

o . 4 2 7 ( r )0 . 0 9 s { I )r /40 - 3 7 ( 8 )0 . 0 6 9

o . 7 ' 7 L ( 2 1o . o e 2 ( 2 )r / 4

- 0 . 1 ( 2 )0 . 0

o . 2 2 1 \ 2 1o . 4 4 ' 7 ( 2 )r /40 . 0 ( 2 )0 . 0

o . 2 7 4 l 2 lo . 162 12)0 . o 3 7 ( r )o , 2 ( 1 )0 . 0 5

0oo0 . 3 7 ( 9 )0 . 0 6 9

0 . 9 9 r ( 1 )0 . 2 8 o ( t )| /4o . 5 2 ( 1 0 )0 . 0 8 1

0 . 4 2 6 ( r )0 . o 9 3 ( r )r /4o . 4 s ( 7 )0 . o 7 5

0 . 7 6 8 ( 2 )o . o 9 7 1 2 \1 /4

o . r ( 2 )0 . 0 3 5

0 . 224 12)o - 4 4 4 \ 2 )r /4o - o ( 2 )o . 0

o . 2 7 6 l 2 l0 . 1 s 8 ( 2 )0 - 0 3 s ( 1 )0 . 4 ( r )0 . 0 7

000 . 2 ( r )0 . 0 5

0 . 9 9 3 ( 1 )0 . 2 8 3 ( r )t / 40 . 6 ( r )0 . 0 8 7

0 . 4 2 8 ( r )0 . 1 0 1 ( r )7 /40 . 7 ( 1 )0 . 0 9 4

o . 7 7 6 \ 2 )

| /40 . 1 ( 2 )0 . 0 3 6

0 . 2 2 8 ( 3 )0 . 4 4 0 ( 3 )| /4o . 4 1 2 )0 . 0 7

0 . 2 8 r ( 2 )0 . 1 5 3 ( 2 )0 . 0 3 2 ( 1 )o . 6 ( 2 )0 . 0 8 7

000o . 4 3 ( 2 )0 . 0 7 4

0 . e 9 1 4 ( 3 )o . 2 ' 7 7 I l t l1, /4o . 4 3 ( 2 \o . o 7 4

o . 4 2 6 2 ( 2 )0 . 0 9 8 e ( 1 )t /40 . 2 7 l L l0 . 0 s 8

o . 1 6 6 4 1 5 \o - 0 9 0 8 ( 3 )t / 4o . 4 3 ( 4 1o . o ' 7 4

o . 2 2 0 9 ( 5 )0 . 4 4 ' 7 4 ( 2 )| /40 . 3 7 ( 3 )o - 0 6 9

o . 2 7 4 4 e l0 . 1620 (2 )0 . 0 3 4 1 ( 3 )0 . 4 r { 2 )o . o 7 2

M ( 2 )

s i

o ( 1 )

o ( 2 ' )

o ( 3 )

zB

2B

YzB

x

zB

X

zB

a ) " I d e a l - " h e x a g o n a l c l o s e - p a c k e d o l i v i n e m o d e l ( s e e F i g . 1 ) .

b ) D a t a o f S m y t h a n d H a z e n ( 1 9 7 3 ) .

c ) P a r e n t h e s i z e d f i g u r e s r e f e ! t o t h e e s d o f l e a s t u n i t s c i t e d .

Linear and volume thermal expansion of pure for-sterite as determined by Smyth and Hazen (1973)from 23" to 900'C (Table lb) and this study (Table1a) from -196o to I l05oC, are illustrated in Figures2 and 3. Data of Skinner (1962) on expansion of asynthetic iron-bearing forsterite (ForuFaon) from 25oto ll27"C are also presented on these graphs. Thedata of Skinner and this study are in close agreementwith respect to the rate of thermal expansion; allexpansion curves are essentially parallel over the tem-perature range25o to I l00oC. However, serious dis-crepancies exist between the data of these two studiesand those recorded by Smyth and Hazen (1973). Thetemperature measurements of Skinner and of thisstudy were made with calibrated thermocouples, andreported temperatures are based on these calibra-tions. However, the thermocouple employed bySmyth and Hazen was not calibrated. The techniqueof mounting a crystal directly on the thermocouplePtlPt-Rh join requires a larger wire diameter (0.150mm ur. 0.050 mm), and significantly larger join bead(diameter : 0.30 mm Dr. 0.075 mm) than that used inthe silica capillary mount of this study. Calibration ofsimilar large thermocouples by T. L. Grove and this

author revealed that in all cases the measured temper-ature was lower than the real temperature, and thatsuch errors may exceed 100'C at 1000"C (in oneexperiment, gold of melting point 1063"C appearedto melt at ^, 950oC). These errors are assumed toreflect the rapid conduction of heat away from thelarge thermocouple join. Such an error in thermo-couple readings could explain the differences betweenthe different sets of cell parameters. If this is the case,then the forsterite refinements reported in Smyth andHazen (1973) were made at -350", -675", and-1000'C. This would explain the deviations in unit-cell parameters at high temperatures, as well as thesimilarity between Smyth and Hazen's 900'C refine-ment and the partial 1020"C refinement of this study.In the remainder of this study Smyth and Hazen'sdata will be assumed to have been obtained at thehigher temperatures.

Bond distances

Bond distances and bond angles are tabulated inTables 4 and 5 respectively. Figure 4 illustrates thevariation of average Mg-O bond distances for thetwo different magnesium octahedra, as well as the

1284 ROBERT M. HAZEN

Teslr 3. Forsterite anisotropic temperature factor coefficientsu

Atomi j o f

R i i - l 9 6 o c 2 i ' C2 3 ' c a f t e r

. - - o ^ - . ̂ ^ ^ o ^6 7 5 - C

- I U O U C h i g h p r e s s u r e- 35ooc

I I2 2J 3

I 2I J

2 3

o . 2 ( 4 )0 . s ( r )0 . 2 ( 3 )o . 2 ( 2 )

- 0 . r ( 3 )- 0 . 2 ( 1 )

0 . r ( 5 )0 . 4 ( r )0 . 4 ( 3 )

- 0 . 0 ( 2 )

- r . 8 ( 3 ) c

0 . 2 ( r )o . 2 ( 2 )0 . r ( 1 )

- o . 2 ( 9 ) c

0 . 5 ( 2 )r . 1 ( 6 )0 . 5 ( 4 )

- 0 . 2 ( 9 ) c

0 . r ( 2 )0 . 8 ( 5 )0 . 0 ( 4 )

- 0 . 2 ( 6 ) c

0 . 4 ( r )o . 2 ( 4 )0 . 0 ( 2 )0 . 3 ( 4 )0 . 0 ( 2 )

2 . 4 ( 3 )r . 0 2 ( 6 )r . 0 ( 2 )0 . 1 ( r )

- 0 . 3 ( 2 )- o . 3 ( r )

2 . 2 ( 3 )0 . 6 ( r )| . 6 ( 2 10 . 0 ( r )

0 . 8 ( 2 )o . 4 3 ( 4 )0 . 8 ( 1 )0 . 0 5 ( 8 )

o . e ( 5 )0 . e ( 1 )2 . 3 ( 3 )o . 2 ( 2 1

2 . 4 ( 5 )o . 7 ( L l2 . 2 ( 3 )0 . 1 ( 2 )

2 . 2 ( 3 )0 . 9 3 ( 7 )L . 6 ( 2 )0 . r ( 1 )

- 0 . 3 ( 2 )0 . 4 ( r )

7 . 5 ( 2 12 . I 1 2 )2 . e ( 3 )

- 0 . r ( 1 )- 0 . 5 ( 2 )- o . 7 ( 2 )

8 . r ( 8 )r . 4 ( r )4 . 5 ( 4 )0 . 0 s ( e )

2 . 6 ( 2 )r - . 0 ( r )2 . 5 ( 2 )0 . 1 ( 1 )

e . 4 ( r . 0 )L . 2 ( 2 )s . r ( 7 )0 . 3 ( 2 )

7 . 3 ( 7 )| . 4 ( 2 )4 . 0 ( 4 )

- 0 . r ( 2 )

8 . 2 ( e )L . 6 ( 2 )3 . 7 ( 4 )

- 0 . 3 ( 1 )- 0 . 2 ( r )

o . 6 ( 2 )

r 0 . 7 ( 4 )3 .9 (z ' , )6 . 8 ( 4 )

- o . 2 ( 2 )- 1 . 0 ( 2 )- 1 . 2 ( 1 )

L 2 . 4 ( L . O ' )2 . 7 ( 2 )e . L ( 7 )o . 2 ( 2 )

3 . 8 ( 4 )| . 8 ( 2 15 . e ( 4 )0 . 3 ( 2 )

9 . 8 ( r . r )2 . 8 ( 4 )8 . 8 ( 9 )0 . e ( 1 )

1 0 . 0 ( r . 0 )2 . 4 ( 3 17 . 8 ( 6 )

- 0 . 1 ( r )

1 r . 2 ( r . r )3 . 0 ( 2 )8 . 0 ( 7 )

- 0 . s ( 1 )- 0 . 5 ( 3 )

L . 2 ( 2 )

r 5 . 4 ( 3 )5 . 9 ( 3 )9 . 2 ( 6 )

- 0 . 4 ( 1 )- 1 . 7 ( 3 )- L . 7 ( 2 )

1 8 . 3 ( r . 5 )3 . 8 ( 4 )

1 2 . 2 ( r . 0 )0 . s ( 3 )

6 . 8 ( s )2 . 7 ( 4 )7 . 7 ( 7 )o . 4 ( 2 )

r 1 . 8 ( r . 8 )4 .3 ( 4 ' , t

r 0 . 8 ( 1 . 2 )4 . 6 ( 4 )

1 3 . s ( r . r )3 . 2 ( 4 )

r 0 . 6 ( 7 )0 . 0 ( 2 )

1 6 . 3 ( r . 4 )4 . s ( 3 )

1 0 . 3 ( r . 2 )- 0 . 3 ( 1 )- o . 5 ( 2 )

1 . 9 ( 2 )

4 . 2 ( 3 )L . 2 6 ( e )2 . 7 ( 3 )0 . 0 ( 3 )

- 0 . 2 ( 3 )- 0 . 3 ( r )

4 . 6 ( 5 )r . r ( 1 )2 . e ( 3 1o . 2 ( 2 )

1 . 8 ( 3 )0 . I ( r )2 . L ( 2 10 . 0 ( 1 )

3 . 7 ( 9 ' , )1 . 3 ( 2 )2 . 7 ( 5 )0 . 3 ( 4 )

4 . 2 ( 9 \0 . 8 ( 2 )2 . 6 ( s lo . 3 ( 4 )

3 . e ( 6 )r . 2 ( 1 )2 . 8 ( 4 )

- 0 . 0 ( 2 )0 . 0 ( 4 )0 . 3 ( 2 )

M ( 2 ) 1 r2 23 3L 2

s i 1 12 2

L Z

o ( r ) 1 12 23 3T 2

o ( 2 ) 1 12 25 5

L 2

o ( 3 ) 1 r2 25 5

l 21 32 3

a)

b )

c )

6 . . x I o - : f o r M ( 2 ) , s i , o ( l ) a n dr J

Parenthesized f igures refer to the

Atom is non-posi t ive def in i te.

B r : = B z : = o '

o f l e a s t u n i t s c i t e d .

o ( 2 )

e s d

average Si-O tetrahedral bond distance. The bondexpansion curve of the M(l) octahedron is remark-ably similar to that of Mg-O in periclase. As dis-cussed in Hazen (1976), periclase Mg-O bonds havean absolute zero bond length of 2.10 A. ttrey expandat an increasing rate to a temperature of approxi-mately 450 K, and above this temperature Mg-Obonds have a relatively constant expansion rate ofabout 2.7 X l0-'A/K. Forsterite Mg(l)-O bondshave an absolute-zero'mean bond distance of x2.09A. ttrey expand at an increasing rate to approxi-mately 400 K, and above this temperature expand ata constant rate of 2.8 X l0-'A/K. The longerMg(2)-O forsterite average bond distance has qual-itatively similar expansion characteristics. The abso-lute-zero bond distance for Mg(2)-O is =2.13 A, andabove 450 K a constant expansion of 3.4 X l0-'A/Kwas observed. It appears from these average bonddata for magnesium-to-oxygen bonds in periclase andforsterite that longer bonds may have a higher rate of

thermal expansion. To test this possibility all individ-ual magnesium-oxygen bonds have been plotted ver-sus temperature in Figure 5. While there is somescatter in these data, longer Mg-O bonds show signif-icantly higher rates of thermal expansion (as much asfour times greater) than shorter bonds, as illustratedin Figure 6. This fact is consistent with the theorythat shorter bonds are stronger bonds.

Analysis of the silicon-oxygen bonds in forsteriteversus temperature is less straightforward. The aver-age Si-O bond length is constant or increases onlyslightly from liquid-nitrogen temperature to roomtemperature, as illustrated in Figure 4. The distance isseen to decrease by over 0.5 percent between 300 Kand 800 K; only above 1000 K does the average Si-Obond appear to expand. This unexpected behavior offorsterite Si-O bonds between -196" and l000oCcannot be explained with a simple bonding model ofSi-O alone. Rather, complex Mg-O-Si interactionsmust be considered before a thorough understanding

- , , ' -Skfnn.r (1962)-F55

* ' , " ; /...o.': Srnyth O Horcn ltgT'l - Fotr,O,?a.FX./

/.x1Thls Studt-Fo,oo

.,.::ifl2x

TEMPERATURE (K)

Frc. 2. Forsterite unit-cell dimensions versus temperature.Data of Smyth and Hazen (1973) have not been corrected forpresumed errors in temperature.

of Si-O bond distance variation in forsterite can beachieved.

The variation between individual Si-O bonds withtemperature is illustrated in Figure 7. Silicon-to-oxy-gen bonds in forsterite provide a striking example ofthe greater thermal response of longer bonds.The shortest Si-O(l) bond shows no variation inlength between -196o and 1000oC, while the longestSi-O(2) bond shows a maximum variation of 1.3percent over this range.

An important and often neglected aspect of bonddistance analysis is the effect of thermal vibrations onmean interatomic separation. Interatomic distances

tnn\e +. Thermally corrected mean separation of Si-O int forsterite (A)

1285

2oo 400 600 800 looo 1200 1400TEMPERATURE (K)

FIc. 3. Forsterite unit-cell volume versus temperature. Data ofSmyth and Hazen (1973) have not been corrected for presumederrors in temperature.

reported in most studies (including this one) are cal-culated as the distance between mean atomic posi-tions. However, Busing and Levy (1964) and Smyth(1973) have demonstrated that a better measure ofinteratomic distance is the mean separation. In gen-eral, when thermal vibrations are large, the meanseparation of two atoms will be greater than theseparation of atomic positions. Thus, bond thermalexpansion based on mean separation may be greater,and may represent a more valid physical situation,than that reported in most studies which quote dis-tance between atomic coordinates.

It will prove useful to analyze thermally-correctedbond distances for Si-O in forsterite as an illustrationof the possible magnitudes of thermal bond distancecorrections. Lower, upper, riding, noncorrelated, anduncorrected bond distances were computed by theleast-squares and errors programs (RnNn and Beo-rne, Finger, 1969). These data for the mean Si-Obond distance in forsterite are tabulated for temper-atures from -196o to 1000"C in Table 4. and are

FORSTERITE

ro

lrlI3)o

JJl!a)

L

; 2 e 4=

z9ozlrt-o

JJUo

F

2D

CentroldT(oc) separa t lon

Non- UpperRlding correlated Bound

Lowerbound

-196

3506 t J

1000

r . oaaR1 . 6 3 0L . O Z )

L , O L q

1 , 5 3 0

1 . 6 3 31 . 6 3 0L . 6 2 5L . O Z +

1 . 6 3 0

I . O J J

I . O J I

L , 6 2 7L . O Z I

I . b J )

L . 6 3 4 1 . 6 3 51 . 6 3 3 L . 6 3 6L , 6 3 2 1 . 6 4 01 . 6 3 8 r . 6 5 2r . 5 5 1 1 . 5 6 5

- Stlnncr (1e62) - Fo96 !,'- Smyth a Hozcn (1S73)-Fopq;' .- This Study - FotOO

/:' i

1286 ROBERT M. HAZEN

Tnsr-E 5. Forsterite bond distances"

2 3 ' c - r 9 6 ' c - : s o ' c b - e l s ' c b - t o o o " c b t o z o " cB o n d I M u l t j - p l i c i t v ] s 0 k b

Si Te t rahedron

s i - o ( t )s i - o ( 2 )s i - o ( 3 )mean S i -O

o ( r ) - o ( 2 )o ( 1 ) - o ( 3 )o ( 2 ) - o ( 3 ) do ( : ) - o ( : ) dmean O-O

M (1) Octahedron

M ( r ) - o ( r )M ( 1 ) - o ( 2 )M ( 1 ) - o ( 3 )n e a n M ( 1 ) - O

o ( 1 ) - o ( 3 ) eo ( 1 ) - o ( 3 ' )o ( 1 ) - o ( 2 ) e

o ( 1 ) - o ( 2 , )o ( 2 ) - o ( 3 ' )o ( z ) - o ( : ) dmean O-O

M(2) oc tahedron

M ( 2 ) - o ( r )N l ( 2 ) - o ( 2 )M ( 2 ) - o ( 3 )M ( 2 ) - o ( 3 " )m e a n M ( 2 ) - O

o ( 1 ) - o ( 3 " )o ( 1 ) - o ( 3 ) eo ( 2 ) - o ( 3 )o ( 2 ) - o ( 3 ' " )o ( 3 ) - o ( 3 ) do ( 3 ) - o ( 3 , , )o ( 3 n ) _ o ( 3 , " )mean O-O

O ( t ) Te t rahedron

o ( r ) - M ( l ) Bo ( 1 ) - M ( 2 ) Bo ( 1 ) - s i Am e a n O ( 1 ) - M

M ( r ) s - M ( 1 ) gM ( I ) e - M ( 2 ) sM ( r ) B - S i A

M ( 2 ) B - S i A

O ( 2 ) T e t r a h e d r o n

o ( 2 ) - M ( r ) Bo ( 2 ) - M ( 2 ) Ao ( 2 ) - s i B

m e a n O ( 2 ) - M

M ( 1 ) B - M ( 1 ) B

M ( 1 ) B - M ( 2 ) AM ( 1 ) B - S i BM ( 2 ) A - S i B

O ( 3 ) T e t r a h e d r o n

o ( 3 ) - s ( r ) Bo ( 3 ) - M ( 2 ) Bo ( 3 ) - M ( 2 ) A

o ( 3 ) - s i B

m e a n o ( 3 ) - M

M ( 1 ) B - M ( 2 ) BM ( 1 ) R - M ( 2 )

"M ( 1 ) e - s i s

M ( 2 ) B - M ( 2 ) A

M ( 2 ) B - S i BM ( 2 ) A - S i B

t1lol12)

t1l121l2lril

lz)l2lt l

Lz)t )tt)L2J12)P]

IU[]l2lt4

t lt2lt lLzltlll2ltll

12)t1ltLl

tilt2l12)tLl

r . e t s 1 : 1 " r . 6 1 6 ( 2 )r . 6 4 0 ( 3 ) r . 6 4 s 1 2 \1 . 6 3 3 ( 2 ) 1 . 6 3 3 ( 2 )1 . 6 3 0 1 . 6 3 3

2 . 7 2 s ( 6 ) 2 . 7 1 0 ( 4 )2 . 7 4 4 ( 5 ) 2 . 7 5 3 1 4 \2 . s 5 8 ( 5 ) 2 . 5 5 0 ( 3 )2 . 5 9 o ( 7 ) 2 . 5 9 O ( 4 )2 . O t 7 2 - b 5 b

2 . 0 8 3 ( 2 ) 2 . 0 8 s ( t )2 . o 7 4 1 2 ] ' 2 . 0 6 9 ( r )2 . \ 4 5 ( 3 ) 2 . 1 2 6 ( 2 )2 . I O I 2 . 0 9 5

2 . 4 6 4 l s ) 2 . a 4 6 ( 4 )3 . r 2 2 ( r ) 3 . 1 0 4 ( 4 )2 . 8 6 3 ( 6 ) 2 - A 4 e G )3 . 0 2 ' 7 ( r ) 3 . o 2 2 1 3 )3 . 3 4 2 ( 2 ) 3 . 3 4 1 ( 3 )2 . 5 5 S ( s ) 2 . s s o ( 3 )2 . 9 6 3 2 . 9 5 2

2 . 1 6 6 ( 3 ) 2 . 1 6 6 ( 4 )2 . 0 4 5 ( 5 ) 2 . O 4 0 ( 3 \2 . 2 O 8 ( 4 ) 2 . 2 o e ( 3 )2 , o 6 4 l 4 ) 2 . 0 6 6 ( 2 )2 . 126 2 . 126

3 . 0 0 8 { 5 ) 3 . 0 r 6 ( 4 )2 - 864 l5 \ 2 . e46 (3 )3 . r 7 7 ( 5 ) 3 . r 2 O ( 4 )2 . 9 2 6 ( 5 ) 2 - 9 2 O 1 3 \2 . s 9 O ( 7 \ 2 - 5 9 0 ( 4 )2 . s 7 s ( 4 ) 2 . 9 9 o ( 3 )3 . 3 e 3 ( 2 ) 3 . 3 8 6 ( 3 )2 . 9 9 0 2 . 9 8 0

2 . 0 8 3 ( 2 ) 2 . 0 8 5 ( 1 )2 . 7 6 6 ( 3 ) 2 . 1 6 6 ( 4 )1 . 6 1 5 ( 3 ) r . 6 1 6 ( 2 )1 . 9 4 7 1 . 9 9 8

2 . 9 9 3 ( 1 ) 2 . 9 8 8 ( 3 )3 . r 9 7 ( 2 ) 3 . r 9 2 ( 3 )3 . 2 5 6 ( 2 \ 3 . 2 5 0 ( 3 )3 . 2 4 9 ( 3 ) 3 . 2 4 7 ( 3 1

2 . O ' ? 4 ( 2 ) 2 . 0 6 9 ( 1 )2 . 0 4 5 ( 5 ) 2 . 0 4 0 ( 3 )1 . 6 4 0 ( 3 ) r . 6 4 9 ( 2 )1 . 9 5 8 I . 9 5 7

2 . 9 9 3 ( L ) 2 . 9 8 8 ( 3 )3 . 6 3 7 1 2 \ 3 - 6 3 3 ( 2 )2 . 6 9 2 1 2 ) 2 . 6 e 0 ( 2 )3 . 2 8 3 ( 2 1 3 . 2 7 8 ( 3 )

2 . r 4 5 ( 3 ) 2 . 1 2 6 ( 2 )2 . 2 0 5 ( 4 ) 2 . 2 0 8 \ 3 )2 . 064 (4 ) 2 . 066 12)1 . 6 3 3 ( 2 ) r - 6 3 3 ( 2 )2 . O 7 2 2 . 0 0 8

3 . r 9 ' 7 ( 2 1 3 . 1 e 2 ( 3 )3 . 5 7 s 1 2 ) 3 . 5 7 9 ( 3 )2 . 6 9 2 1 2 ) 2 . 6 9 0 ( 2 )3 . 4 6 2 e ) 3 . 8 5 6 ( 3 )2 . 7 8 3 ( 3 ) 2 . 7 A r ( 3 13 . 2 4 2 ( 2 ) 3 . 2 ' 7 4 ( 3 \

1 , 6 1 4 ( 3 ) 1 . 6 1 5 ( 5 )1 . 6 3 6 ( 4 ) 1 . 6 3 6 ( 5 )r . 6 2 4 ( 3 ) r . 6 2 3 ( 3 )L 6 2 5 I . 6 2 4

2 . 7 2 0 ( 6 ) 2 . 7 1 9 ( 6 \2 . 7 4 2 ( 5 1 2 . ' 7 4 0 ( 5 )2 . s 3 7 ( 5 ) 2 . s 3 8 ( 5 )2 . s 4 7 ( 6 ) 2 . 5 A 9 ( 7 )2 . 6 4 4 2 . 6 4 4

2 . 0 9 r ( 3 ) 2 . 1 0 r ( 3 )2 . 0 8 0 ( 3 ) 2 . 0 8 7 ( 3 )2 . 1 4 0 ( 3 ) 2 . 1 s 3 ( 3 )2 . \ O 4 2 . 1 1 3

2 . e 7 1 ( 5 ) 2 - 8 9 0 ( s )3 . r o e ( 5 ) 3 . 1 2 3 ( s )2 . 8 6 1 ( 6 ) 2 . e 1 o ( 6 )3 . 0 3 6 ( 3 ) 3 . 0 5 0 ( 3 )3 . 3 7 3 ( 3 ) 3 . 3 8 0 ( 2 )2 . 5 3 7 ( 5 ) 2 . 5 3 8 ( 5 )2 . 9 6 5 2 . 9 7 5

2 . r 9 9 ( 5 ) 2 . 2 r 9 1 6 )2 , O 5 7 ( 5 ) 2 . 0 6 6 ( 5 )2 . 2 3 I ( 4 ) 2 - 2 4 s ( 4 \2 . O 7 3 ( 3 ) 2 . O 4 2 ( 4 )2 . r 4 4 2 . 1 5 6

3 . 0 s 2 ( s ) 3 . o ' 7 4 ( 5 )2 . 8 7 1 ( s ) 2 . 8 9 0 ( 5 )3 . 2 2 4 ( 5 1 3 . 2 4 ' 7 \ 5 12 . 9 2 3 ( 5 ) 2 . 9 3 3 ( s )2 . 547 (6 ) 2 . 549 (1 )3 . 0 1 4 ( 4 ) 3 . 0 2 e ( 4 )3 . 4 r 2 ( 2 1 3 . 4 3 2 ( 2 )3 . 0 1 4 3 . 0 3 I

2 . 0 9 r ( 3 ) 2 . 1 0 r ( 3 )2 . r 9 9 1 5 ) 2 . 2 1 9 ( 6 )1 . 6 1 4 ( s ) 1 . 6 1 5 ( 5 )1 . 9 9 9 2 . O O 9

3 . 0 0 0 ( 3 ) 3 . 0 1 4 ( 3 )3 . 2 r 7 ( 3 ) 3 . 2 3 8 ( 3 )3 . 2 6 4 ( 3 ) 3 . 2 ' 7 6 ( 3 )3 . 2 6 0 ( 4 ) 3 . 2 7 O ( 4 )

2 . 0 8 0 ( 3 ) 2 . 0 S 7 ( 3 )2 . O 5 7 ( 5 ) 2 . 0 6 6 1 5 )1 . 6 3 6 ( 4 ) 1 . 6 3 6 ( 5 )1 . 9 6 3 I . 9 6 9

3 . 0 0 0 ( 3 ) 3 . 0 1 4 ( 3 )3 . 6 4 s ( 3 ) 3 . 6 5 4 ( 3 )2 . 6 e 9 ( 2 ) 2 . ' 7 7 0 ( 2 )3 . 2 8 e ( 3 ) 3 . 3 0 2 ( 4 )

2 . 1 4 0 ( 3 ) 2 - 1 s 3 ( 3 )2 . 2 3 r \ 4 1 2 . 2 4 5 ( 4 )2 . O ' 7 3 ( 3 ) 2 . O e 2 ( 4 )I . 6 2 4 ( 3 ) r . 6 2 3 ( 3 )2 . O I 7 2 . 0 2 6

3 . 2 I 7 ( 3 \ 3 . 2 3 e ( 3 )3 . s 9 2 ( 3 ) 3 . 6 0 3 ( 2 )2 . 6 e 9 ( 2 ) 2 . l r o ( 2 )3 . 8 7 2 ( 3 ) 3 . 8 8 6 ( 3 )2 . 7 9 ' 7 1 3 \ 2 . 8 1 1 ( 3 )3 . 2 8 8 ( 3 ) 3 . 3 0 3 i 4 )

r . 6 1 s ( 4 ) r . 5 3 ( 2 )r . 6 4 9 ( 4 1 r . 6 0 ( 2 )1 . 6 2 8 ( 3 ) r . 6 ' 7 ( 2 )1 . 6 3 0 L . 6 2

2 . 7 3 3 ( 6 \ 2 . 6 0 ( 3 )2 . 7 3 s ( 4 ) 2 . 8 0 ( 3 )2 . s 6 1 ( s ) 2 . 5 0 ( 3 )2 . s e s 1 6 1 2 . 6 6 { 3 )

2 . r r 7 ( 3 ) 2 . r 7 ( 2 )2 . o e ' 7 1 3 ) 2 . 0 6 ( ) . )2 . r B 0 ( 3 ) 2 . \ 3 ( 2 )2 . 1 3 1 2 - 1 2

2 . e r r ( s ) 3 . 0 0 ( 2 )3 . 1 6 0 ( 5 ) 3 . r , 2 ( 2 )2 . 8 8 8 ( 6 ) 2 . 9 o ( 2 )3 . 0 6 7 ( 3 ) 3 . O ' 7 1 6 )3 . 3 8 8 ( 4 ) 3 . 3 7 ( 2 12 - 5 6 1 ( s ) 2 . s 0 ( 3 )2 . 9 9 6 2 . 9 9

2 . 2 3 ' 7 ( 5 ) 2 . 3 2 1 3 )2 . 0 6 8 ( 5 ) 2 . 1 3 ( 3 )2 . 2 5 4 ( 3 ) 2 . 2 5 ( 2 )2 . 0 8 6 ( 3 ) 2 . o a ( 2 )2 . 1 6 5 2 . 1 9

3 . 0 8 0 ( s ) 3 . 0 1 ( 2 )2 . s r r ( s ) 3 . 0 0 ( 2 )3 . 2 5 7 ( 5 ) 3 . 2 6 ( 2 12 . 9 4 7 ( 4 ) 3 . 0 0 ( 3 )2 - 5 s 5 ( 6 ) 2 . 6 6 ( 3 13 . 0 3 r ( 4 ) 3 . 0 5 ( 2 )3 . 4 6 5 ( 2 ) 3 . 3 9 ( 2 )3 . 0 4 3 3 . 0 7

2 . r r 7 ( 3 ) 2 . r ' t ( 2 )2 . 2 3 ' t ( s ) 2 . 3 2 ( 3 )1 . 6 r s ( 4 ) r . s 3 ( 2 )2 . 0 2 2 2 . O s

3 . 0 3 0 ( 3 ) 3 . 0 3 3 ( 3 )3 . 2 6 7 ( 3 1 3 . 2 8 1 2 )3 - 2 8 8 ( 3 ) 3 . 2 e \ 2 \3 . 2 a 3 ( 4 ) 3 . 2 a 1 2 )

2 . 0 9 7 1 3 ) 2 . 0 6 ( r )2 . 0 6 8 ( 5 ) 2 . 1 3 ( 3 )r . 6 4 9 \ 4 ) 1 . 6 0 ( 2 )L - 9 7 4 r . 9 6

3 . 0 3 0 ( 3 ) 3 . 0 3 3 ( 3 )3 . 6 6 5 ( 3 ) 3 . 6 8 ( 1 )2 . 7 2 5 ( 2 ) 2 . ' t 3 ( r \3 . 3 0 9 ( s ) 3 . 3 2 ( 2 )

2 . 1 8 0 ( 3 ) 2 . 1 3 ( 2 )2 . 2 s 4 ( 3 ) 2 . 2 4 \ 2 \2 . 0 8 6 ( 3 ) 2 . O a ( 2 )r . 6 2 4 ( 3 ) L . 6 7 ( 2 )2 . 0 3 8 2 . O 4

3 . 2 6 ' 1 1 3 1 3 . 2 8 ( 1 )3 . 6 r ' 7 1 2 ] ' 3 . 6 1 ( 1 )2 . 7 2 5 \ 2 ) 2 . 7 3 ( r )3 . e l 2 ( 3 ) 3 . e l ( 2 )2 . 8 2 9 l 3 ) 2 . A 4 ( 2 )3 . 3 1 1 ( 5 ) 3 . 3 2 ( 2 )

r . 6 3 ( L ) 1 . 6 2 ( r )| . 6 6 ( 2 ) r . 6 6 \ 2 11 . 6 0 ( 1 ) 1 . 6 0 ( t )1 . 6 2 7 . 6 2

2 . 7 4 ( 2 ) 2 . a O ( 2 )2 . 7 5 ( L ) 2 . 7 3 ( ) , 12 . 5 1 ( 3 ) 2 . 5 0 ( 3 )2 . 5 3 ( 1 ) 2 . 5 s ( r )

2 . 0 6 ( 1 ) 2 . 0 8 ( 1 )2 . 0 4 ( t ) 2 . 0 4 ( ) , )2 . 1 1 ( 1 ) 2 . o 7 ( 2 )2 . 0 7 2 . 0 6

2 . 8 1 ( 3 ) 2 . 7 e ( r )3 . 0 9 ( 3 ) 3 . 0 7 1 4 \2 . B O ( 2 \ 2 . 8 3 ( 2 )3 . 0 1 ( r ) 3 . 0 0 ( 1 )3 . 3 1 ( 2 ) 3 . 2 6 ( 2 )2 . s 1 ( 3 ) 2 . 5 0 ( 3 )2 . 9 2 2 . 9 I

2 . 1 4 \ 2 1 2 . 1 2 ( r )2 . o 4 ( 2 1 2 . o O ( 2 )2 . r e ( 1 ) 2 . 2 2 ( r )2 . 0 8 ( r ) 2 . o 7 l r )2 . L 2 2 , I l

3 . 0 1 ( 3 ) 2 . e 8 ( 3 )2 . 8 1 ( 3 ) 2 , ' 1 9 ( r J3 . 1 5 ( 3 ) 3 . 1 4 ( 3 )2 . e 4 ( L l 2 . e 2 l L )2 . 5 3 ( r ) 2 . 5 s ( 1 )3 . 0 2 ( 3 ) 3 . 0 3 ( 2 )3 . 3 s ( 1 ) 3 . 4 2 ( r l2 . 9 4 2 . 9 7

2 . 0 6 ( r ) 2 . 0 8 ( r )2 . L 4 1 2 ) 2 . 1 2 ( 1 )1 - 6 3 ( r ) r . 6 2 l r )1 . . 9 7 I . 9 ' 7

2 . 9 1 1 l 3 ) 2 . 9 7 0 l 3 \3 . 1 7 ( 1 ) 3 . 1 7 ( r )3 . 2 4 4 ( s ) 3 - 2 3 ( r )3 . 2 3 ( 2 ) 3 . 1 6 ( r )

2 . 0 4 ( 1 " ) 2 . 0 4 ( r )2 . 0 4 ( 2 ) 2 . O o ( 2 )r . 6 6 ( 2 ) L . 6 6 1 2 )r . 9 5 t . 9 4

2 - 9 7 ' 7 ( 3 ) 2 . 9 ' 7 0 ( 3 13 . 6 1 7 ( 7 ) 3 . s 9 ( 7 )2 . 6 8 ' / ( 6 ) 2 . 6 ' 1 r ( 6 )3 . 2 6 ( 1 ) 3 . 2 6 1 r )

2 . 1 1 ( 1 ) 2 . o 1 1 2 )2 . 1 9 ( 1 ) 2 . 2 2 ( l )2 . 0 8 ( 1 ) 2 . 0 7 ( r )1 . 6 0 ( 1 ) r . 6 0 ( 1 )2 . 0 0 1 . 9 9

3 . r 7 ( r ) 3 . r 7 ( 1 )3 . 5 6 5 ( e ) 3 . 5 3 ( 1 )2 . 6 S ' t ( 6 1 2 . 6 ' 7 1 , ( 6 \3 . 4 4 5 ( 2 ) 3 . 4 4 4 ( 2 12 . 7 7 l r ) 2 . 7 8 ( r )3 . 2 5 ( r ) 3 . 2 s ( 1 )

1 . 6 4 ( r ) 1 . 6 1 7 ( 3 )r . 7 6 ( 2 ) r . 6 4 5 ( 3 )r . 5 6 ( 1 ) \ - 6 2 5 ( 2 11 . 6 3 I . 6 2 4

2 . 8 4 l L i 2 . 7 3 6 ( 4 12 . 7 I ( r ) 2 . 7 5 2 ( 3 )2 . 4 9 ( 2 ) 2 . s 3 7 ( 3 12 . 5 9 ( L ) 2 . s 8 r ( 4 )2 . 6 4 2 . 6 4 9

2 . 0 9 ( 1 ) 2 . 0 7 9 ( 2 )2 . o 5 ( 2 ) 2 . 0 6 8 ( 2 )2 . O 3 \ 2 ) 2 . 1 2 6 ( 2 )2 . 0 6 2 . 0 9 L

2 . 7 6 ( 3 1 2 . 8 4 4 ( 3 )2 . 9 s ( 3 ) 3 . 0 9 4 ( 3 )2 . 8 8 ( 2 ) 2 . 4 3 e ( 4 )3 . 0 2 ( r ) 3 . 0 2 2 ( r )3 . 2 4 ( 2 ) 3 . 3 3 9 ( 2 )2 . 4 9 ( 2 ) 2 . 5 3 7 ( 3 )2 . A 9 2 - 9 4 7

2 . 0 6 ( 3 ) 2 . r 7 e ( 3 )r . 9 2 ( 2 ) 2 . 0 5 3 ( 3 )2 . 2 7 \ 2 1 2 . 2 I 4 ( 2 )2 . 0 6 ( 3 ) 2 . 0 7 2 ( 2 )2 . L I 2 . 1 3 4

3 . 0 7 ( 4 ) 3 . 0 2 2 ( r )2 . 7 6 l r ) 2 . a 4 e ( 4 )3 . 2 4 ( r ) 3 . r 9 ' 7 ( 3 12 . 8 5 ( 1 ) 2 . e 2 6 ( 3 )2 . s 9 ( 1 ) 2 . s 8 1 ( 4 )3 . 0 7 ( 3 ) 3 . 0 0 5 ( 2 )3 . 6 4 ( 1 ) 3 . 3 9 8 ( 3 )3 . O 2 2 . 9 9 4

2 . 0 9 ( 1 ) 2 . o 7 9 1 2 )2 . o 6 ( 3 ) 2 . r 7 s ( 3 )r . 6 4 ( 1 ) r . 6 1 7 ( 3 )I . 9 ' 7 r . 9 8 9

2 . e ' 7 ( r ) 2 . e 8 e ( 1 )3 . 1 8 ( r ) 3 . 1 9 6 ( r )3 . 2 4 3 ( 6 ) 3 . 2 5 3 ( 2 13 . 1 e ( r ) 3 . 2 5 r ( 2 )

2 . O 5 \ 2 ) 2 . 0 6 4 ( 2 1L . s 2 1 2 ) 2 . 0 5 3 ( 3 )I - 7 6 ( 2 ) r . 6 4 s ( 3 )1 . 9 4 1 . 9 5 9

2 . 9 ? \ L ) 2 . 9 8 9 ( r )3 . 5 5 ( 2 ) 3 . 6 3 2 ( 2 12 . 7 O ( r ) 2 . 6 9 3 ( t \3 . 2 1 ( 1 ) 3 . 2 8 0 ( 1 )

2 . 0 3 ( 2 ) 2 . L 2 6 ( 2 )2 . 2 1 ( 2 ) 2 . 2 r 4 ( 2 )2 . 0 6 ( 3 ) 2 . o 7 2 ( 2 )1 . s 6 ( r ) r . 6 2 5 1 2 ) '1 . 9 8 1 . 9 8 7

3 . 1 8 ( r ) 3 . 1 9 6 ( r )3 . s r ( 2 ) 3 . s 8 0 ( 2 )2 . 7 0 ( 1 ) 2 . 6 9 3 ( r )3 . 8 s ( r ) 3 . 8 s 8 ( 4 )2 . 7 3 ( r ) 2 . 7 s 0 ( 2 )3 . 2 0 ( r ) 3 . 2 6 3 \ 2 )

12)IUtll

n I

L2)tzlt1l

t1ltrlF]trl

t1lnlt1lD]t!trl

a ) A 1 1 d i s t a n c e s i n A . b ) D a t a o f

c ) P a r e n t h e s i z e d f i q , u r e s r e f e r t o t h e e s d

e ) E d g e s h a r e d b e t w e e n t w o o c t a h e d r a ,

S m y t h a n d H a z e n ( 1 9 7 3 ) .

o f l e a s t u n i t s c i t e d . d ) E d g e s h a r e d b e t w e e n a n o c t a h e d r o n a n d t e t l a h e d r o n .

?.t4

2.t3

2 . t2

2 . t l

2.to

lrloz,

oo

o-zlrl

800 roooTEMPERATURE (K)

Frc. 4. Forsterite Mg-O and Si-O mean bond distances versus

temperature.

plotted in Figure 8. From Figure 8 it is evident thatuncorrected, lower bound, and riding corrections be-have in a similar way, and are not significantly differ-ent from each other. Thus, if metal-oxygen bondshave predominantly parallel correlated vibration mo-

M(2)- o(3)

H(2)- o(l)

:g2 .

2 .18

2.r6

214

2.12

2

2a '

FORSTERITE t287

Mg-o DISTAI {CE (or O K}

Frc. 6. Forsterite Mg-O bond length thermal expansion versus0 Kelvin Mg-O distance.0 K Mg-O distance is extrapolated fromFig. 5.

tion, then the uncorrected distance usually reported isa valid measure of the physical separation of atoms.However, non-correlated and anti-correlated (upperbound) separations are significantly larger than thelower bound, and may give a totally different pictureof bond distances and thermal expansion at elevatedtemperature. If a bond has a large component of anti-parallel or noncorrelated vibrations, the simple cen-troid separation distance will not provide a realisticmeasure of mean atomic separation. It should berecognized that reported bond distances and bondthermal expansions represent the lower limits of thesephysical quantites.

Temperature factorsIsotropic temperature factors of the six atoms in

forsterite's asymmetric unit are plotted versus tem-perature in Figure 9. All six curves show the samegeneral form of a near-zero slope at 0 K, a rapidlyincreasing slope between 0 and 500 K, and an ap-proximately constant slope above 500 K. This is thesame behavior as that recorded for periclase (Hazen,1976). Also, as with periclase, magnesium and oxy-gen atoms have approximately equal magnitudes ofthermal vibrations. However, silicon's isotropic tem-

2o0

versus Ftc. 7. Forsteritetemperature.

TEMPERATURE (K}

individual Si-O bond distances versus

? . 1 6

r.64

M(2 ) -O

-x - - - - - -X- - - - ' . - - s i -o - -x - '1

" ' x . - -_ ___-x - - - - - " - "

!

o'9

x

?l9:j

ul()2

Fo6

o=o@

oq

:E uozFq

o

oq

Frc. 5. Forsteritetemperature.

TEMPERATURE

indiv idual Mg-O

tooo t20(( K )

bond distance

A M(r)_o(c) Ird2Fo(5)

A u ( r ) - o ( r )^ A P.rictox (Mg-o)I No)-o(z)

--!t-a'-

t 288

t . 6

ROBERT M, HAZEN

R BOUND

2.2

€" 2.O(loI '..r

H r.6)k ,.oU.L

! t . 2F

(,E '.o(lF

I o..F

fr o.eJs5 0.4

trl

r .65

r . 64

r.6J

Uoz

Foo

o

o

z

E-

TEMPERATURE (K}

Fig. 8. Forsterite Si-O mean distances versus temperaturecorrected for several thermal-motion models.

perature factor is only about half that of the magne-sium and oxygen atoms. This behavior has been ob-served in many other silicates (Burnham, 1965), andis a consequence of the strong bonds between oxygenand four-valent silicon.

L. Finger (personal communication) has noted apossible source of error in the magnitudes of temper-ature factors reported in this study. No correctionswere made for secondary extinction; when a gemquality crystal such as synthetic forsterite is used

200 400 600 600 tooo t200

TEMPERAfURE (K)

FIc. 9. Forsterite isotropic temperature factors versustem Derature.

several strong reflections with lF"u"l ((lF"nr"l are re-jected from the data set. Consequently, the scale fac-tor will be too high, and the temperature factors willbe too low. This effect explains the nonpositive-defi-nite thermal factors of oxygen and silicon in theliquid-nitrogen temperature case (see Table 3). Thus,while the qualitative aspects of Figure 9 are correct,the absolute values reported may be low.

o,2

Tnrlr 6. Forsterite bond ansles

3 ' c - 3 5 0 " . 1 0 0 0 " 20 kb 50 kbo ( r ) - s i - o ( 2 )o ( 1 ) - s i - o ( 3 )o ( 2 ) - s i - o ( 3 )o ( 3 ) - s i - o { 3 ' )m e a n O - S i - O

o ( r ) - M ( r ) - o ( 3 )o ( r ) - M ( r ) - o ( 3 ' )o t r ) - M ( 1 ) - o ( 2 )o ( 1 ) - M ( r ) - o ( 2 ' )o ( 2 ) - M ( 1 ) - o ( 3 ' )o ( 2 ) - M ( 1 ) - o ( 3 )m e a n O - M ( 1 ) - O

o ( 1 ) - M ( 2 ) - o ( 3 ' )o ( r ) - M ( 2 ) - o ( 3 )o ( 2 ) - M ( 2 ) - o ( 3 )o ( 2 ) - M ( 2 ) - O ( 3 ' " )

o ( 3 ) - M ( 2 ) - ) ( 3 ' )o ( 3 ) - M ( 2 ) - o ( 3 , , )o ( 3 ) - M ( 2 ) - o ( 3 . )m e a n O - M ( 2 ) - O

1 r 4 . 1 ( 3 )1 r s . 6 ( 2 )r o 2 . 7 ( 2 J1 o 4 . 6 ( 3 )t o g . 2

8 5 . I ( 2 )9 4 . 9 ( 2 \8 6 . 8 ( 1 )9 3 . 2 ( 1 )

r 0 s . 4 ( 2 )7 4 . 6 l 2 ' )90

9 0 . 3 ( 1 )8 1 . 7 ( 1 )9 6 . 7 1 2 )9 0 . 8 ( 1 )' t r

. 9 ( 2 )8 8 . 3 ( r )

r r o . 7 ( 2 )a 9 . 9

r I 4 . 2 ( 2 11 r s . 9 ( 1 )r o 2 . 0 ( t )r 0 s . 0 ( 2 )I 0 9 . I

8 5 . 0 ( r )9 s . 0 ( 1 )8 6 - 6 ( 1 )e 3 - 4 ( 1 )

1 0 s . 2 ( r )7 4 . 4 ( \ )90

9 0 . 9 ( 1 )8 1 . r ( t )9 6 . 9 ( 1 )9 0 . 6 ( 1 )7 1 . 8 ( r )8 8 . 7 ( 1 )

r 1 0 . 0 ( r )8 9 . 9

r L 3 . ' 7 ( 2 Jr r s . 7 ( 1 )r o 2 . 2 1 2 )1 0 s . 6 ( 2 )1 0 9 . 2

8 s - 4 ( l )s 4 - 6 ( 1 )8 6 - 6 ( r )9 3 . 4 ( 1 )

r 0 5 . r ( 1 )7 3 . 9 ( 1 )90

9 1 . 1 ( 1 )e 0 8 ( r )9 1 . 4 ( 2 )9 0 - r ( 1 )7 O . 9 ( 2 )8 8 . 8 ( 2 )

r 1 0 . 8 ( r )4 9 . I

1 1 3 . 6 ( 2 )r 1 5 . 6 ( r )r o 2 - 3 ( 2 \r 0 s . 9 ( 2 )1o9.2

8 s - 6 ( 2 )s 4 - 4 ( 2 )8 6 . s ( 1 )9 3 . s ( r )

1 0 6 . s ( 2 )1 3 . s ( 2 190

9 r - 2 ( r )8 0 . 7 ( 1 )e7 . ' ] (2 ' )e 0 . o ( 1 )1 O . 4 1 2 ,8 8 . 8 ( 1 )

r 1 1 - 3 ( 2 )8 9 . 9

I I 3 . 7 ( 2 )r r 5 . 2 ( r )1 0 2 . 8 ( r )t o 5 . 7 ( 2 )I O 9 - 2

8 s . 3 ( 1 )s 4 - ' 7 ( t - )8 6 . 6 ( 1 )9 3 . 4 ( r )

1 0 6 . s ( 1 )7 3 . 5 ( 1 )9 0

9 0 . 8 ( 1 )8 0 . 7 ( 1 )9 7 . 6 ( l J9 0 . 4 ( 1 )7 0 . 1 ( 2 )8 8 . 4 ( 1 )

r r 2 . 3 1 2 )8 9 . 9

r l s . I ( r . 0 )1 1 6 . 5 ( s )r 0 0 . 9 ( 7 )1 0 4 . 6 ( 9 )1 0 9 1

8 4 . s ( 6 )9 s - 5 ( 6 )8 s - I ( 4 )9 4 - 2 ( 4 )

r o s . 7 ( 7 1' 7 4 . 3

l 7 )90

9 1 . 2 ( 5 )8 0 . 8 ( 7 )9 6 . 2 ( 6 )9 r . 0 ( 7 )7 0 . 6 1 6 )8 9 . r ( 4 )

r 1 0 . 6 ( 6 )8 9 . 8

r r 5 . 7 ( 6 )r r 7 . 4 ( r . r )r 0 0 . o ( 7 )r 0 5 . 7 ( 6 )1 0 9 . 4

8 4 5 ( 6 )9 s . 5 ( 6 )4 6 . ? ( 4 19 3 . 3 ( 4 )

l o 5 . 2 ( 7 )1 4 - a ( ' 7 ] ,9 0

e 0 . 9 ( 6 )8 0 . 2 ( 7 )9 6 . 2 ( 6 )9 r - 6 ( 7 )7 0 . 1 ( 6 )8 9 . 6 ( 4 )

r o 9 . 9 ( 6 )8 9 . 8

1 1 s ( 1 )r 1 6 ( 1 )

e 7 ( r )1 1 0 ( r )1 0 9

8 3 . 8 ( 8 )9 6 . 2 ( A )4 6 . 7 l 4 l9 3 . 3 ( 4 )

L O 4 . 9 l 7 l1 s . r ( 7 )9 0

9 2 l L l7 9 ( 1 )s 7 ( 1 )9 L 5 ( 1 . O )6 9 . 4 ( 7 19 0 . 0 ( 7 )

r 0 9 . 7 ( 5 )9 0

1 1 4 . r ( 2 )r 1 6 . r ( l )1 0 r . 7 ( I )1 0 s . r ( I )1 0 9 . r

8 s . 3 ( r )94 .1 l r )8 6 . 4 ( 1 )e 3 . s ( 1 )

1 0 5 . 6 ( l )7 4 . 4 ( t l90

9 r - 2 ( r )8 0 - 8 ( t )9 7 . O ( I )e 0 . 3 ( 1 )7 1 . 3 ( r )8 9 - 0 ( 1 )

r r 0 . r ( 1 )89 .9

f r l

l2l[2]tll

12)t2Jt2lL2lLrlhl

t l121irjt2jtrlt ltrl

1 0 9 , 51 0 9 . 51 0 9 . 5r 0 9 , sr 0 9 . 5

909 09 09 09 09 09 0

9090909 090909 090

1 1 4 ( 1 )r r 7 ( I )9s ( r )

r 0 6 ( r )1 0 9

87 16)9 r ( r )B B ( 1 )e 3 ( r )

r 0 7 ( I )7 3 ( 1 )90

e 0 ( r )8 r ( 1 )9 9 ( r )9 0 ( r )7 2 l r l8 9 ( 1 )

r 0 9 ( r )90

a l

b )

c )

" I d e a l " h e x a g o n a l c l o s e - p a c k e d o l i v i n e m o d e l ( s e e F i g . 1 )P a r e n t h e s i z e d f i g u r e s r e f e r t o t h e e s d o f I e a s t u n i t s c i t e d .D a t a o f S m y t h a n d H a z e n ( 1 9 7 3 ) .

FORSTERITE t289

Because each of olivine's six atoms has an irregularcoordination polyhedron, anisotropic vibrations areexpected, and have been modeled as triaxial ellipsoidsin the refinement procedures. Anisotropic temper-ature factors are presented in Table 3, while corre-sponding magnitudes and orientations of thermal el-lipsoids appear in Table 7. These values agree withdata reported by Brown (1970), and while the magni-tudes of thermal ellipsoids increase with temperature,there is no significant change in ellipsoid orientationwith changes in temperature.

Dats collection: I atm to 50 kbar

The forsterite crystal used in the liquid-nitro-gen-temperature studies was remounted with thegood (010) cleavage parallel to the diamond faces of

the miniature pressure cell. The cell was tightened,and unit-cell parameters were determined on thefour-circle diffractometer as a : 4.743 t 0.005, b =

10.092 + 0.010, c :5.954 + 0.006 A, and volume :

285.0 + 0.5 At, which represents a volume decreaseof 1.65 percent. Pressure calibration was based on avolume compressibility of -7.6 x lO-alkbar for oli-vine as measured by Olinger and Halleck (1975). Itthus appears that the first high-pressure data collec-tion was made at approximately 20 kbar. All avail-able diffractions were collected, but of 969 measuredintensities only 329 were observed (/ < 2o). After tencycles of refinement with isotropic temperature fac-tors, convergence was achieved with a weighted R of7.0 percent (9.4 percent unweighted). Refinementconditions, atom parameters, and isotropic temper-

TlsI-r 7 Magnitudes and orientations of forsterite thermal ellipsoids'

M ( 2 \

b c a

- 1 9 6

2 3

' 3 5 0

- 6 7 5

- l o 0 0

h ishpress ure)

0 . 0 1 ( 2 )0 . 0 r e ( 1 3 )0 . 0 5 4 ( s )

0 . 0 3 7 ( 4 )0 . 0 s 3 ( 3 )o . o ' 7 5 ( 2 \

0 . 0 6 8 ( 6 )o . 0 9 4 ( 5 )0 . 1 1 0 ( 4 )

0 . r 0 r ( 5 )0 . 1 1 s ( 5 )o . t 4 9 ( 4 )

0 . r r 7 ( 4 )o . I 4 0 ( 4 )0 . 1 8 s ( 3 )

o . o 6 6 ( 4 )o . o 7 r ( 4 )0 . 0 8 4

r 2 ( 6 r )e 5 { r 3 4 )

r01 (9 )' 7 4 ( ) .O lr 7 ( 1 0 )e 5 ( s )

8 0 ( r 0 )1 0 ( 1 0 )e 1 ( 1 3 )

5 6 ( 1 3 )3 4 ( 1 3 )s r ( s )s 8 ( 7 )3 2 ( ? )8 e ( 3 )

5 8 ( 3 2 )3 2 ( 3 2 )96 l12)

r o 2 ( 2 7 )r 0 0 ( 3 0 )r 6 4 ( 9 )

7 6 ( 4 \e e ( 6 )

7 3 ( 5 )s 3 ( 1 3 )

r 6 3 ( 6 )

? 3 ( 5 )r o 2 ( 6 1L59 (4 \

1 5 ( 2 )s s ( 4 )

r 6 2 \ 2 1

1 2 ( r 3 )r 0 7 ( 1 6 )r 5 s ( 1 0 )

9 3 ( r 3 4 )1 6 8 ( 3 9 )

7 9 ( 1 0 )

2 r ( 7 )1 0 4 ( 1 0 )

7 s ( 3 )

L e ( 7 )e 9 ( r 0 )7 3 ( 6 \

3 s ( 1 1 )I2 r ( r2 )

6e 14)

3 7 ( 6 )1 2 r ( 6 )

7 2 \ 3 1

3 7 \ 2 3 )1 1 6 ( 2 7 \

6 6 ( e )

0 . 0 1 ( 2 )0 . 0 2 7 ( 1 0 )0 . 0 4 s ( 6 )

0 . 0 s l ( 3 )0 . 0 5 4 ( 3 )0 . 0 s 5 ( 3 )

0 - 0 8 6 ( 5 )0 . 0 9 0 ( 5 )o . o97 (4 )

0 . 1 1 7 ( s )o . r 2 2 ( 5 1o . L29 (4 )

0 . 1 4 0 ( 4 )o . r49 14\0 . r s 1 ( 4 )

0 . 0 6 9 ( 4 )0 - 0 7 2 ( 3 )0 . 0 7 8 ( 4 )

I ( 1 4 )90e r ( 1 4 )

9 ( 4 r )9 8 ( 4 7 )90

9 4 ( 2 3 )90

4 ( 2 3 \

I 4 0 1 4 1 )

90

s 3 ( 1 6 )

90

3 6 ( 1 8 )905 4 ( 1 8 )

e e ( 1 4 )90

r ( 1 4 )

d s ( + r )r 7 2 ( 4 7 \

90

4 ( 2 3 \908 6 ( 2 3 )

r 4 0 ( 4 3 )1 3 0 ( 4 3 )

90

1 4 3 ( 1 6 )L26 \ t -6 )

90

r 2 6 ( r 3 )903 6 ( 1 B )

I23

123

I23

123

I

23

L23

- 9 0

1 8 090

901 8 0

90

9090

0

901 8 0

90

90r80

90

901 8 0

90

9090

0

90180

90

9090

la0

9090

0

901 8 090

0 . 0 2 2 ( 7 ) 3 ( s ) e 3 ( s )0 . 0 3 0 ( 3 ) 9 0 9 00 . 0 3 2 ( 3 ) 8 7 ( s ) 3 ( s )

o . 0 s 4 ( 6 ) 5 ( 1 3 ) e s ( 1 3 )0 . 0 6 7 ( 5 ) 9 0 9 00 . 0 7 3 ( 4 ) B s ( 1 3 ) 5 ( 1 3 )

0 . 0 6 s ( s ) 7 ( 6 ) 9 ' / ( 6 )0 . r 0 0 ( 4 ) 9 7 ( 6 1 1 7 3 ( 6 )0 - 1 0 4 ( 3 ) 9 0 9 0

0 . 0 8 8 ( 4 ) 8 ( 4 ) e 8 ( 4 )0 . r r 9 ( 3 ) 9 0 9 0o . r 2 2 ( 3 \ A 2 ( 4 ) I ( 4 )

o . 0 4 6 ( 4 \ 2 ( 9 ) s 2 ( e \0 . 0 6 2 ( 3 ) 9 0 9 00 . 0 6 5 ( 3 ) 8 8 ( e ) 2 t e l

M ( 1 )

c

- 3 5 0

- 6 7 5

1 0 0 0

h i g hpres s ure, l

1 0 0 ( 8 )90r 0 ( 8 )

r 7 0 { 1 9 )90B 0 ( r 9 )

r 1 5 ( r s )90

2 5 ( 1 5 )

9 7 ( 9 )90

'7 (e )

1 0 6 ( 1 6 )901 6 ( t 6 )

9 0 0 . 0 4 4 t 4 )e 0 0 . 0 s 5 ( 4 )

0 0 . 0 7 4 ( 3 )

9 0 0 . 0 7 3 ( 8 )1 8 0 o . o 9 4 ( 7 )

9 0 0 - 1 o 3 ( 6 )

9 0 0 . r 0 7 ( 7 )9 0 o . r 1 3 ( 6 )

0 0 . 1 4 0 ( 6 )

9 0 0 . 1 2 s ( 5 )9 0 0 . 1 3 7 ( 5 )

0 0 . 1 7 1 ( 5 )

9 0 0 . 0 6 7 ( 5 )1 8 0 0 . 0 6 8 ( 5 )

9 0 0 . 0 8 r ( 5 )

o . 0 3 1 ( 9 ) 1 0 ( B )o . 0 6 4 ( 4 \ 9 00 . 0 6 e ( 4 ) 8 r ( 8 )

0 . 0 7 e ( 1 r ) 8 0 ( 1 9 )0 . 0 9 7 ( 9 ) s 00 . 1 0 5 ( 9 ) r 0 ( l e )

0 . 1 0 1 ( 1 0 ) 2 5 ( 1 5 )0 . r 2 7 ( 8 ) 9 0

o . r 2 8 ( 8 ) 6 4 ( 1 5 )

0 . 1 1 6 ( 7 ) ' 1 ( 9 \

o . r 4 2 \ 7 ) 9 00 . 1 s 4 ( 7 ) 8 3 ( e )

0 . 0 6 3 ( e ) 1 6 ( 1 6 )o . 0 7 1 ( 7 ) 9 0o . 0 8 s ( 8 ) ' 7 4 ( 1 6 )

0 . 0 4 8 ( 6 ) 5 7 ( 3 ' ) ) r 4 7 1 3 7 )0 . o 5 5 ( 6 ) r 4 7 ( 3 ' t ) r 2 3 1 3 - 7 )o . 0 6 2 ( 5 \ 9 0 9 0

0 . 0 8 4 ( 1 0 ) 7 3 ( 6 2 ) r ' 7 ( 6 2 )0 . 0 8 6 ( 1 0 ) 9 0 9 00 . 0 9 2 ( r 0 ) I 1 ( 6 2 \ 1 0 7 ( 6 7 )

o . r 0 7 ( 9 ) 7 1 7 7 ) A 3 ( 1 7 )0 . r l 3 ( e ) 9 6 ( 8 4 ) 6 ( a 4 l

0 . 1 2 0 ( e ) 9 0 9 0

o - r 2 5 ( 7 ) L ( 4 6 ) 9 r ( 4 6 )0 . 1 3 3 ( 7 ) s L ( 4 6 ) r 1 e 1 4 6 l0 . 1 4 r ( 7 ) 9 0 9 0

o . 0 6 2 ( 8 ) 5 1 1 2 7 ) 3 e ( 2 1 ] ,0 . 0 6 8 ( 7 ) 9 0 9 00 . 0 7 3 ( 7 ) 3 9 ( 2 1 ) r 2 9 ( 2 ' 7 ' , )

4 e ( r 6 ) r o s ( 7 ) 4 8 ( 1 3 )4 2 \ 1 6 )

' 7 5 ( 9 ) 1 2 8 ( 1 4 )8 8 ( 7 ) 2 5 ( 7 1 6 5 ( 6 )

e 3 ( 1 s ) 1 2 3 ( 1 3 ) 3 4 ( 1 3 )r 3 9 ( 3 1 ) I 2 2 \ 2 6 ) l r 3 ( 2 0 )1 3 r ( 3 r ) 4 9 ( 2 3 \ 6 7 ( r 7 )

e s ( 4 8 ) 1 3 2 ( 8 ) 4 3 ( 1 9 )165 129) 94 \34) lo 5 ( 37)1 0 2 ( r o ) 4 2 l a ) 5 l ( 8 )

8 6 ( 1 9 ) 1 2 1 ( 5 ) 3 r ( 4 )r 7 4 ( r 3 ) e 6 ( r 2 ) e O ( 1 6 )

9 s ( 6 ) 3 2 ( 5 ) 5 9 ( 4 )

3 ( r s 6 ) 8 8 ( 8 s ) 9 3 ( 1 3 6 )9 3 ( 1 5 8 ) 5 9 ( r 7 ) 1 4 9 ( 2 r )e o ( 1 6 ) 3 r ( 1 6 ) 5 9 ( 1 6 )

90180

90

901 8 0

90

901 8 0

90

901 8 0

90

901 8 0

90

a ) A 1 1 d a t a a t t a t m .

b ) P a r e n t h e s i z e d f i g u r e s r e f e r t o

c ) A t o m i s n o n - p o s i t i v e d e f i n i t e .

t h e e s d o f l e a s t u n i t s c i t e d .

t290 ROBERT M, HAZEN

ature factors are given in Tables lb and 2 respec-tively.

The pressure cell was tightened again for a seconddata collection at higher pressure, but the crystal wascrushed and no further experiments were possible onthe specimen. Another smaller crystal plate approxi-mately 160 X 140 X 80 pm with (010) cleavage wasmounted in the diamond cell, which was tightened asbefore. Cell dimensions of a : 4.734 + 0.005. , :10.02 + 0.01, c : 5.940 + 0.006 A, and volume :281.8 + 0.5 A', indicated a pressure of approximately40 kbar. Only 374 of 964 measured diffractions wereobserved, and these data were refined to an R of 7.3percent (9.6 percent unweighted) after l0 cycles. Thepressure cell was tightened for a third high-pressuredata collection; this tightening was stopped when asmall fracture was observed in the crystal. Sub-sequent precession and cone-axis photographsshowed that diffraction intensities remained fairlysharp (though some spreading of diffraction peakswas noted), and no change occurred in space groupor lattice translations. Unit-cell dimensions of thethird high-pressure data collection were a = 4.712 t0.005, D : 9.967 + 0.010, c : 5.955 + 0.006 A, andvolume 279.7 + 0.5 As, which represents a 3.50 per-cent volume decrease from I atm. At hydrostaticpressure such a volume decrease would imply a pres-sure of approximately 50 kbar. However, closer ex-amination of cell-edge parameters reveals that while aand b have decreased in length, the c cell dimensionincreased between the second and third high-pressuredata collection. Thus. there is serious doubt that thepresumed 50 kbar data collection was made underhydrostatic pressure. It is probable that at this high-est pressure the forsterite crystal was in direct contactwith the diamonds, and this fact may account forboth the small fracture and the anomalous cell pa-rameters of this specimen.

Data collection at =50 kbar (nonhydrostatic) pres-sure included 1429 measured reflections of whichonly 332 were observed. The fact that fewer than 25percent of measured diffractions were observed maybe ascribed in part to absorption of X-rays by thediamond cell. However, a more basic cause is thatpeak-to-noiSe ratio was low for all diffraction max-ima: peaks were broad, and scattered X-radiationfrom the pressure cell was high. Thus, while a weight-ed R of 6.8 percent was obtained for refinement withisotrofic temperature factors, the unweighted R was12.3 percent, indicating a serious lack of precision forthe weak observed data.

Further high-pressure data collection was not at-

tempted due to the beginning of fracturing in theforsterite crystal. This crystal was removed from thepressure cell and was remounted on a standard go-niometer for a final room-pressure and temperaturedata collection. Unit-cell parameters of this crystalwere remeasured, and found to agree within experi-mental error with earlier determinations (see Tablel). 1026 diffraction intensities were measured and 790of these were observed and used in least-squares re-finement with anisotropic temperature factors. Afterl2 cycles of refinement, an unweighted R of 3.9 per-cent was achieved, and all atomic coordinates matchthose of the original room-temperature and pressurerefinement within twice the estimated standard devia-tion, as recorded in Table 2. All observed and calcu-lated structure factors for forsterite are availablefrom the author on request.

Bond distances and angles from refinements usingthe data collected at high pressures are listed in Ta-bles 5 and 6. The estimated standard deviations ofthese distances are large, and small changes in bonddistances with pressure are thus difficult to detect.However, mean M-O bond lengths for both M(l)and M(2) show compression exceeding 1.0 percent atpressures to 50 kbar, as illustrated in Figure 10. Onthe other hand. the silicon tetrahedron shows nobond compression with respect to experimental errorover the same pressure range. Thus, in olivine, mag-nesium-oxygen bonds appear more compressiblethan silicon-oxygen bonds.

Isotropic temperature factors show little systematicchange with increasing pressure. A slight increase inapparent thermal vibrations with increasing pressureis observed for M(2), Si, O(2), and O(3), but thesechanges are small, and perhaps insignificant, consid-ering their estimated standard deviations. Significantchanges in refined thermal vibration parameters areseen in comparisons of room-temperature and pres-sure forsterite refinements before and after high-pres-sure data collection; average apparent root-mean-square displacements are twice as large after high-pressure data collection. One possible explanation forthis change might be an increase in mosaic spreadingof the forsterite crystal, which was signaled by theincrease in diffraction diffuseness at high pressure.

Coordination polyhedral uolumes and bulk moduli

Polyhedral volumes have been calculated for M(l)and M(2) octahedra and the silicon tetrahedron, andthese volumes are given in Table 8. Distorted poly-hedra are constrained to be of smaller volume thantheir ideal counterparts with the same mean bond

FORSTERITE

TlsI-E 8. Forsterite oolvhedral volumes and distortions

t29l

P = l a t m

Si te Parameter

o c t a h e d r a lI V I ( L J

vo Lume

I d e a lVol ume

Bond Ang leS t r a i n

Ang leV a r i a n c e

^ , , . 4 Y > + i ^

F l ^ h d r + i ^ h

n . f : h a d r ^ lM I ' \

Vo I ume

I d e a l

VoI ume

B o n d A n g l e

S t r a i n

A n g l e

V a r l a n c e

n r r r d r i l i ^

F l ^ h d r + i ^ ^

1 2 . 2 9 1 2 . 5 4 1 2 . 7 )

1 2 . 4 2 1 2 . 5 8 1 2 . 9 0

3 2 . 2 3 3 . 0 3 3 . 0

1 0 6 . 2 1 1 0 . 5 r t l . 2

r . 0 6 9 1 . 0 7 5 1 . O 8 1

L 2 . t 1 1 . 8 1

1 2 . 7 1 1 . 8 3

3 4 3 1 . 4

1 0 9 1 0 7 0

r . 0 9 0 t . o s

l r , . 6 1 I 1 . 5 0 1 2 . 1 9

1 1 . 6 5 1 1 . 6 s t 2 . 1 9

3 0 . 4 2 9 . 4 3 r . 2

9 8 . 9 9 4 . 7 1 0 1 . 0

I . O ' 7 1 . 1 1 r , . 0 6 8

3 u"n-o 12 '20 12 '2 r

t u'rn_o :.2.26 12.36

3 0 . 4 3 0 . 8

o o ' 9 7 . 3 g a . j

r . o o r . 0 6 5 1 . 0 6 5

4 a

: d r s _ o 1 2 . 7 - t L 2 . 7 A

l u t n _ o

1 2 . 8 8 1 2 . s B

3 8 . 2 3 8 . 8

o o ' 9 0 . o 9 0 . I

r . 0 0 1 . 0 1 9 1 . 0 2 8

c i T e t r a h e d r a l 1 6 ^ 3 . ) 1 a \ 1Vo lume 916* S i -O

I d e a r l g , 3v o r u m e , 1 l 5 " s t - o

2 ' z + z ' z j

B o n d . A n g l e r 3 . 9 r z . 9

S t r a l n

Ang l e oo , 48 42

V a r i a n c e

o u a d r a t i c r . o o r . o o 7 r . o o 9E l o n q a t i o n

a ) " T d e a l " h e x a g o n a l c l o s e - p a c k e d o l i v i n e m o d e l

b ) D a t a f r o m S m v t h a n d H a z e n ( 1 9 7 3 ) .

1 3 . 0 8 1 3 . 2 5 1 3 . 4 s 1 3 , 6 5

1 3 . 1 4 1 3 , 3 6 1 3 . 5 3 1 4 . O 0

3 9 . 9 4 0 . 9 4 2 . 2 3 7 . 0

9 8 . 3 1 0 3 . 2 1 0 8 . 0 9 2

1 . 0 2 3 1 . 0 2 5 1 . 0 6 3 I . O 4

1 2 . 5 9 1 2 . 4 t \ 2 . 2 2

1 _ 2 . 7 | L 2 - 5 3 1 2 . 5 3

4 0 . 0 0 3 9 . 8 4 0 . 3

9 5 . 8 9 7 . t 1 0 5 . 9

r . 0 2 6 1 . 0 1 5 1 . 0 2 1

1 2 . 9 6

3 8 . 8

9 3 . 3

I . O 2 2

2 " 1 7 2 . I 7

2 . 2 0 2 . 2 0

1 3 . 5 1 3 . 3

4 3 4 2

1 . 0 0 7 1 . 0 0 7

( s e e F i g , 1 ) ,

2 . 2 r 2 . r 7

2 . 2 2 2 . l A

r 2 . 4 1 8

l d / 5

t . O t 2 1 . 0 6

2 . t 9 2 . t 7 2 . 2 0 2 . l A

2 . r a 2 . L A 2 . 2 2 2 . 2 I

1 5 . 6 1 7 . 4 1 9 1 4 . 4

6 0 7 2 5 7 5 0

1 . 0 2 8 1 . O 2 5 1 . 0 9 1 . 0 1 0

distance; both M(l) and Si are only about I percentsmaller, whereas M(2) is 2 percent smaller. While thetwo octahedral sites show considerable expansionand compression, the sil icon tetrahedron shows nochange in volume with changing 7 and P over theranges studied.

Changes in polyhedral volume may be directly com-pared to unit-cell volume changes. From -196o to1000'C the net change of volume in M(l) and M(2) isabout 0.5 A' per octahedron, while the change intetrahedral volume is nil. As i l lustrated in Figure l,each olivine unit cell has four M(l) and four M(2)octahedra. Thus, cation polyhedral expansion ac-counts for 4.0 At volume decrease per unit cell. How-ever, the total unit-cell volume change from - 196. to1000"C is about l3 A3, leaving 9 A, unaccounted forby cation polyhedral expansion. Thus, approximately70 percent of olivine's thermal expansion may beattributed to volume increases of the unoccupied oc-

tahedral and tetrahedral sites (i.e. "voids") in theclose-packed oxygen array. Magnesium octahedronexpansion plays a secondary role, while tetrahedronexpansion has no effect on changes in olivine's vol-ume with temperature to 1000'C.

Similar calculations may be performed for forster-ite compression. At 50 kbar, a volume decrease ofapproximately 0.7 As in M(l) and 0.5 A'in lf12; hasbeen recorded, thus contributing 4.8 A3 of compres-sion per unit cell. The total compression of magne-sium olivine at 50 kbar is about l0 A' per unit cell.Thus, 50 percent of forsterite's compression may beascribed to octahedral sites and 50 percent to voids,whereas silicon tetrahedra again have little effect onvolume change.

Huggins (1974, p. 339 et se4.) has discussed thesignificance of individual polyhedral bulk moduli tothe pressure response of crystalline solids. He statesthat systematic studies of the compressibility of poly-

t292

2.t3

2t2

€ z. ' r

ROBERT M. HAZEN

EozFoo

oIo

z

uE

M(2) - O

Frc. 10. Forsterite ,n"J"::,t"[:r"r'lo]"0 distance versuspressure.

hedra may "enable detailed predictiorrs of crystalstructures to be made at any pressure and may pro-vide valuable insight into crystallochemical criteriafor the stability of structures and into the reasons forphase changes." Bulk moduli of M(l), M(2), and Siare easily calculated from data in Table 8 of poly-hedral volumes versus pressure by applying the equa-tion: Kt : -l/P(V) = -r dP/3 dr, where K2 is thebulk modulus, l3(V) is the volume compressibil i ty, Pis the pressure, and r is the interatomic distance (seee.g. Broecker and Oversby, l97l). The estimatedM(l) and M(2) bulk moduli are thus approximately1200 and 1000 kbar respectively at pressures between20 and 50 kbar. These values for Mg octahedral bulkmodulus are similar to the 1600 kbar bulk modulus ofpericlase.

The silicate tetrahedron of forsterite shows no vol-ume compression within gxperimental error (+ 0.02A'). Therefore bulk modulus must be significantlygreater than 2.2 x 50/0.02 : 5500 kbar. This is inagreement with Huggins (1974) estimate of K"t ",13,900 kbar in andradite.

Polyhedral dis t ortions

Characterization of polyhedral distortions is essen-tial to an understanding of olivine crystal chemistry,and consequently, many authors have attempted toquantify these deviations from ideality. While most

previous studies were designed to define the shape ofolivine polyhedra (Dollase, 1974), or to plot changesin shape with changing composition (Brown, 1970;Robinson et al., l97 l), these methods are equallyapplicable to a study of polyhedral changes with tem-perature and pressure. Three commonly cited dis-tortion indices have been calculated and are tabu-lated in Table 8. These are: (1) bond angle strain, (2)polyhedral angle variance, and (3) mean polyhedralelongation. Each is a measure of the deviation from aregular polyhedral form, and each has been appliedwith good results to the olivine group.

Several generalizations may be made regarding theeffects of temperature and pressure on forsterite poly-hedral distortions, as measured by the three dis-tortion indices. The two magnesian olivine octahe-dral sites display very similar behavior. Both exhibitstrong increasing distortions with increasing temper-ature, but l i tt le, if any, change with increasing pres-sure. This pattern may be explained in terms of thetrends of expansion and compression of individualMg-O bonds, as shown in Figure 6. Since longerbonds wil l expand up to four times more rapidly thanshorter bonds, an increase in temperature wil l resultin increased variance of bond distances, and corre-sponding increases in distortion. Conversely, an in-crease in pressure wil l force a decrease in the range ofbond distances, and perhaps a reduction in octahe-dral distortions. The fact that a definite trend towardsreduction in octahedral distortions was not observedat high pressures may be due in part to non-hydrostatic conditions at high pressure. Further high-pressure structure refinements are needed to clarifyoctahedral distortion behavior.

Tetrahedral distortions are small compared tothose of the two octahedral sites. Virtually nochanges in distortions occur between -196' and1000"C, though some fluctuations (within experimen-tal error) are observed. However, a general increaseoftetrahedral distortions is seen with increasing pres-sure. Whether this increase is due to nonhydrostaticeffects, or to real distortion, is not clear. However, itshould again be emphasized that errors in forsteritebond angles and bond distances are large, and that alldistortion calculations are based on dffirences be-tween these numbers of similar size. Therefore, addi-tional data are required to resolve the nature of sil i-con tetrahedral distortions with pressure.

Conclusions

Development within the past decade of high-tem-perature, low-temperature, and high-pressure diffrac-

. toz

2

FORSTERITE t293

tion apparatus now permits the study of mineral crys-ta l s t ruc tu res over a range o f p ressure andtemperature. This investigation concentrated on thecomplex pressure and temperature response of themagnesian olivine forsterite. Significant differenceswere observed in the thermal expansion and compres-sibility of magnesium octahedra versus silicon tetra-hedra, reflecting the differences in charge and coordi-nation of these two cations. Furthermore, withineach polyhedron a range of individual bond expan-sions was noted, with longer bonds expanding morerapidly than shorter bonds. These data will be com-bined in a subsequent study with data on iron-bear-ing olivines in an effort to predict the structure andstability, as a function of composition, temperature,and pressure for the ferromagnesian olivines.

AcknowledgmentsThe author is pleased to acknowledge the aid and advice of

Professor Charles W. Burnham. Thanks are also due to ProfessorR. G. Burns, Dr. T. L. Grove, Professor J. F. Hays, and ProfessorT. Shankland for their discussions of the ideas in this study, and toDr. L. Finger and Professor C. T. Prewitt for their thorough andthoughtful reviews of the manuscript.

This research was supported by National Science FoundationGrants GA-12852 and GA-41415.

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Manuscripl receiued, December 15, 1975; accepted

for pablication, MaY 28' 1976.