Microscopic derivation of the generalized collective Hamiltonian

The nucleus in the center-of-mass frame is described by <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>3</a:mn><a:mi>A</a:mi><a:mtext>−</a:mtext><a:mn>3</a:mn></a:mrow></a:math> independent coordinates, including three Euler angles and three variables <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>ρ</b:mi><b:mo>,</b:mo><b:mspace width="0.28em"/><b:mi>β</b:mi><b:mo>,</b:mo><b:mspace width="4.pt"/><b:mtext>and</b:mtext><b:mspace width="4.pt"/><b:mi>γ</b:mi></b:mrow></b:math> to describe orientation, size, and shape of the nuclear inertia ellipsoid as well as <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mn>3</f:mn><f:mi>A</f:mi><f:mtext>−</f:mtext><f:mn>9</f:mn></f:mrow></f:math> generalized Euler angles to describe internal motion of the nucleons. Based on the idea that the incompressible nucleus has constant density, we suggest a definition of the deformation parameter <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>β</g:mi></g:math>. It allows us to derive the Hamiltonian, which describes large-amplitude collective motions. At <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>β</h:mi><h:mo>≪</h:mo><h:mn>1</h:mn></h:mrow></h:math> it reduces to a sum of the Bohr Hamiltonian and the Hamiltonian of monopole <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>ρ</i:mi></i:math> vibrations. As a result of straightforward calculations we obtain explicit expressions for all the inertial parameters as functions of the variables <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>ρ</j:mi><j:mo>,</j:mo><j:mspace width="0.28em"/><j:mi>β</j:mi><j:mo>,</j:mo><j:mspace width="4.pt"/><j:mtext>and</j:mtext><j:mspace width="4.pt"/><j:mi>γ</j:mi></j:mrow></j:math>.