Crystal eigenstates

Creation operator, 505 representation of, 507 Critical value, 338 Crystallographic point groups irreducible representations, 726 Crystallographic symmetry groups construction of mixed groups, 728 Crystal, eigenstates of, 725 Crystal symmetry, changes in, 758 Crystals... 

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

Papalexi, N., 356,371,377,381 Paramagnetic crystals point groups for, 737 symmetry properties of eigenstates, 745... 

Symmetry of magnetic structures, 726 Symmetry properties of eigenstates of a paramagnetic crystal, 745... 

As discussed in Sect. 6.2, the electronic states of a paramagnetic ion are determined by the spin Hamiltonian, (6.1). At finite temperamres, the crystal field is modulated because of thermal oscillations of the ligands. This results in spin-lattice relaxation, i.e. transitions between the electronic eigenstates induced by interactions between the ionic spin and the phonons [10, 11, 31, 32]. The spin-lattice relaxation frequency increases with increasing temperature because of the temperature dependence of the population of the phonon states. For high-spin Fe ", the coupling between the spin and the lattice is weak because of the spherical symmetry of the ground state. This... 

The phase and amplitude spectrum of the laser pulse are tailored to create a wave packet with selected properties. The various eigenstates that comprise the wave packet are populated by different frequency components of the laser pulse, each with its specified amplitude and phase. For example, rovibrational wave packets of Li2 in the El E+ state were created, consisting of vibrational levels v = 12-16 and rotational levels J = 11, 19. The phases and amplitudes of the pump pulse shown in Fig. 20 were generated with a 128-pixel liquid crystal SLM. The pulse was tailored to optimize the ionization signal at a delay time of 7 ps. The phases used to maximize or minimize the ionization signal are shown by solid and dashed lines, respectively, and the intensities at the eigenfrequencies of the wave packet are indicated by circles. 

We concentrate on the expression for the covalent part of the crystal field Wffff and transform it to the form coinciding with the AOM and relate the parameters of the latter with the electronic structure of the ligands. To do so, we perform a unitary transformation of the canonical MOs of the /-system Z) - the eigenstates of the Fock operator - to the localized MO L) separately for the occupied and vacant canonical MO ... 

Thus the hamiltonian (2.15) couples the electronic excitations to the vibrations by linear terms in and by quadratic terms in A. The molecular eigenstates of (2.15) are the vibronic states they are different from tensorial products of electronic excitations and undressed vibrations. Even for this simple intramolecular effect, we cannot, when moving to the crystal, consider excitonic and vibrational motions as independent. 

The vibronic exciton approximation restricts H to a subspace corresponding to a given vibronic molecular state. In this subspace the degeneracy of the localized vibronic states is lifted by the interactions JnmB Bm. Using the translational invariance, the eigenstates of the crystal are seen to be the vibronic excitons, or vibrons ... 

The interesting point is that if we know the exact eigenstates of the crystal, we can substitute them into Eq. (3-38) to obtain localized Wannier functions s,> if substituted into Eq. (3-37), the localized Wannier functions will give exact eigenstates of the crystal. 

Consider the operator H of the crystal-plus-impurity system let us denote by i, ) and its eigenstates and eigenfimctions. The (spinless) density matrix operator for the ground state is... 

ForNd3+ion, J = 9/2 and hence the ground state multiplet is tenfold degenerate. Under the influence of a crystal field of hexagonal symmetry, the degeneracy is partially removed. Five Kramer s doublets are formed with the eigenstates, 1/2),... 

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