Direct Observation of Anapoles by Neutron Diffraction

The scope of magnetic neutron scattering has been expanded by the observation of electronic Dirac dipoles (anapoles) that are polar (parity odd) and magnetic (time odd). A zero-magnetization ferromagnet ${\mathrm{Sm}}_{0.976}{\mathrm{Gd}}_{0.024}{\mathrm{Al}}_2$ with a diamond-type structure presents Dirac multipoles at basis-forbidden reflections that include the standard (2, 2, 2) reflection. Magnetic amplitudes measured at four such reflections are in full accord with a structure factor calculated from the appropriate magnetic space group.

Figure 1

Magnetic amplitudes as a function of temperature measured in an interval that embraces the compensation temperature, $T_{\mathrm{comp}}\approx 68.7\mathrm{K}$, of ${\mathrm{Sm}}_{0.976}{\mathrm{Gd}}_{0.024}{\mathrm{Al}}_2$. Panels (a) and (b) show $\gamma =F^{(-)}/F_N$ for (2, $-2$, 2) and (2, $-2$, 6) basis-forbidden Bragg spots attributed to anapoles. Displayed values are a simple average of measurements for ($H_o$, $-H_o$, $L_o$) and $-(H_o,-H_o,L_o)$ that are identical according to Eq. (2). Panels (c) and (d) show $\gamma =F^{(+)}/F_N$ for (1, $-1$, 3) and (1, $-1$, 5) Bragg spots attributed to axial magnetic dipoles.

Figure 2

Top panel: crystal structure (#227, C15 cubic Laves) with Sm ions in yellow and Al ions in green, and cell edges ($a$, $b$, $c$). Bottom panel: dipoles for magnetization parallel to [1, 1, 0] using a magnetic space-group $Im{m}^{\prime }{a}^{\prime }$ (#74.559) [29]. Cubic parent cell outlined in black, and orthorhombic magnetic cell ($\xi$, $\eta$, $\zeta$) with $\xi =(1/2,1/2,0)$, $\eta =(1/2,-1/2,0)$, and $\zeta =(0,0,-1)$ outlined in yellow. Green arrows are axial dipoles parallel to the $\xi$ axis, while blue and red arrows that lie along the $\eta$ axis denote anapoles related by point inversion.

Figure 3

Radial integrals for Dirac multipoles that appear in Eq. (2) derived from an atomic code due to Cowan [38]. Dimensionless variable $w=12\pi a_{\mathrm{o}}s$, where $a_{\mathrm{o}}$ is the Bohr radius, while the standard variable for radial integrals $s$ is derived from the Bragg angle and neutron wavelength $s=\sin \theta /\lambda$. Green curve shows ($h_1$) and blue shows [$w\times (j_0)$]. Note that ($j_0$) is proportional to $1/w$ as the wave vector approaches zero. Atomic wave functions are $4f^5–5d^1$. Also included in the figure is the standard radial integral $⟨j_0⟩$ that appears in the so-called dipole-approximation [Eq. (1)] for diffraction by axial dipole moments. Results obtained with our ${\mathrm{Sm}}^{3+}$ ($4f^5$) wave function are denoted by the continuous black curve, to which we added for comparison four values ($+$) derived from the standard interpolation formula [32].