Two-nucleon transfer reactions as a test of quantum phase transitions in nuclei

A quantal and a semiclassical analysis of two-nucleon transfer intensities is done within the framework of the interacting boson model. The expected features of these quantities for the quantum phase transition (QPT) between spherical, U(5), and axially deformed, SU(3), shapes are discussed. Experimental data for ($p,t$) and $(t,p$) transfer reactions clearly show the occurrence of QPTs in Gd, Sm, and Nd.

Figure 1

Evolution of the classical order parameter ${\beta }_e$ in the U(5)–SU(3) transition for $N=10$ with the inset showing the critical behavior of ${\beta }_e$.

Figure 2

(a) Evolution of the classical element $F_1$ in the U(5)–SU(3) transition for $N=10$ with the inset showing the behavior in the vicinity of the critical point. (b) The same as in (a) but for $F_2$. (c) The same as in (a) but for $F_3$. (d) The same as in (a) but for $F_4$.

Figure 3

Quantal-classical correspondence for the matrix elements of $s, d_{\mu }$ appropriate to $(p,t)$ reaction intensities. The inset in panel (a) shows the critical behavior of ${\beta }_e$.

Figure 4

Quantal-classical correspondence for the matrix elements of $s^{†}, d_{\mu }^{†}$ appropriate to $(t,p)$ reaction intensities.

Figure 5

Comparison between the experimental (symbols) energies of the low-lying levels in the Gd, Sm, and Nd nuclei [15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and the calculated (lines) energies with the Hamiltonian (18). The calculated ${\beta }_e$ values as a function of neutron number are given in panel (d).

Figure 6

Comparison between calculated and experimental $(p,t)$ transfer intensities for Gd. Here the values of the overall scale parameters $t_{a_{\nu }}$ and $t_{b_{\nu }}$ in the transfer operators are obtained by fitting the experimental data and given as $t_{a_{\nu }}=t_{b_{\nu }}=3.46$ (arbitrary units).

Figure 7

Comparison between calculated and experimental $(t,p)$ transfer intensities for Gd. Here the values of the overall scale parameters $t_{a_{\nu }}$ and $t_{b_{\nu }}$ in the transfer operators are obtained by fitting the experimental data and given as $t_{a_{\nu }}=2.24$ and $t_{b_{\nu }}=0.63$ (arbitrary units).

Figure 8

Comparison between calculated and experimental $(p,t)$ transfer intensities for Sm. Here the values of the overall scale parameters $t_{a_{\nu }}$ and $t_{b_{\nu }}$ in the transfer operators are obtained by fitting the experimental data and given as $t_{a_{\nu }}=4.47$ and $t_{b_{\nu }}=3.16$ (arbitrary units).

Figure 9

Comparison between calculated and experimental ($t,p$) transfer intensities for Sm. Here the values of the overall scale parameters in the transfer operators are obtained by fitting the experimental data and given as $t_{a_{\nu }}=2.45$ and $t_{b_{\nu }}=1.73$ (arbitrary units).

Figure 10

Comparison between calculated and experimental ($p,t$) transfer intensities for Nd. Here the values of the overall scale parameters in the transfer operators are obtained by fitting the experimental data and given as $t_{a_{\nu }}=t_{b_{\nu }}=5.48$ (arbitrary units).

Figure 11

Comparison between calculated and experimental ($t,p$) transfer intensities ratios for Nd. Ratios are independent of scale parameter.