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>. <jpa00205822>
HAL Id: jpa-00205822
https://hal.archives-ouvertes.fr/jpa-00205822
Submitted on 1 Jan 1964
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528
rence
est donc due à des raisons
triques
déplacement
-
purement géomé-
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)
=
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 hélimagnétique permet de conclure que les interactions AB
sont prédominantes dans FeCrIS4 vis-à-vis des
interactions B-B, et d’autre part que les interactions négatives Cr-Cr décroissent fortement avec la
distance, l’ion S2- étant considérablement plus
grand que l’ion 02-.
3 j2.
Ce résultat n’est guère susceptible d’amélioration, la précision sur S(Fe) et S(Cr) étant respectivement ~ 0,12 et -~ 0,15 et le facteur de corrélation étant très voisin de l’unité.
Note
accord
Nos résultats sont en
~, la correction.
de SHIRANE (C.), Cox (R. E.) et PICKART
Conférence on magnetism, Atlantic City, nov. 1963,
ajoutée
-
avec ceux
(S, J.),
J. -,,4ppl. Phys.. 964, 35,
et
95IL.
BIBLIOGRAPHIE
[1]
[2]
LOTGERING (F. K.), Philips Research Reports, 1956, 11,
218-249.
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
Lincoln
By N. MENYUK, K. DWIGHT and A. WOLD (1),
Massachusetts Institute of Technology, Lexington 73, Massachusetts,
(2),
Laboratory
U. S. A.
Le chromite de cobalt CoCr2O4 est un spinelle cubique ferrimagnetique aux basses
Résumé.
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 magnétique, 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 diagramme 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
2014
selon [001].
On sait que la
configuration
de
spirale ferrimagnétique
JBB SB/JAB SA &#x3E; 0,98
devient instable pour
(c’est-à-dire pour JBB/JAB &#x3E; 0,98 dans CoCr2O4). Notre résultat indique cependant que la configuration 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 magnétiques 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.
with support from the U. S. Army, Navy and Air Force.
(S) Operated
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01964002505052801
529
Cobalt chromite, CoCr2O4, is a cubic spinel which is ferrimagnetic at low tempeAbstract.
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 parameter 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 proposed 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 diffraction pattern predicted on the basis of the spiral model is completely determined upon fixing
1. 5 on the basis of
the exchange ratio JBB/JAB and the cone axis direction. Taking JBB /JAB
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 &#x3E; 0.98 (i.e. JBB/JAB &#x3E; 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 corroborate the validity of the spiral model for CoCr2O4, and indicate that the molecular field approximation accurately describes the axial component throughout its ferrimagnetic range, but not the
2014
=
azimuthal component.
Cobalt chromite is a cubic
1. Introduction.
spinel with cell length ao 8.332 A [1]. A study
of its magnetic properties has shown it to be ferri97 OK, and a
magnetic with a Curie point
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 temperature 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 ferrimagnetic conical spiral configuration of the type
predicted for cubic spinels by Lyons, Kaplan,
Dwight and Menyuk (LKDM) [3], and first observed by Hastings and Corliss [4]. This interpretation uniquely determines the exchange parameter and direction of easy magnetization.
An unambiguous prediction of the magnetic properties of this material as a function of temperature was obtained from a molecular field calculation, 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 disagreements in others, indicating that the molecularfield approximation yields accurate predictions of
certain magnetic properties, but not of others.
The disagreements are shown to be due to correlation effects, which can apparently play an important role in materials with complicated spin confi-
=
gurations.
II. Low
A. EXPEsamples of cobalt
temperature configuration.
RIMENTAL RESULTS.
--
Powdered
-
chromite can best be prepared from the precursor
COCrO.4CSHN, as described by Whipple and
Wold [5], since the ignition of this complex molecule results in a very finely divided, highly reactive
oxide. However, this precursor is usually found
to be deficient in chromium. Hence, after determining 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 remainder. 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
The neutron wavelength was
an aluminum tube.
1.196 A. For simplicity, all the tables and curves
are normalized to a monitor count of 600,000 neutrons (~ 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 computer which was programmed to normalize to the
530
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 additional magnetic peaks are designated as " satellites ", and the magnetic contributions to the
nuclear peaks are called " fundamentals ".
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 parameter 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 experimental 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.
=
pattern of CoCr2o4’
satellite peak positions are indicated by vertical indices, the fundamental peak positions are indexed
horizontally. The aluminum peaks are produced by
specimen holder.
FIG. 2.
=
TABLE I
COMPARISON
OF
EXPERIMENTAL
AND CALCULATED INTEGRATED NUCLEAR PEAK INTENSITIES
-
4.2 oK neutron diffraction
Magnetic
,
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 ferrimagnetic 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 satellite 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 configuration is completely characterized to within a
rotational degeneracy by a single parameter u,
=
=
where
(1) Corrected for Debye- "’Taller temperatur.e factor.
.38 707.
(z) Normal spinel, oxygen parameter
=
(1)
531
TABLE II
COMPARISON
OF
CALCULATED
(~)
AND EXPERIMENTAL PEAK INTENSITIES AT
~t.2 ~~1
Satellite peak locations for ferromagnetic spiral
fiic. 3.
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 respectively. In a real material the rotation degeneracy
is lifted by anisotropy effects to establish a particular 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
2 in this material [2]. Choosing this
that u
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 comparison of experimental intensities with the values
predicted by the above ferrimagnetic spiral configuration with net magnetization in the [001] direction is given in Table I I. The magnetic intensities
listed are absolute intensities, based on the instrument normalization factor obtained from the room
=
temperature pattern.
It should be noted that every predicted peak is
observed and, conversely, there are no peaks present which cannot be accounted for by the ferrimagnetic 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 fundamental peak intensities were uniformly high by a
P2
(1) u = 2.0, Cone angles : Oi
(P6 900.
1500 ; (Di
1&#x3E;3
~4
Phase angles : 11 = Y2
Y3
Y4 ~ 0 ;
Net magnetization direction along [001].
-
=
=
=
32°;
=
=
=
Y5
=
P6
=
w
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 structure is locally stable over a range of values extending from the boundary of Néel mode stability
(u
8/9) to u N 1.3. For u &#x3E; 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
Under these
more complex configuration below.
-
=
532
circumstances, discrepancies between the calculated and experimental patterns are to be expected,
such as the possible existence of non-spiral peaks
or deviations from the predicted magnetic intensities of the observed spiral peaks. However,
although some descrepancies do exist, as discussed
below, the most striking feature of the experimental 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 calculated 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 relatively high fundamental peak intensities and the
extra peaks observed in Mn[Cr2]O4 cannot satisfactorily be accounted for by merely noting that
u
1. 6 &#x3E; 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 phenomena not considered by LKDM may vary significantly in these two materials. Amongst these
are the near-neighbor A-A and next-near-neighbor
B-B interactions, which might give rise to observable departures from theoretical predictions, such
H owas the wavelength discrepancy in Co[Cr2]04.
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
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
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 magnitudes of the average magnetic moments for each of
the three pairs of sublattices, as well as the magnitude of the propagation vector k, can all be expected to vary continuously with temperature. However, it can be argued from symmetry considerations 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 numerically 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
-
CALCULATION.
-
=
spin magnitudes at forty temperatures, equally
for several
spaced between absolute zero and
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
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
u N 2.
transition from
This behavior
represents
low-temperature ferrimagnetic
spiral to a high-temperature collinear (Néel) configuration. It is worth noting that the length of the
propagation vector at this transition is identical
a
a
with the value
3.647 ~/2 which destabilizes
the Néel configuration in the LKDM ground-state
calculation, independently of u.
=
533
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
This low-field curve is approxicurve in figure 4.
worse agreement with the neutron diffraction results described in the next section. Funthermore, the displacement between our high-field
data and the points calculated for u
2.03 is
consistent with the high-field susceptibility determined for MnCr204 [11].
Thus, the molecular-field calculation for
2.03 gives extremely good agreement with
u
the spontaneous magnetization deduced from the
experimental data shown in figure 1. This calculation predicts a transition from the low-temperature 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 instability of the spiral configuration at such low temperatures. Although direct substantiation of this
tinctly
=
==
=
hypothesis would be desirable, the necessary calculation of the stability of the molecular-field free
energy is beyond the scope of this paper. Nevertheless, such instability has been predicted for the
ground state [3].
4.
Magnetization of cobalt chromite as a function
temperature. The experimental data are shown as
curves ; the values calculated for a spiral model according to the molecular field theory are given as points.
--
of
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 spontaneous
magnetization. Although the magnetization 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 considerations,
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
ratures, whereas those for
curve
r~
=
at low
2.08
tempegive dis-
Since
C. NEUTRON DIFFRACTION PATTERNS.
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 intermediate 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 temperature ~’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 temperature 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. However, this explanation not only contradicts our
analysis of the magnetization data, but also fails to
account for the persistence of considerable shortrange order (as indicated by the sizable liquid
peak) up to at least 77 oK.
-
=
=
534
In addition to the liquid peak, the pattern shown
in figure 5 includes appreciable magnetic contributions to the intensities of the nuclear peaks,
and IV for the collinear model. The serious discrepancy 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, substantiates the existence of a spiral-to-collinear transition at some higher temperatures
86 OK
as predicted by the spiral model).
TABLE IV
COMPARISON
AND
OF EXPERIMENTAL
NUCLEAR PEAK INTENSITIES ~I
COBALT CHROMITE AT 77 OK
CALCULATED
FOR
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.
are given in Table III.
The corresponding
magnetic intensities calculated from our molecularfield treatment of a [110] spiral model (with u = 2
and the cone axes in the [001] direction, as established by the 4. 2 ~K measurements) are also given
in Table III [14]. Here the agreement between
theory and experiment is exceptionally good, being
which
(1)
COMPARISON
FOR
OF
COBALT
CHROMITE
AT
INTENSITIES
50 °K
The measurements at 4.2 OK
D. DISCUSSION.
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 molecularfield calculation normalized to the experimental
Curie point, we find excellent agreement between
the measured magnetization curve and those computed for u gr 2. The significance of this result is
enhanced by the sensitivity of the calculated mag-
(1) Taken
from the
top
of the
liquid peak.
We’* have also considered the case of collinear
in the molecular-field approximation [16].
For the predetermined value u
2, this calculation yields the intensities shown in Tables III
spins
=
liquid peak.
for the combined theoretical intensities of the two
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 longrange azimuthal coherence of a conical configuration [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.
EXPERIMENTAL
NUCLEAR PEAK
of the
lowest-angle
TABLE III
CALCULATED
top
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 configuration. In this connection, it appears significant that the total intensity of the liquid peak in
figure 5 is approximately 4 070, compared to 4 224
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].
AND
Taken from the
,
535
netization to u. In addition, extremely goodg
agreement exists between the measured intensities
of the magnetic contributions to the nuclear dif-
are the Si associated with each magnetic site in the
lattice. The molecular field approximation replaces the true probability function by a product of
peaks at 50,OK and 77 OK and those comfrom the spiral model. This detailed agreement is particularly impressive in view of the
inflexibility of the model: the fitting of theory to
independent, s-ngle-ion probabilit"es :
fraction
puted
experiment
involved
variable parameters.
demonstrate that the
no
Moreover, these findings
molecular-field
approximation
the behavior of the k
=
can
0 Fourier
reliably predict
component of
a
spiral spin configuration.
Only the disappearance of sharp satellite peaks
at 31 OK disagrees with predictions of the molecular-field calculation for the spiral model. Howresults show that this loss of coherent
ever,
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 approximation 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 components with k N ka even at temperatures well below
the Curie point. The above interpretation is supported 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
of
a
component,
spiral spin configuration is involved.
our
=
ACKNOWLEDGMENTS.
We should like to
express our thanks to E. R. Whipple for his assistance 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
system
general expression
for the free energy of
a
is
where the probability operator p is a function of
all the operators which appear in the Hamiltonian.
For the Heisenberg Hamiltonian, these operators
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
and
a
normalizing constant, ~ _ (kT)~l,
Here the exchange integrals are positive for
antiferromagnetic coupling.
Eqs. (3) and (4) can be used to evaluate the
definition
Si &#x3E; = 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 stationary 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
Si &#x3E; satisfying eqs. (4) and (5))
(set of
which minimizes FM, and its rigorous determination would require examination of all such sets
of
Si &#x3E; - 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 :
state
.
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 ferrimagnetic spiral we can write
=
536
in the notation of Ref. 1. Llpon rewriting eq.
in these terms, one finds that the magnitudes
given by the Brillouin functions [20]
while the directional
(5)
are
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,
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
and the free energy of this
unique set of and
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
ARNOTT (R. J.), Lincoln Laboratory Quarterly Progress 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.,
[14] The intensities calculated for
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
[16]
[1]
requirement that all the spins fluctuate independently of each other.
[11] JACOBS (I. S.), J. Phys. Chem. Solids, 1960, 15, 54.
[12] This calculation was based on the spin-only moments
the
[13]
of 3 03BCB for the cobalt and chromium ions.
The fine structure indicated by the 50 °K
appears to undergo modification by 77 oK.
pattern
u
2.08 disagree significantly with our experimental values, showing that
the exchange parameter cannot be that large.
[25] At 4.2 oK, the discrepancies between the experi-
[17]
[18]
[19]
=
mental and calculated fundamental intensities were
all of the order of 4 ~Itotal, which is greater than
the probable statistical error.
Collinear configurations can be stationary states of
the free energy only when both pairs of B-sublattices possess the same average moment, i.e. when
they are Néel configurations.
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.
A similar suggestion has been made by J. M. HASTINGS
and L. M. CORLISS in Ref. 4.
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).