Python real time class methods

Python real time class methods

Python is a great programming language with countless merits. However, one short coming (compared with Matlab e.g.) is that one often has to restart the program in order to refresh the modified the code, which is kind of a hassle. Below, I will show what I did to circumvent this problem. Basically, I define the class method in another file, say methods.py, which is different from the file that contains the definition of the class, say myclass.py. Then I found a way to reload (or redefine) all the instance (or class) methods.

class MyClass():
    def __init__(self):
        import methods as methods;
        self.add_extended_methods(methods);
        pass;

    def add_extended_methods(self, methods):
        import inspect;      
        if not methods.lazy:
            reload(methods);
        self.methods=methods;
        
        modulefile=inspect.getfile(methods);
        modulefile=modulefile.replace('.pyc','.py');
        
        for k in methods.__dict__.keys():
            expr="methods."+k;
            v=eval(expr);
            if inspect.isfunction(v) and inspect.getfile(v)==modulefile:
                cmd="def "+k+"(self,*args, **kwargs):return self.methods."+k+"(self,*args, **kwargs);"
                exec(cmd);
                import new;
                cmd="self."+k+"=new.instancemethod("+k+", self, MyClass)";
                exec(cmd);

One can also call the "add_extended_methods" anytime when he wants to use updated methods.

This saved me a lot of time from restarting the whole Python/Program

Parameters of multiferroics:

Parameters of multiferroics:

Type	Name	TC (K)	TN (K)	Polarization (uC/cm^2)	Critical field	Other critical temperatures
I.1 Lone pair FE	BiFeO3	1100	640	100	20 T quenching spiral, 10 T spiral rotation [Xu2009]	140 K spin reorientation
I.2 Geometric FE	YMnO3	1200	42	5
II.1 Valence order of magnetic ions	LuFe2O4	320	240	25	see phase diagram of [Xu2008]
III.1 Spiral Magnets	TbMnO3	41	28	6e-2
III.2 frustrated collinear M	Ca3CoMnO6	16.5	16.5	9e-2

[Xu2008] Charge Order, Dynamics, and Magnetostructural Transition in Multiferroic LuFe2O4

Author(s): Xu XS, Angst M, Brinzari TV, et al.
Source: PHYSICAL REVIEW LETTERS   Volume: 101   Issue: 22 Article Number: 227602   Published: NOV 28 2008


[Xu2009] Optical properties and magnetochromism in multiferroic BiFeO3

Author(s): Xu XS, Brinzari TV, Lee S, et al.
Source: PHYSICAL REVIEW B   Volume: 79   Issue: 13 Article Number: 134425   Published: APR 2009

classes of multiferroics

category of multiferroics

	Type-I FE, M order from different subsystem Tc>>TN	Type-II FE, M order from same subsystem, but independent origin Tc>TN	Type-III FE caused by M order TC<=TN
SubType:	I.1 Lone pair FE	II.1 Valence order of magnetic ions	III.1 Spiral Magnets
Example:	BiFeO3 TC: 1100 TN: 640 K P: 100uC/cm^2 [Xu2009Ref] Intro: perovskite, A disp along (111)	LuFe2O4 TCO: 320 K TN: 240 K  [Xu2008] P: 25 uC/cm2	TbMnO3 TN=41K TC=28K P: 6e-2 [Kimura2007] Intro: inverse Dzyaloshinskii–Moriya effect. P~Qxe, where three symbols are ploarization, spiral propagation direction and normal vector of the spiral plan
Other Examples:	{BiMnO3, PbVO3}		Ni3V2O6, MnWO4, CuO orthorhombic RMnO3 (R=Tb,Dy)
SubType:	I.2 Geometric FE		III.2 frustrated collinear M
Example	YMnO3 TC: 1200 K TN: 42 K P: 5 μC/cm^2 [VanAken2004] Intro: perovskite, BO6 title, then A goes up and down asymmetrically		Ca3CoMnO6 TN, TC=16.5 K, P: 9e-2 uC/cm2 Intro: magnetostriction
Other Examples:	hexagonal RMnO3 (R=Ho-Lu)		RMn2O5 (R=Pr-Lu, Bi,Y)

[Kimura2007] Spiral magnets as magnetoelectrics
Author(s): Kimura T
Source: ANNUAL REVIEW OF MATERIALS RESEARCH   Volume: 37   Pages: 387-413   Published: 2007


[VanAken2004] The origin of ferroelectricity in magnetoelectric YMnO3

Author(s): Van Aken BB, Palstra TTM, Filippetti A, et al.
Source: NATURE MATERIALS   Volume: 3   Issue: 3   Pages: 164-170   Published: MAR 2004


[Xu2008] Charge Order, Dynamics, and Magnetostructural Transition in Multiferroic LuFe2O4

Author(s): Xu XS, Angst M, Brinzari TV, et al.
Source: PHYSICAL REVIEW LETTERS   Volume: 101   Issue: 22 Article Number: 227602   Published: NOV 28 2008


[Xu2009] Optical properties and magnetochromism in multiferroic BiFeO3

Author(s): Xu XS, Brinzari TV, Lee S, et al.
Source: PHYSICAL REVIEW B   Volume: 79   Issue: 13 Article Number: 134425   Published: APR 2009

Python, global function

Python, global function

Problem

1) One want to organize the code using modules (files)

2) One wan packages to see other packages (at least a global function that is aware of all the definition of classes and can be used by all module)

3) One can not using from xx import * in every package

The solution is dynamic importing (which is one of the beauty of Python)

We can

1) define the global function, say, isa() in __ini__.py in the package's root directory

2) import in the METHOD (not __init__ method or as a module import) where you want to use.

Xpy version 1.47 is released

See http://xpydaov.spaces.live.com/

Electric dipole-dipole interaction

Electric dipole-dipole interaction between f electron atoms

Estimate of electric dipole moment:

According to Yen et al.[1], the electric dipole moment can be estimated as:
 
p~e<r>V_c/D,

where e is the elctronic charge, <r>~10^-10 m is the mean radius of the 4f electronic orbit, V_c~500 cm^-1 is the characteristic crystal field splitting, and D~60000 cm^-1 [2] is the energy separation between (4f)ⁿ and (4f)^n-1(5d)' configuration.

Then p~1.3e-31 C.m

Then the dipole-dipole interaction is

W=\frac{ p_1 p_2}{4\pi \epsilon_0 R^3}

For example, if R=0.42 nm, we get

W=1.2 e-5 eV, or 0.01 meV.

[1]    W. M. Yen et al., Physical Review 140, 1188 (1965).
[2]    G. H. Dieke, and H. M. Crosswhite, Applied Optics 2, 675 (1963).

Transfer helium from (100 Liter) Dewa...

Transfer helium from (100 Liter) Dewar to bolometer

In order to transfer helium smoothly, we need to position the bolometer at a good altitude, which depends on the dimensions of the Dewars and the trasfer line. Here is an illustration of how to determine how high the bottom of the bolometer should be.

From oscillator strength to spontaneo...

From oscillator strength to spontaneous decay rate:

Approach I

Using
quantum mechanics, one can derive the relation between the oscillator
strength calculated from the absorption coefficient and the transition
matrix element |x|². And the spontaneous decay rate is also decided by the matrix element |x|², therefore, one can find the decay rate from the absorption coefficient.

f=\frac{2m\omega}{\hbar}|x|^2

while spontaneous decay rate:
A=\frac{e\omega^3}{3\pi\hbar c^3\epsilon_0 n^2}|x|^2

Therefore,

\frac{A}{f}=\frac{\omega^2e^2}{6m\pi c^3\epsilon_0n^2}

Approach II: Judd-Ofelt theory,

f=\frac{2m\omega}{\hbar}S

A=\frac{\omega^3e^2n^2}{\hbar c^3\pi\epsilon_0}S

Therefore,

\frac{A}{f}=\frac{\omega^2e^2n^2}{2mc^3\pi\epsilon_0}

One can see that, in two approaches, there is an factor of 3n^4.

I am not so sure why two are not the same, but at this moment I tend to believe the second approach because it is specialized just for this case.

Usage

In a real experiment, oscillator strength can be found using partial sum rule:
The dimensionless oscillator strength is:
f=\frac{2c}{N_e\hbar \pi \omega_p^2}\int_{E_2}^{E_1}n\alpha dE

Here
\omega_p^2=\frac{e^2 \rho}{m\epsilon_0}

[Reference:]

Kumar G.A. et al, Journal of Luminescence vol. 99, p. 141-148 (2002)

Conservation of

Conservation of Σλ_i

Description:

For a Helbert space (wave function subspace) {φ_i} defined by the operator O whose eigenvalues are λ_i, corresponding to wavefunction  φ_i.

Simply from linear algebra, Σλ_i is actually the trace of the operator matrix O, which is of course conserved, no matter what unitary transformation of the {φ_i} is taken to form new set of basis, say {ψ_i}.

To generalize this, Σλ_iⁿ, is also conserved, because Oⁿ also defines the space.

Example:

For a 4f electron multiplet, say, Nd³⁺, ⁴I_9/2, ΣJ_z,iⁿ is conserved and of course Σμ_z,iⁿ is conserved too because they differ only by a factor of gμ_B

Just to give a few numbers,

sqrt(3Σμ_z,i²/5)=

3.6 μ_B(theoretical value)
3.4, 3.5 and 3.8 from measurement on crystal field split ⁴I_9/2 multiplet.

Polishing sample

Polishing sample

We have successfully polished a sample down to 45 micron thickness using template method. Following is a summary.

As shown in the figure, the idea is to create a template using much harder materials (such as glass, rasor blade) than the sample. One advantage of this way is that you can measure the sample thickness using a micrometer (basically sample+glass -glass). Everything is glued using paraffin, which is safe and surprisingly strong for polishing and easy for removing.

The steps are:

1) we can start with as thick as 500 micron

2) polish the template down to 75 micron (150 grit, 400 grit)

3) put on sample, using 800 grit to polish easily down to the same thickness of the template

4) continue polishing with 800 grit until thickness is less than 50 micron

5) continue with 1500 grit, at this time one of the templates may be gone, which is ok.

6) polish until the satisfactory thickness, the lower limit may be 30 micron.

LO-TO splitting

effective charge

Born and ionic effective charge:

Compound	q/e	qB/e
NaCl	1.16	0.80
MgO	1.26	0.77
GaAs	1.48	0.34

The data shown in the upper table is calculated from the data of ε(0), ε(∞), ω_TO found in and Ashcroft[1] and Kittel[2].

The formulas used are:
For Born effective charge
q_B^2=\epsilon_0mV\omega_{TO}^2[\varepsilon(0)-\varepsilon(\infty)]

For ionic effective charge
q^2=\epsilon_0mV\frac{\omega_{TO}^2[\varepsilon(0)-\varepsilon(\infty)]}{\eta^2[\varepsilon(\infty)+1/\eta-1]^2}

I only showed a couple of examples for typical materials, basically, as long as you have the data of ε(0), ε(∞), ωLO, you can calculate ionic and Born effective charge.

Here m is the reduced mass, V is the volume of the oscillator, η is the depolarization factor (for cubic system it is 1/3). The formula is in SI units.

[1] N. W. Ashcroft and N. D. Mermin, Solid state physics (Holt, Rinehart and Winston, New York, 1976).
[2] C. Kittel, Introduction to solid state physics (Wiley, New York, 1966).

Energy conversions

Some useful conversion between energy scales

Let's use jouls as our standard

Unit	Jouls
Jouls	1
eV	1.6e-19
μB	9.27e-24
cm-1	1.98e-23
THz	6.63e-22
Kelvin	1.38e-23

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	D2h23	2D2h(1);3C2h(2);D2(2); 3C2v(2);2C2(4);3Cs(4); C1(8)	Graphite
Tetragonal Structures (#75-#142)	tPn,tIn
134	P42/nnm	D4h12	2D2d(2);2D2(4);2C2h(4); C2v(4);5C,(8);Cs(8); C1(16)	MnF2
141	I41/amd	D4h14	2D2d(2);2C2h(4);C2v(4); 2C2(8);Cs(8);C1(16)	TiO2
Trigonal Structures (#143-#167)	hPn,hRn
150	P321	D32	2D3(1);2C3(2);2C2(3); C1(6)	Nd3Ga5SiO14
160	R3m	C3v5	C3v(1); Cs(3);C1(6)	NiS(B13),Moissanite-9R (CSi)
161	R3c	C3v6	C3(2); C1(6)	BiFeO3
166	R-3m	D3d5	2D3d(1);C3v(2);2C2h(3); 2C2(6);Cs(6);C1(12)	alpha-As,LuFe2O4
167	R-3c	D3d6	D3(2);C3i(2);C3(4); Ci(6);C2(6);C1(12)	Corundum (Al2O3,Cr2O3,Fe2O3)
Hexagonal Structures (#168-#194)	hPn
186	P63mc	C3v4	2C3v(2);Cs(6);C1(12)	Wurtzite, SiC(Moissanite)
194	P63/mmc	D6h4	D3d(2);3D3h(2);2C3v(4); C2h(6);C2v(6);C2(12); 2Cs(12);C1(24)	MoS2,WS2,BN(Bk)
Cubic Structures (#195-#230)	cPn,cFn,cIn

Symmetry of crystals

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

Symmetry properties of wave functions in magnetic crystals

Symmetry properties of wave functions in magnetic crystals

	space operator	time reversal	Full space group
Paramagnetic	H	θ	G=H+Hθ
magnetic (below TN)	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
ℋk	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 ni is the dimension of Δ(i)(u) ω=1 for single group,   =-1 for double group	ωχ{Δ(i)(v02)}M/ni	-ωχ{Δ(i)(v02)}M/ni	0
If v02={E|Rn} or {E'|Rn} where E=ωE	ωM	-ωM	0

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