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) [
at low temperature [6,8]. Using those techniques, it was shown also that the ferromagnetic behavior vanishes by increasing
, while the antiferromagnetic one persists up to higher temperatures.
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.
and
(
>
) on alternating sites with competing antiferromagnetic nearest and next nearest neighbor couplings, 
and 
respectively, may be described by the Hamiltonian
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 [
which in any case is an approximation of
.
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 [
is obtained by taking thermal averages [
,
runs over half of Brillouin zone and 
is the coefficient in Bogoliubov transformation; see Ref. [
and 
are bose distribution functions. Specific heat 
is calculated by taking the differentiation of the free energy with respect to
.
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.
and
. At T = 0 it shows the ground state energy of the system. It was founded that dimerization at T = 0 reduces ground state energy [2,16], it does the same with free energy. We found that the decrease of the free energy goes as
.
with a slower decrease for higher dimerization. This decrease shows a power law behavior goes as
.
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.
, 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 [
at least in our range of temperature.
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.
used to enhance it. The same peculiar behavior was observed in the gapless mode at any finite temperature.
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.
pushes it up at intermediate T for higher values of dimerization.
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.
, 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
.
we found that the reduction on magnetization is faster because of phase transition at this point.
is suppressed with the increase of frustration parameter
.
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.
) 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.
(11)
and
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.
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 
. The double peak is observed, not for the system
, at least within our T range, which suggests that even at a high values of
, which we found there is a decrease before this increase as shall be discussed below.
, 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 
, for different 
values after this minimum, we found that even in 2D the system with 
shows antiferromagnetic behavior more than ferromagnetic and it is vis versa for the spin system (
) as clear from 
. While temperature decreases the free energy it got increased with frustration parameter. This behavior is illustrated in 
.
. 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 
. The shocttky-like peak reduced in height with frustration and shifts towards smaller values of temperatures as can be seen from 
shows a minimum which shifts towards smaller T by introducing more frustration in the system indicating that frustration supports the enhancment of antiferromagnetic mode in these systems. After this minimum frustration increases the susceptibility. From this increase we reach to the same conclusion
shows antiferromagnetic behavior more than ferromagnetic and it is vis versa for the spin system (
).
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.