Magnetic Properties of an Antiferromagnetic Spin-1/2 XYZ Model in the Presence of Different Magnetic Fields: Finite-Size Effects of Inhomogeneity Property?

Hamid Arian Zad, Azam Zoshki, and Moones Sabeti

1Young Researchers and Elite Club,Mashhad Branch,Islamic Azad University,Mashhad,Iran

2Alikhanyan National Science Laboratory,Alikhanian Br.2,0036 Yerevan,Armenia

Abstract Magnetic and thermodynamic properties of the anisotropic XYZ spin-1/2 finite chain under both homogeneous and inhomogeneous magnetic fields are theoretically studied at low temperature.Using exact diagonalization method(ED),we study the magnetization,magnetic susceptibility,and specific heat of the model characterized in terms of the finite correlation length in the presence of three different magnetic fields including longitudinal,transverse,and transverse staggered magnetic fields.The magnetization,susceptibility,and the specific heat of the model are investigated under two conditions separately: (i) When the model is putted in the presence of homogeneous magnetic fields.(ii) When finite inhomogeneities are considered for all applied magnetic fields in the Hamiltonian.We show that for the finite-size XYZ chains at low temperature,the evident magnetization plateaus gradually convert to their counterpart quasi-plateaus when the transverse magnetic field increases.Moreover,the influence of the transverse and staggered transverse magnetic fields,and their corresponding inhomogeneities on the magnetization process,magnetic susceptibility,and specific heat are reported in detail.Our exact results illustrate that by altering the inhomogeneity parameters,magnetization plateaus gradually convert to their counterpart quasi-plateaus.The specific heat manifests Schottky-type maximum,double-peak,and triple-peak,as well as,transformation between them by varying considered inhomogeneity parameters in the Hamiltonian.

Key words: spin chain,magnetization,susceptibility,specific heat

1 Introduction

Exactly solved one-dimensional(1-D)spin models represent important milestones in statistical mechanics,since they pave the way to understand several aspects of magnetic materials in the real world.The spinS=1/2 Heisenberg models (XX,XYZ,XXZ) in the presence of a longitudinal magnetic field are paradigmatic examples of exactly tractable models,which not only have been applicable to elucidate generic features of quantum phase transitions,but also have long served as a paradigm for the study of quantum magnetism in low dimensions.[1?2]The study of external homogeneous magnetic field influences on the Heisenberg spin-1/2 models have been encountered with a lot of attentions in terms of both theoretical and experimental condensed matter physics.[3?13]

The nonuniform magnetic field is rarely taken into account.It is obvious that in any condensed matter physics subject,inhomogeneous zeeman coupling has remarkable effects on the energy band gaps as well as thermodynamic parameters of the quantum spin systems.So it is momentous to investigate the thermodynamic behavior of a spin system under a nonuniform magnetic field.Recently,Panti′cet al.[14]studied the effect of inhomogeneous magnetic field on the thermodynamic properties of an isotropic two-qubit XXX spin system.We note that the magnetization behavior for an XXZ spin model in nonuniform magnetic fields either longitudinal or transvesal has not been discussed specifically at low temperature.Although Felicien Capraro and Claudius Gros[15]studied the influence of both homogeneous longitudinal and transverse fields as well as transverse staggered field on opening of a spin-gap in 1-D spin chain,in the theoretical analysis we strongly believe that it is stimulating and should be investigated,the magnetic and thermodynamic properties of the spin chain under inhomogeneous magnetic fields specifically inhomogeneous transverse staggered field.This is the main motivation of this paper.

To investigate the critical points[2,6,16?17]in which phase transition occurs,the magnetic and thermodynamic properties of various metal containing compounds have been studied in the literature.Some of these compounds are very similar to 1-D spin-1/2 models.For instance,Eggert investigated magnetic and thermodynamic properties of material Sr2CuO3in Refs.[18–19].It was demonstrated that both materials Sr2CuO3and SrCuO2can be regarded as 1-D S-1/2 Heisenberg systems by fitting the temperature dependence of magnetic susceptibility with the theoretical calculation by Eggert,Affleck,and Takahashi (EAT) at low temperatures as low as 0.01J.[20]The magnetic properties of rare-earth compound Yb4As3in the absence of external fields,can be investigated by considering such compound as an antiferromagnetic Heisenberg spin-1/2 chain.The 4f-compound Yb4As3in the vicinity of external homogeneous,longitudinal,transverse,and transverse staggered magnetic fields have profoundly been studied.[15,21?22]

The specific heat behavior with respect to the temperature of spinS=1/2 chains has been studied by several groups.[23?29]In Ref.[30],D.C.Johnstonet al.indicated that parameter fluctuation effects play an essential role in the specific heat behavior versus temperature for an insulator NaV2O5,which its magnetic susceptibility is that of a 1-D Heisenberg chain.[31]They demonstrated a good agreement between theoretical results and experimental data.Furthermore,O.Breunig[32]experimentally studied the specific heat of one-dimensional magnetic material Cs2CoCl4with a comparison to the theoretical predictions of the Heisenberg spin chain.Generally,they found a good description of the experimental analysis in high temperature and strong magnetic field,although some differences between theory and experiment were observed at finite magnetic field.The magnetic part of the specific heat can be usually estimated in a certain temperature range by the Schottky theory.The associated round maximum of the specific heat,the so-called Schottky peak(maximum),has been experimentally detected in various magnetic compounds among which one could mention the molecular magnets.Recently,it has been theoretically reported that spin clusters can also depict the mentioned Schottky peak due to a typical competition between antiferromagnetic interactions and magnetic field.[33?34]

In order to figure out the magnetic behavior of the spin-1/2 Heisenberg chains in terms of applied magnetic field and/or exchange couplings between spins,magnetization plateaus have considerably been examined for various copper oxide compounds.[18,20,28?29,35?38]The behavior of uniform magnetization in the different phases with their dependence on the longitudinal(transverse)field for fixed values of other applied parameters was studied by P.Thakuret al.[1]Consequently,they observed that in the presence of the transverse field,the nature of the behavior of the uniform and staggered magnetizations near the critical fields dramatically change.K.Hida obtained the magnetization curve by numerical diagonalization of finite size systems.The result explains the low temperature magnetization data for 3CuCl2·2dnx.It is predicted that the magnetization curve has a plateau at 1/3 of the saturation magnetization if the ferromagnetic exchange energy is comparable to or smaller than the antiferromagnetic exchange energy.[39?40]The magnetization curve as well as magnetic susceptibility has been measured by numerical diagonalization of finite size systems for material 3CuCl2·2dx.

In this work,we will study the magnetic and thermodynamic properties of a 1-D spin-1/2 chain in the presence of various kinds of applied homogeneous magnetic fields such as longitudinal,transverse,and transverse staggered fields at low temperature.Then,we consider a finite inhomogeneity property for all applied magnetic fields and repeat our investigations,and compare our results with the case when the system is in the presence of homogeneous magnetic fields.We will limit our particular attention to a detailed examination of the magnetization,magnetic susceptibility and the specific heat.

The plan of our paper is as follows: In the next section,we briefly discuss the XXZ model in the presence of the desired magnetic fields,and introduce the model by means of a well-understanding Hamiltonian.In Sec.3,we discuss the behavior of thermodynamic parameters such as magnetization,magnetic susceptibility and specific heat and their dependences on the either homogeneous or inhomogeneous external fields.Finally,we end in Sec.4 with a brief summary and outlook.

2 Model

The anisotropicS=1/2 XYZ finite Heisenberg chain(Fig.1) as an exactly solvable system under inhomogeneous longitudinal and transverse,further transverse staggered magnetic fields,can be described by Hamiltonian

The integersj=(1,2,3,...,N) are the number of spins,where under periodic boundary conditions:N+1=1.Jx,Jy,andJzrepresent the Heisenberg exchange interactions between adjacent spinsSj,andSj+1(Sαwithα={x,y,z}are spin-1/2 operators),and the sum is over unique exchange bonds.Bzis uniform longitudinal magnetic field,Bxrepresents transverse field,anddenotes staggered transverse field incorporates all features proposed to be relevant for real materials like Yb4As3.The applied magnetic fields include the gyromagnetic gfactors and Bohr magneton coefficient.Parametersbz,bxandλcontrol the degree of inhomogeneity imposed into the longitudinal,transverse,and transverse staggered fields,respectively.We note that according to our assumption,the inhomogeneity leads to difference in strength of the induced magnetic fields into the odd and even sites of the Hamiltonian.

Fig.1 (Color online) Schematic representations of the spin-1/2 XYZ chain with finite length of (a) 6 particle,and (b) 10 particle,under periodic boundary conditions.J denotes XYZ Heisenberg exchange interaction between each adjacent spins.

The transverse staggered magnetic field applied in the Hamiltonian is directly induced by a staggered Dzyaloshinsky-Moriya (DM) interaction given by[22]

in whichDis the length of the DM vector (here we consider thez-direction).SupposingD=|D|=Jzsin(2θ)the DM interaction can be eliminated by a rotation aroundDby an angleθleading to,which can be interpreted as an effective staggered g-tensor.

It is quite obvious that the effect of a homogeneous longitudinal field likeon the structure of the XYZ spin chain,is not too much.This can be easily understood by noticing that [HXX,]=0,whereHXXis the Hamiltonian of an XX spin chain in the absence of external magnetic field.[41]What is really fascinating is applying an inhomogeneous longitudinal field defined by

for which generic magnetic fieldBz(j) is dependent on the sitej.In this case Eq.(3)does not commute with the total Hamiltonian of the system,namely

By performing some straightforward calculations,one can prove that the sum of all inhomogeneous magnetic fields applied in Eq.(1)does not commute with the total Hamiltonian.Consequently,the important feature of the Hamiltonian(1)is its noncommutativity with the magnetization operator.This non-commutativity leads to a non-linear transverse magnetic field dependence of the spectrum of the model and to the phenomena of quasi-plateau in magnetization curve.[42]Regarding this,we here assume that the system under consideration is in the presence of external inhomogeneous magnetic fields as specified in Eq.(1).

3 Results

In the present work,firstly,we examine in detail magnetic fields dependences of the magnetization,magnetic susceptibility and specific heat of the model introduced by Eq.(1) with the uniform exchange interactions between nearest-neighbor spins.In the second stage,we assume that the system is in the presence of the all introduced magnetic fields consisting of a finite inhomogenity.The magnetization,susceptibility and the specific heat can be straightforwardly calculated from the Gibbs free energyfaccording to the basic thermodynamic relations

The non-conserved magnetization can be directly interpreted using an unusual behavior of the magnetic susceptibility at low temperature.Figure 2 displays exact results for the magnetization and magnetic susceptibility as a function of the longitudinal magnetic fieldBz/Jzfor various fixed values of the transversal fieldBx/Jzat low temperature,where the Heisenberg coupling constants have been conventionally taken asJx=8JzandJy=10Jz(one may consider different values for these parameters in order to more diversity of investigations).In this figure we consider the model under homogeneous magnetic fields (inhomogeneous parametersbz/Jz,bx/Jz,andλ/Jzare equal to zero) with finite lengths ofN=6 andN=10,separately.Figures 2(a) and 2(c) show the magnetization and susceptibility of the spin chain with lengthN=6.At low temperature,weak transverse magnetic fieldBx/Jzand low transverse staggered field withθ=π/30 (black dotted-line),there is a plateau at zero magnetizationM/Ms=0,as well as,two intermediate plateaus atM/Ms=1/3 andM/Ms=2/3.With the increase ofBx/Jz,magnetization plateaus gradually convert to their counterpart quasi-plateaus.Although,the transformation from plateau to quasi-plateau will speed up upon increasing angleθ(the inset of Fig.2(a)).The transverse fieldBx/Jzand angleθincrement leads to delay in reaching saturation magnetization (see blue solidlines).

In Fig.2(c) the magnetic susceptibility for the same set of parameters is shown.The susceptibility behavior evidences the non-plateau nature of the magnetization within the same eigenstates of the model.Interestingly,for the strong fieldBx/Jzone can see that the susceptibility monotonically decreases upon increasing the fieldBz/Jz(blue solid-line).With further increase of the fieldBz/Jz,the susceptibility has a non-monotone behavior with the maximums in those intervals of the longitudinal magnetic field at which quasi-plateaus arise in the magnetization curve.This difference in behavior of the susceptibility for various fixed values of the transverse fieldBx/Jzat low temperature indicates that the system undergoes several phase transitions by increasing longitudinal fieldBz/Jz.

Figures 2(b)and 2(d)illustrate the magnetization and magnetic susceptibility for the chain of lengthN=10 under the same conditions asN=6.Here,in addition to the zero-magnetization plateau,there are four intermediate plateaus at:M/Ms=1/5,M/Ms=2/5,M/Ms=3/5,M/Ms=4/5,then the magnetization reaches its saturation in strong magnetic fields.As a result,while the number of magnetization plateaus increases with increase of chain size,the effect of transverse magnetic fieldBxand angleθis more sensible in this case.Namely,the transformation from plateau to quasi-plateau occurs for the lower amount of applied fieldBx/Jz.To clarify this point,if one compares red dash-dotted lines drawn in Figs.2(a) and 2(b) together,he finds that for the caseN=10 quasiplateaus appear for lower amounts of the transverse field compared with that of for caseN=6.As before,by increasingθquasi-plateaus gradually disappear.

Fig.2 (Color online) Low-temperature (T=0.1Jz) magnetization and magnetic susceptibility as functions of the longitudinal magnetic field Bz/Jz for several fixed values of the transverse field Bx/Jz ,and an arbitrary angle θ =π/30 under the condition bz =bx =λ=0.The system is considered in the presence of external homogeneous magnetic fields for which coupling constants have been conventionally taken as Jx=8Jz and Jy=10Jz.Panels(a) and (c) correspond to the spin-1/2 XYZ model of finite length N=6; panels (b) and (d) correspond to the chain of length N=10.Insets show the corresponding magnetization and magnetic susceptibility curves for different angle θ =π/10.

The magnetic susceptibility of the chain withN=10 as a function of the longitudinal fieldBz/Jzfor several fixed values of the transverse field is depicted in Fig.2(d).When the transverse magnetic fieldBx/Jzincreases,the height of peaks of the susceptibility corresponding to the magnetization jumping between plateaus decrease.As an important result,when the magnetization quasi-plateaus gradually appear by increasing the transverse fieldBx/Jz,accordingly,special peaks of susceptibility will arise.We would like to draw your attention to another interesting effect of the transverse field increment on the susceptibility behavior,i.e.,when the transverse field increases,the zero-field susceptibility has non-monotone behavior for both casesN=6 andN=10.The anomalous magnetic susceptibility behavior at very weak magnetic fieldBz/Jzis a remarkable evidence of existing magnetization quasi-plateau in the magnetization curve at low temperatures.With increase of the transverse staggered field coefficientθ,the zero-field susceptibility gets further than other peaks in both casesN=6 andN=10.For the strong magnetic field regionBz >10Jz,there is a steep decrease in the susceptibility curve for all considered fixed values of the transverse fieldBx/Jz,which denotes the magnetization goes to its saturation value.

In order to accomplish with our discussion concerning to finite-size effects of inhomogeneity property on the thermodynamic properties,we study the behavior of the magnetization and magnetic susceptibility when the system is in the presence of inhomogeneous magnetic fields at low temperature.We have plotted in Fig.3 the magnetization and magnetic susceptibility of the model with the same conditions as Fig.2,but under inhomogeneous magnetic fields (here,inhomogeneous parameters are taken as non-zero fixed valuesbz=0.6Jz,bx=0.3Jz,andλ=Jz).Figures 3(a) and 3(c) display the magnetization and susceptibility with the finite lengthN=6 under inhomogeneous longitudinal,transverse,and transverse staggered magnetic fields.Figures 3(b) and 3(d) are related to the chain of lengthN=10.In this situation for both casesN=6 andN=10,all plateaus have been shifted toward higher values of the magnetization.Hence,we can see that inhomogeneity dramatically affects on the height and position of the low-temperature peaks in susceptibility.When the transverse magnetic field increases,firstly height of the peaks increases,then with further increase of the fieldBx/Jzgradually decreases.Moreover,under inhomogeneous magnetic fields,the susceptibility does not vanish even at zero longitudinal fieldBz=0.Consequently,by imposing weak inhomogeneity property into the all magnetic fields,width of the magnetization plateaus decreases,and there is no zero magnetization plateau as well as zerofield susceptibility for the model under consideration with arbitrary length at low temperature.

Fig.3 (Color online) Magnetization and magnetic susceptibility as functions of the longitudinal magnetic field Bz/Jz for several fixed values of the transverse field Bx/Jz at low temperature (T=0.1Jz) and finite angle θ =π/30.Inhomogeneous property is considered for all applied magnetic fields such that; bz =0.6Jz, bx =0.3Jz,and λ = Jz.Coupling constants have been set as Fig.2.Figures 3(a) and 3(c) are associated to the chain with finite length N=6; panels (b) and (d) correspond to that of the length N=10.Insets show the corresponding magnetization and magnetic susceptibility curves for higher transverse staggered field Bx/Jz by setting θ =π/10.

By altering transverse staggered field intensity (θ=π/10),one can see less variation in the shape of susceptibility for weak longitudinal fieldBz <2Jzcompared with the case when the system is putted in the presence of homogeneous magnetic fields (see insets of Fig.3).It is quite evidence that under inhomogeneity,variations of both transverse fieldBx/Jzand transverse staggered fieldqualitatively affect the quasi-plateaus arisen in the magnetization curves more than when the system is in the vicinity of homogeneous magnetic fields,revealing that the magnetization curves including quasi-plateaus are more monotone than without inhomogeneity.

Finally,we investigate the temperature dependences of the specific heat under both homogeneous and inhomogeneous external magnetic fields.The corresponding plots of the specific heat as function of the temperature for several fixed values of the longitudinal magnetic field are presented in Figs.4 and 5.When the chain of lengthN=6(Fig.4(a))is putted in the presence of homogeneous magnetic fields,it is seen that the specific heat exhibits a double-peak temperature dependence for weak longitudinal fieldBz ≤Jzat low temperatureT=0.1Jz,where other parameters utilized in the Hamiltonian are taken asBx=Jz,bz=0,bx=0,λ=0,andθ=π/30.The height of double-peak monotonically decreases with increasing the fieldBz/Jz.With further increase of the longitudinal fieldBz/Jzdouble-peak merge together and create a broad single Schottky-type maximum with smaller height.In the high longitudinal magnetic fields (Bz >5Jz),one observes that Schottky-type peak convert to a doublepeak (blue and black marked lines of Fig.4(a)).Consequently,the shape of specific heat maxima alternatively change upon increasing the fieldBz/Jz.When the transverse staggered field increases (θ=π/10),the longitudinal field dependences of the specific heat are explicitly impressed(the inset of Fig.4(a)).In other words,varying the angleθresults in arising third peak in the strong longitudinal fields (blue and black marked lines in the inset of Fig.4(a)).

Fig.4 Temperature dependences of the specific heat of the 1-D XXZ spin chain under various fixed values of the longitudinal magnetic field Bz/Jz.Other external magnetic fields and parameters are taken as Bx = Jz,θ = π/30, Jx=8Jz and Jy=10Jz.All applied magnetic fields are also considered as homogeneous fields such that: bz=0, bx=0,and λ=0.(a) The specific heat of the chain with finite length N=6,and (b) N=10.Insets show the specific heat of the model in the presence of higher transverse staggered field as θ =π/10.

For the chain with more sites (N=10),there is a double-peak in the specific heat curve for the rangeBz ≤7Jz).In this case,the specific heat maxima have an alternating behavior upon increasingBz/Jz.Ultimately,we see that two maxima merge together and make a sharp and narrow Schottky-type maximum in the strong longitudinal fieldBz/Jz(black marked line in Fig.4(b)).Increase of the angleθalso alters the shape,position and height of the peaks (the inset of Fig.4(b)).We note that in this case the behavior of specific heat maxima (from change in heights and positions point of view) versus altering longitudinal fieldBz/Jzis more regular rather than the caseN=6.The relationship between Schottky peak and the double-peak can be plausibly identified in terms of the alternations of the inhomogeneity parameters.

Let us now examine the specific heat for the case when the system is in the presence of external inhomogeneous magnetic fields.As shown in Fig.5(a),by imposing inhomogeneity property into the applied magnetic fields asbz=0.6Jz,bx=0.3Jz,andλ=Jz,whereBx=Jzandθ=π/30,one can see a double-peak for the rangeBz ≤5Jz.As we noted for Fig.4(a),by increasing the fieldBz/Jz,double-peak gradually merge together and finally make a single Schottky-type maximum in the range 3Jz < Bz <7Jz.With further increase ofBz/Jz,the double-peak appears again in the specific heat curve.Another important point affecting the maxima of the specific heat is the altering the angleθ.For higher values ofθ,we see that the specific heat displays a double-peak for the strong magnetic fieldBz/Jzand fixedBx=Jz,which merge together by decreasing the longitudinal field.For higher transverse staggered field (θ=π/10),we witness an anomalous triple-peak temperature dependence in the presence of strong longitudinal magnetic fieldBz ≥7Jz.

When the number of sites in the chain increases(Fig.5(b)),there is a sharp Schottky-type maximum for weak longitudinal magnetic field.By increasing the magnetic fieldBz/Jz,a double-peak will appear and remains even for strong longitudinal magnetic fields.When the transverse staggered field coefficientθincreases,the shape of Schottky-type maximum remarkably changes,namely,it gets more sharper and narrow with lower temperature position.When the strength of longitudinal magnetic field increases a double-peak arises for valuesBz ≤7Jz(the inset of Fig.5(b)).

Fig.5 (Color online) Temperature dependences of the specific heat of the 1-D XYZ spin chain under various fixed values of the longitudinal magnetic field Bz/Jz.Other external magnetic fields and parameters are as in Fig.4.Here,all applied magnetic fields have been considered as inhomogeneous fields such that: bz=0.6Jz,bx=0.3Jz,and λ = Jz.(a) The specific heat as a function of the temperature for the chain with finite length N =6,and (b) N =10.Insets show the specific heat of the model versus temperature in the presence of higher transverse staggered field as θ =π/10.

4 Conclusions

The present work deals with the study of magnetization,magnetic susceptibility and the specific heat of the exactly solvable 1-D spin-1/2 XYZ chain under different external magnetic fields including longitudinal,transverse,and transverse staggered.Two small number of particles have been considered for the chain under periodic boundary conditions due to better understanding the finite-size effects of inhomogeneity on the spin models.The thermodynamic parameters of the spin system have rigorously been investigated under two different circumstances: Firstly,for the case when the system is in the presence of homogeneous magnetic fields;Secondly,for the case when all applied magnetic fields have a finite inhomogeneiny property.To consider suitable inhomogeneity properties in the applied magnetic fields,we have implemented inhomogeneity coefficients correspond to the three kinds of applied magnetic fields consisting of longitudinal,transverse,and transverse staggered magnetic fields.As a matter of fact,we have assumed that the strength of the induced magnetic fields is different for the odd and even sites of the chain.Since the magnetization operator does not commute with the Hamiltonian some unusual phenomena have been observed.

The low temperature examinations of the magnetization and magnetic susceptibility for the XYZ model under homogeneous magnetic fields revealed that the magnetization curve undergoes an interesting evolution such that upon increasing the transverse magnetic field,all plateaus convert to their counterpart quasi-plateaus.Moreover,by increasing the staggered field coefficientθ,quasi-plateaus gradually disappear,where the magnetization has a smooth curve versus longitudinal magnetic field for the high values of the transverse field and largerθ.We have observed that the susceptibility curve has also intriguing behavior with respect to the longitudinal magnetic field,when the strength of the transverse and transverse staggered fields change.In a good agreement with the jumps between magnetization plateaus,susceptibility curve has maxima whose shapes and positions are strongly dependent on the strength of all applied fields in the Hamiltonian.The non-monotone behavior of the susceptibility at higher values of transverse field indicates existence of the quasi-plateaus in the magnetization at low temperature.We also found a zero longitudinal field susceptibility upon increasing the transverse field.When the inhomogeneity property was imposed into the magnetic fields,low temperature behavior of the magnetization and magnetic susceptibility versus longitudinal field remarkably varied for low amounts of the transverse magnetic field.As a main result,here we have seen a zero longitudinal field susceptibility for both casesN=6 andN=10 even under the weak transverse magnetic field.

Finally,we have examined the specific heat of the finite-size model as a function of the temperature for various fixed values of the longitudinal magnetic field.We have concluded that,when the system withN=6 particles is putted in the presence of homogeneous magnetic fields,there is a strongly transverse field dependent double-peak which gradually tends to a Schottky-type maximum upon increasing the transverse field.Amazingly,for the strong longitudinal magnetic field,a triplepeak has been appeared in the specific heat curve when the strength of the transverse staggered field increased,while for the caseN=10 we have not observed triple-peak even at high longitudinal and high transverse staggered fields.Our calculations and simulations prove that changes in the specific heat behavior are in an excellent coincidence with the magnetization steps and jumps,accompanying with the magnetic ground-state phase transitions.

For the case when the system is considered under inhomogeneous magnetic fields,the shape and the positions of the specific heat maxima have been remarkably changed specifically for chain ofN=10 particles.Ultimately,we understood that for the finite length chains the altering staggered field has substantial influences on the behavior of the magnetization process,magnetic susceptibility and specific heat under the both different circumstances described above.Our exact results and straight expressions demonstrated in this work are fundamentally applicable for investigating small spin clusters and single molecular magnets with the same size in the presence of different kinds of magnetic fields that are crucial not only in the theoretical condensed matter but also in the experimental activities.