Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 1CHAPTER 12Fast nu lear rotation the ranking model.At very high spin, one expe ts the Coriolis and entrifugal for es to disturb strongly thewavefun tions of many nu leons. As dis ussed in the introdu tion to the previous hapter, itthen be omes desirable to treat all nu leons on the same footing.In this hapter, we will dis uss two models of this kind, namely the ranking model and therotating liquid drop model. Then the ranked Nilsson-Strutinsky (CNS) approa h orrespondingto a ombination of the two models is introdu ed. Within this approa h, phenomena like bandterminations and superdeformed high-spin states are dis ussed.12.1 The ranking modelIn the ranking model, the rotation is treated in the lassi al sense with the rotation ve tor oin iding with one of the main axes of the nu leus. It then turns out that in this system,the nu leons an be des ribed as independent parti les moving in a rotating potential. In fa t,the rotation degree of freedom enters in very mu h the same way as the deformation degrees offreedom whi h were introdu ed in hapters 8 and 9. One important short oming of the rankingmodel is that the wave-fun tions are not eigenstates of the angular momentum operator. Instead,the angular momentum is generally identied with the expe tation value of its proje tion on therotation axis.The mathemati al formulation of a rotating single-parti le potential was rst given by Inglis(1954). With the oordinates in the laboratory system given by x, y and z and those in therotating system by x1, x2 and x3, we get for onstant angular velo ity, ω, around the x1-axis:x1 = x

x2 = y cosωt+ z sinωt

x3 = −y sinωt+ z cosωtApart from some phase-fa tor, the time-dependent wave fun tions in the two systems mustsatisfyψω (x1, x2, x3, t) = ψ (x, y, z, t)whi h leads to (

∂ψ

∂t

)

x,y,z=

(∂ψω

∂t

)

x1,x2,x3

+∂ψω

∂x2

∂x2

∂t+∂ψω

∂x3

∂x3

∂tWith∂x2

∂t= ω(−y sinωt+ z cosωt) = ωx3

∂x3

∂t= ω(−y cosωt− z sinωt) = −ωx2we now nd

∂ψ(x, y, z, t)

∂t=

(∂

∂t− iωℓ1

)ψω (x1, x2, x3, t)where the angular momentum operator ℓ1 is given by

ℓ1 = −i

(x2

∂

∂x3

− x3

∂

∂x2

)= −i

(y∂

∂z− z

∂

∂y

)= ℓxThe equality between ℓ1 and ℓx is easily proven by dire t evaluation.

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 2The relation for the time derivatives implies that the time dependent S hrödinger equationfor ψih∂ψ(x, y, z, t)

∂t= hψ(x, y, z, t)is transformed into

ih∂ψω

∂t(x1, x2, x3, t) = (h− hωℓ1)ψ

ω (x1, x2, x3, t)for the wave-fun tion in the intrinsi system, ψω. In these equations, the Hamiltonian is givenby h to point out that it is a one-parti le operator. This is in ontrast to the total Hamiltonianwhi h is denoted by H (in the pre eding hapters, no su h distin tion has been made and a apital H has been used also for the single-parti le Hamiltonian).The S hrödinger equation in the rotating system an now be solved in the standard way asan eigenvalue problem(h− hωj1)φ

ω = eωφωwhere the orbital angular momentum operator ℓ1 has been generalized to over also parti leshaving an intrinsi spin and has thus been repla ed by j1 (j = ℓ + s). The Hamiltonian in therotating system,hω = h− hωj1is also referred to as the ranking one-parti le Hamiltonian. The eigenvalues eωi are referred toas the single-parti le energies in the rotating system or more properly Routhians. This is sobe ause the Hamiltonian in the rotating system does not overlap with the energy. The rankingone-parti le Hamiltonian may be summed over all the independent parti les of the system toobtain the total ranking Hamiltonian,Hω = H − hωI1Alternatively, the ranking Hamiltonian an be derived by dire t use of the rotation operator

R = exp (−iIxωt) (see e.g. de Voigt, Dudek and Szyma«ski, 1983) or from the Lagranian(problem 11.1, 12.1 in the book).A simple way to obtain the ranking Hamiltonian is to minimize the energy.E = 〈Ψ|H|Ψ〉under the onstraint that the total spinI = 〈Ψ|I1|Ψ〉is xed. The rotational frequen y ω (or rather hω) will then take the role of a Lagrangian mul-tiplier whi h, as seen from the derivation above, an be identied with the rotational frequen y.The energies of the parti les are measured in the laboratory system and are al ulated asei = 〈φω

i |h|φωi 〉where it should be observed that the time-independent wave-fun tions, φω

i , are not eigenve torsof the Hamiltonian, h. Similarly, the angular momentum is al ulated as an expe tation value〈jx〉i = 〈φω

i |jx|φωi 〉 = 〈φω

i |j1|φωi 〉The total energy and the total spin are now given as sums over the o upied orbitals:

Esp =∑

occ

ei

I ≈ Ix =∑

occ

〈jx〉i

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 3In a similar way as for deformed nu lei at spin zero ( hapter 9), these summed quantities an onthe average be renormalized to a liquid drop behaviour, see se t. 12.5.12.3 The rotating liquid drop modelWe will now for a moment ignore the quantal ee ts and onsider the rotation of a nu leusa ording to the laws of lassi al me hani s. In su h a ma ros opi model, the energy is givenbyEmacr(E,N, def., I) = E(Z,N, def.) +

h2I2

2J (Z,N, def.)The energy E(Z,N, def.) is taken as the stati liquid drop energy whi h was treated in hapter4. The variable "def." denotes a number of deformation parameters, e.g. ε, γ, ε4 . . .. For stablenu lei the liquid drop energy has a minimum for spheri al shape. This minimum is aused bythe surfa e energy whi h over omes the deforming tenden ies of the Coulomb energy.For the rotating harmoni os illator (see se t. 12.2 in Shapes and Shells in Nu lear Stru ture)the dynami al moment of inertia was essentially equal to the rigid body value. In the ase ofindependent nu leon motion, this is what is generally expe ted also for other potentials than theharmoni os illator. The fa t that the experimentally observed moment of inertia is smaller thanJrig for low I an be tra ed ba k to the pairing orrelations. At higher spins, however, these orrelations should disappear. As the rotating liquid drop model is relevant only at relativelyhigh spins, we will use the rigid body moment of inertia in onne tion with this model.The rotational energy be omes smaller with in reasing J . Thus, with the rigid body value, ongurations with the nu leons far away from the rotation axis are favoured. This means thatthe rotational energy tries to deform the nu leus and this tenden y will be ome dominating fora large enough value of the angular momentum I.For small values of I, the nu leus will behave in a similar way as the rotating earth andbe ome attened at the poles, i.e. oblate shape with rotation around the symmetry axis. Withthe rotation axis being the 1-axis and with the denitions of ε and γ given in hapter 8, this orresponds to γ = 60 (there exists some onfusion about the denition of the sign of the angleγ and onsequently, oblate shape with rotation around the symmetry axis is sometimes referredto as γ = −60). With in reasing spin, the distortion of the nu leus will be ome larger and εwill in rease (still at γ = 60). The ma ros opi energy of the nu leus 154Sm, rotating with aspin of I = 40, is shown as a fun tion of deformation in g. 12.2.Detailed al ulations, as has been arried out by Cohen, Plasil and Swiate ki (1974), thenshow that at a large enough angular momentum, the stability towards axial asymmetry is lost.For higher spins, most nu lei will for some intermediate spin values have a minimum for triaxialshape (60 < γ < 0) before, for even higher spins the stability in the ssion dire tion is lost.Thus, the nu leus divides into two fragments whi h y apart due to the entrifugal for es. TheI-values where the transition to triaxial shape and where ssion instability sets in, respe tively,are shown in g. 12.3. This gure also shows that for heavy nu lei, these two I-values oin ide.This means that as soon as the oblate regime be omes unstable the nu leus goes to ssion.12.4 An illustrative example of mi ros opi al ulations of high spin states 20NeWhen onsidering rapid rotation in deformed nu lei, 20Ne is a good starting point be ausegeneral features an be illustrated in a simple ase with only a few a tive single-parti le orbitals.Let us rst point out that for su h a light nu leus, already I = 4 or I = 6 are high spin statesin the sense that they orrespond to high rotational frequen ies ( f. problem 11.2, 12.6 in thebook). The al ulated spin zero potential energy surfa e of 20Ne has a minimum for prolateshape with ε ≈ 0.35. Fig. 4 then shows that this orresponds to a situation where the N = 1

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 4

Figure 2: Contour plot in the (ε, γ)-plane of the rotating liquid drop energy al ulated for the nu leus154Sm at I = 40. The rotation axis (dened as the 1-axis) is sket hed for the dierent ases of axiallysymmetri shape ( f. g. 8.6). The same nu lear shapes are formed in the three 60 se tors but therotation axis oin ides with the smaller (γ = 0 − 60), the intermediate (γ = 0 −−60) and the larger(γ = −60 − −120) prin ipal axis, respe tively. The numbers on the ontour lines refer to MeV abovethe energy of a spheri al liquid drop at I = 0 (from Andersson et al. 1976).shell is ompletely lled and there are in addition two protons and two neutrons in the N = 2orbital with asymptoti quantum numbers |Nn3ΛΩ〉 = |220 1/2〉.With the N = 1 shell being ompletely lled, the parti les in this shell do not ontribute tothe spin. Thus, 8 protons and 8 neutrons form an inert ore. The valen e protons and valen eneutrons are in the N = 2 shell, i.e. in the d5/2, s1/2 and d3/2 subshells. The maximum j-valueis thus 5/2. However, two identi al parti les in the N = 2 shell an only ouple to I = 4. This

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 5

Figure 3: Spin values in the rotating liquid drop model where the ssion barrier disappears ”Bf = 0”and is equal to 8 MeV, respe tively. The dashed urve shows the spin value where the stability of theoblate regime (γ = 60) is lost. Thus, between the dashed line and the (Bf = 0) line, the equilibriumshape is triaxial (from Cohen et al., 1974).is most easily seen by onsidering the maximum proje tions on the rotation axis (the x-axis). Ifthe rst parti le has a proje tion of jx = 5/2, the se ond an at most have jx = 3/2, being eithera se ond d5/2 parti le or a d3/2 parti le (or rather a mixture in the deformed ase but with d5/2dominating be ause the d5/2 shell is learly below the d3/2 shell, see g. 4). It is thus possible forthe two protons and two neutrons to ouple independently to I = 4, whi h means that they anhave a total maximum spin of Imax = 8. In this maximum spin state, they have their spin ve torsquantized along the rotation axis whi h means that the nu leus is axially symmetri around thisaxis. Furthermore, the four parti les rotate mainly around the equator of the nu leus giving riseto an oblate nu lear shape (g. 5).The evolution of the proton or neutron single-parti le orbitals for the ground band of 20Neis illustrated in g. 4. To the far left in this gure, the splitting (and mixing) of the spheri alsubshells aused by prolate deformation is illustrated. At ε ≈ 0.35, this leads to the orbitalsappropriate for the ground state of 20Ne where two protons and two neutrons ll the [220 1/2orbital. The potential is now ranked around a perpendi ular axis (the x-axis) leading to asplitting of the doubly degenerate orbitals and new eigenvalues eωi (ω). Without going intodetails, we should mention that apart from parity, one more symmetry (asso iated with rotation,180, around the ranking axis) survives so that the orbitals labelled by + and (signatureα = +1/2 and α = −1/2) in g. 4 remain un oupled.For ranking at a xed deformation, the slope of the orbitals orrespond to the alignment,〈jx〉 = m. This is seen from the relation,

〈jx〉 = −∂eωi∂ω

,whi h an easily be obtained from the ranking Hamiltonian, using the theorem (R.P. Feynman,

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 6

Figure 4: Single-parti le orbitals along a path in the (ε, ε4, γ, ω) spa e as indi ated s hemati ally in thelower part of the gure. The path is hosen to illustrate how the orbitals an be followed when a prolate olle tive band goes to termination at oblate shape. The spheri al origin of the orbitals at a typi allow-spin deformation is tra ed in part A while in part B, rotation is swit hed on at onstant deformation.At a frequen y of ω/0ω0≃ 0.15 orresponding to I ≃ 6 in the 20Ne ground band, the driving for es towardoblate shape be ome important. Thus, in part C the deformation is varied over the γ plane togetherwith hanges in the other parameters as they o ur when a band approa hes termination at γ = 60.In part D, nally, the origin of the aligned oblate orbitals is tra ed illustrating to whi h j shell theymainly belong and their aligned spin. The o upation of sd-shell orbitals in the ground state and in theterminating 8+ state of 20Ne is also indi ated. It is interesting to note how the Z = N = 10 gap stayslarge all the way to the termination (ε ≃ 0.20, γ = 60).Phys. Rev. 56 (1939) 340),

d〈ψ|O|ψ〉

dµ= 〈ψ|

dO

dµ|ψ〉,whi h is valid if ψ is an eigenstate of the operator O.For prolate shape and small ω-values, the two bran hes of an Ω = 1/2 orbital get an alignmentof ±(1/2)a where a is the de oupling fa tor dis ussed in hapter 11 while in lowest order of ω,the Ω > 1/2 orbitals show no alignment (no de oupling fa tor). Then with in reasing rotationalfrequen y ω, the oupling between the dierent orbitals means that all orbitals get a 〈jx〉 dierentfrom zero. Note espe ially that the two orbitals emerging from [220 1/2 be ome strongly alignedat large ω i.e. their slopes orrespond approximately to jx = 5/2 and jx = 3/2. These are theorbitals o upied in the ground band of 20Ne as dis ussed qualitatively above. Their strongalignment orrespond to a polarization toward oblate shape. Thus, in the third se tion of g. 4,the shape is followed through the γ-plane with slightly in reasing rotational frequen y ending upin an oblate nu leus "rotating around its symmetry axis". This orresponds to the state drawns hemati ally to the right in g. 4.It is interesting to note that by hoosing the path in deformation/rotational frequen y illus-

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 7

Figure 5: Cal ulated Iπ = 4+ and 8+ energy surfa es with in lusion of the shell energy for 20Ne togetherwith s hemati illustrations of the ongurations at the minima. The denition of ε and γ is the same asin g. 12.2. The ontour line separation is 2 MeV and the numbers on the lines refer to ex itation energyabove the spheri al liquid drop at I = 0 (from Ragnarsson et al., 1981).trated in g. 4, it is possible to follow in a ontinuous way (no rossings between orbitals) theevolution of 20Ne from its ground state to the aligned 8+ state. In the right part of g. 4, de-formation and the rotational frequen y are varied simultaneously so that the spheri al subshellsare regained at the right edge of the gure.Using the methods dis ussed in the next subse tion, it is possible to al ulate potential energysurfa es also for I 6= 0. An example of su h a al ulation is given in g. 5, where the I = 4 andI = 8 energy surfa es of 20Ne are drawn. The ground state I = 0 shape of 20Ne is al ulatedat ε ≃ 0.35, γ = 0. At I = 4, the disturban e aused by the rotation is rather small with anessentially un hanged deformation at the potential energy minimum. For I = 8, the rotationaldisturban es are mu h larger and, as anti ipated above, the minimal energy shape is oblate withrotation around the symmetry axis (γ = 60).In g. 7, the experimental yrast states are ompared to the al ulated energies within theharmoni os illator and modied os illator models. Furthermore, we show the energy

E =h2

2JrigI2where Jrig is taken as a onstant, namely the rigid moment of inertia of the harmoni os illatorat I = 0 (see problem 12:4 in the book). Ex ept for the 2+ energy, the al ulated yrast lineof the modied os illator potential (Ragnarsson et al., 1981) is in quite good agreement with

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 8

Figure 7: Measured ground band energies of 20Ne plotted vs. I2 ompared to al ulated energies in somedierent approximations. The urve marked "Harm. os ." shows the energies in the simplied solutionof ranked harmoni os illator while "Mod. os ." indi ates the modied os illator (g. 5) model. Forea h spin, the energy is minimized as a fun tion of deformation. The orresponding shapes are shown inthe (ε, γ)-plane in the inset. The urve marked "Rigid rot." gives the energy E =(h2/2Jrig

)I2 where

Jrig is kept onstant equal to the rigid body value at the ground state shape of the harmoni os illator.experiment. It is espe ially satisfying that the relatively low energy of the I = 8 state is repro-du ed. This is in ontrast to the harmoni os illator al ulations whi h give a mu h too largeE8+ to E6+ spa ing. The equilibrium shapes in the two models are also shown in g. 7. Themain feature is that the shape remains essentially prolate up to the I = 6 state and that a large hange in deformation o urs between I = 6 and I = 8.If one goes to higher spins than I = 8 for 20Ne, one expe ts an in rease of the deformation.In the ma ros opi des ription, this is understood as a result of the entrifugal for es. In themi ros opi harmoni os illator model, parti les must be ex ited to higher shells to get spinsabove I = 8. This naturally leads to larger deformations.12.5 The shell orre tion method for I 6= 0When the ground state potential energy has been al ulated at some xed deformation itshould be possible to get the I-dependen e simply by adding the rotational energy as extra tedfrom the ranking model. Thus, for a pres ribed spin I0, the rotational frequen y ωI0 is deter-mined so that

I0 =∑

i

occ

〈jx〉iThen the ex itation energy is obtained asEexc =

∑

i

occ

ei |ω=ωI0

−∑

i

occ

ei |ω=0

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 9For example, in g. 5, the Strutinsky shell orre tion method has been applied to al ulatethe I = 0 energy surfa e while the I-dependen e at a xed deformation has been al ulateda ording to the above formulas. In pra ti e, in ea h mesh points in deformation spa e, the ranking Hamiltonian is diagonalized for a number of ω-values. Subsequently, ωI0 and then Eexcis obtained from interpolation. In the energy surfa e of g. 5, the energy has also been minimizedwith respe t to ε4 deformations.Very often, however, the simple summation to obtain Eexc might lead to undesired features.In general, this is aused by de ien ies in the single-parti le potential so that the averagebehaviour of Eexc is unrealisti . For example, the (unphysi al) ℓ2-term in the modied os illatorpotential orrespond to a velo ity dependen e and leads to an average moment of inertia whi h is onsiderably larger than Jrig. Similarly, in some parametrisations of the Woods-Saxon potential,the radius parameter is dierent from experimentally observed nu lear radii and with J ∝ r2,this might have rather drasti ee ts.It is expe ted, however, that the u tuations are more a urately des ribed by the sums, f. hapter 9. Therefore, it appears reasonable to retain only these u tuations with the average be-haviour governed by the rigid body moment of inertia. To this end we dene (see e.g. Anderssonet al., 1976) a spin-dependent shell orre tion energyEsh (I0) = Σei

∣∣∣I=I0 − Σei∣∣∣I=I0where the smoothed single-parti le sums (indi ated by "∼") are al ulated from a Strutinskypro edure whi h is essentially the same as des ribed in hapter 9. Subsequently, the total energyis al ulated as the sum of the rotating liquid drop energy and the shell energy,

Etot (ε, I) = EL.D. (ε, I = 0) +h2

2Jrig(ε)I2 + Esh (ε, I)where ε is a shorthand notation for the deformation, ε = (ε, γ, ε4, . . .).In the denition of the shell energy, all quantities should be evaluated at the same spin I0, i.e.the smoothed single-parti le energy sum should be al ulated at an ω-value giving a smoothedspin Σmi = I0. Thus, the ω-value in the dis rete sum and in the smoothed sum is generallydierent and it be omes di ult to get any feeling of the variation of Esh from an inspe tion ofa single-parti le diagram. However, it an be shown that the quantity

Equasi−sh(ω) = Σeωi − Σeωiwith all quantities al ulated at the same ω is numeri ally very similar to Esh. An elementarydis ussion of this is given in Ragnarsson et al. (1978). The quantity Equasi−sh is dened exa tlyanalogous to the stati shell energy dis ussed in hapter 9. Thus, ω enters very mu h as adeformation parameter and we an take over all our experien e from the stati ase; spe i allythat gaps in the single-parti le spe trum give a favoured (negative) shell energy while a largelevel density leads to a positive shell energy, i.e. an unfavoured onguration.12.6 Competition between olle tive and single-parti le degrees of freedom inmedium-heavy nu leiWe will now turn to heavier nu lei where, as seen in g. 11.2, the moment of inertia extra tedfrom the measured 2+ to 0+ energy spa ing is less than 50% of the al ulated rigid body value.We have already pointed out that the low value is due to the pairing orrelations (the pairing orrelations are less important in a light nu leus like 20Ne). With in reasing spin, the experi-mental moment of inertia be omes larger (g. 11.13) and for the deformed rare earth nu lei, it omes lose to the rigid body value in the I = 20 − 30 region. This suggests that the pairing

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 10 orrelations are rather unimportant at these spins and the same on lusion is also rea hed frommore fundamental theoreti al onsiderations. The ranking model in the form we applied it to20Ne, with independent parti les in a rotating potential, should then be appli able to heavy nu- lei at high enough spins, let's say I ≥ 30. For su h high spins, the approximation to identify thetotal spin with the proje tion on the rotation axis should also be quite a urate. The result from20Ne that the model seems to des ribe the spe trum quite reasonably all the way down to I = 0or at least I = 2 is in some way surprising. Indeed, the appli ation of a rotating independentparti le model to the I = 0, 2, . . . states of 20Ne an hardly be justied theoreti ally.Cal ulated potential energy surfa es for 160Yb at dierent spin values are exhibited in g. 8.The nu leus is prolate for I = 0 but be omes soft toward γ = 60, with in reasing spin. At

Figure 8: Cal ulated I = 0, 40, 50, 60, 70 and 80 potential energy surfa es in the (ε, γ) plane within lusion of the shell energy for 16070Yb90. The ontour line separation is 1 MeV (from Andersson et al.,1976).

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 11

Figure 9: Evolution of the nu lear stru ture of 158Er with spin. Ex itation energies of a variety ofobserved stru tures are plotted with respe t to a rigid-rotor referen e in order to emphasize the hangesthat o ur along and lose to the yrast line. The strongest new high moment-of-inertia band is in luded,but its exa t ex itation energy is not known. The insert illustrates the hanging shape of 158Er within reasing spin within the standard (ε, γ) deformation plane (from E.S. Paul et al., Phys. Rev. Lett., 98,012501 (2007), see also www.aip.org/pnu/2006/split/807-2.html) for a `popular summary'.I = 50, a shape transition has o urred and the lowest energy is found for oblate shape withrotation around the symmetry axis (γ = 60). The same general trend as in 20Ne is thus obtained.However, the dieren e in mass means that the large shape transitions o ur at very dierentspins, I = 8 in 20Ne ompared to I = 40 − 50 in 160Yb.The energy surfa es of g. 8 should be understood as some average of the states in theyrast region at the dierent spin values exhibited. If the dierent states are onsidered inmore detail, (Bengtsson and Ragnarsson, 1985), it turns out that the shell ee ts leading tothe γ = 60 shape transition are very similar to those in 20Ne. Thus, at high spins, 160Yb an be onsidered as a losed ore of 146Gd and 14 additional valen e parti les in the j-shellsabove the Z = 64 and N = 82 shell losures. (See gs. 11.5 and 11.6 where one notes that theZ = 64 gap is smaller than e.g. the Z = 50 and Z = 82 gaps, and it is only for nu lei with alimited number of valen e nu leons outside the 146Gd ore that it shows any magi properties(Kleinheinz et. al., 1979).) For example, a ording to al ulations for 160Yb, it is possible todene a onguration with six h11/2 valen e protons, six valen e neutrons distributed over thef7/2 and h9/2 shells and the remaining two neutrons in the i13/2 shell. In this ongurationdenoted π(h11/2)

6ν(f7/2h9/2)6(i13/2)

2, it is possible to follow the gradual alignment of the spinve tors till full alignment in an Iπ = 48+ state where the h11/2 protons ontribute with 18 spinunits (11/2 + 9/2 + 7/2 + 5/2 + 3/2 + 1/2), the f7/2 and h9/2 neutrons also with 18 spinunits (9/2 + 7/2 + 7/2 + 5/2 + 5/2 + 3/2) and the i13/2 neutrons with 12 spin units (13/2

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 12+ 11/2). Similarly, it is for example possible to form a 46+ terminating state in 158Er fromthe onguration π(h11/2)4ν(f7/2h9/2)

6(i13/2)2 and a 42+ terminating state in 156Er from the onguration π(h11/2)

4ν(f7/2h9/2)4(i13/2)

2. These terminating states in 156Er and 158Er havebeen observed experimentally and stand out as spe i ally low in energy. This is illustrated ing. 11.12 for 158Er. The present status of the high-spin bands in 158Er is illustrated in g. 9.12.7 Smooth terminating bands.If one follows the yrast states in 158Er, they be ome somewhat irregular when approa hingthe terminating I = 46 state. However, some other terminating bands are mu h more regularand have been referred to as smooth terminating bands. In general, the rotational bands tend toget smoother if they are built not only from parti les outside a losed ore but also from one ortwo holes in the ore. Congurations of this kind are illustrated for 62Zn and 109Sb in g. 10,where the losed ores are 56Ni (Z = N = 28) and 100Sn (Z = N = 50), respe tively. Similarly,it turns out that ongurations dominated by holes in a ore show somewhat similar properties.For example, the 199Pb onguration drawn to the right in g. 10 has 11 holes and two parti lesrelative to a 208Pb (Z = 82, N = 126) ore.

28 28

50 50

g9/2

p1/2

f5/2

p3/2

f7/2

h11/2

d5/2

g7/2

g9/2

protons neutrons protons neutrons

62Zn32 30 109Sb58 51

82

126

i13/2

h9/2

s1/2

d3/2

h11/2

g9/2

p1/2

f5/2

p3/2

i13/2

protons neutrons

199Pb117 82

Figure 10: S hemati ongurations of typi al terminating bands in 62Zn and 109Sb and of a so- alledmagneti band 199Pb . Note that these ongurations do not refer to the pure j-shells but rather to theorbitals in the deformed rotating potential whi h are dominated by these j-shells. The dierent j-shellsare shown, but when xing ongurations, a distin tion is only made between the high-j intruder orbitals(e.g. g9/2 for the N = 4 shell) and the other `low-j' orbitals (g7/2d5/2(d3/2s1/2) for N = 4).The nu leus 62Zn [1 is parti ularly interesting be ause, one an follow in detail how ong-urations with higher and higher maximum spin are formed when parti les are ex ited to highershells. This is illustrated in g. 11 where the bands al ulated with the methods outlined inprevious subse tions are ompared with experiment. The lowest energy onguration in 62Znwill orrespond to a losed ore and the two valen e protons and four valen e neutrons in orbitalsof p3/2 and f5/2 hara ter (when the nu leus be omes deformed, the orbitals from p3/2 and f5/2(and p1/2) mixes so they annot be distinguished in a similar way as we did not distinguishbetween the dierent N = 2 orbitals for 20Ne). In the lowest energy onguration whi h goesto higher spin values than I = 10, one neutron is ex ited from the se ond (p3/2f5/2) orbital tothe lowest g9/2 orbital. The maximum spin for the three remaining (p3/2f5/2) neutrons is then5/2 + 3/2 + 3/2 = 5.5 but it turns out that the energy is more favoured in the band where the

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 1362Zn Exp

0+ 2+

4+

6+

8+

10+

3− 5− 7−

9−

11−

13− 10+ 12+ 14+

16+ 15−

17−

19−

12+

14+

16+

18+

20+

11+ 13+

15+

17+

19+

21+

13−

15−

17−

19−

21−

23−

14−

16−

18−

20−

22−

24−

0

5

10

15

20

25

Exc

. ene

rgy

(MeV

)

CNS calc.

+νg9/2

+πg9/2

+νg9/2

+π(f7/2)−1

0+

2+ 4+

6+

8+

10+

3−

7− 9−

11−

13−

10+

12+

14+

16+

15−

17− 19−

12+

14+

16+

18+

20+

11+ 13+

15+

17+

19+

21+

13−

15−

17−

19−

21−

23−

14−

16−

18−

20−

22−

24−

Figure 11: The low-lying observed bands in 62Zn are ompared with the al ulated bands assigned tothem, all shown to their maximum spin values. In the ground band, the 2 valen e protons and the 4valen e neutrons o upy low-j N = 3 orbitals (orbitals of (p3/2, f5/2) hara ter) while the next threebands are formed from su essive ex itations of a proton or a neutron to the g9/2 orbitals as indi ated inthe right panel. In these bands, only the favoured signature is shown. With one proton hole in f7/2 (theproton onguration in g. 10), the maximum spin in reases by 5h resulting in the bands with maximumspin 21+ and 24−, respe tively. Note that some observed bands tend to get larger energy dieren es loseto termination while other bands show the opposite feature and that these tenden ies are reprodu ed inthe al ulations.maximum spin is 5/2 + 3/2 + 1/2 = 4.5 (this depends on the signature quantum number of thethird (p3/2f5/2) neutron). Combined with the maximum spin for the g9/2 neutron, 4.5, this givesa maximum neutron spin of 9 and thus in luding also the 2 protons, Imax = 13h whi h agreeswith experiment, see g. 11. In the next ex ited onguration, one proton is also ex ited to g9/2,leading to a maximum proton spin of 2.5 + 4.5 and thus a total onguration with Imax = 16h.If one more neutron is ex ited, from (p3/2f5/2), to g9/2 (leading to the neutron ongurationshown in g. 10, three more spin units be ome available and a band with a maximum spin of19h is obtained. Finally, starting from the two ongurations terminating at 16h and 19h, oneproton is ex ited from the f7/2 shell below the Z = 28 to the (p3/2f5/2) orbitals. In the resultingproton onguration shown to the left in g. 10, the maximum spin is 3.5+2.5+1.5+4.5 = 12h.Combined with the two neutron ongurations, it leads to the two highest spin ongurations ing. 11, terminating at I = 21h and I = 24h, respe tively. Note that in these ases, also bandsterminating at I = 20h and I = 23h are shown. They are formed if the proton hole is pla ed inthe other signature bran h of the f7/2 orbital. This is typi al for orbitals high up in shells wherethe two signatures remain essentially degenerate up to high rotational frequen ies ( f. g. 4).Consequently, the I = I0 state in the band of one signature has an energy lose to the average

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 14of the I0 − 1 and I0 + 1 states in the other signature band.Ex ept for the very low-spin states, there is an impressive agreement between al ulationsand experiment in g. 11. A spe ial feature is that the transition energies lose to terminationin rease strongly with spin in some bands while they rather de rease in other bands. Su hfeatures are seen more learly if the energies are shown relative to an average rotational energyas done for 109Sb below.

Calc.

ν(g7/2d5/2)6(h11/2)

2 ν(g7/2d5/2)5(h11/2)

3

π(g9/2)−2(g7/2d5/2)

2(h11/2)1

16 20 24 28 32 36 40 44 48 52

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Spin, I (h)

E −

0.0

13 I(

I+1)

(M

eV)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Exp.

109Sb -1.5

-1.0

-0.5

0.0

0.5

E −

0.0

13 I(

I+1)

(M

eV)

Figure 12: Comparison between the three lowest al ulated and observed rotational bands in 109Sb. Thesame smooth referen e has been subtra ted in both ases. Dashed (solid) lines indi ate negative (positive)parity. The al ulated bands whi h we ompare to experiment (thi ker lines, with proton and neutron ongurations indi ated) are based on 2p-2h proton ex itations a ross the Z = 50 gap, see g. 10. Thebands shown at higher spins are based on 3p-3h ex itations and those at lower spins on 1p-1h and 0p-0hex itations. Terminations are indi ated by large open rings.The onguration for 109Sb in g. 10 is essentially equivalent to that of 62Zn but one shellhigher up. It is easy to al ulate the maximum proton and neutron spins as 4.5 + 3.5 + 3.5 +2.5 + 5.5 = 19.5h and 3.5 + 2.5 + 2.5 + 1.5 + 1.5 + 0.5 + 5.5 + 4.5 = 22h, i.e. Imax = 41.5h for the ombined onguration. The rotational band al ulated [3 in this 109Sb onguration is drawnwith a dashed line in g. 12 where it is ompared with its experimental ounterpart (see [4 for areview). The bands are shown relative to a rotational referen e so the urves show the dieren esrelative to this referen e. The fa t that the ( al ulated and experimental) urve bends up loseto termination shows that the transition energies be ome unusually large at these spin values ina similar way as for the (f7/2)

−1 onguration of 62Zn.

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 15

Figure 13: The gamma-ray spe trum of the superdeformed band in 152Dy as originally identied in the1986 Daresbury experiment (from Twin et al., 1986).When g. 12 was drawn, the observed band was not onne ted with the low-spin level s hemeso the exa t spin values assigned to the band were largely determined from omparisons with al ulations. Subsequently, the experimental band has been onne ted showing that the spinvalues are indeed as suggested in g. 12. In the other al ulated bands drawn by thi k linesin g. 12, one more neutron has been lifted from the (g7/2d5/2)-orbitals to the h11/2-orbitalswhere two signature partners of similar energy are formed depending on the signature of the5th (g7/2d5/2)-parti le. These al ulated bands are identied with two observed bands for whi hneither the parity nor the spin values have been determined. These quantities are thus `tted' toget a good agreement with the al ulations, espe ially suggesting that they are both observed totheir respe tive terminations, Imax = 43.5− and 44.5−, respe tively. Considering the ex ellentagreement, it would be surprising if these spin values will not turn out to be orre t while we an ertainly expe t some ` orre tions' when it omes to the relative energies of the bands. If theserelative energies were determined, they ould be used to t the relative energies of the j-shellsdrawn s hemati ally in g. 10.The 199Pb onguration in g. 10 is similar to the 62Zn and 109Sb ongurations if parti lesare ex hanged by holes. Rotational bands assigned to ongurations of this kind have beenidentied in several Pb isotopes. They show similarities with the terminating bands but alsodieren es. They are often referred to magneti bands be ause the I → (I−1)M1-transitions aregenerally dominating ompared with the I → (I − 2) E2-transitions. At present, in our resear hgroup in Lund, we are trying to get a better understanding of these bands whose ongurationsare known in less detail than those of the terminating bands [2.12.7 Rotational bands at superdeformation.In the pre eeding se tion, we dis ussed the ase of a few valen e nu leons outside losed shellsleading to states of single-parti le hara ter at intermediate spin values. With more parti lesoutside the ore, the nu leus will stay olle tive to higher spins with only small shape hanges.In any ase, however, the entrifugal for e will sooner or later be ome dominating as dis ussed

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 16within the liquid drop model above and illustrated in gs. 12.2 and 12.3. Indeed, for nu lei withmass A = 100−150, the liquid drop energy will be very soft over large regions of the deformationplane for spins I ≈ 50 − 60. This means that the shell ee ts may play a very important role reating minima at small but also at large and very large deformations.An important question is now if one an expe t parti ularly strong shell ee ts at somedeformation. In the harmoni os illator (HO), this is the ase when the longer axis (the z-axis)is twi e the perpendi ular axis. In this ase, there are large degenera ies be ause two quanta inthe z-dire tion an be inter hanged by one quantum in the perpendi ular dire tion. New magi gaps are thus formed as seen previously in g. 8.1. It turns out that these gaps survive to someextent also in realisti potentials but they are shifted to somewhat higher parti le numbers. Thisis similar to the spheri al ase where the magi 40, 70 and 112 gaps of the HO are repla ed by themagi numbers 50, 82 and 126 in realisti nu lear potentials, see g. 6.3. The best example at2:1 shape is seen for the HO gaps at parti le numbers 60 and 80 (g. 8.1) whi h lead to favouredshell ee ts at Z ≈ 66 and N ≈ 86 in realisti nu lear potentials. These general expe tationshave been veried and 152Dy (Z = 66,N = 86) was the rst ase where a high-spin rotationalband at approximate 2:1 shape was observed as illustrated in gs 13, 14. In su h a band theangular momentum is largely built from small ontributions from many parti les whi h meansthat it is very regular (g 13). Furthermore, su h bands show no tenden ies to terminate in thespin range where they an be observed.Today, superdeformed and large deformation rotational bands have been identied in manyregions of the nu lear periodi table. One example is the band labelled `new high moment ofinertia band' in g. 9. However, no band with an axes ratio onsiderably larger than 2:1 hasbeen identied and the sear h for su h bands is one of the hallenging tasks in high spin physi s.Referen es[1 C. E. Svensson, C. Baktash, G. C. Ball, J. A. Cameron, M. Devlin, J. Eberth, S. Flibotte, A.Galindo-Uribarri, D. S. Haslip, V. P. Janzen, D. R. LaFosse, I. Y. Lee, A. O. Ma hiavelli,R. W. Ma Leod, J. M. Nieminen, S. D. Paul, D. C. Radford, L. L. Riedinger, D. Rudolph,D. G. Sarantites, H. G. Thomas, J. C. Waddington, D. Ward, W. Weintraub, J. N. Wilson,A. V. Afanasjev and I. Ragnarsson, Phys. Rev. Lett. 80 (1998) 2558.[2 B.G. Carlsson and I. Ragnarsson, Phys. Rev. C 74 (2006) 044310.[3 I. Ragnarsson, V. P. Janzen, D. B. Fossan, N. C. S hmeing and R. Wadsworth, Phys. Rev.Lett. 74 (1995) 3935.[4 A.V. Afanasjev, D.B. Fossan, G.J. Lane and I. Ragnarsson, Phys. Rep. 322 (1999) 1.

Nu lear stru ture theory, FMF121, FYST11, ht09, h. 12 17

Figure 14: The full spe trum observed for the nu leus 152Dy showing the low-spin non- olle tive yraststates in the middle, a olle tive normal-deformed band to the left and the superdeformed band to theright. The inset in the upper left orner shows E vs. I plotted in a s hemati way for the dierentstru tures (from J.F. Sharpey-S haer, Physi s World, Sept. 1990, p. 31).