Band structure of ${}^{235}$U

Over a period of several years we have performed three separate experiments at Lawrence Berkeley National Laboratory's 88-Inch Cyclotron in which ${}^{235}$U (thick target) was Coulomb-excited. The program involved stand-alone experiments with Gammmasphere and with the 8pi Spectrometer using ${}^{136}$Xe beams at 720 MeV, and a CHICO-Gammasphere experiment with a ${}^{40}$Ca beam at 184 MeV. In addition to extending the known negative-parity bands to high spin, we have assigned levels in some seven positive-parity bands which are in some cases (e.g., $\left[631\right]1/2$, $\left[624\right]7/2$, and $\left[622\right]5/2$) strongly populated by $E3$ excitation. The CHICO data have been analyzed to extract $E2$ and $E3$ matrix elements from the observed yields. Additionally, many $M1$ matrix elements could be extracted from the $\gamma$-ray branching ratios. A number of new features have emerged, including the unexpected attenuation of magnetic transitions between states of the same Nilsson multiplet, the breakdown of Coriolis staggering at high spin, and the effect of $E3$ collectivity on Coriolis interactions.

Figure 1

Coincidence $\gamma$-ray spectra showing the features of the principal bands to high spin in the Gammasphere stand-alone experiment. (a) Coincidence spectrum with a single gate set at $E_{\gamma }=409$ keV, the $43/2^{-}\to 39/2^{-}$ transition in the ground-state rotational band. The $45/2^{-}\to 41/2^{-}$ and $41/2^{-}\to 37/2^{-}$ transitions are not in coincidence with the gate and are absent in the spectrum. The decrease in peak height with decreasing spin represents the increasing conversion coefficient. The decay down this band is seen to spread the population to both signatures very quickly. (b) Coincidence spectrum summed over the gates $E_{\gamma }=241$, $E_{\gamma }=277$, and $E_{\gamma }=311$ keV, respectively, the $23/2^{+}\to 19/2^{+},$ $27/2^{+}\to 23/2^{+},$ and $31/2^{+}\to 27/2^{+}$ transitions in the Nilsson [631]1/2 band. Compared to panel (a), the population is seen to stay more closely to the signature in which the gates were set. (c) Coincidence spectrum summed over the gates $E_{\gamma }=250$, $E_{\gamma }=270$, and $E_{\gamma }=288$, $E_{\gamma }=305$, and $E_{\gamma }=322\text{keV}$, respectively, the $23/2^{+}\to 19/2^{+}$, $25/2^{+}\to 21/2^{+},$ $27/2^{+}\to 23/2^{+},$ $29/2^{+}\to 25/2^{+},$ and $31/2^{+}\to 27/2^{+}$ transitions in the Nilsson [622]5/2 band.

Figure 2

The principal high-spin rotational bands seen in the ${}^{136}$Xe experiments

Figure 3

Examples of the decay from the Nilsson [631]3/2 band to the Nilsson [631]1/2 band with Gammasphere in the ${}^{136}$Xe experiments. The gates shown correspond to the $23/2^{+}\to 23/2^{+},27/2^{+}\to 27/2^{+},$ and $31/2^{+}\to 31/2^{+}$ transitions. The spectra illustrate where each of these gates feed into the [631]1/2 band, and also weak cascade transitions lying above the gates at $E_{\gamma }=308$, 343, 371, and 395 keV in the [631]3/2 band (cf. the detailed level scheme in Fig. 20).

Figure 4

Portion of the spectrum of $\gamma$ rays in coincidence with $E_{\gamma }=168\text{keV}$, the $17/2^{-}\to 13/2^{-}$ transition in the ground-state rotational band. Data are taken in the Gammasphere experiment with ${}^{136}$Xe at 720 MeV. With a few exceptions, the peaks may be identified with transitions from the [752]5/2 and [734]9/2 members of the $j_{15/2}$ multiplet together with the $K=3/2$ and 11/2 $\gamma$-vibrational bands. For the $K=3/2$ and 5/2 bands, the signatures are labeled separately in this figure (cf. the detailed level schemes in Figs. 20, 21, 22).

Figure 5

Portion of the spectrum of $\gamma$ rays in coincidence with the CHICO detectors with a ${}^{40}$Ca beam at 184 MeV. The cascade transitions indicated in the rotational bands built on the Nilsson [631]1/2 (cf. Fig. 23) and [622]5/2 (cf. Fig. 24) levels are those used to measure the cross section and derive $E3$ matrix elements as discussed in the text. The peaks labeled at $E_{\gamma }=317$ and 339 keV are decays of the Nilsson [624]7/2 band to the [622]5/2 band originating from spin 7/2${}^{+}$ and 9/2${}^{+}$ states (cf. Fig. 24). These $\gamma$ rays are relatively intense in this figure because the other transitions depicted originate from much higher spins. Gamma rays labeled only by their energy are cascades of the ${}^{235}$U ground-state band ranging from $E_{\gamma }=232\text{keV}$, $23/2^{-}\to 19/2^{-}$ to $E_{\gamma }=333$ keV, $33/2^{-}\to 29/2^{-}.$

Figure 6

Portion of the spectrum of $\gamma$ rays in coincidence with the CHICO detectors with a ${}^{40}$Ca beam at 184 MeV. The peaks are labeled by their energy (keV) and are grouped together in bands which belong to the negative-parity [752]5/2 and [734]9/2 (cf. Fig. 27) members of the $j_{15/2}$ multiplet together with the $K=3/2$ and 11/2 $\gamma$-vibrational bands (cf. Fig. 26). The labeled peaks are decays to the ground-state band in all cases. Note that the [734]9/2 Nilsson band is crossed by the $K=11/2$ $\gamma$-vibrational band at spin 17/2.

Figure 7

Portion of the spectrum of $\gamma$ rays in coincidence with $E_{\gamma }=168\text{keV}$, the $17/2^{-}\to 13/2^{-}$ transition in the ground-state rotational band, taken from the $E_{\gamma }·E_{\gamma }$coincidence matrix gated by the CHICO detector. The beam was ${}^{40}$Ca at 184 MeV incident energy. In comparison with the corresponding spectrum taken in the ${}^{136}$Xe experiment it can be seen that the highest spins seen in the ${}^{40}$Ca experiment overlap the lowest spins identified in the ${}^{136}$Xe experiment.

Figure 8

Ratios of $B\left(M1\right)$/$B\left(E2\right)$ values derived from measured $\gamma$-ray branching ratios in these experiments. The experimental values have been corrected for the small effect of $E2$/$M1$ mixing in the ($J\to J-1$) transitions based on a simple model. In panels (a) and (b), solid lines correspond to the rigid-rotor model with parameters chosen to best reproduce the results. For $K=5/2$ there is only one free parameter, namely, ($g_K-g_R$)${}^2$, and values are discussed in the text. For the special case $K=1/2$, there is an additional parameter usually called the magnetic-decoupling parameter, $b$, which governs the magnitude and phase of the oscillation in $B\left(M1\right)$ values. Two calculations are shown in panel (c). The lower fitted curve is a rigid-rotor calculation with the one free parameter ($g_K-g_R$)${}^2$ fitted at low spin. The higher fitted curve is a particle-rotor calculation with attenuated Coriolis matrix elements (cf. text).

Figure 9

Goodness of fit to the data shown for the $K=1/2$ band in Fig. 8. For the seven data points and one free parameter, the un-normalized chi-squared, shown should minimize at the value 6. The lower minimum is therefore statistically favored.

Figure 10

Derived $B\left(M1\right)$ values from analysis of $\gamma$-ray branching data in the Xe experiments. In all cases, the final state refers to the ground band. Transitions going ($J\to J$) are shown as circles: transitions going ($J\to J+1$) are shown as squares; and transitions going ($J\to J-1$) are shown as diamonds. The lines are to guide the eye. Note that the [734]9/2 Nilsson band is crossed by the $K=11/2$ $\gamma$-vibrational band at spin 17/2.

Figure 11

Accumulated yields normalized to the 8${}^{+}$ state in ${}^{238}$U present as a contaminant in the ${}^{235}$U target. The $E2$ matrix elements were taken from Eq. (5) with $Q_o=11.1e$b. The $E3$ matrix elements connecting the ground-state rotational band to the $K=0$ octupole band were taken from Eq. (21) with $Q_t(\lambda =3)=2.3e$ 10${}^{-36}$cm${}^3$ chosen to fit the intensity of the octupole band.

Figure 12

Measured and calculated Coulomb excitation yields for the negative-parity bands observed in ${}^{235}$U in the CHICO experiments. The $E2$ matrix elements in the ground-state band (GSB) and in the excited bands were taken from Eq. (5) with $Q_o=9.75e$b. The $E2$ matrix elements linking the ground-state band with the excited bands were taken from Eq. (21) for the $K=5/2$ and $K=9/2$ bands and from Eq. (22) for the $K=3/2$ and 11/2 $\gamma$-vibrational bands. The parameter $M_2/M_1$ in Eq. (22) was fixed from analysis of the present and previously measured $E2$ $\gamma$-ray branching ratios between the vibrational bands and the ground-state band. The parameter $M_1$ or $Q_t$ was varied for each band to give a best fit to the relative yield. The measured yields in the ground-state band are the directly observed $E2$ $\gamma$-ray intensities normalized to the $19/2^{-}\to 15/2^{-}$ transition, $E_{\gamma }$ $=$ 189.6 keV, with $I_{\gamma }=100$; the calculated yields were derived from the Winther–de Boer code folding in the decay properties of the band as explained in the text and are also normalized to the $19/2^{-}\to 15/2^{-}$ transition. The excited bands were treated differently: in these cases the total yield of the state obtained by summing the intensities of all $\gamma$-ray transitions depopulating the state is shown and compared with the same calculated quantity. Note that the [734]9/2 Nilsson band is crossed by the $K=11/2$ $\gamma$-vibrational band at spin 17/2.

Figure 13

Measured and calculated Coulomb excitation yields for some positive-parity bands observed in ${}^{235}$U in the CHICO experiments. The $E2$ matrix elements in the ground-state band (GSB) and in the positive-parity bands were taken from Eq. (5) with $Q_0=9.75e$b. The $E3$ matrix elements linking the ground-state band with the excited bands were taken from Eq. (21). The parameter $Q_t$ was varied for each band to give a best fit to the relative yield. The measured yields in the ground-state band and in the [631]1/2 and [622]5/2 bands are the directly observed $E2$ $\gamma$-ray intensities normalized to the $19/2^{-}\to 15/2^{-}$ transition, $E_{\gamma }$ $=$ 189.6 keV, with $I_{\gamma }=100$; the calculated yields were derived from the Winther–de Boer code folding in the decay properties of the bands as explained in the text and are also normalized to the $19/2^{-}\to 15/2^{-}$ transition. The [624]7/2 and [631]3/2 bands were treated differently: in these cases the total yield of the state obtained by summing the intensities of all $\gamma$-ray transitions depopulating the state is shown and compared with the same calculated quantity.

Figure 14

Measured and calculated Coulomb excitation yields for some positive-parity bands observed in ${}^{235}$U. The $E2$ matrix elements in the ground-state band (GSB) and in the positive-parity bands were taken from Eq. (5) with $Q_o=9.75e$b. The $E3$ -matrix elements linking the ground-state band with the excited bands were taken from Eq. (21). The parameter $Q_t$ was varied for each band to give a best fit to the relative yield. The measured yields in the ground-state band are the directly observed $E2$ $\gamma$-ray intensities normalized to the $19/2^{-}\to 15/2^{-}$ transition, $E_{\gamma }$ $=$ 189.6 keV, with $I_{\gamma }=100$; the calculated yields were derived from the Winther–de Boer code folding in the decay properties of the band, as explained in the text. For the positive-parity bands the total yield of the state obtained by summing the intensities of all $\gamma$-ray transitions depopulating the state is shown and compared with the same calculated quantity.

Figure 15

The rotational constant calculated for members of the $j_{15/2}$ multiplet with the parameters of Ref. [1]. The filled points were known and fitted in Ref. [1]; open points are from the present data. Note that the [734]9/2 Nilsson band is crossed by the $K=11/2$ $\gamma$-vibrational band at spin 17/2.

Figure 16

The crossing of the Nilsson [734]9/2 band with the $K=11/2$ $\gamma$-vibrational band at spin 17/2. The reference energy $E=5.5J(J+1)$ keV has been subtracted from the experimental values in order to make the crossing easier to visualize. We assume that the unperturbed 17/2 levels are essentially degenerate, in which case the interaction energy at spin 17/2 is one half of the observed separation energy. The dashed lines sketch a plausible locus for the unperturbed bands in the region of the crossing.

Figure 17

The rotational constant, $A$, vs $J^2$ for the $K=3/2$ $\gamma$-vibrational band at 638 keV. The linear relationship, with nearly equal slopes and a common intercept, is typical of a $K=3/2$ band perturbed by a $K=1/2$ band through a Coriolis interaction.

Figure 18

The energies of low-spin states in the $K=3/2$ $\gamma$-vibrational band and the Nilsson [752]5/2 band after subtraction of a rigid-rotor reference energy of 5.37 $J(J+1)$ keV. In the text we discuss the possible deviations from systematic behavior that might give some clues to the interactions between the bands. We also note the remarkable isospectral character between corresponding signatures, which is at the level of about 1 to 2 keV over this low-spin range.

Figure 19

The lowest-order diagrams for the quasiparticle-vibration coupling for octupole phonons. The quasiparticle energy is modified by the phonon coupling.

Figure 20

Partial level scheme based on $\gamma -\gamma$ coincidences from the 8Pi and Gammasphere stand-alone experiments showing transitions within the [743]7/2 ground-state band (Tables  and ), within the [631]1/2 band (Table ), within the [631]3/2 band and its decay to the [631]1/2 band (Table ), and within the $K=3/2$ $\gamma$-vibrational band and its decay to both the [631]1/2 band and to the ground-state band (Table ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $21/2^{-}\to 17/2^{-}$ transition of the ground-state band.

Figure 21

Partial level scheme based on $\gamma -\gamma$ coincidences from the 8Pi and Gammasphere stand-alone experiments showing transitions within the [743]7/2 ground-state band (Tables  and ), within the [622]5/2 band and its decay to the ground-state band (Tables  and ), and within the [734]9/2 band and its decay to the ground-state band (Table ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $21/2^{-}\to 17/2^{-}$ transition of the ground-state band.

Figure 22

Partial level scheme based on $\gamma -\gamma$ coincidences from the 8Pi and Gammasphere stand-alone experiments showing transitions within the [743]7/2 ground-state band (Tables  and ), within the [752]5/2 band and its decay to the ground-state band (Tables  and ), and the decay from the $K=11/2$ $\gamma$-vibrational band to the ground-state band (Tables  and ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $21/2^{-}\to 17/2^{-}$ transition of the ground-state band.

Figure 23

Partial level scheme based on the Gammasphere and CHICO data showing transitions within the [631]1/2 band (Table ), the decay from the $K=3/2$ $\gamma$-vibrational band to the [631]1/2 band (Table ), the decay from the [631]3/2 to the [631]1/2 band (Table ), and the decay from the [624]7/2 to the [631]1/2 band (Table ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $19/2^{-}\to 15/2^{-}$ transition of the ground-state band.

Figure 24

Partial level scheme based on the Gammasphere and CHICO data showing transitions within the [622]5/2 band (Table ), the decay from the [633]5/2 band to the [622]5/2 band (Table ), and the decay from the [624]7/2 band to the [622]5/2 band (Table ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $19/2^{-}\to 15/2^{-}$ transition of the ground-state band.

Figure 25

Partial level scheme based on the Gammasphere and CHICO data showing transitions within the [743]7/2 ground-state band (Table ), the decay from the [633]5/2 band to the ground-state band (Table ), the decay from the [624]7/2 band to the ground-state band (Table ), the decay from the [615]9/2 band to both the [624]7/2 band and the ground-state band (Table ), the decay from the [613]7/2 band to both the [624]7/2 band and the ground-state band (Table ), and the decay from the presumed $\beta$-vibrational band to both the [624]7/2 band and the ground-state band (Table ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $19/2^{-}\to 15/2^{-}$ transition of the ground-state band.

Figure 26

Partial level scheme based on the Gammasphere and CHICO data showing transitions within the [743]7/2 ground-state band (Table ), the decay from the $K=3/2$ $\gamma$-vibrational band to the ground-state band (Table ), and the decay from the $K=11/2$ $\gamma$-vibrational band to the ground-state band (Table ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $19/2^{-}\to 15/2^{-}$ transition of the ground-state band.

Figure 27

Partial level scheme based on the Gammasphere and CHICO data showing transitions within the [743]7/2 ground-state band (Table ), the decay from the [752]5/2 to the ground-state band (Table ), and the decay from the [734]9/2 to the ground-state band (Table ). The width of the arrows is proportional to the observed intensity of the transitions and was normalized to the $19/2^{-}\to 15/2^{-}$ transition of the ground-state band.