American Mineralogist, Volume 59, pages 1083-1093, 1974

Distortions in the Goordination Polyhedra of fi Site Atomsin Olivines, Clinopyroxenes, and Amphiboles

MrcHesr E. FrnEr

Department ol Geology, Uniuersity ol Western Ontario,Lon:don, Ontmio, C anada

Abstract

Distortions resulting from shared edges in the coordination polyhedra of metal (M) siteatoms in olivines, c2/c clinopyroxenes and (Ca),Mg,Fe'?* amphiboles have been investigatedusing published crystal structure data. In the clinopyroxenes there is qualitative agreement be-tween the apparent stretching of M(l)-M(l) and M(2)-Si distances associared with sharedpolyhedral edges and the calculated relative electrostatic potentials due to the M site atoms.The stretching is attributed to interatomic repulsion and decreases in the sequences, Al ; Fe"*> (Cf-, In) ) Mg ) Mn'* for M(l) atoms and Ca ) Li ) Na for M(2) atoms. In Caamphiboles, in agreement with calculated data, the repulsive force due to Mg is apparentlygreater than that due to Fd*. However, in olivines the repulsive force due to Fe'* appears to begreater than that due to Mg.

Although the t / orbitals on transition metal atoms must project toward octahedral edges,octahedra coordinated about transition metals do not show a consequent reduction in shared-edge-related distortions. Thus, 3d electrons in tn, orbitals do not appear to screen the eftectiveatomic charges along M-M and M-Si directions.

In olivines variations with composition of M(2)-Si distances, individual M-O bond distancesand O-M4 and O-Si-O bond angles indicate slight departures from the linear trends expectedfor ideal solution.

fntroductionIn ferromagnesian minerals the nearest-neighbor

octahedra and eight-fold coordinated polyhedra ofmetal (M) site atoms share edges with each otherand, frequently, with the coordination tetrahedra ofatoms on adjacent tetrahedral (Z) sites. Sharedoctahedral edges are usually shorter than the idealvalues for regular octahedra and, since the corre-sponding M-M distances are usually greater thanthe ideal values, this effect is generally attributed torepulsion between the positively charged metalatoms. Pauling (1929) summarized the effects ofshared coordination edges on complex ionic crystalsin the following rule, "The presence of shared edges,and particularly of shared faces, in a coordinatedstructure decrease its stability; this effect is large forcations with large valence and small coordinationnumber. . . ." Thus, the repulsion between fuI atomsrelated through shared coordination edges should in-crease with increase in effective atomic charge (2"6)and with decrease in ionic radius (R). The repulsiveforce experienced by an M atom from an edge-related M atom must be proportional to the electro-static potential of the second M atom at the nucleus

of the first, which is given by 2"6/(M-M), whercM-M is the interatomic separation. In general, theeftective force due to an M atom coordinated withoxygen must be proportional to the electrostaticpotential Z.ff/(M-O).

The dependence of polyhedral distortions in in-organic crystals on simple electrostatic bond char-acteristics has been supported by Baur (1970). Theimportance of the repulsion between cations relatedby shared polyhedral edges was stressed in this studyalthough quantitative estimates of the extent of theserepulsions were not attempted.

In the present study on some olivines, clino-pyroxenes, and amphiboles, those polyhedral distor-tions which may be referred to repulsion betweenedge-related M atoms are outlined, and an attemptis made to correlate the extent of these distortionswith estimates of the repulsive forces involved. Theaims of this study, then, are quite limited, and fullrationalizations of the structural characteristics ofthese phases will not be attempted. The M atomsconsidered are Li, Na, and Ca in eight-fold coordi-nated sites in clinopyroxenes, Al, Fe3*, Cr3*, In, Mg,and Mn2* in octahedral sites in clinopyroxenes, and

1083

1084

Mg and Fe2* in octahedral sites in olivines and amphi-boles. The distortions are documented throughanalysis of specific bond angles and interatomic dis-tances. Estimates of overall polyhedral distortionfrom variance of polyhedral angles and quadraticelongation (Robinson, Gibbs, and Ribbe, 1971) andfrom mean-square relative deviations from averagebond lengths (Brown and Shannon, 1973) are notsuited to the present investigation because repulsiveinteractions between metal atoms are not the onlycauses of distortion in the various polyhedra ana-lyzed, and also these interactions are directionallydependent.

There are possible additional complications foredge-related interactions between transition metalatoms. In ideal octahedral coordination, the 3d elec-trons on transition metals will be distributed be-tween the stabilized set of three /2n orbitals and adestabilized set of two es orbitals. Tlte en orbitalsproject along the M-O directions, toward the ligands,but the /20 orbitals project toward the octahedraledges, that is along the M-M directions betweenedge-related M-site atoms. With more polarizableligands (S2*2,As2-2, and so on) the lpe orbitals extendoutward to the extent that attractive, bonding reac-tions may occur between singly occupied orbitals onedge-related atoms (Fleet, 1973), and repulsive re-actions may occur between filled orbitals; for ex-ample, in compounds with the marcasite structureinteractions between /2o orbitals appear to have beenthe dominant factors controlling distortions in thecoordination octahedra of the metal atoms (Pearson,1965; Nickel,1968; Brostigen and Kjekshus, 1970).Although /ro orbitals are assumed to be deeply buriedin oxygen coordination polyhedra, electron transferbetween overlapping /2, orbitals has been proposed toexplain the Fe2* -+ Fe3* charge transfer bands inferromagnesian silicates (Burns, l97O), and. there isa possibility that t2o orbital interactions might be afactor in the distortion of edge-related polyhedra ofM site atoms. However, these t2o orbitals more likelycontribute to the distortions of these polyhedra byscreening the nuclear charge on transition metalatoms along the M-M directions. The screeningefficiency of d orbitals must have a pronouncedangular dependency, and electrons in /2, orbitals, byprojecting in M-M directions, might tend to neu-Iralize the nuclear charge in these directions. Thishypothesis can be critically evaluated in the presentstudy.

It should be noted that the high spin 3d electron

configurations expected for CrBt, Mn2*, Fez*, and Fe3*in octahedral coordination with oxygen do not resultin unequal electron population of eo orbitals. Con-sequently, axial Jahn-Teller distortions are not an-ticipated in the nearest-neighbor environments ofthese cations.

Electrostatic Potentials

There has been much discussion in recent years ofcovalent bonding forces in silicate compounds (for

example, Gibbs, Hamil, Bartell, and Yow, 1972;Louisnathan and Gibbs, 1972a, 1972b). Accept-ance of a model involving molecular orbitals on theoxygen atoms to account for features in the Z-Obonds requires that these orbitals must project towardthe M atoms also, resulting in some degree of orbitaloverlap in the M-O bonds. Thus, the effective chargeon a metal atom (Z"rt) is less than that indicated byits valence.

An estimate of Z"y may be obtained for any Matom from the relationships,

2.11 : iZ,

where i is the fractional ionic character of the M-Obond and Z isthe ideal ionic charge, and

i : | - e-o '25AX2 (Paul ing, 1960),

where aX is the difference in electronegativity be-tween 02- and M. Batsanov ( 1 968 ) has recommendeda revised expression for the relationship between I andL,X, i = | - ro.le aX2, which is calibrated against idata obtained from the dipole moments of the hy-drogen halides. However, the earlier expression,although based on an erroneous i value for HF,clearly provides a better fit to the measured i datapresented by Pauling ( 1960, Fig. 3-8 ). Accordingly,values of 2.6 (Table 1) have been calculated withelectronegativity data from the recent compilation ofBatsanov (1968, Table 5) using the original Paulingexpression. The resulting values of the electrostaticpotential Z"ff/ (M-O) for the M atoms in octahedralcoordination are for ideal M-O distances calculatedusing the data for effective ionic radii of Shannonand Prewitt (1969). These data, used qualitatively,

suggest that interatomic M-M and M-T repulsionshould decrease in the sequence, Al ) Cr3* > In- Fe3* - Mg'* ) Mn2* ) Fe2*. The analogous elec-

trostatic potential data for the eight-fold coordinated

M site atoms (Table 1) allow the prediction thatrepulsion with a common M or T atom should de-

crease in the sequence Ca ) Li > Na.

MICHAEL E. FLEET

M SITE ATOMS IN OLIVINES. CLINOPYROXENES. AND AMPHIBOLES 1085

However, it is well-known that atomic charges cal-culated from electronegativity data have only ap-proximate significance. This is ernphasized by theconsiderable scatter in the calibration data at highvalues of aX (Pauling, 1960, Fig. 3-8 ). This calibra-tion is based largely on the dipole moments ofgaseous diatomic halide compounds, whereas thepresent data are for six- and eight-fold oxygen co-ordinated complexes in solids. Also, no attempt hasbeen made to correct the electronegativity data usedfor charge effects and hybridization of the oxygenbonding orbitals, and it might be expected by con-sideration of the electroneutrality principle (Pauling,1960) that a proportion of the atomic charges wouldbe neutralized by o-bonding. Clearly, before accept-ing this 2.11 data, consideration must be given tocharge data obtained by other methods.

A compilation of effective charge data for metalatoms in oxides obtained by a variety of experimentaland theoretical methods (Bartenev et al, 1972)shows a wide variation. The ranges for the cationsincluded in this study are Mg (1.0 to 1.76), Al(1 .23 to 2 .43 ) , Mn (1 .18 to 1 .34 ) , Fe ' . ( 1 .03 to1 .32 ) , Fe3 . (1 .58 to 2 .15 ) , and Ca (1 .18 to 1 .66 ) .The charges from electronegativity data in Table 1for Mg, A1, Mn, and Ca fall within the respectiveranges of the Bartenev et al (1972) data but thecharges in Table 1 for Fe'z- and FeS* are lower thanthose reported in their study. Other relevant, com-parative charge data are: Mg in MgSiO3 (1.0O), Alin various silicates (0.90 to 1.20), and Na in Na2SiO3(0.85), estimated from line shifts in KoX-ray spectra(Urusov, 1967); M site atoms of lunar and terrestrialolivines (0.13 to 0.34), by crystal structure refine-ment with X-ray diffraction data (Wenk and Ray-mond, 1973); Na in kernite (0.43 to 0.57), bycrystal structure refinement with the Extended Z-shell method (Coppens, 1972); Mg in MgO'o-u(1.26) and Al in ,{10,-6 (1.96), from molecularorbital calculations (Tossell, 1973). Clearly, thedata obtained by other methods are fragmentary anddo not provide a consistent basis for comparison withthe charge data from electronegativities. A partialexplanation for the discrepancies in the data mustbe related to the definition of the charge being tnea-sured or calculated in a particular case and the re-lationship of this charge to the atom involved; forexample, Cochran (1961) has cautioned that atomiccharges determined from X-ray diftraction may beambiguous and are dependent on the particularatomic descriotions used.

Tlsr-n 1. Electrostatic Valence Terms

! rm E lec t ronegat iv l t y Deta Pos i t ion MeEhod

Data

l,letal

Lontc

rad lua t

1dea1

ion iccharge(z)

Elec t ro - Ion ic E f fec l -

negaliv- char- ive

i ty ac te r a tomic(x ) o f U-0 charge

bond (Z-++)( i )

E lecCro- E f fec t - E lec l to -

s ta t i c i ve s te t i cpotent- atomic potent-

ia1 , charge ia l, l r t' e f f ' \ -e f f ' ' e f f . '

(M-o) (M-o)

1 . 6 0 . 5 97 . 6 0 , 5 91 . 4 0 , 6 71 . 8 0 . 5 11 , 9 0 . 4 71 . 8 0 . 5 I

1 . 0 0 . 7 90 . 9 0 . 8 21 . 0 0 . 7 9

ltg +ZA1 +3cr +3Mn +2Fe +2Fe +3In +3

'Ila +1ca +2

o . 7 2 00 . 5 3 00 . 6 1 5o . 8 2 0o . 7 7 0o . 6 4 50 . 7 9 0

1 . 1 6L . t 2

+ 1 . 8+ 1 . 8

+ 1 . 0

i{ .8{{ .8+ 1 . 6

o . 6 9o . 9 20 . 8 E0 . 6 00 . 4 70 . 6 90 . 7 0

0 , 3 7o , 3 20 , 6 3

+ 1 . 1 4 0 . 5 4+2,43 r ,26

+1.18 0 .53+1.L2 0 .52+2. t5 r ,05

+1.45 0 .58

Data f rm Shannon and Prewi tE (1969) . t (L i+ ) fo r s ix coord ina l ion .

Thus, although the Z"u data used in this study aresomewhat crude, an alternative, self-consistent set of

charge data covering the range of metal atoms in-

vestigated is not available. Self-consistency is an

important requirement in such data. The present useof effective charges is purely qualitative, and muchmore weight is given to the relative order of the elec-trostatic potential data than to the absolute values

of the charges. In this connection, the errors in the2"6 data used are expected to vary systematicallywithin the groups of chemically related polyhedral

complexes analyzed.In support of this, independentlydetermined chemical parameters frequently showgood correlation with quantities calculated from

electronegativity values; for example, binding energies

determined by electron spectroscopy for groups ofrelated compounds correlate well with calculatedatomic charges (Siegbahn et a|,1967).

Finally, Z* data obtained by an independentmethod, annihilation of positrons (Bartenev et al,1972), are presented for comparison (Table 1).These data indicate relatively higher charges on Al

and Fe3* so that, for octahedrally coordinated M

atoms, the interatomic repulsion is expected to de-crease in the sequence Al ) Fe3* )> Mg : Mn2* )Fe2*.

Olivines

In the olivine structure the oxygen atoms formapproximate hexagonal close-packed arrays normalto the a-axis: one-half of the octahedral intersticesare occupied by M site atoms and one-eighth of thetetrahedral interstices are occupied by Si. The struc-ture has been discussed in great detail by variousworkers (Hanke, 1965; Birle et al, 1968; Kamb,1968; Baur, 1972). There are two crystallographi-

1086

o(3')

Fro. l. M site environments in fayalite; M atoms, smallopen circles; oxygen, large open circles with r coordinates(x tOO;; Si, small full circles.

cally distinct M sites: M(l), point symmetry 1; andM(2), point symmetry m. The coordination octa-hedron of M(I)-site atoms (Fig. 1) shares six edges,two with M(l) octahedra, two wrthM(2) octahedra,and two with Si tetrahedra. The coordination octa-hedron of M(2)-site atoms shares three edges, twowith M( 1 ) octahedra and one with a Si tetrahedron.In both octahedra, the shared edges are associated toform octahedral faces which contribute to the "empty"tetrahedron of oxygen atoms (Hanke, 1965; Kamb,1968). The distortions from ideal octahedral sym-metry result from accommodation of the tetrahedraledges and internuclear repulsion associated with al1of the shared edges. The M(1) octahedron is stretchedas a result of the repulsive forces along one "trigonal"axis. The M(2) atom is displaced away from theoctahedral face formed of shared edges. Mg and Fe2*are generally disordered among the M(I) and M(2)sites in all compositions in the forsterite (Fo)-fayalite (Fa) sdries. The X-ray diffraction study of

Tenre 2. Edge-Related Interatomic Distances (A) andAssociated Bond Angles (') in Olivines

Bitle et ql (1968) did not detect any M site ordering,

and the site ordering reported by other investigators-fe1 g1zrnple, Finger (1969/1970), Wenk and

Raymond (1973), Brown and Prewitt (1973)-is

slight.The present study has been made using the crystal

structure data of Birle et al (1968) for four olivines,forsterite (Mg6.e6Fea 1,0SiO1), hyalosiderite (Mgo.uru

Fes.a56Mnn naoCao.oosSiOr ), hortonolite (Mgo.roFeo +eMnn.61Cae.61SiO4), and fayalite (Feo szM&.oaSiOr).The polyhedral edge-related M-M distances (Birle

et al, 1968) and M-Si distances and the associatedenclosing O-M-O bond angles lor M-M interactions

and O-Si-O bond angles are given in Table 2. The

deviations of the non-equivalent M-O bond distances

from the mean M-O distances for each site are given

in Table 3.In the olivine structure, the dominant cohesive

forces are the Si-O and M-O bonds. The ideal struc-

ture based on hexagonal close-packed arrays of oxy-gen atoms must distort to allow adustment of these

bond lengths to equilibrium values. The principal dis-

tortion in the structure is contraction of the Si-O

tetrahedra (Kamb, 1968). Further structural stabil-ization results through increased separation of the M

atoms, away from the shared coordination polyhedra

edges (Kamb, 1968; Bav, 1972). This is accom-panied by shortening of the O-O distances along the

shared edges, which allows greater M-M and M-Si

separations without undue stretching of the M-O and

Si-O bonds. In the present study O-M-O and O-

Si-O bond angles are used, where applicable, as mea-

surements of relative stretching of. M-M and M-Si

distances rather than the associated shared-edgelengths. Although the latter are related to the repul-

sive forces involved, they are also affected by other

factors, for example, size of the coordinated metal

Tesrn 3. Deviations (Vo) of M-O Bond Distances fromMean Values (A) in Olivines

MICHAEL E. FLEET

F o r s t e r i t e( F o 9 o .

o )

I lyalo-sider iEe( F o s : .

s )

HorEono-

( t o 4 9 . o )

Faya l i te( to4 .

o )

For- Hyalo-s te r iEe s ider i te(Fogo.o) (Fos : .s )

Hortono- Fayalitel iEe (Fo , 6 )

( o o 4 9 . o '

M ( 1 ) - M ( 1 ' )o ( 1 ) - M ( 1 ) - o ( 2 )

I r ( r ) -M(2)o ( r ) -M ( r ) -o (3 )o ( r ) -M (2) -o (3 )

M ( l ) - s io (3 ) -s i -o (2 )

M ( 2 ) - S io (3 ) -s i -o (3 )

8 6 . 6 0

3.20984.868 1 . 1 0

2 , 1 0 4lo2 .0 t

2 .800r04.51

3 . 0 1 98 6 . 6 0

3 . 2 5 L44.498 1 . 0 4

2 . 7 3 r102.O9

2 . 8 3 81 0 5 . 5 9

3 . 0 2 28 6 . 4 7

3 . 2 5 98 4 . 8 08 0 . 5 1

2 . 1 3 4LO2.6l

2 . 8 4 11 0 5 . 6 3

3 . 0 5 0 M ( 1 ) - 0 ( 1 )8 6 . 3 8 - o ( 2 )

_ o ( 3 )

' i :33' ( M(r)-o)8 0 . 6 9

M ( 2 ) - o ( 1 )2 . -168 -O (2 )

1 0 3 . 0 0 - O ( 3 )-o (3 ' )

,ot ' . \ i" ( M(2)-o)

- 0 .6- 1 . 3+1 .9

2.1o3

+2 .O- 3 . 6+3.8- 3 . 0

2 . I 35

- 0 . 5

+ ) 2

2 1 ) 7

+ L . 2- 2 q

+ 5 . 1-4 .3

-L.4-1 .2

2. t29

-3 .9+5 .3-4.42 . t 54

-1 .6- I . 7+3 .2

2.L59

- ? 1

+ q ,

- 4 . 92 . t 7 I

M SITE ATOMS IN OLIVINES, CLINOPYROXENES, AND AMPHIBOLES 1087

atoms. The preceding discussion implies that the Siatoms experience repulsion from the M atoms relatedthrough shared edges; this is supported by relativelysmall O-Si-O bond angles (and relatively short O-Odistances).

The repulsive M-M and M-Si interactions leavethe M(l'l atom in the center of its coordinationoctahedron. All of the M(1)-O distances arestretched by these repulsive interactions and, in gen-eral, the distortions tend to be constrained by thestructure as a whole so as to be partially suppressed.In contrast, an M(2) atom may respond to repulsiondue to edge-related atoms by displacement awayfrom the shared edges. In the fayalite of Birle et al(1968) the mean M(2)-O distances for shared andunshared edges are 2.27 A and 2.08 A, respectively;the difference (0.19A) must largely reflect the re-pulsion resulting from the edge-related Si and M( 1)atoms. Thus, the destabilization resulting from theserepulsions is quite considerable. Variations in theindividual M(2)-O bond lengths may be used di-rectly to estimate the relative repulsion experiencedby various pairs of cations. For the four olivine com-positions used (Table 2), the deviations in M(2)-O(3) distances are appreciably greater than those inM(2)-A(l) distances, so that M(z)-Si repulsion isgreater than M(Z)-M(1) repulsion. However, thismay largely reflect the shorter ideal M(2)-Si dis-tance and cannot be used to infer that the charge ona Si atom is greater than that on either Mg or Feztatoms. The data in Table 2 do suggest, though, thatthe distortions of both M(1) and M(2) octahedraare greatest in fayalite and that Fer*-Si repulsionis greater than Mg-Si repulsion and Fe2*-Fe2* re-pulsion is greater than Mg-Mg repulsion, which iscontrary to the predictions made in the introduction.Of course, this discrepancy may result because ac-commodation of the larger Fe2* atoms may make thestructure more susceptible to interatomic repulsiveforces.

The mean M-O distances and the edge-relatedM-M and M(1)-Si distances show linear variationswith Fo content, which are expected from the knownlinear variations of the unit cell parameters ofolivines with composition and are evidence in favorof ideal solution. Ifowever, the individual M-Obonds (Table 3), the edge-related M(2)-Si dis-tances (Fig.2), and the O-M-O and O-Si-O anglesassociated with the shared edges (Table 2) indicateslight departures from the trends expected for idealsolution. Clearly, the adjustments made by the

- - -O -O ' - " - - ' '

8 0 6 0 4 0 2 0

Mlle loFo

Fro. 2. Variation of M(2)-Si distances in olivines witliforsterite content.

olivine structure to accommodate Fe2* atoms aremore complex than those attributable to merely agradual increase in the size of the unit cell. How-ever. the variations in the M-M and M-Si distanceswith composition are not consistent with possibleneutralization of the nuclear charge on Fe2* by the3d electrons in the t2n orbitals projecting along M-Mvectors. Although it cannot be concluded from thedata available that the departures from ideal be-havior are not due to occupied /2o orbitals on Fe2*,it is rather surprising to observe that these orbitalsdo not screen the nuclear charge on Fe2*.

Clinopyroxenes

The compilation of crystal structure data forC2/c clinopyroxenes of Clark, Appleman, andPapike ( 1969) represents a wide variety of end mem-ber compositions and includes data for jadeite(NaAlSizOa, Prewitt and Burnham, 1.966), Nalns*SigOo (Christensen and Hazell, 1967), johannsenite(CaMn'*Si2O6, Freed and Peacor,'1967), andspodumene (LiAlSi2O6), LiFes-SizOo, ureyite (NaCr'*Si2O6), acmite (NaFe3.Si2O6), and diopside (CaMgSizOo). The Li clinopyroxenes have the probablespace group C2 but were refined in CZ/c to obtainaverage structures. The data for these eight clino-pyroxenes are the basis of the present study.

ln the C2/c clinopyroxene structure, single chainsof Si-O tetrahedra are parallel to the c axis. The twocrystallographically distinct M sites are arranged inplanes parallel to (100) at x = O, L/2. The smallerM(1) coordination octahedron (Fig. 3) is formedentirely of six non-bridging tetrahedral oxygens. Al-though the point symmetry of.the M(I) site is 2, thedistortions of the M(1) octahedron from ideal sym-

d l ^

5" {c v

6 2.s'-

t-' , 2 ' 8

G

1088 MICHAEL E. FLEET

Ftc. 3. M site environments in diopside; M atoms, smallopen circles; oxygen, large open circles with x coordinates( x tOO;; T atoms (Si), small full circles.

metry are not very great. The M(l) octahedron hasfive shared edges, two with M(1) octahedra andthree with M(2) polyhedra. The larger M(2) co-ordination polyhedron is formed of both bridgingand non-bridging oxygens. The bridging oxygens alllie to one side of the plane parallel to (010) passingthrough the M(2) atom, and the M(2) atom is

generally displaced towards the non-bridging oxy-gens. In Na and Ca clinopyroxenes, the M(2) atomsare eight-fold coordinated and the M(2) polyhedron

is quite irregular. It shares three edges with M(I)octahedra, two with M(2) polyhedra, and three with7 tetrahedra, although the two M(2)-M(2) inter-actions and the M(z)-T' interaction involve longM(2)-O(3') distances and the resulting repulsiveforces between these juxtaposed cations must beminimal. Distortions in the M(2) polyhedron thenarise, in part, through accommodation of shortershared tetrahedral and M(1.) octahedral edges andshorter bonds to non-bridging oxygens. Repulsionbetween edge-related atoms must contribute to thedistortion in both M(I) and M(2) polyhedra. Inaddition, Clark, Appleman, and Papike (1969) havenoted that the shortest bonded distances from 7,M(I ) , andM(2) atoms are a l l to O(2) atoms. O(2)is coordinated to only three atoms, and apparentlythe additional bonding is shared among these. In

terms of the ionic bonding model, the extra negativeelectrostatic valence is compensated by extra positive

Tasls 4. Edge-Related Interatomic Distances, Associated Bond Angles, and Deviations from Mean Values of M-O Bond Distancestn C2/c Clinopyroxenes

o(3)

Selected Edge-Related Interatonic Distances (R) rna Associated Boncl Ang1es (o) in CZ/c Clinopyroxenes

11 (1 ) -M( r )o ( t ) - M ( 1 ) - o ( L ' )

M ( 1 ' ) - M ( 2 )M( r ) -M ( 2 )o ( r ) -M ( 1 ) - o ( 1 )

M(2 ) - ro ( 2 ) - r - o ( 3 )

M( 2 ) - r 'o ( 3 ) - r , - o ( 3 ' )

M ( r ) - o ( r )- o ( 1 ' )- o ( 2 ' )

( M( r ) -o )

M(2) -o ( 1 )-o (2 )-o (3)- o ( 3 : )

( u (z ) -o ) *

3 . 0 4 2 3 . 1 1 77 8 , 9 8 0 . 9

3 . 0 2 0 3 . 0 1 83 . 0 9 1 3 . 2 0 6

8 4 . 8 8 1 . 9

2 .862 2 .9LL104 .2 104 .6

3 . 3 0 9 3 . 4 2 11 0 7 , 3 1 0 8 . 9

3 . 0 6 6 3 . 0 8 97 7 . 4 8 0 . 6

3 , 1 5 4 3 . 2 0 13 . 3 8 1 3 . 4 2 6

8 6 . 2 8 5 . 8

2 . 9 8 2 3 . 0 0 01 0 5 . 9 1 0 6 . 1

3 . 1 6 8 3 . 2 0 51 0 6 . 3 1 0 7 . 1

+ 3 . 5 + 2 , L+ 0 . 3 + 0 . 6- 5 . 1 - z . o

r . 9 2 8 1 . 9 9 8

- 0 . 9 - 0 . 8+ r . ) - u . J- 0 , 6 + 1 . I

+ 1 5 . 3 + 1 5 . 32 . 3 7 8 2 . 3 9 7

3 . 0 9 2 3 . L 6 48 4 . 6 8 7 . 1

3 . 2 2 L 3 . 2 5 63 , 5 0 0 3 . 6 3 1

8 2 . 2 1 9 , 2

3 .LO2 3 .1 :26ro3 ,46 103 .80

3 . 3 1 2 3 . 3 8 81 0 4 . 1 7 l O 4 , 7 L

+ 1 . 8 + 2 . 7- 0 . 6 - 0 . 8- 1 . 3 - 1 . 8

2 . 0 7 7 2 . L 7 3

- L . T

- 3 . 0 - 5 . 5+ 5 . 6 + 8 . 2

+ 1 ? . 0 + 1 3 . 12 , 4 2 5 2 , 4 5 0

+ 4 . Lf I . J

- 5 . 31 . 9 1 9

- 4 . 8+ 3 . 0+1 . 8

+42 .22 .2 I I

+ 5 . 3+ 0 . 1- 5 . 4

2 , 031

- c 7

- 3 . 6+ 9 , 3

+41 . 32 .249

+ 0 . 2- 4 . 4

2 . 0 2 5

- 0 , 70 . 0

-o .7+ t7 .3

2 . t t L |

3 .3028 1 . 7 8

3 .2393 . 7 3 1

8 0 . 1 5

3 . 046t06.7

3 . 3071 0 9 . 5 5

+ 3 . 2+ 0 . 7- 4 . 0

2 .L42

- 0 . 9- 2 . 8L t 1

+18 . 52 . 4 6 6

3 .1891 0 a

3 . 1 7 23 . 5 2 7

8 3 . 1

3 . 028t 0 5 . 6

3 . 2 3 81 0 7 . 1

Dev ia t ions (z ) o f M-o Bond D is tances f rom Mean va lues (R) in c2 /c c l inopyroxenes

* Mean of six nearest oxygens

electrostatic valence on the adjacent cations. fn acovalent bonding model the O(2) atom could formo bonds with distorted sp, triangular hybrid orbitals,and the extra bonding could be contributed through,r.bonds betweenthe p" orbital on O(2) and orbitals onthe adjacent atoms.

Selected edge-related M-M and M-? distancesand associated enclosing O-M(l)-O and O-Z-Obond angles are given in Table 4; the M(l)-M(l)distances and all of the bond angles are from Clark,Appleman, and Papike (1969).

The deviations of the non-equivalent M-O dis-tances from the mean M-O bond distances for bothM(l) and. M(2) atoms are given in Table 4; thedata for M(2) atoms are relative to ttyg six shortestM(z)-O distances to allow realistic cornparisonswith the Li clinopyroxene data.

The problem of associating specific distortions incoordination polyhedra with the repulsive forces be-tween individual pairs of cations is more difficult forclinopyroxenes than for olivines because of thegreater complexity of the clinopyroxene structure. Inparticular, the Si-O tetrahedra may not move inde-pendently of one another, and there appears to be alarge variation in M(2)-O bond strengths. More-over, the data for deviations in the M(l)-O(2)bonds and, to a lesser extent, in the M(1)-O(1)bonds (Table 4) do suggest a certain amount ofM(2) control on the configuration of the M(1)octahedron. However, the data for the deviations inthe M(2)-O bond distances (Table 4) do not showsystematic correlations between those pyroxenes withcommon M(l) atoms, and it appears thatthe M(2)polyhedron configuration is largely independent ofthe M(l) atom.

The deviations in the M(2)-O(3) bond distancesand the variations in the O(2)-f-O(3) bond anglesdo suggest that the M(Z)-S| repulsion varies sys-tematically with the type of M(2) atom and de-creases in the sequence Ca-Si > Li-Si ) Na-Si. inagreement with the predictions made in the introduc-tion.

The O(1)-M(l)-O(l) bond angles are appre-ciably less than 90" and decrease fairly systemati-cally with M(I)-M(2) distance (Fig. a). However,the closest approach of M(1) and M(2) atoms isreally defined by the M(7,)-M(2) distances (Fig.5), and the acute angles result, in part, because theM(l) atom is drawn by the multiple bonding towardthe O(2) atoms on the opposite side of the M(l)octahedron. Also, repulsion between M(l) and

1089

8 6 a NoAIa N o C r

. L | A I

a

CoMg r

No Fe

' LiFe

o No ln

C o M n O

3 . 2 3 - 4 3 . 6 3 . 8

M(l)-M(z) Distance (A)

Frc. 4. Variation of O(1)-M(1)-O(1) bond angle in C2/cclinopyroxenes with M(l)-M (2) distance.

M(2) atoms cannot be too significant since bothM(l')-M(z) and M(l)-M(2) distances increasemore or less proportionally to the sums of the ionicradii of the M(l) and M(2) cations.

The O(|)-M(I)-O(1') bond angle varies withthe type of M(l) atom (Table 4), suggesting thatM(l.)-M(l) repulsion decreases in the sequence:Al ) Fe3* t Crs* > In ) Mg ) Mn,.. Distortion dueto M(1,)-M(1) repulsion is illustrated iq. plots ofM(l)-M(l ') against mean M(l)-O bond distance(Fig. 6). The relative stretching of the M( l)-M(l ')distances suggests that M(l)-M(l,) repulsion de-creases in the sequence Al ) Fe3* > In > Ci'- >>Mg > Mn2*. These sequences are fairly consistentwith the data in Table 1.

The M(l) atoms include a variety of group IIand III elements and 3d transition metals. Once againthere is no evidence that occupied t2o orbitals on thetransition metals (Crs", Fe3*, and Mn,.) perturb thecoordination polyhedra containing them, and it ap-pears that electrons in these orbitals do not screenthe nuclear charge very efficiently along M-M di-rections.

8 8

84

82

80

78

M SITE ATOMS IN OLIVINES, CLINOPYROXENES, AND AMPHIBOLES

o

(Dc')c

Ecoco

oI

I

o

o<

Q)Oco-aO

N

I

=

1090

3.8

3.6

Noln o

o CoMn

NoFe t a coMg

NoCr o

NoAl o

o LiFe

O L |AI

3.O 3.2 3.4M (l' )-kI (2) Distance (A)

FIc. 5. Variation ot M(l)-M(2) distance in C2/c clinopyroxenes with M (l' ) -M(2 ) distance.

Amphiboles

The amphibole minerals are complex chemically,and while a great deal of crystal structure work hasbeen done on them there is insufficient data to allowa worthwhile comparison of edge-related distortionsin either ideal end-member compositions or in repre-sentatives of a single solid solution series. However,structural refinements have been made on three(Ca),Mg,Fe'z* amphiboles, tremolite (Papike, Ross,and Clarlg 1969), actinolite (Mitchell, Bloss, andGibbs, l97l), and grunerite (Finger, 1969), and apreliminary study on the sites efiectively dominatedby Mg and Fe2* may be mentioned.

The tremolite structure consists of double chains,or bands, of Si-O tetrahedra parallel to the c axiswith the four non-equivalent M sites (Fig. 7) ar-ranged in strips parallel to (100). The M(4) atomcoordination polyhedron is essentially equivalent inalmost all structural details to the M(2) polyhedronin clinopyroxene, but it is occupied by Ca in thetremolite studied, by Ca(Na) in the actinolite and by

N o l n o

NoFe 'o LiFe o CoMn

NoCrNoAl '

aO L iA I

. CoMg

3.4

3.2

3.O

3 .3

3.2

3 .O

3 .1

2.22 .12.OMean M(l)-O BondLength (A)

Frc. 6. Variation ot M(l)-M(l') distance ia C2/c clino'pyroxenes with mean M(l)-O bond length.

Fe'-(Mg) in the grunerite so that realistic compari-sons of distortions within these polyhedra cannotbe made. T}rre M(l), M(2), and M(3) octahedra(Fig. 7) are all approximately equivalent to theM(1) octahedron in clinopyroxene, being only slightlydistorted and formed of non-bridging oxygens, al-though (OH)- groups do contribute to thre M(l)and M(3) octahedra. The M(1) site has point sym-metry 2: the M(1,) octahedron shares six octahedral

Frc. 7. M site environments in tremolite; M atoms, smallopen circles; oxygen, large open circles with r coordinates(x 100); hydroxyl gxoups' stippled; r atoms, small fullcircles.

MICHAEL E. FLEET

o {

(Doc(f

Fad

I

c / 2

M SITE ATOMS IN OLIVINES, CLINOPYROXENES. AND AMPHIBOLES 1 @ 1

edges, two vrrth M(2) and, M(3) octahedra and oneeach with a M(l) octahedron and a M(4) ply-hedron. T\e M(2) site has point symmetry 2 also;the M(2) polyhedron shares five octahedral edges,two with M(I) octahedra, two with M(4) polyhedra,and one with a M(3) octahedron. The M(3) sitehas point symmetry 2/m; the M(3) octahedronshares six octahedral edges, four with M(l) octa-hedra and two with M(2) octahedra. The O-M-Obond angles associated with shared edges are allsignificantly less than 90o and, as a result of therepulsion between the edge-related M site atoms,the M(l), M(2), and. M(3) octahedra are com-pressed parallel to [100], each being essentiallytrigonally distorted.

When the edge-related M-M distances are plottedagainst mean M-O bond lengths (Fig. 8), certainstructural controls on the polyhedral distortions arequite evident. In particular, the M-M distances paral-lel to the D axis (M(l)-M(1, ' ) , M(1)-M(4), andM(2)-M(3) ) are stretched the most and must re-flect the relative rigidity of the Si-O tetrahedral bandsalong their lengths (parallel to the c axis). Also, theM(l,)-M(I' ) distance is dependent on the nature ofthe M (4) cat ion but the M ( l )-M (2), M (I)-M (3),and M(2)-M(3) distances appear not to be. How-

ever, the data do show the expected general increasein size of coordination polyhedra with increase inFe2* content. In the Ca amphiboles, though, theM(l)-M(4) and M(2)-M(4) distances do not in-crease with Fe'?* content; one may conclude, tenta-tively, that electrostatic repulsion from Mg exceedsthat from Fe2* in these minerals.

Conclusions

The apparent magnitudes of the M(l)-M(I) andM(z)-Si repulsions of edge-related coordinationpolyhedra in C2/c clinopyroxenes show partial cor-relation with the electrostatic potentials due to theM atoms calculated from electronegativity data. Theprincipal exception to this relates to the electrostaticpotentials calculated for Fe3* in LiFeSizOe and acmite(NaFeSi2O6) and for CrS* in ureyite (NaCrSi:Oa);when the O(l)-M(l)-O(1') bond angle is plottedagainst electrostatic potentials calculated from bothelectronegativity and positron Z"s data (Fig. 9) thecorrelation is substantially improved.

The M-M repulsive distortions in (Ca),Mg,Fe2-amphiboles tend to be constrained by the doublechains of Si-O tetrahedra. However, in Ca amphi-boles, M(1)-{a and M(2)-Ca distances do suggest,in asreement with the calculated data. that the elec-

"S( l) 3.3Uco6O

3 . 2I

Z . U t ) 2 . t 42 . r o2 . 1 42 . to 2 .O6 2.O6 2 . to 2 . 1 4

Meon M-O Bond Lenqth (A)Frc. 8. Variation of. M-M distance with M-O bond length for M(l), M(2), and M(3 ) sites

of (Ca),Mg,Fe2. amphiboles; tremolite, open circles; actinolite, half-open circles; grunerite, fullcircles; Mg and Fe'* data refer to individual site occupancies.

e g0valvt+lo-66,"

'tnco

o

n

-M('to/ t-M.,AZd

-M(3)"

o . 9 . o

=b tY Y F

j -

; gS mP i\i

o

. e

M@-M(4) o1!e

o o . ot L u= c o o )Y n c o-

o - o -o o

N =s / \o -

o

-ae--M(l) o-

to92

88 o-{ CoMn

H coMg

o- NoAl--o

86

84

82

80

78

0.6 0 .8 l .o 1 .2

z.ffl(M-o\

FIc. 9. Variat ion of a(l)-M(l)-O(1') bond angle inC2/c clir'opyroxenes with electrostatic potential, Z"n/(M-O) calculated frorn electronegativity data (solid circles)and positron method data ( open circles ) .

trostatic potential due to Mg is apparently greaterthan that due to Fe2*. In contrast, in olivines, distor-tions related to both M-M and M-Si repulsive inter-actions indicate that the repulsion due to Fe2* isgreater than that due to Mg.

Quantitative comparisons between M-M and M-Tdistortions and calculated electrostatic potentialshave not been attempted in this study because of theuncertainty in the calculated potentials and the dif-ficulty in isolating the effects of specific distortingforces in complex structures. For example, in clino-pyroxenes it has been noted that there is some M(2)atom control on the distortions of the M(1.) alomoctahedron. It is expected that M(1.')-M(2) repul-sions will tend to negate the effects of. M(L)-M(I')repulsions (Fig. 3) and that this tendency will de-crease in the M(2) site occupancy sequence Ca )Li > Na: this is largely confirmed in plots of theO(I)-M(I)-O(1') bond angle against Z.rr/ (M-O)(F ie .9 ) .

However, the present approach does demonstratethat, for olivines and C/2c clinopyroxenes, 3d elec-

trons in l2o orbitals do not appear to screen theeffective atomic charges of transition metal atomsalongM-M and M-Si directions.

Acknowledgments

This work was supported by a National Research Councilof Canada operating grant.

References

BmreNnv, G. M,, A. Z. Vlnrsov, V. I. GoroeNsxtr, E. P.PRoKopEv, lNo A. D. Tsvc,tNov (1972) Determinationof the effective charges of anions in media of the ionictype by the positron method. Russ. Chem. Reu. 41r 305-3 1 3 .

BerseNov, S. S. (1958) The concept of electronegativity.Conclusions and prospects. Russ. Chem. Reo' 37r 332-351-

Beun, W. H. (1970) Bond length variation and distortedcoordination polyhedra in inorganic crystals. Trans. Am.Crystallogr. Assoc. 6, 129-155.

(1972) Comptrter-simulated crystal structures ofobserved and hypothetical Mg,SiO. polymorphs of lowand high density. Am. Mineral. 57,709-731.

Brnre, I. D., G. V. Gnss, P. B. Moone, eNp J. V. Surrs(1968) Crystal structures of natural olivines. Am. Min-eral. 53, 807-824.

Bnosrrcex, G., lxr A. K.rsrsnus (1970) Bonding schemesfor compounds with the pyrite, marcasite, and arsenopyritotype structures. Acta Chem. Scand' 24, 2993-3012.

BnowN, G. E., eNo C. T. Pnewrrr (1973) High-tem-perature crystal chemistry of hortonolite. Am. Mineral'58, 577-587.

BnowN, I. D., eNo R. D. SneNNoN (1973) Empirical bond-strength bond-length curves for oxides. Actt Crystallogr.A29,266-282.

BuRNs, R. G. (1970) Mineralogical Applications of CrystalField Theory. Cambridge University Press, Cambridge,United Kingdom.

CnnrsrnNsex, A. N., ,c,No R. G. Hezerl (1967) The crystalstructure of NaIn(SiOs)". Acta Chem. Scand. 21, 1425-1429.

Cunx, I. R., D. E. Apprtrvrlx, eNo J. J. Plrtre (1969)

Crystal-chemical characterization of clinopyroxenes basedon eight new structure refinements. Mineral. Soc, Am.Spec. Pap.2,3l-50.

CocnneN, W. (1961 ) "Effective" ionic charge in crystals.Nature,191, 60-61.

CorrrNs, P. (1972) Alternatives in the comparison oftheoretical and experimental charge densities: Differencemaps, net atomic charges, and population parameters.Trans. Am. Crystollogr. Assoc. E, 93-lll.

FrNcnn, L. W. (1969) The crystal structure and cation dis-tribution of a grunerite. Mineral. Sttc. Am. Spec' Pop.2, 95-100.

(1969/lnD) Fe/Me ordering in olivines. CarnegieInst. Wash. Year Book, 691 3O2-3O5.

Frrnr, M. E. (1973) The crystal structure of maucherite(NL'As"). Am. Mineral. 5E, 203-210.

Fneeo, R. L., eNo D. R. Pucor. (1967) Refinement of thecrystal structure of johannsenite. Am. Mineral. 52, 7O9-720.

MICHAEL E. FLEET

o

c)cftc

!coco

--

oI

-

I

o

Grnns, G. V., M. M. Henrrr,, L. S. Blnrplr, eNo H. Yow(1972) Correlations between Si-O bond length, Si-O-Siangle, and bond overlap populations calculated usingextended Hiickel molecular orbital theory. Am. Mineral.57.1578-t613.

HeNre, K. (1965) Beitriige zu kristallstrukturen vomOlivin-Typ. Beit. Mineral. Petogr. 11, 535-558.

Ketvrs, B. (1958) Structural basis of the olivine-spinel sta-bility relation. Am. Mineral. 53, 1439-1455.

LoursNerrreN, S. J., euo G. V. Grsss (1972a') The effectof tetrahedral angles on Si-O bond overlap populationsfor isolated tetrahedra. Am. Mineral. 57,1614-1642.

AND _ (1972b) Variation of Si_O distancesin olivines, soda melilite, and sodium metasilicate as pre-dicted by semi-empirical molecular orbital calculations.Am. Mineral. 57, 1643-1663.

Mrrcnerr-, J. T., F. D. Bross, lNo G. V. GrBBs (1971)Examination of the actinolite structure and four otherC2/m amphiboles in terms of double bonding. Z. Kristal-logr. 133,273-300.

Ntcrnr, E. H. (1968) Structural stability of minerals withthe pyrite, marcasite, arsenopyrite, and lcillingite struc-ixes. Can. Mineral. 9.311-321.

PAIIKE, I. J., M. Ross, eNo J. R. Crenx (1969) Crystal-chemical characterization of clinoamphiboles based onfive new structure refinements. Mineral. Soc. Am. Spec.Pap. 2, 117-136.

PeurrNc, L. (1929) The principles determining the struc-ture of complex ionic crystals. l. Am. Chem. Soc. 51,1010-r026.

1093

(196O) The Nature ol the Chemical Bond,3rd. ed.Cornell University Press, Ithaca, New York.

PrersoN, W. B. (1965) Compounds with the marcasitestructure. Z. Kristallogr. l2l, 449462.

PREwrrr, C. T., eNo C. W. BunNneu (1966) The crystalstructure of jadeite, NaAISLO". Am. Mineral. 51, 9'56-975.

RorrNsox, K., G. V. Gnes, exo P. H. Rnnr (1971)

Quadratic elongation: A quantitative measure of distor-tion in coqrdination polyhedra. Science,172, 567-570.

SercnanN, K., C. Nonorrxc, A. FlnrrueN, R. Nononenc,K. Hlrvrnrr, J. HpoIrrlN, G. JoneNssoN, T. Bencr"renx,S.-E. KenrssoN, I. LTNDGREN, exo B. LtNorrnc (1967)Esce. atomic. molecular and solid state structure studiedby means of electron spectroscopy. Noua Acto RegiaeSoc. Sci. Upsaliensis, Ser. IV, Vol. 20.

SneNr.toN, R. D., euo C. T. Pnewrrr (1969) Effectiveionic radii in oxides and fluorides. Acta Crystallogr.B.25,925-945.

Tosserr, I. A. (1973) Interpretation of K X-ray emissionspectra and chemical bonding in oxides of Mg, Al, andSi using quantitative molecular orbital theory. Geochim.Cosmochim. Acta, 37, 583-594.

URUsov, V. S. (1967) Chemical bonding in silica andsilicates. Geochem. Int. 4, 35V362.

WrNr, H.-R., eNo K. N. RevuoNo (1973) Four new struc-ture refinements of olivine. Z. Kristallogr. 137' 86-105.

Manuscript receioed, August 13, 1973; occepted

for publication, April 24, 1974.

M SITE ATOMS IN OLIVINES. CLINOPYROXENES. AND AMPHIBOLES