170Yb Mössbauer study of the crystalline electric field and exchange interaction in YbFe2

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170Yb Mössbauer study of the crystalline electric field and exchange interaction in YbFe2

C. Meyer, Y. Gros, F. Hartmann-Boutron, J.J. Capponi

To cite this version:

C. Meyer, Y. Gros, F. Hartmann-Boutron, J.J. Capponi. 170Yb Mössbauer study of the crystalline electric field and exchange interaction in YbFe2. Journal de Physique, 1979, 40 (4), pp.403-415.

�10.1051/jphys:01979004004040300�. �jpa-00209121�

170Yb Mössbauer study ^{of the} crystalline ^{electric field and} exchange interaction in YbFe2

C. Meyer. ^{Y. Gros. F.} Hartmann-Boutron

Laboratoire de Spectrométrie Physique (*), Université Scientifique ^et Médicale de Grenoble, B.P. 53X. 38041 Grenoble cedex, France

and J. J. Capponi

Laboratoire de Cristallographie ^du C.N.R.S., B.P. 166X, 38042 Grenoble cedex, ^France.

(Reçu ^{le 28} septembre 1978, révisé le 29 novembre 1978, accepté ^{le 4 décembre} 1978)

Résumé.

Nous avons étudié l’effet Mössbauer de 170Yb dans YbFe2 ^{entre 4,2 et 60 K. Les} spectres hyperfins

sont compatibles ^{avec une direction} de facile aimantation suivant [100]. Les variations thermiques ^du champ hyperfin ^{et du} couplage quadrupolaire électrique suggèrent que les effets de champ cristallin sur l’ion ytterbium

sont réduits, ^et les courbes s’interprètent ^{bien en le} supposant seulement soumis à un champ d’échange (03BCB, Hex/kB

^{111 ±} 4K);

la contribution des électrons de conduction au champ hyperfin ^est Hc

²⁴⁰ ± 100 ^{kOe et} le facteur d’écran de Sternheimer est RQ

^{0,22 ± 0,01. En relation avec la petitesse du champ} cristallin, nous examinons l’influence de la magnétostriction ^{sur le} potentiel cristallin vu par ^un ion de terre rare dans les composés RFe2.

Abstract.

The Mössbauer study ^{of 170Yb in} YbFe2 ^was performed ^{in the} temperature range 4.2-60 ^K. The

hyperfine spectra ^{are in} agreement ^{with an} easy direction of the magnetization along [100]. The thermal variations of the hyperfine field and of the electric quadrupole coupling suggest ^{that the} crystalline ^{field effects on the} ytter- bium ion are rather small and we get good least square fits with pure exchange only (03BCB Hex/kB

^{111 ± 4} K) ;

the corresponding values of the conduction electron field and of the Sternheimer shielding ^{factor are} respectively Hc

240 ± 100 kOe and RQ

0.22 ± 0.01. In connection with the smallness of the crystalline field, ^{we discuss} the influence of the magnetostriction ^{on the} crystalline potential ^seen by ^{a rare earth} ion in the RFe2 compounds.

Classification

Physics ^{A bstracts}

76.80

75.30

1. Introduction.

In a previous paper [1] ^we

described the synthesis ^and magnetic properties ^of

the Laves phase compound YbFe2. Magnetic ^measure-

ments and 57Fe Môssbauer spectroscopy ^{show that} this material is ferrimagnetic ^{and that at low} tempe-

rature it is magnetized along ^a [100] direction. Above 50 K the magnetization ^{seems to} slightly ^deviate

from this direction. In [1] we ^{evaluated the} magnetic

moment of Yb3 + in YbFe2 ^{with the} help ^{of the} crystal-

line field and exchange parameters determined ⁱⁿ [2]

by Yanovsky et ^{al. from the} Môssbauer effect of l’°Yb

as an impurity ^{in the similar} compounds TmFe2 ^and

(*) ^{Associé au C.N.R.S.}

TmxHo1-xFe2 ^(x

^{0.1 ;} 0.2) [3]. ^However TmFe2, magnetized along [111], ^and TmO.2Hoo.8Fe2l ^magne-

tized along [ 110] ^{below 40} K, are expected ^{to exhibit} important magnetostriction ^{effects at low} tempera-

tures and this _may influence the crystalline ^field.

Also the RE-Fe and RE-RE exchange interactions may exhibit some anisotropy, whose manifestations

are similar to a crystalline field and which depends

on the pair ^{of atoms involved (for} example ^Tm-Yb

in TmFe2 : ^{Yb ; Yb-Yb in} YbFe2). ^{For these reasons}

we performed ^{the Môssbauer} study ^{of l’°Yb in} YbFe2 ^to try ^{to determine} directly ^the crystalline

field and exchange interactions in this compound

from the hyperfine data, ^{with a view of} comparing

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004004040300

with the values of [2] ^{and with} magnetic anisotropy

data. The whole paper and its appendix ^{1 are devoted}

to this problem. ^In appendix ^{II we use the results}

obtained in order to slightly ^{correct some estimates}

of reference [1] ^{which were based on the data of} [2].

Also, in connection with recent experiments ^on NdFe2, ^{we discuss the} possible role of an anisotropic hyperfine ^{structure of the iron atoms in the inter-} pretation ^{of the 5’Fe Môssbauer spectra of YbFe2}

which were reported ⁱⁿ [1].

2. Experiment. ^{- The Môssbauer} y of 17°Yb has

an energy of 84.4 keV, and it is _necessary to cool both the source and the absorber. The apparatus which we use is similar to that of reference [4], ^which

is characterized by ^a horizontal movement transmitted to the absorber under vacuum. The 20 mCi source

of 170Tm in TmAl2 ^{is cooled to} approximately ^{15 K} through ^flexible copper wires connected to the helium bath. The minimum linewidth obtained is 2.5 mm/s

with respect ^{to an} absorber of YbAl2 ^{at 4.2 K. The}

scintillation detector is a 5 mm thick NaI crystal.

The absorber of YbFe2 (about 800-900 mg covering 3 cm2) ^{is a mixture of} powders from thirteen different

preparations ^{under 80} kbars. From X-ray powder

data we expect the presence of impurities ^{in the form}

of both Yb metal (which ^was put ⁱⁿ slight ^excess

for the preparation) ^{and YbO (because of unavoi-}

dable oxidation) (’). Finally ^{about 40} % ^{of the} pre-

parations ^{also contained a very small amount of} Yb6Fe23. ^{As this did not} appear in the 57Fe spectra

Fig. ^1.

^Môssbauer spectra ^of 17°Yb in YbFe2 ^{at 4.2} K, ^{20 K} and 50 K. The full curve is a theoretical fit with a central impurity

line and with an l’°Yb spectrum composed of five lines with equal height ^{and width.}

(1) ^{Most of the unknown} X-ray lines mentioned in reference [1]

correspond ^{to YbO. We are indebted to} Dr. J. F. Cannon for

suggesting this identification.

we could forecast that it would be absent from the 17°Yb spectra.

Experiments ^were performed ^{at 4.2 K, 12 K, 20 K,}

34 K, 50 K and 60 K . We give ⁱⁿ figure ^{1 the} spectra

at 4.2 K, ^{20 K and 50 K.} They ^{exhibit a} well defined five lines magnetic hyperfine pattern corresponding

to YbFe2, together ^{with an} additional broad line at

zero velocity. ^{We notice that the lines of} YbFe2 ^are

narrow, which indicates that our preparation ^under high pressure is reproducible. On the other hand,

as far as the statistics can show, ^the broad central line seems to be best fitted with two more or less

superimposed single ^{lines, one with a} rather normal width and one very broad. Their negative ^isomer

shifts with respect ^to YbAl2 ^are characteristic of Yb2+ and we can probably assign ^{them to Yb and} YbO, ^{but our limited} accuracy does not allow a

detailed study. Anyway ^these diamagnetic impurities

do not affect the spectrum ^of interest, ^{i.e. that of} YbFe2.

3. Interpretation ^{of the Môssbauer spectra of} YbFe2-

-

3. 1 VALENCE STATE OF THE YTTERBIUM IN YbFe2.

The lattice parameter ^of YbFe2 (7.244 A) ^{is very}

similar to that of TmFe2 (7.247 A) ; ⁱⁿ addition it falls on the straight line which gives ^{the lattice} para- meter of the other RFe2 ^{versus the atomic number}

of the (trivalent) ^{rare earth R :} this indicates that

ytterbium ^is trivalent in YbFe2, ^{as confirmed} by ^the

existence of a compensation point ^{in the} magnetiza-

tion curve [1].

[Roughly speaking, ^{when Yb} is trivalent in this kind of intermetallic compounds, ^{the lattice} para- meter is _very similar to that of the homologous ^{Tm3 +} compound ; when Yb is divalent the lattice parameter is about 0.1 1 to 0.14 A larger ; when Yb has an inter- mediate valency it is about 0.06 A larger.]

3.2 HYPERFINE CHARACTERISTICS OF (17OYb)3+.

The 84.4 keV _y transition takes place ^{between an}

excited nuclear state I = 2, ^{and a} ground ^state

Ig=0.

In the nuclear state 1

2 a (magnetic ⁺ quadru- polar) hyperfine ^{structure is} present. ^{For the free} Yb3 + ion (4f13,

2F7/2’ J

^{7/2) the hyperfine} Hamiltonian is [5] :

with :

In these expressions (1

RQ) ^{is the} Sternheimer

shielding ^{factor of the 4f} shell ; J il N Il J >, J Il ex )) J ) ^are reduced matrix elements to be found in Abragam ^and Bleaney [5] ; ye is ^the nuclear _gyro-

magnetic ^{ratio in state I = 2} ( ge ^{= 0.334 whence} Ye/2 n

^{0.254 6} kHz/Oe) ; finally A s A’ ⁺ A’ ^is

the sum of a relativistic contribution A r ^{and of the}

core polarization contribution A. ([5], ^p. 296-300).

If the RE ion is now at a cubic site in a magneti- cally ^ordered alloy (like ^{Yb3 + in} YbFe2), ^{and if}

the electric field gradient ^{has axial} symmetry ^around the direction OZ of the hyperfine ^field, eq. (1) ^is replaced by ^the effective Hamiltonian :

with

where H.", vii characterize the contributions of the Rare Earth ion and He ^{is an} additional contri- bution due to the polarization ^{of the conduction}

electrons.

When the rare earth is fully polarized ( J z > = - J)

It is not very easy ^to determine a value for HFI’

According ^{to the} estimates of table 5.5 of [5] (based

on ionic crystals data) (2) :

-

on the other hand, ^{in the} cubic salt CaF2 : Yb,

(AJ)exp

^{300 MHz for 17°Yb} and 879.57 MHz for 1 71 Yb [6b]. ^After correcting for the difference between

(AJ)CaF2exp exp ^{and A J J for 171Yb} this would yield

whence HFI

^{4 160 kOe.}

Finally there exists a relativistic calculation which leads to HFI

^{4 232 kOe} [6c]. ^{In what follows we}

will tentatively ^{assume that :}

In a metallic matrix one must also take account of the conduction electron field He ^whose origin

(2) ^In principle HFI ^{defined as} AJ/nYn ^should slightly depend ^on

the isotope because of the hyperfine anomaly (j 4 ( % ^0.5 % ^{in the}

RE : see [5] p. 252).

has been discussed by various authors but is not yet quite ^clear [7]. ^{It may involve both a} self polarization

of the conduction electrons by ^{the rare} earth itself and also transferred fields associated with the pola-

rization of the conduction electrons by neighbouring

and distant magnetic ^atoms (Fe ^{and other} RE).

The total contribution HcRE associated with the rare

earths should _vary like Jz >, i.e. it should give ^rise

to an apparent variation of the hyperfine coupling

constant, while the contribution HcFe ^associated

with the irons should stay ^{constant below 60 K. In} the interpretation ^{of the} experiments ^we thus defined

a field HFI by :

and varied HFI ^and HF’. ^{In terms of these two} quanti- ties, ^{the total} conduction electron field He ^{is :}

Let us now look at the quadrupole interaction.

With the values :

we arrive at the maximum value in the fully polarized

state (JZ

J) :

On the other hand, according ^{to table 3 of} [7a], RQ

defined as

should be of the order 0.21 in metallic compounds.

3.3 HYPERFINE DATA FOR l7°Yb IN YbFe2.-

The Môssbauer spectra ^{between 4.2 K and 60 K are} well fitted with the simple Hamiltonian _eq. (2) : ^the

full line curves in figure ¹ represent ^least square fits with five lines with equal amplitude ^{and widths}

Table 1.

Hyperfine parameters of 17°Yb ⁱⁿ YbFe2.

The precision ^{is about + 60 kOe} for ^the hyperfine

field and + ^{60 MHz} for ^the quadrupole coupling.

(FWHM

^2.5 mm/s ^{at 4.2} K). ^{Table 1 shows the} corresponding ^{values of} Heff ^{in MOe and e2} vii Q

in MHz. (One should however mention that the Môssbauer spectra ^{are not} very sensitive to a small asymmetry ^{of the} E.F.G., il 0.5.)

In YbFe2 ^the ytterbium ion is submitted both to

crystalline ^field effects and to exchange ^{fields, the}

main contribution to the exchange being ^{due to RE-Fe} exchange ^{which we will assume to be} isotropic ^for simplicity. ^{Then, if the} exchange ^{field is} parallel ^to [100] ^or [ 111 ], ^the hyperfine ^field Heff ^{is also} parallel ^to [100] ^or [111] ^{and the} E.F.G. has axial symmetry around the direction of the hyperfine field. Otherwise the gradient ^{is not axial} (in particular along [110] ^the

E.F.G. cannot be specified by ^a single number). ^In YbFe2, ^the gradient being apparently axial around

H,,ff, ^this suggests ^{that the} ytterbium ^{moments can} only ^lie along [100] ^or [111], ⁱⁿ agreement with the data of reference [1] ^{for 57Fe which indicate that the} magnetization ^is along ^{a direction} [100] up ^to 50 K ; in addition the deviation from this direction above 50 K must still be _very small at 60 K since it is not detectable in the Môssbauer spectra of 17°Yb. [100]

and [111] being high ^order symmetry ^axes this discus- sion would remain valid even in the presence of

anisotropic exchange.

3.4 INTERPRETATION OF THE THERMAL VARIATION OF THE HYPERFINE PARAMETERS.

We first attempted

to fit our curves for Heff ^{and e2} VZZ Q ^{versus T} by using ^the crystalline ^{field and} exchange parameters determined by Yanovsky et ^al. [2] ^{for 17°Yb as an} impurity ⁱⁿ TmFe2 ^and TmxHo1-xFe2 (x

0.1, 0.2).

The exchange ^and crystalline field Hamiltonians are

taken as :

with :

with these values, ^when Hex ^is along [100] ^{we find that}

at 0 K and at 4.2 K : and

(i.e. ^{we are} almost in the fully ^{saturated case}

Jz > = - 3.5 and 3 Jz2 - J(J + 1) > = 21). If we

take account of these results in order to compare the experimental ^{values at 4.2 K :} Heff

^{4.44 MOe}

Fig. ^2.

a) Hyperfine ^{field, and} b) ^electric quadrupole coupling,

for 170Yb in YbFe2 (heavy dot-dashed curve joining ^the experi-

mental points 0) compared with theoretical curves computed ^for

the three principal directions using ^the crystalline ^{field and} exchange

values of reference [2]. ^{Dashed curves and} experimental points A D :

results of reference [2] a : ^result of reference [9]. ^{Full curves :}

recomputed ^as indicated in text.

and e2 Vii Q

^{3 210 MHz,} with the maximum theo- retical values at 0 K :

we find that :

These figures ^seem reasonable. In particular RQ

compares favorably ^{with the} estimates of reference [7a].

By comparison the results of [2] ^were

(deduced ^{from an} extrapolated ^value

in figure ^{2 of} [2], ^{which seems to be in} disagreement

with the value 2 400 + ^{250 MHz} mentioned in the

abstract). ^{The value} RQ ~ ^{0.36 seems to be anoma-} lously large. ^{Because of these} differences, ^we decided to entirely recompute ^{the set of curves} giving Heff ^and

e2 Vii ^{Q as a function of T, with the same} crystalline

and exchange parameters ^as before, ^but with

He

320 kG and RQ

^0.22. The results of this calculation [8] ^are represented by ^{the full line curves in} figure ² while the results of Yanovsky et ^al. [2] ^are represented by the dashed lines. Notice that the two sets are not related simply by ^a translation and/or ^an affinity. Indeed the results of our diagonalization

seem to differ _very slightly from those of [2] ; ^for example along [111] ^{the distance between our two}

lowest electronic states is 1.57 K instead of 2.8 K in reference [2] ^{and the next} electronic state is at 50 K from the ground ^state instead of 55 K in [2]. ^{In addi-} tion, ^as already mentioned, ^we have found that in the

[110] direction the E.F.G. is far from axial (at 0 K, Vzz ^{oc +} 15.48, Vyy ^{oc -} 1.94 ; Vxx ^{oc -} 13.54) (3);

therefore the quadrupole splitting ^{cannot be characte-}

rized by ^{its sole} component Vzz along ^the exchange field, ^{and the} experimental points ^{relative to} Tmo.2Hoo.8Fe2 ^{at low} temperature ^{should not be} compared ^{to the} [110] ^{curves of} figure ^2.

If we now look at our experimental points ^relative

to YbFe2 (dot-dashed curve) ^{we see that} they ^{do not}

fall at all on the curve computed ^{for the} [100] ^direction using the values of [2] : ^{this means} that these values

are not appropriate ^to YbFe2 and that the parameters relative to this compound ^must be determined directly by ^least square fitting ^{of the} hyperfine ^data. Many

least _square fit calculations were made, with different numbers of adjustable parameters. The final results

are characterized by ^{the fact that} they ^{all lead to}

similar r.m.s. deviations with respect ^{to the} experi-

mental data : 47 to 58 kOe for the hyperfine ^field,

25 to 35 MHz for the quadrupole coupling, ^which

are less than the experimental uncertainties.

(3) OZ // [110], ^{OX ll} [001 ], ^{0 ye} [110].

Fig. ^3.

Least square fits of the hyperfine ^field Heff ^{(3a) and}

electric quadrupole coupling ^e2 V4fzz Q (3b) ^with only ^an exchange

field MB Hex/kB

^{109.64 K.}

Least _square fits were made :

-

with _pure exchange only (Fig. 3) : IÀB Hex adjus-

-

with exchange ⁺ crystalline ^{field :} JUB Hex, A4 r4 >, A6 ( r6 ) adjustable

-

and also (although ^{such a} contribution is not

expected ^{in the} [100] ^{direction even} in the presence of

magnetostriction, ^see Chap. ⁴ below) ^{with an addi-}

tional second order crystalline ^{field :}

In all cases it is found that the crystal field coeffi-

cients are very small

while the exchange ^is comparable ^{to that in} TmFe2 :

Yb :

These conclusions remain unaltered when we vary

H§j ^{between 4 100 and 4 400 kOe.}

On the other hand, ^when HF, ^{varies, the value at 0 K}

of the conduction electron field He (eq. (6)), always

remains in the _range 200-270 kOe : what this means

is that our accuracy is sufficient to determine Hc ^but

not to separate HcRE ^and HcFe. ^{When we take account}

of the uncertainty ^on HFI, ^{we may} say that

Finally RQ ^{is always comprised between 0.21 and}

0.23.

3.5 COMPARISON WITH OTHER DATA.

We must now try ^to interpret ^these results : existence of a large

fourth-order crystalline ^field A4 r4 > ⁱⁿ TmFe2 : ^Yb

and of a small field in YbFe2. ^{As a matter of fact the}

small crystal ^{field in} YbFe2 is in contradiction not

only ^with TmFe2 : ^Yb [2], ^{but also with the} large magnetic anisotropies observed in several RFe2

and with the assumption ^{of constant} A4 ^and A6

which was generally ^{used in the} interpretation ^of magnetic anisotropy ^data.

Before going ^{further we must however make a}

remark concerning TmFe2 : ^{Yb. In fact the Môssbauer} spectrum of 170Yb at 4.2 K has also been obtained by ^a

second _group [9] ^and from their spectrum it follows that Heff

^{4 140 k0e} instead of 3 750 k0e in [2]

and that (e2 Vzz Q )[111] ] = 2 500 MHz instead of 1 700 MHz in [2]. The authors of [9] ^{have informed us}

that their sample ^was probably ^not very good [10].

Nevertheless the fact that two different preparations

can lead to such differences in the hyperfine para- meters suggests ^{that new Môssbauer} experiments ^on TmFe2 : ^{Yb are} perhaps necessary before detailed

comparisons ^can be made with YbFe2.

On the other hand, ^{if we look at the} magnetic anisotropy data, ^{we see that the estimates} rely ^{on two}

methods :

-

study ^{of the} magnetization ^{direction as a func-}

tion of composition ⁱⁿ ternary compounds

-

anisotropy measurements on single crystals.

Concerning ^{the first} method, examination of the

figures of references [11a] ^and [11 b] shows that the boundaries in the ternary diagrams ^are relatively

insensitive to A4 ^{and to} /lB Hex : A4 = _kB ---- to 50 4 K

a4o a4o in reference [lla] (4) ^where only major symmetry

idere d, A4 ^{36 K}

1 1b] h directions were considered, i- = ^B 36 a4 K 0 ^{in [11b] where} other directions ^{were also} considered; similarly

YB Hexl kB

^{100 to 150 K in [11a],}

^{150 K in} [11b].

(4) ao is the radius of the first Bohr orbit uo

0.529 A. These values are deduced from the experimental ^data A4 r4 > 4f by using the ,.-+ of Freeman and Watson [13]. ^In practice

In a more recent paper [12] magnetostrictive ^contribu-

tions associated with the external stress (see ^next chapter) ^were incorporated ^{into the} calculations ; ^com- parison ^{with the} experimental ^{data on} HoxTbl _xFe2

then leads to (in ^{our notation, and} neglecting ui) A4/kB

¹⁸ K/aÓ, J1B H,,.IkB

^{172 K.}

As for direct anisotropy measurements on single crystals, they ^{have been} performed ^on ErFe2 [14] ^and TmFe2 [15] ^{as a} function of T and on TbFe2 ^and DyFe2 ^{at room} temperature [16]. ErFe2 ^and TmFe2

are magnetized along [111] ] ^{and have a rather} large anisotropy ^{at low T ; for this reason the} anisotropy ^at

0 K is only ^an extrapolated ^{value. In} TmFe2 ^the extrapolated ^{fourth order} anisotropy constant at 0 K is K - 5 ^x 10+ 8 ergs/cm3 [14], i.e., ^{for one}

RE ion, K/Tm3 + - ^{120 cm -1} (see the definition of K below : _eq. (27), (26), (23)). ^{If K were} entirely ^due

to the fourth order crystal field, ^the corresponding

value of A4 ^r4 >/kB ^{for Tm3+ in} TmFe2 ^{would be}

36 K (see eq. (26) below).

All these results, although suggesting ^that A4 r4 >4f ^is usually fairly large, ^{are not} precise enough ^to prove the constancy ^{of either} A4 ^{or J1B} Hex

in the RFe2 ^series.

In addition the possible influence of magnetostric-

tive effects on the effective crystalline potential ^seen by the Môssbauer atom has never been investigated.

As we will show in the next chapter ^the existence of a

large ^and anisotropic (111 1 » Â, 00) magnetostriction

in the RFe2 compounds [19] ^{has two} closely ^related

effects : first the total anisotropy constant K contains a

contribution KME ^{which is} probably ^not negligible

with respect ^{to the} contribution Kl ^{due to} the crys- talline field V4 ; second, ^{the RE ion sees a second} order crystalline ^field V2 - KME ^which depends ^{on the}

orientation of the magnetization and vanishes only

when it is parallel ^to [100]; ^{this term} V2 ^{should be}

taken into account in the interpretation ^{of the} hyper-

fine parameters derived from the RE Môssbauer spectra in materials which are not magnetized along [100].

4. Comparison ^{of the} effects of crystalline ^{field and} magnétostriction ^on magnetic anisotropy ^measure-

ments and ^on the Môssbauer spectra.

^{41 1 SIMPLE}

MODEL OF MAGNETOSTRICTION WITH ONLY EXTERNAL STRAIN.

Let us assume that we have a rare earth atom in cubic surroundings, ^for example ^{that it}

is at the centre of a cube of eight charges q ^{in a cubic} crystal ^whose symmetry ^axes coincide with those of the cube.

Let us now slightly ^{distort the} crystal according ^to

the law

where xi, yi, zi ^are the coordinates of one of the eight charges ^with respect ^{to the centre of the cube.}

It is _easy to show that this will create at the RE

atom a second order (5) crystalline potential ^{of the}

form (to ^order E) :

where xe, y,, ze ^are the coordinates of the eth electron 4f.

In the particular ^{case of a cube of} edge ^{2a one has}

that

On the other hand it is clear from symmetry ^that V2

will retain a form of the type (13) whatever the nature

of the cubic neighbourhood.

Inside a state J, v2 ^{can be} replaced by

where J a Il J > ^{is a Stevens} coefficient.

Let us now derive a magnetoelastic coupling ^{at 0 K}

from _eq. (15). ^{For this we assume that the RE is also}

submitted to a coupling ^{with a} large magnetic ^field (IÀB H » V2) with direction cosines oc,, a2, (13 (in practice ^{H is the Fe-RE} exchange field) ^{and that in this}

field it is fully polarized ^{in the} state ! 1 J z

J >

quantized along ^{the field. It} is easy ^to check that :

Let us now define the strains (Kittel’s ^conven-

tion [17]) :

(5) ^In this standard approach ^one neglects ^tne variations of the fourth and sixth order terms in the crystalline potential, eq. (8).

This amounts to neglecting contributions of order six or more in a to the magnetostrictive anisotropy energy EMË, ^eq. (22) below. This

approximation ^seems reasonable because of both the experimental

smallness of the sixth order anisotropy energy and the lack of detailed information on the crystalline potential.

and set :

Eq. (16) then reduces to the standard for magneto- elastic coupling :

When one adds to this the elastic _energy Xel’

standard treatment of J6ei ⁺ HME ^{leads to} [18] :

whence the magnetostriction constants :

(C 11, Cl 2, C44 ^{are the elastic stiffness} constants).

Inserting eqs. (20) ^{back into} eq. (19) ^we get ^the magnetoelastic anisotropy energy

with

This contribution to the magnetic anisotropy ^adds

to the standard contribution arising ^{from the} crystal-

line field created by ^the undistorted neighbourhood :

.f corresponding ^to

As recalled in appendix 1 the method used for

extracting _JCME from V2 ^{can also be used for establish-}

ing relationships ^between K1, K2 ^{at 0 K and} A4 r4 >,

Because K2 « KI in all ^{the very} anisotropic RFe2

studied _up to now, measurements of the magnetic anisotropy essentially yield ^the fourth order anisotropy

constant :

As shown by ^{this formula,} if KME is ^{not known} and/

or is not small, ^{this does not} give ^any direct informa- tion about Kl, ^i.e. A4 r4 ).

Let us now look at the Môssbauer case. The hyper-

fine parameters depend ^{on the wave} functions and

energies ^{of the} ion, ^{which are} determined by ^{the total} crystalline potential ^{to which it is} submitted :

where, ^{for a} given ^direction al, a2, a3 of the magneti-

zation the magnetostrictive contribution V2 ^is given by

eq. (15) ^{in which one} replaces E« ^and _eij by ^their

i #i

expressions eqs. (17), (20). ^{From now on we will}

assume that À,100

⁰ (this will be discussed in the next paragraph). ^{Then :}

and :

We see that v2 ^{is zero} only ^{in the} [100] ^direction.

Otherwise it is non zero and its strength ^is directly

related to the magnetostriction energy. We may try ^to characterize it by ^the magnitude ^{of its} effects

when the magnetization ^is along [111]. If we quantize ^J along [111] we have that :

which gives ^{rise to} effects of the order KmE.

Let us now try ^{to evaluate} KME. ^A first method is to use the formula

In reference [14], KME ^was evaluated for TbFe2 ^{at room}

temperature using À111 1

2.4 x 10-3. If we assume

that C44 ⁱⁿ TmFe2 ^{at 0 K is the same and use the} experimental value À111 ^{= - 3.5 x 10-3} [15] ^we

arrive at KME

^9.2 cm-1/ion. Alternatively, ^if

we take : C44

^{5.38 x 1011} ergs/cm3 ^{as in refe-}

rence [12] ^we get ^that KME

^{7.2 cm -1. We may}

also use the formula

into which we put ^the experimental À111- ^{while we}

compute B2 ^{for a} tetrahedron of four charges 3+,

which represent ^{half of the} charges of the cube consi- dered above and correspond ^{to the four RE nearest} neighbours ^{of a RE in the} RFe2. We take their dis-

tance to the central atom to be 3.14 A as in TmFe2.

We then find that KME

^{+ 14.1} cm-1/ion. ^{All these}

estimates are open ^to criticism because the formulas

they ^{use do not take account} of the internal mode to be discussed in the next paragraph. ^In addition, ^as

concerns the third one, B2 being negative, eq. (21)

should lead to a positive À111 ^{hence to a} negative KME

while the experimental À11l ^is negative ; ^this is pro-

bably ^{due to} the fact that our estimate of B2 ^{uses a} point charge ^{model and does not take account of the}

iron nearest neighbours.

4.2 MORE ELABORATE TREATMENT OF MAGNETO- STRICTION IN THE RFe2. - This is done in refe-

rences [19, 20] ^{where it is} argued ^{that the} great ^aniso- tropy ^{of the} magnetostriction ^{observed in the} RFe2 (À111 ~ ⁵⁰ À100) ^can only be understood if one assumes that the external strain e is accompanied by an

internal mode ^u coupled ^{to e.}

In the notations of [20] ^{one has that :}

(where VIII is ^{some kind} of internal magnetostriction)

and :

The magnetoelastic coupling ^terms in eq. (2) ^of [20]

are :

(similar ^{to our} eq. (19) ^with B,

0). ^The correspond- ing second order crystalline potential ^{is :}

and when we substitute _eqs. (35) ^and (36) ^into (39)

and use eq. (37) ^{we recover the} general eq. (31) ^obtain-

ed in the preceding paragraph which relates V2 ^and KME. ^{The only} difference is that now KME depends ^on

the properties ^{of the} internal mode (CO, g) ^{and on its} coupling ( f ) ^{to the external stress which are} presently

not known. We may nevertheless hope ^{that the nume-}

rical estimates of the preceding paragraph ^remain roughly valid in the _presence of the internal mode.

We have found that KME ~ - ¹⁰ cm -1 /ion, ^which

may well be in error by ^{at least a factor of two or four.}

If we compare these estimates with the extrapolated

value at 4 K : K- - 120 cm -1 /ion ^{for Tm3 + in} TmFe2, ^{we see that} magnetostriction might ^well

account for something ^{like one tenth to one third}

of the magnetic anisotropy ^{of this} compound.

Another interesting remark is that in a mixture

(R(1)Fe2-R(2)Fe2)’ ^the contributions to the magne- toelastic coupling coefficients b2 and g ^are expected

to be additive, ^{but not those to} KME (since it involves

bi, g2, etc...). This could perhaps ^{be a} way ^to estimate the relative contributions to K of Kl (additive) ^and KME (non additive).

5. Discussion.

According ^{to the} preceding para-

graph ^the crystalline potential which determines the

hyperfine parameters ^{of the Rare Earth ion in a} Môssbauer experiment ^{is :}

V4 ⁺ V6 ^{in a} RFe2 magnetized along [100] (YbFe2), V2 ⁺ V4 + V6 in a RFe2 magnetized ⁱⁿ any other direction in particular [111] and [110] (TmFe2).

On the other hand the magnetic anisotropy ^constant

of order four is, ^{in all} RFe2 :

V2 (eq. (31)) ^and KME ^are magnetostrictive ^contribu-

tions and the effects of V2 ^{are of order} KME.

These expressions show that the quantities ^observed

in different RFe2 ^or by ^different techniques ^{are not} necessarily ^{the same.} Comparison ^{would make sense} only ^if KME ^{could be} evaluated with precision, ^which

is not possible ^at present because of the unknown contribution of the internal mode.

We have estimated _very roughly ^{that in} TmFe2, 1 KME 1 ^could perhaps ^{be of the order - 10,}

-

40 cm -1 /Tm3 + . Comparison ^{with the} total ani- sotropy : Kl(V4) ⁺ KME

¹²⁰ cm-1/Tm3+, sug- gests that in this compound ^the crystalline ^field V4

at the Tm3+ ion, ^{while not} responsible ^{for the} totality

of K, ^is certainly ^{not small. We must} therefore admit that V4 ^{for Yb3+ in} YbFe2 ^is effectively smaller than

V4 ^{for Tm3+ in} TmFe2. ^One may think of several

explanations for this : either A4 varies and depends

both on the RFe2 matrix and on the RE ion in it or,

A4 being ^constant, V4 contains another contribution due to anisotropic exchange, ^{whose effects are similar}

to those of a crystalline ^{field ; the most} likely ^candidate

would be Fe-RE exchange ^{which is} large ; ^{on the}

contrary ^RE-RE exchange, ^while probably ^anisotro-

pic, ^{seems too} small, ^{as shown} by ^the ordering ^tem- peratures ^{of the} RNi2 where Ni is non magnetic :

What also comes out from all these considerations is the interest of Môssbauer spectroscopy for obtain-

ing information on the crystalline ^{field at low} tempera-

tures where, at least in high anisotropy compounds, magnetic anisotropy measurements are no longer possible. ^{There are however some} difficulties : first,

the hyperfine parameters ^are mainly ^{sensitive to the} exchange ^{field and} only accessorily ^{so to the} crystal-

line field ; ^therefore precise measurements at a number of temperatures ^{are needed. As an} illustration of this fact, old Môssbauer measurements on 161 Dy ⁱⁿ

DyFe2 [23] ^{and 169Tm in} TmFe2 [24] ^were fitted with pure exchange only ÛlB Hex/kB

^{202 K and 165 K} respectively) ^{while at} least for Tm3 + in TmFe2 ^the crystalline ^{field is rather} large, ^{as discussed} above ; these fits were probably ^made possible by ^{the small}

number of points and the limited accuracy. A second

difficulty is that in the RFe2 magnetized along [111]

one should perform ^{a four} parameters ^{fit for the} hyperfine ^data ( V2, V4’ Y6, MB Hex) ; ^{this is a source of} uncertainty, ^{but it} should nevertheless yield ^some

information about the magnitude ^of V2 + V4 ⁺ V6.

Notice that when a RE impurity ^R(1) is substituted into a matrix R(2)Fe2’ V2 ^{at the} impurity ^{will not be} proportional ^to KME, contrary ^{to the} Y2 ^{at R (2).}

6. Conclusion.

In conclusion we have discussed the role of magnetostrictive effects in magnetic anisotropy and Môssbauer measurements. While these effects should be taken into account in the

interpretation ^{of Môssbauer} experiments ^{on the} RFe2 magnetized along [111 ], they ^are apparently ^{too small}

to account for the difference between the crystalline potentials observed in YbFe2 ^{on the one} hand and in

TmFe2 ^or ErFe2 ^{on the other} hand. This difference

seems therefore to arise either from a non constancy of the true crystalline potential V4 + V6 in ^the RFe2

series or from anisotropic ^Fe-RE exchange. ^Addi-

tional magnetic anisotropy and Môssbauer measure-

ments would be of interest for determining ^more exactly ^the magnetostrictive anisotropy KME ^{and the} crystalline ^field V4 + V6, ^{as well as} their variations

throughout the series. Note that for isotopes ^like 170Yb the accuracy of the Môssbauer data could be

considerably improved by resorting ^to high precision spectroscopy ^{based on several Ci sources} [25].

Acknowledgments.

^{We wish to} thank Mr. M.

Merlin and Dr. M. Lombardi for their help ^{with the} computer calculations. Mr. Perroux of the Labora- toire de Cristallographie kindly assisted with the

preparation ^{of the} samples.

Note added in proof.

^{In ref.} [19] the assertion that

1 Â 111 1 » 1 À100 I was ^{based on a} comparison ^between TbFe2 (Â 111 (OK) = ^{4 400 x} 10-6) ^and DyFe2 (Àloo(OK) = - ^{70 x} 10-6). ^The magnetostriction ÀlOO ^of HoFe2 ^has very recently ^{been measured} (R. Abbundi and A. E. Clark, Proceedings ^{of the} Magnetism Conference, Cleveland, ^nov. 1978, ^{to be} published ^{in J.} Appl. Phys.) and has been found to be much greater than that of

Nevertheless 1 îloo IHoFe2 is still six times smaller

than 1 Â ¹¹¹ ITbFe 2and the considerations of chap. ⁴

remain valid.

APPENDIX 1

Correspondence ^between crystalline ^{field and aniso-} tropy.

^{Let us assume that the RE} ion is submitted

to a crystalline field, represented by ^the equivalent operator

where Okq (J) ^{is the operator} equivalent [26] ^of

Let us assume that in addition to Hc, ^{the RE ion}

is also submitted to a large exchange field with polar angles (0, w)

and that kB T Hc « Jez, ^{i.e. the atom is in the state} 1 Jz = - J ) quantified along Hex. By rotating Hc

it is possible ^to demonstrate that [27, 28] :

with

Ean ^{is the} anisotropy energy and eq. (A. 5) gives ^us

the correspondence ^{between the} crystalline potential

and the anisotropy.

Let us now assume that we have cubic symmetry :

and that the exchange ^field has cosines ai a2 a3.

We find that :

We now want to put ^{this into the usual} form for cubic symmetry :

We get ^{that :}

We see that V6 contributes non only ^to K2 ^{but also}

to Ki . ^{This is due to the fact that the two} polynomials

in _{al, (X2’} a3 in eq. (A. 9) ^{do not} belong ^{to an ortho-} gonal ^{basis set} [29].

In practice ^the contribution of V6 ^to Ki ^{is eleven}

times smaller than its contribution to K2. ^{It follows}

that as long ^as K2 KI, Ki ^is essentially ^{due to} V4. ^{This is} particularly ^{true in the most} anisotropic RFe2 ^where experimentally K2 ^{is found to be} very small with respect ^to Kt.

APPENDIX II

Small corrections to référence [1] ] ^and possible ^role

of an anisotropic hyperfine ^{structure of the irons in 57Fe}

Môssbauer spectra ^of YbFe2.

^{1 . SMALL CORREC-}

TIONS.

In référence [1] ^several quaiitities ^relative

to YbFe2 ^were estimated with the help of theoretical values of the Yb 31 moment computed ^{with the data}

of reference [2] ^{obtained on} TmFe2 : ^{Yb ; in} parti-

cular the moment of the iron atoms was estimated

by writing ^{that at the} compensation temperature JlFe

1/2 JlYb. We ^{must now correct} these quantities

on the basis of the information obtained in the present

paper with the Môssbauer effect of 17°Yb in YbFe2

itself.

Before doing ^{so, we note that the error bars of}

reference [1] ^{for the} compensation temperature oc

^{31 ± 7 K were} probably ^too high. ^{Here we}

will admit that ec

³¹ ± ^{2 K} which corresponds

to ± 2 % variations of the 17°Yb hyperfine ^field

in figure ^3.

In these conditions we find that the Yb3+ moment at the compensation temperature ^is comprised ^between

the values _{3.23 YB} (obtained by interpolation ^{of the} experimental ^{Yb3 +} hyperfine ^{fields at 20 K and 34 K}

after deducing He

²⁴⁰ kG) ^{and 3.35} YB (obtained

by calculating Jz > ^{in a pure} exchange ^field

YB Hex/kB

^109.64 K). This leads to

at low T instead of 1.8 YB in [1].

At 85 K we estimate from figure ^{3 of the} present paper thatyyb 1-99 /lB instead of 2.03 _/lB computed

in [1]. ^{We have shown in} [1] ^{that the} theoretical

dipolar parameter Ad ^{which comes into} play ^{in the} interpretation ^{of the 57Fe Môssbauer} spectra ^is given by (eqs. (6), (14) ^of [1]) :

With the new values of the moments we find that :

-

at 4 K, Ad

⁵ 000 Oe instead of 5 200 in [1 ],

- at 85 K, Ad

³ 650 Oe instead of 3 900 in [1].

Also we had found that at room temperature, 285 K (where crystalline field effects are negligible),

J1Yb

0.59 _J1B. We must correct this value for the

change ^{in the} exchange ^field (uB Hex/kB

^{111 K} instead ^{of 116 K in} [2]). This leads _{to J1Yb}

0.56 _J1B.

If we assume in addition that _{J1Fe oc} HnF’, ^the expected

saturation moment per formula unit at 285 K is :

instead of 2.6 _YB in [1 ].

Despite ^{these small} corrections the conclusions of reference [1] ^remain unchanged. ^{We must however}

make some remarks relative to the detailed inter-

pretation ^{of the 57Fe Môssbauer} spectra ⁱⁿ [1].

2. POSSIBLE ROLE OF AN ANISOTROPIC HYPERFINE INTERACTION IN THE 57Fe MÔSSBAUER SPECTRA OF

YbFe2.

^{In reference} [1] ] the evaluation of tipping effects in the interpretation ^of the Môssbauer spectra

at 5.2 K was made with the help ^{of the} computed

value of the dipolar parameter (,/2-A, ^{= 7.3 kOe).}

New fits using ^{a least} square program lead ^{to a} very

slight improvement ^{of the} agreement ^{if one uses the} empirical parameter : /2- A ^{= 11.9 kOe, whence}

A

8.43 kOe. The corresponding ^{value for} e2 2 qQ

is

0.52 mm/s ^{instead of} the tentative value

-

0.50 mm/s of reference [1].

This empirical ^{value of A at 5 K is} appreciably higher ^{than the} theoretical dipolar ^{value 5 kOe.}

On the other hand with a higher ^{value the reduction}

factors associated with AH and DE at 85 K (see ^dis-

cussion in top ^of p. 1454 of [1]) would become more

similar.

In addition it should be mentioned that recent

57Fe Môssbauer experiments ^on NdFe2 [30] ^lead

us to suspect ^{that A is} probably ^not entirely ^of dipolar origin. ^{If we assume that with} respect ^{to its local} trigonal symmetry axis 111> ^the hyperfine ^structure

of an iron atom is anisotropic with axial symmetry :

and take account of the four directions 111 ) ^we

arrive at splittings of the Môssbauer spectra ^which have exactly ^{the same behaviour as those} arising

from dipolar effects, the combined effect of both of them being characterized by ^the parameter :

This can be easily ^checked by transforming eq. (B. 2)

into the usual axes parallel ^{to the} cubic cell axes

and by comparing ^the result with the dipolar ^tensors

Ad associated with the four 111 > directions [1].