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hokhani
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Do any band structure (in absence of any external field) in general, is symmetric with respect to k? In other words, do we always have E(k)=E(-k).
Your statement is not true in general.Henryk said:Energy is a quadratic function of k.
By this statement do you mean that to have E(k)=E(-k), both "time reversal and inversion symmetry" are necessary or one of them suffices?radium said:In regards to your statement, it is true if you have time reversal and inversion symmetries.
##(\psi _k(r) = e^{ikr} \phi_k (r))^*=e^{-ikr}\phi_k^*(r) ##. This only equals ##\psi_{-k}(r)## if ##\phi_k^*=\phi_{-k}##. The other point is that in the presence of spin orbit interaction or magnetic fields, U may not be real because the spin matrix ##\sigma_y## is imaginary.Henryk said:Suppose, we have a Bloch wavefunction ##\psi _k(r) = e^{ikr} \phi (r) ## corresponding to energy E.
Now, we can simply take a complex conjugate of the above equation !
The complex conjugate of ## \psi _k(r) ## is ## \psi _{-k}(r) ##,
Of course, but we already said that E(k)=E(-k) may fail to hold if there is no inversion symmetry.hokhani said:I think at systems with inversion symmetry, an electron moving towards one direction see the same environment as the electron moving in the opposite direction. Therefore it seems E(k)=E(-k) to be held regardless of whether the time reversal exists or not.
The concept of symmetry in band structure refers to the repeating patterns or symmetries in the energy levels and wave functions of electrons in a crystal lattice. This symmetry is a result of the periodic arrangement of atoms in the crystal.
Symmetry plays a crucial role in determining the electronic properties of materials. It affects the energy levels and wave functions of electrons, which in turn determine properties such as conductivity, magnetism, and optical properties of materials.
Studying symmetry in band structure allows scientists to understand and predict the electronic properties of materials. It also helps in designing and engineering materials with desired properties for various applications in fields such as electronics, photonics, and materials science.
Symmetry in band structure diagrams is represented by different shapes and labels. These shapes and labels correspond to the symmetry operations of the crystal structure, such as rotation, reflection, and translation. They help in identifying the symmetries present in the energy bands and band gaps.
Some techniques used to study symmetry in band structure include X-ray crystallography, neutron diffraction, and electron microscopy. These techniques provide information about the crystal structure and symmetries present, which can then be used to understand the electronic properties of materials.