Normal persistent currents and gross shell structure at high...

Normal persistent currents and gross shell structure at highspin

S. Frauendorf∗, M.A. Deleplanque† and V.V. Pashkevich∗∗

∗Physics Department, University of Notre Dame, Notre Dame, Indiana 46556†Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720

∗∗Joint Institute of Nuclear Research, Dubna, Russia

Abstract. The magnetic response of metal clusters and the rotational response of nuclei are determined by strong normalpersistent currents, the influence of which can be understood in terms of classical periodic orbits. The semiclassic PeriodicOrbit Theory also explains the gross shell structure of the binding energies and the shapes of nuclei and clusters. The relevantproperties, which are measured for nuclei, are explained.

Keywords: persistent currents, shell structure, nuclear rotation, magnetic response, metal clusters, mesoscopic electron systemPACS: 21.10.Re, 21.10.Dr, 23.20.Lv, 24.10.Pa, 36.40.Cg, 73.22.-f,73.23.Ra

RELATION BETWEEN ROTATIONAL AND MAGNETIC RESPONSE

A macroscopic system of electrons has only a weak magnetic response, the Landau diamagnetism. In contrast, amesoscopic system responds strongly. The response oscillates as a consequence of the quantization of the electronicmotion in the confined volume. As an example, fig. 1 shows the magnetic susceptibilityχ of a two-dimensionalelectron gas confined to a square billiard. The quantization generates oscillations as a function of the number ofelectrons, which is determined bykF a. The electron states bunch into shells and the oscillations reflect how of theFermi level passes through the shells. The number of electrons in the system is several thousands, which calls for asemiclassical interpretation. The figures compares the exact quantal calculation with the semiclassical results obtainedby means of the Periodic Orbit Theory (POT), which we also will use. In the frame of POT, the closed periodic orbits ofthe classical motion generate the quantal oscillations, the frequency of which isk FL, wherekF is the Fermi momentumandL the length of the orbit. The susceptibility also oscillates as function of the magnetic fluxΦ that penetrates theorbit. The magnetic response of mesoscopic systems has been reviewed in [1].

The oscillations have been observed in an assembly of quantum dots (nano scale metal layers) on an insulatingsurface [3]. The strong magnetic moment of each dot is generated by an electric current generated by the electronson the periodic orbits. Since the magnetic moment remains constant as long as the external magnetic field does notchange, the currents are persistent. Persistent currents are induced when a superconductor is brought into a magneticfield. They do not die out because the superconductor does not have an Ohmic resistance. The quantum dots are in thenormal, non-superconductingstate. The reason that the current does not die is the quantization of the electronic motion.Its consequences become more and more significant with shrinking size of the system. Hence the name “quantum sizeeffects” is used in the solid state community.

We have studied the magnetic response of unsupported three-dimensional metallic clusters, which freely adjust theirshape. The shape is determined by the quantal states of the conduction electrons, which move (nearly) free throughthe cluster. They are confined by the average potential generated by the positive ions and the conduction electrons.There is a far-reaching analogy between metal clusters and nuclei (see e.g. [4]). The nucleons move freely inside thenucleus. The difference is that they are confined by the average potential generated by only the nucleons themselves.However this difference is not important for the following. The shape of nuclei is determined by the quantal statesof the nucleons (shell structure) as the one of the clusters by the quantal states of the conduction electrons. It is notpossible to measure the magnetic response of nuclei with the neccessary accuracy. However it is very easy to measurethe rotational response of nuclei. There are systematic data available, which we used for our analysis. The data on themagnetic response of free mass-selected metal clusters is sparse, because the measurements are difficult. However,the following discussion should also be relevant to the more practical case of ensembles of clusters on a surface orembedded in a matrix material. The material of this talk is published in [5].

Larmor’s theorem states the equivalence between a system in an external magnetic field and a system rotating with

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http://dx.doi.org/10.1063/1.1996872

FIGURE 1. Magnetic susceptibilityχ of a two-dimensional electron system in a square billiard. The susceptibility of the Landaudiamagnetism of an extended two-dimensional electron systemχL is used as unit. The susceptibility is show as function of thenumber of electrons (∝ kF a) and of the magnetic fluxΦ ∝ ωL ( denoted byϕ in the figure). One classical periodic orbit is shown.The full lines show the result of the exact quantal calculation, the dashed ones show the results of the semiclassical POT. From [2].

the angular velocity corresponding to the Larmor frequencyhωL = µBB, whereB is the external magnetic field andµB the Bohr magneton. More carefully, the Hamiltonian for a nucleus rotating with the angular velocityω about thez-axis is

HR = H −ωlz, (1)

wherel is the orbital angular momentum of the nucleons; and we disregard the spin. The Hamiltonian of the conductionelectrons in the presence of the magnetic field is

HM = H +ωLlz +mω2

L

2(x2 + y2). (2)

In linear order the two Hamiltonians are equivalent, which is the content of Larmor’s theorem. The quadratic term isthe negative centrifugal potential. It compensates the centrifugal potential, which is only present in the case of rotation.We restrict ourselves to the case of slow rotation. Then we can compare with the magnetic case by applying first orderperturbation theory to the quadratic term (ωL = ω),

ER = EM− <mω2

2(x2 + y2) > . (3)

The second term is the rotational energy in the rotating frame of reference, if the system rotates like a rigid body:

ER = Eo − ω2

2Jrig (4)

Jrig = m∫

ρ(x2 + y2)dτ , (5)

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FIGURE 2. Yrast lines for a rotational nucleus160Er (top) and for a non-rotational nucleus150Dy (bottom) together with theirfit for the ten highest spins, which are larger than 10h. The angular momentum is given asJ2 = I(I +1)h2.

whereρ is the particle density. In this case there are no currents in the rotating frame, ie.

jlab = jbody +ρωxr, (6)

andjbody = 0. The finite size effects are manifest by the existence of normal persistent currents. They generate a strongmagnetic response of the cluster, which is described byHM. In the case of rotation, the persistent currents appear ascurrents in the rotating frame, ie.jbody = 0 . They cause a deviation of the rotational energy from its value for rigidrotation. If the rotational energy remains a quadratic function ofω these currents cause a deviation of the moment ofinertiaJ from the rigid body value. The quadratic dependence is equivalent with the linear relationJ = J ω betweenthe angular momentumJ and the angular frequencyω and the quadratic expession for the energy

E(J) = Eo +J2

2J. (7)

As I will discuss later, this is reasonably well fulfilled for the considered range ofω. Thus the magnetic responseχVof clusters (V being the volume) should be compared with the deviations of the moment of inertia of nuclei from itsrigid body value,Jshell = J −Jrig.

MOMENTS OF INERTIA OF UNPAIRED NUCLEI

The nuclear moments of inertia are modified by the pair correlations. Since we were interested in their values in theunpaired state, we extracted them at high spin, where the pair correlations are suppressed. Assuming that the energy

57


FIGURE 3. Experimental shell moments of inertia ( deviations from the rigid-body moments of inertia) as a function of neutronnumber for small and normal deformation. The different symbols refer to different proton numbers, the legend being given in Fig.8.The rigid body values are included for comparison.

depends quadratically onJ, the moments of inertia were found by fitting the straight lineE(J 2) = Eo + J2/2J tothe last ten levels of the yrast sequence (levels of minimal energy for given angular momentum ). Fig. 2 shows twoexamples: the well deformed nucleus160Er, for which the yrast states belong to regularly spaced rotational bands,and the near-spherical nucleus150Dy, for which the yrast levels are irregularly spaced. We carried out a systematic fitto the available high spin spectra. From the experimental moments of inertia obtained in this way we subtracted therigid body value, which was calculated by means of expression (5) assuming that the nucleus has the same shape asin the ground state. The results are shown in Fig. 3 as a function of the neutron numberN. Only nuclei for which thehighest measured angular momentum is larger than 20h are included. Another fit with the less stringent restriction thatthe maximal angular momentum should be larger that 15h gives essentially the same dependence onN, however witha somewhat negative bias. This is attributed to the residual pair correlation, which become more important at lowerangular momentum . The shell structure dramatically changes the rotational response of nuclei in the unpaired state.The deviations of the moments of inertia from the rigid-body values are in the same order of magnitude asJ rig. Thereis also a contribution to the moment of inertia that is analogous to the Landau diamagnetism, which is of the order of1% of Jrig. Comparing with this, we find that the normal persistent currents in nuclei are as strong as in the abovediscussed example of a quantum dot.

In order to investigate if the experimental deviations from the rigid-body can be explained by the quantization ofthe nucleonic motion we calculated the moments of inertia quantum mechanically by means of the Strutinsky shellcorrection method for rotating nuclei [6]. The single-particle levels were calculated at the rotational frequencyω bydiagonalizing the cranked deformed Woods-Saxon potential

h′ = t +V(α ,α4)−ω ·j, (8)

wherej is the single-particle angular momentum . The axially symmetric shape was parametrized by the parametersα

58


FIGURE 4. Upper panel: Calculated shell moments of inertia (deviations of the moment of inertia from the rigid-body value) as afunction of neutron number for optimal orientation of the rotational axis. Lower panel: Experimental deviations from the rigid-bodymoments of inertia.

andα4, which are close to the usual quadrupole and hexadecapole deformation parametersε andε 4 for the consideredmoderate deformations. The shell energy (Strutinsky shell correction)E sh(α ,α4,ω) was obtained from the single-particle levels by the Strutinsky’s averaging procedure. The deformation parameters were determined by minimizingwith respect toα andα4 the total Routhian (energy in the rotating frame)

E ′(α ,α4,ω) = Esh(α ,α4,ω)+ ELD(α ,α4)− ω2

2Jrig,ν (α ,α4), (9)

at the given rotational frequencyω. We considered the two possibilities that the rotational axisω was perpendicular tothe symmetry axis (ν =⊥) and that it was parallel (ν = ‖). The moments of inertia were obtained from the expectationvalue of the angular momentum projection on the rotational axis,

Jν (ω) =< ω| jν |ω >

ω, (10)

where ν = ⊥ or ‖ and |ω > is the lowest configuration in the rotating Woods-Saxon potential. We calculatedthe deviation of the moment of inertia from the rigid-body value, which we call theshell moment of inertia, forω = 0.3 MeV/h in the range of nuclei for which there is experimental data. The nucleus was allowed to rotate aroundthe symmetry axis and around an axis perpendicular to the symmetry axis, and the shape parameters were optimizedseparately for prolate and oblate shape. Out of the four calculations, the mode with the lowest total energy was chosento calculate the optimal shell energies and the optimal shell moments of inertia. The results are compared in Figs. 4with the deviations of the experimental moments of inertia from the rigid-body value.

The variation of the shell moments of inertia with neutron number is very similar in the experiment and in thecalculation: there are minima at the spherical neutron magic numbers 50, 82 and 126, a peaks atN = 88, then lower

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values aroundN = 94, and another peak atN = 106, which is the region of high-K isomers. We will see later thatthese features can be related to particular properties of the nucleus (such as deformation, closed shells and axis ofrotation). Overall, the experimental shell moments of inertia are shifted by about 10% to the negative side comparedto the calculated ones. We attribute this shift to the residual pair correlations, which are fluctuations of the pair fieldaround zero.

INTERPRETATION IN TERMS OF THE PERIODIC ORBIT THEORY

In the following we interpret the gross structure of the ground state shell energy and of the shell moment of inertia inframe of the Periodic Orbit Theory (POT). A detailed presentation of the Periodic Orbit Theory was given in the bookby Brack and Bhaduri [7]. The level density is decomposed into an oscillating part, which represents the shell structure,and a smooth background. The oscillations show up in related quantities such as the energy and the moments of inertia.POT calculates the oscillating part of the level density and of the derived quantities in terms of the periodic orbits of theclassical system with the same Hamiltonian. More specifically, it aims at the oscillating part of the level density thatis averaged over a certain energy interval. For a sufficiently wide averaging interval, only the gross structure remains,which is described by a few short orbits. This tremendous simplification makes POT a powerful tool for interpretingthe gross shell structure.

Shell structure and deformation of the ground state

For understanding the magnetic and rotational response, I need to discuss the gross shell structure of the groundstate first. The level densityg is decomposed into the smooth part ˜g and an oscillating partg sh, which contains theshell structure,

g = g+ gsh. (11)

The oscillating part is given bygsh = ∑

βgβ (12)

whereβ labels the periodic orbits that contribute. For simplicity we discuss the case of a cavity , i.e. of an infinitepotential step at the surface. Then the classical orbits are composed of straight lines and specular reflection on thesurface. The energye is conveniently expressed by the wave numberk =

√2me/h. For the spherical cavity, the

periodic orbits are equilateral polygons. The upper panel of Fig. 5 shows the triangle and the square, which are thesimplest. Each polygon belongs to a three-fold degenerate family, which corresponds to the possible orientations ofthe polygon in space, which are generated by the three Euler angles. All orbits of a family have the same lengthL,the circumference of the polygon. As seen in Fig 5, there are two types of orbits in the case of a cavity with moderatespheroidal deformation. One type, the equator orbits, are the regular polygons in the equatorial plane, which belongto a one-fold degenerate family, generated by the possible rotations of the polygon in the plane. The other type, themeridian orbits, are the polygons in a plane that goes through the symmetry axis. This is a two-fold degenerate family,which is generated by the rotation of the meridian plane about the symmetry axis and a shift of the reflection pointson the surface within the meridian plane. It seems surprising that the meridian orbits of one type of polygon have allthe same length, because the shape of the polygon changes..

The contribution from each family to level density is given by

gβ(k) = Aβ(k)sin(Lβ k +νβ)D(

kLβ

γRo

), (13)

whereRo is the radius of a sphere with the same volume as the spheroidal cavity. The relation between radius and theparticle numberN for constant densityρ is Ro = roN1/3, wherero = 3/(4πρ). To be brief, we shall refer to the wholefamily of such degenerate orbits as the triangle, square, etc. As a function of the wave numberk, each term in the sumoscillates with the frequency given by the lengthLβ of the orbital. The Maslov indexνβ is a constant phase, whichtakes into account that each bounce at the surface and each turn around the center changes the phase by a constant (see[7]). The amplitudeAβ depends on the degeneracy of the periodic orbit: the more symmetries a system has, the greaterthe degeneracy, and the more pronounced are the fluctuations of the level density. The damping factorD(kL β/γRo)

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FIGURE 5. Classical periodic orbits in a spheroidal cavity at small deformation.

is a decreasing function of its argument. Its concrete form and the value of damping parameterγ depend on how thelevel density is averaged overk, in order to filter out its gross structure. The wider the averaging intervalγ the morethe long orbits are suppressed. In our study did not explicitly average. However following the changes as function ofN by eye disregarding the details corresponds to some kind of averaging.

The shell correction to the energy is given by

Esh(kF) = ∑β

Eβ (14)

Eβ =(

hτβ

)2

Aβ(kF)sin(Lβ kF +νβ)D(

kF Lβ

γR

)= ∑

β

(hτβ

)2

gβ(kF). (15)

The period of revolutionτβ of a particle moving with the Fermi momentumhkF =√

2meF on the orbitβ is given by

hτβ

=h2kF

mLβ=

2eF

bRo

LβN−1/3. (16)

The radius of the sphere isRo = roN1/3 andkF ro = b = 1.92.

Basic shell and supershell structure

Let me discuss some features of the shell structure of the spherical cavity [8]. The most important orbits are thetriangle (β = 3) and the square (β = 4), to which we restrict the sums overβ . They are the shortest orbits with thelengthsL3 = 5.19Ro andL4 = 5.66Ro. Since,L3 ≈ L4, one hasτ3 ≈ τ4 ≈ τ , A3 ≈ A4 ≈ A andD3 ≈ D4 ≈ D. Usingthe addition theorem for the sine function one finds

Esh = 2

(hτ

)2

A sin(kF L+ ν)cos(kF∆L+∆ν)D, (17)

with L = (L3 + L4)/2 = 5.42Ro and∆L = (L4−L3)/2 = 0.24Ro and the analogous definitions forν and∆ν . A wellknown phenomenon is encountered: the superposition of two oscillations with similar frequency results in a beatmode. The fast oscillation represents basic shell structure and the slow beat mode was called supershell structure. Thephenomenon of supershell structure was observed in Na clusters [9], almost three decades after it was predicted [8].Nuclei become unstable against fission when the seventh shell is encountered, which is too early for observing supershell structure. Supershell structure is also discussed in other talks.

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FIGURE 6. Shell energy at zero rotational frequency for cavity-like potential. Forα = 0, the shape is spherical, forα = 0.5, theshape is prolate with an axis ratio of

√3, and forα = −0.5, the shape is oblate with an axis ratio of 1/

√3. The dashed and full

lines correspond to sin(L4kF +ν4) = 1, −1, respectively. The figure is relevant for alkali clusters.

Deformation

Let me now discuss the basic shell structure more in detail. It is determined by the average of the contribution fromthe triangle and the tetragon. For the following qualitative discussion it is sufficient to consider only the tetragons(which are a bit simpler than the triangles from the point of view of geometry).

In the middle between the closed shells,Esh > 0 for spherical shape. Nuclei and alkali clusters reduce this shellenergy by taking a non-spherical shape, i.e., they avoid a high level density near the Fermi level. This is analogous tothe Jahn-Teller effect in molecules. Due to symmetry, the electronic levels of a molecule may be degenerate. If sucha degenerate level is incompletely filled, the molecule changes its shape such that the degeneracy is lifted. Whereasthe final shape of the molecule is determined by the balance between this driving force and the restoring force ofthe chemical bonds, in the case of nuclei and clusters a shape is attained that minimizes the level density near theFermi surface. The optimal shapes are described by the few lowest multipoles. This has been known for a long timefor nuclei, where it is experimentally confirmed. Also in the case of alkali clusters, several multipoles are needed todescribe the equilibrium shapes (see, for example the calculations in [10]). We have calculated the shapes includingthe multipoles necessary for a completely relaxed axial shape.

Fig.6 shows the shell energy ofN fermions in a spheroidal cavity with deformationα . The volume of the cavityis constant,V = 4πR3

o/3. We included in the figure the linesL4kF + ν4 = π(2n + 1/2) (dashed) andL4kF + ν4 =2π(2n− 1/2), which correspond to the minima and maxima of the sin function in eq. (15). At spherical shape, theshell energy oscillates as function of the particle numberN. The minima atN = 58,92,136 are the spherical shellclosures. The magic numbers are well reproduced by sin(5.42bN 1/3+ν4) ≈ sin(k f L4 +ν4). (The number 5.42 is theaverage of the triangle and square, which gives the shell spacing more accurately than 5.66 of the square.) When thethe cavity takes a prolate deformation (α > 0), the meridian orbits become longer. In order to keep the length constant,the size of the cavity must decrease, which corresponds to a smaller particle numberN, because the the volume isproportional toN. Hence the lines of constant length of the meridian orbits are down sloping. The equator orbits

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FIGURE 7. Shell energy at zero rotational frequency for Woods-Saxon potential with a spin-orbit term. The dashed and full linescorrespond to sin(L4kF +ν4 +δ) = 1, −1, respectively. The phase shiftδ is chosen such that the the minima at spherical shapeagree with the magic numbers 50, 82, 126 in the presence of a spin-orbit term. The figure is relevant for nuclei.

become shorter at prolate deformation. In order to keep the length constant,N must increase, and the lines of constantlength are up sloping. Fig. 6 shows the interference pattern between the meridian and equator orbits. There is a systemof down sloping valleys and ridges that originates from the minima and maxima at spherical shape which follow thelines of constant length of the meridian orbits. Superimposed is a system of up sloping valleys and ridges that followsthe lines of constant length of the equator orbits. The contribution from the meridian orbits is stronger because they aretwo-fold degenerate, whereas the equator orbits are only one-fold degenerate. On the oblate side, the meridian orbitsbecome shorter and the equator orbits longer, and the particle numberN must increase and decrease, respectively, inorder to keep the length constant.

Fig. 7 shows the shell energy for the realistic nuclear potential. The structure of the landscape is similar to the cavitycase, except that the spin-orbit term changes the magic numbers for spherical shell closure toN = 50,82,126. Hence,we can use interpretation in terms the of the interference between the meridian and equator orbit in a cavity if weconsider the shell structure as function of the degree of the shell filling.

The shape is spherical (α = 0) for closed shells (N = 58, 92, 136 for clusters andN = 50, 82, 126 for nuclei).Taking particles away, the equilibrium shape is located in the valley on the prolate side that is generated by the meridianorbits. The smooth increase of the deformation with decreasingN is experimentally well established in nuclei. In thecase of clusters the measurements of the deformation are not accurate enough to see this. IfN decreases further,the equilibrium shape moves over the saddle on the ridge that is generated by the meridian orbits. The deformationdecreases abruptly after the saddle. This sudden loss of deformation is also experimentally well known for nuclei.Ref. [11] first pointed out that the asymmetric development of the deformation through a shell reflects the downsloping meridian valleys. The superimposed valley-ridge structure generated by the equator orbits is less important forthe deformation, because it is weaker. It determines the location of the saddle on the meridian ridge. In addition, itmodulates the bottom of the meridian valley. There is a saddle where the equator ridge crosses the meridian valley.

The nuclear experimental shell energies at zero spin in Fig. 8 show quite clearly the interference between the equatorand meridian orbits at normal deformation. Going down from the closed shell atN = 126 one climbs up the bottom of

63


FIGURE 8. Experimental shell energy of nuclei at zero rotational frequency. The shell energy is obtained by subtracting theenergy of a droplet of nuclear matter from the experimental ground state energy. The expression for the droplet energy is given inRef. [13].

the valley generated by the meridian orbits, crossing the ridge generated by the equator orbits, and reaches the regionof most stable deformation (N = 98), which is generated by the constructive interference of both the meridian andequator valleys. One then follows the valley generated by the equator orbits. Along the bottom of this valley one hasto go over the ridge generated by the meridian orbits (N = 90) in order to reach the spherical minimum atN = 82.Though less pronounced, the same pattern is seen in the shell 50≤ N ≤ 82. The traverse of the meridian and equatorridges is also is clearly seen in the shell energies of Na clusters calculated in Ref. [10] (cf. Fig. 2 therein, minimizationwith respect toα only).

The meridian ridges and valleys are less down-sloping for oblate shape than for prolate shape. The different slopecan be explained in terms of geometry. In the lower panel of Fig. 5, consider the meridian square orbit that has two ofits sides perpendicular and two parallel to the symmetry axis of the cavity. Ifa is the length of the symmetry semi-axisandb the length of the other two axes, then the length of the orbitL 4 ∝ a+b. The volume of the cavity isV = 4πab2/3.It has to be the same as for spherical shapeV = 4πR3

o/3. This meansL4 ∝ b/Ro +1/(b/RO)2. This function has a largernegative slope forb/Ro < 1 (prolate) than forb/Ro > 1 (oblate). In fact, the slope becomes positive forb/R O > 21/3.The system tries to avoid the mid-shell mountain at spherical shape by taking a deformed shape. It is energeticallyfavorable to go to the prolate side, because the argument of the sin function changes more rapidly. This is H. Frisk’sexplanation [12] for the preponderance of prolate over oblate nuclei.

At large deformation a new system of ridges and valleys develops. It is generated by the orbits shown in Fig. 9,which appear forα > 0.3. Since the orbits are longer, the sequence of ridges and valleys is more closely spaced.The structure becomes more clearly visible forα > 0.5, which is not shown in Figs. 6 and 7. Nuclei with such largedeformation (the axes ratio is about 1:2) are called superdeformed. The distance inN between regions where they areobserved is consistent with the greater length of the orbits. The relation between superdeformation and the periodicorbits has been discussed in Ref. [14].

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FIGURE 9. Classical periodic orbits in a spheroidal cavity at large deformation.

Shell structure at finite rotational frequency

As discussed in the beginning, a mesoscopic electron system in an external magnetic field and a rotating nucleus aredescribed by the Hamiltonian (1), whereω = −ωL in the former case. We consider the case of slow rotation, whichmeans that the periodic orbits are close to the discussed polygons. The influence of the term−ωl z can be taken intoaccount by the version of perturbation theory developed in Ref. [15] for POT.

The change of the action due to rotation is given in first order by

∆Sβ = ω∫

βldt = ωτβ lβ ≡ hΦβ . (18)

The integration runs over the unperturbed orbit in the non rotating cavity. In the case of the spherical cavity, the angularmomentumlβ of the orbit is conserved and the integration is trivial. In case of the spheroidal cavity,l β is conservedfor the equator orbits but not for the meridian. For the latter,l β is the average angular momentum of the orbit whichmust be found by evaluating the integral (18). Sinceωl = m(r×v) · ω,

hΦ(θ) = m∫

β(r×ds) · ω = 2m

∫β

df ·ω = 2mAβωcosθ ≡ hΦβ cosθ, (19)

whereAβ is the area enclosed by the orbit andθ is the angle between the normal of its plane and the axis of rotation.Hence,Φ is the “rotational flux” in units ofh, of the vector field 2mω enclosed by the orbit. In the analogous case ofparticles in a magnetic field,Φβ is the magnetic flux in units of the elementary flux quantum.

The rotation manifests itself in the appearance of an additional modulation factorM in the expressions for the leveldensity (12) and the shell energy (14),

∑β

gβ →∑β

Mβ gβ , ∑β

Eβ → ∑β

Mβ Eβ . (20)

The modulation factor is given by the average of exp(iΦ) over all orbits of the same length, which belong to the familyβ ,

Mβ(Φβ ) =1

4π

∫eiΦβ (θ,φ)dΩ. (21)

Here we denote the maximal flux through the orbit byΦβ , which corresponds to perpendicular orientation of theorbital plane to the rotational axis. In the case of a spherical cavity the modulation factor becomesM ©(Φβ ) = jo(Φβ ),where jo is the spherical Bessel function. In the case of the spheroidal cavity, we distinguish between the rotationalaxis being parallel or perpendicular to the symmetry axis. For perpendicular rotation only the meridian orbits carry

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FIGURE 10. Modulation factors as functions of the rotational fluxΦ. The dashed lines show the quadratic approximation.

flux andM⊥(Φβ ) = Jo(Φβ ), whereJo is the Bessel function. For parallel rotation, only the equator family carriesrotational flux, andM‖(Φβ ) = cos(Φβ ). Fig. 10 shows the three types of modulation factors.

In the analysis of the experiment and the microscopic calculations we assume that the rotational energy has theform ω2J /2, which corresponds to an expansion ofM up to second order inΦ β . Using the expansions of cos(Φβ ),Jo(Φβ ) and jo(Φβ ), one finds

M = 1−aΦ2β , (22)

where, respectively,a‖=1/2 anda⊥=1/4 for the equator and meridian orbits in the spheroidal cavity anda ©=1/6 for thespherical cavity. As seen in Fig. 10, 1−Mβ deviates from its quadratic approximation by less than 35% forΦβ < 2,where the quadratic approximation is better forJo(Φβ ) and jo(Φβ ) than for cos(Φβ ).

The microscopic calculations corresponds to fluxes ofΦ = 0.6 and 1.1 forA = 80 and 200, respectively. Hence,the quadratic approximation is rather accurate. In the case of the experimentΦ = 2.0, 1.6, 1.5 for the regionsA = 80, 160, 200, respectively. Hence, for the bulk of the data the quadratic approximation is not very accurate,but still acceptable. It becomes problematic for the light nuclei and for some the cases that reach very high spin. Aparabola that approximatesM (Φ) at large values ofΦ has a lower curvature than the parabola that approximatesthe low-Φ part (see Fig. 10). Hence, the deviations ofM (Φ) from the quadratic form will tend to reduce the shellmoments of inertia that we derive from the data.

Moments of inertia

For the discussion, I continue keeping only one term in the sum over the periodic orbits. Then the moments of inertiaare given by:

Jsh‖ = l2‖A (kF)sin(L‖kF + ν)D

(kFL‖γR

), (23)

and

Jsh⊥ =12

l2⊥A (kF)sin(L⊥kF + ν)D

(kF L⊥γR

). (24)

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FIGURE 11. Calculated shell moments of inertia (deviations of the moment of inertia from the rigid-body value) as a function ofneutron number for different orientations of the rotational axis with respect to the symmetry axis (perpendicular - left and parallel- right).

The moments of inertia are related to the ground-state shell energy (15):

Jsh‖ =h2

e2F

(k2F A‖)2Esh‖, (25)

and

Jsh⊥ =h2

2e2F

(k2F A⊥)2Esh⊥. (26)

The relation explains the similarity between theN dependence of the shell moment of inertia in Fig. 3 and the ground

state shell energy in Fig. 8. The relative scale can be estimated as follows:J sh ∼(

hb2

2eF

)2N4/3Esh ≈ h2

400 MeV 2 N4/3Esh ≈h2

1000MeV 2 A4/3Esh. Fig. 8 is scaled by the factor 1000A4/3 and can directly be compared with Fig. 3.However, there is a difference between the POT expressions for the ground state energy and the moment of inertia.

Only the orbits that carry rotational flux contribute to the deviation of the moment of inertia from the rigid-body value,that is only the meridian orbits contribute if the rotational axis is perpendicular to the rotational axis, and only theequator orbits contribute if the rotational axis is parallel to the rotational axis. This is indicated by the subscripts‖and⊥ for the parallel and perpendicular orientation of the rotational axis. On the other hand, both the meridian andequator orbits contribute to the ground state shell energy, i.e.E sh = Esh‖ + Esh⊥.

Let me first discuss the region aroundN = 126 by means of Figs. 3, 4, and 7. The neutron shell is closed. The protonnumberZ is not very different from the closed shell value 82. The shell moment of inertia is strongly negative for bothprotons and neutrons. The moments of inertia become very small, because strong persistent currents flow opposite tothe rotational motion. The minimum atN = 82 is caused by the closed neutron shell, which makes the shape spherical.The proton number is around 56, which lies far in the open shell. The proton contribution for spherical shape is positivemid-shell, which partially compensates the negative neutron contribution. This makes theN = 82 minimum much lessdeep than the minimum atN = 126. Below theN = 82 minimum there is a spike atN = 80. Inside this interval theshape is spherical. The neutron shell contribution becomes less negative when moving away from 82. WhenN goesoutside the interval, the deformation suddenly jumps to a substantial value and the positive proton shell contributiondrops, which generates the spike.

Now I discuss theN dependence of the shell contribution to moments of inertia in the open shell. Most nuclei rotateperpendicular to the symmetry axis, because the moment of inertia is larger than for parallel rotation. This orientationof rotational axis is reflected by the appearance of regularly spaced rotational bands. The meridian orbits determinethe shell moment of inertia, because only they enclose rotational flux. Starting at theN = 82 minimum and followingthe east path of lowest elevation around theN = 100 mountain, one goes over the meridian pass, which shows up asthe maximum of theJsh⊥. Then one follows the valley to theN = 126 minimum. Fig. 11 shows the microscopiccalculations ofJsh⊥. There is the maximum atN = 88, which is caused by the meridian ridge. The shell contribution

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is positive. The moment of inertia is larger than the rigid-body , because there are persistent currents with the samedirection as the rotational motion. In contrast toEsh, there is no bump caused byN = 106 ridge, because it is generatedby the equator orbits, which do not carry rotational flux. For rotation about the symmetry axis,J sh‖ is determined bythe equator orbits. As seen in Fig. 11, following the same path, one goes through a minimum atN = 98, which is causedby the equator valley, and over a maximum atN = 106, which is caused by the equator ridge. The maximum caused byequator ridge appears also inEsh, because it is not flux selective. If the rotational axis is parallel to the symmetry axis,the angular momentum is built up by particle-hole excitations. Instead of being a regularly space rotational band, thesequence of yrast levels (levels of minimal energy for given angular momentum ) is irregularly spaced and containsmany isomeric states. Therefore, the direction of the rotational axis is well known from experiment.

The experimental deviations of the moments of inertia from the rigid-body must be compared with with the onescalculated for the favored orientation of the rotational axis, which are shown in Fig. 4. Most of the figure correspondsto rotation perpendicular to the symmetry axis. The meridian maximum is clearly seen in experiment. On the quantumlevel, the positive shell contribution is caused by the alignment of the angular momentum of high-j single particlewith the axis of rotation. Only aroundN = 106 the large value ofJ sh‖ makes it preferable that the nucleus rotatesabout the symmetry axis. Therefore the maximum for parallel rotation appears in the figure. The maximum is seen inthe experimental shell moment of inertia as well. The nuclei in this region show indeed many high-K isomeric states,which indicates that rotation about the symmetry axis is energetically favored.

The orbits that cause the shell structure of prolate superdeformed nuclei do not carry rotational flux if the rotationalaxis is perpendicular to the symmetry axis. This is evident for the orbits in the equator plane. For the butterfly orbit inFig. 9 one must take into account that the rotational flux has a sign. If the rotational axis points out of the page, it ispositive if the particle runs counterclockwise around the enclosed area and it is negative if it runs clockwise. Therefore,the flux has opposite sign for the two wings of the butterfly and the total flux is zero. In superdeformed nuclei the axisof rotation is perpendicular to the symmetry axis. Therefore, their moments of inertia should be equal to the rigid bodyvalue. The experimental deviation from the rigid-body value is about 5% or less in theA = 150 region.

If the axis of rotation was parallel to the symmetry axis the equator orbits would carry rotational flux. Sincethey cause the strongly negative shell energy of the superdeformed nuclei, their shell contribution to the momentof inertia would also be strongly negative. This means that the appearance of high-K isomers is strongly disfavored insuperdeformed nuclei. So far no isomers have been reported.

Shell structure at high spin

So far I have discussed the case of small rotational frequency, which allows a quadratic approximation of themodulation factor. At higher rotational frequency, yet small enough that the curvature of the orbits remains smallwithin the size of the system, the modulation factor oscillates (cf. Fig. 10). The modulation factorM ‖(Φβ ) = cos(Φβ )also appears in a two-dimensional electron gas in an external magnetic field. It has been observed as oscillations ofthe electric current through a circular quantum dot. Fig. 12 shows the current as function of the gate voltage and themagnetic field strength. The figure shows also a calculation of the level density of the gas as function of the sizeand the magnetic field strength. The gate voltage controls the effective size of the dot. Therefore the length of theorbits changes with the gate voltage. The size of the figures is chosen such that the change of the gate voltage on theright is converted into a change of the size on the left. Since the argument of the sine function changes withk F Lβ ,the wavelength of the oscillations is equal toa/kF , where the factor of proportionality is a combination of the ratiosbetween the lengths of the diameter, triangle, and square and of the size of the system. In this way the oscillations withrespect to the gate voltage are determined by theorbit length. On the other hand the wavelength of the oscillationswith respect to the the magnetic field strength is determined by theorbit area, because cos(Φβ ) = cos(2mAβω/h).Here appears only a combination of area of the the triangle and the square, because the diameter does not carry flux(see [7, 16]).

The qualitative behavior of the shell contribution to the level density at high rotational frequency and moderatedeformation is illustrated in Fig. 13. The POT result for the tetragonal orbits is shown, where the change of the actiondue to both deformation and rotation is treated by perturbation theory [15]. Taking into account the change of thelength of the orbit in linear order of the deformation parameterα , the contributions of deformation and rotation to theaction are given by

∆S‖/h = −12

kL4P2(cosθ)α −Φ4cosθ (27)

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FIGURE 12. Left: Level density of a two-dimensional electron gas confined by a circular infinite potential. Right: Current througha circular quantum dot as function of the gate voltage and the magnetic field. The gate voltage can be converted into an effectivesize of the dot. The size on the left is chosen such that the figures can be directly compared.

FIGURE 13. POT level density (arbitrary units) generated by a the family tetragonal orbits in a rotating spheroidal cavity. Onlycontributions of first order inα to the orbit length are taken into account.

∆S⊥/h = −12

kL4P2(cosθ)α −Φ4sinθ cosφ, (28)

where the Euler anglesψ,θ,φ describe the orientation of the tetragon. We denote the length of the square in the sphereby L4 and the flux through it byΦ4. The changes of the action due to deformation and rotation give rise to the moregeneral modulation factor

M =1

4π

∫ 2π

0

∫ π

0ei∆S(θ,φ)/h sinθdθdφ, (29)

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which describes the changes due to both deformation and rotation. It is numerically evaluated. Clearly this will notaccount for the superdeformed shell structure.

At large deformation, one may evaluate the integral (29) using the stationary phase approximation. The derivativeof the Legendre polynomialP2(cosθ) is zero forθ = π/2, which corresponds to the meridian orbits, and forθ = 0, π,which corresponds to the equator orbits. The modulation factor becomes proportional to the rotational modulationfactorsM⊥ andM‖ dicussed before. The structure of the level density is the consequence of the interference of thesetwo families, as already discussed. The interference pattern is still recognizable at moderate deformations, where thestationary phase approximation becomes problematic. This justifies the interpretation of the shell structure in terms ofthe interference of the meridian and equator orbits.

Only the orbits that carry flux enter the modulation factor, and their contribution to the shell energy changes withω. The contribution of the other orbits remains the same. This simple observation is the key for understandingthe evolution of shell energies with frequency. With increasing frequencyω, the valley-ridge system generated bythe meridian orbits is attenuated and mostly gone athω = 0.6 MeV . What remains is the up sloping valley-ridgesystem generated by the equator orbits. ForN = 100,α = 0.3, and hω = 0.6 MeV , the fluxΦ⊥ = 2.5, which givesM⊥(2.5)=−0.06 (cf. Fig. 10). The contribution of the meridian orbits has essentially vanished. For rotation parallel tothe symmetry axis, only the contribution from the equator orbits is modified. ForN = 100,α = 0.3, and hω = 0.6 MeV ,the fluxΦ‖ = 1.9, which givesM‖(1.9) =−0.3 (cf. Fig. 10). Now the equator orbits contribute with the opposite sign,i. e. valleys become ridges and vice versa. TheN = 100 mountain, which forω = 0 is generated by the constructiveinterference of the meridian ridge and the equator ridge, becomes a saddle due to the destructive interference. Forthe same reason, theN = 126 minimum also becomes a saddle. The topology does not change forN = 50 becausethe fluxes are smaller. The microscopic calculations for the cavity and for the Wood-Saxon potential follow the samepattern.

SUMMARY

There is a far-reaching analogy between the anomalous magnetic susceptibility of an electron gas confined to amesoscopic volume and the deviation of nuclear moments of inertia from the rigid-body value. We have studied theconsequences of the shell structure, which generates normal persistent currents in metal clusters and the deviation fromthe rigid-body flow pattern in nuclei. The Periodic Orbit Theory provides a simple interpretation of these phenomena.The basic shell structure is qualitatively described by one family of periodic orbits in a spheroidal cavity, e.g. thetetragons. The length of these orbits determines the period of the oscillations of the shell energy as function of theparticle number or the size of the system. It is also responsible for the equilibrium shape at zero rotational frequency.The shape of nuclei and of free liquid metal clusters is mainly determined the tetragons in the meridian planes. Thecorresponding oscillations of the shell energy interfere with weaker oscillations generated by the tetragons in theequator plane. The interference pattern is seen as the gross structure of the shell energy as function of the particlenumber. The rotational/magnetic flux through the orbit determines the oscillations of the shell energy as function ofthe rotational/Larmor frequency. For a given frequency, the flux is proportional to the area of the orbit. Oscillations asfunctions of Larmor frequency have been observed in quantum dots. Nuclei fission before the first period is completed.We studied the low-flux limit, where the consequences of the normal persistent currents can be cast into a shellcontribution to the moment of inertia. This shell moment of inertia is generated by only the orbits that carry flux,which cannot lie in a plane that contains the rotational axis. The dependence of the shell moment of inertia on theparticle number is determined by the orbit length. This explains the observed strong correlation between the shellenergy in the ground state and the shell moments of inertia . The orientation of the axis of rotation depends on whichorbit gives the larger shell moment of inertia . Usually the dominant meridian orbits win, and nuclei rotate about anaxis perpendicular to the symmetry axis. In this case one observes regularly spaced rotational bands. The equatororbits may win against the meridian ones for particle numbers where their contribution has a maximum, for exampleN ≈ 106. Then the axis of rotation is parallel to the symmetry axis and the sequence of levels with increasing spinis irregularly spaced, which leads to the appearance of isomeric states. Deformed nuclei behave this way: For mostof them, the sequence of states with minimal energy for given angular momentum is a nice regular rotational band.However aroundN = 106 the irregularly spaced high-K isomers enter this sequence.

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ACKNOWLEDGMENTS

This work was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office ofHigh Energy and Nuclear Physics of the U.S. Department of Energy under contract No. DE-AC03-76SF00098 andDE-FG02-95ER40934, and by a EU grant, INTAS-93-151-EXT.

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