# Ising ferrimagnetic models. II

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<ul><li><p>This content has been downloaded from IOPscience. Please scroll down to see the full text.</p><p>Download details:</p><p>IP Address: 134.153.184.170</p><p>This content was downloaded on 31/07/2014 at 20:31</p><p>Please note that terms and conditions apply.</p><p>Ising ferrimagnetic models. II</p><p>View the table of contents for this issue, or go to the journal homepage for more</p><p>1974 J. Phys. C: Solid State Phys. 7 1189</p><p>(http://iopscience.iop.org/0022-3719/7/6/017)</p><p>Home Search Collections Journals About Contact us My IOPscience</p><p>iopscience.iop.org/page/termshttp://iopscience.iop.org/0022-3719/7/6http://iopscience.iop.org/0022-3719http://iopscience.iop.org/http://iopscience.iop.org/searchhttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/journalshttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/contacthttp://iopscience.iop.org/myiopscience</p></li><li><p>J. Phys. C : Solid State Phys., Vol. 7 . 1974 Printed in Great Britain. 0 1974 </p><p>Ising ferrimagnetic models : I1 </p><p>G M Bell Chelsea College, University of London, Manresa Road, London SW3 6LX </p><p>Received 23 July 1973, in final form 4 October 1973 </p><p>Abstract. If, in the triangular 1 :3 ferrimagnet of part I, the exchange energy between sites of the larger (b) sub-lattice is put equal to zero, then the b sites decorate the bonds between the sites of the other (a) sub-lattice. At any temperature Tand external field H the properties of a decorated model are derivable from those of a ferromagnet on the a sites at an effective temperature and field which are functions of T and H. An expression for the zero-field sus- ceptibility is found and illustrated by that of the 1 :3 model a t various values of the magnetic moment ratio pa/pb. The behaviour of the model in the (T, H) plane is derived from the curves on which, respectively, the effective field on the equivalent ferromagnet is zero and its effective temperature has the critical value. It is found that where a compensation point exists a curve of discontinuity in the (7: H) plane intersects the Taxis there. If a direct a-a interaction is intro- duced the curves of discontinuity are found tO start from the H axis. The critical temperature of the triangular 1 :2 ferrimagnet with no b-b interaction is found by a star-triangle transforma- tion. The application of the work of parts1 and II to three-dimensional systems is discussed and some results derived for a decorated diamond model. The properties of the triangular 1:3 model with no b-b interaction are contrasted with those when the b-b and a-b exchange energies are of equal magnitude. The accurate results for this model are compared with those of the molecular field approximation and a conjecture is made about its behaviour for general values of the exchange constant ratio. </p><p>1. Introduction </p><p>This paper continues the study of king model ferrimagnets (FIM) started in part I (Bell 1974) and uses the same notation. There are Na sites occupied by spins of magnetic moment pa and N , sites occupied by spins of moment ph. The exchange energy Jab = - J for a nearest-neighbour a-b pair is negative, tending to align the a and b spins in opposite directions, and the condition for ferrimagnetic states at low absolute temperature T and external field H is that Napa # Nhph. After considering a one-dimensional model part 1 was mainly concerned with the spin-reversal case with exchange energies of equal magnitude though not sign. Below a critical temperature the a and b sub-lattice relative magnetizations are then given by ma = - mb = m, m being the relative magnetiza- tion of the ferromagnet (FM) on the same lattice. Just above T, the (zero-field) suscepti- bility x is proportional to that of the FM but at higher temperatures there are considerable differences between the x- curves for the FIM and FM. </p><p>Two FIM models, termed (i) and (ii), based on the triangular lattice were introduced (see I, figures 3 and 4). As pointed out at the end of I, if Jbb = 0, model (ii) becomes a triangular lattice of a sites, termed the primary lattice, with the a-a nearest-neighbour </p><p>1189 </p></li><li><p>1190 G M Bell </p><p>bonds decorated by b sites. The spontaneous magnetization of a decorated FIM has been studied by Syozi and Nakano (1955), Hattori et ul(1965). Hattori (1966) and Nakano (1968); the latter obtained a zeroth-order approximation for the susceptibility. (For a recent review, see Syozi (1972).) In $ 2 we obtain an accurate expression for the suscepti- bility above T, and in $ 3 curves for the overall magnetization M and inverse susceptibility of model (ii) are given for various values of the ratio r = pa/pb. One term in the suscepti- bility expression contains the susceptibility of the FM on the primary lattice and the numerical results for model (ii) thus depend on the Pade formula of Baker (1961) for the FM triangular lattice. We are also able (see figure 2) to compare spontaneous magnetiza- tions M for model (ii) with pa = pb in three cases, firstly with a-b interactions only, secondly with Jbb = J a n d thirdly with Jbb = 0 and a direct a-a interaction. For decorated models a compensation point where M = 0, due to cancellation of the magnetization of the a and b sub-lattices, appears at a certain value of r . It is shown that at this value of r the critical index ~ ( F I M ) = ~ ( F M ) + 1 and the critical index ~ ( F I M ) = Y(FM) - 2. Hence the transition at is of the third order with the tangent to the curve of M against T becoming horizontal while the susceptibility remains finite at T, + 0, for both two and three-dimensional models. </p><p>The state of a decorated model (eg model (ii) with Jbb = 0) at any Tand H can be related to that of the primary lattice FM at a reduced temperature and field which are functions of T and H (Fisher 1959). The connection between FIM and FM models is thus closer than in the spin-reversal case where it only exists at H = 0. In $4 it is shown that for certain parameter values the field on the equivalent FM is zero on curves in the (?; H ) plane besides the axis H = 0. These curves are boundaries on which ma in the FIM changes from positive to negative and may contain segments across which the magnetization is discontinuous. For instance, at a compensation point the segment of discontinuity (0, T,) on the H = 0 axis is crossed normally by a truncated curve of discontinuity. With a-b interactions only, the point of discontinuity where ma changes from - 1 to 1 on the T = O axis remains an isolated singularity. If however a direct interaction is introduced between the nearest- neighbour pairs on the primary lattice this point becomes the terminus of a curve of discontinuity. (In model (ii) this is equivalent to introducing a second-neighbour inter- action between the a sites of the original lattice). </p><p>If Jbb = 0 in model (i) then each b spin interacts only with a triangle of three a spins. In $ 5 the system is shown by a star-triangle transformation to be equivalent to an FM on the a site lattice. Unfortunately much more information is required about the latter than in the decorated case obtained from model (ii). However the critical temperature of model (i) with Jbb = 0 is obtained. </p><p>Most theoretical analysis of ferrimagnetism, starting with the pioneer work of Nee1 (1948, 1953) has used the molecular (Weiss) field approximation. Hence it is of consider- able interest to compare molecular field results with accurate results for the system, model (ii), in which we have obtained the largest range of the latter. This is done in 5 7 below. </p><p>2. Decorated lattice ferrimagnet; magnetization and susceptibility </p><p>We now consider a primary lattice of Na a sites with each a-a nearest-neighbour pair link decorated by a b site. If the coordination number of the primary lattice is z , then N , = $zaNa. (An example is model (ii) with Jbb = 0, where each nearest-neighbour a-a pair on the primary lattice is a second-neighbour pair on the original lattice of a and b </p></li><li><p>Ising ferrimagnetic models: 11 1191 </p><p>sites). Labelling a spins oriented along and against the external field H with the indices 1 and 2 we denote the numbers of a spins of types 1 and 2 by N , and N , and the numbers ofa-a nearest-neighbour pairs oftypes 1-1,2-2 and 1-2 by N , ,, N,, and N , , respectively, </p><p>N , , + i N , , = f z a N , N,, + f N l z = f z a N , = f za(Na - N,) . (2.1) Introducing a direct interaction with exchange energy 4:) between nearest-neighbour a-a pairs (Nakono 1968) we define </p><p>(2.2) K - _ J2 - k T Now for each 1-1 pair there is a factor exp(2K - L,) + exp( -2K + L,) in each term of the partitition function due to the two possible orientations of the decorating b spin, where I.,, and K are defined by equation I (2.4). With similar factors for 2-2 and 1-2 pairs the partition function for the assembly can be written </p><p>(PF) = ([exp (2K - L,) + exp (-2K + L,)] [exp (2K + L,) + exp ( - 2K - L,] x [exp (L,) + exp ( - Lb)]2)*zaNa(PF)e </p><p>g w , > NI,) exp [ U N , - 4 1 1 exp [K,(N,, + N,, - 4 2 1 1 </p><p>(2.3) </p><p>(2.4) </p><p>where (PF), is the partition function of an equivalent FM given by </p><p>(PF), = NI, N12 </p><p>where g is a primary lattice configuration number. The effective reduced temperature Kp and reduced field Le are given by </p><p>exp(4Ke) = [exp (2K - L,) + exp (- 2K + L,,)] [exp (2K + L,) + exp ( - 2K - L,)] Cexp + exp ( - </p><p>x exp (4K;) (2.5) </p><p>La being defined by equation 1 (2.4). For H = 0, </p><p>When Tal = 0 the second relation of (2.7) can be expressed as Le = 0, </p><p>w = tanh K , = tanhK = t2, </p><p>K e = f In (cosh 2K) + K;. (2.7) </p><p>(2.8) where the first member of (2.8) defines w. It is useful to define a reduced interaction energy and (zero field) susceptibility for the primary lattice by </p><p>Using equations (2.3), I(2.7) and (2.1) the relative magnetization of the spins is given by m, = tanh L, - fm,(tanh (2K + L,) + tanh (2K - L,)) </p><p>+ $(1 - e*) {tanh (2K +L,) - tanh (2K - L,) - 2 tanh L,}. </p><p>M = Napb(fza tanh (2K) - r} (-ma), </p><p>(2.10) </p><p>(2.1 1) </p><p>Hence the overall magnetization when H = 0 is given by </p></li><li><p>1192 G M Bell </p><p>r being the ratio p a / p b , which is the result derived by Syozi and Nakono (1955). It can be seen that there is spontaneous magnetization where T < T = J/ (kK, ) , K c being the critical value of K obtained by substituting into (2.7) the critical value Kec of K , for the FM on the primary lattice. The critical index has the same value as for the FM except for r = i z , tanh 2Kc, when it is equal to ~ ( F M ) + 1, giving a third-order transition, </p><p>We now consider the (zero-field) susceptibility X. Where necessary zero-field values are distinguished by the index 0. Now, from (2.6), (s)o = - (2) [ i z , tanh (2K) - r ] . (2.12) Then differentiating (2.10) with respect to H and putting H = 0, </p><p>e:) tanh2 (2K) + tanh (2K) { i z , tanh (2K) - r}t*]. (2.13) Using equations I(2.9) and (2.12) we then have </p><p>X J G = </p><p>Nap: + N b p i N,[1 - (1 - ez)2t2/(1 + t2)2] + N,[(zat/l + t2) - rI2t* </p><p>N, + N,r2 = K , (2.14) where the first member defines the reduced susceptibility G. The critical index y is the same as for the FM on the primary lattice except for r = i z , tanh (2Kc) where it is equal to ~ ( F M ) - 2. Thus the susceptibility remains finite as T -+ T, + 0 in this case, where we have seen that there is a third-order transition. </p><p>Equation (2.14) is a special case of equation (43) of Fisher (1959), valid for a general decorating system placed between each pair of a sites. However Fisher gives no detailed derivation and we shall need to use some of the intermediate results stated here. </p><p>3. Applications and numerical results </p><p>3.1. One-dimensional decorated model </p><p>The one-dimensional loose-packed model discussed at the end of 14 3 can be looked at as a linear lattice of a sites decorated by b sites. Equations I (3.5) with K , = K , T(3.10) and (2.9) then give </p><p>l + w 1 + t 2 </p></li><li><p>Ising ferrimagnetic models: II 1193 </p><p>Hence with a-b interactions alone equation (2.7) gives cosh (2Kc) = $, equivalent to K c = 0.5731, k q / J = 1.745. Removing the b-b interactions thus reduces the critical temperature considerably from the value 3.641 (J /k) found when Jbb = J. 1n figure 1 </p><p>0.5 I k T l J </p><p>Figure 1. Triangular FIM (ii) with a-b interactions only: magnetization in the limit H --t + O against temperature. Each curve is labelled with the value of r = pa/pb. </p><p>the overall magnetization M in the limit H + +O is plotted against reduced temperature for several values of r, using equation (2.11) with z , = 6 and Na = i N . The magnitude (mal is derived from formula 1 (4.4) for m, with t replaced by w = t 2 . The sign of ma is chosen to make M positive. Using the critical value K c just derived </p><p>3 sinh (2Kc) cosh (2KJ </p><p>$za tanh (2KJ = = $. (3.3) </p><p>For r = $it can be seen from figure 1 that the tangent to the magnetization curve at the critical point is horizontal, although it is vertical for the other values of r. Thus for r = $ there is a third-order transition to the state of spontaneous magnetization at T = i", inaccordancewithg2above.Forr < $,31mb/>lmal,mb >Oandma </p></li><li><p>1194 G M Bell </p><p>0.2 t- </p><p>1 0 I 2 3 </p><p>kTlJ </p><p>1 4 I </p><p>Figure 2. Triangular FIM (ii); magnetization in the limit H + + O against temperature with = P, . Curve a. J,, = -.Jab = J : Curve !3, J,, = 0, a-b interactions only: Curve y , J,, = 0, </p><p>4; = -Jab = J . </p><p>3.3. Triangular ferrimagnet (ii): susceptibility. </p><p>With a-b interaction only the zero-field susceptibility is derived by putting za = 6, N, = 3Na in equation (2.14) above. A formula (due ultimately to Wannier 1950 and Houtappel 1950) for the reduced zero-field energy e* of the FM on the primary sites at temperatures above the critical value was obtained from Domb (1960): </p><p>the parameter 1 in the elliptic integral of the first kind being given by </p><p>16w3(1 + w3) 1 = </p><p>(1 - w)6(1 + w)3 (3.4b) </p><p>and w defined by equation (2.8). No exact closed form exists for the FM reduced suscepti- bility t* occuring in equation (2.14) and the Pade expression for xt due to Baker (1961) was used, Bakers U being replaced by w. Curves of inverse susceptibility for five of the values of r used in figure 1 are shown in figure 3. For r = ,,/is the inverse susceptibility curve rises from a minimum to a finite value at T = T,. For r = 1 the curve lies below its high-temperature asymptote and has only one point of inflection. We have shown in 15 5 that a similar behaviour is likely for the same value of r with full b-b interaction. For other values of r the curve lies above its asymptote and, except for r = $, there are two points of inflection before the region of ferromagnetic behaviour just above T = T, is reached. </p></li><li><p>Ising ferrimagnetic models: II 1195 </p><p>2 3 4 5 6 k r l J </p><p>Figure 3. Triangular FIM (ii) with a-b interactions only; inverse susceptibility against tempera- ture. Each curve is labelled with the value of r. The broken line is the high-temperature asymptote for both the r = 1 and r = 3 curves. </p><p>3.4. Triangular ferrimagnet (ii): high-temperature susceptibility series </p><p>From equation ( 3 . 4 ~ ) it can be shown...</p></li></ul>