= Solution
{c}
A consistent mechanism is a transition to G-type <antiferromagnetism> at the <Néel temperature>. On the manganese <primitive cubic lattice>, take the <magnetic moments> to be
$$
\mathbf m_{j_xj_yj_z}=\mathbf m_0(-1)^{j_x+j_y+j_z}.
$$
Every nearest neighbour has the opposite <magnetic moment>. The nuclear <crystal structure> can retain its original <lattice constant>, while the <magnetic order> introduces the propagation <wavevector>
$$
\mathbf Q=\frac{2\pi}{a}\left(\frac12,\frac12,\frac12\right).
$$
Summing the <scattering amplitude> from the <magnetic moments> over sites gives peaks at $\mathbf q=\mathbf G+\mathbf Q$, where $\mathbf G$ is a nuclear <reciprocal lattice vector>. Thus all three components of the new <Miller indices>, expressed relative to the original <unit cell>, are half-integers. The two smallest squared magnitudes in units of $(2\pi/a)^2$ are
$$
\frac34\quad\text{for }\left\{\frac12,\frac12,\frac12\right\},\qquad
\frac{11}{4}\quad\text{for }\left\{\frac32,\frac12,\frac12\right\}.
$$
The first predicts $2\theta\simeq21.29^\circ$, agreeing with the new ring. The next squared magnitude is $19/4$, outside the measured angular range. Taking the observed first magnetic ring as the calibration, <Bragg's law> predicts
$$
\sin\theta_2=\sqrt{\frac{11}{3}}\sin(10.65^\circ),\qquad
\boxed{2\theta_2\simeq41.45^\circ\simeq41.4^\circ.}
$$
Using the independently fitted <lattice constant> gives $41.42^\circ$, within the input rounding. One ring alone does not uniquely determine a <magnetic order>; this proposed nearest-neighbour <antiferromagnetic> pattern explains both the observed ring and the count of two new rings in the specified interval.
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