Discovery of collective states in 208 Pb by complete spectroscopy

. Most neutron and proton bound states in 208 Pb are described in the shell model as one-particle one-hole conﬁgurations. Besides precise data obtained with the Q3D magnetic spectrograph of the MLL (Garching, Germany) an important reaction is the inelastic proton scattering via isobaric analog resonances in 209 Bi. It yields amplitudes of neutron one-particle one-hole conﬁgurations with relative signs in each state of 208 Pb. The orthogonality, normality, and sum rule relations allow to investigate the completeness of the transformation matrices of one-particle one-hole conﬁgurations describing the states in 208 Pb with spins from 0 − to 14 − and 0 + to 12 + . By this method amplitudes of unobservable one-particle one-hole conﬁgurations can be determined. The comparison of spin, parity, and dominant particle-hole components thus derived in up to 30 states of a certain spin to shell model calculations allows to identify states described as collective excitations of the entire nucleus.


Introduction
208 Pb is the heaviest nucleus where more than 500 states are rather well known [1]. The excitation energies of most neutron bound states (S (n) = 7368 keV) and part of the proton bound states (S (n) = 8004 keV) are known with an uncertainty of less than 1 keV. Most states are one-particle one-hole (1p1h) configurations but three dozen states consist definitely of other configurations [2][3][4][5].
Knowledge of spin, parity, and dominant configuration up to a certain excitation energy defines complete spectroscopy. It is rather good for states at E x < 7.2 MeV. States at 6.2 < E x < 7.2 MeV [6,7] are being analyzed from reconstructed data presented in [8]. States at E x < 6.2 MeV are completely known with spin, parity, and dominant configuration [3]. Complete spectroscopy allows to find collective states in the heavy nucleus 208 Pb not described as 1p1h configurations.

Experimental observations
Spectroscopy of 208 Pb started immediately after the discovery of radioactivity. Rather soon the high excitation energy of the lowest state in 208 Pb was recognized. The energy of two and a half MeV is the almost the highest across the whole nuclear chart. It is now known with a precision of 10 eV [1]. In 1954 the spin of the lowest state was determined as 3 − [9] which is again rather unique across the whole nuclear chart.
netic spectrograph (Fig. 2) at the Maier-Leibnitz Laboratorium (MLL) at Garching (Germany) higher excited states (4.0 < E x < 8.0 MeV) could be studied. The resolution of 3 keV is barely sufficient to resolve the states in 208 Pb with a mean distance of 10 keV. The resolution is limited to 3 keV because the M-electrons in lead have such binding energies. L-electrons and K-electrons with binding energies of 15 and 88 keV produce annoying satellites to each peak.
Using the Q3D magnetic spectrograph from 2003 until now most negative parity states at E x < 7.2 MeV and nearly all states at E x < 6.2 MeV became known. Five states were described as neutron and proton vibrations [2]. Ten states were recognized as tetrahedral rotations and vibrations [4]. Two dozen states were described by the coupling of 1p1h configurations to the 3 − yrast state [5].
The question how to discern other collective states in 208 Pb from 1p1h states can be only solved by complete spectroscopy. It means that up to a certain limit in excitation energy all states must be found and for each of them spin, partity and dominant structure must be determined.

Complete spectroscopy
Complete spectroscopy is approached by studying several different reactions, especially 208 Pb(p, p ′ ), 208 Pb(d, d ′ ), 208 Pb(α, α ′ ), 207 Pb(d, p) and 209 Bi(d, 3 He). However, the most important reaction is the inelastic proton scattering via IARs in 209 Bi. It allows to determine amplitudes with relative signs of all neutron one-particle one-hole configurations in each state.
In 208 Pb there are 44 excess neutrons. By adding one proton an IAR in 209 Bi is created with a typical energy of 15-20 MeV and a width of about 300 keV. In the subsequent proton decay all neutron particle-hole configurations in each state of 208 Pb are excited. The decay protons act in a coherent manner to create interference patterns in the angular distributions (Fig. 1).
The knowledge of some amplitudes with relative signs allows to investigate the orthogonality relations among the states. Together with the normality and sum rule relations amplitudes of unobservable configurations can be determined. They are unobservable either because the cross section is too low or because there is no target as is the case for proton particle-hole configurations built with a particle in a higher orbit.  2 MeV to shell model calculations. One calculation (SDI) uses the surface delta interaction [12], another one (M3Y) the Michigan 3-Yukawa interaction [13]. The SDI calculates the multipole splitting and assumes no further residual interaction, the M3Y employs the mixing among 1p1h, 2p2h, and 3p3h configurations.  Both by SDI and M3Y describe the excitation energies for most states with spins 3 − and 5 − with a mean deviation of 30 keV, also for most states with spins 0 − , 1 − , 2 − , 4 − , 6 − , 7 − , 8 − , 12 − , 13 − , 14 − , and 3 + , 5 + , 7 + , 8 + , 9 + , 10 + , 11 + , and the 12 + yrast state. Fig. 5 shows level schemes for the 6 + states at E x < 6.2 MeV and for the 12 + states at E x < 10 MeV (see also Figs. 5 and 9 in [3]). Fig. 3 in [5] shows the level scheme for all positive parity states at E x < 6.5 MeV, see also [14]. The 6 + yrast and 12 + yrare states evidently appear in addition to 1p1h configurations. Figure 5. Comparison of level schemes for the 6 + and 12 + states at E x < 6.2 MeV to shell model calculations with SDI [3] and M3Y [13]. (The energies are shown relative to the yrast state.) The 6 + yrast and 12 + yrare states are not explained by any existing model. Two 6 + states at E x ≈ 6.0 MeV (1.6 MeV above the yrast state) are described by the coupling of g 9/2 p 1/2 to the 3 − state. 2p2h configurations with spin 12 + are predicted more than 1.5 MeV above the yrast state (wavy lines).

Summary
In the heavy nucleus 208 Pb several different classes of excitations are observed, -overwhelmingly 1p1h configurations, -two dozen states with 1p1h configurations coupled to the 3 − yrast state, -five neutron and proton pairing vibrations, -ten members of tetrahedral rotating and vibrating bands, -the 6 + yrast and 12 + yrare states unexplained, -one more 3 − state and two more 5 − states in the region at 5.2 < E x < 6.05 MeV not explained by any existing model.
Still a few positive parity states below E x = 6.2 MeV are not yet identified and above E x = 6.2 MeV most positive parity states are unknown. A mysterious question is about a 2 + state near the ground state predicted as the coupling of both intruders j 15/2 and i 13/2 to the 3 − yrast state [5].

Outlook
Urgently needed is the theoretical description of dodecahedral and icosahedral configurations similar to the tetrahedral configurations by some algebraic cluster model.
On the experimental side the neutron capture on 207 Pb and the investigation of the subsequent γ-cascade is needed. Modern equipment exists but studying γtransitions among the more than 500 states needs a great effort.