PHYSICAL REVIEW B 67, 224422 共2003兲

Weak ferrimagnetism, compensation point, and magnetization reversal in Ni„HCOO… 2 "2H2 O
H. Kageyama,1 D. I. Khomskii,2 R. Z. Levitin,3 and A. N. Vasil’ev1,*
1

Materials Design and Characterization Laboratory, Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,
Chiba 277-8581, Japan
2
Solid State Physics Laboratory, Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
3
Physics Faculty, Moscow State University, Moscow 119992, Russia
共Received 28 October 2002; revised manuscript received 13 March 2003; published 18 June 2003兲
The nickel 共II兲 formate dihydrate Ni(HCOO) 2 •2H2 O shows a peculiar magnetic response at T⬍T N⫽15.5 K. The magnitude of the weak magnetic moment increases initially below T N , is zero at T * ⫽8.5 K,
and increases again at lowered temperature. The sign of low-field magnetization at any given temperature isdetermined by the sample’s magnetic prehistory and the signs are opposite from each other at T⬍T * andT * ⬍T⬍T N . This behavior suggests that Ni(HCOO) 2 •2H2 O is a weak ferrimagnet and T * is a compensation
point. The magnetic properties of Ni(HCOO) 2 •2H2 O can be understood by taking into account the features of
its crystal structure. There are two nonequivalent Ni sites in the monoclinic P2 1 /c structure of this compound,forming two magnetic subsystems. Each of these subsystems is weakly ferromagnetic due to the canting ofpredominantly antiferromagnetic sublattices. The weak ferromagnetic moments of these subsystems are oriented opposite from each other, resulting in weak ferrimagnetism. The model proposed explains the mainfeatures of the experimentally observed phenomena.
DOI: 10.1103/PhysRevB.67.224422

PACS number共s兲: 75.30.⫺m, 75.50.⫺y, 75.60.⫺d

I. INTRODUCTION

The phenomenon of ferrimagnetism is associated with apartial cancellation of antiferromagnetically aligned magnetic sublattices with different values of magnetic moments
and/or different temperature dependencies of magnetization.
It has been well studied in ferrites1,2 and recently in various
inorganic3,4 and molecular transition-metal complexes.5–10
Ferrimagnetic behavior is observed frequently in compounds
containing different magnetic ions. It can also be seen inmaterials containing only one type of magnetic ions but witheither different valence states or different crystallographic
positions.11,12 In the latter case the origin of ferrimagnetism
lies in the difference of molecular fields acting on nonequivalent magnetic sites.
In some ferrimagnets 共according to the Néel’s classification, it is N-type ferrimagnets13兲 the total magnetization of a
substance becomes zero at a certain compensation temperature T * . Both above and below this temperature the magnetization of different sublattices prevails. In this case magnetization reversal can be observed. In weak magnetic field
共less than the field of coercivity兲 the magnetization changes
sign at the compensation temperature. The metastable ‘‘diamagnetic’’ state at a certain temperature range can be fixedby magnetocrystalline anisotropy.2
Much rarer is the observation of magnetization reversal incanted antiferromagnets, where the weak ferromagnetic moment is due to the tilting of sublattice magnetizations. The
single-ion magnetic anisotropy itself can fix the directions ofsublattice magnetizations tilted from antiparallel alignment.14
On other hand, the canting of antiferromagnetic sublattices
can be due to antisymmetric Dzyaloshinsky-Moriya
interaction,15–17 which favors perpendicular orientations of
the sublattice magnetizations. Very few canted antiferromagnets show compensation point and magnetization reversal.
These phenomena were observed in YVO3 single crystals
0163-1829/2003/67共22兲/224422共7兲/$20.00

and were treated on the basis of competition between the
single-ion
anisotropy
andDzyaloshinsky-Moriya
interaction.18,19 In orthoferrites Y(FeCr)O3 the ferrimagneticlike behavior and compensation point were ascribed tocompetition of the Dzyaloshinsky-Moriya interactions between different magnetic ions.20
Herein, we report the observation of weak ferrimagnetism, compensation point, and magnetization reversal in
the nickel 共II兲 formate dihydrate Ni(HCOO) 2 •2H2 O. The
magnetic and thermal properties of isostructural dihydrated
formates of transition metals were extensively studied in the
1970s. The interest was due to the specific layered arrangement of metal ions in the crystal structure of these compounds. Within layers the metal ions are either linked byformate groups or they are not. This alternate stacking of
coupled and uncoupled metal ions suggested the reduced dimensionality of the magnetic subsystem. The interlayer interaction through the formate groups was taken into accountalso, but was considered to be much weaker than the intralayer one. The predominantly antiferromagnetic ordering at
Néel temperature T N ⫽3.72 K in Mn(HCOO) 2 •2H2 O, 21–24
at T N ⫽3.74 K in Fe(HCOO) 2 •2H2 O, 24 –27 at T N ⫽5.1 K in
Co(HCOO) 2 •2H2 O, 24 and at T N ⫽15.5 K in Ni(HCOO) 2•2H2 O 共Refs. 24,25, and 27兲 was established. In the Nibased formate long-range ordering was suggested to occur
only within strongly coupled layers at T N , while the metal
ions in neighboring layers were considered to remain essentially independent even at low temperatures.25 It wassuggested24 that the metal ions within magnetically coupledlayers were set into canted antiferromagnetic states below
T N . It was argued,27 however, that either ferrimagnetism or
weak ferromagnetism of the canted antiferromagnet could be
responsible for the spontaneous moment at low temperatures.
Either of these situations is compatible with the crystal structure of dihydrated formates but the origin of the spontaneous
moment in Mn- and Fe-based formates remains unclear. It

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PHYSICAL REVIEW B 67, 224422 共2003兲

KAGEYAMA, KHOMSKII, LEVITIN, AND VASIL’EV

FIG. 1. The crystal structure of Ni(HCOO) 2 •2H2 O in polyhedral representation. The octahedrons NiO6 belonging to Ni1 and Ni2
subsystems are of light and dark shades, respectively. Note thateach subsystem contains two kinds of octahedrons tilted with respect to each other. The dark circles denote the C ions, while the Hions are not shown for clarity. The solid lines indicate the exchangeinteraction links between Ni ions.

was not established explicitly whether the Ni-based formate
possesses a nonzero moment below T N .
This paper is organized in the following manner. First, thecrystal structure of Ni(HCOO) 2 •2H2 O is described. Then,the results of magnetization are presented and it is concluded
that this compound is a weak ferrimagnet. A model of themagnetic structure of nickel formate dihydrate is suggested
and a molecular-field theory for this model is given. A comparison of the results of calculation with experimental data
and a discussion are provided.
II. CRYSTAL STRUCTURE

Ni(HCOO) 2 •2H2 O crystallizes in the monoclinic P2 1 /cspace group and includes 4 f.u. in the unit cell with a⫽0.860 nm, b⫽0.706 nm, c⫽0.921 nm, and ␤ ⫽96.5°. 22 It
is of light blue color and presumably insulating. The structure, shown in Fig. 1, contains two kinds of Ni2⫹ ions: theNi1 with six oxygen atoms of different formate ions as nearest neighbors, the Ni2 with four water molecules, and two
formate oxygen atoms coordinated. In both Ni1 and Ni2 subsystems the octahedrons coordinating the Ni ions are tilted
with respect to each other. The fourfold axes of NiO6 octahedrons are oriented differently with respect to the crystal
lattice axes. In the Ni1 subsystem the fourfold axes lie
mainly in the a-b crystal plane and are tilted for ␺ 1 /2
⫽22.5° in opposite directions from the a axis. In the Ni2
subsystem the fourfold axes also lie mainly in the a-b plane
and are tilted for ␺ 2 /2⫽57.5° from the a axis. The slightdeviations of NiO6 fourfold axes from the a-b plane will betaken into account below.

FIG. 2. The temperature dependencies of magnetization ofNi(HCOO) 2 •2H2 O at H⫽0.01 T taken in zero-field-cooling andfield-cooling regimes.

range up to 5 T. X-ray analysis confirmed the single phase of
the title compound in the measured sample.
The temperature dependencies of magnetization of theNi(HCOO) 2 •2H2 O powder sample taken in zero-fieldcooling 共ZFC兲 and field-cooling 共FC兲 regimes at H⫽0.01 T
are shown in Fig. 2. At heating in the ZFC regime the sample
shows first a large ‘‘paramagnetic’’ response at low temperatures, then the magnetic moment gradually decreases withincreasing temperature, changes to ‘‘diamagnetic’’ at T *⫽8.5 K, and once again becomes paramagnetic above 15.5
K. At subsequent cooling in the FC regime the magnetic
behavior of the sample at low temperatures appears to be
mirrorlike with respect to magnetization sign as comparedwith ZFC measurements. The magnetization reversal was observed only at low magnetic field. At higher fields the magnetic moment monotonously increases at lowering temperature, showing only weak singularity at 15.5 K. This is incomplete accordance with earlier observations.25
The M vs H curves in the range 2–15 K are shown in Fig.
3. These curves are ferromagneticlike, i.e., they show a spontaneous magnetic moment. The temperature dependence ofthe spontaneous magnetic moment of Ni(HCOO) 2 •2H2 O isshown in Fig. 4. The saturation magnetization at low temperatures is two orders-of-magnitude smaller than that corresponding to the parallel alignment of Ni2⫹ magnetic moments. The ferromagneticlike character of the curves shown
is supported by hysteretic behavior of magnetization. The

III. EXPERIMENT

Magnetic properties of commercially available 共Waco
Company兲 powder samples of Ni(HCOO) 2 •2H2 O were
measured by the Quantum Design superconducting quantum
interference device magnetometer in a 2–300-K temperature

FIG. 3. The magnetization curves of Ni(HCOO) 2 •2H2 O. The
inset shows theoretical 共solid line兲 and experimental (〫) M vs H
dependencies at T⫽2 K.

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WEAK FERRIMAGNETISM, COMPENSATION POINT, . . .

FIG. 4. The experimental (䊊) and theoretical 共solid line兲 temperature dependencies of the spontaneous magnetic moment ofNi(HCOO) 2 •2H2 O.

magnetization loops of Ni(HCOO) 2 •2H2 O taken at several
temperatures are shown in Fig. 5, and the low-field region isenlarged in Fig. 6. Evidently, the remanent magnetization
decreases with increasing temperature, is close to zero in
vicinity of T * , increases again at increasing temperature, andvanishes above 15.5 K.
Therefore, the experimental data presented suggest that
below T N ⫽15.5 K a weakly ferrimagnetic state inNi(HCOO) 2 •2H2 O is realized, and T * ⫽8.5 K is a compensation temperature. The temperature dependence of highfield magnetic susceptibility ␹ in the magnetically orderedstate obtained from the slope of magnetization curves at H
⬃1 T is shown in Fig. 7. This dependence is quite unusual
for magnetically ordered compounds, i.e., ␹ increases at decreasing temperature.
The paramagnetic susceptibility ␹ para of Ni(HCOO) 2•2H2 O was measured in a wide range from the Néel point toroom temperature. At high temperatures ␹ para follows theCurie-Weiss law with paramagnetic Curie temperature ⌰⫽⫺15.5 K and an effective magnetic moment ␮ e f f⫽3.14␮ B . An experimental value of ␮ e f f is a typical one for
Ni2⫹ ions in various compounds and exceeds the spin only
part 2.82␮ B . This means that the orbital moment is not completely quenched in a crystal field. A negative value of ⌰implies that the main exchange interaction in Ni(HCOO) 2

FIG. 5. The magnetization loops of Ni(HCOO) 2 •2H2 O taken atvarious temperatures at T⬍T N .

FIG. 6. The low-field regions of hysteresis loops ofNi(HCOO) 2 •2H2 O taken at various temperatures at T⬍T N .

•2H2 O is antiferromagnetic. When approaching T N , the␹ para deviates from the Curie-Weiss law as shown in Fig. 8.
IV. MODELThe crystal structure considerations indicate that the superexchange pathways between Ni ions can be realized
through Ni–O–C–O–Ni links. As noted earlier, both Ni1and Ni2 subsystems contain two kinds of Ni ions with different orientations of octahedral environments. Consequently, it is natural to represent each Ni subsystem as consisting of two sublattices, namely, Ni1⬘, Ni1⬙ and Ni2⬘, Ni2⬙.
The superexchange Ni–O–C–O–Ni links exist between Ni1⬘
and Ni1⬙ and also between Ni1⬘ and Ni2⬘, and Ni1⬙ and Ni2⬙.
Therefore, each Ni1⬘ ion interacts with four-nearestneighboring Ni1⬙ ions and two-nearest-neighboring Ni2⬘ ions.
The same is true for Ni1⬙ ions. There is no Ni–O–C–O–Nisuperexchange pathway within the Ni2 subsystem. Thismeans that in the absence of Ni1 - Ni2 exchange interactions
the Ni2 subsystem can be considered paramagnetic. The
same exchange interactions pattern was used to analyze themagnetic properties of Ni(HCOO) 2 •2H2 O in Ref. 24.
Within Ni–O–C–O–Ni links the angles of superexchangeinteraction lie in the range 122°⫺137°, suggesting the predominance of antiferromagnetic interaction.28,29 The param-

FIG. 7. The experimental 共line⫹symbol兲 and theoretical 共solidline兲 temperature dependencies of high-field magnetic susceptibilityof Ni(HCOO) 2 •2H2 O at T⬍T N .

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FIG. 8. The inverse paramagnetic susceptibility ofNi(HCOO) 2 •2H2 O at T⬎T N . The experimental data are shown by
symbols. The solid line represents the Curie-Weiss law. The dottedline represents the results of calculation by Eq. 共11兲.

eters of different pathways, i.e., angles and interatomic distances, are close to each other. In the first approximation theexchange interactions through these links can be consideredequal. This approximation significantly simplifies the description of magnetic properties of Ni(HCOO) 2 •2H2 O, but
is not critical. The magnetic structure corresponding to thepattern of exchange interactions described is shown in Fig. 9.
The superexchange pathways are shown by solid lines. This
structure is drawn in simplest exchange approximation anddoes not take into account the crystal-field effects leading to
the single-ion anisotropy. The crystal-field effects fix the orientations of different Ni magnetic moments in different directions with respect to crystallographic axes and result in
tilting of these moments from antiparallel alignment. The
model suggested is oversimplified in the sense that it assumes a unique exchange interaction parameter within themagnetic system of Ni(HCOO) 2 •2H2 O. Another assumption is that the principal axes of NiO6 octahedrons lie in thea-b plane and the model neglects their slight deviations from

FIG. 10. The model of the weak ferrimagnetic structure ofNi(HCOO) 2 •2H2 O. The magnetic moments M 1 ⬘ , M 1 ⬙ , M 2 ⬘ , andM 2 ⬙ of different Ni ions are assumed to lie in the a-b plane. The
dotted lines show the orientations of the principal axes 共easy directions of magnetization兲 Z 1 ⬘ , Z 1 ⬙ , Z 2 ⬘ , and Z 2 ⬙ of different NiO6octahedrons. The angles between easy directions of magnetizationand the a axis are ␺ 1 ⬘ /2⫽ ␺ 1 ⬙ /2⫽ ␺ 1 /2 and ␺ 2 ⬘ /2⫽ ␺ 2 ⬙ /2⫽ ␺ 2 /2.
The deviations from antiparallel alignment of magnetic moments
within Ni1 and Ni2 subsystems are designated as ␧ and ␦ , correspondingly. The angles ␸ 1 ⬘ ⫽ ␸ 1 ⬙ ⫽ ␸ 1 and ␸ 2 ⬘ ⫽ ␸ 2 ⬙ ⫽ ␸ 2 give thedeviations of corresponding magnetic moments from easy directions of magnetization.

this plane. The model does not take into account an antisymmetric Dzyaloshinsky - Moriya interaction, which can leadalso to weak ferromagnetism.
The properties of Ni(HCOO) 2 •2H2 O in the framework of
the model suggested above can be described using the following expression for free energy:
E⫽4␭M1⬘ M1⬙ ⫹2␭ 共 M1⬘ M2⬘ ⫹M1⬙ M2⬙ 兲 ⫹K Ni1 sin2 ␸ 1 ⬘⫹K Ni1 sin2 ␸ 1 ⬙ ⫹K Ni2 sin2 ␸ 2 ⬘ ⫹K Ni2 sin2 ␸ 2 ⬙⫺H共 M1⬘ ⫹M1⬙ ⫹M2⬘ ⫹M2⬙ 兲 ,FIG. 9. The proposed magnetic structure of Ni(HCOO) 2•2H2 O. The pathways of exchange interaction are shown by solidlines. The Ni ions belonging to two different sublattices within bothNi1 and Ni2 subsystems are represented by four different symbols.

共1兲

where ␭⫽Ig 2 ␮ B2 is the molecular-field coefficient, I is theexchange interaction parameter, g is the g factor, and 兩 M1⬘ 兩⫽ 兩 M1⬙ 兩 ⫽M 1 and 兩 M2⬘ 兩 ⫽ 兩 M2⬙ 兩 ⫽M 2 are the magnetic moments of Ni1 and Ni2 sublattices. The first term in Eq. 共1兲
describes the exchange interactions within the Ni1 subsystem, and the second term describes the exchange interactions between Ni1 and Ni2 subsystems. The next four terms
describe the energy of magnetic anisotropy. The K Ni1 andK Ni2 are constants of the single-ion magnetic anisotropy ofdifferent Ni subsystems. The angles ␸ 1 ⬘ ⫽ ␸ 1 ⬙ ⫽ ␸ 1 and ␸ 2 ⬘⫽ ␸ 2 ⬙ ⫽ ␸ 2 are the angles between the axes of anisotropy andmagnetic moments of corresponding Ni ions 共see Fig. 10兲. It
is natural to assume that the axes of anisotropy coincide withthe principal axes of NiO6 octahedrons. The last term in Eq.
共1兲 is the Zeeman energy.

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WEAK FERRIMAGNETISM, COMPENSATION POINT, . . .

The minimization of Eq. 共1兲 with respect to ␸ 1 and ␸ 2angles gives the equilibrium orientations of Ni magnetic moments. The type of the resulting magnetic structure depends
on the signs and values of K Ni1 and K Ni2 . At K Ni1⬎0 and
K Ni2⬍0 the Ni magnetic moments lie in the a-b plane asshown in Fig. 9. According to this arrangement each Ni subsystem is weakly ferromagnetic and these weakly ferromagnetic moments are oriented antiparallel to each other beingaligned along the b axis. Therefore, the resulting magnetic
structure of Ni(HCOO) 2 •2H2 O is a weakly ferrimagnetic
one. The reversal of K Ni1 and K Ni2 signs results in a 90°

␧⫽⫺

␦⫽

rotation of the magnetic structure in the a-b
plane. Below we consider a magnetic structure shown in
Fig. 10.
Assuming that the crystal-field energy for Ni ions is muchsmaller than the energy of the exchange interaction, deviations from the antiparallel alignment will be small. In Ni1
and Ni2 subsystems these deviations will be denoted ␧ and␦ , correspondingly 共see Fig. 10兲. The minimization of freeenergy given by Eq. 共1兲 at H⫽0 allows to obtain the deviations of magnetic moments ␧ and ␦ from antiparallel alignment within both Ni1 and Ni2 subsystems:

m 共 q 1 sin ␺ 1 ⫺q 2 sin ␺ 2 兲 ⫹2q 1 q 2 sin ␺ 1 cos ␺ 2,
2 关 m 共 2⫹q 1 cos ␺ 1 ⫹q 2 cos ␺ 2 兲 ⫹2q 2 共 2 cos ␺ 2 ⫹q 1 cos ␺ 1 cos ␺ 2 兲兴m 共 q 1 sin ␺ 1 ⫺q 2 sin ␺ 2 兲 ⫹2q 2 共 2 sin ␺ 2 ⫹q 1 cos ␺ 1 cos ␺ 2 兲.
2 关 m 共 2⫹q 1 cos ␺ 1 ⫹q 2 cos ␺ 2 兲 ⫹2q 2 共 2 cos ␺ 2 ⫹q 1 cos ␺ 1 cos ␺ 2 兲兴

The angles ␺ 1 and ␺ 2 are the angles between axes ofanisotropy within Ni1 and Ni2 subsystems, q 1 ⫽K Ni1/2IM 21 ,q 2 ⫽K Ni2/2IM 21 , and m⫽M 1 /M 2 . The resulting weak ferrimagnetic moment in this model is

M total ⫽M total ⫽M 1 共 ␧⫹m ␦ 兲 /2

共2兲
共3兲

共4兲

or taking into account the expressions for ␧ and ␦ ,M 1 m 2 共 q 1 sin ␺ 1 ⫺q 2 sin ␺ 2 兲
4 关 m 共 2⫹q 1 cos ␺ 1 ⫹q 2 cos ␺ 2 兲 ⫹2q 2 共 2 cos ␺ 2 ⫹q 1 cos ␺ 1 cos ␺ 2 兲兴⫺

M 1 m 共 q 1 sin ␺ 1 ⫹3q 2 sin ␺ 2 ⫹2q 1 q 2 cos ␺ 1 cos ␺ 2 兲
4 关 m 共 2⫹q 1 cos ␺ 1 ⫹q 2 cos ␺ 2 兲 ⫹2q 2 共 2 cos ␺ 2 ⫹q 1 cos ␺ 1 cos ␺ 2 兲兴⫺

M 1 2q 1 q 2 sin ␺ 1 sin ␺ 2.
4 关 m 共 2⫹q 1 cos ␺ 1 ⫹q 2 cos ␺ 2 兲 ⫹2q 2 共 2 cos ␺ 2 ⫹q 1 cos ␺ 1 cos ␺ 2 兲兴This formula describes the evolution of the weak magnetic moment of Ni(HCOO) 2 •2H2 O with temperature. To
calculate the temperature dependence of M total , the M 1 (T)and M 2 (T) dependencies as well as K Ni1(T) and K Ni2(T)dependencies should be used. The temperature dependencies
of molar sublattice magnetizations M 1 and M 2 can be calculated within the exchange approximation through the corresponding Brillouin functions B S ,M 1 ⫽gS ␮ B N A B S 关 gS ␮ B I 共 4M 1 ⫹2M 2 兲 /k B T 兴 ,共6兲

M 2 ⫽gS ␮ B N A B S 关 gS ␮ B I 共 2M 1 兲 /k B T 兴 ,共7兲

where S⫽1 is a spin of Ni2⫹ ions and N A is Avogadro’s
number.
The constants of single-ion anisotropy change with temperature as follows:30

共5兲

K Ni1 共 T 兲 ⫽K Ni1 共 0 兲 Î 5/2兵 L ⫺1 关 M 1 共 T 兲兴 /M 共 0 兲 其 ,共8兲

K Ni2 共 T 兲 ⫽K Ni2 共 0 兲 Î 5/2兵 L ⫺1 关 M 2 共 T 兲兴 /M 共 0 兲 其 ,共9兲

where Î 5/2 is a normalized Bessel function; L ⫺1 is an inverse
Langevin function; K Ni1(0), K Ni2(0), and M (0) are singleion anisotropy constants; and the magnetic moment at T
⫽0 K.
Equations 共6兲 and 共7兲 give the following formulas for theNéel temperature:
T N ⫽2CI 共 冑2⫹1 兲 ,共10兲

where C⫽g 2 S(S⫹1)N A ␮ B2 /3k B .
The paramagnetic susceptibility above the Néel point T N
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␹ para ⫽C
T⫹4CI⫺4C 2 I 2 /T

.

共11兲

Clearly, the ␹ para deviates from the Curie-Weiss law atlowering temperature, reflecting the existence of two different magnetic subsystems in Ni(HCOO) 2 •2H2 O. Equation
共11兲 represents a simplified expression for paramagnetic susceptibility of the ferrimagnet 共Néel’s law兲. The simplificationarises from the facts that both magnetic subsystems are constituted by the same ions and the exchange interaction within
one of them is absent. The inverse paramagnetic susceptibility becomes zero at⌰ ⬘ ⫽2CI 共 冑2⫺1 兲 .

共12兲In exchange approximation it is possible also to calculate
the molar transverse magnetic susceptibility in the magnetically ordered state, i.e., below the Néel point,␹ anti f ⫽N A 共 1⫹m 兲 2
.
16I

共13兲

Note that the temperature dependence of ␹ anti f is nottrivial. It increases at lowering temperature due to the factthat the Ni2 subsystem itself is paramagnetic in the absenceof exchange interaction with the Ni1 subsystem.
V. DISCUSSION

The experimental data available for Ni(HCOO) 2 •2H2 O
can be compared with results of analytical and numericalcalculations within the model proposed. Equation 10 allows
to estimate the exchange interaction coefficient I⫽3.9 K
from the value of the Néel temperature T N ⫽15.5 K. 31 Using
this value of the exchange interaction coefficient the paramagnetic Curie temperature extrapolated from the hightemperature region is ⌰⫽⫺4CI⫽⫺12.7 K 关see Eq. 共11兲兴,
which is close to the experimentally determined value⌰⫽⫺15.5 K. The temperature dependence of inverse paramagnetic susceptibility ␹ ⫺1
para calculated from Eq. 共11兲 isshown in Fig. 8 by the dotted line. This theoretical dependence coincides with the experimental dependence in a
broader temperature range than that of the Curie-Weiss law.
The extrapolated value ⌰ ⬘ ⫽3.6 K is close to the theoreticalestimate ⌰ ⬘ ⫽2.6 K.
The temperature dependencies of sublattices magnetization M 1 (T) and M 2 (T) were calculated from Eqs. 共6兲 and共7兲. These dependencies along with the ratio m(T)⫽M 1 (T)/M 2 (T) are plotted in Fig. 11. The difference inbehavior of Ni1 and Ni2 subsystems is clearly seen fromthese dependencies. While the magnetization of the Ni1 subsystem closely follows the behavior of the typical ferromagnet or antiferromagnet, the magnetization of the Ni2 subsystem strongly deviates from this behavior. This is due tothe fact that the exchange interaction within the Ni2 subsystem does not exist and it orders magnetically only due tothe exchange interaction with the Ni1 subsystem.
The temperature dependence of transverse magnetic susceptibility ␹ anti f in the magnetically ordered state calculated

FIG. 11. The temperature dependencies of reduced magnetizations M 1 (T)/M (0) of Ni1 (䊊) and M 2 (T)/M (0) of Ni2 (〫) sublattices along with the ratio m(T)⫽M 1 (T)/M 2 (T) (䊉).

from Eq. 共13兲 is in correspondence with experimental data,
as shown in Fig. 7. To make this comparison we assumed
that the magnetic susceptibility of the powder sample at highmagnetic field is that of the transverse the magnetic susceptibility of the single crystal. The theoretical and experimental
curves are qualitatively similar, but experimentally foundvalues are somewhat larger than the calculated ones. Thiscan be due to the presence of a small amount of paramagnetic impurities in the sample.
The physical scenario for the appearance of the compensation point at temperature dependence of weak magnetization of Ni(HCOO) 2 •2H2 O in the model presented is as follows. At cooling below the Néel temperature the weakmagnetization of the Ni1 subsystem exceeds that of the Ni2
subsystem. This occurs because M 1 (T) dependence is muchsteeper than M 2 (T) dependence in the vicinity of T N . At
further cooling, the values of Ni1 and Ni2 sublattice magnetization approach each other. If magnetic anisotropy in theNi2 subsystem is ‘‘larger’’ than that of the Ni1 subsystem, at
low temperatures the weak magnetization of the Ni2 subsystem will prevail. The temperature dependence of the weak
ferrimagnetic moment M total is determined not only by sublattice magnetizations M 1 (T) and M 2 (T) but also by singleion anisotropies K Ni1(T) and K Ni2(T) 关see Eq. 共5兲兴. To determine M total vs T dependence the absolute values of K Ni1(0)and K Ni2(0) were calculated from the values of M total ⫽0 at
a compensation temperature T * and M total at saturation ofboth Ni1 and Ni2 magnetic subsystems at T⫽0 K, i.e., atm⫽1. Taking into account that the experimental data wereobtained from a powder sample we will assume that the
weak ferromagnetic moment observed corresponds roughly
to one-third of that of M total . At T⫽0 K, the anisotropy
constants K Ni1(0)⫽6.43⫻108 erg/mole and K Ni2(0)⫽⫺1.86⫻108 erg/mole. M total (T) calculated from Eq. 共5兲
is shown in Fig. 4. Good correspondence between experimental and theoretical curves is clearly seen.
The field dependencies of magnetization of Ni(HCOO) 2•2H2 O at low temperatures are nonlinear and differ fromthat expected for a typical two sublattice antiferromagnet.

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WEAK FERRIMAGNETISM, COMPENSATION POINT, . . .

The calculations within the framework of our model indicatethat this behavior is due to the presence of two nonequivalentinteracting magnetic subsystems. The theoretical M total vs H
dependence for H parallel to the b axis of the single crystal atT⫽2 K is represented in the inset to Fig. 3. Though theexperimental data are obtained from the powder sample andcalculations are performed for a single crystal, the nonlinearity of both M vs H curves is clearly seen.
VI. CONCLUSION

This paper presents the results of experimental and theoretical studies of magnetic properties of the nickel 共II兲 formate dihydrate Ni(HCOO) 2 •2H2 O. At T N ⫽15.5 K thiscompound undergoes a transition to a magnetically orderedstate, exhibiting peculiar magnetic properties: a weak spontaneous ferromagneticlike moment, compensation point, andthe phenomenon of magnetization reversal. It is concluded
that Ni(HCOO) 2 •2H2 O is a weak ferrimagnet at low temperatures.
Weak ferrimagnetism arises from the competition of weakferromagnetic moments of two nonequivalent canted antiferromagnetic Ni1 and Ni2 subsystems. The deviations fromantiparallel alignment of magnetic moments within thesesubsystems are caused by single-ion anisotropy. The compensation point is due to the difference in temperature dependencies of weak ferromagnetic moments of Ni1 and Ni2 subsystems.
In the framework of molecular-field approximation a
simple model is proposed which accounts for the main features of the magnetic behavior of this compound. It describes
qualitatively the temperature dependence of the weak ferrimagnetic moment and magnetic susceptibility both aboveand below the Néel temperature.
Some quantitative disagreement of calculated results withexperimental data can be due to the fact that the measurements were done with powder samples, so the obtained data
are averaged over different crystallographic directions. The
limitations of the model itself are mentioned above.

ACKNOWLEDGMENTS

The authors acknowledge useful discussions with Professor Y. Ueda and the assistance of Dr. T. Uchimoto and Mr.
T. Okuyama in numerical calculations. This work was
supported by the Netherlands Foundation for the Support of
Science 共NWO兲 through Project No. 008-012-047.
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*On leave from Low Temperature Physics and Superconductivity

15

Department, Moscow State University, Moscow 119992, Russia.
Electronic address: vasil@lt.phys.msu.su
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