MnO — Type-II antiferromagnet#

MnO orders below \(T_N = 118\) K with Mn²⁺ moments in {111} planes coupled antiferromagnetically along [111] [3]. Space group \(Fm\bar{3}m\) (No. 225), propagation vector k = (½, ½, ½). The correct k-maximal MSG belongs to the R_I-3m family (Bilbao notation); the present library returns only BNS 2.7 (P\(\bar{1}\)) for this case because the C₃[111] axis in the cubic basis is not recognised as equivalent to the hexagonal C₃ in the library’s W-matrix comparison. Use analyze_msg() with the BNS number from MAXMAGN or your refinement software once the MSG is known.

Setup#

import numpy as np
from msgjson import magnetic_structure_factors

K = [0.5, 0.5, 0.5]   # L point, FCC Brillouin zone

FCC conventional cell — four Mn sites#

With k = (½, ½, ½) the phase \(e^{2\pi i\mathbf{k}\cdot\mathbf{r}}\) is +1 at (0, 0, 0) and −1 at the three face-centred positions. Physical refinements find moments in the {111} plane; here we take m ‖ [1, −1, 0]:

m_amp = 4.58                                    # μ_B, measured Mn²⁺ moment
m_dir = np.array([1.0, -1.0, 0.0]) / np.sqrt(2)
m = m_amp * m_dir

site_moms = [
    (np.array([0.0,  0.0,  0.0 ]),  +m),   # phase +1
    (np.array([0.0,  0.5,  0.5 ]), -m),    # phase −1
    (np.array([0.5,  0.0,  0.5 ]), -m),    # phase −1
    (np.array([0.5,  0.5,  0.0 ]), -m),    # phase −1
]

Three-to-one sublattice split (type-II AFM); net moment zero.

Structure factors#

Satellites at Q = τ + k for FCC-allowed τ (h, k, l all even or all odd):

a = 4.445
lattice = a * np.eye(3)
B = 2 * np.pi * np.linalg.inv(lattice).T

hkl_list = [(0,0,0), (1,1,1), (1,1,-1), (-1,1,1), (2,0,0), (0,0,2)]
F2 = magnetic_structure_factors(
    site_moms, hkl_list, K, lattice=lattice, ion="Mn2+"
)

print(f"  {'τ':12s}  {'Q = τ+k':20s}  {'|F_M|²':>8s}  {'sin²α':>6s}  {'|F_M⊥|²':>9s}")
for hkl_i, f2 in zip(hkl_list, F2):
    Q_frac = tuple(h + ki for h, ki in zip(hkl_i, K))
    Q_cart = B @ np.array(Q_frac)
    Q_mag  = np.linalg.norm(Q_cart)
    s2     = 1 - np.dot(Q_cart / Q_mag, m_dir)**2
    qstr   = "({:.1f},{:.1f},{:.1f})".format(*Q_frac)
    print(f"  {str(hkl_i):12s}  {qstr:20s}  {f2:8.3f}  {s2:6.3f}  {f2*s2:9.3f}")

τ             Q = τ+k                 |F_M|²   sin²α    |F_M⊥|²
(0, 0, 0)     (0.5,0.5,0.5)          272.508   1.000    272.508
(1, 1, 1)     (1.5,1.5,1.5)           62.626   1.000     62.626
(1, 1, -1)    (1.5,1.5,-0.5)          98.939   1.000     98.939
(-1, 1, 1)    (-0.5,1.5,1.5)          98.939   0.579     57.280
(2, 0, 0)     (2.5,0.5,0.5)           62.626   0.704     44.070
(0, 0, 2)     (0.5,0.5,2.5)           62.626   1.000     62.626

All FCC-allowed τ give constructive interference (phases 1, −1, −1, −1 sum to 4 for the 1:3 sublattice split), so the only intensity variation comes from the Mn²⁺ form factor and sin²α. The variation in sin²α across equivalent |Q| peaks (e.g. (1, 1, −1) vs (−1, 1, 1)) reflects the in-plane anisotropy of the moment direction and can be used to determine the moment orientation.