OPTICAL SPECTROSCOPY OF MAGNETIC INSULATORS

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OPTICAL SPECTROSCOPY OF MAGNETIC INSULATORS

S. Hüfner

To cite this version:

S. Hüfner. OPTICAL SPECTROSCOPY OF MAGNETIC INSULATORS. Journal de Physique Col-

loques, 1971, 32 (C1), pp.C1-710-C1-717. �10.1051/jphyscol:19711249�. �jpa-00214078�

OPTICAL SPECTRO SCOPY OF MAGNETIC INSULATORS

by S. HUFNER

Fachbereich Physik Freie Universitat Berlin

RBsumb. - On passe en revue les recherches sur les proprietes optiques des isolants magnetiques. Jusqu'ici ce type de mesures a fourni une quantite #informations sur les interactions de differents types de niveaux d'excitons (electroniqde et magnktique), et a aide, a un degre moindre, la dktermination des constantes d'echange, qui sont necessaires a la des- cription de proprietks microscopiques. Le cas le plus simple est celui des orthoferrites et des orthochromites de terres rares, oh les ions de terres rares sont soumis seulement a des interactions magnetiques faibles, qui peuvent Ctre decrites par un modele de champ moltculaire ; on donne comme exemple les resultats sur ErCrO3. Un autre cas simple est celui de paires couplees magnetiquement, isolees dans une matrice diamagnetique tel que Cr3+ dans A1203 (rubis). Les interac- tions que l'on deduit dans ce cas sont en bon accord avec celles rtellement trouvkes dans Cr203. Les interactions mag&- tiques dans les systkmes comportant de fortes interactions, conduisent & des transitions optiques additionnelles nouvelles, les bandes lathales dues aux ondes de spin. Elles ont ktk, pour la plupart, complktement BtudiCes dans MnF2, et pour une transition particulikre, les phenomenes qui lui sont lies (dispersion des excitons, interaction magnon-exciton) ont bte entikrement compris du point de vue quantitatif. Dans les systemes magnetiques avec de fortes interactions et plusieurs sous-reseaux magnetiques, I'interaction des differents etats crte une separation magnktique de Davidov, comme dans le spectre optique de Cr20 3.

Abstract.

-

The optical investigation of magnetic insulators is reviewed. So far this type of measurements has yielded much information on the interaction of various types of exciton levels (electronic and magnetic) and has aided less to the determination of exchange constants, which are needed for a description of macroscopic properties. -The simplest case is that of the rare earth orthoferrites and orthochromites, where the rare earth ions experience only weak magnetic interactions, which can be described by a molecular field model ; as exemple data on ErCrO3 are presented.

-

The next simple case is that of isolated magnetically coupled pairs in a diamagnetic host, such as e. g. Cr3+ in A1203 (ruby). The interactions deduced in that case are in fair agreement with those actually found in CrzOs. - Magnetic interactions in systems with large interactions lead to new additional optical transitions, the spinwave sidebands. They have most thoroughly been investigated in MnF2, and for one particular transition the phenomena connected with it (exciton dis- persion, exciton magnon interaction) are fully understood quantitatively.

-

In magnetic systems with strong interactions and several magnetic sublattices the interaction of the various states creates a magnetic Davidov splitting as e. g. in the optical spectrum of Cr20 ^3.

I. Introduction. - The investigation of magnetic materials with light gives two kinds of information.

For one thing, the difference in intensity of phase before and after going through the material allows to determinate the energy levels in the magnetic mate- rial. Secondly, the process of the interaction of elec- tromagnetic radiation with magnetic materials can be studied in itself. This summary will be mainly concerned with the first topic. Also from the various possibilities to study the energy levels in a magnetic crystal by optical means, e. g. absorption and emission spectroscopy, Raman scattering, Faraday effect mea- surement, etc., we shall mainly concentrate on the first method.

In the experiments of emission and absorption spec- troscopy, transitions between the groundstate and excited state energy levels are observed. Therefore always in addition to the energies of a magnetic material in the groundstate, those of an excited state can be measured in an optical experiment. The groundstate energies can be used to come to a micros- copic description of macroscopic magnetic properties (e. g. magnetization, anisotropy, etc.) ; the energies measured in the excited states can be used to test the properties derived from the groundstate energy levels o r to give additional data to determinate all para- meters which are necessary to describe a magnetic material from a microscopic point of view.

To understand the energy levels in a magnetic insulator, one may start out with the energy levels of a free atom or ion in a magnetic field. Here we have the Zeeman effect and the splitting of the optical

p . H in the ground- and excited state respectively.

This is a very simple experiment with an unambigous interpretation yielding the magnetic properties one needs to know for an atom.

The same kind of experiment is possible in a para- magnetic crystal. Here the zero field energies are those of the free ion modified by the electrical crystal field. If they carry a magnetic moment, an external magnetic field can remove the degeneracy ; the split- ting of the absorption and emission lines then gives the magnetic moments of the optical states. These lines mentioned so far are understandable in a pure one ion picture. There are of course magnetic interac- tions, dipolar and exchange, between the various ions, which eventually at sufficiently low temperatures drive the crystal into a magnetically ordered state.

These interactions we are after and they describe the magnetically ordered state. There are two ways t o determine them :

(1) to look for additional lines, which d o not come from the single ion transitions in the paramagnetic state (which has been very successful in some Ising type rare earth antiferromagnets) or by incorporating some of the magnetic ions into a diamagnetic host and investigate the energy level diagram of magnetically coupled pairs (as e. g. in the case of Cr3+ in Al,O,, namely ruby) ;

(2) to study the lines in the magnetically ordered state and to deduce information about this state from the additional features then observed as compared to the paramagnetic state.

lines give directly the magnetic interaction energy The easiest, and of course most desirable case is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711249

OPTICAL SPECTROSCOPY OF MAGNETIC INSULATORS C 1

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that when the magnetic interactions on the ion under consideration can be described by a molecular field approximation, i. e. the energy levels are split by the interaction in the same way as by an external applied field. Then the magnetic interactions can be deduced in the same way as for a paramagnetic ion in an exter- nal field. There are two classes of compounds, which can nicely be described by that picture, namely the rare earth ions in the rare earth orthoferrites and ortho- chromites and Ni2+ impurities in MnF,, RbMnF,, and KMnF,. Equally simple is the case of two magne- tically coupled ions in a diamagnetic lattice. The energy level diagram is then determined in a simple way by the spins of the two ions and by the interaction cons- tants, which in turn can be determined from the spec- tra. More complicated is a case like e. g. pure MnF, ; here the magnetic interactions between the Mn2+

ions are so strong that one has to abandon the one ion model, which is very effective in describing the examples mentioned above. In a first approximation one can treat the electronic and magnetic system sepa- rately in a many ion picture ; the excitations in these two systems are then excitons and magnons. If there exists a coupling between the two systems (e. g. via spin orbit coupling), then in optical absorption and emission experiments simultanous excitations in the two systems can be observed : spin wave sidebands, which yield essential information on the magnon- exciton coupling. A further complication may arise if there are different magnetic sublattices with inter- sublattice interactions. Then the energy levels on the various sublattices are no longer degenerate. The splitting of the energy levels of the various sublattices by that intersublattice interaction is called Davidov splitting.

In the following sections we shall try to give examples and details of the principles just mentioned.

We will only be able to emphazise some of the expe- riments, which we think are relevant, and we want to apologize already here to all those whose work will not be mentioned and may be equally or even more important than the presented one.

11. Molecular field approach (ErCrO,).

-

The examples which are nearest to the free atom in an external magnetic field are those which can adequa- tely be described by the molecular field approximation.

From a number of examples it is perhaps ErCrO,, which for the Er3+ ion gives the spectra showing most clearly the magnetic effects. This compound belongs to the class of perovskites with the general formula ABO,, where A stands for a rare earth ion or yttrium and B for Fe, Cr or Mn. The magnetic properties of the ACrO, series have been summarized by Bertaut et a1 [I]. ErCrO, has been investigated by neutron diffraction [2], Mossbauer-effect [ 3 ] , magnetization [4], optical experiments [5], and specific heat [6]. From these measurements it is known that the Cr ions order at T = 133 OK in a Gx mode (using Bertaut's nota- tion). The Er ions show cooperative effects only in the liquid helium temperature region. It is also known from the neutron diffraction measurements that in the liquid helium temperature region the Cr moments change from a Gx mode to a G , mode.

It will be shown now what kind of information on

the Er ions one can extract from an optical experi- ment [7]. To a good approximation the Er ions can be visualized as sitting in a molecular field produced by the Cr ions which acts as an external field and therefore one can describe the energy levels by a molecular field acting in addition to the crystal field interaction. Data are shown (Fig. 1) for the

FIG. 1. - Spectra of ErCrO3 at 4.2 O K and 77 OK and corres- ponding energy level diagram.

transition around 18,500 cm-l. At T = 0 OK only the lowest crystal field state of the 41,,12 groundstate is populated and the transition therefore reflects immediately the crystal field splitting of the 4S31, excited state. By warming up the crystal, additional crystal field levels of the groundstate get populated and transitions originating from them are observed giving the energy of the first excited crystal field states at 46 cm-I and 113 cm-l. We shall now concentrate on one of the lines originating from the lowest crystal field state of the groundstate. It shows a splitting, which is clearly due only to the splitting of the groundstate which can be seen from the intensity variation of the two lines with temperature. The splitting of the excited level is hidden in the linewidth, and therefore the line splitting gives directly the groundstate splitting, which is the total interaction energy of an Er ion in the magnetic state. Figures 2 and 3 show the tempera- ture dependence of this splitting. At high temperatures

FIG. 2.

-

Temperature dependence of the line with the lowest energy (Ia) of the 4 1 ~ 512 + 4 s 312 transition.

A EIBSCHUTZ et al.

FIG. 3. - Groundstate splitting of Er3+ in ErCrO3 as a funo tion of temperature ; also shown are the results of Mossbauer

effect measurement (see Ref. 3).

(T

>

40 OK) it follows the Cr ion sublattice magne- tization. Below T z 40 OK the splitting deviates from that expected from the Cr ion sublattice magnetiza- tion. This is due to an Er-Er interaction. The Er ions get polarized by the Er-Cr interaction and thus the Er-Er interaction produces an additional splitting.

At 9.7 OK the splitting undergoes a sudden change. This

Er-Cr interactions actually lead to the increasing Er- Er interaction.

One comment to whether the Er ions show coope- rative order or not should be made. The case just described is analogues to the one of an antiferroma- gnet in an external field ; then there is also no sharp transition temperature manifesting itself in a 1- anomaly in the specific heat. Yet there is of course still cooperative order, which sets in gradually ; we suggest that the order of the rare earth sublattices in all the systems of the type RBO, is of the same kind.

The sharp optical lines split by the internal magnetic interactions and/or by an external magnetic field have been used to study a great number of different com- pounds 181. The optical method proved to be most useful for the study of the anisotropic exchange inter- actions [9]. Here the splitting of the optical lines by the internal interactions give their contribution in the various crystallographic directions, whereas the split- ting in an external magnetic field gives the g-tensor.

In other investigations it has proven possible to study magnetic structures and the directions of the various fields in a multisublattice system 110-121. Finally it should be mentioned that the molecular field approach has proven to be adequate also for dilute impurities (mostly Ni2+) in transition metal fluorides [13-181.

111. Isolated pairs ; Cr : A1203 (ruby).

-

The situa- tion next complicated to that of an ion whose proper- ties can be described by the simple molecular field approach is that of two neighboring ions coupled by an exchange interaction and thus behaving as a unit.

The case most thoroughly studied in this respect is ruby [19-281, though by far not all the details of the very complicated spectra have been analyzed. The object and hope here is to determine the exchange interactions between the ions on the different lattice positions and to calculate the macroscopic magnetic properties of the pure substance, namely Cr203, from the exchange integrals derived in ruby. This object has not yet been reached.

The Heisenberg interaction between two ions A and B is :

where SA and S, are the spins of the two ions and J is the exchange integral. The Cr3' ion has a ground- state 4A2, which is an orbital singulet and fourfold spin degenerate, therefore SA = SB = 312. The ground- state of a chromium pair will then be split into a quartet level (S = SA

+

SB ; S = 0, 1, 2, 3) separated according to the Land6 rule. The energy level diagram of the groundstate of a Cr pair is shown for ferro- and antiferromagnetic exchange in figure 4. Since all the pair lines observed so far are in the vicinity of the R lines, it would be desirable to know the energy level diagram for a pair in the excited state. This is a little more involved than for the groundstate because the crystal field splitting of the two 2E level may or may not be comparable to the exchange splitting. Not much information and understanding on the excited states of the pairs have been obtained so far and there-

OPTICAL SPECTROSCOPY OF MAGNETIC INSULATORS C ¹

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7

FERROMAGNET&C COUPLING I J I

J P O

-:

^IJI-S=O

ANTIFERROMAGNETIC COUPLING J t O

FIG. 4. - Exchange splitting of a Cr3f ion pair groundstate (from Ref. 23).

fore they will not be considered here any more (but see Ref. 28).

Relative to a particular ion there is a number of neighbor positions possible. As a rough approxima- tion it can be assumed that the size of the interaction decreases with increasing distance between the neigh- bors. So far only spectra due to first, second, third, and fourth nearest neighbors have been identified. Two main methods have been used to attribute a particu- lar observed line to a certain neighbor configuration.

Mollenauer and Schawlow [23] measured the posi- tions of the lines under the application of uniaxial stress for various crystal directions. All the neighbor configurations have different axes of symmetry and by comparing the measured symmetry of the linepatterns with that expected from geometrial reasons (see Fig. 5) it was easy to attribute the various lines to specific neighbors. As an example we show in figure 6 the line

FIRST

-

FIG. 5. -Location of pair axes and angular convention as used in Ref. 23.

ANGLE OF APPLIED STRESS, 8

FIG. 6 .

-

Observed shift of line at 7,540 A ; the experimental points fit rather closely to a curve of the form A cos 2 (0-90°)

+

B, the component of stress along the fist neighbor axes (from Ref. 23).

position of a line due to a first nearest neighbor ; this line should be symmetric to the C3 axis, because the first nearest neighbors lie along the C, axis. This is indeed observed. Kisliuk et al. [27] on the other hand used the temperature dependence of the inten- sity of the various transitions to attribute them to particular neighbor configurations.

Table I gives the exchange integrals (negative sign means antiferromagnetic exchange) for the various neighbors. To improve the fit of eq. (1) to the energy levels, a term of the form j(SA SJ2 was added [28].

Except of the second neighbors this term represents only a minor correction.

Exchange parameters for ruby and Cr203 (in cm-l)

Near neighbor Ruby c r z o ,

pair type J

- - -

^j

-

^J

First

-

240 ?

-

128

Second

-

83.6

-

^9.7

-

54.0

Third

-

11.59 0.06

-

3.6

Fourth

+

^{6.99 0.14}

-

0.8

One might be tempted to use the numbers given in Table I to calculate the macroscopic properties of Cr203. Fortunately recently the exchange constants have been obtained directly for Cr203 by fitting a Hamiltonian containing the exchange integrals [29]

to the measured magnon dispersion relations. The results of this procedure are also given in Table I.

It can be seen that they are similar though not equal to those determined from the spectra in ruby. This might have been expected from the slightly different lattice constants of A1203 and Cr203 ; yet the degree of disagreement is surprising. This also cautions the hope to obtain from experiments in diluted samples

IV. Spin wave sidebands : M a , .

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Having seen the unsatisfactory result of the preceding paragraph, one might be tempted to determine the exchange constants directly from spectra of concentrated mate- rials. To summarize the results of all these studies, it is perhaps fair to remark that all these investigations have added little to the knowledge to exchange para- meters but have revealed a great number of phenomena interesting in itself. In concentrated systems with large interactions one can no longer view the electro- nic states by a simple one ion picture. The eigenstates are then rather those of the total crystal and are cal- led excitons for the pure electronic states and magnons for the spin states. The simplest systems, which fall in this group of materials are those, which have essen- tially no magnetic Davidov splittings. The system in that class studied most thoroughly is MnF,, and we shall start our discussion with it [30-341.

The energy level diagram relevant to the discussion of the absorption and emission spectrum in MnF, is shown in figure 7. The dot at the bottom of the

FIG. 7,

-

E(k) diagram of MnF2 ; explanation of arrows (1) to (5) is given in the text (from Ref. 33).

figure shows the groundstate of the crystal. Also shown is the magnon dispersion relation for MnF,. In addi- tion E 1 and E 2 mark two excitons, which are made up from the two lowest spin orbitals of the 4T,, derived from the 4G free ion energy level. The work of Dietz et al. 1331 has shown that these excitons have practically no dispersion. The simplest type of transi-

sidebands) are in absorption those which involve simultanously transitions of type (1) and (3), whereby the exciton and magnon are created simultanously and a magnon exciton interaction may occur giving rise to an additional energy shift. Contrary, in pro- cess (5) the exciton decays directly to a magnon, the exciton and magnon do not coexist in time and space, and the known magnon dispersion relation may there be used to study the exciton dispersion.

Dietz and collaborator [33] designed a very nice experiment to demonstrate the exciton magnon interac- tion in MnF,. They stressed a crystal of MnF, in the [OOl] direction and observed the change in the magnon sidebands, in emission and absorption as shown in figure 8. Large changes are seen in absorption, but

EMISSION ABSORPTION

-64 - 4 8 -32 -16 0 + I 6 t 3 2 t 4 8 +64

WAVENUMBERS FROM El l k = O )

FIG. 8.

-

Effect of (100) stress on intrinsic absorption and emission, demonstrating that (1) E 1 and E 2 have thesame dis- persion and (2) the absorption exciton-magnon sidebands are

strongly renormalized (from Ref. 33).

practically none in emission. The stress applied to the crystal presumably mixes the E 1 and E 2 states thus changing their orbital character. This change in orbi- tal character of these states obviously changes the exciton magnon interaction, thus changing the side- band shape in the absorption experiment.

In order to derive the exciton dispersion relation from the experimental magnon sideband, one has to calculate the magnon sidebandshape and compare it with the experiment. Dietz and collaborators [33]

have determined the exciton magnon coupling from intensity measurements. Taking the magnon density of states as derived in a neutron diffraction experi- ment, the sidebandshape as shown by the full curve in figure 9 can be obtained assuming negligible exciton dispersion. The good agreement with the experimental data suggests, at least in this case, a very little exciton dispersion. Meltzer et al. [32] have investigated a great number of other states and their magnon side-

OPTICAL SPECTROSCOPY OF MAGNETIC INSULATORS C 1

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715

A X I A L P O L A R I Z A T I O N

I I I I I I I

18,365 18,381 18,397 18,413 18,429

WAVENUMBER

FIG. 9.

-

Fit of theoretical sidebandshape to experimental data (from Ref. 33).

bands in MnF2. They find sizable exciton dispersion up to 75 cm-l.

The case of the magnon sideband in absorption is more involved. Here the exciton and the magnon are created at the same time very near to each other.

Therefore they may interact strongly with each other and this interaction has to be taken into account to account theoretically for the observed sidebandshape.

Incidently this problem is analogous to that of the two spin wave excitations observed in Raman scattering experiments in magnetic crystals. The most complete data on this object are so far available in RbMnF, 13.51.

Figure 10 shows drastically how the theoretical curve

FIG. 10.

-

The full curve shows the 6A1 --f 4T1 absorption spectrum of RbMnF3. The dashed curves are the theoretical curves with (J' = 0.5J, St = 312, where the primed values refer to the excited state) and without (J' = J, S' = S) exciton-

magnon interaction (from Ref. 35).

comes more into agreement with experiment when the exciton magnon interaction is taken into account.

Theory here suffers from a scarce knowledge of the exciton wavefunctions. So the two magnon Raman scattering spectra of the same material are much more convincing [36]. Figure 11 shows how a theory taking into account the magnon-magnon interaction

F R E O U E N C Y ( e m - ' )

FIG. 11.

-

TWO magnon Raman spectra of RbMnF3 (full curve). Theoretical curves are dashed (without) and dotted

(with) magnon-magnon interaction (from Ref. 36).

can almost completely account for the observed line- shape, whereas the theory neglecting this interaction gives only a very poor fit to the experiments.

Similar results along these lines have been recently obtained by Belorizky et al. [37] in a rare earth com- pound, namely Er,Al,O,, (erbiumaluminumgarnet)

.

They forced the compound into a ferromagnetic state by very strong external magnetic fields (up to 90 kOe). In this forced ferromagnetic state they obser- ved besides the transitions originating from the nor- mal Zeeman effect some, which originate from an additional spin flip in the groundstate.

V. Magnetic davidov splitting : Cr203.

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The case least understood and least investigated is the one of magnetic concentrated crystals with a large magnetic interaction between excitations on different sublattices leading to a magnetic Davidov splitting [38].

The matrix-elements responsible for this splitting are in principle the same, which are responsible for the transfer of excitation energy in a crystal. That means there is a coupling between the groundstate of an ion on sublattice (1) and the excited state of an ion on sublattice (2) via exchange interaction. If one applies these ideas to the case of the 4A2 groundstate and the ,E excited state of Cr ions in Cr203, the following matrix element is responsible for the Davidov split- ting :

Since Cr203 has four ions per unit cell, every single sublattice energy level is in principle fourfold degene- rate, and therefore it splits into four different levels via the Davidov splitting. The final number of energy levels actually observed is yet determined by taking into account the symmetry properties of the magnetic group. These ideas result in the energy level diagram for Cr,O, as shown in figure 12, including the electric dipole selection rules. The absorption spectrum of Cr203 in the region of interest is shown in figure 13, where 1,2, 3, and 4 are the lines originating from tran- sitions to the levels numbered in the same way in

FIG. 12. - Davidov split k = 0 exciton states in Cr203.

Electric dipole selection rules (F. means forbidden) (from Ref. 38).

figure 12. Allen et al. [38] give arguments that the two lines at 13,743 cm-I and 13,926 cm-I observed in the absorption spectrum of Cr203 are due to tran- sition from the A,. groundstate to the two lowest E states of the 2E excited state. The size of the magnetic Davidov splitting may be obtained from the following consideration. The two 'E levels are the Davidov split levels of an ion and its counterparts on a fourth nearest neighbor site. The exchange integral to fourth nearest neighbors has been estimated to 10 cm-I from the work of Mollenauer and Schawlow [23].

Since there are six fourth neighbors, each level is displaced by 60 cm-I giving a total splitting of the two 'E levels of

-

120 cm-I in good agreement with the experimental value of 180 cm-l. There can be given arguments [39] that it is appropriate to use in that estimate the exchange constants determined in ruby rather than those in Cr203, which are a factor of ten smaller. But a full understanding of this subject requires further work.

VI. Summary. - Some of the experiments dealing with the optical investigation of magnetic insulators

FIG. 13.

-

Zeernan effect in unpolarized absorption spectrum of 4.42 -f 2E excitons in Cr203. Polarization and labelling are

given in right hand columns (from Ref. 38).

have been described. It was shown that magnetic interactions can be obtained from these experiments in rare earth compounds and from spectra of pairs of magnetic ions in a diamagnetic host (e. g. ruby).

In transition metal compounds, especially the absorp- tion and light scattering experiments have given evi- dence for strong exciton-magnon and magnon-magnon couplings. In addition large Davidov splittings have been found in a few cases. The status of these experi- ments may perhaps be characterized by saying that the gross features of the experiments described are rather well understood but that work is still needed to clear up a number of details, from which much information for a microscopic quantitativ understan- ding of magnetic properties of the materials may revolve.

Acknowledgement : I would like to thank J. W.

Allen, R. Courths, R. E. Dietz, P. A. Fleury, G. F.

Imbusch and L. F. Mollenaner for the permission to use figures from their work.

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717 1201 TOLSTOY (N. A.) and ABRAMOV (A. P.), Opt i Spec- [30] SELL @. D.), J. appl. Phys., 1968, 39, 1030. This troskiya 1963,14,691 ; Engl. trans. : Opt. Spectry. paper contains many references to earlier work.

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(R.

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