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The description of nuclear states by shell model particle-hole configurations in the lead region needs the inclusion of collective excitations at already very low excitation energies. For the two isotopes 206Pb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{206}\mathrm{Pb}$$\end{document} and 208Pb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{208}\mathrm{Pb}$$\end{document} a rather good agreement of excitation energies and configuration mixing is observed for states at 3.7<Ex<4.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<E_x< 4.7$$\end{document} MeV.

The increasing fragmentation of nuclear matter starts with the breakup of two fragments in 208 Pb at excitation energies of 3.2 MeV and at 0 < E x ≤ 1.8 MeV for 206 Pb, 210 Po, 206 Tl, 210 Bi. The fragmentation into three parts starts at E x = 5.8 MeV and with four parts at E x ≈ 10 MeV. Fragmentation with more than four parts near A = 208 is not yet recognized. An open question is the starting energy for fission and ternary fission.
Pairing vibration starts with excitation energies of 4.8 MeV for the doubly magic nucleus [47,48] and 0.2 MeV for semi-magic nuclei [48]. It became known with experiments studying triton beams and enumerate to a handful.
Collective excitations with spin 2 + start to appear for the doubly magic nucleus 208 Pb at E x = 4.1 MeV and semimagic nuclei at E x = 0.2 MeV. Collective excitations with spin 1 − , 2 − , 4 + , 6 + , 12 + start to appear at higher excitation energies, but still less than 6 MeV.
Coupling collective excitations to particle-hole configurations starts at excitation energies of E x = 5.8 MeV [17]. Coupling two different modes of collective excitations to particle-hole configurations starts at E x ≈ 10 MeV [18].
The splitting of particle-hole multiplets amounts up to one MeV in many cases [5,17,69,70]. A refinement of the calculations is provided by including the surface δ interaction (SDI) [4,5,71]. The residual interaction in 208 Pb is thus reduced to typically 50 keV [16].
Many nuclei in the lead region are well described by the coupling of two nucleons. Few states are described with difficulty. Especially higher lying states in 206 Pb are not yet understood. A new theory based on SDI describes particlehole states in 206 Pb [72] but is not yet used. The concept of generalized neutron particle-hole (GNPH) configurations was introduced instead to describe states in the N = 82 region [12][13][14][15]36]. Levels described as GNPH configurations in 206 Pb resemble one-particle three-hole configurations but also contain the coupling of particle-hole configurations to collective states.
Two amplitudes with relative sign were first determined for the 4 − yrast state by Bondorf [35] in 1968. Spin and configuration mixing for a dozen GNPH states in N = 82 isotones were determined [13,14,36] in 1969. Several spins and the configuration mixing for negative parity states in 208 Pb were determined [21,22,24,26] in 1969-1973. A new method described in [46] allows to determine up to five amplitudes of neutron configurations in each state. Angular distributions for 206 Pb(p, p ) and 208 Pb(p, p ) were thus studied.
In 208 Pb two dozen particle-hole states with excitation energies 3.2 < E x < 4.7 MeV were first identified in 1969-1973 [24]. States with spins from 2 − to 7 − at E x < 4.7 MeV were described by rather complete orthonormal transformation matrices for two dozen states to configurations. A few spin assignments derived in 1973 needed to be exchanged. The reason for the exchanges in 1982 was the unknown role of the Coulomb interaction for the proton particle-hole configurations [74,75].
Experiments with the 209 Bi(d, 3 He) reaction revealed the correct spin assignments and state identifications [76,77]. The wave functions deduced from γ -spectroscopy in 1999 [78] agree with the results from 1982. Amplitudes for positive parity configurations determined in 2010 [1] were found to agree similarily.
All states with spins from 1 − to 7 − in 208 Pb at E x < 4.7 MeV were identified in 1982. (The 8 − yrast state was identified in 2006 [61]. The new analysis of the 1 − yrast state was included in 2020 [67].) The exception is the collective state at E x = 4.14 MeV recognized by Glöckner in 1972 [28,79] but identified as the 2 − yrast state only in 2017 [51].
Amplitudes of 1p1h configurations in 208 Pb were determined with dominant g 9/2 p 1/2 , g 9/2 p 3/2 and g 9/2 f 5/2 components and admixtures of g 9/2 f 7/2 . In addition admixtures from 1p1h configurations with other particles than g 9/2 were determined. The results obtained in 1982 provided by Table 4 in [51] are sufficient for the comparison to the isotope 206 Pb.
A refined analysis including admixtures from g 9/2 h 9/2 is still awaited.
The nearly exhaustive identification of states in 208 Pb at E x < 6.2 MeV [16] provides a basis for the comparison of states in 206 Tl, 210 Bi, and 206 Pb. The comparison of three dozen states in 206 Pb and two dozen states in 208 Pb reveals the strength distribution of particle-hole configurations in 208 Pb and GNPH configurations in 206 Pb at 3.7 < E x < 4.7 MeV to be similar in a remarkable manner. Specific differences are related to two neutrons missing from the doubly magic nucleus 208 Pb.
Section 2 discusses the shell model description in the lead region. Section 3 reminds to theoretical descriptions used for the analysis of 1p1h states at E x < 7.2 MeV in 208 Pb [46] and extends them to analyze GNPH states in 206 Pb at 3.7 < E x < 4.7 MeV. Section 4 shortly describes experimental data. Section 5 presents methods to identify states, assign spin and parity, and to determine amplitudes of particle-hole configurations. Section 6 discusses the structure differences of states in 206 Pb and 208 Pb at 3.7 < E x < 4.7 MeV.

Description of nuclei in the lead region by the shell model
Bromley and Weneser [80] pointed out that several nuclei in the lead region are extremely well described by the shell model. The comparison of particle-hole states in 206 Pb and 208 Pb allows to query this statement in detail. The strong binding of nucleon pairs produces low lying 0 + and 2 + states in each even-even nucleus. They are well described by the pairing vibrational model [47,53,81,82] introduced by Bohr and Mottelson [48].
Excitation energies of 1p1h configurations in 208 Pb are described with similar precision as calculations with realistic forces [9][10][11]. In 208 Pb the SSM explains most low lying 1p1h states already with good reliability [1].

Composition of the ground state in 208 Pb
Weak admixtures of the three lowest excited 0 + states to the g.s. may be assumed. The 4868 0 + state in 208 Pb became known as neutron pairing vibrational configuration from (t, p) and (p, t) experiments [47,48]. The 5666 0 + state was recognized as proton vibrational configuration [53,68]. The 5241 0 + state was interpreted as tetrahedral vibration [51]. An interpretation as phonon excitation was given in 1968 [48,56] and in 2000 [10,11].
The weakness of admixtures to the g.s. in the doubly magic nucleus can be deduced from the interference pattern observed in the inelastic proton scattering via IARs observed at 19 < E p < 21 MeV [83] and described by the coupling of the g 9/2 particle to the three lowest excited 0 + states in 208 Pb. The resonance energies in 209 Bi are E res = 19.8, 20.2, 20.6 MeV.
The excitation functions of 208 Pb(p, p ) for the 5 − yrast and 5 − yrare states indicate an enhancement near these three proton energies for the single scattering angle Θ = 165 • used in the 208 Pb(p, p ) experiment ( Fig. 2 in [83]). The structure of the 5 − yrast and 5 − yrare states is precisely described by orthonormal transformation matrices with rank 9 ( Table  4 in [51]). The logarithmic dependence of the s.p. width on the proton energy [39] (Fig. 8 in [29]) enhances the cross section for the excitation of g 9/2 l j for l2 j = p1, p3, f 5 by a big factor with the differences in proton energy of 5-6 MeV. The analysis of the data provided by Fig. 2 in [83] yields an estimate for admixtures of the three lowest excited 0 + states to the g.s. Less than 5% admixture are deduced. The absence of a resonant behaviour for the 4 − yrast state is explained by the different composition [24,51] and the different energy dependence of the s.p. widths [29].
Interference patterns near the same proton energies are observed in the excitation of the 3 − yrast state (lower frame of Fig. 1 [33].
Excitation functions for 14 < E p < 20 MeV in 205 Tl are similar [84]. The coupling of the s 1/2 particle to the 3 − yrast state is explained by the weak coupling model [2]. A ratio 6 : 8 of the cross sections is observed [84].
The additional pattern in the excitation of the 3 − yrast state observed near E p = 19.0 MeV is interpreted by the coupling of the 4086 2 + yrast state to the g 9/2 particle. (The resonance energy in 209 Bi would be E res = 19.01 MeV.) It is interpreted by the primary excitation of the j 15/2 ⊗|3 − 1 component in the 9/2 + IAR followed by the proton decay g 9/2 ⊗|2 + 1 → g 9/2 + |3 − 1 . In the elastic scattering the raise and steep decrease across the resonance observed for the g 9/2 resonance at E p = 14.92 MeV [85] is visibly repeated near E p = 19.0 MeV (upper frame of Fig. 1 in [83]). It is typical for a resonance with spin J = L + 1 2 [86,87].

Two-hole configurations in 206 Pb
The lowest states in 206 Pb at E x < 3.3 MeV (Table 4) can be explained by the coupling of two of the first three neutron holes (p 1/2 , f 5/2 , p 3/2 ) to the 0 + g.s.
First calculations by True and Ford [91] indicated admixtures of about 15% both of f 5/2 −2 and p 3/2 −2 to the dominant configuration p 1/2 −2 in the g.s. of 206 Pb. Most states at E x < ∼ 3.5 MeV contain more than 80% of a single two-hole configuration. Calculations with the pairing model yielded similar wave functions [82].
Two nucleon configurations in the odd-odd 208 Bi studied by Maier et al. [45] and in 210 Bi studied by Cieplicka-Oryńczak et al. [8] revealed a rather simple structure for many multiplets at E x < 1.7 MeV. An essay to describe the multiplet splitting of the lowest configurations by the SDI succeeded mostly rather well but fails for the two lowest states in 210 Bi. The inversion of the 0 − and 1 − yrast states was understood by the strong core-polarization [7].
Multiplets in the odd-odd 206 Tl studied at E x < 1.6 MeV revealed a large configuration mixing [9]. One reason is the small separation between the active orbitals.
Finally the unusually long lifetime of the 10 + yrast state with 0.5µs and the large γ -transition strength from the 10 + yrast to the 7 − yrast state together with the incomplete g 9/2 f 5/2 strength in the 4037 7 − state (with only about 70%, see Table If in [24]) may be related to the observation of three non-1p1h configurations with spin 3 − , 5 − and 2 − or 5 − at 5.3 < E x < 6.0 MeV. The 5318 3 − and two 5 − states (5705, 5993) are identified by complete spectroscopy as configurations of unknown type [93]. The 5993 state may have spin 2 − [46] and not 5 − [16].

Collective excitations of the whole nucleus
The 3 − yrast state in 208 Pb is rather unique among the whole nuclear chart. In a doubly magic nucleus another 3 − state as the lowest excited state is known only in 146 Gd [98]. Three low-lying states were identified in the 1920's [99,100]. The spins were determined in 1954 by the study of angular correlations. By assuming the g.s. to have spin 0 + , the lowest state in 208 Pb was assigned spin 3 − [49].
The high energy of the 3 − yrast state was already noticed by Rutherford with the penetration of radioactivity for Thorium material being a factor 3 1 2 stronger than for Uranium material ( Fig. 2 in [101]). In the 1920's the excitation energy of 2.6 MeV for Thorium material was used as an etalon [102]. The energy calibration for the 2.6 MeV 3 − yrast state changed by 8 keV after 1969 [21,103]. Hence all energies determined by Moore et al. [34] had to be adjusted [63,64], see Eq. (B.1) in [29]. The energy of the 2.6 MeV 3 − yrast state is now known with an uncertainy of 10 eV [23].
The 3 − yrast state in 208 Pb is widely considered as an octupole phonon vibration of the entire 208 Pb nucleus [56]. Besides the states in 208 Pb and 206 Pb other nuclei in the lead region are described with octupole phonons, too. The excitation energies with E x = 2.6 MeV are similarily large for most nuclei in the lead region within 0. An alternate interpretation as a tetrahedral rotor was proposed [51]. Besides the knowledge of the charge radius for states in the lead region the algebraic cluster model [51,105] needs no further parameters for an explanation of the large value BE(E3). The BE(E1) value for the 4.97 MeV 1 − state in 206 Pb [41] and for the 4841 1 − state in 208 Pb [23] is extremely small. The relation P 1 − 1 2 = 0 with the Legendre polynomial P 1 explains the vanishing strength [51,105].
In 206 Pb and 208 Pb the reduced transition strengths BE(Eλ) for the 4MeV 2 + and 4 + states [23,41] can be explained by tetrahedral configurations similarily.
With the 4 + yrast states in 206,208 Pb [23,41] and the 4113 state in 204 Hg [50] the second member of the assumed rotational vibrational g.s. band may be identified at E x = 4.1 MeV. Because of the poor knowledge of spin, parity and structure no other state in the lead region is identified as the tetrahedral member [88]. Solely in 208 Pb ten tetrahedral states at 2 < E x < 8 MeV were identified (Sect. 2.6).

States coupled to non-1p1h configurations
The g 9/2 state in 209 Pb is not pure (Sect. 2.2). Hamamoto and Siemens studied the coupling of the j 15/2 and g 9/2 particle to the 3 − yrast state interpreted as octupole phonon [56]. (The similar coupling of the f 7/2 nucleon to the 3 − yrast state in the doubly magic 146 Gd was studied by Kleinheinz et al. [106].) Two dozen states were found in 209 Pb by the 207 Pb(t, p) 209 Pb reaction with E t = 20 MeV [107]. Some of them may be described by the coupling of the g 9/2 particle to 3 − yrast state.
The coupling of particles to the 3 − yrast state in 208 Pb was studied by Rejmund et al. [108] for 207 Pb, 209 Pb, 207 Tl. They found out that the coupling is pronounced if two orbitals satisfy the Δj ≡ Δl ≡ 3 rule. This is the case for g 9/2 and j 15/2 in 209 Pb, f 7/2 and i 13/2 in 207 Pb, d 5/2 and h 11/2 in 207 Tl. Two states in 205 Tl are described by the coupling of the s 1/2 particle to the 3 − yrast state [84]. States in 207 Tl were studied through β-decay [109]. An experiment at the ISOLDE Decay Station observed the population of a 17/2 + state in 207 Tl at E x = 3813 keV starting from E x = 7.0 MeV [110]. The γtransition from the 3813 17/2 + state to the 1348 11/2 − state is determined as E3. The 3813 17/2 + state may be described by the coupling of the h 11/2 proton to the 3 − yrast state in 208 Pb.
Two-neutron states in 210 Pb and the coupling to the collective 3 − state was studied by Broda et al. [111]. The coupling of a nucleon to the 3 − yrast state in 206 Hg, 206,207 Tl,206,207,208,209 Pb, 209 Bi, 210 Po was studied by Broda et al. [112].
High-spin states up to 17 MeV in 208 Pb were studied with deep inelastic scattering by Broda et al. [11]. The γ -transitions end intermediately in the 9091 17 + state which transits by E3 to the 6744 14 − state described by j 15/2 i 13/2 [113]. The spin of the 14 − state is confirmed [11,96]. The 9091 17 + state may be described by the coupling of the stretched configuration j 15/2 i 13/2 to the 3 − yrast state. The coupling of tetrahedral and other collective configurations to 1p1h states may explain most states populating the 9091 17 + state at E x < 17 MeV [18].
Two dozen 1p1h configurations coupled to the 3 − yrast state in the doubly magic nucleus in 208 Pb were identified [17]. Most of them have positive parity.

Two-nucleon states in the lead region
With the experiments performed in 1965-1969 at the MPIK on the inelastic proton scattering the comparison of particlehole configurations in two heavy nuclei 206 Pb and 208 Pb can be achieved. Whereas about 250 states in 208 Pb are well described by the SSM (Sect. 3.1) no particle-hole states were known in 206 Pb before [41].
Methods for the study of inelastic proton scattering via an IAR in the doubly magic nucleus 208 Pb (Sect. 3.1) and in 206 Pb where two neutrons are missing from the doubly magic nucleus 208 Pb (Sect. 3.2) allow to find spin, parity, and structure of particle-hole states. Inelastic proton scattering via an IAR is equivalent to a neutron pickup reaction on the g.s. or an excited state in the parent nucleus [40]. For 206 Pb(p, p ) the parent states are in 207 Pb, for 208 Pb(p, p ) the parent states are in 209 Pb. Thirty-two particle-hole states in 206 Pb are identified at 3.7 < E x < 4.7 MeV through the 206 Pb(p, p ) experiment performed in 1969 at the MPIK [20]. In the same region 3.7 < E x < 4.7 MeV twenty-three 1p1h states were identified in 208 Pb. Thus 60% more states in 206 Pb are identified and indications for possibly twice the number of states are given. Results from the prior 208 Pb(p, p ) experiment performed for low lying particle-hole states in 208 Pb discussed in 1973 [24] and after 1982 are refined and extended. Yet the results shown in Table 4 in [51] suffice for this work.
The lowest states in 208 Pb and 206 Pb 3.7 < E x < 4.7 MeV allow to discuss comparable shell model configurations in the lead region (Sect. 2) in a quantitative manner (Sect. 6.4).

Description of 1p1h states in 208 Pb
Most states in 208 Pb are described as (1p1h) configurations HereẼ x denotes the state in a unique manner by the known excitation energy rounded to 1-2 keV. I π is spin and parity. L , l are the angular momenta and J, j spin of particle and hole, respectively. Other than 1p1h configurations are discussed in Sect. 2.6.
After the first attempts of an analysis of the resonant 208 Pb(p, p ) reaction [24] a thorough analysis of the states with dominant 1p1h configurations involving the g 9/2 particle was not further pursued. Complementary data obtained in the USA were not used [115,116]. Some of them were discussed later [29]. Table 4 in [51] yields results from an update done in 1982 and slightly improved in 2017. These data are used by this work in the comparison to 206 Pb. Essentially, similar wave functions (including signs of amplitudes) were obtained in 1999 [78].

Particle-hole states in 206 Pb
The lowest states in 206 Pb at E x < 3.5 MeV ( The proton decay of the g 9/2 IAR in 207 Bi strongly excites two dozen states in 206 Pb at 3.7 < E x < 4.7 MeV. The total mean cross section of two dozen states is 3 mb/sr. The value equals the total mean cross section found for the proton decay of the g 9/2 IAR in 209 Bi into the states in 208 Pb in the same range of excitation energies (Table 6). A reduction factor of 0.80±0.02 has to be included [29]. (Note that the correlation of configuration strength with the cross section is strongly distorted by the logarithmic dependence of the s.p. widths on the angular momentum and the bombarding energy [29,39].) The number of states in 206 Pb is about twice the number of states in 208 Pb. The number may be even higher because of the insufficient resolution of about 15 keV [20]. The mean spacing of states is 9 keV in 208 Pb [16] and estimated with about 4 keV in 206 Pb.
Twenty-two levels are observed in 206 Pb (Tables 4, 5, 8). The analysis of angular distributions excited by the proton decay of the g 9/2 IAR in 207 Bi identified thirty-two states in 206 Pb. In the following we denote a state by the excitation energy E x varied within 4 keV 1 from the value E x given by [20] for the level.

Generalized neutron particle-hole configurations
The concept of GNPH configurations was introduced by studying inelastic proton scattering via an IAR in 141 Pr [12][13][14]. It explains several states in the N = 82 isotones 136 Xe, 138 Ba, 140 Ce, 142 Ne, 144 Sm [117]. The method of studying (p, p ) via an IAR allowed to determine spin, parity, and structure of states [13,14]. A theory explained the GNPH configurations by coupling a collective state to 1p1h configurations [15].
The model is used to explain the states observed by Solf et al. [19,20,73]. Negative parity states at 3.7 < E x < 4.7 MeV are described by the coupling of 1p1h configurations to the 0 + g.s. and the 2 + yrast state as Here other configurations denoted as other I − i comprise especially the proton 1p1h configurations h 9/2 s 1/2 ⊗ p 1/2 −2 and h 9/2 d 3/2 ⊗ p 1/2 −2 . Only the configurations described by L J = g 9/2 are exploited in this work. Another theory applying the surface δ interaction (SDI) is in preparation [72].

Orthonormality and sum-rule relations and center of gravity
The amplitudes c Here L J describes a GNPH configuration with the parameter α [Eq. (2)]. The centroid energy is obtained as

Number of states excited by 206 Pb(p, p )
The proton decay of the g 9/2 IAR in 207 Bi is expected to populate negative parity GNPH states in 206 Pb at excitation energies 3 < E x < 5 MeV. The g.s. with dominant structure p 1/2 −2 is assumed to contain admixtures of configurations l j −2 l j −2 . The coupling of the 0 + g.s. to the 803 2 + yrast and g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 p 3/2 1p1h configurations is interpreted as GNPH configurations.
The GNPH configurations are expected with four states at E x = 4.4 MeV with dominant g 9/2 p 3/2 and six states at E x = 4.0 MeV with dominant g 9/2 f 5/2 strength in correspondence to 1p1h configurations in 208 Pb. In addition ten states with spins from 2 − to 7 − at E x = 4.3 MeV and structure g 9/2 p 1/2 ⊗ 803 2 + 1 are expected ( Table 2). Thirty more states with spins from 0 − to 8 − structure (g 9/2 f 5/2 ⊗ 803 2 + 1 at E x = 4.8 MeV and twenty more states with spins from 1 − to 8 − with the structure g 9/2 p 3/2 ⊗ 803 2 + 1 at E x ≈ 5.1 MeV are predicted. Therefore in total about twenty states with a g 9/2 particle are expected in the region 4.0 < E x < 4.5 MeV for 206 Pbtwice the number as for 208 Pb. (Here the isospin is not considered.) The mixing with other configurations not containing the g 9/2 particle (i 11/2 l j, d 5/2 l j and the proton configurations h 9/2 s 1/2 , h 9/2 d 3/2 ) increases the number of GNPH configurations.

States resonantly excited on the g 9/2 IAR
Sixteen negative parity states exist at 3.9 < E x < 4.5 MeV in 208 Pb ( Table 4 in [51]). Among them there are six states with dominant proton configurations h 9/2 s 1/2 , h 9/2 d 3/2 and two states with i 11/2 p 1/2 . Solf et al. [20] observe 27 levels at 3.7 < E x < 4.7 MeV in 206 Pb resonantly excited on the g 9/2 IAR. Four levels resonantly excited by 206 Pb(p, p ) were observed with low resolution at Θ = 90 • [118]. The resolution in the experiment performed at the MPIK was 13-15 keV. The large ratio R of the on-to-off resonance cross sections at 3.7 < E x < 4.7 MeV proofs the presence of several unresolved states (Fig. 10).
The mean distance between any two states in 208 Pb is 9 keV [16]. The number of states in 206 Pb is certainly larger because of the two missing neutrons (Sect. 3.2.3). Therefore within 15 keV often more than one state is concealed. The result that 32 states are discerned in 22 observed levels [20] can be thus understood (Tables 4, 5, 8).

Experiments performed in 1968-1969
Experiments on the inelastic proton scattering performed in 1968-1969 at the MPIK are shortly described in [29]. Two targets of 208 Pb and 206 Pb isotopes were used with an enrichment of 99.98% and 97.38%, respectively. Protons were accelerated using the HVEC-MP Tandem in a scattering chamber equipped with 8-12 ion-implanted Si(II) detectors. The counters were cooled to 170 • K in order to reduce the reverse current. A resolution of 13-15 keV was obtained for 208 Pb [28][29][30] and for 206 Pb [20,73]. By turning the chamber different detectors were placed at the same scattering angle. By this means the solid angle for all 8-12 detectors was measured with a precision of 2%. Absolute and relative cross sections were determined by Rutherford scattering at E p = 5 MeV using the same experimental setup in the scattering chamber. Spectra for 206 Pb(p, p ) were taken for E p = 14.935 (on g 9/2 IAR) and 14.40 MeV (off IAR) at Θ = 125 • , see reproduction in Fig. 1. Additional spectra were taken for E p = 14.935 at Θ = 85 • and 110 • . Spectra taken for 208 Pb(p, p ) were used for the calibration of excitation energies [29]. The uncertainty of the excitation energies is 4 keV ( Table 4).
Similar 206 Pb(p, p ) spectra with low resolution were taken by Temmer and Lenz in 1968 at Θ = 90 • (Fig. 18 in [118]). Levels observed off and on the g 9/2 IAR (E p = 14.50 and 14.97 MeV) correspond to the 27 levels determined by Solf et al. [20]. The cross section from the elastic scattering is a factor hundred larger than the group of four levels from the inelastic scattering. The ratio of the cross section on-resonance to off-resonance is about a factor twenty as expected (Fig. 10).
The  [20]. The unresolved level 32 at E x = 4.21 MeV is evident. Level 45 at E x = 4.50 MeV is near a contamination peak from 12 C(p, p ).
The widest range of excitation functions covered the region 14.0 < E p < 21.8 MeV [83]. The excitation function was measured for 208 Pb(p, p ) and Θ = 165 • (lower frame of Fig. 1 in [83]
Excitation functions for the isotope 206 Pb were taken for the region 11.5 < E p < 20.0 MeV with Θ = 165 • and the excitation energies E x = 0.803 (upper frame) and 1.47 MeV (lower frame) [83]. Excitation functions for (p, p ) with the isotopes 204,206,207 Pb were measured in the energy range 13.5 < E p < 18.0 MeV for scattering angles Θ = 140 • , 165 • [118,120]. Excitation functions for the isotope 207 Pb were measured in the energy range 13.5 < E p < 18.0 MeV for Θ = 140 • and 160 • [121] and in the energy

Excitation functions for 205 Tl
Excitation functions were measured for 205 Tl at the MPIK in 1969 [84]. Proton energies covering 13.8 < E p < 20.0 MeV and scattering angle Θ = 160 • were used. Two states at E x = 2.61 and 2.69 MeV yielded two almost identical excitation functions.
The excitation functions resemble those measured for 208 Pb. Expecially the similarity to 208 Pb(p, p ) taken for Θ = 165 • is striking (lower frame of Fig. 1 in [83] For the study of the 208 Pb(p, p ) reaction in 1968 six beam energies were used [29]. Among them the proton energy E p = 14.99 MeV was used to measure angular distributions near the g 9/2 IAR [28][29][30]. The maximum cross section for 208 Pb(p, p ) on the g 9/2 IAR was later determined with E p = 14.918 ± 0.006 MeV [114]. The reduction of the mean cross section from the maximum is calculated with 0.80 ± 0.02 near E p = 14.99 MeV [29,114].
In comparing data for the two isotopes 206 Pb and 208 Pb only the angular distributions for 208 Pb(p, p ) measured at E p = 14.99 MeV are of interest [29]. Here scattering angles from Θ = 60 • to 165 • in 5 • steps were used. A full evaluation of the analysis of the angular distributions taken near the g 9/2 IAR in 209 Bi is still awaited. The results obtained in 1982 provided by Table 4 in [51] however are sufficient for the comparison to the isotope 206 Pb.

Identification of states in 206 Pb and 208 Pb
Experimental data for the inelastic proton scattering via IARs in 207 Bi and 209 Bi still exists [30,73]. Here we use the evaluated data [19] for 206 Pb(p, p ) and the data reconstructed from scans of spectra [29,46] for 208 Pb(p, p ). Tables 4, 5, 8 present the data analyzed by this work for 206 Pb.
Information about identified states in 206 Pb and angular distributions from the 206 Pb(p, p ) reaction via the g 9/2 IAR is shown in Sect. 4.5. Comparative data for 208 Pb are cited in Sect. 4.6. Tables 6 and 7 compare the results from this work to 208 Pb ( Table 4 in [51]).

Tables
The most recent source of information about states in 206 Pb derives from Nuclear Data Sheets [41]. The discussion of negative parity states at 3.7 < E x < 4.7 MeV in 206 Pb is a main topic of this work.
- Table 1 shows positive parity states at E x < 1.7 MeV discussed in Sect. 2. - Table 2 shows calculations of 1p1h configurations by SDI [4,5,69]. Excitation energies calculated by the SSM [16] for states expected by the coupling to the 2 + yrast state are included. Cross sections for 1p1h configurations near the g 9/2 IAR both in 207 Bi and 209 Bi are shown. - Table 3 characterizes each angular distribution in order to allow the comparison of the shape with calculated angular distributions of various configuration mixings (Figs. 6,7,8,9). -Finally determined spin assignments are given in Tables 4, 5, 8. The correspondence of known states [41] to states identified by this work is discussed in Sect. 6.2.1. -A detailed comparison of calculated angular distributions to best fits is done in Table 5. - Table 6 compares the strength distribution for three ranges of excitation energies (3.0 < E x < 3.7, 3.7 < E x < 4.17, 4.17 < E x < 4.7 MeV) in the two lead isotopes (Sect. 6.4). - Table 7 compares the results from this work to 208 Pb in detail (Sect. 6.4). - Table 8 tabulates the amplitudes of the fit ordered by the assigned spin and the excitation energy. The finally  Table 4). The thick line shows the best fit [19], the 1σ uncertainty is shown by dashed lines. The maximum is arbitrarily set to 1. The x-axis is given by the Legendre polynomial of second degree running from P 2 (cos 90 • ) to P 2 (cos 180 • ); the values P 2 (cos 120 • ) and P 2 (cos 140 • ) are marked at bottom accepted spin (Sect. 6.2.9) is printed bold face, discarded spins italic. For states within doublets (Sect. 6.2.6) or with alternate spin assignments (Sect. 6.2.7) the other spin assignments are shown, too, printed bold face or italic as discussed in Sect. 6.2.9. The amplitudes are given as obtained from the fit, especially for states with unique spin assignments (Sect. 6.2.5) and for doublet states (Sect. 6.2.6). For states with alternate spin assignments (Sect. 6.2.7) the amplitudes refer to the shown cross section. Sect. 6.2.8 discusses the discrimination of spins. For a discarded spin (Sec. 6.2.9) the strength is shown in parentheses. For each level the mean cross section is given as used for determining the amplitudes. For recognized doublets (Sect. 6.2.6) the division into two or three parts is indicated by the factor 1/2 and 1/3. For each spin the centroid energy E x [Eq. (4)] is printed bold face. The sum of the strength c 2 for the configurations g 9/2 p 3/2 and g 9/2 f 5/2 is determined for two ranges of excitation energies, E x < 4.17 MeV and E x > 4.17 MeV. [19] show angular distributions for 206 Pb(p, p ) fitted by Legendre polynomial P K with K = 0, 2, 4 [19]. In total 29 angular distributions were measured.  (Table 4). A special method uses the Legendre polynomial P 2 (cos Θ) as abscissa (see Fig. 3 in [46]). The level number is shown at top, the excitation energy in units of keV at bottom. - Figure 5 shows as an example the angular distribution for level 23 in two variants (top frame) the best fit [20] and the uncertainties of the fit by Legendre polynomials P K , K = 0, 2, 4 in relative units. The maximum is arbitrarily set to 1. (bottom frame) a fit with Legendre polynomials P 2 (cos Θ); the mean cross section is normalized to unity (dotted line).
In order to linearize the angular distributions as much as possible [46], in Figs. 11, 12, 13, 14, 15, 16, 17 and 18 the x-axis is given by the Legendre polynomial of second degree running from P 2 (cos 90 • ) to P 2 (cos 180 • ). The values P 2 (cos 120 • ) and P 2 (cos 140 • ) are marked at bottom. The ordinate is given in values relative to the mean cross section shown by a dotted line. The scale runs up to a maximum of 3.0; the values 0.5, 1.0, 1.3, 1.5, 2.0 are shown at right. The line at left shows the range up to 1.5 thus illustrating extremely large slopes. The description of the x-axis is omitted for clarity.  Table 4 because of rounding uncertainties. For recognized doublets (Sect. 6.2.6) each member is assumed to contribute equally (σ/2 or σ/3). For accepted doublets (Sect. 6.2.5) the factor 2 or 3 is included. In a next line the amplitudes multiplied by a factor hundred are given for the GNPH configuration g 9/2 l 2 j with l j = p 1/2 , p 3/2 , f 5/2 , f 7/2 , h 9/2 . The amplitudes for the levels recognized as doublets (Sect. 6.2.6) are already multiplied by the factor √ 1/2 or √ 1/3.   In the angular distributions the differential cross section is linearized by choosing the abscissa as the Legendre polynomial P 2 (cos Θ) (see Fig. 3 in [46]).
The ordinate is omitted for clarity. Scales from 0 to 1.0 and values 1.3, 1.5, 1.8, 2.0 are shown at right. A scale from 0 to 1.5 allows to estimate the steepness of the angular distribution.
For each frame the uniquely defined energy label E x , spin I π and mean cross section σ in units of μb/sr normalized to unity by c 2 = 1 are shown in the first line of the legend. Because of rounding in the calculations the displayed value σ for the cross section deviates from the value in Table 8. The value dσ rel /d = 1 [Eq. (7)] is shown by the dotted line.    Table 8 shows results for 206 Pb, see also Table 5. Table4 in [51] and Table 2 in [29] are used to summarize results for 208 Pb The next line in the legend shows the configuration mixing g 9/2 l j, l2 j =p1, p3, f5, f7, h9. The third line shows the amplitudes l2 j multiplied by a factor 100. In each case two configurations differing by one sign are depicted, one amplitude is given in parentheses.
For each frame five curves are shown. The drawn curve shows the angular distribution with the amplitudes g 9/2 l2 j yielding a best fit. The dotted curve the shows the fit with one reverse sign, the amplitude with the reverse sign is given in parentheses. The three doubly-dash-dotted curves show the angular distribution measured by Solf et al. and fitted by Legendre polynomials of even order K = 0, 2, 4 and 1σ deviations. Figures 11, 12, 13, 14, 15, 16, 17 and 18 show the fit for 46 states in 22 levels. Figure 11 shows the fit for one state with unique spin assignment but different configurations (a) with two similar   (3)]. Fig. 12 show the fit for four states with unique spin assignments. Figures 13, 14 and 15 show the fit for eleven levels with different spin assignments in each pair from (a, b) to (g,h). In Figs. 13a, f, g, 14a, e, g, 15a, c, the spin assignment is discarded by regarding the orthonormality and sum-rule relations [Eq.
In this work only states at 3.1 < E x < 4.7 MeV are mentioned. Table 4 in [51] and Table 6 show the data used in this paper. The evaluation is based on experimental data for 208 Pb(p, p ) taken in 1968 at the MPIK [28,46] and reconstructed in 2017 [29,30].

Methods of analysis
The main tool to assign a spin to a state in 206 Pb or 208 Pb and determine amplitudes of particle-hole configurations is  Fig. 4a in [20] for levels 27-26/29 displays three angular distributions of 206 Pb(p, p ). The symmetry for |90 • − Θ| is mostly rather well realized. The fits shown in Fig. 2 use only the range 90 • < Θ < 180 • the inelastic proton scattering via an isolated IAR. In the analysis of 208 Pb(p, p ) other available experimental data is used, see especially [23,76,77]. The values for the scattering angles are not equidistant, see Fig. 3 in [46]. The differential cross section with dominant configuration g 9/2 p 3/2 is linearized by the fit with Legendre polynomials P 2 (cos Θ). Deviations from the linearity by admixtures of configurations g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 thus become more pronounced, see Fig. 12  Yet the primary tool is the resonant proton scattering because of its high sensitivity and the opportunity to determine relative signs of amplitudes on each IAR [31,40].

Theory of the inelastic proton scattering via an IAR
Here a short reminder to the theory of the inelastic proton scattering via an IAR is given. It is described in detail in [1,13,36,38]. Eqs. 7-10 in [46] are adapted to describe needed qualities for the analysis of the angular distributions for (p, p ) taken on an IAR.
Here the factor a l j describes the IAR [13]. The single particle widths Γ s. p.
L J and Γ s. p. l j are taken from [29]. Table 4 shows the excitation energies of the configurations g 9/2 f 5/2 and g 9/2 p 3/2 predicted by the SDI together with the mean cross sections σ calc . The angular distribution near the g 9/2 IAR are described by a series of Legendre polynomials The parameter Λ describes the population of the resonance [46].

Methods of spin assignment to states in 206 Pb
Because of the two neutrons missing from the doubly magic nucleus 208 Pb admixtures of g 9/2 p 1/2 to GNPH configurations are expected to be weak. The excitation energies of the configurations g 9/2 p 3/2 are expected to be about 300 keV higher than for g 9/2 f 5/2 . Calculations of excitation energies  Fig. 6 but for mixed configurations g 9/2 l j for l2 j = p1, p3, f 5 with dominant g 9/2 p 3/2 and g 9/2 f 5/2 components and spin 5 − . The angular distributions differ by the choice of two sets of configuration mixing and two signs. The differential cross section is linearized and variations thus become more pronounced by SDI [5,69] and mean cross sections are shown in Table 2.
The spin assignment and the determination of GNPH amplitudes are strongly correlated across all states. The fit of an angular distribution of 206 Pb(p, p ) via the g 9/2 IAR in 207 Bi is done in several major steps. The following assumptions are regarded.
1. The slope and the curvature of the angular distribution of a state with a dominant GNPH configuration is related to the spin; for low admixtures of other GNPH configurations slight changes are expected (Figs. 3, 4, 5, 6, 7, 8, and 9). Some spin assignments can be ruled out by the slope and the bending. 2. The cross section is much higher for g 9/2 p 3/2 than for g 9/2 f 5/2 whereas strengths of g 9/2 p 1/2 and g 9/2 p 3/2 are similar. Here the interference pattern allows to distinguish the two configurations. 3. The distant configurations g 9/2 f 7/2 and g 9/2 h 9/2 contribute less than about one percent in strength. Yet the interference pattern is sensitive to such low admixtures. In cases where three small amplitudes admix to one dominant configuration even a fifth particle-hole configuration distorts the calculated shape of the angular distribution more than only marginally, especially at scattering angles Θ > ∼ 160 • where the Legendre polynomials P K differ most [Eq. (6)]. 4. The excitation energies predicted by the SDI for 208 Pb are assumed to be valid for 206 Pb. States with dominant configurations g 9/2 p 1/2 are expected to be absent because in the SSM the p 1/2 orbits are empty. Because of the low orbital momentum (l = 0) in the (p, p ) reaction considerable cross sections for a weak g 9/2 p 1/2 admixture may be expected. In 206 Pb additional GNPH configurations g 9/2 p 1/2 ⊗2 + 1 , g 9/2 f 5/2 ⊗2 + 1 , g 9/2 p 3/2 ⊗2 + 1 , and h 9/2 s 1/2 ⊗2 + 1 , h 9/2 d 3/2 ⊗2 + 1 are expected to be present. Their excitation energies are higher 2 by 803 keV than the excitation energies calculated by the SDI for 1p1h configurations [5,69].

Sequence of iterations
The study of angular distributions in twenty-two observed levels yields good fits in a few major steps by guessing spin assignments and varying configuration amplitudes (Sects. 6.2.5-6.2.7). In another (not consecutive) step the presence of more than one state in each level is discussed (Sects. 6.2.6, 6.2.9).
Categorizing the shape of the angular distribution. In a first step the shape of the angular distribution with relative values of 1-3 major configurations is investigated (Table 3).
Adjusting the mean cross section. In a second step the measured mean cross section is adapted by applying a common factor to all amplitudes. Because of numerics the cross section shown in Figs. 11, 12, 13, 14, 15, 16, 17 and 18 slightly differs from the measured value shown in Tables 4, 8.
Comparing the mean cross section and the excitation energy to calculations. In a third step the mean cross sections and the excitation energies are compared to calculations of GNPH configurations (Table 2).
Investigating sum rules and centroid energies. In a fourth step spins for alternate assignments (Sect. 6.2.7) or alternate particle-hole compositions (Sects. 6.2.5-6.2.7) are chosen which approximate the unity value of the sum rules for g 9/2 f 5/2 and g 9/2 p 3/2 and assume the sum rule for g 9/2 p 1/2 to be much below unity. Many iterations were tried until reasonable results were obtained. Section 6.2.9 discusses the arguments for the final choice of spins. Table 6 summarizes the results. The final spin assignment for the 32 states in the 22 levels shown in Table  8 may need in future another explanation. The cross sections for the doublets N s certainly are not evenly distributed, the ratio R 20 hints to more unresolved doublets, some discarded assignments N dscd have to be changed.

Major steps in determining spin and structure of states
Using Table 2 and Figs. 6, 7, 8, and 9 the shape alone allows to exclude certain spins for many states. The minimum and maximum or the relative cross section dσ rel /d [Eq. (7)] is important. Table 3 shows characterizing values. In the following, conditions are enumerated which allow to find possible spin assignments.   Table 4). The ratio R = 9 for level 24 may be explained by assuming the triplet to consist of the 5 − , 6 − , 7 − members with cross sections σ/2, σ/4, σ/4 (denoted by dotted lines) three configurations (g 9/2 p 1/2 , g 9/2 p 3/2 , g 9/2 f 5/2 ) and additional weak admixtures from a fourth configuration g 9/2 f 7/2 . Because of the weak s.p. width contributions from g 9/2 h 9/2 can be neglected. 5. For states with spin 4 − and 5 − admixtures of g 9/2 p 1/2 may be sizeable because of the impurity of the g.s. (Sect. 2.1.2). 6. For states with spin 4 − and g 9/2 p 1/2 , g 9/2 p 3/2 , and g 9/2 f 5/2 components the slope is rather flat and the bending is upwards (downwards) for a negative (positive) sign for g 9/2 f 5/2 and positive sign of the g 9/2 p 1/2 amplitude. 7. For states with spin 4 − and g 9/2 p 1/2 , g 9/2 p 3/2 , and g 9/2 f 5/2 components the slope is rather flat and the upward bending becomes increasingly more expressive with negative sign of the g 9/2 f 5/2 amplitude and for each sign of g 9/2 p 1/2 . 8. For states with spin 5 − and g 9/2 p 1/2 , g 9/2 p 3/2 , and g 9/2 f 5/2 components the slope is downwards and the bending is straight (upwards) for a negative (positive) sign for g 9/2 f 5/2 with a positive sign of the g 9/2 p 1/2 amplitude. 9. For states with spin 5 − and g 9/2 p 1/2 , g 9/2 p 3/2 , and g 9/2 f 5/2 components the slope is rather flat and the upward bending is downward (upward) for a positive (negative) sign for g 9/2 f 5/2 with a negative sign of the g 9/2 p 1/2 amplitude. 10. For states with spin 6 − and a dominant g 9/2 p 3/2 and g 9/2 f 5/2 components the bending is nearly straight until the g 9/2 f 5/2 amplitude becomes large and if the signs differ. The maximum reaches 1.3-1.7 at Θ = 180 • . 11. For states with spin 6 − and a dominant g 9/2 f 5/2 component the bending is downward with a weak g 9/2 p 3/2 admixture of positive sign. 12. For states with spin 7 − the bending is upwards and the maximum is about 1.5 at Θ = 180 • . There is a high sensitity to g 9/2 f 7/2 admixtures. 13. For all states admixtures of the distant configurations g 9/2 f 7/2 and g 9/2 h 9/2 should be negligible. 14. The total strength c 2 should not approach unity because proton particle-hole configurations and neutron configurations with another particle are certainly present in each state [Eq. l j ) [29] allow for deviations in the order of several percent.

Discussion
In Sect. 6.1 results for 208 Pb at shortly cited. In Sect. 6.4 states in 206 Pb are compared to states in 208 Pb for the region at 3.7 < E x < 4.7 MeV.

Results for 208 Pb
First results for 208 Pb discussing orthogonal ensembles of 1p1h configurations were presented in 1973 [24]. All of the thirty-four lowest particle-hole negative parity particle-hole states at E x < 5.2 MeV in 208 Pb were investigated after 1982. Results are shown in Table 4 in [51]. An update is being done but still awaited. Only results for the 1 − yrast state are included [67].

Comparison to nuclear data sheets
The correspondence of known states [41] to the states determined from the measured levels [20] by this work is discussed in the following. In view of the resolution of 15 keV equal to four times the mean distance of states at any spin (about 4 keV) the correspondence is rather uncertain.  (Fig. 14f, g) discarded, see Table 8 The excitation energies agree within the uncertainties for all states except for the 4094, 4096 states in the 4094 level. They were not observed before. Note that in order to obtain uniqueness the excitation energies differ from the experimental values [20] by up to two keV and some values in [41] are reported without an uncertainty. An agreement of the excitation energy given in Table 4 with excitation energies from [41] (Fig. 15a, c) discarded, see Table 8 and 4215, 4239 and 4241, 4292, 4340, 4431, 4452, 4453, and 4454, 4500, 4540, 4592 and 4593, and 4680 states. The assignments of spin 5 − are confirmed for the 3772, 4060, and 4452 states corresponding to E x = 3776.1 ± 0.09, 4066 ± 3, 4459 ± 3 keV. A tentative assignment of spin 6 − is confirmed for the 4500 state corresponding to E x = 4496 ± 5 keV.

Starting point to identify particle-hole states in 206 Pb
The starting point to identify particle-hole states in 206 Pb is the observation that the total sum of the cross sections for 22 levels at 3.7 < E x < 4.7 MeV in 206 Pb excited by 206 Pb(p, p ) via the g 9/2 IAR in 207 Bi is 2.7 mb/sr. It corresponds to 2.4 mb/sr found for 20 states at 3.7 < E x < 4.7 MeV in 208 Pb excited by 208 Pb(p, p ) via the g 9/2 IAR in 209 Bi. Note that the chosen proton energy for 208 Pb(p, p ) was E p = 14.99 MeV which is 20% off the resonance maximum [29]. The agreement with the total sum calculated for the   Figure 11 continued. Spin assignment for 4051 5 − , 4095 5 − (Fig. 17c, e) discarded, see Table 8 6

.2.3 Major steps of iterations
Two major iterations allow to identify 32 states in 22 levels.
In a first iteration only the shape dσ rel /d [Eq. (7)] is considered (Sects. 6.2.5, 6.2.6, 6.2.7). The levels are discussed within each subsection in consecutive order. In Sect. 6.2.9 the levels are finally discussed by considering the orthonormality and sum-rule relations [Eq. (3)] and the expectation from calculations by SDI (Table 2).  Figure 11 continued. Spin assignment for 4240 5 − (Fig. 18c) discarded, see Table 8 First iteration. A few steps lead to rather convincing spin assignments with the assistance of Table 3 and Figs. 11, 12, 13, 14, 15, 16, 17 and 18. The high sensitivity of the (p, p ) reaction via an IAR needs to include admixtures of g 9/2 p 1/2 , g 9/2 f 7/2 , g 9/2 h 9/2 . (i) The shape of the angular distribution (Fig. 2) in comparison to calculations (Figs. 6,7,8,9) is considered in the beginning. Often two or three spin assignments are similarly probable.
(ii) Admixtures of g 9/2 f 7/2 , g 9/2 h 9/2 should be less than one percent in strength because of the large distance of the configurations from g 9/2 f 5/2 and g 9/2 p 3/2 ( Table 2). (iii) Admixtures of g 9/2 p 1/2 should be less than half the strength if the g.s. of 206 Pb is rather pure (Sect. 2).
Second iteration. (iii) The centroid energies for states with dominant configurations g 9/2 f 5/2 and g 9/2 p 3/2 should be close to the energy E S DI x (I π , l j) for l j = f 5/2 , p 3/2 , respectively ( Table 2). (iv) The spacing between any two states of the same spin should be larger than 20 keV. The minimum distance between any two states of the same spin in 208 Pb is about 35 keV [16,46]. It is explained by the level repulsion in the theory of chaotic spacing [123,124]. The level density in 206 Pb is twice larger as shown by this work (Sect. 6.4).
This argument is not used explicitly, arguments (i)-(iii) suffice. The final result shows that the minimal spacing between any two states of the same spin in 206 Pb with 15 keV is half of the corresponding value for 208 Pb [16]. Note that the uncertainties of the excitation energies are 4 keV [20]. Tables 4,5,8 show the final spin assignments. Figures 11,12,13,14,15,16,17 and 18 (Sect. 4.5.2) the amplitudes. In the following, the spin assignments and the identification of states are discussed. Figures 6,7,8,and 9 show the shape of the angular distributions for mixtures of the configurations g 9/2 p 1/2 , g 9/2 p 3/2 , g 9/2 f 5/2 . Table 3 characteristizes the angular distributions. In many cases admixtures of g 9/2 f 7/2 and g 9/2 h 9/2 turn out to be sensitive, too. The method described in Sect. 5.2 allows to find solutions in a 4-or even 5-dimensional space.

Shape of the angular distribution
In first iteration considering the shape of the angular distribution alone, some levels contain one state with a unique shape of the angular distribution (Sect. 6.2.5). Some levels are shown to certainly contain two or three states (Sect. 6.2.6). Some levels may contain two or three states (Sect. 6.2.7).
In a second iteration, other considerations discard some spin assignments (Sect. 6.2.9).
This work tries to decipher the experimental data with the utmost accuracy. However Fig. 10 together with Table 4 points to possibly more unresolved states.
A triplet with spins 5 − , 6 − , 7 − is deduced for level 24. The assumption of equal cross sections for the three members yields an extremely low ratio R. The 6 − member contains a large g 9/2 f 7/2 admixture which is unlikely; the cross section of the 7 − member may be weaker, and the five amplitudes in the 5 − member may lead to a mixture with less g 9/2 f 7/2 . By assuming the cross section for the 5 − member to be twice as large as for the 6 − , 7 − members the ratio R would assume a ratio nearer to R = 20 expected for the on-to-off resonance cross section (Fig. 10).
A third iteration is needed in future (Sect. 6.2.10) after the first and second iteration (Sect. 6.2.9).

Unique spin assignments
Some angular distributions have a unique shape dσ rel /d [Eq. (7)] thus firmly assigning a certain spin and a major GNPH configuration to the state. Nevertheless the level may indicate the presence of another unresolved state covered by the level because of discrepancies of the angular distribution at scattering angles Θ < The 3960 level is assumed to contain a 4 − state (Fig. 11).
The shape of the angular distribution steeply decreases from 1.4 at Θ = 90 • to 0.7 at Θ = 180 • with a slight upward bending. The curvature clearly differs from the shape for spin 6 − with dominant g 9/2 f 5/2 . Spin 5 − is excluded because of the different bending at Θ > ∼ 160 • . The fit with four configurations yields either a strong g 9/2 f 5/2 component with unique signs (3960) or an even mixture of g 9/2 p 3/2 and g 9/2 f 5/2 with admixtures from g 9/2 p 1/2 and g 9/2 f 7/2 (3961). The 3713 level is assumed to contain a 2 − state (Fig. 12). The shape of the angular distribution is extremely steep raising from 0.7 to 2.0 with a slight upward bending. The deviation from the factor 2.6 at Θ = 180 • for a pure g 9/2 f 5/2 configuration yields an admixture of g 9/2 f 7/2 with a unique sign; admixtures of g 9/2 h 9/2 are weak. The 4011 level is assumed to contain a 6 − state (Fig.  12). The shape of the angular distribution is very peculiar; it raises from Θ = 90 • to 1.2 at Θ = 130 • and then decreases to 0.25 at Θ = 180 • . The shape excludes all other spins but 6 − . The coincidence with the characteristic bending for a pure g 9/2 p 3/2 configuration yields weak admixtures of g 9/2 f 7/2 and g 9/2 h 9/2 with unique signs. The 4540 level is assumed to contain a 5 − state (Fig. 12). The shape of the angular distribution is peculiar, it raises from a minimum of 0.8 with a slight upward bending to a maximum of 1.5. Spin 3 − is excluded because the slope is much smaller. Spin 7 − is excluded because the pronounced down-bending near Θ = 140 • cannot be fitted. The 4680 level is assumed to contain a 6 − state (Fig. 12). The shape of the angular distribution is rather straight. The angular distribution resembles that for the 4453 level with 10% of its strength. The slope is larger favoring assignment of spin 6 − . In view of the weak cross section no other spin assignment was tried.

Doublets
The cross section of some levels exceeds the value calculated for a single state ( Table 2). Clearly a doublet with two or three states with different spins is present. Table 8 indicates doublets by including the factor 1/2 or 1/3 to the cross section for the two or three members. The amplitudes and strength are determined with the reduced cross section.
States with spin 4 − or 5 − may contain weak g 9/2 p 1/2 admixtures because of the impurity of the g.s. (Sect. 2.1.2). Therefore angular distributions of 4 − or 5 − states are difficult to fit because at least three configurations admix rather strongly; in a few cases even five configurations contribute.
Because of the wide range of shapes angular distributions of 4 − or 5 − states sometimes resemble that with another spin.
The 3773 level has the large cross section of 154μb/sr. It is assumed to consist of the 3772 5 − , 3771 6 − , 3774 7 − states (Fig. 16). The shape of the angular distribution has a strong bending starting from about 1.0 at Θ = 90 • up to 1.3 at Θ = 180 • . Spins 2 − and 3 − are excluded because the slope is too low. The fit assuming spin 5 − or 6 − reproduces the bending. For spin 6 − the g 9/2 f 5/2 amplitude is much larger than g 9/2 p 3/2 , the g 9/2 f 7/2 , g 9/2 h 9/2 admixtures are negligible. For spin 5 − the g 9/2 p 3/2 and g 9/2 p 1/2 amplitudes are equal, admixtures from g 9/2 f 5/2 and g 9/2 f 7/2 are negligible. Assuming spin 7 − the angular distribution deviates largely at Θ < 130 • but the fit at Θ > 130 • is reasonable. The excitation energy is not far from the prediction by SDI. The 3977 level has the large cross section of 191μb/sr. It is assumed to consist of the 3977 4 − and 3978 5 − states. The fit of the level yields similar results for spin 4 − and 5 − (Fig. 13). The angular distribution decreases from 1.2 at Θ = 90 • to 0.5 at Θ = 180 • with a slight downward bending. A fit with spin 4 − alone needs a g 9/2 f 5/2 amplitude larger than unity violating the sum rule relation [Eq. (3)]. A fit with spin 5 − alone needs large g 9/2 f 5/2 and g 9/2 p 3/2 amplitudes. The normality relation is nearly violated [Eq. (3)]. A doublet with equal cross sections is assumed. For spin 4 − a dominant g 9/2 f 5/2 component with strong g 9/2 p 1/2 , and weak g 9/2 p 3/2 , and g 9/2 f 7/2 admixtures are determined. For spin 5 − g 9/2 p 3/2 and g 9/2 f 5/2 components of similar size with weak admixtures from g 9/2 p 1/2 and g 9/2 f 7/2 are determined. The 3993 level is assumed to consist of the 3992 3 − and 3992 or 3994 6 − states (Fig. 16). The shape of the angular distribution is rather steep; it raises by from 0.9 at Θ = 90 • to 1.5 at Θ = 180 • without a bending. The fit assuming spin 3 − and dominant g 9/2 f 5/2 strength reproduces the angular distribution quite reasonable at 130 • < Θ < 180 • with a maximum of 1.35 at Θ = 180 • . Weak g 9/2 p 3/2 , g 9/2 f 7/2 and a negligible g 9/2 h 9/2 admixture are derived. The straight increase starting with 0.9 at Θ = 90 • is better fitted by assuming spin 6 − , either the 3992 state with dominant g 9/2 f 5/2 or the 3994 state with mixed g 9/2 p 3/2 and g 9/2 f 5/2 . The 4214 level has the large cross section of 410μb/sr. It is assumed to consist of the 4214 4 − and 4215 5 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 14). The shape of the angular distribution is steep starting from 1.3 at Θ = 90 • down to 0.3 at Θ = 180 • with a slight bending. The fit for spin 4 − with four configurations yields a strong g 9/2 p 3/2 component with unique signs and weak g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 admixtures. The fit for spin 5 − with four configurations yields a strong g 9/2 p 3/2 component with unique signs, a considerable g 9/2 f 5/2 contribution and weak g 9/2 p 1/2 , g 9/2 f 7/2 admixtures. The 4240 level has the large cross section of 136μb/sr. It is assumed to consist of two states, the 4239 3 − and 4241 4 − states (Fig. 18). The angular distribution starts from 1.0±0.1 at Θ = 90 • ending with 1.4±0.2 at Θ = 180 • with a clear upward bending. The spin assignments 4 − and 5 − yield a reasonable interpretation of the angular distribution. The 5 − state alone would contain half of the g 9/2 f 5/2 strength and a strong g 9/2 p 1/2 admixture. By assuming spin 4 − the bending is well fitted. The fit with four configurations yields a strong g 9/2 p 3/2 component and g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 admixtures of similar size. The fit for spin 3 − with a strong g 9/2 p 3/2 component and weak g 9/2 f 5/2 , g 9/2 f 7/2 admixtures reproduces the steep raise at Θ > 140 • rather well but fails at Θ ≈ 90 • by a factor two. The 4453 level has the large cross section of 297μb/sr. It is assumed to contain the 4454 4 − , 4452 5 − , 4453 6 − states (Fig. 18). Spin 2 − is excluded because of the large cross section and because the slope is not very steep. Spin 3 − is excluded because of the large cross section and because there is no bending. Spin 7 − is excluded because of the large cross section. The shape of the angular distribution is straight. It raises from 0.8 at Θ = 90 • to 1.3 at Θ = 180 • in near coincidence with the factor 1.35 for pure g 9/2 p 3/2 for spin 6 − . It thus yields weak admixtures of g 9/2 f 7/2 and g 9/2 h 9/2 with unique signs. The 4500 level is assumed to consist of the 4499 4 − and 4500 6 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 15). For spin 4 − a dominant g 9/2 p 3/2 component with a strong g 9/2 p 1/2 , and weak g 9/2 f 5/2 , g 9/2 f 7/2 admixtures are determined. For spin 6 − a dominant g 9/2 f 5/2 component with weak g 9/2 f 7/2 , g 9/2 h 9/2 admixtures are determined.

Alternate spin assignments
Several levels can be fitted with two alternate spin assignments. A level may contain two states, either the first or the second energy label or the first and the second energy label is assumed, in few cases even three energy labels. Table 8 indicates spins discarded by further reasoning (Sect. 6.2.9). The strength is given in parentheses, in Table 4 the discarded assignments are denoted by N dscd , see also Table 5.
A factor 1/2 is included in the strength if two spins are assumed to be valid. The amplitudes are calculated with the full cross section.
The 3833 level is assumed to consist of the 3832 6 − and 3833 4 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 13). For spin 4 − a dominant g 9/2 p 3/2 component with weak g 9/2 p 1/2 , g 9/2 f 5/2 , and g 9/2 f 7/2 admixtures are determined. For spin 6 − a dominant g 9/2 f 5/2 component with weak g 9/2 p 3/2 , g 9/2 f 7/2 , and g 9/2 h 9/2 admixtures are determined. The 4040 level is assumed to consist of the 4040 5 − and 4041 4 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 13). The angular distribution has an expressive upward bending starting from 1.0 at Θ = 90 • to 0.8 at Θ = 180 • with a maximum of 1.1. The fit with four configurations yields a strong g 9/2 p 3/2 component in the 4 − state with a weak g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 admixtures. The fit with four configurations yields a strong g 9/2 p 3/2 component in the 5 − state with a considerable g 9/2 p 1/2 , g 9/2 f 5/2 and a weak g 9/2 f 7/2 admixtures. The 4050 level is assumed to consist of the 4049 4 − and 4051 5 − states. Both spin assignments yield a reasonable interpretation of the angular distribution, for spin 4 − two different sets of amplitudes are possible (Fig. 17). The angular distribution smoothly decreases from 1.2 at Θ = 90 • to 0.5±0.2 at Θ = 180 • without a bending; the uncertainty at Θ > 160 • however is large. An assignment of spin 6 − is excluded because the slope is straight and not large. The fit for spin 4 − with five configurations yields either a strong g 9/2 f 5/2 with a weak g 9/2 p 3/2 or a strong g 9/2 p 3/2 with a weak g 9/2 f 5/2 component with negligible admixtures from g 9/2 p 1/2 , g 9/2 f 7/2 , g 9/2 h 9/2 . The fit for spin 5 − with four configurations yields similar amplitudes for the configurations g 9/2 p 1/2 , g 9/2 p 3/2 , g 9/2 f 5/2 and a weak g 9/2 f 7/2 admixture. The 4060 level is assumed to consist of the 4060 5 − and 4061 4 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 15). The angular distribution decreases from 1.2 at Θ = 90 • to 0.7 at Θ = 180 • without a bending. The fit with four configurations yields a strong g 9/2 p 3/2 component in the 4 − state with weak g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 admixtures. The fit with four configurations yields a strong g 9/2 f 5/2 component in the 5 − state with considerable g 9/2 p 1/2 , g 9/2 p 3/2 , and a weak g 9/2 f 7/2 admixtures. The 4094 level is assumed to consist of the 4094 4 − , 4095 5 − , and 4096 6 − states. The spin assignments yield a reasonable interpretation of the angular distribution (Fig. 15), however among all angular distributions these fits are worst. The shape of the angular distribution is very steep starting from 1.4 at Θ = 90 • down to 0.3 at Θ = 180 • with a peculiar double bending. The fit with four configurations for spin 4 − yields strong g 9/2 p 1/2 and g 9/2 f 5/2 components and weak g 9/2 p 3/2 , g 9/2 f 7/2 admixtures. For spin 5 − the fit with four configurations yields a strong g 9/2 p 3/2 component, admixtures of g 9/2 p 1/2 , g 9/2 f 5/2 with unique signs are weak, with g 9/2 f 7/2 negligible. The fit with three configurations for spin 6 − yields a strong g 9/2 f 5/2 component and weak g 9/2 p 3/2 , g 9/2 f 7/2 admixtures. The fit assuming spin 4 − does not reproduce the rather straight angular distribution. The fit assuming spin 5 − reproduces the data at Θ < 130 • even worse. The fit assuming spin 6 − reproduces the data at Θ > 140 • best but completely fails for lower scattering angles. The 4160 level is assumed to consist of the 4161 4 − and 4160 5 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 15). The angular distribution decreases from 1.2 at Θ = 90 • to 0.4 at Θ = 180 • without bending. The fit with four configurations yields a strong g 9/2 p 3/2 component in the 4 − state with weak g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 admixtures. Similarly, the fit with four configurations yields a strong g 9/2 p 3/2 component in the 5 − state with weak g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 admixtures. The 4320 level is assumed to consist of the 4321 4 − and 4320 6 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 15). The angular distribution raises from 0.8 at Θ = 90 • to 1.2 at Θ = 180 • with a slight downward bending. The fit with four configurations yields a strong g 9/2 p 3/2 component in the 4 − state with a weak g 9/2 p 1/2 admixture. The fit with two configurations yields a strong g 9/2 p 3/2 component in the 6 − state. The 4340 level is assumed to consist of the 4341 4 − and 4340 5 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 16). The angular distribution smoothly decreases from 1.2 at Θ = 90 • to 0.5 at Θ = 180 • without a bending. The shape of the 4340 level is similar to shape of the 4050 level. The fit for spin 4 − with four configurations yields a strong g 9/2 p 3/2 component and weak g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 f 7/2 admixtures. The fit for spin 5 − with four configurations yields similar amplitudes for the configurations g 9/2 p 1/2 , g 9/2 f 5/2 and weak g 9/2 p 3/2 g 9/2 f 7/2 admixtures. A fit with similar relative amplitudes as for the 4050 level fails for spin 4 − because of the large cross section whereas for spin 5 − the ratio of the amplitudes is about a factor two. The 4430 level is assumed to consist of the 4430 4 − and 4431 6 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 16). For spin 4 − a dominant g 9/2 p 3/2 component with a considerable g 9/2 p 1/2 , and weak g 9/2 f 5/2 , g 9/2 f 7/2 admixtures are determined. For spin 6 − a dominant g 9/2 f 5/2 component with a considerable g 9/2 p 3/2 , and weak g 9/2 f 7/2 , g 9/2 h 9/2 admixtures are determined. The 4592 level is assumed to consist of the 4592 7 − and 4593 6 − states. Both spin assignments yield a reasonable interpretation of the angular distribution (Fig. 16). The shape of the angular distribution is extremely steep; it raises from 0.8 at Θ = 90 • to 1.5 at Θ = 180 • . The coinciding factor 1.55 for a pure g 9/2 f 5/2 with spin 7 − yields weak admixtures of g 9/2 f 7/2 and g 9/2 h 9/2 with unique signs. The fit for spin 6 − yields a strong g 9/2 p 3/2 component with a considerable g 9/2 f 5/2 admixture and a weak g 9/2 f 7/2 admixture.

Discrimination of spins
Two or three spins can be assigned to an unresolved doublet. The clearest example is the level number 33 with E x = 4094 keV. The shapes of Fig. 17 d-f differ considerably. The spin 5 − is discarded because the excitation energy is low and the g 9/2 p 3/2 strength is large (Fig. 17 e, Sect. 6.2.9). The fit of the 4094 level with even order Legendre polynomials (Figs. 2, 17) deviates at Θ > 140 • by 20% for spin 4 − (d) and at Θ < 120 • by 50% for spin 6 − (f).
In other cases the deviations are smaller but still significant. In Fig. 13 the fit with Legendre polynomials deviates by 20% at Θ < 120 • for spin 4 − by 10% at Θ > ∼ 160 • both for (e) and (f). In Fig. 16 the fit with Legendre polynomials deviates by 20% at Θ < 120 • and for spin 7 − (c); for spin 3 − the bending matches badly throughout (f); especially the deviations at Θ < 120 • are large. The deviations for spin 3 − in Fig. 18a are similar to Fig. 14f. In both cases the bending is typically for the low spin.
Two levels are assumed to contain three states with equal cross sections for each pair. Six levels are assumed to contain two states with equal cross sections for each triple.

States identified in 206 Pb in first and second iteration
In the following the identification of GNPH states in 206 Pb is discussed. Some spin assignments compatible with the shape (Sects. 6.2.5, 6.2.6, 6.2.7) have to be discarded in second iteration (Sect. 6.2.3) alternating with the first iteration. Table 2 is used as a guide for the discussion. Tables 4, 5, 8 assist in locating the levels and states. For each level dominant configurations are mentioned ( Table 5).

Possible identification of more states in 206 Pb in third iteration
Three levels are recognized as triplets with two or three different spins, six levels as doublet with two states (Table  4); they are marked in Fig. 10. Six levels have a ratio R ≈ 50, three levels a ratio of around R ≈ 35. Without almost any exception all these levels were recognized as doublets which were discarded by the investigation of the orthonormality and sum-rules [Eq. (3)] (Sect. 6.2.7).
Obviously some spin assignments (Sect. 6.2.9) must be revised by a third major reconsideration. 15 states are discarded by regarding orthonormality and sum-rule relations [Eq. (3)] (N dscd in Table 4). 15 additional states are suggested to be identified from the ratio R 20 with R = N s shown in Fig. 10 (N R in Table 4). The number of discarded states N dscd is underestimated and N R − N s more states are suggested.
In third iteration the number of identified states at 3.7 < E x < 4.7 MeV in 206 Pb may thus increase to twice the number of states in 208 Pb.

Strength distribution
In order to check the strength distribution for the configurations g 9/2 p 3/2 and g 9/2 f 5/2 in 206 Pb the states are divided into two groups with excitation energies less and larger than E x = 4.17 MeV, a boarder dividing the g 9/2 f 5/2 strength from the g 9/2 p 3/2 strength ( Table 2).
Admixtures from g 9/2 f 7/2 and g 9/2 h 9/2 are less than about one percent in the 32 states. However the g 9/2 p 1/2 strength for spins 4 − and 5 − does not vanish as expected, 35% and 33% g 9/2 p 1/2 strength for spin 4 − and 5 − are found. The centroid energy of the g 9/2 p 1/2 strength is E x ≈ 4.2 MeV. It well corresponds to E x = 4.23 keV predicted by the weak coupling model [2] for 2 + 1 ⊗ g 9/2 p 1/2 . It thus shows the g.s. of 206 Pb to contain weak p 1/2 −2 admixtures with dominant 6.3.1 Distribution of configurations with the g 9/2 particle in 206 Pb Table 8 shows the strength distribution for the configurations g 9/2 f 5/2 and g 9/2 p 3/2 in 206 Pb. Most spin assignments are considered to be valid. Yet some results are problematic. In the following we discuss the results in detail. Figures 19 and 20 compare the distribution of the GNPH configurations in 206 Pb to 1p1h configurations in 208 Pb, both excited near the g 9/2 IAR. Fig. 21 show the 1p1h configurations complementing the configuration strength in the states at E x < 4.8 MeV in 208 Pb. Fig. 22 shows excitation energies calculated by SDI for the 1p1h configurations in 208 Pb; the GNPH configuration g 9/2 p 1/2 ⊗2 + 1 is included. In Sect. 6.3 details of the comparison are discussed. As a general result the strength distribution for GNPH configurations in 206 Pb (Fig. 19) is found to be similar to the well known distribution for 1p1h configurations in 208 Pb.
One 2 − state is observed in 206 Pb. The 3713 state contains about half of the g 9/2 f 5/2 strength. The excitation energy being 0.5 MeV lower may be explained by a strong mixing between the 1p1h configuration g 9/2 f 5/2 E S DI x = 4304 and 2 − 1 ⊗ g 9/2 p 1/2 E calc x = 4230 ( Table  2). Two 3 − states are observed in 206 Pb. They contain 60% of the g 9/2 f 5/2 and 90% of the g 9/2 p 3/2 strength. The excitation energies are similar to E S DI x . Ten 4 − states are observed in 206 Pb. The 4050 state contains the major g 9/2 f 5/2 fraction, the g 9/2 f 5/2 strength is widely distributed. The centroid energies [Eq. (4)] agree with E S DI x . Seven 5 − states are observed in 206 Pb. Both the g 9/2 f 5/2 and the g 9/2 p 3/2 strength is widely distributed. The 4215 state contains a major g 9/2 p 3/2 fraction. The centroid energies agree with SDI. Five 6 − states are observed in 206 Pb. The 4011 state contains a major g 9/2 f 5/2 fraction, the 4453 state contains a major g 9/2 p 3/2 fraction. The centroid energies agree with SDI. Two 7 − states are observed in 206 Pb, each states contains about 30% of the g 9/2 f 5/2 strength. The centroid energy roughly agrees with the SDI calculation. The splitting is explained by a strong mixing between g 9/2 f 5/2 ⊗0 + g.s. and g 9/2 f 5/2 ⊗2 + 1 . Table 5 compares calculated angular distributions for configurations g 9/2 l j with l2 j = p1, p3, f 5, f 7 to best fits. It shows calculated angular distributions with dominant g 9/2 f 5/2 and spin 2 − and 7 − and weak admixtures of g 9/2 f 7/2 in Fig. 6, for spin 3 − and 6 − with dominant g 9/2 f 5/2 and g 9/2 p 3/2 and weak admixtures of g 9/2 f 7/2 in Fig. 7, for dominant g 9/2 f 5/2 and g 9/2 p 3/2 and admixtures of g 9/2 p 1/2 with spin 4 − with in Fig. 8 and with spin 5 − in Fig. 9, for dominant g 9/2 p 1/2 and g 9/2 p 3/2 and weak admixtures of g 9/2 f 5/2 with spin 4 − with in Fig. 6e and with spin 5 − in Fig. 6f. Best fits with spins and configuration mixing derived by investigating sum rules and orthogonality relations are printed bold face in Table 8. Best fits of angular distributions for spin 4 − and two different sets of amplitudes are shown in Fig. 11, for unique spin assignments in Figs. 12 and 13, for different sets of amplitudes but similar shapes of angular distributions in Figs. 14, 15, 16, 17 and 18. Spins in parentheses denote assignments which are discarded by the investigation of sum rules and orthogonality relations [Eq. (3)]. In Table 4 discarded assignments are denoted by N dscd . The dominant configuration or the two major configurations out of the three configurations g 9/2 p 1/2 , g 9/2 p 3/2 , g 9/2 f 5/2 are shown in Cols. 9-11.

Comparison of calculated angular distributions to best fits
The shape of angular distributions varies with spin and amplitudes of particle-hole configurations in a charactistic manner. The relative cross sections vary by factors from about 0.2 to 3 for configurations with amplitudes near one (pure configurations). Figure 4 in [46] shows examples. The mixing of several configurations varies the relative maxima and minima even more ( Table 3). The shape of configurations with one dominant amplitude and admixtures of less than about 10% strength already vary the relative cross section by factors up to 2.
In order to reduce the variation a linearization is introduced with the function P 2 (cos Θ), see Fig. 3 in [46]. By this means the cross sections near Θ = 120 • and 140 • change less, marks indicate the introduced non-linearity in Figs. 3 and 4 in [46] and in Figs. 6, 7, 8, 9, 14, 15, 16, 17 and 18. Admixtures of g 9/2 f 7/2 to states with dominant configuration g 9/2 f 5/2 and spin 2 − or 7 − change the shape of the angular distribution sensitivily. The change is more pronounced for spin 7 − than for 2 − . Variations of g 9/2 f 7/2 admixtures to states with dominant configurations g 9/2 f 5/2 and g 9/2 p 3/2 for spin 3 − or 6 − follow the ratio of the two amplitudes from −1 to +1 in charactistic manner. The change is more pronounced for spin 3 − than for 6 − . Variations of g 9/2 f 7/2 admixtures to states with an already strong g 9/2 p 1/2 component and dominant configurations g 9/2 f 5/2 and g 9/2 p 3/2 for spin 4 − or 5 − follow the ratio of the three amplitudes from −1 to +1 in a systematic manner which however leads to extreme changes.
In addition to the variation of the shape of the angular distributions the strong dependence of the s.p. width Γ s. p. L J , Γ s. p. l j on the proton energy enhances the cross with decreasing proton energy [39] (Fig. 8 in [29]). The change of the s.p. phase ξ s. p. l j with the proton energy is weak [13] and ignored for the lead isotopes. For ξ s. p. h 9/2 a crude guess is sufficient because admixtures from g 9/2 h 9/2 are relevant only in few cases.
In Table 5 Col. 5 shows the number of the calculated angular distribution in Figs. 6, 7, 8 and 9. The drawn curve is denoted by the figure number, the dotted curve by the overlined figure number (Cols. 5-7). In several cases two or three calculations are similar. For spin 4 − and 5 − often no good agreement is found. Only the nearest approximation is shown.
6.4 Comparison of particle-hole configurations in 206 Pb and 208 Pb Table 4 shows excitation energies calculated by SSM and SDI and calculated mean cross sections. The range of excitation energies is limited to 3.7 < E x < 4.7 MeV.
- Table 4 in [51] shows amplitudes of 28 negative parity states at 2.6 < E x < 5.2 MeV in 208 Pb. The results from an update done in 1982 were slightly improved in 2017. Among the 43 neutron 1p1h configurations most larger amplitudes were measured for all IARs (L J = g 9/2 , i 11/2 , j 15/2 , d 5/2 , g 7/2 , d 3/2 ). Amplitudes for g 9/2 h 9/2 may be determined from data taken in 1968 [30] but are not yet evaluated except for the admixture in the 1 − yrast state [67]. Cross sections involving the intruder i 13/2 are vanishingly small.
-Sect. 6.3 discusses the strength distribution in 206 Pb. The expected absencs of the configuration g 9/2 p 1/2 is proven by spectra taken at 2.6 < E x < 6.0 MeV [20,73]. The 2648 3 − yrast state is weakly excited. Half a dozen weak levels show up at 3.0 < E x < 3.7 MeV and 4.7 < E x < ∼ 5.6 MeV.

Global comparison
Energies near 3.5 MeV predicted for dominant g 9/2 p 1/2 components are expected to be absent in 206 Pb. Configurations g 9/2 f 7/2 are observed at E x ≈ 5.7 MeV [65]. They contribute weak admixtures to states at 3.7 < E x < 4.7 MeV. Table 6 shows the sum rules for the three configurations g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 p 3/2 in (top) 206 Pb and (bottom) 208 Pb. Three ranges of excitation energies are chosen, E x < 3.7 MeV, 3.7 < E x < 4.17 MeV, E x > 4.17 MeV. In 208 Pb the full sum rules are observed within the uncertainties of the IAR parameters. The only exception is found for the 7 − state (Sect. 2.6). Note that part of this achievement is obtained by the orthonormality and sum-rule relations as a constraint [24]. The splitting of the g 9/2 p 1/2 , g 9/2 f 5/2 , g 9/2 p 3/2 multiplets is rather well reproduced by SDI calculations.
The analysis of angular distributions for 206 Pb(p, p ) started with the finding of similar sums of cross sections for 206 Pb and 208 Pb (Sect. 3.2). Indeed the total cross sections for spins from 2 − to 7 − agree for 206 Pb and 208 Pb in detail (Cols. 19, 20 in Table 6). The sum rules for 206 Pb and 208 Pb and spins from 2 − to 7 − agree mostly. Yet there are clear differences.
In 206 Pb the sum rules for g 9/2 p 3/2 are overestimated for spins 4 − , 6 − and for g 9/2 f 5/2 underestimated for spins 2 − , 3 − , 6 − . One reason is the arbitrary partitioning of doublets by assuming equal cross sections for each spin. For spin 2 − and 3 − another reason is the low cross section because of the spin factor. The g 9/2 p 1/2 sum rule is a measure of the impurity of the g.s. (Sect. 2).
The centroid energies for g 9/2 f 5/2 and g 9/2 p 3/2 in the two chosen energy ranges and in total are reproduced by the SDI calculations (Table 2) in a reasonable manner.
The comparison of particle-hole configurations in 206 Pb and 208 Pb show remarkable similarities and clear differences as shown in the following. The correspondence of levels observed in 206 Pb to non-1p1h states in 208 Pb may be given for a few states. The 4110 level is identified with the 4116.7±1.8 keV 2 + state (Table 4). It apparently corresponds to the 4086 2 + state with tetrahedral configuration [51]. The 4340 level may contain besides the 5 − state a 4 + state corresponding to the 4324 4 + state with tetrahedral configuration [51]. The 4500 level may contain besides the 6 − state a 6 + state corresponding to the 4424 6 + state with suggested icosahedral configuration [52].

Detailed comparison
Figures 19 and 20 compare the strength distribution for configurations with the g 9/2 particle in 206 Pb and 208 Pb. Fig. 21 shows the distribution for the other configurations in 208 Pb, especially the proton configurations and configurations excited on the i 11/2 IAR. Fig. 22 displays distributions calculated by SDI and weak coupling. Table 8 gives excitation energies and configuration strengths for 206 Pb, Table 4 in [51] for 208 Pb. Finally assumed spin assignments are shown in Tables 4, 5, 8.  Table 7 correlates specific states in 206 Pb to states in 208 Pb with similar excitation energies and similar relative strengths for g 9/2 f 5/2 and g 9/2 p 3/2 .
Similarities and differences are discussed in the following.
1. The number of states differs for 206 Pb and 208 Pb in the region 3.7 < E x < 4.7 MeV.
(a) One 2 − state is observed in 206 Pb at E x = 3.7 MeV. The excitation energy is 0.5 MeV lower than predicted by SDI. The down shift may be explained by the mixing among the configurations g 9/2 f 5/2 ⊗0 + g.s. and g 9/2 f 5/2 ⊗2 + 1 . Apparently higher 2 − states are not observed because of the weak cross section. Two 2 − states are known at E x = 4.2 MeV in 208 Pb, the 4140 2 − state interpreted as tetrahedral configuration and the 4230 2 − state with dominant g 9/2 f 5/2 and sizeable d 5/2 p 1/2 admixture. The 5038 state is the next one with major d 5/2 p 1/2 and f 7/2 d 3/2 . Summing up, for spins 4 − , 5 − , and 6 − the number of states in 206 Pb is about twice the number of states in 208 Pb. The additional states are explained by the GNPH configurations at E x = 4.23 MeV. For spin 2 − the excitation energy of the single observed state and for spin 7 − the two excitation energies differ much from 208 Pb. 2. The summed g 9/2 p 1/2 , g 9/2 p 3/2 , g 9/2 f 5/2 strength differs for 206 Pb and 208 Pb in the region 3.7 < E x < 4.7 MeV for some spins.
The reason for the missing strength for spins 2 − , 3 − , 7 − is related to the difficulty to find levels with weak cross sections and the difficult separation of doublets. The total g 9/2 p 3/2 and g 9/2 f 5/2 strengths approach unity for spins 4 − , 5 − , 6 − . 3. A close correspondence between 206 Pb and 208 Pb is found for several states. The energy difference ΔE x = E x ( 206 Pb) − E x ( 206 Pb) is sometimes remarkably small. Summing up, small energy differences ΔE x are found in astonishingly many pairs of states in the two lead isotopes. Several times the correspondence to some state in 208 Pb is given by two states in 206 Pb. to the g.s. some strength is expected. The 5 − state with dominant i 11/2 p 1/2 strength at E x = 4.1 MeV should be absent; the configuration i 11/2 p 1/2 ⊗2 + 1 is expected 0.8 MeV higher. The mixing between the configurations g 9/2 f 5/2 ⊗0 + g.s. and g 9/2 f 5/2 ⊗2 + 1 separated by 0.8 MeV creates GNPH configurations at E x = 4.23 MeV. Indeed at 3.9 < E x < 4.6 MeV seven states are found whereas in 208 Pb four states are known. (b) The 4206 6 − state in 208 Pb with almost the complete i 11/2 p 1/2 strength has no correspondence in 206 Pb. (c) The absence of the configuration i 11/2 p 1/2 for spin 6 − and the additional GNPH configurations at E x = 4.23 MeV explain the observation of nine 6 − states at 3.8 < E x < 4.7 MeV. In 208 Pb only four states are known. Summing up, the number of states in 206 Pb being larger than in 208 Pb is explained by the appearance of the configuration g 9/2 l j ⊗ 2 + 1 for l j = p 1/2 amidst the region E x ≈ 4.17 MeV which divides g 9/2 f 5/2 from g 9/2 p 3/2 .
The missing correspondence to 4206 state in 208 Pb with the complete i 11/2 p 1/2 strength is expected by the two missing p 1/2 neutrons in 206 Pb. The g 9/2 p 1/2 strength for spins 4 − and 5 − amounts to about one half.

Summary and conclusion
The proton decay of the 9 2 + IAR in 207 Bi at E p = 14.935 MeV populates 22 levels in 206 Pb [19,20,73]. Thirtytwo states in 206 Pb are identified at 3.7 < E x < 4.7 MeV in 22 levels (Tables 4, 7, 8). Two levels contain three states, six levels two states. The cross sections are assumed to be equal for each pair or triple of the states.
The comparison of angular distributions of the inelastic proton scattering via an IAR for the two isotopes 206 Pb and 208 Pb is more interesting than details for 208 Pb(p, p ).
The strength distribution for the configurations g 9/2 p 3/2 and g 9/2 f 5/2 determined for 32 states in 206 Pb (Fig. 19) resembles the corresponding strength distribution for 24 states in 208 Pb (Fig. 20). In both nuclei almost the complete 1p1h strength for g 9/2 p 3/2 and g 9/2 f 5/2 is localized except for some spins (2 − and 3 − in 206 Pb and 7 − in 208 Pb). The 1p1h strength is distributed in 206 Pb among twice the number of states than in 208 Pb.
The g 9/2 p 1/2 strength in 206 Pb is found to be weak but not absent. The g.s. of 206 Pb is shown to contain sizeable p 3/2 −2 and f 5/2 −2 admixtures to the dominant p 1/2 −2 .
The detailed comparison of three dozen states in 206 Pb and two dozen states in 208 Pb at excitation energies of 3.7 < E x < 4.7 MeV yields remarkable similarities and clear differences. In a few cases one state with a dominant 1p1h configuration in 208 Pb corresponds to a close pair of states in 206 Pb with similar excitation energies and similar configuration mixings. In other cases the presence of a low lying 2 + at E x = 803 keV in 206 Pb state splits one state with a dominant 1p1h configuration in 208 Pb into two largely separated GNPH states in 206 Pb.
The mixing of three hole pairs in the g.s. of 206 Pb explains the appearance of states with considerable g 9/2 p 1/2 components not expected in the simple schematic shell model.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.   19 Distribution of GNPH configuration strengths with the g 9/2 particle at 3.1 < E x < 4.9 MeV in states of 206 Pb involving holes l j = p 1/2 , f 5/2 , p 3/2 . See Sects. 6.3, 6.4 for details Fig. 20 Distribution of 1p1h configuration strengths with a g 9/2 particle at 3.1 < E x < 4.9 MeV in states of 208 Pb involving holes l j = p 1/2 , f 5/2 , p 3/2 . See Sect. 6.4 and Table 4 in [51] for details  Fig. 20 for other 1p1h configurations but g 9/2 l j with l j = p 1/2 , f 5/2 , p 3/2 (see Table 4 in [51]). Admixtures from d 5/2 p 1/2 in the 2 − and 3 − states are not shown for clarity. The 4230 2 − state contains 4% d 5/2 p 1/2 strength. The 4051, 4255, 4698 3 − states contain 0%, 1%, 21% d 5/2 p 1/2 strengths  Table 2 shows 1p1h configurations in 208 Pb and all GNPH configurations in 206 Pb at E SSM x < 4.8 MeV. Vertical lines delineate the region 3.7< E x < 4.7 MeV discussed in this work. The GNPH configurations 2 + 1 ⊗ g 9/2 p 1/2 , 2 + 1 ⊗h 9/2 s 1/2 , 2 + 1 ⊗i 11/2 p 1/2 are shown in a simplified manner by including calculations with SDI. They lie mostly outside the discussed region. The interference pattern observed for 206 Pb(p, p ) on the g 9/2 IAR in 209 Bi does not allow to distinguish configurations g 9/2 l j ⊗ 2 + 1 from g 9/2 l j ⊗ 0 + g.s. . Only configurations involving the g 9/2 particle are discussed in the analysis of 206 Pb(p, p ). The twelve configurations i11/2 p 1/2 and g 9/2 p 1/2 ⊗2 + 1 at 4.115, 4.290 and 4.918, 5.093 MeV, respectively, are expected to not exist in the simplified model but seven of them are shown