Lattice constant

Lattice constant of some triangular lattice materials

Material	Lattice Constant	Crystal Structure
LuFe2O4	3.441 (LuO) / 2.086 (FeO)
SiC	a=3.086; c=15.117	Wurtzite
Si	5.43095	Diamond
C	3.56683	Diamond
GaN	a=3.189; c=5.185	Wurtzite
ZnO	3.249	P 63 m c
SiO2	4.916	Quantz

Effective number of oscillators and B...

Effective number of oscillators and Born effective charge

The operational definition of effective number of oscillators is:

n_{eff}=\frac{2}{\pi\omega_p^2}\int_{\omega_1}^{\omega_2}{\epsilon_2\omega d\omega

where 

\omega_p^2=\frac{e^2}{V_0m\epsilon_0}

here e is electronic charge, V_0 is the unit cell volume, m is the reduced mass of the oscillator and \epsilon_0 is the vacuum dielectric constant.

If we introduce (Born) effective charge q_{eff}, we also get

n_{eff} =\frac{Nq_{eff}^2V_0\mu\epsilon_0}{e^2V_0m\epsilon_0} = N(\frac{q_{eff}}{e})^2 \frac{m}{\mu}

LFO

LuFe2O4

BFO

LFO

Introduction to multiferroics

Introduction to multiferroics

Landau theory of charge order: symmet...

Landau theory of charge order: symmetry breaking

Landau theory is a good way to describe the symmetry breaking of of charge order using group theory.

Suppose the charge pattern can be described as

\rho(\vec{r})=\rho_0(\vec{r})+\delta\rho(\vec{r})

where ρ is the total charge density, ρ₀ is the charge density of the high temperature phase corresponding to symmetry group G₀ and δρ correspond to one or more irreducible representations (except the identity) of G₀.

At high temperature δρ=0, giving the high symmetry phase. While at low temperature, δρ≠0, giving the low symmetry phase.

A good example of cubic to tetragonal structural phase transition can be very revealling.

In this case:

\rho(\vec{r})=\sum_i\rho_0(\vec{r}-\vec{R}_i)+\rho_1(\vec{r}-\vec{R}_i+z_0)

where R_i represents the cubic lattice and z₀ is the distortion along z direction.

One can look at the character table (below) and recognize that the z₀ distortion corresponds T_1u irreducible representation (IR).



A closer look at the character table shows that only C₄, σ_h and σ_d leave z₀ invariant. Therefore, in the low temperature (low symmetry phase), the group has E, C₄, σ_d and σ_h, which gives a new group C_4v, corresponding to the symmetry of the low T phase.

Useful special symbols

Useful special symbols

Greek letters	α	β	γ	δ	ε	ζ	η	θ	ι	κ
	λ	μ	ν	ξ	ο	π	ρ	ς	σ	τ
	υ	φ	χ	ψ	ω
	Α	Β	Γ	Δ	Ε	Ζ	Η	Θ	Ι	Κ
	Λ	Μ	Ν	Ξ	Ο	Π	Ρ		Σ	Τ
	Υ	Φ	Χ	Ψ	Ω
Roman letters	ℂ	℃	ℇ	℉	ℐ	ℑ	ℒ	ℏ	℞	Ω
	ℋ	ℛ	ℱ	Å	ℰ
	𝓖
Additional Greek letters	ħ	ϴ	ϵ
Arrows	↔	↕	￫	￩	￬	￪	→	←	↓	↑
	↙	↘	↗	￨
Operators	±	∓	×	÷	∙	†	≈	≣	≡	≠
	∫	∏	∑	∆	∇	∂	∅
	√	∛	∜
	∝	∞
Geometry	‖	⊥
Fractions	¼	½	¾
Units	Å

Conductivity of conductor

Conductivity of conductor

How to define a conductor? The very natural way to say that a conductor is a material that conducts electric current. But for a physicist, that's not enough, most materials do have a measurable conductivity, just the numbers vary by 20 order of magnitude. It is hard to draw the lines between conductors, semiconductors and insulators. However, some good examples (300 K) should be able to at least give us some idea.

Material	Conductivity Sm-1 or 1/(Ωm)	Ref
Silver	63.0 × 106
Copper	59.6 × 106
Gold	45.2 × 106
Mercury	1.0× 106
Carbon	2.8 × 104
Fe3O4	103
LuFe2O4	~1	[1] After breakdown 60V/cm-1
Germanium	2.2
LuFe2O4	10-2	[1] Before breakdown 10V/cm-1
Silicon	1.5× 10-3
BiFeO3	~1× 10-3	[2] 1kV/cm-1
Deionized water	5.5 × 10-6
Glass	10-10 -10-14
Paraffin	10-17
Teflon	10-22 - 10-24

Reference:
[1] Title: Nonlinear current-voltage behavior and electrically driven phase transition in charge-frustrated LuFe2O4
Author(s): Zeng LJ, Yang HX, Zhang Y, et al.
Source: EPL   Volume: 84   Issue: 5 Article Number: 57011   Published: DEC 2008

[2]Title: Switchable Ferroelectric Diode and Photovoltaic Effect in BiFeO3
Author(s): Choi T, Lee S, Choi YJ, et al.
Source: SCIENCE   Volume: 324   Issue: 5923   Pages: 63-66   Published: APR 3 2009

Frustrated spin triangles

Frustrated Ising spin triangles

When we talk about spin liquid, one often invoke the frustrated spin system, for which the classical example is the Ising spin triangle with antiferromagnetic interaction.

Let's then look at such a model system see how it behaves, which will be very revealing for understanding more complicated system.

1. Hamiltonian and basis

Suppose there is a (localized) spin triangle of sites A,B,C, in the language of second quantization, there are 8 possible states, which we take as the basis:

$\phi_1 =|\uparrow_A\uparrow_B\uparrow_C>$;
$\phi_2 =|\uparrow_A\uparrow_B\downarrow_C>$;
$\phi_3 =|\uparrow_A\downarrow_B\uparrow_C>$;
$\phi_4 =|\uparrow_A\downarrow_B\downarrow_C>$;
$\phi_5 =|\downarrow_A\uparrow_B\uparrow_C>$;
$\phi_6 =|\downarrow_A\uparrow_B\downarrow_C>$;
$\phi_7 =|\downarrow_A\downarrow_B\uparrow_C>$;
$\phi_8 =|\downarrow_A\downarrow_B\downarrow_C>$;

The spin Hamiltonian will be:
$H=\frac{1}{2}[\sum_{i \ne j}^{}J_{ij} S^z_iS^z_j+(S^+_iS^-j+S^-_iS^+_j)/2]$

2. Eigenstates

We can diagonalize the Hamitonian and get the eigenstates and eigenenergies.

$\xi _i=\sum_{j}^{}c_{ij}\phi_{j}$


Table:

	E=-3/4J				E=3/4J
	ξ1	ξ2	ξ3	ξ4	ξ5	ξ6	ξ7	ξ8
ci1	0.000	0.000	0.000	0.000	1.000	0.000	0.000	0.000
ci2	-0.816	0.000	0.000	0.000	0.000	0.577	0.000	0.000
ci3	0.408	0.707	0.000	0.000	0.000	0.577	0.000	0.000
ci4	0.000	0.000	0.000	0.816	0.000	0.000	-0.577	0.000
ci5	0.408	-0.707	0.000	0.000	0.000	0.577	0.000	0.000
ci6	0.000	0.000	0.707	-0.408	0.000	0.000	-0.577	0.000
ci7	0.000	0.000	-0.707	-0.408	0.000	0.000	-0.577	0.000
ci8	0.000	0.000	0.000	0.000	0.000	0.000	0.000	1.000

It turns out that the system become two energy subspaces.

The first subspace corresponds to degenerate ground states with energy -3/4J.
One can see that |S_z|= 0 for those states.

The second subspace corresponds to a spin quartet with S=3/2.

3. Field and temperature dependence:  magnetization plateau

By varying temperature and magnetic field, one can study the magnetization change.
Above is the result we found for different J value. Starting from left are J=0,1,2,3,4,5,6,7,9,10,11

1) J=0 corresponds to isolated spins, which is actually Brolluvin function
2) J>=9: there is a clear plateau at low magnetization which corresponds to the saturation magnetization of the first subspace of the eigenstates.
3) J=1-7: intermediate cases

4. Field and temperature dependence.

In real experiment, one can not vary J unfortunately. Instead, we only have control over B and T. Next, we show the B,T dependence of the magnetization with fixed J.

In this picture, we can see that at very low temperature, the two plateau is obvious.

One can show that a system with 6 Ising spin which form triangular lattice also has similar behavior, in the sense that there is a low magnetization plateau of 1/3 of the magnitude. The 10 Ising spin system is not calculable for me however due to the computational difficulty but one can imaging the similarity. The key is that for the frustrate the spin system, there is a ground state with huge degeneracy which can behave like a paramagnetic system with reduced spin magnitude.

5. specific heat

As shown in the above figure (J=1,2,3 from the left), the specific heat has a peak at a energy scale proportional to the exchange interaction.

This is actually very typical behavior of two-level system.

6. specific field in magnetic fields

From left to right, the curves correspond to different magnetic fields. One can see that at intermediate fields, there are two peaks in the specific heat.

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