THE APPLICATIQN OF SHUBNEKQV fiROUM TO THE
DETERMWATICN 0F ANTLFERRGMAGNEWC
STRUCTURES

Thesis for fhe Degree of hi. D.
MtCHIGAN STATE UNIVERSITY
E. Pam Riede!

1961

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ABSTRACT

THE APPLICATION or SHUBNIxov GROUPS r0
‘IHB DITERMINATION or ANTIFERROMAGNETIC STRUCTURES

by E. Paul Riedel

Following a brief historical account of the development
of black-white group theory in one, two and three dimensions,
the results of proton-resonance and xbray studies are com-
bined with the Shubnikov group theory in enumerating the
possible arrangements of the magnetic moments in‘the anti-
ferromagnetie state of asurite Cu3(603)2(0H)2. Four anti-
ferremagnetic symmetry groups are found which describe all
such possible arrangements of the magnetic moments.

A description of twinning by merohedry and by retieular
merohedry in the lh ordinary and 22 black-white three-
dimensional space lattices is presented. All of the possible
merohedry and reticular merohedry twin groups for these
lattices are then constructed and listed. Included in this
list are lh2 new merohedry and LO new'retieular merohedry
twin groups.

The antiferromagnetic 'T wall" twins in N10 are shown to

be pseudo-merohedry twins.

THE APPLICATION OF SHUBNIKOV GROUPS TO THE
DETERMINATION OF ANTIPIRROHAGRBTIC STRUCTURES

3?

\6 L3
31‘ Paul Riedel

A THESIS

Submitted to
unchigan State University
in partial fulfillment of the requirements
for the degree of

Doctor of Philosophy

Department of Physics and Astronomy

1961

302399 0
(9-12-64
ka1~\k&W\

.Acknowiedgment

The author expresses his gratitude to
Professor R. D. Spence for proposing the problems
discussed herein and for his encouragement and

indispensable aid in their solution.

11

 

TABLI OF CONTENTS

SECTION I
Suez-Imus GROUPS
- Page
Introduction..................l
TheBrsvaisLatticeHethod . . . . . . . . . . .6
One-dimensional Space Groups . . . . . . . . . . ll
Two-dimensional Space Groups . . . . . . . . . . l6
Three-dimensional Space Groups . . . . . . . . . 21»

SECTION II
m POSSIBL: urmomcmmc 3mm! GROUPS OF
AZURITI A ' A

Introduction . . . . . . . . . . . . . . . . . . 30
ProtonResonanceData.............32
Point-Group Operations . . . . . . . . . . . . . 38
I-Ray Structure . . . . . . . . . . . . . . . . b0
Point-Group Selection . . . . . . . . . . . . . bl
Space-Group Requirements . . . . . . . . . . . . 1+3
Space-Group Selection . . . . . . . . . . . . . 45

Possible Magnetic Structures . . . . . . . . . . 52

iii

 

SECTION III . .
WINNING BY WOHEDRI AND RITICULAR HIROHEDRT .
h A i i ' Page
Introduction.........i.........59
Twinningbyuerohedry............. 61
a. OrdinaryLattices. . . . . . . . . . . 61
b. Black-white Lattices . . . . . . . . . 6k
Twinning by Reticular Merohedry . . . . . . . . 6'.
a. OrdinaryLattices. . . . . . . . . . . 65
b. Black-White Lattices . . . . . . . . . .67
MerohedryMnNotation. . . . . . . . . . . . .69
Reticular Merohedry Twin Notation . . . . . . . 70
Construction of the Twin Groups . . . . . .V . . 70
TwinsinAntiferromagneticNiO . . . . . . . . . 77
Possible Merohedry Twin Groups . . . . . . . . .107

Possible Reticular Merohedry Twin Groups
for‘RhmbOhedral Canal. e e e e e e e e e e e 113

Possible Reticular Merohedry Twin Groups
forcubiccrystals, ,.............lll.

”Pmn O O O O O O O O O O O O O O O O O O. O O O 0115

WC“ 0 O O O O O O O O O O O O O O O O O O O O 116

iv

 

Table Number

1

10

Tables

Title Page

One-dimensional Space Groups based
on Lattice 1p 12

”Point Groups corresponding to the one-

dimensional Space Groups 15
The two—dimensional uneolorsd Space

_ Groups and.their.corresponding Point

Groups 19
The two-dimensional blackdwhite Space
Groups based on ordinary Lattices and

their corresponding Point Groups 20
The two-dimensional Space Groups based

on black-white Lattices and their
corresponding Point Groups 21
The triclinic and Monoclinic Shubnikov
Groups 28
The number of ordinary, black-white

and gray Space and Point Groups in

one, two and three dimensions 29
Local Magnetic Field vectors at

Proton Positions 36
.aneclinie Heesch Point Groups #0
Space-GroupSSSlection #5

V

 

Thble Number

11

12

15
16
17
18
19

2O
21

22
23

2h

Title

thnetie Field Transformations
under Pas

magnetic Field Transformations
under P321 '

Inagnetie Pield Transformations
under Pbc

magnetic Pield Transformations
under Gas

Magnetic Structure: Pas Symmetry

magnetic Structure: Pa21
JMagnetic Structure: Pbc Symmetry
Magnetic Structure: Cac Symetry
The 90 non-gray Heesch Groups and
the derived Group Types {a1,a;} ,
{a1,a3*} and {a1,a3,a:,ai*}
Groups of Type {a1,a3,a;,a3I}
The Groups of the Set {3p} listed
according to their Heesch subgroup
of order two

Possible Herohedry Twin Groups
Possible Reticular Mbrohedry Twin
Groups for Rhombohedral Crystals
Possible Reticular Herohedry Twin

Groups for Cubic Crystals

Vi

.Page

#9
50
50
52
55
56

57
58

81
98

101

107

113

11k-

 

Figure Number
1

6b

6c

Figures

Title
Diagrams of the one-dimensional
Space Groups based on Lattice 1p
One-dimensional Space.Groups based
on Lattice 19b
The five ordinary and five black-
uhite plane Lattices
Diagrams of the.Space Groups pmfg
and Pans
The 1# original and 22.black-uhite
three-dimensional Lattices
Antiferremagnetic Resonance Diagram
He in the al- c (afperpendiculsr to s)
plane, T: l.6°K
Antiferromagnetic Resonance Diagram H0
in the b-c plane, T=l.6°K
Antiferromagnetic Resonance Diagram
ll° parallel to b-al plane, T=l.6°l£
Stereographic Projection of the Local
Field vectors in Table 8
Point Symmetry of a Magnetic Moment
Vector in the Antiferromagnetic State
of Asurite

vii

Page

13

1k

17

23

25

33

3b

35

37

37

Figure Number

10

11

12

1h
15
l6
17

18

19

20

21

22
23

Title

The Effect of some symmetry
Operations on Magnetic Moment ‘Vectors

'Sy-otry of an Axial. Vector. under the

Triclinic Hoosch Point Groups
W .‘s an Axial. Vector under the
)Ionoslinic. Hoosch Point Groups
Diagram of. P‘21/c

Diagram of P‘c

Diagram of Pazl

Diagram of Pbc

Diagram of C‘c

Magnetic Cell: Pac Symetry
Magnetic Cell: P821 Symmetry
Hexagonal Prism and Rhombohedral
Lattice R

The Rhombohodral Primitive Coll of
the Face-Centered Cubic F Lattice

The Rhombohedral Primitive Cell RI of
the face-Centered Cubic Lattice Fe
Antiferromagnetic Structure of 1110
"T Walls" in N10

viii

Page

38

39

$8383

50
51
53
51+

65

33:38

Section I
Blackéwhite Groups

Introduction

The material in Section I is intended to serve generally
as an introduction to the existing theory of black-white
crystallographic groups in one, two and three dimensions and
specifically as a starting point.for the material in Sections
II and III. As part of the introduction to Section I, a
brief historical outline of the development of these groups
and some of their applications is presented. Following this,
a Bravais lattice method is described for the construction
of the three-dimensional black-white space groups. This
method is then illustrated by outlining the derivation of the
black-white space groups in one and two dimensions. The three-
dimensional triclinic and monoclinic Space groups are listed.
The point groups which correspond to the space groups are also
discussed. _

Ordinary crystallographic group theory is concerned with
describing the symetry of Objects in space. During approx-
imately the last thirty years an important extension of this
theory has been developed which consists essentially of des— '
cribing the symmetry of these objects when a sign + or - is
assigned to them. Such an extension of the theory therefore
requires the introduction of new crystallographic operations
Galled antioperations which transform an object in the same

1

 

way as an ordinary crystallographic operation but change its
sign. Groups which contain antioporations are called anti-
groups. Change of color of the Objects instead of change of
sign is usually more convenient to use in diagrams represent-
ing the symmetry of these groups. Antioperations are then
called colored operations and ordinary operations are called
uncolored operations. Groups which contain one or more
colored Operations but not the colored identity operation are
then called black-white groups. Groups which.contain the
colored identity operation are called gray groups. Groups
which do not contain colored operations are the ordinary
crystallographic groups.

The first extension of ordinary crystallographic group
theory was made in two dimensions in 1928, and 1929 by
- Alexander.and Herrmann.in connection with a study of the
pessible symmetries of liquid crystals.l’2 If only one side
of a plane is considered, the number of possible periodic
symmetries of objects in the plane is described by the
~ordinary 17 two-dimensional crystallographic space groups.
Alexander and Hermann derived all of the two-dimensional
space groups when both sides of the plane are considered to
‘be distinct. They found that 63 new'greups as well as the
original 17 are necessary to describe all such possible
Bymetries. ‘ '

‘ -

WAlexander and K. Herrmann z. Kristallo r.
235 (192%) ' ' 3 92'

.Alexander and K. Herrmanm Z. Kristallo r.

In 1930 Heesch introduced a "fourth coordinate" to
three-dimensional crystallographic group theory.3 This co-
ordinate had no numerical value associated with it: instead
it is represented only by a-+ or'— sign. Thus, the groups
which he called “the four-dimensional groups of three-
dimensional space" describe the space symmetries of objects
in three-dimensional space and in addition label the objects
-+ or - . These groups are today called the Shubnikov Groups.
Heesch derived all such new triclinic and menoclinic space
groups but did.not.work out explicitly those of the remain-
ing crystal systems. He did, however, derive 90 new point
groups which correspond to the new space groups (all the
black-white and gray Shubnikov groups) in the same way as the
ordinary 32 point groups correspond to the ordinary 230 space
groups. .The 122 point groups (the 90 derived by Heesch plus
the original 32) will be called the Heesch point groups in
the present work.’

Shubnikov“ (1951) rederived the Heesch point groups
_and also extended the theory to the study of the symmetries
of three-dimensional figures using both crystallographic
and non-crystallographic operations.

 

3- a. Heesch, z. Kristallogr. 21, 325 (1930).

. * They are usually called the Shubnikov point groups
in the literature.

‘4. v. Shub'ntkon We;
[inite Piggggg-(in Russian), Mbscow: Academy of Sciences

(1951).

I,

In 1952 Cochran5 rederived the blackdwhite plane groups
- send suggested their applicability to the study of the symmetry
caf'real periodic functions used in crystallography.

The complete extension of the blackawhite group theory
two three dimensions was first accomplished by Zamorzaev6'7
by a mathematical method in 1953. There are 11.21 new space
groups, 1191 of which are black-white while the remaining
230 are gray. Zamorsaev named these as well as the original
230 space groups the Shubnikov groups.

The new space groups were rederived by a Bravais lattice
Inethod and listed by Belov et a1.3 in 1955. A revised list
of these groups was published by the latter authors9 in 1957.

In their discussion of the symmetry of magnetic crystals,
‘Tayger and Zaitsev (1956) derived 58 magnetic point groups.10

7-D. Cochran, Acta Cryst. 1, 630 (1952).

6'A. M. Zamorsaev 'A Generalisation Of the Fedorov
Groups." Dissertatioadn Russian), Leningrad (1953).

7°A. m; Zamorsaev, Soviet Physics Cryst., Vol. 2, no. 1,
10 (1957).

8'11. V. Belov, N. N. Neronova and T. S. Smirnova, Trudy

Inst. Krist. Akad. Nauk 8.3.3.3. _1_._, 33 (1955).

9‘11. V. Belov, H. N. Neronova, and T. S. Smirnova,
Soviet Physics Cryst., VOl. 2, No. 3, 311 (1957).

10-3. A. rev er and v. M. Zaitsev, J. Exptl. Theoret.
Phys. U.S.s.a. 1_, 56h (1956).

{the 58 point groups derived by Tavger and Zaitsev are
isomorphic to the 58 black-white Heesch grOups. A

The first application of Shubnikov groups to the deter-
mination of magnetic structures was given by Denney et al.11
in 1958. They proposed a systematic method employing neutron
cliffraction data in conjunction with the Shubnikov groups to
cistermine the magnetic structure of ferromagnetic and anti-
ferromagnetic crystals. They applied the method to the anti-
ferromagnetic crystal chalcepyrite - (CuFesz). A table
showing the effects of ordinary syn-Retry and antisymetry
operations on magnetic moment vectors is also included.

The Heesch groups were first applied in 1958 to the
<iescription of certain types of twinning in crystals by

Curien and Le Come.”-

The possible point symmetry groups of ferromagnetic
and antiferromagnetic crystals were discussed by Tavger in
1959. The point groups imply that only certain directions
are possible for a macroscopic magnetic moment with respect
to the crystal axes. These directions are listed. Tavger

also lists the magnetic point groups for which piesomagnetism
is possible.14

 

’1 G. Donnay, L. M. Corliss J. D. H. Donnay, N. Elliott,
and J. M. Hastings, Phys. Rev. fig, 1917 (1958).

12‘H. Curien, and I. Le Corre, Bull. Soc. franc. Minér.
Grist. 81,126 (1958).

13.3. A. Tavgcr, Soviet Physics Cryst., Vol. 3, No. 3.
3&1 (1959).

111.3. A. Tavger. Soviet Physics Cryst., Vol. 3. NO- 3.
3hh (1959lo

 

 

 

6

The possible space groups and their corresponding point
Egroups were listed for ferromagnetic and ferroelectric crystals
in 1960 by Heronova and Belov.15

No attempt has been made abOve to include all the papers
Clealing with black-white groups. fibre extensive references
sappear in the review papers by Hockey16 and Lo Corre.l7

jgthBravaiseggggiggpygghgg

The method employed by Belov et al. in deriving the
black-white and gray ShubnikOv groups consists of the follow-
ing general steps:

1. From the 15 original Bravais lattices which describe
all the ordinary translational symmetry groups in
three dimensions, 22 new black-white lattices are
derived by coloring each translational element in
turn in a particular Bravais lattice, considering
all possible combinations of colored and uncolored
translational elements and eliminating those result-
ing combinations Of elements which lead to identities
or do not lead to groups.

2. All combinations of colored and uncolored symmetry
elements added to each of the 36 lattices are then

 

15's. u. Neronova and N. v. Belov, Soviet Physics
cry’to' Vol. I." NOo a 769 (1960)o

16". L. “okay, Ac“ crySt. E, 51.3 ‘1957)0

l7-I. Le Corre, Bull. Soc. franc. Minér. Crist. QI,
120 (1958).

considered. When these sets Of elements which

possess the properties of a group and various

identities between groups are recognised, the 1651

Shubnikov groups result. Ten theorems presented in

reference 8 and there employed in this process are

listed below.

Theorem‘;. The reflection in a plane and then a
translation in a direction perpendicular to the plane is
equivalent to a reflection (of the same character, i.e.,
mirror or glide) in a ”derived" plane which is parallel
to the initial one, but-located at half the translation
from it. The reflection will be uncolored if the first
reflection and translation are both uncolored or both
colored; and colored if one of the component operations
is colored and the other uncolored.

Theorem.g. If a colored translation is parallel to
a plane of symmetry, (m,n,c,g) the plane will simultan-
eously be colored. If this plane coincides with the cell
face 10! then in the presence of a color translation?
parallel to the x axis asa', nab', can. and bzn': in
the presence of 3' parallel to y one obtains mzb', nsa',
asn', bsm'. If the color translation lies on a diagonal
cut maul, nsm', asbl and heal.

If color translations are available along both axes

them mansa's b' and bsasm'a m'.

‘

 

llThe color (indicated by a prime) translation, t',
magnitude is half that of theduncolored translation in the
same direction, i.e., zlt;|=|tx|.

Theorem 1. If the translation is disposed obliquely
relative to a symetry plane, resolve the translation
into components perpendicular and parallel to the plane:
The first component determines the derived plane shift-
ing it intact parallel to itself just half its length,
the second component determines the additional glide
component. The derived plane is colored if the plane
and translation giving rise to it differ in color and
uncolored if both are the same color.

m A. _ _An axis of n fold order and a perpendi-
cular translation give rise to a parallel derived axis
of the same order and character (rotation, screw,
inversion, screw translation). The derived axis passes
through the apex of a triangle with an apex angle of
360°/n. The position of the apex is given by the basic
construction in the rotation plane. The color of the
axis is determined as previously by the color of the
elements giving rise to it.

mgogem 5. If there is a rotation or screw axis
and a colored translation parallel to it, the axis appears
simultaneously as a colored rotation or screw axis.

 

2xt.'.=2(21) oxt.'.=o(o;)
letu=21 (2) 63xt....63 (6)

3 x t..= 3 (6’)* 61 x t..- — 61 (6,.)
hxthbuz) 65 xt..=65 (6?)

hats: #1 (1.5) 621t£=62 (65)
(023%=102Ue:l 6hxt,.=6h(61)
bxbvh‘h’

‘1 6r is anew symetry element. 1t represents a rotation

of 120° followed by a colored? translation t” . This element is
therfore of 63:}; order. I:ggf‘t operation 1 repeated three
times, the result is}! :t + ..+ t...I f,L6 ,and P are a
six-fold rotation, six-fold rotation invors on and P a minor

plane perpendicular to L6, then Lg: -12 and Li: P.

9

Theorem 6. If a translation is directed obliquely
to an axis resolve the translation into components per-
pendicular and parallel to the axis. The first trans-
fers the axis to the apex of a triangle with an apex
angle of aeoo/h. The position of the apex is giVen by
the basic construction in the rotation plane. The
second component merges with the resultant axis.

Remarks: 1. The axis of odd order can only be
either uncolored or gray.‘ 2. An axis of.3 or 6-fold
order combined with a non-parallel color translation
can only be gray.

Theorg! 2, The planes (mirror or glide) intersect-
ing in 30°, h5°, 60° or 90° produce a rotation or screw
axis of 6, k, 3 or 2-fold order in the line of inter-
section or at a distance and parallel to it; the result-
ant axis is colored if both planes are of the sane color
and different if they are of different color.

. Theorem g, If two rotation axes of the second order
intersect in an angle of 30°, A5°, 600 or 90°, there
appear resultant axes of 6, k, 3 or 2-fold order perpen-
dicular to the plane of the axes producing them and pass-
ing through the point of intersection.of the axes pro-
ducing than. The resultant axis is uncolored if both
the axes producing it are of the same color and colored
if they are of different color. If one or both of the
original axes are screw'axes, the resultant axis is dis-

placed l/L of the translation along each glide direction.

 

.a‘ '-
44.4

.f-

 

10

Remarksan Since one can consider the 6-fold axis as
the sum of a three and two-fold axis: 6==3+2, we may
write for the colored 6-fold axis 6'5.- 3+2'. Similarly:

6" i 6'— + '
6%:31‘? ? 6": 324-2
63= 3 + 21

Theorem.2, As a result of the intersection of three
mutually perpendicular planes of symmetry or the inter-
section of an axis and a perpendicular plane of symmetry,
there arises a center of symmetry. If there are an even
number of intersecting elements containing half-translation
components parallel to coordinate axes, the center is
not quite symmetrically located along these axes. The
center is colored if the number of colored generating
elements is not even and uncolored if the number is even.
Associated with the intersection of three planes (or an
axis and planes) not in a right angle there appears an
inversion axis colored or uncolored depending on the
number of colored generating elements.

Thegrem.;g,, The combination of a center of symmetry
with a translation produces a resultant center halfway
between the translation related centers. The resultant
center of symmetry is uncolored if the initial center
and translation are both of the same color and colored

if one is of a different color.

 

 

 

ll

One-digensional.§p§§3,§£ggpg

In order to illustrate the general method outlined above,
consider the problem in one dimension. There is only one
ordinary lattice in one dimension. It possesses just one
translational element of symmetry. The diagram of this
lattice is shown below where the open circles represent un-
colored lattice points and t represents the ordinary opera-
tion of translational symmetry.

<3__%_ :9 <3 <3

The only black-white lattice which can be found from

this lattice by the above procedure is shown below'where

dark circles represent black lattice points.
0 .2.) . O O

_ _u-)

Note that the definition of a colored translation implies
that for translations in the same direction

I E I = 21°31.
This is obviously true in two and three dimensions also.

The two translation lattices shown above may be denoted
by the symbols 1p and 1pb, where the subscript 1 denotes one
dimension, p primitive lattice type and b a colored trans-
lation in the b direction.

Consider now'step two of the process outlined above.
The only uncolored symmetry Operation which exists in one
dimension besides the identity and translation is the inver-
sion center i. Graphically this is denoted by a small
circle<3, while it shall be denoted byta and a gray center
(i.e., an uncolored and colored center superimposed) by CD .

 

12

Using the 103h_theorem and the lattice 1p, we find five
groups which are listed in Table 1.

Table l. One-dimensional Space Groups based on
on Lattice 1p.

Space Group Symmetry Operations
Symbol or Group Elements .
1p]. 3,;
lpl E,t,i
151-}. 3,-1.5,1"
1p1' 3.33%.“?
1p'il' 3,3' ,E,‘E‘,i,i'

The groups in Table l are shown graphically in Fig. l
where the triangles are used as objects in general position
to show the effect of the group operations.

 

13

Fig. l. Diagrams of the one-dimensional Space
Groups based on Lattice 1p

11'1
A A A A A
_t.__.,

lii

b. 0.43 QBQA o Bod o 5.041

-—-——t-———~

lfi‘

.94] Q ~QdQL®AQ Lead

-——-——‘t——r

1:1 .

--I
1911.

ma4pmamemadamwfl
t :

 

.__‘{O—._p

 

1h

Consider next the blackdwhite lattice lpb’ From.this two
groups are found, lpbl with elements 8 and.t' and.1pfil with
elements E,t',i and 1'. These groups are shown in Fig. 2.

Fig. 2. One-dimensional Space Groups based on
Latti ca lpb

1%1

[L04] Q ‘0‘ [7 Bod

In the symbol lpfil only the uncolored symbol for the inver-
sion, l, is given; the colored inversion centers being im-
plied by the other elements of symmetry. This method of
group notation is used for the Shubnikov groups which have
black-white lattices, i.e., only the uncolored elements of
symmetry are shown to the right of the lattice symbol. The
colored elements in the group are implied by the lattice
symbol and the uncolored elements. This method of notation

will be used in two dimensions also.

15

The point group of a crystal describes the directiOnal
symmetry of the crystal. The translational part of the
symmetry operations contained in the space group do not -

. change directions. Hence, the point group corresponding to
any space groupumay'be.obtained from,the space group.by rep.
placing the translational part, if any, of every symmetry
operation by the identity operation. The translation't is
therefore replaced by E, if by E', a glide plane in the c
crystallographic direction .c .by m a mirror plane, c’ by
my, etc. in obtaining the elements of the point group. The

five-point groups obtained in this way from.the one-dimensional

space groups are listed in Table 2.

Table 2. Point Groups corresponding to the one-
dimensional Space Groups

Space Group Corresponding Point Symmetry Operations

Symbol Group Symbol or Group Elements
lpl l 3

1p; ll E,i

lp'l" I? s,i'

1pl' 1' 3,3.

1p'i'l' i'l', s,1,a',1'

lpbl If 8,3.

1pr I17 3,3',i,i'

 

16

Egg-dimensional.Spgggtggggpg-

There are 17 ordinary, l7 grayand £6 black-white plane
groups. In the following description of these groups, we
shall continue to follow the methods and, unless otherwise
stated, the notation developed by Belov et al.

From the five ordinary lattices in two dimensions five
blackdwhite lattices can be derived. These 10 lattices are
shown in Pig. 3. Consider, for example, the oblique p
lattice. A colored translation along an edge in the a
crystallographic direction gives the lattice Pa‘ The only
other possibility is that of a lattice with colored trans-
lations along both edges. This gives the lattice shown below'
(solid lines) which is seen to be identical with pi (broken
lines).

 

_..-_-..-._ i,,_n,__ ”U

The notation for the ordinary lattices is the same as
that given in the ”International Tables for X-Ray Crystall-
ography.*18 The black-white lattice notation is the same as
that used by Belov et al. except that small instead of capi-

tal pf” and c?“ are used in order to denote two dimensions.

 

18’InternationaleTables.for X-Raz szgt., vol. 1,
Kynoch Press, Birmingham (1952).

17

Fig. 3. The five ordinary and five black-white
plane Lattices

4 Z ;

 

  

oblique

 

 

 

 

 

 

 

 

 

 

square

hexagonal

 

 

18

The lattice symbols have the following meaning:

1. p‘: primitive lattice with a colored translation

along the a direction.

2. p1: primitive lattice with a colored translation

along the spacial diagonal of the cell.

3. ca: centered lattice with colored translations

along two non-parallel edges.
This notation is also the same as that used for the Shubnikov
groups, except that in this case capital letters such as P
and C are used to indicate the lattice types.

The full and short international symbols for the 17
uncolored two-dimensional space groups are shown in Table 3
along with their corresponding point groups. The right-hand
column consists of the symbols used in reference 8 for these
groups except that in Table 3 the capital P'8 and C" have
been replaced by small letters as mentioned above.

From the original 17 .groups 17 gray groups are found by
coloring the identity element. Then every element in the
group is simultaneously colored and uncolored. These groups
are written as pl', p21', pml', etc.

The 17 original groups also give rise to 26 black-white
groups based on ordinary lattices when the elements to the
right of the lattice symbol, excluding the identity, are
colored in all possible combinations. These groups are shown
in Table A along with their corresponding point groups.

 

19

Table 3. The two-dimensional uncolored Space Groups
and their corresponding Point Groups

stem and . Corresponding Space Group Symbols Belov et al.
3,333” “1‘“ Gm" an; snort SP3§fibSE°“P
oblique l p1 pl 131
p p211 p2 p2
imectangular m plml pm pm
p 9131 pa pa
and clml cm cm
c 21-! 92m pm pm
9225 plus pus _
9238 pas 933
chm. cmm cmm
square h Pk Ph P#
P has phmn phm phmn
plosm pkg phsm
hexagonal 3 93 P3 P3
p ’ 3m p3m1 P3111]. P3111
palm. pBIm p31m
6 'b6 p6 p6
6m . pém p6m p61“:

20

Table 1.. The two-dimensional black-white Space Groups
based on ordinary Latticesand their corresponding

Point Groups
System and Corresponding Belov et al.
Lattice Symbol Point Group Space Group Symbol
oblique 2' p2'
P
T
rectangular m Pm.
v
p P8
T
on
and c v t‘ e
m m pm m
c ‘- a a
as a
T T
P” 8
O 0
cm m
m' pm‘
pas'
I”?
M
cmm'
hexagonal 3m. p3m' .
p , p3}!
6 P6
6m' ' p6m'm'
6'm'm p6'm'm
square ‘0’" ph'.
p 1m. ' plm'm'
Plrs'm'
b'm' pk'lllln'
9' a
pit m m
0
pl» 9'

21

The 20 blackdwhite groups based on blackdwhite lattices
are listed in Table 5 along with their corresponding point

mupa e

Table 5. The two-dimensional Space Groups based on black-
white. Lattices and their corresponding Point Groups

System and Corresponding Belov et al.
Lattice Symbol Point Group Space Group Symbol

oblique l p a1
Pa 21 - 1’a2

rectangular m1 p m
Pa Pall
. pas
I pals
a PI"!
PIS

can

nnl' pgmn
pies
pine
pass
Pr'“
PI“
PI"

cam

square 1,1 p‘IL
p luml' piles-
pIhsn

22

In order to illustrate the meaning of the symbols for

9
the space groups, the groups pm g and p‘mg are shown in
There are as yet no standardised symbols for the
They will

Fig. A.
graphical representation of the antioporations.
here be represented by cross hatching the uncolored operations

as far as is practicable. The uncolored operations will be

represented as in ”the International Tables for X-Ray
Crystallography." For convenience however, all symbols will

be defined as they are introduced.

23

Fig. 1.. Diagrams for the Space Groups pm'g and p.mg.

. r b an
a 90 e gig—colored two-fold axis

 

D 0%....” uncolored glide plane
()3 é 0%
0
.22.- : ' a at s i ‘-
50 f i :0 the spike indicates
a e G —

mirror image.

__.- colored mirror plane

  
    

3-x-“ uncolored two-fold axis
9...“..-colored mirror mi and
uncolored glide plane g

superimposed .

 

 

uncolored mirror m.

 

2‘»

We ‘ inns onal éaaee 922m;
A reproduction of the 1!. original and 22 black-white

lattices shown in reference 9 is shown in Pig. 5. The
lattice symbols are described below. The numbers to the
loft refer to the lattice number in Fig. 5.

2. Pa colored translation along one edge

1.. Pb " " ' an edge in the b direction
5. pa w m s w m s w a w
6. PC " " " the diagonals of the 0 face
8. Ce " " in the c direction
9. Ca " " along edges in the a and b
. directions
11‘. Pc " " along an edge in the c direction
1213’ P A " " " the diagonals of the A face
13. PI " " " ' spacial diagonals
15b. As " " " " a direction
16b. Ac " " " edges in the b and c
directions
17b. ‘0 ' " " :he diagonals of the C
ace
19. F s " " ' the three edges
21. Ic " " in the c direction
29. C c " 5 along the c edge

31. RI I I fl " spacial diagonals

25

‘The IL original and 22 black-white

Fig. 5.

three-dimensional Lattices

 

 

Trlcllnir

SYSIC m

 

 

Monocllnic

system

 

 

Orthor'nombic

     

system

 

 

 

26

 

 

Orthorhombtc system (continued)

 

 

Tetragonal system

 

Hexagonal
(Rhombohedral) syuem

    

ubic system

  

 

C

 

 

 

 

27

The Shubnikov groups of the triclinic and monoclinic
systems are shown in Table 6. Table 6 is a replica of the
first part of the complete list in reference 9. Specific
examples of a few space groups will be illustrated later in
connection with the discussion of asurite.

To the ordinary 230 space groups there correspond 32
point groups. In addition, there are 32 gray and 58 black-
white point groups, or 122 Heesch groups in all. All Heesch
point groups which contain antioperations have the same
general structure. They contain a subgroup of index 2 of
unprimed elements, the remaining elements being primed. All
122 Heesch point groups are tabulated in the discussion of
the twin problem in the last section. Some examples will
also be used in the next section on azurite.

In suhary, the number of point and space groups in
various categories are listed in Table 7.

28

Table 6. The Trielinic and monoclinic Shubnikov Groups

Triclinic
System

c1

1. Pl

2. Pl'
3. Ps1

ci

a..Pl'
5.fihf
6. 91’

7. Pal
Mbnoclinic
System
02
1. P2

2. P21'
0

3. P2

5. 282

5. Pb2

6. Pc2

7. P21
9

I
9. p21
10. Pa21
11. szl
12. P021
13. oz
9

it. 021‘

15.
16.
17.

c.
18.
19.
20.
21.
22.
23.
25.
25.
26.
27.
28.

29.

.30.

31.
32.
33.
3b.

35.
36.

37.
38.
39.
#0.

02'
0,2
caz

Pm

Pnlf
me
Pam

Pbm
P03
Pc

Pcl

Pc'

Ccl'
Cc’

Ccc

us Cac

67. Pz'/t

C
52.
as.
it.
45.
#6.
47.
28.
49.
so.
51.
52.
53.
54.
55.
56.
57.
53.
59.
60.
61.
62.
63.
6t.
65.
66.

2h

P2/m
Pz/nlf
P2'/m
P2/mf
sz/nf
P.2/m
sz/n.
PCZ/m
le/n

t
P21flml.

I
le/m
le/I'.

' O
rzl/n.
2,21/a
cz/n
cz/nlf
CZf/m
CZ/mf
62'/m'
CcZ/m
Ca2/m
P2/c
22/61'

68.
69.
70.
71.
72.
73.
7h.
75.
76.
77.
78.
79.
80.
81.
82.
83.
8h.
85.
86.
87.
88.
89.
90.

91.

P2/c'
P2'/c'
PaZ/c
sz/c
PcZ/c
9,2/o
pcz/o
P21/c
P21/cl'
PZi/c
P2}/c:
P21/c
P.21/c
Pb21/c
Pczllc
9,21/6
2621/6
cz/a
c2/.lf
02'/c
CZ/c'
02'/cf
Caz/c
c,2/.

Table 7.

ordinary
black-white
gray

total
ordinary
black-white
8P8!

total
ordinary
"blackdwhite
SP3!

total

29

The Numbers of ordinary, black-white and

gray Space and Point Groups in one, two

and three Dimensions

Number of Number of
Dimensions Point Groups
1 2
l l
l 2
5
2 10
2 ll
2 10
31
3 32
58
3 32
122

Number of

Space Groups

230
1191
230
I651

Section II

The Possible Antiferromagnetic Symmetry Groups of Asurite

Laggoductign

Host of the material in this section has recently been
pUblished by the present author and Professor R. D. Spence.19

Antiferremagnetic crystals have been shown by neutron
diffraction techniques to be periodic structures where the
chemical cell which displays only the x-ray symmetry is re-
placed by the magnetic cell which displays the neutron dif-
fraction symmetry.20 The magnetic cell represents the symmetry
of the magnetic moments, associated with certain ions, as well
as the atomic positions.

For any particular magnetic moment direction, there
exist two possible senses for the magnetic moment vector.
Given a magnetic moment with a certain direction and sense
in a unit magnetic cell of an antiferromagmetie crystal,
there must also exist in the same cell another magnetic
moment with the same magnitude and direction but with oppo-
site sense. Hence, a magnetic lament is subject to both
ordinary symmetry and antisymmetry eperations where the change
' of sign operation which characterises an operation of anti-
symmetry is here interpreted as an operator R which reverses

Ig’ge 1e fifeaol and He 5e Waco, Phys—17:3 _26, II,‘ (1960’s

zo'See for example: C. G. Shull and E. O. Wollan Solid
325. Ph s c edited by r. Belts and D. Turnbull, (Academi-
ress, ew' ork, 1960), vol. 2, p. 137.

 

 

30

31

the sense of a magnetic moment vector. Thus, an antioperation
transforms a magnetic moment vector in the same way as an
ordinary crystallographic operation followed by a reversal

of the sense of the magnetic moment vector.

Two properties of the R operator are implied by the
above. R must commute with every symmetry operation and
82:3. '

The first published application of Shubnikov groups to
the description of an antiferromagnetic crystal appeared in
a paper by Donnay et al.21 Here x-ray and neutron diffrac-
tion data were used together with the Shubnikov group theory
to find the magnetic structure of chalcopyrite, CuPeSZ.

Proton resonance studies of the monoclinic crystal
asurite Cu3(003)2(0H)2 have shown it to be antiferromagnetic
below‘l.86°K.22 In this section, the results of recent pro-
ton resonance* and xpray studies will be combined with the.
theory of-Shubnikov groups in order to enumerate the various
possible arrangements of the magnetic moments in the anti-
ferromagnetic state.

The general method used to do this is new. Briefly, it
consists of the following. First, the point group symmetry
of the local magnetic field vectors at the proton positions

 

21'Donmay et al., op. cit.

22.R. D. Spence and R. D. Ewing. Ph131°81 3‘75 ALE:
1544 (1958).

* performed by Professor R. D. Spence

32

is found in the antiferromagnetic state. The magnetic field
is an axial vector. The operations of a point group however
.must transform gl;_axial vectors in exactly the same way.
Therefore, the point symmetry of the local field vectors must
also be the point symmetry of the magnetic moment vectors
(which are also axial vectors) associated with the magnetic
ions in the crystal. The Shubnikov space group which repre-
sents the symmetry of the magnetic cell must also transform
all axial vectors in the same way. It must therefore have
as a corresponding Heesch point group one which permits the
observed axial vector point symmetry. This space group must
also permit the same number of protons which are observed to
experience different local magnetic fields. The imposition
of these requirements on all Shubnikov groups which do not
violate the observed x-ray symmetry of the crystal leads in
the case of asurite to four possible antiferromagnetic syms
metry groups.

ngton Rgsonggge 23;;

Figs. 6a, 6b and 6c show’the angular dependence of the
proton resonance lines in the antiferromagnetic state of
asurite in‘the af-c, b-c, b-af (a' perpendicular to c) planes.
From the figures, it is clear that in the magnetic unit cell
there exist eight protons all with different local magnetic
fields. The local fields arising from.the copper ions were

found by fitting the data to the relation
z y °
V’Vo :{14-(fié) + ZHL Cod/9}2 _/
‘95 Ho Ho
where')% is the frequency of the free proton resonance in

 

33

 

 

.
8. o: ow. co. co an
O

\“

 

.0
e e 1| q
. use H,ae omoao
.o o» neasofiunomscd .ov o u.s on» as ex summowe oncogenes ofioosmssonmewaum< .so .mfih

 

 

 

 

31+

 

 

 

 

 

 

3

£00..” n H. .053 o .. a on» 5 on snowman oocmsonon osposmaaosnounofi do .mE

 

 

\

 

 

 

'Il-rl

'DN

 

35

 

4 n..-

-o..-

 

 

 

 

 

£90.” u a .233 .m .. n 3 .3393.“ a: menus: 3:338 933353835. .3 .mi

 

 

36

the dc magnetic field no and ® is the angle between Ho and
the local field Hi“ A derivationof this equation is given
in the appendix. The data in Figs. 6a, 6b, and 6c were
‘tmflken with no: 3430 gauss and r: 1.6°x. The magnitudes and
orientations of the local. fields are given in Table 8, where

9 is measured from the c axis and¢ from the a'f-c plane.

Table 8. Local Magnetic Field Vectors at Proton

Positions
2 H (sauce) 9 ¢ .
1 580 23° 62°
2 580 23° 298°
3 580 157° 242°
4 580 157° 118°
5 545 27° ‘ 56°
6 545 27° 304°
7 545 153° 236°
8 545 153° 124°

The angular relations between the directions of the
local field vectors _in the crystal can be conveniently rep-
resented by a stereographic projection as shown in Pig. 7.
The usual crystallographic convention has been employed in
F'13. 7, i.e., vectors in the upper hemisphere are projected
through the south pole and indicated by solid circles while
Vectors in the lower hemisphere are indicated by open circles.
In considering the smetry of Fig. 7, one must bear in mind
that it represents angular relations between axial. vectors

rather than polar vectors which are comonly shown in

37

Fig. 7. Stereographic Projection of the Local
Field Vectors in Table 8.

 

 

.%‘_- -’

crystallographic applications of stereographic projections.
Since the magnetic moment vectors of the copper ions are
axial vectors, they must be transformed by the symmetry
operations of the crystal in exactly the same way as are
the local field vectors. Hence the point symmetry of any
particular magnetic moment vector in the antiferromagnetic

state of asurite may be represented as shown in Fig. 8.

Fig. 8. Point Symmetry of a Magnetic moment vector
in the Antiferromagnetic State of Asurite.

 

38

221.112 9.1222 229—2...“ on!

Diagrams showing the effect of some ordinary and anti-
symmetry operations on magnetic moment vectors have been
constructed by Denney et al.23 Some of these are shown in
Pig. 9. (A few slight changes in the graphical symbols of

some of the operations have been made).

Fig. 9. The affect of some syuetry Operations on
Magnetic Moment Vectors

 

 

 

 

«OJ'G'

The effect of the symmetry operations of the Heesch
Peint groups of the triclinic and monoclinic systems on a
Bill'igle arbitrarily oriented axial vector is shown in Figs. 10
and 11 respectively.

 

23‘Donnay et al., op. cit. p.l9l8

39

Fig. 10. Symmetry of an Axial vector under the
Triclinic Heesch Point Groups

/,» O O
.‘ 0 ®
0 X e _ e \ e
\1' 1' - ‘1" \‘1"
0
e
.
h"

Fig. ll. Symmetry of an Axial Vector under the

monoclinic Heesch Point Groups
“mg 2+2' Mo‘. ' A . 6’“ or

I. i 3 ‘ (e ,— ‘0

C ' ,1
a g / e e e O '
\X . ;
’\ .I - A .’
\ It" ‘

 

 

 

 

 

 

#0

The symmetry operations comprising each of the triclinic

point groups have been listed in Table 2.

Those for the

monoclinic system are listed in Table 9.

Table 9.

Point Group
(in Belov notation)
2

20

£11115; Structure

Moneclinic Heeach Point Groups '

Symmetry operations
or group elements
2,3
2f,z
2,2',E,s'
mJ
m'J
m,m'.,E,E'
2,m,i,E
' C

2,mi,i ,E

v v
2.0.91 ,3
2',m',1,n

. v t v c
2,m,i,E,2 ,m,i ,1

The chemical space group of azurite is P21/c at room

:emperature. 2" The ounit cell dimensions are: a0 = 5.00,

o: 5. 35, co = 10.35;, (3: 92°20.

divided in two sets.
Occupy the special position

The copper ions may be

In the first set are the ions which

o'o'o;0'*’£.

 

2‘" G. Gattow und J- Zemann, Acta Cryst.

g, 866 (1953) .

1.1

The second. set occupy the general position

3”." 2.1.33 £+YO é": X, é’Ys *+'s
where 220.252, y: 0.1.95 and z : 0.085. There are also four

hydroxyl radicals in general position.

“Point W-

There are two possibilities to consider at l.6°K.
l. The smetry of the atomic positions
remains le/c.
2. Statement number 1. is false.

If atomic positions are altered so that the x-ray sym-
metry is no longer P21/c in a crystal with as complicated a
structure as asurite, it is almost certain that the symmetry
change would result in a triclinic crystal. If the x-ray
syn-etry is triclinic at 1.6°x, then one of the triclinic
Shubnikov space groups must represent the symmetry of the
magnetic cell. Then this group must correspond to one of
the triclinic Heesch point groups in Fig. 10. However, none
or the triclinic point groups of Fig. 10 have the observed
aRial point group syuetry shown in Fig. 8. Therefore, we
311311 seems in the future that the x-ray smetry of asurite
‘t l.6°K is le/c. There exist however, four other possibil-
ities. Before proceeding further these will be discussed.

1. If the x-ray symmetry at l.6°K is triclinic,
it would still be possible to observe the local
field pattern shown in Fig. 7., as the point
groups 12? and ll' in Fig. 10 permit the

#2

observed symmetry provided that vectors 1 and

2 and 5 and 6 of Fig. 7 are unrelated to each
other by any symmetry operation. This would
imply that the observed symmetry of orienta-
tion which for example seems to relate vectors
1, 2, 3 and b is accidental. This possibility
is unlikely especially in view of the fact that
the magnitudes of the vectors 1, 2, 3 and h are
the same.

2. Suppose the xrray symmetry at l.6°K is a mono-
clinic space group other than P21,Pc or P21/c.
Again, this would require a considerable re-
arrangement of ions and therefore lead almost
certainly to triclinic symmetry.

3. If the xpray symmetry at l.6°K belongs to some

system other than triclinic or monoclinic25,
then the Heesch point group might also be one
of higher symmetry than those in the monoclinic
system. An investigation of all of the point

. groups of these higher symmetry systems shows
none which predict the observed axial vector
symmetry.

4. In the event that the xpray symmetry is P21, or

. Po at l.6°K, none of the arguments to follow
need be altered. The only change would appear

 

25"1'he possibility, although unlikely, exists: see
L. D. Landau.and E. M. Lifshits, Statistical? hsics(Addison-
wesley Publishing Company, Inc., Massachusetts, p.k33.

 

#3

in the final conclusion. If, for example, the
xpray symmetry at l.6°K were P21, then instead
of the four Shubnikov groups found in Table 10
as representing the possible magnetic space
group symmetries, P321 could be immediately
selected as the only possible antiferromagnetic
symmetry group. That is, it would be the only
one of the four groups which would have the

correct xbray symmetry.

In view of the previous remarks, the point group in

the antiferromagnetic state must satisfy the following

'conditions:
1.

2.

The point group must transform an arbitrary
axial vector in the same way as is permitted

by the resonance pattern. (Fig. 8 for asurite).
The ordinary crystallographic point group ob-
tained by replacing_all antioperations of the
point group by their corresponding ordinary
operations must be either a proper or improper
subgroup of the ordinary crystallographic point
group of the crystal. For asurite, this is 2/h.

From Fig. ll it appears that the only point groups which
'
satisfy these requirements are 2./a, 2/5', 2/51', 21' and ml'.

Space-Group Egggirement ‘
If the crystalstructure.and resonance data (previously

described for asurite) are available, then one may proceed

M

to select as possible antiferremagnetic.space groups those
groups in Table 6 which satisfy the following requirements.

1.

2.

3.

h.

The operations of the Shubnikov space group
must leave unaltered the positions of the ions
as determined by the xsrey data. This implies
that if a symmetry operation of the Shubnikov
group transforms the ion position‘? into'§,"
then the chemical space group must contain a'
_corresponding symmetry operation which will
transform é into h. The converse ig,ng§ true
and therefore there may exist pairs of ions
whose positions are related by symmetry opera-
tions of the chemical space group but which
are not related by a symmetry operation of the
Shubnikov'group.

The Shubnikov space group must have as its
point-group one of the possible point groups
predicted by the data. For asurite, this is
one of the five point groups feund above.

The Shubnikov group must permit a magnetic unit
cell which contains the same number of protons
with different local magnetic fields as there
are resonance lines in the nuclear magnetic
resonance pattern. For asurite, this number
is eight.

A magnetic ion may not occupy an anticenter.

An anti-inversion moves an axial vector through

#5

the anticenter and then reverses its sense.
Hence, the magnetic moment of an ion located

at an anticenter is zero.

SE§¢69020§£.30;g¢t;Qnu
The left-hand column in Table 10 consists of all those

monoclinic groups in Table 6 which satisfy the previously
discussed requirement number 1. An asterisk to the right
of a group under one of the columns indicates that that
particular requirement number which heads the column.i§,ng§
satisfied by the group. In the last column at the right,
the corresponding point groups are listed.

Table 10. Space Group Selection

Shubnikov Requirement
Point Group
Space Group 2 3 h
'
Pazl/c * Z/hl'
Pzi/cf * * 2'/m
le/cf .. t Z/mf
PZi/c * * 2'/m
P21/c t t ' 2/m
0
08¢ ml.
1
Pbc ml
0
Pac ml,
Pc ‘ * =0: m'
Pc * S m
9
P821 21
' v
a a
P21 2'

921 a: * 2

#6

Consider Fig. 17 page 53. Disregarding for the time
being the numbers in the circles, it may be seen from the
x-ray data that the circles represent the positions of the
copper ions in two chemical cells placed next to each other '
along the x direction where the x,y and s axes of Fig. 17
coincide with the a,b and c crystallographic axes in the
x-ray section. Let us now investigate the group Pazl/c in
Table 10. Fig. 12 is a diagram of the group where the
solid rectangle represents the outline of the projection of
the double chemical cell of Fig. 17 in the x,y plane. A
double chemical cell is needed for Pazl/c since the group
contains an antitranslation in the a direction.‘ The unit
cell of Pazl/c is therefore Just twice the volume of the
chemical cell. The unit cell of this group therefore con-

tains eight protons. However, as may be seen from the

Fig. 12. Diagram of Pazl/c.

 

colored diagonal glide
plane n' superimposed

on an uncolored glide

 

 

1.7

following they do not possess different local fields. Suppose
the components of the local magnetic field at one proton
located at (x,y,s) is Ex,fly,H‘. The anticenter at (£,£,£)

in the magnetic cell implies that the proton at (é-x,y,s)

will experience a local field the components of which are
‘§¥,§; and E, where the bar indicates a negative quantity.

The two protons as well as the local field vectors are trans-
lated to (x+ é,y,s) and (25,?) respectively by the antitrans-
lation in the a direction. The local field components at
these positions are then Evil-53: and £198,413. Thus only

two of these four protons experience different local magnetic
fields. A similar argument applied to the remaining four
protons leads to a similar conclusion, indicating thatthere
are in all only four proton positions at which there are
different local fields in the unit cell of 9321/6.

The group Pac, the diagram of which is shown in Fig. 13,
also contains an antitranslation in the a direction.* There-
fore, the magnetic unit cell of this group contains eight
protons. These may be divided into two sets of four each in
such a way as to make those in one set completely independent
of those in the other set as far as the operations of the
group Pac are concerned. Let the components of the local

field vector at the proton‘ position (x,y,z) in one set

*In the diagrams of the space groups Pac, Pa21, Pbc and
one to follow}_as well as thegroup Pazl/c, the a, b, and c
crystallographic axes coincide with the x,y and a cell axes
with one exception. In the diagram of the group P c, the c
axis is directd-£rom.cell coordinates; (0.04))» (3,0,1).

#8

Fig. 13. Diagram of Fee.

 

 

 

 

 

fe- 0+ '2' Oink 1: ! 0+
a 31’ if f
{ '1?
i- {L
f 1
~15." “1 1 0+
1
1. :1, 1
. '1
0+ ’ Oi+ 0+

be Hx,fiy and Hz and those of a proton in the other set at
(x. ,y',s') be Bx" Ii", and Hz! where the positions (x,y,s) and
(x',y’,s') are related by the operations of the space group
le/c but are unrelated by those of Pas. The existence of
eight different local fields is predicted by the group Opera-
tions as shown in Table 11 where c,n, and t8 represent a
glide reflection (reflection followed by a translation of
half the length of the magnetic cell in the c direction), a
diagonal glide reflection and a translation in the a direction
respectively.

Similar magnetic field transformation tables may be
constructed for the groups P821 and Pbc. Their diagrams
are shown in Figs. lb and 15.

#9

Table 11. Magnetic Field Transformations under

Pac

F c n7 t;
197.: 1.! - v.8 + i x + Liv - v.2 -1 i: 2H tam

3,, fix ax fix

By 3’ Hy "y

a, 3, Hz If,

x',y',s' x',é - y',s'+ is xf+ bi - y',2'+i x'+§,y',z'

ax. if; ax. Fix.

Hy} 3y! 4 Rye 5y;
“5’ Hz! H39 Hz!

Fig. la. Diagram of Pa21°

 

 

 

 

.-
1, J. We VIA—L
I 4 0+ (3+
( pain—'1
3 "J 4
O- I
I 4.x-
"‘ <7“ 4
4 0+ .+
I . LA;
4‘ ‘”‘ 4
a-
_L <+ ill—32L
4 0+ 0+

50

 

Fig. 15. Diagram of Pbc
~~+b ___O+ Oi’ .+ ‘4‘“ 0+
4 ; '
a 1 ( /
I ' v
: f ,+<_n___c
3 % ;
0+ 0:11 L 0+“ 09 -1 0+

 

 

 

For brevity, only the local fields at the proton posi-
tions belonging to one of the two sets of protons are shown
in Tables 12 and 13. The other set is obvious as in the case

of Pas.
Table 12. Magnetic Field Transformations under Pazl

'
i: 2 2 "
1 1 *3,

X.Y.z i-X.Y+£.i-z 1-X.Y+£.i-z X+imz

Hx ' Hx Hx Hx
H. E. *5 5
Hz Ha as as

Table 13. JMagnetic Field Transformations under Pbc

I t

E c p c tb
sz _ Li - 3m + i 1% - v.2 + i any + 1.2

”an Hx 3:

‘4:

tr

f
fiJI-‘olxtnl

51

The antitranslation in both the a and b crystallographic
directions in the group cac leads to a unit magnetic cell
four times the volume of the chemical cell. The diagram is

shown in Fig. 16.

Fig. 16. Diagram of cac

 

 

 

 

 

 

 

.145 0‘1 01” T 019105 1 -1 0+
sf ’1. 21 ’1
1 1 :’ ’.
Er ;- :3 ,
3'1 ,3- 1 A
“31”" 1. (“101* ,2 0+
3' 1 {L ,1 =
21 :1 .L 1 i
11’ ’Ii :, x 1
e l 3L , :. :1 ‘ 1 _J
0* 01+ 0+ 01+ 0+

The sixteen protons in this magnetic cell may be
divided into two sets of eight each, again in such a way as
to make those in one set unrelated by the operations of
Cat to those in the other set. is above, only the local

.fields at the proton positions in one set are shown in

Table 1h.

52

Table 1h. Magnetic Field Transformations under Gas

3 c _ c 11
am 21.14.2111 X.%-7.z+i an him“ 1
Ex i; Hx ax
“y E, “v f,
Hz Hz Hz Hz
1 o ' t t
n th ta tatb
3“ iek’yez12 X,Y+§,Z x+2eYez 1+ *eY‘f £93
Hx Hx x Hx
Hz 5, a; a,

Possible Magetic Structures-

Two possible magnetic structures with the syuetry of
the groups Pac and P821 are shown in Figs. 17 and 18 respec-
tively. Tables 15 and 16 accompany the figures. Magnetic
structures for the two remaining groups26 Pbc and age are.
illustrated in Tables 17 and 18. The first number in the
symbols in the left-hand column of each of the tables 15, 16,
17, and 18 designates one of the four magnetic moments which
belong to that particular family of ions which are all con-
nected by the group operations. The second number in the
same symbol indicates to which one of the three families
(1,2, or 3) the ion belongs. The copper ions in special

*

2!)..th structure a ested by W. Van der Lugt and N. J.
5°“11’ (P1373133 35.- 1313 1959)) is permitted by the group
ac.

 

 

 

53

 

 

 

 

 

 

 

gen—she
use ”Haas espouse: .ea .msm

 

 

 

54

 

 

 

 

 

 

 

 

 

 

 

 

”Zoo «Sense: .3 .m:

 

55

Table 15. Magnetic Structure: Pac symmetry

Numbers in symbol Ion position Magnetic Ion in

at copper ion (x,y,z) in Moment set
position magnetic cell Components - number“)
1,1 ' 0,0,0 “lictulyvulz l 1
1.1 1.0.0 Elx'fily'fils 1
2,1 0.1.1 uu'fily'“1s 1
2,1 1,4,; $111,111,512 1
1,2 .126, .005, .585 uzx,u2y,u2, 2
1,2 .626, .005 , . 585 321,523,523 2
2,2 .126,.l.95, .085 “2x'32y0‘122 2
2,2 .626, .195, .085 fizxenZye-fiZg 2
1.3 3711,1995..le u3x,u3y,u33 2
1,3 .8711, .995 , .415 33:53,.‘63, 2
2,3 .371, .505 , .915 u3x,‘113y,u3z 2:
2,3 .871. .505, .915 63x,u3y,ii3, 2

*) See section on x-ray data

Position (set number 1 in the right—hand column of the above
1‘ our tables) are not connected by the operations of the group,
1' or which the particular table is constructed, to those in
Senoral position (set number 2).

If every copper ion which is linked to another copper
1011 through a common oxygen ion is coupled antiferromagneti-
ealily by a super-exchange mechanism to the other copper ion,
th. resulting magnetic structure forbids an anti-translation

56

Table 16. .Magnetic Structure: 2,21 symmetry

Numbers in symbol Ion position Magnetic Ion in
at copper ion (x,y,s) in Moment set

position magnetic cell Components number

1,1 0,0,0 ulx,u1y.ui5 l

1,1 £,0,0 “lx'uly'a1s 1

2,1 0,},f “1x,aiy,“13 l

2,1 2.2.2 21:,u1y'fii‘ 1

1,2 .126,.005,.585 112:,”an8 2

1,2 .626,.005,.585 influx” 2

2,2 .371...505,.915 fizxmzyifiz. 2

2,2 .871», .505, .915 u2x,‘i2y,u2, 2

1.3 o37h.o995.ohl5 “3;,u31,u3, 2

'1',3 .871...995..1.15 631,3”,33, 2

2,3 .126,.z.95,.085 331,113,,33, 2

2,3 .626, .195 , .085 u3x,33y,u3z 2

in the b direction. In such a case, the groups Pac and

P82 are the only possibilities.

1 It should be noted that except for the fact that they
are not zero no use has been made in the above analysis of
the magnitudes of the local field vectors listed in Table 8.
In principle it may be possible to obtain a complete des-
cription of the magnetic structure by combining the known
local fields in the antiferromagnetic state with the posi-

tions of the protons obtained from proton resonance in the

57

Table 17. Magnetic Structure: Pbc symetry

Numbers in symbol Ion position Magnetic Ion in
at copper ion (x,y,s) in Moment set
position magnetic cell Components umber

1,1 0,0,0 uh,u1y,u1, 1

1,1 0,1,0 . enamel, 1

2:1 0.1.2 313.“).y0-‘ilz 1

2,1 0,137fi ullenun 1

1,2 .252, .0025, .585 unwary, 2

1,2 .252,.5025,.585 321,122,332, 2

2,2 .252,.2A75,.085 91.9,,32, 2

2,2 .252,.7t.75,.085 1121,32,,112, 2

1.3 .7118, .2525, .915 u3x,u3y,u3z 2

1.3 .7618, .7525, .915 Era-3,0332 2

2.3 .7118, .4975 , .1115 u3x,23y,u3, 2

2,3 .7w,.9975,.z.15 33x,u3,.'63, 2

paramagnetic phase and the known crystal structure. However,
the work of Poulis at al. on Cucn.2 . 21120 has clearly indicated
that this program is extremely difficult if not impossible.

58

HHHHNNNNNNNN

senses

see me noH

ems.hna.xns
umMyhms.Nm“
mm:.hw“rxms
mmMrhmmrwa
suntan: anus.
nus.hWMrw~s
muflahmmsxmfl
sways“
was.has.xas
mdashdflaflaa
umwrhasawmm
avenues“

senescence

99—0303

unease“:

hhoommhm omo

.mae..meme..eee....mas..meom..esm.v
.mns..mesm..sem...Amas..mees..sam..
.mam..mum~..¢as...xmam..n~na..een..

RS. .32... .43. .. Ram. .33. 35m. 9.

.

.mwo..mee~..e~e...Ammo..meee..o~a..
Ammo..meee..o~e....nwo..mee~..oea..
new“..m~oo..m~e....mmm..m~om..n~a..
.mmn..m~om..e~e....nmm..mmoo..ema..
Acacias.
sameness.
.o.m.o...o.o.h.
.o.m.m...o.o.o.

Aden oaaemwes ca
Am.hawv noavaeoa mOH

«onsposaem eavesmmz .wa manna

&

a: o: N. on
Q 1 Q
“—4 0: IN H

N
a

r4 .4 ri'r4
F‘lf4 cu lo: .4

coauwmoa
so“ menace
no Honahm

ea opossum

Section III

Twinning by.Merohedry and ReticularlMerohedry

Inggggugtiggr
1 .Two identical crystals (or individuals) are said to be
in twin relation to each other if they are joined together in
some way such that the pair possesses an element of macro-
scopic symmetry which is not a symmetry element of either
individual. This element of’macrescopic symmetry is the same
as that dealt with in crystal morphology. A plane of symp
metry which relates two twinned individuals to each other
(twin plane) must be parallel to a lattice plane in both
individuals. A twin axis must be parallel to a lattice row
common to both individuals.
‘With rare exceptions, all known twins can be described
in one of four ways (known as Friedel's rules of twinning.)27
1. The individuals of the twin are related by a
twin symmetry element which is an element of
symmetry of the lattice of each individual but
which is not a symmetry element of the crystal
(i.e., either individual). This type of twin-
ning is called twinning by merohedryi'i

 

,. 27°See1the review'by R. U. Cahn,,idvances in Phys. 3,

* The term merohedry, of Greek origin, implies that

each individual of the twin has fewer faces than it would
have if it had the full symmetry of its lattice.

59

1 60

2. The individuals of the twin are related by a
twin symmetry element which is an element of
symmetry of what is called a coincidence lattice
but which is not a symmetry element of the
crystal. The coincidence lattice extends from
one individual to the other without modifica-
tion but consists of only some of the lattice
points of each. This type of twinning is called
twinning by reticular (or lattice).morohodry.

3. The individuals of the. twin are related by a
twin symmetry element which is g1g3g5;a cym-
notry element of the lattice (i.e., a pseudo-
symmotry element of the lattice). This method
of twinning is therefore Just an extension of
the first method and is called twinning by
pseudo-norohedry.

L. The individuals of the twin are related by a
twin symmetry element which is giggg§,an ole-
mont of symmetry of a coincidence lattice.

This extension of the second method is «11.4
twinning by pseudo-reticular norohedry.
The formalism of the Hoosch groups has only recently
been employed by Curien and Le Corre28 to describe all of
the possible norohedry and reticular norohedry twin relations
in crystals the symmetry of which is completely described by '

 

 

28. H. Curien, and 1. Le Corre, op. cit.

61

the position of the ions. These relations are called twin
groups. '

Although twinning in the antiferronagnetic crystal liO
has been observedf, so far as is known to the author no one
has yet described the possible.merehedry and reticular*mere- I
:hodry twin relations in antiferremagnetic crystals. These
twin groups are constructed and listed in this section. A
total of 177 nerehodry twin groups are described; of these,
35 are the sane as the 35 lerehedry twin groups found by
Curicn and Le Corrc while the remaining 1&2 groups are new;
In addition, 51 reticular nerehodry twin groups are listed;
of these, ll are the same as the ll reticular nerehodry twin
groups found by Ourien and Le Corre while the remaining 1.0

groups are new;

mum

a. Ordinary Lattices

If the symmetry of a crystal is lower than the symmetry
of its lattice, twinning by nerehodry is possible. This is
because there are then various possible orientations of the
atoms of the crystal relative to its lattice. A possible
twin operation is an operation which relates one of the pos-
sible orientations of the atone to another. These operations
aro.the point group symmetry operations of the lattice which
are not contained in the point group of the crystal. But

fDiscussed_at the end of this section

62

what are the point group symnetry operations of the lattice?
In ordinary crystallography, these operations conprise the
highest point group symmetry possible for the particular
sy-etry system to which the lattice belongs (This is called
the holohedral point group.) . For exanple, for the mono—
clinic lattices, the holohedral point group is 2/n. For our
purposes, it is helpful to think of the holohedrnl point
group as resulting fren the fact that the sy-etry of the
neneclinic lattices permit the simultaneous existence of a
two-fold axis (or two-fold screw axis) and a plane (or glide
plane). These two operations also imply a center of sym-
metry and hence the group 2/n contains 1. elements. Consider
now a crystal which has point sy-etry 2. According to tb
above, it is therefore possible fron the standpoint of sym-
metry for the crystal to twin by nerehodry by either a twin
plane or an inversion center (it will later be shown that
these two are equivalent in this case). Ourien and Le Corre
express this fact by using the Heesch group 2/n'. which they
call a twin group. The twin group therefore expresses the
point sy-etry of the crystal (2) and also identifies the
equivalent twin operations (n'. and i'.) throughthe use of a
prime (see Thble 9). In the next two paragraphs the notation
in Section II for the Hoesch groups will be used to express
twin groups. However, a prine will only denote a colored
operation which nay or may not be a twin operation. A twin
operation will be these operations not contained in the point
group of the crystal but which.are included in the 'helehedral'

63

point group. In all work which follows the next two para-
graphs, an asterisk will be.used—to.explicitly indicate twin
operationsvand the twin group notation will be changed. For
example, Curios and Lo Gerre's.twin.group.2Amf above will be
denoted by 2:19;. ‘ ' ‘
Suppose that the point group of a crystal under considera-
tion is 2’. The lattice of this crystal is the same as the
previous crystal, i.e., an ordinary uncolored lattice. How»
evcr, as the list of Shubnikov groups show; the uncolored
lattices of the monoclinic system permit the simultaneous
existence of combinations of colored axes with colored and
uncolored symnetry planes which result in two 'helohedral'
point groups as far as the crystal with point group 2) is‘
concerned. These are 2'/n and zffim'. These two groups imply
two possible twin groups and therefore two possible distinct
twins. The equivalent twin operations implied by the first
group (2f/h) are an uncolored mirror plane and a colored in-
version centcr, while those implied by the second group are
a colored mirror plane and an uneolorcd center.
Let us return to the example where the crystal has point
symmetry 2. If the crystal is a type for which its symmetry
is completely expressed by the position of its ions, then the
only twin group which exists is that found by Curios and
Le Corre, discussed above. If the crystal possesses some
other characteristic (such as magnetic moments) so that it
belongs to a general category of crystals (such as antifcrre-
magnetic crystals) the symnetry of which can be described by'

6b

the Shubnikov group theory, then there exists the possibility
of colored twin operations even though the point group of the
crystal is uncolored. Proceeding then, in the same way as

in the case of point symmetry 27(where the possibility of:
colored twin operations must obviously be considered) leads
to another “holohodral“ point group Z/hf. This twin group
implies that the crystal can twin by a colored plane or
equivalently by a colored inversion center.

It is therefore evident that the point symmetry of the
lattice must be modified as described above in extending the
idea of twinning to types of crystals subject to the Shubnikov
theory whose space groups are based on ordinary lattices.

b. Egg-M Lattiggs

For twinning in blackuwhite lattices there is always
only one holohodral point group. This group is Just the
gray Hecsch group which may be constructed from the ordinary
(or uncolored) holohodral Hccsch group by introducing the
element 3’. For example, 2/m-r2/ml'.

mission 9.: 1mm 5mm
When twinning by reticular norohedry takes place, the

lattices of the two members of a twin are not parallel in
orientation. There exists, however, what is called a coinci-
dence lattice which consists of some of the lattice points of
both individuals and continues from one individual to the
other without modification. The remarks made above concern-

ing the 'helohedral'.point groups of uncolored (or ordinary)
lattices and the holohodral point group of blachdwhite

65

lattices are also true for uncolored and black-white coinci-
dence lattices.

a. Ordigggz Latticeeeg_i coincidence lattice may be
defined for rhembohedral and cubic crystals. This coincidence
lattice consists of some of the lattice points of the rhombo-
hedral lattice for a rhembohcdral crystal and some of the
lattice points of the cubic lattice for a cubic crystal. For
rhembehodral crystals, the coincidence lattice is only those
points of the rhombohedral lattice which generate those
lattice points described by the symmetry of a hexagonal
prism. The hexagonal prion and the rhombohedral lattice R
is shown on page 20 in the 'Thc.International Tables for
IéRay Crystallography" and redrawn for convenience in Fig. 19.

Fig. 19. Hexagonal Prism and Rhombohedral Lattice R.

 

""‘"“"5?

 

 

 

 

"F

 

 

66

The hexagonal prism is Just a multiple of the C lattice of
the hexagonal system in Pig. 5. Hence, possible reticular
nerehodry twin operations for rhombohedral crystals will be
point group operations contained in the hexagonal system
which are not symmetry operations of the rhombohedral crystal.
For cubic crystals, the coincidence lattice consists of
only those points of the cubic lattice which generate the
lattice points described by the symmetry of a hexagonal
prism. This hexagonal prism is the same as the one generated
for rhombohedral crystals and arises in exactly the same
way, i.e., from a rhombohedral lattice. The origin of the
rhombohedral lattice in cubic crystals is clear when it is
remembered that cubic lattices are Just special cases of
rhombohedral lattices. As an example, consider the face-
centored cubic P lattice. It may also be described by a
rhombohedral primitive coll shown in Pig. 20.

Fig. 20. The Rhombohcdral Primitive Coll of the
the Face-Centered Cubic I Lattice

 

T

 

 

 

 

 

 

67

The operations available for possible twinning by roti-
cular norohedry in cubic crystals are therefore those opera-
tions of the hexagonal system which are not symmetry opera-
tions of the cubic crystal.

Sineorthere are four space diagonals of a cube, the
rhombohedral primitive cell can be oriented in four ways.
This implies.four possible orientations ofthe hexagonal.
lattice witherespect to the cube. Then a.posaible reticular
norohedry-twin operation.can be.orionted in four possible
directions.- The sithold rotation.axis.for example can.
have the direction.of any one of the four space diagonals of
the cube. .

b. Blackéihite Lattices. If the rhombohedral lattice
in Fig. 19 ia.body centered with a black lattice point, it .
becomes the black-white.rhombohedral lattice RI in Pig. 5.
The hexagonal prism then has a colored translation in the
c direction and is Just a multiple of the Cc lattice of the
hexagonal system in Fig. 5. This implies that the coincidence
lattice for rhombohedral crystals with lattice RI consists
of those points of the rhombohedral lattice RI which gene-
rate the lattice points of a hexagonal prism.with a colored
translation in the c direction. The possible operations of
reticular norohedry twinning for crystals with lattice RI
are then those point group operations compatible with the
symmetry of the black-white hexagon which are not point
symmetry operations of the rhombohedral crystal. The holo-
hodral point group of the black-white hexagon is 6/mmlf.

68

As mentioned above, the lattice P of the cubic system
is a special case of the lattice R of the rhombohedral system.
Similarly, the lattice PI of the cubic system is a special
case of the lattice RI. 'The only other black-white lattice
in the cubic system, I", is also a special case of RI. This
is shown in Fig. 21.

Fig. 21. The Rhombohedral Primitive Cell RI of the
race-Centered Cubic Lattice P.

“’0

 

\

V ._._..b
/ 4

 

J"

a

For cubic crystals with black-white lattices, the coin-
cidence lattice is therefore the same as the coincidence
lattice for rhombohedral crystals with lattice RI. The
possible reticular norohedry twin operations are therefore
these point group operations compatible with the. symetry of
the blacks-white hexagon which are not point sy-etry operations
of the cubic crystal.

69

gm.un flotation. ‘
Let a twin operation be denoted by qf. This operation

may be thought of in the following way. The operation q .is
(the operation which transforms one individual of the twin
at position #1. to the other individual at position #2. The *
then colors the individual at position #2 black while the
other individual is left uncolored. If t is this coloring
operation we may write qf: tq. If the coloring takes place
before the operation q, the result is still the same, i.e.,
q*-_-. qt, or t comutes with q. Since there are only two
colors, t2: 3. Thus t has the same mathematical properties
as the reversal operator (see Section II).

A twin group gt is defined as a set (of elements of the
‘ general type (9109:} satisfying the group properties with
the following structure: the set { p1} is a subgroup of index
two of 3t and ipflEH where H is the Heesch group of each of
the two individuals of the twin. Since I! is an invariant
subgroup of ‘t' the group g‘ may be written as

5t: 11,qu or
3t: (11,3;B

where for example, the elements in the left coset q'fB are
q*p,qu2, ... , and where q*p1§ (1;. A group of the type gt
is therefore constructed from H and one element q not in 1!.
Since all of the p1 belong to 3, all the twin elements q:

in 3t are equivalent. That is, they all imply two individuals
of a twin related to each other in exactly the same my.
Therefore, the group ‘t so defined expresses the symmetry

70

of each of- the twoindividuals. of.thc.twin and-enumerates a
possible twin operation as well as. all possible twin opera:-
tions equivalent to this twin Operation. In order to make

the notation as explicit as possible, the group gt will be
written in the general form ngf'

 

. Except for a slight modification for cubic crystals
which will be discussed later, the .nctationncmployed for
twinning by reticular norohedry is the same as that used
for twinning by norohedry.

Con of m
The problem of finding all of the possible norohedry
and reticular norohedry twin groups may be broken into two
parts:
1. The construction of the set of point groups
{gp} from which a set of point groups { gt}
may be selected where the set { ‘t) are those
groups which describe all of the possible
point symmetry twin relations.
2. The selection of the set { g J .
l. The Shubnikov space groups which are not gray may
be divided into three general types: 81,82, and 33. To
describe these, lot 1'1 represent a pure translation operation

of the symmetry. Lot 11 represent any one of the other
operations of the space group synotry.

71

The three general types may then be written symbolically
in terms of the types of operations they contain:

81:: A1,T1
32 = A1,A3 ,1'1
33: A1,A3,T1,T3
where the identity I may be thought of as one of the opera-
tions of the set {A1} .

The point group operation a1 corresponding to A1 may be
found by replacing the translational part, if any, of A1,.
with the identity 8. The point group operations correspond-
ing to A331 and T3 are then a3,3, and If respectively.

Consider now the Hecsch point groups corresponding to
groups of type 81. These groups are the well known 32 point
groups any one of which may be written as in“ . Therefore,
any point group belonging to the sot { gp‘g which could des-
cribe a twin in a crystal with space group type 81 can con-
tain only two kinds of operations: a1 and a; where a; is
a possible operation which carries one individual of a twin
edifice into another. This group type may be written as

{ are?) . Groups of this type are isomorphic to the 58
black-white ficesch point groups. Groups of the type {‘i";}
are not considered as possible twin groups since they contain
the operation 3". These are isomorphic to the 32 gray acesch
groups.

A Heesch group corresponding to a group of type 82 con-
tains only operations of the type a1 and a3 and may be written

symbolically as {eras} . Therefore any point group in the

72

set {gp} which could describe a twin in a crystal with space
group type 32 can have only operations of the type: a1,a3,a:
and a'f'. There are two general kinds of these groups: one
contains a subgroup of index two of the type {a1} with the

rest of the operators of the type or. A group of this kind

may be written as {a1,a3"_‘} . There are 58 of these since
they are isomorphic to the 58 black-white Heesch. groups. The
second kind of group is that which has as a subgroup of in-
dex two one of the black-white Hcesch groups. This kind con-
tains all fem! possible types of operations and may be written
as {a1,a3,a:,a;f . There are 111 such groups. These groups
are constructed in the following way. Start with the Heeeeh.
group {sung} . Find all Heesch groups for which ferns)“

a subgroup. of order two. In those groups place an asterisk
over those elements which are not contained in the original
Heesch group brag} .

Finally, consider those Heesch groups corresponding to
groups of type 83. Any one of these may be written “(31";,(°
Therefore, any point group in the set fgphhich could. describe
a twin in a crystal with space group type 33 can contain only
operations of type a1 ,a; ,a; and a3; that is, groups. which
contain 3' but not 8*. The following argument shows that

the only possible groups,“ this type may be written as
'*

J

3108;0339‘

73

1. According to the above, every conceivable group
must contain operations of the type a1 and a;. Suppose those
of type a; are also in the group. Then Bfa3:a3*. Since no
other types of operations satisfy the group requirements, the
operations aéf above may be written as the special type a3*.

2. Suppose we start with only the operations a1,a; and
a3‘. Then E'a a3‘: a; and again no other operations satisfy
the group requirements.

The groups of type {a1,a1,a;,a3*} may be constructed
by introducing the element I. into the group {a1,a: I. There
are therefore Just 58 of the groups {avaLagJB’f‘ .

The following table summarises the above’.

Shubnikov Space Corresponding Hoesch Group Types

Group Type Point Group Type in the Set {gp}
31 ‘1'? (‘1)32 58 {‘1"J)58
S2 A1A3,T1{a1,aé} {a1,a3* }“ :{a1,a3,a;,a‘ ‘m
'fSB A1,A3,Ti,T3 {a1,a1}32 {a1,aJ.a1,a3 m}

Specific and general examples of the multiplication
tables for those groups in the right-hand column above, are
shown on pg. 7b.

From the above, the set {gp} contains 285 groups. The
original 90 non-gray Hoesch groups are listed in Thblo 19 at

 

?) }“ denotes I groups of type {'}.

'T'Included in S3. is the group P .1, even though it con-
tains no elements Ad.

Grou Type
and Syi’obe of
Specific-Group

{‘10‘3)
2;P"
y

r 0*
1‘1": }
2;P"'l

{a1 ”3,4335%

*
l';Py

 

710»

§8§°m° imp le of _ General
tiplication Table Multipbi-cation
Ta e

*

3&3. P*c

. ' we]
I 31.2 Py d" —‘—J—,
LhyL ::c"""1>"‘['l~1](a1)I
P! y
0*

2y I

 

{some}?
{spa}, {a1 "3)

 

_(‘Li'g‘g } H931

[‘i-‘J {era’s}

75

the end of Section III along with the group types {“1083} ,
{avaS'Uand {a1,a3,a:,a'*} derived from them.

In Table 20, the groups of type {a1,a3,a;,a3f}are
listed.

The notation used in writing the symbols for groups in
the set {gp} (except for those in Table 21.) is the following:
the Hecsch subgroup B of index two is written followed by a
semi-colon after which is written a symbol a? or In"? which
represents respectively either an uncolored or colored twin
operation contained in the twin group.

2. The first step in the selection of the set of pos-
sible twin groups {gt} from the. set {gpg may be accomplished
by choosing for each Heesch group all those groups in { ‘13)
for which that Heesch group is a subgroup of order two. This
listing is shown in Table 21.

The norohedry twin groups corresponding to each Hoesch
group may be found from the above listing. They are those
groups to the right of the fleeach group which belong to the
same synetry system as that particular Hccsch group. The —
norohedry twin groups are listed in Table 22.

There are two kinds of norohedry twin groups for crystals
with Hecsch groups of the type {a1} . One kind has the form
Half where H is the uncolored Heesch group { a1} which repre-
sents the point syuetry of any individual of the twin edi-
fice. These twin groups are the same as found by Curien and
Le Corre. The other kind has the form 11:14" where every twin
operation is a. colored operation. ( . '

76

The norohedry twin groups for crystals with Heesch
groups of the type (91033} are of only one kind but have
symbols of the torn and} and am"? where a is the black-
white Hoesch groupjavaS} . ' The form Hm)"I has been used
rarely and only for the purpose of sledding identical sym-
hole for two different groups.

The: norohedry twin groups for crystals with Reesch
groups of..the type (“1"?) arise from twinning possibilities
in black-white lattices. These twin groups have symbols of
the form 8:14",I where B is the gray Heesch group {span .

It is interesting to note that in this case for every possible
twin operation If its colored companion 11'} also exists in
the twin group.

The reticular norohedry twin groups for rhombohedral
crystals are listed in Table 23 . The classification of these
groups by the type of their Hecsch subgroups of index two is
the same as above for norohedry twin groups . For example,
those groups of the form 8;)? where H is an uncolored Hoesch
group are the same as found by Curien and Le Corre.

The possible reticular norohedry twin groups for cubic
crystals are the same as those fororhombohedral crystals.
Following Curien and Le Corre, the cubic case is distin-
guished from the rhombohedral case by writing in front of
the rhombohedral twin group that cubic group which contains
the rhombohedral Heesch group as a proper subgroup. These
groups are listed in Table 21..

77

M-&WM

The only antiferromagnetic twins which seem to have
been studied to date are those investigated by Roth29 and
Slackao in N10. This crystal is cubic above the Nécl tempera-
ture of- 523°K. Below this temperature the cube contracts
slightly along one of its four-body diagonals. . The M
ofthe imsthenbecsmesrhombehedralwitharhomhohedml
angle ”90°13 for. a multiple rhombohedral. cell. containing
Asia. The magnetic moments associated witth hi. ions .form
ferromagnetic sheets which are perpendicular. to the contrac-
tion axis. If the contraction axis is [111], the sheets are.
parallel to (lll). The spin, direction is then. [110] and the
spins in adjacent sheets have opposite senses. This cell is
shown in Big. 22 (Pig. 1 in Roth's paper) where the open

circles represent oxygen atoms.

Fig. 22. Antiferromagnetic Structure of 310
[00!]

 

 

k
y C I ‘0‘“
/[:ool

”paw. L. Roth, J. of Appl. Phys. 1;, 2000 (1960) .

30-6. A. Slack, 'J. of Appl. Phys. 3_l_, 1571 (1960).

 

78

The 'T domain walls'I which are the observed twin planes
for the antiferromagnetic twins are restricted by the con-
ditions given on page 2002 in Roth's paper:

a. “The ferromagnetic sheets in adjoining domains
intersect along a common magnetization direc--
tion and

b. the domain wall contains this direction and is
parallel to a mirror plane in the original
cubic crystal.‘

Two examples are shown in Fig. 23 (Fig. 5 in Roth's paper)

fig. 23. T walls in Rio

 

 

 

 

 

 

 

 

 

 

 

1' \
i T
1
‘"F
I 1
i
- .. ---.J\
A .

 

In an attempt to apply the twin theory developed in this
section to these Rio twins, three general possibilities must
be investigated:

1. Twinning by norohedry

79

2 . Twinning by reticular norohedry

3. Twinning by some other. method

1- newsroom The point croup syn-om of
each member of the twin (in this case each domain such as
that shown in Pig. 22) must first be found. This can be
done in this case since the magnetic structure is known.

The space syuetry of the ions is rhombohedral, however ,_ the
magnetic moments do not have the three-fold synetry required
by the rhombohedral cell. This implies a triclinic Shubnikcv
group symetry. Therefore, the Heesch point group symetry
is also triclinic. Since no triclinic lattice contains a
symmetry plane, the twin planes in Rio are not norohedry

twin planes.

2- m 321 noises: means: one possibility
is also ruled out since this type of twinning does not take
place in triclinic crystals.

3. Twinning EZ§2£2 mm The twinning in 1110
can be described by twinning by pseudo-nerehodry. Possible
pseudo-norohedry twin planes are planes which are parallel
to planes which are pimps; planes of symmetry of the lattice.
The planes which are almost planes of symmetry of the anti-
ferromagnetic lattice of Rio are the planes of symmetry of
the cubic lattice I". This lattice contains anti-translations
along all three edges. ’11 the rhombohedral distortion is
neglected, there are only tool planes compatible with the
lattice P. which are planes of symmetry of Fig. 22 and which
also satisfy the requirements a and b above. One of these

80

is an antimirror plane parallel to (001) and the other is
an antiglidc plane parallel to (110). These planes are Just
those domain walls shown in Fig. 23.f I

If the magnetic axis is changed to any one of the other
observed directions shown in Table I,-psge- 2002, of Roth's
paper, there again.exist Just two planes similar to the two
found above which may be chosen from I. and which satisfy
conditions a and b above. These planes are the domain walls
listed in Thble I under that particular direction of the
magnetic axis.

L

*Tho point group equivalent of the glide plane will
also produce the observed twin.

Group

Table 19.
Group Types {a1, a3},

. ‘ U ‘ O . U . C U . . Q . C C O

.

NMMMMMNMMMNHWMMHMNNNM

Elements

81

The 90 nonegray Heesch Groups
and the, erivcd , e

vs
{git a1.) andfai, a3, ak, a1 }

The symbol definitions given
below are the same as those in
reference 10

L2, L3, L51, L4, Lzl, L6, L31
are rotations by 180, 120, ~120,
90, -90, 60 and -60 degrees res-
pectively. s3, 351, 34' szl, 36,
831 are respectively rotations by
120, -120, 90, ~90, 60 and -60
degrecs together with reflection.
in a plane at right angles to the
axis of rotation. The inversion
is C and P the reflection plane.

The subscripts x, y, 2, xy, 3,
indicate the direction

1
of the axis of rotation; the same

on L1 and S

subscripts on P indicate the
direction of the normal to the
plane of reflection.

The subscript.Lorllindicatcs
that the axis of rotation through

180° or the reflection plane is

perpendicular or parallel to the
main crystal axis of three-fold or

higher symmetry.

82

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Table 20‘!I Groups or Type {air a3, a;, a3*}

Group Elements
1 t a:
1 :6 E, c , 8r. '
v I! at
1 3L2: 3' I‘Zz' 8“ '
' at a: '
l :P E, P , 8r. '
Y y »
21';P"' E, 1.2 P’, 0*, 8:. v
.‘ z " I . -
mlzzy E, Py, Lzy,cv,&.'
- v at a: all
11_;1.2y E, c, 1.2,, Py, as"
0 It at s '
21.3 x 3' L29 1'va LZy! 8? '
21'-P* E P“ P" a v
.9 x 9 L23. *x’ I, . ‘
c. a: .
I]... 2: B, Py' L23' Px, 8c ' .
‘sl: It at at
22212151 E, 1'2:- L2,, 1.2,, P‘, Py, P,, c , a. c
at an: s t
‘1';sz Es L23: sz Pys L219 L2yo PLV 0 8° '
v » s an:
2/n1 n.3,: E, L2,, P,” c, Lgx, 1.33,, PX, Py, a; '
an: '- It
21.11;: E' 1’23' thv L 1,9 35 ' .
v 'at - at s at
41;sz 3: L1,”. LA}: L239 1'sz L2,, Lazy. LZfi! & '
v * - 1*
2221 :1.“ 3. L2,. L2,. L2” LT... L H. L2”. 1.3—. as '
0 s - a: ‘ .. a:
1.1 331.2 E, L“, 1.1.}, 1.2,, Pa, 333, sky, c , e .
‘ 1 all ' a: .. .. at '
2/nl :1.“ E, 1.2,, P2, 0, 1.“, 1.1%“, 3Z3, 31.; , as '
"' v as - at _ * at
41 3101‘: E, L22, 81." Ski, 1.1.2, Lhi." P3. 0 ’ & '
c s .. at an: * at
41 31’): n, L“, 1.4;, 1.23, P1, Py, ny, P-i-i, 8c '
'. * ‘ at .. a: it at ‘ ‘
Ill 6L1}: E, 10221 PX. Py, Ll“, Lag , ny. P-xy-y 8r. '

 

 

n eae , te enentsinMarenot
written explicitly for each group. Instead, only half the
elenents in each group are explicitly written. the retaining
half are collectively indicated by the symbols as '. T0111“
these elenents explicitly, write the slenentcto thvleft Of
a; and then prise each of then. For supple the four elements
of the first group (1':C*) ,are'E, 0* E , c”.

2221 :31“,
t *
ml 331.3
31 :13;
31.31,"
31'ssz,
azlfgsg,
9. *
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3-1'; ,3;
31'»;
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61 iLzl
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s s as
no LWL 1:39 1‘23: P8: 81,39 3;;s C, L23: L2,: L2”.

1 * t
Px, Py, PH, PL, a P

P

a:

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L-l

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3" Pr ny' P17: 173‘, 1;"
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L2”, 1.2;, 11:, 87;, 3“ , Cf, & '

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123:? 1’ . PLP 3;... 3351‘ '

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Lax,1.2’,P*P;yP;y, claw

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E, 1.2,, 3:1,, 31.11:: a: P
* t t

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3P 12:» I'nyP 1'23, 3:2. 3;}:9 P; P; 8‘ '
3P I‘2st xP pyP 3:” 33?: Lnyo LEE,-
3,13,, :1, 31%,, as P

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3' L3:" 1'3!” 33! P Says-1*P *3L21,0*, ‘1‘ '
B La» 1.5:, 86.. 831 c 31%;“. 31;, P '

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3" *‘
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P L3,. ‘1. guns-1* 4.: P'
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3' 13v L3,. 312.1! L6,. 1'62‘P 1‘2;- 31'2» 1‘ '

100

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3311‘, c"_‘, 8'. P

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1’1- 6‘3 as P

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- *
433% , P:, 8: P

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3- 13v Psi. 3P..- L3... 16?. 132.. 3P3“. as P

3' L3I' LigP 3L21' 33;: 35;, 33.. Pan L32: La:*,
L3v 3L;lP 32‘: 33:*P 33f: O*P &.' .
EP L3sP LiaP L63' Lat. L23, 36;: 33;. S3,, 35%.
P”, c, 61.3,, 6P,’,", a; P

2. Lung-1.31.26 s... 83:. n... c. L2,. Lara
1%.. 31-2.. 33;, 35?. 311?. Pi- PP

EP LBsP LitP L6zP LE:P L23. 6L21P 332. 3g;*, 3;,:

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533.3 6P”: Pg: 0.9 & '
-13:

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E; L33P LBsP L63; LfiiP LZsP 6fil' 6L21P SésP 36: P

_ ‘
33v 33?? P2P ‘1 P ‘1‘ '

a, 31-2. 1.1.3. 1.1.51. usz, usg1“. 3P“. 0*. as P

s, 31.2. 1.1.3. 1.1.51, 6P") 38:, 33;”, a; P

E, 31.2, 1.1.3, 4.1.51, 31.x, 31;”, 61;, a P

1:, 31.2. 1.1.3. LL51, 3L“. 31.31, 61.2, 38:, 33;;1“,

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61;, 33:, 33;“, 6P"; a. P

101

Table 21. The Groups of the Set{g§}listed

according to their Heesch Subgroup of Order Two

Heesch

Group

Triclinic

Monoclinic
2

Groups in{gp}uhich contain the Heesch Group
as a subgroup of Order Two

1;c"' 1:69" 1:13, 13.5: 1;P; 19;?
ifgcf 11; *, 11;P;
_ * .. e.
if”! 1332’.
l ; y 1‘;Py
., ,
ll 3L2,
'* i t
29; 2gp, 2:1.2x 2:14; 29; 2;P;* 2:1.“
0* , s '*
Z’Lk‘ 2,3“: 2,8h'
2'; "' 2';c* 2';ng 2';P;*

21';p; 2113Lgx 21'3P; 21.31.:3 21';8:,
nng’ ngLé; ”L;a nzLé:
{a}; “he“ (a; I'3P
.113Lgy ‘1'3ng

2/n;L;x 2/-;L§; 2/nsz. Z/nzLL';
Zf/P:L;z

2/i'3sz 2/h13LZz 2/3'33:s Z/h';P;
27";ng

261131.33c 2/-17=LZ.

*
Y

102

Orthorholbic
222 222;P; 222;P;f 22231:; 2223LL; 222gsz,
‘ 222:3L;
2'2'2 2'2'23P; 21212;P: zfzfszz‘ 2'2'2;sz,
2'21zgsz:
2221' 22211gP; 2221';LZi 2221';3Z,
.- ...ng an; ngLZ, ..ng: mags}; nag;
n'nf 1'1“;ng Ifnsz: n'n'flt, {11.33221 ' 3'1.ng
{11131.};
In. “21;. “0:01:
unl' nml' .L;, .mi' :Lz, unl';sZ,
III III3LZ‘ *nnnzLi:
Ilia..." n _n 3.31.2.
ammf ‘fln'3Lzs "'1;SZz
n'n'n l'n'mngs

Tetragonsl
1. 6:13,, 4:14,; MP: MP? MP; MP”
4' #1:L;x asz: h':sZ, h'BP;
In" 411qu 1.11:8; HEP;
42 hzzP; 1.29;:
«2' 1.27:1": 1.251»;
a'z ufng; A'Z;Sts
#21. #21';P;
u/n. ulna-Z; Mug;
h/nf h/n'agx IMP-3?;

13/-
13/."
u/i11

IPIPPI

G

‘ .

E'EJEJE'E'PE

.103

.- _'*_*-
u;LZ, u:L,, usP, a:P;f

.... * ..
13:1.“ PUP, Pug” A'sP;

:1'11'23 :1.qu
22:92; 1.22:ng

"’ I *

1'2“ 31%:

“12‘1“"; *
a'z-ngz #2.th
.. , *

1.2.1 a,“

Rhonbohedral

3

31
32

32

321

" 9
I

Hexagonal

61
6I2

31:12ng:

10h

331-3; mil 3:??? 3:11?“ 3:32. 333%:
azsg, 3:35: 331;; 33L3:

31131; 311:??? 3133;, 31533, 3131.3,
32;sg, 32;sg; 32:8;, 32:85: 32;LZ, 32;Lg:
32:;11: 32383, 3258;, 32':ng 3251-2,

32 ;L i ‘

321132;, 321133;; 321'3133

3mzsz, amzsg: 3m:8§. 3-385: 3msz, 3-PL6:
3n:;L;J. 3n';szz 3-f:3;, 3-1385: 3-':L2.
33,:3?

331133;: 3M1';s;s 3'111in

3:37 3:137.” 331-3. 5316:

3'1ng: 3";11’.“ 5';3;, 551.3,

51'3L21 Si'PLZs

3-33. 3-1133:

5""s;z 35'“;

313:8; 3532,

31-31:; 3.3.313;

5P1'3L62

.. - - t

6:13; 6:141 631-6.

6'3??? 631-21 5'3ng 6.3362
61'3L21. 31'313.

6.2;ng

 

 

311'2'
3‘22'
"'0 O

612

621

105
2".231’22
"t t a:
6"? “'2
6! 2;L68

34231-31
35233;:
3.11.2: 8;,
2;“2151‘22 ‘
6:L;2. 6=L£I 6:82. 6:22: 6:83: 6:3?
6.31;; 693;: 6382: 6'33?

61.31.;l 6138;, 61.9:

623822 62:83:

62';P.’:‘ 62.282,

6f2;8§, 6'2;ng

621382z

6/131;1 6/n3L';J_

67:121.;

mun;L shun»:

67"..31'32

6/n1';L;J_

6n=322 6mg:

6111553; 6mfnf;L;l

65.23;ng Cat-1332‘

“#382:

w
6;?”

106

Cubic

23 23:83‘ 3338;? 23;P* 23;??? 233L: 23:LL*
231* 231?;32 231728: 231'2LZ
m3 ' :20sz nan-If

n'3 .73ng {3:3}:

n31' 10151.:

7:3- Z3m3LZ .1:ansz

23-. 231;“: 2311:3922

Ian-1' Kan-1'37;

23 43:8: 2328;?

2'3 1:3;3}: 25:32

431? 231'282

I13- -

31.31:. -

Ian" -

3'3; «-

107

Table 22. The Possible Herohedry Twin Group.

Hench Poeeible Merohedry Twin Groups

Group
Triclinic
a: u:
1 1;c 1;c,_
*
1' 1'30
I -
'1" ..
Il' -
Monoclinic
a:
2 22;?Jr 2:15;"
2' 25?; 220*
t ' *
2 .
1 21 1;, '
n 1n;L2y nsz;
m' n'; all m';C*
9 I *
n1 ml 3.2,
2/n ' -
2./I -
2/m' __
2'/n' -

Orthorhombic

222
' I
2 22
22211
I.

.

'P.

108

222:2; 2223;?
2'272;1>; 222:1):

2221732;
mgr“! 31:15;
0 ' t v 2 *
an; x tm 3P:
c a: c n-
“ 3 z- ”.30.
I a:
I ”1.;1‘21
. t . 1* . t . 0* t '*
[”sz 1”sz ’09P; “9?; thx h3Px
I * I * 0 * f *
it 3102: 12.3}; 10351.2 5.3Px
c c c an:
2121.; 2123:, u 3";
1223?; 223??
3|! at
REP: 1.231}
a: c
2'29: I. 233:2
2213?;

Mir-31;; wag;
Min-UL; u/nfgp;
{final-3x

tr]. 3?;
h/nlfsbzx

 

-. * -. '* -. * .
4'2LZ, 4'2 P 2'21.“
21' 31-7.; {$21.25;

122' ;L2:
I. 21:11:;

-2 an:
‘12 2n,1.:: 1232:;sz

m1 “:3
* '* 0
331-22 321.21 32?: 32P.._",‘ 3 .66, 3 .83:
31':L§1, 31'23f 31’3322
322322 3228;;
32311:? 32'282,
O *
321 :26: '
O O *
3"“;1 3"..3322

 

fin “_r I —- I!

3nl
”‘9

31

3'22:

110

321.31 33;: 331.22

3";1’: gfng‘L '5';th 3:82;
.6-1'31-3; Elfin;

sum-2, 3-2222
35'2'2Lgl Eh'z'ng,

33253;; Z'm2'2LZ,

3.12.252; 3.113231%;

332131.25

631;; 6&5: égsgz 63;: 6:82;;
6'3L;L 6'5;z 633; 6'9:
61'21;¢ 61'232z 61f22f

6223;II ézgsg;
62'23f 62'233,
6'223;a 6722s;z
621';82‘

6/n;L;J_ 6/nzLé:

an:
631’“

6:

'*
fl:

 

6' /I
6/I

6 /I
6/l1 .

6.".
6'..-

6/mm
6 /m
6/u
6' /n'n
6/3'3'
6/m'n
6/Im1

Cubic

111

6713.5;

6/n';L;.L 6/"51’2:
6'fl-13Lz;

H.131;

ongsz’ Gum-é:
6.333;, 6I'.I'.3L;.L
6335;, 6flfmfiszz
“’33::

23282 2328? 2322* 2331’" 2331': 23‘le

231';sg 231';P*' 231':LZ
. III . I

“3-14. “3-103

n'3;Lz H.333:
! *

3.31:1.1'... 2*

2.3221,: 23:21”.

Tania: 332-232

.. *

12.331..ng

“age: #333L*

1.73:3: 5.3332

112

Table 23.

Heesch

Group

321

113

The Possible Reticular Merobedry Twin Groups

for Rho-bobedral Crystals

Possible Reticular Merohedry Twin Groups

321;; 321.3 32P.‘." 3 .19."? 3233', 3285:
331‘68 3 L6:

31312131 .3PIL 31'. 3:3: 31.32;:
322 332 322 33, 32 .16, 3221.6:
32' 2332 32 232: 32' 21.6, 32' L222
321' 233, 321' 216,

3.233: 32.233: 3.21.2, 3.21.3:
3-'_ 23;: 33' 233; 322231.22 3.22,?
3.123;, 32.1-21.6,

331’: 3311." 3'13: 53:5:

-'3L;.L 3 3P: 3' 3332 5.31%:

E “3333 5"31‘62
3n' 3L2, 3.312;...
._ ,. *

 

114

Table 2‘». Th. Possible Rsticulsr Morohodry Twin Groups
for Cubic Crystals
Hoosch Possible Roticulsr Morohodry Mn Groups
Group
23 23,321.31 23,321.51 23.3235." 23,322."? 23.3233.
23.3283: 23,321.22 23.32ng
231' 2313313135; 231331512: 231’231'2332
231'.31f21g2
.3 .3523: .3322? $521.22 .3533;
3'3 133.33%; £3,339: {3.553% 113353322
331' $133131; 01'231'3Lz.
2'3- 23-3-2832 23.,3.233: 33.3.21; Ina-21.2,:
33-' 23.33.7233, 2322336235: 33.23-31.22
Baal-3311'."
Kan-1' I3-1',3n1'2s§2 Z3m1',3n1'216'2
1.3 23.322333; «3.32235: 1.3.3232. 1.3.3233:
3'3 33.32728; 1.33.3255; 133.32'2LZ2 1.5.3236;
1231' 2313321385. 431333213333:
:31: wry-3.113132 133.51%;
333:3 I'Bn'23-333383; .3333,3'n3;Lz2
I333 n3; 233-31138}; IBD' 333:1.22
n'Bn' 113333,; 33,3333L2’ 133.3,31133L3‘L
n3n1' 113-13 5111' 31.22

 

Appendix

Let g be the total magnetic field at the proton eite.
Then the Hamiltonian/V for the proton nay be mitten an

A! = - 573 I ' .H
where g is the alt-emetic ratio, ,6 the nuclear nagneton

’2 t/Z MC and ; the total nuclear angular momentum in units
of ‘k . Then '

0 7'} (a L2) {11 0)

_ L<HZ HK'CHY)
_- Z HX+£HY —Hz 0

Let H 5.- {10+ H" where 8.. is the applied dc field and H“ is
the a component of the local field H1. Then ,‘(W = E W and

H +3275 HK-iH,

Hx+iHY -HZ+§7

 

-.: :: g3 /(H,,+H,,)"+(Hf+ Hf) and

 

= A1: = gflflzz+2HaHn+H2

Let AVG: jfl HO . l'hen since H,,=H£Co<1®)

/{-1‘-Z.H_£Coa@— -/.

115

 

 

 

 

Liet of References

Alexander, B. and Herman, K., 2. Kristallogr. 99, 285 (1928).

Alexander, I. and Herman, L, 2. Krintallogr. 19 328 (1929).

Belov, N. 7., Noroaova, N. H. and Shiraova, '1'. 8., Trudy Inet.
Kriet. Akad. Nauk S.S.S.R. LL. 33 (1955).

Belov, ll. 7., Nerenova, N. N. and Slirnova, ‘1'. 3., Soviet
Physics Cry-t., Vol. 2, No. 3, 311 (1957).

Cahn a. in, Advances in Phys. 3.. 363 (195A). ’

Cochran, W., Akta Cryet. 2, 630 (1952).

Curien, 3., and Lo Corro, 1., Bull. Soc. franc. Minor. Oriet.
1;, 126 (1958).

Denney, 0., Corliee, L. H. Donnay, J. D. 11., Elliott, H. and
Haetinge, J. 14., Phye.IRev. 1L2, 1917 (1958).

Gattov, G. and Zonann, J. Acta Cryot. y” 866 (1958).

Hoooch, 3., z. Krietallogr. a, 325 (1930).

Miami; for 14-91. 95191., Vol. 1, Kynoch Preee,
Birninghan (1952).

Landau, L. D. and Lifahitt, E. 11., Statistical theicg
(Addison-Wesley Publishing Conpany, Inc., Maeeachueette,
l958). pg. #33.

Lo Corro, L, Bull. Soc. franc. Min/er. Grist. _8_l_, 120 (1958).

Hackay, A. L., Aota Cryet. L0. 5L3 (1957).

Neroaova, 11.. N. and Belov, N. V. Soviet Physics Cryst., Vol. 1..
No. 6, 769 (1960).

Riedel, I. P. and Spence, R. D., Physica g, 1171. (1960).

116

 

 

117

Roth, W. L., J. of Appl. Phys. 3_1_, .2000 (1960).
3mm... A. v. m-WW Luis:
limes in Russian lloscow: Academy of Sciences (1951).
Skull, 0. c. and v1.11», 3. 0., M§2§3thsics edited
by Seits, P. and Tumbull, D. (Acadenic Press, low York,
1960). Vol. 2, pg. 137.
Slack, G. A., J. of Appl. Phys. 1;, 1571 (1960).
Spence, R. D. and Ewing, R. D., Phys. Rev. 11;, 15% (1958).
Tan», B. A. and Zaitsov, V. IL, J. kptl. Theoret. Phys.
0.3.3.3. 19. 561. (1956). ' ‘
Tavgor, B. A., Soviet Physics Cryst., 11.1. 3, Ho. 3. 31.1 (1959).
Tavger, B. A. Soviet Physics Cryst., Vol. 3, No. 3, 3M. (1959).
Van der Lugt, w., and Poulis, n. J. Physics _2_5_, 1313 (1959).
Zanersaev, A. 14., "A Generalization of tho Podorov Groups.”
Dissertation ‘in Russian Leningrad (1953).
Zamorsaev, A. M., Soviet Physics Cryst., Vol. 2, No. 1,
10 (1957).

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