MATR316, Nuclear Physics, 10 cr

Fall 2017, Period II

Pertti O. Tikkanen

Lecture Notes of Tuesday, Nov. 14th and Thursday, Nov. 16th

Department of Physicspertti.tikkanen@helsinki.fi

1

Nuclear Structure and Nuclear Models

Introduction

Liquid drop model

Fermi-gas model

The Nuclear Shell model; experimental evidence

Nuclear Collective Motion, CriteriaMultipole Modes

The Nuclear Shell Model

Single-particle SM

Central potential:Square well

The harmonic oscillator potential

Summary of sp-potentials

The spin-orbit coupling: describing real nuclei

Woods-Saxon potential

Pairing Interaction

1

Nuclear Structure and NuclearModels

Introduction

Introduction

The nucleus as a many-body system from a number of viewpoints:

• Collective motion of nucleons→ collective models to describe smooth systematicvariation, like binding energy (LDM, RLD, vibration/rotation, deformation)

• Single-Particle motion→ Fermi-gas model (the other extreme), nuclear shellmodel

• Single particle in a mean field, L− S coupling, shell structure

• Mean-field potentials: harmonic oscillator (spherical/cartesian), Woods-Saxon,etc. for a single nucleon

• Deformed single-particle potential→ Nilsson model

• Many-particle shell model, pairing correlations

• Predicted quantities: total binding energies, level energies, transition rates,compared with experiments

2

Literature

Books

• Bertulani Ch 5

• K.S. Krane

• Bohr & Mottelson, Nuclear Structure,vols 1 & vol 2

• K. Heyde, Basic Ideas and Conceptsin Nuclear Physics, Ch. 7-10

Journals etc.

• B. A.Brown: Lecture notes 2011

• Advances in Nuclear Physics

• Annual Review of Nuclear ParticleScience

• Reviews of Modern Physics

• MIT, TALENT, youtube, vimeo

3

Liquid drop model

Historically the first model to describe nuclear properties, based on observation thatnuclear forces exhibit saturation properties:

• The binding energy per nucleon BE(A,Z )/Arelatively constant,

• instead of the form A(A− 1)/2 predicted by anon-saturated 2-body nucleon-nucleoninteraction

• Also, the nucleus presents a lowcompressibility,

• and a well defined nuclear surface.

• liquid drop model taken as a fully classicalmodel, cannot be extrapolated too far in theatomic nucleus:

• spikes on BE(A,Z )/A appear at A = 4n,presenting a favoured α-particle like structure

4

• average distance of nucleons much larger than in a classical fluid (2-bodypotential min. at 0.7 fm) because they are fermions→ Fermi liquid. Collisions veryrare. Resulting system is a weakly interacting Fermi gas. No allowed levels toscatter to (Pauli exclusion). Mean free path of the order of nuclear diameter.

• Saturation in binding results is almost constant BE(A,Z )/A ≈ 8 MeV independentof A, Z .

5

Volume, surface and Coulomb contributions

1. The volume term expresses the factthat nuclear force is saturated with themain contribution as

BE(A,Z )V = aVA. (147)

2. The surface effect: nucleons close tosurface feel less binding, thecorrection being ∝ 4πR2,

BE(A,Z )S = −aSA2/3. (148)

3. Coulomb effect, a charge of Ze present within nuclear volume. For ahomogeneously charged liquid drop with sharp radius R and density

ρc =Ze

43πR3

(149)

6

Use a classical argument:

The Coulomb energy needed to add a spherical shell, to the outside of the sphere withradius r , to give an increment dr becomes

U′c =1

4πε0

∫ R

0

43πr3ρc4πr2ρc

rdr (150)

Using the above charge density, the integral becomes

U′c =35

Z 2e2

R1

4πε0(151)

7

But wait, the protons’ spurious self-energy must be subtracted:

Using the proton smeared charge density

ρp =e

43πR3

(152)

and a self-Coulomb energy, for the Zprotons, as

U′′c =35

Ze2

R1

4πε0(153)

the total Coulomb energy correction becomes

Uc = U′c − U′′c =35

Z (Z − 1)e2

R1

4πε0= acZ (Z − 1)A−1/3 (154)

The three contributions together then give

BE(A,Z ) = aV A− aSA2/3 − acZ (Z − 1)A−1/3, (155)

or, per nucleon,

BE(A,Z )/A = aV − aSA−1/3 − acZ (Z − 1)A−4/3. (156)

8

Left panel: Squares show experimental BE/A, curve the contribution of volume,surface and Coulomb terms.

Right panel: Contribution of separate terms, surface term largest at small A, Coulombcorrection (estimated by Z = A/2) largest for heavy nuclei (many protons, A2/3 effect).

9

Non-spherical shape

• Surface and Coulomb energy corrections will change.

• Lowest deformation multipoles via the (θ, ϕ) expansion

R = R0 [1 + α2P2(cos θ) + α4P4(cos θ)]⇒ (157)

• Correction functions for surface energy g(α2, α4) and for Coulomb energyf (α2, α4) appear in the expression for

BE(A,Z )/A = aV − aSg(α2, α4)A−1/3 − ac f (α2, α4)Z (Z − 1)A−4/3. (158)

• Example: ellipsoidal deformation with major axis

a = R(1 + ε),

(ε =

√1− b2/a2

), (159)

and the minor axisb = R(1 + ε)−1/2, (160)

with volume V = 43πab2 ' 4

3πR3.

10

• In terms of parameter of deformation ε,

• surface and Coulomb energy terms become

ES = aSA2/3(

1 +25ε2)

EC = acZ (Z − 1)A−1/3(

1−15ε2). (161)

• The total energy change, due to deformation

∆E = ∆ES + ∆EC = ε2[

25

A2/3aS −15

aCZ (Z − 1)A−1/3]. (162)

• If ∆E > 0, the spherical shape is stable if Z 2/A < 49, as can be “proved” bysimplifying Z (Z − 1) ≈ Z 2 and using the best-fit values (Wapstra 1971), aS=17.2MeV and aC=0.70 MeV.

11

Deformation, a doorway to fission?

Dashed line: potential energy versusnuclear distance between two nuclei. Thepoint at ε = 0 corresponds to the sphericalnucleus. For large separation, callingr = R1 + R2 (with R1 and R2 the radii ofboth fragments), the energy variesaccording to the Coulomb energy

UC =Z1Z2e2

4πε0r(163)

Deformation complicates the preciseevaluation of the full total potential energyand requires complicated fissioncalculations.

12

Symmetry energy, pairing and shell corrections

The specific nucleon (Dirac-Fermi statistics) properties of the nuclear interior modify anumber of results. Manifestations of the Pauli principle governing the occupation of thesingle-particle orbitals in the nuclear, average field and the nucleonic residualinteractions that try to pair-off identical nucleons to 0+ coupled pairs.

• Symmetry energy :Binding energy largest when Z = N = A/2, and any repartitionN = (A/2) + ν,Z = (A/2)− ν involves lifting nucleons from occupied to emptyorbitals.

• Energy loss due to repartition (∆ is the inter-orbital energy difference):

∆Ebinding = ν(

∆×ν

2

)=

18

(N − Z )2∆

with ν = (N − Z )/2.

• The energy spacing between single-particle states scales as A−1

13

• The final result, expressing the loss of symmetry energy due to the Pauli effectwhich blocks the occupation of those levels that already contain two identicalnucleons, becomes

BE(A,Z ) = aV A− aSA2/3 − aCZ (Z − 1)A−1/3 − aA(A− 2Z )A−1. (164)

• Better understanding from Fermi-gas model (later derivation of the coefficient aA)

• Pairing energy contribution:Nucleons preferentially form pairs coupled to J = 0, resulting in a pairing energycorrection δ = aPA1/2:

∆Epair =

+δ (even-even)

0 (odd-even)

−δ (odd-odd)

(165)

14

After all pieces are put together, we have finally: semi-empirical mass equation(Bethe-Weizsäcker formula)

BE(A, Z ) = aV A− aSA2/3 − aCZ (Z − 1)A−1/3 − aA(A− 2Z )2A−1

+δ

0−δ

(166)

The parameters fitted to experimental values by Wapstra (1971) and more recently byBertsh et al :

Param. Values from fit by[MeV] Wapstra Bertsh et al

aV 15.85 15.74063aS 18.34 17.61628ac 0.71 0.71544aA 23.21 23.42742ap 12 12.59898

Note that Bertulani defines theantisymmetry term differently which impliesthat aA(“Bertulani”) = 4×aA(“ours”)

• but you can make your own parameter fit to the most recent atomic masses [G.Audi et al, Chinese Physics 36, 1603, 2012]!

• The smooth cross behaviour is reproduced nicely, but large deviations occur nearclosed shells:

15

Figure 28: Nuclear (liquid drop) masses and the deviations with respect to the nuclear data andthis, as a function of proton and neutron number. The shell closure effects are shown mostdramatically (Myers 1966).

The relation between the nuclear binding energy and nuclear mass is

BE(A,Z )/c2 = MN(A,Z )− ZMp − (A− Z )Mn (167)

16

In case the difference between atomic and nuclear mass is important, use the relation

MNucl.(A,Z ) = MAtom(A,Z )− Z ×mel. + Bel(Z )/c2, (168)

where the total binding energy of the atomic electrons can be approximated by

Bel(Z ) = 14.4381Z 2.39 + 1.55468× 10−6Z 5.35[eV]1. (169)

The most precisely known masses (2012) are listed in the table next page forreference. Who knows, when you need them!

In the literature, the atomic mass is often given as mass excess (either in mass units uor in eV90)

∆ = M − A

The reason is that the mass M(A,Z ) rounded to nearest integer equals A.

1For high-accuracy work, it’s important to specify what definition of the unit volt, V, is used when converting massunits [u] to energy units [eV]. The latest Atomic Mass Evaluation (2012), recommends the maintained volt V90, thatis traceable to primary standards. 1 uc2 = 931 494.0023± 0.0007 keV90

17

18

0 20 40 60 80 100 120 140 1600

20

40

60

80

100 Z = N

Z = A/21+0.0077A2/3

Neutron number N

Pro

ton

num

berZ

Figure 29: Nuclide chart of all the nuclei with experimentally known atomic masses (2012). Theblue curve shows the valley of stability as predicted by the LDM. The horizontal lines show themagic proton numbers at Z = 2, 8, 20, 28, 50, 82 ( c©P.O.T. 14.11.2017).

19

Nuclear stability: the mass surface and line of stability

Nuclear mass equation from the Bethe-Weizsäcker formula:

Mnucl.(AZ XN )c2 =ZMpc2 + (A− Z )Mnc2

−aV A + aSA2/3+aCZ (Z − 1)A−1/3 + aA(A− 2Z )2A−1

−δ+0+δ

(170)

representing a quadratic equation in the proton number Z , i.e.,

Mnucl.(AZ XN )c2 = xA + yZ + zZ 2

+δ

+0−δ.

(171)

The most stable nucleus for each Z is then approximately (show!)

Zmin(A) =A/2

1 + 14 aC/aAA2/3

=A/2

1 + 0.0077A2/3. (172)

20

Several interesting results can be derived

• Parabolic behaviour of nuclear masses for a given A: (i) Members of an odd-Achain can decay—towards a single stable nucleus—either by β− or β+/EC, butnot both. (ii) For even-A nuclei, both even-even and odd-odd nuclei can occur and(because of the δ value) two parabolae are implied by the mass equation. In manycases, more than one stable element results (double-beta decay!) There are evencases with three ‘stable’ elements which depend on the specific curvature of theparabola and the precise location of the integer Z values:

21

42Mo 43Tc 44Ru 45Rb 46Pd0

2

4

6

Z

LDM

mas

s(M

eV)

Figure 30: Mass parabola for A = 101

22

• Constraints on the range of possible A,Z values and stability against radioactivedecay processes can be determined. Spontaneous α-decay (Sα = 0) follows fromthe equation

BE(AZ XN )−

[BE(A−4

Z−2YN−2) + BE(42He2)

]= 0. (173)

• The condition Sn = 0 (Sp = 0), indicates the borderline where a neutron (proton)is no longer bound in the nucleus. It’s called the neutron (proton) drip line.

• The energy released in nuclear fission, in the simple case of symmetric fission ofthe element (A,Z ) into two nuclei (A/2,Z/2) is

Efission = Mnucl.(AZ XN )c2 − 2Mnucl.(

A/2Z/2YN/2)c2

= [−4.78A2/3 + 0.26 Z 2A−1/3]c2 (174)

and becomes positive near A ' 75, reaching a value of about 185 MeV for 236U.2

2For these fission products, neutron-rich nuclei are obtained which will decay by the emission ofn → p + e− + νe . So, a fission process is a good generator of νe , antineutrinos of the electron type; the fusionprocess on the other hand will mainly give rise to νe electron neutrino beta-decay transitions.

23

Two-neutron separation energies

• Nuclear masses and a derived quantity, S2n, may reveal the presence of extracorrelations on top of a smooth liquid-drop behaviour:

S2n(A,Z ) = BE(A,Z )− BE(A− 2,Z ) (175)

• Can be written as

S2n ≈ 2(aV − aA)−43

aSA−1/3 +23

aCZ (Z − 1)A−4/3 + 8aAZ 2

A(A− 2)

where the surface and Coulomb terms are only approximated expressions.

• Insert Z0, that maximizes the binding energy for each given A (this is the definitionfor the valley of stability),

Z0 =A/2

1 + 0.0077A2/3, (176)

to obtain, for large values of A,

S2n = 2(aV − aA)−43

aSA−1/3 +

(8aA +

23

aCA2/3)

14 + 0.06A2/3

24

25

• Behaviour of S2n along the valley of stability for even-even nuclei, together with theexperimental data.

• The experimental data correspond to a range of Z between Z − 1 and Z − 2.

• Overall decrease and the specific mass dependence are well contained within theLDM.

• How well the experimental two-neutron separation energy, through a chain ofisotopes, is reproduced in LDM.

• Besides the sudden variations near mass number A = 90 (presence of shellclosure at N = 50) and near mass number A = 140 (presence of the shell closureat N = 82), the specific mass dependence for a series of isotopes comes closer tospecific sets of straight lines.

• LDM is able to describe the almost linear behaviour of S2n for series of isotopes;expand the various terms (volume, surface, etc.) in terms of X = A− A0 andε = X/A0 near a fixed A to see their separate contributions. It turns out that theslope is determined mainly by the asymmetry term.

26

Fermi gas model

• Approximate distribution for kinetic energy of nucleons confined to the nuclearvolume V (= 4

3πR3) with density ρ = A/V .

• Strong spatial confinement results in widely spaced energy levels with only thelowest being occupied, except for very high energies.

• Appropriate model, degenerate Fermi gas, up to excitation of ∼10 MeV

• Under conditions present in nuclear matter but approximates also heavy nuclei

• Assume lowest energy state allowed by Pauli exclusion principle.

• Start with symmetric nuclear matter, Z = N

• At low excitation energy (temperature T = 0 limit), levels pairwise filled up toFermi level, beyond that levels fully unoccupied: degenerate Fermi gas.

• At high temperature, occupancy partially redistributed,

• At very high temperature, all levels partially filled.

27

Neutrons (left) and protons (right) in nuclear squarewell. Degenerate T = 0 Fermi gas.

Level occupancy at T = 0(shaded), higher T > 0 and veryhigh T 0 temperatures.

Density of states for free Fermi gas confined to volume V : 3

n =2V

(2π~)3

∫ pF

0d3p =

V p3F

3π2~3, (177)

3See BAB for an alternative derivation with a cube of side L.

28

• resulting in the Fermi momentum

pF = ~(

3π2 nV

)1/3= h

(3

8πnV

)1/3, (178)

where n/V is the density contributing to a quantum mechanical pressure.

• Apply to atomic nucleus to estimate the depth of the potential well, U0:

pF,n =~r0

(9πN4A

)1/3, pF,p =

~r0

(9πZ4A

)1/3, (179)

and for self-conjugate nuclei, N = Z = A/2:

pF,n = pF,p =~r0

(9π8

)1/3≈

300r0

MeV/c. (180)

• The corresponding Fermi (kinetic) energy with r0 = 1.2 fm becomes

EF =p2

F,n

2m=

p2F,n

2m' 33 MeV. (181)

29

• This energy corresponds to the kinetic energy of the highest occupied orbit(smallest binding energy).

• Given the average binding energy B = 8 MeV, we can make a good estimate ofthe nuclear well depth of U0 ' 41 MeV.

• In more realistic calculations, values of this order are obtained.

• The average kinetic energy per nucleon (for non-relativistic motion) is

〈E〉 =

∫ pF0 Ekind3p∫ pF

0 d3p=

35

p2F

2m=

35

EF ' 20 MeV (182)

• In the degenerate Fermi-gas model, even at zero excitation energy, a largeamount of ‘zero-point’ energy is present and, owing to the Pauli principle, aquantum-mechanical ‘pressure’ results.

• Given a nucleus with Z protons and N neutrons, the total, average kinetic energybecomes

〈E(A,Z )〉 = Z 〈E〉p + N 〈E〉n =3

10m

(Zp2

F,p + Np2F,n

)=

310m

~2

r20

(9π4

)2/3 (N5/3 + Z 5/3)A2/3

30

Fermi gas: Nuclear symmetry potential

Expand the formula of the average energy at N = Z = A/2 to derive an expression ofthe symmetry energy and a value for the constant aA:

• Set Z − N = ε to have Z = A/2(1 + ε/A); N = A/2(1− ε/A) and with ε/A 1obtain (using binomial expansion):

〈E(A,Z )〉 =3

10m~2

r20

(9π4

)2/3A(

12

)5/3 ((1− ε/A)5/3 + (1 + ε/A)5/3)A2/3

=3

10m~2

r20

(9π8

)2/3(

A +59

(N − Z )2

A+ . . .

)

• The first term is proportional to A↔ volume energy.

• The next term has exactly the form of the symmetry energy A,Z -dependent termin the Bethe-Weizsäcker mass equation.

• Inserting the values of the various constants m, ~, r0, π, we obtain the result

〈E(A,Z )〉 = 〈E(A,Z = A/2)〉+ ∆Esymm

∆Esymm ' 11 MeV(N − Z )2A−1. (183)

31

The numerical result is about one-half the value obtained earlier (or fitted). Thedifference arises from the fact that the well depth U0 also depends on the neutronexcess N − Z .

Higher-order terms with a dependence (N − Z )4/A3, . . . naturally derive from themore general 〈E(A,Z )〉 value.

• The Fermi-gas model can be applied successfully to various different problems:

• Stability of degenerate electron gas: white dwarfs stars

• Stability of degenerate neutron gas: description of neutron stars

• Balancing the gravitational energy with the quantum-mechanical Fermi-gas energyto obtain equation of state for the neutron star. (Bertulani 12.11, Heyde 8.3)

32

The Nuclear Shell Model

Single-particle motion cannot be completely replaced by a collective approach wherethe dynamics is contained in collective, small amplitude vibrations and/or rotations ofthe nucleus as a whole.

A simple independent-particle approach required by the Fermi-gas model does notcontain enough detailed features of the nucleon-nucleon forces, active in the nucleus.

Experimental evidence for nuclear shell structure is seen as deviations from otherwiserather smooth behaviour of various quantities as a function of N and Z :

• Neutron and proton separation energies Sn and Sp and two-neutron separationenergies S2n (you can make your own plots!),

• Extra stability at neutron number 8, 20, 28, 50, 82, and 126, or at proton number8, 20, 28, 50, and 82. (no Z = 126 element known!)

• Excitation energy of the first 2+ state in even-even nuclei,

• Reduced transition probabilities of electric quadrupole transitions,B(E2; 0+ → 2+)↑, in even-even nuclei.

33

The left panel shows the excitation energies of the first 2+ states in even-even nuclei,the right panel the reduced transition probabilities B(E2; 0+ → 2+).4

4The vertical lines show the location of magic proton numbers at Z = 8, 20, 28, 50, and 82.

34

As a function of neutron number: (a) Eex(2+), (b)B(E2; 0+ → 2+)

(c) Deformation parameter β/βsp (d) EWSR(II) = Energy WeightedSum rule

35

As a function of proton number: (a) Eex(2+), (b) B(E2; 0+ → 2+)(c) Deformation parameter β/βsp (d) EWSR(II) = Energy WeightedSum rule

36

Nuclear Collective Motion

Before jumping into the details of the nuclear shell model, a few words on collectivemotion of nucleons and on the criteria to collectivity.

Nuclear collective motion is fairly distinctive in physics. There is some similarity withthe phenomena of oscillations of a classical liquid drop and with electron gasoscillations, but there are also essential differences. Unlike the liquid drop, the nucleusis a quantum system, and its collective parameters differ considerably fromhydrodynamic values. The most obvious difference with the plasma is that the nucleusis a relatively small system, so that its collective modes are characterized by angularrather than linear momenta, and also by parity. A further difference arises from theexistence of two types of nuclear particle. These may oscillate in phase or out ofphase. In the language of isospin, we say that the two types of oscillation are isoscalar(T = 0) and isovector (T = 1).

37

Nuclear collective modes are thus classified by L = (T , L, π) (isospin, angularmomentum, parity). We refer to L as the multipole order. Experimentally the onlywell-established modes are those with parity π = (−1)L, i.e., in electromagneticterminology, “electric multipoles” such as E3 signifying L = 3, π = −. For such modesit is convenient to define “multipole operators”:

QTLM =

∑i rL

i YLM (Ωi ) forT = 0∑i τ3i rL

i YLM (Ωi ) forT = 1(184)

Note that Q0LM is the “mass multipole operator" and the “charge multipole operator”(which refers to protons only) is

QELM =12

(Q0LM − Q1LM )

38

Criteria for Collective Motion

In surveying the data on nuclei in a given mass region, there are two basic criteria usedfor deciding whether collective phenomena occur:

1. Regularities in spectra characteristic of collective modes. The simplest are theuniform spacing of the vibrator and the J(J + 1)-type spectrum of the rotator. Aless decisive criterion is the systematic occurrence of a given type of level indifferent nuclei.

2. Strong electromagnetic matrix elements, either diagonal (e.g., E2 moments) ornon-diagonal (e.g., EL transitions). By “strong” here, we mean at least severaltimes larger than a typical value for a single proton. The matrix element for aproton transition between a particle state of orbital angular momentum L and oneof L = 0 is:

〈LM|Q0LM |00〉 =1√

4π

∫ ∞0

[uL(r)rLu0(r)

]r2dr

where the u’s are radial wavefunctions. For a rough estimate, we may replace theintegral by the Lth moment of the density distribution ρ(r) of the nucleusconcerned: ⟨

rL⟩

=

∫rLρ(r)dr/

∫ρ(r)dr .

39

Sum Rules

An alternative version of criterion (2) is that the transition exhausts at least a fairfraction (say & 5 per cent) of a sum rule. There are two sum rules that are relevant; inobvious notations, these are ∑

n| 〈n|QTL0|0〉 |2 =

⟨0|(QTL0)2|0

⟩≡ STL

NEW

∑n

(En − E0)| 〈n|QTL0|0〉 |2 =12〈0| [QTL0, [H,QTL0]] |0〉 ≡ STL

EW

We call these the non-energy-weighted (NEW) and energy-weighted (EW) sum rules.The only sum that can be evaluated exactly (i.e., without reference to a model) is theEWS for T = 0:

S0LEW =

~2A8πM

L(2L + 1)⟨

r2L−2⟩

The only assumption is that H contains no explicitly velocity-dependent forces(exchange forces are permitted). Strictly, this value is obtained when the ground statehas spin 0, but any correction for non-zero spin is small. It is fortunate that mostobserved modes appear to be (or are assumed to be) T = 0.

40

• Assuming a uniformly charged spherical nucleus with a sharp radius R0 = r0A1/3,the energy-weighted sum for the E2 multipole

SI ≡ S02EW =

∑n

(En − E0)B(E2)↑= 30e2 ~2

8πmAR2

0 (185)

where m is the nucleon mass.

• The isoscalar part of the full sum is given by

SII ≡ S(I)(ZA

)2. (186)

• The percentage fraction of E × B(E2; 0+ → 2+)↑ of SII, for only the first 2+ stateonly in even-even nuclei is shown in the above figures.

• The deformation parameter (there are several definitions in the literature!) shownin the figures is related to the B(E2)↑ by

β =4π3

ZR20

[B(E2)↑ /e2

]1/2, (187)

with the corresponding single-particle (Weisskopf) estimate βsp = 1.59/Z .

41

The Nuclear Shell Model

The Nuclear Shell Model

Literature

• Bertulani, Ch 5.4,

• BAB: Ch 8, Ch 9 The harmonic oscillator, Ch 10.1 The Woods-Saxon potential,Ch 11, The general many-body problem for fermions

• As an introduction to the subject, read BAB: Ch 8 Overview of the nuclear shellmodel

• A. deShalit and I. Talmi

• A. Bohr and B. Mottelson, Nuclear Structure, vols. I & 2

42

Single-particle shell model

• The 1st approximation to the hard many-nucleon problem.

• Nucleons move in a central average potential, but otherwise independently⇔

• Single-particle shell model or extreme shell model:

• Central potential U(r) plus spin-orbit interaction of the form f (r)~l ·~s

• Choice of central potential: 1) square-well, 2) simple harmonic oscillator (SHO), 3)Woods-Saxon, depending on which one is the most convenient (analytical,symmetry, etc. properties)

• Woods-Saxon approximates the nuclear density distribution. Nuclear force ofshort range and potential should follow the density.

• Non-central potential: Nilsson model, assumes the potential to be deformed fromthe beginning

43

Central potential: Square well

• Separate the 3-d Schrödinger equation in spherical coordinates

• Spherical harmonics solve the angular part, radial equation depends on thepotential

• For the square-well potential model

U(r) =

−U0, r ≤ R∞, r > R

(188)

the corresponding Schrödinger one-particle equation becomes[−

~2

2m∆ + U(r)

]Φ(~r) = EΦ(~r), (189)

with

∆ =d2

dr2+

2r

ddr−

l2

~2

1r2,

l2 denotes the angular momentum operator, with eigenfunctions Yml (θ, ϕ) and

corresponding eigenvalues l(l + 1)~2. Separating the wavefunction as

Φ(~r) = Rnl (r)Yml (θ, ϕ) (190)

44

• the corresponding radial equation becomes

d2R(r)

dr2+

2r

dR(r)

dr+

[2m~2

(E − U(r))−l(l + 1)

r2

]R(r) = 0. (191)

The solutions become the solutions of the Bessel differential equation (for theregular ones which we concentrate on)

Rnl (r) =A√

krJl+1/2(kr). (192)

These solutions are regular at the origin (the spherical Bessel functions of the firstkind) with

jl (kr) =

√π

2krJl+1/2(kr). (193)

• The energy eigenvalues

Rnl (R) = 0 or Jl+1/2(knl R) = 0. (194)

For these knl R coincides with the zeros of the spherical Bessel function,

Enl =~2k2

nl2m

=~2X 2

nl2mR2

(195)

with Xnl = knl R.

45

The harmonic oscillator potential

• The radial problem for a spherical, harmonic oscillator potential can be solvedeither

• (i) Using a Cartesian basis

U(r) = −U0 +12

mω2(x2 + y2 + z2). (196)

The three (x , y , z) specific one-dimensional oscillator eigenvalue equationsbecome [

d2

dx2+

2m~2

(E1 +

U0

3−

12

mω2x2)]

φ1(x) = 0 (197)

withE = E1 + E2 + E3. (198)

The three eigenvalues are then

Ei = ~ω(ni + 1/2)− U0/3, i = 1, 2, 3 (199)

orE = ~ω(N + 3/2)− U0,

(N = n1 + n2 + n3 with n1, n2, n3 three positive integer numbers 0,1,. . . ).

46

• The wavefunctions for the one-dimensional oscillator are the Hermite polynomials,characterized by the radial quantum number ni , so

φ1(x) = N1 exp(−

mω2~

x2)

Hn1 (νx),

(ν =

√mω~

)(200)

andΦ(x , y , z) = φ1(x)φ2(y)φ3(z), (201)

N1,N2,N3 are normalization coefficients.

• (ii) Separate the radial variable r from the angular (θ, ϕ) ones. In this case, theradial equation reduces to

d2R(r)

dr2+

2r

dR(r)

dr+

(2mE~2

+2m~2

U0 −m2ω2

~2r2 −

l(l + 1)

r2

)R(r) = 0. (202)

47

• The normalized radial solutions are the Laguerre polynomials

Rnl (r) = Nnl (νr)l exp

(−ν2r2

2

)Ll+1/2

n−1 (ν2r2), (203)

and ν =√

mω/~, describing the oscillator frequency. The Laguerre polynomials(Abramowitz and Stegun 1964) are given by the series

Ll+1/2n−1 (x) =

n−1∑k=0

alk (−1)k xk . (204)

• In BAB there are given expressions that show all explicitly. Also see Bertulani.

• (2(n − 1) + l) can be identified with the major oscillator quantum number N weobtained in the Cartesian description. The total wavefunction then corresponds toenergies

EN = (2(n − 1) + l)~ω +32~ω − U0. (205)

• For a specific level (n, l) there exists degeneracy relative to the energycharacterized by quantum number N, i.e. we have to construct all possible (n, l)values such that

2(n − 1) + l = N. (206)

48

Table 7: 3-d simple harmonic oscillator energies.

N EN (~ω) (n, l)∑

nl 2(2l + l) Total

0 3/2 1s 2 21 5/2 1p 6 82 7/2 2s, 1d 12 203 9/2 2p, 1f 20 404 11/2 3s, 2d, 1g 30 705 13/2 3p, 2f, 1h 42 1126 15/2 4s, 3d, 2g, 1i 56 168

An alternative parametrization uses HO length parameter b = 1/ν =√

~/mω. Theradial wavefunctions are then given as5

Rα(r) =

√2l−nr +2(2l + 2nr + 1)!!√π(nr )!b2l+3[(2l + 1)!!]2

r l+1e−r2/2b2

×nr∑

k=0

(−1)k 2k (nr )!(2l + 1)!!

k!(nr − k)!(2l + 2k + 1)!!(r/b)2k (207)

5Note that in this parametrization, nr = n − 1.

49

Figure 31: Illustration of the harmonic oscillator radial wavefunctions rRnl (r). The upper partshows, for given l , the variation with radial quantum number n (n = 1: no nodes in the interval(0,∞)). The lower part shows the variation with orbital angular momentum l for the n = 1 value.

50

• Using ~c = 197 MeVfm and mc2 = 938 MeV, we have b2 = 41.4 fm2/(~ω)

• Virial theorem applied to UHO0 (r) = 1

2 mω2r2,

〈φα|T |φα〉 = 〈φα|U|φα〉 =12

mω2⟨φα|r2|φα

⟩together with

〈φα|U|φα〉+ 〈φα|T |φα〉 =

(N +

32

)~ω

gives the diagonal matrix elements of r2:⟨φα|r2|φα

⟩=⟨

N|r2|N⟩

= (N +32

)b2 (208)

and〈φα|T |φα〉 = (N +

32

)~ω2

(209)

• Maximum number of spin-1/2 fermions in state with given N isDN = (N + 1)(N + 2), and number of states that can be occupied up to Nmax is

Nmax∑N=0

DN =13

(Nmax + 1)(Nmax + 2)(Nmax + 3) (210)

51

• The harmonic oscillator parameter ω reproduce the observed mean-squarecharge radius,

• related to the mean-square radius for protons in the nucleus whose harmonicoscillator levels are filled up to Nmax⟨

r2⟩

p=

∑NmaxN=0 DN

⟨N|r2|N

⟩∑NmaxN=0 DN

=34

(Nmax + 2)b2. (211)

• Compare to the observed proton mean-square radius as obtained from measuredcharge mean-square radius by⟨

r2⟩

p=⟨

r2⟩

ch−⟨

r2⟩

o(212)

where⟨r2⟩

o = 0.77 fm2 is the mean-square charge radius of the proton.

• For example, for the nucleus 16O, where the eight protons fill the N = 0 and N = 1major shells, Nmax = 1 and

⟨r2⟩

p (16O) = (9/4)b2.

• The experimental root-mean-square (rms) charge radius for 16O is√⟨r2⟩

ch = 2.71 fm, and hence⟨r2⟩

p = 6.57 fm2, b2 = 2.92 fm2 and ~ω = 14.2MeV for the point protons.

• Since there are equal numbers of neutrons and protons in 16O and since theyshould have about the same rms radii, it is a good approximation to use the sameoscillator parameter for the neutrons as was obtained for protons.

52

CHAPTER 8. OVERVIEW OF THE NUCLEAR SHELL MODEL 101

singl

e-pa

rticle

ene

rgy

(MeV

)

-50

-40

-30

-20

-10

0

10

[ 2] 2 N=0

[ 6] 8 N=1

[12] 20 N=2

[20] 40 N=3

[30] 70 N=4

[42] 112 N=5

[56] 168 N=6

[ 2] 2 0s

[ 6] 8 0p

[10] 18 0d

[ 2] 20 1s

[14] 34 0f

[ 6] 40 1p

[18] 58 0g

[10] 68 1d[ 2] 70 2s

[22] 92 0h

[14] 106 1f[ 6] 112 2p

[26] 138 0i

[18] 156 1g[10] 166 2d[ 2] 168 3s

[ 2] 2 0s1

[ 4] 6 0p3[ 2] 8 0p1

[ 6] 14 0d5[ 4] 18 0d3[ 2] 20 1s1

[ 8] 28 0f7

[ 6] 34 0f5[ 4] 38 1p3[ 2] 40 1p1[10] 50 0g9

[ 8] 58 0g7[ 6] 64 1d5[ 4] 68 1d3[ 2] 70 2s1[12] 82 0h11

[ 8] 90 1f7[10] 100 0h9[14] 114 0i13[ 4] 118 2p3[ 6] 124 1f5[ 2] 126 2p1

[10] 136 1g9[12] 148 0i11[ 6] 154 2d5[ 2] 156 3s1[ 8] 164 1g7[16] 180 0j15[ 4] 184 2d3

HarmonicOscillatorPotential

Woods-SaxonPotential

Woods-SaxonPlus Spin-OrbitPotential

Neutron Single-Particle Energies

2

8

20

28

50

82

126

Figure 8.2: Neutron single-particle states in 208Pb with three potential models, harmonic oscillator(left), Woods-Saxon without spin-orbit (middle) and Woods-Saxon with spin orbit (right). Thenumbers in square brackets are the maximum number of neutrons in that each level can contain,the following number is a running sum of the total. In addition the harmonic oscillator is labeled bythe major quantum number N = 2n + ℓ, the Woods Saxon is labeled by n, ℓ and the Woods-Saxonwith spin-orbit is labeled by n, ℓ, 2j.

Figure 32: Neutron single-particlestates in 208Pb with three potentialmodels. The numbers in squarebrackets are the maximum number ofneutrons in that each level can contain,the following number is a running sumof the total. In addition the harmonicoscillator is labeled by the majorquantum number N = 2n + l , theWoods Saxon is labeled by n, l and theWoods-Saxon with spin-orbit is labeledby n, l, 2j .

53

• For heavy nuclei, use the large Nmax approximation for the sum,Z ≈ (Nmax + 2)3/3, to obtain ⟨

r2⟩

p≈

34

(3Z )1/3b2 (213)

• For example, for 208Pb (Z = 82) the experimental rms charge radius is 5.50 fm,giving

⟨r2⟩

p = 29.4 fm2, b2 = 6.27 fm2 and ~ωp = 6.6 MeV for the point protons.

• Experimentally, the proton and neutron rms radii in 208Pb are about equal⟨r2⟩

p ≈⟨r2⟩

n.

• The oscillator parameter for neutrons in 208Pb, obtained by using an rms radius of5.5 fm and N in place of Z , is ~ωn = 7.6 MeV.

• Along the valley of stability the rms proton radii are approximately given by1.08A1/3. Rewriting Eq. (213) in terms of A ≈ 2.5Z , one obtains for heavy nuclei~ωp ≈ 39A−1/3 MeV.

• A better smooth approximation for the proton oscillator parameter in both light andheavy nuclei is ~ωp ≈ 45A−1/3 − 25A−2/3 MeV

54

Summary

• In both the square-well potential and in the harmonic oscillator potential 2, 8, 20are the common shell-closures and these numbers correspond to the particularlystable nuclei 4He, 16O, 48Ca and agree with the data. The harmonic oscillatorpotential has the larger degeneracy of the energy eigenvalues and thecorresponding wavefunctions (the degeneracy on 2(n − 1) + l is split for thesquare-well potential).

• Use of a more realistic potential does not help to solve the discrepancy!

• Spin-orbit interaction needed:It is clear that something fundamental is missing still. We shall pay attention to theorigin of the spin-orbit force, coupling the orbital and intrinsic spin, that gives riseto a solution with the correct shell-closure numbers.

55

The spin-orbit coupling: describing real nuclei

• It was pointed out (independently by Mayer (1949,1950) and Haxel, Jensen, andSuess (1949,1950)) that a contribution to the average field, felt by each individualnucleon, should contain a spin-orbit term.

• The corrected potential then becomes

U(r) = −U0 +12

mω2r2 −2~2αl · s. (214)

The spin-orbit term, with the scalar product of the orbital angular momentumoperator l and the intrinsic spin operator s, can be rewritten using the totalnucleon angular momentum j , as

j = l + s, (215)

andl · s =

12

(j2 − l2 − s2

)(216)

• The operators l2, s2, and j2 form a set of commuting angular momentumoperators, so they have a set of common wavefunctions.

56

• These wavefunctions, characterized by the quantum numbers corresponding tothe four commuting operators l2, s2, j2, and jz result from angular momentumcoupling the orbital (Ym

l (θ, ϕ)) and spin (χms1/2(σ)) wavefunctions. We denote

these as

φ(~r , n(l 12 )jm) ≡ Rnl (r)

∑ml ,ms

⟨lml ,

12 ms|jm

⟩Yml

l (θ, ϕ)χms1/2(σ), (217)

(with m = ml + ms). These wavefunctions correspond to good values of (l 12 )jm

and we derive the eigenvalue equation

l · s φ(~r , n(l 12 )jm) =

~2

2

[j(j + 1)− l(l + 1)−

34

]φ(~r , n(l 1

2 )jm) (218)

• Since we have the two orientations

j = l ±12

(219)

we obtain

l · s φ(~r , n(l 12 )jm) =

~2

2

l j = l + 1/2

−(l + 1) j = l − 1/2

φ(~r , n(l 1

2 )jm). (220)

57

• The effective potential then becomes slightly different for the two orientations i.e.

U(r) = −U0 +12

mω2r2 + α

−l

+(l + 1)

, j =

l + 1

2l − 1

2(221)

• The potential is more attractive for the parallel j = l + 1/2 orientation, relative tothe anti-parallel situation. The corresponding energy eigenvalues now become

εn(l 12 )j = ~ω

[2(n − 1) + l +

32

]− U0 + α

−l

l + 1

, j =

l + 1

2l − 1

2(222)

and the degeneracy in the j = l ± 12 coupling is broken. The final single-particle

spectrum now becomes as given in figure

58

Figure 33: Single-particle spectrum up to N = 6. The various contributions to the full orbital andspin-orbit splitting are presented. Partial and accumulated nucleon numbers are drawn at theextreme right. (Taken from Mayer and Jensen, Elementary Theory of Nuclear Shell Structure,c©1955 John Wiley & Sons. Reprinted by permission.) 59

Woods-Saxon potential

• A convenient phenomenological choice for the one-body potential, for modellingproperties of bound-state and continuum single-particle wavefunctions.

• cannot be used for the total binding energy since it is not based upon a specifictwo-body interaction

• Parameters from best fit of nuclear single-particle energies and nuclear radii (O =orbital, SO = spin-orbit):

U(r) = UO(r) + USO(r )l · s + UC(r), where

UO(r) = UOfO(r) =UO

1 + exp[(r − RO)/aO],

spin-independent centralpotential of Fermi shape

USO(r) = USO1r

ddr

fSO(r), the spin-orbit potential with

fSO(r) =1

1 + exp[(r − RSO)/aSO],

and UC(r) is the Coulomb potential for protons:

60

UC(r) =Ze2

4πε0

1r for r ≥ RC

1RC

[32 −

r2

2R2C

]for r ≤ RC

• The radial parameters RO,RSO, and RC are usually expressed as:

Ri = ri A1/3.

• Average proton-neutron potential is stronger than the average neutron-neutron (orproton-proton) potential:

UO,p = U0 +N − Z

AU1 for protons

UO,n = U0 −N − Z

AU1 for neutrons

• Typical values: U0 = −53 MeV, U1 = −30 MeV and USO = 22 MeV, for thestrengths, and rO = rSO = 1.25 fm and aO = aSO = 0.65 fm for the geometry. Forthe Coulomb term the radius is a little smaller with rC = 1.20 fm.6

6One can find in the literature many other sets of parameters which are better for specific nuclei or mass regions.

61

Figure 34: Woods-Saxon potential for a 1d5/2 proton or neutron with a 16O (left) or 208Pb (right)core, together with the total potential and the contributing potentials as described in the text.

62

Woods-Saxon neutronsingle-particle energiesfor nuclei near the valleyof stability as a function ofA. With increasing A,more single-particlestates become bound butthe energy of the mostloosely bound filled orbitis always around 8 MeV(Bohr & Mottelson 1969).

63

Pairing Interaction

• The extreme single-particle shell model applies to some rather simple situations,where a single un-paired nucleon (or hole) is outside a closed core. A morerealistic description is needed.• Many properties are influenced by pairing correlations:

• Gap in the excitation spectra of even-A nuclei Odd-even mass differences• Rotational moments of inertia and particle alignments, etc. . .

64