AmericanMineralogist, Volume61, pages1280-1293,1976
Effectsof temperatureandpressureon the crystalstructureof forsterite
Rosnnr M. HlzeNl
Haruard Uniuersity, Department of Geological Sciences
Cambridge, M assachusetts 02I 38
Abstract
The crystalstructureof syntheticforsteritehas beenstudiedas a function of temperature
diffractiondata
and pressure.Crystal structureshave been refinedfrom three-dimensional
to 50 kbar. Volume
collectedat temperatures
from -196'C to 1020'C and at pressures
compressibilityand thermalexpansionof individualmagnesiumoctahedraarecomparableto
thoseof bulk foresterite.However,the silicon tetrahedronneitherexpandsnor compresses
within the rangeof conditionsinvestigated.
Thus, changesin Mg-O bond distancesaccount
for virtually all of forsterite'svolumeresponseto changesin temperatureand pressure.Short
Mg-O bonds vary significantlyless with temperatureand pressurethan do long bonds,
implying that shorter bondsare stronger.Octahedraldistortionsincreasewith temperature,
but show no strongtrendswith increasingpressure.
Introduction
and Anderson(1967).More recently,variationsof
The characterizationof mineralsat high pressures crystalstructurewith temperaturehavebeenreported
(Smyth and Hazen,
and temperaturesis an essentialpart of the earth for pure forsteriteand FoseFaoz
(Brownand Prewitt,1973),and pure
sciences.
Many physicalpropertiesof mineralsin- 1973),FoogFar'
cluding compressibility,thermal expansivity,and fayalite (Smyth, 1975).Compressibilitystudieson
electricalconductivity are functions of temperature olivineare numerous,and data of Bridgman(1948),
and pressure.
Sincea solid'sphysicalpropertiesarea Olingerand Duba (1971),and Schocket al. (1972)
direct consequenceof its atomic arrangement,the are frequentlycited. Recentstudiesof Olinger and
continuousvariation of atomic positionsresulting Halleck(1975)offer improvedvaluesfor linear and
from changesin ? and P may thus be usedto under- volume compressibilities
of olivine.
standmore fully the variation of physicalproperties
Olivine has two crystallographrcallydistinct ocwith theseintensiveparameters.The principal objec- tahedralsites,one tetrahedralsite,and threedistinct
tive of this study is to determinethe atomiccoordi- oxygen positions.It displayssignificantdeviations
natesand thermalvibration pararneters
of magnesian from idealclose-packing
of oxygens,and thesedeviaolivine at a variety of temperaturesand pressures
in tions are most convenientlyvisualizedin terms of
order better to definethe equation of state of this distortionsof the octahedraland tetrahedralcation
coordination polyhedra from their regular forms
importantmineral.
(compare
Fig. la and Fig. lb). The M(l) site has
The crystalstructureof olivinewasfirst determined
(1926);
i
symmetry,
and its coordinationpolyhedronapsubsequentrefinem€nts
by Braggand Brown
proximates
(1951),
an
were reported by Belov e, al.
Hanke and
octahedronflattenedalong a threeZemann(1963),Birle et al. (1968),and Brown (1970). fold axis. The M(2) octahedron,which has mirror
Lattice parametersof magnesianolivine have been symmetry,has larger averageM-O distancesthan
evaluatedby many authors(e.g. Louisnathanand M(l), and does not approximateany simple disSmith, 1968).Thermalexpansionof olivinehasbeen tortion model (Dollase,1974).As noted by Brown
studiedby severalinvestigators
including Kozu et al. (1970), the three olivine cation polyhedra have a
(1934),Rigby et al. (1946),Skinner(1962),and Soga numberof oxygen-oxygen
edgesin common,including two betweenadjacentM(l)'s, two betweenM(l)
I Present address:
Geophysical Laboratory, 2801 Upton Street, and M(2), two betweenM(l) and Si, and one beN. W., Washington,D. C. 20008.
tween M(2) and Si. Distortions of the hexagonal
I 280
FORSTERITE
l2El
compositionof 99.99*'percent MgzSiOr,and th€
indicesof refractionared = 1.635+ 0.002,P : 1.650
to
+ 0.002,and 7 : 1.670+ 0.002,whichcorrespond
reported values for pure magnesianolivine (Heinriques, 1957).The crystalsare optically free of fracturesor other visibledefects,and a good (010)cleavageis present.An irregularcrystalof approximately
400 X 300 X 300 pm maximum dimensionswas selectedfor preliminaryX-ray study. A suite of a-, b',
and c-axisprecessionphotographswas obtainedto
confirm previousunit-celland symmetrydeterminations. A diffraction symbol of mmmPbn- was
observed,which correspondsto the reportedolivine
spacegroup Pbnm.
Unit-cell dimensionsof forsteritewere determined
on the oriented single crystal from precisionbackreflectionWeissenbergphotographs.Refinementof
82 diffraction spot pairs yielded orthorhombic cell
parametersof a : 4.7535+ 0.0004,b : 10.1943+
0.0005,and c : 5.9807+ 0.0004A, with a resulting
unit-cellvolumeof 289.80+ 0.05A'. Thesevalues
by Yoder and Sacloselyagreewith determinations
hama(1957).
T
I
A.
! _{
l._
B.
Frc. l. Olivine crystal structure:A-ideal HCP model, Bactualstructure,(after Brown, 1970).
Dala collectionand inslrumentation
Proceduresfor data collectionfrom singlecrystals
at room temperature,liquid-nitrogen temperature,
close-packed oxygen array have been related to
and high temperatureare describedby Hazen(1976),
cation-cation repulsionsacrossthese sharededges while high-pressure
X-ray diffractiontechniqueswere
in agreementwith Pauling'sthird rule.
similarto thoseof Hazenand Burnham(1974,1975).
High-pressurediffraction data have been corrected
Experimental
for diamondpressurecell absorptionaswell as specimen absorption(Hazen, 1976).
Specimendescription
Colorlessand transparentsinglecrystalsof synResults
thetic forsteritehavebeenmanufacturedby the Crystal ProductsDivisionof Union CarbideCorporation, Room temperalure
T. J. Shank- A referenceroom-temperatureand pressurestrucand werekindly providedby Professor
pure
This
forsterite
has
a reported ture refinementwas madewith 956 observeddiffracland.
essentially
TABLE lA.
T( t c )
23
180
305
450
610
750
900
1020
I105
* Parenthesized
Forsterite unit-cell parameters from 23" to I 105.C
o
b (A)
o
a (A)
_o.
c (A)
vol
o^
(A J)
4.752(3)x
4.758(s)
4.76s(5)
4 . 7 6 7( 3 )
1 0 . 19 3( 8 )
1O.272(7O)
ro.231(1O)
1 o . 2 4 8( 8 )
5 , 9 7 7( 5 )
5 . 9 9 r( 6 )
5 . 9 9 8( 6 )
6 . o o 9( s )
28e.s(3)
2 9 r , r( 5 )
2 9 2 . 4( 5 )
293.6(3)
4.775(5)
4.780(3)
4.785(7')
4 . 7 e 7( 7 )
4.80s(7)
10.273(10)
ro.292(8)
1 0 . 3 3( 1)
1 0 . 3 5( 2 )
6 . o 2 3( 6 )
6.o32(5)
6 . o 4 7( 9 )
6 . o 5 8( 8 )
6 . 0 6 9( 8 )
2 e 5. 5 ( 5 )
296.8(3)
2 9 8 . 9( 8 )
30o.7(12)
fisures
refer
to
the
esd of
least
units
cited.
1282
ROBERT M. HAZEN
tion maximarepresenting
all hkl's in one octantof thermocoupleby naturaladhesionto hot platinum.
reciprocalspacefrom 5o to 80o2d(MoKa). Of 1096 Unit-cell constantsand intensitydata werecollected
measuredreflections,ll6 including all 90 space- first at 23"C and then at reported temperaturesof
group extinctreflections,
wereunobserved.
In addi- 300o,600o,and 900' (seebelow).
tion, 24 of the strongestreflections,for which lF,u"l
In the presentstudy, the crystal was mounted
(( lF"rt"l, wererejecteddueto presumedsecondary within a silica capillary,and unit-cellconstantsof
extinction.A calculatedlinear absorptioncoefficient forsteriteweremeasuredat temperatures
to I105'C
of 10.594cm-r wasapplied,and transmission
factors (seeTable la). An attemptat a 1020'Cdata collecvariedfrom 78 to 83 percent.Refinementof forsterite tion was then made,but fewer than 150diffractions
included40 variables,representingI I atomic posi- were measuredbeforethe heaterburned out. During
tionalparameters,
28 anisotropicvibrationalparame- the rapid cooling following heaterfailure, the silica
ters, and a scalefactor. A weightedR-factor of 4.8 capillary fracturedand the crystalwas lost. Thus, it
percent(4.9 percent unweighted)was obtained for was not possibleto recheckthe crystalat room temthis refinement. Forsterite refinement conditions, perature.
positional parameters,and temperaturefactors are
Another forsterite crystal of approximatelyreclistedin Tables1,2, and 3. Room temperatureand tangularshape,200 x 180X 120pm, wasmounted
pressurevaluesare in closeagreementwith previous on a coppar pin; unit-celland intensitydata were
23"C forsteriterefinementsby Birle et al. (1968) and collected at liquid-nitrogen temperatureusing the
Brown(1970).
Cryo-Tip system.Unit-cell parameterswere redeterminedfollowing the low-temperature
study,andwere
Cell dimensionand structural factor measuremewqr
previous
room-temperature
measunchanged
from
-196" to I 105"C
urements.In each of the six structurerefinements,
Forsteriteunit-cell parametersand structurefac- m a d ea t - 1 9 6 o , 2 3 o 3, 0 0 o , 6 0 0 o , 9 0 0aon,d 1 0 2 0 " C ,
tors were measuredat severaltemperaturesfrom all Pbnm spacegroup-extinctreflectionswere unob-196"C to above1000'C.Preliminaryresultsof the served.From 6 to 24 of the strongestreflectionswere
high-temperature
portion of this study werereported rejectedfrom each refinementdue to presumedsecby Smyth and Hazen(1973).The crystalchosenfor ondary extinction(i.e.,lF"6"l(lF""r"l). Resultsare
high-temperature
work by Smyth and Hazen was presented
in Tables1,2, and3, in the sameformatas
mountedin air directlyon thejoin of the PtlPtroRhro that of Brown (1970)for easeof comparison.
Forsterite unit-cell parameters and refinement conditions
TABLE 18.
T
^
("c)
P
conditions
No.
-
rlil
"til
'or (i3)
cfi)
of
.8.(%1" .Bl%)'
No.
of
oi""i".a
No.
of
*.:."t.a"
h
zJ
'
?t'
- 350
-
1.7535(4r r0.1943(s)
aoo^!
; : : ; T " ' " ' o t " " n . r u r 1 r 1 r 0 . 2 4( r )
5.9807(4) 289.80(s)
4.8
4 9
5 .e e e( 6 )
2 9 2 . 6( 5 )
5.4
5.6
7
676
564
6
686
565
8
12-9-
135
t08
9
5.5
742
646
,'c
4.7ia(5\
10.29(1)
6.0r7(6)
296.0(s)
5.7
s.9
1000
nc
4,795(5\
ro.36(r)
6.060(6)
3 0 0 . 8( 5 )
5.s
s.9
4.79e(5)
10.36(r)
6.06S(6)
301.4(5)
rl -t-
^
10.r8(1)
5.9'76(6)
288.6(5)
4.9
capSilica
.,,
ltlarymount
,,
"
ar\r^-f
-I96
i ^
mount
23
2okb
23
40kb
23
50kb
23
1 atm
4.743(51
10.0e(r)
5.e54(6)
28s.0(5)
7-o"
9-4"
969
329
3
";ii-*-,
4.734(5)
r0.02 (r)
5 . 9 4 0( 6 )
28r.8 (s)
;. :d
s. ed
964
374
6
"
4.7r2(5)
9.97(1)
5 . 9 5 5( 6 )
2 7 9 . 1( 5 \
9.8-
12.3-
962
r0.19(1)
5.980(6)
2e9.5(S)
3.9
4-2
LO26
St a n d a r d
mount (after
hiqh p)
a)
l l " , o . -l l r " " r " l'l: . 0
Parenthesized
figures
Data
and
d)
Isotropic
1^^15)
DFc<c,rra
b)
of
d d
956
569
- 675
1020
1096
Snyth
4-74915)
refer
Hazen
temperature
to
(1973),
factors
the
wirh
only,
esd
of
least
corrected
units
cited.
E e h p e r a r u r e e s c l m aE e s .
l3
790
2
l 283
FORSTERITE
isotropictemperaturefactors,and r. m. s. equivalents
Tnslr 2. Forsteritepositionalparameters,
1
Pa ra-
Atom
meter
-rooo"cb
1 02 0 ' c
40 kb
20kb
50kb
ter
atm
af-
high
P
o
o
0
0.26(1)
0.057
o
0
0
0.67(3)
0.092
0
0
0
r.20 l4l
o.I22
0
0
0
r . 7 11 4 )
0.150
0
o
0
r.8(3)
0 .1 5
0
0
0
0 . 1 6( 9 )
o.046
0
o
o
0.37(9)
0.069
0
0
0.2(r)
0.05
0
0
0
o.43(2)
0.074
0 . 9 9 1 4( 3 )
o . 2 ' 7 7 (2r l
l/4
0.08(2)
0.031
0 . 9 9 r 5( 2 )
o - 2 7 7 4l t \
t/4
o.22(rl
0.053
0 . 9 9 r s( s )
0 . 2 7 8 0( 2 )
r/4
0 . 6 6( 4 )
0.091
0.9919(5)
o . 2 1 4 5l 2 l
l/4
r.19(4)
o.r22
o . 9 9 2 4( 4 )
o . 2 7 7 2( r )
t/4
1 . 7 0( 4 )
o-r47
0 . 9 9 r( 3 )
0.280(1)
7 /4
1-8(3)
0 .r 5
0 . 9 9 2( r )
o . 2 7 7( r )
r/4
0 . 3 7( r 0 )
0.069
0.99r(1)
0.28o(t)
| /4
o . 5 2( 1 0 )
0.081
0.993(1)
0.283(r)
t/4
0.6(r)
0.087
0 . e 9 1 4( 3 )
o . 2 ' 7 7 Il t l
1,/4
o . 4 3( 2 \
o.o74
3/a
0.o833
r/4
o.4261(2)
0.0939(1)
t/4
-o.02(21
0.00
o . 4 2 6 2( r )
0 . 0 9 4 0( r )
r/4
0.08(1)
0.031
o . 4 2 s 7( 4 )
0.0939(2)
r /4
o . 3 3( 3 )
0.065
0 . 4 2 s ' 7l 3 l
0 . 0 9 4 r( r )
t/4
0.66(3)
0.091
o . 4 2 6 3l 3 )
0.0943(1)
r/4
0.97(3)
0 .1 1 1
o . 4 2 ' 11 2 ) o . 4 2 7( r )
0 . 0 9 4( r ) 0 . 0 9 s { I )
r /4
L/4
0 - 3 7( 8 )
1.0(2)
0.069
O.II
0.426(r)
0.o93(r)
r /4
o. 4 s( 7 )
0.o75
0.428(r)
0 . 1 0 1( r )
7 /4
0.7(1)
0.094
o . 4 2 6 2( 2 )
0.098e(1)
t/4
0.27 lLl
0.0s8
3 /4
0.0833
r/4
0 . 7 6 6 1( 6 )
0.0919(3)
r/4
0.12(4)
o.039
0 . 7 6 s 7( 3 )
0.0913(2)
t/4
o . 2 6( 2 1
0.0s7
0.7657(10)
0 . 0 9 1 0( s )
t/4
0 . 7 0( 6 )
0.094
o - ' 1 6 3 1( 9 )
0 . o 9 0 6( 4 )
),/4
t.t2 \'7)
0.119
0.7631(8)
0 . 0 9 1 4( 4 )
t/4
r - 5 1( 6 )
o.L3'7
o . 7 3 1 1 2 ) o . 7 ' 7L ( 2 1
0 . 0 8 8( 2 ) o . o e 2( 2 )
r/4
| /4
-0.1(2)
1.0(s)
0
.0
0.11
0 . 7 6 8( 2 )
o . o 9 71 2 \
o . 7 7 6\ 2 )
o.r(2)
0.035
| /4
0.1(2)
0.036
o . 1 6 6 41 5 \
o-0908(3)
t/4
o . 4 3( 4 1
o . o ' 74
t/4
o.4\67
t/4
o . 2 2 0 2\ 6 1
0 . 4 4 5 9( 3 )
1-/4
0 . 0 6( 4 )
0.026
0.2215(4)
o.447412)
l/4
o . 2 4( 2 )
0.055
0 . 2 1 7 8( 9 )
o . 4 4 9 2( s )
r /4
0. 60{6)
0.087
o . 2 1 7 8( 9 )
o . 4 4 e 1( 4 )
r/4
r-02(7)
0.113
0 . 2 r 7 8( e )
0 . 4 4 9 7( 4 )
r/4
1.40(6)
0.133
o . 2 2 4( 4 )
0 . 4 s 8( 3 )
r /4
r . 0 (5 )
O.II
o . 2 2 1\ 2 1
o . 4 4 ' 7( 2 )
r/4
0.0(2)
0.0
0 . 2 2 41 2 )
o - 4 4 4\ 2 )
r/4
o-o(2)
o.0
0 . 2 2 8( 3 )
0 . 4 4 0( 3 )
| /4
o. 4 1 2 )
0.07
o . 2 2 0 9( 5 )
0 . 4 4 ' 7 4( 2 )
| /4
0.37(3)
o-069
X
t/4
o . L667
z
0
o . 2 1 7 ' 71 4 )
o.t62A12)
0.0333(3)
0.07(3)
o.o29
o . 2 7 1 7\ 2 )
0.1628(1)
0.033r(2)
o.28(21
0.060
o . 2 a 2 21 6 )
o . 1 6 1 9( 3 )
0 . 0 3 4 7( 6 )
o . 6 5( 4 )
0.090
o . 2 4 2 2( 6 1
0 . 1 6 r 9( 3 )
0.03s2(s)
r.1s(s)
0. 120
o . 2 4 4 3( 5 1
0.1629(3)
0.03s9(s)
r.67(s)
o.L44
0 . 1 6 0( 2 )
0 . 0 3 0( 3 )
r.8 (4)
o.r5
o . 2 7 4l 2 l
o . 162 12)
0.o37(r)
o , 2( 1 )
0.05
o . 2 7 6l 2 l
0.1s8(2)
0 - 0 3 s( 1 )
0.4(r)
0.07
0 . 2 8 r( 2 )
0.153(2)
0 . 0 3 2( 1 )
o. 6 ( 2 )
0.087
o . 2 7 4 4e l
0 . 1 6 2 0( 2 )
0 . 0 3 4 1( 3 )
0 . 4 r {2 )
o.o72
z
B
M(2)
z
0
t/4
| /4
B
si
2
B
o(1)
Y
z
B
x
z
B
o(3)
-arr""b
0
0
0
o.og(z)"
0. 034
M(1)
o(2')
-:so""b
23' c
HCP
B
hexagonal
a)
"Ideal-"
b)
Data
c)
Parenthesized
of
close-packed
Smyth and Hazen
figures
olivine
model
(see Fig.
1 /4
1 ).
(1973).
refe!
to
the
esd of
least
units
cited.
Linear and volume thermalexpansionof pure for- authorrevealedthat in all casesthe measuredtempersterite as determinedby Smyth and Hazen (1973) ature was lower than the real temperature,and that
from 23" to 900'C (Tablelb) and this study(Table such errors may exceed100'C at 1000"C(in one
gold of meltingpoint 1063"Cappeared
1a)from -196o to I l05oC,areillustratedin Figures experiment,
^,
950oC).Theseerrors are assumedto
2 and 3. Data of Skinner(1962)on expansionof a to melt at
rapid
conductionof heat away from the
(ForuFaon)
reflect
the
from 25o
syntheticiron-bearingforsterite
join. Such an error in thermothermocouple
graphs.
large
The
to ll27"C are also presentedon these
between
could
explainthe differences
readings
couple
dataof Skinnerand this studyarein closeagreement
is
the
case,
parameters.
If
this
of
cell
sets
the
different
all
with respect to the rate of thermal expansion;
and
in
Smyth
reported
refinements
parallel
forsterite
temthen
the
over
the
expansioncurvesareessentially
peraturerange25oto I l00oC. However,seriousdis- Hazen (1973) were made at -350", -675", and
existbetweenthe data of thesetwo studies -1000'C. This would explainthe deviationsin unitcrepancies
and thoserecordedby Smythand Hazen(1973).The cell parametersat high temperatures,as well as the
temperaturemeasurementsof Skinner and of this similarity betweenSmyth and Hazen's900'C refineand ment and the partial 1020"Crefinementof this study.
study weremadewith calibratedthermocouples,
reported temperaturesare based on thesecalibra- In the remainderof this study Smyth and Hazen's
tions. However, the thermocouple employed by data will be assumedto have been obtainedat the
Smyth and Hazenwas not calibrated.The technique higher temperatures.
of mounting a crystal directly on the thermocouple
Bond distances
PtlPt-Rh join requiresa largerwire diameter(0.150
and bond anglesare tabulatedin
Bond distances
largerjoin bead
mm ur. 0.050mm), and significantly
(diameter: 0.30mm Dr.0.075mm) thanthat usedin Tables4 and 5 respectively.Figure 4 illustratesthe
the silicacapillarymountof thisstudy.Calibrationof variation of averageMg-O bond distancesfor the
by T. L. Grove and this two different magnesiumoctahedra,as well as the
similarlargethermocouples
1284
ROBERT M. HAZEN
Teslr 3. Forsteriteanisotropictemperaturefactor coefficientsu
ij
of
Rii
-l 96oc
2i'C
II
22
o. 2 ( 4 )
0 .s ( r )
23
0.2(3)
o.2(2)
-0. r (3)
-0.2(1)
2. 4 ( 3 )
r.02 (6)
r.0 (2)
0.1(r)
-0.3 (2)
-o.3 (r)
M(2)
1r
22
33
L2
0. r (5)
0.4(r)
0.4 (3)
-0.0(2)
2.2 (3)
0.6(r)
| .6 (21
0 . 0 (r )
8.r (8)
L 2. 4 ( L . O ' ) 1 8 . 3 ( r . 5 )
r . 4 (r )
2. 7 ( 2 )
3.8(4)
4. 5(4)
e.L (7)
12.2(r.0)
0.0s(e) o.2 (2)
0. s (3)
4.6
r. r
2.e
o.2
si
11
22
0.8(2)
o.43(4)
0.8 (1)
0.05(8)
2 . 6( 2 )
r-.0(r)
2 . 5( 2 )
0.1(1)
3.8 (4)
| .8 (21
5.e(4)
0.3 (2)
1.8(3)
0. I (r)
2.L(21
0.0(1)
e.4(r.0)
L . 2( 2 )
s.r (7)
0.3(2)
9.8(r. r)
2. 8 ( 4 )
8.8 (9)
0.e(1)
r1.8(r.8)
4.3 (4',t
r0.8(1.2)
4.6 (4)
3.7
1.3
2. 7
0.3
1 3 .s ( r . r )
4.2 (9\
0.8(2)
2. 6 ( s l
o. 3(4)
Atom
J3
I2
I J
- 35ooc
.--o^
675-C
7. 5 ( 2 1
2 .I 1 2 )
2 .e ( 3 )
-0.r (1)
-0.5(2)
-o.7(2)
r0.7(4)
3. 9 ( z ' , )
6.8 (4)
-o.2
-1.0
-1.2
(2)
(2 )
(1)
LZ
-r.8(3)c
0.2(r)
o.2 (2)
0. r (1)
o(r)
11
22
33
T2
-o. 2 (9)c
0.5 (2)
r.1 (6)
0.5 (4)
o.e (5)
0. e (1)
2.3(3)
o.2 (21
o(2)
11
22
- 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 (5)
o.7 (Ll
2.2 (3)
0.1(2)
7.3(7)
| .4(2)
4.0(4)
-0. r (2)
1 0 . 0 (r . 0 )
2.4 (31
7.8(6)
-0.1(r)
2.2 (3)
0.93(7)
L.6 (2)
0. r (1)
-0.3(2)
0.4(r)
8 .2 ( e )
1 r . 2( r . r )
55
L2
o(3)
1r
22
55
l2
13
23
a)
6r.J. x I o - :
b)
Parenthesized
c)
Atom is
for M(2), si,
figures
non-positive
L.6 (2)
3. 7 ( 4 )
-0.3(1)
-0.2(r)
o.6 (2)
o(l)
refer
to
3.0(2)
8.0(7)
-0.s(1)
-0.5(3)
L.2 (2)
and o(2)
-
. ^^^o^
IUOU C
r5.4 (3)
5.9 (3)
9. 2(6)
-0.4(1)
-1.7 (3)
-L .7 (2)
6.8(s)
2.7 (4)
7.7(7)
o.4 (2)
3.2 (4)
r0.6(7)
0.0(2)
16.3(r.4)
4.s(3)
10.3(r.2)
-0.3 (1)
-o.5 (2)
1.9 (2)
23'c after
high
pressure
4.2 (3)
L.26(e)
2.7 (3)
0.0(3)
-0.2 (3)
-0.3(r)
(5)
(1)
(31
(2)
(9',)
(2)
(5)
(4)
3.e(6)
r.2 (1)
2. 8 ( 4 )
-0.0 (2)
0.0 (4)
0.3 (2)
Br: = Bz: = o'
the e s d o f
least
units
cited.
definite.
average Si-O tetrahedral bond distance.The bond
expansion curve of the M(l) octahedron is remarkably similar to that of Mg-O in periclase.As discussedin Hazen (1976), periclaseMg-O bonds have
an absolutezero bond length of 2.10 A. ttrey expand
at an increasing rate to a temperature of approximately 450 K, and above this temperature Mg-O
bonds have a relatively constant expansion rate of
about 2.7 X l0-'A/K.
Forsterite Mg(l)-O bonds
have an absolute-zero'meanbond distanceof x2.09
A. ttrey expand at an increasing rate to approximately 400 K, and above this temperatureexpand at
a constant rate of 2.8 X l0-'A/K.
The longer
Mg(2)-O forsterite average bond distance has qualitatively similar expansioncharacteristics.The absolute-zerobond distancefor Mg(2)-O is =2.13 A, and
above 450 K a constant expansionof 3.4 X l0-'A/K
was observed. It appears from these average bond
data for magnesium-to-oxygenbonds in periclaseand
forsterite that longer bonds may have a higher rate of
thermalexpansion.
To testthis possibilityall individual magnesium-oxygen
bondshavebeenplotted versus temperaturein Figure 5. While there is some
scatterin thesedata,longerMg-O bondsshowsignificantly higherratesof thermalexpansion(asmuch as
four timesgreater)than shorterbonds,as illustrated
in Figure 6. This fact is consistentwith the theory
that shorterbondsare strongerbonds.
Analysisof the silicon-oxygenbonds in forsterite
versustemperatureis lessstraightforward.The average Si-O bond length is constantor increasesonly
slightly from liquid-nitrogen temperatureto room
temperature,asillustratedin Figure4. The distanceis
seento decreaseby over 0.5 percentbetween300 K
and 800K; only above 1000K doesthe averageSi-O
bond appearto expand.This unexpected
behaviorof
forsterite Si-O bonds between -196" and l000oC
cannotbe explainedwith a simplebondingmodelof
Si-O alone.Rather,complexMg-O-Si interactions
must be consideredbeforea thoroughunderstanding
1285
FORSTERITE
- Stlnncr (1e62)- Fo96
!,'
- Smytha Hozcn(1S73)-Fopq;'
.
- This Study- FotOO
/:' i
ro
(1962)-F55
-,,'-Skfnn.r
*',";/
...o.':Srnyth O Horcn ltgT'l Fotr,O,?a.FX./
/.x1Thls Studt-Fo,oo
z
9
o
z
lrt
-
lrl
I
3
)
o
.,.::ifl2x
o
J
J
l!
J
J
U
o
a)
F
;2e4
L
=
2
D
TEMPERATURE(K)
Frc. 2. Forsterite unit-cell dimensionsversus temperature.
Data of Smyth and Hazen (1973)have not beencorrectedfor
presumederrorsin temperature.
of Si-O bond distancevariation in forsteritecan be
achieved.
The variation betweenindividual Si-O bondswith
temperatureis illustratedin Figure7. Silicon-to-oxygen bonds in forsteriteprovide a striking exampleof
the greater thermal response of longer bonds.
The shortestSi-O(l) bond shows no variation in
lengthbetween-196o and 1000oC,while the longest
Si-O(2) bond showsa maximum variation of 1.3
percentover this range.
An important and often neglectedaspectof bond
distanceanalysisis the effectof thermalvibrationson
mean interatomicseparation.Interatomic distances
tnn\e +. Thermally corrected mean separationof Si-O in
t
forsterite(A)
T(oc)
-196
Centrold
separatlon
r. oaaR
Lower
bound
Rlding
350
L.OZ)
1.633
1.630
L.625
6tJ
L, OLq
L.OZ+
L.OZI
1 ,5 3 0
1.630
I.
1.630
1000
I.
OJJ
I.
OJI
L,627
bJ)
Noncorrelated
Upper
Bound
L.634
1.633
L,632
1.638
r.551
1.635
L.636
1.640
r.652
1.565
2oo
400
600 800 looo 1200
T E M P E R A T U R E( K )
1400
Data of
FIc. 3. Forsteriteunit-cellvolumeversustemperature.
Smyth and Hazen(1973)have not beencorrectedfor presumed
errorsin temperature.
reportedin most studies(includingthis one) are calculated as the distancebetweenmean atomic positions. However,Busingand Levy (1964)and Smyth
(1973) have demonstratedthat a better measureof
interatomicdistanceis the meanseparation.In general, when thermal vibrations are large, the mean
separationof two atoms will be greater than the
separationof atomic positions.Thus, bond thermal
expansionbasedon meanseparationmay be greater,
and may representa more valid physicalsituation,
than that reportedin most studieswhich quote distancebetweenatomic coordinates.
It will prove usefulto analyzethermally-corrected
bond distancesfor Si-O in forsteriteasan illustration
of the possiblemagnitudesof thermalbond distance
corrections.Lower, upper,riding, noncorrelated,and
uncorrectedbond distanceswere computed by the
least-squares
and errors programs(RnNn and Beorne, Finger, 1969).Thesedata for the mean Si-O
bond distancein forsteriteare tabulatedfor temperaturesfrom -196o to 1000"Cin Table 4. and are
1286
ROBERT M. HAZEN
Tnsr-E5. Forsteritebond distances"
Bond
-r96'c
23'c
IMultj-plicitv]
-:so'cb
-els'cb
-tooo"cb
tozo"c
s0 kb
Si Tetrahedron
si-o(t )
si-o(2)
si-o (3)
mean Si-O
t1l
ol
r.ets1:1"
r.640(3)
1 . 6 3 3( 2 )
1.630
r.616(2) 1,614(3)
r . 6 4 s 1 2 \ 1 . 6 3 6( 4 )
1 . 6 3 3( 2 ) r . 6 2 4 ( 3 )
1.633
L625
1.615(5)
1 . 6 3 6( 5 )
r.623(3)
I.624
r . 6 1 s( 4 )
r.649 (41
1.628(3)
1.630
r.53(2)
r.60 (2)
r . 6 ' 7( 2 )
L.62
r.63(L)
|.66 (2)
1.60(1)
1.62
1.62(r)
r.66\21
1.60(t)
7.62
1.64(r)
r.76(2)
r.56(1)
1. 63
1.617(3)
r.645(3)
\-625(21
I.624
o(r)-o(2)
o ( 1 )- o ( 3 )
o(2)-o(3)d
o(:)-o(:)d
mean O-O
t1l
121
l2l
ril
2 . 7 2 s( 6 )
2.744(5)
2.s58(5)
2 . 5 9 o( 7 )
2 . 7 1 0( 4 )
2.75314\
2 . 5 5 0( 3 )
2 . 5 9 O( 4 )
2.60(3)
2.80(3)
2.50(3)
2.66{3)
2.74(2)
2.75(L)
2 . 5 1( 3 )
2.53(1)
2.aO(2)
2.73(),1
2 . 5 0( 3 )
2.5s(r)
2- b5b
2 . 7 1 9( 6 \
2.'740(5)
2 . s 3 8( 5 )
2.5A9(7)
2.644
2.733(6\
2.73s(4)
2.s61(s)
2.ses161
2.Ot7
2 . 7 2 0( 6 )
2.742(51
2 . s 3 7( 5 )
2.s47(6)
2.644
2.84lLi
2. 7 I ( r )
2.49(2)
2.59(L)
2.64
2 . 73 6 ( 4 1
2 . 75 2 ( 3 )
2 . s 3 7( 3 1
2.s8r(4)
2.649
2.083(2)
2.o7412]'
2.\45 (3)
2.IOI
2.08s(t)
2.069(r)
2 . 1 2 6( 2 )
2.095
2 . 0 9 r( 3 )
2.080(3)
2 . 1 4 0( 3 )
2.\O4
2 . 1 0 r( 3 )
2.087(3)
2 . 1 s 3( 3 )
2.113
2.rr 7(3)
2 . o e ' 71 3 )
2 . r B 0( 3 )
2.131
2.r7(2)
2.06().)
2.\3 (2)
2-12
2.06(1)
2.04(t)
2.11(1)
2.07
2.08(1)
2.04(),)
2 . o 7( 2 )
2.06
2 . 0 9( 1 )
2.o5(2)
2.O3\2)
2.06
2 . 0 7 9( 2 )
2.068(2)
2.126(2)
2.09L
12)
M( 1 ) O c t a h e d r o n
M(r)-o(r)
M ( 1 )- o ( 2 )
M(1)-o(3)
nean M(1)-O
lz)
l2l
o(1)-o(3)e
o(1)-o(3')
o ( 1 )- o ( 2 )e
Lz) 2 . 4 6 4 l s )
t ) 3. r22(r)
tt) 2 . 8 6 3 ( 6 )
L2J 3 . 0 2 ' 7( r )
12) 3 . 3 4 2( 2 )
P] 2 . 5 5 S ( s )
2.963
2 . a 4 6( 4 )
3 . 1 0 4( 4 )
2-A4eG)
3.o2213)
3 . 3 4 1( 3 )
2.sso(3)
2.952
2.e71(5)
3.roe(5)
2.861(6)
3 . 0 3 6( 3 )
3.373(3)
2 . 5 3 7( 5 )
2.965
2-890(s)
3.123(s)
2.e1o(6)
3 . 0 5 0( 3 )
3.380(2)
2.538(5)
2 . 9 75
2.err(s)
3.160(5)
2.888(6)
3 . 0 6 7( 3 )
3.388(4)
2 - 5 6 1( s )
2.996
3.00(2)
3.r,2(2)
2.9o(2)
3 . O ' 71 6 )
3 . 3 7( 2 1
2 . s 0( 3 )
2.99
2.81(3)
3.09(3)
2 . B O( 2 \
3 . 0 1( r )
3.31(2)
2.s1(3)
2.92
2.7e(r)
3.0714\
2.83(2)
3 . 0 0( 1 )
3.26(2)
2.50(3)
2.9I
2.76(31
2 . 9 s( 3 )
2 . 8 8( 2 )
3 . 0 2( r )
3.24(2)
2.49(2)
2.A9
2.844(3)
3 . 0 9 4( 3 )
2 . 4 3 e( 4 )
3.022(r)
3.339(2)
2 . 5 3 7( 3 )
2-947
IU
2 . 1 6 6( 3 )
2.045(5)
2 . 2 O 8( 4 )
2 , o 6 4l 4 )
2. 126
2 . 1 6 6( 4 )
2.O40(3\
2 . 2 o e( 3 )
2 . 0 6 6( 2 )
2. 126
2.r99(5)
2,O57(5)
2.23I (4)
2 . O 7 3( 3 )
2.r44
2.2r916)
2.066(5)
2 - 2 4 s( 4 \
2 . O 4 2( 4 )
2.156
2 . 2 3 ' 7( 5 )
2.068(5)
2 . 2 5 4( 3 )
2 . 0 8 6( 3 )
2.165
2. 3213)
2.13(3)
2.25(2)
2.oa(2)
2.19
2 . 1 4\ 2 1
2.o4(21
2.re(1)
2 . 0 8(r )
2.L2
2 . 1 2( r )
2 . o O( 2 )
2.22(r)
2.o7lr)
2,Il
2 . 0 6( 3 )
r.92(2)
2 . 2 7\ 2 1
2.06(3)
2.LI
2 . r 7 e( 3 )
2.053(3)
2 . 2 I 4( 2 )
2.072(2)
2.134
tl
t2l
tl
3 . 0 0 8{ 5 )
2- 864 l5\
3.r77(5)
2.926(5)
2 . s 9 O( 7 \
2.s7s(4)
3.3e3(2)
2.990
3.0r6 (4)
2. e46(3)
3.r2O(4)
2 - 9 2 O1 3 \
2 - 5 9 0( 4 )
2.99o(3)
3.386(3)
2. 980
3.0s2(s)
2.871(s)
3 . 2 2 4( 5 1
2.923(5)
2 . 5 4 7( 6 )
3.014(4)
3.4r2 (21
3.014
3.o'74(5)
2.890(5)
3 . 2 4 ' 7\ 5 1
2.933(s)
2 . 5 4 9( 1 )
3.02e(4)
3 . 4 3 2( 2 )
3.03I
3.080(s)
2.srr(s)
3 . 2 5 7( 5 )
2 . 9 4 7( 4 )
2-5s5(6)
3.03r(4)
3.465(2)
3.043
3.01(2)
3.00(2)
3 . 2 6( 2 1
3 . 0 0( 3 )
2.66(31
3.05(2)
3.39(2)
3.07
3.01(3)
2 . 8 1( 3 )
3.15(3)
2.e4(Ll
2.53(r)
3.02(3)
3.3s(1)
2.94
2.e8(3)
2,'19(rJ
3.14(3)
2.e2lL)
2.5s(1)
3.03(2)
3.42(rl
2.97
3.07(4)
2 . 76 l r )
3 . 2 4( r )
2.85(1)
2 . s 9( 1 )
3.07(3)
3.64(1)
3.O2
3.022(r)
2 . a 4 e( 4 )
3 . r 9 ' 7( 3 1
2.e26(3)
2 . s 8 1( 4 )
3.005(2)
3.398(3)
2.994
12)
t1l
tLl
2.083(2)
2.766(3)
1.615(3)
1.947
2.085(1)
2.166(4)
r.616(2)
1.998
2.09r(3)
2.r9915)
1.614(s)
1.999
2.10r(3)
2.219(6)
1.615(5)
2.OO9
2. rr7 (3)
2 . 2 3 ' t( s )
1 . 6 r s( 4 )
2.022
2. r ' t ( 2 )
2 . 3 2( 3 )
r . s 3( 2 )
2.Os
2.06(r)
2 . L 41 2 )
1- 6 3(r )
1.97
2.08(r)
2 . 1 2( 1 )
r.62 lr)
I .9'7
2.09(1)
2.o6(3)
r.64(1)
I.9'7
2.o7912)
2.r7s(3)
r.617(3)
r.989
til
t2l
12)
tLl
2.993(1)
3.r97(2)
3.256(2\
3 . 2 4 9( 3 )
2.988(3)
3 . r 9 2 (3 )
3 . 2 5 0( 3 )
3 . 2 4 7( 3 1
3 . 0 0 0( 3 )
3 . 2 r 7( 3 )
3.264(3)
3 . 2 6 0( 4 )
3.014(3)
3.238(3)
3.2'76(3)
3 . 2 7 O( 4 )
3.030(3)
3 . 2 6 7( 3 1
3 - 2 8 8( 3 )
3 . 2 a 3( 4 )
3.033(3)
3.2812)
3 . 2 e\ 2 \
3.2a12)
2 . 9 1 1l 3 )
3 . 1 7( 1 )
3.244(s)
3.23(2)
2 . 9 7 0l 3 \
3 . 1 7( r )
3 -2 3( r )
3.16(r)
2 . e ' 7( r )
3.18(r)
3.243 (6)
3 . 1 e( r )
2 . e 8 e( 1 )
3.196(r)
3 . 2 5 3( 2 1
3.25r(2)
12)
IU
tll
2.O'?4(2)
2.045(5)
1.640(3)
1.958
2.069(1)
2 . 0 4 0( 3 )
r.649(2)
I.957
2 . 0 8 0( 3 )
2 . O 5 7( 5 )
1.636(4)
1.963
2 . 0 S 7( 3 )
2.06615)
1.636(5)
I.969
2 . 0 9 71 3 )
2.068(5)
r.649\4)
L-974
2.06(r)
2 . 1 3( 3 )
1.60(2)
r.96
2.04 (1")
2. 0 4( 2 )
r.66(2)
r.95
2. 04 (r)
2 . O o( 2 )
L . 6 61 2 )
t.94
2.O5\2)
L.s212)
I-76(2)
1. 9 4
2.064(21
2.053(3)
r . 6 4 s( 3 )
1.959
nI
2.993(L)
3 . 6 3 71 2 \
2 . 6 9 21 2 )
3.283(21
2.988(3)
3 - 6 3 3( 2 )
2 . 6 e 0( 2 )
3.278(3)
3.000(3)
3 . 6 4 s( 3 )
2 . 6 e 9( 2 )
3 . 2 8 e( 3 )
3.014(3)
3 . 6 5 4( 3 )
2 . ' 7 7 0( 2 )
3 . 3 0 2( 4 )
3 . 0 3 0( 3 )
3.665(3)
2.725(2)
3.309(s)
3 . 0 3 3( 3 )
3.68(1)
2.'t3(r\
3.32(2)
2 - 9 7 ' 7( 3 )
3 . 6 1 7( 7 )
2 . 6 8 ' /( 6 )
3.26(1)
2 . 9 ' 7 0( 3 1
3.s9(7)
2 . 6 ' 1 r( 6 )
3.261r)
2 . 9 ?\ L )
3.55(2)
2 . 7 O( r )
3 . 2 1( 1 )
2 . 9 8 9( r )
3.632(21
2 . 6 9 3( t \
3 . 2 8 0( 1 )
t1l
trl
F]
trl
2. r45 (3)
2 . 2 0 5( 4 )
2 . 064 (4)
1.633(2)
2.O72
2 . 1 2 6( 2 )
2.208\3)
2 . 066 12)
r -633(2)
2.008
2 . 1 4 0( 3 )
2.23r\41
2 . O ' 7 (33 )
I.624(3)
2.OI7
2 - 1 s 3( 3 )
2.245(4)
2 . O e 2( 4 )
r.623(3)
2.026
2.180(3)
2.2s4(3)
2.086(3)
r . 6 2 4( 3 )
2.038
2.13(2)
2.24\2\
2.Oa(2)
L . 6 7( 2 )
2.O4
2.11(1)
2 . 1 9( 1 )
2 . 0 8( 1 )
1 . 6 0( 1 )
2.00
2.o112)
2 . 2 2( l )
2 . 0 7( r )
r.60(1)
1.99
2 . 0 3( 2 )
2 . 2 1( 2 )
2 . 0 6(3 )
1.s6(r)
1.98
2 . L 2 6( 2 )
2.2r4(2)
2 . o 7 2( 2 )
r.62512)'
1.987
t1l
3.r9'7(21
3.57s12)
2 . 6 9 21 2 )
3.462e)
2 . 7 8 3( 3 )
3.242(2)
3.1e2(3)
3.579(3)
2 . 6 9 0( 2 )
3.856(3)
2.7Ar(31
3.2'74(3\
3.2I7(3\
3.s92(3)
2.6e9 (2)
3 . 8 7 2( 3 )
2 . 7 9 ' 71 3 \
3.288(3)
3.23e(3)
3.603(2)
2. lro (2)
3 . 8 8 6( 3 )
2 . 8 1 1( 3 )
3.303i4)
3.26'1131
3 . 6 r ' 71 2 ] '
2 . 7 2 5\ 2 )
3 .e l 2 ( 3 )
2.829 l3)
3.311(5)
3.28(1)
3 . 6 1( 1 )
2 . 7 3( r )
3 .e l ( 2 )
2 . A 4( 2 )
3.32(2)
3 .r 7 ( r )
3 . 5 6 5( e )
2 . 6 S ' t( 6 1
3 . 4 4 5( 2 )
2.77lr)
3.25(r)
3 .r 7 ( 1 )
3 . 5 3( 1 )
2 . 6 ' 7 1(,6 \
3 . 4 4 4( 2 1
2 . 7 8( r )
3.2s(1)
3 . 1 8( r )
3.sr (2)
2.70(1)
3.8s(r)
2.73(r)
3.20(r)
3 . 1 9 6( r )
3 . s 8 0( 2 )
2.693(r)
3.8s8(4)
2.7s0(2)
3.263\2)
tl
o(1)-o(2,)
o(2)-o(3')
o(z)-o(:)d
mean O-O
M(2) octahedron
M ( 2 )- o ( r )
N l ( 2 )- o ( 2 )
M ( 2 )- o ( 3 )
M ( 2 )- o ( 3 " )
m e a n M( 2 ) - O
[]
l2l
t4
o(1)-o(3")
o(1)-o(3)e
o(2)-o(3)
o(2)-o(3'")
o(3)-o(3)d
o(3)-o(3,,)
o(3n)_o(3,")
mean O-O
O(t)
Lzl
tll
l2l
tll
Tetrahedron
o(r)-M(l)B
o(1)-M(2)B
o(1)-siA
mean
O(1)-M
M(r)s-M(1)g
M(I)e-M(2)s
M(r)B-SiA
M (2) B-SiA
O(2)
Tetrahedron
o(2)-M(r)B
o(2)-M(2)A
o(2) -siB
mean O(2)-M
M(1)B-M(1) B
M(1)B-M(2)A
M (1 ) B-SiB
M (2)
A-SiB
O ( 3) Tetrahedron
o(3)-s(r)B
o(3)-M(2)B
o(3)-M(2)A
o(3) -siB
L2)
tzl
t1l
mean o(3)-M
M(1)B-M(2)B
M(1)R-M(2)
"
M(1)e-sis
M(2)B-M(2)A
M(2) B-SiB
M(2)A-SiB
nl
t1l
D]
t!
trl
a)
A11 distances
in
c)
Parenthesized
fiq,ures
e)
Edge
shared
A.
between
refer
two
b)
Data
to
the
octahedra,
of
esd
Smyth
of
and Hazen
least
units
(1973).
cited.
d)
Edge
shared
between
an
octahedron
and
tetlahedron.
t287
FORSTERITE
?.16
!
A M(r)_o(c) Ird2Fo(5)
M(2)-O
o
'9
?.t4
lrl
o
z,
o
o
o
-
^
2.t3
?l
A P.rictox (Mg-o)
I No)-o(z)
9:
2.t2
j
2.tl
Mg-o
2.to
DISTAI{CE
(or O K}
Frc. 6. ForsteriteMg-O bond lengththermalexpansionversus
0 Kelvin Mg-O distance.0K Mg-O distanceis extrapolatedfrom
Fig. 5.
z
lrl
r.64
-x------X----'.-si-o
- - x" - ' 1
"'x.--_ ___-x-----"800
rooo
TEMPERATURE (K)
Frc. 4. Forsterite Mg-O and Si-O mean bond distancesversus
temperature.
plottedin Figure8. From Figure8 it is evidentthat
lowerbound,and ridingcorrections
beuncorrected,
havein a similar way, and arenot significantlydifferent from each other. Thus, if metal-oxygenbonds
parallelcorrelated
vibrationmohavepredominantly
M(2)-o(3)
H(2)- o(l)
:g2.
ul
2
() 2 . 1 8
F
o
6
A u ( r ) - or )(
x
2.r6
tion, then the uncorrecteddistanceusuallyreportedis
a valid measureof the physicalseparationof atoms.
However,non-correlatedand anti-correlated(upper
are significantlylargerthan the
bound) separations
givea totally differentpicture
may
and
lower bound,
expansionat elevated
thermal
and
distances
of bond
If a bondhasa largecomponentof antitemperature.
vibrations,the simplecenparallelor noncorrelated
troid separationdistancewill not providea realistic
measureof mean atomic separation.It should be
recognizedthat reported bond distancesand bond
thelowerlimitsof these
represent
thermalexpansions
physicalquantites.
Temperaturefactors
Isotropic temperaturefactors of the six atoms in
forsterite'sasymmetricunit are plotted versustemperaturein Figure9. All six curvesshow the same
generalform of a near-zeroslope at 0 K, a rapidly
increasingslopebetween0 and 500 K, and an approximatelyconstantslopeabove500K. This is the
samebehavioras that recordedfor periclase(Hazen,
and oxymagnesium
1976).Also, as with periclase,
of
equalmagnitudes
gen atomshave approximately
thermalvibrations.However,silicon'sisotropictem-
o 214
=
o
@
o
2.12
q
:E
2
2
--!t-a'-
a'
u
o
z
F
q
o
o
q
tooo
TEMPERATURE ( K )
Frc. 5. Forsterite i n d i v i d u a l M g - O
temperature.
t20(
b o n d d i s t a n c e versus
2o0
TEMPERATURE
Ftc. 7. Forsterite individual
temperature.
Si-O
(K}
bond
distances
versus
t 288
ROBERT M, HAZEN
t.6
2.2
€
"
(l
2.O
o
Ir '..
U
o
z
H r.6
)
r.65
kU ,.o
.L
F
o
o
!
(,
'.o
E
(l
o
F
z
-E
t.2
F
r.64
o
I o..
r.6J
F
fr o.e
s5 0 . 4
J
R
BOUND
trl
o,2
TEMPERATURE (K}
200
Fig. 8. Forsterite Si-O mean distancesversustemperature
correctedfor severalthermal-motionmodels.
peraturefactor is only about half that of the magnesium and oxygenatoms.This behaviorhasbeenobservedin many other silicates(Burnham,1965),and
is a consequence
of the strongbondsbetweenoxygen
and four-valentsilicon.
L. Finger (personalcommunication)has noted a
possiblesourceof error in the magnitudesof temperature factors reportedin this study. No corrections
were made for secondaryextinction; when a gem
quality crystal such as synthetic forsterite is used
400
600
600
TEMPERAfURE
tooo
t200
(K)
FIc. 9. Forsterite isotropic temperature factors versus
temDerature.
severalstrongreflectionswith lF"u"l ((lF"nr"l are rejectedfrom the data set.Consequently,
the scalefactor will be too high,and the temperature
factorswill
be too low. This effectexplainsthe nonpositive-definite thermal factors of oxygen and silicon in the
liquid-nitrogentemperaturecase(seeTable 3). Thus,
while the qualitativeaspectsof Figure 9 are correct,
the absolutevaluesreportedmay be low.
Tnrlr 6. Forsteritebond ansles
3'c
o ( r ) - s i - o (2 )
o ( 1 )- s i - o ( 3 )
o(2)-si-o(3)
o ( 3 )- s i - o { 3 ' )
mean
frl
l2l
[2]
tll
O-Si-O
o ( r ) - M ( r ) - o ( 3 ) 12)
o ( r ) - M ( r ) - o ( 3 ' ) t2J
o t r ) - M ( 1 ) - o ( 2 ) t2l
o ( 1 ) - M ( r ) - o ( 2 ' ) L2l
o ( 2 ) - M ( 1 ) - o ( 3 ' ) Lrl
o ( 2 ) - M ( 1 ) - o ( 3 ) hl
mean 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 . )
mean O-M(2) -O
al
"Ideal"
b)
Parenthesized
c)
Data of
tl
121
irj
t2j
trl
tl
trl
hexagonal
.1000"
20 kb
rI4.2 (21
1rs.9(1)
ro2.0(t)
r 0 s . 0( 2 )
I09.I
90
90
90
90
90
90
90
85.I (2)
94.9 (2\
86.8(1)
93.2(1)
r0s.4(2)
74.6l2')
90
90
90
90
90
90
90
90
90
90.3(1)
81.7(1)
96.712)
90.8(1)
'tr
.9 (2)
88.3(r)
rro.7 (2)
a 9. 9
8 5 . 0( r )
8 s -4 ( l )
8 s -6 ( 2 )
8s.3 (1)
87 16)
84. s (6)
9 s . 0 ( 1 ) s 4 -6 ( 1 )
s 4 - 4( 2 )
s4-'7 (t-)
9 r( r ) 9 s - 5 ( 6 )
8 6 -6 ( 1 )
8 6 -6 ( r )
8 6 .s ( 1 )
86.6(1)
B B( 1 )
8s- I (4)
e 3 -4 ( 1 )
9 3 . 4( 1 )
93.s(r) 93.4(r)
e3(r)
94-2 (4)
1 0 s . 2( r ) r 0 5 . r ( 1 ) 1 0 6 .s ( 2 ) 1 0 6 . s ( 1 ) r 0 7 ( I ) r o s . 7 ( 7 1
7 4. 4 ( \ )
73.9(1) 13.s(21 73.5(1)
7 3 ( 1 ) ' 7 4 . 3l 7 )
90
90
90
90
90
90
9 0 . 9 ( 1 ) 9 1 . 1( 1 )
9r-2(r)
90.8(1)
e 0( r )
91.2(5)
81.r(t)
e0 8(r)
8 0 .7 ( 1 )
80. 7 (1)
8r (1)
80.8(7)
9 6 . 9( 1 )
9 1 . 4( 2 )
e 7. ' ] ( 2 ' )
9 7. 6 ( l J
9 9( r )
9 6. 2 ( 6 )
9 0 . 6 ( 1 ) 9 0 - r ( 1 ) e 0 .o ( 1 )
90.4(1)
9 0( r )
9r.0(7)
7 1 . 8 ( r ) 7 O . 9( 2 )
1 O . 41 2 , 7 0 . 1 ( 2 )
72lrl
70.6 16)
8 8 . 7( 1 )
8 8 . 8( 2 )
8 8 . 8( 1 )
88.4(1)
8 9( 1 )
89. r (4)
r 1 0 . 0( r ) r 1 0 . 8( r ) r 1 1 -3 ( 2 ) r r 2 . 3 1 2 ) r 0 9( r ) r 1 0 . 6 ( 6 )
89.9
4 9 .I
89.9
89.9
90
89.8
close-packed
figures
- 350"
1r4.1(3)
1rs.6(2)
r o 2 . 7( 2 J
1o4.6(3)
tog.2
109, 5
109.5
109.5
r09, s
r09.5
refer
to
Smyth and Hazen (1973).
olivine
the
model
esd of
r L 3 . ' 7( 2 J
rrs.7(1)
r o 2 . 21 2 )
1 0 s . 6( 2 )
109.2
(see Fig.
Ieast
units
1 1 3 . 6( 2 )
r15.6(r)
r o 2 -3 ( 2 \
r 0 s . 9( 2 )
1o9.2
1)
cited.
II3.7 (2)
1 1 4 (1 )
rr5.2(r) rr7(I)
1 0 2 . (8r ) 9 s( r )
to5.7 (2)
r06(r)
IO9-2
109
rls.I(r.0)
116.5(s)
r00.9(7)
104.6(9)
109 1
50 kb
rr5.7 (6)
rr7.4(r.r)
r00. o (7)
r05.7 (6)
109.4
1 1 s( 1 )
r16 (1)
e7(r)
1 1 0( r )
109
1 1 4 .r ( 2 )
r 1 6 .r ( l )
1 0 r . 7( I )
1 0 s .r ( I )
1 0 9 .r
84 5 (6)
9s.5(6)
46. ? (41
93.3(4)
lo5 .2 (7)
14-a('7],
90
83.8(8)
96.2 (A)
46.7 l4l
93.3(4)
LO4.9l7l
1s.r(7)
90
8s.3(r)
94.1 lr)
8 6 . 4( 1 )
e 3 .s ( 1 )
1 0 5 . 6( l )
74.4(tl
90
e0. 9 (6)
80.2(7)
96.2 (6)
9r-6(7)
70.1(6)
89.6 (4)
ro9. 9 (6)
89. 8
92 lLl
79(1)
s7(1)
9L 5(1.O)
69.4(71
90.0 (7)
r09. 7 (5)
90
9r-2(r)
80-8(t)
9 7 .O( I )
e0.3(1)
7 1 . 3( r )
89-0(1)
r r 0 .r ( 1 )
89.9
t289
FORSTERITE
Becauseeachof olivine'ssix atomshasan irregular the miniature pressurecell. The cell was tightened,
coordinationpolyhedron,anisotropicvibrationsare and unit-cell parameterswere determinedon the
expected,and havebeenmodeledastriaxial ellipsoids four-circlediffractometeras a : 4.743t 0.005,b =
in the refinementprocedures.Anisotropic temper- 10.092+ 0.010,c :5.954 + 0.006A, and volume :
a volumedecrease
ature factors are presentedin Table 3, while corre- 285.0+ 0.5 At, which represents
calibrationwas basedon a
spondingmagnitudes
and orientationsof thermalel- of 1.65percent.Pressure
lipsoidsappearin Table7. Thesevaluesagreewith volumecompressibility
of -7.6 x lO-alkbarfor olidatareportedby Brown(1970),andwhilethe magni- vine as measuredby Olingerand Halleck(1975).It
data collectudesof thermalellipsoidsincrease
with temperature, thus appearsthat the first high-pressure
20
kbar.
All availthereis no significantchangein ellipsoidorientation tion was made at approximately
measured
were
but
of
969
with changesin temperature.
collected,
able diffractions
intensitiesonly 329wereobserved(/ < 2o). After ten
Dats collection:I atm to 50 kbar
cyclesof refinementwith isotropic temperaturefacwas achievedwith a weightedR of
The forsterite crystal used in the liquid-nitro- tors, convergence
gen-temperaturestudies was remounted with the 7.0 percent (9.4 percent unweighted).Refinement
good (010)cleavage
and isotropictemperparallelto the diamondfacesof conditions,atom parameters,
TlsI-r 7
Magnitudes and orientations of forsterite thermal ellipsoids'
M(1)
M(2\
b
- 196
I
2
3
0 . 0 1( 2 )
0.0re(13)
0.054(s)
23
1
2
3
0 . 0 3 7( 4 )
0 . 0 s 3( 3 )
o.o'75(2\
'7 ().Ol
4
r7(10)
e5(s)
'3 50
I
2
3
0 . 0 6 8( 6 )
o . 0 9 4( 5 )
0 . 1 1 0( 4 )
-675
1
2
3
0. r0r (5)
0 . 1 1 s( 5 )
o. t 49(4)
I
2
3
L
2
3
- lo00
hish
press ure)
c
r 2 ( 6 r ) ro2 (27)
e 5 { r 3 4 ) r00 (30)
r 0 1( 9 )
r64(9)
a
93 (r34)
1 6 8( 3 9 )
79(10)
0 . 0 1( 2 )
0.027(10)
0 . 0 4 s( 6 )
I (14)
90
er (14)
ee (14)
90
r (14)
76(4\
e e( 6 )
2r (7)
1 0 4( 1 0 )
7s(3)
0.0sl(3)
0 . 0 5 4( 3 )
0.0s5(3)
9 (4r)
98 (47)
90
r72(47\
90
90
90
0
80(r0)
10(10)
e1(13)
7 3( 5 )
s3(13)
r63(6)
L e( 7 )
e9 (r0)
73(6\
0-086(5)
0. 090(5)
o . o97 (4)
94 (23)
90
4 (23\
4 (23\
90
8 6( 2 3 )
90
180
90
5 6( 1 3 )
34(13)
sr(s)
? 3( 5 )
ro2 (61
L59(4\
3s(11)
I2r (r2)
6e 14)
0.117(s)
o. r22(51
o . L29(4)
0. rr7 (4)
o. I40 (4)
0.18s(3)
s8(7)
3 2( ? )
8e(3)
15(2)
ss (4)
r62 \21
3 7( 6 )
12r(6)
7 2\ 3 1
0.140(4)
o . r49 14\
0. rs1(4)
o . o 6 6( 4 )
o.o7r (4)
0. 084
58(32)
32(32)
96 l12)
12(r3)
r07(16)
r5s(10)
37\23)
1 1 6( 2 7 \
66 (e)
0 . 0 6 9( 4 )
0-072(3)
0.078(4)
ds(+r)
90
180
90
-90
0.022(7)
3(s)
0.030(3) 90
0.032(3) 87(s)
e3(s)
90
3(s)
o.0s4(6)
5(13)
0. 067(5) 90
0.073(4) Bs(13)
e s ( 1 3 ) 90
90
180
5 ( 1 3 ) 90
180
90
r40 (43)
90
1 3 0 ( 4 3 ) 90
90
la0
0.06s(s)
0. r00(4)
0-104(3)
7(6)
97(61
90
9'/(6)
1 7 3( 6 )
90
90
90
0
90
1 4 3( 1 6 )
L26\t-6)
90
90
90
0
0.088(4)
0.rr9(3)
o. r22(3\
8(4)
90
A 2( 4 )
e8(4)
90
I (4)
90
180
90
36(18)
90
54(18)
r26(r3)
90
36(1B)
90
180
90
o . 0 4 6( 4 \
0.062(3)
0.065 (3)
2(9)
90
88(e)
s2(e\
90
2 tel
90
r80
90
I 40 141)
90
s3(16)
c
-350
- 675
1000
high
pres s ure,l
a)
A11 data
b)
Parenthesized
c)
Atom
is
at
t
o . 0 3 1( 9 )
o.064(4\
0 . 0 6 e( 4 )
1 0( B )
90
8r (8)
0.07e(1r)
0. 097(9)
0 . 1 0 5( 9 )
80(19) r70{19)
s0
90
r0 (le)
B 0( r 9 )
0 . 1 0 1( 1 0 )
0.r27(8)
25(15) r15(rs)
90
90
1 0 0( 8 )
90
r0 (8)
90
180
90
0.048(6)
0.o55(6)
o . 0 6 2( 5 \
90
180
90
0.084(10)
0.086(10)
0 . 0 9 2( r 0 )
73(62)
90
I1 (62\
90
180
90
25(15)
o. r07(9)
0. rl3 (e)
0 . 1 2 0( e )
97(9)
90
' 7( e )
90
180
90
o. r28(8)
64(15)
0.116(7)
o.r42 \7)
0.1s4(7)
90
83(e)
0.063(e)
o.071(7)
o. 08s(8)
1 6 ( 1 6 ) 1 0 6( 1 6 )
90
90
'7 (16)
4
16(t6)
90
180
90
the
least
'1(9\
57(3'))
r47(3't)
90
90
e0
0
0 . 0 4 4t 4 )
0.0s5(4)
0.074(3)
r ' 7( 6 2 )
90
1 0 7( 6 7 )
90
180
90
7177)
96(84)
90
A 3( 1 7 )
6 (a4l
90
o-r25 (7)
0 . 1 3 3( 7 )
0.14r (7)
L(46)
sL(46)
90
9r (46)
r1e146l
90
o. 062(8)
0 . 0 6 8( 7 )
0.073 (7)
51127)
90
39(21)
90
3e(21],
180
90
r29(2'7',) 90
atm.
figures
non-positive
refer
definite.
to
esd
of
units
cited.
r47 137)
r2313-7)
90
4e(r6) ros(7)
' 7 5( 9 )
48(13)
42\16)
88(7)
25(71
1 2 8( 1 4 )
65(6)
0 . 0 7 3( 8 )
o . o 9 4( 7 )
0 - 1 o 3( 6 )
e3(1s)
r39(31)
13r(3r)
1 2 3( 1 3 )
I22\26)
49(23\
3 4( 1 3 )
lr3(20)
67(r7)
90
90
0
0. r07(7)
o. r13(6)
0 . 1 4 0( 6 )
es (48)
165 129)
102(ro)
1 3 2( 8 )
94 \34)
42la)
43 (19)
lo 5 (37)
5l(8)
90
90
0
8 6( 1 9 )
0 . 1 2 s( 5 )
0.137(5) r74(r3)
9s(6)
0.171(5)
1 2 1( 5 )
e6(r2)
32(5)
0.067(5)
0.068(5)
0. 08r (5)
3(rs6)
93(158)
eo(16)
8 8( 8 s )
5 9( r 7 )
3r(16)
3r (4)
eO(16)
59(4)
9 3( 1 3 6 )
1 4 9( 2 r )
59(16)
t290
ROBERT M, HAZEN
ature factors are given in Tables lb and 2 respec- tempted due to the beginning of fracturing in the
tively.
forsteritecrystal.This crystalwas removedfrom the
The pressurecell was tightenedagainfor a second pressure
cell and was remountedon a standardgodatacollectionat higherpressure,
but thecrystalwas niometerfor a final room-pressureand temperature
crushedand no furtherexperiments
werepossibleon data collection.Unit-cell parametersof this crystal
the specimen.Another smallercrystalplate approxi- were remeasured,
and found to agreewithin experimately 160X 140 X 80 pm with (010)cleavage
was mental error with earlier determinations(seeTable
mountedin the diamondcell,whichwastightenedas l). 1026diffractionintensitiesweremeasuredand 790
before.Cell dimensionsof a : 4.734+ 0.005., :
reof thesewere observedand usedin least-squares
10.02+ 0.01,c : 5.940+ 0.006A, and volume :
finementwith anisotropictemperaturefactors.After
281.8+ 0.5 A', indicateda pressure
of approximately l2 cyclesof refinement,
an unweighted
R of 3.9 per40 kbar. Only 374 of 964 measureddiffractionswere cent was achieved,and all atomic coordinatesmatch
observed,
and thesedatawererefinedto an R of 7.3 thoseof the original room-temperature
and pressure
percent(9.6 percentunweighted)after l0 cycles.The refinementwithin twicethe estimatedstandarddeviapressurecell was tightenedfor a third high-pressure tion, as recordedin Table2. All observedand calcudata collection;this tighteningwas stoppedwhen a lated structure factors for forsterite are available
small fracture was observedin the crystal. Sub- from the author on request.
sequent precession and cone-axis photographs
Bond distancesand anglesfrom refinementsusing
showed that diffraction intensitiesremained fairly the data collectedat high pressuresare listedin Tasharp (though some spreadingof diffraction peaks bles 5 and 6. The estimatedstandarddeviationsof
wasnoted),and no changeoccurredin spacegroup thesedistancesare large,and small changesin bond
or lattice translations.Unit-cell dimensionsof the distanceswith pressureare thus difficult to detect.
third high-pressure
datacollectionwerea = 4.712t
However,mean M-O bond lengthsfor both M(l)
:
0.005,D 9.967+ 0.010,c : 5.955+ 0.006A, and andM(2) showcompressionexceeding1.0percentat
volume279.7+ 0.5 As, which represents
a 3.50per- pressures
to 50 kbar, as illustratedin Figure 10.On
cent volume decreasefrom I atm. At hydrostatic the other hand. the silicon tetrahedronshows no
pressure
sucha volumedecrease
would imply a pres- bond compressionwith respectto experimentalerror
sureof approximately50 kbar. However,closerex- over the samepressurerange.Thus,in olivine,magaminationof cell-edgeparametersrevealsthat whilea nesium-oxygenbonds appear more compressible
and b have decreased
in length,the c cell dimension than silicon-oxygenbonds.
increasedbetweenthe secondand third high-pressure Isotropictemperaturefactorsshowlittle systematic
data collection.Thus.thereis seriousdoubt that the changewith increasingpressure.A slight increasein
presumed50 kbar data collectionwas made under apparentthermal vibrationswith increasingpressure
hydrostaticpressure.
It is probablethat at this high- is observedfor M(2), Si, O(2), and O(3), but these
estpressurethe forsteritecrystalwasin directcontact changesare small,and perhapsinsignificant,
considwith the diamonds,and this fact may account for ering their estimatedstandarddeviations.Significant
both the small fractureand the anomalouscell pa- changesin refinedthermal vibration parametersare
rametersof this specimen.
seenin comparisonsof room-temperatureand presData collectionat =50 kbar (nonhydrostatic)
pres- sureforsteriterefinementsbeforeand after high-pressure included 1429 measuredreflectionsof which sure data collection; averageapparent root-meanonly 332wereobserved.
The fact that fewerthan 25 squaredisplacementsare twice as large after highpercentof measureddiffractionswere observedmay pressuredatacollection.Onepossibleexplanationfor
be ascribedin part to absorptionof X-rays by the thischangemight be an increase
in mosaicspreading
diamondcell. However,a more basiccauseis that of the forsteritecrystal, which was signaledby the
peak-to-noiSe
ratio was low for all diffraction max- increasein diffraction diffuseness
at high pressure.
ima: peaks were broad, and scatteredX-radiation
from the pressure
cell was high. Thus, while a weight- Coordinationpolyhedraluolumesand bulk moduli
ed R of 6.8 percentwasobtainedfor refinementwith
Polyhedralvolumeshavebeencalculatedfor M(l)
isotrofic temperaturefactors,the unweightedR was and M(2) octahedraand the silicontetrahedron,and
12.3percent,indicatinga seriouslack of precisionfor thesevolumesare given in Table 8. Distorted polythe weak observeddata.
hedra are constrainedto be of smallervolumethan
Further high-pressure
data collectionwas not at- their ideal counterpartswith the same mean bond
t29l
FORSTERITE
TlsI-E 8. Forsteriteoolvhedralvolumesand distortions
P=latm
Parameter
Site
octahedral
IVI( LJ
vo Lume
3 u"n-o
Ideal
Vol ume
t
u'rn_o
Bond Angle
St rai n
Angle
Variance
^,,.4Y>+i
Fl ^hdr+
MI'\
oo'
^
i ^h
n.f:hadr^l
: drs_o
Vo I ume
Ideal
Angle
Strain
Angle
oo'
Varlance
ci
12'2r
12.29
12.54
12.7)
L2.t
1 1. 8 1
lr,. 61
I 1. 50
12. 19
:.2.26
12.36
12.42
12.58
12.90
12.7
11.83
11.65
11.6s
t2.19
30.4
3 0 .8
32.2
33.0
33.0
34
31.4
30.4
29.4
3r.2
97.3
ga.j
106.2
110.5
109
107 0
98.9
94.7
101.0
r.065
1.065
12.7-t L2.7A
l u t n _ o 1 2 . 8 8 1 2 .s B
VoI ume
Bond
r .oo
4a
12'20
nrrrdril
i ^
Fl ^hdr+
i ^^
r.069
rtl.2
1.075
1.O81
r.090
t.os
I.O'7
1.11
r,.068
13.08
13.25
13.4s
13,65
12.59
12.4t
\2.22
13.14
13,36
13.53
14.O0
1 _ .27 |
L 2- 5 3
12. 53
1 2. 9 6
37.0
40.00
39.8
40.3
38.8
92
95.8
97.t
105.9
93.3
38.2
38.8
39.9
40.9
42.2
90.o
90. I
98.3
1 0 3 .2
108.0
r.00
1.019
1.028
1.023
1.025
1.063
I.O4
r.026
1.015
1.021
I. O22
Tetrahedral
Volume
16 ^3
916* Si-O
.
a \1
2"17
2.I7
2.2r
2.r7
2.t9
2.t7
2.20
2.lA
Idear
vorume
lg,3
,1l5"st-o
2'z+
z'zj
2.20
2.20
2.22
2.lA
2.ra
2.LA
2.22
2.2I
Bond.Angle
Straln
Angl e
Variance
oo,
ouadratic
Elonqation
r.oo
hexagonal
)1
r3.9
rz.9
13.5
13.3
r2.4
18
15.6
17.4
19
14.4
48
42
43
42
ld
/ 5
60
72
57
50
r.oo7
a)
"Tdeal"
b)
Data from Smvth and Hazen (1973) .
close-packed
r.oo9
olivine
model
1.007
(see Fig,
distance;both M(l) and Si are only about I percent
smaller,whereasM(2) is 2 percentsmaller.While the
two octahedral sites show considerable expansion
and compression,the silicon tetrahedron shows no
change in volume with changing 7 and P over the
ranges studied.
Changesin polyhedralvolume may be directly compared to unit-cell volume changes.From -196o to
1000'C the net changeof volume in M(l) and M(2) is
about 0.5 A' per octahedron, while the change in
tetrahedral volume is nil. As illustrated in Figure l,
each olivine unit cell has four M(l) and four M(2)
octahedra. Thus, cation polyhedral expansion accounts for 4.0 At volume decreaseper unit cell. However,the total unit-cell volume changefrom - 196. to
1000"C is about l3 A3, leaving 9 A, unaccountedfor
by cation polyhedral expansion.Thus, approximately
70 percent of olivine's thermal expansion may be
attributed to volume increasesof the unoccupiedoc-
1.007
t.Ot2
1.06
1.028
1.O25
1.09
1.010
1 ),
tahedraland tetrahedralsites(i.e. "voids") in the
oxygenarray. Magnesiumoctahedron
close-packed
expansionplaysa secondaryrole, while tetrahedron
expansionhas no effecton changesin olivine'svolto 1000'C.
ume with temperature
Similar calculationsmay be performedfor forsterof
At 50 kbar, a volume decrease
ite compression.
approximately
0.7As in M(l) and 0.5A'in lf12; has
beenrecorded,thus contributing4.8 A3 of compresof magnesion per unit cell.The total compression
sium olivineat 50 kbar is about l0 A' per unit cell.
Thus, 50 percentof forsterite'scompressionmay be
ascribedto octahedralsitesand 50 percentto voids,
whereassilicon tetrahedraagainhave little effecton
volumechange.
the
Huggins(1974,p. 339 et se4.)has discussed
significance
of individualpolyhedralbulk moduli to
of crystallinesolids.He states
the pressureresponse
that systematicstudiesof the compressibilityof poly-
t292
ROBERT M. HAZEN
2.t3
2t2
€ z.'r
M(2)- O
E
o
z
F
o
o
o
I
o
z
.to
2
z
u
E
Frc. 10. Forsterite,n"J"::,t"[:r"r'lo]"0
pressure.
distanceversus
hedra may "enable detailed predictiorrs of crystal
structures to be made at any pressure and may provide valuable insight into crystallochemical criteria
for the stability of structures and into the reasonsfor
phase changes." Bulk moduli of M(l), M(2), and Si
are easily calculated from data in Table 8 of polyhedral volumes versuspressureby applying the equation: Kt : -l/P(V) = -r dP/3 dr, where K2 is the
bulk modulus, l3(V) is the volume compressibility,P
is the pressure,and r is the interatomic distance(see
e.g. Broecker and Oversby, l97l). The estimated
M(l) and M(2) bulk moduli are thus approximately
1200 and 1000 kbar respectively at pressuresbetween
20 and 50 kbar. These values for Mg octahedral bulk
modulus are similar to the 1600kbar bulk modulus of
periclase.
The silicate tetrahedron of forsterite shows no volume compressionwithin gxperimentalerror (+ 0.02
A'). Therefore bulk modulus must be significantly
greater than 2.2 x 50/0.02 : 5500 kbar. This is in
agreement with Huggins (1974) estimate of K"t ",
13,900kbar in andradite.
Polyhedral dist ortions
previous studies were designedto define the shape of
olivine polyhedra (Dollase, 1974),or to plot changes
in shape with changing composition (Brown, 1970;
Robinson et al., l97 l), these methods are equally
applicable to a study of polyhedral changeswith temperature and pressure.Three commonly cited distortion indices have been calculated and are tabulated in Table 8. Theseare: (1) bond angle strain, (2)
polyhedral angle variance, and (3) mean polyhedral
elongation.Each is a measureof the deviation from a
regular polyhedral form, and each has been applied
with good results to the olivine group.
Severalgeneralizationsmay be made regardingthe
effectsof temperature and pressureon forsterite polyhedral distortions, as measured by the three distortion indices. The two magnesian olivine octahedral sites display very similar behavior. Both exhibit
strong increasing distortions with increasing temperature, but little, if any, change with increasingpressure. This pattern may be explained in terms of the
trends of expansion and compression of individual
Mg-O bonds, as shown in Figure 6. Since longer
bonds will expand up to four times more rapidly than
shorter bonds, an increasein temperaturewill result
in increasedvariance of bond distances,and corresponding increasesin distortion. Conversely,an increasein pressurewill force a decreasein the rangeof
bond distances,and perhaps a reduction in octahedral distortions. The fact that a definite trend towards
reduction in octahedraldistortions was not observed
at high pressures may be due in part to nonhydrostaticconditions at high pressure.Further highpressure structure refinements are needed to clarify
octahedraldistortion behavior.
Tetrahedral distortions are small compared to
those of the two octahedral sites. Virtually no
changes in distortions occur between -196' and
1000"C,though some fluctuations(within experimental error) are observed.However, a general increase
oftetrahedral distortions is seenwith increasingpressure. Whether this increaseis due to nonhydrostatic
effects, or to real distortion, is not clear. However, it
should again be emphasized that errors in forsterite
bond anglesand bond distancesare large, and that all
distortion calculations are based on dffirences between these numbers of similar size. Therefore, additional data are required to resolvethe nature of silicon tetrahedraldistortions with pressure.
Characterizationof polyhedral distortions is essenConclusions
tial to an understandingof olivine crystal chemistry,
and consequently,many authors have attempted to
Development within the past decadeof high-temquantify these deviations from ideality. While most perature, low-temperature, and high-pressurediffrac-
FORSTERITE
t293
tion apparatusnow permitsthestudyof mineralcrys- FTNGER,
L. W. (1969) Determinationof cation distribution by
X-ray data. Carnegie
refinementof single-crystal
least-squares
t a l s t r u c t u r e so v e r a r a n g e o f p r e s s u r ea n d
61, 216-217.
Book,
Wash.
Year
Inst.
temperature.
This investigation
concentrated
on the
der KristallstrukHnNrr, K eNo J. ZsMeNN(1963)Verfeinerung
complex pressureand temperatureresponseof the
enschaften 50' 9 | -92'
tur von Olivin. N aturwiss
magnesianolivine forsterite.Significantdifferences HrznN, R. M. (1976)Effectsof temperatureand pressureon the
wereobservedin the thermalexpansionand compres- cell dimensionand X-ray temperaturefactorsof periclase'Am.
sibilityof magnesium
octahedraversussilicontetra- Mineral. 61,266-271.
ANDC. W. BunNseu( 1974)The crystalstructuresof gillesin chargeand coordihedra,reflectingthe differences
pite I and II: a structuredeterminationat high pressure.lz.
nation of thesetwo cations.Furthermore,within
M ineral.59, 1166-l176.
eachpolyhedrona rangeof individualbond expan- -,
(1975)The crystalstructureof gillespiteII at 26
ANDkilobars:Correctionand addenda.Am. Mineral.60' 937-938.
sionswas noted,with longerbondsexpandingmore
A. (tSSZ)The effectofcations on the opticalproperrapidlythan shorterbonds.Thesedata will be com- HErNRreuEs,
cell
dimensionsof knebeliteand olivine'Arkiu Mineral.
ties
and
bined in a subsequent
studywith data on iron-bearGeol.2, 305-3I 3.
ing olivinesin an effort to predict the structureand HucctNs, F. (1974)M'Lssbauerstudiesof iron mineralsunderpresstability, as a function of composition,temperature, sures of up to 200 kilobars. Ph. D. Thesis, MassachusettsInand pressurefor the ferromagnesian
olivines.
stituteof Technology,Cambridge,Massachusetts.
of
(1934)Thermalexpansion
Kozu, S.,J. Uenl .cNoS. TsunuNlr
olivine. Imp. Acad Japan Proc. 10,83-86.
of
The author is pleasedto acknowledgethe aid and adviceof LoursNlts,qx,S. J. ,cNoJ. V. SuIrs (1968)Cell dimensions
olivine. Mineral Mag 36' 1123-1134.
ProfessorCharlesW. Burnham.Thanksare also dueto Professor
of olivineto 100
R. G. Burns,Dr. T. L. Grove,Professor
J. F. Hays,and Professor Olrucnn, B. lNn A. Dune (1971)Compression
kilobars.J. Geophys.Res.76, 2610-2616.
T. Shanklandfor their discussions
of the ideasin this study,and to
of the relative
ANDP. M. Hlr-lrcr (1975)Redetermination
Dr. L. Fingerand ProfessorC. T. Prewitt for their thoroughand Res 80, (in
of the cell edgesof olivine.J. Geophys'
compression
thoughtfulreviewsof the manuscript.
press)
This researchwas supportedby National ScienceFoundation
Rrcsv. G. R.. G. H. B. LovELLeNo A. T. GnrlN (1946)The
GrantsGA-12852and GA-41415.
thermalexpansionand otherpropertiesof somemagreversible
ferrous
silicates.Trans.Brit. Ceram.Soc.45,237-250
nesian
References
K., G. V. GIgss rNo P. H. Rtssr (1971)Quadratic
RoBrNsoN,
Bsr-ov,N. V., E. N. Brlov, N. N. ANonrlrovr lNo R. F. Slrnelongation:a quantitativemeasureof distortionin coordination
in the olivine
Novn (1951)Determinationof the parameters
polyhedra.Science,172' 567-570.
(forsterite)structurewith the harmonic 3-D synthesis.C. R.
Scuocr. R. N.. B. OI-lNcsnnNpA. Dusr (1972)Additionaldata
Acad Sci. USSR,8l, 399-402.
of olivine to 140kilobars.J. Geophys'Res.
on the compression
Brnls, J. D., G. V. Gress,P. B. Moonr ANDJ. V. SMIrH(1968)
71, 382-384.
Crystafstructuresof naturalolivines.Am. Mineral.53,807-824.
SxtNNrn,B. J. (1962)Thermalexpansionof ten minerals.U. ,S.
Bnrcc, W. L. eNoG. B. BnowN(1926)Die StrukturdesOlivines. Geol Suru.Prof. Pap.450D' 109-112.
Z. Kristallogr 63, 538.
structureup to 850'C. ,42.
J. R. (1973)An orthopyroxene
Srravrs,
to
of 177substances
BnrpcveN,P. W. (1948)Roughcompression
Mineral. 58, 636-648.
40,000kglcm'z.Proc.Am. Acad Arts Sci.76,7l-87.
-(1975)
High temperaturecrystalchemistryof fayalite.Am.
Bnoecrrn, W. S.ruo V. M. Ovrnsav(1971)Chemicalequilibriain
.
M ineral. 60, 1092-1097
theEarth.McGraw-Hill,New York, 318p.
-AND
R. M. HnzPN(1973)The crystalstructuresof forsterite
BRowN,G. E. (1970)Crystal Chemistryof the Oliuines.Ph. D.
up to 900o'Am. Minand hortonoliteat severaltemperatures
Thesis,Virginia PolytechnicInstitute,Blacksburg,Virginia.
eral 58,588-593.
ANDC. T. Pnswtrr (1973)High-temperature
crystalchemelasticity
Socn, N. rNo O. L. ANoensoN(1967)High-temperature
istry of hortonolite.Am. Mineral 58, 577-587.
and expansivityof forsteriteand steatite.J. Am. Ceram'Soc 50,
parameters
of silicatecrystal
Bunnuln, C. W. (1965)Temperature
239-242.
structures(abstr) Am Mineral. 50,282.
(1957)OlivineX-ray determinaYoDER,H S. eNoT G. Sen.lnre
(1964)
The effectof thermal
Busruc. W. R. nNo H. A. Levv
tive curve.Am. Mineral. 42,475-491.
motionon theestimationof bond lengthsfrom diffractionmeas11, 142-146.
urements.A cta Crystall.ogr.
December15, 1975;accepted
Manuscriplreceiued,
Doluse, w. A. (1974)A methodof determiningthe distortionof
coordinationpolyhedra.Acta Crystallogr.A30, 513-517.
for pablication,MaY 28' 1976.
Acknowledgments