Internal consistency of the Heisenberg-Dirac model for antiferromagnetism

J. Phya. Chant. Sol&

Pergamon

Press 1959. Vol. 11. pp. 97-104.

Printed in Great Britain.

INTERNAL CONSISTENCY OF THE HEISENBERGDIRAC MODEL FOR ANTIFERROMAGNETISM* J. S. SMART? Chemistry

Department,

Brookhaven

National Laboratory,

Upton,

L.I.,

N.Y.

(Received 29 Jamtary 1959)

Abstract-An attempt is made to investigate the internal consistency of the Heisenberg-Dirac model for antiferromagnets with nearest-neighbor interactions only. The Bethe-Peierls-Weiss approximation and the results of BROWNand LUTTIZ~GER are used to obtain relations between the N&e1 temperature and the exchange integral, J, and between the susceptibility at the N&l temperature and J. By combining these relations with experimental values of T,v and X(TN) for a given compound, two distinct values of J can be deduced. These values should agree if the theory is internally consistent. Comparisons have been made with experimental data on six antiferromagnetic compounds which seem to satisfy the nearest-neighbors-only restriction. The agreement is about as good as could be expected, considering the estimated experimental error. Approximate values of /Jl/k deduced for the compounds MnFz, FeFe, LaFeOs, LaCrOs, KFeFs and KCoFs are 1.8, 3.0, 27, 28, 6.0 and 12”K, respectively.

1. INTRODUCI’IUN

Heisenberg-Dirac model(lf of co-operative magnetic phenomena has as its basis the assumption that the interaction between a pair of magnetic atoms i and j is

THE

VQ = -2J$&

* S$

kl-

The sum over k is over sets of neighbors, from the nearest to the most distant (N’), while the second sum is over all pairs of kth nearest neighbors. Jk is the exchange interaction between kth nearest neighbors, As the magnitude of the exchange interaction generally diminishes rapidly with distance, only a limited number of values of k need be considered. In practice there are many cases where only one term is sufficient and very few where more than two are necessary. Although it has not been possible to obtain exact solutions for this model, various approximations which have been tried have yielded results in good qualitative agreement with experiment. For example, the simplest approximation of all, the molecular-field method, gives a description of ferro-, antiferroand ferrimagnetism which is generally good at all temperatures, although wrong in some details. Improved agreement can be obtained by using other approximations, which are more precise but are valid only in a limited temperature range. Thus, high-temperature

(1)

where Jfj is an exchange interaction between the ith and jth atoms and St and Sj are the spin operators. Thus the Hamiltonian operator for a system of N atoms is

If the system is a crystal which contains only one type of magnetic atom, it is possible to take advantage of the inherent symmetry and rewrite the Hamiltonian in the form -. . * Research performed under the auspices of the C‘S

Atomic Energy Commission. A preliminary account of this work was given at the Washington Meeting of the American Physical Society, May 1958. t Present address: Office of Naval Research, ment of the Navy, Washington 25, D.C. c

Lkth neighbors

Depart97

J.

08

S.

approximations such as the ~CVCHI(~), BJZHEPEIERLS--WEISS@) and constant-coupling@) methods take account of short-range-order effects which are neglected in the molecular-field treatment. Perhaps the biggest defect of the Heisenberg-Dirac model is its apparent failure to predict an ordered ground state in antiferromagnetism at absolute zero;(5f the relation of this point to the results of this paper will be taken up in the discussion. Despite the apparent successes of the Heisenberg-Dirac model, a number of objections have been raised against it;* the objectors have suggested that the agreement with experiment is only superficial, mereiy reflecting the fact that any model which introduces an apparent spin intcraction to account for the effects of the Pauli Principle will have some measure of success. There exists in the literature considerable controversy over the validity of the model, and the matter is by no means settIed at the present time. In principle, a pragmatic test could be made for any specific system by calculating the relevant exchange integrals, evaluating the partition function, and combining these results to derive such properties as the Curie temperature, spontaneous magnetization, susceptibility, etc. In practice, neither of the two results required can at present be obtained with any overall degree of accuracy. The calculation of exchange integrals is still in a primitive state, and for some systems it is not even clear which integrals are the important ones. The second step, the evaluation of the partition function (or its equivalent operation, the diagonalization of X) is on somewhat better footing; although, as previously stated, no exact solution has been obtained, the various approximations do lead to detailed and specific predictions which can be correlated and compared with experiment. Thus, the situation now existing is that arguments based on first principles have failed to obtain any measure of agreement about the validity of the Heisenberg-Dirac model, while a straightforward test of the type described in the previous paragraph is impossible because of mathematical difficulties. However, to set our sights on a much more modest goal, it should in principle be possible at least to check the infernal corrsistency of the -----. -__ * See SLATER(~)for a detailed exposition of the objcctions to the Heisenberg-Dirac

model.

SMART

model. As an example, consider an antiferromagnetic system with nearest-neighbor interactions only. Given an adequate theory, the value of the exchange integral J could be obtained from experiment in any one of a number of ways; examples of measurements which would provide a value of J are: (a) N&l tempcraturc, (6) susceptibility at the N&l temperature, (c) inelastic neutron scattcring,(i) (d) specific heat@) and (e) excitation of spinwave resonances.@) If the theory is self-consistent, the values of J obtained from the diffcrcnt measurements should be the same within experimental error. The desirability of such comparisons is obvious, but very little has been done along these lines in the past. Probabiy the principal reason is that the more rigorous theoretical treatments have generally been limited to spin .i and to nearestneighbor interactions only, while practically all of the experimental data are on more complex systems. Of the methods mentioned in the previous paragraphs whereby J may be evaluated, appreciable experimental data exist only on the See1 temperature, the susceptibility and the specific heat. HOFMAZIN et u/.(s) have previously used the specific-heat data on ferromagnetic and antiferromagnetic systems to make estimates of exchange interactions. In the present paper, we shall see what results can be obtained by using the NCel temperature and susceptibility data. For the necessary theoretical relations, either the BethcPeicrls-Weiss or the cr~nstant-coupling approximation to the Heisenberg-Dirac model might be used ; these two methods are generally regarded as the best approximations for the high-temperature range (T >3 TX) and they give essentially identical results, both for the NCel points and the susceptibilities. We have chosen to use the B-P-W method, mainly because BROWN and Lu~~INGER(~~) have already extended this method to obtain the partition function and the NCel temperature for arbitrary spin. Thus only the susceptibility is needed, and their first result makes its calculation a straightforward, though tedious, matter. Howcscr, like BROWN and LUTTIISGER, we have retained the nearest-neighbors-only restriction, as formulation of the B-P-W method for both arbitrary spin and more than one set of interactions leads to equations of staggering complexity. BROWN and LUTTINGI~H have also derived a cfassical approximation of the

INTERNAL

CONSISTENCY

OF

THE

B-P-W method in which the spin operators are regarded as classical vectors of length [S(S+l)]*. It will be instructive to use the results of this method, and of the molecular-field method as well, to determine values of J and to make comparisons with the values determined from the B-P-W method itself. The relevant equations are given in the next section.

HEISENBERG-DIRAC

(c) Bethe-Peied-Weiss

MODEL

99

theory

BROWNand LUTTINGER give the partition

function for a B-P-W cluster of a center atom and its n nearest neighbors in the form Z(j, ho, hr) = &%j)+&(j&?+

+2h(j)bh+&2(j)h2

(6)

Here 2. EQUATIONS FQR THE N&L ‘jl?MPERATURE AND SUSCEPTIBILITY AT THE NEEL TEMPERA-

(a) Molecular-field theory The molecular-field theory gives the following well-known results:

iJI -=

~TN

k

(44

2nSo(So+ 1)

IJI

3cM

-z.z k

4nSo(so+

(‘+b)

(b) Classical Bethe-Peieds-Weiss theory The relation between J and TN is

IJI -=

(5a)

2so(f50+ 1)

k

4(n-

l)so(so+

1)X&

Zoo(j) =

9&j> =

2

44)

WI TN)

2

(74 (7b) The sum over 5’ is from I&--SO] to &+Ss, and the sum over Sl is from 0 to nSs, except for the term Sr = SO, S = 0, which is taken care of by the 9~’ term. Also of ways of forming spin & from

R spins SO. 00 =

~(~+~~+~l(~l+~)--so(so+~)

Ql

=

~(~+~)--l(~l+~)+~o(so+~)

6~0

=

Kronecker

delta

Then

Yn(l)+Yn’(l)

= 978{[s(S+ ~)]-‘(uo~~+ uoai[S(S+ ~)]-1j)}-49n'(j) 12&l(j) = Yd[S(S+ I)]-I(u0ulj2ooui[S(S+ l)]-lj))+4Yn'(j)

!

I

exP(jooW+llf(j)

s

S‘

w(Sr) = number

where 9-1 is the inverse Langevin function. BROWNand LUTTINCER do not give an expression for the susceptibility, but by the usual methods we find

IJI -=

The functions Z&j) are perhaps best defined by first defining the sums

I)%

In these equations, T*r is the Neel temperature, CM the molar Curie constant, XM the molar susceptibility, n the number of nearest neighbors for each atom and Se the spin of each atom in units of ri.

k

= JikT (note that j < 0 for antiferromagnetism) ho = gWo!J h =giWlJ HO = applied magnetic field HI = effective field acting on shell of nearest neighbors p = Bohr magneton

j

(84

24220(j)

(8b)

24&(j)

(8~) (84

= ~~8{[S(S+l)]-~(a~j~+osor[S(S+l)]-~j)}-49~’(j)

100

J.

S.

SMART

The Bethe-Peierls-Weiss theory does not provide an explicit relation between J and TN analogous to equations (4a) and (Sa), but jh: = J/kT_I: is determined from the relation n&(jN)+2ZOZ(jN)

=

0

(94

Values forjN for different values of tl and SO are tabulated in reference (11). Again, by extending BROWN and LUTTINGER’S results, we find for the relation between J and XM( T,v)

J-

_= -

k

~~02(j~)~20(jN)-~112(jhr) ---

---

jNz~(jN)~2~02(jN)-“Zll(jN)}x

3cM

X sO(s0-t 3. COMPARISONS

WITH

EXPERIMENT

The gist of the previous section theory gives relations of the form

IJI K

=

(9b)

ljXM(jN)

is that

#(n, So)Ts

each

(104

and

IJI=

I’@,

c*M

So)

k

(10”)

x~(T~)

Values of 4 and I’ for tl = 6 and 8 and for SO = l/2, 1,3/2,2 and S/2 are given in Table 1. Because

of the labor involved, values of P were computed only for SO = l/2, 3/2, S/2, the values for SO = 1, 2 being obtained by interpolation. This procedure may introduce a small error, if, as seems likely, there is a small integral-half integral alternation in I; however, calculations indicate that the error introduced in this way is less than 1 per cent and thus considerably smaller than the average experimental error in the available data. In selecting experimental data for use in the consistency check, we must be careful to choose only compounds which satisfy the nearestneighbor-interactions-only restriction. Moreover, as the B-P-W method is limited to cases in which nearest neighbors of a given atom are not themselves nearest neighbors, we must also include this restriction. The list of such compounds for which adequate data on T.v and X&TN) exist is extremely limited. There is, of course, no hard and fast rule for deciding how many sets of interactions are important in a particular crystal, and at best the decision rests on indirect evidence. Some indications can usually be obtained from the crystal structure and magnetic structure; perhaps the most quantitative information comes from the ratio B/T_,?,where 0 is the constant in the high-temperature approximation to the susceptibility, x(T) = Ci(T+o)

Table 1. Values of 4 and r as given by the molecular-field (MF), the classical Bethe-PeierLs- Weiss (BPWc), and the Bethe-Pet&& Weiss (BP W) theories. The top number of each pair is + and the bottom number is r. -_ __ . .~_ .-

I

.-

SO

MF

I

_-

n=8

n=6 BPWc

BPW

i

BPWc

MF

0.289 0.124

BPW

0.316 0.115 -

112

0.3333 0.1667

0.410 0.164

0.499 0.1445

0*2500 0.1250

1

0.1250 0.0625

0.154 0.0615

0.162 0.0587

0.09375 0.04687

: 0.1085 1 0.0465

0.1107 oG451

0.08203

0.08396 0.0321

0~05000 0.02500

j

0.05869 0.0245

312

0.06667

, 0.0328

2

0.04167 0.02083

0.05128 0.0205

512

0.02857 0.01429

0~03515 0.01406

..___

0.05787 0.0248

0.03648 0.0153

0.05200 0.0202

1 :::::;9

j :::;g

~-::::iy

1 ::E$ -

INTERNAL

CONSISTENCY

Table 2. Vales

z-- ._

Compounds

OF

of O/TN

I

THE

WTC

~-

Temperature

I

I ;

1.18. l-487

range

BPW theory (nn only)

!

1.781

I

101

for someantiferromagnetic com~nd~

Exp. MnFz FeFa LaFeOs LaCrO3 KFeFs KCoFs

MODEL

HEISENBERG-DIRAC

::35i.i

I

1.16 1.17 1.24 1.26 l-25 1 a26

4-S 2-3

TN TN

2-3 3-4 3-4

TN TN TN

1 I

_Reference (17). t H. B~ZETTEand B. TSAI, C.R. Acad. Sci., Puris212,119 1 Reference (20). 8 N. ELLIOTT. Private communication.

--_ --_

l

For example, for the simplest case of two sublattices, the ratio should lie in the range 1.1-1.5 if only one type of interaction is present (the exact value depending on n and Se). However, as Lx(“) has pointed out, if 0 is deduced from measurements at temperatures not too high compared with Tc, then values larger than the theoretical one must be expected. Six compounds-MnFs, FeFs, LaFeOa, LaCrOa KFeFs and KCoFs-have been chosen for the comparison. MnFs and FeFs both have the rutile structure, in which the magnetic lattice is bodycentered tetragonal. Strictly speaking, each magnetic atom has two nearest neighbors, one in each direction along the c-axis. However, it Seems likely that the important interactions are super-exchange interactions between a “central” atom and its eight corner neighbors, and the neutron-diffraction result@) support this view. The other four compounds all have pseudo-perovskite structures,(raJ4) in which the magnetic lattice is approximately simple cubic. Each magnetic atom has six nearest neighbors and is separated from each of them by an intervening anion. The neutron-diffraction data(lSJs) are consistent with the idea that the principal magnetic interaction in these compounds is a superexchange interaction between nearest neighbors. Table 2 gives the available information on the t?/Tc values. The last column shows the range of temperatures in which the susceptibility measurements from which 0 was determined were made.

(1941).

The experimental data which are needed, in conjunction with equations (1Oa) and (lob), to determine J are given in Table 3. In view of the results of BIZ~E and T&l@ on the effect of particle reorientation on the magnetic susceptibility of antiferromagnetic compounds, it would be preferable to use only data on single crystals. Unfortunately, such data have been obtained only for MnFa; however, for this compound there are two different sets of data, due to FONER@‘) and to BIZETTEand TsAI.(‘~) We have quoted only FONER’S data in the table, but the two sets are in good agreement and give essentially the same results for J. The results on the other four compounds have been taken on polycrystalline samples. The accuracy of these results is difficult to assess, but it is probably around f70 per cent, as compared to an estimate(l7) of f2 per cent for the MnFs data. The LaFeOa values are perhaps the least reliable, inasmuch as there is some evidence(a0) of a first-order phase transition in this material near the Neel temperature. The last column of Table 3 gives the values of CM which were used in the calculation. In principle, it would be better to use experimental values of CM, as this helps to take into account certain effects, such as the orbital contribution to the susceptibility, which cannot be estimated accurately by theoretical methods. However, if the data are not taken at very high temperatures, the measured values may be considerably in error. We have compromised by using the theoretical spin-only

102

J. S. SMAR’I Table 3. flata on a~t~~r~~t~c

compoundf --=

Compounds

n

: I

-‘-‘MnF2 1 8 : FeFz 8 ; 6 LaFeQs i 6 ! LaCrOs KFeF3 ; 6 1 KCoFs 6 , I .-.-. -.-. _!. --. ~...--__-..__ * Reference (17). f li. BIZETE and

So

I T&‘K)

’

S/2 2 512 312 ;,2

j

68f 79t 7401

1 0.0251* ! 0.0196t 0~002ll~

I 4.38 ; 3+38t 4.38

;;yi

j ~;~~~~;

z!;

-.-..-.

-

I _.

XM(TN) (cgsu/mol)

c.11

--.

B. TSAI,CR. Acad. Sci., Paris (1941). t: Reference (20). $ Reference (14). !I N. ELLIOTT.Private communication.

values where they seem to be applicable (for MnFa, LaFeOa, LaCrOa) and the experimental values in the other three cases. As the evaluation of /f//K by the three different theories shows much the same pattern for each of the six compounds, it will be sufficient for comparison of the theories to show the detailed results for a single system only. Table 4 gives such results Table 4. Values of 1Jl/k for FeFz ___-L--: -~

Table 5 gives the jJ]/k values determined for all five compounds by the B-P-W equations. The last column, marked W, Iists [f/,/k values proposed by HOFMAN et aZ.(*)from their analysis of the specific heats of antiferromagnetic compounds. Table 5. Values of /J//k determined by B-P-W theory ____~__ -Compound

X0-N)

2.47 i 3.12 MF BPWc ! 2.86 / 3.10 3.06 BPW ; 2.88 I _-.--_ --_.-.-. _ for FeFa. First of all, we notice that the discrepancy between the corresponding values of 1Ji/k decreases as we go from the molecular-field theory to the B-P-W theory; this gives some confidence that the more rigorous theory does give quantitatively better results. Also, the molecular-field theory gives too high a value for JJI by the T,v method and too low a value by the X(T,v) method; from equations (lOa) and (lob), one can see that this behavior is a direct consequence of the fact that the molecular-field theory overestimates both the NCel temperature and the susceptibility at the Gel temperature.

Values of

I

MnFz FeFs L&e03 LaCrOa KFeF3 KCoFs

1J i/k from

! pNi

IJl/k from

i TN ’

212, 119

: ! 1

W’N)

1.70 2.88 26.3 26.9 5.98

j

1.87

; !

3.03 28.9 30.8 6.04

11.3

13.6

iw

/ i I

1.73. 3.01* -

I * Reference 8. 4. DISCUSSION

The agreement between the values of Jobtained by different methods on the same compound is seen to bc reasonably good. The discrepancy that dots exist may be attributed to one of four causes: (a) a defect in the Heiscnberg-Dirac model itself, (b) a defect in the B-P-W approximation, (c) the existence of more than one kind of interaction in the compounds studied, and (d) experimental error. We have no way of separating these various contributions, but it is worth noting that the

INTERSAL

CONSISTENCY

OF

THE

average discrepancy is little, if any, larger than would be expected from the estimated experimental error alone. Obviously, no general conclusions can be drawn from an analysis of data on only six compounds, but the results are sufficiently encouraging to suggest that further investigations of this type be made. It would be desirable to extend the comparisons by studying more compounds and by making several different kinds of observations on the same compound. We would like to remark again that the results recorded in Table 5 bear only on the self-consistency of the Heisenberg-Dirac model and not on its general validity. What the results do suggest is that for each of the six compounds there exists a number, J, which, when inserted in the appropriate B-P-Wequations, will give a reasonably quantitative (say, f 10 per cent) description of the hightemperature properties of these systems. It is, of course, in keepingwith the spirit of the HeisenbergDirac model to interpret the J’s as exchange integrals, and those who are interested in the calculation of such integrals may find useful information in a comparison of the magnitudes of the values of J in Table 5. For example, it appears that in the iso-structural perovskite-type compounds, the superexchange through oxygen is appreciably larger than that through fluorine. This result is consistent with general ideas about the relation between superexchange and electronegativity of the intervening anion.(lslsl) Moreover, it is tempting to attribute the difference in J value between FeFs and KFeFs, where the two compounds have the same anion and the same magnetic cation, to the smaller Fe’+-F--Fe++ bond angle in FeF2. These remarks, however, must be regarded only as speculations. In connection with the general validity of the Heisenberg-Dirac model of antiferromagnetism, MARSHALL@) has studied the ground state predicted by this model for spin 4 and for simple lattices with nearest-neighbor interactions only. His results indicate that the ground state is disordered in all the cases investigated, and thus antiferromagnetism does not occur. This result, however, is not as incompatible with ours as might seem at first glance. For instance, the Heisenberg-Dirac model may still produce antiferromagnetism for SO > 3. Also, the ground-state investigations refer to the low-temperature pro-

HEISENBERG-DIRAC

MODEL

103

perties of the system, while our comparisons have been concerned with the high-temperature properties; there is no reason in principle why the model should not give correct predictions in one temperature range and not in the other. As an example of a way in which this might happen, MARSHALL has noted that there are probably some ordered states which have energies almost as low as the ground state, and thus the introduction of some other small terms, notably anisotropy, into the Hamiltonian might be sufficient to produce an ordered ground state without affecting the hightemperature properties very much. The calculations of TAKETA and NAKAMURA(~~) lend some support to this argument.

Acknowledgements-I am indebted to L. CORLISS, N. ELLIOT, J. HASTINGSand S. FONER for permitting me to quote some of their unpublished experimental results ; to W. MARSHALLfor some comments on the discussion; and to Miss D. L. DUFFY for her invaluable assistance with the computations.

REFERENCES 1. See, for instance, VAN VLECK J. H., The Theory of Electric and Magnetic Susceptibilities p. 316. Oxford University Press (1932). 2. OGCCHI T., Progr. Theor. Phys. 13, 148 (1955). 3. WEISs P. R., Phss. Rew. 74, 1493 (1948). 4. YVON J., Cah. Phys. Nos. 31-32 (1948). NAKAMCRA T., Busseiron-Kenkyu No. 63, 12 (1953). KASTELEIJNP. W. and VAX KRANENDONK J. Physica s’Grav. 22, 317, 367 (1956). KIKUCHI R., Ann. Phjx, Neu York 4, 1 (1958). 5. MARSHALLW., Proc. Roy. Sot. A232, 48 (1955). 6. SLATER J. C., Rev. Mod. PhJls. 25, 199 (1956). 7. BRCJCKHOUSE B. N., Phys. Rev. 106,859 (1957). I-CAR P. K. and BROCKHOUSE B. N., Bull. Amer. Phys. Sot. 3, 195 (1958). 8. HOFMAXNJ. A., PASKINA., TAUER K. J. and WEISS R. J., J. Phys. Chem. Solids 1, 45 (1956). 9. KITTEL C.. Phvs. Rev. 110. 1295 (1958). 10. BROWN H.‘A. and LUTTIN&R J. M:, Phis. Rev. 100, 685 (1955). 11. I>I Y.-Y., Phys. Rev. 84, 721 (1951). 12. ERICKSONR. A. and SHULL C. G., Phys. Rm. 83, 208 (1951). 13. YAKEL H. L., Acta Cryst., Camb. 8, 394 (1955) (LaFeOs and LaCrOs). 14. MARTIN R. L., NYHOLM R. S., and STEPHENSON N. C., Chem. and Ind. (Rev.) 83 (1956) (KFeFs and KCoFs). 15. KOEHLERW. C. and WOLLAN E. O., J. Phys. Chem. Solids 2, 100 (1957) (LaFeOs and LaCrOs).

104

J.

S.

16. CORLI~~ L. M. and HASTINGSJ. H., private communieation (KFeFa and KCoFs). 17. FOXER S.. private communication. 18. BIZETIXH:, Ann. Phys., Paris 1, 233 (1946). 19. BIZ~ H. and TSAI B., C.R. Acad. Sci., Paris 238,

1575 (19.54). 20. JONKER G. H., Physica 22,707

(1956). Results in fair

SMART agreement with JONKER’Sare given by WATANABE H., Sci. Rep. Res. Inst. Tohoku lJm.v. A8, 14 (1956). and BENO~T R.. C.R. Acad. Sci., Paris

240, i&9 (1955). 21. AXDEXSONP.W.,P&s. Rew.79,350 (1950). 22. TAKETA H. and NAMMURA T., J. Phys. Sac. Japan 11, 919 (1956).