Experimental arguments for a non-zero transition temperature in RKKY Heisenberg spin glasses without anisotropy

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Experimental arguments for a non-zero transition temperature in RKKY Heisenberg spin glasses without

anisotropy

A. Fert, N. de Courtenay, H. Bouchiat

To cite this version:

A. Fert, N. de Courtenay, H. Bouchiat. Experimental arguments for a non-zero transition temperature in RKKY Heisenberg spin glasses without anisotropy. Journal de Physique, 1988, 49 (7), pp.1173-1178.

�10.1051/jphys:019880049070117300�. �jpa-00210799�

Experimental arguments ^{for a non-zero} transition temperature ^{in RKKY} Heisenberg spin glasses ^without anisotropy

A. Fert, ^{N. de} Courtenay ^{and H. Bouchiat}

Laboratoire de Physique ^{des Solides,} Université de Paris Sud, ⁹¹⁴⁰⁵ Orsay, ^France (Reçu le 17 novembre 1987, accepté ^sous forme définitive ^{le 23} fivrier 1988)

Résumé.

Alors que l’existence d’une transition de phase ^à température ^{non nulle est} bien admise pour les

verres de spin ^de type Ising ^{à 3} dimensions, ^{des travaux} théoriques ^récents suggèrent que Tc ^est nul pour des

verres de spin Heisenberg ^sans anisotropie ^et que le Tc ^{non nul des} systèmes ^{réels est dû à} l’anisotropie. ^Pour

éclaircir ce problème, ^nous analysons l’influence de l’anisotropie

^le comportement critique ^{et la} température de transition de phase d’alliages CuMn ^et AgMn dopés ^{avec des} impuretés ^{d’or ou de} platine.

Cette analyse suggère plutôt ^un Tc fini pour les ^verres de spin Heisenberg ^RKKY dans la limite d’anisotropie

nulle.

Abstract.

While the existence of

phase transition at finite Tc is well admitted for Ising spin glasses ^{in 3-} dimensions, ^recent theoretical studies have proposed ^that Tc ^{is zero in} Heisenberg spin glasses and that the finite Tc ^{of real} systems is induced by ^the anisotropy. ^To clear up this question,

analyse the influence of the

anisotropy

the critical behaviour and the transition temperature of CuMn and AgMn alloys doped ^{with Au}

or

Pt. This analysis ^rather suggests

^finite Tc ^{for RKKY} Heisenberg spin glasses ⁱⁿ the limit of

anisotropy.

Classification

Physics ^Abstracts

75.30 - 75.30G - 75.30H - 75.40

1. Introduction.

There is now a strong experimental evidence for a

phase transition at finite T, ⁱⁿ spin glass alloys [1].

On the theoretical side, ^{it seems well admitted} that, in three-dimensional systems, ^a phase ^transition

takes place ⁱⁿ Ising spin glasses [2] ^{but not in short}

range Heisenberg spin glasses [3]. ^{The case of}

RKKY Heisenberg spin glasses ^{is not} completely

clear. On the basis of scaling arguments, Bray et ^al.

[4] ^have proposed ^{that a RKKY} spin glass ^{in 3d is at}

its lower critical dimension, ^so that the transition at finite Tc ^observed in metallic spin glasses ^{should be}

due to the existence of anisotropy. Bray et ^al. [4]

predict ^{for the} variation of 7c ^{with the} anisotropy ⁱⁿ

a RKKY vector spin glass ^with Dzyaloshinsky- Moriya (DM) ^or dipolar anisotropy:

which gives :

J and dJ =- D are the exchange ^and anisotropy strengths respectively (A ^{is a constant of order} unity

and yD = 3/2 for 3-dimensional systems). ^{As d tends}

to zero, Tc ^{tends to zero but the} logarithmic depen-

dence of equations (1) ^can explain ^that T, ^{does not} change very much in the experimental range of d.

Computer simulations on RKKY Heisenberg sys- tems have been recently published [5]. They ^are compatible ^{with the} predictions ^of Bray et ^al. [4] ^but

cannot rule out completely ^the possibility ^{of a non-}

zero Tc. ^In this paper ^we present ^some arguments for

a non-zero Tc ^{in RKKY} Heisenberg spin glasses.

These arguments ^are derived from experimental

data on CuMn and AgMn ^{alloys in which the DM} anisotropy ^{is made to} vary by adding ^{Au or Pt} impurities [6]. ^More precisely ^our arguments ^are based on :

(a) ^our previously published analysis of the influ-

ence of anisotropy ^on the critical behaviour [7] ; (b) ^an analysis ^of experimental ^{data for} Tc(d) [8],

with T, (d) extrapolated in the limit d --+ O.

We discuss the two preceding points in sections 2 and 3 respectively.

2. Influence of anisotropy ^on the critical behaviour.

de Courtenay et ^al. [7] ^{have studied the critical}

behaviour of the non-linear susceptibility ^{for series}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070117300

1174

of CuMn and AgMn alloys doped ^{with Au} impuri-

ties. We show in figure ^{la an} example ^{of the}

influence of the DM anisotropy strength (fixed by

the concentration of Au) ^{on the} non-linear suscepti- bility KNL ^{at T =} T,, : ^{two different} regimes, ^corre- sponding ^to exponents 8 I = 3.1 ^{± 0.2 and} 8 H =

4.5 ± 0.2, ^{are observed} in the low and high ^field

limits and the field of the cross-over between the two

regimes ^{increases with the} concentration of Au, ^i.e.

with the DM anisotropy strength (1). ^{A similar} anisotropy dependent cross-over between two differ- ent values, y I and y H, of the exponent y is also

Fig. 1.2013(a) Non-linear susceptibility KNL

^{H at} r==7e ^for CuMnO.2 at% alloys doped ^{with Au} [7].

KNL ^{is defined} by ^M

XH(1 - KNL). ^With KNL ~ H2/6,

the slopes ^{at low and} high ^fields correspond ^to 5j

3.1 ± 0.2 and 8 H

^{4.5 ±} 0.2. Similar results have been found in AgMnAu ^alloys [7] ; (b) K NL/ a * (d)

H/H*(d) for series of CuMnAu and AgMnAu alloys ^at T - Tc, ^with a * (d ) ^and H* (d ) varying

^{d1.7 and} dO.9 respectively [7]. ^This scaling expresses equation (2b).

(I) Notation : KNL

(X - M/H)/X ^with X

(M/ H)H -+0’ KNL - h2/s ^{for T h2,} KNL - h2/T y ^for

h2 Tlp, ^with

r/7, - 1, ^h

gJ-LB H/kB Tc, CPA

anisotropy cross-over exponent, cp H

ï’ H s H/ ( s H -1 ).

observed in the temperature dependence ^{of the non-}

linear susceptibility ^at low fields above Tc (see ^Ref.

[7] ^{for a detailed} report).

These experimental results, ^{which show an ani-} sotropy ^induced cross-over between two critical

regimes [9], ^are accounted for in reference [7] by using ^a three-parameter scaling ^function (1) :

with values of the reduced anisotropy d calculated as a function of the concentration of gold [7]. ^For example, ^{we show in} figure Ib how all the data at

Tc for series of CuMnAu and AgMnAu ^{can be} gathered ^{on a} unique ^curve by plotting KNL/ a * (d ) (with a ^{* -} d0.9 ^versus H/H* (d ) (with ^{H* -} dl-1),

in agreement with the limit of equation (2a) ^foi

T - 0 :

with

This analysis ^has given ^the exponent ^{of the}

anisotropy ^induced cross-over, (P A =z 0. 8 -t 0. 1 ^for

both the CuMn and AgMn systems, ^{and two sets of} exponents (cp I, 8I, yI) ^and (cp H, 03B4H, yH) characteriz-

ing ^the scalin function in the

two ^limits, d >

T I’P A, d> , (PA ’OH ^{and d «} T, d .r. h’,"PA qH ^re-

spectively [7]. ^{We note that one value} Of 0 A

accounts for the dependence ^{of both a* and}

H* on d, which checks the selfconsistency ^{of the} picture.

Obviously, ^{if a} scaling function in the form of

equations (2) exists and if its low anisotropy ^{limit for}

d much smaller than T 1/’PA and h 1/’OA ’OH is ^well

established, ^{this means} that, in the limit of zero

anisotropy, ^RKKY Heisenberg spin glasses present

a transition at Tc =F 0 ^{with the set of} exponents 03B4H, YH9 ’PH’ ^etc. The experiments of reference [7]

have convincingly shown the existence of an ani- sotropy dependent scaling and the d ---> 0 limit of the critical behaviour appears ^{to be} clearly observed, ^see

e.g., the right ^{side of} figure ^{1. However, even for the} sample ^{with the lowest} anisotropy, ^{CuMn without}

Au (d

0.036), ^the Heisenberg regime ^{is observed} only ^for h > 2 x 10- 2, 7 > 0.1. This _{range of}

corresponds ^{to a} correlation length ^{smaller than}

100 A. In this length range deviations from the

RKKY behaviour (R - 2 instead of R - 3) 4re-expected

[10] and this could be a plausible explanation ^{for a}

critical behaviour different from that expected by Bray ^{Moore and} Young [4] ^{for RKKY} Heisenberg spin glasses. ^{Whence we} infer that the analysis ^{of the}

influence of the anisotropy ^{on the} critical behaviour suggests ^{the existence of a} transition at finite

Tc ^{for zero} anisotropy Heisenberg systems ^but

cannot be absolutely conclusive (of ^{course, the}

existence of an Heisenberg regime ^{would be} proved

more strongly if there would exist a spin glass ^{with a}

lower anisotropy ^and Heisenberg regime extending

to smaller values of h and T).

3. Shift of the transition temperature ^induced by ^the anisotropy.

The most direct _way to see if Tc (d = 0 ) ^{has a zero or}

non-zero value is to analyse the variation Tc(d) ^and

to extrapolate ^{to d}

^{0. A} difficulty arises from the additional variation of Tc ^{due to} the reduction of the

isotropic exchange by ^{mean free} path ^{effects. Fortu-} nately Vier and Schultz [8] (VS) ^have studied the variation of Tc in very extensive series of CuNin and

AgM alloys doped with several types ^of impurity,

Au and Pt giving ^{rise to} anisotropy-induced positive shifts, ^and In, Sn, ^Sb giving ^{rise to} negative ^mean free-path ^{effects. Moreover VS} have used their data

on alloys doped ^with In, ^{Sn or Sb to derive an} empirical ^law describing accurately ^the variation of

Tc ^{due to mean free} path (MFP) ^effects [11] :

where _p is the resistivity ^{of the} alloy ^{at low} tempera-

tures and

This empirical ^{law can be used} to correct the data of VS on CuMn and AgMn doped ^{with Au or Pt for the}

mean free path ^effects.

In figure ^{2 we} show the variation of Tc ^measured by ^{VS in CuMn4 at% and} AgMn5.5 ^at% alloys doped ^{with Au or Pt. From} equation (3) ^{the shift of} T, ^due only ^{to the} reduction of the mean free path by

the Au or Pt impurities ^can be written :

We have calculated A Trp by using the values of po and [Tc(p

0 ) - Tc (p

oo )] quoted ^{above and}

Fig. 2. - Tc ^{versus y for} C_uMn4 at%Pty%, CuMn4 at%Auy% ^and AgMn5.5 ^{at%Auy% alloys : (0 )} experimental Tc ^found by Vier and Schultz [8] ; (.) Tc corrected for mean free path ^effects (see text).

p (CuMn4%Auy% ) ^{=10.8 + 0.49} y [14] ,

p ( AgMn5.5% )

^8.63 03BC03A9cm [8] ,

p (AgMn5.5%Auy% )

^{8. b3 + 0.38} y [13] .

The data of VS corrected for the mean free path

effects are represented by ^black symbols ⁱⁿ figure ^2.

For the CuMnPt and AgMnAu ^{series the} corrections for mean free path ^{effects are} relatively small, whereas, ^for CuMnAu, ^the experimental ^{shift and}

the correction have approximately ^{the same} magni-

tude. We will therefore limit our analysis ^{to the}

CuMnPt and AgMnAu systems for which the exper- imental data ^cannot be significantly ^affected by ^an

error in the calculation of the mean free path

contribution [14].

We now calculate the anisotropy ^{d as a function of}

the concentration of Au or Pt. For, say CuMnPt, ^we write [15, 7] :

where Vrn ^and Vr are ^{the DM} coefficients associat- ed with Mn-Mn-Mn and Mn-Mn-Pt triangles respect-

ively, Vo ^{is the RKKY} coefficient, yAu and xmn ^are the concentrations of Au and Mn and Rc ^{= 42} A,

ro

2.55 A (in AgMn, Rc

³⁹ A, ro

2.89 A) [15, 7]. The values of the ratios V1/Vo ^{have been derived}

in previous ^works by combining ^{ab initio} calculations and experimental ^data [6, 7, 16, 17] ^{and are listed in}

table I. One can rely ^{on this set} of values which

appeared to account consistently for several spin-

orbit induced properties, from ESR shifts ^to changes

in the critical behaviour. The corresponding ^values

1176

Table I. - Relative strengths of ^{the DM and RKKY}

interactions in CuMn or AgMn alloys doped ^{with Au}

or Pt : (a) from ^{ab initio} calculations of the spin-orbit scattering by Pt, ^see reference [6] : (b) ^obtained from

( ^{v 1 N} by comparing ^the macroscopic ^ani-

( ^{0 CuMn}

sotropy energy of CuMnPt with that of CuMn, ^see reference [16], ^{or that} of CuMnAu, ^see reference [13] ; (c) ^obtained from ( ( v0 v 1 Mn ^{by cumn comparing}

the ESR shift of ^{CuMn and} AgMn alloys, ^see reference [7].

-

of d derived from equation (5) range from 0.0036

(CuMn) ^{to 0.33} (AgMn5.5%Au5%), ^{see table II.}

We now compare the variation of Tc ^induced by

Au or Pt impurities ^{in CuMn or} AgMn ^{with the}

variation expected in several models :

.

(a) the variation expected by Bray, ^{Moore and} Young [4] (BMY) ^{for RKKY vector} spin glasses ^can

be derived from equation (1a). ^We have calculated

Tc ^{for the CuMnPt and} AgMnAu ^{series of VS} [8] by using the values of d

DIJ calculated above, A =1, YD

3/2 and by choosing ^{J to obtain} Tc = 24 K for CuMn4 at% (d

0.036) ^and T, =

18.5 K for AgMn5.5 ^at% (d

0.095). The values of

Tc calculated for the series CuMn,Pty ^{are shown in} figure 3, ^{see curve} BMY, ^{and can} be _compared with the corrected experimental Tc (i.e. ^the experimental Tc of VS corrected for the mean free path effects, ^as explained above). ^{It turns out that} equation (1)

Fig. ^3.

T,

y for CuMn4 at%Pty% alloys. ^{VS :} experimental values of Viers and Schultz [8] corrected for

mean

free path effects. BMY : T,, calculated from equation (1). ^{MCMB :} Tc calculated from equation (6). ^{KS :} Tc calculated from equation (7).

predicts ^too large shifts of Tc. ^{We can lower the}

calculated shifts by lowering ^{the values of d uni-} formly. ^{When we} lower the factors of d by ^{about an}

order of magnitude, the calculated Tc ^{come closer to}

the experimental Tc but it is hard to obtain a good ^fit throughout ^the whole concentration range. This ^can be clearly understood from figure ^{4 where} j2 ITC’ 2 ^is

Fig. ^4. - J2/ T;

^In (dJ / Tc). ^The straight line is the variation predicted by equation (1). ^The experimental T,

those obtained by Vier and Schultz [8] corrected for

mean free path effects. In order to fit the experimental

curve with the calculated line in the left side of the figure,

we

have taken values of d lowered by

factor of 8 (with respect ^to their values of Tab. II), ^{and J}

^{63 K for}

CuMn4 at% or J

44.6 K for AgMn5.5 ^at%.

Table II.

The transition temperature Tc ^measured by VS, reference [8], ^the transition temperature ^corrected

for ^mean free path effects, T§", ^{and the} anisotropy parameter d calculated from equation (5) ^{with the}

parameters V1/Vo of ^{table I} for ^series of CuMnPt and AgMnAu alloys.

°

plotted ^{as a} function of In (dY/Tc). ^Whereas equation (1) gives ^a straight line in such a plot, ^the experimental ^variation departs definitely ^{from a} straight ^line (in Fig. ^{4 we used} A/yD

4/3, ^{values of}

d lowered by ^{a factor of 8} [18] ^{and J}

^{63 K for}

CuMn4 at% or J

44.6 K for AgMn5.5 at%, ^which

allows us to obtain an approximate agreement between the calculated and experimental ^{data in the}

left side of the figure). Alternatively, by lowering

the value of A and keeping the values of d of table II, ^the calculated and experimental ^{data can be} brought closer but again, ^as expected ^from figure 4,

the form of the calculated and experimental ^curves

are different. We can however remark that, ^{if the} data of figure ⁴ for CuMn and AgMn respectively

are considered independently, they ^are approxi- mately consistent with two straight ^lines (excepted

for the most Pt-doped ^CuMn alloys, Cm ^{= 1 % and} 1,96 %). ^{This means that an} approximate agreement

can be obtained only by introducing ⁱⁿ equations (1)

different values of A (by ^a factor of about two) ^for

the CuMn and AgMn systems ;

(b) the variation of T,*expected for short range vector spin glasses in 3 dimensions is written [3] :

The values of 7c calculated in this way for the series

CuMn.,Pty ^{(with a value of J fixed to obtain}

Tc

^{24 K} for the value d

0.036 of CuMn4 at%)

are shown in figure 3, see curve MCMB. The calculated shifts are much larger than the exper- imental ones ;

(c) the variation of Tc predicted ^{in the mean field}

model of Kotliar and Sompolinsky [19] is written:

This gives ^{the curve KS in} figure ^{3. The} predicted

values of the shifts induced by ^{Pt are much too small.}

(d) ^The scaling function, equation (2), ^{can be}

written :

which implies for the variation of Tc ^{with d :}

As discussed in section 2, ^the analysis ^{of the ani-} sotropy ^{induc d} cross-over between two critical

regimes ^{in CuM u and} AgMnAu ^has given [7] :

We have plotted ^the experimental Tc ^{of VS} [8]

(corrected ^{for mean free} path effects) ^as function of

do.8, ^see figure ^{5. It} appears that ^{all the} Tc gather ^on

two straight lines. These straight ^lines intercept ^the

vertical axis at Tc (d

0 )

^{22.4 K for CuMn4 at%}

Fig. ^5.

T,, versus d0.8 for the CuMnPt and AgMnAu

series (T, ^from VS, ^Ref. [8], after correction of the mean

free path effects ; d calculated as described in the text).

and T,(d

0) ^{= 15.8 K for} AgMn5.5 ^at%.

Moreover, ^when Tc/Tc(d

0) ^is plotted ^{as a func-}

tion of dO.s, ^see figure 6, ^{all the} experimental ^{data of}

the two series gather ^{on a} unique straight ^line corresponding ^{to the} equation :

with

=1.13 ± 0.1. One could wonder whether the

non-zero limit of T,, ^obtained by extrapolating ^to

zero DM anisotropy ^cannot be accounted for by ^a

residual non-DM anisotropy. Actually it would be necessary ^{to assume} that the residual non-DM

anisotropy has the order of magnitude ^{of the DM} anisotropy ^{of CuMn,} which is ruled out by ^the

definite DM character (unidirectional) ^{of the macro-} scopic anisotropy ^{of CuMn.}

Fig. ^6.

Tc/Tc(d

0) versus do.8 for the CuMnPt and

AgMnAu ^series. Tc (d = 0) ^{is derived from} figure ^{5 :} Tc(d

0)

22.3 K for CuMn4 at%, T,,(d

⁰⁾

^{15.7 K}

for AgMn5.5 ^at%.

4. Conclusions.

We have analysed experimental ^{data on} the effect of

the DM anisotropy ^{on the} critical behaviour and the

transition temperature ^of AgMn ^{and CuMn} alloys

doped ^{with Pt or Au} impurities. ^{Our aim was to}

clarify the role of anisotropy ⁱⁿ Heisenberg spin

glasses : ^{is there a} transition at finite temperature ⁱⁿ

three dimensional RKKY spin glasses ?

1178

The experimental ^{data for our} series of CuMnPt and AgMnAu alloys fit with the following picture :

(i) the effect of anisotropy ^{on the} critical behavior of the non-linear susceptibility ^can be described by ^a scaling function of the 3 parameters ^T, h and d. We

obtain cfJ A === 0.8 ^{for the exponent} describing ^the

cross-over between the low and high anisotropy regimes ;

(ii) ^a good ^{fit is obtained with} Tc(d

0) =1= ^{0 and} Tc(d)

Tc(d

0)(1 ^{+ 1.13} d0.8 ^{for both the CuMn}

and AgMn systems. ^An approximate ^{fit can also be}

obtained with the picture ^of Bray, ^{Moore and} Young [4], ^{T -} [1 + 2 A IYD In (Tc/dJ)]-1I2, ^but only ^when

different values of A (by ^a factor of about two) ^are

taken for the CuMn and AgMn ^series.

In conclusion, ^our analysis ^{of the} experimental

data cannot rule the picture ^of Bray, ^{Moore and} Young [4] ^{but a better} agreement is obtained with

Tc (d = 0 ) =A 0 ^and A Tc - d . ^{It thus} brings ^new

arguments ^{in favour of a} transition at finite tempera-

ture in RKKY Heisenberg spin glasses ^without anisotropy. ^We hope ^{these new} data will lead to further theoretical and experimental arguments.

Acknowledgments.

We acknowledge fruitful discussions with M. A.

Moore and P. Young ^who suggested ^the plot ^of figure ^4.

References

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BRAY, ^{A. J. and} MooRE, ^M. A., Phys. ^{Rev. B 34} (1986) ^6561.

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[8] VIER, D. C. and SCHULTZ, S., Phys. Rev. Lett. 54

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[9] ^{The existence of}

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FISCHER, ^{D. S. and} SOMPOLINSKY, H., Phys. ^Rev.

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YESHURUN, ^{Y. and} SOMPOLINSKY, H., Phys. ^Rev.

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KOTLIAR, G., Phys. Rev. B 16 (1987) ^8646.

[10] ^{LEVY, P. M. and} ZHANG, Q., Phys. Rev. B ³³ (1986)

665.

[11] ^The following theoretical articles have

justified

the non-zero limit of Tc ^for 03C1 ~ ^~ which charac- terizes equation (3): JAGANNATHAN, A.,

ABRAHAMS, E. and STEPHEN, M. J., Phys. ^Rev.

B 37 (1987) 436 ;

ZYVZIN, A. Y. and SPIVAK, B. Z., JETP Lett. 43

(1986) 234 ;

BERGMANN, G., Phys. ^{Rev. B 36} (1987) ^2469.

[12] SCHMITT, R. W. and JACOBS, ^I. S., ^J. Phys. ^Chem.

Solids 3 (1957) ^324.

[13] GOLDBERG, ^S. M., LEVY, ^{P. M. and} FERT, A., Phys.

Rev. B 31 (1985) 3106 and references therein.

[14] ^As

^{matter of} fact, ^{there is some} discrepancy

between the variation of Tc

^a function of the DM strength in CuMnAu on one hand and

CuMnPt

AgMnAu

the other hand. We suppose this is due ^{to an error} in the correction of the significant

^free path effects of CuMnAu.

[15] ^{LEVY, P.} M., MORGAN-POND, ^{C. and} RAGHAVAN, R., Phys. ^{Rev. B 30} (1984) ^2358.

[16] LEVY, ^P. M., MORGAN-POND, C. and FERT, A., ^J.

Appl. Phys. ⁵³ (1982) ^2168.

[17] ^DE COURTENAY, N., FERT, ^{A. and} CAMPBELL, I. A., Phys. Rev. B 30 (1984) ^6791.

[18] ^{An error} by

uniform factor of 8 in the calculated values of d cannot be ruled out absolutely.

Whereas the relative values of d for series of

doped ^{CuMn and} AgMn ^{alloys has been verified} by

^{number of} experiments (ESR, ^transverse susceptibility, anisotropy dependence of the criti- cal exponents), their absolute value is fixed by only the ab initio calculations of Levy ^and

coworkers [6,16] expressing ^{the DM} interactions induced by ^non magnetic transition metal impuri-

ties (e.g. Pt, Pd...)

function of the number of d electrons and their spin-orbit ^constant 03BBd. ^For

^series

have used the calculation for Pt impurities ^to fix the absolute values of d.

However there is some possible

^the

number of d electrons of Pt and also

the

prefactor ^of equation (5). Finally,

^error by

factor of 8

the absolute values, although

unlikely, ^{cannot be ruled out} completely.

[19] ^{KOTLIAR, G. and} SOMPOLINSKY, H., Phys. ^{Rev. Lett.}

54 (1985) ^1063.