THE STRUCTURES AND CRYSTAL CHEMISTRY
OF BUSTAMITE AND RHODONITE
BY
DONALD RALPH PEACOR
ck- rN
I
I
4
S.B., Tufts University
N4
(1958)
S..,
Massachusetts Institute of
Technology
(1960)
SUBMITTED IN PARTIAL FULFtLUENT
OF THE REQUIREMEIC FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August 1
Signature
of
.
Author.
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1962
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0
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0
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a
a
ppartment of Geology, August 20, 1962
Certified by
.
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I
Accepted
by.
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0
0
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0
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o
a
I
1. .
a
*
0
*
0
0
0 .
Thesis Supervisor
.
.
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.
.
.
.
.
.
Chairman, Departmental Committee
on Graduate Students
THE STRUCTURES AND CRYSTAL CHEMISTRY OF
BUSTAMITE AND RHODO1ITE
3y
Donald R. Peacor
Submitted to the Department of Geology and Geophysics
on August 20, 1962, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
ABSTRACT
Bustamite (CaMnSh%) has spage group F1 an# unit
c w 13.824A, o. w
cell parameters a = 15.412A, b = 7.157A,
89*29', p - 94*51' , ym 102*56', with 2 = 12. Three-dimensional intensity data were gathered with the single-crystal,
Geiger-counter diffractometer. Application of the minimum
function to P(xs) and comparison of the unit cells of bustamite and wollastonite (CaSio 3 ) yielded a trial structure
which was refined with least-squares.
Rhodonite (Mn CaSi Oa5)
has space group P1 and ,
c = 6.707A.,
b = 11.818f,
o. s 92*21', A 93*57' , Y = 105*40' , with Z = 2. Interpretation of the three-dimensional Patterson function led
to the identification of (Mn,Ca) inversion peaks whicb were
used to form the minimum function. Interpretation of this
function resulted in a trial structure which was refined
by means of least-squares.
unit cell parameters a
7.682A,
The structures of bustamite and rhodonite are
similar to those of other metasilicates having chains of
silica tetrahedra. Oxygen atom are arranged approximately
in close-packing. Planes of Mn and Ca ions occupying octahedral interstices alternate with planes of tetrahedrally
coordinated Si ions, between oxygen sheets. Silica tetrabedra each share two vertices to form chains with a repeat
unit of three tetrabedra in bustamite and five in rhodonite.
111
TABLE OF CONTENTS
List of Figures
List of Tables
PART II
A. Determination and refinemt of the crystal
structure of bustamite, CaMni 2 06
Abstract
Pae?
vi
viii
2
2
Introduction
Unit cel and space group
Intensity data
Structure determination
Determination of x and z coordinates
Determination of y coordinates
4
4
7
8
8
16
Refinement
20
Description of the bustamite structure
Acknowledgement$
References
24
38
39
B. Comparison of the crystal structures of
bustamite and wollastonite
Abstract
Introduction
Description of structures
Arrangement of SiO 3 chains
Substructure relations
Cation ordering
Interatomic distances
Pseudomonolinic cells
41
41
42
44
51
61
61
63
67
Acknowledgements
68
References
69
iv
TABLE OF CONTENTS cont.
PART I
1. Review of literature on phase relations and crystal
structures in the system CaSiOgMnSiO3
Phase relations
X-ray crystallography
Tthe determination and refinement of the crystal
structure of rhodonite
72
72
79
84
Introduction
84
Unit cell and space group
Preliminary considerations
Comparison of wollastonite and rhodonite
Interpretation of the Patterson function
86
88
92
93
P(xyz)
Refinement
Description of the rhodonite structure
Sheet of octabedrally coordinated cations
Coordination polyhedra and cation ordering
Interatomic distances
Conformity of rhodonite to Pauling's rules
Temperature factors
3. Crystal chemical relations among some pyrozenes
and pyroxenoids
other pyrosenes and pyroxenoida
Classification of pyroxenes and pyrozenoide
Thase tranfoations
APPENDIX I DCgLL; IM 709/7090
Program for the transformation of unit cell parameters and coordinates
103
128
135
135
143
14$
146
148
159
161
161
164
TABLE OF CONTENTS cont.
Directions for operation of DCELL
FORTRAN listing of DCELL
FORTRAN listing of subroutine
SYTRY for Pt space group
165
165
170
Comparison of observed and calcuAPPENDIX II
lated structure factors of bustamite
171.
Coparison of observed and calcuAPPENDIX III
lated structure factors of rhodonite
19S
Acknowledgemente
Biography
237
239
References
241
LIST Of FIGURES
Part I
Al
A2
A3
A4
Page
Projection along b of the structure of
wollastonite
10
Busamitey fl.(zz)
4
Bustaaite, f(ns)
12
Coaparison of unit cells of bustamite
and wollastonite
17
Projection along
14
the pea'"s o01
o
(o (Cyz)
A6
Interpretation of
A7
Structure of bustamite !rom x
projected
2S
(xys)
&ae
piaai; die
_to-
p
4
31
by the
axes b and c zsin
AS
Projectiont.
bustamite
along c of the structure of
33
Projectioa along b of the structure of
wollastonite
47
B LB
Proj ection along b of the structure of
bastaraite
49
B 2A
Projection along a of the structura of
wollastonite
52
B 2B
Projecti
bustamite
L 1A
along a of the structure of
Projection along C of the structure of
wollastonite
56
vii
LIST 0) FIGUES cont.
I 3B
part %
1.1
2.1
Projection along c of the structure of
hustamite
58
Phase relations in the subsolidus region
of the system CaS±03 - Msio3 as a funetion of teperature and coposition
77
Selected peaks of P(xyx) projected onto
94
(001)
2.2
Peaks ot ?4,(xys) projected onto (001)
2.3
Coordination polyhedra projected onco
(001)
e
(xys) projected onto (001)
101
2.4
Peaks of
2.5
Projection on (001) of the structure of
rhodonite
Projection along a of the structure of
rhodtniza
131
Projection along a of the structure of
150
2.6
3.1
129
133
Clinoe statite
3.2
Projection along
fiot7
of the structure
152
j of the structure
154
of wollAstoite
3.3
Projection along f[0u
f rbodanita
viii
LIST OF TABLES
Part I
Al
Synetry and unit call data for bustamite
and wollastonite
A2
Interatooic distances and cation peak
heights after cycle 3 of refinemnt
22
A3
Coordinates and isotropic temperature
factors for atom of bustamite
25
A4
Interatomiz distances i bustamite
3D6-
31
Sym$etry and unit cell data for bustamiteu
and vollastonlte
43
B2
Coordinatas and isotropic temperature
factors of atom in wollastonite and
45
bustamite
Part I1
1.1
Catinorygen interatomic distances
64
Unit cells and space groups of Ca, Mn
80
metasilicates
2.1
Lattice constants of rhodonite
85
2.2
Comparison of unit cells of som pyroxenes
and pyroenoids
90
2.3
Atom coordinates from the minima function
104
and after refinemt
2.4
Outline of least-squares refinement
110
LIST OF TABLES cont.
122
2.5
Peak beights at cation positions of Ap(xyz)
and Mn-O interatomic distances after cycle
32
2.6
Comparison of coordinates of Liebau
and this refinemnt
2.7
Cation-oxygen interatomic distances
136
2.8
Oxygen-cation interatomic distances
138
2.9
Oxygenu-ovygen interatomic distances
140
t al.
125
1
Part I
Papers which have been prepared for
publication.
Part
I-A
Determination and refinement of the crystal structure
of bustamite, CaMnSi
2
6
By Donald R. Peacor and M.J. Buerger
Massachusetts Institute of Technology
Cambridge, Massachusetts
Abstract
Bustamite is triclinic, space group Fl, with
cell dimensions a = 15.412 A,
89*29',
f
94*51',
b = 7.157 A, £ - 13.824 A,
e = 102*56'.
ideally contains 12(CaMSi 2 06).
The unit cell
Three-dimensional inten-
sity data were gathered with the single-crystal Geigercounter diffractometer.
function to _P(_M),
Application of the minimum
and comparison of the unit cells of bust-
amite and wollastonite yielded a trial structure which was
refined with least-squares.
Planes of approximately close-packed oxygen atoms
are oriented parallel to (101).
Planes containing Ca and Mn
atoms in octahedral coordination alternate with planes of Si
3
atoms in tetrahedral coordination between oxygen planes.
Ca and Mn are ordered.
Si tetrahedra are linked to form
chains parallel to b with a repeat unit of three tetrahedra.
Introduction
Several authors have noted the similarity of
the properties of bustamite, CaMnSi20 6 , to that of
wollastonite, CaSio 3 .
On the basis of a similarity of
optical properties of the two minerals, SundiusI a
Schaller
proposed that bustamite is Mn-rich woplas-
tonite.
Berman and Gonyer4 accepted this view when they
determined the unit cell of bustamite and noted its
correspondence to that of wollastonite (Table Al).
Buerger5 and Lebau e at.
found, however, that the
unit cells are different although closely related (Table Al).
Because of the recent interest in the structures
of the minerals of the wollastonite family, it is important to understand the relation of bustamite to other
triclinic metasilicates.
In this paper the results of an
investigation of the structure are given.
Unit cell and space group
Specimens of bustamite from Franklin, N.J., were
kindly provided by Professor C. Frondel of Harvard University.
The unit cell obtained by Buerger was confirmed with zero,
first, second, and third-level b axis Weissenberg photographs.
-i 1,i 4
, 'llm
95.ZOI
sSt.TOI
*9ZSOI
,*t,6
Izs,*
1"Vt93
Z91Z
91'1£
11 Iot
IV*5I
V ULZ
V +160L
9
0
etang(auo
oona)
fcl0m
w3
nwpn)
(xaltwng)
pms unaga)
s4g £ogOng9*Oisij
(zte~
937=2"1.mw0s4
'flWuO3srrOa PuS njTmnsnq .to; V~j' TIO
I[V alNvL
flufl pu
LneunmAs
Final cell parameters (Table Al) were obtained by leastsquares refinemet of data from a, b, and
-axis zero-
level back-reflection precision Weissniberg photographs.
Refinement was carried out using an IBM 709 program
written by C. W. Burnham.7
Although it is usual to
choose a reduced, primitive triclinic unit cell, the facecentered cell was retained because of its close correspondence to that of wollastonite.5
Using the refined cell parameters and an analysis of Franklin, N.J., bustamite,8 the unit-cell contents
were found to correspond to the following formula:
Ca1 3 .
37
10.57
0.48
s24.00
072.68
Zn 0.20
Fe 0.11
24.73
Since a precise value for the specific gravity was not
available, this distcibution was obtained by normalizing
the formla to 24.00 Si atoms per cell. The excess of Ca.
Mn, etc., relative to Si (24.73 : 24.00) is attributed to
a slight error in the analysis.
The above formla corresponds ideally to
12(Catni2 0 6 ) per face-centered cell, or 3(CaMnSi206) per
-
ta*~t
*
-.
-
-
-
7
primitive cell.
of rank 1 and 2.
Space group jl
contains only equipoints
Three Ca and three Mn atoms can all
occupy general equipoints of rank 2 only if there is disorder between at least one Ca aM one Mn.
Coplete order
of Ca and Mn requires that at least one Ca and one Mn be
on inversion centers of rank
.
The grouping of analyses
of natural material about the ideal composition suggested
the latter.
Since no Ca atoms occupy inversion centers in
wollastonite, there was a strong possibility that this constituted a major difference between the structures.
Intensity data
The specimen used for the intensity measurement
was a prism bounded by (100) and (001) cleavages, 0.233 mm
long and 0.030 x 0.048 mm in cross section.
The crystal
was mounted on the single-crystal diffractometer with the
k-axis as rotation axis.
Cu_ a
Intensities of the reflections with
radiation were measured with an argon-filled Geiger
counter.
Care was taken not to exceed the linearity range
of the Geiger counter; counting rates were maintained below
-
-
a'.
_______
-
-
8
500 c.p.s. by using aluminim absorption foils.
were measured by direct counting.
Intensities
Background was measured
on each side of a peak and the average backgrounr
subtracted
from the value of background plus integrated intensity obtained by scaning through the peak.
Of approximately 1650
non-equivalent reflections in the Cu sphere, 1212 were ree
corded.
These were corrected for the Lorents and polari-
uation factors using an IBM 709 program written by C. T.
Preitt,9 and for absorption using an IBM 709 program
written by C. V. Burnham.
10
Structure determination
Determination of x and s coordinates.
Comparison
of hOJ1 reflections of bustamite and woliastonite shoed that
the distribution and intensities of each were similar.
This
suggested that the projections a the structures along b
are the same.
The Patterson projection P(s) of bustamite
was therefore prepared.
Its similarity to that of P(g) of
wollastonite confirmed that the structures are very probably
the same in projection.
9
The structure of the wollastonite
projected along b
is shown in Fig. Al. Ca1 and Ca2 are superimposed in pro-
jection. On the assumption that Ca and Mn atoms in bustamite have a similar arrangement, a minimum function
M2()
was prepared for bustamite using the equivalent of the superimposed Ca1 + Ca2 inversion peaks of wollastonite. In wollas-v
tonite, the Patterson projection P(x)
the structure as seen from Ca3'
contains an image of
The Patterson projection
and M2(g) were therefore superimposed to obtain the minimum
function
shown in Fig. AZ.
f(xa)
Note that for each atom of
(x)
there is a corres"
ponding peak of the correct relative weight in
bustamite.
M(s) of
There are four extra peaks, labelled A,
and D, in g(fJa).
. C,
Three of these, A. & and C, are near the
peaks corresponding to 07A 08 and Si 3 of wollastonite.
Structure factors I.1,
calculated omitting these atoms,
gave a discrepancy factor R = 31%. Using these structure
factors the electron-density projections
and &p(Xa) were then calculated.
(s), Fig. A3,
As can be seen from a
comparison of Figs. Al and A3, these confirmed that the projections p (_)
the sam.
of bustamite and wollastonite are essentially
None of the false peaks, A,
appeared in
Si 3 do occur.
p (as), while peaks corresponding to 07, 08 and
A second structure factor calculation based on
coordinates from
for all Ebol.
, C and D of g(a)
(x) and
(xs) yielded an R of 21%
10
Figure 1, Part IA
Projection along b of the structure of wollastonite.
Oxygen atoms are represented by large
circles, Ca atoms by circles of intermediate
size, and Si atoms by small circles.
C Sin a
0
0
0C
O0 , 04
QC
3
OCa, ,Ca
2
0
os
a sin
y
12
Figure 2, Part IA
Bustamtts,, M4(xz).
is abaded In.
The high (Ca + Mn) peak
Peaks which do not correspond
to atoms are labelled A, B, C and D.
,/u
s Io
7i
14
Figure 3, Part ZA
Bustamutte,
shaded in.
(xz).
(Ca + Mn) peaks are
uIs
/0
Z)u I s /
16
Determination of X coordinates. The lower part of
Fig. A4 shows the primitive triclinic unit cell of wollastonite
and the upper part shows the face-centered bustamite cell
in corresponding orientation.
Inversion centers coinciding
with lattice points are shown as solid circles and all other
inversion centers as open circles. All inversion centers
lie on (101) planes which intersect a and c at 0 and
in
32
wollastonite and at 0,
and
in bustamite. The distri'4'2
4
bution of inversion centers is the same in both structures,
-
except that those centers on planes parallel to (101) in
1
and a and £ at 3 are
bustamite intersecting a and s at
$
shifted by
relative to those in wollastonite.
Figures Al and A3 show that Si and (Ca, Mn) atoms lie
approximately on planes parallel to (101) which contain the
inversion centers in both structures.
Planes containing Si
atoms alternate with those containing (Ca, Mn).
Figure Al shows that in wollastonite there are inversion centers in the (101) planes at
x
-
0 and 2. Because the
b
sub-structure has period 2, there are also pseudocenters for
the Ca atoms and their coordinating oxygen atoms, which pro1
3
and a. Since the Si
ject at the same xz coordinate, at y
atoms are not in the substructure these pseudocenters do not
apply to them.
17
Figure 4, Part IA
Comparison of unit cells of bustamite (above)
and wollastonite (below).
Lattice points (which
are coincident with inversion centers) are shown
as solid circles.
Other inversion centers are
shown a6 open circles.
0
0
0
0
0
0
*
o
0
o
o
o
~
0
0
o
0
o
0
0
0
0
0
~
0
*
00
0
0
0
00
03
19
A reasonable model for the structure of bustamite
results from combining the above information with the fact
that wollastonite and bustamite are similiar in projection.
Coordinates of Ca and Mn atoms and their coordinating oxygen
atoms in bustamite differ from those in wollastonite in the
following ways:
1.,The pseudoinversion centers in wollastonite at
1, 3 in (101) planes containing Ca become true inversion
centers in bustamite.
2. The inversion centers at y = 0, 2 in wollastonite
become pseudoinversion centers in bustamite.
The relative arrangement of silica tetrahedra is the same
in both structures except that tetrahedra in alternating
b
sheets are displaced by g by the face-centering translation
in bustamite.
The above arrangement of inversion centers requires
that the (Ca, Mn) of bustamite, equivalent to Ca3 of wollastonite, be located on inversion centers; that is, four (Ca,
Mn) atoms are still in two general positions of rank 2, and
two (Ca, Mn) atoms are on centers with rank 1.
This is just
the distribution required for ordering of Ca and Mn atom
which was noted in the section on unit cell and space group.
20
Structure factors were calculated for all 1212 observed reflections using x and x paramteras derived from
the projection, and
x
as required above.
Ca and )fn atom were assumed to be
parameters of wollastonite adjusted
completely disordered, since at thi point there was still
no direct evidence fZor ordering.
L
was only 3,
The discrepancy factor,
suggesting that the proposed structure
was probably corrert.
lfn
t
was carried out using SFLSQ2, a least-
squares IM 709 program usiag the full matrix
.
For the
initial ref iisant, form-factor curves were used for Ca and
Mn assuming couplet* disorder, and a correction made for the
average real component of anomalous dispersion.
were assumed to be half ionized.
All 1212 reflections were
used as input to each cycle of refn
with
All atoms
t, but those reflections
> 0.5 were rejected by the program.
>-k)/4
weighting schme reco
The
d by Hughes 1 3 was used throughout.
Isotropic temperature factors for all atoms were initially
21
set at 0.5, a value consistent with temperature factors of
other silicate structures refined in this laboratory.
The discrepancy factor R decreased to 12.3% after
only three cycles, confirming the correctness of the proposed
structure.
For these cycles, only the scale factor and co-
ordinates were allowed to vary.
Using structure factors cal-
culated from coordinates of cycle 3, a three-dimensional
difference Fourier synthesis 4 p(19)
was computed.
(All
three-dimensional Fourier syntheses were computed using the
IBM 709/7090 program ERFR2. 1)
The only major discontinuities
in the synthesis were negative or positive peaks at the positions of (Ca + Mn) atoms.
Table A2.
The peak heights are listed in
Interatomic distances, also shown in Table A2,
were calculated using ORXFE, an IBM 704 program.15
Mn has five more electrons than Ca, assuming equal
ionization of Mn and Ca.
Thus the positive peaks at the posi-
tions of cations I and 3 clearly indicate that Mn atoms are
located there, while the negative peaks of cations 2 and 4
+2
The radius of Mn
suggest that Ca occupies those positions.
0+2
(0.91 A) is less than that of Ca
0
(1.06 A).
The smaller
average cation-oxygen distances of cations 1 and 3 and the
larger distances of cations 2 and 4 therefore confirm the
ordering indicated by the peaks of Ap(gx).
22
Tabie A2
Interatomic distances and cation peak heights after cycle
3
of
refinement.
Cation
Equipoint
Average cation-
number
rank
oxygen inter-
Peak height
in A p (a&)
Interpre-
tation of
atomic distance
377
Mn, excess Ca
1
2
2.25 A
2
2
2.38
-1170
Ca
2.21
642
Mu
2.39
-1250
Ca
3
4
1
r
23
It was noted above that there is an excess of Ca over
Ma in the material used for this study.
position of cation 1 in
for cation 3.
i
&p (gy)
The peak at the
is slightly lower than that
In addition, the average cation-oxygen distance
slightly larger for cation 1 than for cation 3.
This in-
dicates that the excess Ca substitutes for fn preferentially
in the site labelled cation 1.
Three Structure-factor computAtions were made (taking
account of both the real and imaginary components of anomalous
dispersion of Mn and Ca) for the following distributions of
the excess Ca:
1. all excess Ca assigned to cation I;
1
2
2.
excess Ca assigned to cation 1, 2 to cation 3;
3. all excess Ca assigned to cation 3.
The R value was essentially equal for all three distributions
(#N9.5%) but the comparison of
a ds al was slightly
better with the eess Ca substituting for the Hn of cation I.
Since this confirmed the conclusion reached on the basis of
the peaks of A? (xys) and the interatomic distances, this dittribution of the excess Ca was accepted as being correct.
Refin
t was continued with the Ca and Mn distribution
determined above, taking full account of anomalous dispersion.
The Zn, Fe and Mg reported in the analysis were considered to
24
substitute for Mn, since these atoms are smaller than Mn.
Two more cycles ( 4 and 5 ) were executed varying
only coordinates and the scale factor.
essentially unchanged in cycle 5.
The coordinates were
Individual isotropic
teyperature factors were varied in cycles 6 and 7.
Refine-
ment was concluded with two cycles in the first of which the
coordinates and the scale factor were permitted to vary,
and another cycle in whieb isotropic temperature factors
were refined.
Final parameters and their errors are listed
in Table 3.
Description of the bustamite structure
The electron density p (zW) was computed using the
signs of the structure factors of the final cycle of refinement.
The peaks of this three-dimsnsional function are
shown projected parallel to b in Fig. A5. Figure A6 is an
interpretation of the same projection.
The similarity of
the structures of bustamite and wollastonite (Fig. Al) has
been noted above several times.
The relations between these
structures is discussed in another place.
16
TZble AS
texperature factort for atoms of
Coordinates adtaropc
1000 -.009
.0994
Ca
Iw
2
Ca2
I2t
Si
02
.6725
.0003
.3733
.1583
.0004
.337s
85
4
4
.0884
.0888
. 2158
.2018
03
(3'
.1563
04
.1509
-0000
.0002
.0002
-0002
.0005
.0005
1
2'
.6454
.9805
-9758
.0131
06
.0140
.0005
.0005
.7206
.3964
*8513
-1287
.1240
08
.1364
.7625
09
.0926
.0006
.0005
.0005
.0014
4250
.0014
.0014
.0014
.0014
.0014
.0014
.0016
----
5
01
1
.000
0
.669
0
.03
.56
.04
4
I
4
,1343
.4840
.1338
-0005
00
0
.2003
.0005
05
07
1
0
-1975
Si3
01
.0001
- C)
C0
--
0
.73
.34
.32
.05
.1325
40002
.0218
.0002
.16
.4027
.0002
.68
04
.4069
.0005
.57
.12
.2293
.0005
.0005
.48
.12
.50
,12
.64
.12
*12
.2315
.3549
.3717
.0005
* OO
005
.0
.66
.0393
.0005
.0411
1
.0005
-.1147 --
6
-o34
Oo06
57
.04
* 04
.12
290
.0411
1.34
14
.14
26
Figure 5 Part ZA
Projection along b of the paks of g(xyz).
90
0)
0)oUS/
00
28
Figure 6, Part IA
Interpretatton of
at y
p (ayz).
Inversion ceuters
0, I sherm as solid circles and inversion
centers at y
;* A as open circles.
SD
7ouls cJ
pill
30
In Fig. A6 the arrangemnt of oxygan atoms crudely
approximates close packing, all oxygen atom
except 09
lying in sheets wbicb are parallel to (101). Ca and Ma
atoms and St atoms alternate in layers between sheets of
oxygen atoms, with Ca and Ma in octahedral coordination
and Si in tetrahedral coordination.
$L0 4 tetrahedra all
share two oxygen atoms with other tetrahedra to form a chain
whose repeat unit is three tetrahedra and which is oriented
parallel to the h axis.
The nature of the arrangement of
tetrabedra is best shown in Figs. A7 and A8.
Fig. A7 is
a projection of the structure parallel to I while Fig. AS
is a projection along
£.
The sheet of Ca and Mn octabedra can be oWared to
the sheet of Mg octahedra in brucite.
Octahedra share edges
to form a continuous two4imensinal sheet in brucite.
The
wollastonite arrangement may be interpreted in terms of ideal
close packing of the oxygen atom in sheets coordinating Ca
atoms.
As shown in Figs. A7 and AB, octahedra share edges to
form a band three octahedra wide,
parallel to b.
extending infinitely
Individual bands are separated by a column
of unoccupied octahedrally coordinated voids.
There is
actually ensiderable distortion of the close-packed oxygen
layers, however, particularly around the column of vacant
octahedra.
31
rigure 7, Part TA
Structure of bustamite from *%k
tO
projected onto a plane defined by the axes
b ande sin
%
32
33
Figure 8, Part IA
Projection along c of the structure of
bustamite.
9UISD
owsq
35
Interatomic distances are listed in Table A4.
These
are discussed in detail elsewhere.16 All distances are close
The average Si-O distance is 1.623 A.
to accepted values.
Smith and Bailey17 state that the average Si-O distance of
a silicate is a function of the extent of tetrahedral link*
age.
The distance 1.623 A falls exactly at the value pre-
dicted by them for metasilicates.
There are three non-equivalent Si-0-Si angles in
bustamite, which are:
Si1 -0 -Si2 , 161*,
Si
-07 -S4, 135*,
8±2 -08
137*.
4,Si4
Corresponding angles in wollastonite are 149*
140* respectively.18
139* and
In a survey of Si-0-Si angles, Liebaut 9
found that the average for well-determined structures is
about 140*.
Angles greater than 150* are uncommon and,
under normal conditions, represent a relatively unstable
condition.
The angle Sil-09-Si2 is large in both bustamite
and wollastonite, but unusually so in bustamite.
In addition,
the isotropic temperature factor for 09 of bustamite is considerably larger (1.34) than all other temperature factors
of the structure (see Table A3).
In wollastonite, this
temperature factor (0.68) is only slightly above the average
for the entire structure.
-T
S
36
Table A4
Interatamic distances in bustamite
SiI - 03 1.628 A
05
1.587
Ca
-0
2.437 A
07 1.645
02 2.531
03 2.298
09 1.616
05 2.382
Av.1.619
o6 2.302
08 2.358
Si2 - 04 1.626
09 2.899
06 1.585
Av.2.384 (excluding 09)
08 1.647
09 1.613
n,2 - 20, 2.215
Av.1.618
203 2.154
204 2.241
St3 - 01 1.600
Av. 2.203
02 1.595
07 1.660
Ca - 2D2
2.344
08 1.671
203 2.412
Av.1.632
204 2.421
209 2.891
Av. 2.392 (excluding 09)
37
Table A4
(Conti.)
Interatomic distances in bustamite
mn
- 01
2.499
02
2.286
04
2.163
05
2.144
06
2.041
07
2.335
Av. 2.245
38
Acknowledgements
This work was supported by a grant from the
National Science Foundation.
Calculations were per-
formed in part on the IBM 7090 computer at the M.I.T.
Computation Center.
39
References
N. Sundius. On the triclinic manganiferous pyroxenes.
Am. Mineral., 16(1931) 411-429, 488-518.
Waldemar T. Seballer.
Johannsenite, a new manganese
pyrozene. Am. Mineral., 23(1938) 575-582.
3 Waldemar T. Schaller, The pectolite-scbisolite-serandite
series. Am. Mineral., 40 (1955) 1022-1031.
4
H. Berman and F. A. Gonyer. The structural lattice and
classification of bustamite. Am. Mineral., 22 (1937)
215-216.
5 M. J. Buerger. The arrangement of atoms in crystals of
the wollastonite group of metasilicates. Proc. Nat.
Acad. Sci. 42 (1956) 113-116.
6 F. Liebau, M. Sprung and 1. Thilo. Uber das system
MaSiG -CaMn(SiO)32. Z.Anorg. Allg. Chem. 297 (1958)
213-25.
Charles W. Burnham. WLSQ, crystallographic lattice cone
stant least-squares ref ineent program. (unpublished)
8 Esper S. Larsen and Earl V. Shannon.
Franklin Furnace, New Jersey.
95-100.
Bustamite from
Am. Mineral.,
7 (1922)
0. 7. Prewitt. The parameters T and (P for equi-inclination,
with application to the single-crystal counter diffracto-
meter.
Z. Krist., 114 (1961) 355-360,
10 C. W. Burnham.
The structures and crystal chemistry of
the aluminum-silicate minerals.
M.I.T.
PhD. Thesis, 1961,
40
M. J. Buerger and C. T. Preitt. The crystal structures
of wollastonite and pectolite. Proc. Nat. Aad. Set.,
47 (1961) 1884-1888.
12 C. T. Prewitt. SFLSQ2, an IBM 7090 program for leastsquares refinement.(unpublished)
13 E. W. Hughes. The crystal structure of melamine.
Amer. Chem. Soc., 63 (1941) 1737-1752.
J.
14 W. G. Sly, D. P. Shoemaker and J. H. Van der Iende.
ERR2, IBM 709/7090 Fourier program. (unpublished)
15 W. R. Busing and H. A. Levy. A crystallographic function
and error program for the IBM 704. Oak Ridge National
Laboratory Report No. 59-12-3, 1959.
16 D. R. Peacor. Comparison of the crystal structures of
bustamite and wollastonite. (in press)
17 J. V. Smith and S. W. Bailey. Second review of Al-0 and
(in press)
Si-a tetrahedral distances.
18 C. T. Prewitt and M. J. Buerger. A conparison of the
crystal structures of wollastonite and pectolite.
(in press)
19 Friedrich Liebau. Untersucbungen uber die Grosse des
Si-0-Si Valenwinkels. Acta Cryst., 14 (1961)
1103-1109.
41
Part
I-B
Comparison of the crystal structures
bustamite and wollastonite
Donald R. Peacor
Massachusetts Institute of Technology
Cambridge 39, Massachusetts, U.S.A.
Abstract
The structures of bustamite and wollastonite
differ principally only in arrangamnt of the chains of
tetrahedra. Chains in alternating sheets may be described
by the sequence AAA.... in wollastonite and ABAB.... in
bustamite, Both structures have a pseudomonoclinic cell,
this unit having space group P2I/m in wollastonite and
AZ/m in bustamite.
42
Introduction
On the basis of a comparison of optical properties,
Sundius (1931) postulated that bustamite (CaMnSi 2 0 6 ) is Mnrich wollastonite (Casio 3 ).
Schaller (1938,
1955) also
concluded that bustamite had the wollastonite structure because of a close relationship between the optical properties
of the two minerals.
Berman and Gonyor (1937),
using rotating-
crystal photographs, found that their unit cells were similar
(Table 1B) and concluded that they were related only by
solid solution.
Buerger, however (1956) found that the unit
cell of bustamite (Table 13) is closely related to , but
different from the cell of wollastonite.
He noted that there
is a sort of superstructure relation between the two minerals.
Liebau at al. (1958) confirmed Buerger's unit cell and
guessed that the difference in structures is based only on
a different ordering of chains and cations.
The structure of bustamite has recently been determined and refined (D.R. Peacor and M.J. Buerger, 1962).
The structure of wollastonite was determined by Namedov and
Belov (1956) and refined by Buerger and Prewitt (1961).
These structures are different but bear a very close relationship to one another.
TV
TA
U
s:Ic
0rcioi
etV.Z,g
dnoa8
!1S01
I' kaf
&1£,*06
2 iSo
&Z06
46E
Z6'9S t
LVE'lL
'C
ntqe8j
0rruvflwsnq
iR9
9T et
I ,'
~~9
MISTO9
r
iIwl,
i
9*3,Wo~l0PUOA pile
al
to' t
L
:aLM uoS~f
aelang
xo~; t~wp uas---.eTun VWu XxluaimS
*I
qn
44
Description of structures
The face-centered unit cell of bustamite may be
transformed to an A-centered cell with the following transformation matrix:
-1
-4
0
0
0
1
0
0
-
The A-centered cell bears a close relationship to the wollastonite cell, as does the face-centered call (Table 1B).
Since the relation between the structures of vollastonite
and bustamite is clearer if bustamite is described in terms
of the A-centered cell, all description of bustamite is referred to it in this paper.
Atomic coordinates for both bustamite and wollastonite are listed in Table 2B.
The similarity shows that
the asymmetric units of each structure are essentially the
same.
The structures are shown projected along b in Figs. IA
and I B .
Four wollastonite unit cells and two bustamite
unit cells are shown.
Ca or Mn atoms are shown as large
It
1 3
open circles, inversion centers a# y a 1, 3 as small open
circles, and inversion centers at y
circles.
a
0, 21 as small solid
The structures are the same in projection except
for minor coordinates shifts.
The arrangement of the oxygen
45
Table 2B
Coordinates and isotropic temperature factors of atoms
in wollastonite (upper values) and bustamite (lower values).
For comparison , bustamite a coordinates are multiplied by
2,
Coordinates of atoms in bustamite are given relative to
the A-ecentered unit cell.
Atom
Ca1
x
.1985
M.2018
y
a
B
.4228
.7608
.41
.4284
.7466
.56
Ca2
.2027
.9293
.7640
.45
Ca1
.1988
.9411
.7570
.69
Ca3
.4966
.2495
.4720
.37
Mn2
1/2
1/4
1/2
.56
Ca3
.5034
.7505
.5280
.37
Ca2
1/2
3/4
1/2
.73
Si1
.1852
.3870
.2687
.24
.1768
.3881
.2686
.34
.1849
.9545
.2692
.24
.1775
.9434
.2650
.32
.3970
.7235
.0560
.22
.3950
.7171
.0436
.16
.4291
.2314
.8019
.48
.4316
.2400
.8054
.68
Si2
St3
0
46
A tom
02
03
04
05
06
07
08
09
x
y
zB
.4008
.7259
.8302
o37
.4036
.7178
.8138
.57
.3037
.4635
.4641
.60
.3126
.4725
.4586
,48
.3017
.3017
.9374
.4655
.64
49302
.4630
.50
.0154
.0261
.6254
-7343
.63
.6167
.7098
.64
-0175
.1319
.7353
. 71
.0280
.1627
.7434
.66
.2732
.5118
.0919
.37
.2574
.5047
.0786
.37
.2713
.3717
.0940
.51
.2729
.8739
.0822
.29
.2188
,1784
.2228
.68
.1851
.1676
.2294
1.34
47
Figure IA, Part IB
Projection along b of the structure of
wollastonite (space group Pi).
cells are shown.
Four unit
c
sina
a siny
49
Figure 13, part IS
Projection along b of the structure of
bustamite (space group At) Two unit cell
are sboc.
--- - -----------
csina
atoms of both structures approximates close packing in a
crude way with an obvious layering parallel to (101).
Layers
consisting of Ca (wollastonite) or Ca and Mn (bustamite)
atoms in octahedral coordination alternate with layers composed of Si atoms between the sheets of oxygen atoms.
The
5i0 4 tetrahedra are arranged in chains whose repeat unit is
three tetrahedra and which are oriented parallel to the b
axis (Figs, Z and 3 ).
Arrangement of S103 chains
The primary difference in the structures of bustamite and wollastonite lies in the arrangement
of the Sit0 3
chains. Note first that the arrangmt of Mn and Ca atoms
is approximately the same in both structures over all space.
This is most easily seen in Figs. 2.A and 2-.
This relation
also holds true for all oxygen atoms whieb coordinate Ca or
Mn atoms.
(This relation will be discussed in more detail in
the section on substructure.) -The difference in the structures
then lies principally in the location of 09 and the silicon
atoms.
In successive layers in wollastonite these atoms
are related by the translation 001, and in bustamite by the
52
Figure 2A, Part tB
Projection along a of tb. structure of,
wollastanitte,
Only Ca atoms aM translation-
related chains are shown.
Ausq
54
Flgure 2B, Part 1B
Projection along a of the structure of bustaite. Only Ca ad Mn atom and translationrelated chains are sham.
Xuisq
56
Figure 30, Part 13
t;ojection a1ong c of the structure Of Wollastonitt.
The primitive triclinic cell is out-
lined with a light line.
The pud
nocliie
cell (space group P2 /=) is partially outlined
with a dmtted line, and mirror planos are indi-p
cated with a heavy line.
Ouis q
58
Figure 3B, Part I
Projection along ar of the structure of bustamite.
The A-centered triclWc cell is out-
lined with a li5ht Line.
cell (space group A2/)
The pseudomoocinic
is partially outlined with
a dotted line, and irror planes are indicated
with a heavy line.
Duis q
60
translation 0+4.
This merely mean
that 8±03 chains in
successive layers are shifted by y/2 in bustamite but not
in wollastonite. The arrangement is illustrated in Figs. LA
and 2-1
, where the related structures are sbown projected
along a.
Only translation-related chains and Ca and Mn
atoms are shown.
The arrangement of the SiO3 chains may be
represented by the notation AAA...., in wollastonite, and
by ABAB...O
in bustamite.
Here A symbolises a shift of 0
and B a shift of J in the b direction for 8±03 ebains of
successive layers.
It is interesting to compare this with the polymorphism of MgSi03 described by Brown et at. (1961).
Here
the layers of Mg and Si atoms also alternate between layers
of oxygen atoms arranged approximately in close packing.
8±03 chains in successive layers may be displaced by c/2
(the axis c is parallel to the chains of tetrahedra) giving
rise to stacking relations analogous to those between bustamite and wollastonite.
In the polymorpbs of MgSiO 3 , hwever,
the relative positions of the Mg polyhedra may differ from level
to level.
61
Substructur4 relati4n
Both bustamite and wollastoatte are characterized
by
t
prominent Substructure with period b/2 (Fig. BI).
All
atoms except Si3 and 09 are related to a second atom by a
shift of about b/2.
Note, for example, in Fig. BIA, that
Ca1 and Ca2 fall almost exactly over each other, and differ
in y by 0.51.
The relation is imperfect in some cases, as
with 01 and 02 of wollastonite, where the shift ( A x, A y,
A z) is 0.03, -0.49, -0.03,
mately true.
Nevertheless, it is approxi-
The coordinates of the substructure atom are
very similar in the two structures.
Thus, with the exception
of Si3 and 09, the structures of wollastonite and bustamite
are approximately the same.
Cation ordering
Sehaller (1938) states that minerals in this group
have been found with "MnO ranging from a few per cent. ..to
a mainum of 33 per cent."
No Mn was reported in the wollas-
tonite refined by Buerger and Prewitt (1961).
The bustamite
(ideally CaMnSi 2O6 ) whose structure was refined by Peacor
and Buerger contained a slight excess of Ca relative to M.
62
In Wollastonite the Ca atoms are distributed over three
general positions.
The Ca and Mn are ordered in bustamite,
with one Ca and one Mn inversion centers and two Ca and two
fn in general positions.
In addition, evidence strongly
suggested that the "excess" Ca of bastamite replaces WMat
the position Ma1 rather than the special position Mn2 .
Although all Ca or Mn atoms of each structure
have approximately the same coordinates, there is not camplate equivalence of positions.
The C*2 and Matao
of
bustamite on inversion centers are equivalent to the Ca
3
atom in a general position in vollastonite. The Mn and
Caatoms in general positions of bustamite, hmever, are
not equivalent to CS1 and Ca2 of wollastonite.
Half of
atoms of buastamite occupy positions in the
the Mn
structure similar to those of Ca
of vollastonite, and
half occupy positions similar to Ca2. The same is true
of Ca
of bustamite.
This is reflected in the difference
in distribution of inversion centers in (101) layers containing Ca and Mi at os, as can be seen in Figs. 31A and
811.
Compounds whose compositions are intermediate bet'-
ween bustamite and wollastonite still must be investigated
to determine the limits and mechanism of solid solution in
each.
It is possible, if not probably, that metastable
intermediate compounds exist which have partial or complete
disorder both in Ca and Mn and in BiD 3 chain distributions.
Interatomic distances
All average Ca-0 distances are remarkably similar
(2.388 ± .004 A) in both structures as shown in Table 38,
except for that of Ca1 of wollastonite.
The comparison of
Si0 distances is particularly interesting. The average
of all Si-O distances is 1.623 A in bustamite and 1.626 A
in wollastonite, the difference being well within the
standard deviation. All average Si-0 distances for Si1
and Si2 are almost exactly equal, and Si3-0 distances are
uniformly larger than Si -0 and 8i2 -0 distances in both
structures.
There is an excellent correlation of individual
Si-0 distances with coordination of oxygen.
For instance,
9
all Si-0 7 and
$i-08
being the largest.
0
distances are greater than 1.64 A, 1.673 A
Both 07 and 08 are coordinated to two
Si atoms, from which they receive a total bond strength of
2, and to one Ca or Mn atom, from which they receive a bond
is thus com-
strength of 3. The excess of bond strength
All Si-O5 and
pensated by unusually large Sis-O distances.
0
0
Si.06 distances are less than 1.59 A, 1.572 A being the
Table 3Z
Cation-oxygen interatomic distances
Wollastonite
Bustamite
1 01
2.499
02
2.286
0
2.272
Ca
-sO
2.302 A
04
05
2.163
0:
2.324
2.144
0
2.302
06
2.041
Of
2.272
2.335
0
2.412
2.245
Av r.
2.314
07
Av
Ca-0
2.437
02
03
05
06
08
A9
Av.
Ca2 - 0
2.316
2.531
02
2.369
2.298
04
2.421
2,382
2.302
05
2.358
'6
08
2.899
Av.
2.384(excluding 09)
2.501
2.316
2.406
2.388
65
Bustamite
mn2
Wollastonite
g
2.439
2.154
02
2.349
204
2.241
03
2.429
Av.
2.203
0'o3
2.335
04
2.441
04'
2.349
09
Av.
2.642
2 1
2.215
203
Ca3
2.390(excluding 09)
Ca2 -202
2.344
Ca3 - 01
2.439
203
2.412
02
2.349
204
2.421
03
2.429
209
2.891
03
2.335
2.392(ezcluding 09)
04
Av.
Si
2.441
04,
2.349
09
Av.
2.642
2.390
1.618
03
1.628
05
1.587
05
07
1.645
07
1.659
09
1.616
09
1.647
Av.
1.619
Av.
1.624
Si-
03
1.572
66
Wollastonite
Bustamite
si 2 - 04
0
1,626
0
09
1.613
Av.
1.618
02
07
08
Av.
1.600
086
Os
A9
AV#.
Si3
01
1.595
02
1.660
07
1.671
1.632
1.617
1.581
1.585
1.647
Si3 - 0
4
1.650
1.637
1.621
1.599
1.599
1.665
1.673
Av. 1.634
67
smallest. These oxygen atoms are coordinated to one Si
atom from witch they receive a bond of strength I and to two
*a or Mn atoms from which they receive a bond of strength
3M
Thus there is a bond deficiency
,
which results in
unusually short SiO distances.
The only major differences in coordination between
the two structures involve 094
This is coordinated to Si
and Si2 in both structures, but Si-O distances are larger in
wollastonite.
In addition, 09 is coordinated to both Ca
atoms in bustamite, but with very large Ca-0 distances, and
to only one Ca in wollastonite, at a shorter distance.
Pseudomonolinic cells
Ito (1950) noted that the angle c of the wollastonite unit cell is very nearly 90* and that (140), which is
normal to (100),
approximates a mirror plane.
From these
data he suggested that the triclinic wollastonite cell is
made up of "twinned or otherwise juxtaposed" monoclinic
cells.
Individual monoclinic units art related by the glide
b/4 or -b/4. He noted that the monoclinic unit, which be
called protowollastonite, must have either of the space
groups P2/m or P2 /m.
68
Figures 3A and 33 are projections along c of the
structures of wollastonite and bustamite respectively. Prewitt
and Buerger (in press) noted that wollastonite has a pseudomonoclinic cell with space group P2 1 /m.
This cell is partially
outlined on the right side of Fig. *3A with a dotted line,
while the mirror planes are indicated by doubly heavy lines.
The triclinic cell is outlined on the left with a lighter line.
Fig. B33 is an equivalent diagram of bustamite.
The basic re-
peat unit is the &am in bustamite as in wollastonite; i.e.,
two chains related by a 21 axis.
The bustamite pseudomonoclinic
cell is A-centered however, with space group A2/m. In wollastonite 21 axes are aligned along e. In bustamite there is a
similar relation except that 21 and 2-fold axes alternate along
C.
Acknowledge~mnts
I would like to express my appreciation to C.T. Prewitt
for many helpful discussions and for kindly making available unpublished data on wollastonLte, and to Professor M.J. Buerger
for suggesting this problem. This work was sponsored by a
69
Computations
grant from the National Science Foundatton.
were performed in part on the IBM 7090 computer at the
M.I.T. Computation Center.
References
Berman, Harry and Forest A. Gonyar (1937),
The structural
lattice and classification of bustamite.
Mineral.,
Brown, W.L.,
Am.
22, 215-216.
N. Morimoto and J.V. Smith (1961), A structural
explanation of the polymorphism and transitions
of MgSi03.
Buerger, M.J. (1956),
J. Geol,, 69, 609-616.
The arrangemet of atoms in crystals
of the wollastonite group of metasilicates.
Nat. Acad. Sci.,
Proc.
42, 113-116.
Buerger, M.J. and C.T. Prewitt (1961),
The crystal structures
of wollastonite and pectolite.
Proc. Nat. Acad. Sci.,
47, 1884-1888.
Ito, T. (1950),
Tokyo)
X-ray studies ont polymorphism. (Maruzan,
70
Liebau, F., M. Sprns and E. Thilo (1958),
Uber das System
KSto 3-CaMn(Si3) 2. Z. Anorg. Allg. Chem,) 297,
213-225.
Madov, Kb. S. and N.V. Belov (1956),
wollastonite.
Crystal structures of
Doklady Akad. Nauk SSSR, 107,
463-466.
Peacor, Donald R. and M.J. Buerger (1962), Determination
and refinement of the crystal structure of bustamite,
CaZnSi206.
Z. Krist.
Prewitt, C.T. and MJ, Buerger (in press), A compaison of
the crystal structures of wollastonite and pecto"
lite. Am. Mineral.
Schaller, Waldemar T. (1938), Jobennsenite, a new manganese
pyroxene. Am. Mineral., 23, 575-582.
Schaller, Waldemar T. (1955), The pectolite-sebisoliteserandite series. Am., Mineral., 40, 1022-1031.
Sundius, N. (1931), On the triolinic manganiferous pyrotenes.
Am. Mineral., 16, 411-429, 488-518.
71
Fart It
Thesis content not specifically prepared
for publication.
72
Chapter 1
Review of Literature on Phase Relations and
Crystal Structures in the System CaSiO 3 - MnsiO3
Phase-relations. The following naturally occurring and
synthetic ca ounds have been described in that part of the
system CaSio 3 w MnSi0 3 which may be considered to be binary:
wollastonite (Casio3), parawollastonite (Casio3 ), pseudowollastonite (CaSio 3 ), johannsenite (CamnSi 2 06 ), bustamite
(CaMnSi 2 6). rhodonite (MnSi0 3 ),
X -MnSi0 3 *
P -MSi03*
Solid solution limits are not given in the above formula$
since they are imperfectly known.
These relations are dis-
cussed in detail in the following sections.
Wollastonite is the stable low temperature poly-
morph of CaSio 3 and is the form commonly found in contact
metamorphosed, Ca-rich rocks. Wollastonite inverts at approximately 1125*C. (Glasser and Glasser, 1960) to pseudowollastonite, which is the form usually found in slags, but rarely
in nature.
rences.
Paravollastonite is known only from natural occur-
Its stability range, if any, relative to those of
pseudowllastonite and wollastonite is unknown.
Bustamite is the high-temperature form of CaMnSi 2 06.
Johannsenite, the low-temperature form, inverts to bustamite
73
at approximately 830C, (Scball'er, 1938 and Sebiavinato,
1953).
The relations between the polymorphs of MnS±0
3
are less well knovn. Liebau at al. (1958). in an investigation of the system CaMnSi 2 P6 - MnS%03, found three forms,
which they named v,
and
af0-Msio 3
(w#Mnsi03 is
equivalent to rhodonite and is stable at low temperatures.
It inverts at approximately 650*C. to
'MnSiQ
3,
which has
the bustamite structure. The third form, b-MnSiO3 us
metastable at all tewperatures and is produced only as an
intermediate form.
Phase relations in the subsolidus region of the
system Casio3
-
MaiC 3 are poorly known.
The most reliable
information is derived in many cases from surveys of
mineralogical compositions and relations.
Sundius (1931)
surveyed pyroxenoid analyses for those minerals in the
ternary system Fe(Mg)Si0 3 -
asio3 - MnSi3.
Wollastonite,
bustamite and rhodonite were included in this survey.
The
compositions of the natural wollastonites are all very
close to CaSiO3 , with no more than 8 mol. per cent (FeMg)Si0
3
and less than about 3 mol. per cent MnSiC 3 in solid solution.
Bustamite compositions range from 32 to 57 mol. per cent
CaSiO3 and rhodonite compositions 0 to 22 mol. per cent
CaSO 3 .
There is thus no mineral repres
range 22-32 sl.
wollastonite.
mineral
Utve in the
asto
and
per cent CaSLO3 between
ThiE indicates that tha formlas for these
should be written as follows#
wlastonite Ca
bustaite
3
CacI 2t
(St103)2
rhdonite
ta Ca 2
(S1035 , z0.0 to l0
*
z"W"*
to -*8
One smple invtstiated by Sundius contained c.xisting bstamite wIth 33 =a.
per cent CaSi0
3
and rho-
nite with 20 moL oer cent, strongly supporting other evidame
for an l
seihility gap between rhodenite and bustaite.
Because of the similarity in the optics and physical properties
of bustamita and rbdonite
Sundiuo concluded that there is
a cowplete series of solid solutions between them at 1
temper*turs, despite the wide range in interm
te am-
poitios not represented,
Schaller (1938) reported that be had
sauples of bustmtea or wollastonita "with perentgs of
Mn ranging from * few per cent to a maxim of 33 per cent
MnO8 or 61 per can
of Mn&SiOj2.
lie concluded from this,
and * similarity of optical properties of the two vinerats,
that they "srs a%heamognous solid solution of Ca04SO2
(Walla nite) with Mn0'Sit0
75
Voos (1935) investigated the system CaSiO 3 - MOMi3
at temperatures above 1200*C. He recognised only two phases,
pseudooliastonite and a material he called
tion.
P -solid sau-
The latter material has the wollastonite structure and
is represented by a coWplete series of solid solutions between CaSiO3 and MS03.
Pseudowollastonite was found to be
stable only at very high temperatures in the Ca-rich portion
of the system.
The results of Voos are seemingly at vari-
ance with those of Sundius, but on closer examination they
are seen to be caupatible.
Voos used the powder x-ray
pbotograpbic technique to identify phases.
Buerger (1956)
has shown that the diffraction patterns of bustamite and
wollastonite are very similar, except in the case of weak
super structure reflections.
The powder technique would
therefore not be adequate to show the minor differences
in the diffractica
patterns of these two minerals.
Liebau, Sprung and Thilo (1958) investigated
phase relations in the system CaMS 206 - 4n0
above they found, three MnSi
(rhodonite),
3
polyMorphs:
3.
As noted
-Mnsio 3
.-MnSi03 (bustamite structure), c -MnSi0 3
(pseudowellastonite structure, metastable).
They agreed
with Sundius that rhodonite may contain only up to 20 mol.
per cent CaSio 3 .
They found, further, that the temperature
76
of the rhodonite - bustamite inversion decreases with rising
Ca content.
The occurrence of the bustamite structure at
high temperatures throughout this system confirms the work
of Voos.
Glaser (1926) noted a single transition in MnSi3
and Jaeger and van Klooster (1916) noted two transitions at
1120' and 1208*C.
This confirms the work of Liebau with
respect to the rhodonite
- A -Mnsi03 transition.
One of
the three forms found by Jaeger and van Klooster may actually
be the metastable
t-MnSiO3 .
Glasser and Glasser (1960) and Glasser (1958) concluded that rhodonite is the only stable form of "WS10
3
at all temperatures up to the melting point, in a study of
synthetic phases in the system MnO-SiO2 .
Glasser noted
that MnSiO3 synthesized at high temperatures and quenched
exhibits diffuse x-ray reflections similar to those of
heated natural rhodonite.
He suggested that there may be
some disorder at high temperatures and/or a rapid unquenchable
inversion.
Fig. 11 is a disgrannatic representation of the
probable subsolidus phase relations, taken in part from
several of the above references.
only relative features.
It is intended to show
77
Figure 1.1
Phase relations in the subsolidus region
of the system CaSiO3 -MnSi0 3 as a function
of temperature and composition.
78
1100
;-wSS
JSS
900+
700
BSS
BSS
7QQI
RS
~JSSB$
I
RSS
I
Ro
Soo
500I
300
0
32
50
68
80
CaSiO 3
100
MnSiO 3
JSS = Johannsenite solid solution
BSS = Bustamite
WSS = Wollastonite
RSS = Rhodonite
PSS = Pseudowollastonite
79
Z-ray crystallography.
Wollastonite is triclinic, with
space group PT , and unit cell parameters (Buerger, 1956)
as listed in table 1.1.
Paravollastonite is monoclinic,
with a unit cell (table 1.1) very closely related to that
of wollastonite (Tolliday, 1958).
Barnick (1935) proposed
a structure for paravollastonite which contains three-mmbered rings of silica tetrahedra similar to those found in
benitoite.
Ito (1950) seeingly confirmed a similar structure
for wollastonite.
Buerger (1956) showed, however, that the
structures of pectolite, bustamite and wollastonite are
closely related, and that the structure of pectolite is
based on a chain of silica tetrahedra with a repeat unit of
three tetrahedra.
Momedov and Belov (1956) solved the
structure of wollastonite and confirmed that it contains a
chain of tetrahedra similar to that found in pectolite,
Ca2 NaH(Si03 3. Buerger and Prewitt (1961) subsequently refined the structure of wollastonite. Tolliday (1958) has
shown that the structure of parawollastonite is based on a
similar chain, and is closely related to the structure of
wollastonite.
A structure containing three-mebered rings
of silica tetrahedra has been proposed by Boll-Dornberger
(1961) for pseudowollastonite.
Lattice constants are shown
Table 1.1
Unit cells and space groups of Ca, Mn metasilicates
Mineral
Author
Space
group
(1)
(A)
g
(A)
Buerger
PT
7.94
7.32
7.07
90*02*
95*22'
Parawollastonite
TollIday
(1958)
P2i
15.42
7.32
7.07
-m
95*24'
Pseudmollastonite
Jeffrey
and Beller
(1953)
PI or PT
6.82
6.82
19.65
90*24'
9024'
119*18'
Pi or P i
7.64
7.16
6.87
92*08'
94*54'
101*35'
15.46
7.18
13.84
89*34'
94*53
102*47'
16.06
7.11
13.68
-*
c
9.81
9.02
5.26
t
PT7
6.68
7.66
12.20
1
111:,1*
Wollastanite
(1956)
Bustamite
103*26'
and
yer
(1937)
Bustamite
P-Mn0.8Ca0. 2S206
Buarger
(1956)
Liebau et al.
(1958)
Johannsenite
Schavinato
(1953)
Rhodonite
1
Hlmr et a.
t(tS 0
FT
F I or F T
105*
86.0*
93.2*
r-
-I-I
- - - --- -. -j
P
"
-"
;
-,'
81
in table 1.1.
The relations between the structures of the
polymerphs of CaSiO3 have been discussed in detail by
Prewitt (1962).
n and Gonyer (1937) investigated the unit
cell of bustamite by means of rotating crystal photographs,
and found it to be similar to that of wollastonite (table 1.l).
Buerger (1956) found, however, that the a and e axes of
bustamite are doubled relative to those of wollastonite
(table 1.1).
The unit cell, in an orientation similar to
that of voilastonite, is face-centered.
Buerger suggested
that "Bustaite evidently bears a kind of superstructure
relation to wollastonite, and it may be regarded as a double
salt, Casio3 - ?si03". Lebau et al. (1958) confirmed
Buerger's unt cell determination with synthetic material
with the coMosition Ca0 .2Nn0 .8Si 206 (table 1.1).
Liebau
et al. also concluded that the structures of bustamite
and wollastonite are different, but closely related.
They
proposed that the difference is based on a mutual ordering
of chains and/or cations.
Layarev and Tenisheva (1960),
in
a study of the infrared absorption spectra of the pyroxenoids,
concluded that the structure of bustamite is based on chains
of silica tetrahedra with a repeat unit of three tetrahedra,
as in wollastonite.
They concluded, however, that there are
82
"substantial differenes between the crystal structures
of wollastonite and kustamite".
Sohavinato (1953) dtermned the unit cell aid
space group of johannamite (table 1.1) and found that they
fare equivalent to those of topside, CaflgSt 2
eluded that
johaito
0
Ut enT-
is isotypie with 4topside.
Two sehools have proposed difereat, but very
similar, structures for rhodonite.
and D
Thilo
Hilr, L
ger-Sebiff (1956) and Lithau, Hilmer aid
(1958) proposed a structure based an chain of
silica tetrabedra with a repeat urit of five tetralwra.
This structure is very similar to those of otbr known
pyrmens A pyrtagenids.
Oxygen atom are srrang&d
appnointely in close packing.
Planes of cations in
octahedral Ioordination alternto with planet of Si atoms
in tetrbedral crdination.
Silica tetrahedra all sbare
two vertices with other tetrhdra to f
infinite chaina.
This structure may be incorrectly doterMe,
atw c
or at bt
ates ar* any very apprnimtely know,
can be sea in the followin:
R, was reported for only mo
for whieh data was available.
as
1. The. discrepancy factor,
of the two projections, (hO1),
The R-factor is extremely
high, being 37Z for the 58 strongest reflections, and 51%
for all
eflections.
neved
2. No y coordinates are
r
83
reported for the oxygen atoms. 3. The structure was solved
only with two projections.
The final Fourier projection
show the undulating contours expected of data with a high
R-factor.
Some oxygen coordinates appear to have been com-
pletely guessed.
4. The ordering af the Ca relative to Mn
was guessed merely by analogy with the structure of NaAsO
Although the R-factor was higher with Ca in its assigned
position, this position was still accepted as beirg correct.
Mamedov (1958) proposed a second structure for
rhodonite.
It may be described in the same general terms
as the "Liebau-type".
this structure aIN4,
Only two projections were used for
despite the fact that the structure
solution is one of at least moderate difficulty. The
following points show that this structure is also subject
to question.
.. Coordinates of Mn satms were determined
from two Patterson projections.
0 ato
Attemts to locate SI and
with Harker-Kasper injqualitiea and Zachariarsen
statistical methods failed.
2. All Si and 0 coordinates
were guessed on the basis of the predetermined Mn positions.
3. No refinemt was attempted.
4. No. R-factor is given.
Mamedov merely states that there is "good" agreement betwean observed and calculated structure factors.
Chapter 2
The Determination and Refinement of the
Crystal Structure of Rhodonite
Introduction.
Preparations were made to solve the crystal
structure of rhodonite in the crystallographic laboratory
several years ago. Specimens of rhodonite from Pajsberg,
Sweden were kindly provided by Professor C. Frondel of
Harvard University. Buerger determined that rhodonite is
triclinic with the reduced unit cell whose dimensions are
listed in table 2.1.
Intensities were measured by Dr. N.
Niieki using a cleavage fragment mounted on the singlecrystal, Geiger-counter diffractomater.
settings
The diffractometer
T and T were graphically determined.
Inten-
sities were obtained by integrating recorded peaks with a
planimeter. All reflections were corrected for the Lorents,
and polarisation factors.
The corrected values of F2 were
then used to compute the three-dimensional Patterson function,
?(xys), in sections normal to c, and the Patterson projection
P(x,y).
These functions were computed using the Whirlwind
computer in intervals of 1/50 along a and b, and 1/20 along
c for the three-dimensional function.
All of the data described above, including all
preliminary notes, were turned over to the author by M.J.Buerger.
-
-n-
-fi
-
-a-
'a-.
t t
09t9
C
-
-
Tzfm~Xeag
Tznsased
inqs
xAqo iv
*IEZ 0fl
0599*9O"
0.6 -S
lot019
SSf *Z6
al 'TI
0?'?!
i-*I
5909
(9S61)
1; ,sMT
*ra ti '2fl)av.j
1000gV
P,
.
,6ZTfl
it' 11
OL 9
Ayo(v)
a
a
*
y)q: (y)e
e
uo
toLnany
ds
I'* S 'qvj
86
Unit cell and space group.
The unit cell obtained by Buerger
(Table 2.1) was confirmed with a series of c-axis Weissenberg,
photographs, except that minor changes were found in the magnitudes of the translations.
This unit cell is the reduced
cell, but is different from the cell obtained by Hilmer St al.
(1956) and kamedov (1958)
(Table 2.1).
only in the choice of axis of length
The cells differ
N
12 A.
this axis in order to obtain the reduced cell.
Buerger chose
The other
axis was probably chosen because the two most prominent
cleavages are parallel to it.
Since it had been proposed that
rhodonite contains chains of silica tetrahedra, and since
cleavages usually parallel the tetrahedral "unit", this was
a natural, choice.
The matrix of the transformation from the
Hilmer cell to the Buerger cell is
A specimen
0
-o
0
0
-l
1
0
1
0
bounded by 100, 010, and 001 clea-
veges (relative to the Hilmer cell) was selected since it
could be easily oriented with three principal axes as rotation
axes with the optical goniometer.
Back-reflection, 0-level
precision-Weissenberg photographs were obtained with each
of the three principal axes as rotation axes.
Data obtained
87
from these films was used in conjunction with the IBM
709/7090 program LCLSQ3 (Burnham, unpublished) to obtain
refined lattice parameters (Table 2.1).
An IBM 709/7090
program (described in Appendix 2) was then used to obtain
lattice parameters corresponding to the Buerger cell (Table
2.1).
Five analyses of Pajsberg rhodonite are tabulated
by Doelter (1897).
An average of these five analyses follows.
Sio2
MnO
F7O
CaO
MgO
Al 2 0 3
Total
46.34%
43.97
1.61
7.03
1.00
0.11
100.06%
This data was used to compute the unit cell contents in the following way. Since a precise value of the
specific gravity was not available, the cell contents were
normalized to 10.00 Si + Al atoms per unit cell.
From this
a specific gravity of 3.84 g/cc was calculated. This value
was used to compute the unit cell formula, which follows.
Mn
Ca
8.01
1.621
Mg
0.321
Fe.
0.29
10.24
Si
A
9.97
0.003
10.00
0
88
In Chapter 1 it was noted that Sundius (1931) coneluded that the formula of rhodonite should be written
Mn4 +, Ca 1, Si5915; x = 0 - 1. Liebau at al. (1958)
came to a similar conclusion in another review of rhodonite
analyses.
It is interesting to compare the unit cell contents listed above with the results of Sundius and Liebau
et al.
Assuming that the space group is P1 ( and not P1)
and that all cations occupy general positions,
five cation positions of rank 2.
there are
The analysis is in good
agreement with this, indicating that 8.01 Mn atoms may
occupy four positions, and that Ca + Mg + Fe (2.23 per cell)
occupy one position. The departure of the total cation
content (10.24) from 10.00 is due to errors in chemical
analyses or to an incorrect assumption in calculation of
the specific gravity. This distribution assumes substitution
of Mg and Fe for Ca.
Since Mg and Fe ions are smaller than
Mn ions, which are in turn smaller than Ca ions, this conclusion is questionable.
The only conclusion that can be
reached is that Ca may occupy only one position of the five
positions which the large cations may occupy.
The distri
bution of Mg and Fe cannot be ascertained.
Preliminary considerations.
Liebau (1956) has shown that
the unit cells of those metasilicates having structures
89
shown or proposed to be based on chains of silica tetrahedra
are related. The chains have repeat units of two, three,
five or seven tetrahedra (xwier% drier-, funfer- and
siebenketten).
The translation parallel to the chains in
pyroxenes (swiferketten) is about 5.2 A and in bustamite
(drierketten) 7.16 A. Rhodonite has a translation of
12.23 A and pyroxmangite a translation of 17.45 A (Liebau,
£957).
Thus, if a period of approximately 5 A is assumed
to be associated with two linked tetrahedra, as indicated by
pyroxenes, rhodonite and pyroxmangite may be assumed tc have
five and seven tetrahedra respectively as repeat units.
Structures supposedly confirming this have been proposed by
Hilmer at al. (1956) and Liebau at al. (1958), and Mamedov,
(1958) for rhodonite and Liebau (1957, 1959) for pyroxmangite.
There is a further relation among the unit-cell parameters
of the metasilicates, however. All pyroxenes and pyroxenoids
have two perfect cleavages parallel to the clMss, as noted
above.
If two axes are chosen similarly wizL. respect to these
cleavages (other than the axes parallel to them) a close correspondence in their magnitudes is found.
These relations
are shown in Table 2.2, the two translations being chosen
so that they are approximately normal to the cleavages.
In
those minerals whose structures are well known the cause of
Table 2.2
Coparison of unit cells of some pyroxenew
and pyroxenoids
10......
*
Mineral
Author
Translation approximately
normal to chains
Protoenstatite
Smith
(1959)
2 x 6.36 A
Johannsenite
CaMnSi 206
Bustamite
CsMnSi 2 O6
Schiavinato
(1953)
This thesis
2
2
x 6.36
Translation
along chain
5.32 A
6.67
6.67
5.26
x 6.91
2 z 7.71
7.16
Pectolite
Ca2 NH (S 13 3
Buerger
(1956)
7.02
7.99
7.04
Rhodonite
(MnCa)S03
This thesis
6.71
7.68
12.23
Pyroxmangite
(FeMnCa,Mg)Sio 3
Liebau
(1957)
6.77
7.56
17.45
the similarity is easily seen. Each structure contains sheets
of approximately close-packed oxygen atoms with Si atoms and
octahedrally coordinated cations alternating in planes between
them.
The two axes described above intersect the close-packed
sheets at angles of about 45*.
From the above discussion it can be inferred that
the structure of rhodonite may be described as follows:
1.
It contains sheets of approximately close-packed oxygen atom.
2. Silica tetrahedra are arranged in a chain parallel to lf_1_.
The chain has a repeat unit of five tetrahedra.
3. Ma atms
are arranged in sheets of edge-sharing octahedra.
Further evidence pointing to the above conclusins
is found in a comparison of optical properties of rhodonite
and bustamite.
At the time of this investigation the
structure of bustamite was not determined.
As discussed in
Chapter 1, Buerger (1956) and Liebau at al. (1958) had shown
that it was closely related to that of wollastonite, and
that it very probably was based on drierjyetten, Hey (1929)
showed that the optical properties of rhodonite and bustamite
are very similar.
On the basis of optical properties of in-
termediate solid solution members, he incorrectly concluded
that there is an isomorphous series between rhodonite and
bustamite. This indicates that the structures may be very
closely related.
92
The structures proposed by Hilmnr et al. (1956)
and Liebau et al. (1958) and Mamedov (1958) are both consistent with the above description.
Since these are different,
and both very poorly determined, it must be concluded that
little more is definitely known about the structure of
rhodonite than can be tentatively inferred from an elementary
consideration of physical properties and unit cell relations.
Copaoson of volastonite and rhodonite. The discussion of
the last section was based only on a cmarison of the unit
cell and physical properties of rhodonite with those of other
pyroxenes and pyroxenoids.
The conclusions reached can
therefore be considered to be tentative at best.
There are
relations between the intensities of reflections of rhodonite
and wollastonite which confirm these results, however,
Comparison of ce-axis, o-4evel Weissenberg photographs of wollastonite and rhodonite shows that both have a
substructure of approximately equal dimensions.
This corres-
pondence is of course also apparent in a comparison of the
Patterson projections, P(xy),, of each.
This sublattice, in
projection at least, has the following translations relative
te the rhodonite unit cell given above:
A* -$a+
5
IB
5b
7, TOb
There are many substructure relations in wollastonite.
The most important involves pairs of atoms related
by the subperiod b/2.
This is chiefly caused by chains of
octahedra parallel to b sharing edges, the subperiod b/2
corresponding to an octahedron edge. The magnitude of b/2
in wollastonite is approximately equal to the magnitude of B
in rhodoite.
Thus there may also be a chain of edge-sharing
octahedra in rhodonite, oriented parallel to B.
Interpretation of the Patterson function, P(xyz).
Fig. 2.1
is a projection on (001) of some of the highest peaks of
P(xyz).
It is clear that this resembles a diagram of a chain
of octahedra in which each octahedron shares an edge with
each of two adjacent octahedra.
For simplicity, peaks cor-
responding to the upper ad lower vertices of each octahedra
are not shown, although they are present.,
In each octahedron
the relative peak heights correspond to weighting expected
for an octahedron composed of oxygen atoms coordinating W;
that is,
the central peak is much higher than the four coor-
dinating peaks.
That these peaks correspond to a chain of
Mn octahedra is further shon by distances between peaks,
which correspond closaly to values predicted from Mn and 0
radii.
When all such peaks in the full cell are considered
it is clear that the octahedra define a plane parallel to (il1).
94
Figure 2.1
Selected peaks of P(xyz) projected onto (001).
Each peak is labelled with its approximate
height and the level, in 20tb,
appears.
on which it
1'
(I-)
95
96
This is further evidene for the conclusions reached in
the two preceeding sections.
The structures of other pyroxenes and pyroxenoids
have a similar "octahedral sheet"
as discussed above.
In
each of these the sheets are parallel to plan". of inversion
centers.
The plane of central cations, in fact, coincides
with the plane of inversion centers.
If a brucite-like
octahedral sheet is oriented similarly in rhodonite, such
that chains of edge-sharing octahedra are orinted as in
P(xyz), it
is seen that the period between inversion centers
is approximately the same as the period between centers in
the "ideal" sheet.
It was therefore assumed that such an
arrangement exists in rhodonite.
Examination of a brucite-like sheet of edge-sharing
octahedra shows that there are only two kinds of positions
in the Ha plane where centers may be placed.
These corres-
pond to the center of a shared edge and to the center of an
octahedron at a cation position.
Examination of the struc-
tures proposed by Hilmer et al. and Liebau at al., and
aumdov shows that these structures differ principally in
having each of these two arrangements respectively.
The Mamedov structure was thougit most likely to
be incorrect, since this required two Mn atoms to be on inversion centers.
The structure is based on a chain of nine
edge-sharing octahedra with the central Mn atom on an inversion center at 000,
at 00j.
The remaining Mn atom is on the center
Peaks supposedly corresponding to Mn-Mn inversion
peaks were easily located in the Patterson function.
Peaks
corresponding to nine of the tan Mn atoms in the unit cell
were found in each minimum function formed.
A suitable peak
was not found at 04, or at any other center, which might
correspond to the remaining Mn atom, however. This indicated
that the structure was incorrect.
A second attempt was later made to verify this
structure.
Structure factors were calculated for all re-
flections, using only the Mn atoms in general positions and
one Mn at 000 (after correcting two Mn coordinates for incorrectly published values) the R-factor was high (60%) but
there was fair agreement between F0 and Fe for substructure
reflections.
An electron density function P(xy:) was then
computed using only those reflections showing reasonable
agreement between F. and F
and having large values of F.
(The IBM 709/7090 program ERFR2 (Sly, Shoemaker and Van der
Hende, 1962) was used to compute all Fourier syntheses in this
investigation).
Although peaks apparently corresponding to
Si and 0 atoms were present, there was no peak representing
the remaining Mnatom at 00j, or on any other center, This
was conclusive proof that the structure is incorrect.
98
The second possible arrangement of Mn octahedra
was investigated in the following way.
Inversion peaks
corresponding to each of four Mn atoms were located in
the Patterson function, P(xys), and the minimum functions
M2(xyz) contoured.
Two of these were combined to form
the function M4(xys), the peaks of which are shown projected onto (001) in Fig. 2.2.
Each peak is labelled with
the level, in 20ths, on whicb it appeared.
Fig. 2.3 is an
interpretation of this figure, showing coordination polyhedra.
The peak heights and positions of the remaining
two minimum functions closely resemble those shown in Fig.
2,2,
Six peaks labelled as Mn atoms are shown in Fig. 2.2.
This nomenclature is used merely because six peaks appeared
in the Mn chain, although there are only five Mn atoms in
the structure.
One of these is obviously a false peak,
caused by superposition in P(xya) of substructure vectors.
Since the peak labelled Mn6 was the smallest of the six in
all minimun functions, this was considered to be a false
peak.
From a consideration of interpeak distances and
peak heights, peaks of M(xyt) were correlated with all
oxygen atoms with the exception of 09, 013' '14,
resulting Mn coordination octahedra are shomn
015.
The
in Fig. 2.3.
I
r
'-
".. - "..,
.-.-
99
Figure 2.2
Peaks of M(xyz) projected onto (001).
Each
peak is labelled with the name of the atom
which it represents and with the heights, in
20ths, on which it appears.
Peaks comresponding
to Og, 013* 014 and 015 are not present.
I
t
;""; ,
-i7
-
gG
o
z
89u
II
g*~
y
6 a
t-,
9
-i
sc2-K<II
-tp cu AJ
12is
gj0
-- 9,
S§egV.A
6
101
Figure 2.3
Coordtnation polyhedra (octahedra and tetrahedra) projected onto (001),
obtained from
an interpretation of fig. 2.2.
of &tou
0 A
09, 013' OP
Th. positions
015 were predicte
from an interpretation of the
function.
They are labelIed with the lvel, in 20ths, on
which they occur.
CJ
0
q-0
016
i 11-
-,
. -
-,
@-WA4
103
Note that the coordination of fn5 is not completely octabedral.
Peaks corresponding to Si 2 - 5 were easily identified.
The location of Si was uncertain, however.
There are two
peaks, labelled Si 1 and $i1,
one of which corresponds to $14.
unThe positions of four oxygen atoms r
Assuming that Si is tetrahedrally coordinated,
detrmined.
and that these tetrahedra are linked in a chain, these positions (Fig. 2.3) were easily predicted.
The validity of this
proposed structure was checked by a consideration of Pauling
bond strengths, all of which were reasonable. Coordinates
are tabulated in Table 2.3.
The peak labelled Si
was
initially assigned to Sil.
Refin
nt.
The intensity data were first corrected for
absorption in the following way. The crystal used in the
measurement of intensities was a triangular prism approximately
0.1 mm long with triangular face edges of lengths 0.12, 0.13
and 0410 -,.
The mass absorption coefficient,
and the average value of *r
2.5.
, is 489 cm
All reflections were cor-
rected for absorption using an IBM 709/7090 program written
by C.W. Burnham (1962).
Structure factors calculated for hko reflections
were in reasonable agreement with observed structure factors.
Those reflections showing the best agreement were used in
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108
the cooputation of the Fourier syntheses
(xy) and
(xy)
These indicated that the structure was essentially correct;
i.e. that the fiva Mn atoms had been correctly placed with
respect to the six observed substructure peaks in the miniaum functions.
Two cycles oi three-dimensional refinmnt were
therefore carried out.
(The IBM 709/7090 program SFLSQ2
(Prewitt, 1962) was used for all structure factor calculations and least-vquares refinement in this investigation.)
Form factors were used assuming ball-ionization of all atoms.
Complete disorder oi Mn, Ca, M&and Fe was assumed and the
average cation scattering function corrected for the average
real component of snomalous dispersion.
All reflections
having (70-F)/F > 0.5 were rejacted in the refinement.
by Hughes (1941) was used
A weighting scheme rec
in a slightly revised version. This weighted reflections
in the following way.
F
-W
4in
Fv~4 ~
*
r=Fsin
rW
F-**0
where
-
4
tw(l/r
15.0
)i
0
.r
109
The scale factor was varied in cycla ; and the scalte factor
and all coordinates in cycle 2. Table 3.4 is an outline
of the least-squars refineseac procedure.
The value of the discrepancy tactgr, R, was eztreely high (76%).
It was suspected that this was partially
caused by wrongly guesse
coordinates lor the oxygen) atoms
0 15. Therefore two cycles (cycles 3 and 4)
09, 013' 014 a
were executed with these atoms outtad. The value of R
decreased to only 69t however.
agreement between F
and F
but bad for all others.
and FI (kl),
however.
It was then noted that
was good for hko reflections
Comparison was good between F(hkl)
This indicated that the aes
b,
or c, had souhw become reversed in the calculation of
F(xyf).
Therafore refinement was contizued with all z
coordinates replaced by D-a.
Structure factors were computed for all reflections
with F
#
0, for which i was only 517.
Approximately 500
reflections having good agrees*n between F, and Fe were
used to compute the Fourier synthesis
09,
013* 01
and 0
p (xys).
The atoms
were not included in this computation.
The electron density function indicated that most atom positions were correct although no peaks corresponding to the
atoms 0.
013 , o4 and 015, could be identified.
Since
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Table 2.4
28
920
20,9
cont.
K and 0
coord.
29
18.8
K and Si
and Mn coot
30
16.4
K, and 0
coord.
31
15.2
K, and Ui
and Mn coot
32
14.4
K and 0
coord.
33
12.3
5 0 and Si
coord, and
34
12.1
K, and Si
Hughes
F
11.9
K and 0
coord.
11.7
36
37
"f
10.8
Isotropic
tew factors.
"#
0
Cations ordered
and Mn coord.
35
1FAO.re
iI
*rrdox~eeiQtl
0Z6
CV
90000 1h
sTX0jpwTX VA
O~w;p~o
liv
9<
se~:
it
-u;PZOoa yfj
Sr.I
07[
6c
07V91
9c
rft"Onl rz?
7 eclej
115
most of the reflections used in the calculation of
(Xy's)
were substructure reflections, and since these oxygen atoms
are in the complement structure, this was not unexpected.
In addition, peaks of
P (xys)
at the peak labelled Si
indicated that Si was located
in Fig. 2.2, not at Si
as pre*
viously concluded.
Cycles 5 and 6 were executed with coordinates
derived from the minimum function. Coordinates of SiI
were substituted for those of Si
and new coordinates for
re09, 013, 014, and 015 (Table 3.1) were obtained from a
interpretation of the minimum function. The value of R
decreased only to 49%. The scale factor, K, attained a
.5. The value of K should adjust such that
value of
F a
F . The magnitude of K calculated by
equating these terms was 0.67, which is considerably larger
than the value obtained from least-squares refinement.
The
scale factor was therefore set at 0.67 and not varied in
cycles 7-9.
The program SFLSQ2 required more than five minutes
running time on the IBM 7090 computer at the M.I.T. Computation Center when all reflections were used and all coordinates
varied.
Such problems are assigned low priority and may re-
quire as much as one week for execution. Problems requiring
116
less than five minutes computation time are quickly expedited
however.
The number of reflections was therefore reduced
to about 920 for some cycles, to reduce computation time to
less than five minutes.
Cycles in which the limited number
of reflections were used are designated with subscript n.
For these cycles only Mn and Si coordinates or all oxygen
coordinates were varied in order to maintain an acceptable
ratio of the number of reflections to the number of variables.
Cycle 7n was executed varying only Si and Mn coordinates, The value of R increased to 55%, probably because
of the increase in K noted above.
Since the decrease in R
predicted for this cycle was negligible, the Bunn synthesis
(xy)
was computed using all reflections with Fj
0.
This function showed considerable variation, but little significance could be attributed to it. A peak at the Mn substructure
position labelled *it in Fig. 2.2 indicated that it may be
occupied by Mn. Since peaks in the minimum functions corresponding to Mn5 were the smallest of the remaining five, it
was assumed that this position was vacant.
The value of R for
cycle 8n increased to 59%, however, indicating that this Mn
distribution was incorrect.
Since no other reasonable dis-
tribution of Mn atoms could be formulated through interpretation of the Bunn error synthesis or the minimu
functions,
the original Mn distribution was assumed to be correct.
.7.
-
a..
9
117
Only Si and Mn coordinates were varied in cycle 9*
The values
the scale factor being held constant at 0.67.
of
t
F0 a
E
Te
from this cycle indicated that K
should be decreased to 0.63.
The least-squares value of
K had formerly been thought to be too low.
There was then
a possibility that refinemet had been hindered by using an
incorrect value of K, and holding it constant.
In cycle 10,
it was allowed co vary, as were all oxygen coordinates. The
discrepancy factor, R, decreased to 49%.
Refinement was not progressing satisfactorily.
The following tests were therefore made of the validity of
the structure.
If an atom is in an incorrect position, the
value of its temperature factor may refine to an unusually
large value. All isotropic temperature factors were therefore varied in cycle 11
.
Although temperature factors varied
in an unpredictable way, some attaining negative values,
none refined to exceptionally large values.
Cycle 12n was
executed with all coordinates altered by -.02 from their
refined values of cycle 10.
were varied.
Only Si and Mn coordinates
If coordinates were refining into substructure
positiors such shift* might alter this trend. All coordinates
shifted toward their original values, however, so the results
were inconclusive.
118
All evidence indicated that the proposed structure
was correct0
(Rw44%) and 14
Refi
t was therefore continued. Cycles 13n
(R44%) were executed with only the scale
factor, K, and Si and Mn coordinates varyin.
tion
of these results showed that the rejection scheme was not
operating correctly. Only those reflectios with (F -V
F? . 0.5 were being rejected while those with \ F
should be rejected.
,I 0.5
The rejection test was there fore,
ar-
rected, and the critical rejection value chaed to 0.6 so
that more reflections might be included in the refinemt.
All coordinates were varied in cyele 15.
creased sligtly to 47%.
The R-factor in-
Refinemnt may not converge pro-
perly if inappropriate weighting or rejection saches are
used.
Therefore, in subsequent cycles the weighting, and
the rejection routine changed so that all reflections were
included in the refinemt ecoept those with F,
0.
(Only a very few of these reflections were Wing included
as input to each cycle in order to decrease computation
time.)
Si and Mn coordinates and the scale factor were
in which R decreased to
varied in cycles 16U and 17,,
42%.
In two me cycles in which all coordinates were
varied R was 44% and 42%.
The slight increase in R
from cycle 17n to cycle 18 is caused by the inclusion of
about 700 more reflections in cycle 18.
Thetet were chiefly
rT
119
reflections of high sin 6 , for which discrepancies between
F0 and F, are usually greater than for reflections with low
sin E .
Since the refinement still was converging very
slowly, an attempt was made to ascertain if there was a
basic error in the structure.
the atemn Mn1
4
All evidence indicated that
were correctly located.
There was some
evidence (see above) that Mn5 might be in an incorrect position.
The relatively heavy Mn atoms control many phases.
Structure factors were therefore computed using only
1 4
#.
Those structure factors of medium or large magnitude
having good agreement between F
and Fe were used to com(xyz).
pute the Fourier synthesis
This contained the
same ambiguities as the minimum function however.
This
was probably caused by the use only of reflections of at
least moderate intensity.
These are cbiefly substructure
intensities and reproduce only the substructure in
p(xys).
Reflections with small magnitudes very probably have incorrect signs, if R is large as in this case, ant should
not be used in Fourier calculations.
This problem As, of
course, inherent in substructure problems.
Examination of the atom coordinates shoved that
0 1was not correctly located.
In cycle 20, therefore, it
120
was shifted to a position consistent with the mininm
In addition, the use of the Hughes weighting
function.
routine was reintroduced.
This scheme gives a higher
weight to small F's and a lower weight to large F's.
This is particularly valuable in a substructure problem
since it may prevent refinement into a substructure.
Following cycle 20 reexamination of the set of coordinates
showed that 01
had refined into a position inconsistent
with accepted values of interatomic distances and coordlnation.
This was corrected in cycle ?1, in which all coordina-
tes and the scale factor were varied.
Structure factors for all Fhkl with F
-
0 were
calculated using the refined coordinates of cycle 21.
were used to compute the Bunn error synthesis &g(xyz).
These
This
indicated that the c coordinates of Mn4 and Mn5 should be
.03 and
changed by
-. 07 respectively.
with a difference Fourier synthesis,
This was confirmed
A p(xy), calculated
using all reflections except those with F0
0.
The discrepancy factor, R, decreased to 35% after
cycles 22
and 23 , in which only K, and Si and Mn coordinates
were varied.
After cycle 24, in which all coordinates were
varied, R was still 35%.
This was expected, however, since
the number of reflections was much larger in cycle 24 (see
above).
121
The refinement converged rapidly with further
cycling, in which oxygen coordinates, and Si and Mn
coordinates were varied in alternating cycles (see table
2.4 for details).
Convergence was almost complete after
cycle 32, for whch -Awas 14.4%, so an attempt was made
to determine the ordering, if any, present among the Ca,
Mn, Fe and Mg atoms.
Form factors computed assuming
complete disorder of these atom had been used throughout
the refinemnt.
Structure factors were computed for
1620 reflections for which F
was non-zero, and these
were used in the computation of the difference Fourier
synthesis, Af (xys).
The only major features in this
synthesis were peaks at the positions of Mn -Mn5 .
heights of these peaks are listed in table 2.5.
The
The peak
heights of Mny Mn2 and Mn3 are all positive and approximately equal.
Since Mn ions have more electrons than the
average atom used in the refinement, this suggests that only
Mn, and not Mg, Fe or Ca, occupies these positions.
The
negative peaks at Mn44 and Mn5 suggest that Mg and Ca occupy
these positions, since they contain fewer electrons than the
average atom.
It is also very likely that Fe occupies the
same positions as Mg since their ionic radii are very similar,
and smaller than either Ca or Mn, and they are known to
substitute commonly for each other.
UK
UK
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5+1191
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17
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t1V
UK
ftAx)S V u
guTaep.%o jo
zm -f:1 v:j*adav,4uj
stourh#l
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puv (X&X)dV ;0 OtWTjZSOd U03O2 ~tgo
S*Z
vlqvX
123
Assuming that Mn occupies the positions Mn,
Mn2 and Mn3 2
all Mg and Fe the position Mn, all Ca the position Mn5y
and the remaining Mn the positions 14a and Mn5, peak heights
for Ae(xys) of -100 and 0400 were predicted for Mn4 and Mn5
respectively.
This is in good agreement with the peaks
actually observed and thus indicates that this ordering
schem is correct.
Interatomic distances were then calculated between
Mn-
5
and oxygen atoms.
The average for each polyhedron is
shoin in table 2.5.
Average distances for Mn1 , Mn2 and Mn
3
are between 2.18 and 2.27 A, in good agreement with known
Mn-O distances. Average distances for Mn4 and Mn5 are greater
than those for Hn1 , that of Mn5 being the largest (2.43 A).
Since Ca is the largest cation of Ca, Mn, Fe and Mg, this
indicates that most
if not all, of the Ca occupies the posi-
The average distance for Mn, is large only because
5,
one of the six distances for which the average was computed is
tion Mn
very large.
Two Mug- distances are amon
all Mn-O distances,
this position.
the smallest of
This indicates that Fe and Mg occupy
Thus the interatomic distances verify the
conclusions on ordering reached on a basis of the peaks in
Ap(xyz).
Mn occupies the positions Mn,
Mn2 and
3*
and Fe the position Mn4 , Ca the position Mn5 , with the remaining Mn at Mn4 and Mn5 .
r
124
Refinement was continued with form factors computed
for Mn-
assuming the orderinj decermi
5
above, and taking
account of ancaious dispersion ot Mu, tc, and Ca.
Oxygen,
and Si and Mn coordinates were varied tn cycles 33
- 35g.
Since refinement of coordinates had almost completely converged at this point, isotropic temperature factors were
refined in cycles 36n and 37n.
fRefin ent
waa completed
witb cycles 38 and 39, in which all coordinate3 and the
scale factor were varied, and cycle 40, in which all isotropic temperature factors were varied.
The
inal para-
meter variations were all less thaa tua standard deviations.
Final coordinates and temperature factors, and their standard
deviations , are listed in table 2.3 for cmuparison with
values obtained from the miniMum function.
Final coordin-
ates are also listed in tabla 2.6 where they are compared
with coordinates obtained by 41ebau at al (1958).
The
values obtained by Liebau at at. are reasonably close to
the values of this investigation, except il
instances:
0,z
-
Ax
Mn,
.26;
0
,sy
.08; Mn3 &
W.08;
0
the following
.08; Mn4 ,
,x&
.10; 0
as
1
a
s
.10;
-
.09.
The coordinates given by Liebau at aL. have been transformed,
of course, to a unit cll
f the Buarger type.
In order to
125
Table
2.6
Comparison of atom coordinates of Liebau at al. (1959)
(below) and this refinement (above). Liebau coordinates
transformed to the Buerger cell and values of t replaced
by 1-l.
Atom
y
.963
.8517
.889
.9697
.998
Kat,
.6827
.710
.554e
.567
.8748
.875
Mn 3
.4916
.8109
Kat.
.505
.2700
.309
Mn 4
Rat.
,3018
.273
.9767
.998
.7967
.700
Mn 5
Kat.
.0457
.107
.6938
.691
.6389
.615
Si 1
Si 3
.2191
.145
.1246
.105
.4956
.533
S12
Si 1
.2687
.281
.4701
.462
.6375
Si 3
Si 2
.4610
47393
.7092
.541
.778
.743
Si 4
Si 4
.7446
.728
.0891]
.059
.7538
.794
Si 5
Si 5
.9263
.943
.3466
.8450
.338
4864
0 1
0 15
-9544
.6772
.643
.9628
.971
Mn 1
Kat.
.8819
Mn 2
.734
.611
126
Table 2.6
0 2
.601t
0 7
0 3
.3981
o
cont,
.7312
.8655
3.734
3895
0398
.8858
.626
.4375
.8074
55.453
01
.5485
0
01129
19N
0 6
0 7
06
a.874
06
.32
0 98524
'937
912
.802
0975
.828
.1318
.7374
.141
.766
,8149
.7438
,749
.882
.6591
i627
.560
.9962
.998
.4459
-7457
.5871
.5846
3
40414
.9433
0 12
.5819
4000
o7790
0 1..739
013
.3191
o 1
0 04,7
02
0 11
00
02
.8430
04
0
-154E
0 13
.8607
.576
5764149
66142
641
.451
.590
942
.5154
.530
.6300
.665
44360
.7006
.364
.599
.2221
0209
.7024
790
127
obtain satisfactory comparison it was necessary to change
all a coordinates of Liebau et al. to 1-,
Liebau has noted that all z coordinates of Mn
and Si atoms as published (Liebau et al. 1958) should be
changed to 1-g, and a of oxygen atoms to j-t.
This transo
formation was accordingly applied before transforming coordinates to the Buerger cell.
A satisfactory correlation
of coordinates of Liebau et al. and this investigation could
not be found, however.
This indicates that the change recom-
mended by Liebau must also involve a change in the choice
of unit cell axes, since a reasonable correlation of coordinates was found, as noted above.
Such a change in the unit
cell would result in a change in the matrix of the transformation of coordinates.
The shisfactory correlation of coordinates may
be obtained if all x coordinates of Liebau et al. are replaced by values of l-x, retaining the unit cell.
The change
recommended by Liebau is obviously incorrect, however, although the structure determined by him is basically correct.
Structure factors were calculated following cycle
40, for which the discrepancy factor, R, was
reflections with F
t
10.s%
for all
0, and i5.4-4 for all reflections.
ture factors are listed in Appendix 4.
Struce-
128
Description of the Rhodonite Structure
Peaks of
(xys), computed with structure factors
from cycle 37, are shown in Fig. 2.4 projected along c onto
(001).
Only those peaks from: - 0 - 0.5 are shown. Mn
atoms are represented by shaded rather than contoured peaks.
Fig. 2.5 is an interpretation of the corresponding projection
(xy), and Fig. 2.6 is an interpretation of the projection
(xa) (relative to the Hilmar unit cell).
The unusual
latter projection has been chosen since it is parallel to
the chains of silica tetrahedra.
The structure, at least in major features, is
similar to the structure proposed by Hilmar et al. (1956)
and Liebau at al. (1958), as can be seen from the comparison
of coordinates in table 2.6. Planes of approximately closepacked oxygen atoms are arranged parallel to (111).
Planes
of Mn, Ca, Mg and Fe ions in octahedral coordination alternate between planes of oxygen atoms with planes of Si ions
in tetrahedral coordination.
The silica tetrahedra each
share two vertices with other tetrahedra to form a chain
ztending parallel to
of five tetrahedra..
flOI~J.
This chain has a repeat unit
129
Figure 2.4
Peaks of
(xys), calculated after cycle 38
of refinement, projected onto (001).
Only
peaks of the asymmtric unit, from Z = 0 to
are showr.
Peaks are labelled with the level,
in 100ths, on which they appear.
Mn5 64
O,14 7o0
0 674(>
2 64
Mn4 so
0 774
0481
04
Mn
3
6
(~N013
0Oe
0
0
Mn2 87@
S4 75
:OD
03 89
0
70
(02
02
9
01252
0
>a
Mn,
9
-r
131
Figure 2.5
Projection on (001) of the structure of
rhodonite.
Ma and Ca atoms, and Si coordina-
tion tetrahedra are shom.
EUiS D
r
,
-
-
133
Figure 2.6
Projection along a of the structure of rhodonite.
Mn And Ca atoms, and Si coordination
tetrahedra are shown4
-
I -
- MONW:
,
,
4us
(0JV[001 11 UIS [Oil]
135
gheet!of octhedrllcooiatedeton.
Fig. 3.3,
Chapter 3, to a projection of part of the structure of rhodonite onto the plane parallel to sheets of oxygen atoms
described above. One octahedral layer is shoon below,
with sections of the silica chains in the overlying sheet.
The coordination polyhedra of M1-5 approximate octahedra.
These octahedra each share two edges with adjacent tetrahedra to form a chain ten octahedra long. The shared edge
at the midpoint of this chain lies on an inversion center so
that the two halves of the chain are controsymetrically related. These chains are bonded to similar chains through
further edge sharing in a staggered manner to form a band
of octahedra extending parallel to /101.7.
One complete
band is shown in Fig. 3.3 with the outer edge of a translation related band on the left.
Note that these bonds are
separated by a rift of unoccupied octahedrally coordinated
sites.
These relations are treated in more detail in
Chapter 3.
Coordination polyhedra and cation orderin.
All silica tetra-
hedra are regular, as can be seen from the Si-fO and 0-0
distances of tables 2.7 and 2.9 respectively. The average
Of all such 0-0 distances is 2.65 A. Individual distances
vary from this average by no more than 0.20 A.
All Si-0
136
Table
2.7
Cation-oxygen interatomic distances
Mn - 12.6276
Mn2-0
2.219 A
02
2.249
0
2.335
06
2.112
03
2.149
0
2.14
04
2.253
01
2.353
04
2.266
o011
2.175
2* 175
010 2.065
Ave.
2.219
Mn3-0
2.13
Ave. 2.215
MlggO
5 2.037
03
2.107
06
2.23
04
2.28 2
07
1.97
05
2.196
08
2.87
06
2.411
09 2.386
012 2.333
Ave. 2.228
'011 2.120
Ave. 2.272
137
Table 2.7
Mn5
Ave. excluding
09, 2.150
1
2.32
07
226
08
2.267
0
2.30 4
Sig.06
1*644
013 2.526
08
1.599
014 2.635
09
1.64
015 2.605
012 1.650
Ave.
2.418
6i 2 e0 4
1.589
Ave.
1.634
S13 '0 2
1.616
010 1.593
07
1.590
013 1.646
012 1.660
014 1.673
013 1.624
Ave.
1.625
S±4i05
1.596
09
1.640
Ave.
cont.
Ave.
1.623
Si5-0
03
1.608
1.616
Oil 1.633
014 1.638
015 1.649
015 1.656
1.630
Ave.
1.628
138
Table
2.8
oxSen-cation interatomic distances
01-Si5
1.60 A
02
i3
1.616 A
mn
2.276
®a
2.24
Mn
2.219
M2
2.335
Mn 5
2.329
Mn3
2.139
03± 5
1.616
04-812
1.589
Mn2
2.14
Mn2
2.253
Mn3
2.107
Mn2
2.266
n3
2.282
A
1.64
05 ±4
1.59
06
Mn3
2.196
m'i
2.112
Mn4
2.03
Mn3
2.411
Mr
V.23
07-±3
1.590
Mn4
1.978
mni
2.148
Mn
2.260
Mn
2.878
Mn35
2.26 7
08
1.599
- :, -
""
" Ii 'co 0
139
Table 2.8 cent.
0 -Si
Mn
1.64
010OS2 1.593
1.64
Mn-2 2.50O6
2.386
Mn5 230
1.63 3
0 12
1
1.65
m4 2.35 3
3±3 1.66
Mn I 2.175
Mn3 2.23.
Mn 2.120
013~4 2 1.646
S±3 1.624
Mn 5
015.'4
2.526
1.649
S's 1.% 6
Mn5 2,60
5
0140-82 A.673
St5 1.63
Mn5 2.63
140
Table
2.9
Oxygen-oxygen interatomic diattlanes
of silica tetrahedra (in Angstroms).
Distances of shared edges are underlinaed.
81
8
06
2.81
08
0
2,61
2
2.68
2.70
2.61
0
Si2
010
0
2.75
0 10
013
2.68
2.63
2.
2.59
07
02
07
012
014
2.64
0 13
Si3
19
2.75
012
2.64
013
2.72
2.71
2.45
2.59
141
Table 2.9
St
cont.
01
4
05
09
2.72
09
4
2.65
015
2.61
2.65
2.62
Oi
st 5
2.71
03
01
3
014
2.75
0 14.5
2.67
2.63
2.67
2.65
14%
distances are also very close to values found in other silicate structures.
The average of all Si-O distances (1.628 A)
is close to the value predicted by Smith and Bailey (in press)
for metasiUcates (1.623 A) and to those in bustamitc (1.623 A)
and wollastonite (1.626 A, Prewitt, 1962).
The coordination polybedra about Mn, Ca, Mg and Fe
display some interesting features, however. First, note in
table 2.7 that the MnO distances of M1
3
k
occupies these positions) are all close to the average values
for these cations, with no large deviations,
This La an
indication that these polyhedra are reasonably regular
octahedra.
Distances for m 4 (Mg and Fe occupy ths position
with Mn) are irregular however..
Three of the six distances
are short ( N 2.13 A), indicating of course that Mg and Fe
are distributed there.
The Mng-O8 distance (2.88 A) is
ezceptionally long though.
This indicates that the coordina-
tion of this cation actually approaches only five oxygen ions.
The average &4-0 distances for these five oxygen ions is
0
0
only 2.150 A, at least 0.07 A less than that for any of the
Mn ions (M1-3)*
The coordination polyhedron of Mn5 (Ca occupies
this position with so= Mn) lies at the ends of the chain of
143
ten edge-sharing octahedra.
This coordination polyhedron
is irrogular, and consists of seven oxygen ions.
mnt is unusual in that O
07,
% and O1
The arrange-
(which are each
coordinated to one Si ion and one or two other Mn ions)
form four vertices approximting one side of an octahedron.
However, 013, (l4Oand 015 (which are each also coordinated
to two Si ions) torm the other, irregular, side oi the polybedron.
In Fig. 3.3 it can be seen that this polyhedron
has vertices in two adjacent close-packed planes, of
oxygen atoms, wIth four vertices in one plane and three
in the other.
Interatoaic distances.
As noted above, the cationeoxygen
interatomic distances (table 2.7) of the full refinement
substantiate the ordering assumed at an earlier stage. The
average cation-oxygen distances of Mn, tn2 and Mn3 are very
similar, lying between 2.215 and 2.228 A. Since these dis~
tance& are almovt equal, and since they agree well with
ther
published average Mn-O distances (e.g. 2.203 and 2.24S a in
bustamite) it is very probable that Mn occupies these
positions.
Although the average distance (2.27 A) for MfngjO
is higher than those of Mn1 - 3 , indicating that Ca (r = 1.06 A)
rather than Mg (r w 0.78 A) or Fe (r
0.78 A) is distributed
there, one distance of the six is extremely large. The average
Mn4 -0
144
0
distance excluding this is 2.150 A, in good agreement with
the assumption that Mg and Fe occupy this position.
The average distance for Mn5-0 (2.42 A) is greater
than for Mn1 4 -0, indicating that Ca is distributed there.
Note that the three distances involving 013, 014, 015 (each
coordinated to two Si ions) are all much larger than distances
involving 01, 07, 08 and 010 (each coordinated to only one Si
ion and one or twc other Mn ions).
The long Mn5 *0 bond
lengths and the unusual coordination of this cation clearly
shows why Ca occupies this position, and explains why rhodonite may contain up to, but no more than 20 mol. per cent
CaSiO 3.
The analyses of Pajsberg rhodonite used in this
investigation was actually an average of five analyses in
which FeO, for instance, varied from 0.36 to 3.31%. There
is therefore no guarantee that the specimen used actually
contained cations in the ratios of the average analysis.
Despite this, the conclusions on cation ordering are still
valid and within the accuracy permitted by the data.
Al-
though this ordering is certainly not as complete or ideal
as assumed from the peaks of the difference synthesis or
interatomic distances, it
is accurate in principle.
145
Conformity of rhodonite to Paulings rules.
Several of the
Pauling's valency bonds show deviatios from ideal values.
All oxygen-cation distances are listed in table 2.8.
oxygen ions are of three types;
Si is
and one "
ion.
ion and three "W' ions.
ion and two "Wn"
ions.
The
A. Those coordinated by two
B. Those coordinated by one Si
C. Those coordinated by one Si
If all Mn ions are considered to be
octahedrally coordinated each Mn-O bond has strength 3,
while each Si-O bond has strength 1. Thus all type A oxygen
atoms have an excess bond strength of 3'
, those of type B
are neutral, and those of type C have a deficiency of .
This is compensated, in general, by a lengthening of cationoxygen bonds of type A and a contraction of bonds of type C.
For instance, the average Si-O distances for bonds of type A,
B and C respectively are 1.648 A, 1.615 A and 1.599 A. A
similar situation exists with Mn-O bond lengths, for which
the averages are 2.477, 2.280 and 2.137 A respectively.
A feature of structures of pyroxenes and pyroxenoids is edge sharing between tetrahedra and octahedra.
(Chapter 3).
Tn rhodonite there are five such edges shared.
In table 2.9 the distances along the tetrahedra edges
are
underlined. The cation polyhedra and the shared oxygenoxygen edges are as follows:
146
Si
Mn
3i1 M
;
3
; 06 - 012
Si2 Mn5 ;
£i3
W
Si
Mn5
5
5
This edge sharing results in
of the edge involved.
6 ' 09
;
;
0
r0
014
7
13
O4'- i1
contraction of the length
The average length of all tetrahedra
edges is 2.653 A, while the longest shared edge is only
2.61 A.
Tampeture factors
The teperature factors in general
(table 2.3) have values which are consistent with those of
other silicate structures refined in the crystallographic
laboratory.
That is,
those of Si art- smallest (wo. 2),
the average of 0 are intermediate (00. 5) and those of the
octahedrally coordinated cstiont largest (N O.7).
Reliance
cannot be placed or the absolute values of temperature factors, since they show a wide variation in structures refined
in different laboratories with data gathered in different
manners, but the relative values of this refinenat seem to
have som
meaniug.
Mn5 (Ca) is 0,93.
For instance the temperature factor of
This is the largest of the Ma teperature
factors, as expected from the unusual coordination of this
cation.
147
The temperature factors of those four oxygen atoms
with a deficiency of bond strengthb (01 05' 07, and
type C Cbove) are tb
four largest oxygen temperature factors.
Those for oxygen atoms with an excess
0 12*
1, O
and 0;
f bond strength (09g
type A above) are all smaller than
the average, and Include the smalest oxygen temperature
factors.
As noted above, bond distances may also be cor-
related with values of PaulingP valenr bonds.
Thus there
iA good correlation between deficiences in bond strength,
osmal
boi
lengths and large temerature's
factors on one band,
and an excess of bond strength, targe bond lengths and small
temperature factors on th
other.
148
Chapter 3
Crystal Chemical Relations Among Some
Pyroxenes and Pyroxenoids
Mineralogists have long recognized the close relationship of the minerals of the pyroxene and pyroxenoid
groups.
Structure determinations have shown that these
minerals all contain chains of silica tetrahedra.
Liebau
has recently systematized these relations (see Chapter 2)
and shown that all pyroxenes contain twierketten, and
pyroxenoids drierketten (e.g. wollastonite), funferketten
(e.g. rhodonite) or siebenketten (pyrosmangite).
Several
of these structures have now been refined in detail and it
is possible to find further relations among them.
It was noted in Chapter 2 that a unit cell with
two axes of related magnitudes, and a third of length proportional to the tetrahedral chain, may be chosen for pyroxenes and pyroxenoids.
This is an expression of the
structural similarity of these minerals.
All are based on
an approximately close-packed array of oxygen atoms.
Planes
of octahedrally coordinated cations alternate with planes
of tetrahedrally coordinated Si ions, between sheets of oxygen
atoms.
The tetrahedra each share two vertices to form a chain.
149
Figs. 3.1, 3.2, and 3.3 are interpretations of
the structures of clinoenstatite (zwierketten) (Morimoto
et al., 1960) wollastonite (drierketten) (Buerger and Prewitt,
1961) and rhodonite (funferketten) respectively.
a projection onto the planes of oxygen atoms.
Each is
Oxygen atoms
are represented by vertices of polyhedra, all of which occur
in only three planes parallel to the plane of the diagram.
Tetrahedra are drawn with heavy lines, and two chains in
one sheet are shown above the underlying sheet of octahedra.
In each chain n + I tetrahedra are shown, where n is the
number of tetrahedra in the repeat unit.
In addition,
the chains are not shown in the upper halves of the diagrams,
in order to show the nature of the underlying sheet of
octahedra more clearly.
The magnitudes and directions of
the unit translations are shown in the lower left of each
figure.
The structures 'obviously differ in the number of
tetrahedra in the repeat unit of the chain.
In addition,
the arrangement of cations in the octahedral layer is
different in each, although there are basic similarities.
Each octahedral sheet is defined by two layers of closepacked oxygen atoms.
The cation occupancy of the octahedral
sites is different in each.
In the clinoenstatite structura,
150
digure 3.1
?rojection along & of the structure of
clinoanstatitt
Coordination polyhedra
are shown for one octahedral layer (below)
and for part of the tetrahedral layer (above,
heavy lines).
Unit translations are shown in
the lower left corner of the diagram.
151
/
%
-le
Ai
152
Figure 3.2
Projection along L101
of wollastonite,
of the structure
Coordination polyhedra
are shown for one octahedral layer (below)
and for part of the tetrahedral layer (above,
heavy lines).
Unit translations are shwn
in the lower left corner of the diagram.
153
/
pr--
154
Figure 3.3
Projection along
of rhodonite,
10i
of the structure
Coordination polyhedra
are shown for one octahedral layer (below)
and for part of the tetrahedral layer (above,
heavy lines).
Unit translations are shown
in the lower left corner of the diagram.
155
156
of the octahedrally coordinated sites are occupied, while
in wollastonite and rhodonite, the fractions are 4 and
respectively.
In clinoenstatite octahedra share edges to
form a chain four octahedra long.
Such chains are bound
together through octahedral edge sharing in a staggered
manner to form a band whose axis paraLlels the chain of
tetrahedra.
Similar bands of octahedra are found in wollas-
tonite and rhodonite.
In wollastontite they are formed of
three infinitely long chains of edge sharing octahedra
which are bound together through further edge sharing.
The chains ot octahedra in rhodonite are ten octahedra
long and they are bound together in a staggered manner to
form the band.
All three structures are similar, then, in having
a band of edge-sharing octahedra arranged parallel to the
chain of tetrahedra, and separated from other bands by a
rift of unoccupied octahedrally coordinated sites.
In each
diagram a single band is shown separated on the left from
a portion of another band by the rift.
These structures
differ, however, in the arrangement of filled and vacant
octahedrally coordinated sites.
Two chains are shown in each of Figs. 3.1, 3.2, and
3.3.
In each, the chain on the right has vertices pointing
r
157
down, bonded to the octahedral sheet.
The chain on the left
has vertices pointing up, with the triangular bases of
the tetrahedra in the upper plane of oxygen atoms coordinating
the cations.
sheet.
This chain lies above the rift in the octahedral
There is, in fact, in all three structures, a second
chain of tetrahedra arranged along the lower side of the
rift.
Tetrahedral chains are thus arranged "back-torback"
in each, separated by a column of vacant octahedral sites,
The bands of octahedra are knit together by the chains of
tetrahedra in this way.
An can be seen in the righthand chain of each
f5gure, each chain fits over the octahedral sheet in a
similar manner.
That is, lower vertices of paired tetra-
hedra are bound to the octahedral sheet at either and of
an octahedral edge.
Thus, in each structure, the vertices
of the tetrahedra follow, in zig-zag fasion, a continuous
line of octabedral edges.
Liebau (1956) proposed that the
length of the repeat unit in the tetrahedral chain depends
only on the size of the cations in the octahedral sheet.
Thus clinoenstatite with a small Cation (r a .78 A) conMg
tains %wierketten and wollastonite with a large cation
(rCa = 1.06 A) cosntains drierketten.
Rhodonite, with an
average cation size less than that of wollastonite, contains
158
funferketten and pyroxmangite with still smaller cations
(but still larger than the average of pyroxenes) contains
siebenketten,
It therefore appears that clinoenstatite and
wollastonite represent two extreme cases.
If the average
cation size is intermediate, the chain tends to have a
longer length the smaller the catin.
(1960) support this.
He noted that in Mg metasilicates
paired tetrahedra can
edges.
(Fig. 3.1)
Belov's observattons
"fit over" the smaller octahedron
In wollastontte three tetrahedra
(Fig. 3.2) may correspond to a single edge.
Rhodonite
and pyroxmangite have chains which are combinations of
these cases,
The three structures pictured in Figs. 1-3 have
other features in common.
None have Pauling's valency
bonds exactly satisfied. This is caused principally by
another unusual feature, the sharing of edges between
octahedra and tetrahedra, shown in the lefthand chain of
each figure.
In clinoenstatite each tetrahedron shares
one edge with an octahedron.
In wollastonite, the tetra-
hedron projecting out from the chain shares two edges.
Five edges are shared in rhodonite.
Note that the top
tetrahedron of the chain on the left shares two edges
with Mn octahedra. The three tetrahedra immediately be-
r
159
low each share one edge with the polyhedron at the and of
the chain of octahedra.
This coordination polyhedron is
unusual, however, in being an irregular polyhedron of
seven oxygen atoms.
In each example of edge-sharing above, one of
the two oxygen atoms is coordinated to two Si atoms and
one other cation.
It thus receives bonds of total strength
two from the Si atoms, and N$ from the other cation.
results in an excess of N
in each case.
This
This is usually
compensated by Si-O interatomic distances whicb are larger
than average. Each structure also contains other examples
of deviations of bond strengths from ideal values.
These
relations show that Pauling's rules should never be applied
too rigidly, for considerable deviation from ideal cases
is to be expected.
Other pyroenes and pyroxanoids.
The relations described
above are generally applicable to all minerals of these
groups.
Protoenstatite and enstatite-type structures merely
represent different stacking of units approximately similar
to that shown in Fig. 3.1
(Morimoto et al., 1960).
The
diopside structure is merely a distortion of the clinoenstatite
structure (Morimoto at al. 1960).
The distortion causes co-
ordinatiujg of the cations (Ca) on the edge of the band to change
160
from 6 to 8.
defined.
The rift of vacant sites is much less well
This structure is unrefined, however, and de-
tailed refinement may alter this interpretation.
Other pyroxenoids with drierketten include bustamite (CaMnSi 2 0 6 ) and pectolite (Ca2NafHSi30 9 ), schisolite
((CaMn) 2 NAHSi 3 0 9 ) and serandite ((Mn,Ca) 2 aHSi 309), the
latter three forming an isomorphous series.
Bustaite
is very similar to wollastonite as pictured in Fig. 3.2,
except that chains of tetrahedra are displaced in alternating layers.
Pectolite, etc., (Buerger and Prewitt, 1961)
differ principally in the arrangemet of cations in the
octahedral layer.
#It
Babingtonite (CaFe
it
Fe HSi50 1 5 ) is probably
the only other known pyroxenoid with funferketten.
(1937)
Richmond
found that the unit cell of babingtonite is similar
to that of rhodonite.
Liebau (1956) proposed that babing-
tonite contains double-chains of tetrahedra since it
tains (OH),
con-
as opposed to the single chains in rhodonite.
Weissenberg c-axis photographs of each show an identical
arrangement of reflections and
a similarity in intensities,
It is thus highly likely that babingtonite contains funferketten.
The presence of hydrogen probably results in a
different distribution of cations in the octahedral sheet
161
as is the case in the wollasconite - pectolite relation.
Classification of pyroxenes and pyroxenoids.
It is obvious
from the relations discussed above that the classification
of these minerals into pyroxene and pyroxenoid groups is
an artificial arrangement.
Liebau has circumvented this
with his classification of structures according to the repeat unit of the tetrahedral cbain (zwier-, drier-, funfer-*,
and siebenketten).
Such a classification is perfectly
natural, as opposed to the old system which divided the
minerals into groups with twierketten and those with chains
with a longer repeat unit.
The terms used by Liebau have
disadvantages, but are the best proposed until this time,
and should be retained.
Phase transformations.
Glasser and Glasser (1961) postui
lated a structural mechanism for the rhodonite-wollastonite
transformation. They used crystals of Franklin, N6J.
rhodonite, which upon heating transformed to "wollastonite".
Using oscillating-crystal and optical goniometer studies of
cleavage fragments before and after heating they noted that:
I. The external morpiiology of the crystals are preserved.
2. The two unit cell axes approximtely normal to the silica
chains (see Chapter 2) are retained.
3. There is about a 9*
difference in the orientation of the axis parallel to the
162
chains.
They concluded that this proves that Si-O bonds
are broken and reformed while (Ca,Mn)-0 bonds remain
intact.
They further show that the planes of octahedra
in both wollastonite and rhodonite are very similar and
that one transforms into the other through simple distortion.
Both structures involved in the transformation
have now been well refined and it is possible to review
these conclusions in detail.
First, it is obvious from
the literature on the system CaSiO3 - MnSi
3
of Chapter 1
that rhodonite transforms to bustamite, not wollastonite.
Since b-axis photographs of both are very similar, and
since the transformed crystals were imperfect, this error
is easily explained.
Since bustamite and wollastonite
are very similar this should not alter their conclusions
however. Glasser and Glasser, postulate two possible
mechanisms for the transformation:
1. Reorientation of
silica chains and reconstruction of the (Mn,Ca) layers.
2. Reconstruction of the chains with simple distortion
of the (Mn,Ca) layers.
mechanism is
correct.
They conclude that the latter
The silica chains almost certainly
undergo reconstruction, based on the evidence presented.
However, it was noted above that the arrangement of filled
163
and vacant octahedrally coordinated sites is entirely
different in these structures. A simple distortion is
far from sufficient as a transformation meebanism. At
least some of the (MnCa)4- bonds must be broken and
reformed, with migration of (MnCa) ions.
Glasser and
Glasser present a detailed mechanism involving migration
of only 7 of 15 Si ions and 4 0ions. It can only be
concluded that although their conclusion on the reconstruction of the chains is probably correct, that the
mechanism for the transformation is extremely complex
and cannot be deduced from a simple comparison of the
structure involved.
A similar situation exists in the johannsenttebustamite transformation. Here, too, the axes approximately
normal to the chains are probably retained while the ebain
orientation changes.
The arrangement of cations in the
octahedral sheet is also different in each, as noted above.
This transformation, then, must also involve reconstruction
of both the sheets of octahedra and the silica chains.
k4
164
Appendix I
DCELL:
IBM 709/7090 program for the
transformation of unit cell parameters and coordinates
When direct and reciprocal cell parameters and atom
coordinates are known for one unit cll
setting but desired
for anotber, hand calculation or graphtcal determination
may be time consuming under the following conditions:
2. Transformtions of
1. The unit cell is triclinic.
coordinates are required for many symmetry related atoms.
An IBM 709/7090 program has therefore been written in
FORTRAN to perform these calculations.
Calculations are performed in the following way.
Transformed unit cell vectors a' b' c' are derived where
(S#) (5)
a
as
and a, b, c are unit vectors of the original cell and (S)
is the transformation matrix.
The magnitudes of a' ,b' ,c'
are computed from relations of the type
a'
-
(a'
a')
and interaxial angles from relations of the type
0CI
-
CO-
b'-.c
b x a
a
mum
Er.
-
e
165
Reciprocal transformed cell constants are then coMuted
with relations of the type
co
(coo
-co/cos
-co
coo
)
aI
a &in lb
$in
The program then calls the subrautine
SYNTRY which
calculates the set of coordinates of all atoms in the
original cell symmtry related to some atom I whose coordinates are supplied to the program.
Transformed co-
ordinates for this set of atoms are Obtained with the relation
ji)
where S L is the reciprocal of the transform of the matrix
(S).
This calculation is repeated for each atom, 1, sup-
plied as input data.
Directions for operation of DCZLL
The following data eards must be supplied:
1. Title card. Format (12 A6) Cols. 1-72; Any identification information may be included.
166
2. Cell card.
Fwat (3(F7.4, 3Y)2 3(F7.3, 3X))
parMeter
cos.
a A
1-7
li-17
b
21-27
C
31-37
oC degrees
"
41-47
51-57
3. Transformation matrix card.
(S)
Format (9(F7.4))
is the matrix of the call transformation
1-7
$11
8-14
s12
15-21
s13
22-28
821
29-35
822
3642
s23
43-49
S31
50-056
s32
57-63
s33
4. Coordinate cards.
Format (3(F6.4, 4X), 11)
One card per atom
167
Colo.
prmta
1-6
7
114-16
y
21-26Z
31
Slank in all ezcpt the
final card; integer 1 in the
final card
168
FORTRAN LISTING OF MAIN PROGRAM DCELL
9
10
11
12
121
122
COMMON K
DIMFNSION TITLE(12),DX(3),DY(3),DZ(3),AXNU(?),ZI(3,3),X(200),Y(200
1) ,Z (200)
READ INPUT TAPE 4,9q(TITLE(I),I=1,12)
FORMAT(12A6)
WRITE OUTPUT TAPE 2,9,(TITLE(I),I=1,12)
READ INPUT TAPE 4,10,ABCALPHABETAGAMMA
FORMAT(3(F7.4,3X),3(F7.3,3X))
WRITE OUTPUT TAPE 2,10,ABCALPHABETAGAMMA
12,DX(1),DY(1),DZ(1),DX(2),DY(2),DZ(2),DX(3),D
READ INPUT TAPE 4,
1Y(3) ,DZ(3)
FORMAT(9(F7.4))
WRITE OUTPUT TAPE 2,121
FORMAT(24HOCOORDINATES OF NEW AXES)
WRITF OUTPUT TAPE 2,12,DX(i),DY(1),DZ(1),DX(2),DY(2),DZ(2),DX(3),DF)
1Y(3) ,DZ (3)
WRITE OUTPUT TAPE 2,122
FORMAT(24HONEW CELL A B C AL BE GA)
ASQ=A**2
BSQ=B**2
CSQ=C**2
PI=.017453
14
19
21
22
23
C
25
ALPHA=ALPHA*PI
BETA=BETA*PI
GAMMA=GAMMA*PI
AB=A*B*COSF(GAMMA)
AC=A*C*COSF(BETA)
BC=B*C*COSF(ALPHA)
DO 14 K=1,3
AXNU(K)=SQRTF (DX (K)**2*ASQ+DY(K)**2*BSQ+DZ(K)**2*CSQ+2.0*DX (K)*DY(
1K)*AB+2.0*DZ(K)*DX(K)*AC+2.0*DY(K)*DZ(K)*BC)
CONTINUE
DO 22 K=1,2
DO 21 J=2,3
IF(K-J)19,21,19
ZI(KJ)=DX(K)*DX(J)*ASQ+DY(K)*DY(J) *BSQ+DZ(K)*DZ(J)*CSQ+(DX(K)*DY(
'J)+DY(K)*DX(J))*AB+(DZ(K)*DX(J)+DX(K)*DZ(J))*AC+(DY(K)*DZ(J)+DZ(K)
2*DY (J)) *BC
ZI(KJ)=ACOSF(ZI(KJ)/(AXNU(K)*AXNU(J)))
ZI(KJ)=ZI(KJ)/PI
CONTINUE
CONTINUE
WRITE OUTPUT TAPE 2,23,AXNU(1) ,AXNU(2) ,AXNU(3) ,ZI(2,3) ,ZI(1,3) ,ZI(
11,2)
FORMAT(4X,6(F7.3,2X))
ROUTINE TO TRANSFORM CELL AXES AND COORDINATES
WRITE OUTPUT TAPE 2o25
FORMAT(35HONEW RECIPROCAL CELL A B C AL BE GA)
T23=DX(1)*DY(3)-DX(3)*DY(1)
1,2 )*P I
169
ZI(1,3)=ZI( 1 , 3 )*P I
ZI(2,3)=ZI( 2 -)3 ) P I
COSA =COSF( ZI (2,3))
COSB=COSF(Z I(1,3)
COSC=COSF(Z 1(1,2)
SINA=SINF(Z 1(2,3))
SINB=SINF(Z I(1,3))
SINC=SINF(Z I(1 2)
ALS=ACOSF( COSB*COSC-COSA) SINB*SINC)
BES=ACOSF(( COSA*COSC-COSB) SINA*SINC)
SINA*SINB)
GAS=ACOSF(( COSA*COSB-COSC)
AS=1*0/(AXN U(1)*SINF(BES)* INC)
BS=I.0/(AXN U(2)*SINF(ALS)* INC)
CS=1.0/(AXN U(3)*SINF(ALS)* INB)
ALS=ALS/PI
BES=BES/PI
GAS=GAS/PI
TAPE 2, 2 6,ASB SCS ,ALSBESGAS
WRITE OUTPU
(F
8.5 ,2X) 0 3 (F7.3 ,3X)
FORMAT(4X,3
26
T11=DY(2)*D Z( 3 )-DY( 3 *DZ (2)
T12=DX(2)*D
3) -DX (3 *DZ (2)
Y(
3 )-DX (3 *DY (2)
T13=DX(2 )*D
Z(
3)-DY (3 *DZ (1)
T21=DY(1)*D
Z(
3) -DX (3 *DZ (1)
T22=DX(1)*D
Z(
T31=DY(1)*D Z( 2 )-DY( 2 *DZ (1)
Z(
2 )-DX( 2 *DZ (1)
T32=DX(1)*D
Y(
T33=DX(1)*D
2 )-DX (2 *DY (1)
*D
Y(2) *DZ 3 )+DY( 1)*DZ(2)*DX(3)+DZ(1)*DX(2)*DY(3)-DX(3) *
DELTA=DX(1)
Y (3 )*DZ 2 )*DX( 1)-DZ(3)* DX(2)*DY(1I)
10Y(2)*DZ(1)
WRITE OUTPUT TAPE 2t261
FORMAT(41HONEW COORDINATES FRACTIONAL AND ANGSTROMS)
261
27
READ INPUT TAPE 4,28,X(1)tY (1) ,Z(1),IEND
FORMAT(3(F6.4,4X),I
28
CALL SYMTRY(XYZ)
WRITE OUTPUT TAPE 2 ,28,X(1),Y(1)0Z(1),)IEND
DO 31 I=1,K
XX=Tll*X(I)-T12*Y(I )+T13*Z(
YY=-T21*X(I)+T22*Y( I )-T23*Z
ZZ=T31*X(I)-T32*Y(I )+T33*Z(
XX=XX/DELTA
YY=YY/DELTA
ZZ=ZZ/DELTA
XXX=XX*AXNU(1)
YYY=YY*AXNU(2)
ZZZ=ZZ*AXNU(3)
WRITE OUTPUT TAPE 2,30,XXYYZZXXXYYYZZZ
FORMAT(5X,3(F8.4,2X),1OX,3(F8.3,2X))
30
CONTINUE
31
IF( IEND)32,27,32
CALL EXIT
32
END
7 I(1, 2 )=
170
FORTRAN LISTING OF SUBROUTINE SYMTRY
THIS EXAMPLE IS FOR SPACE GROUP P 1BAR
SUBROUTINE SYMTRY(XY,Z)
COMMON K
DIMFNSION X(200),Y(200),Z(200)
K=2
X(2)=1.O-X(1)
Y(2 )=] *O-Y( 1)
Z(2)=1.O-Z(1)
RETURN
END
171
Appendix II
Comparison of observed and calculated structure factors of bustamite..
172
AOBS
BOBS
K
L
FOBS
FCAL
0
4
756.18R
-7609
0
0
6
40.49
77. 46
37.30
0
0
8
122.99
137.3
122.54
0
0
0
0
0
0
10
12
14
42.47
51- 18
17.05
38.78
50.09
14.11
42.26
-51 -06
16.48
0
0
16
38.05
37.12
36.68
1-l2
31.41
106.45
111.63
26.44
97.80
92.19
10.81
6-09
5.44
11-05
-4.01
H
2
2
2
0
0
0
2
-4
4
28.56
97.99
92.35
2
0
-6
115.40
2
0
n
0
0
0
0
0
6
16.09
2
2
2
2
2
2
-8
-7.13
8
-10
10
-12
-12
20.09
14._10
15.73
61.80
61.80
33.88
53.89
12
-120.3
113.56
.? 1-
40.24
114.87
-15.58
-709
-7
4.48
10.48
4.24
-3.43
4.38
7-5
1.73
-1. 06
10.06
5.78
5.78
19.73
14.47
17.07
63.65
63.65
20.02
1376
-12.09
61.53
61.53
.32.68
33-33
6.10
53.35
52.92
10.17
-?.98
2
0
2
0
2
0
14
12.76
12-?A
12-41
2
2
4
0
0
-16
16
0
20.58
7.23
-21.06
-7.42
-36. 16
1.52
154
-1.42
0
-2
21.11
7.58
36.19
'A- 7R
33.16
31.97
8.80
0
4
56.37
59.10
55.46
10.08
0
0
0
n0
0
6
8.03
41, 81
12.00
9.33
-3.23
7.35
-14
4L
4
60.16
A2-.1733.24
L
4
4
4
4
4
4
4
4
4
0
0
8
-41 - 74
9.32
A66 2?a
-11.65
-35--2-L-
-2.86
-- 474-0.42
10
26.75
24.80
-26.75
12
-14
75.96
20-AA
74.61
75.35
9.61
8 4
14
12.19
10.57
8.69
85 5
173
-16
16
0
6.05
20.28
74.60
7.37
18.
78.07
-5.87
-0.18-97
73.93
-1.46
6
0
0
0
6
0
2
38.33
5.42
28.3?
-0.68
6
0
16.77
15.73
6-4?
76.06
13.86
-5877
-72.30
4
4
6
6
6
6
6
6
6
6
6
6
6
8
8
8
8
8
8
8
8
0
-2
-4
4
58.81
72.33
0
0
0
0
0
0
0
0
0
9.44
-?27
-2.07
-A'- 3 -4
0i
0
0
0
10.00
6
-A
8
=010
-12
12
-14
14
39.66
66-2 9
86.81
0.
36.75
AR-27
85.52
0.92
9-4
12.66
11.24
?2-74-
38.56
65 -4 6
86.23
-9-0
a
-1.*86
-12.48
-2.13
-'3-
59
15.26
14.83
4
49.41
55.84
17.58
22.45
0.
45.10
52.03
16.52
19-64
1.94
48.88
54.50
17.23
22.a8
-0.
-6
l2. 10
27.3(-,
6
27.46
26.31
12 H1
1?
9
55.01
26.42
38.73
-2
-4
*
10.00
-0.
0.
16.99
0
9.27
10-a4'A
27.26
-w1 0.4
57.41
25.73
38.95
-O-56
8.30
7.22
12- 17
-3.50
0.
-3.32
A.10
6.54
-2.91
10.82
8
8
8
0
0
0
8
-10
10
57.78
25.89
40.42
8
0
-12
6.93
5.8
8
0
12
9.23
7.75
-9.10
1.57
8
0
23.64
20.64
20-87
11.10
14.31
-3.06
-14
6.73
8
0
14
14.63
12.93
10
0
0
67.74
67.28
10
10
10
10
10
10
10
10
(
10
10
10
10
10
1?
12
0
0
0
0
0
0
0
0
0
0
0
0
0
2
77.27
79.70
77.17
-2
11.84
12.41
11.28
-4
4
-6
6
-8
A
12
0
0
0
-10
10
-12
12
-14
0
2
.- 2
-4
12
0
4
12
0
12
0
12
0
12
-6
6
-8
97.11
10,44
76.84
5.43
52.78
4ci a41
7.04
3941
50.46
6051
24.26
4.79
13.60
A82,s43-8A4.63e19.95
98.43
115-05
75.24
381
50.64
42.45
5.53
37.17
48.18
5998
23.92
44169
14.44
20.84
-67-64
96.46
104-93
76.73
1*76
-52.66
L45.9(
1.67
-3.66
3.99
3.60
11.26
10.39
4.12
i14
-3.54
- 3 -f-5
5.42
39.21
49.30
59o65
23.99
45:39
13.26
8 1.075
-19.88
-30-18
-13.99
4.50
4*01
10.75
10.16
3.64
6.05
-3.04
100
1.68
1.77
-3.67
30-?3
29-65
14.46
39.51
64.44
37.72
37.95
11.00
63.68
64.19
5.72
12.24
174
12
12
12
14
14
14
14
14
0
0
0
0
0
n
0
0
14
0
14
14
0
0
14
0
14
14
14
0
0
0
16
16
16
16
0
0
0
16
0
0
16
0
16
0
lb
-1
18
0
01
18
18
0
0
-1
-1
1
1
-1
-1--1
1
1-I
1
-1
1
-1
1
-1
1
-1
1
41.30
-5.-2-
10
7.75
40.16
-5--987.00
-3-24+-
C)-
~~1
-2
-4
17.06
18.18
17.06
5 - 1-6
1-197
4
-A
25.06
24.75
-24.96
5.68
140-97-3.25
-10.40
23 -2 5
--- L-06-
6
20.97
21.88
8
19.14
17.99
20.97
16.58
-0.26
-2.20
-5-5c3-
-0.26
9.57
-10
10 28
10
-12
0
20.53
13.3930.14
19.22
15 .73
27.59
19.14
13.a0
-30.07
?2.79
?1 -51A-
-2
-4
4
-6
6
23.05
25.44
?l .27
-23*04
54.27
53.80
53.55
8.81
16. 13
14.66A
4.10
1.15
-3.32
-2.40
0
34.79
0.
15.83
34.33
34.77
1.11
28.55
;0.757
18.07
25.70
11.24
5.40
-2.70
3
-5
11.15
3.85
28.42
-30.58
-17.99
-5
56.-44
55.a55
5
5
-7
-7
7
43.91
43.09
3. 12
10.00
-2
-4
-3
1. 25
9
.58
-1.98
R , 17
-0.60
A8-69
13.36
16.09
-43*90
24
-1.67
0.83
-4--15
25.32
22.91
32.66
30.84
-32.61
-1-+-
-q
-4-01
11.65
4--4+4025.32
12.76
7.42
-l
8-78
65.o94
-5.21
-1.77
-0*05
A- 1 a
"9
13.98
14.62
-13.60
24-6c2
-27-Q5
20.74
12.19
9.40
20.36
8.48
20.69
1 21, 5~
8.78
5.03
3.89
-4.95
11-?25
-3.23
0.71
1.44
-3.36
-3.*5 7
1
-13
- 1q
1
1~
13
13
-15
27
2
?709
1
15
1
0
%"
24.88
24.85
15
3
7.04
26.97
,
-1
40.91
-5-7-6,9-
0
1
-1
-1
8
1
-0.88
-26.9
oQ
-20.21
0.49
20.22
20.25
5.33
4.63
2.64
-4.63
4.95
9.23
1-n(
1.27
-n.21
21.69
0.25
9.32
1806
18-A
21.69
19.88
175
"-1
3
-3
3
1
-1---
3
1
-3
-3
3
27.27
-54.20
-25-5?
26.93
5.54
2.53
-5.38
8.01
4.27
15.39
15.74
8.01
-17.65
17.41
-1-05
11-75
-0 , S9
-1--5
55
I
-7
~~~~1~~
-7
3
1
7
~~~1~~
7
1
3
1
3 -I-----
~~~~1~~
-9
1
~~~~1~~ -9
-3
9
1
3-3
3
-11
1
1
1
11
-3
1
3
-13
1
3 ~~~~1~~
13
-33
1
-3
-15
-15
3
1
15
15
3
1
-5
1
-5
-1
-5
1
-1 -5
-3
1
-35 ~~~~1~~
3
5
1
-5
-5
5
1
-5
5
1
-7
~-~1-5
-7
1
3
17.66
1 1.76
10 -08
1.08
0.
1 5.70
1L4 .?9
29.73
21.17
20.32
29.79
19.75
19.87
24.41
24.679
0.
2.02
7.51
11.57
20.37
4.00
12.11
20.57
9.65
8.92
5.86
1.95
1.34
0.23
-2.97
0.
*1 9
-29.51
-3.61
-?0-76
-4-13
-20.14
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51,20
18.88
-26.47
33.77
51.87
1.36
-2.24
-0.
1.24
5
5
3
35.25
35.65
-35.04
-3.86
-5
5
-5
5
5
5
5
5
-5
-5
5
5
44,66
27.03
14.11
16.61
41.58
25.41
13.12
1&.21
44.63
-26.67
-14-01
16.59
1.58
-4.40
-1.66
-0.80
-5
5
-7
15.92
15.44
15.92
0.03
5
-5
5
-5
5
5
5 _
5
5
5
-7
7
7
-9
-9
21.40
31-04
11.70
25.30
14.08
-20.95
~?9.16
11.35
-27.94
15.22
-2.92
-3.53
1.09
-4-63
1.71
-5
5
9
20.60
-20.05
-1-4A
5
5
9
21.15
?9.37
11.40
28.32
15.32
20.10
0.
1.41
0.
-0.
191
5
5
-5
5
6.62
22.51
-11
-11
6.40
23.50
-5.92
2250
-2.97
0.6
0.94
-3.30
5
11
14.31
14.17
14.28
-5
-5
5
5
5
11
-13
Ila
31.50
5.18
4.9n
33,p
5.52
- -31.33
4.96
A6,3~45
-7
5
1
32.01
29.86
-31.99
1.04
1
4-14
47.88
18-73
-6-00
47.98
-18.77
2+32
0.18
-1.02
-5
5
1.48
7
-7
7
5
5
5
-1
-1
6,43
47.98
18.80
-7
5
-3
12.59
14.22
1?.17
-3.23
7
5
5
5
5
5
-31
3
3
-5
-5
5.33
18.90
4.11
2.23
2.54
4.42
19.07
2-53
4.68
1.94
1.20
-18.74
-3*56
-1.61
-0.58
-(-122
-2.42
-2.05
-1.54
-247
5
5
18.81
16.82
18.36
-4.07
5
5
5
5
5
5
5
5
5
5
5
5
5
-7
-7
7
7
-9
-9
9
9
-11
-11
11
9.10
40.47
5.84
19.43
20.81
33.99
5,95
4.52
10.88
14.30
15.59
18.20
9.60
38.49
3.43
18.35
19.54
33.48
7*16
5.17
8.70
14.10
18.16
19.16
8.35
-40.46
-4.86
19.42
-20.74
33.98
5*94
-4.44
-10.81
13.87
-15.10
-18.14
-3.62
1.00
3.23
-0.70
-1.75
0.92
-0.35
0.84
1.23
-3.50
-3.86
-1.43
5
13
1.35
4.58
-0.90
1
1
-1
-1
-3
-3
3
21.46
15.92
8.54
16.08
21.66
30.15
4.28
21.98
14.16
6.75
17.71
18.74
29.82
5.17
3
7.75
:7.93;
3.75
22.03
17.24
10.37
7.02
3.33
21.07
16.84
10.55
5.73
210
3.28
2235
23.09
4063
24.68
23.18
4.47
11.60
18.62
34,00
34.81
-7
7
-7
7
-7
7
-7
7
-7
7
-7
7
-7
7
-7
7
-7
-7
-9
9
-9
9
-9
9
-9
9
-9
9
-9
9
-9
9
-9
9
-9
9
-9
-9
-9
-11
11
-11
11
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
-5
-5
5
5
-7
-7
7
7
-9
-9
9
-11
11
1
1
-1
-1
3.28
1.34
20.72
25.07
.,63
25.78
24.22
6.10
12.93
16.44
36.37
33.49
21.23
15.76
8.52
-15.63
-21.63
-30.14
3.35
7 a66
-1.80
21.95
-17.24
-10.36
-5.20
2.2q
1.31
?-0.48
25.07
-0165
25.67
-24.17
4.10
-l263
-16.13
-36.15
33.49
-1.01
-3.14
-2.22
0.52
-3.76
1.07
-0.83
-2.66
1.a21
-3.29
1.93
0.14
-0.4
-4.72
-2.36
0.27
-3.35
0.20
-31;7
-2.43
1,59
-4.52
-2,79
-3.17
-3.99
0.11
192
-2.31
-1.96
28.21
-11
11
11
5
5
5
-3
-3
3
2.36
204
28.22
1.91
5 15
28.45
1l
5
3
12.59
14.82
-5
24.23
21.44
24.18
q.8
8.R5
-A . rQ
5
-11
115
11
-11
-11
-5
5
5
2.02
0.69
-12,37
-0.46
0.57
0.76
-2.32
1.58
-9
0.32
1.99
5
-7
15.55
12.61
-15-36
-2-45
-11
5
5
5
7
-9
9
24.13
25.00
6.81
22.31
2239
4.09
-23.91
-24-57
4.80
-11
5
1
( a.q
1
-1
-3
15.65
3
-1
5
-13
-D
-13
5
5
-13
5
-13
-5
-1 3
-13
-13
"13
5
5
5
5
5
-07
7
9
-15
5
1
-15
5
-15
5
-15
5
-15
-15
17
-17
0
0
0
0
0
0
0
-l
-17--01
-0.64
4.44
-6.30
9
-
-1.85
11
-
14.35
6.81
3.*18
15.55
7.23
5.56
-14.28
-6.48
3.16
1.41
-?.31
0.37
25.Q4
24.78
25-00
-1.49
9.92
6.86
9.77
1.6 9
1 0.:Z
8.Aa84
-A
34.88
-32.61
-3.37
14.45
-16.56
-4,46
5
5
5
5
6
6
6
6
6
5
7
1
-3
-4
4
-6
6
-8
10.44
6.79
21.19
9.340
34.67
13.94
24.60
4o38
40.36
9.33
8-18
23.34
8 - 87
31.24
14.47
22.62
7*39
37.90
-10.32
-M6.75
20.96
92
-34.50
~13.57
24.32
3.70
38.92
-1.60
0.75
-3.08
C-A
-3.43
-3.19
3o.9
*34
10.67
6
-8
76a09
83.277..-.4
19.18
16.73
3-29
10.07
18.68
63.8
64.8-7
6.610.16
7o97
30,55
40.35
28*69
7.63
55.0?2
33.07
7.09
27.61
37.02
27.84
3.04
53.-85
31.51
-7.27
29.80
39.91
2 .-3.97
5.4-7
31.26
13.47
14-.1A
3.34
1.s45
1.14
14-n?2
9.7--7
~"13-0-76
6
-10
1n
0
2
-2
-4
-4
4
-6
-6
6
2
6
6
6
-2
2
-2
2
-2
17.07
6.57
17.15
6
6
6
6
6
6
6
6
6
2
6.22
17.02
0.
32.78
3
6
-2
A -99
-15.61
-0.
-0.94
-5
2
2
2
-2
2
2
-2
2
-2
0
7 -01
16.51
2-44
3.97
-3.26
-4.62
-4.83
) - 11
1.07
-0.
-6.15
6
6
6
6
6
-8
-8
6
8
-10
-10
10
4.?72
9.88
3.91
30.19
25.62
1, 96
16.40
2.43
27.6,(7
22.94
2.66
14.73
Q
1-06
9.70
-1 '-.17
3.37
nil1I-Lo
23.41
Q.78
-16.10
4.33
4'14
le86
-3.26
6.75
5.97
c;.6 9
-6.51
1 0-2n
10.79
-2-
A2
3.14
2.:)-70
1099
97
10.42
-1.80
-3o13
193
2
4
4
6
6
6
10
4
6
-?
35.44
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
A
6
6
6
A
6
6
6
-4
66.86
24.11
18-Z
9.48
33.23
62.66
-4
58.3Q
552
39.82
44-07
4.52
20--48
17.36
35.65
6.80
17.13
14.30
12.38
7.70
34*97
21.88
40.36
12.65
31.27
9.39
24.33
26.57
23,79
30.49
11.02
0.
11-08
73.21
34.04
27.54
12.91
42.67
29.42
8.13
3584
63.62
9.27
0.
2.59.23
2.59
6.21
31. a7
61.24
45.98
12.97
35.73
41.56
14.16
17.84
16.23
33.77
3.63
17-55
15.26
11.42
6.25
36,09
22.17
37.72
11.10
27.89
8.17
24.15
24.62
22.01
26.26
10.27
3.88
10.62
72.70
a0.59
23.66
12.80
41.87
26.97
8.29
36.53A
64.59
8 .67
2.06
19.87
-4
4
-4
4
-4
4
-4
4
-4
4
-4
4
-4
4
-4
6
6
6
-6
6
-6
6
-6
6
-6
6
-6
6
-6
-6
-6
-8
8
8
-8
8
-8
8
-8
8
-8
8
-8
-A
-8
-8
6
-8
6
-10
-10
6
6
-10
6
-10
6
0
2
4
4
-6
-6
6
6
-8
-8
8
8
-10
-10
10
0
2
-2
-4
-4
4
4
-6
-6
6
6
-8
-8
8
-10
10
0
0
23
-2
-2
-4
-4
4
4
-6
-6
6
8
8
-10
10
0
2
--2
-4
24.30
19-89
10.50
1.-A
22.21
-19-76
10.50
34*56
66.14
-57.6l
38.43
42.94
..
3.77
20-48
17.35
34.82
-6.18
-17.05
-14.20
-12.17
-7.69
34,37
20.54
39.26
12.63
30.03
-9.12
-24.29
-26.47
-23.73
29.25
11.01
0.
7-68
72.66
32-81
25.33
-12.91
42.02
28-74
-5.72
34 ,6 0
62.80
8.89
0.
-'-33
9.86
-2-23
-0.08
7.84
9.79
9.5i
10.44
9-90
2.49
0.09
-0.66
7.64
-2.84
-1.70
-1.73
-2.26
-0.43
6.45
7.55
9,38
-0.74
8.72
-2.22
-1.45
-2.27
-1.75
8.62
-0-53
-0.
7.98
8.95
9.08
10.80
-0-03
7.43
6.27
5.78
9-,45
10.18
-2.63
0.
--- 2..-1.2-2.30
2.41
-3.64
955
10.38
6.66
1 .14
4.65
302 9
60.70
8.94
-0-96
-5.03
30 .20
60.35
42.49
ILS.A2
ri78
13.43
11.82
17-91
16.99
5.34
10.30
6.47
5.13
-5.32
-3.69
27.70
14.44
30.87
15.54
-27*55
14.01
-2*91
3.50
38.50
36.6A
38.32
48.18
43.98
46.84
-372
11.27
194
-10
-10
-10
4
-6
60.76
13a15
37.92
6
6
6
-10
-120
-10
6
8
26.67
-12
-12
6
0
?
-2
23.74
-1?
-12
I2
-12
-172
-12
-14
-14
-14
-14
-14
-14
25.48
6
6
6
6
6
6
6
6
66
6
6
6
18,A4
6.30
59.78
10.87
34.51
2l54
26.72
17.a6
22.00
37.73
-25-26-26.47
3.81
-9-6
-3.22
22.92
6.20
46-.56
4:7-07
10-37
2- 88
-1.75
-6.05
11.33
-12.76
1.57
58.98
12-52
1-058
1-o
4
12.86
-6
6
8
0
-z
-2
-4
4
6
29,649
9.25
12.25
25.44
16.33
24-61
12.43
11.67
21.67
27.80
14.75
92A-55
-11.65
24.78
13.11
12.42
20.90
5,75
9.73
-0.48
5.71
-1-78
13.6?2
2.62
3.97
223*97
-3*79
6.02
22
.37
22-37
-2.49
22
66f
-0.81
7 . 83
195
Appendix III
Comparison of observed and calculated structure factors of rbodonite.
196
3.45
10.63
1.99
0
0
0
0
0
0
0
0
0
-1
1
-l
1
-l
1
-1
1
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4.93
18.89
12.79
35.42
57.06
13.34
1.96
7.06
8.57
19.77
2.42
20.87
25.77
35.46
44.28
24.40
12.63
11.85
-10.34
-40.91
30.24
69.49
-108.93
-35.72
-15.61
-18.84
-18.15
38.33
-74.52
-38.96
-50.19
-63.33
-81.68
-47.92
22.60
-24.16
-2.84
1.64
3.64
2.22
-21.36
15.71
3.04
-1.59
-0.13
7.46
-15.74
-0.55
-3.46
-1.24
-5.53
-5.23
-0.54
-9.38
1
0
23.26
19.06
40.55
0
9.98
7.63
4.89
0
5.15
5.40
0
10.91
0
n
0
0
0
0
31.69
0
0
0
0
0
0
0
0
0
0.
40.83
138.28
76.55
22.44
7.45
12.08
5.06
13.42
1
5
-1
1
6
7
-1
1
7
8
8
9
0
1
1
2
2
3
3
4
4
5
5
BOBS
6.19
6.14
23.43
17.43
39.79
63.53
22,33
9.10
10.82
10.39
22.35
43.59
22.30
28.79
36.25
46.85
27.59
12.94
14.83
0
6
AOBS
FOBS
K
1
2
3
4
5
6
7
8
9
1
1
2
2
3
3
4
4
5
FCAL
L
H
-1
1
-l
2
-2
2
-2
2
-2
2
-2
2
-2
2
8.n8
18.61
6.3
8.20
3.52
8-93
26.67
6.42
12.99
-7.44
-18.91
54.57
2.68
-16.74
5.06
244
9.37
13.78
13.48
7.02
1.24
29.59
10.63
-14.31
-2.45
-0.72
-5.64
0.87
43.76
159.67
83.40
20.57
5.13
10.20
5.68
11.86
0.
-71.23
-240.61
132.35
38.77
9.85
20.93
-8.76
-23.25
-0.
-4.01
-22.13
19.35
5.88
-8.52
-2.72
-1.17
-3.05
4.56
285
197
6
6
7
7
8
-2
2
-2
2
-2
8
2
-2
9
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
0
1
1
2
2
3
3
4
0
0
0
0
0
0
0
-3
3
-3
12.72
11.93
1A.S0
7.55
-7.88
4*68
35.49
0
20.31
0
17.62
19.45
0
47.41
51.31
3a26
0.
3
-3
3
-3
3
-3
3
-3
3
-3
3
-3
11.48
23-.05
18.11
31-33
25.19
8.15
1.30-4
21.68
3
22.71
-44.07
34.87
67*32
-46.48
13.03
25*22
19.99
40.-3
27.53
27.97
0
017.99
42.76
0
14.82
0
8.70
0
6.91
0
9.70
0
0,
0
4.25
0
25.07
0
23.02
0
-29.99
82.78
0
Ra.19
48.57
30.14
12.00
44.99
12.42
9.28
4.74
9.99
0.92
3.84
19.54
22.46
3
0
9005
6.30
-3
4
-4
4
-4
4
-4
4
-4
0
0
0
0
0
0
0
0
0
29.42
9.74
100.20
6.85
18.00
28.13
8.26
13.32
17.89
23.99
7.82
135.88
5.11
18.73
2285
8.52
12.72
17.70
341,21
73.18
25.40
-15.05
-10.69
16.75
0.
7.25
-43.60
-39.91
1.64
0.626A
1.99
20.24
-12.39
-12.32
10.58
3,25
0.47
-6.95
3.28
n.
5.40
-A57
15.06
5.06
-2.17
5.62
2.62
0.
-1.60
-4.24
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237
ACKN WLMDGDYENTS
I am indebted to Professor M. J. Zuerger who
brought this problem to my attention and supervised the
investigation.
All intensity data for rhodonite, which
were collected by Dr. N. Nitteki, was turned over to me
by Professor Buerger.
In addition, he made available
his notes on preliminary investigations of both rhodonite
and bustamite.
Discussions with Mr. Charles Prewitt, who worked
on the related minerals pectolete and wollastonite, resulted in many valuable ideas.
Mr. Prewitt kindly made
avAilable his program for least-squares refinement of
crystal structures and many times helped in revisions of
this program.
In addition, he kindly made available un-
published data on vollastonite.
All other members of the
crystallographie laboratory, including Mr. Wayne Dollase,
Mr. bernhardt Wuensch, and Mrs. HiLda Cid-Dresdner, have
been encouraging and helpful at all times,
Specimens for this investigation were kindly provided by Dr. Clifford Frondel of Harvard University.
Dr.
Waldemar Sehaller of the U.S.G.S. made available many other
specimens of Ca, Wt ntasilcates.
IBM 709/7090 programs,
r
238
as well as advice on their use, were contributed by Dr.
C. W. Burnhaa of the Geophysical Laboratory and Dr. Donald
Wright of the M.I.T. Chemistry Departet.
My wife, Dorothy, helped in countless ways toward
the coqpetion of this projact.
She contributed her time
and effort as well as providing constant eacourqgenat.
Miss Dborah Jope typed the final draft of this
thesis, and Mrs. Shirley Veale and Miss Gay Lorraine the
final drafts of the published papers.
Fellowship support provided by Standard Oil of
California is gratefully acknowledged.
239
BTGRAPY
The author was born February 15, 1937 at
Smerville, Massachusetts to Carroll and Leona Paicor.
He has two brothers, C. Norman, and Robert W.
ie attended gra4e school and high school in
Stonebam, YAssachusetts from 1942 to 1954.
Institutions
attended include Tufts University (1954-1958), where he
received the degree of Bachelor ot Science* Magna Cum
Laude, and M4l.T. (1958-1962) where he received the degree
of Master of Science in 1960. At Tufts University he was
a laboratory assistant for three years in the Department of
Geology, and both a teaching and research assistant in the
Department of Geology & Geophysics at M.I.T.
Professional experience includes the following:
Summer of 1955 as a geological field assistant; Sumer of
1957 as a field assistant in Greenland involved in permafrost research; Sumers of 1959 and 1960 research assistant
in crystallography in the Departwent of Geology of M.I.T.;
Summer of 1961, staff member at Lincoln Laboratory.
The author received the Standard oil of California
fellowship at M.I.T.,
1961-1962, and is a member of Sigma Xi.
He is a co-author of the following publications.
240
1. M. J. Buerer, C. W. Birnham and D. R. Peacor
(1962) Assessment of the Several structures proposed for
tourmaline. Acta Cryst., 15, 583-590.
2. D. R. Peacor and M. J. Buerger (1962) The determination and refinement of the structure of narsarsukite,
Na2Tt
40 10 .
Am. Mineral.
3. D. R. Peacor and C. T. Prewitt (1962)
coordinates for crystal structure models.
Drilling
Rev. Sci. lustr.,
33, 530-551.
He wae married to Mis Dorothy Contaris of Milford,
Massachusetts on February 1, 1959.
David, born February 1, 1962.
They have a son, Scott
The author has accepted a
position as instructor in the Departmant of Geology and
Mineralogy, University of Michigan, beginning Septeter, 1962.
241
Reftrences
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2
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3
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22, 215-216.
4
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5
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6
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7
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structures of wollastonite and pectolite. Proc. Nat'l.
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8
Burnham, C.W. (unpublished) LCLSQ3, IBM 709/7090
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9
Burnham, C.W. (1962) Absorption corrections for
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242
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13
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16
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18
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20
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21
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22
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23
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24
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26
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244
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29
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30
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31
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32
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33
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34
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