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Algebraic Solution of the Coincidence Problem in Two and Three Dimensions

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Abstract

The coincidence problem is analyzed in an illustrative fashion for some lattices and modules in two and three dimensions which are important for crystals and quasicrystals. We give a complete description of the groups of coincidence rotations with their associated indices and encapsulate their statistics by means of generating functions.
A lg eb ra ic S olu t i o n of th e Co inc id enc e P ro b l e m
in Tw o an d T hr e e Dim en sio n s
Michael Baakeal and Peter A. B. Pleasantsb
a Institut für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany
b Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia
Z. Naturforsch. 50a, 711-717 (1995); received March 27, 1995
Dedicated to Wolfram Prandl on the occasion of his 60th birthday
The coincidence problem is analyzed in an illustrative fashion for some lattices and modules in
two and three dimensions which are important for crystals and quasicrystals. We give a complete
description of the groups of coincidence rotations with their associated indices and encapsulate
their statistics by means of generating functions.
Introduction
The concept of coincidence site lattice (CSL) arises
in the crystallography of grain and twin boundaries
[1]. Different domains of a crystal across a boundary
are related by having a sublattice (of full rank) in com
mon. This is the CSL. It can be viewed as the inter
section of a lattice with a rotated copy of itself, where
the points in common form a sublattice of finite index
(and we will not discuss any situation other than that).
So far CSLs have been investigated only for true lat
tices, for example cubic or hexagonal crystals [2,3,4].
With the advent of quasicrystals infinitely many new
cases arise, since quasicrystals also have grain bound
aries and one would like to know the coincidence site
quasilattices [5, 6] and, more specifically, which of
them can form twins (or multiple twins, where the
angle between the grains is a rational multiple of 7r).
Added impetus is given by the experimental progress
made in recent years [7, 8], in particular on the recon
struction of fine structure details. So an extension of
the CSL analysis to all discrete structures is desirable.
We will attack this problem in two steps.
In this paper, we shall present an algebraic ap
proach to the coincidence problem for planar struc
tures with 4-, 6- and 12-fold rotation symmetry and
for 3D structures with cubic and icosahedral sym
metry. This extends existing work [5, 9, 6] and puts
1 Heisenberg-fellow
Reprint requests to Dr. M. Baake.
it in a simpler, more unified setting. The results on
non-crystalline symmetries are needed for quasicrys-
talline T-phases which are quasiperiodic in a plane
and periodically stacked in the third dimension and
for 3D quasicrystals with icosahedral symmetry. Both
require a two stage treatment: not only do we have to
find the coincidence isometries (the universal part of
the problem) but also the specific modifications of the
atomic surfaces (also called windows or acceptance
domains) that are needed to describe the set of co
inciding points. While we present the universal part
in detail, we only briefly comment on the acceptance
corrections.
In Part 1 of the article we set the scene by starting
with the coincidence problem for the square lattice
Z2. The set of coincidence transformations for Z
forms a group, the generators of which can be given
explicitly through their connection with Gaussian in
tegers [10]. Simultaneously, the so-called 27-factor or
coincidence index can be calculated for an arbitrary
CSL isometry. We then show how a completely anal
ogous procedure gives the results for the triangular
lattice A2. Though this is not new, the approach we
use here can be generalized to quasiperiodic planar
patterns with N-io\d symmetry [11]. We give explic
itly the results for twelvefold symmetry and discuss
their relation to the previous two cases.
Part 2 of the article deals with the 3D case. Again
we start with the known case, in this instance the cubic
lattices [12, 2, 3], but formulate the results in an al
gebraic way that allows an immediate generalization
to the coincidence problem for the 3D icosahedral
0932-0784 / 95 / 0800-0693 $ 06.00 © - Verlag der Zeitschrift für Naturforschung, D-72072 Tübingen
712 M. Baake and P. A. B. Pleasants Algebraic solution of the coincidence problem
modules of rank 6, the most important ones for the
description of icosahedral quasicrystals [10]. As 3D
rotations do not commute in general, the coincidence
rotations form a non-Abelian group whose structure
is more complicated than in the 2D case. Nevertheless
the main results are simple and are given below.
Part 1: The two-dimensional case
A lattice (or module) r may admit certain rotations
R (special point symmetries) which bring R r into
perfect coincidence with r . In addition to these one
can also find rotations such that R r n r is a sublattice
of r of finite index, the so-called coincidence site
lattice [5, 13, 2, 3], CSL. The index is called the
coincidence index (or S-factor) of the rotation R:
S(R) = [ r : ( r n R r ) ] . (1)
One has S(R) = E (R ~ l). The set of all such rotations
forms a group which we call S O C (r). In the planar
case this group is a subgroup of SO(2), so Abelian.
All this is illustrated in the three examples to follow.
The extension to reflections (and thus general point
isometries) is straightforward but not discussed here.
1.1. Coincidence rotations fo r the square lattice
The square lattice Z 2 consists of all integral linear
combinations of the two vectors e\ and e2. A rotated
copy R Z2, with
R = R(tp) = cos(</?)
sin(</?)
- sin(y)
cos (v?) <E SO(2, IR), (2)
results in a CSL of finite index if and only if both
cos(<£>) and sin(<p) are rational. This gives the well-
known relation between coincidence rotations and
primitive Pythagorean triples [13]. The group of co
incidence rotations is thus given by
SOC(Z2) = SO(2,Q ). (3)
To investigate this group, we notice that, with i =
>/- F , we can identify Z 2 with the ring of Gaussian
integers (the integers of the quadratic field Q(i)):
Z2 = Z[i] = {ra + ra | r a , n G Z} , (4)
The ring Z[i] has a finite group of units isomorphic
to C\ (namely, i and its powers) and has unique prime
factorization up to units [14].
In this setting, a rotation R(ip) G SOC(Z2) cor
responds to multiplication by a complex number
ei(p G Q(i). This number can be written as e1^ = a / ß
with a , ß G Z[i] coprime and of equal norm. As a
consequence of the unique factorization one can show
[11] that every coincidence rotation can be factorized
as
P= 1(4)
(5)
where np G Z (only finitely many of them ^ 0), £ is a
unit in Z[z] (a power of i), p runs through the rational
primes congruent to 1 (mod 4) and the cjp, tJp are the
(complex conjugate) Gaussian prime factors of p (i.e.,
ujpLüp = p). We thus have
SOC(Z2) = SO(2,Q) ~ C4 0 Z «0 (6)
with generators i for C4 and up/tJ p withp = 1 (4) for
the infinite cyclic groups.
The coincidence index U(R) is the norm of the
denominator of (5), that is,
s m = n p ]nr
P= 1(4)
(7)
In particular, it is 1 for the units (i.e. true symmetry
rotations) and p for the generator u p/cJp. The first few
generators with 27 > 1 are
4 + 3« 12 + 5 i 15 + 8z 21+20i
13 '
35 + 12z
37
17 '
40 + 9 i
41
29
etc.
These have been normalized to have denominator £
(a prime = 1 (mod 4)) and argument in (0, 7t/4). All
other CSL rotations are obtained by combinations, as
indicated in eq. (5). We have dealt here with rotation
coincidences only. For the easy extension to reflec
tions see [11].
It is convenient to summarize the possible coin
cidence indices and the number o f rotations with a
given index by means of a generating function. To do
so, let 4 /(m ) denote the number of CSL rotations of
index m. It turns out [11] that /(m ) is a multiplicative
713 M. Baake and P. A. B. Pleasants Algebraic solution of the coincidence problem
function (i.e., }{m\m2) - f(jni)f(m 2) for coprime
mi, 7712) so a Dirichlet series #(3) is an appropriate
generating function. We find f(p r) = 2 for a prime
power pr (p = 1 (mod 4), r > 1) and obtain
' m s
m= 1
= n , 2 2 \ TJ 1 + p~s
1 + + -=- + ••• = 1 -
Vs VZs J ±± 1 v~a
p=l(4) x y F 7 p=l(4) y
_ 2 t 2 | 2 | 2 [ 2 | 2
5s 13s 17s 25s 29s 37s
2 2 2 4 2
+ 41s + 53s + 61s + 65s + 73s +
This generating function is not only a succinct way of
representing the statistics of CSL indices, it is also a
powerful tool for determining their asymptotic prop
erties [11]. For example, we have used it to show that
the number of CSL rotations of 1? with index < X
is asymptotically AX/it. The possible CSL indices
are precisely the numbers m with all prime factors
= 1 (mod 4) and we have f(m ) = 2a, where a is the
number of distinct prime divisors of ra. Each CSL is
itself a square lattice, with the index as the area of its
fundamental domain.
1.2. Coincidence rotations for sixfold symmetry
Before treating twelvefold symmetry (the main aim
of Part 1 of this article) we look at sixfold symmetry.
The triangular (or hexagonal) lattice consists of all
integral linear combinations of the two vectors e\
and |( e i + -/3e2). It is (up to a scale factor y/2)
the root lattice A2 [15]. A rotated copy RA 2 with
R G SO(2) results in a CSL of finite index if and
only if cos(<p) G Q and sin(<£) 6 v^Q . This defines
SOC(A2) as a subgroup of SO(2, Q(v^)). To further
describe SOC(A2) we notice that A2 can be written
as
~^=A2 = {m + riQ I m ,n G Z j = Z[g] (8)
v2
with g = 1 + iV 3). The lattice A2/V2 is there
fore the ring of integers of the imaginary quadratic
field Q(y^3), the so-called Eisenstein (or Eisenstein-
Jacobi) integers [14]. It has a finite group of units iso
morphic to Ce (namely, g and its powers) and unique
prime factorization up to units.
A rotation R(<p) e SOC(A2) corresponds to multi
plication by a complex number et<p Q (\/-3)- That
number can be written as el(fi = a /ß with a, ß G Z[g]
coprime and of equal norm. As a consequence of the
unique factorization one can again show [11] that ev
ery coincidence rotation can be factorized, this time
as
n ( £
P= 1(3)
(9)
where np G Z (only finitely many of them 4 0), z
is a unit in J.[g] (a power of g), p runs through the
rational primes congruent to 1 (mod 3) and the u p, ujp
are the (complex conjugate) Eisenstein prime factors
of p (i.e., ujpüjp = p). We thus have
SOC(A2) = { R e SO(2) I cos(<p) G Q,
sin(t^) G V3Q } (10)
with generators g for Ce and uip/cJp with p = 1 (3)
for the infinite cyclic groups.
As in the previous example, the coincidence index
is 1 for the units and p for the other generators, so for
a rotation R(p) factorized as in (9), we have
E(R) = J} p\n» I .
P= 1 (3)
OD
The first three generators with S > 1, normalized
to have denominator E (a prime = 1 (mod 3)) and
argument in (0, 7t/6), are
5 + 3 q 8 + 70 16 + 5p
13 19
Finally, if 6 f(m ) denotes the number of CSL rota
tions of index m, /(m ) is multiplicative and one finds
the Dirichlet series generating function [11]
m= 1
f(m )
n
P=l(3)
1 + p~ s
m9 1 -p ~ s
2 2 222
T7+ 13s + 19s + 3P + 37s
22 4 2
49* + " + 79s + 9P + 97s
43*
The possible coincidence indices are precisely the
numbers m with all prime factors = 1 (mod 3) and
714 M. Baake and P. A. B. Pleasants Algebraic solution of the coincidence problem
we have /( m ) = 2a , where a is the number of distinct
prime divisors of m. Each CSL A2 fl RA2 is itself a
scaled version of A2, the scale factor being the square
root of its index U(R). The number of CSL rotations
with index < X is asymptotically Z s/lX /it.
1.3. Coincidence rotations fo r twelvefold symmetry
Let us consider a 2D quasicrystal with twelvefold
symmetry, the Stampfli [16] or the square triangle
tiling [17], say. As mentioned earlier, the coincidence
problem splits into two parts: first the coincidence
problem for the underlying Z-module M \2 and sec
ond the correction, due to the acceptance domain,
of the coincidence indices obtained in this way [11].
Here we discuss in detail only the first part.
In the complex plane, the twelvefold module [18]
(of rank 4 over Z) can be written as the direct sum
M n = © Z -f © Z £ 2 © Z £ 3 with£ = e2/12.
Thus M \2 is the ring of integers in the cyclotomic
field K = Q(£)- Again prime factorization is unique
up to units. Every coincidence rotation (written as
el<p K) can be factorized as [11]
* n ( f ) " !' ( $ f " n ( * ) - .
p=l(12) V p ' X p ' p=±5(12) P/
(12)
where e is one of the 12 roots of unity in K. The
factorization is more complicated than before, and
we have two independent generators of index p when
p = 1(12) since these primes have four prime factors
in K which form two pairs of complex conjugates.
The index of a rotation R is
u(R ) = n p(in>,)Hn*2)|) n p 2inpi <i3>
p=l(12) p=±5 (12)
and the group of coincidence rotations has the form
SO C(M i2) ~ C \2 ® . (14)
Thus, in spite of the more complicated factorization,
the structure of the coincidence group remains simple,
as indeed it does for all other planar symmetries [11].
If 12/(m) denotes the number of coincidence ro
tations of index ra, the Dirichlet series generating
function for /(ra ) is
^ , V^ /(ra)
= >
' ms
m=l ^
= n ( i n ( i _ p- 2 s )
P= 1(12) V F ' p=±5(12) V F '
_ 4 2 4 2 4 4
+ l3 7 + 257 + 377 + 497 + 617 + 73s
_4_ _ 4 _ _ 4 _ _ 8 _ 4
+ 977+ 1Ö97 + 1577 + 169s + 181s + ' " '
All coincidence modules are scaled versions of M \2
and the number of CSL rotations with index < X is
asymptotically 12\/3 ln(2 + y /3)X /ir2.
Let us compare these findings with those of the pre
vious two cases. One would expect the CSL rotations
of 4- and 6-fold symmetry to reappear here (though
possibly with a different index), and this is indeed so
since Q(i) and Q (> /-3 ) are subfields of K. To go into
more detail, except for 2 and 3, all primes are = ±1
or ±5 (mod 12). For the square lattice, the generators
of coincidence rotations come from primes p = 1 (4),
therefore = 1 or 5 (mod 12). And for the triangular
lattice, the generators of coincidence rotations come
from primes p = 1 (3), therefore = 1 or -5 (mod
12). It turns out that SOC(Z2) and SOC(A2) together
generate the subgroup of SOC(M i2) consisting of all
coincidences whose indices are squares.
How do these results apply to the coincidence prob
lem for twelvefold symmetric tilings? Let us assume
that such a tiling is obtained through projection with a
certain window, a dodecagon say [16]. A coincidence
in the set of vertex points occurs if there is a coinci
dence in the module M \2 such that the image point in
internal space lies both in the original window and in
an appropriately rotated window. A consequence of
this is that the coincidence group of the tiling is still
SO C (Xi2) but the index of each group element is
normally larger than its index in A412 by a correction
factor close to 1 (depending on the group element). It
also explains why the set of coinciding points forms a
tiling of slightly different type from the original one,
a small proportion of the points of the original tiling
being missing from it. In fact the term "index" for
the reciprocal of the fraction of coinciding points is
perhaps inappropriate in this setting as it no longer
has a purely algebraic interpretation and is no longer
715 M. Baake and P. A. B. Pleasants Algebraic solution of the coincidence problem
an integer. Details of this and the determination of the
rotation angle in internal space by means of algebraic
conjugation are given in [11].
Part 2: Examples in three dimensions
The coincidence problem in three dimensions is
more involved. In particular, the SOC-group, being a
subgroup of SO(3), is in general no longer Abelian.
In what follows, we present two examples where we
can give a complete description of the SOC-group
and its indices through Cay ley's parametrization of
0(3). Since 0(3) = S0(3) <8>{±1}, the extension to
reflections is trivial and will not be discussed below.
2.7. Coincidence rotations for cubic lattices
The simplest object to start with is the primitive
cubic lattice, Z3. (The bcc and fee lattices require
only a minimal modification mentioned at the end of
the section.) If we write Z 3 = Zei © Ze 2 © Ze3 it
is immediate that the coincidence group is
SOC(Z3) = S0 ( 3 ,Q ). (15)
(other rotations might also lead to coincidences, but
not to a CSL of full rank 3). The subgroup of rotations
with index 1 is the rotation group of the cube [20] of
order 24, O = SO(3, Z).
To determine the index of a rotation R £ S0(3, Q),
we use quaternions [21,22] and Cay ley's parametriza
tion [22] with 4 coprime integers /c, A, p, v
R(K,\,P,V ) =
Sx
I 2KV + 2An
2kv + 2\H 2/c/z + 2Aus
2/cA + 2pv
\ 2kh + 2XU 2 n\ + 2pv 8V )
where o = k2 + A2 + p2 + v2, 6\ = k2 + A2 p 2 v2,
6ß = K2 - X2 + ß2 - v2, and bv - k2 - A2 - p2 + v2.
This gives a double cover of S0(3, Q) (since R(q) =
R(-q)). One can show [23] that the coincidence index
U(R) is the denominator of R, defined as den(R) =
gcd(r IM | rR integral). It is the "odd part" of cr:
E(R ) = denCRfa, A, p, u)) = a / 2 s, (16)
where s is the largest integer such that 2s divides a. In
particular, we reproduce the result [12, 2] that E (R)
runs precisely through all odd integers.
Cayley's parametrization has the nice property that
(A, p, v) gives the rotation axis of R(k, A, p, u),
R(k, A, p, v) (A, p, v? = (A, p, v)1 , (17)
while the rotation angle follows from tr(i?) = 1 +
2 cos(<£>), so
cos(</?) = K2 - \ 2 - p 2 V2
K2 + A2 + p2 + V2 (18)
One can easily construct all solutions for small indices
explicitly [3,4,23], while the case k = 0 gives all CSL
rotations through 180° as described by Lück [13].
If 24/(m ) is the number of CSL rotations of index
m, it is known [24, 23] that /(1 ) = 1, /(2 n) = 0,
f(p r) = (p + l ) p r_1 for odd primes, and f ( m n) =
/( m )/(n) for m, n coprime (multiplicativity of /) .
The corresponding generating function reads
f( m )
m=l mi 11 1 _ !-<
p odd
4 6 8 12 12 14 24
+ 3s + 57 "1" I s + 9s + 11s 13s 15s
18 20 32 24 30
+ 177 + I9 7 + 2 P + 23s + 25s + ' " '
and the number of CSL rotations with index < X is
asymptotically 72X2/7r2.
This is not the end of the story. Coincidence rota
tions of given index m can be collected into equiva
lence classes of rotations related by the action of O
[24]. This double coset analysis will be described in
[23]. For example, truly different CSL's of Z 3 with
the same index occur for the first time at U = 13.
Also, describing the fine structure of a coincidence
rotation requires an analysis of the lattice planes
perpendicular to the rotation axis. For example, the
(unique) equivalence class for 27 = 3 can be repre
sented by q = (0,1,1 ,1), i.e. a rotation through n
around (1,1, I f. Here, three layers are stacked pe
riodically, with perfect coincidence in one (which is
therefore an ideal grain boundary), but none in the
other two. So this coincidence rotation represents a
change in the stacking sequence.
Finally, let us mention that it is precisely in this
layer structure that the three cubic lattices (primitive,
bcc and fee) differ, although the SOC-group and the
index formula are the same for all three.
716 M. Baake and P. A. B. Pleasants Algebraic solution of the coincidence problem
Icosahedral quasicrystals are of particular interest,
and one would like to know their coincidence struc
ture in detail [6, 25]. We restrict our discussion to the
investigation of the 3 different 3D icosahedral mod
ules [26] (of rank 6 over Z) and again omit the deter
mination of the window correction [23]. We will call
the modules M-&, M p, M y for body-centred, primi
tive and face-centred, respectively. They are spanned
by the orthonormal basis e\, e2, e 3 with coefficients
a { G Z[r], r = (1 + V5)/2, as follows:
M b = ( E l i a ie i I + r a 2 + a 3 = 0 (2)},
M P = {a; M B | a i + a 2 + c*3 = 0 or r (2)}, (19)
M f = {x e M b I ct\ + a 2 + a 3 = 0 (2)}.
Cayley's parametrization can again be used. Our first
assertion is that the coincidence group is the same for
all three modules [23], namely
SOC(M b) = SO C (Xp) = SOC(Mf) (20)
= SO(3, Q(r)).
The unit quaternions (1, 0, 0, 0), |(1, 1, 1, 1), and
\{ t, 1, -1/r, 0) together with all even permutations
and arbitrary sign flips form a group [27,19] Y of or
der 120 which is the usual double cover of the icosa
hedral group Y = {R G SO(3, Q(r)) | 27(£) = 1}.
The icosian ring [19] I consists of all integral linear
combinations of elements in Y and is a maximal order
with unique (left- or right-) factorization. One finds
the relation SO(3, Q(r)) = {R(q) \q G I }, and our
second assertion is the index formula for a coinci
dence rotation Ro G SO(3, Q(r)), again for all three
modules:
27(7*0) = min {N(\q\2) \ q G I, R(q) = Ro} (21)
where the argument |g|2 on the right hand side is
always a number in Z [r] and its norm is defined by
N(m +nT) = m 2+m n n2. The indices 27 run through
all positive integers of the form m 2 + m n n 2 with
integral ra and n. These are the numbers all of whose
2.2. 3D icosahedral modules of rank 6 prime factors congruent to 2 or 3 (mod 5) occur with
even exponent only. (They can also be characterized
as the positive numbers of the form 5x2 y2 with
integral x and y, as used in [25].) For 27 < 100, one
finds the list of the numbers 1,4, 5, 9, 11,16, 19, 20,
25, 29, 31, 36, 41, 44, 45, 49, 55, 59, 61, 64, 71, 76,
79, 80, 81, 89, 95, 99, 100 - which covers the known
cases [6].
If 60 f(m ) is the number of coincidence rotations of
index m, the generating function for the multiplicative
function / (ra ) starts as
/(ra ) = 1 + 5
_
_i 6 10 24
i _
m s 4s
_
__
5s 9s + 11*
20
16s +
40
19s +
30
20s
30
+ 25s +
60
29s +
64
31s
50
36s +
84
41s +
120
44*
60
+ 45s +
50
49s +
(where the denominators show all indices < 50) and
the number of CSL rotations with index < X is
asymptotically 1350\/5 ln (r)X2/7r4. A more com
plete description of the icosahedral case, which re
quires a more mathematical treatment, will be given
in [23].
Concluding rem arks
Various examples of coincidence problems in two
and three dimensions have been presented together
with their solutions in algebraic terms. Although
proofs have been omitted, we hope this illustrates
the usefulness of the algebraic approach. Many more
cases can be treated similarly though there are open
questions concerning the detailed way a coincidence
is realized in three dimensions (e.g., its layer struc
ture).
There remains the question of what can be said
in higher dimensions. In dimensions > 4 orthogo
nal transformations can no longer be parametrized by
quaternions. Nevertheless, two problems look solv
able: the cubic lattices in arbitrary dimension and the
module HA in 4-space which is related to a highly
symmetric 4D quasicrystal [28, 29]. We hope to re
port on these soon.
717 M. Baake and P. A. B. Pleasants Algebraic solution of the coincidence problem
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... This has been investigated in great detail in connection with the classification of grain boundaries and twins in materials science, compare [8,13]. The connection with the algebraic problem we are interested in here emerges via the so-called coincidence site lattices (CSLs), see [5,2,19] and references therein. These are finite index sublattices of a given lattice Γ that are intersections of the form Γ ∩ RΓ with an isometry R from the commensurator. ...
... It is well known [13,5] ...
... which is clear from Γ bcc ∼ Z 3 and the corresponding statement for Z 3 , compare [2,5]. From now on, we shall restrict ourselves to rotations, i.e., to orientation preserving isometries. ...
Preprint
The coincidence site lattice (CSL) problem and its generalization to Z-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.
... A CSL is a lattice defined by the nodes common to two individual lattices, one of which is rotated with respect to the other by an angle not belonging to the holohedry [27]. The CSL theory plays an important role in the derivation of low-energy grain boundaries in both crystals [28] and quasicrystals [29]. Several kinds of twin are based on a CSL too (see below). ...
... The mutually perpendicular directions in X bisecting the angles j and p À j are both reflection lines for the CSL, which has at least symmetry 2mm. The reciprocal of the density of common nodes defines the coincidence index or factor [29], which depends upon j and is the two-dimensional counterpart of the twin index. For the square and the hexagonal nets, the minimal coincidence indices are 5 and 7, respectively [29,32]. ...
... The reciprocal of the density of common nodes defines the coincidence index or factor [29], which depends upon j and is the two-dimensional counterpart of the twin index. For the square and the hexagonal nets, the minimal coincidence indices are 5 and 7, respectively [29,32]. In Fig. 1 a plot of vs. j for the hexagonal net is given for 100 and 0 j 60 . ...
Article
Full-text available
Oriented crystal associations with a low degree of restoration of lattice nodes are discussed and it is shown that they are based on a large coincidence-site lattice (CSL). In this type of associations the oriented crystals are related by symmetry or pseudo-symmetry elements expressed by high indices in the crystal lattice. The relative orientations of the crystals correspond to rotations belonging to non-crystallographic point groups. In spite of the low number of common lattice nodes, these associations are not uncommon, especially in layer compounds. Since this kind of associations does not enter either in the definition of twinning or in that of cell-twinning, we propose to introduce the definition of plesiotwinning. It is suggested that plesiotwins may form when two or more crystals coalesce or exsolve. A plesiotwin orientation may require a minor adjustment of the relative orientations, whereas a more stable configuration, such as a parallel growth or to a twin, may require a larger, kinetically more hindered rotation.
... This has been investigated in great detail in connection with the classification of grain boundaries and twins in materials science, compare [8, 13]. The connection with the algebraic problem we are interested in here emerges via the so-called coincidence site lattices (CSLs), see [5, 2, 19] and references therein. These are finite index sublattices of a given lattice Γ that are intersections of the form Γ ∩ RΓ with an isometry R from the commensurator. ...
... It is well known [13, 5] that (S)OC(Γ bcc ) = (S)O(3, Q ) , which is clear from Γ bcc ∼ Z 3 and the corresponding statement for Z 3 , compare [2, 5]. From now on, we shall restrict ourselves to rotations, i.e., to orientation preserving isometries. ...
... It is well known [13, 5] that (S)OC(Γ bcc ) = (S)O(3, Q ) , which is clear from Γ bcc ∼ Z 3 and the corresponding statement for Z 3 , compare [2, 5]. From now on, we shall restrict ourselves to rotations, i.e., to orientation preserving isometries. ...
Article
The coincidence site lattice (CSL) problem and its generalization to ℤ-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.
... To our knowledge, no example of diperiodic twins has been reported so far. In a diperiodic twin the Σ factor, which corresponds to the density of common nodes in the composition plane (the two-dimensional counterpart of the twin index: Baake and Pleasants, 1995), would be small (often 1), but the individuals would have no common period along the third direction. Friedel (1926, pp. ...
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The direct and indirect evidences for the oriented attachment of pre-formed crystals as a mechanism of twin formation are reviewed and discussed. Although rarer than the nucleation-stage formation, this mechanism, often overlooked, has been demonstrated in laboratory experiments, with the artificial production of twins. Moreover, the role of the oriented attachment in the formation of natural twins, although more likely to occur in environments where crystals are free of moving and interacting, is supported also by the occurrence of monoperiodic twins and of plesiotwins, whose origin cannot be explained by a nucleation-stage formation. The so far observed absence of diperiodic twins is discussed in terms of two-dimensional site-coincidence and twin obliquity.
... How is the torus parameter of F 0 related to that of F? Well, if F has parameter t then the point on the centreline of S that projects onto the origin of E is t + ( =2 p 5)e int , so F 0 has parameter M(t + 2 p 5 e int ) = Mt ? e int 2 p 5 : (4) As F varies, the in ation given by enclosing the strip for F in a strip times as wide with the lower edges coinciding is represented on T 2 by t 7 ! Mt + c, where c = ?e ...
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The torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or ination properties. The method is explained in an illustrative fashion for some widely used tiling classes with golden mean rescaling, namely for the Fibonacci chain (1D), the triangle and Penrose patterns (2D) and for Kramer's and Danzer's icosahedral tilings (3D). We obtain a rather complete picture of the orbit structure within these classes, but discuss also various general results.
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A reciprocity relation for Bollmann's O-lattice is introduced. This result completes the existing Grimmer's reciprocity results between coincidence sites and displacement shift complete lattices. We show that the lattice generated by ai* - bi* (i = 1,2,3) is reciprocal to the O-lattice. This result, supported by Multislice calculations, indicates that it is possible to observe the O-lattice under an electron microscope using annular apertures, thus allowing the study of strain fields existing in interfaces or between a thin film growing onto a crystalline substrate.
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