Analytical comparison between X(3) and X(5) models of the Bohr Hamiltonian

Abstract

The 3-D Bohr-Mottelson Hamiltonian for <i>γ</i>-rigid prolate isotopes is solved via inverse square potential having only one free parameter, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math>. The exact form of the wave functions and the energy spectra are obtained as a function of the free parameter that determines the changes in the spectra ratios and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi><mml:mo>(</mml:mo><mml:mi>E</mml:mi><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:math>. The solutions are compared with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:math> model and some new set of equations that show the relationships between the two models are stated. The entire solutions from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> to ∞ are compared with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:math>. The solutions spread from the region around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:math> over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:math> and approach <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:math> at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>∞</mml:mo></mml:math>. The exact solutions obtained via variational procedure are compared with some existing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:math> models in the literature. The strong agreement between the present model and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:math> via infinite square well potential is discussed. Twelve best critical point isotopes, ¹⁰²Mo, ^104-108Ru, ^120-126Xe, ¹⁴⁸Nd, ^184-188Pt are chosen for experimental realization of the model and moderate agreements are recorded. An excellent agreement which appears in the first <i>β</i>-excited state in the comparison of the present model with three <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>90</mml:mn></mml:math> isotones: ¹⁵⁰Nd, ¹⁵⁴Gd, and ¹⁵⁶Dy, known to be <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:math> candidates, suggests that the present model compensates the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mo>(</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:math> models whose predictions are excellent in the ground states but moderately bad in the first <i>β</i>-excited states.