Part A Fundamental Nuclear Research · of use for light nuclei is a mirror pair, which is a pair of nuclei with N and Z interchanged. An example of such a pair would be 23Na 12 and

1

Part AFundamental Nuclear Research

Encyclopedia of Nuclear Physics and its Applications, First Edition. Edited by Reinhard Stock.© 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

3

1

Nuclear Structure

Jan Jolie

1.1 Introduction 51.2 General Nuclear Properties 61.2.1 Properties of Stable Nuclei 61.2.2 Properties of Radioactive Nuclei 71.3 Nuclear Binding Energies and the Semiempirical Mass Formula 81.3.1 Nuclear Binding Energies 81.3.2 The Semiempirical Mass Formula 101.4 Nuclear Charge and Mass Distributions 121.4.1 General Comments 121.4.2 Nuclear Charge Distributions from Electron Scattering 131.4.3 Nuclear Charge Distributions from Atomic Transitions 141.4.4 Nuclear Mass Distributions 151.5 Electromagnetic Transitions and Static Moments 161.5.1 General Comments 161.5.2 Electromagnetic Transitions and Selection Rules 171.5.3 Static Moments 191.5.3.1 Magnetic Dipole Moments 191.5.3.2 Electric Quadrupole Moments 211.6 Excited States and Level Structures 221.6.1 The First Excited State in Even–Even Nuclei 221.6.2 Regions of Different Level Structures 231.6.3 Shell Structures 231.6.4 Collective Structures 251.6.4.1 Vibrational Levels 251.6.4.2 Rotational Levels 261.6.5 Odd-A Nuclei 281.6.5.1 Single-Particle Levels 281.6.5.2 Vibrational Levels 28


4 1 Nuclear Structure

1.6.5.3 Rotational Levels 281.6.6 Odd–Odd Nuclei 281.7 Nuclear Models 291.7.1 Introduction 291.7.2 The Spherical-Shell Model 301.7.3 The Deformed Shell Model 321.7.4 Collective Models of Even–Even Nuclei 331.7.5 Boson Models 35

Glossary 40References 41Further Readings 42

5

1.1Introduction

The study of nuclear structure todayencompasses a vast territory from the studyof simple, few-particle systems to systemswith close to 300 particles, from stablenuclei to the short-lived exotic nuclei, fromground-state properties to excitations ofsuch energy that the nucleus disintegratesinto substructures and individual con-stituents, from the strong force that holdthe atomic nucleus together to the effec-tive interactions that describe the collectivebehavior observed in many heavy nuclei.

After the discovery of different kinds ofradioactive decays, the discovery of thestructure of the atomic nucleus beginswith the fundamental paper by ErnestRutherford [1], in which he explainedthe large-angle alpha (α)-particle scatteringfrom gold that had been discovered earlierby Hans W. Geiger and Ernest Marsden.Indeed, Rutherford shows that the atomholds in its center a very tiny, positivelycharged nucleus that contains 99.98%of the atomic mass. In 1914, HenryMoseley [2, 3] showed that the nuclearcharge number Z equaled the atomicnumber. Using the first mass separators,Soddy [4] was able to show that onechemical element could contain atomicnuclei with different masses, forming

different isotopes. With the availability ofα-sources, due to the works of the Curies inParis, Rutherford [5] was able to performthe first nuclear reactions on nitrogen.The first attempt at understanding therelative stability of nuclear systems wasmade by Harkins and Majorsky [6]. Thismodel, like many others of the time,consisted of protons and electrons. In1924, Wolfgang Pauli [7] suggested thatthe optical hyperfine structure might beexplained if the nucleus had a magneticdipole moment, while later Giulio Racah[8] investigated the effect on the hyperfinestructure if the nuclear charge were notspherically symmetric – that is, if it had anelectric quadrupole moment.

All of these structure suggestionsoccurred before James Chadwick [9] discov-ered the neutron, which not only explainedcertain difficulties of previous models (e.g.,the problems of the confinement of theelectron or the spins of light nuclei), butopened the way to a very rapid expan-sion of our knowledge of the structure ofthe nucleus. Shortly after the discovery ofthe neutron, Heisenberg [10] proposed thatthe proton and neutron are two states of thenucleon classified by a new spin quantumnumber, the isospin. It may be difficultto believe today, 60 years after Chadwick’sdiscovery, just how rapidly our knowledgeof the nucleus increased in the mid-1930s.



Hans A. Bethe’s review articles [11, 12], oneof the earliest and certainly the best known,discuss many of the areas that not onlyform the basis of our current knowledgebut that are still being investigated, albeitwith much more sophisticated methods.

The organization here will begin withgeneral nuclear properties, such as size,charge, and mass for the stable nuclei,as well as half-lives and decay modes(α, β, γ, and fission) for unstable systems.Binding energies and the mass defectlead to a discussion of the stability ofsystems and the possibility of nuclearfusion and fission. Then follow detailsof the charge and current distributions,which, in turn, lead to an understanding ofstatic electromagnetic moments (magneticdipole and octupole, electric quadrupole,etc.) and transitions. Next follows thediscussion of single-particle and collectivelevels for the three classes of nuclei:even–even, odd-A, and odd–odd (i.e.,odd Z and odd N). With these mainlyexperimental details in hand, a discussionof various major nuclear models follows.These discussions attempt, in their ownway, to categorize and explain the mass ofexperimental data.

1.2General Nuclear Properties

1.2.1Properties of Stable Nuclei

The discovery of the neutron allowed eachnucleus to be assigned a number, A, themass number, which is the sum of thenumber of protons (Z) and neutrons (N)in the particular nucleus. The atomicnumber of chemistry is identical to theproton number Z. The mass number A isthe integer closest to the ratio between the

mass of a nucleus and the fundamentalmass unit. This mass unit, the unifiedatomic mass unit, has the value 1 u =1.660538921(73) ×10−27 kg = 931.494061(21) MeV c−2. It has been picked so thatthe atomic mass of a 12C6 atom is exactlyequal to 12 u. The notation here is AXN,where X is the chemical symbol for thegiven element, which fixes the number ofelectrons and hence the number of protonsZ. This commonly used notation containssome redundancy because A = Z + Nbut avoids the need for one to look upthe Z-value for each chemical element.From this last expression, one can seethat there may be several combinationsof Z and N to yield the same A. Thesenuclides are called isobars. An examplemight be the pair 196Pt118 and

196Au117.Furthermore, an examination of a table ofnuclides shows many examples of nucleiwith the same Z-value but different A-and N-values. Such nuclei are said tobe isotopes of the element. For example,oxygen (O) has three stable isotopes: 16O8,17O9, and

18O10. A group of nuclei thathave the same number of neutrons, N, butdifferent numbers of protons, Z (and, ofcourse, A), are called isotones. An examplemight be 38Ar20,

39K20, and40Ca20. Some

elements have but one stable isotope (e.g.,9Be5,

19F10, and197Au118), others, two,

three, or more. Tin (Z = 50) has the mostat 10. Finally, the element technetium hasno stable isotope at all. A final definitionof use for light nuclei is a mirror pair,which is a pair of nuclei with N and Zinterchanged. An example of such a pairwould be 23Na12 and

23Mg11.The nuclear masses of stable isotopes

are determined with a mass spectrometer,and we shall return to this fundamentalproperty when we discuss the nuclearbinding energy and the mass defect inSection 1.3. After mass, the next property

1.2 General Nuclear Properties 7

of interest is the size of a nucleus. Thesimplest assumption here is that the massand charge form a uniform sphere whosesize is determined by the radius. Whilenot all nuclei are spherical or of uniformdensity, the assumption of a uniformmass/charge density and spherical shapeis an adequate starting assumption (morecomplicated charge distributions arediscussed in Section 1.4 and beyond). Thenuclear radius and, therefore, the nuclearvolume or size is usually determined byelectron-scattering experiments; the radiusis given by the relation

R = r0A1/3 (1.1)

which, with r0 = 1.25 fm, gives an adequatefit over the entire range of nuclei nearstability. An expression such as Eq. (1.1)implies that nuclei have a density indepen-dent of A, that is, they are incompressible.A somewhat better fit to the nuclearsizes can be obtained from the Coulombenergy difference of mirror nuclei, whichcovers but a fifth of the total range ofA. This yields r0 = 1.22 fm. Even if thecharge and/or mass distribution is neitherspherical nor uniform, one can still definean equivalent radius as a size parameter.

Two important properties of a nuclide arethe spin J and the parity π , often expressedjointly as Jπ , of its ground state. These areusually listed in a table of isotopes and giveimportant information about the structureof the nuclide of interest. An examinationof such a table will show that the groundstate and parity of all even–even nucleiis 0+. The spin and parity assignmentsof the odd-A and odd–odd nuclei tell agreat deal about the nature of the principalparts of their ground-state wave functions.A final property of a given element is therelative abundance of its stable isotopes.These are determined again with a mass

spectrograph and listed in various tables ofthe nuclides.

1.2.2Properties of Radioactive Nuclei

A nucleus that is unstable, that is, it candecay to a different or daughter nucleus,is characterized not only by its mass, size,spin, and parity but also by its lifetime τand decay mode or modes. (In fact, eachlevel of a nucleus is characterized by itsspin, parity, lifetime, and decay modes.)The law of radioactive decay is simply

N(t) = N(0)e−λt = N(0)e−t/τ (1.2)

where N(0) is the number of nuclei initiallypresent, λ is the decay constant, and itsreciprocal τ is the lifetime. Instead of thelifetime, often the half-life T1/2 is used.It is the time in which half of the nucleidecay. By setting N(T1/2) = N(0)/2 in Eq.(1.2), one obtains the relation

T1/2 = ln(2)τ = 0.693τ (1.3)

The decay mode of ground states can beα, β, or spontaneous fission. Excited statesmostly decay by γ-emission. More exoticdecays are observed in unstable nuclei farfrom stability where nuclei decay takesplace by emission of a proton or neutron.

In α-decay, the parent nucleus emitsan α-particle (a nucleus of 4He2), leavingthe daughter with two fewer neutrons andprotons:

AXN → A−4YN−2 + 4He2 (1.4)

The α-particle has zero spin, but it cancarry off angular momentum. In β-decaythe weak interaction converts neutronsinto protons (β−-decay) or protons intoneutrons (β+-decay). Which of the two


decays takes place depends strongly onthe masses of the initial and final nuclei.Because a neutron is heavier than aproton, the free neutron is unstable againstβ−-decay and has a lifetime of 878.5(10) s.The mass excess in β−-decay is releasedas kinetic energy of the final particles.In the case of the free neutron, the finalparticles are a proton, an electron, and anantineutrino, denoted by ν. All of theseparticles have spin 1/2 and can also carryoff angular momentum. In the case ofβ+-decay, the final particles are a boundproton, an antielectron or positron, anda neutrino. Finally, as an alternative toβ+-decay the initial nucleus can capturean inner electron. In this so-called electroncapture decay, only a neutrino, ν, is emittedby the final nucleus. In general, the decayscan be written as

β−-decay : AXN → AYN−1 + e−+ν (1.5a)

β+ -decay : AXN → AYN+1 + e++ ν(1.5b)

β+-decay (ec) : AXN + e−→AYN−1 + ν(1.5c)

One very rare mode of decay is doubleβ-decay, in which a nucleus is unable toβ-decay to a Z + 1 daughter for energyreasons but can emit two electrons andmake a transition to a Z + 2 daughter. Anexample is 82Se48 → 82Kr46 with a half-lifeof (1.7 ± 0.3) × 1020 years. Double β-decayis observed under the emission of twoneutrinos. Neutrinoless double β-decay isintensively searched for in 76Ge because itis forbidden for massless neutrinos withdefinite helicities. Enriched Ge is henceused as it allows the use of a large singlecrystal as source and detector (for a reviewsee [13]).

In spontaneous fission, a very heavynucleus simply breaks into two heavypieces. For a given nuclide, the decay modeis not necessarily unique. If more thanone mode occurs, then the branching ratiois also a characteristic of the radioactivenucleus in question.

An interesting example of a multi-mode radioactive nucleus is 242Am147. Itsground state (Jπ = 1−, T1/2 = 16.01 h) candecay either by electron capture (17.3%of the time) to 242Pu148 or by β

− decay(82.7% of the time) to 242Cm146. Onthe other hand, a low-lying excited stateat 0.04863 MeV (Jπ = 5−, T1/2 = 152 years)can decay either by emitting a γ-ray (99.52%of the time) and going to the ground stateor by emitting an α-particle (0.48% of thetime) and going to 238Np145. There is anexcited state at 2.3 MeV with a half-lifeof 14.0 ms that undergoes spontaneousfission [14]. The overall measured half-life of 242Am147 is then determined bythat of the 0.04863 MeV state. Such long-lived excited states are known as isomericstates. From this information on branch-ing ratios, one easily finds the severalpartial decay constants for 242Am147. Forthe ground state, λec = 2.080 × 10−6 s−1and λβ− = 9.944 × 10−6 s−1, while for theexcited state at 0.04863 MeV, λγ = 1.439 ×10−10 s−1 and λα = 6.639 × 10−13 s−1 andfor the excited state at 2.3 MeV, λSF =49.5 s−1.

1.3Nuclear Binding Energies and theSemiempirical Mass Formula

1.3.1Nuclear Binding Energies

One of the more important properties ofany compound system, whether molecular,

1.3 Nuclear Binding Energies and the Semiempirical Mass Formula 9

atomic, or nuclear, is the amount of energyneeded to pull it apart, or, alternatively, theenergy released in assembling it from itconstituent parts. In the case of nuclei,these are protons and neutrons. Thebinding energy of a nucleus AXN can bedefined as

B(A, Z) = ZMH + NMn − MX(Z, A) (1.6)

where MH is the mass of a hydrogenatom, Mn the mass of a neutron, andMX(Z,A) the mass of a neutral atom ofisotope A. Because the binding energy ofatomic electrons is very much less thannuclear binding energies, they have beenneglected in Eq. (1.6). The usual units areatomic mass units, u. Another quantity thatcontains essentially the same informationas the binding energy is the mass excess orthe mass defect, = M(A) − A. (Anotheruseful quantity is the packing fractionP = [M(A) − A]/A =/A.) The most inter-esting experimental quantity B(A,Z)/Ais the binding energy per nucleon,which varies from somewhat more than1 MeV nucleon−1 (1.112 MeV nucleon−1)for deuterium (2H1) to a peak near

56Fe30 of8.790 MeV nucleon−1 and then falls slowlyuntil, at 235U143, it is 7.591 MeV nucleon

−1.Except for the very light nuclei, thisquantity is roughly (within about 10%)8 MeV nucleon−1. A strongly bound lightnucleus is the α-particle, as for 4He2 thebinding energy is 7.074 MeV nucleon−1. Itis instructive to plot, for a given massnumber, the packing fraction as a func-tion of Z. These plots are quite accuratelyparabolas with the most β-stable nuclideat the bottom. The β− emitters will occuron one side of the parabola (the left orlower two side) and the β+ emitters on theother side. For odd-A nuclei, there is butone parabola, the β-unstable nuclei pro-ceeding down each side of the parabola

until the bottom or most stable nucleusis reached. For the even-A nuclei, thereare two parabolas, with the odd–odd onelying above the even–even parabola. Thefact that the odd–odd parabola is abovethe even–even one indicates that a pair-ing force exists that tends to increase thebinding energy of the even–even nuclei.See Figure 1.1 for the A = 100 mass chain.Other indications of the importance of thispairing force are the before-mentioned 0+

ground states of all even–even nuclei andthe fact that only four stable odd–oddnuclei exist: 2H1,

6Li3,10B5, and

14N7.For even-A nuclei, the β-unstable nucleizig-zag between the odd–odd parabolaand the even–even parabola until arriv-ing at the most β-stable nuclide, usuallyan even–even one. If the masses for eachA are assembled into a three-dimensionalplot (with N running along one long axis,Z along a perpendicular axis, and M(A,Z)mutually perpendicular to these two), onefinds a ‘‘landscape’’ with a deep valley run-ning from one end to the other. This valleyis known as the valley of stability.

The immediate consequence of thebehavior of B(A,Z)/A is that a very largeamount of energy per nucleon is to begained from combining two neutrons andtwo protons to form a helium nucleus.This process is called fusion. The release ofenergy in the fission process follows fromthe fact that B(A,Z)/A for uranium is lessthan for nuclei with more or less half thenumber of protons. Finally, the fact that thebinding energy per nucleon peaks near ironis important to the understanding of thosestellar explosions known as supernovae. InFigure 1.2, the packing fraction, P =/A,is plotted against A for the most stablenuclei for a given mass number. Notethat P has a broad minimum near iron(A = 56) and rises slowly until lawrencium(A = 260). This shows most clearly the


38 40 42 44 46 48 50

Z

−0.95

−0.9

−0.85

−0.8

−0.75

−0.7

−0.65

Pac

king

frac

tion

(Δ/A

) M

eV

44Ru56

45Rh55

41Nb59

43Tc57

40Zr60

39Y61

42Mo5846Pd64

48Cd52

47Ag53

Even ZOdd Z

A = 100

Figure 1.1 The packing fraction /A plotted against thenuclear charge Z for nuclei with mass number A = 100.Note that the odd–odd nuclei lie above the even–evenones. The β− transitions are indicated by - • -, the β+

transitions by ---, while the double β-decay 10042 Mo58 →10044Ru56 is denoted by • • •. Data from [14]. The double β-decayfrom [15].

energy gain from the fission of very heavyelements.

1.3.2The Semiempirical Mass Formula

The semiempirical mass formula may belooked upon as simply the expansion ofB(A,Z) in terms of the mass number.Because B(A,Z)/A is nearly constant, themost important term in this expansionmust be the term in A. From Eq. (1.1)relating the nuclear radius to A1/3, we seethat a term proportional to A is a volumeterm. However, this term overbinds the sys-tem because it assumes that each nucleon

is surrounded by the same number ofneighbors. Clearly, this is not true forsurface nucleons, and so a surface termproportional to A2/3 must be subtractedfrom the volume term. (One might iden-tify this with the surface tension found in aliquid drop.) Next, the repulsive Coulombforces between protons must be included.As this force is between pairs of protons,this term will be of the form Z{Z − l)/2, thenumber of pairs of Z protons, divided by acharacteristic nuclear length or A1/3. Twoother terms are necessary in this simplemodel. One term takes into account that,in general, Z ∼ A/2, clearly true for stablelight nuclei, and less so for heavier stable

1.3 Nuclear Binding Energies and the Semiempirical Mass Formula 11

−2

0

2

4

6

0 50 100 150 200 250

Mass number (A )

Pac

king

frac

tion

(Δ/A

) M

eV

Figure 1.2 The packing fraction /A plotted against themass number A for all nuclei from 21D1 to

260103Lr157. Data

from [14].

nuclei where more neutrons are neededto overcome the mutual repulsion of theprotons. This term is generally taken tobe of the form asym (N − Z)2/A. The otherterm takes into account the fact, noted inSection 1.3.1, that even–even nuclei aremore tightly bound than odd–odd nucleibecause all of the nucleons of the formerare paired off. This is done by adding aterm δ/2 that is positive for even–evensystems and negative for odd–odd sys-tems and zero for odd-A nuclei. Thus, thetwo parabolas for even A are separatedby δ. From Eq. (1.6), the semiempiri-cal Bethe–Weisäcker mass formula thenbecomes M(A,Z) = ZMH + NMn − B(A,Z)with

B(A, Z) = avA − asA2/3 − acZ(Z − 1)A−1/3

− asym(N − Z)2A−1 +δ

2(1.7)

Originally, the constants were fixedby the measured binding energies andadjusted to give appropriate behavior withthe mass number [16]. Myers and Swiatecki[17] (see also [18]) have included otherterms to account for regions of nucleardeformation, as well as an exponentialterm of the form −aaAexp(−γ A1/3), forwhich they provide no physical explanationbeyond the fact that it reduces thedeviation from experiment. Their modelevolved into the macroscopic-microscopicglobal mass formula, called the finite-range droplet model (see [19]) and theDZ-model proposed by Duflo and Zuker[20], and more microscopic models, calledHFB [21]. The many adjustable parametersof the available mass formulas are thenfitted to masses of 1760 atomic nuclei[22]. The formulas fit binding energiesquite well with errors below 1%, but still


have problems to predict masses far fromstability. As those are important for nuclearastrophysics, the measurement of massesof exotic nuclei is an important field today.

A number of consequences flow fromeven a superficial examination of Eq. (1.7).The fact that the binding energy pernucleon, B/A, is essentially constant with Aimplies that the nuclear density is constantand, thus, the nuclear force saturates.That is, nucleons interact only with asmall number of their neighbors. Thisis a consequence of the very short rangeof the strong force. If this were not so,then each nucleon would interact with allothers in the given nucleus (just as theprotons interact with all other protons),and the leading term in B(A,Z) would beproportional to the number of pairs ofnucleons, which is A(A − l)/2 or roughlyA2. This would imply that B/A would goas A. Thus, not only does the nuclear forcesaturate (the Coulomb force does not) butit is also of very short range (that of theCoulomb force is infinite) as the sizes ofnuclei are of the order of 3.0 fm (recallEq. (1.1)).

1.4Nuclear Charge and Mass Distributions

1.4.1General Comments

In his 1911 paper, Rutherford was able toconclude that the positive charge of theatom was concentrated within a sphere ofradius

1.4 Nuclear Charge and Mass Distributions 13

the electron, the radii of the muon orbits arereduced by that amount, so that electron-shielding problems are much reduced.)The energies of the photons are such thatthe 2pl/2 –23/2 splitting is easily measured(in 116Sn66, it is 45.666 keV). Thus, bothnuclear charge radii and isotope shiftsare quite accurately determined. In somerecent experiments, root mean square(RMS) charge radii have been measuredwith a precision of 2 × 10−18 m. As electronscattering and muonic atoms are the twomethods of measuring characteristics ofthe nuclear charge radius most susceptibleof the greatest accuracy, they will bediscussed in turn.

1.4.2Nuclear Charge Distributions from ElectronScattering

In any scattering experiment, what ismeasured is the differential cross section(dσ /d�). Rutherford developed an expres-sion for α-particle scattering that can beused for low-energy, spinless particles inci-dent on a spinless target. Both incident andtarget particles are assumed to be pointparticles. The differential cross section forscattering relativistic electrons off point-charged particles leads to the expression forMott scattering, while, if the target particlehas nonzero spin (there is then a mag-netic contribution), one obtains the Diracscattering formula for (dσ /d�). However,real nuclei are not point particles, so oneneeds to make use of the charge form factorF(−→q ) with −→q the transferred momentum.The form factor F(−→q ) is then the Fouriertransform of the charge density ρch(

−→r ):

F(−→q ) =∫

ρch(−→r )ei−→q •−→r dτ (1.8)

If one restricts the problem to spheri-cally symmetric distributions, the angular

integration of the Fourier integral followsat once, so that

F(q) = 4πq

∫ρch(r) sin(qr)rdr (1.9)

If the target nucleus has zero spin(applicable to all even–even nuclei), thenthe differential cross sections for a pointtarget and a finite-sized target are relatedby

dσ

d�=

(dσ

d�

)Ruth.

F(q)2 (1.10)

With the charge form factor deter-mined experimentally, the inverse trans-form yields the radial charge density

ρch(r) =1

2π2r

∫F(q) sin(qr)qdq

(1.11)

If the target nucleus is not of spinzero, then an additional term containingthe so-called transverse form factor, FT(q),is needed. (The form factor definedin Eq. (1.9) is sometimes called thelongitudinal form factor.) In any event, thecharge distribution must be normalized tothe number of protons (Z) in the targetnucleus.

At this point, there are two ways toproceed. The first is a model-independentanalysis of the form factor, or, second,one can assume a model with severalparameters and fit these to the data.Limiting oneself to small momentumtransfers, one can obtain the form factor asa power series in q2 by expanding sin(qr) inEq. (1.9) in a power series of its argument.Keeping only the lowest term of order q2,one obtains

F(q) = Z(

1 − 16

q2〈r2

〉)(1.12)


with 〈r2

〉 = ∫ r2ρch(r)4πr2dr (1.13)the RMS radius of the charge distribution.It should be noted that this is not thenuclear radius R, which is usually taken asthe radius of the constant-density sphere.This yields〈

r2〉 = 3

5R2 (1.14)

Data compilations [23] show that formost stable nuclei,

R ≈ 1.25A1/3 (1.15)

As was stated in Section 1.2.1, a betterway to describe the charge distribution isto use a Fermi distribution which takesaccount of the constant charge density,ρ0, at the center of the nucleus and thegradual decrease near the surface. This isachieved by

ρch(r) = ρ01

1 + er−R1/2

a

(1.16)

with R1/2 the radius at half density anda the diffuseness parameter indicatingthe distance at which the density fallsfrom 90 to 10% of the constant densityρ0. For heavy nuclei, the followingparameterization holds:

ρ0 = 0.17Ze

Afm−3

a = 2.4 fmR1/2 = 1.128A1/3 − 0.89A−1/3 fm

(1.17)There are enough experimental electron-

scattering data available throughout allregions of the stable nuclei that quiteaccurate charge parameters exist for almostall of the systems. The compendium byde Vries et al. [23] lists these parametersfitted to the data for several distribution

functions in addition to the two-parameterFermi functions.

1.4.3Nuclear Charge Distributions from AtomicTransitions

During the last decades, tremendousprogress was obtained in the study ofatomic transitions using high-precisionlaser spectroscopy. This allows themeasurement of nuclear charge radii andalso of nuclear moments for stable andeven unstable isotopes. This is becausethe difference between a point nucleusand a finite-size nucleus causes a verysmall change in the Coulomb potentialthe atomic electrons feel. A small energydifference on the atomic levels resultswhen we assume that the nucleus is asphere with constant charge density. For1s electrons one obtains

E1s =2

5

Z4e2

4πε0

R2

a30(1.18)

with a0 the Bohr radius. Because no pointnucleus exists and the theoretical calcula-tions are not accurate enough to calculatethe small shift exactly, one generallymeasures isotope shifts as the frequencydifference of atomic transitions measuredin two isotopes of a given element. Thisthen yield the differences in the nuclearradius. Starting from known radii of stableisotopes, it is then possible to determinethe radii of unstable nuclei on which onecannot perform electron scattering. It isalso possible to measure isotope shiftsusing optical transitions. Because theseare caused by the outermost electrons, theshifts are very small in the order of parts permillion. As indicated above, they are stillwithin reach of modern laser techniques.

The small shift for 1s binding energiesis related to the large difference between

1.4 Nuclear Charge and Mass Distributions 15

the Bohr radius and the nuclear radius.On replacing one electron by a muon,the muonic orbits shrink by a factor of207, the mass difference between theheavy muon and the light electron. Atthe same time, the muon binding energyis increased by a similar factor, makingthe transition energies in the mega elec-tron volt region. The energies are so high(in 238U146, the measured 2p–1s transitionis about 6.1 MeV) that one must generateDirac solutions for the muon moving ina Coulomb potential generated by a non-point-charge distribution. To these initialDirac solutions, one must add corrections,which, in order of size, are vacuum polar-ization, nuclear polarization, the Lambshift, and relativistic recoil. Electronscreen-ing corrections are often included, but theyare very tiny (for the ls muonic state in238U146, this correction has the value of11 eV).

As many nuclei are not spherical, severalstudies have used as the appropriatecharge distribution a slightly modifiedform of Eq. (1.16), which includes thedeformations

ρch(r) = ρ0

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1 + exp⎛⎜⎜⎜⎜⎝

r − R1/2(

1 +∑n=1

βnYn0 (θ , φ)

)a

⎞⎟⎟⎟⎟⎠⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

−1

(1.19)

Here the βn are deformation parametersthat determine the nuclear shape. As anexample, the nuclear mean square radiusmay be expressed as

〈r2

〉deformed

≈ 〈r2〉sph (1 + 54π (β22 + β24 + · · ·))

(1.20)

Experiments to fit a, c, and, in deformedregions, βn have been made throughout theperiodic table with results consistent withthe electron data. However, to combine theresults of electron-scattering experimentswith those from muonic atoms, it isnecessary to use the so-called Barrettmoment

〈rke−αr

〉= 4π

Z

∫ ∞0

ρ(r)e−αr rk+2dr

(1.21)where k and α are fitted to the experimentaldata. The muonic data are equivalent todata from electron-scattering experimentsat low momentum transfer. The inclusionof the muonic Barrett moment improvesthe overall fit by reducing normalizationerrors. This then reduces the uncertaintiesover what would be obtained by fittingeither the electron-scattering or the muonicatom data alone. Extensive tables of datafitted by various charge distribution modelsas well as model independent analyzes canbe found in de Vries et al. [23].

1.4.4Nuclear Mass Distributions

While the measurement of the chargedistribution can be made using electro-magnetic probes, this is not possiblefor the mass distribution because of theuncharged neutron. Instead, the nuclearstrong force has to be used. This is more


complicated as mostly both Coulomb forceand strong force are present. Nevertheless,from α-scattering experiments, informa-tion of the mass distribution is obtained.There are also indirect ways in which onecan get information on nuclear mass radii.One example is the dependence of α-decayrate on the nuclear radius that defines theCoulomb barrier. In deformed nuclei, thiscauses an anisotropy because the Coulombbarrier is lower in the direction of thelongest axis, making the tunnel probabilityenhanced. A second way is to use pionsinstead of muons. These interact with thenucleus through both the Coulomb forceand strong force, which, in comparisonto muonic atoms, causes an extra shiftthat allows the determination of the massradius. The result of these experimentson stable nuclei finds that the charge andmass radii are equal to within about 0.1 fm.This somewhat surprising result can beunderstood as a balance between the protonCoulomb repulsion that tends to push theprotons to the outside and a strongly attrac-tive neutron–proton strong force that tendsto pull the extra neutrons to the inward.

Recently, the common opinion that theradii scale with A1/3 was found to be heavilyviolated in more exotic nuclei.

Especially in light nuclei with a largeneutron number, so-called halo-nuclei,strong deviations were observed (an earlyreview is given in [24]). Using theradioactive beam techniques, very neutron-rich He, Li, and Be isotopes can be createdand studied in the laboratory. It turnedout that these loosely bound nuclei showvery extended neutron radii whereby twoneutrons are moving at radii similar to theradii of Pb isotopes. Moreover, as is the caseof 11Li8, the bound system consists of threeentities: two neutrons and a 9Li6 that cannotexist two by two, as the dineutron and10Li7 are unbound. The research on exotic

nuclei is still in its infancy and more exoticfeatures such as proton halos or neutronskins are expected. They are of importanceas they may influence the creation of theelements under astrophysical conditions.

1.5Electromagnetic Transitions and StaticMoments

1.5.1General Comments

Static electromagnetic nuclear momentsplayed an important role in the unscram-bling of the detailed measurements ofatomic optical hyperfine structure wellbefore the gross components of atomicnuclei were in hand. Almost a decadebefore the discovery of the neutron, Pauli[7] suggested that the optical hyperfinesplitting might be due, in part, to the inter-action with a nuclear magnetic moment(μ). This suggestion lay fallow until 1930,when Goudsmit and Young, using thespectroscopic data of Schiller and ofGranath, deduced the nuclear magneticmoment of 7Li to be μ = 3.29μN, wherethe nuclear magneton equals

μN =eh̄

2Mp= 5.050789 × 10−27 J

T

(1.22)This value is quite close to the currently

accepted value (μ = 2.327μN). Because ofthe existence, by then, of extensive hyper-fine optical spectroscopic data, Goudsmit,in 1933, was able to publish a table ofsome 20 nuclear magnetic moments rang-ing from 7Li to 209Bi. In 1937, Schmidtpublished a simple, single-particle modelof nuclear magnetic moments and sup-ported it with the experimental momentsof 32 odd-proton nuclei and 15 odd-neutron

1.5 Electromagnetic Transitions and Static Moments 17

nuclei. This simple model yields whatis now known as the Schmidt limits,within which almost all nuclear magneticmoments lie (see the following).

The suggestion that the nuclear electricquadrupole moments (Q) might also playan important role in optical hyperfinestructure was again made before thediscovery of the neutron. Racah [8] wasthe first to work out the theory associatedwith ‘‘nuclear charge asymmetry’’ andthe interaction with the atomic electrons.Casimir [25], sometime later, developedthe theory of nuclear electric quadrupolehyperfine interaction and applied it to151Eu and 153Eu. In this paper, Casimirmentions work by Schiller and Schmidt,who determined Q for 175Lu. A shorttime later, Gollnow [26] obtained Q = 5.9b for this nucleus, quite close to thecurrently accepted value of 5.68 b. Thisvery large quadrupole moment (very muchlarger than can be accounted for by thesingle-particle shell model) was to provide,20 years later, strong impetus for thedevelopment of the collective model of thenucleus. In 1954, Schwartz [27] extendedthe theory of nuclear hyperfine structure toexamine the magnetic octupole hyperfineinteraction and calculated the first fournuclear magnetic octupole moments (O)from data of the hyperfine structureof the nuclear ground states. The nextnuclear moment is the hexadecapole (H);however, no direct measurements of suchstatic moments exist. What is knownabout these moments comes mainly fromelectromagnetic transitions of electronsand negative muons. For an in-depththeoretical study of all of these momentsand how they can be used to test variousnuclear models see, in particular, the textby Castel and Towner [28].

Nowadays, the measurement ofmoments is still very important to assess

the single-particle structure of exoticnuclei and several powerful techniqueshave been developed in this domain [29].Most information is, however, gatheredvia the determination of electromagnetictransitions by γ-ray spectroscopy. This isto a large extent due to the availability oflarge-volume semiconductor detectors forγ-ray detection and the high computingpower that allows one to analyze moreand more complex measured spectrausing coincidence conditions. The recentdevelopment of γ-ray tracking detectors outof segmented Ge-detectors offers very highperspectives in the field of exotic nuclei [30].

1.5.2Electromagnetic Transitions and SelectionRules

Without going into a detailed discussionon how matrix elements are calculated, wereview here the calculation of electromag-netic transitions. The interested reader canfind more details in Heyde [31]. The cal-culation of transitions and also momentsinvolves the wavefunctions of nuclearstates and forms a very sensitive probefor nuclear structure research. On theother hand these transitions and momentsare electromagnetic in nature makingthe interaction very well understood.Using the long-wavelength approximationλ� R and a multipole expansion of theelectromagnetic operators, the transitionrates per unit of time can be expressed as

T(L) = 2ε0h̄

L + 1(L[(2L + 1)!!]2)

(ωc

)2L+1B(L)

(1.23)with L the multipolarity, ω the angular fre-quency of the radiation such that h̄ω ∼= Ei −Ef up to a small nuclear recoil correction,and B(L) the reduced transition probability


B(Ji → Jf ; L)

=∑

Mi ,Mf

∣∣∣〈αf ; Jf Mf ∣∣∣ O(LM) ∣∣αi; JiMi〉∣∣∣2(1.24)

in units of e2bL and μNbL−2 for electric

and magnetic B(LM) values. The labelsα identify the initial and final states andO(LM) is the electric or magnetic multipoleoperator of rank LM. As all states havegood angular momentum, one can now usefor the transition rates the Wigner–Eckarttheorem to remove all reference to the Mprojections. This yields

B(Ji → Jf ; L)

= 12Ji + 1

∣∣∣〈αf ; Jf ‖O (L)‖αi; Ji〉∣∣∣2(1.25)The electric multipole operator for a

number of point charges becomes

O(E; LM) =∑

i

eeff (i)riLYL,M(θi, φi)

(1.26)with eeff(i) the effective charge of the ithnucleon. Here it is anticipated that owing tocore polarization effects and truncations ofthe model space, other values than the freecharges +e(0) for proton (neutron) need tobe used. Instead of the operator (Eq. (1.25)),one can also use a similar operator butusing the nuclear electric charge densityρch and an integration over the nuclearvolume. These are used as several modelsdescribe the nucleus as a droplet (Section1.7). For the magnetic multipole operator,we have

O(M; LM) = √L(2L + 1)μN∑i

riL−1

×{

eeff (i)

e

2

L + 1 [YL−1 ⊗ ji](L)

+(

gs (i) −eeff (i)

e

1

L + 1)

× [YL−1 ⊗ si](L)} (1.27)

with the effective gyromagnetic ratio gswhich also may differ from the free ones.

The multipole expansion and thefact that states in atomic nuclei havegood angular momentum and parityleads to several selection rules. The firstone is related to the vector couplingof the angular momentum and states|Ji − Jf | ≤ L ≤ Ji + Jf . The second is due tothe parity of the operators, which clearly is(−1)L for the electric and (−1)L−1 for themagnetic operator. Owing to this, electricand magnetic transitions of order L cannottake place at the same time between states,and moments such as the electric dipolemoment are forbidden. While the selectionrules allow the determination of the spinand parities of nuclear excited states, theyare also (at a higher level) invaluable to testnuclear models (Section 1.7).

Weisskopf has estimated the so-calledsingle-particle values or Weisskopf units(W.u.) by assuming that a single-particlemakes a transition with multipole L froma state with spin L + 1/2 toward a statewith spin 1/2, that the radial part of thewavefunction can be approximated by aconstant value up to the radius R, and thatcertain values for the effective charges hold.This leads to the following estimates forthe half-lives corresponding to the single-particle values:

T1/2(E1) =6.764 × 10−6

E3γ A2/3

(s)

T1/2(M1) =2.202 × 10−5

E3γ(s)

T1/2(E2) =9.527 × 106

E5γ A4/3

(s)

T1/2(M2) =3.102 × 107

E5γ A2/3

(s)

T1/2(E3) =2.045 × 1019

E7γ A2

(s)


T1/2(M3) =6.659 × 1019

E7γ A4/3

(s) (1.28)

One notices that transition rates of thelower multipolarities are faster than thehigher by orders of magnitude and thatfor a given multipolarity, the electric onesare about 100–1000 faster as the magneticones. Owing to the selection rules andenhanced quadrupole collectivity, only E2and M1 transitions happen on similartimescales. In this case, one has a transitionof mixed multipolarity.

The Weisskopf estimates are very crucialto determine whether a transition iscaused by a single nucleon changing orbitsor by several nucleons acting in a collectiveway. The measurement of transition ratesof excited states delivers very importantinformation on nuclear structure, but isalso quite involved. One needs to measurethe lifetime of a state, the (mixed) multi-polarity, and the energy and intensities ofthe transitions deexiting a given state. Tothis end, γ-ray arrays consisting of severalGe-detectors are appropriate. Besides this,electromagnetic decay can also take placewith the emission of conversion electrons.These electrons allow one to determine themultipolarity as well as to observe the byγ-emission forbidden E0 transitions (dueto the fact that the photon has spin 1 withprojection +1 and −1).

1.5.3Static Moments

In contrast to the transition rates, themultipole moments are generally definedas the matrix element of the M = 0component of the moment operator fora single state with magnetic projection

M =+J. Of the moments, the two lowestare the most important.

1.5.3.1 Magnetic Dipole MomentsIf one assumes that the magnetic prop-erties are associated with the individualnucleons, then the magnetic moment isdefined as

μ = 〈α, JJ∣∣∑i

gl(i)lz,i + gs(i)sz,i∣∣α; JJ〉(1.29)

where the sum extends over all of theA nucleons. Generally, this will not beneeded; for instance, the magnetic momentof an odd-A nucleus will be generated bythe last neutron and proton as the adjacenteven–even ground state has no magneticmoment. In Eq. (1.29), g l and gs are theorbital and spin gyromagnetic ratios. Theyare often chosen as 0.7gfree where the free-particle values are g l = l, gs = 5.587 forprotons and g l = 0, gs = −3.826 for neu-trons (all in μN). It is instructive to calculatethe magnetic moment for a single nucleon(which, as explained earlier, is a goodapproximation for the ground state of odd-A nuclei). Using the total angular momen-tum and the Wigner–Eckart theorem,one deduces the single-particle magneticmoments for aligned and antialignedorbital momentum and spin:

μ(j = l + 1/2) = j(

gl +(gs − gl

)2

)(1.30a)

μ(j = l − 1/2) =(

jgl −j

j + 1

(gs − gl

)2

)(1.30b)

The magnetic moments plotted as afunction of spin form the so-called Schmidtlines. It is interesting that, when plotted,almost all of these moments lie betweenthe Schmidt lines. The moments that


lie outside these limits occur mainlyfor some very light nuclei. One mayconclude that the single-particle modeldoes possess some validity. Another setof limits, the Margenau–Wigner (M-W)limits, is obtained by replacing the free-particle values for the orbital gyromagneticratios, g l, by the uniform value Z/A. Thejustification for this is that one is ineffect averaging over all states that lead tothe correct nuclear spin. This calculationrepresents an early attempt to account forcore contributions to the dipole-momentoperator. Figure 1.3 and Figure 1.4 showplots of a number of the ground-statemagnetic moments for odd-Z (Figure 1.3)and odd-N (Figure 1.4) nuclei.

Besides the ground state, excited statescan have magnetic moment and their

measurement is often used to extractinformation on the underlying single-particle structure. Common in nuclearstructure physics is the use of g-factorsthat are analogous to the single-particlegyromagnetic ratios, except that they aredimensionless. They are defined as

μ = gIμN (1.31)

One way to determine the g-factor isto measure the Lamor frequency whenexcited nuclei are placed in an externalmagnetic field B. Then,

ωL =gBμN

h̄(1.32)

The main problem hereby is to alignthe spins of an ensemble of atomic nuclei.

0 1 2 3 4 5 6 7

I (Odd Z )

−2

0

2

4

6

8

µ/µ

N

M-W

M-W

I − L − 1/2

I = L + 1

Schmidt

Schmidt

15N

100Tm

64Mn

100Ag 197Au

27Al

153Eu176La

181Ta

139La

93Nb

165Ho

51V

123Sb

200Bi

113In

187I

193Ir

37Cl

36Cl

159Tb

23Na

165,167Re

19F

31P

71Ga

7Li

Figure 1.3 Nuclear magnetic moments in units of thenuclear magneton (μ/μN) plotted against the nuclear spin(I) for a number of odd-Z nuclei. The Schmidt limits, aswell as the Margenau–Wigner (M-W) limits, are shown assolid lines. Data from [14].


0 1 2 3 4 5 6 7

I (Odd N )

−2

0

2

4

µ/µ

N

Schmidt

Schmidt

I = L + 1/2

I = L + 1/2

M-W

M-W

3He

125Te 9Be91Zr

67Zn

97Mo

136Ba

132Ba

61Ni145Nd47Ti49Ti

105Pd

161Dy167Er

43Ca

163Dy

201Hg

177Hr

157Gd

73Ge83Kr87Sr

33S

123Te

29Si

183W

207Pb

13C

17O

Figure 1.4 Nuclear magnetic moments in units of thenuclear magneton (μ/μN) plotted against the nuclear spin(I) for a number of odd-N nuclei. The Schmidt limits, aswell as the Margenau–Wigner (M-W) limits, are shown assolid lines. Data from [14].

This has to be done by the nuclear reactionused, cooling in an external magnetic field,or via the observation of changes in angularcorrelations.

1.5.3.2 Electric Quadrupole MomentsAs a general definition of the quadrupolemoment, we have the expectation value of(3z2 − r2). Using the proportionality of thequantity with Y20(θ ,φ) one gets

Q(J) =√

16π

5

×〈α; JJ

∣∣∣∣∣∑i

eeff (i)

eri

2Y2,0(θi, φi)

∣∣∣∣∣α; JJ〉

(1.33)

or, using the Wigner–Eckart theorem,

Q(J) =√

16π

5

√J(2J − 1)

(2J + 1)(2J + 3)(J + 1)

×〈α;J

∥∥∥∥∥∑i

eeff (i)

eri

2Y2(θi, φi)

∥∥∥∥∥α; J〉

(1.34)

The quadrupole moment can also becalculated for a single nucleon in an orbitj = J. This yields

Q(j) = − (2j − 1)(2j + 2)

eeff (i)

e

〈r2

〉(1.35)

One thus obtains a negative quadrupolemoment for a single nucleon. If the orbit


is filled up to a single hole, a quadrupolemoment as in Eq. (1.35) but with a pos-itive sign is expected. We would like toillustrate the application of Eq. (1.35) withexamples near the doubly magic nucleus16O8 (see also Section 1.7). The orbit thatthe ninth nucleon can occupy has j = 5/2and we can use Eqs. (1.13) and (1.14) to esti-mate . This yields for 17F8 using thefree-proton charge Q =−5.9 fm2, which isin excellent agreement with the experi-mental absolute value given in Firestoneet al. (1996) of |Q| = 5.8(4) fm2. For the oddneutron nucleus 17O9, one finds exper-imentally Q = −2.578 fm2, which showsthe need for effective charges and canbe reproduced using 0.44e as neutroneffective charge. Finally, if we place fiveneutrons in the j = 5/2 orbit, we havethe N = 13 isotones and expect momentsof Q = +2.6 fm2. Experimentally, onefinds Q = 20.1(3) fm2 in 25Mg13 andQ = 10.1(2) fm2 in 23Ne13. This observa-tion of much larger quadrupole momentsoccurs for most atomic nuclei having sev-eral nucleons in an orbit or in several orbits,indicating that the model is too simple andthat all of the electric charges must be con-sidered. This holds especially if the core isnot spherically symmetric and the motionbecomes collective.

1.6Excited States and Level Structures

1.6.1The First Excited State in Even–Even Nuclei

The most obvious characteristic of thevarious nuclei is that all of the even–evennuclei have ground-state spins and paritiesof 0+. This not only categorizes one largegroup of nuclei but also indicates thatthe nuclear force is such that it couples,

preferentially, pairs of like nucleons toangular momentum zero. The secondobservation is that the first excited statein even–even nuclei is almost always a2+ excitation. This can be understood bythe combination of good total angularmomentum and the Pauli principle. If onecouples two nucleons in orbit j and impliesthe antisymmetrization, one obtains

ϕ(j, j; JM) = 12

∑mm′

(jmjm′|JM)(ψ1(jm)ψ2(jm′)

−ψ2(jm)ψ1(jm′)) (1.36)

with the Clebsch–Gordan coupling coeffi-cient for the angular momentum coupling.This can, however, be rewritten as the m-values are in the summation. Using thesymmetry properties, one arrives at

ϕ(j, j; JM) = 1

2

(1 − (−1)2j−J

)×∑mm′

(jmjm′|JM)ψ1(jm)ψ2(jm′)

(1.37)

The phase factor, and the fact that2j is an odd number, imply that stateswith odd values of J do not exist. Onemight wonder whether this is an essentialproperty of fermions, but surprisinglyit is not. Consider bosons with integerangular momentum l. Then, owing tosymmetrization, Eq. (1.36) becomes

ϕ(l, l; JM)= 12

∑mm′

(lmlm′|JM)

×(ψ1(lm)ψ2(lm′)+ψ2(lm)ψ1(lm′)) (1.38)

which yields after reordering,

ϕ(l, l; JM)= 12

(1 + (−1)2l−J

)∑mm′

×(lmlm′|JM)ψ1(lm)ψ2(lm′)(1.39)

1.6 Excited States and Level Structures 23

but now 2l is even and again all states withodd J-values disappear. The fact that bothbosons and fermions yield similar resultsis very important for nuclear models.It means that one can often describeeven–even atomic nuclei using eitherfermionic nucleons or collective bosonssuch as phonons.

1.6.2Regions of Different Level Structures

The existence of a low-lying 2+ excita-tion can be explained in different ways,that is, short-range interaction, a collectivequadrupole vibration, or as a first excitedstate of a rotational band. Which interpre-tation is right or better and which mixtureof interpretations is right depends a loton where on the Segre chart the nucleusis located and how the higher excitationsbehave. This is illustrated in Figure 1.5,which shows the known energies of thefirst excited states in the even–even Cdand Hf isotopes. One notices huge differ-ences in the absolute excitation energiesbut also a quite smooth behavior. If onenow also considers the next excited state,which is a 4+ excitation and plots the ratioR4/2,

R4/2 = Eexc(4+)

Eexc(2+)

(1.40)

(see also Figure 1.5) more informationcan be obtained. In the Hf isotopes, aclear saturation near R4/2 = 3.33 occursindicating that a rotational band, with itstypical I(I + 1) energy dependence, is builton the ground state. For Cd, we observevery high 2+ excitation energies and lowR4/2 ratios at the beginning and end of theisotope chain. This is a clear evidence that atthe extremes, a shell closure occurs, whenN = 50 and 82. In the middle, one finds,

compared to Hf, much higher 2+ excitationenergies and R4/2 ratios slightly above2, indicating an anharmonic quadrupolevibrational nature at mid shell.

While the strong pairing of like nucleonscan explain the ground state of even–evennuclei, it does not yield predictions forthe ground-state spins and parities of odd-A and odd–odd nuclei. Nevertheless, itsimplifies this task a lot, as in most casesone can conclude that the ground-state spinand parity of an odd-A nucleus is equal tothe one of the last odd nucleon. This formsan important testing ground for the shellmodel, which predicts the single-particleenergy, spin, and parity of the subsequentorbits.

1.6.3Shell Structures

Various sets of nuclear data indicatethat certain numbers of nucleons, eitherneutrons or protons, correspond to thefilling of angular momentum ‘‘shells.’’These are similar to the atomic shells oftendenoted as the K-shell, L-shell, and so on.In nuclei, the shell filling is similar but notidentical, and the principal shell closingsoccur at experimentally observed numbers.These are for either N or Z equal to 2, 8,20, 28, 50, 82, and 126. These are themagic numbers. They occur where there aredrastic changes in neutron cross sections,nucleon separation energies, and so on.An interesting, but small, group of nucleicomprises the doubly magic ones – thatis, nuclei with both neutron and protonnumbers magic. The five stable ones are4He2,

16O8,40Ca20,

48Ca28, and208Pb126.

Note that no stable doubly magic nucleiwith 50 and 82 exist. What is notableabout the doubly magic nuclei is thattheir first excited states are at a very


0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Cd

Cd

4+1

4+1

2+1

2+1

Hf

Hf

90 100 110 120 130 140 150 160 170 180 190Λ

1

1.5

2

2.5

3

3.5

4

R4/

2E

(K

eV)

(a)

(b)

Figure 1.5 The known energies of the first excited statesin the even–even Cd and Hf isotopes (a). The R4/2 ratiosare also given (b). The left side represents the Cd isotopesand the right side the Hf isotopes. Arrows mark the closedshells at A = 98, 130, and 154.

high energy compared with their non-doubly magic neighbors. The energy inmegaelectronvolts, spin, and parity of thesefirst excited states is 20.1, 0+; 6.05, 0+; 3.35,

0+; 3.83, 2+; and 2.61, 3−, respectively. Alsothe observed spins (and parities) of the firstexcited states is very different from that ofall other even–even nuclei.


Because of these large gaps or changesin nuclear properties, they divide the low-lying excited states of nuclei into roughlythree regions. These are, broadly, thenuclei in the neighborhood of the magicnumbers that are dominated by shell orsingle-particle structures and those furtheraway, which show collective behavior. Ofparticular interest are the ones foundbetween these two, as they may giveclues to the onset of collective motionin atomic nuclei and other systems. Wementioned earlier that 126 is a magicnumber, but strictly speaking, this holdsonly for neutrons as the unstable elementwith Z = 126 has not yet been observed.Nowadays, there is a strong interest to studywhether magic numbers still stay valid inexotic nuclei far from stability and whichis the next magic number for superheavyisotopes.

1.6.4Collective Structures

Within major divisions (even–even, odd-A,and odd–odd), the excited levels are furtherdivided into groups that are single-particleor collective (vibrational or rotational) innature. However, because of the strongpairing force, even–even nuclei do notshow single-particle excited levels butrather one- or two-particle-hole excitationsor two-quasiparticle excitations. Here, weintroduce the most general features of thetwo major classes of collective nuclei. Werestrict ourselves to even–even nuclei forthe sake of simplicity.

1.6.4.1 Vibrational LevelsThe intermediate systems between thetightly bound magic nuclei and thedeformed nuclei are the so-called vibra-tional nuclei. Here the number of nucleonsoutside of a deformable core is small,

and the zero-point energy of the lowestoscillations is greater than the energy ofdeformation, so that the shape of the coreis not stabilized. The motion then is ofquantized surface oscillations with angu-lar momentum two. One can introducecreation and destruction phonon operatorsQ+LM and QLM which fulfill the boson com-mutation rules:

[QLM, Q+ĹḾ] = δLĹ δMḾ (1.41)

and for which all other commutatorsare zero. The angular momentum of thephonon is L, its magnetic projection Mand its parity π = (−1)L. The magnetic pro-jection is mostly not of great importance.In this representation, the number opera-tor counting the number of L phonons isgiven by

N̂L =∑

M

Q+LMQLM (1.42)

Using the number operator, the simplestHamiltonian becomes

Ĥ =∑

L

h̄ωL

(N̂L + L +

1

2

)(1.43)

where the additional factors are chosensuch that the Hamiltonian corresponds tothe quantized harmonic oscillators for Lphonons. Each of the phonons has energyh̄ωL. Because most even–even nuclei haveas first excited state a 2+ state, the L = 2quadrupole phonons are dominant. Athigher energy, there is evidence for thepresence of L = 3 octupole phonon makinga 3− state. Using the energy solution of theHamiltonian equation (Eq. (1.43)),

E(NL, L) =∑

L

h̄ωL

(NL + L +

1

2

)(1.44)


and the proper angular momentumcoupling and symmetrization given inEq. (1.39), one arrives at a very simpleprediction. The one-quadrupole phononstate has L = 2 and Eexc = h̄ω2 and thetwo-phonon states have L = 0, 2, 4 andEexc = 2h̄ω2. This exactly yields the R4/2ratio of 2 observed in Figure 1.5.

A nice example of a ‘‘good’’ vibrationaleven–even nucleus is 110Cd62. The firstexcited 2+ state is at 0.6578 MeV followedby a 0+, 2+, 4+ states at 1.473, 1.476,and 1.542 MeV. The ratio R4/2 = 2.35 issomewhat larger than the one the simplemodel gives.

1.6.4.2 Rotational LevelsIf the number of nucleons outside thedeformable core is such that the zero-point oscillations are much less than theenergy of deformation, then the systemwill have a stable but deformed shape andone must quantize a rigid rotator. Fromclassical considerations, we know that, ifthe system has a permanent, nonsphericalshape, there exists a body-fixed system inwhich the inertial tensor I is diagonal andis related to the laboratory-fixed systemby an Euler transformation. We denotethese two coordinate systems as (1,2,3)and (x,y,z), respectively. The inertial tensorthen has components I1, I2, and I3, so thatthe Hamiltonian is just

Ĥ = 12

(L̂21I1

+ L̂22

I2+ L̂

23

I3

)(1.45)

with L̂1,2,3 the body-fixed angular momen-tum. The simplest system is the one forwhich all moments of inertia are equal:I1 = I2 = I3 = I. Then the energies aredirectly found:

Ĥ = h̄2

2I(L(L + 1)) (1.46)

The system specified by Eq. (1.45) pos-sesses the symmetry properties belongingto the point group D2 for which four repre-sentations exist. The Hamiltonian operator(Eq. (1.45)) does not mix different repre-sentations. The basis functions are thoseof a symmetric top, |LMK> being diagonalin L2, Lz, and L3 with the usual eigen-values L(L + 1), M, and K, respectively.Here K is the projection of the angu-lar momentum on the body-fixed frame.Applying this formula for the ground-staterotational band with K = 0, we find nowR4/2 = 3.33. However, states with differ-ent K-values are degenerated. This is notfound in real deformed nuclei.

If the inertia tensor is such thatI1 �= I2 �= I3, an asymmetric top problemresults. In this case, the wave functions areto be expanded in terms of the symmetric-top functions:

|LM〉 =+L∑

K=−LAK |LMK〉 (1.47)

The 1, 2, 3 labels of the momentalellipsoid’s semiaxes are quite arbitraryand can be chosen in 24 different waysfor right- (left-) handed systems. These24 relabelings can be produced by threedifferent relabeling transformations T1,T2, and T3 [32], which correspond to labelinterchanges and operate on the |LM>wave functions. They have the properties

T21 = T42 = T33 = 1 (1.48)

From these relations, one can generateTable 1.1, relating the four representationsand the values of angular momentumand parity allowed in each. Note inparticular that the A and B2 representationsare associated with positive-parity (π =+)states, while B1 and B3 representationsare associated with negative parity (π =−)


Table 1.1 The representation of the point group D2 to whichthe states belong, and the allowed values of the angularmomentum and parity associated with them.

Representation K Parity Allowed L-values

A Even + L = 0,2,2,3,4,4,4,5,5B1 Even − L = 1,2,3,3,4,4,5,5,5B2 Odd + L = 1,2,3,3,4,4,5,5,5B3 Odd − L = 1,2,3,3,4,4,5,5,5

states. From this table, we note that onlythe rigid rotator systems belonging tothe A representation have a zero angularmomentum state. They must be associatedwith the lowest levels of even–even nucleiand form the K = 0 ground-state band withK = 0 and L = 0, 2, 4, 6, . . . followed by theγ-vibrational band with K = 2 and L = 2, 3,4, 5, . . . .

Negative-parity states in even–evennuclei could be associated with either theB1 or B3 representations. If one argues that,given two (or more) possibilities, the lowestset of states will be the most symmetric,then because the B1 representation arisesfrom sums over even K quantum numbers,one should associate this representationwith the lowest-lying negative-parity statesin even–even systems. Furthermore, inthe deformed regions, the negative-paritybands in these nuclei not only lie above theground-state rotational band but they alsohave as their band head a 1− level. Thisis another indication that these negative-parity rotational bands belong to the B1representation.

The problem of solving the asymmetric,rigid rotor is far more involved thanthe symmetric rotor as all of the latter’seigenvalues can be given in closed form. Toproceed along this line, one sets I0 = I1 =I2 �= I3. The Hamiltonian of the system

then becomes

Ĥsym =1

2

(L̂21 + L̂22

I0+ L̂

23

I3

)

= 12

(L̂2

I0+

(1

I3− 1

I0

)L̂23

)(1.49)

and both of the operators appearing arediagonal. Thus, we obtain immediately theeigenvalues

Esym =h2

2

(L (L + 1)

I0+

(1

I3− 1

I0

)K2

)(1.50)

This expression is model independentand only depends upon the ratio of themoments of inertia I0/I3. There have beenrecent observations of ‘‘superdeformation’’for rare earth nuclei with spheroidal-axesratios of 2 : 1. For symmetric rotationalsystems with K = 0, the ratio R4/2 is10/3 rather than 2 for the vibrationalsystems. Thus, this ratio differentiates wellbetween rotational and vibrational systems.In recent years, there was quite someinterest in how these shape changes occur[33, 34], as they can be treated as quantum-phase transitions. Atomic nuclei that havean R4/2 ratio of around 2.9 lie at thecritical point of the spherical-deformedphase transition.


1.6.5Odd-A Nuclei

1.6.5.1 Single-Particle LevelsNear the magic numbers, the groundand lower excited states appear to bemainly single-particle states, that is, statesthat do not seem to possess collectiveproperties. For instance, near the doublymagic nucleus 16O8, one finds that both15N8 and

15O7 have 1/2− ground states and

nearly identical lower excited states. Thisshows that the N = Z = 8 shell is filled bya p1/2 orbit. Above the closed shell are17F8 and

17O9 both of which have theexpected 5/2+ ground state from the d5/2orbit. These data clearly support the notionthat angular momentum shells are filled ina manner similar to the electronic shells ofthe elements.

1.6.5.2 Vibrational LevelsThe nucleons all possess intrinsic spin1/2 and angular momentum l, so thatthe simplest model for vibrational odd-Anuclei that one might construct is to adda single nucleon to an even–even core inwhich the nucleons of each kind pair tozero angular momentum. If the core is agood vibrational nucleus (R4/2 ∼ 2), thenthe ground state of the odd-A system willhave the angular momentum properties ofthe odd nucleon. Furthermore, by couplingthis nucleon to the excited states of thecore, one should expect to be able todetermine the angular momenta of theodd-A nucleus’s excited states. An exampleis the coupling of a neutron to thevibrational nucleus 110Cd62 (Section 1.6.2)to yield the odd-A nucleus 111Cd63, whichhas a ground-state spin of 1/2+. One mightexpect that there would be two excited statesbuilt on the first excited 2+ state with spins3/2+ and 5/2+. This nucleus does indeedhave a 3/2+, 5/2+ pair as its first excited

states, which are at 245.4 and 342.1 keV.Coupling of the 1/2+ neutron with the two-phonon states would lead to excited statespins from 1/2+ through 9/2+. Such a setof states does appear at higher energies.However, it is difficult to pick out whichstates might belong to the two-phonon corestates. Below them there is a first negativeparity state with spin 11/2− at 396.22 keV,which indicates that one has the presenceof a low-lying h11/2 neutron orbit.

1.6.5.3 Rotational LevelsOne might expect that a simple modelfor the odd-A rotational levels wouldfollow that of the vibrational levels.However, the experimental data do notshow such a similar structure. As anexample we consider 157Gd93 which has a3/2− ground state. Following the reasoningfor vibrational nuclei, one expects from thecoupling with the first excited 2+ state amultiplet with spins 1/2−, 3/2−, 5/2−, and7/2−. Instead, one observes a rotational-like band with the spin sequence: 5/2−,7/2−, 9/2−, 11/2−, . . . . This is due to thefact that the ground-state band has alsoa given value of K and we conclude thatK = 3/2.

1.6.6Odd–Odd Nuclei

As noted in Section 1.6.1, there are onlyfour stable odd–odd nuclei; however, agreat deal of data exist for all of thisclass of nuclei. Again, one might feel thatthese nuclei would be well represented byan even–even core plus a neutron–protonpair. Even if these are coupled to a spin-zerocore, there is still considerable ambiguityas to how these couple to form the ground-state spin of the system. For instance, if theproton and neutron angular momenta arejp, jn, respectively, then their vector sum

1.7 Nuclear Models 29

can lead to the range of J values∣∣∣jp − jn∣∣∣ ≤ J ≤ jp + jn (1.51)One can obtain the values of jp and

jn from the ground-state spins of theneighboring odd-A nuclei with one lessneutron or one less proton, respectively. Inorder to determine more exactly the rangeof the possible ground-state spins of thesenuclei, one must make use of Nordheim’srules, which are the following:Strong rule,for η = 0:

J =∣∣∣jp − jn∣∣∣ (1.52)

Weak rule, for η = ±1:

J =∣∣∣jp − jn∣∣∣ or jp + jn (1.53)

with

η = (jp − lp) + (jn − ln) (1.54)

In general, these do not work all of thetime. To predict the excited states spinsis even more difficult, making odd–oddnuclei the most difficult to describe.

1.7Nuclear Models

1.7.1Introduction

The nucleus, being a many-body collectionof A particles interacting through a shortrange but strong force, is not easilydealt with. From a theoretical point ofview, making use of an accurate many-body calculation is impossible becausethese are beyond the scope of currenttheoretical methods. Thus, calculationsbased upon various simple models are

able to provide a good deal of physicalinsight as well as making predictions ofmeasurable properties, providing a wayto categorize the vast amounts of dataavailable from nuclear physics laboratories.While these approaches may seem to beorganized in an historical order, all are togreater or less extent being continuouslyrefined.

The nuclear many-body problem isnot solved definitely, owing to the largenumber of strongly interacting particles.To illustrate the difficulties, we take theexample of 208Pb. If we consider up to two-body interactions between the nucleons,the general Hamiltonian is given by

H =208∑i=1

ti +1

2

208∑i �=j

Vij (1.55)

The radius of this nucleus is 7 fm, whilethe strong force between two nucleonsvaries significantly on the scale of 0.2 fm.If one would calculate only the two-bodyinteraction on a mesh, one should foreseefor each nucleon at least 100 mesh points ineach direction and for all 208 nucleons thenumber of possible combinations wouldbe 1003.208. For a fast computer today 109

operations per second are feasible, whichgives 1016 operations per year. Thus toperform the calculation one would need101232 years, while the age of our universeis only 2 × 1010 years.

As an exact treatment is impossible,one of the major characteristics of nuclearstructure calculations is the choice ofthe right approximations and truncationsneeded to make a calculation feasible whilekeeping the essential physics. Of great helpthereby is the extensive use of symme-try concepts, which allow one to simplifythe many-body problem and make thecalculations tractable. Many quite differ-ent approaches have turned out to be


successful for certain classes of atomicnuclei. Most theoretical models rely onspherical symmetry, which allow the useof the very powerful ‘‘Racah machinery,’’originally developed for atomic physicsby Guilio Racah. The introduction ofirreducible tensors and coupling coeffi-cients then provides the necessary tech-nical skills to perform nuclear structurecalculations.

The first approach is based on theexperimental observation of magic num-bers and leads to the nuclear shellmodel as described in Section 1.7.2. InSection 1.7.3 we introduce the deformedshell model, which allows the descriptionof medium-heavy nuclei that are deformed.Section 1.7.4 is devoted to the collectivemodel in which medium heavy and heavynuclei are described by a quantum liquiddroplet.

In Section 1.7.5, we will deal with anew theoretical approach to many-bodyproblems originating from nuclear physics:the interacting boson approximation (IBA).Although the IBA is a method that wasdeveloped to describe the atomic nucleus,it has since been applied to molecules,quarks, and fullerenes. The major assump-tion of the interacting boson model (IBM)is twofold. First, because of the shell struc-ture evinced by many experiments, onlythe valence nucleons are considered tobe important for the low-energy excitedstates of the atomic nucleus. Those moveto a first approximation in orbits formedby their commonly created average field,which is spherical symmetric. Secondly,between like nucleons, either protons orneutrons, there is a dominant pairing forcewhen they occupy the same orbit. Thisallows to replace coupled fermion pairsby real bosons that are simpler to dealwith.

1.7.2The Spherical-Shell Model

One of the first and best understoodnuclear models is the nuclear shell model.The major assumption of this model is thatnucleons up to a good approximation moveindependently in a spherically symmetricaveraged field U(ri) created by the othernucleons. This means that the many-bodyHamiltonian is rewritten as

H = H0 + Vres (1.56)

with

H0 =N∑

i=1ti +

N∑i=1

U(ri) (1.57)

and the residual interaction V res is givenby

Vres =1

2

N∑i �=j

Vij −N∑

i=1U(ri) (1.58)

The shell model uses the ansatz(Eq. (1.57)) in which the single parti-cles move in a potential well provided bythe other particles. While it might seemimprobable that nuclei made up of a largenumber of strongly interacting particlescould in any way be represented by sucha model, one must recall that the Pauliprinciple prevents most interactions of anucleon in the nucleus. In general, scatter-ing cannot take place, as most of the statesinto which a low-energy nucleon can scatterare filled. This implies that the mean freepath for nucleon motion is quite long, anda single-particle model is worth exploring.

One obvious and important characteris-tic of odd-A nuclei is their ground-statespins and parities. Starting with the light-est nuclei, the general order of odd-Aground-state spins and parities is 1/2+ for


neutron and proton and after the stronglybound 4He 3/2−, 1/2− then, after 16O8,it becomes 5/2+, 3/2+, 1/2+ and then,after 40Ca20, 7/2

−, 5/2−, 3/2−, 1/2− anorder reminiscent of the atomic shells inwhich the alternating parities are relatedto the orbital angular momentum sug-gesting a sequence: s-p-d and s-f and pto which a spin one-half is coupled. Toobtain a model with such an ordering ofangular momentum l-values requires solv-ing a single-particle Schrodinger equationwith a reasonable potential function. Theform of this potential is not too impor-tant because only minor changes in l-valueorder occur using one or another r depen-dence. The two simplest to calculate arethe isotropic harmonic oscillator and theinfinite square well. The latter is not realis-tic because the nucleon separation energyis quite finite and well-known for mostnuclei. (One might also consider a potentialfunction related to the charge distribu-tions of Eqs. (1.16) and (1.19); however,these would require extensive numericalcomputing to extract the eigenvalues.) Theangular momentum order of the levels andthe number of particles in each for theisotropic harmonic oscillator is 1s (2); 1p(6), 1d, 2s (12); If, 2p (20); lg, 2d, 3s (30);and so on. (The semicolons separate thelevels with different quanta of energy h̄ω,and most are degenerate.) The total num-ber of levels thus becomes 2, 8, 20, 40, 70,112, 168, which only agrees with the lowestobserved ‘‘magic numbers’’ 2, 8, 20, 28, 50,82, and 126 (cf. Section 1.6.1).

It took the genius of Maria GoeppertMayer [35] and others [36] to realize that byadding a spin–orbit term of the form

Vls = −V(r)−→l •−→s (1.59)

the theoretical order could be brought intoagreement with that observed. The minus

sign guarantees that the j = l + 1/2 levelwill lie below the j = l − 1/2 level, asobserved and the dependence on l makesthat the effect increases with l leavingthe shell closures at 2, 8, 20 unaffectedwhile introducing a new shell closure at28 due to the lowering of the 1f7/2 orbit.Further, the observed closures at 50, 82,126, are obtained in a natural way. Alsothe spins of the odd-A ground statesis mostly correctly predicted. The shellmodel explains via large gaps between thesingle-particle orbital energies the largebinding energies of nuclei having Z orN a magic number making the number ofstable isotopes/isotones particularly large,for example, there are 10 stable Sn isotopeswith Z = 50. For the excited states, magicnuclei show them at very high excitationenergies as one needs to break the closedshell and lift one particle over the largeshell gap. As an example in the doublymagic nucleus 208Pb, an energy of 2.6 MeVis needed to create the first excited 3− stateand even 4.085 MeV for the 2+ state.

The robustness of double-magic nucleiallows a major truncation for shell modelcalculations. It turns out that they can infirst instance be treated as inert cores,whereby the spins of all of the particlesin the core pair off to zero. Thereby,only the so-called valence nucleons needto be considered. This is of importanceas with increasing number of nucleonsand possible orbitals, the shell modelallows for many different configurations.As angular momentum and parity areconserved, the residual interaction usedcan be diagonalized separately for agiven spin and parity. Nevertheless, thedimensions become quickly huge.

Although the shell model is one of theolder nuclear models, it has become ofincreasing use in the last decades. Thisis due to two key aspects. The first is


the tremendous increase in computingpower allowing large-scale shell modelcodes to be developed, including quan-tum Monte Carlo diagonalization. Thesecond is the construction of effectivein medium interactions starting from thenucleon–nucleon scattering data. Herebythe elimination of the hard core repulsionof the nucleon–nucleon interaction wascrucial and produced a universal, so-calledV low-k interaction. The recent progressopens the direct way from quantum chro-modynamics (QCD) to the effective inter-actions, which is in particular very muchneeded for the study of exotic nuclei. Themost involved calculations are done forlight nuclei up to 11B. Here no-core shellmodel calculations are able to describeaccurately binding energies and excitedstates for very light nuclei [37].

1.7.3The Deformed Shell Model

As the very large quadrupole moments ofthe nuclei in the middle of the highershells cannot be explained by the simplespherical model, their explanation requireslarge contributions from the even–evencore, which carries all (or almost all)of the charge. It can be shown thata single-particle moving in a nonrigidpotential well will have a lower energyif the well is not spherical, rather thanif it is spherical. As the lowering ofthe particle energy is proportional to theeccentricity, such a system will assume adeformed equilibrium shape. The nuclearHamiltonian will now contain a term thatproduces deviations from spherical:

H = H0 + Hd − Vls(r) + cl2 (1.60)

where H0 is the isotropic harmonicoscillator Hamiltonian and V ls(r) is the

spin–orbit term, both of which have provedso successful for the spherical-shell model.(The I2 term helps give the proper levelorder in the isotropic limit.) Expandingthe nuclear surface in spherical harmonicsgives

S(θ , φ) = R0

⎡⎣a0 + ∑λ>1,μ

aλμYλμ (θ , φ)

⎤⎦(1.61)

where R0 is the radius of the sphericalnucleus and a0 is unity to quantities ofsecond order and assures that the volumeremains constant for small deformations.Taking the lowest order, one has that

S(θ , φ) = R0[

1 ++2∑−2

a2μY2μ (θ , φ)

](1.62)

[This expression should be comparedwith the nuclear charge distribution ofEq. (1.19).] It is customary to take onlythe symmetric term a20 in this equationto be nonzero. This then leads to thedeformation term in the Hamiltonian

Hd = −Ka20ω20r2Y20(θ , φ) (1.63)

The full Hamiltonian can now bediagonalized using a set of spherical shellmodel basis states, and the resultingdeformed, single-particle states are calledNilsson states after Sven Gosta Nilsson [38],who first carried out this diagonalization.The operators in Eq. (1.60) do not connectoscillator states differing by one principaloscillator quantum number, N, but doconnect states differing by two. Nilssonneglected matrix elements not diagonal inN. This is a reasonably good assumptionas oscillator states differing by N = 2 arerather far apart, except for states belongingto large N and large deformation (i.e.,


nuclei with large A and large quadrupolemoments). The Nilsson levels are labeled�|N,n3,�]. The symbol � is the projectionof the particle angular momentum j on thethree axes of the body-fixed axis system andis a good quantum number. In the limit ofvery large deformations, the single-particleenergy is given by

E =(

n3 +1

2

)h̄ω3 + (n⊥ + 1)h̄ω⊥

(1.64)where the symbol ω3 is the oscillatorfrequency along the symmetry axis andthe symbol ω⊥ is the oscillator frequencyin the perpendicular directions. Thus,in the limit of very large deformations,one has N = n3 + n⊥. In this same limit,the projection of the orbital angularmomentum on the symmetry axis becomesthe last label,

N = ±n⊥, ±n⊥ − 2, . . . , ±1 or 0(1.65)

This scheme is used to label not onlythe ground states of deformed nuclei butalso the excited state rotational band heads.The level order will now be L, L + 1,L + 2, . . . , as a result of core rotation,so that, when a level does not fit into thissequence, it must belong to a differentsingle-particle band. One now proceedsto assign quantum numbers to the statesby determining the odd-nucleon number;then, because the Nilsson energy levels areplotted against the core deformation, oneassigns the deformation by finding thatplace where a Nilsson level appears withthe measured ground-state spin. The nextNilsson level up should give the excited-band head, and so on. As an example,consider the nucleus 173Lu102, which has a7/2+ ground state. Following the procedureabove, the appropriate level is the 7/2[404],

with excited members 9/2+, 11/2+, 13/2+.Just above the 9/2+ level is a close doubletwith spins 5/2−, 1/2−, which has beenassigned to the 1/2[541] Nilsson level. The3/2− level belonging to this band lies abovethe 9/2+ state. This nonuniform orderof a 1/2 band is an example of Corioliscoupling, which mixes the level order of�= 1/2 bands.

1.7.4Collective Models of Even–Even Nuclei

In Section 1.6.4.2, where the rotational-level structures of even–even nuclei werediscussed, it was pointed out that themoments of inertia play a crucial role todescribe the atomic nucleus as a rigid-bodymomental ellipsoid. Bohr [32] proposeda model of the momental ellipsoid andthus of deformed even–even nuclei andthe cores of odd-A and odd–odd nuclei.Several important assumptions were madein order to extract these moments ofinertia. The first is that the nuclear coreis incompressible, so that the nucleardensity is constant. Second, the flow isassumed to be irrotational, so that a velocitypotential exists and satisfies Laplace’sequation. Finally, it must be assumed thatthe motion is such as to preserve theprincipal axis system. The moments ofinertia for quadrupole deformations thenare

I2k = 4B2β2sin2(

γ − 2π3

k

)(1.66)

Here B2 and β are the quadrupolemass and deformation parameters, whileγ is an asymmetry parameter whoserange is 0 ≤ γ ≤ π/6. There are similarrelations for other deformations such asoctupole (Y3), hexadecapole (Y4), and soon. The Hamiltonian for this system isnow


Hquad =1

2B2(β̇

2 + β2γ̇ )

+ 1B2β

2

3∑k=1

L2ksin2

(γ − 2πk

) + 12

C2β2

(1.67)

If the system is not rigid, then thedeformation and asymmetry parametersare no longer fixed, and the system canexecute either deformation (β) vibrationsor asymmetry (γ ) vibrations, or both. The β-vibrational bands have angular momentum0 – they are akin to molecular ‘‘breathing’’modes – and the γ-vibrational bands haveangular momentum 2. Thus, nuclei willexhibit ground-state bands where nβ = 0and nγ = 0, β-bands nβ = 1, nγ = 0), γ-bands (nβ = 0, nγ = 1), and higher orderbands. Although the β-vibrations arethe simplest, they are not really wellestablished [39], much better establishedare γ-bands of which double excitationswith nγ = 2 are found in nature [40].

In his original paper, Bohr [32] arguedthat the nucleus would stabilize aroundγ = 0, so that the momental ellipsoid wassymmetric (long and thin, cigarlike), whilethe surface itself was spheroidal. Then I3 =0. This leads to difficulties as Eq. (1.50)shows that the energy eigenvalues becomeinfinite unless K = 0. Thus, the ground-state bands of even–even nuclei have K = 0,and the angular momentum sequencebecomes 0, 2, 4, 6, . . . , which is whatis observed. These models automaticallyconform to the A representation. Again, ifthe system is not rigid, against β-vibrations,there will be one or more β-bands above theground-state rotational band that mimicthe ground-state band in spin sequence,but because will be greater than forthe ground state, the levels will be closertogether.

Finally, in this model if γ is notconstant, γ-vibrations can arise with

angular momentum 2. (One might thinkof these as equatorial bulges movingaround the nuclear equator. They are akinto the well-known Jacobi ellipsoids of aself-gravitating, rotating fluid.) These γ-vibrations then produce a γ-band that isdifferent in spin sequence from the β- andground-state bands. This occurs becauseKγ = 2, not 0. The γ-band sequence is2, 3, 4, 5, . . . . There can be several γ-bands with different K values and differentspin sequences. The lowest one has Kγ = 2,while the next will have Kγ = 0 and 4, andso on.

It must be made clear that the Bohrmodel is only applicable for the low-lyingpositive-parity levels. The negative-paritylevels belonging to the B1 representationhave moments of inertia that are consid-erably more complicated as they representan octupole or pearlike shape.

An example of a well-developedrotational system is 154Sm92 whereR4/2 = 3.25. This nucleus has a well-developed ground-state rotational bandwith the spin sequence 0+, 2+, 4+, . . . ,12+ with the first excited state rather lowin energy at 81.99 keV. It has a β-bandwith a 0+ band head 1099.28 keV, followedby the first excited 2+ state at 1177.78 keV(E = 78.52 keV). This is a strikingexample of the increase in the moment ofinertia due to the quantum of β-vibrationalenergy. Furthermore, the K = 2 γ-band,with sequence 2+, 3+, 4+, . . . , starts at1440.05 keV.

Also to be seen is an octupole bandstarting with the 1− level at 921.39 keV,followed by the 2−, 3−, 5−, . . . in theexpected order. Thus, one sees four well-developed rotational bands fitting quitenicely into the model scheme.

Further, the collective behavior mani-fests itself in the quadrupole moments andelectric quadrupole transitions. The axially


symmetric core collective model yields fora state with quantum numbers K and I alaboratory quadrupole moment Q :

Q = 3K2 − I(I + 1)

(I + 1)(2I + 3) Q0 (1.68)

with an intrinsic quadrupole moment Q0given by

Q0 =3√5π

ZR20β(1 + 0.16β) (1.69)

The fact that by definition K has to belower than I (Eq. (1.74)) produces almostalways a change of sign between intrin-sic and laboratory quadrupole moments.Related to the intrinsic quadrupolemoments are the B(E2) values. In betweenstates belonging to a given K band, they aregiven by

B(E2; Ii → If ) =5

16πe2Q20

〈IiK20|If K

〉2(1.70)

When comparing Eq. (1.70) directly withthe data, one often extracts the transitionquadrupole moment Q t as being the Q0value obtained for a given transition. Avery useful relation is obtained for B(E2)ratios between states belonging to bandshaving different K quantum numbers.These relations are called the Alaga rulesand yield

B(E2; Ii → If )B(E2; Ii → Íf )

=⎛⎝

〈IiK2K|If Kf

〉〈IiK2K|Íf Kf

〉⎞⎠2

(1.71)

whereby K = Kf − K as is expected for anangular momentum projection quantumnumber. Using the Alaga rules it is possibleto determine the K values of rotationalbands.

If one does not follow the Bohr assump-tion that the nuc

Documents

Part A Fundamental Nuclear Research · of use for light nuclei is a mirror pair, which is a pair of nuclei with N and Z interchanged. An example of such a pair would be 23Na 12 and