Synchrotron X-ray Imaging of the Electron Density in RFeO₃ (R = Y, Ho) Using an APD Detector

Structure factors for small hydrothermally grown yttrium and holmium orthoferrites, RFeO₃ (R = Y, Ho), were measured with focused synchrotron radiation at wavelengths of 0.70 and 0.84 Å using both scintillation and high-speed avalanche photodiode (APD) detectors. Resulting APD Δρ images showed striking correlations between aspherical electron densities around Fe and rare-earth metals. Approximate high symmetry in the Δρ images indicates that cations deform the electron density far more strongly than the O atoms. The Ho—Fe magnetic interactions appear to affect the electron density distribution of the Fe atoms and the magnetic phase transitions. Space group Pnma, orthorhombic, YFeO₃ (APD): M_r = 192.76, a = 5.5931 (3), b = 7.6102 (4), c = 5.2806 (3) Å, V = 224.77 (2) ų, Z = 4, D_x = 5.695 Mg m⁻³, μ_0.84 = 15.56 mm⁻¹, F(000) = 356, T = 293 K, R = 0.045, wR = 0.073, S = 4.83 (9) for 1282 unique reflections; HoFeO₃ (APD): M_r = 268.78, a = 5.5922 (3), b = 7.6157 (5), c = 5.2798 (3) Å, V = 224.86 (2) ų, Z = 4, D_x = 7.939 Mg m⁻³, μ_0.84 = 61.98 mm⁻¹, F(000) = 356, T = 293 K, R = 0.036, wR = 0.037, S = 3.07 (6) for 1284 unique reflections.

1. Introduction

Synchrotron X-radiation combines high flux with high-quality diffracted radiation. It has the potential to provide major advances in disciplines that depend on the distribution of electrons depicted in materials. Given adequate precision in the diffraction images, detailed information on the structural stability and some properties of materials can be derived.

Lattice-type crystals of materials containing heavy elements, including rare-earth (RE) elements, produce intense diffraction reflections. Even for small samples, photon rates for strong low-angle reflections are often beyond the linear dynamic range of conventional detectors. For example, the Tl-doped NaI scintillation detector has a counting linearity up to some 10⁴ counts s⁻¹. This requires the use of attenuators and dead-time corrections for the measurement of strong reflections. Since those corrections depend on the X-ray energy and the experimental set-up, considerable error can be introduced into the measurement of strong reflections. Extinction-like effects are often related to this error. Since the accuracy of strong intensities at low angles seriously affects the electron density images, the need for higher-speed detectors with a wide dynamic range, when using high-flux synchrotron radiation, is urgent.

Recently, a new fast X-ray photon detector with stacked avalanche photodiodes has been developed (Kishimoto, 1995; Kishimoto et al., 1998). The main feature of the detector is its relatively high efficiency, coupled with a dynamic range of 10⁻² to more than 10⁸ counts s⁻¹. The present paper reports the application of this APD detector to the imaging of the deformation electron density (Δρ) for two RE orthoferrites RFeO₃ (R = Y, Ho). Δρ represents the redistribution of the electron density of free atoms when they form a crystal. The goals of this paper are twofold. First, the avalanche photodiode (APD) detector measurements of diffraction intensities for two crystals are compared with those obtained by using a conventional scintillation detector (SD). The SD measurements for YFeO₃ were performed recently by du Boulay et al. (1995). The second goal is to relate density images to the magnetic properties of RE orthoferrites. Although strong magnetic effects are due primarily to electron spin density, charge density and spin density are related by the effect of uncompensated spins on the electron probability density. Exchange correlation between electrons occurs when atoms overlap and they strongly influence the Δρ density image.

Crystals of YFeO₃ and HoFeO₃ are structurally very similar due to the almost identical ionic radii for Y and Ho. However, substitution of the diamagnetic Y atom by the paramagnetic Ho atom introduces new terms into the Hamiltonian for the system, including those due to interactions between unlike magnetic atoms of Fe and Ho. In these compounds the isotropic Fe—Fe exchange exceeds the R—Fe interactions and leads to the high T_N (Néel) temperature of 644 K and 639 K for YFeO₃ and HoFeO₃, respectively (Belov et al., 1987). However, the R—Fe interaction together with anisotropic non-Heisenberg (Dzyaloshinski-Moria) Fe—Fe exchange plays an important role in determining the anisotropic properties of the system. The anisotropy energy is related to the difference in the splitting of the ground state of the R ion. The absolute value of the splitting of the ground state of the R ion determines its degree of polarization and, consequently, the influence of the R ions on the Fe system. All mechanisms of anisotropy for the Fe cations stabilize the spontaneous magnetic structure, which combines antiferromagnetic ordering along the Pnma c axis and the weak ferromagnetic moment along the b axis. The R—Fe interaction contributes considerably to the weak ferromagnetic moment at T << T_N and determines the mechanism of the spin reorientation transitions. Over the temperature range 53–59 K, HoFeO₃ exhibits a continuous second-order phase transition of the spin orientation in the Pnma bc plane, stabilizing weak ferromagnetic moments along the c axis. YFeO₃ does not show any spin reorientation transitions. The present study focuses on the effects of the Fe—Fe and R—Fe interactions on the Δρ density images. This should assist our understanding of magnetic properties of RE orthoferrites.

2. Experimental

2.1. Samples

Single crystals were prepared by hydrothermal methods. Previous studies of orthoferrites by du Boulay et al. (1995) and Maslen et al. (1996a,b) showed that the mosaic distributions for hydrothermally grown crystals were more uniform than those for flux-grown crystals. This allowed secondary extinction corrections to be determined reliably. The RFeO₃ (R = Y, Ho) specimens were precipitated from a solution of R₂O₃ + Fe₂O₃ (0.2 wt%)–NaOH (53 wt%) in H₂O, in a sealed 2 ml Pt tube (fill factor ∼0.7), inserted into an autoclave for 2 d at ∼650 K. Small dark red crystals were removed after the tube was cut open. The crystals chosen for synchrotron X-ray diffraction measurements were bounded by two {010}, two {101} and two faces with distances 2 × 5.2, 2 × 10.3 and 2 × 5.6 µm for YFeO₃ and 2 × 2.8, 2 × 4.5 and 2 × 3.3 µm for HoFeO₃ from the crystal centre. Those dimensions were measured and the faces were indexed using optical and scanning electron Electroscan E3 ESEM microscopes. The faces of the crystals correspond to the ideal pseudo-cubic perovskite crystal faces. This reflects that the crystal habit of RE orthoferrites is influenced more strongly by the pseudo-cell than by the true crystallographic cell.

2.2. Data collection

Diffraction intensities for HoFeO₃ were measured at room temperature with 0.70 and 0.84 Å synchrotron X-radiation using the BL14A four-circle diffractometer (Satow & Iitaka, 1989) at the Tsukuba Photon Factory. The data set at 0.70 Å was collected with a conventional Tl-doped NaI SD. Then, the same sample was used for data collection at 0.84 Å with an APD detector. For a more extensive comparison of the two detectors, the diffraction intensities for YFeO₃ were remeasured at 0.84 Å with the APD detector. An SD data set for a different sample of YFeO₃ was obtained at 0.75 Å by du Boulay et al. (1995).

Vertically polarized synchrotron radiation from a vertical wiggler was monochromated by a double Si(111) crystal monochromator, and focused using a curved mirror. The polarization ratio, i.e. the fraction of the total incident beam intensity with its electric vector vertical, was 0.95 for a wavelength of 0.7 Å, and 1.0 at 0.84 Å. An incident beam slit, 0.4 mm in diameter, installed before the monitor ion counter with Ar gas flow, and a collimator, 0.2 mm in diameter, placed after the monitor chamber, provide an intense and adequately homogeneous beam. By positioning the crystal specimen slightly off-focus, changes in the synchrotron radiation intensity due to particle beam instabilities were minimized. Since the stored charged particle beam decays exponentially with time, an ion chamber was used to monitor the incident beam intensity. To stabilize the intensity of the incident beam, the alignment of the first monochromator crystal and of the vertical translations of the mirror were optimized automatically by flux maximization every 20 min during data collection. A 3 × 3 mm receiving slit was placed in front of the SD counter. The higher harmonics in the beam from the monochromator were removed by the curved mirror and discriminators.

A new high-speed APD detector with counting linearity up to 10⁸ counts s⁻¹ was used for data collection at a wavelength of 0.84 Å. The estimated dead time was less than 4 ns. The noise level is about 10^–2 counts s⁻¹. The detector mounted on the 2θ arm of the BL14A four-circle diffractometer contained four epitaxic silicon avalanche photodiodes stacked perpendicular to the diffracted X-radiation. Each photodiode has a detection area of 2.8 mm in diameter with a thickness of ∼120 µm. The total intrinsic detection efficiency of the APD detector is 67.5% at 14.7 keV (0.84 Å) compared with 47.6% at 17.6 keV (0.70 Å). The longer 0.84 Å wavelength was preferred for the APD data collection. This wavelength was determined by the resolution obtained at 0.7 Å wavelength and the maximum values of the diffractometer 2θ angles. A significant increase in diffraction intensities at 0.84 Å allowed the reflection scan speed to be doubled. The pulse-height distribution at high photon rates was measured using sequential single-channel discrimination.

Lattice constants at 0.84 Å wavelength were evaluated from the diffractometer angles for three groups of symmetry related reflections (0 ±4 ±8), (0 ±12 ±2) and (±10 ±2 0) (total 12 reflections) at 2θ values of 84.68, 85.86 and 98.75° for YFeO₃ and 84.68, 85.79 and 98.77° for HoFeO₃. Lattice constants for HoFeO₃ at 0.7 Å were evaluated from the diffractometer angles for three pairs of Friedel-related reflections at 2θ values of 44.03 (±6 0 0), 63.92 (0 0 ±8) and 66.82° (0 ±12 0) (total six reflections) similar to the SD measurements for YFeO₃. For every data set the reflection intensities were measured systematically using ω/2θ scans for a complete sphere of reciprocal space with −12 ≤ h ≤ 12, −16 ≤ k ≤ 16, −11 ≤ l ≤ 11. Intensities for the SD data sets were corrected for dead-time counting losses using the polynomial expansion by Hester et al. (1993). The NaI SD dead time was measured to be 1.2 ms. Six standard reflection intensities were remeasured every 100 reflections to monitor the incident beam stability. Further experimental details are given in Table 1.

Table 1
Experimental and refinement data for YFeO₃ and HoFeO₃

Chemical formula	YFeO3	HoFeO3	HoFeO3
Chemical formula weight	192.76	268.78	268.78
Space group	Pnma, orthorhombic	Pnma, orthorhombic	Pnma, orthorhombic
a (Å)	5.5931 (3)	5.5922 (3)	5.6021 (5)
b (Å)	7.6102 (4)	7.6157 (5)	7.6278 (7)
c (Å)	5.2806 (3)	5.2798 (3)	5.2899 (4)
V (Å^3)	224.77 (2)	224.86 (2)	226.05 (3)
Dx (Mg m^−3)	5.695	7.939	7.897
Z	4	4	4
Absorption coefficient, μ (mm^−1)	15.56	61.98	37.80
F(000)	356	468	468
Crystal volume (mm^3)	0.24 × 10^−5	0.33 × 10^−6	0.33 × 10^−6
Radiation, λ (Å)	PF†, 0.84	PF†, 0.84	PF†, 0.70
Monochromator	Si(111)	Si(111)	Si(111)
Detector	APD	APD	Scintillation
Scan speed (° min^−1)	32	32	16
Peak scan width, 2θ (°)	0.6	0.6	0.5
Maximum 2θ (°); sinθ/λ (Å^−1)	133.42; 1.0934	133.45; 1.0941	99.40; 1.0943
Maximum intensity variation (%)	9.5	3.5	1.1
of standards (hkl)	±(400), ±(0120), ±(006)	±(200), ±(040), ±(004)	±(200), ±(040), ±(004)
Reflections measured	9355	9203	9320
Transmission range, minimum; maximum	0.7801; 0.8589	0.6366; 0.7294	0.7626; 0.8209
Independent reflections, F^2> 0	1282	1284	1268
Extinction‡, r*	|−r*| < σ(r*),	|−r*| < σ(r*),	r* < σ(r*),
	no extinction applied	no extinction applied	no extinction applied
Rint (F^2), before; after absorption	0.0970; 0.0969	0.0747; 0.0650	0.1004; 0.0100
R	0.0446	0.0358	0.0326
wR	0.0730	0.0368	0.0293
S	4.83 (9)	3.07 (6)	1.27 (3)
Maximum shift/s.u.	0.2 × 10^−4	0.9 × 10^−4	0.3 × 10^−4
Δρ, minimum; maximum; σ(Δρ)§ (e Å^−3)	−4.15; +4.91; 0.24	−4.78; +6.21; 0.19	−2.90; 3.31; 0.26

†The BL14A four-circle diffractometer (Satow & Iitaka, 1989) at the Tsukuba Photon Factory, Japan.
‡As proposed by Maslen & Spadaccini (1993).
§Mean s.u. value (Cruickshank, 1949).

2.3. Data reduction and structure refinement

Measured intensities were normalized using the incident beam intensities monitored during each scan. Integrated intensities were further modified and structure factor variances from counting statistics were adjusted for source instability as indicated by the standards (Rees, 1977).

Lorentz and polarization corrections were applied. Absorption correction factors (Alcock, 1974) were evaluated analytically using linear absorption coefficients at wavelengths of 0.7 and 0.84 Å evaluated from the atomic absorption coefficients calculated by Creagh (1996). The reference state for all structure-factor calculations was the independent atom model (IAM) evaluated using spherical atomic scattering factors from International Tables for X-ray Crystallography (1992) with dispersion corrections Δf′, Δf′′ of −1.745, 0.682 at 0.84 Å for Y; −0.174, 4.576 at 0.7 Å and −0.432, 6.193 at 0.84 Å for Ho; 0.343, 0.826 at 0.7 Å and 0.373, 1.146 at 0.84 Å for Fe; and 0.011, 0.006 at 0.7 Å and 0.017, 0.009 at 0.84 Å for O calculated by Creagh (1996).

Before refining the structural parameters, the extinction correction for the full data set based on intensities for symmetry equivalent reflections with different path lengths (Maslen & Spadaccini, 1993) was attempted. The extinction parameter determined with different subsets of strong reflections varied from slightly negative for the APD to positive for the SD data sets by amounts less than the s.u. No extinction correction was applied at this stage. Symmetrically equivalent reflections were averaged and variances consistent with Poisson statistics were retained. Those for the other reflections were increased according to the scatter of equivalents following a Fisher test (Hamilton, 1964).

More precise, though potentially biased, extinction corrections were also attempted by refining an extinction parameter as part of a least-squares optimization of the structural model. The widely invoked formula of Zachariasen (1967) was included in refinement following the implementation of Larson (1970), yielding a maximum extinction correction of 2% for measured F(002) structure factors from the SD data set for HoFeO₃. In this case, extinction refinement altered the scale factor and the Δρ density appreciably, resulting in the change of the Fe cation polarity as indicated by the atomic charges calculated from Δρ by Hirshfeld (1977) partitioning. Larger extinction parameters are expected for the longer 0.84 Å wavelength of X-radiation. However, refined extinction parameters for the APD data sets at 0.84 Å were negligibly small. No extinction indicated by comparison of equivalent reflections for extremely small samples suggests that the extinction parameter determined as part of the structure refinement over-corrects the structure factors. No extinction was applied.

For each data set, 27 independent structural parameters, including all anisotropic vibration tensor elements, were refined by full-matrix least-squares methods based on |F| with w = 1/σ²(F₀) weights for all measured structure factors. Further details of data refinements are given in Table 1.¹ All subsequent calculations utilized the Xtal3.4 system of crystallographic programs (Hall et al., 1995) implemented on DEC ALPHA work stations.

3. Results and discussions

3.1. APD versus SD data

The instrument instability affected the APD data collection for YFeO₃ as seen from the variation of standard reflections in Table 1. Even though this effect was significantly decreased by scaling on standards, the reliability of the data is reduced. Nevertheless, combined with the SD measurements for YFeO₃ by du Boulay et al. (1995), they can serve as confirmatory data to more accurate measurements for HoFeO₃.

Distribution of the ΔF = F_o − F_c residual structure factors after a least-squares IAM refinement versus sinθ/λ for the HoFeO₃ APD and SD data are depicted in Fig. 1. The ΔF values for the APD-measured low-angle reflections tend to be positive, though those for the SD data set are predominantly negative. Consequently, a positive extinction parameter that reduced the negative ΔF values at low angles was refined by least-squares methods for the SD data set. The distributions of the ΔF structure factors for YFeO₃ data sets are similar. This can be related to the SD's dead-time losses for the strong low-angle reflections. Even careful dead-time corrections appeared to be insufficient. Since low-angle reflections contribute strongly to the deformation density function, substantial effects on Δρ images and the structural parameters are expected.

Figure 1 ΔF = Fo − Fc versus sinθ/λ for the HoFeO3 APD and SD data sets.

3.2. Structural parameters

The orthorhombic perovskite-like HoFeO₃ structure is depicted in Fig. 2. The refined positional and vibrational parameters are listed in Table 2, with selected bond lengths and Fe octahedral angles in Table 3. Detailed characteristics of the structural geometries of the RE orthoferrites were described fully by Marezio et al. (1970) and more recently by du Boulay et al. (1995) and Maslen et al. (1996a,b).

Table 2
Fractional coordinates and anisotropic vibration parameters (U_ij) (Ų) with s.u.'s in parenthesis for (I) YFeO₃ (APD), (II) HoFeO₃ (APD) and (III) HoFeO₃ (SD)

T = exp{−2π²[U₁₁(ha^*)² + . . . + 2U₁₂ha^*kb^* + . . .]}.

	x	y	z	U11	U22	U33	U12	U13	U23
Y (I)	0.06874 (7)	1/4	−0.01795 (8)	0.0038 (1)	0.0107 (1)	0.0085 (1)	0	−0.00047 (9)	0
Ho (II)	0.06821 (4)	1/4	−0.01791 (4)	0.00411 (6)	0.00653 (7)	0.00454 (6)	0	−0.00044 (5)	0
Ho (III)	0.06819 (3)	1/4	−0.01796 (4)	0.00501 (5)	0.00663 (6)	0.00593 (5)	0	−0.00041 (5)	0
Fe (I)	0	0	1/2	0.0047 (2)	0.0084 (2)	0.0071 (2)	0.0008 (1)	0.0002 (1)	−0.0001 (1)
Fe (II)	0	0	1/2	0.0044 (2)	0.0048 (2)	0.0032 (2)	0.0007 (2)	0.0003 (2)	0.0000 (2)
Fe (III)	0	0	1/2	0.0059 (2)	0.0049 (2)	0.0047 (2)	0.0009 (1)	0.0004 (1)	−0.0002 (1)
O(1) (I)	0.4624 (6)	1/4	0.1102 (6)	0.0052 (8)	0.0088 (9)	0.0099 (9)	0	−0.0014 (8)	0
O(1) (II)	0.4614 (7)	1/4	0.1103 (7)	0.006 (1)	0.003 (1)	0.007 (1)	0	−0.0007 (9)	0
O(1) (III)	0.4612 (6)	1/4	0.1111 (6)	0.007 (1)	0.0034 (9)	0.008 (1)	0	−0.0013 (8)	0
O(2) (I)	−0.3056 (4)	−0.0575 (3)	0.3081 (4)	0.0044 (6)	0.0119 (7)	0.0085 (6)	−0.0013 (5)	−0.0012 (5)	0.0013 (5)
O(2) (II)	−0.3048 (5)	−0.0562 (4)	0.3077 (5)	0.0049 (9)	0.009 (1)	0.0041 (8)	−0.0013 (7)	−0.0020 (6)	0.0007 (7)
O(2) (III)	−0.3050 (4)	−0.0567 (3)	0.3082 (4)	0.0059 (7)	0.0067 (8)	0.0057 (7)	−0.0001 (6)	−0.0015 (5)	0.0012 (6)

Table 3
Selected interatomic distances (Å) and angles (°) for RFeO₃ (R = Y, Ho) (APD)

Results for HoFeO₃ follow those for YFeO₃ for each table entry.

R—R^i (2)	3.7184 (6)	Fe—Fe^viii (2)	3.8051 (2)
	3.7181 (4)		3.8079 (2)
R—R^ii (2)	3.8867 (2)	Fe—Fe^vii (4)	3.8460 (2)
	3.8881 (3)		3.8454 (2)
R—R^iii (2)	3.9786 (6)	Fe—O1^v (2)	2.001 (1)
	3.9776 (4)		2.003 (1)
R—Fe^iii (2)	3.0736 (3)	Fe—O2^ix (2)	2.005 (2)
	3.0764 (2)		2.004 (3)
R—Fe^iv (2)	3.2011 (4)	Fe—O2 (2)	2.035 (2)
	3.2014 (2)		2.030 (3)
R—Fe (2)	3.3538 (4)	O1—O1^v (2)	3.162 (4)
	3.3538 (2)		3.162 (6)
R—Fe^v (2)	3.7078 (4)	O1—O2^ii (2)	2.792 (4)†
	3.7055 (3)		2.796 (4)†
R—O1^v	2.234 (3)	O1—O2^iii (2)	2.812 (3)†
	2.234 (4)		2.800 (4)†
R—O1	2.304 (3)	O1—O2^x (2)	2.873 (3)†
	2.300 (4)		2.870 (4)†
R—O1^i	3.184 (3)	O1—O2^xi (2)	2.894 (4)†
	3.184 (4)		2.901 (5)†
R—O1^vi	3.458 (3)	O1—O2^xii (2)	3.514 (4)
	3.460 (4)		3.519 (5)
R—O2^vii (2)	2.271 (2)	O2—O2^xiii (2)	2.850 (3)†
	2.279 (3)		2.842 (4)†
R—O2^ii (2)	2.500 (2)	O2—O2^vii (2)	2.863 (3)†
	2.504 (3)		2.862 (4)†
R—O2^iii (2)	2.683 (2)	O2—O2^xiv	2.930 (3)
	2.679 (3)		2.951 (5)
R—O2 (2)	3.581 (2)	O2—O2^xv	3.099 (3)
	3.570 (3)		3.102 (4)

Iron octahedra angles

O1^v—Fe—O2	88.4 (1)	O1^v—Fe—O1^xvi	180 by
	88.0 (1)		symmetry
O1^v—Fe—O2^ix	88.4 (1)	O2^xiii—Fe—O2^ix	180 by
	88.5 (1)		symmetry
O2^xiii—Fe—O2^xii	89.74 (9)	Fe^iii—O1—Fe^xi	144.0 (2)
	89.6 (1)		143.9 (2)
		Fe—O2—Fe^vii	144.4 (1)
			144.9 (2)
Symmetry codes: (i) x−1/2, 1/2−y, −1/2−z; (ii) −x, 1/2+y, −z; (iii) 1/2+x, 1/2−y, 1/2−z; (iv) x, y, z−1; (v) x−1/2, 1/2−y, 1/2−z; (vi) x−1, y, z; (vii) −x−1/2, −y, z−1/2; (viii) −x, 1/2+y, 1−z; (ix) −1/2−x, −y, 1/2+z; (x) 1+x, y, z; (xi) 1/2−x, −y, z−1/2; (xii) −x, −y, 1−z; (xiii) 1/2+x, y, 1/2−z; (xiv) x, −1/2−y, z; (xv) −1−x, −y, 1−z; (xvi) 1/2−x, −y, 1/2+z.

†O—O distances in the FeO₆ octahedra.

Figure 2 Orthorhombic unit cell for HoFeO3. Vibrational ellipsoids are shown at the 99% probability level.

A lack of agreement between atomic fractional coordinates from Table 2 for the HoFeO₃ APD and SD data sets is consistent with differences of the cell parameters for those data sets in Table 1. This can be related to a more accurate determination of the cell parameters for the APD data sets by using 12 higher-angle reflections. The APD unit-cell parameters are close to those reported by Coppens & Eibschutz (1965) for YFeO₃ and by Marezio et al. (1970) for HoFeO₃. However, the b dimensions are still larger for the APD studies of two structures. The unit-cell parameters for YFeO₃ and HoFeO₃ APD data sets differ by Δa = 0.0009 (4), Δb = −0.0055 (6) and Δc = 0.0108 (4) Å. Substitution of diamagnetic Y by paramagnetic Ho introduces significant deformation of the unit cell in the bc plane of magnetic ordering. Atomic positions in Table 2 for the two structures are similar, though all unconstrained YFeO₃ fractional coordinates are displaced from the ideal values by slightly more than their HoFeO₃ equivalents. Selected interatomic R—O, Fe—O and O—O distances and angles for YFeO₃ given in Table 3 closely resemble those for HoFeO₃, also given in Table 3. However, intermetallic distances for the two structures are different, reflecting the unit-cell deformations. The Ho—Hoⁱⁱ and Fe—Fe^viii distances in the [010] direction are increased significantly, while Ho—Hoⁱⁱⁱ and Fe—Fe^vii distances in the ac plane are reduced within three s.u.s compared with the equivalent distances for YFeO₃. Although the shorter Ho—Feⁱⁱⁱ contacts in the ab plane are increased and the longer Ho—Fe^v contacts are reduced, the intermediate R—Fe^iv and R—Fe distances in the bc plane are in striking agreement for the two structures. It seems that magnetic Ho—Fe interactions regularize the Ho atoms substructure by reducing the displacement of Ho from the ideal positions.

3.3. Vibration parameters

Vibration ellipsoids for HoFeO₃ (APD) at the 99% probability level are depicted in Fig. 2. Vibration parameters appeared to be affected by the use of different counting systems for measurements of diffraction intensities. On average, the atomic vibrations for the APD data sets tend to be larger than those for the SD data sets from Table 2 and reported by du Boulay et al. (1995). The anisotropy of atomic vibrations for HoFeO₃ differs from that observed for YFeO₃. Vibration amplitudes in the b and c directions for all atoms of HoFeO₃ are notably smaller than those for YFeO₃, suggesting that the HoFeO₃ structure packing is more rigid in the bc plane of magnetic ordering. Y atoms which move further from the b axis in the (010) plane vibrate more strongly along the [010] direction.

3.4. Atomic charges

Atomic charges determined by projecting Δρ onto atomic density basis functions (Hirshfeld, 1977) are shown in Table 4. The charge signs, being consistent with electro-negativities for all the atoms, suggest electron transfer from cations towards the O anions. Atomic charges for the APD data sets are systematically smaller than those for the SD data sets from Table 4 and reported by du Boulay et al. (1995). This reflects their dependence on strong low-angle reflections, measurements of which were especially affected by the performances of photon counters. The sensitivity of atomic charges to the values of the low-angle reflections increases with atomic number. An overestimated extinction correction by a least-squares refinement for the HoFeO₃ SD data tended to reduce the polarity of the structure. A similar effect was observed for YFeO₃ by du Boulay et al. (1995).

Table 4
Atomic charges in electrons from the Hirshfeld (1977) partitioning of Δρ for RFeO₃ (R = Y, Ho)

	YFeO3 (APD)	HoFeO3 (APD)	HoFeO3 (SD)
R	0.9 (2)	0.6 (2)	1.7 (4)
Fe	0.4 (3)	0.3 (2)	0.9 (4)
O1	−0.7 (2)	−0.3 (2)	−1.2 (4)
O2	−0.3 (2)	−0.3 (1)	−0.7 (3)

3.5. Electron density

Deformation density (Δρ) sections in the (010) and approximate (100) planes of Fe—Fe and intermediate Fe—R (R = Y, Ho) contacts are shown in Fig. 3. The 0.5 and 0.75 e Å⁻³ contour intervals for YFeO₃ and HoFeO₃, respectively, are more than twice the σ(Δρ) values listed in Table 1.

Figure 3 Δρ for YFeO3 and HoFeO3. (a), (c), (e) In the (010) plane through the Fe atoms. (b), (d), (f) In the plane through the intermediate Fe—R (R = Y, Ho) contacts. (a), (b) APD data for YFeO3. (c), (d) APD and (e), (f) SD data for HoFeO3. For atoms deviating from the plane, their deviations are shown in italics. Map borders are 6.6 × 6.6 Å and 8.0 × 6.6 Å. Contour intervals are 0.5 e Å^−3 in (a) and (b), and 0.75 e Å^−3 in (c)–(f); positive contours, solid lines; negative contours, short dashes.

The Δρ map topography in the plane of the Fe—Fe contacts for the HoFeO₃ SD data shown in Fig. 3(e) is generally consistent with that for the APD data set shown in Fig. 3(c). However, Δρ densities around the heavier Ho atoms in Figs. 3(d) and 3(f) are significantly different, as expected from the ΔF distributions for the APD and SD low-angle reflections in Fig. 1. The sensitivity of Δρ densities to the values of the low-angle reflections increases with atomic number, as indicated by the atomic charges. Similarly, a remarkable agreement between the APD and SD Δρ images in the Fe—Fe plane for YFeO₃ shown in Fig. 3(a) and reported by du Boulay et al. (1995) is contrasted by discrepancies between the Δρ maps in the plane of the Fe—Y contacts for the APD in Fig. 3(b) and the SD data (not shown). Complexity of topography and stronger features of the electron distribution near the heavier Y and Ho atoms for the APD Δρ densities justify greater confidence in the reliability of the APD measurements. Therefore, further discussions will be focused only on the APD Δρ images.

The Δρ electron distribution near the Fe atoms in Figs. 3(a) and 3(c) is closely approximated by mmm symmetry. This approximate symmetry is higher than the inversion centre expected from the exact structural geometry around the Fe atom. It matches the mmm symmetry of the nearest-neighbour Fe cations with the shorter Fe—Fe vector along [001] more closely, in stark contrast with the lower symmetry of the O-atom positions projected in Figs. 3(a) and 3(c). The six positive Δρ density lobes formed around the Fe atom in YFeO₃ (Figs. 3a and 3b) are retained near this atom in HoFeO₃ (Figs. 3c and 3d), but rotated by 45° around the [100] axis. There are only two positive density lobes in the [100] direction for HoFeO₃ (Fig. 3c), compared with an approximate fourfold density cloud near the Fe atoms in YFeO₃ (Fig. 3a). Similarly, the Δρ density near the Fe atoms in the (010) plane of SmFeO₃ (Maslen et al., 1996a,b) has also only two positive lobes; however, these are in the [001] direction across the map. The reduced symmetry in the (010) plane may thus be attributed to the perturbing effect of the magnetic Ho and Sm cations rather than the neighbouring O anions. The Ho atoms projected in the planes of Figs. 3(a) and 3(c) are located away from the central Fe atom at 3.0764 (2) and 3.7055 (3) Å in the [100] direction down the map and at 3.2014 (2) and 3.3538 (2) Å in the [001] direction across the map. The Fe—Ho vectors differ significantly in length, and the Δρ symmetry is affected by that length difference.

The approximate (100) sections that contain the two intermediate-length R—Fe (R = Y, Ho) contacts are shown in Figs. 3(b) and 3(d), with the [001] direction of antiferromagnetically ordered spins across the maps. For both structures the Δρ topography around the R and Fe atoms is strongly correlated. However, the sharp increase of positive Δρ near the Ho atom has a more complex fourfold shape which resembles that near the Fe atom in Fig. 3(d). The stronger Ho—Fe coupling indicated by Δρ maps in the (100) plane correlates with the spin reorientation transition in this plane for HoFeO₃ at low temperatures.

Finally, there is a correlation between the Δρ topography near the Ho and Fe atoms and the differences in cell dimensions between YFeO₃ and HoFeO₃. The HoFeO₃ c dimension in the direction of antiferromagnetic order, along which the density is more depleted, is significantly 0.0108 (4) Å smaller than in YFeO₃. The dimension in the [010] direction of weak ferromagnetic moment alignment is 0.0055 (6) Å larger for HoFeO₃. Along this direction the electron density is less depleted. These changes from diamagnetic Y to paramagnetic Ho are consistent with what would be expected if the changes in interatomic distances resulted from magnetostrictive forces due to interactions between the electron spins for the paramagnetic Ho atoms.

4. Concluding remarks

The advantages of high-flux synchrotron radiation combined with a fast photon counter for imaging of electron density has been illustrated by example. The fast APD detector developed recently at the Photon Factory beamline 14A provided superior measurements for the diffraction intensities in the dynamic range span of 10¹⁰ counts s⁻¹ due to its low intrinsic noise and high maximum counting rate.

The structure factors for strong low-angle reflections tend to be lower when measured with a conventional scintillation counter. This indicates that the applied dead-time corrections were insufficient, resulting in the extinction-like effects on data for shorter wavelengths. The type of X-photon counting system affects strongly the electron density images, especially in the regions near heavier atoms, since the sensitivity of Δρ densities to the values of the low-angle reflections increases with atomic number.

The APD measurements revealed a striking correlation of Δρ images around unlike cations of Fe and R in RFeO₃ (R = Y, Ho). This correlation is more pronounced for magnetic Fe and Ho atoms. The APD Δρ images indicate that electron density is perturbed significantly by metal–metal interactions, an argument supported by other synchrotron radiation studies of orthoferrites: YFeO₃ (du Boulay et al., 1995) and SmFeO₃ (Maslen et al., 1996a), and the C-type sesquioxides Y₂O₃, Dy₂O₃ and Ho₂O₃ (Maslen et al., 1996b). When spin dependent, they play an important role in magnetic ordering and phase transitions. This is consistent with the hypothesis that the structures of oxides are governed by the cation subarrays and can be described as oxygen-stuffed alloys (O'Keefe & Hyde, 1985; Ramos-Gallardo & Vegas, 1997) as an alternative to the traditional model of cation-centred anion polyhedra.

Footnotes

¹A list of structure factors has been deposited with the IUCr (Reference: MA0001). Copies may be obtained through The Managing Editor, International Union of Crystallography, 5 Abbey Square, Chester CH1 2HU, England.