Introduction

It has been suggested on the basis of the realistic nuclear mean-field calculations, Ref. [1], that there should exist atomic nuclei whose shapes are tetrahedral-symmetric. Theoretical calculations, Ref. [2] and references therein, suggest that to a first approximation, the nuclei whose shapes are characterized by the exact tetrahedral symmetry have vanishing multipole moments except for the $Q_{3\pm2}$ one, the next order multipoles allowed by the tetrahedral symmetry are $Q_{7\pm2}$ and $Q_{7\pm6}$ and such contributions are expected to be very small if not totally neglegible in the nucleus studied. Thus, unlike in rotational bands of quadrupole-deformed nuclei where the E2 transitions dominate, in tetrahedral bands the E2 transitions are predicted to vanish or to be very weak, because the quadrupole moments go to zero when the tetrahedral symmetry becomes exact. According to ENSDF, Ref. [3], the nucleus $^{156}$Gd has been studied in over 15 different excitation modes with varying target-beam combinations, beam energies, and detection systems. Although a regular sequence of odd-spin negative-parity states has been established down to $I^\pi=3^-$, the intra-band E2 transitions below the $I^\pi=9^{-}$ state have never been seen. The energies of the corresponding states have been measured exclusively through the inter-band E1 transitions to the ground-state band. Such a behavior is expected to be a consequence of tetrahedral symmetry [2]. Already in the early eighties, Konijn and co-workers, Ref. [4], carried out an experiment using an $\alpha$-particle beam and measured the ratios of the reduced transition strengths, $B$(E2)/$B$(E1), for two negative-parity bands in $^{156}$Gd - at that time interpreted as octupole vibrational bands. The $B$(E2)/$B$(E1) ratios were found to be about a factor 50 lower in the odd-spin, as compared to those in the even-spin negative-parity bands. More recently, Sugawara, Ref. [5], measured the branching ratios of these two negative-parity bands by using the reaction $^{150}$Nd($^{13}$C,$\alpha$3n). In the case of the odd-spin negative-parity bands, a minimum in the $B$(E2)/$B$(E1) ratios at intermediate spins was reported and some upper limits of branching ratios at low spins were measured. These measurements have been carried out at best by using the $\gamma-\gamma$ coincidences with a population of $^{156}$Gd that may not have been enough to observe the low-intensity transitions. The main goal of this experiment was to search for the E2 transitions forbidden by the tetrahedral symmetry with high statistics, to determine the $B$(E2)/$B$(E1) ratios, and to search for any signs of cross-feeding involving the odd-spin negative-parity band.