= Magnetic neutron diffraction
<Magnetic neutron diffraction> uses scattering of <neutrons> by ordered <magnetic moments> to infer <magnetic order>. <Neutrons> also scatter from <atomic nuclei>, so a pattern can contain nuclear and magnetic contributions. A <Bragg scattering> peak encodes periodicity; it is not a photograph of an individual <spin>.
A minimal original calculation makes the extra periodicity visible. Consider identical nuclear scatterers at positions $x_n=na$, with equal neighboring separation $a$, and ordered <magnetic moments> alternating between $+m_A$ and $-m_B$. A two-site repeating block of <length> $2a$ has nuclear and magnetic <scattering amplitudes>, apart from common geometric factors,
$$
F_N(q)=b(1+e^{iqa}),\qquad
F_M(q)=m_A-m_Be^{iqa},\qquad
q=\frac{\pi h}{a},\quad h\in\mathbb Z.
$$
Here $b$ is the common nuclear scattering coefficient and $q$ the component of the scattering <wave vector> along the row. The magnetic expression assumes equally scattering point <magnetic moments> perpendicular to the scattering <wave vector>; real magnetic scattering measures the perpendicular component and has a <momentum>-dependent amplitude. This one-dimensional calculation is a periodicity illustration, not a finite-<temperature> solution for a chain.
For even $h$, $F_N=2b$ and $F_M=m_A-m_B$. For odd $h$, $F_N=0$ and $F_M=m_A+m_B$. Thus equivalent opposing <magnetic sublattices>, $m_A=m_B$, cancel in forward scattering but reinforce at the odd positions of the doubled period. The <intensity (physics)> is proportional to the squared magnitude of the <scattering amplitude>, so changing an amplitude's sign does not make a negative <intensity (physics)>.
\Image[/magnetic-diffraction-sublattice-selection.png]
{title=An alternating <magnetic moment> pattern creates extra <reciprocal lattice> peaks}
{description=Original exact two-site <crystal basis structure factor> illustration, not historical measurements or a simulation of MnO. Nuclear strengths use $b=1$. Ordered magnetic strengths use $(m_A,m_B)=(1,1)$ or $(2,1)$; each channel is in its own arbitrary units, so their absolute heights cannot be compared. Thin solid nuclear sticks and dashed magnetic sticks coincide at even $h$ in the ferrimagnetic case. The top panel stipulates loss of long-range magnetic order while retaining nuclear periodicity; diffuse magnetic scattering is omitted, rather than asserted absent. Only positive nonzero $q$ is shown.}
This calculation also exposes a useful difference between <antiferromagnetism> and <ferrimagnetism>: with $m_A=2$ and $m_B=1$, magnetic squared amplitudes are $1$ at even $h$ and $9$ at odd $h$. Extra magnetic peaks can therefore occur in a <ferrimagnet> too. Identifying a real magnetic structure needs the full <crystal basis structure factor>, multiple reflections, and the allowed moment orientations, rather than a single extra peak.
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