Ameican MinnalooistVol. 57, pp. 709-737 e97Z)

COMPUTER-SIMULATED CRYSTAL STRUCTURESOF OBSERVED AND HYPOTTIETTCAL

Mg,SiOo POLYMORPHS oF Low AND HIGH DENSITYWm,Npn H. Beun, Dep_artment of Geolog'ical Scicnces, Uniuersi,ty of

Illinois at Chicago, Chicago, Ittinois 606g0, USA

Assrn.{cr

Iwrnouucrrou

rn the past ten years a large amount of highly precise exp€rimentaldata on the crystal structures of inorganic compounds has beengathered. various attempts have been made to systematize this knowl-edge according to different viewpoints. But, short of doing an experi-

709

7IO WERNER H. BAAR

mental investigation, we still are not able to give a definite concrete

answer to the question: what is the crystal structure of a cerbain

chemical compound at, given P, ? conditions? Using the available

empirical knowledge we "can

supply partial answers and predict with

reasonable sucoess interatomic dist'ances and coordination numbers

of anions around cations. To take as a specific example the compound

Mg2SiOa, we can say that at atmospheric pressure and room tempera-

ture the Mg atoms should be surrounded octahedrally by oxygen

atoms (although Mg is known to occur in 4-,-5-,1"d.S:coordination

against o*yg.t ) *hil" th" Si-atoms should be tetrahedrally coordi-

,tut"d (ho*.oer six coordinated silicon is also known) ' Assuming

octahedrally coordinated Mg and tetrahedrally coordinated Si we

can predici that the interatimic distances Mg-O and Si-O should

be about 2.1 and,1.6 A, but we have no way of predicting in which

way the Mg and Si-coordination polyhedra should be connected to

each other. out of the multitude of possible polyhedral linkages, three,

the olivine type, the B-MgSiO+-type and the spinel phase are known

tobes tab lea tvar iousp, " . .o " " . .The i rc rys ta ls t ruc tu reshavebeendetermined. But we couid not predict from the structure types

what

the details of the structures would be' Our knowledge of bonding

mechanisms is so incomplete that we cannot predict a pri'ori which

distortions of interatomic distances and of bond angles should occur in

a given crystal structure and more importantly which effect these

dislortions would have on the stability of a structure. The particular

problems of the polymorphism of MgSiOa and/or of the polyhedral

distortions in olivine have been discussed by Hanke (1965) ' Kamb

(1968), Birle et at' (196$, and Moore and Smith (1970)' These

authors used Pauling's mles for ionic compounds and mostly qualita-

tive geometrical argriments in their interpretations of the observations'

In ttis paper I am trying to put the geometrical arguments on a

^o"" quinti'tatiue basis than has been done before'

OssERvPo PolYn'tonPHs or M&SiO+

Three polymorphs of MgrSiOn are known presently (T'b]: 1)' OnIy

one of thes", for.l"rite or !-ry1grSiO., occurs as a mineral. The crystal

structure of forsterite can conveniently be described as consisting of

un upp"o*i*ately hexagonal close packed (h'c'p') arrangement of

o*vg., 'atomsinwhichonehalfof theoctahedralcoordinat ionsi tesis o-ccupied by Mg-atoms and one eighth of the tetrahedral sites by

Si-atoms. This hexagonal close packing of the oxygen atoms was reoog-

nized by Bragg arrd-Bro*r, (1gt6) and helped them to solve the crystal -

structure of olivine. Although a spinel phase of the composition

POLYMORPHS OF MsilSiO,, 7Ll

T a b l e l . 0 b s e r v e d . a n d h y p o t h e t i c a l 1 4 g r S i 0 r s t r u c t u r e r y p e s .x : e x p e r i m e n t a l u n i t c e l I d i t a ; t : u n i t c e l I f i " o m D L S ;S : s - y n t h e t i c ; N : n a t u r a l l y o c c u i i n g ; H : h y p o t h e t i c a l ;z : T o r m u l a u n i t s p e r - u n i t c e l l ; V : v o l u m e p e r I r l g r S i 0 u ;D i d e n s ' i t y f o r c h e m i c a l c m p o s i t i o n ( t ! . t n . n r e O . i ) r S i O O .

c o m p o u n d z c r o u p d b

F o r s t e r i t e ( 4 : 4 ) s N 4 p b n m x l ) q . z o z R 1 o . 2 z 5 AL 4 . 7 9 1 0 . 1 9

B - l 4 g 2 s i 0 4 s 8 I b m m x z l a . z q e 1 1 . 4 4 4L 8 . 1 6 1 1 . 6 8

y - M g r s i 0 o ( n o r m a l ) ( s N ) 8 F d 3 m x 3 ) g . o s t 8 . 0 9 i

L 8 . ] 2y - M g r S i 0 O ( i n v e r s e ) H 8 F d 3 m L 8 . 1 7M o d e l I ( 7 : 1 )

M o d e l I I ( 6 : 2 ) H

4 P 2 / n L l 0 . t j

4 C 2 / n L I 0 . 0 5

c V D

5 . e 9 4 4 7 3 . 0 4 3 3 . 3 4 9 . c f r 3

5 . 8 5 7 1 . 4 3 . 4 2

5 . 6 9 6 6 7 . 2 3 . 6 3

5 . 7 1 6 8 . 0 3 . 5 9

8 . 0 9 ' , 1 6 6 . ? 3 . 6 9

8 . 1 2 6 6 . 9 3 . 6 5

8 . 1 7 6 8 . 2 3 . 5 8

4 . 7 0 6 8 . 6 3 . 5 6

4 . 8 7 7 0 , 4 3 . 4 7

4 . 5 2 6 5 . 4 3 . 7 3

2 . 7 5 6 0 . 6 4 . 0 3

1 0 . 4 5 6 4 . 5 3 . 5 3

l v l o d e l I I I ( 8 : 0 ) H 4 C Z / n L 1 0 . 0 3S r 2 P b 0 4 - t y p e

K 2 l , l g F 4 - t y p eH 2 P b a m L 4 . 9 eH 2 I 4 l m m m L 3 . 5 1

8 . 1 28 . 1 75 . 7 7

5 . 7 78 . 8 53 . 5 1

i ) B i r l e e t a l . , ' 1 9 6 8 ;

2 ) M o o r e a n d s m i t h , 1 9 7 0 ; 3 ) R i n g w o o d a n d M a j o r , 1 9 7 0

(Mgo.rrFeo.r6)rsion has been described as a mineral (ringwoodite, Binns,1970) pure z-MgrSiO+ has not been found in nature nor has it beensynthesized yet. The Mg-richest spinel in the system MgzSiOn-FegSiO,synthesized so far is (Mgo.rFeo.r)rSiOn, and it is conceivable, but not atall certain, that pure 7-MgrSioa might form at still higher pressures(see the discussion by Ringwood and Major, lg70). The 7-lMg, Fe),SiOnstructure appears to be the normal spinel type (Kamb, 1g6g). It isbased on a cubic close packed (c.c.p.) a,nangement of oxygen atoms,as is the structure of p-MgqSiOa, the other high-pressure polymorphof Mgrsioa. Moore and smith (1970) determined the crystal structureof the p-phase from a powder pattern of a sample of composition(Mgq.rNio.r)rSiOn, while Morimoto et al. (lg70) studied, by singlecrystal methods, the isostructural compounds MnrGeOa and CorSiOn.The p-MgrSiOn phase should be formulated as MgaOSirOT, since itcontains discrete Si2Or6- groups and one of the oxygen atoms is bondedonly to the octahedrally coordinated Mg-atoms. According to Ringwoodand Major (1970) the F-MgrSiOa structure is the stable phase at. pres-sures between 120 kb and 200 kb for compositions between Mgrsionand (Mgo. sFeo.r)rSiOn.

HyporHnnrcar, Por,yMoRpHS oF MgSiO+rn order to shed further light on the reasons for the relative stabil-

ities of @-, F-, and 7-MgSiOa phases it is useful to investigate otherhypothetical phases which are geometrically possible but which havenot yet been observed. Starting with an h.c.p. array (ABA) of oxygenatoms and without limiting the size of the unit cell, there is an infinite

712 .WERNER H. BAUR

the Mg-atoms per unit cell are approximately at a height of zero

(between the 24. and B layers) and the remaining four Mg-a:toms are

at a height, of. 1/2 (between the B and the 'A layers)' The Si-atoms

are distributed over such remaining tetrahedral voids which do not

share faces with the Mg coordination octahedra. Model I is con-

Model II is identical with the "hypothetical structure with olivine

stoichiometry" mentioned by Birle et aI. (1968).

the SrsPbO* type.

POLYMOHPHS OF Ms"SiAr lrc

The six hypothetical structures selected and discussed here are ofcourse not presented in an attempt to enumerate all conceivable MgzSiOastructure types. They are merely used as examples to illustrate theapproach discussed in this paper.

Gpourrnrc Dnm,nMrNerroN oF THE CRysrAr, STRUcTuRESThe nine crystal structures of MgSiOr were refined geometrically

by distance least squares (DLS) using the program written by Villiger(1969). A DLS refinement of a crystal structure is possible wheneverthe interatomic distances used as input to the refinement can be pre-dicted with sufficient accuracy. The procedure followed these steps:

1) Average Mg-O and Si-O distances were derived from the em-pirical effective ionic radii tabulated by Shannon and Prewitt (1969),taking due account of both the coordination of oxygen atoms aroundthe cations and the cation coordination around the oxygen atoms. Foreasier comparison with the forsterites refined by Hanke (1965) and byBirle ef ol. (1968), which both had the composition Foa.eFa6.1, a 10percent Fe-component was assumed throughout, resulting in an imputvalue of d(Mg,Fe-O) of 2.105 A for Mg in six and O in four coordi-nation (instead of 2.10 A for pure Mg-O bonds). Therefore in thefollowing discussions "Mg" always means ,'Mgo.nFeo.r". For four co-ordinated magnesium the value d(Mg-O) : 1.g55 A, determined inKzMgrSirzOsg w&s used (Khan, Baur, and Forbes, 1972). The radiusof magnesium in trigonally prismatic coordination was assumed to bethe same as in octahedral coordination. The distance MS-O for nine-coordinated Mg was taken to be 3 percent larger than for eight-coordinated Mg. Shannon and Prewitt (1969) have pointed out thatin highly symmetrical structures, such as perovskites, the calculatedinteratomic distances are systematically longer than the observeddistances. The same effect can be observed for compounds crystallizingin the KrMgFn type, as a survey of the o cell parameters of 22 suchcompounds showed. Accordingly the Si-O distance of six coordinatedsilicon in the K2MgFa type was taken to be 3 percent smaller thanthe calculated sum of the radii of silicon and oxygen.

2) The individual Mg-O and Si-O distances were calculated fromthe extended electrostatic valence rule (Baur, ISZO; Ig7la) for thoseoxygen atoms for which Pauling's (1960) second rule for ioniccrystals was not satisfied. In the equation for the individual cation-oxygen distances

d(M-O) : (d(M-O)-"," * bApo) A ( l )b was taken as 0.12 A,/v.u.forMg and as 0.091 A/v.u. for Si (Baur,

714 WERNER H. BAUR

1970). This calculation was only necessary for B-MpSiO+ and for theK2MgFa type. No attempt has been made (for the high pressurephases) to account for the possible shortening of bonds under hydro-static pressure. While the compressibilities of many minerals areknown, the effects of high pressure in complicated structures on thebonds themselves are not known. For pressures around 100 kb theshortening of cation-anion bonds may, very well amount to less than'1 percent of the bond length at atmospheric pressure (as can beestimated from the compressibility values tabulated by Birch, 1966).

3) In order to account for the well known shortening of shared edges(Pauling, 1960, rule 3 for ionic compounds), any edge shared betweentwo coordination polyhedra was assumed to be 0.15 A shorter than anunshared edge. The shortening of shared edges has no effect on thesum of the tengthd of all edges in a tetrahedron or octahedron as hasbeen pointed out by Drits (1971). Consequently care'was taken tolengthen the unshared edges by the same amount by which the sharededges were shortened. This calculation was straightforward for thetetrahedra because in the structures considered they either share noedges or hatf of their edges. For the octahedra the sum of the lengthsof the edges (2) was obtained from (>) : l2\/2d (M-O)-"," (Drits,1971). The length of an unshared edge was then

d(o-o)"""b - Q) - qd(o-J)b ! o'r\n (L) Q)

where q and d (O-O) I are the number and the length of the tetrahedraledges shared with the octahedron, and m and n are the numbers of theunshared edges and those shared with other octahedra, respectively.

4) Each of the M-O distances was assigned a weight which wasproportional to the Pauling electrostatic bond strength of the cor-responding bond. All the O-O distances were given the same weight,which was set arbitrarily at 0.07 of the weight assigned to the tetra-hedral Si-O bond. This weighting scheme (Baur, 1971b) gives sub-stantially different results from the weighting applied by Meier andVilliger (1969) who assign to the O-O distances weights from 0.3 to0.5 relative to a weight of 1.0 for (Si, Al)-O. With such relativelyhigh values for the weights of the O-O distances the coordinationpolyhedra tend to remain rather regular, while the smaller valuesused here allow distortions in O-O distances of the same magnitudeas observed experimentally in olivine and comparable crystal struc-tures.

5) The calculated M-O and O-O distances with their assignedweights were the input for a distance least-squares refinement (Shoe-

POLYMORPHS OF Ms'SiOa 715

CoupenrsoN oF OBSERvED AND D-srRUcruRE oF aN Or,rvrNE

- A detailed comparison of the geometricaily refined structure offorsterite with the experimental results of Birle et at. (lg6$ showsthat the mean deviation between the carculated and the observed inter-atomic distances is 0.04 A (Tabre B). The si-o distances of the D-

WERNER H. BAUR

T a b l e 2 . c a l c u l a t e d p o s i t i o n a l c o o r d i n a t e s ( x I O 4 ) r o r t h r e e e x D e r i m e n t a l

a n d s i x n v p o i t ' " i i i i i I ' 1 9 - r s i 0 { s t r u c i u r e s ' - E x D e r i . m e n t a l v a l u e si d " i o i s i L i " i i " - i " o t n s i i i " e t a l ' ( 1 9 6 8 ) ' f o r B - t l g 2 S i 0 4 f r o n H o o r e; ; ; s ; i t i ( 1 9 7 0 ) , s t a n d a r d d e v i a t i o n s i n p a r e n t h e s e s '

x

F o r s t e r i t es i ( D L S ) 0 9 4 5

( E x D ) 0 7 3 1 ( 3 )

M s ( 1 ) 0 o 0 o( E x p ) 0 0 0 0

M s ( 2 ) 0 0 5 6( E x p ) o l o 3 ( 3 )

0 ( l ) 2 4 9 s( E x p ) 2 s 4 2 ( 7 )

0(2) 2832( E x p ) 2 7 9 s ( 7 ,

0 ( 3 ) 2 3 1 2( Ex p ) 221s (5 )

B - M g 2 s i 0 4

s i ( D L S ) r 3 0 3( E x p ) 1 3 1 6 ( 1 2 )

M s ( l ) 2 2 9 7( r x p ) 2 2 7 1 ( 1 3 )

M g ( 2 ) 2 s 0 0( E x p ) 2 5 0 0

M e ( 3 ) 0 0 0 0( E x p ) 0 0 0 0

o ( r ) 0 3 o o. ( E x p ) 0 4 s 6 ( 3 e )0 (2 ) 47ss

( E x o ) 4 7 0 2 ( 3 7 )o ( 3 ) o o 4 o

( E x p ) s 7 7 9 ( 2 5 )

o ( 4 ) 2 5 5 7( r x p ) 2 5 5 0 ( l e )

1 - M o r S i 0 o ( n o r m a l )

s iM g

0

4078 25004 o s 7 ( l ) 2 s o 00 0 0 0 0 0 0 00000 00002632 75002774(2) Tsoo0 9 0 7 7 5 0 0o g l e ( 4 ) 7 5 o o0 5 7 8 2 5 0 0

0 s 2 2 ( 4 1 2 5 0 0s370 47073 3 6 s ( 3 ) 4 6 s 7 ( 5 )

I 2 5 5 2 5 0 0' l 2 6 8 ( 8 ) 2 5 o o0 0 0 0 7 5 0 00 0 0 0 7 5 0 02 5 0 0 7 5 0 02 5 0 0 7 5 0 03 7 0 1 0 0 0 03 6 7 5 ( 7 ) 0 o o o0 0 0 0 2 5 0 00 0 0 0 2 5 0 00 0 0 0 7 5 0 00 0 0 0 7 5 0 02 3 4 8 2 5 0 02 3 5 3 ( l 8 ) 2 s o o3 7 5 1 0 1 6 8

3 7 5 7 ( r 2 ) o 2 4 s ( 2 6 )

1 2 5 0 1 2 5 0

5 0 0 0 5 0 0 0

2416 2416

xM o d e l Is i ( 1 ) 8 4 2 0s i ( 2 ) ? 2 2 6M s ( 1 ) 5 0 0 0M s ( z ) o o o o! ' t s (3 ) 0000M s ( 4 ) 5 0 o ous(9) 2482o ( 1 ) 3 1 5 4o ( 2 ) 1 6 4 2o ( 3 ) 8 3 0 5o ( 4 ) 3 3 3 0o ( 5 ) 0 8 s I0 ( 6 ) s e 3 2

l ' lodel I Is i 3 3 3 0M s ( l ) 0 0 0 0M s ( 2 ) s o o oM s ( 3 ) 2 5 0 00 ( r ) 8 l e 40 ( 2 ) 3 3 4 1o ( 3 ) o e o s

M o d e l I I Is i 3 3 0 9M s ( 1 ) 0 0 0 0l v l s ( z ) 5 0 0 0

M s ( 3 ) 2 5 0 0o( l ) 82e2

o ( 2 ) 3 3 3 7

0 ( 3 ) 0 8 6 4

SrrPb0 O- tvpe

s i o o o 0l , lg 0819

0 ( r ) 2 1 6 4

0 ( 2 ) 3 e 2 5

KTMSFU- type

5000 87520000 88245000 0000

0000 50005000 50000000 5000

2 5 2 7 4 9 9 0

5 0 0 0 2 1 0 55000 77570 0 0 0 2 3 3 40000 23092601 22872281 2210

0000 89870000 00000 0 0 0 5 0 0 02500 50000000 22200 0 0 0 2 3 5 32703 2253

0000 85380 0 0 0 5 0 0 00 0 0 0 5 0 0 0

2 5 0 0 5 0 0 00000 22190 0 0 0 2 1 6 32606 2196

0 0 0 0 0 0 0 03329 5000

0448 5000

3 0 8 7 0 0 0 0

0000 00000 0 0 0 3 7 4 0

5 0 0 0 0 0 0 0

1 2 5 0

5 0 0 0

241 6

' y - M s r S i 0 o ( i n v e r s e )

M g I 2 5 0

M g , S i 5 0 0 0

0 2 6 3 0

I 2 5 0 I 2 5 05000 5000

2630 2630t { g 0000

0 ( l ) 0 0 0 0

S i 0 0 0 0

717POLYMO&PHS OF Ms"SiOt

. r c <o f o @ @ @ N N O @ O s ! - F OX 6 - o r O N O O @ r O . + O

U Od O O N N N N O N O N O O

F <O = O O e - O N O O @ N o OJ o ( f O F r F - O O F O O ( t lo o

d O O N N N N ( ' N O N O N

P JS O N N O O O O N N N N N No l o o o o o o o o o o o o

. F C

o . r O o o O O o O O O o O O=q q

P J N N n 6 A O @ @ c | @ @ @f O F F O O O O O @ O o O oo o c o o o o oE O

H P O O N N N N o N O O O O

EU

N O Oo o v m o o o o ov v o o o oo o l r r r o o o o o c |

l { ^ t t t t t l^ ^ N N N NF N v T F N N o Ov v O O O U )o o = = = = o o o o o oX X X X X X X X X

N N ( \ I N N N N N N

P G (. F F o i + = f , € A o O € + O @

C - 5 @ 6 A N N n b O O F O @X @

U O F F F N N $ l ( \ l N N N O NE

p <- . + + s l - O o N h O O T N Q

@ = @ A A @ N @ @ r F r C ) OJ @6 0 r r r N N N N N N N o N

F N

= E =! | J ^ ^ ^ | J

a F N O ^ ^

N O O O v v v N N

- O OF N O c ' O O O t t t O O

t t t r ^ ^ ^ t lo o o

l l l r r N O vv o o o v vq o 6 0 0 0 0 = = = o o

x x x x x x x xN N N N N N N N

P Js o o o c ) N T N N O O O N No o o o o o o o o o d ) o o

. F C

o . F T F T o O O o O O O O O=@ c (

P J O O O N N N N O b @ N NJ O O o O o O O O O O O e @O @ @ A N N @ t O - r F F oq o

H { J F F F N N N N N N N ( ' N

€ , g pE O O E

. F . F s ( u> # ' - E

. F 6 ( F OF g O C

O ' F E . F

! l ! oo o c L

t s o 60 0

@ @ EP L C J Pr o o

@ o . P4 ) oI o J F L

o 6 ( uF C L ( F6 d E O .p P O ! og o s oO . r > , o f

O r d! s c J >o o q

O E O Fx l - o l - 6o ( u t r P

! 6 @ Co E S J O

s f o o EP q

o o t< o P S Op s 6 F. r p o x= ' - o

o E .E P C E OO ' F O s! L ! + )6 0 @ 6q q - P s P

E O O @ 6o t Jo o o p

o 6 ! qo l - t 6 6

P O O p

@ @f F O (u E@ o ! o oO . r 6 ! Ot p f oI F O E

q = @ O r

o s p <o . r

E F o O C6 F . r O

O . A = 6

J ^ E Eo @ > s o

@ O P Eo o

p F o S E

p - F s r= . O rq F L oc 6 . E O

c op o s + )

- o L > . F( l , -OF cs o o o s

> t . > s. r ' r r ( U PF O O S . F

O v o P =

(l,

6F

718

T a b l e 4 . 0 L S - i n p u t v a l u e sh e x a g o n a l c l o s eT h e m u l t i P l i e r so c t a h e d r o n .

WENNER H. BAAN

f o r 1 1 o , S i 0 , D o l v m o r p h s b a s e d o n c u b i c a n dp a i r . i n i l s t r a i ^ e d - p o l ! n e a r a l e d g e s a r e i d e n t i f i e d .i e f e r i o t h e n u m b e r o f d i s t a n c e s p e r c o o r d l n a t l o n

B - 1 4 g 2 S i 0 4

s i - o | ) r . 6 5 8 8

s i - o ( 3 ) t 6 2 8

2 x s i - 0 ( 4 ) I 6 2 8

0 ( r ) - 0 ( 3 ) 2 . 6 7 1

2 x 0 ( l ) - 0 ( a ) 2 . 6 7 1

2 x 0 ( 3 ) - 0 ( 4 ) 2 . 6 7 1

0 ( 4 ) - 0 ( 4 ) 2 6 7 t

t t g ( l ) - 0 ( l ) 2 . 1 4 5

M g ( 1 ) - 0 ( 2 ) 2 . 0 6 5

1 - t 4 g r S i 0 O ( n o r m a l )

4 x S i - O 1 . 6 3 9 8

6 x 0 - 0 2 . 6 7 7

^ y - l t l S r S i 0 O ( i n v e r s e )

4 x l 1 g - 0 1 . 9 5 5 R

6 x 0 - 0 3 . 1 9 3

M o d e l I

s i ( r ) - o ( 1 ) r . 6 3 e f fs i ( 1 ) - o ( 2 ) 1 . 6 3 e

2 x s i ( l ) - 0 ( 5 ) 1 . 6 3 9o ( l ) - o ( 2 ) 2 . 7 5 2

2 x 0 ( 2 ) - 0 ( 5 ) 2 . 7 5 22 x 0 ( I ) - 0 ( 5 I 1 , 4 5 1 2 . 6 0 2

o ( 5 ) - 0 ( 5 f , r Y 3 l 2 . 6 0 2s i ( 2 ) - 0 ( 3 ) 1 . 6 3 9s i ( 2 ) - 0 ( 4 ) r . 6 3 e

2 x s i ( 2 ) - 0 ( 6 ) , l . 6 3 9

0 ( 3 ) - 0 ( 4 ) 2 . 7 5 ?

2 x 0 ( a ) - 0 ( 6 ) 2 . 7 5 22 x 0 ( 3 ) - 0 ( 6 I 1 " 1 5 1 2 . 6 0 2

o ( 6 ) - o ( 6 I M 4 l 2 . 6 0 22 x M 9 ( l ) - 0 ( l ) 2 . 1 0 5a x i l g ( l ) - 0 ( 6 ) 2 . 1 0 5

M o d e l I I

s i - o ( r ) l . 6 3 e Rs i - 0 ( 2 ) I . 6 3 e

2 x s i - 0 ( 3 ) 1 . 6 3 9o ( 1 ) - o ( 2 ) 2 . 7 5 2

2 x 0 ( l ) - 0 ( 3 ) [ M 3 ] 2 . 6 0 22 x 0 ( 2 ) - 0 ( 3 ) 2 . 7 5 2

0 ( 3 ) - 0 ( 3 ) t M 2 l 2 . 6 0 2

2 x l t g ( l ) - 0 ( 1 ) 2 . 1 0 5a x M g ( l ) - 0 ( 3 ) 2 . 1 0 5

M o d e l I l l

s i - o ( ] ) r . 6 3 9 Rs i - 0 ( 2 ) 1 . 6 3 9

2 x s i - 0 ( 3 ) 1 . 6 3 9o ( r ) - o ( 2 ) 2 . 7 5 2

2 x 0 ( I ) - 0 ( 3 ) [ r , 1 3 ] 2 . 6 0 22 x 0 ( 2 ) - 0 ( 3 ) 2 , 7 5 2

o ( 3 ) - o ( 3 ) [ M 2 ] 2 . 6 0 2

2 x M s ( 1 ) - 0 ( l ) 2 . 1 0 5

4 x M s ( l ) - 0 ( 4 ) 2 1 0 5 [

a x O ( l ) - 0 ( a ) 3 . 0 5 2

4 x O ( 2 ) - O ( 4 ) [ r 4 3 ] 2 . 9 0 2

2 x 0 ( 4 ) - o ( 4 ) [ r , 1 2 ] 2 . s 0 2

2 x 0 ( 4 ) - 0 ( a ) 3 . 0 5 2

2 x M 9 ( 2 ) - 0 ( 3 ) 2 . 1 0 5

4 x M 9 ( 2 ) - 0 ( 4 ) 2 . 1 0 5

a x o ( 3 ) - 0 ( a ) [ M 3 ] 2 9 0 2

a x 0 ( 3 ) - 0 ( 4 ) 3 . 0 s 2

6 x M g - 0 2 . 1 0 5 R

6 x o - 0 [ ! ] l 2 . 9 0 2

6 x ( M s , S i ) - 0 1 . 9 4 0 R

6 x 0 - 0 [ r u ] 2 . 6 6 9

a x o ( r ) - o ( 6 ) 2 . s 7 7 8 '

a x o ( 1 ) - 0 ( 6 ) 2 . 9 7 7

2 x 0 ( 6 ) - 0 ( 6 ) 2 . 9 7 7

2 x 0 ( 6 ) - 0 ( 6 ) 2 . 9 7 7

2 x M s ( 2 ) - 0 ( 3 ) 2 . 1 0 5

a x M e ( 2 ) - 0 ( 5 ) 2 . , ] 0 5

4 x 0 ( 3 ) - 0 ( 5 ) 3 ' 0 s 2

4 x 0 ( 3 ) - o ( 5 ) [ M 5 ] 2 . e o a

z x o ( 5 ) - 0 ( 5 ) 3 . 0 5 2

2 x o ( 5 ) - 0 ( 5 ) [ I ' 1 3 ] 2 . 9 0 2

2 x l 4 s ( 3 ) - 0 ( 2 ) 2 . 1 0 5

a x M s ( 3 ) - 0 ( 5 ) 2 . 1 0 5

a x 0 ( 2 ) - 0 ( 5 ) 3 , 1 4 2

4 x 0 ( 2 ) - 0 ( 5 ) [ M 5 ] 2 : 9 9 2

2 x M g ( a ) - 0 ( 4 ) 2 . l o s

a x M s ( a ) - 0 ( 6 ) 2 . 1 0 5

4 x 0 ( l ) - 0 ( 3 ) 2 , s 7 ' 1 44 x 0 ( l ) - 0 ( 3 ) 2 . 9 7 7

2 x 0 ( 3 ) - 0 ( 3 ) 2 . 9 7 72 x 0 ( 3 ) - 0 ( : ) 2 . 9 7 7

2 x M 9 ( 2 ) - 0 ( 2 ) 2 ' 1 0 5

4 x M g ( 2 ) - 0 ( 3 ) 2 . 1 0 5

a x 0 ( 2 ) - 0 ( 3 ) [ M 3 ] 2 . 9 6 2

a x o ( 2 ) - 0 ( 3 ) 3 . 1 1 2

4 x M s ( l ) - 0 ( 3 ) 2 . 1 0 5 8

a x o ( 1 ) - 0 ( 3 ) 3 . 0 5 2

a x 0 ( l ) - 0 ( 3 ) [ M 3 ] 2 . 9 0 2

2 x 0 ( 3 ) - 0 ( 3 ) [ M 2 ] 2 . 9 0 2

2 x 0 ( 3 ) - 0 ( 3 ) 3 . 0 s 2

z x M g ( 2 ) - 0 ( 2 ) 2 . 1 0 5

4 x M g ( 2 ) - 0 ( 3 ) 2 . 1 0 5

a x o ( 2 ) - o ( 3 ) [ t , | 3 ] 2 . 9 9 2

z x o ( a ) - o ( 4 ) 3 . 0 5 2 f f

z x M s ( 3 ) - 0 ( 2 ) 2 . 0 7 9

2 x t 4 s ( 3 ) - 0 ( 3 ) 2 ' 1 1 8

z x M s ( 3 ) - 0 ( 4 ) 2 . I 1 8

o ( 2 ) - o ( 2 ) [ 1 4 3 ] 2 . 9 1 5

2 x o ( 2 ) - 0 ( 3 ) [ i l 3 ] 2 . 9 1 s

2 x 0 ( 2 ) - 0 ( a ) 3 . 0 6 5

o ( 2 ) - o ( 3 ) 3 . 0 6 5

2 x 0 ( 3 ) - 0 ( 4 ) 3 . 0 6 5

6 x 0 - o 3 . 0 5 2 R

6 x O - 0 2 . 8 1 9 4

a x o ( a ) - o ( 6 ) 3 . l l 2 R

4 x 0 ( a ) - 0 ( 6 ) [ M s ] 2 . s 6 2

2 x 0 ( 6 ) - 0 ( 6 ) 3 . 1 1 2

M s ( 5 ) - 0 ( l ) 2 . 1 0 5

I t s ( s ) - 0 ( 2 ) 2 ' 1 0 5

M s ( 5 ) - 0 ( 3 ) 2 . 1 0 5

i l s ( 5 ) - 0 ( a ) 2 . 1 0 5

M s ( 5 ) - 0 ( 5 ) 2 . 1 0 5

M s ( 5 ) - 0 ( 6 ) 2 . 1 0 s

o ( l ) - o ( 6 ) 3 . 1 2 7

0 ( 2 ) - 0 ( 6 ) 3 . 1 2 7

o ( 4 ) - o ( 5 ) 3 . 1 2 7

o ( l ) - o ( 2 ) [ M s ] 2 . s 7 7

o ( 1 ) - o ( 4 ) 3 . 1 2 7

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

0 ( 3 ) - 0 ( 4 ) [ M 5 ] 2 . e 7 7

2 x 0 ( 3 ) - 0 ( : ) 3 . l l 2 n

z x M s ( 3 ) - 0 ( l ) 2 . 1 0 5

z x r r l s ( 3 ) - 0 ( 2 ) 2 . 1 0 s

2 x M s ( 3 ) - 0 ( 3 ) 2 . ' l 0 s

2 x 0 ( 2 ) - 0 ( 3 ) 3 , 1 r 2

2 x 0 ( i ) - 0 ( 3 ) 3 . 1 1 2

2 x o ( 1 ) - 0 ( 2 ) [ M 3 ] 2 . 9 6 2

2 x 0 ( l ) - 0 ( 2 ) 3 . 1 1 2

2 x 0 ( 2 ) - 0 ( 3 ) 3 . i 4 2 R

2 x l , 4 s ( 3 ) - 0 ( 1 ) 2 . 1 0 5

z x M s ( 3 ) - 0 ( 2 ) 2 . 1 0 s

2 x M g ( 3 ) - 0 ( 3 ) 2 . 1 0 5

2 x 0 ( 2 ) - 0 ( 3 ) 3 : 1 4 2

2 x o ( l ) - 0 ( 2 ) [ M 3 ] 2 . 9 e 2

? x 0 ( l ) - 0 ( 2 ) 3 . 1 4 ?

POLYMOHPHS OF MszSiOr ZIg

Mg(2)-O distance in the D-structure is slighily higher (2.LZLA) thanthe mean Mg(l)-o distance (z.r0g a). The same is true of the ob-served structure, where the values are 2.rBE and 2.108 A, respectively.This parbicular similarity between the D-structure and the actualstructure lends further credence to Kamb,s (196g) interpretation ,,thatstructural strains set up in the distorbion from idear crose-packedgeometry are such as to distend the Mg(2) octahedrar site". The D-structure however does not reflect at all the distribution of individualMe-o distances observed in forsterite, which vary from 2.06 to 2.t2 ain length. This variation is not connected with variations in the electro-

oxygen atoms which participate in edges shared between different

cations), while the anion-anion distances within the shared edEe areshortened in comparison with unshared edges. This allows u

"*id",

separation of the cations and increases therefore the stability of thecompound. The stabilizing influence of cation recoil is demonstrated

cations, it is not surprising that the Mg-o distances do not reflectvariations which are due to these ionic forces.

and 3.35 A, and for the shorb edges which without exception are also,shared edges. This is unexpected because shared edges are generallythought to be short because of the electrostatic inteiactions betweenthe cations. The DLS refinement shows however that part of theshortening must be attributed to the adjustment stresses due to the

7N WERNER H. BAAR

fitting of the polyhedra to each other (Baur, 1971b)' The observed

difference between the average lengths of the shared and the un-

shared edges in the octahedra is 0.26 A (Table 5), while in the D-

structure it i. O.tZ A. This can be taken to mean that 2/3 of. Lhe

difference is due to adjustment siresses, and only 1/3 due to electro-

static interactions. For the tetrahedron the differences (Table 5) are

0.18 A and 0.09 A respectively, that is l/2 of the difference can be

attributed to adjustment stresses. It cannot be argued that the DLS-

results merely reflect the plS-input values, because even when all

o-o distances in the octahedra were entered with the same length,

the computed shared edges still tended to be short. In order to obtain

shared and unshared tetrahedral edges of the same length, the input

values of the shared. edges had to be O.15 A, Ionger than those for the

unshared edges. Furthermore the experiences with the DlS-refine-

ments of B-Mgsio+ and the hypothetical structures show that the

small input values for the shared edges can easily be overridden when

the geometric adjustments so require.

D-srnucrunns or Ornnn Mg2I6l Si t1l Oa Polvuonnrs

The geometric refinement of F-MgzSiOa, report€d by Baur (1971b)

is considered to be more precise than the X-ray structure determina-

tion by Moore and smith (1970), which is based on powder data.

The DLS positional parameters reported in Table 2 are slightly dif-

ferent from those given in the earlier publication (Baur, 1971b), be-

cause the older geometric refinement was based on d(Mg-O)mean =

2.085 a and the mean value of the lengths of the polyhedral edges was

not kept constant, as actually is required (Drits, 1971) ' The mean

deviation between the D-structure bond lengfhs and the experimental

bond lengths for B-Mgsior is 0.073 A. This deviation must be partly

due to experimental error in the powder data refinement since an

analogous calculation for F-cozsio+ gave a mean deviation between

D-structure and observed structure of only 0.027 L, which apparently

is related to the fact that p-cozsio+ was determined with higher

precision by single crystal diffraction. An inspection of the averaged

dimensions of some key distances in B-MgSiO+ (Table 5) shows that

the only major difference between the D-structure and the experi-

mental structure concerns the length of the shared and the unshared

edges in the Mg-coordination octahedra: the observed avelage lengths

are equal, the calculated lengths are longer for the shared edges. The

difference between observed and calculated structure is of the same

type as for forsterite and seems to be connected with the fact that the

721

o

@ @ 6 N N N - - - " O -

Q 6 6

N N N O O O N N O N @

n 6 b e

€ a a a 9 @ @ @ N A 6

o N oo 6 0

N N Nx x x

D b o

o o o

POLYMOHPHS OF Mg"SiOq

L IO F

o sU qog E <

L O = r 6 O O + + O O O N Oo L l N O N o o m o o o o o

a 6 Ft 9 0 0 0 0 0 0 0 0 0 0@ x x x x x x x x x x x

€ o a o o o N N @ @ 9

E O N r O @ rO = N @ @ r + o o! l6 F N N N N N Ns x x x x x xq O O o O o o

1 ! O 9 $ + @@ 9 E

N N r N NC > lL 6 = O O O O Oo € x x x x x

o @ a @ Q o @

os

!p

o

dL

!o<6

a

!o

O O b N O T T N @ O $ - bS I N N N @ 9 @ e | _ @ N O6 C tC N N N N N N N O N N N

= X X X X X X X X X X Xq * o o o @ @ 9 @ a o O o

; ' ! r N o N o o ou d o o Q o r @ Ns

o N N N N N N

! < + s o o N + @ o o o o oO T O O O O O 6 @ O @ @! =6 T O O O N N N N N N N N

s = x x x x x x x x x x xq o o o s b o € 9 o o 9

@ a a +

o > o > >> < J J X J J J J J J J

+ < fo o

N N - r ro o o o o

= = ! E !r r o o o> > - E = E

@ N O O b O O @ O N O NI O T O O O O o @ O O O O6 l

€ O O O O N N N N N N O NO X X X X X X X X X X XC 6 O o - F F N N N A O=

o

o

d

!

o

z

P

oo

o E O qo o r Q N N 6 c o @ + o o o < +o L O @ O O O O O O A O O O

- OJ s N N N N N N o N o o o

z . HoP +

o! . r@ @

P No o! =o r

! L O

oE l . F

p N @ Oo F o J o! = = o

< 9 . . OL O O O . r

o F p o

C . J d S

O d < . ro o F o =o > 5 Eo X E . r P

o o o o$ q - o E

E O X Oo o o c! o g

s ' F x oo g ! L< o o . .= o o q c

q o o! | 4 0c o o > Ed o o x o

o E o O! r O 6O O . . !

t o r - J

! o oD @ 9 . r . .c o g P >

' F d 6 O J> ! P O

q . ^O O . F r o> ! ! o P

o ! 6

@ o p s oO ' r d Co < + ! oc = o + o6 0

P . ! P Fo J o

. r O a ' . O! ' r E F o

q =

O N C . ^ ro o c 6o = o o PL s . r co q - P P O> o o E6 0 0 . F

o p It + - < r o

! @ L Xc o s ! oo E o o@ > \ ! t E

d o! o o P6 A ! O !o o o oE o ' r 6O s F . . 6

o P o = o

o

o

F

7T2 WERNER H. BAUR

geometric model neglects the electrostatic interactions. It was already

observed by Moore and Smith (1970) that "there is no systematic

pattern of shortening of shared edges in the p-structure". However

they did not discuss this point further because of the low accuracy of

their crystal structure determination. The present geometric refinementgives additional evidence on this point, as does the structure deter-

mination of F-cozsio+ by Morimoho et al. (1970) who observed that in

this structure the mean lengths of the shared and the unshared edges

are also approximately equal.The shared edges are longer than the unshared edges in the D-stmc-

ture of the normal spinel model of 7-MgSiOa, and in the three hypo-

thetical models I to III (Table 5). For these cases we have no observed

structure with which we could compare the results and, except for

normal y-MgSiOa, we have no experimental cell parameters which

could be kept constant. The comparison of the results of the cell

parameter data ltabte 1) however is reassuring' The DlS-refined

individual cell constants and the unit cell volumes do not deviate by

more than 2 percent from the observed values, and the important

mean distances show the same trends. we can therefore conclude that

among the structures studied here only forsterite and the inverse

spinel type can provide for the energetically advantageous shorten-

ing of th" sha""d edges. At this point it should be stressed that the

DiS-refinement of forsterite and F-MgzSiOr does provide a partial -

implicit recognition of ionic interactions as long as the geometric

refinement is constrained to the observed unit cell constants (see IC

in Table 5) because the overall dimensions of the observed structure

are forced upon the D-stmcture. The refinement with variable cell

constants (LV in Table 5) is free of this constraint and is therefore

a model which actually does not "know" anything about electrostatics,

except for the shortening by 0.15 a of the input values of the shared

polyhedral edges.

CnYsrer, Clrpurcer, SrcmuclNcn

The use of the weighted distance least squares method, as proposed

here, with high weights for the cation-anion distances and relatively

smail weights for the anion-anion distances, is similar to the applica-

tion of a simple force model in which the lengths of the bonds cor-

respond to the lengths of elastic springs and the weights are pro-

porlional to the strengths of the springs (to the restoring forces). Since

lhe only distances used as input are within the coordination polyhedra(central cation-oxygen distances and oxygen-oxygen edges), one could

also say that the model deals with elastically flexible coordination

POLYMARPHS OF Ms,SiOt 7ZJ

polyhedra. The lengths of the edges define the angles subtended bythe cation-cation distances at the central cation, the flexibility of thepolyhedra is defined by the assigned weights. The mutual adjustmentof the different polyhedra results in defomations of these polyhedrawith consequent changes of the cation-anion and anion-anion distancesand anion-cation-anion angles, that is the distances and angles arestrained in response to the adjustment stresses. The advantage of theleast squares method is that all the different distances can be con-sidered simultaneously even in a complicated structure. The surpris-ingly good agreement found for the case of forsterite between the ob-served and the D-structure prorres that this approach can be a usefultool for the prediction of the details of crystal structures.

The refinement, of the forsterite and spinel phases of MgSiOr fullysupports Kamb's (1968) reasoning regarding the relative stabilities ofthese two structures: "While the contraction of the SiO+ tetrahedron[in forsterite] leads naturally and perhaps inevitably to the shor0eningof shared polyhedral edges, just the opposite is true in the spinelsttucture". As we can see from a eomparison of the lengths of sharedand unshared edges in the Mg2SiO+ structures (Table 5) the forsteritestructure is unique in this respect (except for those spinel structureswhich have relatively large tetrahedral cations). An inspection of thetotal numbers of shared edges and corners (Table 5) shows howeverthat forsterite is not unique with regard to the number of shared poly-hedral elements. According to Pauling's third rule coneerning thedestabilizing influence of shared polyhedral edges and faces at leastmodel II is energetically just as favorable as forsterite. The differencebetween model II and olivine is that in the latter the shared edgescontract simultaneously and naturally because of the adjustment ofthe polyhedra to each other, while in model II the shared edges remainlong, thus bringing the eations into close contact. The shortening ofshared polyhedral edges in polyhedral network structures should notbe viewed solely as a consequence of the electrostatic repulsion ofcations. Instead one can say that,:

Ionic structures with shared polyhedral edges and faces can only be stableif their geometry allows the shortening of the shared polyhedral edges. Whenadjustment stresses force a shared edge to be long, this is a partieularlSr de-stabilizing feature of the crystal structure.

This extension and amendment of Pauling's third rule is proposed onthe basis of few data (Table 5). Before it ean be fully accepted weneed further studies of the kind performed here on the MgSiOn polV-morphs.

Crystal structures with shared polyhedral edges and faces in which

724 WERNER H. BAAR

all shared edges can simultaneously contract cannot be easily in-

vented. Models, I, II, and III fail in this regard because the oxygenatoms are in such positions that the shortening of one shared edgeresults in the lengthening of another shared edge. The geometric fea-

ture of the olivine structure which facilitates simultaneous shorteningof all shared edges is the -L-tetrahedron (1, for "leer") the importanceof which was pointed out by Hanke (1965) and Kamb (1968)' The

six edges of the z-tetrahedron contain all the edges shared in the

olivine structule between different coordination polyhedra. The central

site of the /,-tetrahedron is not occupied. A movement of oxygenatoms toward the center of the z-tetrahedron contracts all shared

edges simultaneously. Models I, II, and III have also Z tetrahedra, butin the case of I and III, there are additional shared edges outside of

the .L-tetrahedra, so that a concerted adjustment of all shared edges

is not possible. In model II all shared edges are concentrated in an L-

tetrahedron, but simultaneous contraction of all shared edges never-

theless cannot be achieved, presumably because the arrangement of

the Z-tetrahedra relative to each other is not favorable.since the o-si-o and o,Mg-O angles are partly functions of the

o-o distances, these angles will be distorted from their ideal values

in response to the adjustment stresses as the o-o distances change.

Therefore it would seem that bond angle theories of the kind proposed

by Gillespie (1963) could not be extended to such polyhedral frame-

work structures as we are treating here, but instead should be tested

on isolated coordination polyhedra only. However the question arises

whether or not a coordination polyhedron observed in a crystallinesubstance can be considered to be sufficiently isolated from any of its

neighbors to be really unperturbed by them. Possibly the only statein which we could observe the unperturbed influence of the electronicstructure on polyhedral geometry is the gaseous state.

Purnor,ocrc Inpr,rc.trroNs

Phases of a bulk composition close to (Mgo.str'eo 1)2SiO4 are thought

to be a major component of the mantle (Ringwood, 1970). Therefore

it is of considerable geochemical interest to discuss the meaning of the

caleulated densities and formula voluines for the different hypothetical

MgSiO+ polymorphs (Table 1). It must be emphasized again that all

volumes discussecl here pertain to zero pressure' even when a hypo-

thetical high pressure phase is discussed. some of the arguments made

could be invalidated in case that similar interatomic distances respond

in a different manner in different structure types to the same kind of

external pressure.All polymorph$ considered here have smaller formula volumes than

POLYM0RI'HS 0l/ Ms"S' iO+ 725

the forsterite-type. Model II, which appears to be energetically favor-able both in regard to the number of shared edges and the fact that atleast the shared tetrahedral edges are shorter than the unshared edges(Table 5), has a volume which is only 4 percent smaller than thevolume of the low pressure olivine phase. It is therefore no surprisethat B-M9SiO+ which has a volume 8 percent smaller than olivine isthe preferred high density phase. Model I has a higher density thanmodel II, but is energetically even more unfavorable because it hasmore shared edges (Table 5). Model III has the smallest formulavolume of all structures considered here (except for the SrePbO+ andKzMgF+-type) but is also energetically most unfavorable in termsof the number .and length of its shared edges. Since no attempt wasmade to try the impossible and enumerate all possible Mg2SiO+ poly-morphs the comparison of these three models with the observed struc-tures does not prove anything beyond the fact that a reasonableinterpretation can be offered why these three types have not beenobserved yet. It is conceivable, though not likely, that a hypotheticalmodel IV can be constructed which obeys Pauling's rules and the"shared edge length" criterion better than the models discussed here.Such a model would be of interest especially if its formula volumeshould calculate to be less than 66 A3, because this would be apotential post-spinel phase. Incidentally: should any of the threemodels be subsequently observed in an actual crystal structure I wouldexpect it to be model II.

Cubic close packed and hexagonal close packed arrangements ofoxygen atoms have different properties in regard to shared polyhedralelements when their voids are filled with cations, a point which hasbeen discussed by Moore and Smith (lg70). For the case of theMgSiOa stoichiometry c.c.p. allows arrangements in which no edgesare shared between the Si-tetrahedra and the Mg-octahedra. In h.c.p.however the tetrahedra must share edges with the octahedra. Once weget into high pressure phases, which are more dense than olivine, butalso have unfavorably long shared edges, c.c.p. structures (B and7-Mg2SiO+) appear to be more stable than the h.c.p. structures becausethe long shared edges involve only the magnesium coordination octa-hedra and not both octahedra and tetrahedra as in models I, II, andIII (Table 5). The inverse spinel structure (MgS,i) t61Mgt41o4, whichwould have the advantage of shorter shared edges, has however so farnot been reported to occur for MgSiOa at high pressures. The reasonfor this may be that its formula volume is larger than B-MgSiO+ ornormal 7-M9SiO+ (Table 1) and that Si and Mg would be sharingoctahedral edges. The B-MgSiOr and the y-Mg2SiO4 types occur both

7% TryERNER H. BAUR

at high pressures in stable phases despite the fact that the sharedpolyhedral edges in these structures are long. Apparently the free

energy change resulting from the decreased volume of these phases isgreater than the destabilizing influence of the long shared edges (as

already pointed out for spinel by Kamb, 1968).Reid and Ringwood (1970) and Ringwood (1970) have suggested

that Mg2SiO+ may crystallize at high pressures over 200 kbar withthe strontium plumbate structure type or (less likely) the KzMgF+type. The D-structure of MgSiOr in the strontium plumbate typeindeed is very reasonable both in terms of the formula volume whichis l7 percent below the volume of forsterite, as predicted by Reid andRingwood (1970), and in regard to the calculated interatomic distances(Table 6). Even the rather short distance O(1)-O(1) of 2.30 A oc-curring in the edge shared by two si coordination octahedra fits wellinto our knowledge of six coordinated silicon. The correspondingshared edge in stishovite (rutile-type SiOe) has a length of' 2.29 Fr(Baui and Khan, 1971). The edge shared SiO6 octahedra formingchains of composition SiOr are a common feature of both stishoviteand strontium plumbate type MgSiO'+. In both structures the chainsextend parallel to the crystallographic c-direction. In stishovite eachsio+ chain is connected via common corners to four neighboring chains,while in strontium plumbate type MgeSiOr the sio+ chains are isolatedfrom each other by intervening edge shared chains of Mg coordination

T a b l e 6 . M g r S i 0 + i n s r 2 P b o q - a n d K z M g F r - t y p e s . D L S - i n o u ta n d D L S - r e s u l t s .

S r2Pb04 - t ypeI n p u t R e s u i t I n P u t R e s u l t

4 x s i - 0 ( l ) t . z s o n l . z g n M s - o ( l ) z . o o o R z . e c tz x s l - 0 ( z ) I . 7 8 0 1 . 7 8 2 x M g - 0 ( 2 ) ? . 1 1 0 2 . 0 8

2 x o ( l ) - 0 ( l ) 2 . 5 9 ? 2 . 7 5 2 x M g - 0 ( 2 ) 2 . 1 I 0 2 . 0 9

2 x 0 ( l ) - 0 ( i ) l s i ] 2 . 4 4 7 ? . 3 0 0 ( l ) - 0 ( l ) f M l 2 . 8 0 s 2 . e 3

a x o ( 1 ) - 0 ( 2 ) [ M ] 2 . 4 4 7 2 . 4 9 2 x 0 ( l ) - 0 ( 2 ) 2 . s ' e 2 - 8 5

a x 0 ( r ) - 0 ( 2 ) 2 . 5 9 2 2 , 5 6 2 x 0 ( 2 ) - 0 ( 2 ) [ n ] 2 . 8 0 e 2 ' 7 5

z x s i - s i 2 ' 7 5 2 x 0 ( 2 ) - 0 ( 2 ) [ M ] 2 ' 8 0 e 2 ' 7 0

4xs i - i l g 2 .90 2xMg-149 2 ' 75

M s - o ( l ) 2 . 1 1 0 2 . 1 2 z x M g - M s ? . 8 9

M g - O ( l ) 2 . 1 1 0 2 . 1 3 M g - l u l g 3 . 0 7

KzMgF4- tYPe I npu t Resu l t

4 x s i - 0 ( ' l ) I . 7 6 0 f t . z o 8 4 x l t g - 0 ( 1 )z x s i - 0 ( 2 ) 1 . 7 2 0 1 . 7 2 M 9 - 0 ( 2 )

a x o ( l ) - 0B x o ( I ) - 0 ( 2 ) 2 . 4 7 5 2 . 4 6 l ' { g - M g

8 x S i - M g ? . 8 1

I n p u t R e s u l t

z . t soR z . zo {2 . 1 8 0 2 . ' t 9

2 , 4 9 0 2 . 5 22 . 6 4

POLYMORPHS OF Ms"SiOn 7tI

polyhedra. Six Mg polyhedral chains surround every SiOe chain andshare with it polyhedral corners. The arrangement of the chains (notof the oxygen atoms) is close packed when viewed along c, and thereare twice as many Mg chains as SiOr chains. The geometric analorybetween stishovite and this hypothetical MgSiOe phase is encour&gring, not so much because it would supply direct evidence for theexistence of the strontium plumbate type at higher pressures, butbecause it is aesthetically pleasing and may be a hint towards acompletely new high pressure silicate crystal chemistry based on thecondensations of silicon coordination octahedra through corners andedges. Another hint is provided by the high pressure transfonnationof KAlSiaOe into the hollandite type structure (Ringwood, 1970).

The D-structure of MgrSiOn in the KrMgFn type has a volume only12 percent smaller than the volume of forsterite. Reid and Ringwood(1970), who compared relative volumes of many substances in theKrMgFa type with the volumes of the constituent oxides, suggestedthat the volume should be 25 percent smaller. The reason for thisdiscrepancy has to be sought in the relatively small size of the mag-nesium atom in nine coordination and in the particular geometry ofthe KrMgFn type. Since the o cell constants are determined by the

a shared square face of the Mg coordination polyhedra. The c-cellconstant could be only shortened, and the volume decreased, by makingthis Mg-Mg distance even shorter. Therefore the KrMgFr structuretype appears to occur only for chemical compounds in which the 9-coordinated cation is large enough to have nine contacts to oxygenatoms of approximately the same'length. It does not seem'to be alikely structure for MgrSiOn under any conditions.

The difficulty in finding a strueture type (or types) for Mg2SiOa,which has a volume smaller than the volumes of the constituent oxides(periclase and stishovite) may be indicative of a deeper seated problem(no pun intended). While a substantial decrease in volume can beattained easily by going from four to six coordination the same is nottrue for going from six to eight coordination. A hypothetical MgO inthe sphalerite-type structure would have a formula volume of 23.0 A3,the halite type periclase has a volume of 18.8 A' 1*hi"h is a reductionof 18 percent). A change to the CsCl type, based on an Mg-O distanceof 2.31 A in eight coordination, would leave the formula volume es-

7?3 WERNER H. BAI;R

sentially unchanged at 19.0 A. th" reason for the constant volume isthe increase in bond length with higher coordination and, more im-portantly, the fact that the oxygen atoms in periclase are cubic closedpacked (12 neighbors at 2.98 A) while in the CsCl-type, a less efficientsimple cubic packing occurs (6 neighbors at 2.67 A). A contractioncould be achieved however in going from stishovite to CaFz-type SiOr.Assuming the distance Si-O to be 1.83 A, the volume for SiOz wouldbe 18.9 A'(as compared. to 23.3 A'for stishovite). This is a small changeif compared with the formula volume of 34.3 A' in coesite type SiOr.Therefore an MgrSiOn composition consisting of NaCl lype MgO andCaF, type SiO" would have a formula volume of 56.5 At or 7 percentbelow the volume of periclase plus stishovite (60.8 A'). A change tothe coordination numbers g, 10, or 11 does not give promise of closepacked arrangements either, as the experience with the KzMgFr typeshows us. A coordination number of 12 could not be attained for astoichiometry MgrSiOn because it is too cation-rich and would involve'cation-cation

contact (unless of course the bonding character wouldchange under high pressures so drastically as to allow cation-cationcontacts). Of most promise would be S-coordinated structures but itis a curious fact that no Mrt"tXtulOu or MrtEtXLstO4 structures seemto be known at present. This may be related to the inefficiency ofpacking of S-coordinated polyhedra. The volume of 56.5 A'is 23 percentbelow the volume of forsterite. Since it is difficult to envision muchcloser packed atomic arrangements it has to be assumed that furthercompressions must be achieved by a reduction in the lengths of thedistances between the atoms. A volume 23 percent lower than thevolume of olivine seems therefore to be near the limiting volume whichcan be achieved by a rearrangement of coordination numbers. It appearspossible that MgzSiOr phases at pressures higher than the assumedstability limit of the hypothetical strontium plumbate type returnto arrangements based on oxygen close packing but with appreciablyshortened oxygen-oxygen and cation-oxygen distances. This shouldimply that density increases with increasing pressure will be smallerfor the post strontium plumbate phases than for smaller pressrresand is consistent with the smaller seismic velocity-depth gradientsassumed in the lower mantle (see discussion by Ringwood, 1970).

CoNcr,usroN

The work reported above shows that it is possible to predict certaindetails of complete crystal structures by starting with chemically rea-sonable interatomic distanees and adjusting these distances to eachother by distance least squares refinement. This method makes use of

POLYMORPHS OF Ms,SiO+ 729

a computer for the modeling of crystal structures and allows therdforea rapid application of the type of geometric analysis which was per-formed by Kamb (1963) for olivine and y-Mg2SiOa. The crystal struc-tures investigated here are all of the same type: they represent ratherdense polyhedral linkages with extensive sharing of polyhedral edgesand corners. Meier and Villiger (1969) had investigated (under a verydifferent viewpoint) the open tetrahedral frameworks, mostly of feld-spars and zeolites, where only corner-sharing is occurring. Shoemakerand Shoemaker (1967) applied the method to an intermetallic com-pound. The approach should be extended to further classes of com-pounds. rt could also be refined by allowing explicitly for electrostaticinteractions. This could be done either by straight electrostatic calcula-tions, or by introducing pseudo-electrostatic terms into the DlS-refine-ment: for instance the Mg-Mg repulsion could be approximated by aninput distance d(Mg-Mg) which is deliberately chosen at a largervalue than actually observed in known crystal structures.

The fact that the DlS-refinement method allows us to investigatecrystal structures geometrically without recourse to any bondingmodels is its strength rather than a weakness. This is so becausedifferent investigators may disagree (and do disagree very often)about the bonding theory applicable to a given crystal structure. Therecan be however very little disagreement about the lengths of observedinteratomic distanees even when the interpretations in terms of bond-ing models do differ. Therefore the use of the geometric refinementmethod may be termed crystal chemical positivism. In this applicationthe positivism has proved to be fruitful. In our present state of knowl-edge, neither the ionic theory (electrostatic potential calculation) northe covalent approach would have allowed the calculation of unit eellconstants and positional coordinates of the MgrSiOn polymorphs.

The simulation of crystal structures by eomputer opens the way forthe investigation of erystal structures at simulated conditions underwhich their study is experimentally difficult or impossible. ft mayaid in deciding between different models for a particular structurewhen the experimental evidence is inconclusive. Structure simulationwill also allow the testing of hypothetical structures or of hypothesesabout, the behavior of erystal structures under varying conditions. Itmay help in predicting which isomorphic substitutions in a parbicularstructure should be possible and even more, at which point a struc-ture should become unstable and show a transition. In short: crystalstructure simulation appears to have many potentially interestingand/or useful applications.

730 WERNEN H. BAAR

ACKNOWI,EDGMENTS

I thank Dr. w. c. Forbes for the critical reading of the manuscript, him and

Drs. W. A. Dollase and P. B. Moore for fruitful discussions, Dr' W' M' Meier for

a cop]r of the DLS program, and the Computer Center of the University of

Illinois at Chicago for computing time.

RnrunoNcns

Baun, W. H. (1961) uber die verfeinerung der Kristallstrukturbestimmung

einiger vertreter des Rutiltyps. III. btr Gittertheorie des Rutiltyps. .4clo

C rg s tall.o gr. r4, 209-213.- (1970) Bond length variation and distorted coordination polyhedra in

inorganic crystals. Trotts. Amer' Crystal,Xogr- Assoc. 6, 129-155'- (lg7la) The prediction of bond length variations in silicon-oxygen bonds.

Amer. M'ineral. 56, 1573-1599.- (1971b) Geometric refinement of the crystal structure of p-MgSiOi.

Nature, Phus. $ci.233, L35-137.eNr A. A. KneN (1971) On Rutile-type Compounds' IV' SiO',-GeO' and

a comparison with other Rutile-type structures. Acta crgstallny. 827,

2t33-2139.Brwws, R. A. (1970) (Mg, Fe)*SiOa spinel in a meteorite. Phys. Ea.rth. Plnnet.

Interinrs,3, 156-160.Brncn, F. (1966) Compressibility; elastic constants. GeoL Soc' Amer' Mem' 97,

97-r73.Brnr,o, J. D., G. V. Grrns, P. B. Moonn, eNn J. V. Srrrfir (1968) Crystal structures

of natural olivines. Amer. Mineral. 53, NV-824.Bna,cc. W. L.. eNn G. B. Bnowr (19%) Die Struktur des Olivins. Z. Kristalbgr.

63. 528--556.Bnnur,on, B., elln H. G. F. WrNrr,nn (1954) Die Struktur des K"MgF,. HeidcL

berger B ei,tr. M'ineral. Petrogr. 4, 6-11.Dnrrs, V. A. (1970) The relationship between the average a,nion-anion distances

and anion-cation distances in the simplest crystal structure units: Tetra-

hedra and octahedra. Kristathgra:fiyq 15, 913-917 [Transl. Sou. Phgsics

C rystallo gr. 15, 795-7981.Grr,r,nsprp, R. J. (1963) The valence-shell electron-pair repulsion (VSEPR) theorv

of directed valency. J. Chem. Educ.4o,295-301'I[,rNro, K. (1965) BeitrEge zu Kristallstrukturen vom olivin-Typ. Beitr. Mineral

Petrogr.11, 535-558.KeMs, B. (1968) Structural basis of the olivine-spinel stability relalion. Amer'

Mineral. 53, 1439-1455.Krraw, A. A., W. H. Baun, ervo W. C. Fonsns (1972) Synthetic magnesian merri-

huit€, dipotassium pentamagnesium dodeca-silicate: a tetrahedral magnesio-

silicate frarnework crysta.l structure. Acta Crgstallogr. 828, %7-n2.Mnron, W. M., eNn I[. Vnr,rcon (1969) Die Methode der Abstandsverfeinerung

zur Bestimmung der Atomkoordinaten idealisierter Geri-iststrukhtren' Z.

Krtstallogr. 129, 411423.Moonn, P. B., eNn J. V. Sntrru (1970) Crystal structure of ,e-MgbSiO': crystal

chemical and geophysical implications' Phys. Earth Planet' Inter'iots, 3,

166-177.Monnroro. N.. S. Arrrtoro, K. Koto, eun M. Tororelrr (1970) Crystal struc'

POLYMORPHS OF Ms,SiOa 7gl

tures of high pressure modifications of MnzGeOn and Co"SiO+. Phys. EarthPlanet. Interi,ors, S, 161-16b.

Peur,rxo, L. (1960) The Natwe of, the Chemical B,ond, (Brd. ed.), Chapter 18,p. 505, Comell Univ. Press, Ithaca, New york.

Rnrn, A. F., ewn A. E. Rrxcwooo (1g70) The Crystal Chemistry of Dense MsOrPolymorphs: Eligh Pressure CanGeOn of KrNiF. Structure Type..I. Sol. StateChem. r,557-565.

RrNcwooo, A. E. (1970) Phase transformations and the constitution of the mantle.Phys. Earth" Plmtet. Interinrs,3, l0g'1b.5.

ervn A. Me.ron (1910) The system MgSiO._FeaSiOr at high presures andtemperatures. Phgs. Earth Planet. Interiors,3, 89-108.

SneNNor.r, R. D., ewo C. T. Pnnwrrr (1g69) Effective Ionic Radii in Oxides andFluorides. Acta C rystallogr. 825, g2E-946.

Suonntaron, C. B., awo D. P. Ssonruernn (1g67) The crystal structure of the Mphase Nb-Ni-Al. Acta C rg stalla gr. 23, 2BL-ZJ8.

Tntirvrnr,, M. (1965) Zur Stmktur der Verbindungen vom SrzPbOcTyp. Naturwis-senschal ten, 52, 492493.

Vrr,r,rcnn, H. (1969) DI"S-MonuaI, ETII, Ztirich..Weosr,or A. D., A. F. Rprp, eNo A. E. Rrwcwoou (1g68) The high pressure form

of MnuGeO., a member of the olivine grovp. Acta CrAstallngr. B24r 7tL0-742.

Marruscri,pt recei,ued, Actuber 22, 1g71; accepted for publi,cation, Febru,ary g,1972.