Correlation tables

Correlation tables

Oh		D2h	C3v
A1g		Ag	A1
T1g		B1g+B2g+B3g	A2+E
T2g		B1g+B2g+B3g	A1+E

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	σv1 σv2 σv3	σv1R σv2R σv3R	Pseudo vector	polar vector
A1	1	1	1	1	1	1		z
A2	1	1	1	1	-1	-1	Sz
E3/2+	1	-1	1	-1	i	-i
E3/2-	1	-1	1	-1	-i	i
E	2	2	-1	-1	0	0	(Sx,Sy)	(x,y)
E1/2	2	-2	-1	1	0	0

IR product table

	A1	A2	E3/2+	E3/2-	E	E1/2
A1
A2		A1	E3/2-	E3/2+	E	E1/2
E3/2+		E3/2-	A2	A1	E1/2	E
E3/2-		E3/2+	A1	A2	E1/2	E
E		E	E1/2	E1/2	A1+A2+E	E3/2++E3/2-+E1/2
E1/2		E1/2	E	E	E3/2++E3/2-+E1/2	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	C2x C2xR	C2y C2yR	C2z C2zR	i	i R	σh σhR	σvx σvxR	σvy σvyR	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	Sz
Γ3+	1	1	1	-1	-1	1	1	-1	1	-1	Sx
Γ4+	1	1	-1	1	-1	1	1	-1	-1	1	Sy
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

Irreducible representation notations

Irreducible representation notations

		Bethe	BSW	EWK
Oh (m3m)
		Γ1		A1g
		Γ1'		A1u
		Γ2		A2g
		Γ2'		A2u
		Γ12		Eg
		Γ12'		Eu
		Γ15'		T1g
		Γ15		T1u
		Γ25'		T2g
		Γ25		T2u
Td (-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
C2v		Γ1	A1
		Γ3	A2
		Γ2	B1
		Γ4	B2

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

Flying frog

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

and

Hence,

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.

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)

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

Brillouin function

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})

where 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

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

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

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

\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

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).

Useful scientific weblinks

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

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

Useful LaTEX Sites

To write formula

http://sixthform.info/steve/wordpress/?p=59

http://latex.liveserver.com/

http://thornahawk.unitedti.org/equationeditor/equationeditor.php

http://www.texify.com/links.php

http://www.codecogs.com/components/equationeditor/equationeditor.php

http://www.bgoncalves.com/online/latex/

http://www.tlhiv.org/ltxpreview/

http://mathtran.open.ac.uk/index.html

http://www.sitmo.com/latex/

http://out.l3s.uni-hannover.de:9080/Equation/

http://code.google.com/p/jstexrender/

<pre lang="eq.latex">
int_{0}^{1}frac{x^{4}left(1-xright)^{4}}{1+x^{2}}dx
=frac{22}{7}-pi
</pre>

http://mathurl.com/

To write document
http://dev.baywifi.com/latex/

http://nirvana.informatik.uni-halle.de/~thuering/php/latex-online/latex.php?sprachauswahl=2&aufruf=22103

http://www.sciencesoft.at/index.jsp?link=latex&lang=en

http://tex.mendelu.cz/

http://www.latexlab.org/

Some understandings of ferroelectricity

Some conceptional paradox with ferroelectricity (FE).

Figure 1

1) As shown in Fig. 1, there is a one-dimensional system that consists equally N anions with charge -q and N cations with charge +q, separated by distance a. This system definitely has spontaneous permanent dipole moment p=Nqa, but why it is not FE?

Note that normally FE is defined for bulk (infinite) system, in which case not so many systems have FE. However, for a finite non-metallic system, spontaneous dipole moments are not rare at all. Figure 1 is actually a good example. Then the paradox is that the real systems are finite anyway, how come they are not normally FE?

The answer has to do with the imperfectness of the real system. One can see in Fig. 1, the direction of the dipole moment depends on the type of the atoms on the surface, which is random for a real system. For example, we don't normally have a NaCl sample that has a surface only with Na atoms and the opposite surface with only Cl atoms. This randomness will eventually cancel out these "spontaneous dipole moments", making the total dipole moment of the sample negligibly zero.

Figure 2

2) The system shown in Fig. 2 is trully FE.

For this system in panel A, the dipole moment is p=Nqa. But if the surface atoms change to panel B, the dipole moment is p=-Nqb. Therefore the randomness of the surface can not cancel the dipole moment, which on average is p=Nq(a-b). This is why the system is FE.

3) To summarize the cases we discussed above, to determine if a system is FE. We have to test if the system (infinite) is central-symmetric. In other words, whether you can find an unit cell that has an inversion symmetry. For the system in Fig. 1, it is central symmetric. But for system in Fig. 2, it is not. That's why it is FE.

Low frequency dielectric constant

Starting from Lorentz model:

md²x/dt²+mΓdx/dt+mω₀²x=qE

which is the equation of motion for a bond electric charge q with mass m in an oscillating field E, where ω₀ is the frequency of the Harmonic potential of the charge and Γ is the dampling parameter.

Take the oscillating form of x and E:

x=x₀e^-iωt and
E=E₀e^-iωt

one get the solution:

x=q/m/(ω₀²-ω²-iΓω).

The the dielectric constant will be (notice that P=xq):

ε=1+P/(Eε₀)=1+q²/mε₀/(ω₀²-ω²-iΓω)

For a low frequency measurement ω₀>>ω,which is valid for normal dielectric measurement because the lowest ω₀ is normally at the 10¹² Hz range.

In this case, we can ignore the high order term ω²,

ε=1+q²/mε₀/(ω₀²-iΓω).

Imaginary part:

The imaginary part is

ε₂=q²Γω/mε₀/(ω₀⁴+Γ²ω²).

Experimentally, a maximum is always observed for the temperature dependence of ε₂, here we can calculate the maximum by solving

dε₂/dT=0.

Note that the only temperature dependent parameter is Γ.

Then one gets:

Γ=ω₀²/ω at dε₂/dT=0.

Now let's look at the parameter Γ. It is a dampling parameter, which should be proportional to relaxation time τ, which satisfies the Arhenious relation:

τ=τ₀e^E_a/(k_BT)

where Ea is activation energy (or energy barrier to overcome the).

We can even define:

τ=Γ/ω₀².

Then

τ₀e^E_a/(k_BT)=1/ω should give us the temprature where the maximum ε₂ is, which is

E_a/(k_BT_max)=-log(τ₀ω).

Experimentally, fitting the the relation between T_max and ω, one gets the activation energy E_a, which is what most paper shows.

Real part:

The real part dielectric constant is

ε₁=q²ω₀²/mε₀/(ω₀⁴+Γ²ω²).

One can see that it increases monotonic with temperature simply because of the temperature dependence of Γ.