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Introduction
The heaviest doubly magic nucleus 208 Pb is fascinating as it offers the opportunity to study different classes of nuclear excitations. Most states in 208 Pb are described in the shell model (SM) by one-particle one-hole (1p1h) configurations. More than two dozen states below the neutron and proton threshold (S(n) = 7368 keV, S( p) = 8004 keV [1]) are recognized to be not described as 1p1h configurations.
At E x ≈ 6.0 MeV configurations built by the coupling of the 3 − yrast state with 1p1h configurations were identified in eighteen states [2]. Ten states at 2 < E x < 8 MeV were recognized as tetrahedral configurations [3,4]. Four states at E x < 5.9 MeV are pairing vibrations [5][6][7]. Four states at E x < 6.2 MeV are suggested to be icosahedral configurations [8]. a e-mail: A.Heusler@mpi-hd.mpg.de (corresponding author) Negative parity states below E x = 7.0 MeV are almost completely identified by this work. It allows to find positive parity states and non-1p1h states [9] by comparison to SM calculations [10][11][12][13].
The surface delta interaction (SDI) was introduced by Green and Moszkowski [14] and extensively discussed by Talmi [15]. It provides a simple method to understand the interaction among two particles in a complex nucleus [2,[15][16][17][18].
The essential tool to determine amplitudes of 1p1h configurations in states of 208 Pb derives from the study of isobaric analog resonances (IAR) . It allows to obtain information about the residual interaction among one-particle one-hole configurations [31][32][33][34].
The proton decay of IARs allows to determine amplitudes of 1p1h configurations including the sign from the interference pattern in the angular distributions of (p, p ) [32,33,43,45,47,48]. The orthogonality and sum rule relations allow to determine amplitudes of unobserable configurations (either because of vanishing cross sections or because of the absence of a target state) [9,43,49].
Because of the large amount of information an appendix separated from the main text shows more figures, excitation functions from 1968 [28], spectra taken in 1969 with semiconductor detectors [41,42], and spectra taken in 2003-2016 with the Q3D magnetic spectrograph [51,52,57,58]. Finally the appendix includes a survey of the excitation energies of states at 6.20 < E x < 7.15 MeV with spin and parity assignments complementing Table VI in [50].

Isobaric analog resonances
IARs in heavy nuclei were discovered in 1963 [19]. Accelerators with the capability to excite IARs in the lead region with proton energies of 14-20 MeV became available shortly later. The Max-Planck-Institut für Kernphysik (MPIK) at Heidelberg (Germany) bought a MP Van-de-Graaf accelerator for this purpose in 1967.
The high potential of the investigation of IARs in heavy nuclei was recognized by Robson as he wrote at the end of the discussion during the conference on "Isospin in Nuclear Physics" [65] (p. 507): On the experimental side it is safe to predict that there is a gold mine of information still to be dug out. It is becoming clear that more accurate data as well as sophisticated measurements will be what we need. ... Nevertheless, there appears to be a tremendous amount of work still to be done before we will have a reasonable complete set of data and a satisfying understanding of it.
This paper is the first one to identify three dozen states with the structure g 7/2 p 3/2 , g 7/2 f 5/2 and d 3/2 p 3/2 , d 3/2 f 5/2 , and spins from 1 − to 6 − . The excitation energies are in the range 6 < ∼ E x < 7 MeV. Fourteen states with spins from 3 − to 6 − are newly identified. Admixtures of other 1p1h configurations to states near E x = 5.9 MeV with dominant d 3/2 p 1/2 strength and spins 1 − , 2 − and with dominant g 7/2 p 1/2 strength and spins 3 − , 4 − are determined with high precision from angular distributions [39] and excitation functions [28]. Essential 207 Pb(d, p) data were obtained with the Q3D magnetic spectrograph at the MLL in the 1990s using polarized deuterons [55,56] and in 2004-2018 with unpolarized deuterons [57,58]. The 208 Pb(d, d ) reaction [57,58] exciting all states without any preference for spin, parity, and structure helped to resolve ambuigities in the interpretation of angular distributions for 208 Pb(p, p ).
The 209 Bi(d, 3 He) reaction [60,61] does not excite states at E x > 6 MeV with measurable cross sections. No data from the reaction 208 Pb(α, α ) exciting preferentially natural parity states exist for E x > 6 MeV [71].

Description of the proton decay of IARs
Inelastic proton scattering via an IAR is described in detail in [32,33]. Complementary descriptions are given in [43,45,47,48]. Computer codes to calculate angular distributions are provided [72,73]. Here a short summary is given.

Description of 1p1h states
Most states in 208 Pb are described by a superposition of 1p1h configurations L J +1 l j −1 (shortly L J l j) where L J describes the particle with orbital angular momentum L and spin J and l j the hole accordingly, The amplitudes c obey the orthonormality and sum-rule conditions L J l j L J l j cẼ x ,I π L J l j cẼ x ,I π L J l j = 0, L J l j cẼ x ,I π L J l j 2 = 1, Ẽ x Ẽ x cẼ x ,I π L J l j cẼ x ,I π L J l j = 0, Ẽ x cẼ x ,I π L J l j 2 = 1 (3) for each spin I π . The mixing with non-1p1h configurations is negligible, non−1 p1h cẼ x ,I π non−1 p1h For the 2615 3 − yrast state interpreted as a pure tetrahedral rotor [3,4], however, weak admixtures of 1p1h configurations are observed ( Table 4 in [3]). They produce interference patterns with the non-resonant (p, p ) excitation across all IARs in 209 Bi at 14 < E p < 22 MeV ( Fig. 1 in [21], Fig. 3 in [22], and Fig. 1 in [74]). A first excited state with spin 3 − is a rarity among all nuclei. Besides the 3 − yrast state in 208 Pb the only other known example is in the doubly magic 146 64 Gd 82 [75]. For all positive parity states resonantly excited by the j 15/2 IAR [45] and for all known negative parity states except for the 3 − yrast state, the interference for the 208 Pb(p, p ) reaction via an isolated IAR with the non-resonant excitation is negligible. The asymmetry of the excitation functions is explained by the logarithmic dependence of the single particle (s.p.) widths on the proton energy [76] (Sect. 3.3.6). The asymmetry leads to a rather constant cross section continuing beyond the IAR for several hundred keV [20][21][22][23]45].
In the schematic shell model (SSM) the particle L J and the hole l j couple to 1p1h configurations L J l j without residual interaction [45]. By including the SDI the multiplet splitting of the 1p1h configurations L J l j is explained [17,18]. It varies with −0.73 < E S DI x − E SSM x < +0. 38 MeV. SM calculations with the M3Y interaction [11][12][13] do not yield much differing excitation energies up to E x ≈ 7.0 MeV, but the number of states differs (Sect. 4

.3.3).
The proton decay of IARs in 209 Bi excites all neutron 1p1h configurations in 208 Pb. The 207 Pb(d, p) reaction [1,55,56] excites the 1p1h configurations with all particles L J in 209 Pb coupled to the p 1/2 hole.

Mean cross section and centroid energy
Mean cross section. The mean (angle integrated) cross section for the reaction 208 Pb(p, p ) on a certain IAR L J for a state with pure configuration L J l j and spin I π is l j a L J l j .
The formalism [32,33] derives the factor a L J l j . Table 3 shows relevant values for pure configurations [72,73]. The s.p. widths depend on the energy of the incoming and outgoing protons in a logarithmic manner [76] (Fig. 8 in [40], Fig. 14 in [78]). At a certain proton energy in the range 7 < E p < 14 MeV the ratio of some s.p. width Γ s. p. L J to Γ s. p. L J for L = L + 2 is roughly a factor four for L = 0, · · · , 7 and similarly for l j ( Table 2). Figure 1 shows the strength distribution for all neutron 1p1h configurations at 3 < E x < 7 MeV in a schematic manner. The excitation energies and the cross sections are not to scale. Only the ratios of the cross section within a certain multiplet are to scale.
Centroid energy. The centroid energy is calculated as where states with order number m are selected to contain major fractions of the configuration L J l j. Because of the strong dependence of the s.p. widths on the angular momentum l, weak admixtures of configurations L J p 1/2 and L J p 3/2 to states with dominant configuration L J f 5/2 change the value E x considerably.

Description of angular distributions
The angular distributions near a single IAR are described by a series of even order Legendre polynomials P K [32,33], l j cẼ x ,I π L J l j cẼ x ,I π L J l j .  [25].) The number of states with different spins and their strengths are shown in rough relations, the excitation energies are only approximately to scale. The cross section of the configurations with an i 13/2 hole is unobservable small. The parity of the states is negative except if the one-particle one-hole configuration contains one intruder, either j 15/2 or i 13/2 Here Λ describes the excitation function (Sect. 2.2.4), a K the geometry of the recoupling, Γ s. p. l j the s.p. widths [40], and φ a phase difference. Computer codes doing the calculation of the coefficients a K together with tables and figures are available [72,73].
(9) Table 1 shows the resonance energies E res and total widths Γ tot for all known IARs in 209 Bi. Table 2 shows the s.p. widths Γ s. p. for the particle orbits L J in 209 Pb and the hole orbits l j in 207 Pb.

Description of excitation functions
The excitation function for the reaction 208 Pb(p, p ) via the IAR L J exciting the state Ẽ x , I π M [Eq. (1)] is described by a Lorentzian [Eq. (4a) in [32]] Figure 2 shows calculated excitation functions covering the s 1/2 , g 7/2 , and d 3/2 IARs for a range of proton energies within about ten total widths Γ tot . In reality the shape of the excitation function is highly asymmetric [21,22]. The asymmetry is explained by the logarithmic energy dependence of the s.p. widths [76] (Fig. 8 in [40]).

Shape of angular distributions
Equation (7) describes the angular distribution for the proton decay of an IAR L J into the state |Ẽ x , I π . The shape is isotropic for orbital angular momentum L = 0 (the s 1/2 IAR) or the p 1/2 hole in case no other configuration admixes [Eq. (9)].
The s.p. width for l = 2 is roughly four times stronger than for l = 4 (Sect. 2.2.2). Hence weak admixtures of configurations with a p 1/2 or f 5/2 hole to a stateẼ x , I π with dominant configuration L J p 3/2 creates an angular distribution with a strong P 2 component.
In order to enhance the appearance of a dominant P 2 shape, P 2 is chosen as the abscissa thus leading to an almost linear function in most cases. Figure 3 shows examples with the distributions

Configurations excited by IARs in 209 Bi
IARs in 209 Bi excite neutron 1p1h configurations in each state in 208 Pb (Fig. 1). The coupling of a particle L J to a hole l j creates multiplets with 2-14 members. The cross sections are proportional to the spin factor 2I + 1 [Eq. (7)].
All neutron 1p1h configurations have negative parity except for those built with one intruder, j 15/2 or i 13/2 for neutron configurations, i 13/2 and h 11/2 for proton configurations. The configuration j 15/2 i 13/2 has negative parity.

Configurations at
SM calculations using the SDI [15] predict the multiplet splitting of 1p1h configurations. It also explains the ground state of even-even nuclei to have spin 0 + because of the large down shift in the multiplet splitting with two nucleons in the same orbit [15,16]. The interaction among two nucleons to be strongly localized is verified by the calculation of the wave functions for neutrons in orbits 82 ≤ N ≤ 184 and protons in orbits 50 ≤ N ≤ 126 shown in Figures 3,4 in [79].
The coupling of the 3 − yrast state to 1p1h configurations produces states at E x > ∼ 6.0 MeV [2]. A few more non-1p1h states are suggested at E x < 7.2 MeV [8]. Because most of these states have positive parity, they are of little interest in this paper. Figure 4 shows the shape of the angular distributions for pure configurations with a d 3/2 , g 7/2 particle and a p 3/2 , f 5/2 hole and spins from 0 − to 6 − . The angular distributions for pure configurations with a p 1/2 hole are isotropic. Note the similarity of the shapes for d 3/2 p 3/2 , d 3/2 f 5/2 , g 7/2 p 3/2 with corresponding spins I = L − l, I = L − l + 1, I = L − l + 3 with the exception of I = L − l + 2 for d 3/2 p 3/2 . The shape for g 7/2 f 5/2 with I = L − l + 5 is much steeper than the shapes for d 3/2 p 3/2 , d 3/2 f 5/2 with I = L − l + 3.

Angular distributions on the g 7/2 +d 3/2 doublet IARs
The interference between the different IARs cannot be calculated. (An attempt was made [80].) However, for spins 0 − , 5 − , and 6 − no interference between the two close IARs g 7/2 and d 3/2 and configurations with p 3/2 or f 5/2 holes is possible. For spin 1 − the shape of the angular distribution for the configurations, d 3/2 p 3/2 , d 3/2 f 5/2 , and g 7/2 f 5/2 is similar and hence a distortion of the angular distribution is expected to be weak.
For spins 2 − and 3 − the three angular distributions for d 3/2 p 3/2 , d 3/2 f 5/2 , and g 7/2 f 5/2 are dissimilar and hence a distortion by the interference between different IARs is expected to be large. For spin 4 − the ratio of the s.p. widths Γ s. p. p 3/2 and Γ s. p. Table 2) leads to the expectation that a dominant contribution of g 7/2 p 3/2 should yield a rather undistorted shape of the angular distribution.
The shape of angular distributions for spin 3 − with a strong g 7/2 p 3/2 component does not differ much from the shape for spin 4 − (Fig. 4). Here the SDI calculation of the multiplet splitting helps (Table 3, Sect. 3.4.1).

Excitation functions near the g 7/2 + d 3/2 doublet IARs
The g 7/2 and d 3/2 IARs build a strongly overlapping doublet of IARs (Fig. 2 for pure configurations L J l j (Fig. 2) are expected to be The ratios R 47/40 , R 75/40 determined from the fit of angular distributions can be estimated from Table 4   In 1965-1969 experiments on 208 Pb(p, p ) were performed using semiconductor detectors at Austin (USA) and Seattle (USA) [22][23][24][27][28][29][30]. Angular distributions and excitation functions were taken for levels at 2.6 < E x < 6.9 MeV near all IARs (Table 1). Figures 16,17 reproduce excitation functions taken at Seattle [28].     In 1966 208 Pb(p, p ) spectra were taken with the Elbeck magnetic spectrograph at Los Alamos (USA) with a resolution of about 8 keV [25,26]. 48 levels at 2.6 < E x < 6.1 MeV were detected at 12 proton energies 14.95 < E x < 17.50 MeV covering the g 9/2 IAR with three proton energies, the d 5/2 IAR with four proton energies, the s 1/2 , g 7/2 , d 3/2 IARs with five proton energies. (More original spectra taken at Los Alamos are still available [26].) Cross sections at the single scattering angle Θ = 90 • were measured. Twelve spins of states at E x < 5.5 MeV were shown. All spin assignments are still valid. The assignment of spin 3 − to the first excited state [81] is valid since seventy years. The identity of the 3 − yrast and three higher excited states is known since hundred years [82,83]. The excitation energies E Ek x obtained from 208 Pb(p, p ) data taken with the Elbeck magnetic spectrograph were used in subsequent papers [23,24,[27][28][29][30]. They have to be recalibrated because of the choice of the excitation energy of the 3 − yrast state incorrect by 8 keV (see Eq. (6) in [47]).
M. Martin noted that the discrepancy stems from the problematic calibration of the excitation energy for the 2615 3 − yrast state before 1960. The extremely high ex-citation energy was already observed by Rutherford in 1899 [84].
Relative cross sections at different scattering angles were measured with a precision of 2% [40]. Yet absolute cross sections were not determined [39]. Figures 22,23 show spectra covering the region 6.0 < E x < 7.0 MeV. Figures 24,25,26,27,28,29,30 and 31 show angular distributions for states with spins from 1 − to 6 − and excitation energies 5.8 < E x < 7.0 MeV. The evaluated experimental data shown in the angular distributions are presented in [41,42]. The data taking was described in [40]. In Sects. 3.2 some details are remembered.
In the 1990s experiments on 207 Pb(d, p) with polarized deuterons and on 208 Pb(p, p ) were performed at the MLL (Garching, Germany) using the Q3D magnetic spectrograph and evaluated by Valnion [55,56].
In 1992 experiments on 208 Pb(α, α ) were performed at the MLL (Garching, Germany) [71]. In this paper the data were not used but for inspecting the 4140 2 − state [3].

Reconstruction
As described in [40] data for 208 Pb(p, p ) were taken with the newly installed MP Van-de-Graaf accelerator at the MPIK in 1968-1969. In total about 6 months of beam time were used to cover the g 9/2 , d 5/2 , g 7/2 , and d 3/2 IARs. A group of around twenty students and more than twenty technicians were involved in preparing the experiments and performing the data taking. The evaluation was done by Glöckner [39]. Fotographic reproductions of the computer plots for 77 208 Pb(p, p ) spectra both original and fitted by the triangle method [40] are still available [41]. By help of the Engauge Digitizer code [85] scans of the fotographic reproductions were digitized. Using another computer code spectra were reconstructed as input for GAS-PAN [86]. Details for the description of GASPAN spectra can be found in e.g. [45] or [50]. The resolution of about 12 keV was used as a guide for the reconstruction.
The spectrum analysis code GASPAN [86] was used in the usual manner to fit the spectra. However the level energies were not fitted but had fixed values shown in Table 15. Figures 22, 23 show some spectra. In contrast to the simple triangle method [40] the advanced features of GASPAN were used to fit the contamination peaks from 12 C(p, p ), 16 O(p, p ), 208 Pb(p, d), and 208 Pb(p, t) with different peak widths. By this manner more levels of 208 Pb could be resolved in the neighborhood of the contaminations than found by Glöckner [39].
In general 208 Pb(d, d ) spectra show more states than 208 Pb(p, p ) because of the presence of positive parity states (Fig. 20), 207 Pb(d, p) show markedly different spectra (Fig. 21) because of the selective excitation of configurations with a p 1/2 hole. Even very weak admixtures can be measured [55,56].

Definition of peaks, levels, and states
Spectra with different resolutions were evaluated. Therefore the number of resolved nuclear states differs in different experimental data.
States are distinguished from levels and peaks. We further define a doublet as an ensemble of close lying (mostly two) states. Sometimes a member is known from other experiments only and is not excited at all.
A state is defined by its qualities in some nucleus. A peak is defined as some hump in a spectrum more or less resolved from neighboring humps. A level is defined as the result of a fit by some spectrum analysis tool resolved from its neighboring levels. We are using GASPAN [86].
In the fit of the data obtained in 1968-1969 [41,42] by GASPAN the option of fixed energies [45] was used. By this means a doublet with two or more states in a distance smaller than the resolution is separated into two levels.
For each enumerated level shown in Table 15 the excitation energy corresponding to the shown energy labelẼ x was selected, preferably from [1], otherwise the mean value obtained from the 208 Pb(p, p ), 207 Pb(d, p), and 208 Pb(d, d ) data obtained with the Q3D magnetic spectrograph was used [51,52,57,58].
Because the separation in spectra taken at similar scattering angles is affected by statistical fluctations, up and down jumps are produced in the angular distribution when comparing cross sections at consecutive scattering angles (see examples discussed in Sect. A.5). In a few evident cases the arithmetic mean for two neighboring levels is used instead the values given by the fit.
The evaluation is not yet finished. Only the most prominent levels are considered in this paper, about half of the resolved levels (Table 15). Angular distributions for 40 states in 37 levels were evaluated (Tables 3, 4

Calibration
By combining results from all experiments described in Sect. 3.1 both the excitation energies and the cross sections were calibrated.
Calibration of excitation energies. Calibration of excitation energies. Excitation energies at 6.2 < E x < 7.2 MeV were derived from existing Q3D data taken in 2003-2017 at the MLL [51,52,57,58]. Table 15 shows all known levels. The levels recognized in the 1969 Heidelberg data [39] are enumerated. The high resolution data for 208 Pb(p, p ) with E p = 35 MeV [87] obtained with the Enge split-pole magnetic spectrograph at Michigan (USA) well correlate with the data (Sect. 4.2, Table 15). Some levels not listed in [1] are strongly excited by 208 Pb(p, p ) via the g 7/2 +d 3/2 doublet IARs. The calibration of the excitation energies for the MPIK data was done in two steps. Glöckner [39] assumed a linear response of the semiconductor detectors and used the then available information on 208 Pb for strong peaks [88]. All contamination peaks from 12 C(p, p ), 16 O(p, p ), 208 Pb(p, d), and 208 Pb(p, t) were identified. In a second step the excitation energies of states listed in [1,89,90] with known spins of 1 − , 2 − , and 3 − were used. Table 15 shows all states identified at E x < 7.15 MeV. Calibration of cross sections. The spectra reconstructed from the data taken with semiconductor detectors in 1969 at the MPIK were fitted by GASPAN by using the feature of fixed energies in [45]. Contamination peaks from 12 C(p, p ), 16 O(p, p ), 208 Pb(p, d), and 208 Pb(p, t) [39] were fitted with broader widths and a free fit of the position. The ordinates of the spectra in the fotographic reproductions show only relative values, scales with 4, 6, 10, or 20 steps without any notations are drawn as ordinate [39]. Excitation functions of 208 Pb(p, p ) [28] and spectroscopic factors from 207 Pb(d, p) [55,56] assisted in finding the calibration of the cross section within a factor 0.8 (Sect. 3.4.3). The calibration of the relative cross sections across all scattering angles was done with an uncertainty of 2% [40]. The small uncertainty of the relative solid angles derives from the use of a scattering chamber with up to ten solid state detectors in 10 • distance. By turning the scattering chamber different detectors were positioned at the same angle. The angular distributions taken near the g 9/2 IAR discussed in [40] were well reproduced. By help of the reconstructed data [41] the 4140 2 − state [3,4] first stated by Glöckner [39] was observed in 208 Pb(p, p ) spectra [57,58]. 208 Pb(d, d ) spectra taken with the Q3D magnetic spectrograph at the MLL confirmed the 4140 2 − state interpreted as a tetrahedral rotation and vibration [59]. The final results indicate the correctness of the calibration also for the data not discussed in [40], namely all other IARs than g 9/2 and all off-resonance proton energies. The cross section at E p = 17.40 MeV for states with dominant configurations g 7/2 p 3/2 and g 7/2 f 5/2 is close to that at E p = 17.47 MeV (Eq. (11), Table 4). The g 7/2 resonance has a maximum at E p = 17.43 MeV and a width of Γ tot = 288 keV (Table 1, Fig. 2). The measured cross sections yield angular distributions where often gaps appear at certain scattering angles because of contamination lines or because the correspon-dence to known levels is unclear. Figures 24, 25 A systematic uncertainty of the cross sections in the order of 10% is estimated because of the problematic calibration. The systematic uncertainty of the amplitudes, however, is much lower because of the strong correlations.

Fit of angular distributions by superposition of three 1p1h configurations
The mean distance between any two states in 208 Pb is 9 keV. The distance between two states with the same spin and parity is larger than 30 keV. It is explained by the chaoticity [62][63][64]. The state contributing dominantly to a certain level in the MPIK data could be identified by the high resolution Q3D data (  (Tables 3, 4, 5, 6, 7, 8, 9,10,11). For all states mostly three configurations are sufficient to describe the measured angular distribution. The sensitivity is extremely high, amplitudes corresponding down to 0.1% strength can often be reliably determined together with the relative sign.
The shape of the angular distribution should not vary much with the proton energy E p . In this work three energies are considered, E p = 17.40, 17.47, and 17.75 MeV. The shape is determined either for E p = 17.40 or 17.47 MeV and not varied for the two other proton energies. The excitation energies calculated by SDI [17] (Fig. 4) and the cross sections for pure configurations (Tables 6, 7, 8, 9, 10, 11) are used as a guide to assign spins.
Assuming a weak mixing of the 1p1h configurations the shape should be given by the dominant component (Fig. 4,Sect. 3.4.4). For states with negligible admixtures of configurations with a p 1/2 or p 3/2 hole, configurations with a f 7/2 hole are included as the third configuration. These amplitudes from distant configurations are assumed to contribute less than 1% of the total strength. If because of the spin the p 1/2 or p 3/2 hole is absent configurations with a h 9/2 hole are included.

Sign of amplitudes
The (p, p ) reaction via IAR is highly sensitive to the relative signs of the amplitudes in the state. An example is given in [9]. Up to four amplitudes may be determined in each state if statistics are sufficient. Two methods are used for the analysis of angular distributions, the method used in this paper is described Sect. 3.3.6.
The high sensitivity in determining the sign of amplitudes is better explained by the method used earlier [32]. This other method starts with the fit of the angular distribution by a series of Legendre polynomials, where a 0 σ 0 is the mean cross section (Sect. 2.2.2). The uncertainty of the cross section is given by = δa 0 + δa 2 P 2 (cos Θ) + δa 4 P 4 (cos Θ) + δa 6 P 6 (cos Θ).
The relative sign of three amplitudes c 1 , c 2 , c 3 can be determined by observing how the ratio of the Legendre coefficients a K /a 0 depends on the size of the amplitudes. The A weak fourth amplitude changes the shape slightly, The uncertainty of the coefficients a K increases with K ; for a 6 it is very large and hence the value a 6 is mostly useless. We discuss two examples.
The 6389 state. Figure 5 shows the dependence of the amplitudes on the ratios a K /a 0 of the Legendre coefficients. Fig. 6a shows the fit of the angular distribution with Legendre polynomials. In a second step the mean cross section σ 0 a 0 is determined by adjusting the common factor a 0 in Eq. (15). The final result yields c 6389,5 − g 7/2 p 3/2 = 0.00 ± 0.02, The 6688 state. Figure 6b shows the fit of the angular distribution with Legendre polynomials. Figure 7 shows the dependence of the amplitudes from the ratios a K /a 0 of the Legendre coefficients.
Similarly as shown for the 6389 state the final result yields c 6688,5 − g 7/2 p 3/2 = +0.78 ± 0.01, The results given by Eq. (19), Eq. (20) differ from the amplitudes shown in Table 4 because of different choices of the s.p. widths thus exhibiting the systematic uncertainty (Sect. 3.3.5). The relative amplitudes are similar and especially the signs are the same, however.
The method presented here is more tedious than the method used by this work. It needs two separate steps. Hence iterations need more effort. The method is advantageous as it demonstrates the uncertainty of the amplitudes more clearly in many cases.

Parameter dependence
The fit of the angular distributions by Eq. (7) needs the total width Γ tot for the IAR, the s.p. widths Γ s. p. for the particle and hole orbits, and the phase differences φ (which all depend at the proton energy) and at the resonance energy E res .
The sensitivity of the fit on the resonance energy E res , the total width Γ tot , and phase differences φ is weak. It affects the variation of the mean cross section with the energy. Values for the resonance energy E res were slightly changed within the uncertainties derived by Wharton et al. [28], see Table 1. The improvement of the fits shown in Figs. 24, 25, 26, 27, 28, 29, 30 and 31 change the relative cross sections on the g 7/2 and d 3/2 IARs by about 10%, however the improvement is within the systematic uncertainty of the mean cross section. The resulting amplitudes shown in Table 4 are little affected.
In the fit, the s.p. width Γ s. p. for the particle affects only the absolute cross section and hence a common factor to the size of all amplitudes. In contrast, the s.p. widths Γ s. p. for the holes change all amplitudes in a complicated manner in the fit of the angular distributions.
We did not attempt to construct an orthonormal matrix of the amplitudes [Eq. (3)], but we tried to verify the predictions of the excitation energies calculated by the SDI and the calculations of the total cross sections with the known parameters (Tables 1, 2, 3). The results shown in the left frames of Figs. 8, 9, 10, 11, 12, 13 and Table 3 show the success (Sect. 4).
In the fit, the s.p. widths Γ s. p. for the holes change all amplitudes in a complicated manner. The uncertainty of the value of the dominant amplitude is related to the uncertainty of the corresponding s.p. width; however because of the interference with the other configurations the correspondence is not linear. The logarithmic dependence of the s.p. width for the holes was calculated by Clarkson as described in [76] and shown in [78], see also [72]. In the fit, the phase differences φ [Eq. (7)] affect the results only weakly. Phase differences calculated by Harney [33,72] were used.
In this paper we tried to optimize the parameters Γ s. p. (L J ), and E res (L J ). The values Γ s. p. (l j), Γ tot (L J ) and φ(l j, l j ) were not varied.
Each amplitude shown in Table 4 has unknown systematic uncertainties in the order of several percent, relatively. The relative signs of the amplitudes, however, are rather firm as discussed in Sect. 3.3.4.
The number of particle-hole configurations included in the fit (2-4) was adjusted until the sign of the weakest configuration could be reliably discriminated. The relative signs of the amplitudes shown in Table 4 are firm; the reverse sign given in parentheses changes the fit by more than 2σ in most cases (Figs. 24,25,26,27,28,29).

Iterative fitting process
The iterative fitting process is explained in more detail. Spin assignment and fit of the angular distributions are highly interrelated.
The shape is assumed to not depend on the proton energy. This fact was verified in many cases by comparison of angular distributions taken at proton energies E p = 16.30, 16.45, 16.60 MeV [41].
The constancy of the shape across an isolated IAR explains the absence of non-resonant contributions in all states. An exception is the 3 − yrast state [21,22,74] where all amplitudes are weak because of its dominant tetrahedral structure (Sect. 2.2.1). Together with the logarithmic dependence of the s.p. widths from the proton energy [76] the constant shape explains the evident asymmetry of the excitation functions ( Fig. 10 in [45]) already observed in the very first experiments on 208 Pb(p, p ) via the g 9/2 IAR for the 3192 5 − and 3475 4 − yrast states [21] and more states [22].
The change of the mean cross section from E p = 17.40 to 17.47 MeV predicted by Eqs. (11)-(13) served as a guide to select the particle L J . Only one major choice of the amplitudes for L J l j was done; the amplitudes were not varied for different proton energies E p except for the Lorentz factor [Eq. (10)]. For configurations g 7/2 l j the best fit at proton energy E p = 17.40 MeV is chosen and for d 3/2 l j at E p = 17.47 MeV. The cross section for E p = 17.75 MeV is predicted to be reduced by about a factor 5 in all amplitudes (a factor 25 in cross section) (Fig. 2, Eqs. (11), (12)).
The fit of an angular distribution consists of several steps. A major problem is the fact that no states with spins 4 − , 5 − , 6 − were known before.
1. In first approximation a single configuration is chosen. 2. For states with known spins 1 − , 2 − , 3 − the shape leads to rather good discrimination between the configurations L J p 3/2 and L J f 5/2 (Figs. 24, 25, 26). 3. The shape for states with unknown spin often approaches predictions for pure configurations reasonable (Fig. 4) thus giving an important hint to the spin assignment. 4. The mean cross section excludes lower spins in case the values are exceeding the prediction given in Table 3. By this means states with spin 4 − , 5 − , and 6 − were rather soon identified. 5. As an underlying assumption the major configuration L J l j should correspond to the prediction for the excitation energy by the SDI (Table 3) within less than 200 keV. 6. Especially the major strength L J p 3/2 should lie 0.33 MeV higher than L J f 5/2 but more precisely as predicted by including the multiplet splitting calculated with SDI (Table 3) 9. In case of L J p 1/2 values known from the 207 Pb(d, p) study [55] the given particle L J = g 7/2 or d 3/2 is prefered. The d 3/2 p 1/2 amplitude in the 6088 3 − , 6264 1 − , 6969 2 − states is weak, but the dominant configurations are g 7/2 l j. 10. In a second approximation the amplitude of another configuration L J l j is chosen and the sign determined. 11. In case the relative sign of the two amplitudes could be clearly determined, a third configuration L J l j is included. The change of the sign for the third configuration often leads to different fits. In Figs Table 4 shows the amplitudes c L J l j obtained by the fit, Table 5 the strengths c 2 L J l j . Figures 8, 9, 10, 11, 12 and 13 show the distribution of the detected strengths.

Criterions for a reasonable fit
The fit is optimized to reproduce the values at backward scattering angles Θ > ∼ 130 • because here the contributions from the Legendre polynomials differ most (Fig. 3). Cross sections at scattering angles Θ < ∼ 130 • were allowed to deviate more. For two doublets the fit was done for Θ < ∼ 130 • and Θ > ∼ 130 • separately, see Sects. 3.3.8, 4.1.7, 4.1.8.
A common set of amplitudes was used to optimize the fit of the angular distributions taken with the proton energies E p = 17. 40, 17.47, 17.75 MeV. The square root of the Lorentzian factor defined by Eq. (10) was applied to reduce the amplitudes in the fit of the angular distribution taken near the off-resonance energies (especially E p = 17.75 MeV). However the optimal fit was mainly searched near the top of the IAR, near E p = 17.40 MeV for configurations with the g 7/2 particle and near E p = 17.47 MeV for configurations with the d 3/2 particle, see Sect. 3.3.4 and Table 1.
As an effect of the fit with fixed energies (Sect. 3.2.1) fluctuations in the shape of the angular distributions show up. Fluctuations at some scattering angles could be removed by averaging values obtained from neighboring levels [42].

Fit of the two states in known doublets
Angular distributions of the two states in known doublets were fitted by regarding the backward angles for one state Table 5 Strength distribution for the configurations g 7/2 p 1/2 , g 7/2 f 5/2 , g 7/2 p 3/2 , and d 3/2 p 1/2 , d 3/2 f 5/2 , d 3/2 p 3/2 derived from Table 4  The differential cross section for the doublet is described by the sum of the two components [Eq. (7)] The differential cross section for the two states is described with the factor 0 < α 2 < 1 by The factors α and √ 1 − α 2 are displayed in Fig. 24 for the 6264 1 − state and in Figs. 30, 31.

Identification of states with dominant 1p1h strengths and spin assignment
The fit of the angular distribution by three 1p1h constituents often allows for disparate choices, see Sect. 3.3.4 and Fig. 5. The SSM acts as an important guide to determine the dominant 1p1h configuration and to decide the spin assignment. The calculation of the excitation energies by SDI together with estimates of the cross section (Table 3) refines the expectation (Tables 6, 7, 8, 9,10,11).
In the end the distribution of the sum of the 1p1h strengths and the centroid energies should reproduce this expectation. Figure 1 serves as an important guide to identify states. The distribution of the 1p1h strength across states in a typical range of less than 150 keV should yield a centroid energy close to the excitation energies calculated by the SDI (Table  6, 7,8,9,10,11). Figures 8, 9, 10, 11, 12 and 13 show the SDI energies in the left frames for spins from 1 − to 6 − .
The shape of the angular distribution for a state with one dominant configuration with a d 3/2 or g 7/2 particle is close to the shape shown in Fig. 4. (A complete list for all one-particle one-hole configurations built with particles and holes in the orbits 50 ≤ N ≤ 184 is found in [72].) This often allows to decide the spin and configuration assignment unambigously.

Dominant particle in the 1p1h configurations
The excitation functions of Wharton et al. [28] served as an important guide for the choice of the dominant particle in the 1p1h configurations. The excitation energies were recalibrated by Eq. (14).
For each level all contributing states could be uniquely

Dominant hole in the 1p1h configurations
Strongly excited states must have components of configurations with either a p 1/2 or p 3/2 hole because of the about four times stronger s.p. widths in relation to configurations with a f 5/2 hole [40]. The shape of the angular distribution should be close to the shape of the dominant configuration (Fig. 4).
Configurations with a dominant f 5/2 hole produce angular distributions with large coefficients of Legendre polynomial P 4 [Eq. (9)] thus deviating from the linearity introduced by the choice of P 2 as abscissa.

Presence of 1p1h configurations with a p 1/2 hole
An important guide to fit the angular distributions taken near the g 7/2 +d 3/2 IARs derives from the results for 207 Pb(d, p) with polarized deuterons by Valnion [55,56]. For many states one or two values of the amplitudes cẼ x ,I π L J p 1/2 [Eq. (1)] are determined for J = L ± 1 2 and L = 0, 2, 4, 6. The 207 Pb(d, p) results allow to give a good first guess of the configurations d 3/2 p 1/2 or g 7/2 p 1/2 . However for some states the dominant configuration does not have the d 3/2 or g 7/2 particle ( Table 4).
The 5947 1 − and 5924 2 − are known to bear the dominant d 3/2 p 1/2 strengths, and the 5874 3 − and 5969 4 − the dominant g 7/2 p 1/2 strengths. The angular distributions confirm the values from [55] for all four states. The 6088 state in the 6.09 MeV doublet level is identified to bear the missing g 7/2 p 1/2 strength for spin 3 − . The high sensitivity of the angular distributions reveals weak admixtures of d 3/2 p 3/2 , d 3/2 f 5/2 and g 7/2 p 3/2 , g 7/2 f 5/2 in these five states. The

Discussion
Among more than hundred known states in the region 6.2 < E x < 7.0 MeV this paper discusses half of them. The spin assignments should be taken with precaution; some of them are not entirely firm, however the parity assignments are firm because of the resonant excitation on IARs with known parity.
The spin assignment is based on the selective excitation on the g 7/2 and d 3/2 IARs in 209 Bi. Carefully measured angular distributions for 208 Pb(p, p ) at three proton energies close to the g 7/2 and d 3/2 doublet IAR provide the basis. Both the shape of the angular distribution and the mean cross section were investigated. The high sensitivity of the resonance reaction allows to determine up to four amplitudes of 1p1h configurations for each state. The good knowledge of the s.p. widths [40] is essential. The procedure to fit an angular distribution is described in Sects. 3.3.6, 3.4. An alternative method is described in Sect. 3.3.4.
Data taken with the Q3D magnetic spectrograph at the MLL in 2003-2019 for various reactions with a resolution of 3 keV (Table 15) allowed to find the dominant state for most levels observed with semiconductor detectors at the MPIK in 1969 with a resolution of 12 keV [39].
Data taken for 207 Pb(d, p) with polarized deuterons provided precise amplitudes for two dozen states [55]. They were important to calibrate the absolute cross sections (Sect. 3.3.2).
The purpose of the paper is to present a first survey of the strongest excited negative parity states in 208 Pb at 6.2 < E x < 7.0 MeV. No attempt is made to discriminate the parity for all states. In principle it is possible because negative parity states exhibit a change of the cross section across the IARs and the positive parity states tend to have low cross sections.
SM calculations [11][12][13]17,18] provide a guide to identify new states (Sect. 3.4, Tables 3, 4). The dominant configuration is suggested by the excitation energy being close to the predicted energy E S DI x . The shape of states with weak admixtures should be close to the shape for the pure configuration (Fig. 4). The resulting strength distribution is expected to not deviate much from the predictions (Figs. 8, 9, 10, 11, 12, 13). The final result confirms the expectation. The results explain the excitation functions measured by Wharton et al. [28] and reproduced in Figs. 16, 17 (Sect. 3).
We discuss only states where good results were obtained. A future analysis of available data [53] on 208 Pb(p, p ) will yield more results.
The calibration of the absolute cross sections which was not provided [39] is discussed in Sect. 3.3.2. It introduces systematic uncertainties in the size of the amplitudes but not in the relative signs.

0 − states
The two lowest 0 − states are a perfect example of a two-levels system [91,92]. Less than 3% strength of other configurations admix.
A search for levels with the unique shape of the angular distribution for d 3/2 p 3/2 (Fig. 4) was not successful. The excitation energy E S DI x = 7108 keV calculated with the SDI [17,18] is however above the threshold E x = 7.0 MeV chosen in the data taking of 1969 [41,42].
The fit of the angular distribution assuming configurations with the g 7/2 particle (Fig. 24) needs a strength of g 7/2 f 5/2 close to unity which is unlikely. The data points at Θ ≈ 120 • are not fitted. The decrease of the mean cross section from  Table 15 c Tetrahedral configuration [3] d Not yet identified e No data, the excitation energy is above the chosen limit E x = 7.0 MeV f Tentative assignment Table 7 Same as Table 6 for [keV]  Table 15 c Tetrahedral configuration [3] d From [44] e Not yet identified f This work     Table 4 derive from the fit curves shown in Fig. 24. Apparently the interference between the two IARs changes the shape of the angular distribution at the three proton energies only slightly.
The excitation energy E S DI x = 7015 keV for g 7/2 p 3/2 calculated with the SDI [17,18] is above the threshold E x = 7.0 MeV chosen in the data taking of 1969 [41,42]. Nev-ertheless half of the g 7/2 p 3/2 strength was found (
No angular distribution could be analyzed for 3 − states at 6.2 < E x < 7.0 MeV with order numbers M = 25, 27 because of lack of data. The 7020 3 − state with order number M = 29 [Eq. (2)] is recognized as tetrahedral configuration [3,4]. It is assumed to differ from the SM states with m = 28, 30.

The 6010 3 − , 6012 4 − doublet
The doublet with the 6010 3 − and 6012 4 − states was resolved only late [50]. With the resolution of 12 keV the two states build an unresolved doublet in the 1969 data. Nevertheless the doublet could be fitted (Eqs. Both states have about equal mean cross sections. Only the 6010 3 − state is excited by 207 Pb(d, p) [50]; the 6012 4 − state with unnatural parity is not excited. Valnion [55] determined the g 7/2 p 1/2 strength for the 6.01 MeV doublet. It is entirely assigned to the 6010 3 − state with the larger cross section. The amplitude c 60103 − g 7/2 p 1/2 = 0.53 allows to fix the ratio within about 10% [55,56]. The fit for the 6010 3 − state clearly yields the relative sign of the two amplitudes of g 7/2 p 1/2 and g 7/2 p 3/2 . The fit for the 6012 4 − state yields the three amplitudes g 7/2 p 1/2 , g 7/2 p 3/2 and g 7/2 f 5/2 with low uncertainties (Fig. 30).
Kulleck et al. [30] deduced a large coefficient A 4 from the fit of the angular distribution. It can be explained by a considerable admixture of g 7/2 f 7/2 and a negligible g 9/2 p 1/2 admixture to the 6012 4 − state [Eq. (9)]. A fit with four amplitudes (i.e. including g 7/2 f 7/2 ) was not possible because the 6.01 MeV doublet was not resolved in the 1969 data.
The ratio of the mean cross sections for the 6010 3 − and the 6012 4 − states is determined with 0.57 ± 0.05 : 0.43 ∓ 0.05 in agreement with the estimate [50]. Tables 4, 5 show the results, Fig. 30 the fits.
By neglecting the existence of a doublet the fit of the 6.09 MeV level yields a fit with a d 3/2 p 1/2 component with an amplitude of |c| = 0.3 for the 6086 2 − state in stark disagreement with the value 0.48 measured by Valnion [55]; the mean cross section is a factor two too low.
The existence of the 6086 2 − , 6088 3 − states in the doublet was recently shown [93]. The steep decline at 90 • < Θ < 120 • is explained by a strong d 3/2 p 1/2 component in the 6088 3 − state and the negative sign of a d 3/2 p 3/2 admixture yielding a mean cross section slightly higher than for the 6086 2 − state whereas the smooth increase of th cross section at Θ < 140 • < 170 • is well fitted by a strong d 3/2 p 1/2 component and a d 3/2 p 3/2 admixture with negative sign. In both cases the sign of the d 3/2 f 5/2 admixture is clearly determined (Fig. 31, Tables 4,  5).

New spin assignments
Fourteen states are newly identified or newly assigned a spin.
The progress in the evaluation of states in 208 Pb over hundred years is impressive [25,[81][82][83][84]. In the region 6.2 < E x < 7.0 MeV twenty states were observed in 1967 and no spin was known (Table 12). Now about 150 states are discerned (Table 15). Spins from 1 − to 6 − for 42 states are determined, among them 14 new assignments by this work (Tables 4, 5
The 6337 state is newly identified and assigned spin 3 − (Fig. 26). It is clearly distinct from the 6340 state suggested with spins 1 − , 2 − , or 3 − and an uncertainty in excitation energy of 5 keV [1].
The 6243 state is tentatively assigned spin 3 − because of the considerable cross section near the g 7/2 IAR (Fig. 26).
Other spins in the range from 1 − to 6 − are excluded by the shape of the angular distribution (Fig. 4).

A rather complete system of 4 − states
Twenty states with spin 4 − below E x = 7.0 MeV are known (Tables 3, 4 , 5, 9). States with spin 4 − and dominant g 7/2 p 3/2 or g 7/2 f 5/2 strength were found by the pronounced shape of the angular distribution near the g 7/2 IAR if a weak d 3/2 f 5/2 admixture can be expected (Fig. 4).
The 6739 state was suggested to have spin 4 − because of the large cross section [40] (Fig. 27, Tables 4, 5). The Also the cross section is enhanced in the region at Θ ≈ 120 • . The poor fit of the angular distributions of the 6739 and 6801 states explains the summed strength for g 7/2 p 3/2 to be larger than unity.
The angular distributions of the 6452 state are well fitted by assuming a strong g 7/2 f 5/2 contribution (Fig. 27). The g 7/2 p 1/2 strength is weak, indeed the 6452 state is not observed in 207 Pb(d, p). The 4 − state with order number M = 23 [Eq. (2)] is identified as a member of the tetrahedral rotation-vibration band (010) ( Table 9), the excitation energy agrees perfectly with the prediction [3,4]. The next SM configuration is predicted 100 keV higher (proton f 5/2 d 5/2 configuration).

States with spin 5 −
Nineteen states with spin 5 − below E x = 7.0 MeV are known (Tables 3, 4, 5, 10). States with spins 5 − and 6 − at 6.2 < E x < 7.0 MeV are excited by the proton decay of the g 7/2 IAR only, configurations d 3/2 f 7/2 and d 3/2 h 9/2 are far away and their contribution is negligible. The shape of the angular distributions is pronounced near the g 7/2 IAR (Fig. 4), the cross section of the 5 − states is expected to be large.
The 6688 state was tentatively assigned spin 5 − [40] because of the large cross section; it was assigned spin 5 − [1] in view of the excitation function measured by Whar- a Not yet identified ton et al. [28], see reproduction in Fig. 17. The unique shape of the angular distribution confirms the assignment of the dominant g 7/2 p 3/2 configuration (Fig. 28). The g 7/2 p 3/2 strength is determined with 64% (Tables 4, 5, Fig. 12). Three other 5 − states are identified by the rather unique shape of the angular distributions and the large cross sections ( Table 5, Fig. 28). The 6878 state with 25% g 7/2 p 3/2 strength thus complements the g 7/2 p 3/2 strength to near completeness. Similarly the 6354 and 6389 states together contain nearly the full g 7/2 f 5/2 strength. Still three more 5 − states are expected at 6.2 < E x < 7.0 MeV (Table 10, Fig.12). Apparently their cross sections are weak.

A rather isolated system of 6 − states
Twelve states with spin 6 − below E x = 7.0 MeV are known (Tables 3, 4, 5, 11). States with spin 6 − and dominant g 7/2 f 5/2 strength were found by the pronounced shape of the angular distribution near the g 7/2 IAR (Fig. 4); the cross sections should be rather large because of the spin factor [Eq. (5)]. The SDI predicts 6 − states at 6.2 < E x < 7.0 MeV with order numbers M = 9-14 [Eq. (2)]; the 10th 6 − state is not yet identified.
The SDI predicts the configuration g 7/2 f 5/2 for spin 6 − to be shifted by 160 keV in relation to the SSM due to the Nordheim number N h = +1 similar as for i 11/2 f 5/2 and spin 8 − [17]. The shift in excitation energy is verified (Table 11).
Eight 6 − states are known at much lower excitation energies, there is a gap of 778 keV between the state with order number M = 8 and 9 (Table 11, Fig. 12). The 14th 6 − state is predicted 279 keV higher in excitation energy [17,18]. Hence the 6 − states with order number M = 8-14 excepting the still missing 10th state may be considered as a complete system isolated from the rest (Fig. 13). Admixtures of configurations with order numbers m ≤ 8 and m ≥ 14 are expected to be small.
The poor fit of the angular distributions of the 6462 and 6631 states explains the summed strength for g 7/2 f 5/2 to be larger than unity.
The right parts display the excitation energies and the measured particle-hole strength of all states with spins 1 − , 2 − , 3 − , 4 − , 5 − , and 6 − in the region 5.8 < E x < 7.0 MeV. For states with identified strengths the complement of the strengths (for a configuration not shown in the figure or an unobserved configuration) is shown in the bottom line as "unobs". The sum of the identified strengths are shown with thick triangles in the left parts. Table 3 tabulates identified strengths and centroid energies both calculated and derived from experiment.
The 3 − states. The strengths for d 3/2 f 5/2 , g 7/2 p 1/2 , g 7/2 p 3/2 are nearly completely located. For the missing s 1/2 f 5/2 strength the reason is the lack of data near the s 1/2 IAR and the interference with other IARs leading to non-isotropic angular distributions. Most d 3/2 p 3/2 and some g 7/2 f 5/2 strengths are still missing. Again the reason is the weakness of the cross sections and the interference among the IARs.
The 4 − states. The g 7/2 p 1/2 strength was completely found by Valnion [55,56], note footnotes in Table 4 and Sect. A.2.1. It was crucial in doing calibration of the cross sections (Sect. 3.3.2) and the adjustment of the parameters for the IARs (Tables 1, 2).
The strength for g 7/2 p 1/2 , g 7/2 f 5/2 is located within a few percent, see Sect. 4.2.3. No d 3/2 f 5/2 strength could be found. Again the reason is the weakness of the cross sections and the interference among the IARs. The overestimated g 7/2 p 3/2 strength has the distribution across all states as the reason leading to statistical fluctuations.
The 5 − states. The strengths for g 7/2 p 3/2 , g 7/2 f 5/2 are located within a few percent. The finding proves the good calibration of the cross sections (Sect. 3.3.2) and the adjustment of the parameters for the IARs (Tables 1, 2).
The 6 − states. The strength for g 7/2 f 5/2 is located within a few percent, see Sect. 4.2.5. Again, the finding proves the good calibration of the cross sections (Sect. 3.3.2) and the adjustment of the parameters for the IARs (Tables 1, 2).
The 1 − states. The right part of Fig. 8 shows the distribution of the strengths, the left part the values E S DI x by a dotted line with a diamond at bottom. The centroid energy agrees with E S DI x within 50 keV (about 100 keV) for (s 1/2 p 3/2 ), (d 3/2 p 1/2 ), d 3/2 f 5/2 , and g 7/2 f 5/2 . It deviates by more than 100 keV from E S DI x for d 3/2 p 3/2 . The reason is simply the limiting choice E x < 7.0 MeV in the experimental data. Obviously much d 3/2 p 3/2 strength will be found in the well known 1 − states above E x = 7.0 MeV in future.
The 2 − states. The right part of Fig. 9 shows the distribution of the strengths, the left part the values E S DI x by a dotted line with a diamond at bottom. The centroid energy agrees with E S DI x within 50 keV (about 100 keV) for s 1/2 p 3/2 , s 1/2 f 5/2 , d 3/2 p 1/2 , (d 3/2 f 5/2 ), and also for g 9/2 f 7/2 ;  Table 6 denoted as "other" in a; the excitation energy E S DI x (g 9/2 f 7/2 ) is shown, too, because of its importance in the region 5.5 < E x < 6.2 MeV (and the position for E S DI x > 5.8 MeV marked by a square for spins 2 − , 4 − , 6 − ). (Right frame) Distribution determined from experiment. Sum of unobserved strengths shown denoted as "unobs" in a. In each state only either g 7/2strength or d 3/2 -strength is determined; an interference between components with different particles L J cannot be obtained (Sect. 2). The distribution of the s 1/2 p 1/2 and s 1/2 p 3/2 strengths is also known [46,49]  It deviates by more than 100 keV from E S DI x for d 3/2 p 3/2 , g 7/2 p 3/2 , g 7/2 f 5/2 . For d 3/2 p 3/2 the reason is the limiting choice E x < 7.0 MeV in the experimental data as already mentioned. For g 7/2 p 3/2 , g 7/2 f 5/2 the distinction from d 3/2 p 3/2 , d 3/2 f 5/2 is hindered by the interference among the IARs.
The 3 − states. The right part of Fig. 10 shows the distribution of the strengths, the left part the values E S DI x by a dotted line with a diamond at bottom. The centroid energy agrees with E S DI x within 50 keV for d 3/2 f 5/2 and within about 100 keV for s 1/2 f 5/2 , g 7/2 p 1/2 . It deviates by more than 100 keV from E S DI x for d 3/2 p 3/2 , d 3/2 f 5/2 , and g 7/2 p 3/2 . The distinction between g 7/2 f 5/2 and d 3/2 f 5/2 is hindered by the weak cross sections and the interference among the IARs. For

Number of states at 6.2 < E x < 7.2 MeV
Tables 4 and 5 list the results for 40 states with spins from 1 − to 6 − . All states with spins 1 − , 2 − , 3 − were known before except for two 3 − states. The 6243 3 − state is identified because of the considerable cross section near the g 7/2 IAR, the 6337 3 − state is distinguished from the close lying 6340 state (Table 15).
In the region 6.2 < E x < 7.0 MeV in total the number of observed states agrees with calculations by the SM for 1p1h configurations [12,13,17,18] and with the coupling of 1p1h configurations to the 3 − yrast state [2] rather well, at least for the negative parity states. However several disagreements with the SM are obvious.
We do not discuss the disagreements in this paper because part of them occur with the yrast or yrare states and at E x ≈ 5.5 MeV [9]. Tables 6, 7 Figure 15 compares the distribution of states with negative and any parity to calculations. The SM predicts gaps in the distribution of excitation energies at E x ≈ 4.5, 5.5, 6.1, 6.9 MeV. The gap at E x ≈ 4.5 MeV enabled the study of an almost complete system of one 2 − , one 7 − , three 3 − , four 6 − , five 4 − , and six 5 − states in 208 Pb [34]. The spins were finally settled in 1982, the results are shown in Table 4 of [3].

Comparison of the number of states to calculations
The gap at E x ≈ 6.1 MeV enabled the study of the nearly complete ensemble of states below E x = 6.2 MeV [50]. It thus provided data to study the chaoticity among bound states in a heavy nucleus [62][63][64]. By chance the gap at E x ≈ 6.9 MeV is close to the arbitrary limit E x = 7.0 MeV chosen for the data taken in 1969 (Sect. 2.1).
The mean observed cross section of negative parity states is in the order of 10 µb/sr but approaches 500 µb/sr for a few states ( Table 6, 7, 8, 9, 10 and 11, Fig.17). Positive parity states are generally weaker excited [45]. At 6.2 < E x < 7.0 MeV 126 states are observed (Table 15). About 59 states with spins from 1 − to 6 − are observed (Tables 4, 5 Table VI in [50], data for 6.20 < E x < 7.15 MeV from Table 15. The number of known states includes those predicted but not yet observed (Tables 6, 7 g 9/2 f 5/2 , g 9/2 p 3/2 , i 11/2 p 1/2 , and i 11/2 f 5/2 . The number of observed states which do not have negative parity is much lower. It agrees roughly with the SM calculations with M3Y (Fig. 15). Apparently the majority of positive parity states predicted by the coupling of the 3 − yrast state to 1p1h configurations is not yet identified except for 14 states with the coupling to g 9/2 p 1/2 and four other states [2].

Discovery of non-1p1h states
Complete spectroscopy of 1p1h configurations allows to find non-1p1h configurations [9]. A discrepancy between the order numbers predicted by SDI and the observed order assists in resolving the correlation.  [17,18], drawn curves calculated with M3Y [12]. The lowest curve shows the number of non-1p1h configurations known from experiment and calculated by the SDI for the weak coupling of the 3 − yrast state to 1p1h configurations [2]. Larger gaps in the distribution of excitation energies are predicted at E x ≈ 4.5, 5.5, 6.1, 6.9 MeV. They are visible in the experimental distributions except for the last one where the knowledge diminishes. The gaps enabled the study of rather isolated systems of states [34]  The 1 − , 2 − , 3 − yrast states were recognized as tetrahedral configurations [3,4]. Therefore the order number M of the observed states is one higher than calculated by SDI. At E x ≈ 5.5 MeV three more 3 − states and at E x ≈ 5.8 MeV three more 5 − states are observed than predicted by the SDI [9].
The 7020 3 − and 7134 4 − states are recognized as tetrahedral configurations [3,4]. Details are discussed in the following.
In contrast SM calculations with realistic forces underestimate the number of 1 − states (Fig. 14a). However such calculations have a general problem to predict 1 − states because of handling the center of gravity motion [10][11][12][13]. The problem is solved by introducing one artificial state in addition (see e.g. [10,13]).
Because the 1 − yrast state is described by a dominant non-1p1h configuration the order numbers M of the states are one higher than the order numbers m of the configurations counted by the SDI, M = m + 1 [Eq.(2)] (Fig. 14a, Table 6). One more 1 − state is predicted at 7.1 MeV but not found [93] (Sect. 4.1.3).
The 2 − states. The number of identified states for spin 2 − agrees with the expectation by the SM up to E x = 7.0 MeV except for the additional yrast state (M = 1) (Fig. 14b). Hence the order numbers M of the states are one higher than the order numbers m of the configurations counted by the SDI, M = m + 1 [Eq. (2)] ( Table 7). The 4140 2 − yrast state is identified as a tetrahedral configuration; it is not predicted by any SM calculation.
The 3 − states. The number of states assigned spin 3 − is much larger than predicted by SM calculations (Fig. 14c, Table 8). In addition to the yrast state recognized as tetrahedral configuration [3] disagreements are found at E x ≈ 5.5 MeV [9]. For the states at E x ≈ 5.5 MeV the identification and the spin assignments are rather firm [50], but final doubts cannot be excluded.
The 4 − states. The number of states assigned spin 4 − agrees with the SDI calculations (Sect. 4.2.3, Fig.14d, Tables 9,14). In contrast SM calculations with the M3Y interaction [12] underestimate the number of states slightly.
Just above the limit E x = 7.0 MeV chosen in the data taking of 1969 [41,42] the 7134 4 − state was recognized as tetrahedral configuration [3,4].
The 5 − states. For spin 5 − the number of identified states agrees with expectation by the SM up to E x = 5.7 MeV (Fig. 14e). However the large gap of 0.65 MeV between the two configurations g 9/2 f 7/2 j 15/2 i 13/2 predicted by SM calculations is not present. The 5705 5 − 13 and 5993 5 − 14 states do not have a SDI equivalent. In Fig. 14 in [50] one more state at E x ≈ 5.5 MeV is unexplained. This explains the discrepancy of the order numbers by three (M = m + 3) [Eq. (2)] for E x > 6.0 MeV [9] (Table 10).
The spin assignment to the 5993 state is doubted. An essay indicates that an assignment of spin 2 − would be possible, too. Yet this would mean that the discrepancy with the number of one-particle one-hole configurations would lead to M = m + 2 [Eq. (2)] for the 2 − states at E x > 6.0 MeV ( Table 7). The changed mean spacing between the 2 − states would affect the chaoticity only little [62][63][64].
The 6 − states. For spin 6 − the number of identified states agrees with the SM expectation [Sect. 4.2.5, Fig. 14f, Tables 11,14]. The knowledge of states ceases above E x = 6.7 MeV. At 6.2 < E x 7.0 MeV there are sixteen states with negative parity not yet identified in total, one 0 − state, (no 1 − state), one 2 − state, (no 3 − state), three 4 − states, three 5 − states, and two 6 − states. In addition two 9 − states, two 10 − states, two 11 − states and another 12 − state are missing.
As Table 15 shows there are about 90 states without a spin assignment at 6.2 < E x < 7.0 MeV. Currently available experimental data do not allow further discussion.
Calculations by [2] predict a large number of positive parity states in the region. The lowest curve in Fig. 15b displays the number of non-1p1h configurations known from experiment and calculated by the weak coupling of the 3 − yrast state to 1p1h configurations using the SDI [2].
According to [2] about five dozen states are predicted which nearly agrees with the number of states with unknown spin. The difference between the SDI and the M3Y calculations is thus partly explained.

Summary
Precisely measured angular distributions of 208 Pb(p, p ) at the MPIK (Heidelberg, Germany) for states 6.2 < E x < 7.0 MeV taken 50 years ago are evaluated. In the last fifteen years new data were gathered for the same region of excitation energies and beyond with four times better resolution and with the additional reactions 207 Pb(d, p) and 208 Pb(d, d ) at the MLL (Garching, Germany).
Sixteen states are assigned new spins from 2 − to 6 − . Spins with 1 − , 2 − , 3 − , and 5 − are additionally verified in twentyfour states. Four 2 − , one 3 − , four 4 − , and one 6 − states are newly identified. Amplitudes for up to four 1p1h configurations are determined with relative signs in thirty-four states at 6.2 < E x < 7.0 MeV and in addition in six states at 5.7 < E x < 6.2 MeV strongly excited by the 207 Pb(d, p) reaction.
The centroid excitation energies and the sum of the strengths for the 1p1h configurations g 7/2 f 5/2 , d 3/2 f 5/2 , g 7/2 p 3/2 , d 3/2 p 3/2 is determined for most spins. The deviation of the centroid excitation energies from SM calculations is less then 50 keV at average. The sum of the strengths amounts up to the full strength in most cases.
The purpose of the paper is to find positive parity states and non-1p1h states in the heavy nucleus 208 Pb by studying negative parity 1p1h states at 6.2 < E x < 7.0 MeV. Obviously some of the sixteen new spin assignments in the chosen region are not yet firm. About half of the 90 levels comprising about 120 states observed in 1969 at 6.2 < E x < 7.0 MeV are weaker excited and not thoroughly investigated.
Acknowledgements Open Access funding provided by Projekt DEAL. Immeasureable thanks is given to the late Peter von Brentano who introduced the study of isobaric analog resonances for the analysis of states in 208 Pb. Many thanks are given to C. Fred Moore who coordinated the experiments on 208 Pb(p, p ) and to J. Solf who coordinated the experimental setup in 1968-1969.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: The essential datasets analyzed during the current study are available in the Mendeley repository with [41]. Other important datasets used for the analysis are available with [52,58]. Datasets generated during the current study are available with [42].] 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/.
In the following the states related to the original levels are described. In most cases a single state describes the essential cross section. The 6.232 + 6.255 doublet was already recognized, however the doublet contains four states resolved by the Q3D measurements (Table 15). Five more doublets were resolved after 2007 [47,50]; in ten levels other states do not contribute significantly to the cross section; they mostly have positive parity (Table 15).

A.2.1 Figure 16
The top frame of Fig. 16 reproduces Fig. 7 in [28].  Table 4 in [3]. The admixture observed on the s 1/2 IAR is explained by a weak s 1/2 f 5/2 component. Within the resolution of about 30 keV neighboring 5010 9 + , 5040 2 + , 5069 10 + states are unresolved [50]; they contribute little to the observed excitation function. (5.121) The 5127 2 − 4 state contains 33% d 5/2 p 1/2 strength [55]. The admixture observed on the s 1/2 IAR is explained by a weak s 1/2 f 5/2 component. Within the resolution of 30 keV the neighboring 5093 8 + state is unresolved [50]; it contributes little to the observed excitation function. (5.804) Even with MLL Q3D magnetic spectrograph the doublet with the 5812 2 − 9 and 5813 3 − 15 states is unresolved. It was resolved by a sophisticated analysis of neighboring states [48]. The 5813 3 − state contains 12% g 7/2 p 1/2 strength [55]. It explains the enhancement remarked as being due to the d 3/2 IAR in 40 keV distance. However a weak admixture with a weak d 3/2 f 5/2 component in both states is certainly also present. Therefore the shape of excitation function deviates significantly from a Lorentzian near E p = 17.4 MeV. The very weak enhancement on the s 1/2 IAR may be explained by a small amplitude s 1/2 p 3/2 in the 5812 2 − 9 state and s 1/2 f 5/2 in the 5813 3 − 15 state. Within the resolution of about 30 keV the neighboring 5789 3 + , 5805 1 − , 5819 2 + , 5825 8 + , 5835 8 − , 5844 1 + states are unresolved [50]; they contribute little to the observed excitation function. (5.869 [55]. It was recently resolved [50]. The analysis of the angular distributions in the Heidelberg data from 1969 yields three amplitudes for each state (Sect. 4.1.7, Table 4). The large amplitudes with a g 7/2 particle in both states explains the strong enhancement of the excitation function on the g 7/2 IAR. The admixture observed on the s 1/2 IAR is explained by a weak s 1/2 f 5/2 component in the 6010 3 − 17 state. Within the resolution of about 30 keV the neighboring 5989 6 + , 5993 5 − , 6023 7 + , 6026 8 + , 6037 6 + states are unresolved [50]; they contribute little to the observed excitation function.
Within the resolution of about 30 keV the unresolved 6682 2 − , 6699 3 − , 6708 2 − states and the neighboring 6670, 6673, 6683, 6695 states with unknown spin and unknown parity (Table 15) contribute little to the observed excitation function.
The bottom frame of Fig. 17 reproduces Fig. 11 [86] with a Gaussian and an exponential tail [45]. The background is mostly described by a polynomial of first degree; it is displayed by a blue line. Each peak is fitted by a Gaussian of individual width with an exponential tail of common shape; it is displayed by a green curve. The width is chosen as a linear function for peaks describing nuclear levels in lead isotopes because of the quadratic dependence on the magnetic field and the limited size of the spectrum. The widths for contamination peaks are varied freely. The fit of the total spectrum is shown by a red line.
For the fits shown in Figs. 22, 23 the option of fixed energies [45] was used (Sect. 3.3.1). Original data displaying spectra for 208 Pb(p, p ), 208 Pb(d, d ), 207 Pb(d, p) are available [57,58]. Figure 18 shows spectra taken with the MLL Q3D magnetic spectrograph for 208 Pb(p, p ) on the s 1/2 IAR with E p = 16.96 MeV at 6.3 < E x < 7.3 MeV. Figure 19 shows spectra taken with the MLL Q3D magnetic spectrograph for 208 Pb(p, p ) near the g 7/2 IAR with E p = 17.30 MeV at 6.3 < E x < 7.2 MeV. Figure 20 shows spectra taken with the MLL Q3D magnetic spectrograph for 208 Pb(d, d ) with E d = 22 MeV at 6.3 < E x < 7.1 MeV. Figure 21 shows spectra taken with the MLL Q3D magnetic spectrograph for 207 Pb(d, p) with E d = 22 MeV at 6.3 < E x < 7.0 MeV.

A.4 Figures 22 and 23
Original data displaying reconstructed spectra for 208 Pb(p, p ) fitted by GASPAN [86] are available [41,42]. Figures 22 and 23 show reconstructed spectra taken in 1969 at the MPIK for 208 Pb(p, p ) on the d 5/2 and g 7/2 IARs at 6.0 < E x < 6.5 and 6.5 < E x < 7.0 MeV, respectively. The legend in the middle assists to find the levels corresponding to the states with spins from 1 − to 6 − . Figure 22 shows reconstructed spectra taken for 208 Pb(p, p ) at 6.0 < E x < 6.5 MeV (top) on the d 5 Instead of the scattering angle Θ the Legendre polynomial P 2 (cos Θ) is chosen as abscissa (Sect. 2.2.5). The legend for the abscissa is omitted but the values P 2 (cos Θ) are marked for Θ = 120 • and 140 • similar to Fig. 3.

B Tables
The experimental uncertainties in Tables 1 and 2 are shown in the given references. They are not relevant for this work because of much larger systematic uncertainties which are not discussed in detail. The strengths in Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14 derive from Table 4. Systematic uncertainties are estimated in the order of several percent.  Table 4 shows excitation energies and amplitudes of 1p1h configurations for states discussed in this paper. The amplitudes from the 207 Pb(d, p) analysis [55] with the s 1/2 , d 3/2 and g 7/2 particle and a p 1/2 hole are weak, states at 5.8 < E x < 6.2 MeV with major strength of s 1/2 p 1/2 , d 3/2 p 1/2 , and g 7/2 p 1/2 are included (Cols. [6][7][8][9]. In most cases the fit of the angular distributions for 208 Pb(p, p ) yielded similar amplitudes for g 7/2 p 1/2 and d 3/2 p 1/2 (Cols. 10 and 15, respectively). Table 5 shows the strengths c 2 for the states tabulated in Table 4. Tables 6, 7, 8, 9, 10 and 11 show calculated excitation energies and cross sections for states with pure configurations d 3/2 p 3/2 , g 7/2 p 3/2 . g 7/2 f 5/2 , and d 3/2 f 5/2 . The detected sum of the strengths of states with these con-

Fig. 24
Angular distribution of 208 Pb(p, p ) for states assigned spin 1 − . The description of the x-and y-axes is omitted for clarity. The xaxis is given by the Legendre polynomial of second degree running from P 2 (cos(90 • )) to P 2 (cos(180 • )), see Fig.3; the values P 2 (cos(120 • )) and P 2 (cos(140 • )) are indicated at bottom. The y-axis is in relative units; the scale from 0 to 1.   figurations are shown and illustrated in Figs. 8, 9, 10, 11, 12 and 13. Table 12 compares results obtained fifty years ago to recent results. Table 13 shows the distribution of the g 7/2 p 3/2 and g 7/2 f 5/2 strengths for the 4 − states at 6.0 < E x < 7.0 MeV. Table 14 shows the distribution of the g 7/2 f 5/2 strength for the 6 − states at 5.6 < E x < 7.0 MeV. 6573.2 ± 1.5  The mean magnitude of the uncertainties is given as footnotes. This table continues Table VI in [50] for 0 ≤ E x < 6.2 MeV to 6.2 < E x < 7.15 MeV. Below E x < 7. 15 MeV in total about 300 states are now identified in 208 Pb.