Electron densities from gamma-ray diffraction

Electron density in crystals

The electron density is the fundamental variable in the density functional theory for the ground-state energy of a many-electron system. Its spatial distribution in the unit cell, ρ(r), is experimentally accessible from high-quality x-ray diffraction data. Great advances in speed of data collection have been achieved by the use of area detectors at synchrotrons, allowing charge density studies of increasingly complex organic materials. There are important applications, however, where success is limited by the attainable accuracy rather than data extensiveness. This applies in particular to studies of simple systems with medium or high-Z elements including transition metals or lanthanides. The investigation of such structures poses a special challenge, because the valence electron contribution forms an ever smaller fraction of the total scattering. Furthermore, the high symmetry of small-unit-cell structures leads to only few reflections in the low order region where valence scattering is concentrated, calling again for utmost accuracy if meaningful information is to be obtained. Gamma-ray diffraction is especially suited to resolve such issues where other techniques are intrinsically hampered. A structure-factor accuracy at the 0.1% level is achievable.

Gamma-ray diffraction

At the HZB gamma-ray diffractometer we use the 316.5 keV (0.0392 Ǻ) line of a commercially available ¹⁹²Ir source, actually the highest energy employed in crystal diffraction work. The flux at the sample position from a fresh 200-Ci source (T_1/2 = 73.83 d) is 10⁶photons/s/cm². Since the crystal reflectivity is proportional to λ², large samples (2 mm size) are required to compensate the weak interaction. The diffraction peaks are recorded in ω-step scan mode with an intrinsic germanium detector. The use of 316.5 keV radiation offers two basic advantages in removing major sources of systematic error: (i) the high photon energy is the reason for low absorption, absence of primary and reduced secondary extinction, no dispersion correction, and bulk volume probing making surface effects negligible; (ii) favorable experimental conditions such as the perfect stability, homogeneity, and monochromaticity (Δλ/λ = 10^–6) of the incident γ-ray beam. The absence of any optical devices ensures a uniform instrumental resolution.

Selected applications

Ionic antiferromagnets

Exhaustive γ-ray charge density studies have been performed on archetypal antiferromagnets such as the transition-metal difluorides MnF₂, FeF₂, CoF₂ and NiF₂ [1], as well as on the monoxides CoO [2], MnO [3] and NiO [4]. It has been demonstrated that charge-scattering studies enable valuable insights into the interplay between electronic structure and magnetism, thereby supporting and complementing results from magnetic neutron and x-ray diffraction. Quantitative information has been obtained about 3d-orbital populations, spin delocalization and unquenched orbital angular momentum, as exemplified in the following panel and figure.

Magnetic moments from charge scattering

MnO:   ■ Vanishing orbital angular momentum → spin-only magnetism
           ■ 3d population number from γ-ray charge density → μ_S = 4.75(3) μ_B
           ■ Neutron diffraction (BER II) including zero-point spin deviation: μ_S = 4.72(4) μ_B
           ■ Significantly smaller than the formal integer value of 5 μ_B

CoO:    ■ Isolated-atom spin magnetic moment for 3d⁷: μ_S = 3 μ_B
           ■ Number of unpaired electrons from γ-ray charge density → μ_S = 2.40(8) μ_B
           ■ Neutron diffraction (BER II): μ_total = μ_S + μ_L = 3.98(6) μ_B
           ■ Combination yields unquenched large orbital moment: μ_L = 1.6(1) μ_B

NiO:   ■ From neutron diffraction [5]: μ_total = 2.05(6) μ_B
          apparent agreement with expectation for cubic field: L = 0 for (t_2g)⁶(e_g)²
           ■ From γ-ray diffraction: reduced 3d charge on Ni = 7.63(2) |e|:
          partially filled orbitals upon trigonal distortion → μ_L = 0.31(1) μ_B
          reduced spin magnetic moment μ_S = 1.7 μ_B
           ■ Agreement with magnetic synchrotron x-ray diffraction [6]: μ_L = 0.32(5) μ_B

Asphericity of charge distribution in MnO

Paramagnetic phase: e_g – t_2g crystal field splitting
         ■ Nonspherical distortion of formally half filled d shell
     ■ Preferential occupation of e_g orbitals (peaks in the figure)
         ■ Charge deficiency in t_2g levels (holes)
         ■ Observed population ratio e_g / t_2g = 0.80 is larger than expected ratio of 2:3
         ■ Experimental result is in agreement with theoretical work [7].
         ■ Antiferromagnetic phase: further splitting of t_2g upon trigonal distortion
         ■ Distinct rearrangement in electron subshell population
         ■ Antiferromagnetic MnO is not a quasi-cubic system as commonly assumed

Figure 1:

Top: Aspherical contributions to the electron density in paramagnetic MnO (space group Fmm). Mn is located in the center and the corners. Solid lines: accumulation of density, dashed lines: depleted regions.

Bottom: Same plane as above for antiferromagnetic MnO (space group Rm)

3d transition metals

Despite decades of effort only few medium-accuracy crystal structure factors have been reported for the 3d transition metals up to now. Recent progress with γ-ray diffraction includes Cr [8], Fe [9], Ni [10], and Co [11], where extended high-quality data sets have been measured up to sinθ/λ = 1.9 Å^–1. Two important solid-state effects are outlined in the following panel.

Radial extent of 3d valence shell

■ Refined radial contraction relative to the independent-atom _3d value
bcc Cr
bcc Fe
hcp Co
fcc Ni
Contraction (%)
12.6(2)
8.9(2)
1.22(16)
2.07(5)
■ Considerable difference between body-centered and close-packed structures
■ Solid-state orbital contraction is at variance with earlier experiments and theory
■ Long-standing controversy: itinerant versus localized 3d electrons
■ Charge density: significant contraction rather than expansion
strong support of localized atomic character

Number of 3d electrons

■ Common expectation: metallic configuration is close to 3dⁿ⁺¹4s¹
■ Joint interpretation of charge ρ_charge = ρ_↑ + ρ_↓
and spin density ρ_spin = ρ_↑ – ρ_↓
■ For open-shell atoms: majority-spin wave function ψ_↑ contracts relative to ψ_↓
■ Cause of the observed form factor difference f_3d(spin) – f_3d(charge)
■ Confirmation in the case of Fe: only n_d = 7 is consistent with predicted radial behavior
■ Rejection in the case of Co and Ni: 4s – 3d transfer is incompatible with γ-ray data

References

[1] Jauch, W., Acta Cryst. A60, 397 (2004).

[2] Jauch, W. and Reehuis, M., Phys. Rev. B 65, 125111 (2002).

[3] Jauch, W. and Reehuis, M., Phys. Rev. B 67, 184420 (2003).

[4] Jauch, W. and Reehuis, M., Phys. Rev. B 70, 195121 (2004).

[5] Cheetham, A. K. and Hope, D. A. O., Phys. Rev. B 27, 6964 (1983).

[6] Fernandez, V., Vettier, C., de Bergevin, F., Giles, C., and Neubeck, W., Phys. Rev. B 57, 7870 (1998).

[7] Pickett, W. E., Erwin, S. C., and Ethridge, Phys. Rev. B 58, 1201 (1998).

[8] Jauch, W. and Reehuis, M., Phys. Rev. B 73, 085102 (2006).

[9] Jauch, W. and Reehuis, M., Phys. Rev. B 76, 235121 (2007).

[10] Jauch, W. and Reehuis, M., Phys. Rev. B 78, 235113 (2008).

[11] Jauch, W. and Reehuis, M., Phys. Rev. B 80, 125126 (2009).

Most recent publications:

Jauch, W. and Reehuis, M: Anisotropy of extinction: extrapolation to the kinematical limit by γ-ray diffraction. Acta Crystallographica A 67 (2011), 469

Jauch, W.; Reehuis, M.: Electron density distribution in vanadium and niobium from γ-ray diffraction. Physical Review B 83 (2011), 115102

	bcc Cr	bcc Fe	hcp Co	fcc Ni
Contraction (%)	12.6(2)	8.9(2)	1.22(16)	2.07(5)