Ferrimagnetic spiral configurations in cobalt chromite

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Ferrimagnetic spiral configurations in cobalt chromite

N. Menyuk, K. Dwight, A. Wold

To cite this version:

N. Menyuk, K. Dwight, A. Wold. Ferrimagnetic spiral configurations in cobalt chromite. Journal de

Physique, 1964, 25 (5), pp.528-536. �10.1051/jphys:01964002505052801�. �jpa-00205822�

rence est donc due à des raisons purement géomé- triques

déplacement ^du cryostat dans le fais-

ceau.)

Nous avons construit un diagramme ^différence

dans lequel ^les intensités à la température ^ambiante

ont été renormalisées de façon à rendre nulle la différence des intensités des raies 311 (tableau VI).

Les calculs sont faits avec le facteur de forme de Fe2+ de la référence [2], ~.~(Fe) = 2, ^{et avec le}

facteur de forme de Cr2+ de la référence [3], E(Cr) ⁼ 3 j2.

Ce résultat n’est guère susceptible ^d’amélio- ration, ^la précision ^sur S(Fe) ^et S(Cr) étant respec-

tivement ~ 0,12 ^{et -~} 0,15 ^et le facteur de corré- lation étant très voisin de l’unité.

Il est intéressant de comparer FeCr,S, ^et FeCr104 (Bachella [4], ^Pickart [5]). ^Le fait que la structure FeCr2S4 ^{obéisse au} schéma de Néel tandis que celle de FeCr104 ^est certainement hélimagné- tique permet de conclure que les interactions AB sont prédominantes ^dans FeCrIS4 ^{vis-à-vis des}

interactions B-B, ^{et d’autre} part que les interac- tions négatives Cr-Cr décroissent fortement avec la

distance, ^{l’ion S2- étant} considérablement plus grand que l’ion 02-.

Note ajoutée ^~, la correction.

Nos résultats sont ^en accord avec ceux de SHIRANE (C.), ^Cox (R. E.) ^{et PICKART} (S, J.), Conférence on magnetism, ^Atlantic City, ^nov. 1963,

et J. -,,4ppl. Phys.. 964, 35, ^95IL.

BIBLIOGRAPHIE [1] ^LOTGERING (F. K.), Philips ^Research Reports, 1956, 11,

218-249.

[2] ^SCATTURIN (V.), ^CORLISS (L.), ^ELLIOT (N.) ^{et HASTINGS} (J.), ^Acta Cryst., 1961, 14, ^19.

[3] ^CABLE (J. W.), ^WILKINSON (M. K.) ^{et WOLLAN} (E. O.), Phys..Rev.,1960,11$, 950.

[4] ^BACCHELLA (G. L.) ^{et PINOT} (M.), ^sous presse,1964.

[5] ^PICKART (S.), ^sous presse, ^1964.

FERRIMAGNETIC SPIRAL CONFIGURATIONS IN COBALT CHROMITE

By ^N. MENYUK, ^{K. DWIGHT} and A. WOLD (1),

Lincoln Laboratory (2), Massachusetts Institute of Technology, Lexington 73, Massachusetts, ^{U. S. A.}

Résumé.

Le chromite de cobalt CoCr2O4 ^{est un} spinelle cubique ferrimagnetique ^{aux basses} temperatures ^{avec une} temperature ^{de Curie} Tc ~ ^{97 °K. Un} diagramme ^de poudre ^à l’ambiante, corrigé ^{des effets de} temperature, ^montre qu’il s’agit ^d’un spinelle ^{normal avec un} paramètre d’oxygène égal ^à 0,38707 ± 0,00005. ^A 4,2 ^{oK il} y a, ^en plus des contributions magnétiques ^dans

les raies fondamentales, ^un grand ^{nombre de} satellites magnétiques.

Toutes ces raies additionnelles peuvent être indexées sur la base du modèle de la spirale magné- tique, proposée par Lyons, Kaplan, Dwight ^et Menyuk ^dans lequel les composantes ^en spirale ^des spins ^{sont définies} par ^un vecteur unique k dirigé ^{selon la} diagonale ^{d’une face du cube. La valeur} expérimentale de |k| ^est d’approximativement ⁵ % plus grande que prévue par la théorie. Le dia- gramme neutronique ^{se trouve} complètement déterminé dans cette théorie par la ^{donnée du}

rapport JAB/JBB ^{et de} la direction de l’axe du cone. Prenant JBB/JAB = 1,5 grace ^{à des mesures}

d’aimantation publiees antérieurement, les intensités des satellites sont trouvées être en excellent accord avec les intensités prévues par ^le modèle de la spirale ferrimagnetique, ^{l’axe du cone étant}

selon [001].

On sait que la configuration ^de spirale ferrimagnétique devient instable pour

JBB SB/JAB SA > ^0,98

(c’est-à-dire pour JBB/JAB > 0,98 ^dans CoCr2O4). ^{Notre résultat} indique cependant que la confi-

guration ^{réelle est encore stable dans un domaine de} rapports d’intégrales d’échange ^{bien au-delà}

du début d’instabilité locale.

En chauffant au-dessus de 4,2 oK, ^les raies satellites disparaissent entre 25 °K et 35 °K, pour finalement dégénérer ^{en une bande} large observée à 50 °K et 70 °K. Malgré ^cette disparition, ^il

n’est possible, ^à partir ^{du modèle} collinéaire d’obtenir un accord, ni pour les contributions magné- tiques ^{aux raies} fondamentales, ni pour la variation thermique ^{observée de} l’aimantation. Par contre, ^les valeurs prévues par la théorie du champ moléculaire appliquée ^{au modèle de} spirale ferrimagnétique ^{sont en} bon accord avec les mesures expérimentales. Ces résultats corroborent la validité du modèle proposé pour CoCr2O4, ^et indiquent que l’approximation ^du champ moléculaire décrit fidèlement, dans le domaine ferrimagnétique, l’évolution de la composante axiale, ^mais

non celui de la composante ^azimutale.

(1) ^Present address : Brown University, Providence, Rhode Island.

(S) Operated ^{with support from the U. S.} Army, Navy ^{and Air Force.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01964002505052801

Abstract.

^Cobalt chromite, CoCr2O4, ^{is a cubic} spinel ^{which is} ferrimagnetic ^{at low} tempe-

ratures with ^a Curie point Tc ^{~ 97 oK. The room} temperature powder diffraction pattern ^{on this} material, corrected for temperature effects, ^{shows that it is a normal} spinel ^{with an} oxygen para- meter equal to 0.38707 :f: .00005. At 4.2 oK there are, in addition to the magnetic contributions to the fundamental spinel peaks, ^a large ^{number of} magnetic ^{11 satellite "} peaks.

All of the additional peaks ^can be indexed on the basis of the ferrimagnetic spiral ^model pro-

posed for normal cubic spinels by Lyons, Kaplan, Dwight ^and Menyuk, in which the spiraling components ^{of the} spins ^{are defined} by ^a single ^{k vector} along ^{a face} diagonal. ^The experimental magnitude ^of |k| ^is approximately ⁵ % greater ^than theoretically predicted. The neutron dif- fraction pattern predicted ^{on the basis of the} spiral ^{model is} completely determined _upon fixing

the exchange ^ratio JBB/JAB ^{and the cone axis} direction. Taking JBB /JAB ^{= 1. 5 on the basis} of magnetization measurements previously reported, ^{it is found} that the intensities of the various

peaks ^{are in excellent} agreement with the intensities predicted by ^the spiral model with the cone

axis along ^an [001] ^direction.

The ferrimagnetic spiral configuration ^is known to become unstable relative to small deviations for JBB SB /JAB SA > 0.98 (i.e. JBB/JAB > ^{0.98 in} CoCr2O4). However, ^our results indicate that the true configuration closely approximates ^that predicted by ^{the model over a} range of

exchange interaction ratios which extends well beyond the onset of local instability.

Upon increasing ^the temperature ^above 4,2 °K, the satellite peaks disappeared between 25 °K and 35 oK, apparently degenerating ^{into a broad} plateau observed at 50 oK and 77 oK.

Despite ^this disappearance, ^a self-consistent fit to neither the magnetic contributions to the fundamental peaks ^{nor the} observed thermal variation of the magnetization ^can be obtained from a collinear model. However, ^the predicted values based on a molecular field treatment of the spiral ^{model are in} good agreement ^{with these} measurements. These results further corro-

borate the validity ^{of the} spiral ^{model for} CoCr2O4, ^and indicate that the molecular field approxi-

mation accurately describes the axial component throughout ^its ferrimagnetic range, but not the azimuthal component.

1. Introduction.

Cobalt chromite is a cubic

spinel ^{with cell} length ao ⁼ 8.332 A [1]. ^A study

of its magnetic properties ^{has shown it to be ferri-} magnetic ^{with a Curie} point ⁹⁷ OK, ^{and a} magnetic ^{moment at 4.2 OK} corresponding ^{to 0 . ~.4}

Bohr magnetons (PB) [2]. This value is far below the value of 3 _tJ-B predicted by ^the collinear Neel

theory ^of f errimagnetism.

A neutron diffraction study ^of CoCr204 ^{at room} temperature ^and throughout ^the f errimagnetic ^tem- perature range is presented in this paper. It ^is shown that the resultant diffraction pattern ^at

4. 2 oK can be interpreted ^{on the basis of a ferri-} magnetic ^conical spiral configuration ^{of the} type predicted ^{for cubic} spinels by Lyons, Kaplan, Dwight ^and Menyuk (LKDM) [3], and first obser- ved by Hastings ^{and Corliss} [4]. ^This interpre-

tation uniquely determines the exchange para- meter and direction _{of easy} magnetization.

An unambiguous prediction ^{of the} magnetic pro-

perties ^{of this material as a function of} tempe-

rature was obtained from a molecular field calcu-

lation, ^{and a} comparison ^{between the} observed and

predicted ^neutron diffraction patterns ^is given.

We find striking agreement ^between theory ^and experiment ^{in some} respects, ^and striking ^disa- greements ⁱⁿ others, indicating ^{that the molecular-}

field approximation yields ^accurate predictions ^of

certain magnetic properties, ^{but not of others.}

The disagreements ^{are shown to be due to corre-}

lation effects, ^{which can} apparently play ^an impor-

tant role in materials with complicated spin ^confi- gurations.

II. Low temperature configuration.

^{A. EXPE-}

RIMENTAL RESULTS.

Powdered samples ^{of cobalt}

chromite can best be prepared ^from the precursor

COCrO.4CSHN, ^{as described} by Whipple ^and

Wold [5], ^{since the} ignition ^{of this} complex ^mole-

cule results in a very finely divided, highly ^reactive

oxide. However, this precursor ^is usually ^found

to be deficient in chromium. Hence, ^{after deter-} mining ^the total chromium present ^{in a trial} sample

of CoCr204 prepared from this precursor, sufficient

(NH7)2Cr2Ü4 ^{to correct the} deficiency ^{was dissolved}

in about 10 ml of water and added to the remain- der. This resulted in a corrected Co-Cr ratio.

The mixture was then ignited, ground, ^{fired at}

1 200 ~C for three days, ^cooled slowly ^{to 800} OC,

and then quenched.

Analysis ^{of our final} sample, ^{based on a total}

chromium determination, gave ^a Cr-Co ratio of 2.03 : 1. In addition, x-ray analysis ^{showed the} sample ^to consist of a single spinel phase ^{with no} impurity ^lines present in the diffraction pattern.

The neutron diffraction experiments ^{were carried}

out at the M. I. T. nuclear reactor. The powder sample ^{was contained in a vanadium} tube for the

room temperature spectrum, ^{but the data at all} other temperatures ^were taken with the sample ⁱⁿ

an aluminum tube. The neutron wavelength ^was

1.196 A. For simplicity, ^{all the} tables and curves are normalized to a monitor count of 600,000 ^neu-

trons (~ ⁴ minutes) per 3 minutes of _arc, although

this count was doubled at 4.2 OK. The form factors given by ^Watson and Freeman [6] ^for

cobalt and chromium were used in the data analysis.

The nuclear structure of CoCr204 ^{was obtained}

from the room temperature spectrum ^{shown in} figure ^{1. The} oxygen parameter ^and normality

were obtained with the aid of an IBM 7090 com-

puter which ^was programmed ^{to normalize to the}

rrG, 1.

Room temperature ^neutron diffraction pattern

of CoCr2o4. ^The number of neutrons is based on a

monitor count of 600,000.

total integrated intensity ^{of the} peaks investigated

and obtain ^a best least squares fit by independently varying ^the oxygen parameter, ^the normality, ^and

the Debye-Waller correction. The oxygen para- meter was found to equal 0 . 38707 ~ . ^{0000~ and}

the sample is normal with an uncertainty ^{of 5} %.

This uncertainty ^{is caused} by ^{the relative closeness}

of the scattering amplitude of chromium and cobalt (bcr ⁼ .352, ^{bco =} .25) [7]. ^A comparison

between the theoretical intensities and experi-

mental values after correcting ^for temperature ^is given ^{in Table I. We can therefore} characterize the material as Co[Cr2]04’ ^{with the chromium ions}

in the brackets all on octahedral (B) sites, ^{and the}

cobalt ions all on tetrahedral (A) ^{sites. Thus} Co[Crl]04 ^{has an ordered structure of the} type

assumed in the analysis ^{of LKDM.}

TABLE I

COMPARISON OF EXPERIMENTAL

AND CALCULATED INTEGRATED NUCLEAR PEAK INTENSITIES

(1) Corrected for Debye- "’Taller temperatur.e factor.

(z) ^Normal spinel, oxygen parameter ^{= .38 707.}

The neutron diffraction spectrum ^{obtained at}

4.2 OK is shown in figure ^{2. It is} characterized by

a number of peaks ^{which did not} appear in the

room temperature spectrum, ^{as well as} by magnetic

contributions to the nuclear peaks. ^{The addi-}

tional magnetic peaks ^are designated ^{as " satel-} lites ", ^{and the} magnetic contributions to the nuclear peaks ^are called " fundamentals ".

FIG. 2.

4.2 oK neutron diffraction pattern ^of CoCr2o4’

Magnetic ^satellite peak positions ^{are indicated} by ^ver-

tical indices, ^the fundamental peak positions ^{are indexed} horizontally. ^{The aluminum} peaks ^are produced by specimen ^holder.

Although ^{the satellite} peaks ^cannot be obtained by

,

a simple integral enlargement ^{of the unit} cell, ^a pattern ^{of this} type ^can be obtained from a ferri-

magnetic spiral. ^{In that case} the fundamental contributions arise from the collinear-unvarying (k ⁼ 0) component ^{of the} spins ^{in the z’} direction,

while the satellite contributions ^are due to the

spiraling component ^{which is} perpendicular ^{to z’ in}

the x’y’ plane. ^The dependance ^{of the satellite} peak ^locations upon the magnitude ^{and direction}

of the wave vector k which characterizes the spiral

has been dealt with in detail by ^LKDM [3] ^and Hastings ^{and Corliss} [4].

We attempted ^{to index the various} peaks ^{in the}

low temperature Co[Crl]04 spectrum assuming ^{a k}

vector in the h(i -~- j + k), h(2i ⁺ k), ^{hk and} h(i + j) directions. Agreement ^{with the data}

could not be obtained in the first three cases.

However, ^{for k =} h(i -- j), all the observed satel- lite peaks ^{could be} indexed, ^{as indicated in} figure 3,

for h = 0 . 62. This is the direction predicted by LKDM, ^{but is} approximately ⁵ % ^{above their} predicted ^{values of h = 0.59.}

According ^{to the LKDM} theory, ^the spin ^confi- guration ^is completely characterized to within a

rotational degeneracy by ^a single parameter u, where

(1)

fiic. 3.

Satellite peak ^{locations for} ferromagnetic spiral

in CoCr204 ^{as a function of wave vector k,where}

k = h(i -f- j). Peak locations observed experimentally are shown by ^black squares. The thickness is a measure

of experimental uncertainty.

In the above equation, ^{JnB and JAB} represent

the exchange interaction between near-neighbor

A-B and B-B cations respectively ; ^and ¡SAI ^and

represent ^the magnitude ^{of the} magnetic

moment of the cations at the A and B sites respec-

tively. ^{In a} real material the rotation degeneracy

is lifted by anisotropy ^{effects to establish a} parti-

cular cone axis (z’) direction. This direction must be determined experimentally.

Consistency ^{with the 4.2 0 K} magnetization

value of Co[Cr2]04 ^{with the LKDM} theory requires

that u ⁼ 2 in this material [2]. Choosing ^this

value fixes the cone and phase angles ^{of all six}

sublattices (see figure 2 and eq. (10) ^of reference 3),

as indicated under Table II. Several ^cone axis directions were considered ; ^best agreement ^with

the observed diffraction pattern ^was obtained with the cone axis in the [001] direction. _{The compa-} rison of experimental intensities with the values

predicted by ^{the above} ferrimagnetic spiral ^confi- guration ^{with net} magnetization ^{in the} [001] ^direc-

tion is given in Table I I. The magnetic intensities listed are absolute intensities, ^{based on the instru-}

ment normalization factor obtained from the room

temperature pattern.

It should be noted that every predicted peak ^is

observed and, conversely, ^{there are no} peaks pre- sent which cannot be accounted for by ^{the ferri-} magnetic spiral model. In the latter respect ^our

results differ from those obtained by Hastings ^and

Corliss with manganese chromite, ^as they ^observed

two extra-peaks which could be not accounted for

by ^the spiral theory. They also found their funda- mental peak intensities were uniformly high by ^a

TABLE II

COMPARISON OF CALCULATED (~)

AND EXPERIMENTAL PEAK INTENSITIES AT ~t.2 ~~1

(1) u = 2.0, ^Cone angles : Oi ^- P2 ⁼ 32°;

1>3 ⁼ ~4 ⁼ 1500 ; (Di ⁼ (P6 ^900.

Phase angles : 11 = Y2 ⁼ Y3 ⁼ Y4 ~ 0 ; Y5 ⁼ P6 ^{= w}

Net magnetization ^direction along [001].

factor of approximately ³⁰ % ; ^this discrepancy

between theory ^and experiment ^{does not occur in} CO[Cr2]O4.

B. DISCUSSION.

According ^{to the LKDM} theory, which considers only nearest-neighbor ^A-B

and B-B interactions, ^the f errimagnetic spiral ^struc-

ture is locally ^{stable over a range of values exten-} ding ^{from the} boundary ^of Néel mode stability (u ⁼ 8/9) ^{to u N 1.3.} For u > 1. 3, ^the spiral

is locally unstable relative to a more complex configuration. Furthermore, ^the magnetization

curve of Co[Cr2]O4 ^{shows a} sharp change ⁱⁿ slope

at 27 OK, ^{which has been} interpreted ^{as a transition}

from the spiral ^{model above this} temperature ^{to a}

more complex configuration below. Under these

circumstances, discrepancies between the calcu- lated and experimental ^{patterns are to be} expected,

such as the possible existence of non-spiral peaks

or deviations from the predicted magnetic ^inten-

sities of the observed spiral peaks. However, although ^some descrepancies ^do exist, ^{as discussed} below, ^{the most} striking feature of the experi-

mental diffraction patterns ^{is its} agreement ^with

the pattern calculated on the basis of the spiral theory. ^This agreement permits ^{us to fix the} exchange interaction ratio JBB - 1.5 JAB with considerable accuracy in and strongly

indicates the [001] ^{axis to} be the easy direction of the net magnetization.

The most notable discrepancy ^{between the calcu-}

lated and experimental pattern ^{is the} low-observed

intensity ^{of the} 002(0) ^and 113(-1) ^satellite peaks.

Anomalously low values for the corresponding peaks ^were also found in [4], ^{for which}

u N 1.6. The similarity ^of behavior indicates that this departure ^{from the} spiral ^model may be due to the local instability. However, ^{the rela-} tively high fundamental peak intensities and the extra peaks observed in Mn[Cr2]O4 ^{cannot satis-} factorily ^{be accounted for} by merely noting ^that

u ⁼ 1. 6 > 1. 3, ^since these effects do not occur in

Co[Cr2]O4, ^{for which u = 2.} It therefore appears that the relative importance ^of magnetic pheno-

mena not considered by ^LKDM may vary signi- ficantly ^{in these two} materials. Amongst ^these

are the near-neighbor ^{A-A and} next-near-neighbor

B-B interactions, ^which might give ^{rise to obser-}

vable departures from theoretical predictions, ^such

as the wavelength discrepancy ⁱⁿ Co[Cr2]04. ^{H ow-}

ever, the effect of these interactions should be

qualitatively ^{similar in both} CO[Cr2]O4 ^and Mn[Cr2]O4- ^It is therefore difficult to account for the contrasting ^{nature of the} departures ^{from the} spiral spin configurations ^{in these materials on this}

basis alone.

Another possible ^{source of the} diff erent behavior in these materials arises from the fact that Mn2+

cations on tetrahedral sites contribute less than their spin-only value of 5 [8, 9]. ^{This indicates}

that the Mn2+ cations are not in a pure 6S5/2 ^state.

The effect of admixing additional states may well lead to non-negligible magnetic energy ^terms of a

from other than that of the Heisenberg exchange

terms

Since the LKDM theory ^{assumes that the total} magnetic energy ^{is of this} form, additional terms

can lead to notable departures ^{from the} theory.

In on the other hand, ^{both the and}

Cr3 i cations appear ^to have their spin-only ^values

of 3 The agreement ^{between our} results and those predicted by the LKDM indicates that t,he

Heisenberg exchange ^{term is a valid} representation

of the magnetic interaction in this case, and that this theory ^{is a} good approximation ^{over a} range of values of B-B to A-B exchange interaction ratios which extends well beyond the limit of local stabi-

lity.

III. Thermal effects.

A. MOLECULAR FIELD CALCULATION.

The rigorous minimization of the free energy ^at finite temperatures ^{is at least as}

difficult as the determination of the true ground state, ^{even in the} molecular-field approximation.

The thermal evolution of a stationary ^{state of the}

molecular-field free energy is a more tractable

problem. ^{The LKDM} spiral constitutes such a

stationary ^{state at absolute} zero, despite ^not being

the ground ^state [3]. ^At temperatures slightly

above absolute _zero, random thermal fluctuations of the spins [10] would reduce the apparent

moments of the cations, ^{but the} resulting average moments would still correspond ^{to a} spiral-type configuration. ^{The cone} angles ^{and the} magni-

tudes of the average magnetic ^{moments for each of}

the three pairs ^of sublattices, ^{as well as the} magni-

tude of the propagation ^vector k, ^{can all be} expec- ted to vary continuously ^with temperature. ^Ho-

wever, it ^can be argued ^from symmetry ^conside-

rations that the sublattice phase angles ^{and the}

direction of the spiral propagation ^{vector will}

remain unchanged.

The molecular-field free energy ^can be explicitly expressed ^{in terms of the above} variables, ^{and a} system ^{of seven} transcendental equations ^{in seven}

unknowns can be obtained, ^{as shown in} Appendix ^{A. The} solution of this system by ^the Newton-Raphson ^{method was carried out nume-} rically ^{on an IBM 7090 computer for a series of} increasing ^{values of the ratio} using ^the

LKDM spiral ^{as the trial solution for the lowest} T /Tc, ^the final solution as the trial values for the next etc. The appropriate temperature

scale was then obtained from the experimental

value Tc ⁼ 97 OK.

Using the method described above, ^we computed

the spiral wavelength, ^cone angles, and effective

spin magnitudes ^at forty temperatures, equally spaced ^{between absolute zero and for several}

values of the exchange parameter ^{u. The cone} angles ^{become zero and the} distinction between the two pairs of B-sublattices vanishes when T gr 0 . 9 Tc ^for u N 2. This behavior represents

a transition from a low-temperature ferrimagnetic

spiral ^{to a} high-temperature ^collinear (Néel) ^confi-

guration. It is worth noting ^{that the} length ^{of the}

propagation vector at this transition is identical

with the value ⁼ 3.647 ~/2 ^which destabilizes

the Néel configuration in the LKDM ground-state

calculation, independently ^{of u.}

B. MAGNETIC MOMENT. - The magnetic ^moment

of a pure sample of cobalt chromite has already

been measured as a function of(temperature ^{in an}

external field of 11,000 ^Oe [12], the results being reproduced ^{as the dashed curve in} figure ^{4. We}

have now made a similar measurement (on ^the

same powdered sample ^{used in Ref. [2]) in an}

external field of 500 Oe, ^which gives ^{the solid}

curve in figure ^4. This low-field curve is approxi-

4.

Magnetization of cobalt chromite as a function of temperature. ^The experimental ^{data are shown as}

curves ; the values calculated for a spiral ^{model accor-} ding ^{to the molecular field} theory ^are given ^as points.

mately proportional ^{to the} high-field ^{one, as would}

be expected ^{on the basis of} anisotropy effects, ^and

constitutes merely ^a lower bound for the sponta-

neous magnetization. Although ^the magneti-

zation in 11,000 ^{Oe will} presumably ^not be aff ected

greatly by anisotropy, ^{it is} subject ^to the effects of both paramagnetic ^and high-field susceptibility,

which persists ^{down to absolute zero for canted} spin configurations [11]. From these conside-

rations, ^it would appear that ^{the true} spontaneous magnetization for cobalt chromite probably ^{f ollows}

a curve having ^{the same} shape ^{as the dashed curve}

in figure 1, but lowered by ^an approximately

uniform amount. According ^{to this} model, ^the spontaneous magnetization ^{would be} expected ^to

attain its maximum value at about 77 °K, ^where

the experimental peaks ^occur.

Included in figure ^{4 are three sets of} points, representing ^the magnetization ^{values obtained}

from our molecular field calculation [12] ^{for three}

values of the exchange parameter : u ⁼ 1.98, ^2.03

and 2.08. All three sets have approximately ^the

same shape ^{as the dashed curve.} However, ^since

the peak magnetizations ^{are attained at 82} °K,

78 OK and 73 OK, respectively, ^we conclude that

u ⁼ 2.03 gives ^the best fit. In support ^{of this} conclusion, ^{we note that the} points ^{f or u = 1.98}

.

fall too far below the solid curve at low tempe- ratures, ^{whereas those for r~ = 2.08} give ^dis-

tinctly ^worse agreement ^{with the neutron dif-}

fraction results described in the next section. Fun-

thermore, ^the displacement ^{between our} high-field

data and the points calculated for u ⁼ 2.03 is consistent with the high-field susceptibility ^deter-

mined for MnCr204 [11].

Thus, ^the molecular-field calculation for

u ⁼⁼ 2.03 gives extremely good agreement ^with

the spontaneous magnetization deduced from the

experimental ^{data shown in} figure ^1. This calcu- lation predicts ^a transition from the low-tempe-

rature spiral configuration ^{to a} high-temperature

collinear one at the temperature ^{~’t = 86} oK, ^as

indicated in figure 1, ^{with a} small, ^and probably undetectable, discontinuity ^{in the} slope ^{of the}

M vs T curve. Below 27 oK, ^the experimental ^data

deviates markedly ^{from the} spiral predictions.

This transition appears ^{to be} due to the local insta-

bility ^{of the} spiral configuration ^{at such low} tempe-

ratures. Although direct substantiation of this

hypothesis ^{would be} desirable, the necessary calcu- lation of the stability ^{of the} molecular-field free energy is beyond the scope of this paper. Never-

theless, ^such instability ^{has been} predicted ^{for the} ground ^state [3].

C. NEUTRON DIFFRACTION PATTERNS.

Since the spiral ^model yields ^excellent agreement ^with

the temperature dependence ^{of the} spontaneous magnetization ^of cobalt chromite above 27 oK,

and since the spiral appears ^to be unstable below this temperature, ^one might expect ^{to obtain even}

closer agreement between the experimental ^and

calculated neutron diffraction patterns ^{at inter-}

mediate temperatures ^{than at} 4.2 °K. We found 77 °K and 50 aK to be convenient temperatures,

both being ^{below the} spiral-to-Néel transition tem-

perature ^{~’t = 86 °K} predicted for u = 2.03. The

pattern ^{obtained at} 50 oK is shown in figure 5,

and our findings ^{at 77 °K are similar to these}

results.

The sharp ^satellite peaks ^{observed at 4.2 °K are}

not present ^{in our 50 °K} pattern. ^{The locations}

and peak intensities of some of these satellites are

indicated in figure 5, ^{and .it is evident that the}

dominant satellite (20 ⁼ 1605’) ^{has been} replaced by ^a broad, liquid-type peak which extends from

~.2°30’ to 200 [13]. ^{Most of the} degeneration ^from

a coherent to a liquid peak ^{occurs in the} tempe-

rature interval 30°- 32 OK, ^{as was} determined by scanning ^{over a} 24’ interval centered about 16°5’

while the sample warmed up from 4.2 OK. Such behavior would normally ^be interpreted ^{as a} spiral-to-collinear transition at about 31 oK. Ho- wever, this explanation ^not only contradicts our

analysis ^{of the} magnetization data, but also fails to

account for the persistence of considerable short-

range order (as ^indicated by ^{the sizable} liquid

peak) up ^{to at} least 77 oK.

In addition to the liquid peak, ^the pattern ^shown in figure ^{5 includes} appreciable magnetic ^contri-

butions to the intensities of the nuclear peaks,

FIG. 5. - Neutron diffraction pattern ^{from cobalt chromite}

at 50 OK. The locations and intensities marked with S refer to the principal satellites observed at 4.2 OK in Ref. 2. The monitor unit was 600,000 ^counts.

which are given in Table III. The corresponding magnetic intensities calculated from our molecular- field treatment of a [110] spiral ^model (with ^{u = 2}

and the cone axes in the [001] direction, ^{as esta-}

blished by ^{the 4. 2 ~K} measurements) ^{are also} given

in Table III [14]. ^{Here the} agreement ^between theory ^and experiment ^is exceptionally good, being

well within the statistical uncertainties. The data

presented ^{in Table} II shows that similar agreement exists between the experimental intensities and the

spiral calculation at 77 ~K. Not only this agreement itself, ^{but also its} improvement ^{over the} analagous

results at 4.2 OK accord with the predictions ^{of our} spiral ^model [15].

TABLE III

COMPARISON OF EXPERIMENTAL AND CALCULATED NUCLEAR PEAK INTENSITIES

FOR COBALT CHROMITE AT 50 °K

(1) Taken from the top ^{of the} liquid peak.

We’* have also considered the case of collinear

spins in the molecular-field approximation [16].

For the predetermined ^{value u =} 2, ^{this calcu-}

lation yields the intensities shown in Tables III

and IV for the collinear model. The serious dis- crepancy between the experimental and theoretical intensities of the (111) peak ^at 50 °K is sufficient to

preclude any possibility ^{of a} spiral-to-collinear

transition at 31 OK [17]. ^{Even at 77} oK, ^the

collinear model gives ^less satisfactory agreement

with experiment than does the spiral calculation.

This fact, together with the marked improvement

in agreement ^{at 77 OK over that at 50} OK, ^substan-

tiates the existence of a spiral-to-collinear ^tran-

sition at some higher temperatures ^{86 OK}

as predicted by ^the spiral model).

TABLE IV

COMPARISON OF EXPERIMENTAL

AND CALCULATED NUCLEAR PEAK INTENSITIES ~ ^I FOR COBALT CHROMITE AT 77 OK

(1) ^{Taken from the} top ^{of the} liquid peak.

Consequently, ^{one must} search for an alternative

explanation ^{for the} disappearance ^{of the satellite} peaks ^at 31 OK which does not involve a basic

change ^{in the} spiral ’structure of the spin ^confi- guration. ^{In this} connection, ^it appears signi-

ficant that the total intensity ^{of the} liquid peak ⁱⁿ figure ^{5 is} approximately ⁴ 070, compared ^{to 4 224}

for the combined theoretical intensities of the two

lowest-angle satellites. Such agreement suggests

that the satellite peaks ^{are in essence still} present,

but broadened beyong recognition by ^{the effect of} locally correlated thermal fluctuations on the long-

range azimuthal coherence of a conical confi-

guration [18]. However, ^{it should be} pointed ^out

that the calculated satellite-peak intensity drops

below 1 000 by ⁷⁷ °K, whereas the intensity ^{of the} liquid peak ^remains approximately unchanged.

D. DISCUSSION.

The measurements at 4.2 OK established unambiguously ^that cobalt chromite has an exchange parameter u #* 2, ^{that the} [001]

is the direction of easy magnetization, and that the

ground ^{state is an LKDM} f errimagnetic spiral ^{to a} high degree ^of approximation. Upon extending

the theory ^{to finite} temperatures by ^{a molecular-}

field calculation normalized to the experimental

Curie point, ^we find excellent agreement ^between

,

the measured magnetization ^{curve and those com-}

puted ^{for u gr 2. The} significance of this result is

enhanced by ^the sensitivity ^{of the} calculated _mag-

netization to u. In addition, extremely goodg agreement exists between the measured intensities of the magnetic contributions to the nuclear dif- fraction peaks ^{at 50,OK and} 77 OK and those com-

puted ^{from the} spiral model. This detailed agree- ment is particularly impressive in view of the

inflexibility ^{of the} model: the fitting ^of theory ^to experiment ^{involved no variable} parameters.

Moreover, ^these findings demonstrate that the molecular-field approximation ^can reliably predict

the behavior of the k ⁼ 0 Fourier component ^{of a} spiral spin configuration.

Only ^the disappearance ^of sharp ^satellite peaks

at 31 OK disagrees ^with predictions ^{of the mole-}

cular-field calculation for the spiral model. How- ever, ^our results show that this loss of coherent

scattering ^{from the} ko -=/=- ^{0 Fourier} component ^is

not the result of a spiral-to-collinear transition.

An alternative explanation ^can be derived from the well-known failure of the molecular-field approxi-

mation to account adequately ^for correlations between the thermal fluctuations of neighboring spins. ^{In a} spiral structure, ^these correlations would have the effect of broadening ^{the satellite} peaks by introducing additional Fourier compo- nents with k N ka ^{even at} temperatures ^{well below}

the Curie point. ^{The above} interpretation is sup-

ported by ^the agreement ^{between the} intensity ^of

the liquid peak ^{at 50 OK and that} expected ^from

the satellites, ^{and also} by ^{the small remnance of}

this peak ^{in the} vicinities of the two lowest-angle

satellites as indicated in the diffraction pattern

of CoCr20 4 ^{at 4.2} OK. Thus it appears ^{that the} molecular-field approximation ^can yield good, ^or

poor, predictions ^{for finite} temperatures according

to whether the k ⁼ 0 component, ^{or the k ~ 0} component, ^{of a} spiral spin configuration ^{is invol-}

ved.

ACKNOWLEDGMENTS.

We should like to express ^our thanks to E. R. Whipple for his assis- tance in the preparation ^and analysis of the cobalt chromite sample, ^{to T. A.} Kaplan ^{for his} helpful discussions, ⁱⁿ particular concerning ^{the molecular}

field approximation, ^{to C. G. Shull, R. M.} Moon,

D. Murray ^{and W.} Phillips ^{for their} help ^{in the} experimental ^{work conducted at the M. I. T.}

Reactor, ^{and to J. B.} Goodenough ^{for his} support throughout ^{the course of this} investigation.

APPENDIX A

The general expression ^for the free energy of a

system ^is

where the probability operator p ^{is a} function of all the operators which appear in the Hamiltonian.

For the Heisenberg Hamiltonian, ^these operators

are the Si associated with each magnetic site in the lattice. The molecular field approximation ^re- places ^{the true} probability ^function by ^a product ^of independent, s-ngle-ion probabilit"es :

Then the problem ^{consists of} finding ^{that set}

of _pi which minimizes the free energy of _eq. (1).

The necessary condition that the free energy be

stationary with respect ^to arbitrary small variations of the functions _pi yields [19] ^{the familiar result}

that

where Zi is ^a normalizing constant, ~ _ (kT)~l,

and

Here the exchange integrals ^are positive ^for antiferromagnetic coupling.

Eqs. (3) ^and (4) ^{can be used to evaluate the}

definition Si > = tr . (Si pi) explicitly, giving

where Si represents ^the spin quantum ^{number for}

the ith ion. Eqs. (4) ^and (5) generate ^{a set of} coupled equations ^{whose solutions are} all statio- nary ^states of the molecular-field free energy, which ^can by virtue of the above equations ^be

written in the form

The desired solution consists of that stationary

state (set ^{of Si} > satisfying eqs. (4) ^and (5))

which minimizes FM, ^{and its} rigorous ^determi-

nation would require examination of all such sets of Si > - a formidable problem.

However, ^{it is} relatively simple ^to compare all the possible stationary ^states having ^a specified form, ^{such as that of a} ferrimagnetic spiral. ^For

the purpose ^of explicit calculation, ^{it is convenient}

to further define :

with

where F is a constant with the dimensions of energy and where the J’j ^are dimensionless ratios. (In

the case of a normal cubic spinel ^{with nearest} neighbor interactions, E ^{= 3SA} SB JAB and the J’;;

can be expressed ^{in terms of the} single exchange parameter [3].) Furthermore, ^{for a} ferrimagne-

tic spiral ^{we can write}

in the notation of Ref. 1. Llpon rewriting eq. (5)

in these terms, ^one finds that the magnitudes ^are given by ^the Brillouin functions [20]

while the directional properties require ^that

where and H’v ^{refer to the} magnitudes ^{of the}

axial and radial components ^of respectively.

Given k and the y’J, ^{the above} equations yield ^a unique ^{set of and and the} free energy of this

stationary ^{state can} be evaluated by substitution into _eq. (6). ^The requirement ^{that the} derivatives of the resulting expression ^with respect ^{to the} remaining variables, ^denoted generically by ~,

vanish gives ^{the final set of} equations,

Thus the f errimagnetic spiral which minimizes the molecular-field free _energy can be determined

by ^the simultaneous solutions of eqs. (9), (10) ^and (11).

REFERENCES

[1] ^ARNOTT (R. J.), ^Lincoln Laboratory Quarterly ^Pro-

gress Report, ^{Solid State} Research, July 15,1961.

[2] ^MENYUK (N.), ^WOLD (A.), ^ROGERS (D. B.) ^and

DWIGHT (K.), ^J. Appl. Phys., 1962, supp. 33, 1144.

[3] ^LYONS (D. H.), ^KAPLAN (T. A.), ^DWIGHT (K.) ^and

MENYUK (N.), Phys. Rev., 1962, 126, ^540.

[4] ^HASTINGS (J. M.) and CORLISS (L. M.), Phys. Rev., 1962, 126, ^556.

[5] ^WHIPPLE (E.) ^{and WOLD} (A.), ^J. Inorg. ^{and Nucl.}

Chem., 1962, 24, ^23.

[6] ^WATSON (R. E.) and FREEMAN (A. J.), ^Acta Cryst., 1961, 14, 27.

[7] ^BACON (G. E.), ^Neutron Diffraction, p. 31., ^Oxford University Press, London, 1962.

[8] ^BONGERS (P. F.), Thesis, ^{U. of} Leiden, ^1957.

[9] ^HASTINGS (J. M.) ^{and CORLISS} (L. M.), Phys. Rev., 1956, 104, 328.

[10] ^The molecular-field approximation ^is equivalent ^to

the requirement ^{that all the} spins ^{fluctuate inde-} pendently ^{of each other.}

[11] ^JACOBS (I. S.), ^J. Phys. ^Chem. Solids, 1960, 15, ^54.

[12] This calculation was based on the spin-only ^moments

of 3 _03BCB for the cobalt and chromium ions.

[13] ^{The fine structure indicated} by the 50 °K pattern

appears ^to undergo modification by ^{77 oK.}

[14] ^The intensities calculated for ^{u =} 2.08 disagree signi- ficantly ^{with our} experimental values, showing ^that

the exchange parameter ^{cannot be that} large.

[25] ^{At 4.2} oK, ^the discrepancies between the experi-

mental and calculated fundamental intensities were

all of the order of 4 ~Itotal, ^{which is} greater ^than

the probable statistical error.

[16] ^Collinear configurations ^{can be} stationary ^{states of}

the free energy only ^{when both} pairs ^{of B-sublat-}

tices possess ^{the same} average moment, ^{i.e. when} they ^{are Néel} configurations.

[17] ^The elimination of this hypothesis ^{is also} required by

the large discrepancy ^{below 50 °K between the} experimental magnetization curves ^of figure ^{4 and}

the predictions ^{of the} collinear model.

[18] ^{A similar} suggestion ^{has been made} by ^{J. M. HASTINGS}

and L. M. CORLISS in Ref. 4.

[19] ^KAPLAN (T. A.), ^Private communication. The classical

analogue ^{has been} given by ^KAPLAN (T. A.), Phys.

Rev., 1961, 124, 329, ^and by ^FREISER (M. J.), Phys.

Rev., 1961, 123, ^2003.

[20] (S, Z) ^{in our notations means} Bs(Z).