The quantum theory of lattice dynamics. I

The dynamics of a crystal is examined on the basis of the adiabatic approximation. In part I we examine the form of the dynamical and anharmonic tensors on the assumption that the effective nuclear potential energy can be represented as a simple two-body interaction. In part II we derive expressions for the electronic contributions on the basis of the Hartree-Fock approximation and show that these electron-nuclear interactions are more complex than a simple two-body interaction. In part III we examine these interactions in more detail and find that the two-body approximation is equivalent to a rigid-ion model and that this approximation becomes exact in the limit q = 0.

The electronic contribution to the dynamical tensors is examined in more detail and the following results are obtained. First the electronic contribution is shown to be translationally invariant, secondly the assumption that this term can be approximately represented by a two-body interaction is shown to be equivalent to a rigid ion model, and finally this approximation is shown to become exact in the limit q = 0.


Expressions are derived for the expansion of the total energy of the electrons in their ground state in a crystal in powers of displacements of the nuclei from their equilibrium configuration. The expansion is taken up to third order on the basis of the coupled Hartree-Fock equations and thus one obtains expressions for the electronic contributions to the dynamical and anharmonic tensors.


The total electronic energy for condensed systems is examined as a function of nuclear displacements from the equilibrium configuration. The expansion is derived to all orders both for the exact solutions to the many-electron Schrödinger equation and for the approximate Hartree-Fock solutions. The results are shown to be translationally invariant to all orders. Expressions are derived for the electronic contributions to the force-constant tensors in real space and it is shown that the effective nuclear potential has a simple two-body form for a Bravais lattice.


2012 ◽  
Vol 27 (28) ◽  
pp. 1250162 ◽  
Author(s):  
TUNCAY BAYRAM

Constrained Hartree–Fock–Bogoliubov theory with SLy4 and SLy5 Skyrme forces is used to investigate the shape transition between spherical and γ-unstable nuclei in 38–66 Ti . By examining potential energy curves and neutron single-particle levels of even–even Ti isotopes, 46,52,60 Ti are suggested as possible candidates of the nuclei with E(5) symmetry.


1971 ◽  
Vol 9 (3) ◽  
pp. 185-189 ◽  
Author(s):  
K.V. Namjoshi ◽  
S.S. Mitra ◽  
J.F. Vetelino

1970 ◽  
Vol 67 (4) ◽  
pp. 584-594
Author(s):  
S. Boffi ◽  
G. Campagnoli

1977 ◽  
Vol 83 (2) ◽  
pp. 615-624 ◽  
Author(s):  
S. K. Sarkar ◽  
S. K. Das ◽  
D. Roy ◽  
S. Sengupta

1967 ◽  
Vol 20 (5) ◽  
pp. 495 ◽  
Author(s):  
J Oitmaa

The lattice dynamics of harmonic and anharmonic shell models are reviewed. It is shown that the various dynamical equations for the shell model can be expressed in the same form as those for the rigid ion model, but with modified force constants. The anharmonic shell model leads to higher order contributions to the dipole moment, quadratic and cubic in the normal coordinates, for which explicit expressions are obtained.


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