Clayn and Clay Minerals. Vol. 23, pp. 231-246. PergamonPress 1975. Printed in Great Britain
OD-INTERPRETATION OF KAOLINITE-TYPE
STRUCTURES II: THE REGULAR POLYTYPES
(MDO-POLYTYPES) A N D THEIR DERIVATION
K . DORNBERGER-SCHIFF
Central Institute of Physical Chemistry, Academy of Sciencesof the German Democratic Republic,
Rudower Chaussee 5, Berlin-Adlershof,1199, G.D.R.
and
S. [)UROVIC
Institute of Inorganic Chemistry, Slovak Academy of Sciences, Dfibravskficesta 5,
809 34 Bratislava, Czechoslovakia
(Received 13 November 1973)
Abstract--All MDO-polytypes (i.e. regular polytypes) of trioctahedral, dioctahedral and monoctahedral
1:1 phyllosilicates are derived from their OD-character using the results of our preceding paper. The
terms 'simple polytype', "standard polytype' and 'regular polytype' turn out to be synonyms of the term
'MDO-polytype"which has the advantage of being exactly defined.The method used to obtain the MDOpolytypes is also applicable to more complicated cases.
INTRODUCTION
IN OUR previous paper (Dornberger-Schiff and [)urovi~, 1975) we have given a general outline and analysis
of the symmetry of kaolinite-type minerals*, i.e. the
1 : 1 phyllosilicates. It has been shown that these structures can be considered as OD-structures consisting of
three kinds of OD-layerst and that assuming idealized
symmetrical OD-layers corresponding to Pauling's
model, all observed stacking possibilities of atomic
planes, of sheets and of the complete kaolinite layers
which we call OD-packets follow directly from this interpretation. The OD-character of kaolinites explains
also their tendency to form twins and disordered structures (e.g. fireclays).
The three kinds of OD-layers have the following
composition: The first (called tetrahedral layer) consists of the network of SiO4-tetrahedra and the OHions lying in the plane of the apical oxygen atoms. The
second (called octahedral layer) consists of the plane of
octahedrally coordinated cations. The third (called
OH-layer) consists of the plane of OH-ions completing
the octahedral coordination of the cations. What we
call 'OD-packet' or simply 'packet' contains one ODlayer of each kind, the three OD-layers following in the
order just given. Thus such a packet corresponds to the
kaolinite layer. Any given kaolinite structure consists
of equivalent packets.
Of what use is such an interpretation one may ask
if as a result the already known facts familiar to anybody interested in kaolinites are obtained at the cost
* These will be called ~kaolinites' in lhe following, for
short.
~"See 'Some remarks on terminology' in our previous
paper.
of new terms and definitions to which one must get
accustomed. A similar question could have been asked
concerning the use of the 230 space groups, when Xray analysis tackled the first simple crystal structures.
Nobody will however seriously doubt that X-ray
analysis of complicated inorganic or organic structures
could never have been developed without a knowledge
of space groups, which also proved their worth in enabling us to understand physical properties of matter
which may be described as tensors. Similarly it probably would be rather difficult to treat monoctahedral
phyllosilicates, i.e. phyllosilicates in which the three
octahedral sites per unit mesh of the octahedral layer
are occupied in an ordered way by three different
cations or two different cations and a void, without
any knowledge of OD-theory, whereas this theory of
generalized symmetry facilitates the treatment of even
more complicated cases.
Evidently space group theory can handle properly
only fully ordered structures, i.e. structures in which
the position of any building unit is unequivocally
determined by its relation to neighbouring units.
Although naturally any structure periodical in three
dimensions possesses a space group, knowledge of this
space group does not suffice to explain relevant facts,
such as tendency of the formation of various ordered
and disordered polytypes and of twins, if in such a
structure there is more than one position of a structural unit which leads to equivalent relations to neighbouring units. Obviously, kaolinites are of just this
kind.
As a result of our OD-interpretation of kaolinite
structures in our preceding paper we were able:
1. To show for dioctahedral and trioctahedral kaolinites that only one kind of packet, respectively, is
231
232
K. DORNBERGER-SCHIFFand S. E)uRovx6
consistent with the relative position of adjacent ODlayers as postulated by Pauling's model, and to find all
relative positions of adjacent packets consistent with
this model.
2. To obtain the corresponding results for monoctahedral kaolinites.
The subject of this paper is a systematic derivation
of all structures of maximum degree of order (MDOstructures or MDO-polytypes) for the tri-, di- and
monoctahedral kaolinites (for an explanation of the
concept of MDO-polytypes see the Appendix to our
preceding paper).
SOME SPECIAL PROPERTIES OF MDO~STRUCTURES
As has been shown, any MDO-structure of category
II* consisting of M kinds of OD-layers contains equivalent packets, equivalent packet pairs, equivalent
packet triples and equivalent packet quadruples
(Dornberger-Schiff, in preparation).
In our previous paper the symmetry of a single
packet has been found to be P(1), Clm(1) and P(3)lm,
for mon-, d ~ and trioctahedral kaolinites, respectively. The possible relative positions of adjacent
packets have been schematically presented in Figs. 5,
6 and 7 of that paper. Symbols for all those, packet
pairs have been given. The condition for an MDOstructures of category II not only sufficient but also
necessary for an MDO-polytype. Thus a complete list
P~, moreover PI into P2, and generally P, into P, + 1.
Such a symmetry operation will be called l-operation.
As has been shown (Dornberger-Schiff, in preparation), the existence of such an D-operation is for
structures of category II not only sufficient but also
necessary for an MDO-polytype. Thus a complete list
of all MDO-polytypes containing only one particular
kind of pair of equivalent, adjacent packets can be
obtained in the following way: Take note of all operations converting the first packet of the pair into the
second; such operations will be called the q)-operations of the packet pair. Let one after the other of
these ~b-operations be a total operation converting any
packet P, into its successor P, + 1, i.e. an ~-operation.
In this way all MDO-polytypes containing this particular kind of packet pair are obtained, in some cases
some or all of them more than once.
By applying this procedure to one of the non-equivalent packet pairs of a family after the other, using a
list of all kinds of packet pairs compatible with the
family (as compiled in our preceding paper for all
mon-, di- and trioctahedral kaolinites) a complete set
of all MDO-polytypes may be obtained.
* i.e. OD-structures consisting of polar OD-layers in
which all equivalent OD-layers lie with equivalent sides facing the same way.
there and in the following the sequence of characters of
one repeat unit is placed between vertical bars to indicate
a periodical polytype.
This term is used to include also the enantiomorphous
hand.
We shall proceed in this way to obtain a list of all
MDO-polytypes of monoctahedral kaolinites; we shall
however use the relations between these and the diand trioctahedral kaolinites to obtain the corresponding lists for the latter.
To characterize any polytype, the symbolism introduced in our preceding paper will be used throughout.
The deduction of a complete set of MDO-polytypes
may then be considerably simplified, by exploiting a
special property of the fl-operation of any MDO-polytype and this property will be explained by way of
some examples.
In Fig. 1 three monoctahedral MDO-polytypes are
presented schematically in projection along e0 together
with their symbols giving one repeat unit?, in which
the ~-operations are threefold screw operations. The
three polytypes differ only in the position of the screw
axis relative to the respective origin of the single packet
(which, as described in our preceding paper we assume
as on the hexagonal axis of the tetrahedral layer of the
respective packet). The orientation of P0 was chosen as
0' in all three cases. Irrespective of the position of the
screw axis, the orientation of the packet P] is the same
for the three polytypes, and the same holds for the
orientations of P2, and, generally, of any P, with a
given n. In any polytype symbol the orientation:~ of a
packet is given by a character i' or i". Therefore, disregarding for the moment the displacements v,,, + 1, the
symbols of the three polytypes contain necessarily the
same sequence of orientational characters: 0', 2', 4'.
Any packet is brought into the orientation of its successor by a clockwise 120~ rotation, i.e. by the point
symmetry operation corresponding to the fZ-operation; such an operation is called co-operation in the
different for the three polytypes. Common to them is
the fact that successive displacements are also related
by a clockwise 120~ rotation, i.e. by the co-operation.
A similar situation is shown in Fig. 2 for three
MDO-polytypes in which successive packets are
related by a glide reflection across a plane parallel to
100 2 1
u p'
v~
v~'2
%~=
I
c. i 3
<0)
vol = ~ * )
v23 = <~')
vz~ = O 0
I0,=o l
Pot
v,l~
vo~"
( ~)
v~a - < 2 >
Fig. I. Three monoctahedral MDO-polytypes with space
group P32 with their symbols given on top and with the displacements occurring in them underneath. As in the figures
of the preceding paper, the origin of any packet is on the
corner with the smaller acute angle of the triangle representing it.
OD-interpretation of kaolinite-type structures---II
iO,oO;I
10;0,;i io;o,,i
I
I
,
i
-q-r
i
Table 1. ~b- and co-operations occurring in kaolinite polytypes, and relations between orientational and displacement
characters, respectively, related by these operations. The
result of the operations (3)1 and (6)1 is obtained by reversing
the arrows in the indications under (3)- 1 and (6)- 1, respectively. Numerical characters k outside the range from 0-5
should be replaced by values k + 6 inside this range. For all
operations 9 ~ 9
(N) •
(2)1
(3)- ~
(6)- 1
j--*j + 3
e---* u
J---Li + 2
e---~ e
N---~C
vo~- 0 9
vol-42)
Fig. 2. Three monoctahedral MDO-polytypes with space
group Cc, similar to Fig. 1.
m..(.) . . .
j---*5 - - j
j---*j + 1
e---, u
l~---+u
+---._
vol , ~02
233
m.(.)
+.---~-. . . . .
m(.)...
j--~l-j
j-~3
e-----~lg
e----+u
e---* u
u---*e
+--.+
u---~e
+-.~ +
u---~e
+--.+
_
_
_ ____~
_
____~
_
.
u--*e
+-.-.+
.
t:
___~
m
-j
_
:..(.)m . . . . .
(.).m . . . .
(.)..m
aa and Co. Again the orientations of the packets P0 are
j---* 2 - j
j---* 4 - j
j---~ - j
the same and a similar statement holds for the packets
e---* e
e ---.-~e
e---* e
P1, and, generally, for all P. with a given n. Due to the
U -"-'~ U
U "-"* U
U ----~ U
+--,
_
+-..,
_
+--.,
_
fact that in the three polytypes the position of the
----~+
_--~+
_--~+
planes across which the glide reflection operates, relative to the origin of the packet of reference and/or the
glide components are different, the displacements
v.,.+ t are different in the different polytypes. Com- hedral packet if-Y, 0"-5", there are 12 different ~b-opmon to them is the fact that in any of these polytypes erations converting one particular orientation (e.g. 0')
v._ 1.. is related to v.,.+t by a reflection across a into any one of them: six rotations, namely 1 (identity),
plane parallel to a3 and co, i.e. the o-operation which (2) 1, (3) l, (3)- 1, (6)1, (6)- i, and six reflections across the
also relates the orientations of successive packets. following planes: three planes perpendicular to al, a2,
Therefore the three polytypes shown in Fig. 2 are char- aa, respectively and three planes parallel to co and to
acterized by polytype symbols with the same sequence az, a2, a3, respectively. These reflections are to be
of orientational characters, namely 0', 0", 0', 0"..
denoted by [m..(.)...], [.m.(.)...], [..m(.)...],
(where, as in our preceding paper 7 stands for a packet [...(.)m..], [,..(.).m.] and [...(.)..m], respectively
which is the mirror image and thus the enantiomorph (see Table 1 and Fig. 3). These twelve operations are
of a packet with character i'); furthermore, successive the only ~-operations occurring in mon-, di- or trioctadisplacements v,_ ~,,, v.,,+~ indicated in these sym- hedral kaolinites.
bols are related also by such a reflection, i.e. by the cooperation.
MONOCTAHEDRAL
MDO-POLYTYPES
Generally speaking we may say: if there exists a 4~Due to the fact that the symmetry of the single
operation converting Po into P~, then the corresponding point operation which we shall call ~b-operation monoctahedral packet is P(1), there is, in this case, only
converts the orientation of P0 into the orientation of one Q-operation to any kind of packet pair (P,;P, + 1)
P 1. Furthermore, if a certain Q-operation is an f~-oper- converting P, into P , § 1. Therefore, to any kind of
ation, i.e. converts a n y P , into P , + ~, then the corre- packet pair there exists only one M D O - p o l ~ y p e : the
sponding 4~-operation is an co-operation, i.e, it 'con- polytype in which this q~-operation is the only f~-operverts' any orientational character > S, < into ation of the polytype, and the corresponding ~b-operation the only o~-operation.
>S,+ 1 < and any displacement character >v,,,+ 1 <
into > v,+ l,, + 2 < .
The symbols of the complete set of MDO-polytypes
aj
%
containing a certain kind of packet pair (P0; P~) may
thus be obtained by taking note of all q~-operations
converting >So < into >S~ < . By applying any one of
I
[.m.l,)...J
[...t.).mJ
L..t.)m...1
these ~b-operations in succession to >Vo~ < to obtain
> v 1 2 < , to > S ~ < to obtain > $ 2 < , to > v 1 2 < to
L'rnlJ:';3
obtain > Vz3 < a.s.o, the symbol of an MDO-polytype,
is obtained.For doing this in practice, it is convenient
to have a table of all qS-operations occurring in kaofro.. tJ...3
,r... r
linites and of the pairs of orientational and displacement characters connected by them (Table 1). Corre- Fig. 3. The six possible orientations of co-reflection planes
and their symbols.
sponding to the 12 different orientations of a monoctac'cu
23'3
L
K. DORNBERGER-SCHIFFand S. ~)UROVIr
234
Table 2. Monoctahedral MDO-polytypes
Packet
pail"
symbol
0',0'
MDO-polytype symbol
prelim.
0',
final (with enantiomorph)
basic
vectors
space
group
0',
a, b, Co
CI
a.b, co - b/3
C1
v(2) y(3)
0",
0' 0'
0'_
0'_
3" _
0"
0' 0'
_
(2)
0' +
(2)
3'_
+
0'00'
O'0
O'0
a, b, co - a/3
C1
O"0
0'40'
0' 4
2'0
4
4"0
(4)
2"0
0' _'~0'
0'2
O' 3'
0',3',
4'0
0',3',
al,a2,2co
P21
al,a2, 2%
P2x
0",3",
0' 3'
0'__3'+
_
O' 3'
_
+
0" + 3"_
6
(6)
0"_3"+
0' + 3'
0'+3' _
0'3'0
0'3'
03
0'+3'_
0'3'
03
at, a2.2%
P21
0"03"3
0'43'
0'43' 1
2'05'3
8
4"01"3
2"05"3
0'23'
0'3'
2 5
4'01'3
(8)
5
(5)
OD-interpretation of kaolinite-type structures--II
235
Table 2 (continued)
Packet
pair
symbol
0' 2'
MDO-polytype symbol
prelim.
basic
vectors
final (with enantiomorph)
0'2'4'
0'2'4'
al,a2,
3%
O" 4" 2"
space
group
v(2)
,,(3)
P32
9
7
(9)
(7)
10
8
(10)
(8)
11
9
(11)
(9)
12
10
(12)
(10)
(P30
if' 2" 4"
0' 4'
0' 4' 2'
0'
0' 2' 4'
2'
0' 4' 2'
0' 2' 4'
al, a2, 3%
0" 4" 2"
+
+
P32
(P31)
+
0" 2" 4"
0"
4'
+
0'
+
2'
+
0' 4' 2' [
+ + +I
0' 4' 2'
+
+
0' 2' 4'
+ +
+
0' 2' 4'+
++
al,a2,3r
0" 4" 2"
P32
(P3I)
0" + 2" + 4"+
0'
4'
0' 2'
0
O' 4' 2'
0'02'24'4
0' 4' 2'
I
0'02'24'41
al, a2,
0"4" 2"
042
3%
P32
(P30
]0"02"24"41
0' 4'
0
0' 2'
4
1O,o4,42,21
0'04'42' 2
0' 2' 4'
402
2'04'20'41
13
14"02"40"2[
03)
12"04"20"4 '
0' 4'
2
0' 2'
2
0' 4' 2'
204
0'2'4'
240
14'02'40'21
#0'2'
024
14
2"0"4"
042
4"0"2"
024
0' 4'
4
0'44'22' 0
2' 0' 4'
042
(14)
236
K. DORNBERGER-SCHIFF
and S. ]~)UROVIC
Packet
pair
symbol
O' 1~
prelim.
o,:,,)
Table 2 (continued)
MDO-polytypesymbol
final(withenantiomorph)
0'
**)
basic
vectors
1'
a1,~2,6Co
0" 9 5" , ~
space
group
v(2)
v(3)
P65
15
11
(P61)
(15) (11)
P65
16
(P6,)
(16) (12)
P6s
17
0", 1",1
O' 5 ~
o,:,,1
Or
0' 1"+1
1'
0r,5',l
0'_ 1'+1
ill, f12, 6r
0"+ 5"_ 1
o,+1
0'
5'
0'+5'_1
1'
0'+1' 1
+
0~
+
12
o+~_1
0'+ 1'_1
a l , a2, Oe 0
0"_5"+1
(P60
13
(17) (13)
0"+ 1"-1
0'
5'
0'
1i
0
0' 5'
0
Or 1'
4
o,_,,+1
0'01'11
o,+1
f~
0'05'51
0'41'51
i:o31)t
al,a2,6c 0
1~
I~
i4o3-~1
P65
(P60
18
(18) 04)
19
(19)
0' 5'
2
0' 1'
2
i4o3~tl
0'25'11
to;~)
20
14,,:,,)j
0' 5'
4
0'45'3~
I:,,o,,,tl
IV~ll
14
(20)
237
OD-interpretation of kaolinite-type structures--II
Table 2 (continued)
Packet
pair
symbol
prelim.
final (with enantiomorph)
basic
vectors
space
group
v(2)
v(3)
0', 0"
0",0",
0',0",
a, b, 2r 0
Clcl
21
15
O r 2 vt
0" 2 Jt
2',4",
0'4"
O' 4"
2" 4'
O'
O' _ 0" +
0"_0"+
O' 0"
+
O' + i f ' _
0" _ 0' +
O' 2"
0'_2"+
2'_4"+
4"
0'+4"_
2"_4'+
O' 4"
0'_4"+
4'_2"+
O' 2"
+
0' 2"
+ -
4"_2'+
0' 0"
0' 01'
3' 3 J'
Or'
0"
MDO-polytype symbol
22
a, b, 2Co
Clcl
23
16
24
+
0
0' 2"
4
0' 4"
2
0' 0"
2
0"0"
4
0' 2 It
0
0"4"
0
0 0
0'42"4
0'24"2
3 3
25
a,b, 2eo -a/3
Clcl
5' 1 t'
3'53"1
0'40"2
3" 3'
5 1
01 2 Jt
5 t 1 I,
0 2
5 1
0'04"4
5"51' 1
0'4"
4
0'2"
2
0' 4 r
1' 5 'J
4 0
5 1
0'22"0
1"55'1
O' 3"
O' Y'
0" Y'
0" 5"
0r,Stt,
O' 1"
0" 1"
a, b, 2Co -a/3
Clcl
28
18
29
3O
a, b, 2co
Ccll
31
32
2", 1',
17
27
3 3
5" 1'
3 3
0'20"4
26
19
K. DORNBERGER-SCH1FFand S. DUROVI6
238
Table 2 (contimled)
Packet
pair
symbol
M DO-polytype symbol
prelim,
0' 3"
0' 3"
0'
0"
3"
final (with enantiomorph)
3'+0"+
3"
3"
+
-
0'
o' 5"
/
1"
-
-I
o' 5"
0'
0"+5"+
t'
I
I
35
+1
i" 2'
i +
+
0'45"
0'45" 1
2'01" 3
0" 1"
2
0' l"
25
2" 1'
03
0'
0'3"
2 1
3'50"4
0'3"4
0'3"
45
3"0'
54
0'5"
0
0'05"5
5'4"
5 4
0"1"0
0"1"
0 1
5"4'
54
0'41" 3
1' 2" I
5 4I
0'25"3
20
2"
+
0'3"
0 3
2
33
34
0'3"
0 3
0' 1"
4
0" 5"
Ccll
+1
0'03"
23"
a, b, 2eo -b/3
5" 4'
+ +
1"
I
+
v(3)
+
I +
0"+1"+
1"
-
v(2)
5' 4"
+
o'
space
group
0'
+
0' 5"
basic
vectors
a,b, 2e0
Ccll
36
21
37
a , b , 2Co - b / 3
Ccll
38
22
39
40
l"2'
5 4
Thus the symbols of 72 non-equivalent monoctahedral MDO-polytypes were obtained with the help of
Table 1 from the symbols of the 72 non-equivalent
monoctahedral packet pairs given in our previous
paper, by taking the symbol of one pair after the other
and letting the ~b-operation be an co-operation. The
packet pairs and the MDO-polytypes resulting in this
way from them are listed in the first and second
column of Table 2, respectively. The symbols of the
MDO-polytypes consist of the characters referring to
one repeat unit placed between vertical bars.
These symbols have however to be regarded only as
preliminary symbols because some of them lead to an
orientation of the polytype which is not in keeping
with convention.
j,,
Any one of the 36 packet pairs of the type 0'
gives
rise to an MDO-polytype in which such a packet pair
j"0'
and its enantiomorph fl alternate, and which is
thus its own enantiomorph. For these MDO-polytypes
the final symbol (obtained from the preliminary symbol by an appropriate rotation, see below) is given in
column 3 in the same line as the corresponding preliminary symbol.
0r j'
Any of the remaining 36 packet pairs
gives rise
to an MDO-polytype containing orientational characters i' only; to any one of them there exists an enantiomorphous counterpart containing only orientational characters i". One of the symbols of such an
enantiomorphous couple is given in the third, the other
in the fourth column in the subsequent line, again with
the corresponding preliminary symbol in the same line.
OD-interpretation of kaolinite-type structures--II
239
and replaces any orientational character i' by (i + f)'
and j" by (j - f)", where f = + 2. Thus one and the
same polytype may be described by any of those six
symbols.
Chemically isomorphous polytypes
On the other hand, these six symbols may also be
The set of MDO-polytype.s derived above may refer- thought of as representing six isomorphous polytypes
to a fdmily of pol~types belonging to either of two of the same chemical composition, differing only by the
way Ma, Me and Mi are distributed over the three
classes:
octahedral sites. Therefore, the six symbols related as
a) One octahedral site y~ol is empty, the other two, described will also be called 'isomorphous' in the foly~t) and y~2) are occupied by different cations Me and lowing.
Mi ('chemically dioctahedral' class).
Note again that a polytype symbol isomorphous to
b) The three octahedrai sites ym), ym, y~z) are occu- a particular symbol may or may not denote a polytype
pied by different cations Ma, Me, Mi ('chemically trioc- equivalent to the polytype denoted by the original
tahedral' class).
symbol.
Thus the sets of isomorphous symbols have the folThe origin of the octahedral layer is placed in both
lowing form:
classes in the site ym~.
In class a) (chemically dioctahedral class) there are
i' j' ... , (i + 2 ) ' ( j + 2)' "" lI(i + 4)' (j + 4)'
two possibilities for the description of a given polytype: either Me or Mi may be thought of as denoting
the chemical symbol of a given element occupying one
i"j"o~fl" . . , ( i + 2 ) ' ( j + 2 ) " ~ / / ' " '
of the octahedral sites. Any change from one description
to the other may also be thought of as an interchange
of the positions y~ll and y~2), and this corresponds to
(i + 4)"c~(j + 4)"~ ...
a change of any orientational character i' into i" and
vice versa. The figures i of the orientational characters
and the displacement characters remain unchanged o r
because the origins of the OD-layers are not affected.
Thus one and the same polytype may be described by
i' j"
(i + 2)' (j - 2)"
(i + 4 ) ' ( j - 4 ) " 1
either of the two polytype symbols.
~'
~
~'
On the other hand, these two symbols may also be
thought of as representing two isomorphous polytypes
i"j' , (i+2)" (j--Z)'fi, (i+4)" (j-4)'/~
of the same chemical composition, differing only by
an interchange of Me and Mi. For this reason two
polytype symbols related as described, will also be
Some of these symbols may denote the same polycalled 'isomorphous' in the following.
type or enantiomorphous polytypes. A subset of a set
Note that a polytype symbol isomorphous to a given
of isomorphous symbols containing only non-equivalsymbol may or may not denote the polytype enantioent symbols will be called 'isomorphous set'. Some isomorphous to the polytype denoted by the given symmorphous sets contain 6 symbols, other sets 3, 2 or
only one. Due to the fact that isomorphous symbols
bol. Thus e.g. 0" 0 is isomorphous to 0' 0 and
may be thought of as describing the same polytype, the
denotes the polytype enantiomorphous to it. Similarly basic vectors and space group of polytypes denoted by
isomorphous symbols must necessarily be the same.
0"0'
0'0"
0 0 is isomorphous to
0 0 and both sym- Therefore it is convenient to group the 72 non-equivalbols denote the same polytype (starting off with a dll- ent polytype symbols into the isomorphous sets, take
ferent packet) which is its own enantiomorph. On the one of the symbols to represent it and orient it so that
its basic vectors are in keeping with convention.
other hand 0" 2 is isomorphous to 0"2 but not the
The symbols thus obtained are the final symbols
listed in the third column of Table 2; symbols belonging to the same isomorphous set are listed in succesenantiomorph to it; the enantiomorph being 0" 4 .
sion and placed between horizontal lines. RepresentaIn class b) (chemically trioctahedral class) there are tive symbols are numbered 1-22 in the last column
six possibilities to describe a given polytype, corre- [heading v(3)]. Within any set, those symbols which,
sponding to the six possibilities to identify the three from the point of view of chemically dioctahedral polydifferent cations with Ma, Me, Mi, respectively. Any types are to be considered as isomorphous, are listed
change from one description to another may be in succession, and the representatives of these subsets
thought of as a cyclic interchange of the positions ytm, (containing 2 or only 1 symbol and divided from other
y~l~, y~2~or as an interchange of two of these positions, such subsets--if necessary--by broken horizontal
or as a combination of these. A cyclic interchange lines) are numbered 1-40 in the last but one column
leaves again the displacement characters unchanged under the heading v(2). The enantiomorphs of the
The arrangement of the polytype symbols has been
chosen so as to reveal relations which exist between certain polytypes. These relations will now be discussed.
o
240
K. DORNBERGER-SCHIFF a n d S. I~UROVIC'
listed symbols--if different--are given next to their
counterparts in column 4.
The enantiomorphs of the symbols belonging to an
isomorphous set are isomorphs of one another. They
may or may not be isomorphous to their counterparts.
If they are not, the enantiomorph of the symbol with
reference number v(3) or v(2) is marked by the same
number in brackets in the respective column.
gruent MDO-polytypes. If only one of the kinds of
cations (or a void) is considered as fixed to the origin
of the octahedral layer, there are 40 non-equivalent
and 55 non-congruent MDO-polytypes, if the distribution of cations to the octahedral sites is considered as
arbitrary, there are 22 non-equivalent and 32 non-congruent MDO-polytypes.
DIOCTAHEDRAL MDO-POLYTYPES
Selection of
orientation
representative
polytypes
and
their
The principles applied for the selection of the representatives will be discussed separately for those polytypes in which co is the identity, a rotation and a reflection.
If co is the identity, the polytype symbol with
>v01 < = *, - or 0 is selected as representative, f~ is
a translation and the polytype is triclinic. Corresponding to common usage for the description of the mineral
kaolinite, we refer these polytypes to basic vectors a =
- a ~ and b = a 2 , - al and denote their space group
C1. The basic vector e is chosen in such a way that c
is equal to Co, or Co - a/3 or Co - b/3.
l f co is a rotational operation, ~ is a screw operation
and the space group of the resulting polytype is as follows:
o-operation
(2) 1 (3) 1 (3) -1 (6) 1 (6) -1
space group
P21 P31 P32 P61 P6s
If co is a rotation of order N, then the basic vectors are
al, a2 and e = Neo. Of the polytypes with space group
P21', i.e. polytype symbols 0' 3' fl , those with e = ,, and 0 are selected as representatives. Of the polytypes
with space groups P3E and P65, i.e. polytype symbols
0' 2' 4'
0' l'
respectively, those with
fl ~ and
c~ f l!\ '
= . , - , + and 0 are selected.
I f co is a reflection, the polytype is monoclinic and
its symbol is of the type
0'i"
~ fl . Prelimiimry polytype
symbols with i = 0 and i = 3 and ~ = . , - , 0 and 2
were selected as representatives. These polytypes will
be referred to basic vectors a = - a 3 and b = a2 - al.
Those with 0" have the space group Clcl (second setting), for those with 3" the unique monoclinic axis is
parallel to the basic vector a. We do not, however, propose to interchange the names of basic vectors, but introduce instead a "third setting" of monoclinic space
groups, the space group being in these cases Ccl 1. The
polytype symbols of some representatives had to be
converted corresponding to a rotation by 180 ~ around
Co in order to avoid an acute monoclinic angle and to
make this angle as close to 90 ~ as possible (reduced
cell). As a result, all final polytype symbols are referred
to basic vectors a, b, c, where c is equal to 2c 0, 2c0 - a/3
or 2c0 - b / 3 .
Thus, provided that the distribution of cations to the
octahedral sites is considered as fixed, there are 72
non-equivalent MDO-polytypes and 108 non-con-
A list of all dioctahedral MDO-polytypes could be
obtained in a way similar to the procedure used for the
monoctahedral MDO-polytypes. This would however
be more complicated, due to the fact that in the dioctahedral case there are two ~b-operations to any packet
pair. Besides, proceeding in this way, the homomorphism between the set of monoctahedral and the set of
dioctahedral MDO-polytypes which is of interest too,
would not be revealed. Using the results of the previous chapter, we shall discuss this homomorphism
and use it for a simple derivation of the set of dioctahedral MDO-polytypes.
To any monoctahedral MDO-polytype there corresponds exactly one dioctahedral MDO-polytype which
results from it, if y(l) and y(21 are filled by the same
kind of cation, but to some dioctahedral MDO-polytypes there is more than one monoctahedral M D O polytype from which it may be obtained in this way.
The symmetry of the single packet is Clm(1), all
packets are congruent and figures without dashes are
used as orientational characters. Since any dioctahedral polytype is obtainable in the way described from a
monoctahedral MDO-polytype, a list of the symbols of
all non-equivalent dioctahedral MDO-polytypes may
be obtained from the list of non-equivalent monoctahedral MDO-polytypes by leaving out all dashes and
double dashes in the polytype symbols. Only those
MDO-polytypes of Table 2 bearing a reference
number v(2) need be taken because an interchange of
the two cations of the same kind would obviously not
give anything new. In Table 3 the resulting non-equivalent dioctahedral MDO-polytypes (and in brackets
their enantiomorphs) are listed and to any of them the
number(s) v(2) of the monoctahedral MDO-polytype(s)
to which it corresponds is given. The space group and
basic vectors given in Table 3 may also be easily
obtained from the corresponding information of Table
2.
The 40 monoctahedral MDO-polytypes with reference number v(2) (without brackets) lead to 36 nonequivalent dioctahedral MDO-polytypes. Four of them,
namely 0, , 00 , 0 , 3 , , 0033 , result from two nonequivalent monoctahedral polytypes, respectively. They
are the polytypes with a mirror plane. The space group
of any of these is generated by the symmetry operations
of the corresponding monoctahedral MDO-polytypes
and this mirror reflection. The space group and basic
vectors of any of the remaining 32 non-equivalent
MDO-polytypes are the same as those of the corresponding monoctahedral MDO-polytypes. The space
241
OD-interpretation of kaolinite-type structures--II
Table 3. Dioctahedral MDO-polytypes. The 36 non- equivalent MDO-polytypes (with 16 enantiomorphs in brackets after
their counterparts); the reference numbers v(2)refer to corresponding monoctahedral polytypes listed under these numbers
in Table 2. Zvyagin's symbols and numbers of polytypes are also given
MDO~
polytype
symbol
v(2)
basic
vectors
space
group
Zvyagin's
symbol
number
0,l
1, 21
a,b,eo
Clml
0.3 r3 0.3
XI 1
00
3, 26
a,b,e0 - a/3
C1ml
46ro 0.6
VI 1
0, 3,
5, 31
a,b,2eo
Ccm2
0.3"%0.6z30"3
III 2
J00331
7,36
a,b,2eo
Ccm21
0.3"Co0.6Zo0.s
XII1
0_ I
2
a,b,eo -b/3
C1
0.a~t0.3
IV 2
(13_1)
(2)
0.6"~20.6
IV1
a2z-a2
I1
0.,z+0.,
12
alz2a,%0.t
114
0.~'r.,a2zl0. 5
II5
o-327_0.6T+ 0.3
V3
0.3"~+0.6"c_0.3
V2
P32
0.az10.l...
IX3
(P30
0.3z50.5...
IX2
P32
0.3%0.,...
IV6
(P3t)
0.sZlOs...
IV5
P32
0.s%0.1...
VIII3
1201
(1401)
4(+)
a,b,co -a/3
l
C1
(4)
10_3+1
6
(10+3_1)
(6)
120531
8
(140131)
(8)
]0,2,4,1
9
(10,4,2,1)
(9)
10_2_4_1
10
(10+4+2+1)
(10)
10+2+4+1
11
al,a2,2eo
a,,a2,2eo
al,a2,3Co
al,a2,3c0
al,az,3eo
P21
P21
(1~
vI.
0022441
12
([0044221)
(12)
240
024
13
(]402402[)
(13)
1400224[
14
( 2 0044)2
a l,a 2,3%
al,a2,3e o
al,a2,3c o
(14)
10,1,) I
15
(t0,5,1l)
(15)
al,a2,6Co
P32
o"3zo61...
VI 4
(1~
0.aZo0.s...
VI3
P32
0.sz+0.1...
18
(P31)
0.3z-0.2...
17
P32
0.3z-0.1...
19
(P31)
0.3r+0.5...
I6
P6s
0.3%42...
1114
(P6,)
0.3z44,,...
1113
K. DORNBERGER-SCHIFI~ and S. I~)UROVI("
242
Table 3 (continued)
MDC~
polytype
symbol
l~ '+)I
(!~ 31)
v(2)
16
basic
vectors
aDa2,6eo
(t6)
17
aDa2,6%
(17)
I~
(1~
Iu,)l
(I4o311)
1405,)1
(lolsll)
l,4,t
Ioo+l
t 4+l
14=+1
1'3131
13/,I
I'/,t
18
aDa2,6Co
08)
19
al,a2,6%
(19)
20
aDa2,6r
(20)
space
group
symbol
Zvyagin's
P65
0- 3 T 6 0- 2 . . .
lI 9
(P6~)
0 " 3 "s O ' 4 " ""
II 6
P65
O-3 "C4 0- 2 . . .
lI8
{P61)
0" 3 ~C2 0 " 4 " " "
II 7
P6~
0" 3 T O 0- 2 . . .
X3
(P60
G3TOG 4 . . .
X2
P6~
0"3T_
G 2 , ..
VII 3
(P61)
0- 3 T ~ 0- 4 . . .
VII 2
P65
O - 3 T + 0- 2 . . .
V6
(P61)
0" 3 ~
V5
number
0" 4 . , .
22
a,b,2co
C 1c 1
G 17" 5 G 5 ~ " 1 O" 1
IX 1
23
a,b,2co
C lc l
G 3 "f5 O'3 T I G 3
IV 3
24
a,b.2co
Clcl
(7 1 T 3 0 - 5 773 0-1
IV 4
25
a,b,2co
C Ic i
G l'C 1 O'5 7"5 G 1
VIII 1
27(+ +)
a,b,2eo - a / 3
Clcl
GIT~-
G 1
I4
28
a,b,2co - a/3
CI c 1
0 - 3 2 - + 0 - 3 7 - - 0- 3
I3
29
a,b,2eo - a/3
CI c I
GITOG5TOG 1
VI 2
30
a,b,2c o - a / 3
Clcl
fll T-
I5
32
a,b,2eo
Cc I 1
O- 1 T 2 O-2 "f 1 O" 1
III
33
a,b,2co - b / 3
Cell
G3T4G6~SG 3
II 3
34 ( + + + )
a,b,2c o
Cc 11
G l 7 7 6 G 2 T 3 O- 1
112
35
a,b,2co - b / 3
Ccll
O-4T 10-5 T2 0-4
II
37
a,b,2co
&l 1
0-1 T -
3O
54
38
a,b,2co - b / 3
Ccll
0"377+ 0"6T+
15544 ]
39
a,b,2co - b/3
Cc I 1
0 " 1 T 0 G 2 Z 0 0-1
1524
40
a,b,2co - b / 3
Cell
G5T-
I
ls+4+l
I+LI
12o'1
-
b/3
G5T-
GsT
+ GI
~2 T - 0-1
0-4 T"
0"3
0"5
1
1
VI
V4
X1
VII I
OD-interpretation of kaolinite-type structures--II
group of 16 of these 32 polytypes (originating from
monoclinic monoctahedral polytypes) contain glide
operations. Those of the other 16 MDO-polytypes
contain only screw and/or translational operations. To
any of them there exists an enantiomorphous counterpart corresponding to a monoctahedral MDO-polytype with a v(2) in brackets. In Table 3 the symbols introduced by Zvyagin for these 52 non-congruent
MDO-polytypes are also given.
dashes and double dashes disregarded. The resulting
12 non-equivalent MDO-polytypes (and in brackets
their enantiomorphs) are listed in Table 4.
Amongst these non-equivalent MDO-polytypes
there are 6, any one of which corresponds to a single
monoctahedral MDO-polytype. In 4 of these polytypes the space group is the same as that of the monoceee
tahedral polytype and to two of them: I 0 241 and
eu)
0 1
TRIOCTAHEDRAL MDO-POLYTYPES
In a way similar to that used for the derivation of
the dioctahedral MDO-polytypes, the symbols of all
non-equivalent trioctahedral MDO-polytypes may be
obtained from the 22 monoctahedral MDO-polytypes
with reference number v(3) (without brackets). Even
and uneven orientational characters have to be replaced by the characters e and u, respectively, and the
243
there exists an enantiomorphous counterpart
(v(3) in brackets); in other two: e _ u _
and e _ e +
the space groups R3c 1 andP3 lc, respectively, result from
the symmetry operation of the space group of the monoctahedral MDO-polytype and the threefold rotation
which is common to trioctahedral packets displaced by
< + > or < - >. In the other 6 non-equivalent triocta-
Table 4. Trioctahedral MDO-polytypes. The 12 non-equivalent MDO-polytypes (with 4 enantiomorphs in brackets after
their counterparts); the reference numbers v(3) refer to corresponding monoctahedral polytypes listed under these numbers
in Table 2. Zvyagiffs symbols of polytypes are also given
MDOpolytype
symbol
v(3)
basic
vectors
space
group
Zvyagin's
symbol
e01
3,17
a,b,co - a/3
C l m1
~r6Zoor6
[U5Ul]
18
a,b,2c0 - a / 3
CIcl
o'3z + o'a'r_ o'3
e e e
024
10
al,a2,3c o
P32
asZoasZ+ a3r- a3
( e e0e4) 2
(10)
(P31)
0"3TO0"3T- 0"3~+ O"3
leon3[
6,21
a,b,2co
Ccm21
o'3ro o'6"c0a 3
[a5e4l
22
a,b,2co -b/3
Ccll
0-3"g+ 0-6"[+ o-3
le0Ulll
14
al,a2,6Co
P6s
tr3%a6"c- cr3"c-o'6ro~3r + tr6"c+~73
(]e0u5ll)
(14)
(P61)
a3"c00-61+0-3"c+0-6"~00"3z _ 06T_ o-3
R3
o-6"c2o-6
le+l
(2),(8),9
(]u+l)
2,8,(9)
le_e+[
([e u+l )
16
al,a2,3Co
0"3~'1r
at,a2,2c 0
P31c
5,12,(13)
O3ZxO3Zscr3
0"3~'40-6 l'10"3
r~ I
le,u,I
4,11,(11),19
al,a2,2co
P63cm
o3"c6ofi~3 a3
K. DORNBEROER-SCH1FFand S. I~)UROVI~
244
hedral M D O - p o l y t y p e s the space group may be
t h o u g h t of as generated by the c o m b i n a t i o n of the
symmetry operations of the different monoctahedral
M D O - p o l y t y p e s corresponding to them; to two of
them, namely e + and e + u _
there exists a n enantio-
m o r p h o u s counterpart. At the same time the basic vectors are obtained from the corresponding information
o n the monoctahedral polytypes. The r h o m b o h e d r a l
polytyi~es are presented in the standard, i.e. obverse
orientation.
In Table 4 the symbols introduced by Zvyagin for
the 16 non-congruent trioctahedral M D O - p o l y t y p e s
are also given.*)
CORRESPONDING ZVYAGIN SYMBOLS
In order to obtain the rules governing the relation
between Zvyagin's symbols for d i - a n d trioctehedral
polytypes a n d our symbols, the following facts have to
be kept in mind: Zvyagin (1964, 1967) considers the
sequence of sheets and atomic planes in the direction
opposite to the direction we considered, i.e. the coordination octahedra are followed by the sheet of coordination tetrahedra with which they form one of our
packets. Accordingly the c o m p o n e n t perpendicular to
the basic vectors al, a2 or a, b, of his vector e (called
ezv in the following) points in the direction opposite to
c0. Both coordinate systems are right-handed, if
exactly one of the basic vectors az~, bz~. is opposite to
the corresponding vector a or b. We shall take bz~. = b
a n d az~ = - a .
Zvyagin's characters a denote the displacement of a
tetrahedral layer relative to the octahedral layer which
belongs to the same packet; thus Zvyagin's vector a is
opposite to our vector S,. Similar to our characters
> S, < , the a~ are used to characterize the orientation
of a packet. C o m p a r i s o n of the c o m p o n e n t s of cr~ with
the c o m p o n e n t s of < i > shows that a~ = - < i > for
i = 1,2,...5 a n d a6 = - < 0 > . Therefore the sequence
of indices i of Zvyagin's characters a~ in symbols for
dioctahedral polytypes corresponds to the sequence of
our orientational characters in reverse order, with i =
6 replaced by i = O.
Zvyagin's characters z indicate the displacement of
a n octahedral layer relative to the tetrahedral layer
belonging to the packet which he considers the preceding packet. Thus
Tfi = --(S3n + 1,3n+2 "t- $3n+2,3,+3) = --Vn,n+ 1 -t- S n.
Taking the relations between the basic vectors (a,b,
a n d azt, bzv) a n d the definitions of the orientational
a n d displacement characters i a n d c~ in terms of their
c o m p o n e n t s with respect to a,b (see Table 3 of our preceding paper) a n d the definition of z~ in terms of its
c o m p o n e n t s with respect to az~, bzv into account, the
relations between our characters i a n d ~, a n d Zvyagin's
characters % indicated in Table 5 result.
In order to avoid a unit cell with acute monoclinic
angle, the symbol resulting from a conversion according to Table 5 has to be further converted corresponding to a turn of the polytype by 180 ~ a r o u n d Co in those
cases, in which the c o m p o n e n t s Xo, Y0 of Czv in the directions azv a n d bz~ respectively, are Xo = 0 a n d Y0 =
- 1 / 3 , i.e. if e = Co - b/3 or e = 2% - b/3. Such a n
angle would result from the conversion according to
Table 5, because these values of Xo a n d Yo correspond
to an obtuse monoclinic angle e between bzv a n d ezv
a n d thus to an acute angle between b = bz~ a n d e = ez~. For these polytypes the relations between our symbols and Zvyagin's symbols as listed in Tables 3 a n d
4 correspond indeed to a conversion according to
Table 5 followed by a 180 ~ turn a r o u n d e0. For similar
reasons, the conversion according to Table 5 applies
directly for all polytypes with x0 = - 1/3, Yo = 0, i.e.
with c = e0 - a/3 or with e = 2c0 - a/3. F o r all other
polytypes, i.e. polytypes for which e is in the direction
of co, the choice of symbol is not unequivocal, a n d it
has in some cases to be chosen so that the conversion
according to Table 5 applies directly, in other cases so
that it is followed by a turn by a multiple of 60 ~ a r o u n d
co. This turn has to be 180 ~ (or zero) in those cases
which were referred to basic vectors a, b.
After this conversion the symbol may be given starting off with the same or with a n o t h e r orientational
character.
Thus e.g. Zvyagin's symbol O-IT6O'ZT3O"I would be
converted into
_
2
_ by reversal of sequence a n d ap-
plication of Table 5 (the pair of characters r3a~ leading
1
to
, a.s.o.); this has to be turned by 180 ~ a n d leads
;
+ +"
the symbol given in Table 3 is 5 4
+
For an illustration of this a n d a n o t h e r dioctahedral
M D O - p o l y t y p e see Fig. 4.
to
* Schematic drawings of all di and trioctahedral MDOpolytypes can be obtained from the second author on
request.
1
4+5
Table 5. Relations between the orientational characters > S , < = i, the displacement characters >v,., + 1 < = a used in
this paper, and the the subscripts fi of Zvyagin's characters r~.
In the dioctahedral case i stands for any numerical value, e or u for a specified even or uneven value, respectively.
In the trioctahedral case any even or uneven orientational character has to be replaced by e or u, respectively, after
conversion.
i
i
e
e
u
u
i
e
e
[/
0
+
--
+
--
i
e+2
e--2
:~
i
e+2
e--2
u--2
u+2
*
+
--
u
u+2
--
u
u--2
+
OD-interpretation of kaolinite-type structtkres--II
p,a,
^.,
.
i
x,,,,.*
."l!~
z llN
i
!
i
i
245
.............
::::::::::::::::::::::::::L'-J
P, 2~.1
i
at
i
'
.
C lc1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C ell
a, b, 2co - a/3
a, b, 2Co - b/3
Oickife
Nacrife
Fig. 4. Two examples ofdioctahedral MDO-polytypes: dickite and nacrite. One unit cell is shown in each
case. the projected top edges of the cell are drawn in full lines, the bottom edges in weak lines. The origin
is indicated by a small circle. Dot-dash lines represent interleaved c and n glide planes in orthographic
projection.
As a second example we may take Zvyagin's symbol
o'3r0q3r+o-3z_ 0"3. Reversal of the sequence and application of Table 5 leads to
RRU
5 1 3 and a conversion
corresponding to a turn by 60 ~ to the symbol listed in
eee
our Table 4: 0 2 4 ' For an illustration of this and
two other trioctahedral MDO-polytypes see Fig. 5. It
i s interesting to note that this symbol in Zvyagin's
notation does not denote a dioctahedral regular polytype. This is due to the fact that the rule that only orcharacters a3 and tr6 should occur in symbols for trioctahedral regular polytypes, which is meant to ensure
that the trioctahedral polytypes are denoted by not
more than by one symbol, excludes also symbols of
polytypes with ~o-operations (3) 1, (3)- 1, (6)1 and (6)- 1.
On the other hand unequivocal representation of a
trioctahedral regular polytype is not ensured by this
rule. This is connected with the fact that conversion of
our symbol into Zvyagin's symbol may be done by
identifying our characters e and u with 6, 3 or 4, 1 or
2, 5, respectively, with a subsequent turn to obtain acharacters if6 and cr3. In most but not all cases the
resulting Zvyagin's symbols differ only in a trivial way.
But from our symbol
eOu3
the three symbols
0"3"C00"6~'00"3~ 0"3~'+O'627_0"3 a n d o'3"c_ty6~+o-3 may be
obtained using Table 5, and these were actually given
originally by Zvyagin (1964, 1967).
CONCLUSIONS
The application of OD-theory, and especially the
definition of MDO-polytypes which is a part of it, was
used to single out polytypes of m o n - di- and trioctahedral kaolinites which have (in the case of di- and
trioctahedral kaolinites) been called regular or simple
polytypes by other authors.
The symbolism introduced in our preceding paper
turns out to have the following desirable features:
(i) Any symbol of a polytype determines unequivocally the polytype.
(ii) It is easy to recognize whether a given polytype
symbol is the symbol of an MDO-polytype or
not. For any MDO-polytype the total symmetry
operation and thus the space group may easily
be recognized.
(iii) Although different symbols may be used to describe the same polytype, their mutual relation
is so evident that this is no serious drawback.
(iv) The equivalence, if any, of packet pairs or triples
occurring in different polytypes may easily be
recognized.
(v) To any monoctahedral polytype there exist corresponding dioctahedral polytypes which result,
if one of the three octahedral sites is singled out
and the other two are thought of as being occupied by the same kind of cation; this may
obviously be done in three different ways, the
resulting dioctahedral polytypes may or may
not be described by the same symbol. This cor-
V
P31c
ct~j ~2,2c0
o.1~ otz ~ 3c o
P 3~
o~, az, 2c o
P 6~cm
Cran sledfi/e
Cronstedtite
Ame sire,Cron#ed/#e
Fig. 5. Three examples of trioctahedral MDO-polytypes: cronstedtite and amesite.
246
K. DORNBERGER-SCHIFFand S. I)L;ROVI~
respondence may, in any particular case be
deduced directly from the symbol.
(vi) Similar to (v), there exists to any monoctahedral
and to any dioctahedral polytype a corresponding trioctahedral polytype which results, if the
three octahedral sites are thought of as being
occupied by the same kind of cation. This correspondence may also be directly deduced from
the symbol in any particular case.
Feature (iv) may be of importance for the understanding and perhaps also for the recognition of disordered 'polytypes'. If a certain kind of packet pair is
energetically favourable, and if there is more than one
kind of MDO-polytype containing this kind of packet
pair, then we have to expect that under appropriate
conditions of growth not only these MDO-polytypes
will occur in nature but also disordered samples containing bigger or smaller regions corresponding to one
or more of these MDO-polytypes. This is still more
likely if there are different kinds of MDO-polytypes
which contain not only the same kind of packet pair
but also the same kind of packet triple.
Actually, in monoctahedral kaolinites there is only
one MDO-polytype to any kind of packet pair, but in
disordered kaolinites there are several packet pairs to
which there are two MDO-polytypes which differ in
their packet triples. The MDO-polytypes 0+f~ and
MDO-polytype in the sense discussed above. Then a
polytype with composition between two ideal compositions will probably contain regions corresponding to
these two MDO-polytypes. Deviations from these
expectations would be of special interest to crystallochemists.
The symbols used are rather similar to the symbols
introduced by Zvyagin, as far as dioctahedral polytypes are concerned, and the latter have had a decisive
influence on the development of our symbolism. Zvyagin's symbols have the advantage that the relative position of the cations of adjacent tetrahedral and octahedral sheets is directly expressed by his characters ai
and ~. For dioctahedral MDO-polytypes his symbols
have also the features (i), (ii), (iii) and (iv). For trioctahedral polytypes Zvyagin's symbols have, however
only feature (i), whereas (ii), Off) and (iv) do not hold
in all cases. The relations between trioctahedral and
dioctahedral MDO-polytypes (feature (vi)) are not evident from their Zvyagin's symbols in all cases, either.
It seems to us that the principle underlying our symbolism, i.e. to characterize any polytype of a family by
a string of characters denoting the kind, enantiomorphous hand, absolute orientation and relative displacement of the constituting packets (as far as necessary)
could and should be developed and generalized so as
to be applicable to all other families of polytypes, including other classes of phyllosilicates.
I
0+0
are an example, both with packet pairs equi-
valent to O 0 but with different packet triples 9 0 0
+
+ +
and
0
0
+
0
-
, respectively. In the trioctahedral case
there exist MDO-polytypes, e.g. e 4 u5
and e0u 11 '
which contain even packet triples of the same kind and
differ only in their packet quadruples.
Features (v) and (vi) may also be of importance, as
we may see from the following hypothetical example:
Let us assume that there exists a dioctahedral M D O polytype (chemically trioctahedral) with Me occupying
the sites y(1) and y(2), and Ma in y~0), and a trioctahedral MDO-polytype in which all octahedral sites are
occupied by cations Me. If Me and Ma are rather similar(similar ionic radii) we might expect the trioctahedral MDO-polytype to correspond to the dioctahedral
Acknowledgement~The authors wish to express their gratitude to Drs. B. B. Zvyagin and V. A. Drits (Moscow) who
studied earlier versions of this paper and whose critical
remarks contributed considerably to their improvement;
furthermore to Professor S. W. Bailey (Madison, Wis.), who
helped us to clear up some terminological ambiguities and
drew our attention to some mistakes, in text as well as in
figures and tables. Our thanks are also due to Mrs. L.
Ktihnle (Berlin), Mrs. M. Fellnerov~ and Mr. P. Fejdi (Bratislava) for their technical assistance.
REFERENCES
Dornberger-Schiff, K. and [)urovi6, S. (1975) OD-interpreration of kaolinite-type structures--l. Symmetry of kaolinite packets and their stacking possibilities: Clays and
Clay Minerals 23, 219-229.
Zvyagin, B. B. (1964) Elektronografiya i strukturnaya kristallografiya glinistykh mineralov (Electron diffraction
analysis of clay minerals): Izdat. Nauka, Moskva.
Zvyagin, B. B. (1967) Electron Diffraction Analysis of
Clay Mineral's. Plenum Press, New York.