General parametrisation.

Prior to the study of the different phenomena involved in deformed nuclei, the nuclear surface must be parameterized in some way. One possibility is to describe it by an expansion in spherical harmonics with the length of the radius vector R pointing from the origin to the surface [2], given by

$$R=R(\theta,\phi)=R_0(1+\alpha_{00}+\sum_{\lambda=1}^{\infty}... ...a}^{\lambda} \alpha^*_{\lambda\mu}Y_{\lambda\mu}(\theta,\phi))$$ (3)

where R$_0$ is the radius of the spherical nucleus, Y$_{\lambda\mu}$($\theta,\phi$) are the spherical harmonics [3], $\alpha^*_{\lambda\mu}$ are the expansion coefficients, and the constant $\alpha_{00}$ describes changes of the nuclear volume. Since the incompressibility of the nuclear matter is quite high, the volume is fixed as V=4/3$\pi$R$_0^3$ for all deformations [2]. This defines the constant $\alpha_{00}$ which is given, up to second order, by,

$$\alpha_{00}=-\frac{1}{\sqrt{4\pi}}\sum_{\lambda\mu} \vert \alpha_{\lambda\mu} \vert^2.$$ (4)

In the case of quadrupole deformation ($\lambda$=2), the nuclear shape is a pure ellipsoid (see figure ) and for the case of small quadrupole deformations $\alpha_{00}$=0. The five parameters $\alpha_{2\mu}$ that characterise the nuclear shape reduce to two real independent variables a$_{20}$ and a$_{22}$=a$_{2-2}$, which, together with the three Euler angles, give a complete description of the system []. Instead of a$_{20}$ and a$_{22}$, it is convenient to introduce the so-called Hill-Wheeler coordinates $\beta_2$ and $\gamma$, defined as [4],

$$a_{20}=\beta_2{\cdot}cos\gamma$$ (5)

$$a_{22}=\frac{1}{\sqrt{2}}\cdot\beta_2{\cdot}sin{\gamma}$$ (6)

where $\beta_2$ is the elongation or quadrupole deformation and $\gamma$ is the triaxial parameter. By truncating the general expansion in equation at $\lambda$=2, the following equation is obtained for the most usual case of nuclear deformation,

$$R(\theta,\phi)=R_0(1+\beta_2\sqrt\frac{5}{16\pi}[cos\gamma(3cos^2\theta-1)+\sqrt3sin{\gamma}sin^2{\theta}cos(2\phi)])$$ (7)