Condensed Matter Theory - Science topic

Respected Prof. Shore,

Thank you for your kind suggestions. I am currently re-evaluating the code-implementation along these lines.

Regards,

Koushik R. Das

dear Dolgopolov,

the presence of "spatial" ground states does not necessarily ensures the pairing of electrons.

for simplicity, consider a system with only two electrons: one on the ground state, and the other one elsewhere. a relaxation of the system by decreasing the temperature for instance does not necessarily allow the second electron to end up in the ground state. that electron must have an opposite spin with respect to the spin of the electron already in the ground state, before the pairing. otherwise, that second electron will end up on the state just above the ground's one. the whole system therefore becomes a triplet with no possibilities of pairing according the "pauli exclusion principle".

but the pairing mechanism in superconductivity is deeper than this simple fact. a coupling with phonons is necessary to keep the singlet state in the structure. this is one of the reasons for which not all materials are superconductors even at low temperature.

Dear Dr Debamalya Dutta

Yes the DoS can be calculated for a discrete number of states. For small number of atoms or molecules, when they are in a single quantum level, for example the ground energy level, or the first excited state.

The number of states can be discrete, a few ones or even only one, if the DoS is zero it menas that the number of states is constant and belong to the same type of degrees of freedom.

Please check:

Kind Regards.

Hi Dear Prof. Stanislav Dolgopolov

Thank you for the answer & well it seems to be logical, but you do specify the mechanism implicitly, when you write the statement "the created pairs, which initially didn't participate in the current", because you are saying that new pairs of supercurrent bosons are created somehow. They are created, and there is a superconducting mechanism for their creation, even we do not know which one is, if BCS or another unknown one.

Kind Regards.

Air above the equator is heated more and areas near the equator receive more heat from the sun than those near the poles due to a phenomenon called "solar angle" and the way the Earth's curvature and atmosphere interact with incoming solar radiation. This is primarily caused by the Earth's axial tilt and its spherical shape.

1. Solar Angle: The angle at which sunlight reaches a particular location on Earth's surface is a crucial factor. Near the equator, sunlight strikes the surface more directly and perpendicularly compared to regions near the poles. When sunlight strikes a surface at a steeper angle, the same amount of energy is concentrated over a smaller area, leading to higher temperatures. In contrast, at higher latitudes (closer to the poles), sunlight is spread over a larger surface area due to the oblique angle of incidence, resulting in less heating.

2. Earth's Curvature and Atmosphere: The curvature of the Earth plays a role in how sunlight is distributed. Near the equator, the curved surface presents a relatively small area for the sun's energy to be distributed, concentrating the heat. Additionally, the atmosphere plays a significant role in moderating the amount of solar radiation that reaches the surface. When sunlight passes through a thicker layer of atmosphere, it can scatter and be absorbed, reducing the amount of energy that reaches the surface. Near the equator, the sunlight has to pass through a smaller portion of the atmosphere, allowing more energy to reach the surface and result in higher temperatures.

3. Day Length: Near the equator, the length of day and night remains relatively consistent throughout the year. This means that the sun is up for a significant portion of the day, allowing more time for the surface to absorb and store heat. In contrast, areas closer to the poles experience more extreme variations in day length, with long days in the summer and long nights in the winter. This variation affects the amount of time available for solar heating.

4. Heat Redistribution: The equatorial region receives more heat than it radiates back into space, creating a surplus of energy. This excess heat is then transported toward the poles through atmospheric and oceanic circulation patterns, which help to distribute heat around the planet and regulate global climate patterns.

The combination of the solar angle, Earth's curvature, atmospheric effects, and heat redistribution mechanisms results in the equatorial region receiving more direct and concentrated solar energy, leading to higher temperatures compared to areas closer to the poles.

A question for which am looking for an answer.

what is the relation between free electron model and magnetic suscetibillity

Interesting query, Prof. Stanislav Dolgopolov

I do believe in both quantities Δ(T) and Δ_0, but what is does not seem to be clear at this very moment in the literature is that the ratio Δ₀/k_B T is a universal number, even for BCS in superconducting elements.

Kind Regards.

Hi Prof. Stanislav Dolgopolov

In addition to all interesting posts, the answer is: Yes, they can be annihilated as a coherent boson matter state of a suppercurrents in several ways:

Experimenters do it also, but in different way:

Kind Regards.

As a naive observation on R. Monnier's argument of available final states:

The Fermi-distribution is a continuous function of energy (even though values below 2 kT become very small), so by the argument that pairs break when states become available under the Fermi level, the superconducting phase transition should be continuous, not jump-like as typically observed.

You are most welcome, Prof. Stanislav Dolgopolov

In our group that works on unconventional superconductors with strontium, we have found the zero energy gap parameter Δ₀ to be between 0.1 and 1 meV to reproduce well-established theories in the triplet compound strontium ruthenate using a Wigner distribution approach.

Check please for the one last publication, link to the DOI for the manuscript:

But in HTSC with doped nonmagnetic strontium, Δ₀ can be between 10 meV and almost 70 meV if the nonmagnetic disorder is high using the same Wigner distribution approach.

Check our last electronic publication, the link we the manuscript in the DOI

Best Regards.

Thank you for the reference. The spin-mediated interaction between electrons takes place. However, for the superconductivity the spin interaction seems to be too weak, because the distance between electrons in a pair may be up to 100 nm, much larger than distances of spin-mediated forces.

I humbly suggest this video.. CBH fitting using Origin 2019 software is included

Yes, a qualitatively correct description is a precursor for an accurate approach. For the above considered problem a brief description is : in superconductors there are two electronic components (SC electrons, normal electrons), distinguishable in the momentum space. That is every electron belongs to its component as long as the SC state persists, any interchange between components is impossible. Mathematically this mean we should introduce two Fock spaces or two sets of quantum states, which don’t overlap (i.e. there are not common states).

One important consequence: all derivations of conventional theories should be revised within the two-space-approach.

Prof. Stanislav Dolgopolov

A simple answer, electrons at the Fermi level are given by the equation p_F = ℏ k_F if they are around the Fermi surface then there is a linear approximation to that equation: δp = ℏ ( k - k_F ), i.e., which is consistent for most normal metals and serves well for the Fermi-Dirac distribution, the Sommerfeld expansion, the Fermi liquid theory and the concept of quasiparticles.

In addition, electrons are fermions which means they can only occupy one state with one value for spin +/- 1/2, therefore a Fermi Dirac distribution in momentum space implicitly shows that electrons are separate in momentum space if they are treated using QM and for 3 approximations, the free, the quasi-free, and the tight-binding ones.

Best Regards.

Please, have a look the following lecture: http://qpt.physics.harvard.edu/talks/upenn.pdf

Dear Shlu,

As shown in 4he attached figure , the charge current flows from left to right through a conductor Hall bar. If the charge current is non-polarized (with equal numbers of spin-up and spin-down electrons), the spin imbalance does not induce a charge imbalance or transverse voltage at the Hall cross. If electrons, which are polarized in the direction of magnetization M, are injected from a ferromagnetic electrode while a circuit drives a charge current (I) to the left, a spin imbalance is created. This produces a spin current (I_S) without a charge current to the right of the electrode. Spin–orbit interactions again separate spin-up and spin-down electrons, but now the excess of one spin type leads to a transverse charge imbalance and creates a spin Hall voltage, VSH. As the distance, L, between the electrode and the Hall cross increases, the voltage signal decreases, allowing the decay length of spin currents (spin diffusion length l_sf) to be measured. More details about SQHE will be presented in Chapter 9 of my Book, about spin transport in nanostructures.

Dear Shalu Pathak, in addition to all the interesting answers,

To understand the role of nonlocality between the current j_a(z) and the electrical field applied E_b(z´) to a normal metal, i.e.,

j_a(z) = (integral from 0 to infinite) K(z,z')_ab E_b(z´)

where the radius of the kernel K(z,z')_ab ~ l (the mean free path) please review section 3 of the classical work:

Best Regards.

You distinguish a normal classical Hall efect from a Quantum Hall effect.

Normal size devices exhibit the first, contain considerable number of electrons.

The magetic field acting on the current pushes electrons to one side of the device

and is counteracted by the Hall voltage set up from charge accumulation. Proportionality between magnetic field and Hall voltage for steady current.

Quantum devices contain fewer electrons in narrow or small devices (Nanostructures) . The magnetic field provokes the equivalent of Landau levels that contain the states for electrons. These pass at regular intervals as the magnetic field increases. Thus there are regular jumps

in the electron conductance as magnetic induction increases.(In single electron conductance, or normal quantum hall effect

The fractional quantum Hall effect is believed to be the consequence of electron interactions and quasi particle formation. This is an extremly complicated phenomena, and not nearly as well understood as many would have you believe.

Quantum materials are I believe are those materials that exhibit wave behavior, or equivalently particle-wave duality.

ear

This paper is related to this thread:

The following paper is worth mentioning in this thread:

Dr. A. A. Bespalov in addition to all the interesting & concerning answers to this thread, you could check the original paper by Prof. S. Edwards on non-magnetic impurity scattering in normal metals.

The physics of the approximation was set up in his classical paper: S. F. Edwards (1961) The electronic structure of disordered systems, Philosophical Magazine, 6:65, 617-638.

It gives insights to understand what means an undressed/dressed Green function

and further applications as those in the AGD book.

Best Regards.

i work on condense mater ,many work in my page

i hope to help you

As a preliminary question:

1. which sort of material is your device made of? (metal, semiconductor, organic, other?),

2. what is the topology of the conducing layer (3D, 2D, 1D, powder-like, etc...)?

3. and what is the size of the active layer (cm or mm or µm or less)?

As a matter of fact, in a material with a low number of electrons, even a moderate number of electronic traps can capture electrons and greatly affect the conductivity. This is however unlikely in a metallic device.

Khadka B. Chhetri

You can try with Swissparam to generate required force-fields parameters.

Please follow the link:

Upload your structure in .mol2 format. You can use Avogadro or Jmol to prepare the structure in .mol2 format. Once prepared, run the .mol2 structure and wait for few minutes to get output file from Swiss param based software and then do the required changes for the force-field parameters.

It is because SdH oscillation can only occur for those carriers with sufficiently high mobility, whereas in a Hall measurement all carriers are taken into account. So, in cases where transport happens through multi carriers with both high and low mobilities, you may notice such a difference in the value of carrier density obtained from these two measurements.

Thank you, Prof. Stanislav Dolgopolov, yes in BCS superconductors, they both are the same, T_c and T_BEC, somehow if I remember my course in superconductivity (we used D Gennes and LifPit IX Vol books) the resistivity is zero, so we do have 2e charged particles (a supercurrent), and somehow this is important. Superconductivity is a different phenomenon from BEC. Their ground states are totally different.

You see, these classical books did separate well the two phenomena (also Abrikosov book on metals). Something that I guess, became forgotten with High-Tc compounds. And I guess it has been due to the persistence to fit with one theory, the phase diagram of High Tc.

Thank you for the nice discussion.

The question is right, although is not usual, Prof H.G. Callaway.

Lest consider BCS superconductors:

Why? because they are bound by commutation relations, the same way that r and p operators are bound in quantum mechanics.

Please see:

In addition to the very interesting answers provided before, Prof. Abhinav Nag, I would like to point out that in the optical phonon brach occur an interesting macroscopic relationship for some insulators (ionic crystals), the Lyddane–Sachs–Teller relation (LST) which determines the ratio of the natural frequency of longitudinal optic lattice phonons of an ionic crystal to the natural frequency of the transverse optical lattice vibrations for long wavelengths (or zero) wavevectors.

The ratio is that of the static permittivity to the permittivity for frequencies in the visible range

Wikipedia commons source:

You are right Dear Prof. Dr Aga Shahee. Anyway, thank you so much.

In addition to all interesting answers in this thread Prof. Magnus Schlösser .

I will add 2 classical references for triplet superconductivity, also called unconventional superconductivity:

Dear Prof. Qingyong Ren

In addition to all interesting answers in this thread & for a deeply understanding of the phenomenology & the theory of spin waves & magnons---using the equation of motion of the magnetic moment & from where the concepts you mentioned (spin wave, spin density wave, spin excitation, spin fluctuation & spin gap ) were borrowed, you can studied from these books:

[1] The Nature of Magnetism by M.I. Kaganov & V. M. Tsukernik, Science for everyone, Mir-Moscow, 1995.

[2] Eletrodynamics of continuous media, L. Landau & E. Lifshitz, ch V-#48 pp 167, eq 48.2, Pergamon 1984. They use the phi thermodynamic potential free energy.

[3] Statistical Physics Vol 2 by E. Lifshitz & E. Pitaevskii ch VII Magnetism, Pergamon 1980.

[4] Spin Waves by A. I. Akhiezer, V. G. Bar'yakhtar, and S. V. Peletminskii. North-Holland & Interscience (Wiley) 1968.

Dear Prof. Dr Aga Shahee

Your question is interesting & quite relevant, henceforth I will try to answer partially your question in a couple of posts in your thread:

A sound velocity anomaly can be directly calculated by means of Ehrenfest relations for 2^nd order phase transitions-PT. For instance a jump in sound velocity can be calculated at the superconducting transition temperature T_c. This is an anomaly that have been observed in some materials. See [1] for example to have a better idea, but I can refer to some specific literature if you wish.

[1] https://www.worldscientific.com/doi/pdf/10.1142/9789813274181_0001

This is part of the question, but what I cannot do now is to relate the jump to an structural phase transition which I guess is a 1^st order PT [2] quoting & is associated with a dynamical instability by the soft mode. According to some authors [3] a physical explanation will be "...the soft mode happens when cooling a material from a temperature above T_c, a normal mode of vibration of the crystal decreases to 0 freq when the crystal becomes unstable & distorts to a new structure..."

[2] https://arxiv.org/ftp/arxiv/papers/1102/1102.1085.pdf

[3] https://arxiv.org/ftp/cond-mat/papers/0605/0605489.pdf

Finally, respect to lattice degrees of freedom showing divergence, I guess this is what happens to 1D systems (according to Peierls & Landau [4]), isn´t it?

[4] https://en.wikipedia.org/wiki/Peierls_transition

Dear Prof. Sandipan Dutta , in adittion to all interesting answers of this compelling thread, I will add a link to a book, where the concept of quasiparticles is masterfully explained by to of the creators of the quasiparticle approach:

Dear Hitanshu Sachania and Franklin Uriel Parás Hernández You might take a look at the following preprint. It adresses the question of this thread. Regards.

Hello,

Anderson transition is a one body problem in which localization results from quantum interferences effects. The metal to insulator transition can occur at any fiilling (carrier density) , only the strength of the disorder controls the position of the mobility edge that separates extended from localized states.

The Anderson transition does not open a gap in the one particle spectrum.

However, the Mott transition is a many body feature, that occurs at specific band filling and controlled by the strength of the coulomb electron-electron interaction. At the metal to insulator transition, in contrast to Anderson transition a finite gap opens in the charge spectrum.

Vk Singh

this is due to the erroneous proof of fermi energy level by Dirac, with correct mathematics on the chemical potential fermi level is no more a straight line instead it is a oblique line connecting VB to CB. and this line is the line of transition, so any point chosen on this have meaning...later more on this.

You can follow

https://mathinsight.org/linear_transformations_map_parallelograms_parallelepipeds

The concept of wave packet has undergone many, more or less subtle, variations during the last 100 years. Sure, interference is important, but it is not all. The common motive is that when a system admits a continuous band of wave solutions (classical or quantum, it does not really matter), its dynamics can exhibit localized-interference features that behave differently from the individual waves. They appear as persistent "objects" with strikingly different properties than those of the waves that compose them (for example, their apparent collective - or "group" - velocities can be quite different). Examples are vortices, maelstroems and tsunamis in hydrodynamics and, why not, particles and "resonances" in subatomic physics. The concept is nice, interesting, and probably contains a lot of truth, though it need not be strictly applicable to all the situations where it was tried. In any case, any kind of wave equation (be it Schroedinger or other) leads to this kind of discussions.

Usually, k = -pi/a -> pi/a refers to a one dimensional system and the parabolic dispersion is characteristic of free fermions E = p^2/2m , that is, no interaction is considered.

When you go to higher dimensions, say 2d, you should describe your energy as a surface instead of a curve as it will depend on kx and ky , E(kx,ky). To avoid ploting a complex surface in the case when you have interaction and the surface is not parabolic, you just calculate a path of some values in your domain passing by some high symmetry points, in which we can by this especial names, Gamma, X, M , etc.

Usually Gamma represents the center of your Brillouin Zone. For exemple, Gamma = (kx=0, ky=0) in 2d, and Gamma = (kx= 0, ky=0, kz=0) in 3D.

No.

Dear Bahar Jafari Zadeh ,

I must say, it is a good question. In general Graphite consists of graphene layers with ABAB..... stacking or Bernal stacking. If you go from top to bottom approach for example mechanical exfoliation of graphite, bernal stacked FLG (few layer Graphene) can be obtained. On the other hand, if you want to deposite a single layer on top of the other in order to get FLG again, then it is hard to get ABAB.. stacking rather it may yield twisted graphene layers with some angle. This process can lead to generate van hove singularities in the crystal lattice which then tend to show new and exciting phases of matter under consideration.

Just use tables by Carnall:

Debmalya Roy Sir,

I know this fact actually, since transferring monolayer graphene on to a substrate is prone to degarde the quality of graphene imposing contamination or defects on it. So it was seen that when graphite powder is coated on Ni substate (has high carbon solubility) deposited on glass, graphene layer forms by interfacial segregation. Afterwards Ni is etched away, thus one gets graphene without any transferring. Hence this method is better than the conventioned CVD growth of Graphene on Ni/Cu.

The valence band corresponds to more tightly bound 'orbitals' (or atomic bonds) compared to the conduction band, just because it is located deeper in the 'hierarchy' of all the crystal energy bands - both occupied and unoccupied. This means that energy span (=width in eV) of the valence band is lower. The electron momentum span inside the first Brillouin zone is the same for all bands. Qualitatively, a decreased energy span requres an increased mass of the particle - and this can be most easily understood by comparing the dispersion laws for two free particles with different masses. While the dispersion law for electron inside the crystal has little to do with the free electron case, it's all about the width of the energy band.

Also, typically the mobility value is higher for electrons than for holes.

A short summary: there is no derivation of spin-orbit coupling (or even the spin) in "classical" quantum mechanics by the principle of correspondence. You can of course conclude the phenomenon from experiments.

However, an "ab initio" derivation of the phenomenon is possible if you reduce the relativistic Dirac equation to the matter terms (neglecting anti-matter components). The resulting equation is often referred to as the "Pauli equation" (https://en.wikipedia.org/wiki/Pauli_equation#Relationship_with_the_Schr%C3%B6dinger_equation_and_the_Dirac_equation)and contains spin-orbit coupling (and other corrections like the "Darwin term").

So, since the origin of that term is the relativistic equation, it's called a rekativistic term.

In two dimensions, a saddle point in the electronic band structure leads to divergence in the density of states, also known as a Van Hove-singularity (VHS). When the Fermi energy is close to the VHS, interactions, however weak, are magnified by the enhanced density of states (DOS), resulting in instabilities. In twisted graphene layers, both the position of fermi energy and that of the VHSs can be controlled by gating and rotation respectively, providing a powerful toolkit for manipulating electronic phases.

However, although the band structure of graphene contains a VHS, its large distance from the Dirac point makes it prohibitively difficult to reach by either gating or chemical doping. By introducing a rotation between stacked graphene layers, it is possible to induce VHSs that are within the range of fermi energy EF achievable by gate tuning.

Dear Mukesh,

It is true that you are right that you have four degeneracies of the graphene system assuming the four symmetries of the lattice (not of the electrons)

- Translations in each Bravais sublattice A to A and B to B.

- Mirror reflection between A and B

- A rotation symmetry of sixty degrees ( C pi/3)

But that is independent of what I have been speaking about the isospin and spin or the electronic degeneration. This is only due to the lattice symmetries. Is this what you were thinking about?

Thank you for your nice explanation and Bahadır Ozan Aktaş for the suggestions!

Dear Ekaterina Pärschke,

First about your last question. In the linear spin-wave approximation you are supposed to know the magnetic order (the ground state or a meta-stable state). That is because this approach consider small precession of the spins around their equilibrium orientation. If you have a set of interactions, only nearest neighbors (n.n.), or second nearest neighbors, or even other complex interactions such as the Dzyaloshinskii-Moriya, and you don't know the magnetic order, it needs to be determined first, normally with a spin dynamics algorithm (I use the https://github.com/spirit-code/spirit developed in my institute in Forschungszentrum Jülich, Germany).

Now that you have your magnetic order, you can calculate the spin-wave spectrum with as many neighbors as you want.

On calculating the spin-wave spectrum, we normally perform a Fourier transformation of the hamiltonian (more precisely of the dynamical matrix). This Fourier transformation can be done exactly and efficiently if the interactions are of short range, such as in a nearest-neighbors-only model, where we only have to sum the contribution from the site on the left, the one of the right, and a local contribution if any (this in one dimension), it could look like: H(k) = sum_j J_0j exp (i R_j k) = J exp( -a k ) + J_0 + J exp( a k ). Considering the second nearest neighbors is not much of a problem, we would have: H(k) = J_2 exp( -2 a k ) + J_1 exp( - a k ) + J_0 + J_1 exp( a k ) + J_2 exp( 2 a k ). (sorry my lack of rigor here, I just want to give you a feeling.)

So, if you have a single atom in the unit cell, you are going to get a single spin-wave mode. For two atoms, you have the right to two modes, and so on. Please notice that this has nothing to do with the number of neighbors you considering for your interactions.

I have faced these problems you are working on myself. This paper of mine might help for I give a lot of detail on the appendices:

Let me know if I can further help.

Best regards,

Flaviano dos Santos

here

Citation: J. Chem. Phys. 80, 4403 (1984); doi: 10.1063/1.447218

View online: https://doi.org/10.1063/1.447218

View Table of Contents: http://aip.scitation.org/toc/jcp/80/9

Dear Bahadir,

The Kondo effect describes one state of a magnetic impurity and the electric conductivity. The physical model which explains this effect is pseudogap Anderson's model employed for heavy fermions too and quantum critical transition phases. Broadly speaking, the main feature is that below 10K the resistivity increase instead to decrease as in usual metals happens.

Curiously, the dynamics of the magnetic moments induced in d-wave superconductor ,as the cuprates are doped with non-magnetic impurities, can be described as a psedogap Kondo model.

A very good review where Kondo effect and d-superconductivity appears well related is

Rev. Mod. Phys. 79, 1015 (2007)

arXiv:cond-mat/0606317v3

Since the rotation speed of the electron v around the nucleus is relativistic, the nucleus creates a magnetic field of the form B = vXE / c² where E is the electric field viewed by the electron, which is central E = | E / r | r and v = p / m. In addition, E derives from the potential V : E = (1 / e) dV / dr. Thus, by subtitution, we obtain the following expression for B :B = (1 /me r c²)(dV/dr) L, where L = rXp is the oribtal momentum. Finally, the electron having a magnetic moment due to its spin m=-gu_BS, it interacts with the magnetic field B,(Zeeman like-effect called spin-orbite interaction) as E_SO=-m.B = -a L.S, where a=u_B (1 /me r c²)(dV/dr) is the spin-orbite parameter which can be calculated in the quantum approach :

a~ (gu^_B2Z⁴)/(n³ (L(L+1/2)(L+1)) , n=principal quantum number, L:magnetic quantum number, Z: atomic number)

Dear Behnam Farid, Thank you very much for your detailed explanations. Best regards.

Dear Behnam Farid, Thank you very much for your recommendations. Bests.

Phase transitions often break symmetries (this is the case of the Ising model). In these cases, the high temperature (low coupling) phase has typically the symmetry of the Hamiltonian H, whereas the low temperature (high coupling) phase shows a lower symmetry (and an enhanced order). Physically, this is because not all the states of H are “accessible” anymore by fluctuations energy wins over entropy in other words. The phenomenon that the system chooses one state during the phase transition is called spontaneous symmetry breaking. For example the Hamiltonian of the Ising model has a spin-flip symmetry, while the ferromagnetic state is either spin-up or spin-down. This symmetry is spontaneously broken during the phase transition as the system selects one of the two possible states as the ground state. The thermodynamic behaviour in each of the phases can be characterized by an order parameter, let us call it m (for magnetisation in the Ising model), and restrict to a scalar order parameter for simplicity (this is fine when discussing most order-disorder phase transitions). The order parameter m changes in a non-analytical way at the phase transition point. The order parameter is usually constructed in that way that m = 0 in the disordered state and m = 1 in a perfectly ordered state.

At first order phase transitions, the order parameter jumps discontinously at the transition temperature, typically Tc, from 0 to a finite value. This involves a latent heat ∆Q = Tc∆S: while T remains constant at the critical value Tc during the transition, the entropy S changes. As the energy in terms of the latent heat cannot be instantaneously redistributed, a mix of phases occurs during the transition (e.g. liquid water and ice at the melting point).

At second order phase transitions, instead, the order parameter increases slowly and continuously from 0 to a finite value at Tc. There is a discontinuity in the specific heat at Tc, while the correlation length and the susceptibility diverge at Tc.

Dear all: Many thanks for your interest in this question. Recently, (9 days ago) your comments lead me to study the limits of the quantum number m. I underline that the half quantum contribution, ½h to the angular momentum, has always the sign. It seems to me that this strongly indicate that the notion of rotation does explains the two subshells. Of course not all of you probably share my approach, it always like this in research and it is natural. The discussion allows to clarify our point of view.

I am now in my 82 year and it becomes often difficult for me to follow you. This to say that I am progressively becoming really retired.

Tanks again Yours Xavier

Thanks for your reply Mr.Farmani

Dear Sofia: Thanks for your interesting comment. You say “NOT TRUE! The spin of the electron doesn't indicate a rotation around itself. Trials to explain the magnetic moment as coming from a charged sphere rotation around itself, failed.” I agree but for many people the spin is still an own rotation. So when I say "The notion of spin is closely connected with that of space." I mean the rotation it was first supposed to be a rotation.

So let me say like this “the notion of rotation is closely related to the notion of space!” Do you agree?

Now, how can be two objects without possible interactions than that of their existence? Do we must consider them without relative motion or not? If we consider a motion does this motion is only in a plan of rotation or do we have to consider a motion along the perpendicular direction of the plan of rotation? Kind regards Xavier

Ok thanks Helman, i will try this.

How to use DFT, first of all you should know basics of DFT, beside that ,you should have a software specially used for this pupose. for example VASP, Guassian, etc I have uploaded some lectures about the basics of the DFT, and there is also a lecture,  how to use VASP. There is a paper by E Chagaro etal , they have done DFT simulation for CZTS, CZTSe, CZTSSE etc , good luck   http://kummelgroup.ucsd.edu/pubs/papers_2016/CHagarov%20CZTS-Se%20DFT%20Phase%20stability%20JCP%202016.pdf

Hi Senthil,

It’s been a while since I’ve worked with these theories, hope this helps a little.

My work was about conductive (i.e. dielectric constant = infinity) particles inside an insulating polymer matrix. It might be a little different if the filler particles are not metallic.

There are 2 major shortcomings of Maxwell-Garnett and Asymmetric-Bruggeman. Both assume spherical filler particles. And they do not represent any frequency dependence.

Maxwell-Garnett Formula is based upon the assumption of spatially separated conductive particles. Therefore, it is only valid for the case of small volume fractions of inclusions. It does not have a particular percolation threshold. The material is assumed to be an insulator until all the host material is replaced by the conductive filler.

Asymmetric Bruggeman shows an extremum of the overall permittivity for a filler fraction of 0.33, no matter what the permittivity of the insulating host or the shape of the inclusions may be (it assumes spherical particles).

Depending on your system, you might want to look into percolation theory. I can recommend this literature:

Gantmakher, V.F., Electrons and disorder in solids. 2005, Oxford; New York: Clarendon Press ; Oxford University Press. x, 225 p.

Zallen, R., The physics of amorphous solids. 1983, New York: Wiley. xi, 304 p.

Hi, zhenhua. On the basis of what I have understood from your above text, you want to look for a k \cdot p Hamiltonian describing weyl semimetal in  HgTe. In Nat. Comm. 2016, 7,11136 you mentioned, the bulk Hamiltonian without strain is H=H_{luttinger}+H_{bia}. Note that H_{bia} breaks the bulk inversion symmetry. Since generally we can get weyl semimetal from TI by breaking time-reversal symmetry or inversion symmmetry, thus I think by only tuning the parameter in H_{bia}, the weyl semimetal is available. 

We have, O. Cépas and myself (Phys. Rev. B 76, 020401(R) have studied the effects of disorder in manganites (microscopic approach) , more precisely  we have focused on correlated disorder... and have compared our calculations to Monte Carlo calculations in the case of uncorrelated disorder,... the spirit of our approach is somehow different from  what was done before, we proceed within a two step approach : (1) we calculate the effect of the disorder on the Mn-Mn couplings (disorder effects fully included) and then we treat the disordered Heisenberg model beyond mean field to calculate the magnetic properties TC, spin stiffness,...

Agree with previous researchers but want to add that energy of incident x-ray photon and which electron likely to participate in scattering is decided by the Ease with which momentum is conserved. Depending upon Z of atom and incident photons energy scattering can be from K L M ... shells. Assume visible light rather than x-ray we get photo electric effect in Cs Na etc and at 80 keV interaction with k shell and so. So many factors play role in scattering Z electronic position and energy or frequency of incident photon. X-ray generally exhibit photo or Compton effects depending upon interaction with inner or outer electrons.

Thanks Dr. Murat for your suggestion.

I don't understand what you mean by small one. But in semiconductor physics, the effective mass, is the mass of the pseudo-particle called Bloch electron, and it does not have the same mass as the electron. One extreme example of this is graphene, where the electron have zero mass. In this answer in Quora I exaplin how it work  https://www.quora.com/How-is-it-that-electrons-act-like-massless-particles-in-graphene

FDTD, DDA, BEM, FEM/FEA, or Green Dyadic method. The candidate should also be familiar with Linux environment.

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Dear Dr. Muhammed Shareef,

Dissipation coefficient should be a function of the relative energy since the kinetic energy of relative motion of nuclei is transformed into intrinsic energy of nucleons and excitation of collective motion.  The number of excited degrees of freedom and strength of their amplitudes determine dissipation coefficient for the relative motion. Therefore, theoretical models devoted to study the dissipation of the initial kinetic energy should take into account the possible mechanisms of dissipation as coupling between degrees of freedom describing physical process of heavy ion collisions. For example, in our first work in this field we had found that nucleon exchange  between interacting nuclei is one of main mechanisms [see attached papers].  We found a reason of non-equilibrium sharing excitation energy between interacting nuclei. Conclusion is that dissipation coefficient should be sensitive to the collision energy.