J. Phys. Chem. Solids, 1974, Vol. 35, pp. 1139-1152. Pergamon Press. Printed in Great Brilain

HELIMAGNETISM IN MnP-TYPE COMPOUNDS: MnP, FeP, CrAs AND CrAs~_xSbx MIXED CRYSTALS

A. KALLEL* Laboratoire de Diffraction Neutronique, Centre d'Etudes Nuclraires, Grenoble, France

H. BOLLER Institut fiJr Physikalische Chemie der Universit~it A-1090-Wien, Osterreich

and

E. F. BERTAUT Centre National de la Recherche Scientifique et Centre d'Etudes Nuclraires de Grenoble, France

(Received 1 February 1973; in revised [orm 10 December 1973)

Abstract--The solid solution CrAs~_~Sbx has been investigated by neutron diffraction at various temperatures. The phase boundary between mixed crystals with MnP-structure and those with NiAs-structure moves from 59 tool % CrAs at room temperature to 48 mol % CrAs at liquid helium temperature. The mixed crystals with NiAs-type structure at 4-2 K have the collinear magnetic structure of CrSb. Samples having the MnP-type structure exhibit a "double spiral" spin structure like CrAs. The variation of the magnetic moment and the spiral structure have been studied. Using neutron diffraction data for the magnetic propagation vector and the phase difference of the two helices, the signs and the ratios of the exchange integrals are discussed in a Heisenberg model. The stability regions in parameter space are determined by a matrix method for the ferromagnetic mode observed in MnP above 50 K and for the helical modes observed in MnP below 50 K, and in FeP, CrAs and CrAs~_~Sbx. We have also considered the possibility of antisymmetric and symmetric exchange between nearest neighbours only, using a group theoretical approach.

The high ratios of antisymmetric to symmetric exchange which would be necessary to explain the observed helical structures preclude however antisymmetric exchange as a dominant factor.

1. INTRODUCTION

The pseudobinary system CrAs-CrSb has been re- cently studied [1] by X-rays and magnetic measure- ments: arsenic-rich samples have the MnP- structure like CrAs itself, while products rich in ant imony crystallize with the NiAs-type structure. At room temperature the phase boundary between these two structure types is at about 59 mol % CrAs, no two-phase region has been observed.

Magnetic susceptibility measurements show an- tiferromagnetic behaviour for all compositions. The Nrel temperature at first decreases as As atoms are substituted for Sb atoms, reaching a marked minimum at 43 m o l % CrAs[1] and then increases again for more As-concentrated compounds. In the vicinity of CrAs we have observed a less pro- nounced maximum.

*This work is a part of the "Thrse d'Etat" of Ahmed Kallel, C.N.R.S. n ° A.O. 6030. New address: Laboratoire de Physique-Ecole Normale Suprrieure, 94 bd du 9 avril 1938. Tunis, Tunisia.

CrSb is antiferromagnetic below 450°C and has a collinear spin structure with antiparallel alignment of the magnetic moments along the c-axis of the hexagonal unit cell [2]. Recently it has been shown that CrAs [3-6] is helimagnetic below room temper- ature with a double spiral spin structure similar to that of MnP [7-9]. Furthermore the magnetic transi- tion is of first order and accompanied by a crystallographic transition[4,5] which shows a rather large hysteresis [5].

It was the purpose of this investigation to study by neutron diffraction the competition of the differ- ent magnetic interactions responsible for the CrSb- and the CrAs-type spin structures in the ternary system. We compare our results with all helimagne- tic MnP-type compounds known to-day. In our comparison of helimagnetic structures we have dealt with two possible interpretations. The first one uses only isotropic exchange including fourth neighbours. The second interpretation considers only nearest and next nearest neighbours, but also takes into account antisymmetric exchange.

1139

1140 A. KALLEL, H. BOLLER and E. F. BERTAUT

7~

g E

g

2. PREPARATION OF THE SAMPLES AND EXPEmMENTAL TECHNIQUES

The samples were prepared by heating mixtures of the pure powdered elements in evacuated quartz tubes. After grinding, the products were annealed at 800°C for several days. The lattice constants of the homogeneous samples thus obtained were measured by Debye-Scherrer photographs taken at room temperature.

Neutron diffraction measurements were made on powder samples at room temperature and liquid helium temperature (4-2 K). Some additional meas- urements were made at 100 K in order to see the influence of temperature on the helical structure. The samples which were magnetically ordered at room temperature were also studied at higher temp- eratures in a furnace adapted to the powder diffrac- tometer. The atomic parameters and the magnetic structures were refined by the least-squares method.

160

140

1201

Ni As t y p e /

J ~ Ni As fype

MnP t y p f

/ / - - MnP "type ~ / l l V o l u m e s ot"room

/~/" femperofure /~" 0 Volumes of 4.2 K

rAs 50 CrSb Mol, %

3. CRYSTAL STRUCTURES The lattice constants of the samples studied at

room temperature and 4.2 K are given in Table 1; the phase boundary between the MnP-type and the NiAs-type moves from 59 mol % CrAs at room temperature to 42 mol % at 4-2 K, which is the position of the minimum N6el point. With increas- ing arsenic/antimony substitution the orthorhombic distortion of the MnP-type diminishes. This may also be seen from the atomic parameters, obtained by the least square refinements of the neutron diff- raction diagrams (Table 2).

Table 1. Lattice constants in

Fig. 1. Variation of the unit cell volumes of CrAs,_xSb~ mixed crystals with NiAs- and MnP-type structures.

The transition from the NiAs-type to the MnP- type is accompanied by a volume decrease of about 0.7% (Fig. 1). The volumes of all samples at 4.2 K are smaller than at room temperature (except for CrAs). Therefore the mixed crystals do not exhibit the anomalous volume increase at low temperature of binary CrAs. The volume of CrAs measured at room temperature is much smaller than the value obtained from extrapolation of the volume vs

of CrAs,_x Sbx mixed crystals*

Structure Composition Temp6rature type a (]k) b (/*k) c (.A)

CrAs¢ 295 K MnP 5-637 3.445 6.197 90 K MnP 5.591 3-573 6-128

CrAso.86Sbo.~, r.t. MnP 5.69~ 3.61~ 6"258 4.2 K MnP 5-679 3-634 6"255

CrAso.~2Sbo.28 r.t. MnP 5"735 3 "659 6.355 4.2 K MnP 5-717 3.679 6.287

CrAso.osSbo.34 r.t. MnP 5"745 3-699 6.406 4-2 K MnP 5.732 3"71o 6"346

CrAso.soSbo.so r.t. NiAs 3.78~ - - 5.730 4.2 K MnP 5.789 3.76o 6'474

CrAso.4~Sbo.59 r.t. NiAs 3 "846 5"712 4-2 K NiAs 3-928 5"444

CrAso.3sSbo.62 r.t. NiAs 3.86, 5-686 4-2 K NiAs not

measured CrSb r.t. NiAs 4.103 5-463

*The lattice parameters at room temperatures are taken from Debye-Scherrer photographs, those at 4.2 K from neutron diffraction diagrams.

¢Values taken from [5].

Helimagnetism in MnP-type compounds 1141

Table 2. Atomic parameters of some samples in the system CrAs-CrSb (space group Pnma: chromium and metalloids in position (4c))

Cr (As, Sb)

Composition Temperature x z B (/k-:) x z B (-]k -~') R

CrAs* 295K 0-011_1 0-199_.+1 0.6_+0-2 0.202_+1 0.578_+I 0.4_+0.2 7% 90K 0.004_+3 0.211-+2 0.3_+0.2 0.199_+2 0.584_+1 0.2_+0.15 I0%

CrAso.~zSbo._,~l" 295 K 0-014_+ 7 0-208_+ 4 0-2 -+ 0.2 0-219-+ 1 0-591 +-- 5 1-6-+ 0.2 3% CrAso66Sbo.34]' 295 K 0 .004_+4 0.214_+8 1.1 _+0.4 0.219_+2 0.589_+8 0.9_+0.25 3% CrAso.soSbo.~,~ 4-2 K 0.001 -+8 0.225-+8 1.9-+0-5 0.218--+2 0.598_+4 0.7+--0.3 5%

*From X-rays, Ref. [5]. ]From neutron diffraction measurements.

concentration plot. This observation suggests that the low temperature phase of CrAs is stabilized by the arsenic/antimony substitution.

4. M A G N E T I C S T R U C T U R E S

In the region 0-41 mol % CrAs all samples retain the NiAs type at 4.2 K and have the collinear magnetic structure of CrSb. A sample with 42 tool % CrAs was found to be two-phases at 4.2 K, the NiAs-type coexisting with the MnP-type. The neutron diffraction diagram showed a medium to strong (001)-magnetic reflection indicating a very small region at the phase boundary of the NiAs- type with spins inclined against the c-axis.

This inclination may be due to a change in the magnetic anisotropy and marks the beginning of a transition to the helical arrangement, where the spins are also turned away from the [001[-direction of the NiAs-type.

Samples containing more than 42 tool % CrAs have the MnP-type in the magnetically ordered state and exhibit a double spiral structure like CrAs (Fig. 2). The N6el points do not necessarily coin- cide with the crystallographic transition. In particu- lar the arsenic-rich samples have much lower N6eI points. These products have also rather fiat suscep- tibility curves[l[ , and the maxima of these curves are at considerably higher temperatures than the N6el points determined by neutron diffraction measurements.

The N6el temperatures have a maximum in the ternary region (Table 3a). This maximum could be interpreted by a strengthening of the exchange in- teraction. More probably, however, it is caused by the suppression of the first order crystallographic phase transition of CrAs. In CrAs the classical N6el point is not attained because of the spontaneous loss of the magnetic moment at the crystallographic phase transition [5].

The wave vector q is found to be about 0-40 x

2o,ooo

o

:~ ~0,000

Or A50.66 Sb0.54 4-~, K

; i

I I 5 I0 J5

8

I 20 25

g~ CrAs 0 86Sbo 34455 K

c 20.000

o o

to,ooo (u

I I I 5 tO 15 20 25

e

Fig. 2. Neutron diffraction patterns of CrAs°~Sbo.3, at 4-2 K (orthorhombic Pnma setting) and at 455 K (hex-

agonal).

2~c* for the whole range of composition, which corresponds to a spiral length of 2.5 c. This 2-5- ratio is constant within the limits of error although the absolute value of the periodicity changes from 16.19/~ for CrAs0.sSbo.5 to 15.56 ]k for CrAs0.86Sb0.~4. The wave vector does not vary significantly with temperature. This behaviour is different from that of CrAs, where the wave vector is smaller and increases with temperature [5, 6]. It appears that the helical structure of the CrAs~_xSb,-mixed crystals is commensurable (or nearly commensurable) with 5 unit cells (Table 3b) of the crystallographic struc- ture. Such a commensurability might be caused by a magnetic anisotropy in the ab-plane.

The refinement of the magnetic structure was

J P C S Vol. 35, No. 9---H

1142 A. KALLEL, H. BOLLER and E. F. BERTAUT

Table 3(a). Magnetic properties of CrAs,_, Sb~ mixed crystals

Composition

Ordered magnetic T,, from Magnetic moment at 4-2 K neutron structure (in p.8) diffraction

CrAs Helical (double spiral) 1.67 -+ 0.06 250 K

CrAso.86Sbo,,, Helical (double spiral) 1 +88 -+ 0.05 340 K

CrAso.72Sbo+~8 Helical (double spiral 1.90 -+ 0.06 340 K

CrAso.~Sbo.~+ Helical (double spiral 1.93 -+0.07 310 K

CrAso.~oSbo.~o Helical (double spiral) 2.06 -+ 0"06 175 K

CrAso+,,Sbo.~9 Collinear anti- ferromagnetic 2.25 -+ 0.10 - -

CrAso 38Sbo.+2 Collinear anti- ferromagnetic 2-4 -+0.1 - -

CrSb Collinear anti- ferromagnetic 3.0 720 K*

*From Ref. [2].

Table 3(b). Wave vector and turn angles in helimagnetic MnP-type compounds (calculated from neutron diffraction data)

O~ ~ t ~a - - t ~ l

o r q/21r Aq~* Oz- ~b3 /3 = ~ - ~4 q~2z Ref.

0-350 - 126-+5 * - 116 ° 179" 60+8 ° [5] '°l CrAs 0-353 - 133 + 1" - 120 ° 183.5 ° 51.7 ° [6] r*' CrAso+~Sb~, 0-40 - 110 -+ 5* - 98 ° + 171 ° 60.8* this work CrAso.nSb~ 0-40 - 105 -+ 5 ° - 98* 170 ° 59+9 ° this work CrAso.~Sbo.3, 0.40 - 105 -+ 5* - 95* 167 ° 61.6 ° this work CrAso.soSbo.~ 0.40 - 107-+ 5* - I00" 172 ° 64.8 ° this work MnP 0.112 16.0" 20-4 ° - 0.2 ° 15-8 ° [7,8,9] FeP 0.20 168-8 ° 175-9" - 140 ° 28.5* [9]

t"]at 4.2 K. rblat 80 K. Note: ~ - ~ , - - t ~ + / 3 =~qz; ~2-~bt = 2 a +/3. *For meaning of the phase difference Aq~ used in the literature see Appendix C.

car r ied ou t fo r the who le ser ies o f m ixed crys ta ls accord ing to the c i rcular and the ell iptic double spiral mode l [8]. T h e r e was no significant ind ica t ion that the ell iptic mode l p r o p o s e d fo r M n P by Fo r s y th et al. [8] should be pre fe r red . Table 4 shows the in tensi ty ca lcula t ion fo r CrAs0.~Sbo+3<. T h e re- suits o f the magne t i c s t ruc ture re f inements are l is ted in Table 3a and 3b. The re is a ra ther un i fo rm var ia t ion o f the magne t ic m o m e n t s wi th a d i scon- t inui ty at the magne t ic and crys ta l lographic phase boundary (Fig. 3).

$. MAGNETIC INTERACTIONS IN HELIMAGNETIC MnP-TYPE COMPOUNDS

A deta i led analysis o f the hel ical spin ar range- men t in M n P has been g iven by T a k e u c h i and

Mot i zuk i us ing only i s , t r o p i c e x c h a n g e in te rac t ions [10]. T h e s e au thors a s sume an idealized b o d y cen t e r ed s t ruc ture and cons ide r s e v e n ex- change integrals . M o r e r ecen t ly Ber t au t [l l] has po in ted out, in a short note , that the poss ib le pres- e n c e o f an t i symmet r i c coupl ing p rov ides a di f ferent exp lana t ion o f this spin conf igura t ion with only three paramete rs .

In Pa r t (a) of this sec t ion we p re sen t first a theory taking into accoun t the actual crys ta l s t ruc ture wi th only fou r i s . t r o p i c exchange integrals . The mat r ix me thod d e v e l o p e d by Ber taut [12] and re- po r t ed by Nagamiya[13] is used. T h e deta i led cal- culat ion of r ep resen ta t ion theory inc luding anti- s y m m e t r i c [11] and s y m m e t r i c e x c h a n g e b e t w e e n

neares t ne ighbours is g iven in Pa r t Co).

::L

o E

4-

0

rAs

Helimagnetism in MnP-type compounds

Table 4. Observed and calculated intensities of the magnetic satel-

lites of CrAs~.~Sbo.~4 at 4-2 K

I~.,,, .~ . . . I t

hkl L~,,. Luo.

000 ± 1-73-+0"06 1-60 1 00 = 0"00-+0.1 0"00 I 0 1- 2-05 -- 0"2 2-04 002 3-54-+0-4 3-64 011- ] 101+J 9.23-+0.5 8-95

102- 0-00-0-1 0-00 1 1 0 ~ " 7.07 -+ 1 5.7 l l l - J

3 0 1 1 ~ 3"09 -+ 2 2.60

(obscured) 201-J 002* 0-6-+0.6 0.1 1 1 1 ÷ 0.0-+0.1 0-0 112-~ 20 1 + J 2-0 -+ 1-2 1.96

1 02* 0.00-0.1 0.00 202- 5.46-+0.8 4.89 103- ] 2 1 0 ~ obscured 5-8 211-J 3 00='~ 0.00-4--0-1 0.02 l 13 - J 3 0 r 1 1 03+[ 18"0±4 16-74 004- / 1 20=.1

I J I ~ I I 2 5 50 75 C r S b

Mol, %

Fig. 3. Variation of the ordered magnetic moment at 42 K in CrAs~_~ Sb~ mixed crystals.

(a) ISOTROPIC EXCHANGE

In the MnP structure type, there are four cations in the 4c sites of space group P n m a - D ~ :

x,~, z(1); ~,~, 1 - z(2); -~ x ~ -~ z - ,4,2+ z(3); 1 1 1 ~+ x, ~, ~ - z(4).

1143

The numbering of atoms in the literature is shown in Table 5.

Some authors describe the structure in Pbnm [7, 10, I1]. The most important exchange in- tegrals Zj in this structure type and the relevant distances in the CrAs case ( x - 0, z =0-2) are indicated in Fig. 4. With the notations of the matrix theory [12] a hermitian interaction matrix ~(q) may be constructed with elements

~,j(q) = ~ Jq exp iq(r ,0- r~). (I)

Here r~0 is a fixed reference point and the summa- tion is over all rj belonging to the same Bravais lattice j (] = I, 2, 3, 4). q is the wave vector:

q = qxbl + qybz+ q, b3. (2)

The b~ are the reciprocal vectors of the lattice vectors a~ (i = l, 2, 3). For instance, with the reference points given above, point r~0 = x, ~, z has two equivalent neighbors, belonging to the same exchange integral ]~z at points r,.0 and ~, ~, I - z =

c

,O. Ic 0-4(:: 0,1¢ 0 - 4 c O.Ic 0 -4C0" tC 0 - 4 c O Ic

Fig.4(a). Magnetic interactions between neighbouring atoms in the MnP-structure. In CrAs, at 90K, the distances between atoms numbered 1, 2, 2, 3, 3, 4 are d , = 2.836 .~; d~ = 3-143 A; d,, = 3.573 .~; d,2 = 3.967 ]k; d~3 # d~s = 4-5 ]k. Point and indice 2 refer to the position

~ .

Fig.4(b). Comparison of helimagnetic structures ob- served in this structure-type.

1144 A. KALLEL, H. BOLLER and E. F. BERTAUT

Table 5. Numbering of atoms in MnP-type structure

Coordinates of transi- tion metal

atoms (Prima) ~, ' ,~ Origin x, ' , z ' ' ' - ½ . . . . . + x, ~, z x, ~, ~ + z x, ~, 1 - z Ref.

1 4 3 2 This work 1 4 3 [8] 1 2 3 4 [6] 1 2 3 [71 1 4 3 [4, 11]

r~ +010. r~0 has also two other equivalent neigh- hours, belonging to the exchange integral ~r~ (see Fig. 4) at ~, ~, g = r20 + 00i and at ~, ], Z = rz0 + 0iT, so that

~2 = (J~2 + Jt~exp iq~)(1 + exp iqr) exp iq(rz0 - r20)

= 7/~: exp iq(r,o - r20). (3)

The matrix equation (4) has eigenvalues A~(q), re- lated to the magnetic energy E~ per ion by (5), and four-component eigenvectors T~(q), related to the spins S~(r~o) by (6)

(~(q) - A)T = 0 (4)

E~ = - 2A~; p = 1 ,2 ,3 ,4 (5)

Sp (rio) = Tpj exp - iqrio + T~ exp + i qrio. (6)

With the following t ransformation of eigenvectors

Qpi = T,~ e x p - iqrj0 (7)

one may construct a hermitian matrix ~/(q) such that

( ~ (q) - 3,)Q = 0. (8)

The ~/-matrix does not depend on the atomic coor- dinates x, y, z. Thus neither the eigenvalues A~ nor the eigenvectors depend on atomic coordinates. In

t * A D r/(q) = D* A (9)

LD* 0 B*

o u r c a s e

with

A = 2JI1 cos qy;

B = "0,2 = (1 + exp iqy)(Z2 + J,f exp iq~)

B~ = B e x p - i(qy + q=); D = (1 + e x p iqx)J~4. (10)

The spin Sp (rio) of the reference atoms rj0 (j = 1, 2, 3, 4) belonging to the eigenvalue )tp are given by

S, (rio) = Q~j + Q~. (1 t)

We look for solutions of the form*

_ u + i v _ Qi = ~ ' ~ Qj. (12)

Here u and v are orthogonal unit vectors and Qj is a phase factor; Q and Q only differ by the constant factor S(u + iv)/2. Explicit ly

Q~ = e x p - iO~; Q2 = e x p - i¢2; Q3 = e x p - i~3;

Q4 = e x p - iqs4. (13)

Thus S(r~0) = S(u cos Oi + v sin ~ ) . (14)

It is easily seen that the angle between reference spins, say S4 and S3, is equal to the difference of the corresponding arguments Oj, say ¢ 4 - ¢ 3 . Without loss of generali ty we put q ~ = 0 and find the eigenvectors (13) by the simple identification proce- dure of appendix A with

tP3 = +~(qx + qy + q~); tP4=~2 -½(-qx + qy + q~).

(15)

In a "double helix" the angles between S~ and $4, and on the other hand be tween S., and S3 (see Fig. 4) should be equal. This is however only the case if q x = 0 .

F rom the first line of the matrix equation (8) and f rom relat ions (9), (10) one has:

X , = A + ( B + D e x p i ~ ) e x p - i 0 2 (16)

with the abbreviat ion

*About the definition of Q~ see Appendix A. 3' = -- qx + qy + qz. (17)

Helimagnetism in MnP-type compounds

The reality of A implies that the imaginary part of the second member of (16) is zero wherefrom

]~, sin [ - @= +½(q, + q~)] cos ~q~

+ [jr~ sin ( - @~ + ½ qy + q~)

+jr~2sin(-@2+½qy)]cos½qy=O. (18)

For the real part one has:

+- i ' A ~ - A + [B + D exp ~Y[

= 2 J . cos qy + 2[J~, cos 2 ½ q~

+ (jr~2 + Jr~ + 2jrnJ~_ cos q.) cos 2 ~ qr

+2jr,4(J,2+jr, f) cos½q~ cos½qy cos½qz] m. (19)

Another form for A, real part of (16) and paramet- rized in Oz is

AT(q, @:) = 2jrn cos qy

+ 2 cos ½ qy []~_~ cos (½ q, + q= - @2)

+ 1,~ cos (½ q~ - ~:)]

+ 2 cos ½q,jr,, cos [½(qy + q , ) - ~=]. (19a)

It is easily seen that the condition (18) is equivalent to axt(¢9la~2 = O.

Stabil i ty condit ions

(i) Ferromagnet ic mode. For q = 0 there are three antiferromagnetic modes (cf. appendix B) and the ferromagnetic F - m o d e , observed for MnP by Huber and Ridgley[14] between 50 and 291 K. Its exchange energy is:

EF = - 2A~ = - 4(jru + J,, + jrt2 + jrt~). (20)

It is convenient to introduce the abbreviation:

U = jr, d J,4; V = ] , . . /J, , . (21)

From the relations B - 2 of the appendix B we deduce

U + V > 0. (22a)

The following stability conditions are found from the second derivatives of A'~ with respect to q,, q= and qy respectively

U + V + 1 > 0 (22b)

4 U V + U + V > 0 (22c)

jr,, > - ~(J,: + Jr,.~). (22d)

Among the inequalities of the (U, V) diagram, the hyperbolic inequality (22c) is the most restrictive one. According to (22d), Jrn is not necessarily posi- tive.

1145

(ii) Helimagnet ism qx = qy = 0; q, # 0. The dif- ferentiation of A ~ with respect to q, gives rise to

cos g q, = - ~ ~ + - ~ . (23)

From the second derivative with respect to q, we deduce the important inequality

Jnjr,~<O or U V < 0 (24a)

which is also confirmed by the evaluation of azXl÷laq~ 2. From a=A/aq~ one derives the inequality:

jr,, > - ¼lJ,. .- Jr,~[[1 - (4 U V ) - ' ] 'a (24b)

which also shows that J , is not necessarily posi- tive. Finally expressing the existence of cos ½ qz in the interval ( -1 , +1) one finds using (23) and (24a) the hyperbolic inequalities shown in Fig. 5.

4 U V + U + V < 0 ; - 4 U V + U + V < 0 . (24c)

As long as cos ½ q~ exists with UV < 0 the helical mode is stable and the explicit value of A is

A,o, = 2jr,, + 2V,= - j rd [1 - ~jr],/(jr,=J,~)]'/~. (25)

Eva lua t i on o f U and V : Re la t ion (18) d iv ided by jr,4 reduces here to

sin (@2 - ½q~) + U sin (@z- q.-) + V sin ~,_ = 0. (26)

On the bther hand, the parametrized form of A,(q, ~2) when derived with respect to the propaga- tion vector gives rise to another linear relation

sin (0,. - ½ q. ) + 2 U sin ( ~2 - q~ ) -- 0. (27)

u

,, 9, Helicol \ > 4- -:3 mode \ \ D

N + \ > -2

u.v<o X ~.. \ \ -i

-4UV+U÷V=0 \ \ l ' \ ' , ' , r ' , ' , ' a N \ ' { , \ \ ,h \~" lk I

-4 -3 -2 -I \ \ "

-i " \ \

Ferromognet ic mode

2 3 4 I I V

Helical mode N N N

u.v<o\\ N \

N N

N

Fig. 5. Stability regions in (U, V) diagram of the two observed magnetic modes in MnP-type compounds.

1 1 4 6

Whence the solution

U = - sin (t~2- ½ q~)/[2 sin ( ~ z - q~)]

= - sin ~b4/[2 sin (~4 - ~3)] (28a)

V = - sin ( ~ z - ½ q~)/(2 sin qs2)

= - sin ~4/2 sin ~2 (28b)

provided the coefficients of the relation (26) are non zero.

Thus we have the conditions on angles

s i n (~2 -½q~)~0 ; s i n ( ~ - q . . ) ~ 0 ; s i n ~ 0

or equivalently (28c)

sin ~, # 0; sin ( ~ 4 - ~b3) ~ 0; sin ~.~ ~ 0.

Further conditions on angles: It is useful at this point to introduce the turn angles a between planes at 0. I c and/3 between planes at 0.4c distance with

,~ = ~ , , - ~, = ~ - ¢~; 13 = ~p~- ~,;

a +/3 = + ½ q~ = ~b3. (29)

Neither a nor /3 can reach the values 0 and ~r in virtue of the two first conditions (28c). The last condition (28c) expresses the fact that sin (q~ - / 3 ) must also be different from zero.

From (24a) and (28a) one derives the following inequality

sin ~,~_ sin ( ~ _ - q~) < 0 (30a)

which may also be written

sin (2c~ +/3) sin/3 > 0 (30b)

OF

sin (ct +½qz) sin (~ -½q~) < 0. (30c)

From (30c) we obtain a condition between turn angle a and propagation vector

sin z a < sin~ ½q~. (31a)

In all observed cases ½q~ < 90 °. Thus a is restricted to the intervals (excluding of course the values 0

A. KALLEL, H. BOLLER and E. F. BERTAUT

and It)

- ½ q z < ~ < ½ q ~ ; 7 r - ½ q z < a < ' rr +~_q~. (31b)

According to (28a), U and V may be computed from the knowledge of the angles between spins and the conditions (30) to (31) enable us to review data of the existing literature.

M n P - - O n e has qz = 2~r/9. Unfortunately /3 is not known with a sufficient precision and is given as zero [7-9]. The inequality (30b) shows that/3 should be positive. Thus the experimental determination of f3 should be repeated again. We also conclude from '(31b) that a < 20 °. According to the stability condi- tions we find U > 0; V < 0 and

~'ff> 0; Ii2"<0; Ji4 > 0 . (32)

Takeuchi and Motizuki agree on J~z < 0 (their Jobs in Ref. [10]), but do not specify the other signs.

CrAs- -Wi th the values q~=21r0.353, a = - 120 ° = + 240 °,/3 = + 183°.5 [6] we find U -- + 7.1, V = - 0-52 and

J i f < 0 ; J~2>0; J~4<0 (33) i.e. the three relevant interactions in CrAs have signs opposite to those in MnP.

F e P - - W i t h the data of Felcher and Smith [9] (see Table 3b; qz =2~r/5), we have a = + 176 ° in agreement with the condition a # ~r and /3 = - 1 4 0 - 3 ° satisfying condition (31). We find U = - 0-055; V = + 0.066 with

Jl[~" 0; J12"< 0; d~14 .< 0, (34)

In all instances Ii2 and J~f have opposite signs. Whereas in MnP and CrAs the dominant interac- tion is by far J,~ (positive in MnP and negative in CrAs), i.e. between planes at 0.4c distance, the dominant interaction in FeP is J't4 (negative), i.e. between planes at 0.1c distance. Table 6 summar- ize the ratios of exchange integrals in MnP-type structures discussed here.

Remark 1--Direct calculation shows that

sin2 ½q,/sin 2 a = 1 - (4UV) -~. (35)

Table 6. Magnetic interactions in helimagnetic phosphides and arsenides with the MnP-type structure (isotropic exchange case). The ratios of exchange integrals are calculated from the

neutron diffraction data

r ,, U J i , V = Jl"-~z sgn J,¢. sgn Jl~ sgn Jz, Ref.

CrAs -0.52 +7.1 - 1.35 - + - [6] CrAso,72Sbo.2~ - 1.35 ? ? - + - This work CrAsD.~oSbo.5o - 1-06 ? ? - + - This work MnP - 0-54 ? ? + - + [7, 8] FeP + 0-066 - 0.055 - 0-9 + - - [9]

Hel imagnet i sm in

Thus if the product----4UV is large, sin2a is nearly equal to sin2½ q~ which is the case for MnP and CrAs.

Remark 2 - -Some values in the literature fall outside of the stability range of (31b). This is also the case of the solid solutions in Table 3b where one should have 108 ° < a < + 252 ° or - 72 ° < a < 72 ° . We must confess however that the errors are rather large. One may still question if neglect of the interactions J~3 (see Fig, 4) invalidates either rela- tion (24a) or (31a) or both. We only reproduce here the results of rather tedious calculations, sketching the details in Appendix D. One finds that the inequality (24a) is still valid, but that (31a) is replaced by

sin 2 a < sin 2 ½ q, if 4lJi3/J~41 < 1

and

sin2a>sin2½q~ if 4lJ~/J~4l>l (36)

so that for 7j3 large enough a can be exterior to the intervals (31b). The new values of U, V, A and q~ are given here for the sake of completeness

U = ( - ~ s i n (~s2 - ~ q~ ) + W s i n ½ q~ ) / s i n (~s~ - q~) (37)

V = ( -½ sin (~s2 - ½qz) - W sin ½q~)/sin qsz (38)

1 1 A (.T,~) = 2J,, - 2J,3 ( ~ + V ) + 2[J,~

W 2 ] ~n - J,.~l [(1 - ~ - ~ ) (1 --~--~_ I (39)

1 I 1

I t [ ( 1 - 4 - - ~ V ) / ( 1 - - - ~ - ) ] ' n } (40)

with

W - 2L3 / '14 "

(b) NEAREST NEIGHBOURS ONLY: BOTH ISOTROPIC AND ANTISYMMETRIC EXCHANGE

A group theoretical t reatment has already been sketched by one of the authors [l l] for MnP in space group Pbnm. We now investigate the subject further and in a more detailed way in space group Pnma. We limit the isotropic interactions to the shortest distances (J~4, J~2), but include also anisotropic antisymmetric Dzialoshinski-Moriya exchange (D~, D~2). The method follows Refs.[15, 16]. We define a set of generators for the space group G~ of the wave vector (q = 2wk) and find the irreducible representations of Gk.

MnP-type compounds 1147

We study the transformation properties of "k - components of spins" defined below and construct by projection operator techniques basis vectors which describe helical modes. With the basis vec- tors we build a (classical) spin hamiltonian which, in the isotropic case, leads to the same results as the matrix method considered above. New features come in by the introduction of antisymmetric ex- change.

Generators and irreducible representation o f G~ The wave vector k = [00/] is invariant under the

operations of a twofold axis 2~ and mirror planes my and mx perpendicular to the y and x axes respec- tively. The point-group of k is ram2. The space group G~ of k is Pnm 2~. Note that the inversion T is not an operator belonging to Gk. As generators we choose 2~ in ~0z and m in x~z. One has in the usual notation of space group symbols (al~-,)

2~ = (2~1½0½); m = (myl0½0). (41)

Here 2~ and my are point group elements, passing through the origin. One obtains

(2,,¢ = O1001); m" = (11000)

2,~m = (l[010)m 2,~. (42)

For the wave vector [00/] the translation 001 is represen ted by iexp(2cr i l ) where i is the unit matrix. Calling D(a t% ) the matrix representative of (a[%) one has

D((2,~) 2) = i exp 2~ril; D ( m 2) =

D ( 2 ~ ) D ( m ) = D(m)D(2, : ) . (43)

The commutat ion of D ( m ) and D(2~) implies that the irreducible representations can only be one- dimensional. They follow from

D(2~)=+-expTri l ; D ( m ) = - 1 (44)

and are tabulated in Table 7 with the abbreviation:

a = exp ~ril = exp i!q~. (45)

S k ~ C o m p o n e n t s : transformation properties and basis vectors

A helical spin can be written

S(rs) = S~ (ri) + Sk *(rj) (46)

with [17]

S~ (rj) = ~(u - iv)S exp i~ (47)

~ = 27rkr~ + ~pj. (48)

Comparison with (6) and (7) shows that

S~ (rj) = Q~(r~) (49)

1148

Table 7. Representations of G~ = Pnm2~; k = [00/] and basis vectors

Operators

e 2~ m 2,=m x y z

I xu 1 a 1 a F m 1 - a 1 - a - - I x~ 1 a - 1 - a ~ - ~') 1 - a - 1 a ~+

A. KALLEL, S. BOLLER and E. F. BERTAUT

Thus the angles @~3 and @~.a are imposed by group theory. Only the angles between spins S~ and S4,

Basis vectors @~ = at, are unknown. The relations (54), corres- ponding to (52), reduce at once the r/-matrix (9) into 2 x 2 matrices which determine at and A.

~:- _ Q~ = a ' q , ; Q4 = a q 2 . (54)

- - ~7 Invariants ~F (i) Isotropic case. Helical spin configurations may

be formed with y-components in F m and x- components in P~>. Invariants are formed by pair

.multiplication of basis vectors belonging to two conjugate representations. For instance

I~, = ( ~ , ~ * + ~,~'~)~"~ + (qr,~4* + ~,~'P),::~ (55)

is a scalar invariant formed with x-components of basis vectors of 1V(4) and with y-components of Fm. Direct calculation shows that

s _ _ L4 - 4S, S4. (56)

I<:I~0 One calculates in the same way ~:~ ~.s

Abbreviations: a = exp ~ril; 'sl~(&., + a*S~)~; ,.r/~-. = (&a + a&.4)~.

= x, y, z; ~tL = (&, - a*&.Oo ; ~tL = (&a- a&,)~.

Sk transforms like an axial vector. The transforma- tion properties of spins S~ and Sk,. are illustrated in Table 8.

Basis vectors ko~) are constructed with particular ease according to

at~j~ = ~ X(~)(T)*T&.j~. (50) T

Here X{*~(T) is the character of the operation T in the representation F c'~ (Table 7). TS~.j,, represents the effect of the operation T on the spin component S~j~, with j = 1, 2, 3, 4 and a = x, y, z.

One has for instance

q ~ = ( & . ~ + a*&Oo; ~ ) = ( S ~ . ~ + a&.O~ = a ~ )

~;' = ( &~ + a & ,)~ = a ~,; ';

~, = (&a-+ a*&,)= = a = W4o (51)

for v = 2 and a = y or v = 4 and a = x (cf. Table 7 and Fig. 4).

The assignment of representations to basis vec- tors defines the magnetic structure. If we say that the y components of S~., + a*Sk.3 and S~a + aSk,4 be- long to F m and the x-components to F "), we mean that all other y- and x-components must disappear in representations I ~u and F m wherefrom

S~ = a * S ~ ; S ~ = aS~,~ (52)

and

S~. S~ = S ~ cos ½q. = S ~ cos ~'1 = S~. S~. (53)

x,5-= Y, (,I,,Ve=*- + %,I,*) = ~ (,p, a,re* + %a*'P*).

(57)

In this relation E indicates summation on x- and y-components in F ") and F m respectively. Multiply- ing the invariants by appropriate exchange integrals and expressing them in terms of the two basis vectors "~'~ and ~4 (see (51)), the hamiltonian can be cast into the form

+ ( ~ a ~ * + ~ , a *~4)J,f

+ ( % a * ~ : * + ~ T a % ) I L ~ + ( ~ ' + % ~ * ) J , , ]

(*9 = - 2 ~ (q~$, ~$)(M,,) ~4 " (58)

Here M~ is the matrix (59) which gives exactly rise to the eigenvalue X(19) for q = [00q=].

[ Y,, J , ,+a*J , ,+aJ ,2] M~, = 2 Y~4 + aJl,+ a*Yl2 Y. . (59)

Table 8. Transformation of &-components

Operators Components e 2~, m 21.m e 2,= m 2,~m

x &, -&.~ - & ~ &, &.~ - & , . - & = & , y &: - & , &l -&.~ S~.~ -&... S~.~ - & . . z &, + & , - & , -S,.~ &: &,. -&:_ -&.,,

2,, transforms point 2 in X, I, 1 - z into point 4' in ½ +x,~,~-z: ~ Thus 4' = 4 + 001 and &.4. = exp 2~ril &.,.

Helimagnetism in MnP-type compounds

(ii) A n t i s y m m e t r i c exchange. If we stay in P n m 2~, we can only construct symmetr ic invariants, cor- responding to isotropic exchange and containing basis vectors of F (2> and F ~. These two representa- tions coalesce into one unique representat ion if we retain the only symmetry operat ions identi ty and 2~, omitting the mirror m, say in the wave vector group P2~ (see Table 9). By this symmetry descent to the monolinic group P2h not only the x - and y -components of the basis vectors belong to the same representat ion F '~ (Table 9) but the construc- tion of new vectoriel invariants I v by pair multipli- cation of x with y-components becomes possible.

Table 9. Representations and basis vec- tors in G~ = P2~: ; k = [00l]

Operators Basis vectors

e 2~, x y z F '('~ 1 a ~l- ~ - ~+ F '(z~ 1 - a ~* ~+ ~ -

Notations as in Table 7 and relation (51).

Thus for the two shortest dis tances (nearest neigh- bours) 14 and 12

I v = ~ , ~ - ~ , ~ * + c.c. = 4(S,~S,,. - S,yS,x).

(60) Recall ing that

xIr. = - iW.~ (61)

one has

I v = 2 i ( ~ x ~ * - ~*~, .~) . (62)

In the same way

I~= ~ . ~ - ~ q ~ + c.c.

= 2 i ( ~ a ~ * -@~*a*@,x) . (63)

Thus ant isymmetr ic exchange terms I)~. (S~ x Sj) with j = 4 and 7. can appear in the hamiltonian with Dzialoshinski vectors D along the z-axis (symmet- ric anisotropic exchange terms like ~x~4*y+ • ~ y ~ + c.c. are symmetry- forb idden [18]).

With appropr ia te interaction constants D~4 and D~,: we obtain the interact ion matrix M

with the abbreviat ions

E = J + a T ' + i (D + aD ' ) = IE] exp i4~

J = Jl3; J ' = Jl-~; D = Dl4 ; D ' = D n (65)

1149

here ck is the phase of E (we neglect the interact ion Jn). The eigenvalue A ÷ of M is given by

-~x + = J . + l E I (66)

with

E E * = f i + (j,)2 + D 2 + (D,)2 + 2 cos ~ qz (JJ ' + D D ' )

+ 2 sin ½q~(J'D - JD') . (67)

F rom the first line of M (64) and f rom (65) we get

xlq = 41 exp iqk (68) Thus

~b = $4, = a. (69)

Minimizing Z ÷ (66) we obtain

tg½qz = ( J ' D - JD' ) / (JJ ' + D D ' ) , (70)

and finally from (65) (69) one finds for the phase th, taking into account the equilibrium condit ion (70) the simple result

t g a = tgck = (J' sin ½q: + D + D ' cos ½qz)

/ (J ' cos ½ q, + J - D ' sin ½ qz) = D / Z (71)

This result has been given already i n [ l l ] . I t is remarkable that the turn angle between spins 1 and 4 only depends on D / J and not on D ' and Jr'. F rom the identity:

tg½qz =. tg(tx +/3) = ( tga + tg~8)[(1 - tg~ttg[3) (72)

and comparing to (70) and (71) one finds:

- O t t g ~ = j , . (73)

Thus /3' only depends on D ' and J ' and not on D and 3". It is instructive to consider the two limiting cases

(a) D # 0 ; D ' = 0

(70) reduces to

tg½q~ -- tg~ (74a)

with the two solutions

q~ = ot (74b)

which represents rather well the MnP case and

q~ + 7r = a (74c)

which corresponds to the CrAs case

(b) D = 0 ; D ~ 0

(70), (71) and (73) imply

tg½q~ = - D ' / J ' = tg~ (75)

1150 A. KALLEL, H . BOLLER a n d E. F. BERTAUT

Table 10. Magnetic interactions in helimagnetic MnP-type compounds (isotropic and anisotropic antisymmetric exchange case). The ratios of interactions are calculated from

neutron diffraction data

Dr4 - D~.* Observation D,, Jr4 D,f J,~ t g ~ = ' ~ l ~ tg[3= j,~

+ + ~ 0 + + 0"372 ~ 0 [D,,I < J,, 0 - + - ~ 0 + 0.840 [D,~[ < J,~

MnP FeP CrAs "] and ~

CrAs,_~ Sb~ J ~ 0 - + 1.1732 ~ 0 [D,,[> Jr,

with

tga = 0. (76)

The case a = 0 has not yet been found, but the solution a = ~r and /3 given by (75) represents rather well the FeP case. The eigenvalue 3. is given in all cases by

2 1/2 2 2 tD- ~ A = J H + ( J ~ + D t 4 ) + ( J t f + D l ~ ) . (77)

The stabili ty condit ions are

Ji4J, f+ Dt4D,~> 0; JtzD~4- Jt4D,f > O. (78)

Table 10 summarizes the signs of the interactions. Note that the turn angles a and/3 can be zero or 7r without conflicting with the equilibrium condit ions (78).

6. DISCUSSION

At first sight the inclusion of ant isymmetr ic ex- change seems to explain all the observed features. One must however keep in mind that in transit ion metals ant isymmetr ic exchange is generally, much smaller than isotropic exchange (second order effect (Moriya[19]) whereas in the CrAs case we would have to admit that [D[ > JI. This can only happen if there are strong orbital contr ibut ions which give rise to first order effects [20].

Of course one may still ask the question of the influence of the isotropic exchange J~z neglected here. When including Y12 which adds to E (65) a term a*Jlz, one obtains f rom h vs q, equilibrium

• condit ions which can be expressed as follows

D = tga + 2 V sin (½ q, + c~) (80a) ] cos a

D ' V sin 6q~ + ~) j--r = - tg[3 U cos/3 (80b)

They reduce to (28) for D and D ' equal to zero and to (71) and (73) for ~rt2 = 0. In the CrAs case the last equation (80b) for D' = 0 reduces to the already known result of the isotropic case V = - 0.071 U

which inserted into (80a) gives D I J = 1 . 7 3 - 0-16 U so that D / J can be considerably weakened. U being of the order of + 10. However although the algebra becomes awkward and no analytical ex- pression of cos q~ vs the exchange integrals can be found, the condit ion sin a # 0 and sin/3 # 0 of the isotropic case can be shown to be still valid.

On the other hand the hypothesis of isotropic ex- change considering three exchange integrals 3"1~, Jl2 and 3"t, is sufficient to establish the occurrence of double helix structures. I f it cannot explain the turn angles in the solid solutions CrAs,-~ Sbx which are exterior to the stability interval, we have shown however that this difficulty can be avoided by considering fourth neighbour interactions fJ3 of sufficient strength.

Although the model of only nearest neighbour symmetr ic and ant isymmetr ic interactions is eleg- ant f rom the formal point of view, the high D H values, specially in the case of CrAs make the model improbable.

Fur ther exper iments on single crystals with an applied magnetic field are planned and should give a definite answer.

REFERENCES

1. Sobczak R., Boiler H. and Nowotny H., HI Interna- tional Conference on Solid Compounds of Transition Elements, 16--20 June. 1969, Oslo Norway.

2. Snow A. I., Phys. Rev. 85, 365 (1952). 3. Watanabe H , Kazama N., Yamaguchi Y. and Ohashi

M., J. appl. Phys. 40, 1128 (1969). 4. Kazama N. and Watanabe H., J. phys. Soc. Japan 30,

1319 (1971). 5. Boller H. and Kallel A., Solid State Commun. 9, 1699

(1971). 6. SeRe K., Kjekshus A., Jamison W. E., Andresen A.

F. and Engebretsen J. E., Acta Chem. Scan& 25, 1703 (1971).

7. Felcher G. P., J. appl. Phys. 37, 1056 S (1966). 8. Forsyth J. B., Pickart S. J. and Brown P. J., Proc.

Phys. Soc. Lond. 88, 333 (1966). 9. Felcher G. P. and Smith F. A., Phys. Rev. Bg, 3046

(1971).

Helimagnetism in MnP-type compounds

10. Takeuchi S. and Motizuki K., Z phys. Soc. Japan 24, 742 (1967).

1 I. Bertaut E. F., J. appl. Phys. 40, I592 (1969). 12. Bertaut E. F., J. Phys. Chem. Solids 21,256 (1961); In:

Magnetism (Edited by G. T. Rado and H. Suhl), Vol. III, p. 149. Academic Press, New York (1963).

13. Nagamiya T., In: Solid State Physics, Vol. 20, pp. 305-411. Academic Press, New York (1967).

14. Huber E. E. and Ridgley D. H., Phys. Rev. 135, A, 1033 (1964).

15. Bertaut E. F., Acta crystallogr 24A, 217 (1968). 16. Bertaut E. F., J. Phys., Suppl. 32C, 462 (1971). 17. This definition has been chosen because in a lattice

translation Su(r) transforms like a Bloch wave[ l 1]. 18. Bertaut E. F. to be published. 19. Moriya T., Phys. Rev. 117, 635 (1960). 20. Levy P., Phys. Rev. 177, 509 (1969).

APPENDICES

APPENDIX A

Solution of the ~-matrix by comparison of phases The first line of the matrix equation (8) is explicitely

A - A + B e x p - i t ~ 2 + D e x p - i ~ b ~ = O . (AI)

The third line multiplied by exp + itp~ and taking into account D* = D e x p - iq~ becomes

A - A + B exp i ( - ~ . + ~b3- qy - q ~ ) + D e x p i ( - ~ 2 + q ~ + q ~ ) = O (A2)

wherefrom by identifying the arguments

t~,. = ~ , - ~3 + qr + q.- ; ~04 = t~z-- tp~ + q, (A3)

and the relations (15) of the text. The corresponding eigenvatues are:

a ~ B + D e x p i T [ . (A4)

Another solution is obtained by writing

~02 = ~0,- t~3 + q, + q, + 2~r; ~, = ~b~- t~ + q. (A5)

o r

~ = ~- +~(q, + q~ + q,); ~0, = ~ , . - -rr -~7 . (A5')

The corresponding eigenvalues are:

A + - I B - D e x p i ~ l . (A6) I zt

Writing

B + D exp i ~ = 1K,[ exp ik~

(A7)

B - D exp i~ = IK2] exp ik,.

one sees (of. relation (16)) that

tO2 = + k, for A~+; qs2 = ~r + k, for ht-

tp~ = + k2 for Az÷; ~ = ~r + k2for AC.

The forgoing equations completely specify the eigen- values h and corresponding eigenvectors Q,s (19 = 1, 2, 3, 4; ~ = 1, 2, 3, 4).

Note: Q~ (see relation (12) of text) could also have

1151

been defined by

u - i v Qj = S - - - ~ exp id/j.

It is easy to see that one would get ~3 =-~q~ and c~ +/3 = -½q~, i.e. clockwise rotation for the propagation along the c-axis.

APPENDIX B

q = 0 modes The eigenvatues of the f or 7/-matrix are

A~ = 2(J . + J~2 + Jz~ + J,,); Aa = 2(JH -- Jr2 - J~.~- Jr4)

(BI)

Ac = 2(J,~ + J,~_ + J ~ - J,4); A,, = 2(J,, - J,_~- J,~+ J,~).

The subscripts characterize the ferromagnetic mode F (+ + + +) and the antiferromagnetic modes G ( + - + - ) , C(+ + - - ) and A (+ - - +) where the sign sequence cor- responds to the four reference spins. A~ > Ac and AF > A,~ yield respectively

J~ > 0 ; J~z + J~.~>0. (B2)

APPENDIX C

Phase relations From the relation (7) of the text and defining

T~ = e x p - i¢~ (C1)

one has

~l = q~J + qr~o. (C2)

In the presefit case, spins 1 and 3 belong to one helix, i.e. ~b~ = d~ from (C2) and (15) and spins 2 and 4 belong to the other helix, i.e. q~4 = q~2. The phase angle A~ between the helices is defined by

[3 = t~3 - ~, = + 2q=z + q~3~o4 = + 2q=z + b Aq~. (C3)

Thus all angles may be derived from a knowledge of Atp and the propagation vector q (Table 3b).

APPENDIX D

Influence of Yla Taking into account J,3, the average value of the

interactions shown in Fig. 4 one obtains

X (J,3) = X (0) + 8Y,~ cos ½ q~. (D 1)

Here A(0) is the value of A(19) for J t~=0. From O(J~--~3)/aq.. = 0 and 02A(-f~3)/O~q~ < 0 one derives the in- equality

UV < W 2 (D2)

which is weaker than (24a) putting W = 2J~3]J~,. However setting

4 UV cos ~- qz + U + V = - e (D3)

and solving the equilibrium condition 3)t(J~3)/Oq.. = 0 with respect to ~, one finds

1152

From the positivity of ~2 (D4) and of A (.Tt3) it follows that the terms 1 - ( 4 U V ) -t and 1 - W21(UV) must be sepa- rately positive*. From

1 - W21 U-V = ( U V - W ~ ) t ( U V ) > 0 (DS)

and from (D2) follows again that

*Continuity arguments also show that they cannot be simultaneously negative. Indeed with the conditiola (D2), this would imply 0 < UV < ¼ which conflicts with (D6) and (D7) for W ~ 0.

A. KALLEL, H. BOLLER and E. F. BERTAUT

U-V < 0 (D6)

which can be written

(sin 2 ~x - 4 W 2 sin 2 ½ q:)/(sin 2 ot - sin 2 ½ q:) < 0. (D7)

This condition reduces to (31a) for W = 0. Direct calcula- tion shows that

sin ~ a/sin2½q, = [1 - W2/(UV)]I[i - 4/(UV)]. (D8)

From the relations (I)7) and (D8) follow inequalities (36) of the text.