Friday, February 27, 2009

Spacegrouptable

Table of space groups for calculating phonon structures

	Symbol	Factor group	Positions	Examples
Triclinic Structures (#1-#2)	aPn
1	P₁	C₁		FeS₂
Monoclinic Structures (#3-#15)	mPn,mCn

Orthorhombic Structures (#16-#74)	oPn,oFn,oIn,oCn
69	Fmmm	D_2h²³	2D_2h(1);3C_2h(2);D₂(2); 3C_2v(2);2C₂(4);3C_s(4); C₁(8)	Graphite

Tetragonal Structures (#75-#142)	tPn,tIn
134	P4₂/nnm	D_4h¹²	2D_2d(2);2D₂(4);2C_2h(4); C_2v(4);5C,(8);C_s(8); C₁(16)	MnF₂
141	I4₁/amd	D_4h¹⁴	2D_2d(2);2C_2h(4);C_2v(4); 2C₂(8);C_s(8);C₁(16)	TiO₂
Trigonal Structures (#143-#167)	hPn,hRn
150	P₃₂₁	D₃²	2D₃(1);2C₃(2);2C₂(3); C₁(6)	Nd₃Ga₅SiO₁₄
160	R_3m	C_3v⁵	C_3v(1); C_s(3);C₁(6)	NiS(B13),Moissanite-9R (CSi)
161	R_3c	C_3v⁶	C₃(2); C₁(6)	BiFeO3
166	R_-3m	D_3d⁵	2D_3d(1);C_3v(2);2C_2h(3); 2C₂(6);C_s(6);C₁(12)	alpha-As,LuFe2O4
167	R_-3c	D_3d⁶	D₃(2);C_3i(2);C₃(4); C_i(6);C₂(6);C₁(12)	Corundum (Al₂O₃,Cr2O3,Fe₂O₃)

Hexagonal Structures (#168-#194)	hPn
186	P6₃mc	C_3v⁴	2C_3v(2);C_s(6);C₁(12)	Wurtzite, SiC(Moissanite)
194	P₆₃/mmc	D_6h⁴	D_3d(2);3D_3h(2);2C_3v(4); C_2h(6);C_2v(6);C₂(12); 2C_s(12);C₁(24)	MoS₂,WS₂,BN(Bk)
Cubic Structures (#195-#230)	cPn,cFn,cIn

Friday, February 20, 2009

Symmetry of crystals

Here we are going to summarize the symmetry of crystals, including the translational symmetry, rotational symmetry, the combination of them, and a lot of terminologies.

Translational symmetry:

Translational symmetry is the most fundamental symmetry of a crystal, or the definition of a crystal. In principle, a crystal may not have rotational symmetry, but will always have translational symmetry.

Translational symmetry is described by the primitive lattice, or the basis a1,a2,a3. However, they are not unique. One has may ways to choose a1,a2,a3, because by definition, as long as all the points in the lattice can be reproduced by n1a1+n2a2+n3a3, you have the basis. However, there are some rules for chosen primitive axis to unify the usage of them. (e.g. a1=i, a2=j, and a3=k).

Note that we want to avoid the term sc, bcc, fcc and etc. here because they contain information more than just translational symmetry. Let us emphasize that translational symmetry means only primitive axis. To exagerate, for every point on the lattice, we can put a potato. It will still be a lattice as long as the potato have the same orientations.

Primitive cell and unit cell

Primitive cells are the cells that has the smallest repeating
volume. The axes are defined by a1,a2 and a3. As mentioned above, the choice of primitive cells is infinite. Most of them do not reflect the symmetry of the lattice. For example, for fcc a1=(i+j)/2, a2=(k+j)/2, a1=(i+k)/2. This will generate a primitive cell that has no obvious symmetry.

It is always nice to choose a primitive cell to reflect the symmetry of the lattice. So called W-S cell is to do that. W-S cell has all the symmetry of the lattice.

Sometimes, for simplicity, people also choose unit cell that's larger than the primitive cell, just to retain the symmetry. For example, choose the cube for bcc.

Rotational symmetry:

If we talk about rotational symmetry alone, there are infinite number of rotational symmetry, for example, Cn, where n is an integer. However, since we are talking about crystal, we should focus on the rotational symmetries that are only allowed in crystals. (Note that, here, introduction of rotational symmetry alreay implies the coupling between translation and rotational symmetry.) It is well know only n=1,2,3,4,6 is valid for Cn in a crystal.

We have to keep in mind that this is only the rotational symmetry of the lattice (not the crystal). Only when the crystal have simple lattice, we have have the same rotational symmetry for the crystal too.

Lattice system and Bravais lattice

Continuing the discussion of rotational symmetry, we said n can only be 1,2,3,4,6. This combined with the translational symmetry, will only result 7 lattice systems and 14 Bravais lattices. Just to list them, they are

Triclinic
Monoclinic (s, bc 2)
Orthorhombic (s,bc,bc1,fc 4)
Tetragonal (s, bc)
Cubic (sc, bcc, fcc 3)
Trigonal
Hexagonal

Symmetry group (macroscopic symmetry)

So far, we just talked about the microscopic view of a crystal. Next, we need to talk about macroscopic symmetry, e.g. the symmetry of the physical properties such as dielectric constant.

This has to do with the complexity of the lattice point.

Let's define:

simple lattice: lattice points are spheres (isotropic)
complex lattice: lattice points have structures (e.g. a potato)

To emphasize again, when we talk about the rotational lattice, we did not care about the structure of every lattice point. Here we have to because we care about the rotational symmetry now.

Obviously, for the same lattice, the macroscopic symmetry is very different, depending on if you put a sphere on lattice points or if you put a potato. For example, for sc lattice, simple lattice gives cubic macroscopic symmetry (O_h), complex lattice (potato) gives C1 macroscopic symmetry.

A more academic example is the comparison between diamond and ZnS. They both have fcc lattice. But diamond has O_h macroscopic symmetry and ZnS has T_d.

----------------------------------------------------

Space group

Another effect of coupling the translational and rotational symmetry is the space group. When the unit cell has several atom of the same element, there can be screw axis, glide plane which are combination of point group operator and translations.

For example, for BiFeO3. If we don not consider the glide plane, the point group symmetry is C₃. With glide plane, the point group will be C_3v. the space group is R_3c ={{C₃|R}{C_3v|τ+R}}.

Reciprocal space and Brillouvin zone.

The reciprocal lattice can be found by the definition:

b_i=2πa_jXa_k/V, where V is the unit cell volume.

The Brillouvin zone is chosen to be the W-S primitive cell of the reciprocal space.

The symmetry of wave functions

Monday, February 16, 2009

Symmetry properties of wave functions in magnetic crystals

	space operator	time reversal	Full space group
Paramagnetic	H	θ	G=H+Hθ
magnetic (below T_N)	ℋ		G'=ℋ+ℋa₀(a₀=v₀θ)

Extra timer-reversal degeneracy

u belongs to ℋ
u={σ|τ}
σk=k+K_q

v0={ρ₀|τ₀}
ρ₀k=-k+K_q'

Relation between representation of ℋ and G'

	case 1	case 2	case 3
ℋ_k	Δ⁽ⁱ⁾(u)	Δ⁽ⁱ⁾(u)	Δ⁽ⁱ⁾(u), Δ⁽ⁱ⁾(v₀^-1uv₀)^*=Δ^(j)(u)
G'_k	D⁽ⁱ⁾(u)	D⁽ⁱ⁾(u)	D⁽ⁱ⁾(u)
Relation	D⁽ⁱ⁾(u) =Δ⁽ⁱ⁾(u)	D⁽ⁱ⁾(u) =2Δ⁽ⁱ⁾(u)	D⁽ⁱ⁾(u) =Δ⁽ⁱ⁾(u)+Δ^(j)(u)
degeneracy	no new degeneracy	degeneracy doubled	Δ⁽ⁱ⁾(u) and Δ^(j)(u) are now degenerate
Criteria: ∑χΔ⁽ⁱ⁾{u'u} where u=u={σ\|τ} u'=v₀^-1uv₀ The sum is over σ, and there are M of them. The n_i is the dimension of Δ⁽ⁱ⁾(u) ω=1 for single group, =-1 for double group	ωχ{Δ⁽ⁱ⁾(v₀²)}M/n_i	-ωχ{Δ⁽ⁱ⁾(v₀²)}M/n_i	0
If v₀²={E\|R_n} or {E'\|R_n} where E=ωE	ωM	-ωM	0

Sunday, February 8, 2009

Correlation tables

O_h	D_2h	C_3v
A_1g	A_g	A₁
T_1g	B_1g+B_2g+B_3g	A₂+E
T_2g	B_1g+B_2g+B_3g	A₁+E

Wednesday, January 28, 2009

Double group

Double groups

When analyzing splitting of energy levels in external potential using group theory, one may have the problem of half integer angular moment, because the wavefunctions change with rotation of 360 degree. Here we should use so called double group.

The double group consists two the original group G and its product with operation R (rotation with 2\pi degree), which is why it is called double group.

Here are my collections

C_3v' (C_3v)

	E	R	C₃² C₃R	C₃ C₃²R	σ_v₁ σ_v₂ σ_v₃	σ_v₁R σ_v₂R σ_v₃R	Pseudo vector	polar vector
A₁	1	1	1	1	1	1		z
A₂	1	1	1	1	-1	-1	S_z
E_3/2⁺	1	-1	1	-1	i	-i
E_3/2^-	1	-1	1	-1	-i	i
E	2	2	-1	-1	0	0	(S_x,S_y)	(x,y)
E_1/2	2	-2	-1	1	0	0

IR product table

	A₁	A₂	E_3/2⁺	E_3/2^-	E	E_1/2
A₁
A₂		A₁	E_3/2^-	E_3/2⁺	E	E_1/2
E_3/2⁺		E_3/2^-	A₂	A₁	E_1/2	E
E_3/2^-		E_3/2⁺	A₁	A₂	E_1/2	E
E		E	E_1/2	E_1/2	A₁+A₂+E	E_3/2⁺+E_3/2^-+E_1/2
E_1/2		E_1/2	E	E	E_3/2⁺+E_3/2^-+E_1/2	A₁+A₂+E

Note that it is the direct product instead of Γ^†Γ, which is why E_3/2⁺E_3/2⁺=A₂ instead of A₁.

D_2h' (D_2h)

	E	R	C₂^xC₂^xR	C₂^yC₂^yR	C₂^zC₂^zR	i	i R	σ_h σ_hR	σ_vx σ_vxR	σ_vy σ_vyR	Pseudo vector	polar vector
Γ₁⁺	1	1	1	1	1	1	1	1	1	1
Γ₂⁺	1	1	-1	-1	1	1	1	1	-1	-1	S_z
Γ₃⁺	1	1	1	-1	-1	1	1	-1	1	-1	S_x
Γ₄⁺	1	1	-1	1	-1	1	1	-1	-1	1	S_y
Γ₁^-	1	1	1	1	1	-1	-1	-1	-1	-1
Γ₂^-	1	1	-1	-1	1	-1	-1	-1	1	1		z
Γ₃^-	1	1	1	-1	-1	-1	-1	1	-1	1		x
Γ₄^-	1	1	-1	1	-1	-1	-1	1	1	-1		y
Γ₅⁺	2	-2	0	0	0	2	-2	0	0	0
Γ₅^-	2	-2	0	0	0	-2	2	0	0	0

Product table

	Γ₁⁺	Γ₂⁺	Γ₃⁺	Γ₄⁺	Γ₁^-	Γ₂^-	Γ₃^-	Γ₄^-	Γ₅⁺	Γ₅^-
Γ₁⁺
Γ₂⁺
Γ₃⁺
Γ₄⁺
Γ₁^-
Γ₂^-
Γ₃^-
Γ₄^-
Γ₅⁺
Γ₅^-									Γ₁⁺+Γ₂⁺ +Γ₃⁺+Γ₄⁺	Γ₁⁺

D₄' (D₄)

D₄' (16)		E	R	C₄ C₄³R	C₄³ C₄R	C₂ C₂R	2C₂' 2C₂'R	2C₂'' 2C₂''R
Γ₁	A₁'	1	1	1	1	1	1	1
Γ₂	A₂'	1	1	1	1	1	-1	-1
Γ₃	B₁'	1	1	-1	-1	1	1	-1
Γ₄	B₂'	1	1	-1	-1	1	-1	1
Γ₅	E₁'	2	2	0	0	-2	0	0
Γ₆	E₂'	2	-2	√2	-√2	0	0	0
Γ₇	E₃'	2	-2	-√2	√2	0	0	0

O' (O_h)

O' (48)		E	R	4C₃ 4C₃²R	4C₃² 4C₃R	3C2 3C₂R	3C₄ 3C₄³R	3C₄³ 3C₄R	6C₂' 6C₂'R
Γ₁	A₁'	1	1	1	1	1	1	1	1
Γ₂	A₂'	1	1	1	1	1	-1	-1	-1
Γ₃	E₁'	2	2	-1	-1	2	0	0	0
Γ₄	T₁'	3	3	0	0	-1	1	1	-1
Γ₅	T₂'	3	3	0	0	-1	-1	-1	1
Γ₆	E₂'	2	-2	1	-1	0	√2	-√2	0
Γ₇	E₃'	2	-2	1	-1	0	-√2	√2	0
Γ₈	G'	4	-4	-1	1	0	0	0	0

Tuesday, January 13, 2009

Irreducible representation notations

	Bethe	BSW	EWK
O_h (m3m)
	Γ₁		A_1g
	Γ_1'		A_1u
	Γ2		A_2g
	Γ2'		A_2u
	Γ12		E_g
	Γ12'		E_u
	Γ15'		T_1g
	Γ15		T_1u
	Γ25'		T_2g
	Γ25		T_2u
T_d (-43m)
	Γ₁	Γ₁	A1
	Γ₂	Γ₂	A2
	Γ₃	Γ₁₂	E
	Γ₄	Γ₂₅	T1
	Γ₄	Γ₁₅	T2

D2d (-42m)
	Γ₁	A₁
	Γ₂	A₂
	Γ₃	B₁
	Γ₄	B₂
	Γ₅	E

C_2v	Γ₁	A₁
	Γ₃	A₂
	Γ₂	B₁
	Γ₄	B₂

Thursday, January 8, 2009

Abbreviate

Note on abbrievation

To me, there are two cases we have to do abbrievation:
1) when taking notes
2) when writing a computer program

In the 1) case, we want o gain time by abbreviating words, while in 2) case, we want to avoid clutter in our codes. In both cases, the caveat is to make sure that you will be able to recognize these abbreviations afterward without confusion. We should also take advantage of the context.

Here are a list of tricks (some may not apply for computer programmings)

	Original	Abbreviation
Symbols	equal	=
	*	important
	w/	with

Standard	for example	e.g.

1st syllable	dem	democracy
	integer	int

1st syllable+first letter of the second	demo	demonstration

or so much you can recongnize	intro	introduction

Omit vowels	level	lvl

Use the first and last syllables	energy	engy

use the first and last letters	feet	ft

9．Us the first and last syllable

engy = energy

Monday, December 29, 2008

Flying frog

It is said that the high magnetic field (and gradient) can make a frog fly. Let's see the possibility:

What exist most in biological tissue is water (human: 78% newborn, 65% one year old, 60% adult).

Here we simplify the problem by assuming we want to float water instead.

To make a drop of water fly, one has to counteract the gravity on the water, using, in this case, magnetic force. Therefore:

F_G=F_M, where

$eq=F_G=\rho Vg$

and

$eq=F_M=\frac{\chi V}{\mu_0}B\frac{dB}{dz}$

Hence,

$eq=B\frac{dB}{dz}=\frac{\rho g\mu_0}{\chi }$

Note that it has nothing to do with volume of the biological tissue.

Using the density of water g= 1.0x10³ kg/m³ and the susceptibility χ=−9.035×10⁻⁶, one finds

BdB/dz = 1.4x10³ T²/m.

Here is an list of the parameters of the magnets that I have worked with.

	Stern-Gerlach I	Stern-Gerlach II	DC magnet
B (T)	1	2	33
dB/dz (T/m)	345	50	300
BdB/dz (T²/m)	345	100	9900

We can see that the DC magnet is able to do this in the center of the magnet. For this magnet, the threshold is about 13 T.

Wednesday, November 26, 2008

excitation-types

Confusing term crystal field exictation

The problem is the confusing term "crystal
field excitation" and "pure electronic crystal field excitation".
Normally people call the two peaks of Fe2O3 "crystal field
excitation", which is not completely wrong, but it does not really mean
literally pure electronic excitation as often found in the text book. In
stead, the absorption peaks in Fe2O3 consist of excitations of various
natures that all have to obey selection rules. We hereby talk about some details as the following:

Type 1: Pure crystal field excitations: This is magnetic dipole
transition that does not have to obey parity selection rule. Spin orbit
coupling takes care of spin selection rule. This is purely on-site electronic
excitations. Sometimes it is also called Frenkel exciton. The intensity
is the lowest because of the magnetic dipole nature. The energy positions are expected to be at low energy end
of the spectra, if one can observe this type of excitations. Let me
emphasize that this type of excitation is does not create electric
polarization. In contrast, the other two are both electric type
excitations.

Type 2: Magnon sidebands, pure crystal field excitation + magnons. These
are many-body excitations involving Fe sites of both spin sublattice.
The total spin is conserved. Therefore the spin selection rule is
obeyed. The initial and final states have different parities, so the
parity selection rule is also obeyed. The intensity of this type is
supposed to be much higher than the type 1 but much lower than the type
3, which is why the we often can not see the peaks in spectrum of absorption
spectra directly. However, this type of excitation is sensitive to the
magnetic order, which is why they stand out when the magnetic field is
applied. Nevertheless, the intensity we see in Δα is much
smaller than α itself.

Type 3: Crystal field + odd parity phonons. This type of excitation is
mostly responsible to the total intensity we see in Fe2O3. Here
spin-orbit coupling relaxes the spin selection rules and the addition of
odd-parity phonons makes the initial and final states different parity.
If we do fitting on the absorption spectra, we are getting the profile
of this type of excitations. This type of excitations can further couple
to even parity phonons and magnons. The detail of this type of
excitation can not resolved because of the O-p Fe-d hybridization and
the complex structure of odd parity (IR) phonons. In most literatures, so-called
crystal-field excitations actually refer to this type. After all, most
intensity of the absorption comes from here.

Some references:
Srivastava V. C. et al., Solid State Comm. v11, p41(1972)
Tsuboi T., et al, PRB v45, p468 (1992)

Wednesday, November 5, 2008

Application of Lorentz model in vibration of solids.

Application of Lorentz model in solids for multiple modes

See pdf version: http://cid-2a71f357d126d517.skydrive.live.com/self.aspx/Download/Blog/Lorentz.pdf

Friday, October 17, 2008

Brillouin function

People often talk about Brillouin function in the literature and use that to describe the magnetic property as a function of temperature with a parameter J, the total atomic angular momentum. I got confused at first when I looked up the definition of the Brillouin function in my text book, which says:

B_J=\frac{2J+1}{2J}coth(\frac{2J+1}{2J}x)-\frac{1}{2J}coth(\frac{x}{2J})

$eq=B_J=\frac{2J+1}{2J}coth(\frac{2J+1}{2J}x)-\frac{1}{2J}coth(\frac{x}{2J})$

where x=\frac{\mu B}{k_BT}

$eq=x=\frac{\mu B}{k_BT}$

Then the question is where is the field B when people show their Brillouin function.

The answer has to do with the context of ferromagnetism. In Weiss mean field theory, the ferromagnetism is explained in terms of the molecular field B _m that is proportional to the magnetization of the material M:

B_m=\gamma M

$eq=B_m=\gamma M$

Then the ferromagnetism can be found from the following self consitent eqations (SCE):

\frac{M}{M_s}=B_r(x)

\frac{M}{M_s}=\frac{k_BT}{\gamma M_s\mu}x

$eq=\frac{M}{M_s}=B_r(x)$

$eq=\frac{M}{M_s}=\frac{k_BT}{\gamma M_s\mu}x$

The first equation is the thermal dynamic average of the moment, second one describes the relation between x and the magetization. Here is the list of the meaning of the symbols:

M: magnetization

μ: magnetic moments of the sites μ=gJμ_B

M_s: staturation magnetization M_s=Nμ where N is the density of the magnetic sites.

The solution of the temperatures gives the magnetization at different temperature. The only parameter here is γ, unfortunately a microscopic interaction parameter we don't know. However, we have the knowledge of another important parameter, which is the phase transition temperature T_C. Next, we will find out the relation between T_C and γ and replace γ using T_C in the SCE.

To find T_C, we just have to remember when T→Tc, the magnetization M→0, therefore x→0. Hence, we can use the limit of Brillouin function at x→0, which is:

B_r(x)=\frac{2J+1}{3J}x

$eq=B_r(x)=\frac{2J+1}{3J}x$

Therefore, the relation between γ and T_C is found as:

\frac{k_BT_C}{\gamma M_s\mu}=\frac{J+1}{3J}

$eq=\frac{k_BT_C}{\gamma M_s\mu}=\frac{J+1}{3J}$

With this relation, one can easily change the SCE.

\frac{M}{M_s}=B_r(x)

\frac{M}{M_s}=\frac{T(J+1)}{2T_CJ}x

$eq=\frac{M}{M_s}=B_r(x)$

$eq=\frac{M}{M_s}=\frac{T(J+1)}{3T_CJ}x$

Now all the discussion can be based on the relation between M/M_s and T/T_C with the only parameter J.

However, the so-called Brillouin function is a transcendental function which can only be found from solving the SCE.

In the end, sometime Brillouin function refers to the solution of the SCE: M/M_s=f(T/T_C), in stead of the function B_J(x).

Friday, October 3, 2008

Useful scientific weblinks

Chemistry

Field	Description	Link
Group theory	Point Group Symmetry, correlation table	http://www.staff.ncl.ac.uk/j.p.goss/symmetry/index.html
	Crystal system	http://en.wikipedia.org/wiki/Crystal_system
	Xystal Space group	http://img.chem.ucl.ac.uk/sgp/MAINMENU.HTM http://cst-www.nrl.navy.mil/lattice/spcgrp/trigonal.html
	character table	http://www.webqc.org/symmetry.php
	All	S.C. Miller and W.H. Love, Tables of Irreducible Representations of Space Groups and Co-Representations of Magnetic Space Groups. (Pruett Press, Denver 1967) Much of this material is also available on the web: http://www.cryst.ehu.es/ (Bilbao Crystallographic Server, University of the Basque Country, Bilbao, Basque Country, Spain)
	cystal structure	http://jas.eng.buffalo.edu/education/solid/genUnitCell/index.html#
Crystal structure	AMCSD	http://rruff.geo.arizona.edu/AMS/amcsd.php
Elements	Periodic table	http://www.fact-index.com/p/pe/periodic_table__standard_.html


Semiconductor	Organic	http://ceot.ualg.pt/optoel/theory/2terminal/

Phase Diagram	Alloy (Tenary)	http://www1.asminternational.org/AsmEnterprise/APD/
Cehmical product search	chembook	http://www.chemicalbook.com/Search_EN.aspx?keyword=3-HEXYLTHIOPHENE

Phyiscs

Field	Description	Link
	Introduction to surface chemistry	http://www.chem.qmul.ac.uk/surfaces/scc/
Surface science
	UK surface analysis forum	http://www.uksaf.org/tutorials.html

Thursday, September 11, 2008

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

$eq=k_BT_c = frac{2ZJS(S+1)}{3}<br />sum_{k}^{}frac{(1-gamma_k)}{N}$

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}

$eq=gamma_k = frac{1}{Z}sum_{n.n}^{}e^{ikdelta}$

where δ is the vector between nearest neighbors.

One can see that the parameter γ_kdepends 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]

$eq=k_BT_N = frac{2ZJS(S+1)}{3}<br />frac{N}{sum_{k}^{}(frac{1}{1-gamma_k^2})}$

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}

$eq=G_{FM}=sum_{k}^{}frac{(1-gamma_k)}{N}$

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})}

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

Lattice	G_FM	G_AFM
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}

$eq=hbaromega_k=2ZJSsqrt{1-gamma_k^2}$

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