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W “3““
8064 Michigan State
University

 

This is to certify that the
thesis entitled

CRYSTALLOGRAPHIC SYMMETRY IN THE

MAGNETICALLY ORDERED STATE

presented by

Krishna Kumari Yallabandi

has been accepted towards fulfillment
of the requirements for

PhoDo degree in PhXSiCS

P, D. sto— (:53 9’ GBE’EVE/

Major professors

Date January 16, 1969

 

0-169

 

ABSTRACT

CRYSTALLOGRAPHIC SYMMETRY OF THE
MAGNETICALLY ORDERED STATE

By

Krishna Kumari Yallabandi

For a crystal in a magnetically ordered state, the general
symmetry behavior of its internal magnetic field - considered as
a time-averaged classical axial vector field - and the general
symmetry behavior of the Fourier components of the field in
reciprocal lattice space are properties of interest in nuclear
magnetic resonance and neutron diffraction experiments. In this
thesis, general procedures based on the theory of finite groups
are developed which allow the behavior of the internal field and
of its Fourier components to be deduced from the magnetic space
group (Shubnikov group) of the crystal. The procedures are
illustrated through application to seven particular magnetic space

groups belonging to the tetragonal system.

CRYSTALLOGRAPHIC SYMMETRY OF THE
MAGNET ICALLY ORDERED STATE

By

Krishna Kumari Yallabandi

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

1969

.4 (—7: J-fi n) f
.'g 1'. :9 C‘.. ‘F‘.

6921-; , 07

TO MY PARENTS

ii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . .
CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . .

CHAPTER II. MAGNETIC CRYSTAL GROUPS . . . . . . . . . . .
CHAPTER III. SYMMETRY OF THE INTERNAL MAGNETIC FIELD . .

CHAPTER IV. NEUTRON SCATTERING FROM THE INTERNAL MAGNETIC
F IELD O O O O O O O O O O O O O O O O C C O O 0

CHAPTER V. ILLUSTRATION OF THE CALCULATION OF THE FOURIER
COMPONENTS OF THE INTERNAL FIELD . . . . . . .

CHAPTER VI. SUMMARY . . . . . . . . . . . . . . . . . .

REFERENCES 0 O O O O O 0 O O O O O O E 0 O O O O O O O O 0

iii

Page

iv

17

35

49
57

59

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

LIST OF TABLES

I. Group Multiplication Table for Point Group mm2 .
II. Group Multiplication Table for Point Group
mm'Z' . . . . . . . . . . . . . . . . . . . . . .
III. Group Multiplication Table for Point Group
m'm'2 . . . . . . . . . . . . . . . . . . . . . .
IV. The Number of Ordinary and Magnetic Point and
Space Groups for the Seven Crystal Systems . . .
V. Relation between the Operations (81(23), (dirgi)
and 9i . . . . . . . . . . . . . . . . . . . . .
VI. Elements of D6h (6/mmm) and the Corresponding
Matrices (6i) in Condensed thation . . . . . . .
VII. Elements of Oh (m3m) and the Corresponding
Matrices ($1) in Condensed Notation . . . . . .
VIII. The Elements of the Space Group P4222 . . . .
2'2' .

IX. The Elements of P42'22', P42'2'2, and P42

X. The Additional Elements of PC4222, PC4222, and
P14222 O O O O O O O O O O O O O O O O O O O O O

XI. Coordinates of Equivalent Positions . . . . . .

iv

Page

10

11

13

15

16

20

21

22

24

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Page
XII. Equivalent-Position Subgroups of P4222,
I U ' ' I '
P42 22 , P422 2, and P422 2 . . . . . . . . . . . . 27
XIII. Equivalent-Position Subgroups of PC4222 . . . . . 28
XIV. Equivalent-Position Subgroups of PC4222 and

P14222 O O O O O O O O O O O O O O 0 O O O O 0 O O O 29

XV. The Components of §(£) for P4222, P42'22',
P4 '2'2, and P4 2'2' 0 C O O C O O O O O O O O O O O 31
2 2
XVI. The Components of §(:) for PC4222 . . . . . . . . 32
XVII. The Components of §(£) for PC4222 and P14222 . . 33

XVIII. A11 Possible Shubnikov Group Operations which
Satisfy Equation (4.24) and the Corresponding Invariant
Reciprocal Vector . . . . . . . . . . . . . . . . . . 43

XIX. Solutions §(§) of Equation (4.26) for the
Operations (22,1!) and (ZZIIV . . . . . . . . . . . . 44

XX. Solutions £(E) of Equation (4.26) for the
Operations (32):) and (EZI’EY . . . . . . . . . . . . 4S

XXI. Solutions §(§) of Equation (4.26) for the
Operations (3z[§§ . . . . . . . . . . . . . . . . . . 46

XXII. Solutions {(3) of Equation (4.26) for the
Operations (4zr§) and (Earl). . . . . . . . . . . . . 47

XXIII. Solutions g(§) of Equation (4.26) for the

Operations (62'1’) and (62'11). . . . . . . . . . . . . 48

Table XXIV. Reciprocal Vectors 31 and the Elements of the

Corres pond ing Groups 8: (95! g)
Table XXV. Solutions for the Fourier Components_§(§) of the

Internal Magnetic Field

vi

Page

50

56

CHAPTER I

INTRODUCTION

It is an experimental fact that at sufficiently low temperatures
most paramagnetic crystals become magnetically ordered.1’ 2
Under an interaction which tends to make the ionic or atomic
magnetic moments line up parallel, the paramagnetic substance becomes
ferromagnetic at a temperature characteristic of the material, called
the Curie temperature. Such an interaction may be discussed in terms

of an effective uniform internal magnetic field, called the Weiss
field2 or the exchange field, and the interaction energy of a spin
magnetic moment with the Weiss field must be of the order of magnitude
of the thermal energy of the magnetic ion at the Curie point. Below
the Curie point, the exchange field interaction is able to overcome
the disorder due to thermal energy of the ions, and neighboring spins
will tend to align in parallel even in the absence of an external
magnetic field. In this way a cooperative process is set up which
results in spontaneous magnetization of the material.

The exchange coupling mechanism has no classical analog. It is a

quantum mechanical effect which has its origin in the Pauli exclusion

principle. It is very sensitive to relative spin alignment and it is

only for a very small range of spin separation that the energy is a
minimum when the neighboring spins are parallel. In most cases the
exchange energy turns out to be smallest when neighboring spins are
antiparallel. The alignment of the spins in an antiparallel array is
also a cooperative effect which spontaneously sets in at a definite
temperature known as the Neel temperature, and this phenomenon is
known as antiferromagnetism.

Whereas many substances are magnetically ordered at room
temperature or above, there is also considerable low-temperature
interest in this phenomenon. The reason for this is that if one has
a system of magnetic ions, then, however small the exchange interaction
may be, there must exist some temperature low enough for a
cooperative alignment to take place.

Above the transition temperature the crystal structure is
invariant under a group H of unitary spatial operators which is the
space group of the crystal. In the paramagnetic state the crystal is
also invariant under the time reversal operation T, and the products
of T with the elements of H since we assume that in this state the
crystal possesses a vanishing time-averaged magnetic moment density.
Below the transition temperature, however, the crystal exhibits a
non-vanishing magnetic moment density which reverses direction under
the time reversal operation. Although such crystals cannot therefore
then be invariant with respect to time inversion, they may be

invariant with respect to a product of time inversion and a particular

set of spatial symmetry operations. The full symmetry of such crystals
must therefore be investigated by considering the proper combinations
of time inversion and space group operations. As will be discussed
in Chapter II, introduction of the time-reversal operator leads to a
generalization of the 32 ordinary crystallographic point groups to
122 magnetic crystallographic point groups, and to a corresponding
generalization of the 230 ordinary crystallographic space groups to
1651 magnetic crystallographic space groups.

In this dissertation we shall investigate the following problems:
(a) what is the symmetry of the most general time-averaged classical
internal magnetic field §(£) allowed by a given magnetic space group,
and (b) what is the character of neutron scattering from such a

field'§(£).

CHAPTER II

MAGNETIC CRYSTAL GROUPS

Classical crystallography allocates all possible crystal
morphologies to one of 32 crystal classes (point groups), the
symmetry being characterized by the existence of planes, axes,
and centers of symmetry. If the crystalline lattice is considered
to be of infinite extent, then the crystal also possesses
translational symmetry. The inclusion of this translational
symmetry increases the number of distinguishable geometrical
forms to 230. These 230 ordinary space groups or Fedorov groups
are appropriate for the characterization of the charge density
€(E) in a crystal which is of interest in the analysis of x-ray
diffraction data.3 However, to characterize, in addition, the
symmetries of distribution of internal current densities 1(3),
internal magnetic fields §(£), or magnetization densities 5(5),
a still more general system of symmetry operations is required,
because the time-averaged non-vanishing l<£): §(£), or fi(£) are
not invariant under time reversal. Hence for these cases, the

time reversal operation has non-trivial consequences, and inclusion

4

of the time-inversion operator produces a generalization of the 230
ordinary space groups to the magnetic space groups (Shubnikov
groups) which are 1651 in number.

In 1930, HeeschS broadened the concept of symmetry by introducing
non-spatial double-valued attributes such as sign, color, or even
more general qualities. In 1951, Shubnikov4 developed the theory
of symmetry groups in which an operation interchanging black and
white colors is considered in addition to the usual spatial operators.
This color change can be identified as the time reversal operation
and can therefore be interpreted as producing a reversal in the
direction of the internal current, magnetic field, and magnetic
density. By adding this non-spatial operation in the 32 ordinary
point groups, one obtains 122 magnetic point groups (HeeschS’ 6
groups).

In the magnetic groups there exist two types of elements:
"uncolored" elements gi which do not include time reversal, and

H II = 0
colored elements Tgk gk

which do include time reversal. The
. ' 7

latter are referred to as antioperators as they are antiunitary

operators. The time reversal operator T commutes with all spatial

. . 2

operators and it is of order two, i.e., T = E, where E is the

identity operator. Hence, the product of two colored or two un-

colored operators is uncolored, and the product of a colored with

-)

an uncolored operator is colored. Thus suppose that (g1, gj,

is a set of uncolored elements and (gk', gl', ...) a colored set.

Then,
gigk' = gi'gk = gl'; gigk = gi'gk' = g1. (2.1)
One distiguishes three types of magnetic point groups.8 The
first type is the set of 32 ordinary point groups with no anti-
unitary operators. These are called uncolored point groups.
The second type of point group is the set of 32 formed from an
uncolored group by adding to the uncolored elements those which are
formed by adjoining the time reversal operator to £222 of these

elements, i.e., for all gi in a group G, gi is also in the group
which leads to the fact that T itself is an element of the group,
since TE = T. These groups are called grey groups and they are
denoted by G1' in the international notation. It is clear that
T cannot be a symmetry operation in a magnetically ordered crystal,
since it would reverse the sign of all magnetic moments in the
crystal. Hence the grey space groups are applicable only to dia-
magnetic or paramagnetic crystals which have no time-averaged
non-zero magnetic moments. However, Tiig a possible point group
operation in an antiferromagnetic crystal if it always appears in
combination with a translation connecting two antiparallel spins
in identical chemical environments.

The third type of magnetic point group is that which contains
T only in combination with a spatial rotation or reflection. These

groups, 58 in number, cannot include elements of the type gk' ' Tgk

n
if gk is of odd order n, since that would give (gk') = T.

The following properties of G = {gi, gki}, with G one of the S8
colored groups, can be proved9 with ease: (a) no element gi occurs
both with and without T, i.e., the set {gig is always distinct from
the set {g1}; (b) if T is replaced by E in the colored group G, i.e.,

if the set %= igi, g7; is formed, then is one of the 32 ordinary

k
point groups; (c) if we consider the group G “{gi, gki}, then all
the uncolored elements H = {gi} of G form.an invariant unitary sub-
group of G which again is one of the 32 ordinary point groups; (d)
the number of uncolored elements of G is equal to the number of
colored elements of G, i.e., H is an invariant subgroup of index two.
Considering the above properties of G, one can devise a simple
method for constructing all 122 magnetic point groups starting with
the 32 ordinary point groups. The procedure is as follows. After
choosing an ordinary point group G, one finds all its invariant
subgroups H of index two. For each H one then constructs the

i 1

corresponding magnetic group G

i = H. + T(G-H ). By successively
1 i

considering all 32 groups one finds all 58 colored groups G , and

i
adjoining T to all elements of G, one obtains the 32 grey groups.
Thus, with the addition of the 32 colorless point groups, the total
number of magnetic point groups is 58 + 32 + 32 = 122.

For example, let us construct all magnetic point groups that

derive from the ordinary point group mm2 (in international notation).

This is an abelian group of order four, containing the identity

element E, a two-fold z-axis of rotation 22, and the two perpendicular
reflection planes containing the z-axis, mx and my. These elements form
a group as can be seen from the group multiplication table, Table I.
Now let us consider the colored groups. One is mm'Z' and the
other is m'm'2. Their group multiplication tables are given as
Tables II and III, respectively. From these tables it is seen that
these two groups are also abelian which is not, however, generally
true for colored groups. The grey group is mle' which consists of
eight elements four of which are colored and four are uncolored. It
is also an abelian group which follows from the fact that T commutes
with all spatial operators and mm2 itself is an abelian group.

Specifically we have that, e.g.,

m 'm ' = m 'm ' = 2 , (2.2)
x Y y x z

m 'm = m m ' = m 'm = m m ' = 2 ', etc. (2.3)
x Y Y X Y X X Y 2

In order to construct all possible magnetic space groups, it is
. . . . 11
necessary to derive the appropriate Bravais lattices. This can be

done4, 9, 12

by starting with the Bravais lattices of the Fedorov
groups and by adding colored translations along the edges, diagonals
of the faces, and spatial diagonals of the unit cell. In addition to
the fourteen uncolored Bravais lattices, one obtains 22 colored
translational lattices for the Shubnikov groups. Joining these
lattices in all possible combinations of uncolored and colored

elements of symmetry, one arrives at the 1651 Shubnikov groups. If

S stands for a Shubnikov group S = {F +-Di}, then the uncolored

Table I. Group Multiplication Table for Point Group mm2

 

 

 

mm2 E 2 m m
z x y

E E 2 m m
z X Y

2 2 E m m
z z y x

m m m E 2
x x y z

m m m 2 E

Y Y X Z

 

Table II. Group Multiplication Table for Point Group mm'2'

 

 

 

mlz' E 2' m' m
z x Y

E E 2' m' m
z x y
2 ' 2 ' E m m '

Z Z y X
m ' m ' m E 2 '

x x y 2

my m m ' 2 ' E

y x z

 

10

Table III. Group Multiplication Table for Point Group m'm'2

 

 

 

m'm'2 E 2 m ' m '
z x y
E E 2 m ' m '
z x y
2 2 E m ' m '
z z y X

m ' m ' m ' E 2
x x y z

m ' m ' m ' 2 E

y y x z

 

elements F of S always form one of the 230 ordinary space groups, and
also constitute an invariant subgroup of index two. In fact for any
space group, the set of all its pure translational symmetry operations
is an invariant subgroup of index two. If in S one replaces T by E,
forming the set €f= {F + D}, thenagfis also one of the 230 ordinary
space groups. Thus the algorithm for constructing the magnetic space
groups is similar to that for the magnetic point groups. As an
example, magnetic space groups derivable from the uncolored space
group P4222 will be discussed in Chapter III.

Table IV lists the way in which the number of point and space

groups allocates among the seven crystal systems.

11

Table IV. The Number of Ordinary and Magnetic Point and

Space Groups for the Seven Crystal Systems

 

 

 

 

Number of Number of
Crystal Ordinary Groups Magnetic Groups
System
Point Space Point Space
Cubic 5 36 16 149
Hexagonal 7 27 31 164
Trigonal 5 25 16 108
Tetragonal 7 68 31 570
Orthorhombic 3 59 12 562
Monoclinic 3 13 11 91
Triclinic 2 2 5 7
Total 32 230 122 1651

 

 

 

 

 

12

An axial vector such as, for example, the magnetic field vector

§(£) at two positions £1 and E in the magnetic crystal which are
0

related by the equation

_i = «Si-:0 +2; (2.4)
13

is given by
sci) = 91-13%). (2.5)
where the rotation matrix oi acts on the components of the polar
vector E: and the rotation matrix 9i acts on the components of the
axial vector B. The translationfE; is the vector sum of translational
components (as in glide planes and screw axes) and the location of the
element in the magnetic unit cell. The relationship between éi’ 9i,
andfigg on the one hand, and the elements (sirga) of the magnetic
space group of the crystal on the other hand was first studied by
Donnay and Donnay.14 From their work it is possible to develop the
results summarized in Table V in which n stands for n-fold rotation,
n' stands for n-fold antirotation,'fi stands for n-fold reflection-
rotation, and-3' stands for n-fold antireflection-rotation. Thus,
for a particular choice of a Shubnikov operation (birgk), Table V
gives the corresponding operations oi, 91, andfiE;
of (2.4) and (2.5) specify the behavior of B under that Shubnikov

which with the aid

operation.
The set of elements {$1PEE} generated by a given Shubnikov

group constitutes an ordinary space group C(61LEQ) which is often

13

Table V. Relation between the Operations (sikyi), (¢.rflé), and 91
1

 

 

Description of the Operation Operation Operation
Operations 3, 51 ¢. 9,
1 1 1
n-fold rotation n n n
n-fold antirotation n' n ‘3
n-fold reflection-rotation fi' '3' n
n-fold antireflection-rotation 'fi" 7? ‘7?

 

(but not necessarily) the chemical space group of the crystal or one
of its subgroups. If all‘Eé are set equal to zero in the corresponding
group G(oirE;), one obtains the so-called "underlying point group"
{d£} which is one of the ordinary 32 point groups. It can be shown
that the set {9;} also forms a point group, called the "aspect group"
denoted by C(91).13

All point group operations in the crystallographic point groups
belong either to 6/mmm or to m3m.11 Thus, enumeration of the

operations of 6/mmm and m3m produces the complete catalog of {¢i§. To

specify the rotation matrix (oi) of the operation di, we write

(£1) = («51) (30). (2.6)

14

or more explicitly,

xi $11 0512 $13 X0
Y1 = d21 622 623 yo (2'7)
zi c‘31 ‘32 ‘33 zo

All possible rotation matrices (di) are listed in condensed notation
in Tables VI and VII (in which the operations are given in the Schoenr
flies notation). For example, the matrix for the operation 02 (two-

2

fold rotation around z-axis) is given as'xyz which in the form of

(2.7) is to be understood as

‘i -1 O O x .
7 = o -1 o y (2.8)
2 0 0 1 2

Two more complicated examples are provided by the following: the

5
operation 86 3 IC3 (three-fold rotation followed by inversion) for

which Table VI gives x-y,x:; which stands for

x-y 1 -1 O x ,
x = 1 0 O y (2.9)
E? O O -1 z

and the operation 842 = 1042 (four-fold rotation around z-axis

followed by inversion) for which Table VII gives ya; which stands for

y 0 1 0 x . (2 .10)

:4
II
I
H
O
O
‘4

NI
0
O
I
0—“
N

15

 

Table VI. Elements of D6h (6/mmm) and the Corresponding
Matrices (di) in Condensed Notation

xyz E 'y,x-y,z C3 y-x,§,z C32
§§z C22 y,y-X,z C65 x-y,x,z C6
xyz 5’22 —y',x-y,'z' S3 y-x,§,? $3-1
Ry; I y,y-x;E S65 x-y,x,z S6
YXE C2. (1) x,y-x,'i' C2. (2) x-y,y,':7:' C2' (3)
3;; C2" (1) X.X’Y:E C2" (2) y-x,y,z C2"(3)
73:2 63'“) x,x-y,z G’Z'Q) Y’X2Y92 62.6)
yxz 6‘2"(1) x,y-x,z $2" (2) x-y,-y—,z 6}"(3)

 

 

 

16

Table VII. Elements of Oh (m3m) and the Corresponding

Matrices ($1) in Condensed Notation

 

xyz : E zxy : C3(l) yzx : C3(l')
x92 : C2X 'Exy : C3(2) ‘72? : C3(2')
Eyz : 02y 232'}? : c3 (3) ‘y‘zx : c3 (3')
'xyz : C22 Ex? : C3(4) yEE' : C3(4')
36E? : I ER? : 86(1) 'yzi' : 86(1')
xyz 65g Exy : 86(3) yzx : 86(4')
x'y’z 62y z'xy : S6 (4) y'z’x : S6 (2')
xy'g :6’22 zxy : 36(2) yzY : 86(3')
RE? deI ny : Cydl 'yiz Czdl
xzy de2 zyx : Cydz yxE‘ : Czdz
xzy : C4x 'ny : C4y yxz : C42
xzy : C4x' zyx’ : C4y' yEz : C42,
xzy xdl zyx : ngl yxz : szl
x-}7 cxdz ‘z‘yx : Gde Viz cgdz
'iz? 84x 2?; : 84y y'z : S42
TrEy : S4x' 3.x S4y' 37x? S42.

 

 

 

CHAPTER III

SYMMETRY OF THE INTERNAL MAGNETIC FIELD

In the magnetically ordered crystal, the internal magnetic field
vector §(£) exhibits a repetition pattern governed by the magnetic
space group of the crystal, and it is therefore a periodic function
of the position vector 3. Since §(£) is an axial vector, the most
general §(£) allowed by the symmetry at position E in the crystal
should be determinable from the groups C(éilfé) and C(91) that are
associated with the magnetic space group S(sir§&) of the crystal.

The position vector r is said to be invariant under an operation

(é,r§:) if é}.§ +f§L - E +IE, where t is any lattice translation,
1 1 1 1

i.e., E _ n13 + n b + n3g, where a, b, and g are unit vectors of the
2
magnetic unit cell, and n1, n2, and n3 are any set of integers. The
11
set {6 PTO} of G which keeps r invariant forms a group , G'(¢‘rf')'
1 “'1 - 1 1.1
From this and (2.5) it follows that B at E has to satisfy
B(¢$.-r +rd) = snag) (3.1)
_ 1 _ —i 1 _
for all elements of G,(¢.rU ).
1 1‘1
An immediate consequence of (3.1) is that if G, contains T, then

1
B 2.0, since for T,’§TT) = O, and Tables V and VII give 6(T) = (xyz)

l7

18

and 9(T) = (Ed/‘2). With these (3.1) gives B(_1;) = ‘§(£): hence BE 0.
As another example, if G. is the group of order two containing
1
E and 2 , then 6(E) = (xyz), 6(2 ) = (RYZ), 9(E) = (xyz), 9(22)
2 z

= (3672) and '_C_/(E) ='_'d_(22) = 0. With these, (3.1) gives Bx = B = O:
y

and only B may be non-zero at 3' Finally, if G, is the group of
z 1

order one (containing only E), then all three components of B(£) may
be non-zero, hence B is arbitrary at £° In this manner the behavior
of B can be determined at all points of the magnetic cell.

The set Sk of points (ER, E .) is said to be a set of

1,

equivalent positions with respect to a given magnetic space group
S(s,r§1) of the crystal if each point of Sk is related to all the
1 1

other points of Sk by a set of operations {oil’ffi of C(éil’gi), i.e.,

for every pair Bk, r1 of SR there exists at least one operation

(OOPEE) of G such that for that operation (difgg), Ek’ and E satisfy
1

1

‘61. Ek +1931 = E1'

Thus for every point ER in the crystal, one can find the group of

(3.2)

operations G,(¢.r2&) under which rk is invariant, and hence the
1 1 -
set of operations [C(é.rtl) - G (d,rv,i] under which r generates
1'—1 i 1 “1 ’k
the set Sk of equivalent positionS. In this way the symmetry of the
internal magnetic field can be specifically determined for all points
of Sk from (3.1), and the interrelation between the components of B

at all points of SR is taken into account through (2.5).

19

We now proceed to illustrate the above method with the seven
magnetic space groups that derive from the ordinary space group
P4222 which belongs to the tetragonal system.3 The chemical unit
cell is primitive with vectors a1, a1, and CB, where i, i, and B
are the Cartesian unit vectors along the x, y, and 2 directions,
respectively. The seven magnetic space groups are listed by Shubnikov
and Belov4 as tetragonal groups No. 119, 121 - 126, and are denoted
by P4222, P42'22', P42'2'2, P422'2', PC4222, P04222, and P14222.

All of them are antiferromagnetic except P422'2' which is ferro-
magnetic. Tetragonal group No. 120 is the grey space group which
we omit. In Table VIII we list the elements of P4222 (with origin3
at 4221). We introduce a running index number for these elements
in order to have a more concise alternate designation.

In the magnetic space groups P42'22', P42'2'2, and P422'2' some
of the elements of P4222 become colored, and in Table IX these are
designated by priming the corresponding running index number.

The magnetic space groups PC4 22, PC4222, and P14222 are obtained

2
from P4222 by adding, respectively, the antitranslation (EIEO' with
?= 35cB along the c-edge, the antitranslation (EI’EIY with’g’l

= %a(ifj) along the diagonal of the c-face, and the antitranslation
(Ef$;)' with’Bé = %a(ifi) + %cB along the space-diagonal. These three
magnetic space groups thus contain not only the elements of P4222, but

also these elements joined with the corresponding antitranslations.

The additional elements of these three groups are given in Table X.

20

 

 

Table VIII. The Elements of the Space Group P4222
Running Element Description of the element
1 (BIG) Identity operator
2 (22(0) Counterclockwise rotation about z-axis
through 180°
3 (4 f2) Counterclockwise rotation about z-axis
z
0
through 90 followed by translation’Zf
== Lie-1:
4 (42-133 clockwise rotation about z-axis through
0
90 followed by translation‘E/= %cl<_
5 (2 l0) Counterclockwise rotation about x-axis
x
0
through 180
6 (2yl0) Counterclockwise rotation about y-axis
0
through 180
7 (2 r!) Counterclockwise rotation about (ifi) axis
a
through 1800 followed by translationfigv'
8 (2brgb Counterclockwise rotation about (£51) axis
0
through 180 followed by translationfE/

 

 

 

21

Table IX. The Elements of P42'22', P42'2'2, and P422'2'

 

 

Running index Group Group Group
of P4222 P42'22' P42'2'2 P422'2'

1 1 1 1

2 2 2 2

3 3' 3' 3

4 4' 4' 4

5 5 5' 5'

6 6 6' 6'

7 7' 7 7'

8 8' 8 8'

 

 

 

 

Table X.

22

The Additional Elements of PC4222, PC4

22, and P14 22

 

 

 

2 2
Group PC4222 Group PC4222 Group P14222
Running Running Running
Element Element Element
index index index
1C (Em' 1C (Blip 11 (EIE’ZV
2c (22"§)' 2.C (zzl'g’lr 2I (zzlfiofzr
3C “‘2' 0>' 3 C (azl I+_'c_’1>' 3I (azlziz’zr
4C (42'1|0)' 4C (az'll‘fitc/lr 41 (4z-lg-yg)‘
5C (2J’g’)' 5C (zxyg’lr SI (2x1£2)'
6C (2yl'g’)' 6C (2y|3’1)' 6I gym/2).
7C (2&1 o>' 7C (2a|{+;c’l)' 7I (zaréz’zr
8c (213' O) ' 8c: “bi/9:51). 81 (2b|’§+12)'

 

 

 

 

 

 

23

Given the magnetic space group, one can find the corresponding
group C(é'kgi) from Table V, and then one can write down all possible
1

sets of equivalent positions S for that group C<¢1l£&)' In Table XI,

k
the coordinates of all equivalent positions are given for each set
Sk for the seven groups under discussion. The symbols x,y, and z

are here used for the general coordinates expressed as fractions

of the unit cell edge lengths along the corresponding x, y, and
z~axes. Special points like (%, %, %) are read as the vector

E = %ai + kai +'%cB for our tetragonal system examples. One sees
from Table V that the groups G(oir§£) and Gi(éiizé) that derive from

the four groups P4 22, P4 '22', P4 '2'2, and P4 2'2' are identical.
2 2 2

2
Thus the sets of equivalent positions for these four groups are those
3

of P4222 as given in the International Tables. Here we have listed
these positions in Table XI and we have designated them as S1
through 816'

Additional equivalent positions in Table XI are those generated
by the translations'ES 23, and If: respectively, in the three magnetic

2

space groups PC42 I 2

set generated from 810 by operating on every position of $10 with

22, PC4222, and P 4 22. For example, SlO'(PC) is the

(E‘V ), S '(P ) is the set generated from S by operating on every
-1 10 1 10

position of S

° . i I
10 With (E 3%). The sets 89 (PC) and 89 (PC) turn out

to be identical sets and are simply designated by 89'.

All points in the magnetic unit cell can be covered in terms of

sets of equivalent positions, starting with the points of highest

24

Table XI. Coordinates of Equivalent Positions

 

 

Set Equivalent positions (Origin at 4221)

31 (0.0.0). (0.0a)

82 (%.%.0). (%.%.%)

83 (0.55.0). (35.0.35)

s4 (0.12.35). 05.0.0)

S5 (0,0,k), (0,0,3/4)

36 (Ma), (35.15.3/4)

S7 (0,0,2), (0,0,5), (0,0,%+z), (0,0,%-z)

38 053.2). (35.55.73). (95.559542). (35.13.15-2)

39 (0.35.2). (0.55:5). (35.0.5542). (35.0.%-Z)

810 (x,0,0), X,0,0), (0,x,%), (0:§,%)

S11 (X.%.%). Gc'.%.%). (95.X.0). (5255.0)

512 (x,o,s,), 66,095), (0,x,0), (03,0)

313 (X.%.0). 5.35.0). (5.19%). (hit)

s14 (x.X.%). (mm). (tuna/4), (mm/4)

315 (x,x,3/4), x,Ԥ,3/4), x,x,!r), (x3115)

$16 (X.y.2). ("i—>732). G7.X.%+Z). (with). (X573). (iyfi').
(Y:X.%‘Z)a (373.35%)

89' (95.0.2). (35.0.3). (0.%.%+2). (0.%.’5-2)

 

(continued on next page)

 

25

Table XI (cont'd.)

 

 

Set Equivalent positions (Origin at 4221)

Slo'(PC) (%+X.%.0). (t-X.%.0). (%.%+X.%). (%.%-X.%)

SIO'(PI) (%+X.%.%). (t-X.%.%). (%.%+X.0). (%.%-X.0)

811'(PC) (%+X.0.%). (k-X.0.%). (0.%+X.0). (0.%-X.0)

s11'(PI> (%+x.0,0), (e-x,0.0>. (0,%+x,e). (0.%-x.r>

$12'(PC) = Slo'(PI)

312'(P1) = S10 (PC)

313'(PC) = Sll'(PI)

313'(PI> = s11'(PC>

Sl4'(PC) (%+X.%+X.%). (t-X.%-X.%). (t-X.%+x,3/4). (%+X.%-x,3/4)

$14'(PI) (%+X.%+x,3/4). (t-X.%-X.3/4). (t-X.%+X.%). (%+X.%-X.%)

$15'(PC) = 514'(P1)

$15'(PI) = 314'(PC)

$16'(PC) (X.y.’5+2). (Eitfi). @992). (y.3'<'.2). (flint-2). 630.55%)
(y,x,'z’). 63,3)

316'(PC) (%+X.%+y.2). (t-X.%-y.2). (k-y.%+X.%+2). (%+y.%-X.%+2).
(%+X.%-y55). (t-X.%+YIE). (%+y.%+X.%-2). (k-y.%-X.%-2)

Sl6'(PI) (%+X.%+y.%+z). (k-X.%-y.%+2). (t-y.%+X.z). (%+y.%-X.Z).

 

(%+X.35'Y.%'Z). (k'x:%+}'a%'z)9 (£Ws%fia.z)s (k'y:%'xa-z.)

 

26

symmetry and proceeding through points of intermediate symmetry to the
general point (x,y,z). The number of equivalent positions in the set
which contains the general point (x,y,z) is equal to the order of the
Shubnikov group, and the number of equivalent positions in all other
sets must be less than the order of the group and is, in fact, an
integral divisor of the order of the group.

For every set of equivalent positions one can find from (3.1) the
Shubnikov subgroups with elements (81.2%) of the given magnetic space
group which reduce, with T set equal to E, to the groups Gi(¢ir2£)
under which the equivalent positions remain invariant. These Shubnikov
subgroup operations are given in Tables XII, XIII, and XIV. The first
column gives the sets of equivalent positions, the subsequent column(s)
list the elements (31 23) of the subgroups under which the equivalent
positions of each set remain invariant, and the last column gives the
number of points in the set. All the points in a given set may not be
associated with the same subgroup in all cases. This occurs in Table
XII for the sets 810 through 815. For example, for the sets 810

13

group with elements 1, 5', and the second two points are associated

through S of P42'2'2 the first two points are associated with the

with the group with elements 1, 6'. In Table XII, this is indicated
as: l, 5'; 1, 6'. Similar remarks apply to Tables XIII and XIV.

After determining the groups Gi(éi‘3§) one can find Gi(91) from
Table V for each set of equivalent positions. With this, as discussed

at the beginning of this chapter, one can finally determine the

27

 

 

 

 

 

 

 

 

N H H H H on
A A A A WH .QH
e .m .H ..N .H m .H .N .H .w .H ..N .H .H .N .H m . m
m HH
e .o .H m.m .H .o .H H.m .H o .H mm .H .H an .H Hm .NHm . m .on
. . . a m N
o N H N H N H N .H m . m . m
A A A A A A A A A A A A 0 Am.
N .m .N N H m N N H .N .N N H N N H m m
A A A A A A A A A A +V Am AN H
N .o .m N H .o .m N H o n N H n .N .H m m m . m
uom ca maowuwmoa
N N N N
nunHoo .N.N on N.N. on .NN. no NN en unoHu>Hooo
mo .02 mo muom
N A A N A N
.N.N on one N.N.om .NN. on NN on No neoopmepm uoHanoN-uuoHn>Huom .HHN oHnnH

Table XIII.

28

Equivalent-Position Subgroups of PC4222

 

 

Sets of No. of

equivalent PC4222 points

positions in set
31’ 82, S3, S4 1, 2, 3c’ 4C, 5, 6, 7c’ 8c 2
SS, 36 1, 2, 3c’ 4c, 5C, 6C, 7, 8 2
S7, 88 l, 2, 3C, 4C 4
39+59' 1, 2 8
S10+312’ 11 13 1’ 5; 1’ 6 8
814+615 1, 7; 1, 8 8
3 +5 ' 1 16

 

 

 

29

Table XIV. Equivalent-Position Subgroups of PC4222 and P14222

 

 

Sets of No. of
equivalent PC4222 P14222 points
positions in set
Sf$2,85$4 1,2,5, l,2,5,6 4
354's6 1, 2, 7, l, 2, 7, 8 4
Sf$8 1,2 1,2 8
39+sg' 1, 2 --- 8
$9 --- 1, 2, 31, 4
s is ', 3 +5 '
10 10 11 11 1, 5; 1, 1, 5; 1’ 6 8
I I
S12"312 ’ S13+513
314+814', 315+Sls' 1, 7; 1, 1, 7; l, 8 8
3 +3 ' 1 1 16

16 16

 

 

 

 

3O

allowed components of B at each point of these sets and the relation
between these components of B at all points in the set. The results
of these determinations for all sets of equivalent positions and for
all seven groups are given in Tables XV, XVI, and XVII. The x, y, and
z-components of the internal magnetic field B(£) are denoted by u, v,
and w, respectively, and‘fi'stands for -u, etc. The entries list the
components of the field which symmetry allows to be non-vanishing, and
these components at the various points of a given set are listed in
the same order as the points in Table XI. A zero entry designates
that symmetry does not allow non-vanishing B at any point of the set.
Use of Tables XV, XVI, and XVII is best described with the aid of two
examples:

(1) The entries of Table XV for the set of equivalent positions

S7 should be understood to designate the followin :

For P4222,

Bz(0,0,z) = -Bz(0,0,E) = Bz(0,0,35+2) = -Bz(0,0,35-z); (3.3)
for P42'22',

Bz(0,0,z) = -Bz(o,o,‘z’) = -Bz(0,0,l,+z) = Rz(o,o,!5-z); (3.4)
for P42'2'2,

BZ(O,0,z) = Bz(o,0,’z) = -Bz(o,o,1;+z) = -Bz(o,0,15-z); (3.5)
and for P422'2',

BZ(0,0,z) = Bz(o,o,‘z) = Bz(o,o,15+z) = Bz(o,o,5-z). (3.6)

The x and y-components were determined to be vanishing and are not listed.

 

31

3.3.3.3.3.3.3.3

3.3.3.3.3.3.3.3
'n' "

3.3.3.3.3.3.3.3

.3.3.3.3.3.3.3

 

 

 

 

 

 

>.>.:.=.>. >.>.=.:.>.>. .3 .=.>.>.
3A3A3A3 ”ApAwwAa O mHm AQH
3.3.3.3
>AIN/IAHHIAD .K/‘Ak/APA: m ANHm AHHm AOHW
SAPAM/IAKV
A A A A A A A A A ® Am
3333 BPP3 393 m m
3.3 3.3 o om
3.3 o o om .mm .Nm
mcowufimom
N
.N.N ..Nm .NN.N..Vm NNNAVW HCQHUNrU—"DWQ
mo muom
.NHNqu can .N.N. cm ..NN. am .Nmmqm How Ame.wo muconomaoo ogfi .>x oHan

32

Table XVI. The Components of §(£) for P 4222
c

 

 

Sets of
equivalent PC4222
positions
S through S O
l 8
Sg'PSg' w,w,w,w,w,w,w,w
8104612, 811+S13 u,T1',v,V,F,u,‘v',v
814+815 O
' I
sl6+sl6 816' same as for P4222
1, -
J 816 . as for P4222 with
opposite sign

 

 

33

Table XVII. The Components 0f.§CE) for PC4222 and P14222

 

 

Sets of
equivalent PC4222 P14222
positions
, S +3 S +5 0 0
Sf$2 3 4’ 5 o
S7+S8 w,w,w,w,w,w,w,w w,w,w,w,w,w,w,w
S ' 0
_. _. —. 9'
S +59' w,w,w,w,w,w,w,w
9 39': not defined
3 +3 ', 8 +3 '
10 10' 11 11' u,'fi,v,‘\7,'1I,u,‘\7,v u,fi',v,V,'fi,u,V,v
S +5 S +5
12 12 ’ 13 13
' ' 0 0
S14+S14 ’ S15+515
816' same as for P4222 816' same as for P4222
v 1, - I ,
Sl6+Sl6 816 . as for P4222 with $16 . as for P4222 with

opposite sign

 

opposite sign

 

 

34

(2) As another example, consider 810 of P42'2'2 from Table XIV.

This entry explicitly stands for:

B (x.0,0> = ~13 (3,0,0) = B (03.3) -B (03.3). (3.7)
Y y x x

and

B (x,0,0) = B (§,0,0) = -B (0,X,%) “B (O,§3%). (3.8)
2 z z 2

It is also indicated from Table XV that the components of the field at

. I I
the p01nts of $11 (or $12, or $13) of P42 2 2 obey a set of relations
similar to (3.7) and (3.8), i.e., for $11 one has
B (X9%9%) = '3 5935335) = B ($5,150) = 'B (gs-£30): (309)
Y y x x
and
Bz (Xfli‘a’li) = Bz &:%a%) = '32 (%,X,O') = "'32 (Is-£30). (3°10)

It is important to note, however, that the above magnitudes of the
components of B'in the set 811 cannot, from symmetry alone, be related
to the magnitudes of the components of B in the set 310' This applies,
of course, generally in that the allowed components of B within any.
given set of equivalent positions cannot, from symmetry alone, be
related to the allowed components of B in any other set of equivalent
positions.

We have now completed the description of the solution of the
first problem posed in Chapter I, viz., what is the symmetry of the
most general time-averaged internal magnetic field §(E) allowed by a
given magnetic space group, and we have given examples for purposes of

illustration.

CHAPTER IV

NEUTRON SCATTERING FROM THE INTERNAL MAGNETIC FIELD

In order to study the scattering of a monochromatic beam of
neutrons from the internal field B(£), a knowledge of the behavior of

the Fourier components of this field is required. We therefore write

it) = Erwexpur-n. (4.1)
t

where B(B) is the axial vector amplitude of the k-th Fourier component

of B(£), and B_is a vector in the reciprocal lattice which can be

* *
2flR1§*4mB*+ng*), where 3*, b , and g. are the lattice

written as B
vectors of the reciprocal magnetic unit cell and are related to g, B,
and g which were defined in Chapter III as

are." = 11'2" = 3:9." = 1. (4“?)
and l, m, and n are any set of integers.

If an incoming monochromatic neutron wave interacts with the
internal magnetic field B(£) and is scattered elastically into an
outgoing wave, then the matrix element of the interaction is given
by<+1péntl LPf) , where xint = -F.°B(_1;), and/4.13 the neutron

magnetic moment vector. If the incoming neutron has momentum/fiki, and

35

36

the outgoing neutron wave has momentum 1315f, then 4115f =‘H(Bi-B), and
‘fiekiz/ZM =‘HQBf2/2M, where M is the mass of the neutron. If the spin
state of the neutron is specified by spin index<§2 and the state of
the scattering system by an index n, then a state of the total system

can be written as |_lgon>. Hence the matrix element (411' qui +f>

takes the general form

main! 42> =<5151n11 75c» ten tfoifnf)
Representing incoming and outgoing neutrons by plane waves,
<filitx2nqrff>'

«newt-w? 2w ...[-n-,n._._].,pgnf>
= (51111. -fv-E' Eg'flexp -i@-.1§')‘£ dfl‘éné ’

where the integration is taken over the volume V of the magnetic unit
cell. Carrying out the integration we obtain

damn? -v<6.n.I;-r-®I 61-9- M
The intensity of the neutron beam scattered into a final state with a
particular polarization with an associated change in wave vector
B = Bi-Bf is Proportional tol<¢ian¢9Ia hence by (4.3) it is
proportional to'<6"1§-lfy£@)'6fnt>‘ 2, If, for given B, all three
components of BLk) vanish, then <q’i'aeinth’f> = 0, and there will
be no scattered beam.If two components of B(B) vanish, the scattered
beam may be linearly polarized, and if one component of B(B) vanishes,

the scattered beam may be partially polarized.

37

We now wish to relate B(B) to the macroscopic magnetization M(£)
of the crystal. We start with the well-known relation of classical
electromagnetic theory

a = 11 + W14. (4.4)
where §,15 the magnetic induction field, B is the magnetic field, and
M is the magnetization (magnetic moment per unit volume). For the
Fourier expansion of the magnetization we can write

111(5) = 2 31¢) expfik'fi). (4.5)
k

and by Fourier inversion,
as) =. <1/v>f at) epoer) dd. (4.6)
with the integration taken over the volume of the magnetic unit cell.
Since all internal currents are described in terms of the magnetization
ME).
va = 0 (4.7)
which allows B,to be written as the gradient of a scalar function 0,
g = -v0. (4.8)
From (4.4),
we = +41Yv-u '5 v-g. (4.9)
and since the divergence of the magnetic induction always vanishes,
we obtain
4.va = -v.1_1, (4.10)
and hence from (4.6) and (4.8),

v20 = “(v-n = 41:32 and) exp(iB-£). ' (4.11)
k

38

From this it follows by integrating over the volume of the magnetic

unit cell that

4 = «1’12 [E'E(h)/£'E:IGXP(1E°£% (4.12)
k
and thus _
fl = we) = «VS Ln~e<1>/1<.2]exp(iR-r>. (4.13)
k

and hence

g = 4112 [I - (Q/Rzfl-gg) exp(iB-£), (4.14)
k

where I is the unit dyadic. By comparison of (4.14) with (4.1) we find
that

IQ) = 47([1 - @klszfl-aQ) (4.15)
which is the desired relation between the Fourier components of the
magnetic induction field and the Fourier components of the
magnetization.

We now proceed to the study of the Shubnikov symmetry of B(k).
The Fourier inversion of (4.1) is

£05) = (unfit) exp(-ik°g) cm“. (4.16)
from which we have that

91°B(B) = (l/VXfSi~B(£)exp(-iB{£) d1§ (4.17)
and with the aid of (2.5) we can write

9141(5) = (l/V)J‘B(¢Sio£+3’i) exp(-iB°£) d’d (4.18)
Now we set

(6:; +’£’].L = g' . (4.19)

1

39

and hence
g = éi-l-Q' - i.). (4.20)
1
Thus with the use of (4.20), (4.18) can be written
91'§®= (UVJEQ') exp<-it-¢i'1-r') mag-3133;) cm (4.21)
Replacing B_by B'éi-l and £.bY Ed in (4.16), we obtain
2(3-61") = (1/v)f§@) exp(-iB'¢$i-lo£') (11’, (4.22)
16
and using this in (4.21) we get
191.3(5) = exp(iB'éi-1-'§i) gg-ni‘l). (4.23)

We now consider the set of operations {6 ‘23) of the magnetic

,l
space group S(sjr§3) of the crystal that transform the reciprocal

vector Bi such that

-1 =
51.31 151. (4.24)

This set of operations {ojria} for which (4.24) is valid, i.e., the

operations under which B is invariant, and the corresponding set of

10

i
operations {Oj} of the magnetic space group S(sj|3%) form groups

which we denote, respectively, by 1(6TEO and2?$9). For each group
1(étE) for which (4.24) holds, (4.23) takes the form

[1:81) - exp(-1Bi°‘_'_6’j) 931%,) = o, (4.25)
or

[I - exp(-iBi°’Ej) Oj].B(Bi) = o, (4.26)
where Oj and (éjl’i’j) belong to 31(9) andZ/iwlz), respectively. The
equation (4.26) must be satisfied for each element of 39de and

for every corresponding element ofi;{(9). The most restricted common

40

solution of (4.26), i.e., the "intersection" of the common solutions
of the scattering amplitude equation (4.26), will represent the most
general solution compatible with the symmetry of the crystal for the
three components of £(k).

Since (4.26) is a matrix equation, the number of linearly
independent solutions depends10 on the rank r of the coefficient
matrix A = I - exp (-il<_io'_t’j) Qj. The order of the matrix A is three,
thus the number of linearly independent solutions will be (3-r). If
A is non-singular, its rank is three,land then E‘ki) must vanish
identically. For this 51, there will be no scattered beam. If only
one component of £(Ei) is non-vanishing for given 51, e.g., Fz(§i) =7L,
and Fx(§i) = Fy(hi) = 0, then the corresponding linear polarization
of the scattered neutrons is to be expected. If only one component
of §(§i) vanishes, then the corresponding direction of polarization
is expected to be absent from the scattered beam.

It is found that only a few types of symmetry operations of
Shubnikov groups will satisfy (4.24). These are n-fold rotation or
antirotation which keep invariant the vector 5 along the axis of
rotation or antirotation, and any twofold reflection-rotation or
antireflection-rotation which keep invariant the vector §,in the
plane of reflection or antireflection-rotation. In case S(sif§E) con-
tains an antitranslation (Ek§)', E must satisfy

exp (131’) = -1 (4.27)

41

in addition to satisfying the invariance condition (4.24). For in
this case one must have

2%) = 'EQW, (4°28)
and thus from (4.1)

2‘. :(5') expagky = ~23 yg') expEi£'-(r+_'9]o (4.29)
k' y

Multiplying both sides by exp(-i§{£) and integrating over the volume
of the magnetic unit cell, one obtains (4.27).

The condition (4.27) has an important consequence,as may be seen
from the following. Suppose ’E’= 35(gi'b'tg) and lg = 21f(1§_*+mb*+ng*). Then,
from (4.27) one obtains

exp[2fl’1(1g*+mg*+ng_*)- ggaygfl = -1 (4.30)
which requires (1+m+n) to be an odd integer. Thus, for example, if
flr= £2, then n must be odd and hence one cannot choose §_along the
a-axis or b-axis since that would require n to be zero. Thus, in case
of magnetic space groups containing an antitranslation, (4.27)
represents restrictive conditions on the choice of 3 which are
additional to those demanded by (4.24).

We now present all possible solutions of the scattering amplitude
equation for the Shubnikov group operations which satisfy (4.24) in
general, i.e., without taking into account possible antitranslations.
We take all symmetry operations as passing through the origin, and we

choose the z-axis (c-axis) as the n-fold rotation and antirotation

42

axis and also as the twofold reflection-rotation and antireflection-
rotation axis. The complete catalog of such operations is given in
Table XVIII, and the solutions £(k) of (4.26) for the specific
operations in Table XVIII are given in Tables XIX, XX, XXI, XXII,
and XXIII. The details of calculation of such results will be
illustrated in the next chapter.

As an example of how to use Tables XIX through XXIII, consider
the first row of Table XIX which states that (a) the operations
2z and (zzl'g) for even n, and the operation (2219' for odd n, with
33= g2, keep invariant the vector §'= 2flng*; and that (b) the most
general solution gm) of (4.26) is given by F" (E) =X, and FLOQ
= (0,0), i.e., the component of [(E) parallel to the z-axis (c-axis)
is equal to a constant 34 and the components of £(5) perpendicular

to the z-axis are zero. Other entries can be understood along the

same lines.

43

Table XVIII. All Possible Shubnikov Group Operations which Satisfy

Equation (4.24) and the Corresponding Invariant Reciprocal Vector

 

 

Shubnikov group Possible translational part‘zg' Invariant
operation with the elements passing reciprocal vector
. 4/ . . * * *
(w1th —i=0) through the origin .g = 2fl11§_+mb +ng )
, *
2 , 2 0, 553 h = 211113
2 z
0. ’53. 352. kg
- _ * *
22, 22' mm), 95(13+9_).%(9+3). #(sflfi's) k = 2m; we)
%@th). éflts). Meta). terse)
3Z 0, 3/3, 23/3 )3 = zn‘ng"
*
42’ 42' 0» 342’ ’52: 32/4 t = Mus
*
62, 62' O, g/6, 9/3, £3, 23/3, 53/6 5 '= ang

 

 

 

44

Table XIX. Solutions gg) of Equation (4.26) for the

Operations (22,21) and (2 Z)‘
2

 

 

o a ' *
Operation 22 Operation 22 3(2an )
’EI= O, for all n N
g = 153, for odd n En® =34 31(5) = (0,0)
?= $53, for even n
E/= O, for all n
'{= £c, for odd n (E) = O; F @) = (OCP)
- - _ "II “J.
10,- ’53, for even n

 

 

 

Table XX.

45

Operations (2'3) and (21%).

Solutions 305) of Equation (4.26) for the

 

 

Operation '22 Operation ‘2'?" g [zn’(1§_*+m_p_*)]
El= $53. %@+_c_). 1 even ‘§= £3, get), 1 odd
= 35k: 203%)» I“ even = £2. 350221). In odd
= 0, all values of 1, m = Hang), range), Fug) = (0,0),
= Hafiz). ten-2+2), (Hm) (Hm) odd 11%) = g

even

————————-————q

El= 152. May's). 1 odd
= $2. £Q4‘3), m odd

35 @fil) . 5.5 (a_+l>_+2) .

(1+m) odd

%(£t§): 1 even
= %@:E), m even
= %@th)o 12;th1'3),

(11m) even

'§= saga), 1 odd
= th£)9 m Odd

= kEtE). %@t123£).
(lim) odd

‘_'(:= £3, £@+3), 1 even

£th), 1 even
%;(l;t£). m even
Math). Mathis).

(ltm) even

____________ .4

= 352, 35@‘*‘_C_), m even
% (3.42) , lJQ'HI’E) ,

(1-hn) even

0, all values of l, m

%Qt§)a 1 Odd
= 2&1‘2). m odd

= Math) . £09212).

 

 

(ltm) odd

F" a) = (coil).

11¢) = 0

46

Table XXI. Solutions 3(5) of Equation (4.26)

for the Operations (32"3’)

 

*
Operation 3z 3(2'fl’n3 )

 

?= 0, all values of n

= 2/3, 23/3, n = 3N, Fug) x, 11(5) = (0,0)

(N any integer)

 

_____________ L.._______.____._
’L/= c/3, n - 3N+1 F" (13) — O
_ 1913;: :iN:2_ _ J is: BEETS???"
’Q’= c/3, n - 3N+2 F"@) = 0
= 2c/3, n — 3N+1 F1 (1;) = [31}(1- 53,1]

 

Table XXII.

47

Operations (42"9’) and (42,141):

Solutions 3%) of Equation (4.26) for the

 

 

*
Operation 42 Operation 42' _If_(21’n3)
’_d= 0, all n ?= £3, odd n
= £3, even n =3 £3, 33/4, n = 4N+2 Fug) =5)
= 343, 33/4, n = 4N, F‘LQ) = (0,0)
(N any integer)
____________ 1__________-__.___________
ZJ= £3, odd n '5= 0, all n
= £3, 33/4, n = 4N+2 - £3, even n fig) = O
= £3, 33/4, n 4N
____________ +--—_--—-—_-—'1'_—_-—--—--
§=£3,n=4N+l {=ag,n=4u+3 F"(13)=O
= 33/4, :1 = 4N+3 - 3c/4, n = 4N+1 1103) - 31,1)
_____________ _._______.___.._+.__..._______
15= £0, n = 4N+3 ’§= 542, n = 4N+1 Fug) = 0
=4N+1 =33/4, n =4N+3 F1@)=o((1.i)

 

 

 

48

 

 

Table XXIII. Solutions _EQg) of Equation (4.26) for the
Operations (6z Z) and (62,15):
Operation 6Z Operation 62' 303133")
:C’= 0, all n _EN £3, odd n
= £3, even n 3/6, 53/6, n = 3N+3 Fug) = g
= _c_/3, 23/3, n = 3N, lug) = (0,0)
(N any integer)
= _c_/6, 53/6, n = 6N
El= 5e, odd n If 0, all n
= 3/3, 23/3, n = 3N+1, £3, even n
3N+2 3/3, 23/3, n = 3N EQ‘.) = O
= 3/6, 53/6, n = 6N+2, 3/6, 53/6, n = 6N,
6N+3, 6N+4 6N+1, 6N+5 L
:L/= c/6, n = 6N+5 E/ 3/3, n = 3N+l
= 53/6, n = 6N+1 23/3, n = 3N+2 Fug) = 0
3/6, n = 6N+2 F4(§)
53/6, n = 6N+4 =0([1,£(1+J§i)]
_____________ +————-——--—————————————-
E" 3/6, n - 6N+1 E, 3/3, n - 3N+2
= 5c/6, n - 6N+5 23/3, n = 3N+1 F“ (15) = 0
3/6, n = 6N+4 51(5)

 

53/6, n = 6N+2

= «Elsa-61>]

 

 

CHAPTER V

ILLUSTRATION OF THE CALCULATION OF THE FOURIER COMPONENTS

OF THE INTERNAL FIELD

In this chapter we present all possible solutions of the neutron
scattering amplitude equation (4.26) for the seven tetragonal magnetic
space groups P4222 through P14222 introduced in Chapter III, and we
give some examples of how these solutions were calculated.

Since the elements of the magnetic space groups under discussion
consist only of rotations and antirotations followed by translations,
one expects that the reciprocal vector which will be invariant in
these magnetic space groups should lie along the axes of rotation
and antirotation. In case of PC4222, PC4222, and P14222, the
reciprocal vector 5 = 2flklgfdm3*+n3*) should further satisfy (4.27).
The latter means that 3 must lie along the z-axis with n odd for
PC4222, along the x and y-axes with l and m odd in case of PC4222,
and along the x, y, and z-axes with l, m, and n odd in case of
P14222, taking the a, b, and c-axes, respectively, as the x, y, and
z-axes. In Table XXIV are listed the reciprocal vectors‘ki and the
elements of the corresponding groups 1(6r35 for which thehi
satisfy (4.26) and, if applicable, (4.27). The elements of the
corresponding group32}i(9) can be determined from Table V.

49

SO

 

 

--- --- --- m .H Ho.H-.Hy=N mm
111111111 ..inulnuuLinuuuunu.:luuuuuuuuuuuuuuuu
--- --- --- N .H Ho.H.HV=N om
ooo o poo o
H H H o o o
a. q a. m n— N a. q a. m a. N
.H a a .N NH .0 a N a a a N N .. ml
. H e m --- . H e m N H o m N H An o oypN x
lllllllll L.||llllll1.||ll||l|l.||||ll|lllllllllll
poo a poo a
H H o NI
.. o .. H .o .H ..oo .. H .o .H --- o .H Ho.a.o«PN H
IIIIIIIII +.n.u.u.:.n.u.u.u.-.n.u.1.1.1.1.1.n.1.n.|.|.|.|.u.n.|.1.1.1.1.1.|.|.|.I.I
ooo H ooo H
«H H a a «U a0 a a HI-
. m .. H m H . m . H m H --- m .H Ho.o.HyeN a
Show
N . N e an
N H N U N U .N.N ..Vm N—N- ..Vm “cmco 300 “H Um
NN e N NN.e N NN e N N N .
..NN. em .NN em
acuoo> Hwooumwoom

 

 

 

 

 

 

 

«luv—3 Mmmaoouo wcavooamouuoo 0:... mo

3:08on 2.3

can NM muouoo> Hwooumfioom

(«can man—mm.

51

Thus we have available all quantities needed to solve the
amplitude equation (4.26) for the seven magnetic space groups.
Suppose, as an example, that we start with the reciprocal vector
31 = Zfi1§* and the elements of the group2?{(ér§3 which are given
to be element 1 which is E, the identity, and element 5 which is
2x, the two-fold rotation around the x-axis. These are the only
two operations under which‘lg1 is invariant for the group P42'22'.
From Table VII one has that

E = l O O ,

O l O

and

x
0 -1 0
0 0 -1
. 1! “I * ,
and uSing d = E, 2x; 9 = E, 2x; _’= 0, 0; and 3 = 2 13 in (4.26), one
obtains
(E - E)~£(§) E 0 (5.1)
and
(E - 2x>§(k)'-.EO. (5.2)

The equation (5.1) allows all components of 2(3) to be arbitrary.

52

 

 

However, (5.2) gives
'1 o o - 1 o o) Fx(21l/l_a_*) = o,
o 1 o o -1 o Fy(2n’1§*) o
c0 o 1 o o -1_ Fz(21)’12*) 0
or
o o 0 FX = o (5.3)
o 2 o Fy o
o o 2 F2 0

O O O
0 2 0
O O 2

is two, and thus there exists only one non-trivial solution of (5.3).
It is found by multiplying out the matrix equation (5.3) which gives

0Fx = O; 2Fy = 0; and 2Fz = 0. (5.4)

This means that Fx =a(or the parallel component of fig), i.e., Fug),
is arbitrary; and Fy = Fz = O, or the perpendicular component of
‘§(§), i.e., EL(§), is zero. Such a result will be designated by
E@) = (060.0). (5.5)
*
As a more complicated example, considerg3 = Zflhg and the
invariant groupfiwlg with elements 1 which is E, 2 which is 22,
3 which is (42'), and 4 which is (az'lré), for the same magnetic

=E,

space group P42'22'. The corresponding groupé}§(€) consists of 91

92 = 22, 93 = 42, and

4

9 =2 (where? -
z Z

We substitute these values in (4.26) and thereby obtain the following

equations:
(E - E)°§@) = 0.
(E - 22mg) = 0
fig - exp «Ming.
and
fie - exp(-211’1ng*-

More explicitly these

0001:X

y
OOOF
Z
100-
010
L001

 

 

and
'1 o o -
Ho 1 o
L0 o 1

153) ZZJ'EQ) = O:

’52) 333-202) = 0-
equations read:

= 0 ’

exp ("Trim 0 1 O

exp(-Ifin) o -1 o

1 O O

0 O -1

 

v11

r11

r11

'11

F

F

(5.6)
(5.7)

(5.8)

(5.9)

(5.6a)

(5.7a)

(5.8a)

(5.9a)

The solutions thus depend on the value of n, and we first choose n to

54

be even and obtain

OF = OF = OF = o, (5.
X y z
2F = 2F = OF = o, (5.
x y z
- = + = =
(Fx Fy) (Fx Fy) 2Fz O, (5.
and
+ = - + = = . 0
(FX Fy) ( Fx Fy) 2Fz 0 (5

The only common solution for these four sets of equations is

2(5) = 0- (5.

*
Thus, if n is even and §_= 2flh£ , then £(5) = 0.

Now we take n to be odd, and then we obtain:

0F = OF = 0F = O, (5.
x y z
2F = 2F = OF = 0, (5.
X y 2
(F +F ) = (-F +F ) = OF = O, (5,
x y x y z
and
(F -F ) = (F +F ) = OF = 0. (5.
x y X y z
The common solution is thus Fz =¥ with garbitrary, and Fx = F =
Y
i.e., F"(§) is arbitrary and EL(§) is zero. Such a solution will be
designated by
2(5) = (0,0,X), n odd. (5,

In a similar manner, all the solutions of the scattering

amplitude equation can be obtained for the seven Shubnikov groups

6b)

7b)

8b)

9b)

10)

6c)

7C)

BC)

9c)

0.

11)

55

and for the five types of reciprocal vectors, and the results are
given in Table XXV. The first row of Table XXV gives the reciprocal
vectors 31 and the subsequent rows give the solutions to (4.26)

for our seven tetragonal groups.

56

 

S.o.8b+
8.196

:8.on +
84.83

8.8.3

3.8.8» +
8.7.88

8.7.9..
IIIIIII 4

8.3.8

ccc a Ao.o.ov

a
"U
“U
0
c.‘
A
O
a
O
u
0
V

 

86.9.8.3 n m

86.5.5 u m

 

88.8% a m

 

 

 

 

 

“imam 0.32"qu ngouaH of mo Amvw. 3:259:00 Howusoew 2.3 now mafia—Sow

ccc a 8L8 cco a 8.8.3 «up?
ccc c 8&8 25 a 8.8.8 NNNeom
--- --- NNNecm
IIIIII JIIIIIIIJIIIII
28.8% + 3.88% +
8.8.83 8.18m .N.NNE
IIIIII IIIIIIIIJIIIII
2.8.8» + 3.8.8“?
8.8.9.. 8.18L «.N.NE
848 88.3 889$
llllll L..I.lllllu.ll..l.lllu.l
8L8 8.0.3 NNNE
866% n M 86.5.: n M 96.5
.88 3.5

CHAPTER VI

SUMMARY

Inclusion of time inversion as a possible symmetry operation
for the study of crystallographic structure leads to interesting and
significant generalizations of the ordinary crystallographic point
and space groups. As a symmetry operation, time inversion may occur
by itself or in combination with the spatial rotations, reflections,
and translations. The resulting generalized point and space groups,
the Heesch groups and the Shubnikov groups, respectively, are useful
for the study of magnetically ordered crystallographic states.

We considered the time-averaged internal magnetic field of the
ordered state to be a classical axial vector field. When the
Shubnikov group of the magnetic state is given, it is possible to
develop for all points of the unit cell the symmetry behavior of the
internal field, as well as that of its Fourier components in
reciprocal lattice space. The former is of most direct interest in
nuclear magnetic resonance experiments, and the latter in elastic
neutron scattering experiments.

In this dissertation the general theory for this problem is
developed and discussed, and the procedure is illustrated through

57

58

application to seven different Shubnikov groups all of which belong to
the same chemical space group of the tetragonal system. The general
theory is developed in sufficient detail to permit similar
calculations to be made with ease for any of the 1651 possible
Shubnikov groups.

If for a given chemical space group all possible Shubnikov
groups are studied in this way, it will then be possible from the
results of such a study to predict whether a unique assignment of
the Shubnikov symmetry can be made from the available nmr or neutron
diffraction data, or if not, what additional data one would have to

attempt to produce to make the aséignment unique.

 

10.

11.

REFERENCES

H. M. Rosenberg, Low Temperature Solid State Physics (Oxford
University Press, London, 1963), Chapter IX.

C. Kittel, Introduction to Solid State Physics (Wiley, New
York, 1966), 3rd ed., Chapter XV.

International Tables for X-Rhy Crystallography, Vol. I (Kynoch
Press, Birmingham, 1952).

A. V. Shubnikov and N. V. Belov, Colored Symmetry (Pergamon
Press, New York, 1964).

H. Heesch, Z. Krist. 1;, 325 (1934).

J. A. McMillan, Am. J. Phys. 35, 1049 (1967).

E. P. Wigner, Gropp Theory and its Applications to the Quhhhum
Mechhnics of Atomic Spectra (Academic Press, New York, 1959).
M. Tinkham, GrOpp Theory and Quantum Mechanics (McGraw-Hill,
New York, 1964), Chapter VIII.

A. M. Zamorzaev, Soviet Phys. Cryst. 3, 401 (1958).

B. Higman, Applied Group_Theoretic and Matrix Methods (Oxford
University Press, London, 1955).

G. F. Koster, Space Groups and Their Representations in Sglid

State Physics, Vol. V, F. Seitz and D. Turnbull, eds. (Academic

 

Press, New York, 1957).

59

6O

12. W. Opechowski and R. Guccione, Magnetic Symmetry in Magnetism,
Vol. IIA, G. T. Rado and H. Suhl, eds. (Academic Press, New
York, 1965).
13. R. D. Spence and P. A. van Dalen, Acta Cryst. 524, 494 (1968).
14. J. D. H. Donnay and G. Donnay, Compt. rend. 258, 3317 (1959).
15. J. D. Jackson, gigssicgl_glectrodynamics (Wiley, New York,
1962), Chapter V.
16. R. D. Spence, The Representation of Internal Magnetic Fields in
Crystals, Seventh International Congress and Symposium on Crystal

Growth, Moscow, 1966. Abstract 3.15.