The Interacting Boson Model (IBA-1)

At the end of Section 24, the complexity of the shell model program for nuclei with valence protons and neutrons was demonstrated. In a typical nucleus, ¹⁵⁴Sm, valence protons and neutrons may couple to more than 4 × 10¹³ states with J = 0 and positive parity, more than 3 × 10¹⁴ J = 2⁺ states and more than 5 × 10¹⁴ states with J = 4⁺. To calculate eigenstates and eigenvalues of the shell model Hamiltonian, matrices of order 10¹⁴ should be constructed and diagonalized. To construct the Hamiltonian matrices in this case, 1192 matrix elements of two nucleon configurations and 11 single nucleon energies should be known. In simple cases, such matrix elements are determined from experimental data but this would be beyond hope in the present case. Diagonalization of gigantic matrices of order 10¹⁴ is beyond the capability of present day computers. Even if those difficulties could have been overcome, the results would have been disappointing. There is no way in which properties of a wave function with 10¹⁴ components could be studied. These difficulties are particularly frustrating in view of the fact that many nuclei for which the shell model program is so complex, exhibit remarkably simple regularities. The low lying levels of ¹⁵⁴Sm as well as ¹⁴⁸Sm in Fig. 33.1, may serve as a simple example. To understand these regularities a simple coupling scheme should be 732found, truncating the huge shell model space and exhibiting the simple features of nuclear spectra. Figure 33.1 <italic>Experimental low lying levels of</italic> ¹⁴⁸Sm <italic>and</italic> ¹⁵⁴Sm. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/fig33_1.tif"/>