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Using linear spin-wave theory we have investigated the thermal properties of frustrated dimerized Heisenberg ferri- magnetic system with alternating spins and on one- and two-dimensional lattices. At intermediate temperature the susceptibility and the specific heat shows a minimum and a Schottky-like peak respectively. Frustration enhances the antiferromagnetic aspect in the system by causing a left-shift in the peak and the minimum which indicates that the antiferromagnetic behavior overbalance the ferromagnetic one at earlier temperatures. The effect of dimerization is different for the two form of the coupling constants. While the expanded form; , boosts the antiferro- magnetic behavior of the system by making a left-shift of the peak and the minimum, the distance-variable coupling constant; shifts them to the right opposing, for a while, the appearance of the antiferromagnetic aspect. The slope of after the minimum shows that the aspect of ferrimagnetic system with spins (3/2, 1) is more antiferromagnetic and the system with (3/2, 1/2) is ferromagnetic. Free energy and magnetization decreased by increasing dimerization as well as frustration. Both of them scales with PACS numbers: 75.10.Jm, 75.50.Ge.

Many attentions were made towards the low-dimensional antiferromagnets after Halden made his prediction that the integer-spin chain is massive, whereas the half-oddinteger spin chain is massless. This stimulated also several attempts to investigate the quantum behavior of chains consisting of two types of spins. An integrable model of this type was constructed by De Vega and Woynarovich [

The thermal behavior was also investigated for ferrimagnetic chains [3-13]. Besides verifying the existence of two (gapped and gapless) excitation modes, the specific heat and magnetic susceptibility of ferrimagnetic chains were shown to depend upon temperature as and respectively at low temperatures using modified spinwave theory (MSWT) [6,7], density matrix renormalizetion group (DMRG) [

Modified spin wave theory, which includes Takahashi constraint or non-linear spin wave theory, was also shown to give results in surprisingly good agreement with those from quantum Monte Carlo method in the thermodynamic limit of this system [6,7].

In this paper we will study alternating spin systems formed with a spin values; () using linear spin wave theory. We would like to investigate the thermal properties of the lattice magnetization, specific heat, free energy, and susceptibility. We would like to see how their thermal behavior depends on dimerization and frustration. We shall then study alternating spin systems on a square lattice.

The dimerization of chains with spins and (>) on alternating sites with competing antiferromagnetic nearest and next nearest neighbor couplings, and respectively, may be described by the Hamiltonian

with belongs to the first sublattice with spin and to the second one with spin. is the alternating coupling between two adjacent sites, it gives the displacement of the ith atom through. The exchange coupling constants defined in Equation (1) can have different forms. In order to take into account changing distances in the course of spin-Peierls distortions, we assume that the nearest-neighbor spin-spin exchange coupling is itself distance dependent [

instead of the often-used which in any case is an approximation of.

The above Hamiltonian can be linearized using spinwave analysis with the help of Holstein-Primakoff transformations to bosonic spin-deviation operators. The linearized Hamiltonian can be diagonalized using Bogoliubov transformations in terms of normal modes as

with is the ground state energy and are the acoustic and a optical modes for this system. A detailed study of the above system at zero temperature was discussed in our previous paper [

The free energy of the system is obtained by

Static susceptibility is obtained by taking thermal averages [

The sublattice magnetization is calculated by taking the thermal average of the occupation number as

and

where ,

and runs over half of Brillouin zone and is the coefficient in Bogoliubov transformation; see Ref. [

The behavior of these physical quantities with temperature in presence of dimerization and frustration parameters for different system and shall be discussed below. In chains we will investigate the thermal behavior using the expanded form of the coupling constant first and make a comparison with the behavior when the unexpanded form is used.

In a previous work [

Total magnetization decreases with increasing T and with a slower decrease for higher dimerization. This decrease shows a power law behavior goes as.

It is known that ferromagnetic materials show a decaying susceptibility with temperature and increasing for antiferromagnet. Ferrimagnets are believed to have a combined effect of both behaviors [4,8,9,17-19], ferromagnetic and antiferromagnetic. Susceptibility times temperature decreases at low temperature showing a minimum and them starts to increase showing that the antiferromagnetic behavior is not smeared out after the minimum. Activating the effect of dimerization, we found that dimerization pushes up and causes a shift of this minimum to lower values of T at larger values of for all the three spin systems discussed here. The slope after the minimum varies form spin system to another. While the slop changes minutely with increasing dimerization parameter in the case of () it is much visible for (). This behavior indicates that the spin system () acts like antiferromagnetic system than a ferromagnetic one. It is vise versa for the system () This conclusion is in agreement with earlier one saying that the system with is more ferromagnetic and the system with is more antiferromagnetic.

Specific heat increases at low T, and goes as, till it reaches a maximum where it shows a shocttky-like peak at intermediate temperature and then starts to decrease again. The antiferromagnetic excitation mode plays a major rule to this peak because it is gapped at, showing that after this peak the antiferromagnetic behavior is predominant. This peak is softening with the increase of dimerization parameter and shifts towards lower values of temperature. While at low temperature dimerization enhances the specific heat, reduces it at intermediate T for the three spin systems. We notice that at high specific heat become double peaked. This tell us that the ferromagnetic and antiferromagnetic modes are still gapless and gapped modes [

As mentioned above, using as a coupling constant is not of much important in the chains as it should be for the case of a square lattice. But we will investigate the thermal behavior of chains using the distance dependence coupling constant for the sake of having a complete picture of this form.

At zero temperature analysis [

Free energy as well as total lattice magnetization show qualitatively a similar thermal behavior as the expand coupling constant one. Both were founded to decrease with increasing temperature, regardless the magnitude values of spins and, and both of them decrease with as.

decreases at low temperature showing a minimum and them starts to increase showing that the antiferromagnetic behavior is not smeared out after the minimum. We found that dimerization in this case suppresses and shifts the minimum toward a higher values of T with the increase of for all the three spin systems. The slope after the minimum varies form spin system to another. While the slop changes minutely with increasing dimerization parameter in the case of (), it is more visible for () and it is the most for (). This behavior indicates that the spin system () acts like antiferromagnetic system than a ferromagnetic one. It is vise versa for the system (). Although this form delays the appearance of antiferromagnetic mode but the general behavior of the system remains unchanged. As a conclusion, we found an agreement with earlier calculations that the system with is more ferromagnetic and the system with is more antiferromagnetic.

Specific heat shows the same different behavior with dimerization. Shocttky-like peak at intermediate temperature shifts towards larger values of T. That means: the antiferromagnetic excitation mode, which plays a major rule to this peak, goes to a little higher temperature before it become predominant with the increase of dimerization. While at low temperature dimerization suppresses the specific heat, pushes it up at intermediate T for higher values of dimerization.

The effect of frustration in ferrimagnetic systems was investigated [2,21] at zero temperature. Several characterization of the frustration parameter were reported; mainly, which marks the complete destruction of the long range order in the system and marks the onset of spiral phase transition. At any finite temperature, we found that they are temperature independent. We will study the thermal effect of a frustrated chain in the absence of dimerization.

Free energy decreases with frustration. This decrease is faster for higher, and scales with T as. At the critical value of frustration, the free energy become imaginary and our LSW theory fails to go beyond this value of frustration. It was reported earlier [2,21] that at T = 0.0 the ground states energy increases with frustration parameter and then starts to decrease for values of frustration close to.

Like the dimerization effect in ferrimagnetic chains, the total magnetization of the system decreases with increasing temperature and frustration. At we found that the reduction on magnetization is faster because of phase transition at this point.

The shocttky-like peak at intermediate temperatures is also observed in the specific heat with frustration. Frustration enhances antiferromagnetic mode by causing the peak to shift to lower values of temperature. Below this maximum, is suppressed with the increase of frustration parameter.

Susceptibility time temperature decreases at small to a minimum after which it starts to increase for the three spin systems. As in the case of Schottky-like peak in specific heat, frustration makes this minimum in to shift towards smaller values of T verifying the enhancment of antiferromagnetic excitation mode due to frustration and it also pushes the to higher values for larger values of after the minimum. The slope after the minimum varies form spin system to another and shows an agreement with the previous conclusion: the spin system () shows antiferromagnetic aspect more than a ferromagnetic one and the system () is more ferromagnetic than antiferromagnetic one.

In a two-dimensional lattice, different phonons produce different types of dimerizations. A lattice distortion that involves two phonons, one with wavevector () the other with wavevector (), forming a plaquette lattice, was found to be the most favored one [14,22,23]. We will restrict our investigation to this kind of dimerization.

Using the untruncated exchange coupling defined in Equation (2) for both nearest neighbor and the next nearest neighbor interactions, the Hamiltonian of a ferrimagnetic square lattice can be written as a sum of the nearest neighbor and the next nearest neighbor parts:

(11)

with and

Using HP transformation and Bouglibove tansformation the above Hamiltonian can be diagonalized. The procedure of working out the physical quantities was discussed in detail in Ref. [

Like in the chain, free energy and total magnetizationdefined in Equations (4) and (8) respectively, were founded to decrease with increasing temperature as shown for the system in Figures 1 and 2 respectively. This is true regardless the magnitude values of spins and. At T = 0 the free energy reduces [13,14] to the zero temperature ground state energy. We found that the decrease in both goes as in the three spin systems. Along with temperature, we found that dimerization suppress free energy with larger values of, while it increases the total magnetization at a fixed value of T.

Specific heat shows a shocttky-like peak at intermediate temperatures. At low temperature specific heat goes as This peak is softening with increasing of dimerization parameter and shifts towards higher values of temperature. While at low temperature, say below the peak temperature, dimerization reduces the specific heat, enhances specific heat at intermediate T as shown in

dimerization there is still a gapped and pagless modes in the systems. The above result again shows that the antiferromagnetic aspect is predominant in the case of intermediate temperature.

The susceptibility times temperature was also obtained using the Equation (5) for the square lattice. At zero dimerization it was reported by Yoshihiro et al. [

We found that in, the ferromagnetic mode smeared out at T much earlier than the case of chains. The minimum, in support to the specific heat observations, shifts toward higher temperature, and after the minimum, further suppressed with the increase of dimerization parameter. This behavior is observed in the three spin systems and illustrated in

The free energy found to decrease with increasing frustration and also scales with T as. While temperature decreases the free energy it got increased with frustration parameter. This behavior is illustrated in

Total magnetization shows a decrease with frustration and temperature. This decrease goes with T as. Unlike the case of free energy, frustration increases the reduction in total magnetization as a function of temperature until it reaches the transition value, which after it will make a jump to higher values of magnetization and then starts again to decrease with increasing as clear from

Specific heat shows a shocttky-like peak at intermediate temperatures for different values of frustration. While dimerization at low temperatures reduces the specific heat, frustration push it up. At intermediate T, frustration decreases. The shocttky-like peak reduced in height with frustration and shifts towards smaller values of temperatures as can be seen from

As shown in

that in a square ferrimagnetic lattice the spin system shows antiferromagnetic behavior more than ferromagnetic and it is vis versa for the spin system ().

In summery we have used linear spin wave theory to study the thermal properties of ferrimagnetic system in the presence of frustration as well as dimerization parameters. In a chain, we found that dimerization enhances the antiferromagnetic aspect by taking coupling constat as it delays the mode in the case of . In chains as well as in square lattices we found that the spin system shows antiferromagnetic behavior more than ferromagnetic and the spin system () has a ferromagnetic one more than the antiferromagnetic aspect. We also found that both magnetization and free energy get decreased with frustration as well as dimerization and scales to square low of behavior with T.