The possible antiferromagnetic symmetry groups of azurite

Riedcl,

E. P.

Spence,

Physica

R. D.

26

1174-1184

1960

THE POSSIBLE ANTIFERROMAGNETIC SYMMETRY GROUPS OF AZURITE t) *) by E. P. RIEDEL Ikpartment

of Physics

>lichigan

and R. D. SPENCE State

University

East

Laming,

Michigan

synopsis Starting resonance to

with

determine

crvstal

the magnetic

techniques the

point

group

symmetry

obtained

and the x-ray space group, Shubnikov

possible

antiferromagnetic

symmetry

by nuclear

group theory groups

of

the

magnetic

is employed monoclinic

azurite.

1. Introduction.

Proton resonance studies of the mineral azurite below 1.86”K 1). CU3(C03)2(OH)2 h ave shown it to be antiferromagnetic In the present paper we combine results of recent proton resonance and x-ray studies of azurite with the theory of magnetic groups (Shubnikov groups) in order to enumerate the possible arrangement of the magnetic moments in the antiferromagnetic state. 2. Proton Resonance Data. Figs. la, 1b and lc show the angular dependence of the proton resonance lines in the antiferromagnetic state of azurite in the a’ - c, b - c, b - a’ (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

2f-+L{ 1 +(~~+2~coso)i-

1,

where vo is the frequency of the free proton resonance in the dc magnetic field Ho and 0 is the angle between Ho and the local field HI. The data in figs. la, b, c were taken with Ho = 3430 gauss and T = 1.6”K. The magnitudes and orientations of the local fields are given in table I, where 6 is measured from the c axis and + from the a’ - c plane. The angular relations between the directions of the local field vectors in the crystal can be conveniently represented by a stereographic projection t) Supported by the Office of Ordnance Research, *) To be submitted by E. P. Riedel in partial drgrer

at

Michian

State

University,

East

Lansing,

-

U.S. Army fulfillment of the requirements Michigan.

1174 -

for the Ph. 1).

THE POSSIBLE

ANTIFERROMAGNETIC

Fig. la. Antiferromagnetic

SYMMETRY

GROUPS

resonance diagram Ho in the a’-~(a’ plane, T = 1.6”K.

OF AZURITE

1175

perpendicular to c)

I

J3

Fig. lb. Antiferromagnetic

Fig. 1C. Antiferromagnetic

resonance diagram Ho in the b-c

resonance diagram Ho parallel to b-a’

plane, T = 1,b”K.

plane, T = 1.6”K.

1176

E. P. RIEDEL

AND R. D. SPENCE

TABLE

~~~_____...___~._

1

i- Local maanetic field vectors at moton msitions HI gauss 1 6 z 1 580 62” j 23” 2 ! 580 23” 298” 3 4 5 6 7 8

I ;

580 580 545 545 545 545

/ ;

157’ 157” 27” 27” 153” 153”

I

242’ 118” 56"

304” 236"

I

124”

as shown in fig. 2. We employ the usual crystallographic convention that 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 symmetry of fig. 2 one must bear in mind that it represents angular relations between axial vectors rather than polar vectors which are commonly shown in crystallographic applications of stereographic projections. Since the magnetic moment vectors of the

Fig. 2. Stereographic

projection

of the local field vectors

in Table

1

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 azurite may be represented as shown in fig. 3. 3. Shubnikov Groups. The symmetries of the possible arrangements of the magnetic moments together with the atomic positions in antiferromagnetic crystals are described by the so called Shubnikov groups 2)3)4). These groups differ from the ordinary crystallographic groups in that in addition to containing the ordinary crystallographic symmetry operations they may

THE POSSIBLE

ANTIFERROMAGNETIC

contain so called antioperations,

SYMMETRY

An antioperation

GROUPS

OF AZURITE

transforms

1177

an axial vector

in the same way as an ordinary crystallographic operation followed by a reversal of the sense of the axial vector. Each of the 230 ordinary space groups generates a family of Shubnikov groups which are obtained by replacing some of the ordinary operations with antioperations in such a way as to preserve the group property. This greatly enlarges the number of groups and as a result there are 1651 Shubnikov space groups as contrasted with the ordinary 230 crystallographic space groups. In a similar fashion, the 32 ordinary point groups give rise to 122 Heesch point groups.

Fig. 3. Point

symmetry

of a magnetic

moment

vector

in the antiferromagnetic

state

of azurite.

In writing symbols for various groups we shall use the notation found in the International Tables for X-Ray Analysis 5) and shall attach a prime to indicate an antioperation when necessary. Our notation follows that used in reference 3. In graphical representations of the elements we shall follow the standard symbols given in reference 5 but since there are as yet no symbols for the antielements we shall adopt the convention of indicating them by diagonal cross hatching. The effect of the symmetry operations of the Heesch point groups of the monoclinic system on a single arbitrarily oriented axial vector is shown in fig. 4. The symmetry operations comprising each of the point groups are listed in table II, where E, 2, i and m represent the identity, a-fold rotation, inversion and reflection respectively. An excellent description of how these operations affect the components of an axial vector may be found in reference 6. Point Group Selection. We shall assume that the atomic positions are not altered in the paramagnetic to antiferromagnetic transition. The point group in the antiferromagnetic state must satisfy the conditions: 1. The point group must transform an arbitrary axial vector in the same way as is permitted by the resonance pattern. (Fig. 3 for azurite). 2. The ordinary crystallographic point group obtained by replacing all antioperations of the point group by their corresponding ordinary operations 4.

1178

E. P. RIEDEL

_

AND

R. D. SPENCE

must be either a proper or improper subgroup of the ordinary graphic point group of the crystal. For azurite this is 2/m.

crystallo-

From fig. 4 it appears that the only point groups which satisfy requirements are 2’lm, 2/m’, 2/ml’, 21’ and ml’.

Fig. 4. Symmetry

of an axial

vector

under the monoclinic ‘TABLE

l-

Heesch

point

these

groups

II

Monoclinic Ileesch point groups Point Croup (in Belov notation)

Symmetry operations or PI-0”D elelnents

2 2’ 21’ m .z’ ml’ 2/m 2/m’ 2,/m T/m’ 2/w&l’

-I

2, E 2’, I< 2, 2’, E, I<’ m, I< m’, fi: s-2 8m’ , k.I IS’ 2, m, i, E 21m’ , i’ ,,F 2’ , rx, i’ >I< 2’, m’, i, I:‘ 2, IX,,I,, i F 2’ ~8’0i’ E’

5. X-Ray Structure 7). The chemical space group of azurite is P21/c. The unit cell dimensions are: a0 = 5.00, bo = 5.85, CO= 10.35 A, fl = 92”20’. The copper ions may be divided in two sets. In the first set are the ions which occupy the special position 0, 0, 0; 0, 4, 4. The second set occupy

the general position

x, y, z; x, y, z; x, + + y, 4 -

2; x, 4 -

y, 3 + z

THE POSSIBLE

ANTIFERROMAGNETIC

where x = 0.252, radicals

in general

y = 0.495

SYMMETRY

and z = 0.085.

GROUPS

There

OF AZURITE

1179

are also four hydroxyl

position *).

If the crystal structure and resonance data 6. Space-Group Requirements. (previously described for azurite) are available, then one may proceed to select as possible antiferromagnetic

space groups those groups in the table

of reference 3 which satisfy the following requirements. 1. The operations of the Shubnikov space group must leave unaltered the position of the ions as determined by the x-ray data. This implies that if a symmetry operation of the Shubnikov group transforms the ion position I into R, then the chemical space group must contain a corresponding symmetry operation which will transform r into R.The converse is not true and therefore there may exist pairs of ions whose positions are related by symmetry operations of the chemical space group but which are not related by a symmetry operation of the Shubnikov group. 2. The Subnikov space group must have as its point group one of the possible point groups predicted by the data. For azurite this is one of the five point groups found above. 3. 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 azurite this number is eight. 4. A magnetic ion may not occupy an anticenter. An anti-inversion moves an axial vector through the anticenter and then reverses its sense. Hence the magnetic moment of an ion located at an anticenter is zero. 7. Space-Groztp Selection. The left hand column in table III consists of all those antiferromagnetic monoclinic groups in the table of reference 3 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 is not satisfied by the group. In the last column at the right, the corresponding point groups are listed. Since the group P,2i/c contains an antitranslation in the a crystallographic direction, its unit cell is just twice the volume of the chemical cell. The unit cell of this group therefore contains eight protons. However, as may be seen from the following they do not all possess different local fields. Suppose the components of the local magnetic field at one proton located at (x, y, 2) is HZ, H,, H, where the x, y and z axes are taken along the crystallographic axes a, b and c respectively. The anticenter at (t, ‘4, a) in the magnetic cell implies that the proton at (4 - x, 7, E) will experience a local field the components of which are Hz, H,, and 8, where the bar *) The existence of four protons in the chemical cell follows from a knowledge of only the chemical formula, chemical point group and the proton resonance in the paramagnetic state.

1180

~~~~~~____~_

E.

P.

RIEDEL

AND

‘f.4BL6

Space

D.

SPENCE

_

111

group

selection

Kequirernent

i_

I

R.

Point 2

*

I

3 * 1 * / * *

I 4 ~

Group

2/ml’

-

2’jm’

~ * ~ *

2/m’

~

z/m

indicates a negative quantity. The two protons as well as the local field vectors are translated to (X + 4, y, z) and (Z, 7, Z) respectively by the antitranslation in the a direction. The local field components at these positions are then 8%, HY, Hz, and Hz, H,, H,. 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 that there are in all only four proton positions at which there are different local fields in the unit cell of P,21/c. The group P,c also contains an antitranslation in the a direction. Therefore 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 P,c are concerned. Let the components of the local field vector at the proton position (x, y, z) in one set be H,, H,, and H, and those of a proton in the other set at (x’, y’, z’) be Hz’, H,’ and H,’ where the positions (x, y, z) and (x’, y’, z’) are related by the operations of the space group P2i/c but are unrelated by those of Pac. The existence of eight different local fields is predicted by the group operations as shown in table IV where c, n and ta 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 P,2r and PbC. For brevity only the local fields at the proton positions belonging to one of the two sets of protons are shown in tables V and VI. The other set is obvious as in the case of Pac. The antitranslation in both the a and b crystallographic directions in the group C,c leads to a unit magnetic cell four times the volume of the chemical

THE

POSSIBLE

ANTIFERROMAGNETIC

SYMMETRY

TABLE Magnetic

x’, y’, 2’

x’, 1 -

HZ’

under P.c

HU’

ET,,

HZ’

EL,

HZ’

TABLE Magnetic

x,

Y. I

t

-

x,

HZ

Y +

.x’ + 8,

azz

V

field transformations

under P.21 21’

21

I

Y', 2’ iTz, if?,,

n’ + t, t - Y's2’ + d HZ’

HU’

E

t> d -

1 -

2

x,

Y +

a,

L’ t,

d -

2

x

HZ

+

t,

E7,

alI

HZ

HZ

Efz

Magnetic

Y. 57

r7,

H&J

TABLE

1181

OF AZURITE

IV

field transformations

Y', 2’ + 9 EL,

GROUPS

VI

field transformations

under Pbc

x?;[ ’ x>*-zr+* ’ %-I;+’ 1%,y~~z 1 TABLE Magnetic E

H, HZ n’ n f

t, t -

field transformations

I

n, Y, 2 HZ

Y, 2 +- t

c z, t -

VII

Y> 2 + t i7,

I

under C.c tl

x, a -

Y, 2 + t HZ

HU r7,

if, HZ

tb’

f.

x, v + t> 2

n

I

.cc+ d> a Hll i7z taltb, IT t

4, Y 4

a,

HZ

ii:

Rll

H,

HZ

Rz

HZ

cell. The sixteen protons in this magnetic cell may be of eight each, again in such a way as to make those by the operations of C,c to those in the other set. As fields at the proton positions in one set are shown in

Y, 2 + t

R*

4, 2

divided into two sets in one set unrelated above, only the local table VII.

1182

E. P. RIEDEL

AND

TABLE

R. D. SPENCE

\‘I11

Magrletic structure: ?Jumbers in symbol

I

at copper iorl

I,1 i, I 281 2,l 12

IOUpositiw m magnetic

positiorl

0, 0, t> 0, 0, :, ha t3 ’

(x, y, 2)

Magwtic

cell

:

.126, ,005, ,585 ,626, ,005, ,585

2,2

,126, ,495, ,085

2>2

,626,

1.3

,374, ,995, ,415

I,3 2,3

,874, ,995, ,415 ,374, ,505, .915 ,874, .505, ,915

morrwlt

1011ill set

compoIlerlts

0 0 t t

I,2

23

p’,c symmetry

,495, ,085

llurrlber *)

p12, p1r> p1r /ilZ, /iIt/, p1z p12, p1,, p1z PLZ,ply, Bl.

1

/kz, /kJ, yzz pzz, /k?y, pz* pzz, pz,, j&z

2 2

fizz, ,uY, ,lk.

2

,k% cc3Y, &Jz /Gz, f&, pal p3z> &I. /k fi32, /kw, /.zar

I I I 2

1 I !

4 2 --

*) SW section on x-my dntn

Magnetic Numbers

iI1 symbol

at copper ion position!.

structure:

: 1011position

/

in magnetic

P,21 symmetry

(r, y, 2)

Magnetic

cell

mometlt

IOU ill set

comporlcrlts

Ilumber

1,l i,l

0, 0, 0

PlZV PlS, p1z p1z, p1,, p1,

I

t, 0, 0

281 E,l

0, :> t

PlZ,

I

:> :> t ,126, ,005, ,585

p1z, filz/, p1z

I

yzz, PcL2Y, pzz ,iiPZ, pz,, fizz fizz, FZY, pzz

2

1,2 i,2

,626, ,005, ,585

282

.374, ,505, ,915

pu/, /JlZ

I

2 2

E,2

.874, ,505, ,915

,uzz, pz,,

,uzz

2

123

.374, ,995, ,415

/.kz~ ru3Y>/Qr

2

p3z, pw, /h ,kz, pwr P3L pL3z,/i3y, p3r

2 2

i,3

,874, ,995, ,415

2,3 E3

,126, ,495, .085 ,626, ,495, ,085

, i /

2

8. Possible Magnetic Structures. Two possible magnetic structures with the symmetry of the groups P,c and P,21 are shown in figs. 5 and 6 respectively. Tables VIII and IX accompany the figures. Magnetic structures for the two remaining groups *) Pbc and C,c are illustrated in tables X and XI. The first number in the symbols in the left hand column of each of the tables VIII, IX, X and XI designates one of the four magnetic moments which belong to that particular family of ions which are all connected 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 position (set number 1 in the right hand column of the above four tables) are not connected by the operations of the group, for which the particular table is constructed, to those in general position (set number 2). *) The structure suggested in reference 8 is permitted

by the group C,c.

THE POSSIBLE

ANTIFERROMAGNETIC

SYMMETRY

GROUPS

1

Magnetic structure: Pt.c symmetry

iNumbers in symbol

1,1

I,

1 221 %,I 12

Ion in set

Ion positi (z, y, 2) in magnetic cell

at copper ion position

T

i,2

number

0, 0, 0

1

0, t, 0

I

0, a, t

1

0, $7 t ,252, .0025, ,585

2

,252, .5025, .585

2

1

2

2,2 2,2

,252, .7475, ,085

123 I,3

,748, .2525, ,915

2

.746, .7525, .915

2

,748, .4975, .415

2

2,3 E,3

2

2

-

Fig. 5. Magnetic

Fig. 6. Magnetic

cell:

cell:

P,c

P,21

symmetry

symmetry

1183

OF AZURITE

1184

THE POSSIBLE

ANTIFERROMAGNETIC

TABLE Magnetic Numbers symbol

in

structure:

SYMMETRY

GROUPS

OF AZURITE

XI C,c symmetry

;

-I

I

at

Ion position (x, y, 2)

copper ion

in magnetic

Magnetic

cell

moment

Ion in set

components

position

number

I

1,l

I

(0, 0, O), (4, t, 0)

I

i,i z,i

(4, 0, O), (0, t. 0) (0, a, t)> (4, t, 1)

192

(0, $8 t), (4, t> 4) (.126, .0025, .585), (.626, .5025, ,585) i.125, .5025, .585), (.626, .0025, ,585)

E,l

i,2 222

1

d 1

I

2

2

j.126, .2475, .085), (.626, .7475, .085) (.126, .7475, .085), (.626, .2475, ,085)

E,2 1,3

2 2

(.374, .2525, .915), (.874, .7525, ,915) (.374, .7525, .915), (.874, .2525, ,915)

i,3 2,3

, /

2,3

2

2

(.374, .4975, .415), (.874, .9975, ,415) (.374, .9975, .415), (.874, .4975, .415)

2

2

i

If every copper ion which is linked to another copper ion through a common oxygen ion is coupled antiferromagnetically by a superexchange mechanism to the other copper ion, the resulting magnetic structure forbids an antitranslation in the b direction. In such a case the groups P,c and P,2i are the only possibilities. It should be noticed 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 I. In principle it may be possible to obtain a complete description of the magnetic structure by combining the known local fields in the antiferromagnetic state with the positions of the protons obtained from proton resonance in the paramagnetic phase and the known crystal structure. However, the work of Poulis e.a. on CuCls.2HsO has clearly indicated

that this program

is extremely

difficult

if not impossible.

The authors thank Dr. J. D. H. Donnay Acknowledgement. kindly reading the manuscript and for his very helpful suggestions. Received

for

5-8-60 REFERENCES

1) Spence,

a 3)

Belov,

R. 1). and Ewing, R. I)., Phys. Rev. 112 (1958) 1544. N. V., Neronova, N. N. and Smirnova, T. S., Trudy Inst. Krist. Akad. Nauk S.S.S.R.

11 (1955) 33. Belov, N. V., Neronova,

I

N. N. and Smirnova,

T. S., Kristallografiya

2 (1957) 315.

(translation: Soviet Physics Cryst., Vol. 2, No. 3 (1957) 311). B. A. and Zaitsev, V. M., J, Exptl. Theoret. Phys. U.S.S.R. .70(1956) 564. 4) Tavger, (translation: Soviet Phys. JETP 3 (1956) 430.) 1952. 5) International Tables jar X-Ray Cryst., Vol. 1, Kynoch Press, Birmingham, G., Corliss, L. M., Donnay, J, 1,. H., Elliott, N. and Hastings, J. M., Phys. 6) Donnay, Rev. 11s (1958) 1917. G. und Zemann, J., Acta Cry&. 11 (1958) 866. 7) Gattow, N, J., Physica 25 (1959) 1313. 8) Van der Lugt, W. and Poulis,