Monoclinic symmetry of the hcp-type ordered areas in bulk cobalt

The gradual ferromagnetic spin reorientation in hexagonal close packed cobalt (hcp-Co) phase between $230{}^{\circ }\mathrm{C}$ and $330{}^{\circ }\mathrm{C}$ reported for a Co single crystal [Bertaut et al., Solid State Commun. 1, 81 (1963)] suggests that this phase could not have a hexagonal symmetry. This hypothesis is verified positively by synchrotron radiation diffraction and neutron diffraction on polycrystalline powder cobalt. The analysis of diffraction data has been done by using a specific set of Bragg peaks, which are not sensitive to the stacking faults present in abundance in hcp-Co. The crystal structure of the hcp-type ordered areas of cobalt is described by the monoclinic symmetry with the magnetic space group $C{2}^{\prime }/m^{\prime }$. In this monoclinic crystal structure the former hexagonal [001] axis is no longer perpendicular to the hexagonal layers. The hexagonal [001] and [010] axes make an angle equal $\alpha \approx$ 90.10(1)${}^{\circ }$, while the angle between in-plane [100] and [010] axes equals $\gamma \approx$ 120.11(1)${}^{\circ }$. The monoclinic symmetry provides an approximate description of the crystal structure of the stacking faulted hcp-Co areas coexisting with fcc-Co areas.

Figure 1

In-plane atomic positions in the hexagonal A (solid line), B (dashed line), and C (dotted line) layers (adapted from [2]). The hexagonal unit cell is depicted with solid lines.

Figure 2

Selected parts of neutron diffraction pattern of HCP-Co (a) and SR diffraction patterns of HCP-Co (b) and MIX-Co (c) samples at RT. The peaks due to hcp and fcc phase models are marked with “H” and “C” symbols, respectively. The stacking faults insensitive peak (002) is marked with (*). The very weak C(200) peak is shown with an arrow in panel (b). The scattering vector $Q=2sin\theta /\lambda$, where $2\theta$ is the scattering angle.

Figure 3

Selected parts of the SR powder diffraction patterns measured for the MIX-Co (solid line) and HCP-Co (open circles) are shown in upper plots. The only stacking faults insensitive peak is (004) while the remaining peaks are stacking faults sensitive. The SR powder diffraction pattern calculated for a hexagonal model of stacking faults with the probabilities ${\alpha }_{ST}=0.0067$ and ${\beta }_{ST}=0.03$ (see text) is shown in the lower plot (solid line). The stacking faults sensitive peakshapes calculated with FAULTS using the monoclinic model (not shown) are hard to distinguish from the hexagonal model results. The upper plots were shifted vertically for visualization.

Figure 4

Integral breadths ${\beta }_{RAW}$ (in radians) of the hcp-Co insensitive peaks and fcc-Co peaks observed in the SR diffraction patterns of MIX-Co (a) and HCP-Co (b). The hexagonal and cubic indices are shown. In (c) only the hcp-Co insensitive peaks from MIX-Co and HCP-Co samples are compared with the integral breadths of the ${\mathrm{Na}}_2{\mathrm{Ca}}_3{\mathrm{Al}}_2{\mathrm{F}}_{14}$ reference standard [31] (estimation of the instrumental resolution). Panel (c) is shown to compare the peaks due to the hcp-Co that were already shown in MIX-Co (a) and HCP-Co (b). The lines are shown to guide the eye.

Figure 5

Observed integral breadths ${\beta }_{RAW}$ (in radians) of the hcp-Co insensitive peaks observed in HCP-Co (a) are compared with the results of model calculations with both monoclinic and hexagonal symmetry (b) by the program FAULTS (see text).

Figure 6

Williamson-Hall plots for $\left(00l\right)$ and $\left(hh0\right)$ peak families of the HCP-Co sample. Formulas of linear functions obtained through least squares method for $\left(00l\right)$ peaks and linear extrapolation for $\left(hh0\right)$ peaks are shown in the plot and are expressed in the unit of degrees.

Figure 7

The hexagonal (a${}_h^0$, b${}_h^0$, c${}_h^0)$ pseudohexagonal (a${}_h^1$, b${}_h^1$, c${}_h^1)$, and monoclinic (a${}_m^1$, b${}_m^1$, c${}_m^1)$ axes shown in two projections (a) perpendicular to the hexagonal ${\mathbf{c}}_h^{*}$ reciprocal lattice vector and (b) perpendicular to the monoclinic ${\mathbf{b}}_m^{*}$ reciprocal lattice vector. The z = 1/4, 3/4... are the coordinates of atomic planes and they are the same in all three settings. Solid lines mark the base of the hexagonal unit cell and hexagonal axes, dashed lines are used for the pseudohexagonal vectors, and the dashed-dotted lines denote the monoclinic axes. The $\delta \gamma /2$ and $\delta {\beta }_m$ angles were significantly enlarged for visualization. The distance between adjacent planes is denoted as $d_{002}$ in (b) with the same 002 indices in all three settings.

Figure 8

(Left panel) Values of the fit quality indicator, wRp vs angular deviations $\delta \alpha$ and $\delta \gamma$ obtained for the HCP-Co sample. (Right panel) Plot of the one-dimensional cuts with the value of wRp along the lines: $\delta \alpha =0\left(▵\right)$; $\delta \gamma =0(\square )$; $\delta \alpha =\delta \gamma (•)$. The angle deviation is $\delta \gamma$ for $\left(▵\right), \delta \alpha$ for $(\square )$ and $\delta \alpha =\delta \gamma$ for $(•)$.

Figure 9

Neutron powder diffraction pattern of HCP-Co sample measured with instrument D2B at RT. The refinement was done with the four peaks “insensitive” to stacking faults (see text). The grey sections were excluded from refinements. The difference curve is shown at the bottom.

Figure 10

The values of refined magnetic moment in directions parallel (a) and perpendicular (b) to hexagonal $c$ axis vs the angle ${\varphi }_m$ between spin direction and the $c$ axis. [(c)–(e)] Show the total magnetic moment, the isotropic displacement factor $U_{\mathrm{iso}}$ and the refinement quality indicator wRp.

Figure 11

Temperature dependence of the Bragg peaks measured with neutron diffraction for powder HCP-Co sample in the present study using instrument SPODI peak (002) (a) and peak (100) (b). Similar results obtained from neutron diffraction on a Co single crystal by Bertaut et al. [18, 19] are shown: peak (002) (c) and (100) (d).

Figure 12

Schematic presentation of the arrangement of atoms seen from top of the $A$ and $B$ layers (left panel) and $A$ and $C$ layers (right panel). The Co atoms represented by open circles (A, solid lines; B, dotted lines; and C, dashed lines) are shown at their positions in the hexagonal lattice. The vectors indicate how these atoms move on changing to the pseudohexagonal, i.e., monoclinic crystal structure model. The symbol $+$ or 0 or $-$ next to the vector shows if the vector points below or above (see text). The unique monoclinic axis ${\mathbf{b}}_m^1$ (dash-dotted line) and the monoclinic plane ${\left(010\right)}_m$ (dotted line) are shown.