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The Skyrme Hartree-Fock-Bogolyubov equations

The HFB approximation is based on the use of a trial variational wave function which is assumed to be an independent quasiparticle state $ \vert\Phi\rangle$. This state, which mixes different eigenstates of the particle number operator, is a linear combination of independent particle states representing various possibilities of occupying pairs of single particle states. Following the notations and phase convention of [5] we define the particle and pairing density $ \rho$ and $ \tilde\rho$ matrices by

$\displaystyle \rho({\mathbf r}\sigma q,{\mathbf r}'\sigma'q')=\langle\Phi\vert a^\dagger_{{\mathbf r}'\sigma'q'}a_{{\mathbf r}\sigma q}\vert\Phi\rangle\,,$ (1)

$\displaystyle \tilde\rho({\mathbf r}\sigma q,{\mathbf r}'\sigma'q')=-2\sigma'\l...
...e\Phi\vert a_{{\mathbf r}'-\sigma'q'}a_{{\mathbf r}\sigma q}\vert\Phi\rangle\,,$ (2)

where the operators $ a^\dagger_{{\mathbf r}\sigma q}$ and $ a_{{\mathbf r}\sigma q}$ create and annihilate a nucleon at the point $ {\mathbf r}$ having spin $ \sigma=\pm\frac{1}{2}$ and isospin $ q=\pm\frac{1}{2}$. The symmetry properties of $ \rho$ and $ \tilde\rho$ as well as the relation between $ \tilde\rho$ and the pairing tensor $ \kappa$ (defined for example in [10]) are discussed in [5].

The variation of the energy expectation value $ E=\langle\Phi\vert\hat{H}\vert\Phi\rangle$ with respect to $ \rho$ and $ \tilde\rho$ under the constraints $ N=\langle\Phi\vert\hat{N}\vert\Phi\rangle$ and $ Z=\langle\Phi\vert\hat{Z}\vert\Phi\rangle$ (for neutrons and protons) leads to the Hartree-Fock-Bogolyubov equation which reads in coordinate representation

\displaystyle \int d^3{\mathbf r}'\sum_{\sigma'}
...sigma) \cr
\varphi_2(E,\mathbf r\sigma) \cr

where the particle and pairing fields are given by

$\displaystyle h({\mathbf r}\sigma,{\mathbf r}'\sigma')= \frac{\delta E}{\delta\...
...')= \frac{\delta E}{\delta\tilde\rho({\mathbf r}\sigma,{\mathbf r}'\sigma')}\,.$ (3)

Once again we refer the reader to the article [5] for the discussion concerning the quasiparticle spectrum, its symmetries and the relations between the components of the HFB spinors and the densities.

next up previous
Next: Local densities Up: hfbrad23w Previous: Introduction
Jacek Dobaczewski 2005-01-23