Analytical solutions for the Bohr Hamiltonian with the Woods-Saxon potential
aa r X i v : . [ nu c l - t h ] J un Analytical solutions for the Bohr Hamiltonian withthe Woods–Saxon potential
M C¸ apak , D Petrellis , B G¨on¨ul and Dennis Bonatsos Department of Engineering Physics, University of Gaziantep, 27310 Gaziantep,Turkey Department of Physics, University of Istanbul, 34134 Vezneciler, Istanbul, Turkey Institute of Nuclear and Particle Physics, National Centre for Scientific Research“Demokritos”, GR-15310 Aghia Paraskevi, Attiki, GreeceE-mail: [email protected]
Abstract.
Approximate analytical solutions in closed form are obtained for the 5-dimensionalBohr Hamiltonian with the Woods–Saxon potential, taking advantage of the Pekerisapproximation and the exactly soluble one-dimensional extended Woods–Saxonpotential with a dip near its surface. Comparison to the data for several γ -unstableand prolate deformed nuclei indicates that the potential can describe well the groundstate and γ bands of many prolate deformed nuclei corresponding to large enough“well size” and diffuseness, while it fails in describing the β bands, due to its lack ofa hard core, as well as in describing γ -unstable nuclei, because of the small “well size”and diffuseness they exhibit.PACS numbers: 21.60.Ev, 21.60.Fw, 21.10.Re Keywords : Bohr Hamiltonian, Woods-Saxon potential, Pekeris approximation nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential
1. Introduction
The advent of critical point symmetries [1, 2], related to shape/phase transitions innuclear structure [3, 4], has stirred interest in analytical solutions (exact or approximate)[5] of the Bohr Hamiltonian [6]. In addition to the infinite well potential, used inthe critical point symmetries E(5) [1] and X(5) [2], related to the transition fromspherical to γ -unstable (soft with respect to triaxiality) [7] and to prolate deformed[8] nuclei respectively, solutions involving the Davidson [9, 10, 11, 12, 13], Kratzer[14, 15, 16, 17], and Morse [18, 19, 20] potentials (shown in figure 1) have been given.When applied to the bulk of nuclei for which energy spectra and B(E2) transition ratesare experimentally known [21], all these potentials provide good and quite similar results[12, 13, 15, 16, 17, 19, 20], despite having quite different shapes.The question arises if the success of the above mentioned potentials is due to theform of the Bohr Hamiltonian alone, or if there are potentials which, when plugged intothe Bohr Hamiltonian, will not be able to provide satisfactory results for nuclear data.In this work we consider the Woods–Saxon potential [22], which has beenextensively used in nuclear physics in a different context, namely as a single-particlepotential [23]. The potential, shown in figure 1(c), reads U W S ( β ) = − U e a ( β − β ) + 1 , (1)where U , a and β are non-negative free parameters. This study is motivated by severalreasons.1) For a → ∞ the WS potential reduces to a finite square well potential, whichhas also been used in relation to critical point symmetries [24, 25], in addition to theinfinite well potential [1, 2].2) In contrast to the Davidson [9], Kratzer [14], and Morse [18] potentials, whichpossess a hard core, the WS potential does not have a hard core. We will find out theconsequences of this difference.3) The Woods–Saxon potential is also interesting from the mathematical point ofview, since it has no closed analytical solution for the spectrum, even in one dimensionand for vanishing angular momentum [26, 27]. Only an analytical wavefunction is knownin one dimension and for vanishing angular momentum [27].First we manage to produce approximate solutions in closed form for the WSpotential with a centrifugal barrier, exploiting its resemblance (after applying the Pekerisapproximation [28] to it) to a modified spherically symmetric WS potential [29] knownto possess exact analytical solutions [26]. Taking advantage of these solutions within theframework of the Bohr Hamiltonian we subsequently fit several γ -unstable and prolatedeformed nuclei, pointing out the successes, but also the failures of the WS potential.Numerical calculations involving generalized forms of the Bohr Hamiltonian have along history.1) Extensive early numerical calculations for vibrating axially symmetric deformednuclei have been performed in the framework of the Rotation Vibration Model [30, 31]. nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential Β - - V K H Β L =- Β + Β H a L Β V D H Β L =Β + Β H b L Β - - V WS H Β L =- e H Β- L + H c L Β - V M H Β L = e - H Β- L - e - H Β- L H d L Figure 1. (Color online) The Kratzer [14] (a), Davidson [9] (b), Woods–Saxon [22](c) and Morse [18] (d) potentials, for special values of their free parameters, used forsimplicity. All quantities shown are dimensionless.
2) Numerical solutions have subsequently been provided using a general form forthe potential energy [32, 33], giving emphasis on the restrictions imposed on the formof the potentials by symmetry constraints.3) Extensive numerical results using a general collective model employing generalforms of both the kinetic energy and the potential energy have been obtained initially[34] by using a basis provided by Hecht [35], and subsequently [36, 37] by exploiting thebasis of a 5-dimensional (5D) harmonic oscillator [38, 39].4) The recent clarification [40] of the group theoretical structure of the Davidsonpotential, when used in the framework of the Bohr Hamiltonian, led to the developmentof the Algebraic Collective Model [41, 42, 43, 44], which allows the efficient numericalcalculation of spectra and transition probabilities of nuclei of any shape.In addition to the γ -unstable and the prolate deformed nuclei, the description oftriaxial nuclei within the framework of the Bohr Hamiltonian was very early attempted[45, 46] and is still attracting considerable attention [25, 47, 48, 49, 50].On the other hand, studies on the microscopic foundation of the Bohr Hamiltonianhave a long history, some early efforts reported in [51, 52, 53]. Efforts towards thederivation of the Bohr collective Hamiltonian through the adiabatic approximation ofthe time-dependent Hartree-Fock-Bogolyubov theory (the ATDHFB method), as wellas through the generator coordinate method with the Gaussian overlap appoximation(the GCM+GOA method) have been reviewed in [54]. Recently, the derivation of the nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential
2. Modified Woods–Saxon potential with a centrifugal barrier
The Pekeris approximation [28] is a well known method for finding approximateanalytical solutions for potentials involving exponentials. It has been introducedin relation to the Morse potential [18] with a centrifugal barrier in studies ofrotational-vibrational spectra of diatomic molecules [28]. The basic idea of the Pekerisapproximation is to rewrite approximately the centrifugal term using exponentialsresembling the ones appearing in the rest of the potential, with the aim to be ableto “absorb” the centrifugal term into the potential.This approximation has been used in relation with the Morse potential in diatomicmolecules [28, 59], as well as with the pseudo-centrifugal term for Dirac particles withinMorse potentials [60]. Furthermore, a supersymmetric improvement of the Pekerisapproximation in the case of the Morse potential has been worked out [61]. Numericalresults reported in these papers indicate that the approximation works well, especiallyfor relatively low-lying vibrational and rotational states (see in particular the detailedresults reported in [61]).The Pekeris approximation has further been used in relation to various exponential-type potentials [62], to the Wei Hua oscillator [63], to the Rosen–Morse and Manning–Rosen potentials [64], as well as with the pseudo-centrifugal term for Dirac particleswithin Rosen–Morse potentials [65]. In these cases, various functions appear in theoriginal potentials, as well as in the Pekeris approximation terms.The spherically symmetric Woods-Saxon potential in three-dimensions with acentrifugal barrier reads U ( β ) = U W S ( β ) + U c ( β ) = − U e a ( β − β ) + 1 + l ( l + 1) β , (2)where U W S stands for the Woods-Saxon potential, U c stands for the centrifugal term, l is the angular momentum, while U , a and β are non-negative free parameters.It is known [26] that this potential cannot be solved exactly. However, closedanalytical solutions can be derived using the Pekeris approximation [28]. As it wasmentioned above, the basic idea of the Pekeris approximation is to rewrite approximatelythe centrifugal term using exponentials resembling the ones appearing in the rest of thepotential, with the aim to be able to “absorb” the centrifugal term into the potential.In the present case we wish to write the centrifugal term in the approximate form U c ( β ) = l ( l + 1) β ≈ δ (cid:20) C + C e a ( β − β ) + 1 + C [ e a ( β − β ) + 1] (cid:21) . (3) nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential x = ( β − β ) /β . Then the exact centrifugal term takes the form U c ( x ) = δ (1 + x ) , (4)where δ = l ( l + 1) β . (5)Expanding this function into binomial series one gets U c ( x ) = δ (1 − x + 3 x − x + . . . ) . (6)On the other hand, using the same notation, the approximate form of (3) becomes U c ( x ) ≈ δ (cid:20) C + C e aβ x + 1 + C [ e aβ x + 1] (cid:21) . (7)Expanding the exponentials in Taylor series we get U c ( x ) ≈ δ (cid:20)(cid:18) C + C C (cid:19) − β a ( C + C ) x + β a C x + . . . (cid:21) , (8)where a = 12 a . (9)Equating the coefficients of equal powers of x in (6) and (8) we get C = 1 − a β + 12 a β , C = 8 a β − a β , C = 48 a β . (10)Using the approximate expression of (3) in (2), we obtain U ( β ) ≈ δC − ( U − δC ) e a ( β − β ) + 1 + δC [ e a ( β − β ) + 1] , (11)where indeed the centrifugal term has been “absorbed” in the potential.The accuracy of approximating the potential of (2) by (11) will be discussed indetail in section 5, in relation to the parameter values occurring for the nuclei underconsideration (see figs. 4, 5, 6 and relevant discussion in section 5). On the other hand, the modified spherically symmetric Woods–Saxon potential, whichpresents a dip near its surface [29], has the form V ( β ) = − V e a ( β − β ) + 1 − W e a ( β − β ) [ e a ( β − β ) + 1] , (12)where V , W , a and β are non-negative free parameters. This potential, shown in figure2, is known [26] to possess exact solutions of the one-dimensional Schr¨odinger equationin terms of Jacobi polynomials [66]Ψ n ( β ) ∝ (cid:20) e a ( β − β ) + 1 (cid:21) b/ (cid:20) − e a ( β − β ) + 1 (cid:21) c/ P ( b,c ) n (13) nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential Β - - - V H Β L =- e H Β- L + - We H Β- L A e H Β- L + E W = = Figure 2. (Color online) The Woods–Saxon potential with a dip near its surface,given in (12), is shown for special values of its free parameters, used for simplicity. Allquantities shown are dimensionless. where b = 12 (cid:18) ˜ ρ n + V a ˜ ρ n (cid:19) , c = 12 (cid:18) ˜ ρ n − V a ˜ ρ n (cid:19) , (14)with ˜ ρ n = r Wa − (2 n + 1) , (15)and the condition [66] b, c > − b + c > − . (17)The strengths of the potential of (12) can be expressed in the form [26] V = a ( b − c ) ≥ , (18) W = a [( b + c )( b + c + 4 n + 2) + 4 n ( n + 1)] ≥ , (19)which imply stronger restrictions on b , c . Indeed, from (19) for n = 0 one should have( b + c )( b + c + 2) ≥
0, which implies either b + c ≤ −
2, which is not allowed by (17), or b + c ≥ , (20)which is a stronger restriction. Higher values of n do not impose any further restrictions,since (19) leads to b + c ≤ − n + 1), which is not allowed by (17), or b + c ≥ − n ,which is weaker than (17). It should be remembered that the quantity W , given in(19), cannot vanish, since in that case the potential of (12) would collapse to the usualWoods–Saxon potential, which is known to have no closed analytical solutions for itsspectrum [26, 27].The eigenvalues of the energy are known to be [26] E n = − a (cid:18) ˜ ρ n + V a ˜ ρ n (cid:19) − V . (21) nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential e a ( β − β ) [ e a ( β − β ) + 1] = 1 e a ( β − β ) + 1 − e a ( β − β ) + 1] , (22)the potential of (12) can be rewritten as V ( β ) = − V + We a ( β − β ) + 1 + W [ e a ( β − β ) + 1] . (23) We remark that this potential has the same form as the potential of (11), with V + W = U − δC , W = δC , (24)leading to V = U − l ( l + 1) aβ , W = 12 l ( l + 1) a β . (25)Substituting these expressions in (21) and taking into account the first term in (11)the eigenvalues of the energy for the potential of (11) are found to be E n = l ( l + 1) β (cid:18) a β (cid:19) − (cid:18) ˜ ρ n a (cid:19) − U a − l ( l +1) a β ˜ ρ n − U , (26)where ˜ ρ n = s l ( l + 1) a β − (2 n + 1) . (27)Therefore we have managed to obtain the energy eigenvalues of the WS potentialwith a repulsive barrier, as approximated through the Pekeris approximation, byexploiting the known solutions of the modified spherically symmetric WS potential,without having to solve the Schr¨odinger equation anew.We remark that the condition of (20) leads to severe restrictions for n . Indeed,using (20) we have b + c = ˜ ρ n ≥ , (28)which, using (27), leads to n ≤ − s l ( l + 1) a β ! . (29)Thus in the case of l = 0, only n = 0 is allowed.This result is valid in a 3-dimensional space. Following the same path one can see[67, 68] that the result is also valid in D dimensions, with the angular momentum l replaced by l D = l + D − . (30)In the special case of a 5-dimensional space, l is replaced by L = l + 1 . (31) nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential
3. Bohr Hamiltonian for γ -unstable nuclei The original Bohr Hamiltonian [6] is H = − ~ B (cid:20) β ∂∂β β ∂∂β + 1 β sin 3 γ ∂∂γ sin 3 γ ∂∂γ − β X k =1 , , Q k sin (cid:0) γ − πk (cid:1) + V ( β, γ ) , (32)where β and γ are the usual collective coordinates describing the shape of the nuclearsurface, Q k ( k = 1, 2, 3) are the components of angular momentum, and B is the massparameter. In what follows we are going to consider ~ = 2 B = 1.Assuming that the potential depends only on the variable β , i.e. V ( β, γ ) = U ( β ),one can proceed to separation of variables in the standard way [6, 7], using thewavefunction Ψ( β, γ, θ i ) = 1 β f ( β )Φ( γ, θ i ) , (33)where θ i ( i = 1 , ,
3) are the Euler angles describing the orientation of the deformednucleus in space.In the equation involving the angles, the eigenvalues of the second order Casimiroperator of SO(5) occur, having the form Λ = τ ( τ + 3), where τ = 0, 1, 2, . . . is thequantum number characterizing the irreducible representations (irreps) of SO(5), calledthe “seniority” [69]. This equation has been solved by B`es [70].The “radial” equation yields − d fdβ + (cid:18) U ( β ) + ( τ + 1)( τ + 2) β (cid:19) f ( β ) = E n,τ f ( β ) . (34)The effective potential appearing in this equation coincides with that of (2), with theformal replacement of l by τ + 1. Substituting in (26) we obtain the energy eigenvalues E n,τ = ( τ + 1)( τ + 2) β (cid:18) a β (cid:19) − (cid:18) ¯ ρ n,τ a (cid:19) − U a − τ +1)( τ +2) a β ¯ ρ n,τ − U , (35)where ¯ ρ n,τ = s τ + 1)( τ + 2) a β − (2 n + 1) . (36)The restriction of (29) in this case reads n ≤ − s τ + 1)( τ + 2) a β ! . (37)As we shall see in subsection 5.2, only n = 0 is acceptable for the parameter valuesoccurring in real nuclei. The violation of the condition of (37) even for n = 1 has as aconsequence to push the n = 1 bands too low in energy, making them unphysical. nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential Eq. (25) takes the form U = V + 8 a ( τ + 1)( τ + 2) β , (38)implying that if V is a constant, then U depends on τ , and vice versa. We are goingto consider V as being the constant quantity. Then we can use this equation in orderto eliminate U from (35). Using the notation˜ A = a /β , (39)the equation for the energy becomes E n,τ = ( τ + 1)( τ + 2) β (1 + 12 ˜ A − A ) − ¯ ρ n,τ a − V a ¯ ρ n,τ − V , (40)with ¯ ρ n,τ = q τ + 1)( τ + 2) ˜ A − (2 n + 1) . (41)In order to reduce the number of free parameters, we fit nuclear spectra leaving outoverall scales, as it is done in the E(5) and X(5) models [1, 2]. Instead of fitting the rawexperimental levels E L , where L is the angular momentum, the quantities E L − E E − E (42)are used. In other words, the energy of the ground state is subtracted from all levels,and then each level is divided by the energy of the first excited state, which in even-evennuclei is the energy of the first excited state with L = 2. The connection between L and τ is described in detail in subsection 3.4.In the present case this has the following consequences on (40):1) The last term, − V /
2, plays no role, since it is a constant and cancels out.2) Using the rescaling β = ˜ β √ V , a = ˜ a √ V , (43)a common factor V appears in all terms of (40), which therefore cancels out when (42)above is used.3) Using ˜ A = a /β = ˜ a / ˜ β , a common factor ˜ β can be taken away from thedenominator of all terms of (40).Then the equation to be used for the energy fits becomes¯ E n,τ = ( τ + 1)( τ + 2)(1 + 12 ˜ A − A ) − ¯ ρ n,τ
16 ˜ A − ˜ A ˜ β ¯ ρ n,τ , (44)where ¯ ρ n,τ is still given by (41).This is the equation used in the fits, from which the parameters ˜ A and ˜ β can bedetermined. Then ˜ a is calculated from ˜ a = ˜ A ˜ β . nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential The Woods–Saxon potential is essentially different from zero within its radius, which inthis case is β .One way to fix the scale is to identify the radius of the potential, β , with the β exp value obtained from the experimental values of B ( E
2; 0 +1 → +1 ) [71]. Then from (43)one simply has V = ˜ β β exp . (45)Notice that in this case a can be calculated directly from a = ˜ Aβ exp .It should be recalled at this point that while the B ( E
2; 0 +1 → +1 ) values areexperimental quantities which are model independent, the deformation parameter ismodel dependent [71]. In the present case, a uniform charge distribution out to thedistance R ( θ, φ ) is assumed, with zero charge beyond it, while the nuclear radius istaken to be R = 1 . A / fm. Then the deformation is connected to the transitionprobability by β = 4 π ZR r B ( E
2; 0 +1 → +1 ) e , (46)where B ( E
2; 0 +1 → +1 ) is measured in e b . It should be noticed that β isthe deformation parameter defined in the Bohr framework [6]. A slightly differentdeformation parameter ε is used in the framework of the Nilsson model [72], connectedto the Bohr parameter by ε = 0 . β (see p. 125 of [72], or eq. (2.82) of [73]). The spectrum is characterized by the O(5) ⊃ SO(3) symmetry [69, 70]. τ and L are thequantum numbers characterizing the irreps of O(5) and SO(3) respectively. The valuesof angular momentum L contained in each irrep of O(5) (i.e. for each value of τ ) aregiven by the algorithm [74] τ = 3 ν ∆ + λ, ν ∆ = 0 , , . . . , (47) L = λ, λ + 1 , . . . , λ − , λ (48)(with 2 λ − ν ∆ is the missing quantum number in the reduction O(5) ⊃ SO(3), and are listed in Table 1.The ground state band (gsb) has n = 0 and levels L g = 0, 2, 4, 6, . . . , for which ν ∆ = 0 and L g = 2 τ . Thus within the gsb one has( τ + 1)( τ + 2) = ( L + 2)( L + 4)4 . (49)The quasi- γ band has n = 0 and levels with L γ = 2, 3, 4, 5, . . . , which arecharacterized by ν ∆ = 0 and L γ = 2 τ − L γ being even, or by L γ = 2 τ − L γ being odd. Therefore they exhibit the following degeneracies2 γ = 4 g , γ = 4 γ = 6 g , γ = 6 γ = 8 g , γ = 8 γ = 10 g , . . . (50) nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential β band has levels L β = 0, 2, 4, 6, . . . . There are two choices:1) n = 0, in which the levels have ν ∆ = 1 and L β = 2 τ −
6, leading to the followingdegeneracies 0 β = 6 g , β = 8 g , β = 10 g , β = 12 g , . . . (51)2) n = 1, in which the levels have ν ∆ = 0 and L β = 2 τ . Then no degeneracies ofthis kind occur.
4. Bohr Hamiltonian for prolate deformed nuclei
If the potential has a minimum around γ = 0, as it is the case for prolate deformednuclei, the angular momentum term in (32) can be written [2] as X k =1 Q k sin (cid:0) γ − π k (cid:1) ≈
43 ( Q + Q + Q ) + Q (cid:18) γ − (cid:19) . (52)Exact separation of β from the rest of the variables can be obtained for potentials ofthe form [7, 5] U ( β, γ ) = u ( β ) + v ( γ ) β . (53)Concerning the γ and Euler angles, the solution provided by B`es [70] is not valid anymore, thus we seek, as is customary in the literature (see Chapter IV of the reviewarticle [5] for further details) wave functions of the form ψ ( β, γ, θ j ) = ξ L ( β )Γ K ( γ ) D LM,K ( θ j ) , (54)where θ j ( j = 1, 2, 3) are the Euler angles, D ( θ j ) represents Wigner functions of theseangles, L stands for the eigenvalues of the angular momentum, while M and K are theeigenvalues of the projections of the angular momentum on the laboratory-fixed z -axisand the body-fixed z ′ -axis respectively. In these wave functions the dependence on theEuler angles is entering through the Wigner functions, as in the many examples reviewedin [5], while K is assumed to be a good quantum number, an assumption which is not a priori justified, since strong K mixing can be present.The γ -equation occurring from this separation of variables has been solved in [12],using a potential v ( γ ) = (3 c ) γ , (55)its eigenfunctions written in terms of Laguerre polynomials and its energy eigenvaluesgiven by ǫ γ = (3 C )( n γ + 1) , C = 2 c, n γ = 0 , , , , ..., (56)while the separation constant is λ = ǫ γ − K . (57) nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential γ potential used in the Bohr Hamiltonian hasto be periodic in γ [6, 33, 75]. The potential of (55), used in several solutions [2, 5, 12]is a lowest order approximation, representing the first γ -dependent term in the Taylorexpansion of the proper periodic potential. There has been recently extensive workinvolving proper periodic γ potentials in the Bohr Hamiltonian [5, 48, 76, 77, 78, 79,80, 81].The radial equation takes the form [12] (cid:20) − β ∂∂β β ∂∂β + L ( L + 1)3 β + λβ + u ( β ) (cid:21) ξ L ( β ) = ǫξ L ( β ) . (58)Transforming ξ L into χ L by the relation ξ L ( β ) = χ L ( β ) β , (59)we obtain χ ′′ L ( β ) + " ǫ − L ( L +1)3 + λ + 2 β − u ( β ) χ L ( β ) = 0 . (60)We remark that this equation looks very similar to (34), with L ( L +1)3 + λ + 2 replacing( τ + 1)( τ + 2). Therefore from (35) we see that the energy eigenvalues become E n,L = (cid:16) L ( L +1)3 + λ + 2 (cid:17) β (cid:18) a β (cid:19) − (cid:18) ˜ ρ n,L a (cid:19) − U a − ( L ( L +1)3 + λ +2 ) a β ˜ ρ n,L − U , (61)where ˜ ρ n,L = s (cid:18) L ( L + 1)3 + λ + 2 (cid:19) a β − (2 n + 1) . (62)The restriction of (29) in this case reads n ≤ − s (cid:18) L ( L + 1)3 + λ + 2 (cid:19) a β ! . (63)As we shall see in subsection 5.1, only n = 0 is acceptable for the parameter valuesoccurring in real nuclei. The violation of the condition of (63) even for n = 1 has as aconsequence the failure of the relevant fitting attempts. Following the same rescaling procedure as in the γ -unstable case, the equation used inthe fits becomes¯ E n,L = (cid:18) L ( L + 1)3 + λ + 2 (cid:19) (1 + 12 ˜ A − A ) − ˜ ρ n,L
16 ˜ A − ˜ A ˜ β ˜ ρ n,L , (64)where ˜ ρ n,L = s (cid:18) L ( L + 1)3 + λ + 2 (cid:19) ˜ A − (2 n + 1) . (65) nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential A , ˜ β , and C (appearing in λ ) can be determined. Β - - - V Yb Figure 3. (Color online) Effective potentials for L = 10 for some Yb isotopes, obtainedfrom (23). The parameters are taken from Table 2, while W is given by (25) with l ( l +1)replaced by L ( L +1)3 + λ + 2, where λ = 3 C from (57) and (56). The quantities shownare dimensionless.
5. Numerical results
As a first test, the spectra of 63 nuclei with R / > . σ N = s P Ni =1 ( E i ( exp ) − E i ( th )) ( N − E (2 +1 )) , (66)has been used in the rms fits.The results for 46 nuclei are shown in Table 2. The following comments apply.1) Only the ground state band and the quasi- γ band have been included in thefits. 2) When trying to include the quasi- β band in the fits, trying to correspond it tothe n = 1 case, the fits fail. The failure is due to violation of the condition of (63). nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential − Yb, shownin Table 2, exhibit the following features.1) The depth of the potential, V , exhibits a clear minimum near the middle of theneutron shell ( Yb ), where maximum deformation is observed, as indicated by theclear maximum exhibited by β ≡ β exp , the experimental values of deformation obtainedfrom the B ( E
2; 0 + → + ) [71].2) The highest diffuseness ˜ a occurs nearest to the shell closure, while the lowestappears near the middle of the shell.3) In retrospect, the selection of the parameters used in the fits was physicallymeaningful. ˜ β is related to the “well size” [24], while ˜ A is related to the ratio of thediffuseness parameter a over the average width of the potential β .4) The potentials obtained from the Yb isotopes are shown in figure 3, corroboratingthe above made remarks about maximum depth and diffuseness nearest to the shellclosure, and minimum depth and diffuseness near the middle of the shell. We remarkthat the dip near the surface of the modified spherically symmetric WS potential isbecoming very large, dominating the overall shape of the potential. It is instructive tocompare the shapes of the potentials appearing in figure 3 to these in figure 2, sincethey come from (23) and (12) respectively, which are equivalent. Β U Th L = Β U L = Β U L = Β U L = Β U L = Β U L = Figure 4. (Color online) Exact (dashed lines) and approximate (solid lines) effectivepotentials for L = 0, 6, 12, 18, 24, 30 for Th, obtained from (2) and (11) respectively.The parameters are taken from Table 2, providing the quantities needed from (5), (9),and (10) while U is given by (25) with l ( l + 1) replaced by L ( L +1)3 + λ + 2, where λ = 3 C from (57) and (56). The corresponding energy levels, determined from (61),are also shown. The quantities shown are dimensionless. See subsection 5.1 for furtherdiscussion. The spectra obtained for some rare earths and actinides are shown in Table 3. Goodagreement with the data is obtained.A question arising at this point is the accuracy of the Pekeris approximation usedin the derivation of the relevant formulae. In order to check this, we show in figure 4 nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential L , in the case of Th. In addition, therelevant energy levels, obtained from (61), are shown. The following comments apply.1) Near their bottoms, the exact and the approximate potentials nearly coincide.They exhibit minima very close to each other, while in addition the two potentials formsimilar wells around the minima.2) At higher energies, the approximate potential wells are slightly displaced to theleft, in relation to the exact potentials. However, at the energy of main interest, i.e. atthe energy of the relevant level, the widths of the two potentials are nearly equal.3) The approximation appears to be better at higher values of L , i.e. at higherenergy levels. This is expected, since the fits were obtained through an rms procedure. Β U Yb L = Β U L = Β U L = Β U L = Β U Yb L = Β U L = Β U L = Β U L = Figure 5. (Color online) Same as figure 4, but for L = 0, 6, 12, 18, for Yb (a) and
Yb (b). See subsection 5.1 for further discussion.
The same observations hold for all nuclei shown in Table 2, with two exceptions,
Yb and
Yb, the latter exhibited in figure 5. While
Yb (and the rest of the Ybisotopes shown in Table 2) exhibit the qualitative behaviour described above, in
Yb(as well as in
Yb) we remark that the approximation breaks down, since for thelowest values of L the relevant energy level lies higher than the corresponding barrieron the rhs of the exact potential. In other words, we still get fits of good quality, whichare obtained using the approximate potential, but this potential is not similar to theexact potential any more, preventing us to draw any conclusions regarding the exactpotential. It should be noticed at this point that the approximate potentials (solidlines) for Yb isotopes shown in figures 3 and 5 are identical, up to a displacement bya constant. Indeed, the potentials of figure 3 come from (23), while the potentials offigure 5 come from (11), the latter being displaced in relation to the former by the term δC . This term is included in figure 5, since it is necessary for the comparison to theexact potentials (dashed lines) given by (2). Yb and
Yb are not isolated cases. Similar breakdowns of the approximationhave been found in 17 more nuclei, shown in Table 4. In all cases, good fits of quality nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential β ≤ . a ≤ .
43. Since β appears in the denominator of the rotational term, this result isunderstandable. The lower β becomes, the more important the rotational term, inwhich the Pekeris approximation has been used, becomes, thus the approximation tothe exact potential deteriorates.We therefore conclude that good fits are obtained for all nuclei studied, but only ifthe “well size” ˜ β fulfils the condition ˜ β ≥ . a ≥ .
44 the approximate potentials are quite similar to the corresponding exactpotentials. Β U Xe L = Β U L = Β U L = Figure 6. (Color online) Exact (dashed lines) and approximate (solid lines) effectivepotentials for τ = 0, 3, 6 ( L = 0, 6, 12) for Xe, obtained from (2) and (11)respectively. The parameters are taken from Table 5, providing the quantities neededfrom (5), (9), and (10), while W is given by (25) with l ( l + 1) replaced by ( τ + 1)( τ + 2).The corresponding energy levels, determined from (40), are also shown. The quantitiesshown are dimensionless. See subsection 5.2 for further discussion. γ -unstable nuclei The spectra of 38 nuclei with R / < . , , Ru, , , , , , , , Pd, , , Cd, , , , , , , , , Xe, , , Ba, , Ce,
Gd,
Er, , , , , , , , Pt] have been fitted, using the same quality measure.The following comments apply.1) Only the ground state band and the quasi- γ band have been included in thefits. 2) When trying to include the quasi- β band in the fits, trying to correspond it tothe n = 1 case, one finds it lying too low in energy, due to the violation of the restrictionof (37).3) When trying to include the quasi- β band in the fits, trying to correspond itto the n = 0 case, the levels of the β -band in most cases are overestimated. There isnothing to be done against this, since the degeneracies of (51) fix the position of thetheoretical levels. Thus this failure is due to the fact that the experimental data do notexhibit exactly the O(5) degeneracies. nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential − Xe, shownin Table 5. We remark that in all cases the “well size” ˜ β is much lower than 1.8, whilein addition the diffuseness is much lower than 0.43 . The same holds for all of the38 nuclei considered. Therefore the approximate potentials bear little relevance to theexact potentials.As an example, we show in figure 6 both the original exact potential (broken lines),obtained from (2), and the approximate potential (solid lines) occurring after the Pekerisapproximation, obtained from (11), for three different values of the seniority quantumnumber τ , in the case of Xe. In addition, the relevant energy levels, obtained fromeq. (40), are shown. The following comments apply.1) Near their bottoms, the exact and the approximate potentials nearly coincide.2) The approximation breaks down, since for all values of τ the relevant energy levellies higher than the relevant barrier on the rhs of the exact potential. We still get fitsof good quality, which are obtained using the approximate potential, but this potentialis not similar to the exact potential any more, preventing us to draw any conclusionsregarding the exact potential.We therefore conclude that although we could derive analytical expressions forfitting the spectra, the parameter values coming out from the fits correspond to “wellsizes” and diffuseness for which the approximate potential is not quite similar to theexact potential, i.e. the Pekeris approximation breaks down.
6. Conclusions
The main results are summarized here.1) Approximate solutions in closed form are obtained for the 5-dimensional BohrHamiltonian with the Woods–Saxon potential, using the Pekeris approximation and theexact solutions of an extended Woods–Saxon potential in one dimension, featuring adip near its surface.2) Applying the results to several γ -unstable and prolate deformed nuclei, we findthat the WS potential can describe the ground state bands and the γ bands equally wellas other potentials (Davidson, Kratzer, Morse), if the “well size” and the diffuseness arelarge enough (at least 1.9 and 0.44 respectively), but it fails to describe the β bands,apparently because of its lack of a hard core. Several (forty-four) examples of deformednuclei satisfying this condition have been found, but on the other hand all γ -unstablenuclei considered violate this condition.3) The form of the potentials coming out from the fits exhibits a very large dipnear the surface. In other words, the Bohr equation forces the parameters of the WSpotential to obtain values producing a very large dip near its surface, so that its overallshape around its minimum largely resembles the shape around the minimum of theDavidson, or the Kratzer, or the Morse potential.4) The present results suggest that potentials used in the Bohr Hamiltonian canprovide satisfactory results for nuclear spectra if they possess two features, a hard core nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential γ bands, but destroys theability of the potential to describe β bands.Concerning the position of the quasi- β bands, which are not reproduced by theWoods–Saxon potential, the following comments apply.1) The quasi- β bandheads move to higher energies (normalized to the energy ofthe first excited state) both in γ -unstable and in deformed nuclei, if the left wall of theinfinite well potential used in the E(5) and X(5) critical point symmetries is graduallymoved to the right of the origin of the β -axis, approaching the right wall [82, 83].2) The interlevel spacings within the β - band, which are known to be overestimatedin the framework of the X(5) critical point symmetry, get fixed by allowing the rightwall of the infinite well potential to be sloped to the right [84]. It should be mentionedhere that the identification of a symmetry underlying the X(5) special solution of theBohr Hamiltonian remains an open problem.Analytical wave functions for the Bohr Hamiltonian with the WS potential can bereadily obtained by exploiting the similarity of this Hamiltonian, after using the Pekerisapproximation, to the exactly soluble extended WS potential with a dip near its surface.The calculation of B(E2) transition rates becomes then a straightforward task, to beaddressed in further work.Finally, the solution of the Bohr Hamiltonian with a Woods–Saxon potentialwithin the framework of the algebraic collective model [41, 42, 43, 44] would offerthe opportunity of comparison of the present approximate results to exact numericalsolutions. Acknowledgments
B. G¨on¨ul and M. C¸ apak acknowledge financial support by the Scientific and TechnicalResearch Council of Turkey (T ¨UB˙ITAK) under project number ARDEB/1002-113F218.D. Petrellis acknowledges financial support by the Scientific Research ProjectsCoordination Unit of Istanbul University under Project No 50822.
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Quantum numbers appearing in the O(5) ⊃ SO(3) reduction [74], occurringfrom (47) and (48). τ ν ∆ λ L nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential Table 2.
Comparison of theoretical predictions of the Bohr Hamiltonian withthe Woods–Saxon potential for 46 axially symmetric prolate deformed nuclei toexperimental data [21] of rare earth and actinides with R / > +2 and 2 + γ states. The angular momenta of the highest levels of the ground state and γ bands included in the rms fit are labelled by L g and L γ respectively, while N indicatesthe total number of levels involved in the fit and σ is the quality measure of (66). Allenergies are normalized to the energy of the first excited state, E (2 +1 ). For each band,the R / = E (4 +1 ) /E (2 +1 ) ratio (labelled by 4 / γ band (labelled as 2 γ / γ bands included in the fit (labelled by L g / L γ / β has been obtained from [71]. The theoretical predictions are obtained from (64). Seesubsection 5.1 for further discussion. nucleus ˜ β ˜ A C ˜ a β V a L g L γ N σ / / L g / L g / γ / γ / L γ / L γ / Nd 4.1 0.15 7.6 0.62 0.285 207 43 14 4 9 0.24 2.93 3.02 20.6 20.7 8.2 8.6 10.4 9.9
Sm 4.8 0.20 8.6 0.96 0.306 245 61 16 9 15 0.60 3.01 3.10 27.6 28.0 8.9 10.2 19.5 18.4
Sm 4.2 0.22 14.0 0.92 0.341 152 75 16 7 13 0.53 3.25 3.26 36.2 36.5 17.6 18.6 26.3 25.0
Gd 3.4 0.24 6.9 0.82 0.312 119 75 26 7 18 0.35 3.02 3.14 57.3 57.4 8.1 8.8 14.7 14.2
Gd 2.2 0.24 10.8 0.53 0.338 42 81 26 16 27 0.68 3.24 3.25 74.0 74.1 13.0 14.5 44.9 44.0
Gd 2.8 0.18 10.6 0.50 0.348 65 63 12 6 10 0.08 3.29 3.28 23.5 23.5 14.9 15.1 20.4 20.3
Dy 3.5 0.24 6.0 0.84 0.293 143 70 28 13 25 0.49 2.93 3.08 57.9 57.9 6.5 7.5 23.8 22.9
Dy 2.4 0.24 7.7 0.58 0.326 54 78 28 8 20 0.46 3.21 3.21 75.4 74.8 9.6 10.3 19.1 18.2
Dy 3.0 0.23 9.3 0.69 0.339 78 78 24 23 33 0.88 3.27 3.23 65.1 66.5 11.1 12.5 68.2 69.4
Er 3.0 0.24 5.6 0.72 0.304 97 73 26 5 16 0.52 3.10 3.11 55.9 55.4 6.8 7.3 10.5 10.0
Er 1.9 0.24 7.4 0.46 0.322 35 77 20 12 20 0.89 3.23 3.22 43.7 45.4 8.8 10.1 28.5 27.0
Er 2.1 0.21 6.9 0.44 0.333 40 70 22 16 25 0.61 3.28 3.25 61.8 62.8 9.4 9.8 41.6 43.1
Yb 2.4 0.25 4.0 0.60 0.263 83 66 24 4 14 0.33 2.92 3.05 39.9 39.1 4.8 5.3 7.1 6.9
Yb 2.9 0.23 5.6 0.67 0.290 100 67 18 5 12 0.30 3.13 3.14 35.6 35.6 7.0 7.5 10.9 10.3
Yb 2.4 0.23 7.0 0.55 0.315 58 72 24 13 23 0.82 3.23 3.22 62.3 63.9 9.1 9.6 31.2 30.0
Yb 3.4 0.22 8.7 0.75 0.322 111 71 34 7 22 0.64 3.27 3.22 120.5 119.7 11.2 11.6 18.5 17.9
Yb 2.6 0.22 10.1 0.57 0.326 64 72 20 14 22 0.63 3.29 3.26 52.7 54.1 13.6 14.0 39.3 39.9
Yb 2.2 0.19 13.1 0.42 0.330 44 63 14 5 10 0.11 3.31 3.30 32.0 32.1 18.6 18.8 22.6 22.4
Yb 2.6 0.20 15.1 0.52 0.325 64 65 18 5 12 0.13 3.31 3.30 50.2 50.4 21.4 21.5 25.2 25.0
Yb 1.8 0.20 10.7 0.36 0.305 35 61 18 5 12 0.29 3.31 3.29 48.4 49.0 15.4 15.3 19.0 18.9
Hf 2.2 0.25 4.1 0.55 0.250 77 63 22 3 12 0.24 2.97 3.08 36.9 36.5 5.1 5.5 6.3 6.2
Hf 2.5 0.24 5.7 0.60 0.275 83 66 22 4 13 0.28 3.11 3.16 46.5 46.9 7.1 7.7 9.8 9.4
Hf 3.9 0.22 7.9 0.86 0.301 168 66 34 4 19 0.48 3.19 3.16 105.7 105.3 9.5 10.1 12.2 11.7
Hf 3.4 0.23 9.0 0.78 0.276 152 63 38 6 23 0.69 3.25 3.21 132.8 133.5 11.3 11.9 17.0 16.3
Hf 3.1 0.23 10.2 0.71 0.286 117 66 22 4 13 0.43 3.27 3.24 58.2 58.9 13.5 13.7 15.9 15.5
Hf 2.9 0.22 11.4 0.64 0.295 96 65 18 6 13 0.34 3.28 3.27 45.4 45.7 15.2 15.7 21.1 20.6
Hf 2.2 0.22 9.2 0.48 0.280 62 62 18 6 13 0.41 3.29 3.26 44.2 45.0 12.6 12.9 18.1 17.8
W 2.4 0.24 7.6 0.58 22 5 14 0.31 3.22 3.21 51.8 51.8 9.6 10.2 14.0 13.2
W 2.4 0.23 7.6 0.55 14 2 7 0.14 3.24 3.23 27.0 27.2 10.5 10.4 10.5 10.4
W 2.7 0.24 8.7 0.65 0.254 113 61 24 7 17 0.67 3.26 3.22 60.0 60.4 10.8 11.6 18.7 17.6
Os 4.0 0.23 6.0 0.92 24 5 15 0.37 2.93 3.04 45.5 45.5 6.4 7.3 10.4 9.6
Os 3.8 0.23 6.0 0.87 16 5 11 0.40 3.02 3.06 26.1 26.0 6.6 7.4 10.8 9.9
Os 2.5 0.25 5.9 0.63 0.226 122 57 14 7 12 0.59 3.09 3.14 21.8 22.0 6.6 7.7 14.2 13.0
Os 3.7 0.24 6.7 0.89 0.234 250 56 26 7 18 0.84 3.15 3.10 54.0 53.3 7.0 8.3 14.6 13.4
Os 2.2 0.24 6.2 0.53 0.213 107 51 22 6 15 1.05 3.20 3.19 47.9 49.4 7.9 8.4 13.5 12.8
Ra 3.1 0.24 10.1 0.74 0.217 204 52 22 3 12 0.18 3.21 3.23 53.6 54.0 13.3 13.3 14.1 14.0
Th 3.5 0.24 13.1 0.84 0.230 231 55 18 5 12 0.15 3.24 3.25 41.7 41.6 16.8 17.1 20.3 20.0
Th 2.8 0.22 10.6 0.62 0.244 132 54 18 4 11 0.12 3.27 3.26 45.1 45.3 14.7 14.7 16.6 16.6
Th 2.4 0.23 12.1 0.55 0.261 85 60 30 12 25 0.54 3.28 3.27 104.6 105.9 15.9 16.6 36.5 35.6
U 3.0 0.22 13.3 0.66 0.264 129 58 20 4 12 0.20 3.29 3.28 55.9 56.3 18.2 18.4 20.4 20.3
U 2.7 0.23 15.9 0.62 0.272 99 63 28 7 19 0.36 3.30 3.29 98.8 99.6 21.3 21.8 29.0 28.5
U 3.3 0.22 15.5 0.73 0.282 137 62 26 5 16 0.63 3.30 3.29 89.3 90.6 21.2 21.4 24.9 24.7
U 2.9 0.23 18.3 0.67 0.286 103 66 30 27 40 0.85 3.30 3.29 112.1 114.9 23.6 25.0 112.7 113.5
Pu 2.2 0.22 19.0 0.48 0.289 58 64 26 4 15 0.21 3.31 3.30 95.5 96.0 26.6 26.6 28.8 28.6
Pu 2.3 0.22 17.7 0.51 0.292 62 64 26 2 13 0.51 3.31 3.30 93.7 94.8 24.7 24.8 24.7 24.8
Cm 2.7 0.22 17.2 0.59 0.297 83 65 28 2 14 0.87 3.31 3.30 10.5 10.7 24.2 24.0 24.2 24.0 nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential Table 3.
Spectra of some rare earths and actinides, obtained with the parametersshown in Table 2. Gd Gd Er Er Yb Yb Th Th L exp th exp th exp th exp thgsb4 3.24 3.25 3.28 3.25 3.29 3.26 3.28 3.276 6.57 6.59 6.72 6.60 6.80 6.66 6.75 6.698 10.85 10.86 11.21 10.92 11.43 11.07 11.28 11.1310 15.92 15.88 16.61 16.11 17.06 16.37 16.75 16.4712 21.63 21.56 22.79 22.09 23.54 22.49 23.03 22.6014 27.83 27.78 29.58 28.83 30.63 29.36 30.04 29.4516 34.39 34.50 37.33 36.29 37.92 36.94 37.65 36.9718 41.29 41.66 45.10 44.44 45.18 45.21 45.84 45.1120 48.62 49.23 53.28 53.27 52.66 54.13 54.52 53.8522 56.49 57.18 61.85 62.77 63.69 63.1624 64.95 65.49 73.32 73.0326 73.99 74.15 83.38 83.4528 93.82 94.3930 104.56 105.86 γ nalytical solutions for the Bohr Hamiltonian with the Woods–Saxon potential Table 4.
Same as Table 2, for 17 additional nuclei. See subsection 5.1 for furtherdiscussion. nucleus ˜ β ˜ A C ˜ a β V a L g L γ N σ / / L g / L g / γ / γ / L γ / L γ / Gd 1.8 0.18 9.0 0.32 0.353 26 64 16 8 14 0.10 3.30 3.29 40.0 40.2 13.1 13.1 22.8 22.8
Gd 1.5 0.18 8.2 0.27 14 4 9 0.05 3.29 3.29 31.4 31.5 12.0 12.0 14.1 14.1
Dy 1.9 0.19 7.7 0.36 0.343 31 65 18 14 21 0.23 3.29 3.28 47.6 47.9 11.0 11.1 39.4 39.2
Dy 1.2 0.19 7.1 0.23 0.348 12 66 20 10 18 0.26 3.30 3.28 57.4 58.1 10.4 10.4 25.3 25.3
Dy 1.2 0.15 7.5 0.18 6 5 6 0.02 3.31 3.30 6.9 6.8 11.2 11.2 14.9 14.9
Er 1.2 0.21 6.9 0.25 0.342 12 72 16 14 20 0.30 3.29 3.26 36.8 37.1 9.8 9.9 35.7 36.6
Er 1.2 0.16 6.9 0.19 0.338 13 54 18 8 15 0.20 3.31 3.30 50.0 50.6 10.3 10.3 20.4 20.3
Er 0.7 0.18 8.9 0.13 0.336 4 61 26 19 30 0.85 3.31 3.30 95.8 98.1 11.9 13.0 66.2 65.7
Yb 1.3 0.17 9.9 0.22 6 2 3 0.01 3.31 3.30 6.9 6.9 14.5 14.5 14.5 14.5
Hf 1.0 0.16 8.7 0.16 0.274 13 44 12 5 9 0.05 3.31 3.30 24.3 24.4 12.9 12.9 16.7 16.6
W 1.4 0.20 8.6 0.28 0.251 31 50 18 6 13 0.26 3.29 3.28 47.4 48.0 12.2 12.4 17.7 17.6
W 0.9 0.18 5.4 0.16 0.236 15 43 10 6 9 0.06 3.27 3.28 16.7 16.8 8.1 8.0 13.3 13.4
W 1.0 0.19 4.1 0.19 0.226 20 43 14 6 11 0.09 3.23 3.25 29.1 29.2 6.0 6.1 11.4 11.4
Os 1.7 0.22 4.2 0.37 0.200 72 44 14 13 18 0.20 3.17 3.19 25.9 26.1 5.6 6.1 26.5 26.8
Os 1.4 0.23 3.0 0.32 0.186 57 43 12 7 11 0.18 3.08 3.14 18.4 18.7 4.1 4.4 10.9 10.6
Os 1.4 0.24 2.2 0.34 0.178 62 43 10 6 9 0.20 2.93 3.06 12.6 12.7 3.0 3.3 7.9 7.6
Pu 1.7 0.21 16.5 0.36 0.286 35 60 26 4 15 0.34 3.31 3.30 96.8 97.6 23.3 23.4 25.5 25.5
Table 5.
Comparison of theoretical predictions of the Bohr Hamiltonian with theWoods–Saxon potential to experimental data [21] for 9 γ -unstable Xe isotopes with R / < +2 and 2 + γ states. The angular momenta of the highest levels ofthe ground state and quasi- γ bands included in the rms fit are labelled by L g and L γ respectively, while N indicates the total number of levels involved in the fit and σ is thequality measure of (66). All energies are normalized to the energy of the first excitedstate, E (2 +1 ). For each band, the R / = E (4 +1 ) /E (2 +1 ) ratio (labelled by 4 / γ band (labelled as 2 γ / γ bands included in the fit (labelled by L g / L γ / β has been obtained from [71]. The theoreticalpredictions are obtained from (44). See subsection 5.2 for further discussion. nucleus ˜ β ˜ A ˜ a β V a L g L γ N σ / / L g / L g / γ / γ / L γ / L γ / Xe 0.13 0.18 23 0.265 0.2 48 14 8 13 0.26 2.40 2.34 12.9 13.1 2.8 2.3 7.8 8.0
Xe 0.21 0.17 36 0.291 0.5 50 14 9 14 0.47 2.47 2.37 13.8 13.6 2.7 2.4 9.8 10.8
Xe 0.52 0.14 71 0.259 4.0 35 16 9 15 0.65 2.50 2.32 17.5 17.6 2.5 2.3 9.7 11.1
Xe 0.57 0.16 88 0.212 7.2 33 12 8 12 0.51 2.48 2.32 11.0 10.8 2.4 2.3 8.2 8.2
Xe 0.65 0.18 115 0.188 11.9 33 12 9 13 0.63 2.42 2.28 11.0 10.1 2.3 2.3 9.1 10.1
Xe 0.67 0.21 143 0.184 13.4 39 10 7 10 0.47 2.33 2.20 7.6 6.9 2.2 2.2 6.2 6.9
Xe 0.92 0.22 203 0.169 29.4 37 14 5 10 0.30 2.25 2.11 9.5 9.5 2.1 2.1 4.1 4.7
Xe 0.90 0.24 218 0.141 41.2 34 6 2 3 0.34 2.16 2.05 3.2 3.2 1.9 2.1 1.9 2.1134