Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 5

Choose the four basis-spin signs as $(+,-,-,+)$ in the order of positions listed in the source; reversing all signs changes no diffraction intensity. Apart from the common magnetic form factor and perpendicular-moment factor, the magnetic structure factor isThe first factor is nonzero exactly when $h$ and $k$ have opposite parity, and the second exactly when $l$ is odd. Thus the magnetic Bragg peaks requireWhen both conditions hold, $|F_M|=4|m|$ before geometric and magnetic-form-factor weights. These are magnetic selection rules for the antiferromagnetic component, not conditions forbidding all nuclear diffraction at other indices.

Measure several allowed magnetic reflections from an oriented single crystal by magnetic neutron diffraction, preferably with polarization analysis. A neutron couples to the component of the ordered magnetic moment perpendicular to the scattering vector $\mathbf{Q}$. For a collinear arrangement, the geometric intensity factor isCompare reflections whose $\mathbf{Q}$ point in different crystal directions, correcting for the magnetic form factor and known experimental geometry. Their relative intensities distinguish candidate spin directions; a moment parallel to $\mathbf{Q}$ gives no magnetic signal from that component. Polarized neutrons help separate nuclear and magnetic contributions. Observing only peak positions establishes periodicity, not the spin direction.

Treat the ordered spins as classical vectors of fixed length $S$, as in the requested mean-field energy. Each iron site has six nearest neighbors: four in the $xy$ plane and two along $z$. All are antiparallel in the collinear state. Each exchange bond contributes $-J{S}^2$ and is counted only once, giving $-(6/2)J{S}^2=-3J{S}^2$ per spin.

The easy-axis term contributes $-K{S}^2$ because $S_i^x=\pm S$. The cross product of two collinear spins is zero, so the term proportional to $D$ contributes nothing. Therefore

The canted pattern has opposite $x$ components but a common $z$ component. Write neighboring spins as ${\mathbf{S}}_A=S(\cos \theta ,0,\sin \theta )$ and ${\mathbf{S}}_B=S(-\cos \theta ,0,\sin \theta )$. Their scalar product is $-S^2\cos 2\theta$, and the cross-product magnitude is $S^2|\sin 2\theta |$.

There are three exchange bonds per spin, two eligible $xy$ bonds per spin for the $D$ term, and one single-ion term per spin. HenceFor the branch $\theta \ge 0$ close to the collinear state,Minimization is equivalent to maximizing $(3J+K/2)\cos 2\theta +2D\sin 2\theta$, which givesThe chosen branch is a minimum of the energy because it maximizes that positive-amplitude sinusoid. The resulting state is a canted antiferromagnet with a small common $z$ moment. The source's absolute-value interaction selects a canting magnitude but is symmetric between the two signs of the weak moment; it is not an oriented vector Dzyaloshinskii-Moriya interaction specifying a unique handedness.

Measure magnetization along $z$ at low temperature and use a small field to select a single weak-ferromagnetic domain. After domain alignment, subtract the approximately linear field-induced susceptibility background to obtain the spontaneous weak moment, using the extrapolated intercept of that branch rather than a multidomain zero-field average.

There are four iron ions per orthorhombic cell. If the spontaneous magnetization density is $M_z$ and the cell volume is $V_{\mathrm{cell}}$, the ordered moment and dimensionless spin component per ion areHere spin components are expressed in units of $\hslash$ as in the source Hamiltonian. Alternatively normalize the magnetic moment to the measured number of iron ions. A field too large to preserve the small-canting structure would not measure the zero-field component directly. For high-spin Fe${}^{3+}$ with $S=5/2$ and $g\simeq 2$, the angle above corresponds to approximately $0.026{\mu }_B$ per iron ion.