Geometric factor for FM and AFM

In the spin wave model of ferromagnetism and antiferromagnetism, important parameters are transition temperature (T_c, T_N), exchange interaction parameter (J) and magnon frequncy (ω). Normally, T_c, T_N are the easiest (and the first) to measure, it is possible to guess the other parameters out of them. From Green's function theory, one can derive the transition temperature.

Relation between T and J

FM

k_BT_c = frac{2ZJS(S+1)}{3}
sum_{k}^{}frac{(1-gamma_k)}{N}


where S is the spin, Z is the number of the nearest neighbors, k is the wave vector and 
gamma_k = frac{1}{Z}sum_{n.n}^{}e^{ikdelta}

where δ is the vector between nearest neighbors.

One can see that the parameter γ_k depends on the geometry of the crystal.

AFM

k_BT_N = frac{2ZJS(S+1)}{3}
frac{N}{sum_{k}^{}(frac{1}{1-gamma_k^2})} [Hewson1964]

Geometric factors

One can see that different lattice differ only in the summation of γ_k. Therefore, we define the geometric factors for FM and AFM
G_{FM}=sum_{k}^{}frac{(1-gamma_k)}{N}

this is actually not lattice dependent because the integrations can be done analytically and result unity.

And,
G_{AFM}=frac{N}{sum_{k}^{}(frac{1}{1-gamma_k^2})}

Lattice	GFM	GAFM
SC	1	0.67
FCC	1	0.74
BCC	1	0.84

Relation between ω and J

FM

hbaromega_k=2ZJS(1-gamma_k)

AFM

hbaromega_k=2ZJSsqrt{1-gamma_k^2}

Example:

RbMnF₃

A prototype of antiferromagnetic material with bcc structure (perovskite), with only nearest neighbor interaction (J₂/J₁~0.01). The parameters are:
T_N=82 K, ω_max=102 K. [Windsor1966]
From T_N one finds J=3.5 K
From ω_max one finds J=3.4 K.
Obviously, they are very consistent. indicating that the Green's function theory (with RPA) and spin wave model works very well.

BiFeO3

Bismuth ferrite has a distorted perovskite structure with complicated spin structure. For example, spin cycloid and weak ferromagnetism. However, those structure have much lower energy scale than TN. Considering that fact that the real structure is not so different from perovskite, with a rough approximation (especially under the circumstance that no body else has done only finer approxmiation), we can estimate J and ω_max from T_N, assuming nearest neighbor and perfect bcc lattice of Fe.

J=27.3 K,
ω_max=818 K (70 meV).

Reference

[Hewson1964] HEWSON AC GREEN FUNCTION METHOD IN THEORY OF ANTIFERROMAGNETISM PHYSICA 30 : 890 1964

[Windsor1966] Windsor CG, Stevenson RW. SPIN WAVES IN RBMNF3
PROCEEDINGS OF THE PHYSICAL SOCIETY OF LONDON   Volume: 87   Issue: 556P   Pages: 501-&   Published: 1966

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