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,I 'HE AMERICAN MINERALOGIST, VOL. 50, NOVEMBER-DECEMBER, 196.5
REFINEMENT OF THE CRYSTAL STRUCTUREOF SYNTHETIC PYROPE
G. V. Grnrsr awn J. V. Sltrrn, Deportment rtl the Geophysi,calSciences, Llnitersity of Chicago, Ch.icago, Illi.noi.s
ABSTRACT
Anisotropic least-squares analysis of synthetic pyrope yielded the distances: Si-O, 1.635;Al-O, 1.886; Mg-O, 2.198 and 2.343, all +0.002 A. Shared oxygen-oxygen edges of poly-
hedra (2.496 for SiO+ and MgOs, and 2.618 for AlOo and MgOs) are shorter than the un-
shared edges (2.753 ln 5 iO4,2.716 in AlOe and.2771,2.783 in MgOs, a l l *0.003 A). The
thermal vibration ellipsoid of the Mg atom is markedly anisotropic in conformity with thedistorted nature of the MgOs polyhedron. A geometical analysis similar to that carried out
by Born and Zemann shows that the steric details of the structure are consistent with a
;::t"::-*. betrveen the attractive and repulsive ionic forces between both cations and
INrnonucrroN
The crystal structure of the pyrope variety of garnet (Menzer, 1928) isof interest because of the complex interactions between the polyhedra,and because pyrope is stable only at elevated pressure as evinced both byits geological occurrence (see Trciger, 1959) and its laboratory synthesis(Coes, 1955; Boyd and England, 1959). We carried out an isotropic,three-dimensional, Ieast-squares refinement of a crystal kindly supplied in1957 by Dr. F. R. Boyd, Jr. of the Geophysical Laboratory from a samplegrown at 13000 C. under 23 kilobars pressure. Before publication of ourresults, Zemann and Zemann (1961) described a careful, two-dimensionalrefinement of a svnthetic pyrope crystal grown by Coes. Shortly there-after expanded computer facilities permitted us to carry out an aniso-tropic, Ieast-squares analysis of our three-dimensional data. About thesame time, Zerr'ann (1962) in a crystal-chemical study of garnet consid-ered some of the factors governing the interatomic Cistances. He pointedout the elongation of the SiO+ tetrahedron in both p.vrope and grossular(whose structure has been refined by Abrahams and Geller (1958), andmore recently bv Prandl (1964)), and suggested that the concomitantshortening of the edge shared between the ZO+ and XOs groups was to beexpected from electrostatic considerations. It was also observed that theassociated X-O bonds to the oxvgen atoms comprising a shared edge wereshorter than the other X-O bonds in violation of the expectation of alengthening of the former because of cation-cation repulsion. Zemannthen discussed the question of why there are no garnets with undistortedZO+ tetrahedra and YO6 octahedra. Using reasonable Z-O and Y-O dis-
1 Present address: Department of Geochemistry and Mineralogy, The PennsylvaniaState University, University Park, Pennsylvania.
2023
G, V. GIBBS AND J, V. SMITH
tances he found that regula-rity of the ZOa and YO6 polyhedra demanded ashort unshared edge (2.44 A) in the XOa polyhedron, and concluded thatthe distortions of the tetrahedron and octahedron result from the need forunshared edges to be longer than 2.75 A. Mote recently, Born and Zemann(1963; 1964) have published calculations on distances and lattice-energiesof garnets. They found that the elongation and orientation of the tetra-hedron could not be explained in terms of Coulomb attractive forcesalone, and discovered that it was essentierl to include the repulsive forcesbetween the oxygen anions and the divalent cations. To simplify thecalculations, they assumed a fixed, distorted SiOa tetrahedron and aconstant Al-O distance, and assigned a repulsive force of the type trd-" tothe ox-vgen atom and the divalent cation. Using a fully ionic model, onlymoderate agreement was obtained between the maximum of the partiallattice energy and the observed tetrahedral orientation, while assumptionof partial ionization led to excellent agreement. The effect of tetrahedraldistortion on the partial latt ice energy was not so pronounced, but theobserved distortion was favored in comparison to the ideal shape.
Expnnrlrnnra.r
A crysta l showing {110}, {211\ ,1021 } forms was ground to a poly-hedron with maximum and minimum dimensions ol 0.32 and 0.41 mm.Weissenberg and precession fi lms exposed about the [100], [110] and [111]axes of the crystal possessed Laue and diffraction symmetry consistentwith space grottp Ia3d.. The intensities of 374 non-equivalent reflections(those within the CuK* sphere of reflection) were measured from a scin-tillation-counter equi-inclination Weissenberg diffractometer usingmonochromatized MoK" radiation. Absorption corrections were ignoredbecause the use of penetrating MoK" radiation reduced them to insignifi-cance (about l/6 in intensities). The polarizing effect of the bent qttartzcrl.stal monochromator also was about l/e, and was ignored.
The positional and isotropic temperature parameters determined byZemann and Zemann (Table 1) were used as input in three cycles of iso-tropic least-squares refinement (Gibbs and Smith, 1962). The scatteringcurves used in the calculation were those of Berghuis et al. (1955) forAl3+, Mgz+, Sia+ and of Suzuki (1960) for O2-, each modified arbitrarill.for half-ionization, i.e., Al3+t2, Mg+, Siz+ and O-. The calculations weremade on an IBM 704 computer using the Busing-Levy program OR XLS(1959). Examination of the observed and calculated structure amplitudesfor the strong iow-angie reflections suggested that extinction effects weresubstantial. Consequentl-v, f ive separate refinements were carried out,each of three cycles, using different weighting schemes. These calcula-tions, made on an IBM 7090 computer using the Busing-Martin-Levy
PYROPE STRUCT-URE
Tesre 1. CoupanrsoN ol Arourc Penelrprens lnolr Tnpnr Rnltrsrllnuts
Isotropic L.S IsotroPic L'S.
Zemann and Zemann refinement of refinement
(1961) Zemann and Zemann Gibbs and Smith
data (1962)
XO
yo
zoB oBugB.r tBs isi-oAt-oMg.oMgto
The number in parentheses is the random experimentai error. The refinement of the
Gibbs and Smith data utilized all reflections equally weighted at unity.
program OR FLS (1962) for an zrnisotropic least-squares refinement'yielded the atomic parameters of Table 2. Refinement A used ail observedreflections with unit weights; refinement B omitted the ver1, strong 400
rel lec i ion; ref inement C omit ted a l l re f lect ions wi th , For . " I gre l ter than
200. For refinement D the unobserved reflections were added to those
used in C and given an amplitude oI zero. Three reflections 428, 206, and
6.4. 10 showed especially large discrepancies, and refinement E was made
to test the effect of their omission on the atomic parameters. The parame-
ters for D and E agree within the random experimental error, but thosefor the other refinements show discrepancies up to four times the randomerrors. The major changes result from omission of the 400 and otherstrong reflections, while the inclusion of the unobserved reflections pro-
duces changes, only one of which is greater than the random error. Be-
cause extinction seems to be the major factor in producing the difierences,the atomic parameters from refinements A and B have been ignored. As
there is no good reason to exclude reflections 428,206, and 6'4' 10, refine-ment E has been discounted. Although some of the unobserved reflectionshave finite amplitudes, these are not knownl consequentlv, the resultsfrom refinement D have been chosen in preference to those from C. Thedifierence is so small that ail conclusions drawn from these results are
independent of this choice between refinements C and D. The randomdistribution of peaks in difference maps based on structure amplitudes for
D confirmed the validitv of the least-squares refinement. The differencemaps were prepared with the IBM 1620 using the van der Helm three-
0. 034 (1)0 0s0 (1)0.6s4 (1)0 . 40 . 3o 20 . 31 . 6 21 .892 . 2 02 3 5
0 0334 (s)0.0s08 (s)0.6s31 (s)0 37 (4)
0 .24
1 .63s (6)1 .888 (6)2.206 (6)2 .337 6 )
0.032e (+)o os08 (4)0. 6s31 (4)0. 38 (s)o 67 (8)0. 18 (.5)0 13 (4)1 639 (s)1 .887 (s)2.202 ( .s)2 .33s (s)
2026 G. V. GIBBS AND J. l'. SA,IITH
dimensional Fourier summation program (1961). The final R-factors are:all reflections, equall1. weighted lt.00/6; observed reflections, equali l,weighted 8.57o; observed reflections exciuding (400) 7.8/6; observedreflections, excluded those with F > 200 7 .9%. Tabie 3 contains theobserved and caiculated structure amplitudes.
Because of discrepancies between the results obtained b1 Zemann andZemann from Fo and AF s1'ntheses, and those obtained b1'us from least-squares methods, the structure amplitudes measured by Zemann andZemann r,vere submitted to isotropic least-squares refinement on theIBM 7074 using a modified version of the Busing-Martin-Levy program.As the sil icon and magnesium atoms overlap in projection down z, it wasnecessar\r to ascribe arbitrari ly f ixed temperature factors to these atoms.The values 0.3 given br- Zemann were chosen. Only those reflectionslisted by Zemann and Zemann as being unaffected b1- extinction, and asnot being too weak, were used in the refinement. Table 1 shows agree-ment within the error l imits for the parameters from the new refinementof the Zemann-Zemann data and the original refinement of the Gibbs-Smith data. Thus the two sets of data are consistent and there is no needto suppose that there is anv structural difference between the two synthel-ic specimens. The random arrerngement of peaks in the difference synthe-sis calculated from the data of our isotropic refinement suggests thatrefinement by difference syntheses would have ied to essentiall.v the same
T l'rl,T, 2. Arolrc P'ln.llretens lnou Drtrrtrxr Rnrrxsnrltrs
Orygenxvz
At8zzFn8vFraEzt
AlumintLm
AtBrz
M agnesiunt
BtFzzBzt
Silicon
9tAn
0.03386 (32)10s091 (28)6s282 (28)
.00093 (16)
.00073 (1s)00046 (14)
.00032 (13)
.00001 (12)- .00007 (12)
.000.3s (06)
.0000s (09)
000.s0 (19)00118 (14)00037 (20)
.00034 (12)
.00026 (07)
0 03291.04990.6532800094
.000850006900024
- 00001- 00002
.0005100002
0 0327s (18)0s002 (17)6s329 (16)
.00091 (0s)00101 (09)00070 (08)00016 (07)
- 00006 (07)- .00010 (07)
.000s8 (04)
.00002 (05)
0009s (12).001 i1 (09).00045 (12)
000s7 (07).00015 (04)
(23)(21)(2r)( r2)( 1 1 )( 1 1 )(0e)(0e)(0e)
(05)(06)
0 03284 (19) 0 03285 (17)
.0s014 (18) .0s013 (16)
.6s330 (18) .6s327 i .1 .6 )
00099 (10) .00103 (09)
00103 (10) .00110 (09)
00078 (0e) .0008s (09)
00013 (08) 00011 (07)- 00014 (07) - .00017 (06)- 00009 (07) - 00006 (06)
00060 (01) 00064 (04)
oooo4 (05) .oooo3 (01)
00102 (13) 0010,1 (11)
oo l6 i (0e) .0016s (08)
.00011 (12) .00042 (10)
000.56 (08) 00059 (07)
00046 (05) 000s0 (04)
.00086 (1s . )
.001.s9 (11)00041 (1s)
.000s3 (09)00039 (0s)
t The number in parentheses is the random experimental error
2027PYROPE STRUCTURE
T a b l e 3 .Ob..w.d ad c.lculatcd .rtuctur. .hplitd.. to, pylop.
H F F * I F F
9 3 1 6 2 t 2 1 r 4 3 r 9 zr 7 3 ! 6 7 - E 1 6 . 3 . 1 9 ot 3 3 t 6 6 7 7 - 1 . 1 9 2 1 t 9
6 4 1 6 7 7 E ? 9 . 4 . t 9 3 0 3 t8 4 1 6 8 5 8 5 l r - 4 - r 9 t z - 9
l o , 1 1 6 4 0 3 9 r ! , a , 1 9 6t z - 4 t 6 1 0 { 5 a , i 9 t o t 2
7 . 5 , 1 6 Z O - 2 0 6 , 5 t 9 2 r r 79 . 5 . 1 6 t 0 r 0 6 5 . t 9 4 7
, t - 5 . 1 6 1 1 r 0 5 . t 9 z6 . 6 . 1 6 1 8 t 9 l Z 5 t 9 Z O - 2 26 . 6 . f 6 7 6 t 4 5 t 9 1 6 1 1
r o 6 1 6 r 0 t r 7 . 5 . 1 9 2 6 z 29 . 7 . 1 6 0 9 . 6 . t 9 I - ?8 . 8 . t 6 t t 5 t a 5 i t . 6 , t 9 t E z og r . t 1 t 5 - r 2 1 3 . 6 1 9 z
1 0 . r . t ? 6 a , 7 . r 9 2 ? - 2 3t z t , 7 l E - r E 1 0 . ? . 1 9 t 0 - 3t 1 . l . l 7 7 r z , 7 , t 9 1 4 l 51 6 | r 7 1 9 . 4 , t 9 2 7 3 0
7 Z , r 7 r l - 9 r l . s t 9 t 1 - 1 59 2 r 7 1 6 1 6 r 0 9 . r 9 - 9
r l z , l 7 z z - 2 t t z . o . z o 2 z - z ot t 2 t7 t9 . '13 t1 -o-zo 3z -z?t 5 z r 7 - 9 1 6 . 0 . 2 0 1 6 r 7
6 3 t 7 - S t r . t . z o z r 1 9E 3 r 7 Z l 1 9 l 3 . l . Z O 3
1 0 , t , 1 ? | t 5 ! . 2 0 - 5| Z . t t 7 t 9 . . 2 0 l 7 t . z o l o - 7 17 4 t l 7 2 2 - 2 6 i O . 2 . Z O 1 a 3 l
5 , 1 t ? 3 2 - 2 8 r Z 2 2 0 r Z 67 4 r 7 2 7 - Z t t r 2 , 2 0 4 5 4 l9 1 , l 7 2 A - 2 1 1 6 . 2 . 2 0 t O S
t r 1 r 7 z 9 , 3 , 2 0 - 31 ! a . f ? - 4 l l t . z o - 3
6 ' t 7 5 5 - 4 E r 3 , t - Z O l 3 1 0a 5 t 7 3 9 1 5 . ! , 2 0 1
r o 5 r 7 - 3 8 . a , 2 0 t t - 87 2 5 r 7 6 r 0 1 . Z O t 9 - 1 6
7 6 r 7 - 3 t 2 . 1 Z O 4 7 - 1 7
9 . 6 . 1 7 - a 1 6 . 1 . 2 0 51 1 . 6 . r ? 2 0 - t 6 7 . 5 \ Z O l Z - t 2q 7 1 7 1 1 , 1 r r - 9 z o . 6
t o 7 1 7 t 5 t 5 t t . 5 z o . 39 8 r ? - 4 r 5 . 5 Z O 9
r 0 0 1 a t 4 l o 6 6 . 2 0 6 0 6 4t e 0 t a 9 8 9 0 1 0 . 6 . 2 0 5 0 1 6t { 0 - 1 a - 4 t z , 6 z o z o z zt 6 0 1 8 l 2 - 1 5 1 1 6 z O 1 2 1 2t a 0 t a 1 3 - t l 9 ? z o E I9 r 1 8 t 4 t l 1 t 7 2 0 9 t '
t l t t a l 5 1 4 1 t 1 2 0 5t l r r a - 7 8 . 8 . 2 0 5 3 5 l1 5 t r 8 2 0 2 4 1 0 . 8 . 2 0 3 4 - 3 5t ? I t a - 3 t z 8 . 2 0 1 4 l l
8 2 t A I t t g . Z o - 2
i 0 2 t 8 z s - z a l z . t . z t 9 ! 0r z 2 r E 6 1 5 8 1 5 L Z I Zt 4 z l a l t ? l l 2 2 t l z - 1 ,
t 6 2 r a 6 1 5 Z 2 r 1 6 - 1 1
? r r 8 r 2 7 t 7 Z . Z l I t t t9 3 1 8 f 9 - t 5 r o . 1 Z r t t 1 5
1 t I t a 8 - a 1 4 3 . Z l Z O - Z l
t 3 3 r E r 7 - 1 7 9 . 1 . 2 1 1 3 t 0l 5 I t 8 3 t - 3 ! l l 4 z l 1 5 - t z
6 4 1 8 7 7 7 Z l ! , 1 , 2 r 1 9 Z O8 4 1 8 6 6 - 7 2 S S Z i 1
1 0 . 4 . 1 8 5 6 5 t r Z 5 2 r 1 6 t 51 2 4 r 8 - 7 7 . 6 . 2 1 1 1 3 51 4 4 r a 5 7 5 A l ! 6 - 2 r l l t 0
7 ' la -6 13 6-2 l 21 -27
9 5 t 8 t 1 3 . 6 - 2 l l 8 r 71 1 5 t 8 - Z 9 - A Z t l 7 l a1 3 5 t 8 2 4 - 2 1 t 3 8 Z t 6
8 - 6 r 8 r t l z t o 9 2 r 7 1r o 6 1 8 t 6 t 7 r 2 . 9 , Z l t r - r 5
t 2 6 1 8 5 8 5 9 t t . t o z t E - 6
7 7 l A 1 6 - r 5 t 6 - O 2 2 l O 79 . ? . 1 8 o l 3 . r , 2 z r 5 r z
r i . ? . r 8 z o - z t l l . t . z z r i 1 6r o g r 8 - 5 t 3 , l z z r s 1 69 . 9 . t 8 2 7 3 l 1 5 . ! Z Z r l 1 2
1 o . t i 9 2 6 - z ? t o + . 2 2 6 0 5 0r z . l 1 9 - 2 l 1 1 2 2 5 Z 5 2r 4 . r . t 9 1 6 r 1 l l . 5 . z 2 l a - r 5
r 6 . r . t 9 2 a . 6 . 2 2 6 7t 8 f - t 9 I l Z 6 2 2 3 7 a 09 Z t 9 2 1 1 9 9 . ? - 2 2 1 - Z
t 1 , z , l 9 r o - 9 t t . 1 . z z 1 4 t 8t 3 . z t 9 1 3 1 3 l i . a 2 3 t 3 - 1 5
15 Z t9 15 - t { 13 .1 23 ZO '19
r 7 . 2 1 9 - l l z 5 2 3 r o 78 . 3 1 9 r Z . . 1 2 t l . a 2 3 t a - z o
t o 3 1 9 r 7 - r 7 t Z - 1 ' 2 4 9 l z1 2 3 t 9 - Z
R I F F
9 1 r Z 2 t Z Z4 , 4 , 1 2 9 7 - 9 46 4 l Z t 5 - t 5g 4 r z 2 t - r 97 5 , t Z 76 6 rz r r9 rzo4 . 1 t 3 - t6 . t i 3 3 0 - 2 9a . t . t 3 z z z l
1 0 . t . t l 0l z l . r 3 1 - 4
3 Z . l 1 Z5 2 . r 1 7 6 6 77 Z . l 3 49 Z r t 1 9 z !
t t 2 t 5 54 1 t 3 2 2 - 2 26 l t l 3 1 - t 78 l 1 t 3 4 1 l
f 0 1 1 3 ! 4 t 45 4 t t t 9 z z7 4 1 3 7 5 1 59 4 1 1 l 1 t 46 . 5 1 3 t8 5 . t l 1 2 - ! 17 . 6 , 1 3 ! 4 1 76 0 1 { t ? - 1 76 0 t { J 5 2 9
1 0 0 t 4 l 3 3 l1 2 O 1 4 9 2 9 5t 4 . o 1 4 i 6 - t 3
5 . 1 1 4 6 r - 5 27 7 . 1 4 49 l t 4 z z - 2 3
l t t 1 4 9 - 4! l t t 4 - 44 Z t4 l7Z 1596 2 1 4 i 5 1 5a 2 1 4 2 3 - 1 6
t o 2 t 4 t l - 1 0l Z Z 1 4 5 4 4 9
5 t l 4 47 3 t 4 9 79 J t 4 9
r r 3 1 4 1 3 3 56 4 1 4 9 5 9 4a 4 t 4 9 9 - t o l
r 0 4 t 4 7 Z 7 35 5 1 4 4 2 - t 87 5 1 4 1 4 - 1 69 . 5 t 4 la . 6 t 4 t 4 l 77 7 . 1 4 Z a - 2 86 t r 5 t o 2 98 t t 5 4 7 4 4
t o t 1 5 21 2 | 1 5 1 3 - l l1 4 r 1 5 2 5 Z !
5 2 r 5 Z O Z Z7 Z l 5 - 59 2 1 5 3 0 2 9
t l 2 l 5 t 7 - t 9t 1 z l 5 l 5 1 54 1 . 1 5 A t 25 t r 5 48 3 t 5 1 5 1 4
t 0 I t 5 t 0 - l t1 2 1 1 5 I 6
5 4 1 5 2 7 - 2 67 4 1 5 I - 69 1 1 5 Z r - Z r
t t . 4 1 5 36 . 5 , 1 5 2 5 2 48 . 5 t 5 2 9 2 6
i 0 5 t 5 t o - l z7 6 , r t t l - t t9 6 1 5 5 - 6a 7 . 1 5 z o - r 98 0 t 6 r 9 9 z o t
t 0 o 1 6 r 0 - 1 1t z 0 . t 6 r 0 8 9 91 4 0 1 6 1 6 l lt 6 0 1 6 l z 1 t t 1
? . t - t 6 t 6 - 1 99 . t t 6 l 7 2 0
r i . t t 6 0t 3 . I t 6 - ti 5 . r t 6 \
6 Z 1 6 t Z - 9a . z 1 6 2 0 - 1 7
l o z t 6 z s 2 9l z 2 1 6 6 ? - 6 6t 4 z 1 6 z t t 4
5 . 3 , 1 6 t 4 1 07 . 3 , 1 6 Z Z - 2 4
F
3363 l5 Z
- 4 1- t 7 z
- t-269
-421 3 9
- 8 41 9
1 0 a3 7 2
- 3117-14
- z288
84
- 3 65 Z
2 2 7-9
335- l t66
- t 84Z
- 1 9 0- z l- i z
l 48 t
0
a2 7
- 3 8
- l- 1 0
\ 9 1_ 4 3- 1 4t05
7 5- a
1 4194- t 9
7 76z
- 6t 4 l
1 92
26- 2 7- 1 6
3 4
- 1 6_ 4 6
-54a
1 1 3t z 4- 9 0
- 73 a
65
5t 0 ?rz57 1 7
3 3-43
400 255z t t 3 6zoz 33z t 3 3 82 3 3 1 7 1633zM 24r404 EZzz4 142844 4 lz r 5 E i4 1 5 Z Z325 79206 124406 zsz606 6t l6 !49116 44516 z426 ZA4z r 7 1 84t7 666t1 86327 64527 584t7 34208 59408 266 0 8 7 24 0 8 1 5 1! 1 4 t l5 t 8 5 a? 1 4 1 92ZA 42424 1666 z a z z536 86219 t44r9 92619 5at9 44
, 2 9 2 r729 36419 466!9449 10
z , o t o 9 84 0 . t 0 z z 56 0 r 0 4 68 0 t 0 t 4
t 0 0 t 0 9 1l l l 0 7 25 r l o 9T t t o t 69 t . t o l 34 , 2 r O r 9 46 . 2 l O 3 18 2 : t 0 2 3I I t 0 75 3 t 0 3T t l o6 4 t O r 2 35 - 5 t 0 l 7z l t l4 t l t 4 16 l t l 2 6E I l t 3 0
1 0 t t t t 3J Z t l 2 45 , 2 . 1 1 3 57 2 , l l 49 Z t l E4 a t l t E6 1 t l 4 7a 3 l t 2 55 4 l l7 4 r 1 5 96 . 5 r t 1 04 0 . 7 2 3 l6 0 1 2 1 1 5E 0 t 2 t z r
1 0 . O t 2 9 5l z o l z 91 . 1 t z 4 35 7 . t zT l l z9 l 1 2 l 7
7 r r l 2 4 t4 Z l 26 . 2 t Z i i 3a . z l z r t 4
l o . 2 l z t i 85 . 1 r z 2 57 . 1 l Z 1 6
2028 G. V. GIBBS AND J. V. SMITH
coordinates as those obtained from the anisotropic least-squares refine-
ment .The cell edge 11.459 A found by Skinner (1956) for a synthetic pyrope
was used in the calculations of interatomic distances and angles (Table 4).
The estimated standard deviations, obtained in the final least-squares cai-
culation, are given in parentheses in Table 2. For the D refinement these
Iead to standard deviations 0.002 A for AI-O, Mg-O and Si-O and 0.003
A for O-O. The final anisotropic temperature factors 0i: obtained from
the refinement are also i isted in Table 2. Certain constraints exist among
the temperature factors of the atoms in special positions, i.e', lor magne-
s ium, Bl l lBzz:gas and prr :grs:B4 and for a luminum and s i l icon,gr.: gzz: 0zz and Brr: Brr:pza:0. The three principal axes of the thermal
vibration ell ipsoid for each atom were computed using the Busing-
T-lelp 4. INmnerorrrc DrsreNcnsr auo ANcr-os rN Pvtopn
SiO+ tetrahedron Si-O (4)' 1 635 A
o-o (2) 2.1e6, (4) 2 7s3o-si o (2) 99 6'(2) 114.7
A106 octahedron A1-O (6) 1.886 A
o-o (6) 2.618, (6) 2.716o-Ai o (6) 87 .9' (6) 92 |
MgOs cube \{g.O (4) 2.198 AMs'O (4) 2.343o-o (.2) 2.496, (4) 2 618
(4) 2.7rr , (2) 2.783O-Mg'-O (2) 69.2oO-Ms'O (2) 72 94, Q) 109.4o_o_o (4) 16, (2) 7e, (2) 88
(4) e2, (.2) e+, (2) e4(2) 10r, (.2) ro3, (.2) r12(2) rr9
Mg-Al (4) 3.2ffi AMs-Si (2) 2 86s, (4) 3.s09Al si (6) 3.203si-o-Al 1300Si-O-Mgr 95
Si-O-Mg: 123
Mgr-O-Mgz 101
Al-O-Mgr 103
A]-O-Mgr 98
1 These distances are not corrected for atomic motionsl the corrections are small be-
cause of the low temperature factors. (See Busing and Levy, 1964 )2 The number in parentheses is the frequency of occurrence.
PYROPE STRUCTURE 2029
Tleln 5. Trrn n.u s. Dtspr-,lcrunNTs AND OmnxteuoNs or tur: Pnrxcrp,rr, Ares
ol rnr Ttrenulr, VrnnrltroN Er-lrpsorns
u(rJ 0(r, x) olr, y) A l r z \
A I
I23
I
2?
123
123
0.0708 (36) A0.0807 (36)0.0913 (33)
0.0633 (32)0 0633 (31)0 0693 (47)
0 0832 (46)0.091s (44)0 .1171 (41 )
0.0s73 (21)0.0s73 (36)0.0627 (36)
5 8 . 8 ( 1 2 . 3 ) os6.8 (1s 0 )48.9 (.rr.7)
I
I
J + . /
0 09 0 090. 0
1
1
0 . 0
89.6 (13 4)140 5 (14 1).50 .4 (14 1)
I
1
J 4 /
90 .045 .045 .0
1
1
9 0 . 0
3r .2 ( . r2 .2 )108.e (16 1)1 1 3 8 ( 8 . 7 )
1
I
54.7
9 0 . 0135 .045 .0
I
I
9 0 . 0
Mg
Si
1 fndeterminate (uniaxial thermal ellipsoid).rt, 12, h are the ellipsoid axes, and x, y, z the crystallographic axes.
The experimental errors are enclosed in parentheses.
Martin-Levy OR FFE program (1964). The r.m.s. displacements andtheir orientations rvith respect to the crystallographic axes are given in
Table 5.
DrscussroN
The structure of pyrope consists of independent SiOa and 4106 poly-
hedra (Fig. 1) which share corners to form an alumino-silicate framework
within which each magnesium atom is surrounded by an irregular poly-
hedron of eight oxygen atoms (Figs. 2, 3). The sharing coefficients of the
SiOq and A106 polyhedra are one compared with two for the MgO6polyhedron (ZoItai, unpublished). Accordingly, each oxygen atom is co-
ordinated bv asil icon atom at 1.635 A, by an aluminum atom at 1.8864and b1' two mergnesium atoms at 2.198 and 2.343 A (Fig. 3)' Zemann(1962) has suggested that the polvhedron about the magnesium atom is
best regarded as a distorted cube. Two edges of the sil icon tetrahedron
and six edges of the aluminum octahedron are shared with the magnesiumcube leaving unshared four edges in the tetrahedron, six in the octahedronand six in the cube. In fact, the high density and high refractive indices ofpyrope, and of garnet in general, can be attributed to the large percentage
of shared edges "r'hich
leads to a tightly packed arrangement.
G. V. GIBBS AND J. V. SMITH
The shared edges are shorter than the unshared
shared betrveen SiOa and MeOsshared betr.een AlQ6 and MgOs 2.618;unshared SiO+unshared AlOsunshared MgO3
e d o e s '
2 . 4 9 6 , \ ;
a 1 < 7 .
2 .716 ;2 . 7 r 1 , 2 . 7 8 3 .
This Ieads to considerable distortion of the polyhedra. The sii icon tetra-hedron is a tetragonal bisphenoid elongated along the 4 axis while the
( b )
Frc. 1. Idealized dral.ings of the SiO+ and A106 polyhedra in pyrope vierved down 3 (a) andalong z (b); the MgOe polyhedra have been omitted for clarity and the polyhedra havebeen displaced slightly to avoid superposition.
( o )
PYROPE STRUCTURE
Frc. 2. Part of the pyrope structure projected down the z-axis The rarge open circresrepresent oxygen atoms and the smaller one within the open cube is magnesium. Aluminumand silicon atoms center the octahedra and tetrahedra, respectively, and a magnesiumatom centers the closed cube; these are not indicated. The values along the edges of thepolyhedra refer to O-O distances.
aluminum octahedron is a trigonal antiprism erongated along the 3 axis.The Mg cube is considerably distorted with its plane angles varying from76 to 1190.
rt might be expected that the thermal vibration ellipsoid of a cationwould depend on the distortion of its coordination polyhedron of anions.rndeed, the orientation and the anisotropy of the thermal eilipsoid of themagnesium atom in pyrope conform to the distorted nature of its coor-dination polyhedron. Figure 4(a) is a stereographic projection of theneighbors of the magnesium atom and the axes of the thermal ellipsoid.lThe shortest axis (r.m.s. displacement 0.0gJ2 (46) A) bisects the twoshorter Mgro bonds (2.198 A; *hi.tt make an angle of 69o with oneanother. The intermediate axis (0.091s (44) ) bisects two of the longerMgz-O bonds (2.343 A; whicn make an angle of 73o, while the longestaxis (0.1171 (41)) also bisects longer bonds but at an angle of 109o. rt isof interest that the longest axes of the thermar elipsoids for the sil icon
\ zemann and Zemann (1962) tentativery suggested sright disorder of the Mg atom toexplain anisotropy of the electron density, and approximated the disorder by a statisticaldistribution over the general position (0 125, 0.003, 0.253). This anisotropy is in the samedirection as that found by anisotropic least-squares refinement.
2032 G. V. GIBBS AND J. V. SMITH
QlotrJ - t t t'4 2 z ' ,
rf,olt
o=O O=A l @ =S i O=Mg
Fro. 3. 'Ihe coordination polyhedra of oxygen atoms about Al, Mg and Si in pyrope;
one of the cubes about Mg is omitted (Modified after Abrahams and Geller, 1958).
-o/t
' ooe r | > l f i t ' ' oo '
o 2 t g a
o
o o92^ 2 3 4 3" o o l t T
) 1 ( D i o d )
o
( D i o d )
\o,
PYROPE STRUCTURE 2033
and aluminum atoms are also oriented in conformitv u,ith the distortionsof their polvhedrai; however, the ell ipsoids are not significantly aniso-tropic as the axes differ only by about 1o (r.m.s. displacements and ran-dom e r ro rs a re : S i 0 .0573 (21 ) ,0 .0573 (36 ) and 0 .0627 (36 ) ;A I 0 .0633(32) , 0.0633 (31) and 0.0693 (47) A) . F igure 4(b) shows the re lat ion be-tween the thermal ellipsoid and environment of the oxygen atom. Theanisotropy is statisticaily significant, but no interpretation is offered.Experimental values of thermal vibration ellipsoids necessarilv combinepositional disorder and true thermal vibration. The ell ipsoids for Si, Aland O are similarr in size to those for olivine (Gibbs et aI.,1964) and alumi-nosil icates (Burnham and Buerger, 196l; Burnham, 1962; Burnham,1963) and it seems likely that all of them result onlv from true thermalvibration. The ell ipsoid for Mg, however, is larger than might be ex-pected; indeed, it is significantl l ' larger than that of the oxygen atom. Itis suggested that confl icting spatial requirements result in the magnesiumatom lying in a cavity larger than normal, and that it tends to filI thecavity by a strong asymmetric vibration. Because there are no optical ors-ray anomalies from isometric symmetry, it is l ikel1' that if there arepotential energy minima, the Mg atoms occup)' these at random.
Miyashiro (1953) has suggested that in the pyralspite garnets (i.e.those with a small atom in the eight-coordinated site) four of the eightoxygen atoms wil l be closer to the eight-coordinated atom tending to ap-proach the lower coordination number more commonil ' found for mag-nesium and the other small cations. Indeed the magnesium atom inpyrope does have four close neighbors at 2.198 and four more distantones at 2.343 L. However, this feature is found in all garnets so firr in-vestigated (grossular (Abrahams and Geller, 1958; Prandl, 1964),YsFezFeaOrz (Gel ler and Gi l leo, 1957;Bat t and Post , 1962), Gd3FezFeaOrz(Weidenborner, 196l), YsAl2Al3012 (Prince, 1957), hydrogarnet (Cohen-Addad et al.,1963) and appears to be a general property of the garnetstructure rather than a feature unique to cations which normally have aiower coordination number. For example, calcium tends to have a co-ordination number between 6 :Lnd 9 but it sti i l has four near oxyqenneighbors in both grossular and hydrogarnet.
In a similar manner to Zemann (1962) and Born and Zemann (1964),Ie'e have attempted to understand the structure of pyrope in terms ofionic bonding, following the classical treatment described most recentlyby Pauling (1960). The basic conclusions are similar to those of Born andZerr'ann, but there are some differences in emphasis which we feel are im-portant.
The Si-O bond is strong and its length almost immune to change by theneighboring cations. Smith and Bailey (1963) have shown that the meanSi-O distance in a tetrahedron varies from 1.61 A for a tektosil icate to
2034 G, V. GIBBS AND J. V. SMITH
1.63 A for structures with isolated tetrahedra. Within each g-roup of sil i-cates, the observed values fall within a range of about 0.01 A. The Si-Obond length in pvrope, 1.635 A, is in excellent agreement with valuesrecorded for other nesosil icates which have tetrahedra sharing one ormore edges rvith other polyhedra: for example, fayalite 1.632 (Hanke,1964), spurrite 1.633 (Smith, et a1,., 1960), forsterite 1.634, hortonolite1.638 (Gibbs, et al., in preparation) and grossular 1.637 A (Abrahamsand Geller, 1958). These values are somewhat longer than the value1.628 A found in the nesosil icates andalusite (Burnham and Buerger,1961) and k1'anite (Burnham, 1963), whose tetrahedra do not shareedges with the other polyhedra. It appears that the longer distances inthe former are induced by repulsive forces between cations over a sharededge (Pauling, 1928). (Indeed the variation of Si-O clistance from 1.61 Ain a tektosil icate to 1.63 A in a nesosil icate may result in part from repul-sion between two sil icon atoms sharing er common oxygen atom.) Thus itseems reasonable to expect for all garnet structures an Si-O distancewi th in 0.005 ̂ i o f 1.635 A.
The Al-O distance in the trigonal antiprism, i.886 A, is shorter thanthe mean values recorded for a number of other sil icates: synthetic beryl1.898 (Gibbs and Breck, in preparation), euclase 1.90 (X{rose and Apple-
.:-rrron;1962), kyanite 1.906 (Burnham, 1963), mull ite 1.911 (Burnham,1964), paragonite 1.912 and muscovite 1.916 (Burnham and Radoslovich,1964), jadeite 1.927 (Prewitt and Burnham, 1964),- andalusite 1.934(Burnham and Buerger, 196I) and grossular 1.945 A (Abrahams andGeller, 1958). Because the Al-O octahedral bond is weaker than the SiOtetrahedral bond, the influence of neighbors should be greater, and alarger spread of Al-O distances is to be anticipated. The range from 1.886in pyrope to 1.945 A in grossular is rather large, and indicates that theAl-O distance is being affected seriously b1'neighboring atoms.
The Mg-O bond is relatively weak because of the lower charge of thecation and the higher coordination number. The wide spread of Ca-O,Na-O and K-O distances is well-known, and requires no detailed docu-mentation. For grossular, the difference of 0.16 A between the two Ca-Odistances in the XOs group is in l ine with the variation in other sil icates.By analogf it seems reasonable to expect a difference of at least 0.1 Abetween the tlvo Mg-O distances, and the observed difference of 0.13 Ain p1 rope need cause no surpr ise.
Cation-cation repulsion should be strongest across the shared edgebetween the SiO4 and MgO6 polyhedra. The observed values of 2.50 and2.62 h, respectively, for the shared edges joining MgOs to the SiOa andA106 polyhedra are reasonable when compared with the distortions of theoctahedra in corundum (Newnham and deHaan, 1962) and the TiOz
PYROPE STRUCTURE 2035
: : .
i " ' .o
a . ) ? a
l " l
( D l
Frc. 5. Plot of x ts. z positional parameters of the oxygen atom in pyroper for selectedinteratomic distances: (a) MgrO, MC1O, Al-O; (b) O-O: O-Osrc refers to shared edgebetween SiOa and MgOa polyhedra; O-Ouo. the unshared edge of the A106 polyhedron and
O-Ouc the unshared edge of the MgOs polyhedron. Hachured lines indicate limits of reason-able variation of interatomic distances.
polymorphs (Pauling, 1960). The corresponding values in grossular, 2.56ar.d 2.79 A, are larger in conformity with the longer Ca-Si and Ca-Al dis-tances, and consequent lower repulsion. However, in grossular the un-shared- edge of the AlOo polyhedron is shorter than the shared edge(2.71 A) indicating that the distortion of the octahedron is not deter-mined entirely by cation-cation repulsion.
The lengths of the unshared edges of the polyhedra should depend pri-mariiy on the cation and the lengths of the shared edges. The mezrn cat-ion-oxygen distance is determined primarily by the cation;if the sharededges are shortened b.v cation-cation repulsion, those remaining shouldbe lengthened in order to preserve the mean value required by a constantcation-oxygen distance.
Although a ,crystal structure will not be stable unless all the spatialrequirements are satisfied, it would be remarkable if all the requirementswere satisfied exactly. Some compromise can be expected in aII complexstructures, with the amount of compromise for each requirement beingdetermined by the nature of the bonding forces. In order to see the inter-action between the interatomic distances, we have prepared Fig. 5 whichis a graph of the interatomic distances as a function of the x and z co-ordinates of the oxygen atom. The third coordinate, y, was determinedby assuming tha t S i -O :1 .635 A and tha r a :11 .459 A . Bo . t r and Zemannhave prepared similar graphs on the assumption that Si-O:1.62 h,Al -O:1.89 A and a is var iable. From Fig.5(a) i t may be seen that forpyrope there. is a strong interaction between the Al-O and Mg-O dis-tances. For an Mgr-O distance between 2.1 and 2.3 A, which may be re-
2036 G. V. GI]]BS AND J. V. SMITH
garded as reasonable outer l imits for: a magnesium atom in eight-foldcoordination, and for A1-O distances less than 1.95 A, a reasonable upperlimit for an aluninum atom in six-fold coordination, the N'Igz-O distancen rus t be g rex le r t h r t n 2 .24 A . t I t ne i r ve r rge o f t he two Mg-O d i s tances i sto l ie betrveen2.2 and2.25 A while the AI-O dist:Lnce is close to 1.90 A,the coordinates of the oxygen artom can varv only between narrowlimits. Figure 5(b) shows the effect on the O-O distances as the oxygenatom moves. The control exerted b1' the unshared polyhedral edges isremarkable, a conclusion alread.v leached by Zemann (.1962). Only' threeof the unshared edges are plotted because the one in the tetrahedronchanges only- from 2.7 to 2.8 A over the entire riLnge of Fig. 5 (all threehave been assumed to be not shorter than2.7 A). The edge shaled be-tween the octahedron and cube v:rries only fron 2..5.5 to 2.73 A over theentire range of Fig. 5 and rvas not plotted because all these values arepossible if a tolerance of about 0.1 A is permitted. The edge shared be-tween the tetrahedron and cube should be shortened considerably and avalue near 2.5 A appears reasonable. If a rather wicle range from 2.45 to2.6 A is permitted, about half of the area of Fig.5(b) is permissible forthis shared edge. When all the spatial considerations for the oxygenatoms are combined together, onlv a small area is permitted in Fig. 5(b),and indeed the observed position of the oxvgen atom is at the center ofthis region. Returning to Fig.5(a) it nil l be seen that the small area ofFiS.5(b) is consistent with the lerrger area permitted for the Mg-O andAl-O bonds, and that appJ.ication of the l imits of the ox1'gen co-ordinatesdeduced from the oxygen atoms results in verl restricted ranges for theMg-O and Al-O distances. Zemann (1962) pointed out that cation-cationrepulsion shouid cause the IIgrO distance to the ox1'gen atoms forming ashared edge with the sil icon tetrahedron to be longer than the Mgz-Odistance. Actuall.v it is shorter, and Zemann concluded that this resultedfrom the need for unshared oxygen edges to be longer than 2.75 A.(Theoreticall l- cation-cation distances can increase without change incation-anion distances if the anion-anion shared edges are shortened).From Fig. 5 it can be seen that if MgrO is ionger than llgz-O quite un-reasonable values are produced for both O-O unshared edges and forAI-O. As high coordination, low-charge cations show large variations ofdistances to neighboring anions, the inequality of the two Mg-O distancesis not surprising because the mean vaiue of 2.2 A is quite reasonable.
CoNcrusrou
A similar analysis of the crystal structure can be carried out for anygarnet, and indeed it seems Iikely that such a method rvil l f ield an excel-lent prediction of the coordinates of the oxygen atom. In order to test
PYROPE STRUCTURE 2037
this and other structural features we are preparing least-squarres aniso-
tropic refinements for almandine and other garnets. It is remarkable how
closely the competing spatial factors can be satisfied over such wide
ranges of chemical composition in a structure with such high symmetry
and few variable atomic parameters. The deviertions of observed dis-
tances from those to be expected from consideration of just the first and
second neighbors amount only to 0.1 A for the weaker bonds, and are
much less for the slronger bonds. The general principle that ali bonds
must be satisfied within these toierances, and that the finer details are
determined by a compromise between the attractive and repulsive forces
of both cations and anions, hopefully applies to other complexly-bonded
structures such as aluminosilicates and olivine. However, analyses of
these minerals will be more difficult, and perhaps impossible, because of
the lower symmetry and Iarger number of variable parameters' Born
(1964) has presented a lattice-energy calculation of olivine which gives a
good prediction of the position of the oxygen atom. Although covalent
bonding presumably plays an important role in the garnet structure, the
success of the above treatment based solely on ionic bonding supports
the statement of Verhoogen (1958) that silicates of magnesium and a'lumi-
num behave mostly as purely ionic compounds.
AcxNowr,rocxrENTS
The authors wish to thank Dr. Francis R' Boyd, Jr., of the Geo-
physical Laboratory, Washington, D.C. for donating the crystal used in
the study, Mr. Charles Knowles for aid in the collection and correction of
the data; Dr. Charles W. Burnham of the Geophysical LaborertorY, Dr'
Charles T. Prewitt of the E. I. Du Pont de Nemours Central Research
Department, Professor Adolf Pabst of the Geolog,v and Geophysics
Department at the University of Czriifornia, and Dr. Earle R' Ryba of
the Metallurgy Department at The Pennsylvania State Universitl' for
crit ically reading the paper; Dr. Chester M. Smith of the Comput.ation
Center at The Pennsylvania State University for assistance in modifying
the Busing-\{artin-Levy Least Squares Program for the IBM 7074; and
the National Science Foundation for grants G-t4+67 and GP-443 which
supported this research. Professor J. Zemann has assisted us considerably
bl correspondence.
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PYROPE STRUCTURE 2039
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M anuscriptr received, M arch 1 , 1965 ; accepted f or publ,icati,on, JuJy 28, 1965.
