Pressure effects on NaMnF 4 : Structural correlations and Jahn-Teller effect from crystal-field spectroscopy

This work investigates the optical absorption spectrum of the NaMnF 4-layered perovskite and its variation with pressure. The spectrum basically consists of three broadbands located at 1.916, 2.263, and 2.817 eV, which correspond to the crystal-field ~CF! transitionsB1g→G i (G i5A1g , B2g , andEg) with the Jahn-Teller~JT-! distorted MnF6 32 complex (Mn d configuration!. In addition, there are two spin-flipB1g→B1g peaks at 2.397 and 2.890 eV, which are activated by the exchange mechanism. Their variation with pressure reveals that the JT energy does not change significantly with pressure: ]EJT/]P50.8 meV/GPa. Furthermore, the variation of the JT tetragonal splitting of the parent octahedral eg ndt2g , termedDe andD t , respectively, clearly indicate that]De /]P!]D t /]P, althoughDe'4D t . The CF energies and their pressure shift are explained in terms of local structural changes within the MnF 6 32 complex induced by pressure. The structural correlation analysis reveals that the reduction of the MnF 6 32 JT distortion is smaller than the expected one on the basis of the crystal volume reduction, thus indicating tilt phenomena. This interpretation is supported by the decrease of in-layer Mn-F-Mn superexchange, such as is derived from the optical spectra. We demonstrate that the equatorial and axial distances decrease from 1.839 to 1.808 Å and from 2.167 to 2.107 Å, respectively, in the 0–10 GPa range.


I. INTRODUCTION
Mn 3ϩ compounds have received considerable attention due to the variety of structures and associated properties they present in different application fields: e.g., solid-state laser (Mn 3ϩ -doped Y 3 Al 5 O 12 ) ͑Refs.
1-3͒, dichroism (Tl 2 MnF 5 •H 2 O) ͑Ref.4͒, transparent ferromagnets (CsMnF 4 ) ͑Refs.5-8͒, colossal magnetoresistance (Sm 0.55 Sr 0.45 MnO 3 ) ͑Refs.9-11͒, or metal-insulator transition (LaMnO 3 ) ͑Ref.12͒ are examples of this ample behavior.4][15][16][17][18][19] Depending on the ligand sharing of MnF 6 3Ϫ complexes, the crystal structure shows a different dimensional arrangement.It can be either zero dimensional ͑0D͒ if there is no ligand sharing ͑i.e., isolated MnF 6 3Ϫ units like in Na 3 MnF 6 ), 20 1D linear chains if they share the two axially ligands (Na 2 MnF 5 ), 21 or 2D layers if four ligands are shared within the plane forming layers (NaMnF 4 ). 22The latter case, which corresponds to the title compound, displays an antiferrodistortive structure, in which the in-plane equatorial ligands of a given MnF 6 3Ϫ complex act as axial ligands of two nearest-neighbor complexes ͑Fig.1͒.This situation favors a ferromagnetic Mn-F-Mn superexchange within the layer provided that the Mn-F-Mn tilt angle is not far from 180°. 5,6,23The present atomic arrangement simply explains the occurrence of ferromagnetism below T C ϭ19 K in CsMnF 4 . 5A review on the magnetic properties of Mn 3ϩ fluorides can be found elsewhere. 6MnF 6 3Ϫ displays axially elongated coordination geometry along the Mn 3ϩ compound series due to the JT effect.The associated distortion strongly depends on the crystal dimensionality: the higher the dimensionality, the greater the JT distortion. 4 This structural correlation is important since the optical and magnetic properties rely not only on the superexchange Mn-F-Mn, but also on the JT distortion, which mainly determines the crystal-field ͑CF͒ electron structure.There has been interest in Mn 3ϩ fluorides in the search for new ferromagnets as well as potential systems to induce structural changes associated with the high-spin to low-spin ͑HS-LS͒ electron ground-state crossover ͓ 5 E g (Sϭ2)↔ 3 T 1g (Sϭ1)͔. 25,26According to the Tanabe-Sugano diagram for an octahedral 3d 4 ion, 27 the HS-LS transition should occur for a CF of 10Dq ϭ27B, where B is the Racah parameter of MnF 6 3Ϫ (B ϭ0.097 eV͒. 4 From this value and taking into account the equation of state of this AMnF 4 (AϭNa, K, Rb, Cs͒ compound series 23,24 and on the assumption that 10Dq depends on the Mn-F distance as 10DqϭKR Ϫ5 , such as was found for different transition-metal fluoride complexes, 4,28 -31 we estimate a HS-LS transition pressure of about 10 GPa provided that the MnF 6 3Ϫ coordination geometry is octahedral.The AMnF 4 series provides the shortest Mn-F distance and hence the highest CF, 4 thus favoring a HS-LS transition to occur at moderate high pressures.However, the JT effect preserves the crystal to undergo the HS-LS transition around the estimated pressure.As we will see later, the large JTinduced tetragonal splitting of the parent octahedral Mn 3ϩ e g (3z 2 Ϫr 2 ,x 2 Ϫy 2 ) electron levels into b 1g ϩa 1g , increases the LS electron energy with respect to the HS state and hence the transition pressure required for electronic pairing ͑Fig.1͒.Through this work, we estimate that the critical pressure required to induce a HS-LS transition is increased by the JT effect to about 150 GPa.Interestingly, Mn 3ϩ compounds show also an intense pleochroism, which depends on both the JT distortion as well as the orientation of the MnF 6 This work investigates the CF electronic structure of MnF 6 3Ϫ in NaMnF 4 and its variation with pressure in the 0-10 GPa range through optical absorption ͑OA͒ spectroscopy.The aim is to explore how the JT energy of Mn 3ϩ and its associated coordination structure vary with pressure.These systems are attractive for this purpose since a rich band structure associated with the spin-allowed 5 B 1g → 5 ⌫ i (⌫ i ϭA 1g , B 2g , and E g ) and spin-forbidden 5 B 1g → 3 B 1g transitions are well resolved in the VIS-UV range, thus providing a direct determination of the CF electron structure and, consequently, of the JT energy of MnF 6 3Ϫ .We have developed a perturbed-octahedron complex model in order to establish correlations between the variation of the CF energies and the local structure around Mn 3ϩ , which strongly depends on both the volume and JT distortion of MnF 6 3Ϫ . 4,14,32Knowledge of these correlations is important to elucidate whether the application of pressure reduces the JT distortion, leading to partial disappearance of the in-plane antiferrodistortive structure, or whether it induces out-ofplane tilts of the MnF 6 3Ϫ octahedra ͑Fig.1͒.Moreover, it provides information about local-structure changes around Mn 3ϩ under pressure from OA spectroscopy.This aspect is relevant since such information could not be obtained from x-ray diffraction ͑XRD͒ in previous structural works under pressure performed on Mn 3ϩ fluorides (AMnF 4 ). 22,23Besides XRD, local-structure variations induced by pressure are not easy to accomplish through extended x-ray-absorption fine structure ͑EXAFS͒ under pressure due to the strong x-ray absorption of diamonds at the Mn K edge on dealing with a diamond anvil cell ͑DAC͒. 34inally, the structural changes that we found from the pro-posed model are correlated with variations undergone by the exchange-induced 5 B 1g → 3 B 1g spin-flip transitions, whose intensity is very sensitive to the superexchange Mn-F-Mn angle between adjacent MnF 6 3Ϫ complexes within the layer.

II. EXPERIMENT
Single crystals of NaMnF 4 were grown from cold solutions of MnF 3 and NaF fluorides in hydrofluoric acid following a method reported elsewhere. 35,36This layer compound is monoclinic ( P2 1 /c space group, Zϭ2) with lattice parameters aϭ5.736Å, bϭ4.892Å, cϭ5.748Å, and ␤ϭ108.1°atroom temperature. 22The structure consists of layers of interconnected ͓MnF 2 F 4/4 ͔ 3Ϫ corner-sharing octahedra separated by Na ions.The octahedra display a D 2h symmetry ͑nearly D 4h ) as a consequence of the JT distortion and crystal anisotropy.The in-plane equatorial F ligand of one Mn acts as axial ligand of the nearest Mn, leading to the layered antiferrodistortive structure shown in Fig. 1.The axial and equatorial Mn-F distances are R ax ϭ2.167 Å and R eq ϭ1.839 Å.The equatorial distance actually corresponds to the average of the two equatorial distances, R eq1 ϭ1.808 Å and R eq2 ϭ1.869 Å. 22 The OA spectra under pressure were obtained using a specially designed spectrophotometer.The monochromatic light in the UV-VIS-IR range was obtained by means of a Spectra Pro-300i ARC Monochromator and suitable filters.The light was chopped and detected with a Hamamatsu R-928 Phototube and a SR 830 Lock-in amplifier.The experimental setup is described elsewhere. 37Pressure experiments were done in a DAC ͑Diamond Optics, Inc.͒ using single crystals of NaMnF 4 ͑150ϫ120ϫ60 m 3 ͒.We used paraffin oil as pressure transmitting media in order to prevent oxidation.The pressure in the hydrostatic cavity was calibrated from the ruby R-line shift.The ruby was excited with the 530.9-nm line of a Kr ϩ ion laser ͑Coherent CR-500K͒.

A. Pressure spectroscopy on NaMnF 4
The CF spectrum of Mn 3ϩ ions in the A n MnF nϩ3 (A ϭalkali ion, nϭ1 -3) compound series can be understood on the basis of the JT-octahedral-distorted MnF 6 3Ϫ complex. 4ithin an elongated D 4h coordination, the spectra of these compounds consist of three intense broadbands E 1 , E 2 , and E 3 ͑Fig.2͒, which are resolved in the UV-VIS range.These bands are associated with electronic transitions 5 B 1g → 5 ⌫ i (⌫ i ϭA 1g , B 2g , and E g ) involving states of the same spin, Sϭ2, and their energy strongly depends on the CF.In addition, there are several narrow peaks corresponding to spinflip 5 B 1g → 3 B 1g transitions.These electronic transitions are electric-dipole ͑ED͒ forbidden since they involve different spin states.However, in these compounds they are activated by the exchange mechanism. 4In contrast to the spin-allowed 5 B 1g → 5 ⌫ i transitions, the spin-flip transitions are weakly CF dependent and thus they appear as narrow features in the OA spectrum.
Structural correlations performed on Mn 3ϩ fluorides indicate that the tetragonal splitting associated with the parent octahedral e g and t 2g orbitals, termed ⌬ e ϭE 1 and ⌬ t ϭE 3 ϪE 2 , respectively, is proportional to the JT distortion, which is characterized by the Q normal coordinate in MnF 6 3Ϫ com-plexes with local D 2h symmetry ͑Fig.1͒. 4,14,38On the other hand, E 2 depends only on the equatorial Mn-F distance R eq , and therefore its transition energy provides a very sensitive probe to detect variations of R eq upon pressure.
Figure 2 shows the OA spectrum of NaMnF 4 at ambient conditions.The spectrum of the 1D Na 2 MnF 5 compound is also included for comparison purposes.The three E 1 , E 2 , and E 3 bands appear at 1.916, 2.263, and 2.817 eV, respectively, for NaMnF 4 .Note that these energies shift to lower energies in the less JT-distorted MnF 6 3Ϫ complex in Na 2 MnF 5 .The CF energies as well as the corresponding JT splitting, ⌬ e ϭ1.916 eV and ⌬ t ϭ0.554 eV, reflect the nearly D 4h JT distortion of the MnF 6 3Ϫ characteristic of a 2D layered perovskite AMnF 4 (AϭK, Rb, Cs, Tl͒ with Q ϭ0.373 Å and Q ⑀ ϭ0.061 Å for NaMnF 4 . 22The O h normal coordinates e g (Q ,Q ⑀ ) are derived from the equatorial and axial distances obtained from XRD through Eq. ͑1͒: with R eq ϭ 1 2 (R eq1 ϩR eq2 ).Interestingly, these coordinates are useful when dealing with the e E JT effect in O h systems as well as with structural distortions induced by either hydrostatic pressure or axial stress in the framework of an O h perturbed complex. 32he influence of the MnF 6 3Ϫ distortion on the electron structure is clearly evidenced through the OA spectrum of the 1D Na 2 MnF 5 ͑Fig.2͒.Its tetragonal splitting ⌬ e ϭ1.55 eV and ⌬ t ϭ0.39 eV reflects the smaller JT distortion: Q ϭ0.34 Å and Q ⑀ ϭ0.03 Å, derived from XRD. 39 The same FIG.2. Optical absorption spectra of NaMnF 4 ͑single crystal͒ and Na 2 MnF 5 ͑powder͒.An energy-state diagram of Mn 3ϩ in elongated-D 4h symmetry is also included.The assignment of the spin-allowed 5 B 1g → 5 ⌫ i (⌫ i ϭA 1g , B 2g , and E g ) bands and the spin-flip 5 B 1g → 3 B 1g peaks is indicated by arrows.The expression for the tetragonal splitting of the E g (⌬ e ) and T 2g (⌬ t ) states as well as for 10Dq is given as a function of the normal coordinate Q and R eq , respectively.The expressions are also given as a function of the tetragonal CF parameters D s and D t .
trend is found on comparing the spectroscopic and structural parameters for the 0D Na 3 MnF 6 compound.The parameters are ⌬ e ϭ1.04 eV and ⌬ t ϭ0.20 eV, which corresponds to a MnF 6 3Ϫ distortion: Q ϭ0.16 Å and Q ⑀ ϭ0.035 Å. 40,41 From these values we can directly obtain the JT energy at ambient conditions since E 1 ϭ⌬ e ϭ4E JT .Therefore E JT changes along the Na n MnF nϩ3 series as 0.48, 0.39, and 0.26 eV for nϭ1, 2, and 3, respectively.It is worth pointing out that E JT and the corresponding complex distortion increase with the crystal dimensionality, i.e., with the number of shared ligands.This structural correlation does not only hold for Na compounds, but for the whole compound series, as was reported elsewhere. 4igure 3 shows the variation of the OA spectrum with the pressure in NaMnF 4 .The effect of applying pressure is mainly to shift the three intense bands towards higher energies, whereas a slight redshift is observed for the two spinflip 5 B 1g → 3 B 1g peaks.The variation of E 1 , E 2 , and E 3 shows a linear behavior with the pressure.The corresponding coefficients are given in Fig. 4. The small pressure-induced shift of E 1 is worthwhile.It clearly indicates that E JT does not change significantly with pressure.A slight increase of ‫ץ‬E JT ‫ץ/‬ Pϭ0.8 meV/GPa is derived from E 1 ( P) in Fig. 4.However, this result contrasts with the important increase experienced by the t 2g splitting, ‫⌬ץ‬ t ‫ץ/‬ Pϭ16 meV/GPa, in comparison to ‫⌬ץ‬ e ‫ץ/‬ Pϭ3 meV/GPa, in spite of ⌬ e being larger than ⌬ t at ambient conditions: ⌬ e ϭ3.5⌬ t . 4This puzzling feature is consequence of the different variations of the linear electron-lattice coupling coefficient for the oneelectron e g and t 2g levels (O h scheme͒, aside from the structural changes experienced by the MnF 6 4Ϫ complex under pressure.Similar findings were also observed in the variation of the wave function coefficients N e and N t associated with the mainly metallic molecular orbitals e g and t 2g of MnCl 6 4Ϫ in pressure experiments carried out in NH 4 MnCl 3 . 42The pressure variation of N e is slightly smaller than the variation of N t in spite of N e ϾN t .

B. Structural correlations: Octahedron-perturbed model
From the present results, we are able to extract valuable information on the local structural changes around Mn 3ϩ under pressure, provided that we know how E 1 , E 2 , and E 3 depend on R eq and R ax ͑or R eq and Q ).The use of the Q normal coordinate is important since the JT-tetragonal splittings ⌬ e and ⌬ t within a D 4h complex framework are both proportional to Q for small deviations of the O h symmetry: 4,32,38 FIG. 3. Optical absorption spectrum of the NaMnF 4 single crystal and its variation with the pressure.The peak assignment within a D 4h symmetry is indicated.Vertical dashed lines denote the position of the triplet peaks at ambient pressure.
FIG. 4. Variation of the transition energy for the three intense bands E 1 , E 2 , and E 3 ͑top͒ and the narrow 5 B 1g → 3 B 1g peaks ͑bottom͒ with pressure.Straight lines and corresponding equations are the linear least-squares fits to the experimental data.The energy corresponds to the band maximum.The absolute errors for the energy obtained either from the band maximum or by fitting to gaussians is 10 meV, while the error of the energy variation with pressure is 3 meV for E 1 , E 2 , and E 3 .The corresponding errors for the triplets are 1 and 0.2 meV, respectively.The point size in both plots is slightly bigger than the actual pressure and energy errors.The fitting errors for E and ‫ץ/‪E‬ץ‬ P are 9 mev and 2 meV GPa Ϫ1 for 5 B 1g → 5 A 1g , respectively, 3 meV and 1 meV GPa Ϫ1 for 5 B 1g → 5 B 2g , and 10 meV and 3 meV GPa Ϫ1 for 5 B 1g → 5 E g .For the triplets, the fitting errors are 0.4 meV and 0.1 meV GPa Ϫ1 for 5 B 1g → 3 a B 1g and 0.8 meV and 0.2 meV GPa Ϫ1 for 5 B 1g → 3 b B 1g , respectively.The calculated energy shifts using the developed model coincide with fit lines.Note the different energy scale employed in each representation.

͑2͒
Here K e and K t are the JT electron-lattice coupling coefficients that, in general, depend on the volume per molecule, V, or the pressure P. It must be remarked that the structural correlation reported along a series of Mn 3ϩ fluorides 4,43 indicates that ⌬ e ϭ5.08Q and ⌬ t ϭ1.34Q ͑units in eV and Å͒, with a ratio ⌬ e /⌬ t Ϸ4 at ambient conditions.Note that the measured ⌬ e and ⌬ t values in NaMnF 4 are in good agreement with this figure.We assume a simple D 4h symmetry as appropriate for the model analysis, since the occurrence of orthorhombic CF components is unlike as is evidenced from the lack of splitting associated the high-energy 5 B 1g → 5 E g transition ͑Fig.3͒.
Interestingly, E 2 ( 5 B 1g → 5 B 2g ) only depends on the equatorial Mn-F distance R eq . 38According to CF theory, its variation can be expressed as E 2 ϭKR eq Ϫn with nϭ5.Values of n between 4 and 5 have been obtained from calculations for trivalent ions Cr 3ϩ , 30,44 Fe 3ϩ , 45 and spectroscopy. 4,28,29ssuming an average value of the exponent, nϭ4.5, we can easily derive the variation of R eq under pressure from the E 2 shift.It must be emphasized that the present method provides suitable variations of R eq irrespective of particular n choice either 4 or 5.
From the band shifts of Figs. 3 and 4, we conclude the following: ͑i͒ The equatorial Mn-F distance R eq decreases with increasing pressure as is evidenced by the blueshift: ‫ץ‬E 2 ‫ץ/‬ P ϭ17 meV/GPa.
͑ii͒ The variation of R eq with pressure can be obtained from the pressure derivatives of E 2 : R eq ϭϪ4.5 2.263 1.84 ϭϪ5.53 eV/Å, ͑3͒ so that we get ‫ץ‬R eq ‫ץ/‬ PϭϪ3.1ϫ10Ϫ3 Å/GPa, which means a variation of ⌬R eq ϭϪ0.031Å from ambient pressure to 10 GPa.
͑iii͒ The present spectroscopic procedure provides a suitable method for deriving bond-distance variations that actually improve the x-ray absorption spectroscopy ͑XAS͒ sensitivity.By using Eq.͑3͒, an energy-shift accuracy of 5 meV leads to a bond-distance accuracy of 10 Ϫ3 Å.
͑iv͒ The MnF 6 3Ϫ JT distortion does not change significantly upon pressure.Note that disappearance of the JT effect yielding Q Ϸ0 should induce splitting closure of the 5 E g and 5 T 2g states: ⌬ e ϭ0 and ⌬ t ϭ0 ͑Fig.1͒.On the contrary, we observe that both ⌬ e and ⌬ t increase by 0.032 and 0.161 eV, respectively, from ambient pressure to 10 GPa.Moreover, the variation of ⌬ t in the same pressure range is 5 times the variation of ⌬ e ͑Fig.4͒.
͑v͒ The fact that ‫⌬ץ‬ e ( P)/‫ץ‬ Pϭ3 meV/GPa and ‫⌬ץ‬ t ( P)/‫ץ‬ Pϭ16 meV/GPa implies necessarily that K e and K t must both increase with pressure and ⌬K e /K e Ӷ⌬K t /K t .
The relation between the electron-lattice coupling parameters and the crystal volume can be expressed for small departures of the O h symmetry as follows: 37,38 The electron-lattice coupling and the crystal volume parameters at ambient conditions for NaMnF 4 are K e 0 ϭ5.05 eV/Å, K t 0 ϭ1.46 eV/Å, and V 0 ϭ153.3Å 3 .The results of Figs. 3  and 4 can be explained in the framework of the JT model using effective values n e ϭ0.7 and n t ϭ2.2.These exponents are similar to those from CF theory 38 and also from extended-Hu ¨ckel and X␣ complex calculations 46,47 yielding values of n e and n t around 1 and 2, respectively.However, it must be remarked that the particular choice of n e and n t around those values does not affect significantly, as we show later, the quantitative estimates performed on this work.The selected values provide similar values of the pressure derivatives of the JT electron-lattice coupling parameter, ‫ץ‬K e ‫ץ/‬ P Ϸ‫ץ‬K t ‫ץ/‬ P, in agreement with previous studies on the variation of the wave function coefficients N t and N e associated with the e g and t 2g orbitals with pressure ‫ץ‬N e ‫ץ/‬ P Ϸ‫ץ‬N t ‫ץ/‬ P. 9 The structural constraint inferred from Eq. ͑2͒ requires that ⌬ e ( P)/K e ( P)ϭ⌬ t ( P)/K t ( P) for any tetragonal perturbation Q .It means that, for small departures from O h symmetry, the splitting and associated electron-lattice coupling coefficient of e g and t 2g are related by the equation 1 and according to Eq. ͑4͒, the pressure derivatives of K e and K t are given by 1 and therefore, 1 Taking a bulk modulus B 0 ϭ60 GPa ͑Refs.5 and 23͒ and combining Eqs.͑5͒ and ͑7͒ with the experimental data of Fig. 4, we get n t Ϫn e ϭ1.6.According to Eq. ͑6͒, the ratio on the assumption that ‫ץ‬K e ‫ץ/‬ PϷ‫ץ‬K t ‫ץ/‬ P. Hence we obtain n t ϭ2.2 and n e ϭ0.7, so that we get ‫ץ‬K e ‫ץ/‬ PϷ‫ץ‬K t ‫ץ/‬ Pϭ55 meV/Å GPa, which means an electron-lattice coupling at 10 GPa of K e (10)ϭ5.60 eV/Å and K t (10)ϭ2.01eV/Å.
Consequently, the D 4h JT distortion of MnF 6 3Ϫ in NaMnF 4 at this pressure is In conclusion, the application of pressure induces a reduction of the tetragonal distortion from 0.379 Å at ambient pressure to 0.345 Å at 10 GPa: ⌬Q ϭϪ0.034Å.In terms of bond distances, it means a reduction of R eq and R ax from 1.839 to 1.808 Å for R eq and from 2.167 to 2.107 Å for R ax , e.g., ⌬R eq ϭϪ0.031Å and ⌬R ax ϭϪ0.060Å in the 0-10 GPa range ͑Table I͒.

C. Structural correlations based on the crystal-field theory
A similar structural correlation is obtained on the basis of the CF theory ͑or ligand-field theory following Griffith's criterion͒. 48Within this theory the tetragonal splitting for the e g and t 2g levels is given by ⌬ e ϭ4D s ϩ5D t and ⌬ t ϭ3D s Ϫ5D t . 4,38The advantage of using parameters D s and D t lies on the fact that they depend explicitly on the Mn-F bond distances as 38

͑8͒
Taking the electron matrix elements as semiempirical parameters, analogously to what is usually done on dealing with the CF parameter 10Dq, the dependence of D s and D t on small tetragonal distortions from the O h symmetry is given by

͑9͒
Parameters C s and C t can be empirically determined from structural and spectroscopic data at ambient conditions.According to expressions given in Fig. 4, the pressure dependence of D s and D t is Equation ͑11͒ provides the pressure derivative of the normal coordinate Q using Eq.͑10͒ and the pressure derivative of R eq from Eq. ͑3͒.We obtain an average value It means that Q decreases with pressure by ⌬Q ϭ Ϫ0.034 (11) Å from 0 to 10 GPa, which is similar to the previous finding.Table I summarizes the local structural changes undergone by the MnF 6 3Ϫ with pressure following the structural correlations established in this work.
In conclusion, the effect of pressure is mainly to reduce all Mn-F distances of the MnF 6 3Ϫ complex.The axial distance reduces twice the equatorial distance, thus leading to a partial reduction of the JT distortion.This scenario agrees with previous findings in the antiferrodistortive ͓C 3 H 7 NH 3 ͔ 2 CuCl 4 crystal using XRD and EXAFS under pressure. 32,49A similar result was also reported for LaMnO 3 using Raman spectroscopy and XRD under pressure, 12 although the observed JT reduction with pressure was not detected so far from neutron diffraction under pressure. 50s a remarkable feature, we have demonstrated the increase of the electron-lattice coupling related to the JT effect with pressure.Particularly, the relative pressure variation of K t is greater than that of K e .Finally, it must be noted that from the proposed structural variation, the MnF 6 3Ϫ complex leads to a local bulk modulus B local ϭ161 GPa, and therefore the complex appears to be less compressible than the crystal.This feature likely indicates the existence of MnF 6 3Ϫ tilts upon pressure, in agreement with previous findings in the perovskite layers A 2 CuCl 4 ͑Refs.32 and 49͒ under pressure and along the AMnF 4 series. 4,14

D. Exchange effects on the spin-flip 5 B 1g \ 3 B 1g transition
The pressure dependence of the peak intensity associated the spin-flip 5 B 1g → 3 B 1g transition at 2.397 eV supports the proposed structural changes.Figure 5 shows the variation of the integrated peak intensity as a function of pressure.Whereas the intensity of the spin-allowed CF bands decreases with pressure by about 40%-50%, it falls a factor of 6 from 0 to 6 GPa for the exchange-induced 5 B 1g → 3 B 1g transition.The intensity reduction is noteworthy since it reflects the proposed structural change with pressure.We associate this phenomenon with a decrease of the superexchange angle ͑␤͒ between Mn 3ϩ neighbors due to pressure-induced tilting of the MnF 6 3Ϫ octahedra.The exchange-induced ED mechanism is known to be very sensitive to ␤, 4 in such a way that deviations from ␤ϭ180°yield a reduction of the associated transition oscillator strength.Besides the magnetostructural correlation established between the exchange constant J measured along the 1D Mn 3ϩ compound series by magnetic techniques and the Mn-F-Mn tilting angle ␤ derived from XRD ͑Ref.8͒ indicates that J decreases with ␤.This correlation is consistent with present spectroscopic findings if we assume that the exchange-induced ED is somehow proportional to J. Figure 5 compares the OA data for 5 B 1g → 3 B 1g with the variation of ␤ derived from pressure experiments using the local-structure data given in Table I and the equation of state for this compound series. 5,23Although the so-obtained ␤ values are derived on the assumption of an ideal perovskite layer structure, given the absence of suitable structural data for this compound, 5,23,33 the variation of ␤ with pressure is consistent with XRD data along the AMnF 4 series.In fact, the variation of ⌬␤ϭϪ5°on passing from KMnF 4 to NaMnF 4 is associated with a volume contraction of ⌬V/V 0 ϭϪ11%. 5,6A similar situation is encountered for NaMnF 4 upon hydrostatic pressure.The volume contraction of ⌬V/V 0 ϭϪ10% from ambient pressure to 10 GPa is accompanied by an increase of the tilting angle; i.e., ␤ varies from 141°to 137°in the 0-10 GPa range ͑inset of Fig. 5͒.Therefore the present results support the proposed structure variation with pressure as well as the occurrence of tilts as the main cause for intensity reduction.A project to perform suitable XRD experiments under pressure in NaMnF 4 to confirm the proposed model is currently in progress.

IV. CONCLUSIONS
We have investigated the variation of the CF electron structure of Mn 3ϩ in NaMnF 4 under pressure.Additionally, we have developed a perturbative octahedron-complex model to establish correlations between the electron CF and the local structure around Mn 3ϩ .The pressure-induced structural variation derived from this model agrees with estimations based on the CF theory.The model is advantageous since the observed shift rates can be described in terms of linear electron-lattice coupling coefficients whose values are obtained from structural correlations.The main conclusions of this work are the following.͑1͒ The JT energy does not change significantly, but slightly increases with pressure.͑2͒ The observed variation is interpreted in terms of reduction of the JT distortion (⌬Q Ͻ0), accompanied by pressure enhancement of electron-lattice coupling coefficients (e E JT effect͒.The competition between these two opposite contributions determines the weak increase of JT energy with pressure.͑3͒ The effect of pressure is mainly to reduce the Mn-F distances.R ax decreases about twice the reduction of R eq , thus leading to a partial reduction of the JT distortion and tilts of the MnF 6 3Ϫ octahedra.͑4͒ We associate the strong decrease of the exchange-dependent 5 B 1g → 3 B 1g peak intensity with the occurrence of pressure-induced tilts.They reduce the superexchange interaction responsible for the ED transition mechanism and hence the intensity according to magnetostructural correlations established along the Mn 3ϩ compound series.

FIG. 1 .
FIG. 1. Left: crystal structure of the NaMnF 4 -layered perovskite at ambient pressure.View of the b-c plane ͑top͒ and view of the layer along c ͑bottom͒.Note the antiferrodistortive structure displayed by the corner-sharing MnF 6 3Ϫ octahedra.Right: ͑top͒ view of the nearly tetragonal elongated coordination of the MnF 63Ϫ complex, with the corresponding axial and equatorial Mn-F distances R ax and R eq , respectively.͑bottom͒ Correlation diagram of the d levels of Mn 3ϩ in different symmetries.

TABLE I .
Relevant structural and spectroscopic parameters for NaMnF 4 at 0 and 10 GPa.Here E 1 , E 2 , and E 3 are the CF energies associated with transitions between one-electron orbitals b 1g (x 2 Ϫy 2 )→a 1g (3z 2 Ϫr 2 ), b 1g (x 2 Ϫy 2 )→b 2g (xy), and b 1g (x 2 Ϫy 2 ) →e g (xz,yz), respectively.Parameters R eq and R ax are the equatorial and axial Mn-F distances of the MnF 6 3Ϫ complex.The tetragonal normal coordinate Q is related to these distances as Q ϭ(2/ͱ3) ϫ(R ax ϪR eq ).Structural data at Pϭ10 GPa were derived from the structural correlation established in Sec.III B.