aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Magnetocaloric Properties of the Ising nanotube ¨Umit Akıncı Department of Physics, Dokuz Eyl¨ul University, TR-35160 Izmir, Turkey
The magneticaloric properties of the Ising nanotube constituted by arbitrarycore spin values S c and the shell spin values S s have been investigated by meanfield approximation. During this investigation, several quantities have beencalculated, such as isothermal magnetic entropy change, full width at half max-imum value and the refrigerant capacity. The variation of these quantities withthe values of the spins and exchange interaction between the core and shell isdetermined. Besides, recently experimentally observed double peak behavior inthe variation of the isothermal magnetic entropy change with the temperatureis obtained for the nanotube. Magnetocaloric effect (MCE) is defined as an occurred temperature change inthe material when it is subject to the magnetic field. It was first observed in Iron[1] and theoretically explained after many years [2, 3]. MCE is simply based onthe variation in different contributions to the entropy. The entropy of a magneticmaterial is composed of three independent parts namely, the electronic part,lattice part, and magnetic part. Under adiabatic changes, the total entropy ofthe material is constant. This means that, occurred a change in one part of theentropy should be balanced by other parts. Then if one increases the magneticpart of the entropy by an adiabatic process, the lattice part should decrease (byan assumption of the constant electronic part of the entropy). Decreasing latticeentropy manifests itself as a reduction in the temperature of the material. Inthis way, one can construct a thermodynamical cycle, in which at one step thematerial is at the temperature T and at another step it has the temperature T > T .Refrigerant capacity (RC) is the amount of heat that can be transferredfrom the cold end (at temperature T ) to the hot end (at temperature T )in one thermodynamical cycle. This quantity is one of the quantities whichmeasure the suitability of the magnetic material for magnetocaloric purposes.It is in relation to another quantity namely isothermal magnetic entropy change(IMEC). In order to obtain a large adiabatic temperature change, the materialshould have a large IMEC, and a large RC. On the other hand a good candidatehas sufficient thermal conductivity for the aim of easy heat exchange.The typical behavior of the IMEC by the temperature includes peak at acritical temperature of the material. Generally, bulk magnetocaloric materials [email protected] Dy and Ho can lead to an enhanced magnetocaloriceffect in comparison to the bulk counterparts [6, 7]. Similarly, it has been shownfor La . Sr . M nO nanotube arrays, the bulk sample exhibits higher IMECbut nanotubes present an expanded temperature dependence of IMEC curvesthat spread over a broad temperature range [8, 9].As explained in Sec. 3, IMEC is related to the magnetization change withthe temperature. If the magnetization rapidly changes over some interval ofthe temperature, it is said that large MCE obtained. Nanotubes are promisingmaterials for obtaining efficient MCE. For instance, large MCE, associated withthe sharp change in magnetization of the Gd O nanotubes has been shownexperimentally [10, 11]. Another example of experimental MCE in nanotubesis the structural defect-induced MCE in N i . Zn . F e O graphene (NZF/G)nanocomposites [12].As seen in these examples, experimental studies are up to date for MCEin nanotubes. Although, MCE in nano systems is an active research area forexperimentalists, to the best of our knowledge MCE on nanotube geometryhas not been worked out, theoretically. But, from the theoretical side both ofthe magnetic behavior of these systems well studied. After the first theoreticaltreatments of the Ising model on nanotube geometry [13] by effective field ap-proximation, the first results for the anisotropic Heisenberg model on nanotubegeometry have been obtained within the same methodology [14]. As studied inthis work in terms of the MCE, mixed spin models have been worked out forobtaining the magnetic properties. The magnetic properties of the spin (1/2-1)mixed system on nanotube geometry has been worked out within the improvedmean-field approximation [15] and Monte Carlo simulation [16, 17, 18]. Alsohysteresis and magnetic properties of the spin-1/2 spin-1 nanowire have beendetermined by Monte Carlo simulations [19]. The magnetic and hysteresis be-haviors of the higher spin models are also well known theoretically.The magnetic properties of the spin-1 and spin 3/2 nanotube has been de-termined within the Monte Carlo simulation [20] and quantum simulation treat-ment [21]. The same model on the nanowire geometry has been investigated byMonte Carlo simulation [22]. Spin (1/2-3/2) model on nanotube geometry hasbeen investigated within the effective field theory [23] and on a nanowire geom-etry by Monte Carlo simulation [24, 25]. The magnetic phase transition charac-teristics and hysteresis behaviors of spin-3/2 spin -5/2 model on Ising nanowirehave been determined by the Monte Carlo simulation [26, 27]. Besides, hys-teresis and compensation behaviors of mixed spin-2 and spin-1 hexagonal Isingnanowire have been studied within the Monte Carlo simulation [28].The aim of this work is to determine the MCE characteristics of the magneticnanotube, by solving the Ising model with several different spin values. For thisaim, the paper is organized as follows: In Sec. 3 we briefly present the model2nd formulation. The results and discussions are presented in Sec. 4, and finallySec. 5 contains our conclusions. y x y x Figure 1: Schematic representation of one layer of the nanotube in xy plane.The system periodically extends in z direction.We can see the schematic representation of the one layer of the nanotube inFig. (1). As seen in Fig. (1), one layer of the nanotube consists of two hexagonswhich is called core (inner hexagon) and the shell (outer hexagon). Let the corespins have value S c and shell spins have S s . We can write the Hamiltonian ofthis system as H = − J c X (cid:0) S ci S cj (cid:1) − J s X (cid:0) S si S sj (cid:1) − J cs X (cid:0) S ci S sj (cid:1) − H X i S i (1)where S ci , S si denote the z component of the Pauli spin operator at a site i whichbelongs to the core (c) and shell (s), respectively. J c is the exchange interactionbetween the nearest neighbor core spins, J s is the exchange interaction betweenthe nearest neighbor shell spins, and J cs is the exchange interaction betweenthe nearest neighbor core and shell spins. The former three sums in Eq. (1) are3aken over the nearest neighbor sites, while last summation is taken over all thelattice sites. In Eq. (1), H is the longitudinal magnetic field.In order to include the effect of all exchange interactions, we take four spincluster. We can construct one layer of the nanotube by repetition of this se-lected cluster. The nanotube consists of repeating layers (seen in Fig. (1)) in z direction. The Hamiltonian of this cluster (which consists of red colored spinsin Fig. (1)) is H (4) = − J c ( S S ) − J s ( S S ) − J cs ( S S + S S + S S ) (2) − H X i =1 S i − X i =1 h i S i . Here, h i are the local fields that represent the interaction of the i th spin withnearest neighbor spins that belong to outside of the cluster. These local fieldsrepresent the following spin-spin interactions: h = J c ( S + S + S ) + J cs S h = J c ( S + S + S ) + J cs ( S + S ) h = J s ( S + S + S ) h = J s ( S + S + S ) . (3)Here the spins which are denoted as S ij , where i = 1 , , , j = 1 , S i , which are in the upper and lowerplane in z direction. The thermal average of the quantity Ω can be calculatedvia the exact generalized Callen-Suzuki identity [29] h Ω i = * T r Ω exp (cid:0) − β H (4) (cid:1) T r exp (cid:0) − β H (4) (cid:1) + (4)In Eq. (4) T r stands for the partial trace over all the lattice sites which belongto the selected cluster and β = 1 / ( kT ) where k is the Boltzmann constant and T is the temperature.Let us denote the basis set for this finite cluster by {| φ i i} = | s s s s i ,where s k is just one spin eigenvalue of the operator S k ( k = 1 , , , S k | s s s s i = s k | s s s s i , (5)where k = 1 , , ,
4. Note that, since the system consist of spin- S c core andspin- S s shell particles, number of bases equals to (2 S c + 1) (2 S s + 1) .Indeed calculation of Eq. (4) is trivial, since the matrix H (4) is diagonal forthe Hamiltonian given in Eq. (2), in the chosen basis set. The diagonal elementrelated to the base | s s s s i (which can be obtained by applying operator Eq.(2) to bases according to Eq. (5)) is given by4 φ i (cid:12)(cid:12)(cid:12) H (4) (cid:12)(cid:12)(cid:12) φ i E = − J c ( s s ) − J s ( s s ) − J cs ( s s + s s + s s ) (6) − H X i =1 s i − X i =1 h i s i . Let us denote this element as H (4) ( s , s , s , s ), then we can write Eq. (4)as h S k i = * X { s ,s ,s ,s } s k exp (cid:0) − βH (4) ( s , s , s , s ) (cid:1)X { s ,s ,s ,s } exp (cid:0) − βH (4) ( s , s , s , s ) (cid:1) + , k = 1 , , , . (7)The summations are taken over all the possible configurations of ( s , s , s , s ).The core ( m c ), shell ( m s ) and total ( m ) magnetizations can be calculatedvia m c = 12 ( h S i + h S i ) , m s = 12 ( h S i + h S i ) , m = 13 ( m c + 2 m s ) . (8)Up to this point, all equations are exact. But how can local fields in Eq. (6)be treated? Since our aim is to obtain some general qualitative results aboutthe MCE in nanotube system, it is enough to treat these local fields in a levelof mean field, i.e. by writing operators in Eq. (3) as their thermal averages, h = J c (2 m + m ) + J cs m h = J c ( m + 2 m ) + J cs ( m + m ) h = J s (2 m + m ) h = J s ( m + 2 m ) . (9)Note that, the periodicity of the lattice has been used for obtaining the expres-sions of local fields given in Eq. (9) from Eq. (3). In other words, h S i = h S i = h S i = m h S i = h S i = h S i = m h S i = h S i = h S i = h S i = m h S i = h S i = h S i = m . (10)By using this approximation, Eq. (7) gets the form m k = X { s ,s ,s ,s } s k exp (cid:0) − βH (4) ( s , s , s , s ) (cid:1)X { s ,s ,s ,s } exp (cid:0) − βH (4) ( s , s , s , s ) (cid:1) , k = 1 , , , , (11)5here the definitions of local fields given by Eq. (9) have been used in matrixelements H (4) ( s , s , s , s ). Then, the magnetizations m , m , m , m can befound by numerical solution of the nonlinear equation system given by Eq. (11).Core, shell and the total magnetization can be obtained by using Eq. (8). Notethat, the formulation used here is a generalization of the traditional mean-fieldto a larger cluster. The effect of using larger clusters can be found in Ref. [30].In order to determine the magnetocaloric properties of the system, we calcu-late the isothermal magnetic entropy change (IMEC) when maximum appliedlongitudinal field is H max , which is given by [31]∆ S M = H max Z (cid:18) ∂m∂T (cid:19) H dH. (12)The other quantitiy of interest is the refrigerant capacity which is defined by[32] q = − T Z T ∆ S M ( T ) H dT. (13)Here T and T are chosen as those temperatures at which the magnetic entropychange gains the half of the peak value and this is called as the full width athalf maximum value (FWHM) of the IMEC. This is also an important quantityof the MCE. We want to focus on the magnetocaloric properties of the system. The Hamil-tonian of the system includes four parameters, as one can see from Eq. (1). Inorder to make it possible for investigation, we have to reduce this number ofparameters. For this aim let us choose J c = J s = J . By this unit of energy J (which is positive) we can work with scaled quantities as r = J cs J , h = HJ , t = k B TJ . (14)Note that, h max = 1 . S c = 1 / S c = 7 / S s ) andcore-shell exchange interaction values ( r ), which are shown in the related figure.We can see from Fig. (2) that, when the spin value of the shell increases, themaximum value of the IMEC occurs in higher temperatures, and the peak value(i.e. height of the peak) of the IMEC decreases. At the same time, the curvegets wider, i.e. FWHM increases. This is consistent with the general relationbetween the spin value of the model and IMEC behavior. As demonstrated in6ef. [36], when the spin value of the model increases, the height of the peak inIMEC decreases, but the curves get wider, i.e. FWHM increases.Besides for lower values of r , the double peak behavior of the curve takesattention (see Figs. (2) (a) and (b)). This double peak behavior is depressedwhen the interaction of the core-shell gets stronger (see Figs. (2) (c) and (d)).Very recently, this behavior is obtained for the bilayer system experimentally[33]. Besides, double peak behavior has been obtained theoretically for bilayer[34] and superlattice systems [35].The same double peak behavior can be seen for the system constituted byspins S c = 7 / S c = 1 /
2, for thenanotubes that have S c = 7 / S c and S s . The low temperaturepeak seen in Fig.(2) (a) is related to the system with spin value of S c andother peak is related to the system with spin value S s . Since S s > S c in Fig.(2) (a), it is natural for the peak related to the S s to lie right side of thepeak related to S c in ( | ∆ S M | , t ) plane, due to the relations between the criticaltemperatures of layers that have different spin values.The same reasoning holdsalso for Fig.(3) (a). When the interaction between the core and shell increases,one peak behavior takes place (compare Figs.(2) (d) by (a), and Figs.(3) (d) by(a)). While this transition, the peak that occurs at lower temperature valuessuppressed (compare Figs.(2) (b) by (a), and Figs.(3) (b) by (a)).To take a closer look at the IMEC behaviors with the spin value and thevalue of core-shell interaction, we calculate the maximum value (height of thepeak) of the IMEC for different nanotubes which can be seen in Fig. (4). Atfirst sight, height of the peak of IMEC for a certain S c occurs at S s = S c regardless of the value of r . Thus, as seen in Fig. (4) decreasing trend withrising S s occurs for S c = 1 / S s occurs for S c = 7 /
2. For the values of 1 / < S c < /
2, rising S s rises the height of thepeak of IMEC until S s = S c , after then rising S s causes to a decline in theheight of the peak of IMEC. We can see similar behavior for FWHM in Fig.(5). Except ( S c , S s ) = (5 / , / , (3 , / , (7 / , /
2) nanotubes, rising S s firstdecreases FWHM, minimum FWHM occurs at S s = S c , after then rising S s causes to increment behavior in FWHM.For refrigerant capacity defined in Eq. (13), we depict the same scatter plotin Fig. (6). As we can see from Fig. (6), rising S s cause increasing refrigerantcapacity for spin values of core provide S c <
3. If the core spin value exceeds 5 / S c = 3 and 7 / | ∆ S M | tS c =1/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =1/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 Figure 2: The variation of IMEC with the temperature for selected values of S s = 1 / , , / , , / , , / r = 0 . , . , . , . S c = 1 / The MCE properties of the Ising nanotube constituted by arbitrary core spinvalues S c and the shell spin values S s have been investigated by mean field ap-proximation. During this investigation, several quantities have been calculated,such as IMEC, FWHM and the refrigerant capacity ( q ). The variation of thesequantities with the values of the spins and exchange interaction between thecore and shell is determined.First general conclusions about the variation of the IMEC with the temper-ature has been obtained. As consistently by the conclusions obtained in Ref.[36] for the general spin valued Ising model on a regular lattice, it is observedthat when the spin values of the nanotube increase, the height of the peak inIMEC decreases. This peak occurs at the critical temperature of the system,as expected. Besides, for a chosen spin value for the core, increasing shell spinvalue yields rising height of the peak in IMEC, when S c = S s maximum valueis obtained. After that (i.e. S c < S s ), increasing spin value of the shell yieldsdecreasing behavior in the height of the peak in IMEC. Completely reverse evo-lution occurs in FWHM, when the value of the shell spin increases. On theother hand, refrigerant capacity has increasing trend in the conditions of rising8 | ∆ S M | tS c =7/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.0(a) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.1(b) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=0.5(c) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 | ∆ S M | tS c =7/2 r=1.0(d) S s =1/2S s =1S s =3/2S s =2S s =5/2S s =3S s =7/2 Figure 3: The variation of IMEC with the temperature for selected values of S s = 1 / , , / , , / , , / r = 0 . , . , . , . S c = 7 / | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1/2 1 3/2 2 5/2 3 7/2 | ∆ S M | m a x S c r=0.1r=1.0 Figure 4: The maximum value of the IMEC for nanotubes that consist of spinvalues S c , S s = 1 / , , / , , / , , / r = 0 . , . S c contains number of 7 circles for certain value of r . Eachcircle corresponds to different values of S s , starting from S s = 1 / / S s = 7 / References [1] E. Warburg, Ann. Phys. 13, 141 (1881).[2] P. Debye, Ann. Phys. 81, 1154 (1926).[3] W. F. Giauque, J. Amer. Chem. Soc. 49, 1864 (1927).[4] Akhter, S. et al. J. Magn. Magn. Mater. 2014, 367, 75-80.[5] Chaudhary, V. et al J. Appl. Phys. 2014, 116, 163918.[6] V. Mello, A. L. Dantas, and A. Carrico, Solid State Commun. 140, 447(2006).[7] F. C. M. Filho, V. D. Mello, A. L. Dantas, F. H. S. Sales, and A. S. Carrico,J. Appl. Phys. 109, 07A914 (2011)[8] M. Kumaresavanji et al, APPLIED PHYSICS LETTERS 105, 083110(2014) 10 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 0 5 10 15 20 1/2 1 3/2 2 5/2 3 7/2 F W H M S c r=0.1r=1.0 Figure 5: The value of FWHM for nanotubes that consist of spin values S c , S s =1 / , , / , , / , , / r = 0 . , .
0. Each box labeledby S c contains number of 7 circles for certain value of r . Each circle correspondsto different values of S s , starting from S s = 1 / / S s = 7 / q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 0 0.5 1 1.5 2 2.5 3 1/2 1 3/2 2 5/2 3 7/2 q S c r=0.1r=1.0 Figure 6: The value of the refrigerant capacity ( q ) for nanotubes that consistof spin values S c , S s = 1 / , , / , , / , , / r =0 . , .
0. Each box labeled by S c contains number of 7 circles for certain valueof r . Each circle corresponds to different values of S s , starting from S s = 1 / / S s = 7 //