α-RuCl3 — Zigzag antiferromagnet, k = (0, ½, 1)#

α-RuCl3 orders below \(T_N \approx 7\) K in a zigzag antiferromagnetic pattern with Ru3+ moments in the ab-plane [2]. Space group \(R\bar{3}\) (No. 148), propagation vector k = (0, ½, 1) (hexagonal setting). The k-maximal MSG is already BNS 2.7 (\(P_S\bar{1}\), type IV) — no symmetry breaking beyond the maximal is required to allow in-plane moments.

Setup#

import numpy as np
from msgjson import maximal_msgs, orbit_moments, magnetic_structure_factors

SG      = 148
K       = [0, 0.5, 1.0]
RU_SITE = [0, 0, 1/3]   # Ru^3+ Wyckoff 6c

Magnetic space group#

top = maximal_msgs(SG, k=K, sites=[RU_SITE], centering="R")

for r in top:
    s = r.sites[0]
    print(f"BNS {r.bns_number}  type={r.msg_type}  n_ops={r.n_ops}"
          f"  n_free={s.n_free}  origin={r.origin_shift}")

BNS 2.7  type=4  n_ops=4  n_free=3  origin=(0.0, 0.0, 0.0)
BNS 2.7  type=4  n_ops=4  n_free=3  origin=(0.0, 0.0, 0.5)

Two domain states arising from the two inequivalent positions of the anti-translation τ = (0, 0, ½). Both give the same \(|F_M|^2\). n_free = 3 — any moment direction is symmetry-allowed.

Moment orbit#

r = top[0]
m_amp = 0.5                          # μ_B, Ru³⁺ ordered moment
m_ref = m_amp * np.array([1.0, 0.0, 0.0])   # in-plane, along a*

site_moms = orbit_moments(r, site_idx=0, m_ref=m_ref, k=K, centering="R")

for pos, mom in site_moms:
    print(f"  {np.round(pos, 4)}:  m = {np.round(mom, 3)} μ_B")

[0.     0.     0.3333]:  m = [0.5 0.  0. ] μ_B
[0.     0.     0.6667]:  m = [0.5 0.  0. ] μ_B
[0.     0.     0.8333]:  m = [-0.5  0.   0. ] μ_B
[0.     0.     0.1667]:  m = [-0.5  0.   0. ] μ_B

Four-site zigzag +,+,−,− along c. Net moment zero (antiferromagnet).

Structure factors#

Satellites appear at Q = τ + k where τ satisfies the R-centering condition −h + k + l ≡ 0 (mod 3):

a, c_lat = 5.98, 17.0
lattice = np.array([[a,    0,              0    ],
                    [-a/2, a*np.sqrt(3)/2, 0    ],
                    [0,    0,              c_lat]])
B = 2 * np.pi * np.linalg.inv(lattice).T

hkl_list = [(0,0,0), (0,1,-1), (0,0,3), (0,0,6)]
F2 = magnetic_structure_factors(
    site_moms, hkl_list, K, lattice=lattice, ion="Ru1+"
)

print(f"  {'τ':12s}  {'Q = τ+k':18s}  {'|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_ref / np.linalg.norm(m_ref))**2
    qstr    = "({:.0f},{:.1f},{:.0f})".format(*Q_frac)
    print(f"  {str(hkl_i):12s}  {qstr:18s}  {f2:8.3f}  {s2:6.3f}  {f2*s2:9.3f}")

τ             Q = τ+k              |F_M|²   sin²α    |F_M⊥|²
(0, 0, 0)     (0,0.5,1)             0.844   0.846      0.714
(0, 1, -1)    (0,1.5,0)             0.000   0.800      0.000
(0, 0, 3)     (0,0.5,4)             0.000   0.965      0.000
(0, 0, 6)     (0,0.5,7)             0.147   0.987      0.145

Two independent extinctions reduce the observable peak count: (0, 1.5, 0) cancels because Qz = 0 makes all four orbit phases equal; (0, 0.5, 4) cancels from the +,+,−,− phase combination at Qz = 4. The satellite at (0, ½, 1) from τ = (0, 0, 0) is the primary experimental target.