Unpublished paper: prepared for Proc. Phys. Soc in 1963

Exchange Interactions Between Mn⁺⁺ Ions in the KMgF₃ Lattice

C. G. Windsor*, Clarendon Laboratory, Oxford, England

*Present address: A.E.R.E., Harwell, Berks., England

ABSTRACT

Paramagnetic resonance experiments are described on exchange of coupled pairs of Mn⁺⁺ ions present in the KMgF₃ lattice. The nearest neighbour (nn) exchange interaction JSⁱ.S^j is found to be dominant, with the value J_nn = +4.5 ± 2 ^oK. The interaction between next nearest neighbours (nnn) is found to be so small that pair transitions between states of different total spin, which are normally forbidden, become weakly allowed. From the positions of these transitions the exchange interaction J_nnn  = +0.043 ±  0.005 °K is determined.

Introduction

Paramagnetic resonance experiments on exchange coupled pairs of ions in a dilute magnetic lattice provide a direct means of determining exchange interactions between given neighbours in the lattice. Previous Mn⁺⁺  pair experiments have been performed on the MgO lattice by Coles et al 1960) and Harris and Owen (1963) and on the ZnF₂ lattice by Brown et al (1961). We now describe Mn⁺⁺ pair experiments on the KMgF₃ lattice. Figure 1 shows the structures of nearest neighbour (nn) and next nearest neighbour (nnn) pairs in this perovskite structure. Salts of this structure are particularly suitable for theoretical studies. The magnetic ions lie on a simple cubic lattice although this may distort at low temperatures. Also, as the present results confirm, the magnetic properties are dominated by one type of interaction.

Figure 1: The structure of nearest neighbour and next- nearest neighbour pairs in the KMgF₃  lattice.

The Pair Hamiltonian

We describe our results by a Hamiltonian with axial symmetry along a bond axis

H = J Sⁱ.S^j + De (3S_zⁱ.S_z^j  -  Sⁱ.S^j) +  Hⁱ + H^j                              (1)

where Sⁱ = S^j = 5/2;  De contains a dipolar term ( -g²b²/r_ij³) together with possible anisotropic exchange. Hⁱ,H^j are the Hamiltonians of the isolated ions, each of form

          Hⁱ =  g bH.Sⁱ    + D_c {(S_zⁱ) ² -  1/3 Sⁱ.(Sⁱ + 1) } + F_cO₄⁰(Sⁱ)/180

Here O₄⁰(Sⁱ) represents the usual fourth order crystal field operator (Bleaney and Stevens, 1953). This Hamiltonian may be written in a form diagonal in the isotropic exchange by expressing in terms of the total spin S = Sⁱ + S^j. The Hamiltonian (1) then becomes

                                       H =   Ss H_s                                                          (2)

for S = 0, 1, 2 ... (Sⁱ + S^j ), where

  H_s =  g bH.S    + ½J{ S(S + 1)  -  Sⁱ.(Sⁱ + 1) -  S^j.(S^j + 1) }

+(3a_s D_e +  b_s D_c ) {S_z²  -    1/3S(S + 1) }  + g_s F_cO₄⁰(S)/180                  (3)

Expressions for the numerical constants a_s ,  b_s are given by Owen (1961). The expression for g_s is derived by Windsor (1963):

g_s = [ 35S²(S + 1)² - 180 S(S + 1) + 180 - 120 S(S + 1)Sⁱ(Sⁱ + 1)

+ 48 (Sⁱ)²(Sⁱ + 1)²   +444 (Sⁱ)(Sⁱ + 1) ]/

4(25+3)((25+5)(25-3)(25-1)                                               (4)

In previous Mn⁺⁺ pair experiments, and in the nn pairs reported here, this Hamiltonian is dominated by the isotropic exchange. The dipolar and crystal field terms are non-diagonal in this representation but the admixtures between states of different total spin will be of order D/J, and therefore small in this case. The allowed transitions are given by the selection rule

DS = 0      DS_z = ±1                                                                    (5)

so that these transitions may be found by solving the sub-Hamiltonians (3) for each value of the total spin. This is a very much simpler problem than solving the original Hamiltonian (1) but gives results in error by amounts of order D²/J. The positions of these transitions do not depend on the value of the isotropic exchange, which must therefore be deduced from intensity variations.

          When the condition that isotropic exchange is dominant in the Hamiltonian is relaxed, the matrix elements of the dipolar and crystal field terms between states of different total spin induce extra "forbidden" transitions. In the extreme case when all the terms are comparable, the analysis is formidable. Such a case has been studied by Hutchings (1964) for Gd⁺⁺⁺ pairs in LaCl₃.

However in the intermediate case when the exchange interaction is just a few times larger than the other terms the observed spectrum is still fairly simple to analyse and offers the prospect of measuring the exchange interaction from line positions alone. Thus the dipolar interaction has matrix elements connecting states for which DS = ± 2      DS_z = ± 0 . The resultant admixture of states gives rise to forbidden transitions whose intensity is of order (De/J)² of the allowed transitions, and which obey the selection rule

DS = ± 2      DS_z = ±  1                                                             (6)

If the exchange is still relatively large, these transitions give rise to nearly isotropic groups of lines on each side of the main line. These groups are then split by the other terms in the Hamiltonian. The line positions can be found to first order in D_e/J and D_c/J from Hamiltonian (3). With the external field parallel to the pair axis, we find the positions of the transitions where S<-> (S-2), S <= ZSⁱ, and where  ±S_Z <-> ±(S_z - 1), S_Z<=S,  are given by

gbH = hn  m   (2S-1)J + A_s,sz D_e +   B_s,sz D_c

where

A_s,sz =  [-18(S_z - S)² + (6S_z -4S - 1)(2S - 1)(S - 3)(S + 2) –

12Sⁱ (Sⁱ + 1)(4S_z - 2S - 1)(2S_z - 1)]/

 2(2S+3) (2S-1) (2S-5)

and where

B_s,sz   = -2/3 A_s,sz  +  1/3 (6S_z - 4S - 1)                                                 (7)

This is the situation we report here for nnn Mn⁺⁺ pairs in KMgF₃. The isotropic exchange is some four times bigger than the dipolar interaction, other terms being negligible. The forbidden lines thus have intensity of order 1/16 of the ordinary pair lines, and are definitely observable.

Lastly we consider the effect on the pair spectrum of other possible terms in the exchange Hamiltonian. An anti-symmetric exchange term D. Sⁱ x S^j has been shown by Moriya (1960) to arise on the super-exchange model when orbital admixtures are included, and may lead to canting of spins. It is found that the term has no matrix elements between pair states of the same total spin and so has no first order effect on the pair spectrum when the isotropic exchange is dominant. If the term were appreciable compared to the isotropic exchange then second order shifts of order ïDï²/J and forbidden transitions where DS = ±1 may be observed. Moriya (1960) has shown that if the centre of the pair is also a centre of inversion then anti-symmetric exchange must vanish. This condition applies to both nn and nnn pairs in KMnF₃, showing that the canting in this material must be from some other cause.

          Biquadratic exchange -j(SⁱS^j)² has been shown by Harris and Owen (1963) to be of importance in MnO. The term is isotropic and so does not effect the positions of allowed transitions. Their intensities are modified however and this is discussed later.

Measurements on Nearest Neighbour (nn) Pairs

Figure 2 The paramagnetic resonance absorption (differentiated) from a KMgF₃ crystal containing nominally 5% of Mn⁺⁺, as the magnetic field is swept in a (100) direction.  The microwave frequency was 9500 Mcs (X-Band), and the temperature 20°K.  The spectrum shows transitions from nearest neighbour pairs both parallel and perpendicular to the field direction.  The arrows show the calculated line positions for the parameters of table 1.

Figure 2 shows the experimental spectrum observed along a (100) direction where transitions from nn pairs with axes both parallel and perpendicular to the external field are seen. The figure shows clearly the large linewidth of order 400 Gauss, caused by unresolved manganese and fluorine hyperfine structure which has severely limited the accuracy of both line position and intensity measurements. The anisotropy constants of table 1 were found by fitting the positions of some fifteen transitions, measured both at X and Q band frequencies, and at several temperatures, to the Hamiltonian (3). Figure 2 also shows the theoretical line positions calculated according to these parameters. The sigma of the nn anistropy constants were found from the temperature variation of the intensities (Windsor, 1963). Table 1 also shows the calculated dipolar interactions for the host and concentrated lattices. It is seen that the observed interaction is consistent with mainly dipolar anisotropy.

Table 1. Spin Hamiltonian Parameters for Mn⁺⁺ Pairs in KMgF₃

Figure 3: The logarithm of the intensity ratio of pair transitions with S=2 and with S=1 plotted against inverse temperature.  The measurements apply to the two lowest field lines observed with the external field in a (100) direction, and at 36,000 Mcs frequency (Q band).

The isotropic exchange was deduced from a comparison of the intensities of transactions between states of different total spin.  The value given in the table was deduced from measurements at Q band frequencies of the intensity ratio of lines from S=2 and S=1 states over the temperature range 300 - 2.2 ^oK. These measurements are plotted in figure 3.  The intensities of transactions from higher spin states were consistent with this value, but not sufficiently accurate to determine any biquadratic term j(SⁱS^j)²  in the exchange Hamiltonian.

Measurements on Next-Nearest Neighbour (nnn) Pairs

 Figure 4: The spectrum along the (110) axis of next nearest neighbour pairs recorded under the same conditions as in figure 2.  The intense lines at low fields are from allowed transitions.  The weaker lines seen at high fields are ascribed to nnn forbidden transitions.  The arrows show the calculated positions and relative intensities of these lines for the parameters of table 1.

Figure 4 shows the experimental spectrum observed along the (110) axis of nnn pairs.  The two intense transitions seen just below the main line show just the anisotropy expected from the two outer transitions of nnn pairs.  Fitting the positions of these two lines to Hamiltonian (3) gives the values of D_e and D_c shown in table 1.  The figure also shows several weaker lines seen best at high fields, whose splittings from the main line are too large for them to be ascribed to either nn or nnn allowed transitions.  From the intensities and anisotropy of these lines we ascribe them to forbidden transitions of the type described earlier.  The exchange interaction was adjusted to give the best fit with the spectrum taking the values of De and Dc found above.  Figure 3 also shows the positions and relative intensities of the forbidden transitions calculated with the parameters of table 1.  The two outer groups of lines are not resolved in the figure, but an examination with better resolution shows these to be groups of lines as predicted.  It is seen that there is completely satisfactory agreement within the errors of order 50 Gauss caused by our neglect of second order terms in calculating the line positions.

Conclusions

We regard the main result of this work as the verification that only nearest neighbour interactions are of importance in this perovskite structure. It would be interesting if the experimental ratio J_nnn/J_nn= 0.010 ± 0.004 could be derived from theoretical considerations.

In the light of the above conclusion it is likely that the Bethe-Peierls-Weiss calculation of the nn exchange interaction from the Néel temperature will be particularly accurate, since this method relies on a one interaction supposition. Table 2 shows the value calculated by Smart (1963) using this method compared with determinations by other methods.

Table 2. The Nearest Neighbour Exchange Interaction in KMnF₃

The pair result seems distinctly low compared with the other values, especially since the increase in lattice constant from 4.00A in KMgF₃ to 4.198 in KMnF₃ would suggest a still lower value for comparison with results on the concentrated salt.

It is possible that the discrepancies may be resolved by including a biquadratic exchange term -j(SⁱS^j)² in the pair Hamiltonian. The pair experiment measures the quantity J(1 + 13.5j/J) and is thus modified by quite small values of j/J. Preliminary calculations by Windsor (to be published) suggest that the value of exchange deduced from the Curie-Weiss ( q ) is given by J(1 - 0.5 j/J)  while the values deduced from the Néel temperature and from paramagnetic neutron scattering are not appreciably changed. The discrepancies between the various results are then resolved by assuming j/J = -0.025  ±0.01 with J   =6.5±1 °K.

This ratio of j/J is less in magnitude than that found by Harris and Owen (1963) for Mn⁺⁺ pairs in MgO, although the change in sign is unexpected.

`The authors are very grateful to Drs. Owen, Griffiths and Harris for help and encouragement in this work.

References

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