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Ising ferrimagnetic models. II

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1974 J. Phys. C: Solid State Phys. 7 1189

(http://iopscience.iop.org/0022-3719/7/6/017)

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J. Phys. C : Solid State Phys., Vol. 7 . 1974 Printed in Great Britain. 0 1974

Ising ferrimagnetic models : I1

G M Bell Chelsea College, University of London, Manresa Road, London SW3 6LX

Received 23 July 1973, in final form 4 October 1973

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.

1. Introduction

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.

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

1189

1190 G M Bell

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.

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).

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.

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.

2. Decorated lattice ferrimagnet; magnetization and susceptibility

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

Ising ferrimagnetic models: 11 1191

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,

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

(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

(PF) = ([exp (2K - L,) + exp (-2K + L,)] [exp (2K + L,) + exp ( - 2K - L,]

x [exp (L,) + exp ( - Lb)]2)*zaNa(PF)e

g w , > NI,) exp [ U N , - 4 1 1 exp [K,(N,, + N,, - 4 2 1 1

(2.3)

(2.4)

where (PF), is the partition function of an equivalent FM given by

(PF), = NI, N12

where g is a primary lattice configuration number. The effective reduced temperature Kp and reduced field Le are given by

exp(4Ke) = [exp (2K - L,) + exp (- 2K + L,,)] [exp (2K + L,) + exp ( - 2K - L,)]

Cexp + exp ( -

x exp (4K;) (2.5)

La being defined by equation 1 (2.4). For H = 0,

When Tal = 0 the second relation of (2.7) can be expressed as Le = 0,

w = tanh K , = tanh’K = t2,

K e = f In (cosh 2K) + K;. (2.7)

(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

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,))

+ $(1 - e*) {tanh (2K +L,) - tanh (2K - L,) - 2 tanh L,}.

M = Napb(fza tanh (2K) - r} (-ma),

(2.10)

(2.1 1)

Hence the overall magnetization when H = 0 is given by

1192 G M Bell

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,

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,

e:) tanh2 (2K) + tanh (2K) { i z , tanh (2K) - r}t*]. (2.13)

Using equations I(2.9) and (2.12) we then have

X J G =

Nap: + N b p i

N,[1 - (1 - ez)2t2/(1 + t2)2] + N,[(zat/l + t2) - rI2t* 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.

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.

3. Applications and numerical results

3.1. One-dimensional decorated model

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

l + w 1 + t 2 <* = exp (2Ke) = - - - - l - w 1 - t 2 '

e* = - w = - t 2 0

Substitution into (2.14) yields

(1 + r2) cosh (2K) - 2r sinh (2K) (3.2) 2xkT ___- N p ; - 1 - t 2 -

- -

which is equivalent to T(3.16).

3.2. Triangular ferrimagnet (ii): magnetization.

Since the a sites themselves form a triangular lattice the critical value of exp (2KJ is J3.

Ising ferrimagnetic models: II 1193

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

0.5 I k T l J

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.

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

3 sinh (2Kc) cosh (2KJ

$za tanh (2KJ = = $. (3.3)

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 <Oforall T < q. For ,J7 < r < 3 a similar state occurs when T is less than the compensation point tem- perature given by 3 tanh 2K = r. However at the compensation point 31mbl = /mal, making M = 0, while above it 3/mbl < Ima/ and the signs of ma and mb are reversed. In 54 below this behaviour on the H = 0 axis will be related to the situation in the (?; H) plane as a whole.

When the b-b interaction is of equal magnitude to the a-b interaction a magnetiza- tion curve can be obtained from equation 1 (4.5). With an a-a interaction equal to the a-b interaction ( K ; = K)'equations (2.7) and (2.1 1) yield a magnetization curve. These two curves are compared with the one for the same model with a-b interaction only in figure 2, with r = 1 in all three cases.

1194 G M Bell

0.2 t-

1 0 I 2 3

kTlJ

1 4 I

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,

4;’ = -Jab = J .

3.3. Triangular ferrimagnet (ii): susceptibility.

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):

the parameter 1 in the elliptic integral of the first kind being given by

16w3(1 + w3) 1 =

(1 - w)6(1 + w)3 (3.4b)

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, Baker’s 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.

Ising ferrimagnetic models: II 1195

2 3 4 5 6

k r l J

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.

3.4. Triangular ferrimagnet (ii): high-temperature susceptibility series

From equation ( 3 . 4 ~ ) it can be shown that at high temperatures e* = - w + O(w2) = - t 2 + O(t4). Again, from Sykes (1961) (equation (5.16) of our paper I),

c* = 1 + 6w + O(w2) = 1 + 6t2 + O(t4).

Substitutution in equation (2.14) yields

4 ~ k T / ( N p 2 ) = 3 + rz - 12rt + (6r2 + 30) t2 - 60rt3 + .. . (3.5)

The same result follows by putting x = 0, y = -t in the general high-temperature relation I (5.22). Inverting and retaining second-order terms

k T 12r 6(r2 - 1)(15 - r2 ) J - (3 + r2)2 k T " ' (.J-1-_ +- + - J 3 + r Z

For 1 < r < v'i-s the last term is positive, in accordance with the numerical results presented in figure 3. For r = 1 the term in J / k T disappears but it can be shown that the next term is - 13(J/kT)'.

4. Decorated lattice ferrimagnet: behaviour in the T, H plane

From (2.3) the partition function is equal to a continuous function of H and T multiplied into the partition function of an FM. Noting that La = rLb, where r is a constant for given

1196 G M Bell

values of the dipole strengths pa and p,, equations (2.5) and (2.6) define a mapping from the ( K , L,) plane of the decorated model onto the (Ke , Le) plane of the FM. The only curve of singularities in the latter plane is the segment of the Le = 0 axis from K e = Kec to K e = CO, where the magnetisation is discontinuous on crossing the axis, Kec being the critical value of Ke. If the mapping were one-one then, since the system is symmetrical with respect to change of sign of L,, the only curve of discontinuity in the ( K , L,) plane would be a segment of the L, = 0. axis. However the mapping is not one-one and the interesting features of the ( K , L,) plane arise from this fact. When drawing diagrams it is convenient to use K-’ = kT/J and K - I L , = p,H/J as axes rather than K and L,. Since K - and K - ‘L, are reduced temperature and field respectively we are then working in the (T, H ) plane and the situation on the T = 0 axis can be seen clearly.

We first consider the case where there are no a-a interactions (e: = 0, K ; = 0). From equation (2.5) the locus in the (T, H ) plane which maps into the line K e = Kec is given by the relation

sinh (2K) CoshL -

- {exp (4K,,) - l}””

which for L, = 0 is equivalent to equation (2.8) at the critical point. This curve leaves the T = 0 axis at H = 2J/pb, meets the H = 0 axis at T = T, (critical temperature for the decorated model) and, being symmetrical about the H = 0 axis, then attains the T = 0 axis again at H = -2J/p,. At all points inside the curve K e > Kec and at all points outside K , < Kec. (The broken curves in figures 4 and 5 represent the K e = Kec

We must now discuss the locus on the (T , H ) plane which maps into Le = 0. By equation (2.6) all points on the axis H = 0 (La = L, = 0) give Le = 0. However where

locus).

Figure 4. Triangular FIM (ii) with a-b interactions only; the (T H) plane for r = 1. The full curve is the locus Le = 0 on which ma changes sign. The broken curve is the locus K , = Kec. The thickened segment is the interval of discontinuity (0, T).

Ising ferrimagnetic models: I1 1197

I 2

kT lJ

Figure 5. As for figure 4 with r = 2.7. The thickened portions are segments over which the magnetization changes discontinuously.

r < $za there is another branch mapping into Le = 0 and given by the relation

exp [(4z; l r + 2) Lb] - 1 exp (24,) - exp (4.2, ' rLb)

exp (4K) =

The upper half of the curve (4.2) leaves the T = 0 axis at the point H = zaJ /ya where, as shown in Ig4, the magnetization ma on the T = 0 axis changes from - 1 to 1. It meets the H = 0 axis where

za + 2r exp (4K) = ~

za - 2r *za tanh 2K - r = 0 (4.3)

the latter relation being easily obtainable from the former. From (2.12) Le (and hence ma) are negative just above the H = 0 axis for temperatures below that given by (4.3) and positive for higher temperatures. Since ma = 0 only when Le = 0 it follows that, in the H > 0 half-plane, ma < 0 in the region between (4.2) and the H = 0 axis and ma > 0 outside (4.2). All points on the curve (4.2) move to an infinite distance from the origin as r -+ 0 at a given value of pb while as r -+ i z a the curve shrinks to a segment of length 4J/pb on the T = 0 axis.

The curves of discontinuity in the (T, H) plane depend on the relationship between the curves (4.1) and (4.2). There are two possible situations, the first of which occurs when r < t z a tanh 2Kc and which is illustrated in figure 4 for triangular FIM (ii), where i z a tanh 2Kc = $?-The branch (4.2) of the Le = 0 locus lies entirely outside the K e = Kec locus. Hence when H # 0 and T > 0 changes in the sign of ma take place where K e < Kec and thus occur continuously. By equation (2.10), mb is also continuous away from the H = 0 axis. On the T = 0 axis Le is no longer a continuous function of H and we have isolated singularities at H = +zaJ/p , where Le changes from - CO to + CO, and ma from - 1 to 1. Away from the H = 0 axis the 'phase diagram' is very similar to that of the one- dimensional FIM shown in T figure 2 although on the H = 0 axis we now have a segment 0 < T < T, where ma and mb change discontinuously.

1198 G M Bell

The situation when i z a tanh 2Kc < Y < +za is most easily explained by referring to figure 5 , which illustrates such a case for triangular ferrimagnet (ii). The magnetization ma changes continuously from negative to positive over the part AB of the curve (4.2) lying outside the K e = Kec locus. The point A (H = 6J/,uJ where the curve leaves the H = 0 axis is thus still an isolated singularity. However when the part BC of the curve (4.2) is crossed Le changes sign in a region where K e > K e c so that ma must undergo a

f 1 2 . 7 1 i l l

0 2 4 6

k TlJ

Figure 6. As for figures 4 and 5 but with an a-a interaction = -Jab = J .

finite discontinuity. Hence BC and its mirror image in the H = 0 axis form a curve of discontinuity crossing the segment of discontinuity on the H = 0 axis at the point C. At C, Y = +za tanh 2K by equation (4.3) and hence C is the compensation point discussed in 8 3 above where M = 0 and the spontaneous magnetizations ma and mb change sign. There must be a region lying above and to the right of C where ma > 0 and m, < 0. Since, by equation (2.10), m, > 0 at the point B, where ma = 0, it follows that some point on the right hand side of BC is the terminus of a curve on which m, = 0. The point where this curve meets the H = 0 axis has been calculated from equation (2.13) and is at

When 4:) > 0 ( K ; > 0) the Le = 0 locus for given Y is unchanged but the locus K e = Kec alters drastically, giving rise to completely different behaviour in the (T , H) plane. Equation (4.1) changes to

kT/J = 4.13.

sinh (2K) cosh L, =

(exp [4(Kec - K;)] - 1)1’2* (4.4)

The critical value of K , obtained by putting L, = 0, decreases from K c to a new value which we may denote by Kb. However a more important change is that, instead of curving round to meet the T = 0 axis at a finite value of H, the curve (4.4) now never reaches the T = 0 axis but goes to H = kc0 with the line K ; = (4:) /5)K = Kec as

Ising ferrimagnetic models: I I 1199

asymptote. The points H = &z,J/p, on the T = 0 axis are no longer isolated singularities since curves of discontinuity now leave the axis at these points. If r -= $za tanh 2K', these curves terminate on the locus (4.4). (See figure 6 for triangular ferrimagnet (ii) with K ; = K, giving K: = 0.2255, $za tanh 2Kb = 1.2682). Similar truncated segments offirst-order transition points occur in certain metamagnetic models (Stanley and Harbus 1973) but the segments are joined by a curve of second-order transition points, which is not the case here. If iza tanh 2Kk < r < $za then the entire branch (4.2) of the Le = 0 locus forms a curve of discontinuity in the region Ke > K,. and intersects the segment of discontinuity 0 < T < Tb on the H = 0 axis (see figure 6).

There is a finite discontinuity Am, everywhere on the curves of discontinuity discussed in this section except at end-points not lying on the axes. Since, by the symmetry proper- ties of the FM, N,, + N 2 , and m,, are continuous across the curves on which Le = 0 it follows from (2.10) that the discontinuity in mb is given by

Amb = -${tanh (2K + L,) + tanh (2K - L,)) Am,. (4.5) From equations (2.3) and (2.4) there is a corresponding discontinuity in the configura- tional energy given by

= Na[$zaJ{tanh(2K + Lb) - tanh(2K - L,)}

+ tza{tanh (2K + Lb) + tanh (2K - L,)} pbH - rp ,H] Ama. (4.6) This is zero on the H = 0 axis, as is necessary from symmetry considerations, but non- zero on the curves of discontinuity which do not lie on this axis.

5. Triangular ferrimagnet (i) with no b-b interactions

If Jbb = 0 each b spin in the triangular FIM (i) interacts only with a triangle of three a spins (see I figure 3). The zero-field partition function can then be related to that of an FM on the a sites alone by the star-triangle transformation recently reviewed by Syozi (1972). For our purposes it is convenient to introduce it in rather a different way. We denote the number oftriangles ofa sites with all three sites occupied by spins of species 1 by NI, and similarly define N,, , , N , , , and N,,,. Then denoting the nearest-neighbour a-a pair numbers by N,,, N,, and N , , , as in 62 above, there exists for any distribution on the a sites the relations

+NI,, + $" = N , , ;N,,, + $ N , , , = N,, NI,, + N , , , = NI,. (5.1) With Jbb = 0 the partition function may be written

(PF) = 2 g exp [ ( N , - N,) La] (exp (3K - L,) + exp ( - 3K + Lb)}N1ll

x {exp (3K + Lb) + exp ( - 3K - Lb))N222 {exp ( K - Lb)

+ exp ( - K + Lb)}N112 {exp ( K + Lb) + exp ( - K - Lb)}N122 (5.2) where g is the number of configurations corresponding to given values of N , , , , N,,,, N,, , and N,,,. At H = 0 it follows from (5.1) and (5.2) that

(PF) = [exp ( K ) + exp ( - K)]3Nai2 [exp ( 3 ~ ) + exp ( - 3K)INal2 (PF), (5.3)

1200 G M Bell

where (PF), is given by (2.4) with Le = 0 and

exp (3K) + exp (- 3K) exp ( K ) + exp ( - K ) exp (2K,) = = 2cosh2K - 1. (5.4)

The critical temperature is obtained by substituting exp (2K,,) = 4 3 on the left hand side. We find Tc = 2.405 (J /k ) . As for triangular model (ii) this is a considerable reduction from the value T, = 3.641 ( J / k ) found when Jbb = J. It should be noted that K , is twice the value given by Syozi (1972, equation (30)). This is because Syozi, concerned with transforming from the triangular to the honeycomb lattice, only places b spins in alternate triangles of a spins whereas here there is a b spin in all triangles.

From (5.4), ma may be obtained as a function of T , using equation I (4.4). Unfortun- ately other results are difficult to obtain because they depend on information about three-particle correlations in the FM. For instance from (5.2) it can be shown that at

N,m, = - (Nl l l - N,,,) tanh 3K - (N,,, - N,,,) tanh K

H = O

= - Nbma tanh 3K - (w,,, - N,,,) (tanh K - 5 tanh (3K)) (5.5)

where equations (5.1) and (2.1) with za = 6, have been used for the last expression.

6. Three-dimensional models

We have considered two-dimensional models here because for one of them, triangular FIM (ii), we are able to treat both the cases of full b-b interaction and zero b-b interaction. This has not been possible for a three-dimensional model and, in addition, it must be remarked that the crystal structures of real FIMare highly complex (Martin 1967). Howeber. many of the results obtained are applicable to three-dimensional models. In 1 $ 4 models with all exchange integrals of equal magnitude (though not sign) in which equilibrium states are related to those of the FM by reversing the spins on one sub-lattice were dis- cussed. Equation I (4.12), implying that just above the critical point the susceptibility is proportional to that of the FM on the same lattice is valid in three-dimensional models. A particular case is the loose-packed FIM with equivalent sub-lattices carrying spins of unequal magnetic moment. The results plotted in T figure 5 and derived from equation I (4.1 3) are in fact for a three-dimensional loose-packed lattice, the body-centred cubic lattice.

The work of $ 2 above, including equation (2.14) for the susceptibility of the decorated FIM model, is applicable to three-dimensional lattices. Hence the conclusion that, except at one value of r, the critical index y is equal to that for the FM on the primary lattice (a sites) is valid in three dimensions. The conclusion that, at the value of r where the compensation and critical points coincide, the value of y is reduced by two is also valid. The theory of the (T, H ) plane 'phase diagram' for decorated FIM given in $ 4 has general applicability in three dimensions. It may be of interest to consider a decorated diamond model in which a b site is placed on each bond between two a sites (diamond lattice sites) giving N , = 2Na. In this case KG1 = 2.704 (Essam and Sykes 1963) and it can be seen from figure 7 that the (T, H ) plane properties are similar to those illustrated for a two- dimensional model in figures 4, 5 and 6. With a-b interactions only the entire curve Le = 0 for r = 1.5 lies outside the K , = K e c curve. Hence ma changes continuously from negative values inside the Le = 0 curve to positive values outside, except at the isolated

Ising ferrimagnetic models: I1 1201

singularity D on the H axis. For r = 1.9 however there is a first-order transition across the portion BC of the Le = 0 curve, C being the compensation point in the segment (0, T,) on the T axis. Except at A, ma still changes continuously across the portion AB. When the a-a interaction 592) = 0.35 is introduced and r = 1.5 the point D ceases to be

0 0.5 I I .5 2

kTlJ

Figure 7. The decorated diamond FIV; the (T H) plane. Curve E : the locus Le = 0 for r = 1.5. Curve /3: the locus L, = 0 for r = 1.9. Curve y : the locus K e = Kec for a-b interactions only. Curve 6: the locus K e = KcL with an a-a interaction 42,) = -&J.

an isolated singularity and becomes the terminus of a segment DE across which a first- order transition occurs. For r = 1.9 the entire Le = 0 curve becomes a curve of dis- continuity.

7. Comparison with molecular field method

This method has been much used and a comparison with exact results for the model where the largest number of the latter have been obtained is of interest. Accordingly we give the configurational free energy Fc for the triangular FIM (ii) with no interaction between the a spins;

+- 2

2

- N[-Jb,mt - $Jmamb + $paHm, + ipbHmb].

L6

1202 G M Bell

The equilibrium e,uations may then be written

- 0, --- 1 + m a 12J 2P H +--b - a - - ln-

8 dFc NkTdma 1 - m a kT kT

(7.2)

The condition for stability of any energy state is

where

kT 2

1 - m:

12J 6 kT 1 -mi kT

(7.4)

The non-magnetized state ma = mb = 0 is a solution of (7.2) when H = 0 for all T but it ceases to be stable when A(0,O) = 0. This gives critical temperatures T, = 6(J/k) for Jbb = J and Tc = v m ( J / k ) for Jbb = 0, as compared with the accurate values of T, = 3.641 (J /k) and Tc = 1.745 (Jlk). (See I 5 4 and 83 above).

It is not difficult to show from (7.2) that the susceptibility is given by

(rZ + 3) kT - (4r2Jb, + 12rJ) (kT)zA(o, 0)

x = 3Npi

After some manipulation (7.5) can be inverted to give

(7.5)

(7.6) 12{(rZ - 3) J - 2rJbb)’

J(r2 + 3) [ ( r2 + 3) kT - (4r2J,, + 12rJ)I’ - kT 12(rJ - Jbb)

J + o-l =- ( r2 + 3) J

Equation (7.6) gives a curve concave to the T axis over the whole range T, < T < r3

which approaches the T axis with a nonzero slope as T + Tc + 0. This contrasts with the accurate results for both Jbb = J and Jbb = 0 (see 1 4 4 and 8 3 above) where it has been shown that though the curve is concave to the T axis in a certain temperature range it becomes convex near Tc, where the tangent to the curve bends round to become horizontal at T, + 0. The difference just above T, can be expressed by comparing the values of the critical index y which is 1 for the molecular field theory and accurately, just as in the FM. (For the latter see, for instance, Stanley 1972). The molecular field theory gives correct paramagnetic Curie temperatures for the FIM but for some parameter values there are qualitative differences between the curve given by (7.6) and the accurate curve at high temperatures. The curve (7.6) always lies below its asymptote at high temperatures. However from equation (3.6), with a-b interactions only, the accurate curve lies above its asymptote for 1 < r < ,,/rS implying that it is convex to the T axis at high temperatures. From equation I (5.27) the accurate curve for Jbb = J behaves similarly for 1.5120 < r < 8.8126.

Where a solution ma = 0, mb # 0 of the equilibrium relations is stable the regions

Ising ferrimagnetic models: I1 1203

of the (T , H) plane where ma > 0 and ma < 0 respectively are divided by a curve on which

6 Jm, H=-, p a

(7.7)

kT In {%} - 4 {2Jbb + 3 J - mb,

This curve meets the T = 0 axis where H = 6 J / p , and m; changes from - 1 to + I at this point when r < 3. However it can be shown that for very low temperatures (7.7) gives

Hence A(0, m,), as given by equation (7.4), is positive on the curve (7.7) at low temperatures, apart from T = 0. The point H = 6J/pa on the T = 0 axis is thus an isolated singularity for r e 3, as in the accurate theory for Jbb = 0. The curve (7.7) meets the H = 0 axis where

-- kT Jbb - 4- + 6r-l . J J

For T > T, it can be verified that h a p H changes sign at this point. The temperature given by (7.8) decreases as r increases and becomes equal to the critical temperature when

the last expression being obtained by putting A(0,O) = 0. Rearranging (7.9) gives

(7.10)

We show below that the compensation point, where the spontaneous magnetizations of the a and b sub-lattices cancel exactly, coincides with the critical point when r = ro. For Jbb < J a compensation point exists for ro < r < 3, as in the accurate theory for Jbb = 0 in $4 above.

So far the properties of the ma = 0 curve (7.7) have appeared similar to those of the Le = 0 curve for Jbb = 0 discussed in the accurate theory of $ 4 above. However for T < T, a solution of the equilibrium equations with ma = 0 is unstable on the H = 0 axis. Hence, for r > ro, equation (7.7) does not give the whole of the ail t'; dividing the regions ma e 0 and ma > 0 respectively in the (T , H) plane though for r < 3 it always gives a low-temperature portion of this curve extending from the H axis to a point where A(0, m,) = 0. To obtain the compensation point where the dividing curve meets the segment (0, q) of the H = 0 axis we must substitute ma = - 3mbr-', H = 0 directly into the equilibrium equations (7.2). We obtain the pair of relations

(7.1 1) kT In {( 1 + ?)I( 1 - +)} - 12Jm, = 0,

- 0. (7.12)

1204 G M Bell

For given values of Jbh/J and r, equation (7.12) may be solved for mh and the result substituted into (7.11) to give the compensation temperature. It is found that where Jhb < J solutions of (7.12) exist in the range ro < r < 3, the compensation temperature tending to T, as r -+ ro + 0 and to zero as r -+ 3 - 0. In the interval from T = 0 to the compensation temperature, ma < 0 in a region above the T axis while mh < 0 above the T axis between the compensation temperature and T = 2r(J/k) > T,, where am,jaH = 0. This behaviour resembles that found in the accurate theory for Jbb = 0 where a com- pensation point exists. For Jbh > J there is a compensation point in the range 3 < r < ro and now mb < 0 above the T axis below the compensation point with corresponding changes at higher temperatures. At J,, = J the ‘compensation point’ range of r shrinks to the point r = 3, where the overall magnetization M is zero throughout the temperature interval (0, T,).

In the molecular field approximation the critical index f l = $, as for the FM, except at r = ro where a = $,, This corresponds to a third-order transition, as in the accurate results for Jbb = 0. For r = ro it also follows from (7.10) that the third term on the right hand side of (7.6) is zero so that the susceptibility remains finite at T = T,. However the inverse susceptibility becomes a linear function of T which is unlike the accurate function found for Jbb = 0 (see figure 3).

The introduction of an a-a interaction involves the addition of - 12J’,2,)majkT to the middle term of the first relation of (7.2) while the second relation of (7.2) and equation (7.7) remain unchanged. However a new term - 12J$)/(kT) in the top left-hand element of A(ma, mh) means that A < 0 near the point where the curve (7.7) joins the T = 0 axis. This means that the point T = 0, H = 6J/pa is no longer an isolated singularity. The introduction of an a-a interaction was found to have a similar effect in the accurate theory for Jbb = 0 (see figure 6) . From this and previous results it appears that over the whole parameter range the molecular field approximation gives a correct qualitative picture of behaviour in the ( T , H ) plane when compared with the accurate theory for Jbh = 0. However it should be noted that it may give quite wrong results for a particular parameter value. For instance, where 43 < r < 46, the molecular field method predicts a compensation point on the T axis when J,, = 0, but from the accurate theory we know that there is no compensation point in this range of r.

8. Discussion

With certain exceptions, mentioned below, a common shape of inverse susceptibility (x- curve has been found in the various FIM models investigated. By comparing figures 1 and 3 above it can be seen that, in triangular model (ii) with a-b interactions only, this shape is more stable to changes in the value of r than is that of the magnetization curve. The nature of the singularity at the critical temperature T, is the same as for the FM of the same dimensionality and hence there is a range of ‘ferromagnetic’ behaviour in the x - l curve just above T, with the curvature positive. However with a paramagnetic Curie temperature T, < T, the curvature has to change its sign above T, in order for the curve to approach its asymptote. The x-’ curve often rises from the T axis much more abruptly than for the FM (see figure 3 above, I figure 5 or equations I (4.14) and (4.15)). At high temperatures there are two kinds of behaviour associated respectively with negative and positive signs for the T-’ coefficient in the expansion of x-l. In the first kind the curve remains below its asymptote while in the second kind it crosses the asymptote and there is a second point of inflection, with the curvature becoming positive

Ising ferrimagnetic models: II 1205

again. This second change in curvature can be seen on the r = 2 curve of figure 3 but in other cases (eg r = 2.7, r = 3 on figure 3) it is far from obvious on the graph. In fact both the characteristic behaviour just above the critical point and at high temperatures may be hard to see on a reasonably small scale graph and from the latter alone it might easily be concluded in certain cases that the x-’ function has the form predicted by molecular field theory (eg the r = 2.7 and r = 3 curves in figure 3 and curves A and B in I figure 5).

Where the compensation point in decorated models coincides with T, the susceptibility remains finite at T, + 0 and in figure 3, r = & it has a form similar to that found in antiferromagnets. This behaviour is unlikely to be of physical importance as it occurs only at a single special value of r related to the exchange constant. For low values of r in two cases (one-dimensional 2:l model, r < iJbb/J, see I figure 1, curve C ; ‘spin reversal’ model (ii), r < 0.0874, see I @ 5) we have found T, > T,, as in the FM. However in both cases the x-l curve crosses its asymptote and hence the high-temperature behaviour of 1- is unlike that found in the FM.

In contrast to the susceptibility, there are considerable differences in magnetization and (T, H) plane behaviour between Jbb = J and Jbb = 0 for triangular model (ii). In the ‘decorated’ (Jbb = 0) case the form of M varies with r and for a range of r there is a compensation point below T, where the magnetizations of the a and b sub-lattices cancel exactly to give m = 0. As T increases through the compensation point ma (H = +0) and mb (H = +0) change signs as a result of a curve of discontinuity in the (T, H) plane intersecting the T axis normally at the compensation point. In the ‘spin reversal’ (Jbb = J) case on the other hand, the magnetizations of the a and b sub-lattices are firmly locked together and M has the same form as a function of T for all values of r. In consequence there can be no compensation points and no segments of discontinuity in the (T, H) plane crossing the T axis. A question which cannot be answered so far from accurate results is which of these contrasting types of behaviour is ‘normal’. However from molecular field results in $7 above there is a range of r in which a compensation point exists for all values of Jbb/J except Jbb/J = 1, where the range becomes of zero length. Hence it may be conjectured that, rather surprisingly, it is the ‘decorated’ model be- haviour which is ‘normal’.

If a positive a-a interaction is introduced into model (ii), with Jbb = 0, the point on the H axis where ma changes from -1 to +1 ceases to be an isolated singularity and becomes the terminus of a curve of discontinuity. There could conceivably be behaviour of this type in the ‘spin reversal’ model, where there is a positive b-b interaction, although the curve of discontinuity could not bend round to meet the T axis as it does for a certain range of r with an a-a interaction. However molecular field theory indicates that the singularity on the H axis remains isolated as long as there is no a-a interaction, whatever the value of Jbb/J.

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1206 G M Bell

Martin D H 1967 Magnetism in Solids (London: Iliffe) Nakano H 1968 Prog. theor. Phys. Japan 39 1121 Nee1 L 1948 Ann. Phys., Paris 3 137 ~

Stanley H E and Harbus F 1973 Pad4 Approximants and their Applications ed P R Graves-Morris (London:

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Syozi I and Nakano H 1955 Prog. theor. Phjs Japan 13 69 Wannier G H 1950 Phys. Rev. 79 357

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