Journal of Magnetism and Magnetic Materials 331 (2013) 174–182

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Journal of Magnetism and Magnetic Materials

0304-88

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Calculation of magnetization curves and probability distributionfor monoclinic and uniaxial systems

Hala A. Sobh a,n, Samy H. Aly a, Sherif Yehia b

a Department of Physics, Faculty of Science at Damietta, Damietta University, P.O. Box 89, New Damietta 34517, Egyptb Faculty of Science, Department of Physics, Helwan University, Helwan, Cairo, Egypt

a r t i c l e i n f o

Article history:

Received 28 August 2012

Received in revised form

18 October 2012Available online 23 November 2012

Keywords:

Partition function

Anisotropy constant

Probability distribution

Magnetization curve

53/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.jmmm.2012.11.017

esponding author.

ail address: hala.amala@gmail.com (H.A. Sobh

a b s t r a c t

We present the application of a simple classical statistical mechanics-based model to selected

monoclinic and hexagonal model systems. In this model, we treat the magnetization as a classical

vector whose angular orientation is dictated by the laws of equilibrium classical statistical mechanics.

We calculate for these anisotropic systems, the magnetization curves, energy landscapes and prob-

ability distribution for different sets of relevant parameters and magnetic fields of different strengths

and directions. Our results demonstrate a correlation between the most probable orientation of the

magnetization vector, the system’s parameters, and the external magnetic field.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Methods of classical statistical mechanics have been used instudying different magnetic systems [1–5]. The classical partitionfunction is constructed and a host of physical quantities are evaluatedfrom it, e.g. magnetization and magnetic heat capacity. For example,interesting spin reorientation and in-plane anisotropy phenomenonhave been reported in the Nd2Fe14B system [1]. Cubic and mixedanisotropy systems have been discussed in Ref. [5], where systems ofmixed uniaxial and cubic anisotropies were involved. These systemsdid exhibit a magnetic behavior that depends on the competitionbetween the different anisotropies. Size-dependent properties of Gdparticles, where ferromagnetic and super paramagnetic behaviorsshowed up, were studied using the same model [2]. Analyticalmethods have been used to analyze the anisotropy energy in termsof sign and magnitude of the anisotropy constants [6–9]. The analysisdemonstrates that magnetic systems with anisotropy constants ofdifferent signs and magnitudes do vary in their magnetic behavior e.g.they exhibit easy-axis, easy-cone or easy-plane features. In thepresent paper, we direct our attention to the monoclinic systemwhich, up to our knowledge, has not been studied using the simplemodel we use. The angular dependence, on the polar and azimuthalangles of the anisotropy energy in the monoclinic and uniaxialcrystals is different [8,10]. We report our calculation of the magne-tization curves, energy landscapes and probability distribution formonoclinic and hexagonal selected model systems and discuss

ll rights reserved.

).

how these three provide a rather integrated picture of themagnetic behavior of the considered systems.

2. Model and computation

The total energy density for a magnetic system, in general, isgiven by:

E ym,jm,H� �

¼ Ea ym,jm,H� �

�H!:M!

ð1Þ

where �H!:M!

, is the Zeeman energy and Ea(ym,jm,H) is themagneto-crystalline anisotropy energy, which depends on theanisotropy constants. The anisotropy constants are usually tempera-ture dependent, in addition these constants may have magnetic fielddependence as well, for example see Ono [11]. In the present work,we consider only the temperature dependence of the anisotropyconstants. We use the spherical polar coordinates and the subscriptsm and h to differentiate between the polar and azimuthal angles forthe magnetization and the magnetic field respectively.

Therefore, the magnetic field H!

is given by:

H!¼HsinyhcosjhiþHsinyhsinjh JþHcosyhk ð2Þ

and the magnetization M!

is:

M!¼MsinymcosjmiþMsinymsinjm jþMcosymk ð3Þ

So, the Zeeman energy is:

H!:M!¼MH sinymsinyhcosjmcosjh

�þsinymsinyhsinjmsinjhþcosymsinyh

�ð4Þ

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182 175

After several simplifications Eq. (4) becomes:

H!:M!¼MH sinymsinyhcos jm�jh

� �þcosymsinyh

� �ð5Þ

So,

E H,ym,jm

� �¼ Ea ym,jm

� ��MH sinymsinyhcos jm�jh

� ��þcosymsinyh

�ð6Þ

The magneto-crystalline anisotropy energy for the hexagonalcrystal [8] is given by:

Ea ¼ K1sin2ymþK2sin4ymþK3sin6ym

þK4sin6ymcos6jm ð7Þ

And for the monoclinic crystal [10] it is given by:

Ea ¼ K1sin2ymþK2sin2ymcos2jm

þK3sin2ymsin2jmþK4sin4ym ð8Þ

where K1 to K4, are the anisotropy constants. We choose the z-axisas the c-axis for hexagonal crystals. In cubic crystals [5], we haveused the appropriate direction cosines of the magnetic field tocalculate the magnetization curves along specific crystal directionsof experimental interest e.g. [111]. For uniaxial crystals, a welldefined c-axis is perpendicular to a basal plane and magnetization iscalculated for magnetic fields either parallel or perpendicular to thec-axis where in-plane anisotropy may exist [1]. For a monocliniccrystal, however, only two of the three crystal axes are perpendi-cular to each other [12]. We choose the spherical polar coordinatesystem such that, the x-axis of the coordinate system is along theb-axis of the monoclinic crystal which is perpendicular to the yz-plane. The z-axis is at some angle s off the a-axis of the monocliniccrystal, see for example Fig. 1. in Ref. [13]. In order to calculate themagnetization curves we choose the direction of magnetic field tobe either along the z-axis or within the xy-plane for different valuesof jh in the angular range 01–901 (see Section 3.1.1).

The classical partition function is given by:

Z T ,Hð Þ ¼1

4p

Z 2p

0

Z p

0e�E T ,H,ym ,jmð ÞVbsinymdymdjm ð9Þ

where V is the volume of a spherical magnetic particle (of radius80 A in this work) with a uniform magnetization and b ¼ 1

kBT

where kB is the Boltzmann constant, and T is the absolutetemperature which we choose to be 300 K throughout this work.

Fig. 1. Magnetization curves for a monoclinic crystal with K1 ¼

The magnetization is given by:

M¼1

b@ðln ZÞ

@Hð10Þ

The probability that the magnetization vector has angularorientations ym and jm is:

P H,ym,jm

� �¼

e�E H,ym ,jmð ÞVb

Zð11Þ

We have used, throughout this work, the Mathematica pack-age [14] to do all the calculations involved.

3. Results and discussion

3.1. Monoclinic crystal

3.1.1. The magnetization curves

Figs. 1–3, show the calculated magnetization curves for differentdirections within the monoclinic crystal. The anisotropy constantsused in the calculation are displayed in the figure captions. For fieldsbelow H � 185 kOe, a magnetization is easily achieved in thefield’s directions specified by yh¼901 and jh¼01, 451, as comparedto the other three directions. The plane perpendicular to the z-axis isnot completely anisotropic e.g. the two magnetization curves foryh¼901 and jh¼711, 901 are identical. For fields above 185 kOe,however the magnetization curve for yh¼01 becomes the easiestdirection to magnetize along as evident from reaching magneticsaturation.

The main features of Fig. 2 are different from those of Fig. 1 inthat the former shows a stronger in-plane anisotropy, which isevident from the different magnetization curves for yh¼901 andjh ¼ 711 and 901. Moreover, magnetic saturation takes place forfields Z200 kOe for which the hardest magnetization directionyh¼01, becomes the easiest one. The system in Fig. 3 is differentfrom those in Figs. 1 and 2 in that all the four anisotropyconstants are positive. It is clear that a small field of � 10 kOeis sufficient to set in a magnetic saturation in the yh¼01 direction.The other four directions, specified by yh¼901 are magneticallyharder.

�107 , K2 ¼�6� 107 , K3 ¼�6� 107 , K4 ¼�105 erg=cm3.

Fig. 2. Magnetization curves for a monoclinic crystal with K1 ¼�6� 107 erg=cm3 , K2 ¼�5� 107erg=cm3 , K3 ¼�4� 107 erg=cm3 , K4 ¼�3� 107 erg=cm3.

Fig. 3. Magnetization curves for a monoclinic crystal with K1 ¼ 6� 107 erg=cm3 , K2 ¼ 5� 107 erg=cm3 , K3 ¼ 4� 107erg=cm3 , K4 ¼ 3� 107 erg=cm3.

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182176

3.1.2. The probability distribution and energy landscape

Figs. 4 and 5, show respectively, the dependence on ym and jm

of the probability and the corresponding energy, at zero magneticfield, for a system with the same parameters as in Fig. 1.The probability plot in Fig. 4, shows a peak at ym � 1:54 rad andjm ¼ 0:5 rad. The corresponding energy minimum in Fig. 5,is consistent with the probability peak location. For the systemof Fig. 1 , we expect a stronger in-plane anisotropy in zero field.This anisotropy energy is responsible for having the magnetizationvector rigidly oriented in the easy plane i.e. at ym¼901. We have alsocalculated the probability and its associated energy landscape in thepresence of different magnetic fields in order to study the effect ofthe magnetic field on the evolution of the probability distribution.For example, a magnetic field H¼30 kOe applied along the yh¼901,jh¼901 direction led to a probability peak at ymE1.54 rad,jm¼3.5 rad. This behavior is consistent with the fact that themagnetic field is applied in an easy plane of magnetization.

Fig. 6 displays the probability angular distribution, for the systemof Fig. 2, in a magnetic field of H¼80 kOe at yh¼901, jh¼451. Thelocation of the maximum probability is at ymE1.54 rad, jm¼0.5rad which indicates that this field is strong enough to rotate themagnetization vector appreciably to its own direction. The globalminimum in the energy landscape of this system (Fig. 7) is locatedvery close to the location of the probability peak of Fig. 6. We haverepeated our calculation using fields of different strengths anddirections. For example, for a field of 300 kOe applied in theyh¼01 direction, we notice that the probability distribution peaksup at ym¼01 for all values of jm as shown in Fig. 8 . This behavior issupported by the magnetization curves of Fig. 2 which indicates thatat magnetic fields around 300 kOe and higher, the magnetizationreaches saturation for fields applied at yh¼01 off the z-axis, whilebelow this field this same direction is the hardest for this crystal.

In Fig. 9, we show the probability distribution upon the applica-tion of a magnetic field of 200 kOe at yh¼901 and jh¼01 in the hard

Fig. 4. The angular dependence of the probability, in zero field, for a monoclinic crystal with the same parameters as in Fig. 1.

Fig. 5. The energy landscape for the monoclinic crystal in a magnetic field H¼0 Oe with the same parameters used in calculating the probability of Fig. 4.

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182 177

plane of the system of Fig. 3. The easiest magnetization direction isalong the yh ¼ 01 direction, i.e. perpendicular to the hard planewhose yh is 901. Although the field is relatively high the torque it

exerts on the magnetization vector, against the axial anisotropy, is notenough to have the magnetization vector oriented in the hard plane.The probability has its maximum at ymE2.4 rad, jm¼6 rad.

Fig. 6. The angular dependence of the probability for a monoclinic crystal at H¼80 kOe applied at yh¼901, jh¼451 direction with same parameters as in Fig. 2.

Fig. 7. The energy landscape for the monoclinic crystal at H¼kOe applied at yh¼901, jh¼451 direction, with the same parameters used in calculating the probability of Fig. 6.

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182178

3.2. Hexagonal crystal

3.2.1. The magnetization curves

Figs. 10 and 11 show the magnetization curves for easy-axisand easy-plane hexagonal systems respectively. The magnetic

field is taken along and perpendicular to the c-axis for differentvalues of jh. The parameters which have been used in thecalculation are shown in the captions of these figures. In Fig. 10,we applied the magnetic field along the c-axis and in differentdirections within the basal plane of the hexagonal crystal. We

Fig. 8. The angular dependence of the probability for a monoclinic crystal in H¼300 kOe applied at yh¼01 direction with same parameters as in Fig. 2.

Fig. 9. The angular dependence of the probability for a monoclinic crystal, with same parameters as in Fig. 3, for H¼200 kOe applied at the hh ¼ 901 , jh¼01 direction.

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182 179

note that it is easier to magnetize the crystal along the c-axis,therefore the c-axis is the easy axis in this case, and the basalplane is the hard plane for magnetization. The magnetizationcurves are different for different field directions within the hard

plane which indicates the presence of magnetic anisotropy in thebasal plane. The magnetization curves for an easy-plane system(the four anisotropy constants are negative) are shown in Fig. 11.These same parameters are used to calculate the magnetization

Fig. 10. The magnetization curves, for fields applied along and perpendicular to the c-axis, for an easy-axis hexagonal crystal. K1 ¼ 6� 106 , K2 ¼ 2� 105 ,

K3 ¼ 105 , K4 ¼ 2� 107 erg=cm3.

Fig. 11. The magnetization curves for an easy-plane hexagonal crystal, K1 ¼�107 , K2 ¼�6� 107 , K3 ¼�6� 107 and K4 ¼�105 erg=cm3 for field applied along and

perpendicular to the c-axis.

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182180

curves for a monoclinic crystal (Fig. 1). For fields r150 kOe, thebasal plane is the easy plane of magnetization but is completelyisotropic. This trend is reversed for fields 4150 kOe, wheremagnetic saturation takes place along the c-axis. The ratherdifferent behavior evident in these two figures could be explainedin terms of the different angular dependence of the anisotriopyenergy for monoclinic and hexagonal crystals. An analytical studymay be useful (which we are currently undertaking) to elucidatethe role of the sign and magnitude of the anisotropy constants indetermining the most energetically favorable orientation of themagnetization vector.

3.2.2. The probability distribution and energy landscape

Fig. 12 shows the probability plot calculated for 40 kOe alongthe c-axis as a function of ym and jm for the system of Fig. 10.The probability has its maximum at ym¼0 rad for all values of jm.

This shows that the magnetization vector is favorably orientedalong the c-axis for this easy-axis system as expected. Fig. 13shows the probability distribution for the system of Fig. 11, in afield applied along the c-axis. The probability plot shows severalpeaks at ymE1.54 rad but at different values of jm in consistencywith Fig.11.

4. Conclusions

We present an application of a simple model, based onclassical statistical mechanics, to model systems of monoclinicand hexagonal crystals. We have used this model to calculate themagnetization curves, probability angular distribution of themagnetization vector and the corresponding energy landscape.Our calculation shows that the main features of the magnetiza-tion curves are consistent with the probability distribution for

Fig. 12. The angular dependence of the probability for an easy-axis hexagonal crystal in a field of 40 kOe applied along the c-axis with the same parameters as in Fig. 10.

Fig. 13. The angular dependence of the probability for an easy-plane hexagonal crystal in a field of 20 kOe applied along the c-axis with the same parameters as in Fig. 11.

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182 181

a given set of system parameters. The angular location of theprobability peak is dependent on both the magnitude and direc-tion of the magnetic field. We also demonstrate that monoclinic

and hexagonal systems of the same anisotropy constants behaverather differently because of the different angular dependence oftheir anisotropy energy.

H.A. Sobh et al. / Journal of Magnetism and Magnetic Materials 331 (2013) 174–182182

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