Kronig-Penney Model 11 Source: Dr. M. Fedawy's Lecture notes . Lecture Notes Prepared by: Lecture Dr. Amr Bayoumi, Dr. Nadia Rafat Solid State Electronics EC210, Fall 2014 Arab Academy for Science and Technology AAST - Cairo, 8: Band Theory: Kronig-Penny Model and Effective Mass 12 Fig 4.54 S a E k [ 11] 3 E k 1 [ 10] B andA way out of this problem was suggested by Kronig and Penney. They proposed a simplified version of the actual potential experienced by an electron and solved the Schrijdinger equation. You will learn the details of Kronig-Penney model now. 10.2.1 Kronig-Penney Model Kronig and Penney proposed that the potential experienced by an electron in a 1-DExplain a problem in solid state physics that is resolved by the methodology of the Kronig-Penney model (band model) (put references if used) Expert Answer. Who are the experts? Experts are tested by Chegg as specialists in their subject area. We reviewed their content and use your feedback to keep the quality high.Considering the Kronig-Penney (KP) model we propose a new type of δ-function transformation which replaces rectangular potentials by δ-walls (or δ-wells). This approach leads to an energy-dependent δ-potential strength in contrast with the original KP calculations, where this quantity was assumed to be independent of the electron energy. ...A one dimensional Kronig-Penney model for a salt like Na DNA is given. The helical periodicity is treated in a manner suggested by Tinoco and Woody. Using data on the semiconductor band gap, we estimate the strength of the potential barrier. The energy limits of the ten bands filled by 20π electrons per unit cell are calculated and exhibited ...This exercise leads students through how to obtain the allowed momentum / energy states (band structure) for the Kronig-Penney model, a simple model for materials with periodic structure. The main computational challenge is to use numerical root finding to find solutions to the transcendental equation arising from the boundary conditions and ...A WIP library of algorithms and methods developed for the parallel solution of various Kronig-Penney models. rust newton-method kronig-penney-model. Updated on Jun 11. Rust. GitHub is where people build software. More than 100 million people use GitHub to discover, fork, and contribute to over 330 million projects.In the Kronig Penney model, the discret energy level scheme of isolated atoms, the energy bands would be infinitely continuous. In generally is observed a region of forbidden energies between the two successive bands. An energy bands is almost cantered around its parent level. Kronig and Penney2,3(1930) demonstrated after the Bloch theorem1to7 ...We introduce a two-dimensional (2D) model which combines a checkerboard potential, alias the Kronig-Penney (KP) lattice, with the self-focusing cubic and self-defocusing quintic nonlinear terms. The beam-splitting mechanism and soliton multistability are explored in this setting, following the recently considered 1D version of the model.Several models such as the Kronig-Penney Model 2 and Density Functional Theory 2 are used to model band structures in solid state physics. In the strong confinement scenario, the band gap can increase when the quantum dot is smaller than the exciton bohr radius as energy levels become split 4. Total emission energy is increased, as well as emission at …Long Answer Questions (8-marks) Derive an expression for the fermi energy of a free electron gas (Sommerfield theory). Obtain expression for density of states for a free electron gas in 3 dimension. show the variation of density of state with energy. Discuss the Kronig-Penney model and show that how it explain the forbidden gap.Introduction KRONIG-PENNEY MODEL • An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms. • An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms. • Kronig and …Using the transfer matrix method the Kronig‐Penney model is generalized to superlattices formed by whatever successions of layers. Imposing the boundary conditions, the miniband structure as well as the envelope wave functions are obtained. As examples, the "enlarged well in a superlattice" problem and the Fibonacci superlattices are ...The Kronig-Penney model describes more rigorously about formation of a... The concept of allowed and forbidden energy bands are clearly described in this video.The Kronig-Penney model It is worthwhile to reproduce the most essential fragment of the calculations of the original paper of Kronig and Penney [1]. They consider a one-dimensional lattice, described by a periodic potential of rectangular barrier (see fig. 1). The 180 H. Puszkarski / Kronig-Penney model with energy-dependent potential ,v(x ...Kronig-Penney Model 5-5. Effect of Crystal Size & Fine Structure on Energy Bands 5-6. Analogy with Circuit Theory 5-7. Energy Band Structure Calculation Methods 5-7.1. Energy Band Structure ...Considering the Kronig-Penney (KP) model we propose a new type of δ-function transformation which replaces rectangular potentials by δ-walls (or δ-wells). This approach leads to an energy-dependent δ-potential strength in contrast with the original KP calculations, where this quantity was assumed to be independent of the electron energy. ...In the Kronig Penney model, the discret energy level scheme of isolated atoms, the energy bands would be infinitely continuous. In generally is observed a region of forbidden energies between the two successive bands. An energy bands is almost cantered around its parent level. Kronig and Penney2,3(1930) demonstrated after the Bloch theorem1to7 ...Time independent Schrödinger equation (SE) within the envelope function approximation was applied in connection with Kronig-Penney model to calculate the electronic band structure of the InAs QD ...Kronig-Penney Model is a unique model which exhibits many of basic characteristics of the electronic structure of real crystals, generally regarded as an idealized and a one-dimensional model of a crystal. It is among the simplest possible models to describe electrons in a periodic lattice. It has been proved that a band gaps and hence energyKronig- Penney model potential. The origin of peak in photofield emission current in the valence band is explained with the help of result of density of state calculated. 1. IntroductionAnother simple periodic profile is the Kronig-Penney potential, shown in Fig. 31b, which gives relatively simple analytical expressions for the characteristic curves. Its advantage is a more realistic law of the decrease of the Fourier harmonics \(U_{l}\) at \(l \gg>1\) , and hence of the energy gaps in the weak-potential limit:Kronig-Penney model . The essential features of the behavior of electrons in a periodic potential may be explained by a relatively simple 1D model which was first …The separation between the conduction band and valence band on the energy band diagram is known as the forbidden energy gap (band-gap, E g ). The width of the energy gap is a measure of the bondage of valence electrons to the atom. The greater the energy gap, the more tightly the valence electrons are bound to the nucleus.Condensed Matter Physics: Lecture 7 (Normal) In the last lecture we encountered the Kronig-Penney model, a simple onedimensional system in which the Schrdinger equation contains a periodic square-well potential. While you are familiar with the solutions of the individual square-well potential, the periodic case is significantly different. Todays …Kronig-Penney Model. Learn more about kronig_penney_model Can anyone provide me the MATLAB code for Kronig-Penney model to draw band structure for 1-D periodic potential well structure?Find step-by-step Physics solutions and your answer to the following textbook question: State two results of using the Kronig-Penney model with Schrodinger's wave equation.. ... Also, the Kroneg-Penney model gives us a relation between the parameter k k k, the energy of the electron, and the potential.The Kronig-Penney model demonstrates that a simple one-dimensional periodic potential yields energy bands as well as energy band gaps. While it is an oversimplification of the three-dimensional potential and band structure in an actual crystal, it is an instructive tool to demonstrate how the band structure can beSection 4.4 depicts the Kronig–Penney model for the one-dimensional (1-D) periodic crystal lattice. Section 4.5 describes the nearly free electron (NFE) ap-proximation for a three-dimensional (3-D) crystal lattice. The NFE method can be used to find the electronic energy states for the outer-shell valence electrons in which the periodic potential of the …This model requires a given material’s band gap between its valence and conduction bands as well as dipole matrix elements between the bands. In this thesis we follow the Kronig-Penney model to develop a 1D -function potential model of solids to obtain these properties required of the two-band model.Bootstrapping the Kronig-Penney Model Matthew J. Blacker(a;c), Arpan Bhattacharyyab and Aritra Banerjeec a Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom b Indian Institute of Technology, Gandhinagar, Gujarat-382355, India c Okinawa Institute of Science and …Kronig - Penny Model The Kronig -Penney model is a solvable problem in quantum mechanics that can either be viewed as an approximation of an electron in a 1D crystal potential or a generalization of a potential-barrier problem to a 1D chain of potential barriers. The problem consists of aA WIP library of algorithms and methods developed for the parallel solution of various Kronig-Penney models. rust newton-method kronig-penney-model. Updated on Jun 11. Rust. GitHub is where people build software. More than 100 million people use GitHub to discover, fork, and contribute to over 330 million projects.https://www.patreon.com/edmundsjIf you want to see more of these videos, or would like to say thanks for this one, the best way you can do that is by becomin...The superspace is realized by introducing a lattice shift Δ, which displaces the barriers to new positions (dotted lines). from publication: Topological states in the Kronig-Penney model with ...The U.S. Department of Energy's Office of Scientific and Technical InformationIn solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation.The model enables understanding and …General results from Kronig-Penney model: - if potential barrier between wells is strong, energy bands are narrowed and spaced far apart (Corresponds to crystals in which …The Kronig-Penney one-dimensional model . Purpose: to demonstrate that in solids, where many atoms stay closely, the interference between atoms will create allowed and …Kronig-Penney Model. Follow. 117 views (last 30 days) Show older comments. Mou Mahmood on 9 Feb 2021. Vote. 0. Link. Commented: Walter Roberson …The main computational challenge is to use numerical root finding to find solutions to the transcendental equation arising from the boundary conditions and Bloch's Theorem. The periodic potential for the Kronig-Penney model is given by: V (x) = 0,0 < x < a V (x) = V 0,−b < x < 0 And the potential is periodic, V (x +c) = V (x).The repulsive Kronig-Penney (KP) model [1] is a model in which an electron is in a one-dimensional (1D) lattice of negative ions with periodic repulsive potentials. This simple model has had an enduring pedagogical in uence in presenting the electronic band structure and associ-ated properties of solids as described in textbooks and other books ...Kronig-Penney Model. Learn more about kronig_penney_model Can anyone provide me the MATLAB code for Kronig-Penney model to draw band structure for 1-D periodic potential well structure?The Kronig-Penney model extended toarbitrary potentials via numerical matrix mechanics R. L. Pavelich∗ and F. Marsiglio† Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2G7 (Dated: November 14, 2014) The Kronig-Penney model describes what happens to electron states when a confining potential is repeated indefinitely.Kronig‐Penney model - pg 3 Or, if you translate back to the first BZ, it looks like this: Disclaimer: these plots are not really of the boxed equation above. They are plots of the situation where 𝑏→0, 𝑉 4→∞, and 𝑏⋅𝑉 4 Lconstant L 7 6 ħ . à ÔThe potential acting on electrons in crystalline materials can be modeled simply by the Kronig-Penney model. In 1-dimensional materials, the Kronig-Penney model can be represented by the following figure The wave function equation when the electron is in potential V 0 is given by: ψ 1 = A e αx + B e − αx With α = ℏ 2 2 m (E − V 0 ) .We introduce a two-dimensional (2D) model which combines a checkerboard potential, alias the Kronig-Penney (KP) lattice, with the self-focusing cubic and self-defocusing quintic nonlinear terms. The beam-splitting mechanism and soliton multistability are explored in this setting, following the recently considered 1D version of the model.The Kronig-Penney model evaluates the electronic levels of a crystal structure in a onedimensional periodic potential (Kittel, 2005). This model has been modified to be used in PC and takes into ...2.INTRODUCTION • The Kronig - penney model is a simplified model for an electron in a one- dimensional periodic potential. The possible states that the electron can occupy are determined by the Schrodinger equation, • • In the case of Kronig-Penney model, the potential V(x) is a periodic square wave.kronig penney modelkronig penney model (hindi)kronig penney model bsc 3rd yearkronig penney model nptelkronig penney model e-k diagramWhat is Kronig-Penney m...The Kronig-Penney (K-P) model is one of the few solvable models in quantum mechanics which makes it possible to investigate the properties of electronic transport in real solids. In their seminal paper [1], Kronig and Penney discussed the dispersion relation characterizing the transmission of an electron through a periodic potential in a one-This single-particle Kronig–Penney model cannot predict the full fractional conductance plateaus, which are understood to require the presence of strong electron–electron interactions 29,30,31 ...Periodic Potential Lab - Kronig Penney Model - New Interactive Version. By Daniel Mejia, Gerhard Klimeck. Periodic Potential Lab - Kronig Penney Model - Interactively explore bandstructure and wavefunctions with different potentials. Launch Tool. Version 0.2.4a - published on 20 Sep 2023.Bootstrapping the Kronig-Penney model. Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this ...Dec 22, 2020 · The Kronig-Penney Model. Kronig and Penney examined the behavior of electrons in a periodic potential by considering a relatively simple and one-dimensional model. It is assumed that the potential energy of an electron has the shape of a square well as shown in fig. The period of potential is (a+b) Kronig‐Penney Model. by Dr. Colton, Physics 581 (last updated: Fall 2020) First, two notes about the solution to the Schroedinger Equation for regions where the potential is constant. If the energy is higher than the potential, then the wavefunction is like. The Kronig-Penney model considers a periodically repeating square potential defined in one cell by \(V (x) = 0 (0 < x < b); V (x) = V_0 (b < x < l)\), then we can solve for \(u(x)\) in one cell. Like the finite square well, this is a tedious boundary condition problem where matching value and slope of the wavefunction at the potential edge ...KRONIG-PENNEY MODEL 2.1 Preview Kronig-Penney Model is a unique model which exhibits many of basic characteristics of the electronic structure of real crystals, generally regarded as an idealized and a one-dimensional model of a crystal. It is among the simplest possible models to describe electrons in a periodic lattice.The Model Kronig-Penney (1931) adalah model satu-dimensi dari kristal yang menunjukkan bagaimana elektron dalam kristal tersebar ke band diperbolehkan dan dilarang oleh hamburan dari array linier diperpanjang atom. Teori pertama (1931) dari EXAFS adalah setara dengan tiga-dimensi dari model ini. Teori ini menunjukkan bahwa elektron foto ...The Kronig-Penney model considers a periodically repeating square potential defined in one cell by \(V (x) = 0 (0 < x < b); V (x) = V_0 (b < x < l)\), then we …Question: Study the matlab code shown in the appendix and make you understand the correspondence of the script to the Kronig-Penny (KP) model introduced in our lecture. (a) Write a function to calculate the band structure for free electrons and compare with the band structure obtained by the KP model with U_eV=1; a=3e-10; b=4e-10.Details of the Kronig-Penney model . The KP model is a single-electron problem. The electron moves in a one-dimensional crystal of length L. The periodic potential that the …The picture below shows the Kronig-Penney model and the solutions of the Schrdinger equation in the two regions of the system. As we saw previously, by applying both the continuity condition and Blochs theorem we arrive at a somewhat intimidating expression at the top of the right-hand slide. To make life easier, we take the delta-function limit in …1947 Ford Models - The 1947 Ford models were little changed from 1946, and not all the changes were good. Learn more about the 1947 Ford models. Advertisement The factory-suggested prices for the 1947 Ford models went up $120 on average. De...By default, the potential is an array of square wells (the Kronig-Penney model). Below that you will see the probability distribution of the particle's position. Below the particle's position you will see its band structure , displayed as a dispersion curve; this is a graph of energy versus the crystal momentum k, with the energy bands shown as gray in the background.The Kronig-Penney model uses a periodic array of rectangular wells to deduce the existence of semiconductor energy bands from the solution of the corresponding Schroedinger equation. As part of this procedure, it is necessary to then solve a messy transcendental equation; this difficulty was surmounted by replacing the rectangular array with delta function potentials. The author describes two ...A two-dimensional δ-potential Kronig-Penney model for quasi-one-dimensional (Q1D) disordered systems is used to study analytically the influence of a constant electric field on the inverse localization length (LL). Based on the Green's function formalism we have calculated LL as a function of the incoming energy E, electric field F, length L ...About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators ...We utilize this analysis in a nonlinear Kronig–Penney model and investigate transmission and band modification properties. We discuss the modifications that are effected through an electric field and the nonlinear Wannier–Stark localization effects that are induced. Several applications are described, such as polarons in one dimensional ...Kronig Penney model • In the free electron theory a constant potential was assumed inside the solid. • In reality the presence of the positive ion cores gives rise to a varying potential field. In a simple model the potential can be assumed (‘a’ is the lattice spacing and ‘w’ is the width of the potential).Kronig Penny Determinant Solution - Free download as PDF File (.pdf), Text File (.txt) or read online for free. Many books do not provide step by step determinant solution for Kronnig Penny model. Here it is how I solved it.Photonic analogues of the relativistic Kronig-Penney model and of relativistic surface Tamm states are proposed for light propagation in fibre Bragg gratings (FBGs) with phase defects. A periodic sequence of phase slips in the FBG realizes the relativistic Kronig-Penney model, the band structure of which being mapped into the spectral response of the FBG. For the semi-infinite FBG Tamm surface ...derivation of the localization length for the 1D Kronig-Penney model, relating it to the properties of transmis-sion through a finite number of disordered barriers. Our analytical results are compared with the experi-mental data obtained for a single-mode microwave guide. We show that in spite of the standard restrictions of an-Question: 4.5 Single Impurity Consider now a periodic Kronig-Penney model with a single impurity represented by the 8-function potential ti fe81) 2m (4.36) located at site n=0. Using the transfer matrix technique developed in chapter 1, we calculate the energy of the bound state of the quantum particle localized at the impurity site.In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation.The model enables understanding and calculation of the electronic band structures ...Long Answer Questions (8-marks) Derive an expression for the fermi energy of a free electron gas (Sommerfield theory). Obtain expression for density of states for a free electron gas in 3 dimension. show the variation of density of state with energy. Discuss the Kronig-Penney model and show that how it explain the forbidden gap.The calculation of bands in inorganic solids is demonstrated with the Kronig-Penney model. The contrast with polymers is then made with a discussion of the linear combination of atomic orbitals, which underlies the molecular physics of polymers. A discussion of solitons and doping in polyacetylene to exemplify its mechanisms of conduction ...Kronig-Penney model. I am studying the Kronig-Penney model as treated in the book by Kittel: Introduction to Solid State Physics. In this model one considers a period potential which is zero in the region [0, a] [ 0, a] (define as region I), U0 U 0 in the region [a, a + b] [ a, a + b] (define as region II) and again zero in the region [a + b ...Kronig-Penney delta function model. 12 12 ' Require that ( ) ( ') and require that. xa x a. xx dd dx =− =εε dx + Ψ=Ψ ΨΨ = Ψ. 2 (x') 22. 0 [ ] 22 2 n 2. sin( ) cos( ) cos( ) 2 and lim b is width of barrier (see earlier slide) when P 0, α=K (free electron case) when P , E = n (bound electron case) 2. o. bo U. a Ka a P a mE ma P bU ma. αWhat is the origin of band structure and band diagram : I explain in this video using the Kronig Penney Model2. (a) How does the energy of electrons depend on the wavenumber in the free electron theory? Include a drawing of the energy dispersion relation of electrons to illustrate your answer. (b) Illustrate using a sketch, what happens to the form of the energy dispersion relation for the Kronig-Penney model within the nearly free electron theory.
The tight-binding model 4.1 Introduction In the tight-binding model we assume the opposite limit to that used for the nearly-free-electron ap- ... THE TIGHT-BINDING MODEL Kronig-Penney Model, which is a very simple illustration of the formation of bands; see e.g. Quantum Mechanics, by Stephen Gasiorowicz (Wiley, .... Molecular bioscience
What are the findings of Kronig Penney model? Hence we conclude that the total number of possible wave functions in any energy band is equal to the number of unit cells N. We have concluded in the Kronig-Penny model that the energy discontinuities in a monatomic one-dimensional lattice occur when k = (nπ/a), where n is any positive or …1 ก.พ. 2526 ... Kronig–Penney model in reciprocal lattice space;. Singh, Surjit · American Journal of Physics , Volume 51 (2) – Feb 1, 1983. Download PDF.Bloch theorem and Kronig Penney ModelQuestion: This exercise leads students through how to obtain the allowed momentum / energy states (band structure) for the Kronig-Penney model, a simple model for materials with periodic structure. The main computational challenge is to use numerical root finding to find solutions to the transcendental equation arising from the boundary conditions and Bloch's Theorem.Lecture 5: The generalized Kronig-Penney model and complex band structure. Lecture 6: Introduction to the tight-binding method. A single s-band in a one-dimensional lattice. Density of electron states. A one-dimensional lattice with a two-atom basis, the example of trans-polyacetylene. Complex band structure in the tight-binding approximation.The Kronig-Penney model is the simplest model for an electron in a one-dimensional periodic potential. The possible states that the electron can occupy are determined by the Schrödinger equation, The periodic potential in a one-dimensional lattice of spacing ...Kronig Penny Model in Telugu for B.E/B.Tech/ B.Sc studentsFullscreen (disabled) The Kronig-Penney model for an electron in a one-dimensional crystal can be solved in closed form for a sinusoidal potential of wavelength . Consider the Schrödinger equation (in atomic units ) [more] Contributed by: S. M. Blinder (August 2022) Open content licensed under CC BY-NC-SA.Kronig-Penney model. In order to simplify the problem the potential function is approximated by a rectangular potential: Using Bloch's theorem, we only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function u(x) is also continuous and smooth.Considering a single period of the potential:A bipartite Kronig Penney model with Dirac-delta potential scatterers Thomas Benjamin Smith and Alessandro Principi-The relation between the effective band mass in a solid and the free electron mass G Forcade, D van Paridon, R L Pavelich et al.-Non-additive quantum mechanics for a position-dependent mass system: Dirac delta and …能带理论概述. 能带理论 —— 研究固体中电子运动的主要理论基础 —— 固体物理最核心的内容 能带理论 —— 定性地阐明了晶体中电子运动的普遍性的特点• Kronig-Penney model • Numerical solutions: - Real space - Fourier space 1/30/2013 ECE 595, Prof. Bermel . Recap from Monday • Application Examples - Electrostatic potential (Poisson's equation) • 1D array of charge • 2D grid of charge - Arrays of interacting spinsView Kronig penney model PowerPoint PPT Presentations on SlideServe. Collection of 100+ Kronig penney model slideshows.In this work we use the Kronig-Penney model with delta function potentials as a one dimensional model of solid in order to study the effects of structural and chemical disorder in the plane wave representation. Structural disorder takes into account the variation on the potential position and chemical disorder is produced by changing the potential intensity and keeping fixed the distance ...Periodic potentials - Kronig-Penney model Electrons in a lattice see a periodic potential due to the presence of the atoms, which is of the form shown in Figure 1. a Figure 1. Periodic potential in a one-dimensional lattice. As will be shown shortly, this periodic potential will open gaps in the dispersion relation,30 ม.ค. 2564 ... and NOWAK, Hans. LMC-COMPLEXITY IN KRONIG-PENNEY MODEL: COMPARISON OF EFFECTS BETWEEN STRUCTURAL AND CHEMICAL DISORDER. Momento [online]. 2021, ...KRONIG-PENNEY MODEL Potential and electron energy functions of a single, non-interacting, one-electron atom • potential (V) is inversely proportional to distance from positively charged nucleus. • electron is negatively charged. E=-eV energy is negative (which means the electron is attracted to the nucleus) •At infinite distance from nucleus …Kronig-Penney Model. Learn more about kronig_penney_model Can anyone provide me the MATLAB code for Kronig-Penney model to draw band structure for 1-D periodic potential well structure?Question: Kronig Penney Model: Consider a single electron Schrodinger equation to solve the electron wavefunction and energy states for a 10 periodic lattice. The periodic potential is shown below. In this model, known as the Kronig Penny model, the periodic potential of a 1D crystal lattice is replaced by a delta function at each lattice site.I'm writing a report for a computer lab where we ran simulations of the wavefunction of an electron in an array of square wells as per the Kronig-Penney model and I'm just looking for some verification of my interpretation of Bloch's Theorem as it applies to the solutions of the schrodinger...This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Question: The potential of an electron in a one-dimensional lattice is of the same type as that used in the Kronig-Penney model. Assuming V ab << ?/m, prove that the energy band gap at k = n/a is 2V bla..
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