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Chapter 6: Nuclear Structure. Abby Bickley University of Colorado NCSS ‘99. Additional References: Choppin (CLR), Radiochemistry and Nuclear Chemistry, 2nd Edition, Chapter 11 Friedlander (FKMM), Nuclear and Radiochemistry, 3rd Edition, Chapter 10. The Nucleus.

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Chapter 6 nuclear structure

Chapter 6: Nuclear Structure

Abby Bickley

University of Colorado

NCSS ‘99

Additional References:

Choppin (CLR), Radiochemistry and Nuclear Chemistry, 2nd Edition, Chapter 11

Friedlander (FKMM), Nuclear and Radiochemistry, 3rd Edition, Chapter 10


The nucleus

The Nucleus

  • As chemist’s what do we already know about the nucleus of an atom?

    • Composed of protons and neutrons

    • Carries an electric charge equivalent to the number of protons & atomic number of the element

    • Protons and neutrons within nucleus held together by the strong force

  • Any model of nuclear structure must account for both Coulombic repulsion of protons and Strong force attraction between nucleons


Empirical observations

Empirical Observations

  • Chart of the nuclides:

    • 275 stable nuclei

    • 60% even-even

    • 40% even-odd or odd-even

    • Only 5 stable odd-odd nuclei

      21H, 63Li, 105B, 147N, 5023Va (could have large t1/2)

  • Nuclei with an even number of protons have a large number of stable isotopes

    Even # protonsOdd # protons

    50Sn:10 (isotopes)47Ag: 2 (isotopes)

    48Cd: 8 51Sb:2

    52Te: 8 45Rh:1

    49In:1

    53I: 1

  • Roughly equal numbers of stable even-odd and odd-even nuclei


Implications for nuclear models i

Implications for Nuclear Models I

  • Proton-proton and neutron-neutron pairing must result in energy stabilization of bound state nuclei

  • Pairing of protons with protons and neutrons with neutrons results in the same degree of stabilization

  • Pairing of protons with neutrons does not occur (nor translate into stabilization)


Chart of the nuclides

Chart of the nuclides

  • Light elements: N/Z = 1

  • Heavy elements: N/Z  1.6

    • Implies simple pairing not sufficient for stability

  • Neutron Rich: (N>Z)

    • N>Z: nucleus will - decay to stability

    • N>>Z: neutron drip line

  • Proton Rich: (N<Z)

    • N<Z: nucleus will + decay or electron capture to achieve stability

    • N<<Z: proton drip line

      (very rare)

Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.


Implications for nuclear models ii

Implications for Nuclear Models II

4.Pairing not sufficient to achieve stability

Why?

Coulomb repulsion of protons grows with Z2:

Nuclear attractive force must compensate  all stable nuclei with Z > 20 contain more neutrons than protons

Eq. 1


General nuclear properties i

General Nuclear Properties I

8.9

  • Binding energy per nucleon approximately constant for all stable nuclei

7.4

Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.


General nuclear properties ii

General Nuclear Properties II

  • Nuclear radius is proportional to the cube root of the mass

    r = r0 A1/3Eq. 2

  • Experimental studies show ~uniform distribution of the charge and mass throughout the volume of the nucleus

dl = skin thickness

Rl = Half density radius

Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.


Liquid drop model 1935

Liquid Drop Model (1935)

  • Treats nucleus as a statistical assembly of neutrons and protons with an effective surface tension - similar to a drop of liquid

  • Rationale:

    • Volume of nucleus  number of nucleons

      • Implies nuclear matter is incompressible

    • Binding energy of nucleus  number of nucleons

      • Implies nuclear force must have a saturation character, ie each nucleon only interacts with nearest neighbors

  • Mathematical Representation:

    • Treats binding energy as sum of volume, surface and Coulomb energies: Eq. 3


Liquid drop model components i

Volume Energy

Surface Energy

Liquid Drop Model Components I

  • Volume Energy:

    • Binding energy of nucleus  number of nucleons

    • Correction factor accounts for symmetry energy (for a given A the binding energy due to only nuclear forces is greatest for nuclei with equal numbers of protons and neutrons)

  • Surface Energy:

    • Nucleon at surface are unsaturated  reduce binding energy  surface area

    • Surface-to-volume ratio decreases with increasing nuclear size  term is less important for large nuclei

c1 = 15.677 MeV, c2 = 18.56 MeV, c3 = 0.717 MeV, c4 = 1.211 MeV, k = 1.79


Liquid drop model components ii

Liquid Drop Model Components II

  • Coulomb Energy:

    • Electrostatic energy due to Coulomb repulsion between protons

    • Correction factor accounts for diffuse boundary of nucleus (accounts for skin thickness of nucleus)

  • Pairing Energy:

    • Accounts for added stability due to nucleon pairing

    • Even-even:  = +11/A1/2

    • Even-odd & odd-even:  = 0

    • Odd-odd:  = -11/A1/2

Coulomb

Energy

Pairing

Energy

c1 = 15.677 MeV, c2 = 18.56 MeV, c3 = 0.717 MeV, c4 = 1.211 MeV, k = 1.79


Problem 1

Problem 1

  • Using the binding energy equation for the liquid drop model, calculate the binding energy per nucleon for 15N and 148Gd.

  • Compare these results with those obtained by calculating the binding energy per nucleon from the atomic mass and the masses of the constituent nucleons.


Problem 1 answers

Problem 1: Answers

  • 15N = 6.87 MeV/nucleon

  • 148Gd = 8.88 MeV/nucleon

  • 15N = 7.699 MeV/nucleon

  • 148Gd = 8.25 MeV/nucleon


Mass parabolas

Mass Parabolas

Eq. 4

  • Represent mass of atom as difference between sum of constituents and total binding energy:

  • Substitute binding energy equation for EB and group terms by power of Z:

  • For a given number of nucleons (A) f1, f2 and f3 are constants

  • Functional form represents a mass-energy parabola

    • Single parabola for odd A nuclei ( = 0)

    • Double parabola for even A nuclei ( = ±11/A1/2)

Eq. 5


Mass parabolas example 1 a 75 or 157

Mass Parabolas Example 1A = 75 or 157

  • Parabola Vertex:

    • ZA=[-f2/ 2f1]Eq. 6

    • Minimum mass & Maximum EB

  • Used to find mass and EB difference between isobars

  • Nuclear charge of minimum mass is derivative of Eq. 5 => not necessarily integral

  • Comparison of Z = 75 and Z = 157

    • Valley of stability broadens with increasing A

  • For a given value of odd-A only one stable nuclide exists

  • In odd-A isobaric decay chains the -decay energy increases monotonically

Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.


Mass parabolas example 2 a 156

Mass Parabolas Example 2A = 156

  • Eq. 5 results in two mass parabola for a given even value of A

  • For a given value of even-A their exist 2 (or 3) stable nuclides

  • In this figure both 156Gd and 156Dy are stable

  • In even-A isobaric decay chains the -decay energies alternate between small and large values

  • This model successfully reproduces experimentally observed energy levels

  • BUT…….

Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.


Problem 2

Problem 2

  • Find the nuclear charge (ZA) corresponding to the maximum binding energy for:

    A = 157, 156 and 75

  • To which isotopes do these values correspond?

  • Compare your results with the mass parabolas on slides 15 & 16.


Problem 2 answers

Problem 2: Answers

  • Find the nuclear charge (ZA) corresponding to the maximum binding energy for:

    A = 157, 156 and 75

    ZA = 64.69, 64.32, 33.13

  • To which isotopes do these values most closely correspond?

    15765Tb, 15664Gd, 7533As

  • Compare your results with the mass parabolas on slides 15 & 16.


Magic numbers

Magic Numbers

  • Nuclides with “magic numbers” of protons and/or neutrons exhibit an unusual degree of stability

  • 2, 8, 20, 28, 50, 82, 126

  • Suggestive of closed shells as observed in atomic orbitals

  • Analogous to noble gases

  • Much empirical evidence was amassed before a model capable of explaining this phenomenon was proposed

  • Result = Shell Model

Friedlander, “Nuclear and Radiochemistry, 3rd Edition, 1981.


Atomic orbitals history

Atomic Orbitals History

  • Plum Pudding Model: (Thomson, 1897)

    • Each atom has an integral number of electrons whose charge is exactly balanced by a jelly-like fluid of positive charge

  • Nuclear Model: (Rutherford, 1911)

    • Electrons arranged around a small massive core of protons and neutrons* (added later)

  • Planetary Atomic Orbitals: (Bohr, 1913)

    • Assume electrons move in a circular orbit of a given radius around a fixed nucleus

    • Assume quantized energy levels to account for observed atomic spectra

    • Fails for multi-electron systems

  • Schrodinger Equation: (1925)

    • Express electron as a probability distribution in the form of a standing wave function


Atomic orbitals

3s

2p

2s

Atomic Orbitals

  • Schrodinger equation solution reveals quantum numbers

    • n = principal, describes energy level

    • l = angular momentum, 0n-1 (s,p,d,f,g,h…)

    • m = magnetic, - l l, describes behavior of atom in external B field

    • ms = spin, -1/2 or 1/2

  • Pauli Exclusion Principle: e-’s are fermions  no two e-’s can have the same set of quantum numbers

  • Hund’s Rule: when electrons are added to orbitals of equal energy a single electron enters each orbital before a second enters any orbital; the spins remain parallel if possible.

  • Example: C = 1s22s22px12py1

1s


Shell structure of nucleus historical evolution

Shell Structure of NucleusHistorical Evolution

  • Throughout 1930’s and early 1940’s evidence of deviation from liquid drop model accumulates

  • 1949: Mayer & Jenson

    • Independently propose single-particle orbits

      • Long mean free path of nucleons within nucleus supports model of independent movement of nucleons

      • Using harmonic oscillator model can fill first three levels before results deviate from experiment (2,8,20 only)

    • Include spin-orbit coupling to account for magic numbers

      • Orbital angular momentum (l) and nucleon spin (±1/2) interact

      • Total angular momentum must be considered

      • (l+1/2) state lies at significantly lower energy than (l-1/2) state

      • Large energy gaps appear above 28, 50, 82 & 126


Single particle shell model

Single Particle Shell Model

  • Assumes nucleons are distributed in a series of discrete energy levels that satisfy quantum mechanics (analogous to atomic electrons)

  • As each energy level is filled a closed shell forms

  • Protons and neutrons fill shells and energy levels independently

  • Mainly applicable to ground state nuclei

  • Only considers motion of individual nucleons


Shell model and magic numbers

Shell Model and Magic Numbers

  • Magic numbers represent closed shells

  • Elements in periodic table exhibit trends in chemical properties based on number of valence electrons (Noble gases:2,10,18,36..)

  • Nuclear properties also vary periodically based on outer shell nucleons


Pairing

Pairing

  • Just as electrons tend to pair up to form a stable bond, so do like-nucleons; pairing results in increased stability

    • Even-Z and even=N nuclides are the most abundant stable nuclides in nature (165/275)

    • From 15O to 35Cl all odd-Z elements have one stable isotope while all even-Z elements have three

    • The heaviest stable natural nuclide is 20983Bi (N=126)

    • The stable end product of all naturally occurring radioactive series of elements is Pb with Z=82


Shell model evidence abundances

Shell Model Evidence - Abundances

  • The most abundantly occurring nuclides in the universe (terrestrial and cosmogenic) have a magic number of protons and/or neutrons

  • Large fluctuations in natural abundances of elements below 19F are attributed to their use in thermonuclear reactions in the prestellar stage


Shell model evidence abundances1

Shell Model Evidence - Abundances


Shell model evidence stable isotopes

Shell Model Evidence - Stable Isotopes

Stable Isobars

  • The number of stable isotopes of a given element is a reflection of the relative stability of that element. Plot of number of isotopes vs N shows peaks at

    • N = 20, 28, 50, 82

  • A similar effect is observed as a function of Z

# of Isobars

# of Neutrons


Shell model evidence alpha decay

Shell Model Evidence - Alpha Decay

  • Shell Model predictions:

    • Nuclides with 128 neutrons =>

      • short half life

      • Emit energetic 

    • Nuclides with 126 neutrons =>

      • Long half life

      • Emit low energy 


Shell model evidence beta decay

Shell Model Evidence - Beta Decay

  • If product contains a magic number of protons or neutrons the half-life will be short and the energy of the emitted  will be high

N = 19

N = 20

N = 21

Z = 21

Z = 20

Z = 19


Shell model evidence neutrons

Shell Model Evidence - Neutrons

  • Neutrons do not experience Coulomb barrier  even thermal neutrons (low kinetic energy) can penetrate the nucleus

  • Inside nucleus neutron experiences attractive strong force and becomes bound

  • To escape the nucleus a neutron’s KE must be greater than or equal to the nuclear potential at the surface of the nucleus

  • Observation: the absorption cross section for 1.0MeV neutrons is much lower for nuclides containing 20, 50, 82, 126 neutrons compared to those containing 19, 49, 81, 125 neutrons


Shell model evidence energy

Shell Model Evidence - Energy

  • The energy needed to extract the last neutron from a nucleus is much higher if it happens to be a magic number neutron

  • Energy needed to remove a neutron

    • 126th neutron from 208Pb = 7.38 MeV

    • 127th neutron from 209Pb = 3.87 MeV


Shell model evidence nucleon interactions

Shell Model Evidence - Nucleon Interactions

  • Every nucleon is assumed to move in its own orbit independent of the other nucleons, but governed by a common potential due to the interaction of all of the nucleons

  • Implication: in ground state nucleus nucleon-nucleon interactions are negligible

  • Implication: mean free path of ground state nucleon is approximately equal to the nuclear diameter

  • Experimental data does not support this conclusion!!!


Shell model evidence nucleon interactions1

Shell Model Evidence - Nucleon Interactions

  • Scattering experiments show frequent elastic collisions

  • Implication: mean free path << nuclear radius

  • Explanation: Pauli exclusion principle prohibits more than two protons or neutrons from occupying the same orbit (protons and neutrons are fermions)

  • Why Pauli?: nucleon-nucleon collisions result in momentum transfer between the participants BUT all lower energy quantum states are filled  occurrence forbidden

  • Severely limits nucleon-nucleon collision rate


Nuclear potential well

Nuclear Potential Well

  • Nucleon orbit = nucleon quantum state

    • Similar to quantum state of valence electron

      BUT

    • Nucleon “feels” total effect of interactions of all nucleons

    • Implication: nuclear potential is the same for all nucleons

  • Strong Force

    • All nucleons (regardless of their electrical charge) attract one another

    • Attractive force is short range and falls rapidly to zero outside of the nuclear boundary (~1 fm)


Nuclear potential protons

Nuclear Potential - Protons

  • Protons do experience a Coulomb barrier  a proton must have kinetic energy equal or greater than ECoul to penetrate the nucleus

  • If Eproton< ECoul proton will back scatter

  • Inside nucleus proton experiences attractive strong force and becomes bound

  • To escape the nucleus a proton’s kinetic energy must be greater than or equal to ECoul (in the absence of quantum tunneling)


Nuclear potential neutrons

Nuclear Potential - Neutrons

  • Neutrons do not experience Coulomb barrier  even thermal neutrons (low kinetic energy) can penetrate the nucleus

  • Inside nucleus neutron experiences attractive strong force and becomes bound

  • To escape the nucleus a neutron’s kinetic energy must be greater than or equal to the nuclear potential at the surface of the nucleus


Nuclear potential well1

Nuclear Potential Well

Depth of well represents

binding energy


Nuclear potential functions

Nuclear Potential Functions

  • Square Well Potential

  • Harmonic Oscillator Potential

  • Woods-Saxon

  • Exponential Potential

  • Gaussian Potential

  • Yukawa Potential:

Note: R = nuclear radius

r = distance from center of nucleus


Nuclear potential functions1

Nuclear Potential Functions

Exact shape of well is

uncertain and depends

on mathematical function

assumed for the interaction

Yukawa

Exponential

Gaussian

Square Well


Neutron vs proton potential wells

Neutron vs Proton Potential Wells

Coulomb repulsion

prevents potential well

from being as deep for

protons as for neutrons


Quantized energy levels

Quantized Energy Levels

  • Schrodinger Equation: developed to find wave functions and energies of molecules; also can be applied to the nucleus

  • Choose functional form of nuclear potential well and solve Schrodinger Equation:

    H = E 

  • Wave equation allows only certain energy states defined by quantum numbers

  • n = principal quantum number, related to total energy of the system

  • l = azimuthal (radial) quantum number, related to rotational motion of nucleus

  • ms = spin quantum number, intrinsic rotation of a body around its own axis


Angular momentum

Angular Momentum

  • Associated with the rotational motion of an object

  • Like linear motion, rotational motion also has an associated momentum

  • Orbital angular momentum:

    pl = mvrr

  • Spin angular momentum:

    ps = Irot

  • A vector quantity  always has a distinct orientation in space


Magnetic quantum effects spin

Magnetic Quantum Effects - Spin

  • A rotating charge gives rise to a magnetic moment (s).

  • Electrons and protons can  be conceptualized as small magnets

  • Neutrons have internal charge structure and can also be treated as magnets

  • In the absence of a B-field magnets are disoriented in space (can point any direction)

  • In the presence of a B-field the electron, proton and neutron spins are oriented in specific directions based upon quantum mechanical rules


Spin angular momentum

Spin Angular Momentum

No External B-field

Applied External B-field

{

Project spin angular momentum onto the field axes

Allowed values are units of hbar

ps(z) = hbar ms


Spin angular momentum1

Spin Angular Momentum

  • Quantum mechanics requires that the spin angular momentum of electrons, protons and neutrons must have the magnitude

  • s is the spin quantum number

  • For protons and neutrons (just like for electrons) spin is always 1/2


Magnetic quantum effects orbitals

Magnetic Quantum Effects - Orbitals

  • The orbital movement of an atomic electron or a nucleon gives rise to another magnetic moment (l)

  • This magnetic moment also interacts with an external B-field in a similar manner to the spin magnetic moment

  • Quantum mechanics governs how the orbital plane may be oriented in relation to the external field

  • The orbital angular momentum vector (pl) can only be oriented such that its projection onto the z-axis (field axis) has values

    pl (z) = hbar ml

  • Where ml = magnetic orbital quantum number

    ml = -l, -l+1, -l+2….0…. l-2, l-1, l


Orbital angular momentum

B-field axis

pl

pl(z)

3 h

2 h

1 h

0 h

-1 h

-2 h

-3 h

Orientations of ml in a magnetic field

Orbital Angular Momentum

  • Project orbital angular momentum onto the field axes

  • Allowed values are units of hbar

  • pl(z) = hbar ml


Orbital angular momentum1

Orbital Angular Momentum

  • Quantum mechanics requires that the orbital angular momentum of electrons, protons and neutrons must have the magnitude

  • l is the orbital quantum number

  • Allowed values of l:

    • Nucleons: 0  l

    • Electrons: 0  l< (n-1)

  • For nucleons (but not electrons) l can exceed n


Orbital angular momentum2

Orbital Angular Momentum

  • The numerical values of the orbital angular momentum quantum number (l) are designated by the familiar spectroscopic notation

  • Remember: l can only have positive integral values (including 0)


Quantum states

Quantum States


Energy level diagram

nl nucleons [total]

26

138

1i

112

6

3p

14

106

2f

22

92

1h

2

70

3s

10

68

2d

18

58

1g

6

40

2p

14

34

1f

2

20

2s

10

18

1d

6

8

1p

2

2

1s

Energy Level Diagram

Isotropic Harmonic Oscillator Levels


Spin orbit coupling

Spin Orbit Coupling

  • Spin and orbital angular momenta are vector quantities  vector coupling occurs to form a resultant vector pj

  • Total angular momentum:

    pj = pl + ps

  • Coupling splits degeneracy of orbital angular momentum states


Spin orbit coupling1

Spin Orbit Coupling

  • For nucleons and electrons the orbital and spin angular momenta add vectorially to form a resultant vector (pj)

    pj = pl + ps

  • The resultant is oriented towards an external magnetic field so that the projections on the field axis are

    pj(z) = hbar mj

  • The magnitude of pj is

    pj = hbar [j(j+1)]1/2

    j = l s

  • j is the total angular quantum number of the particle


Total angular momentum

Total Angular Momentum

  • Total angular quantum number (j) can have two different values for each orbital quantum number (l)

  • However, j can only have positive values!!

  • Implication: when l = 0, only j=1/2 is allowed

  • All allowed values of j are half-integers

    j = 1/2, 3/2, 5/2, 7/2…


Total angular momentum example

Total Angular Momentum Example

  • What are the allowed values of j for a nucleon with

    l = 1, s = 1/2


Total angular momentum example1

Total Angular MomentumExample

  • What are the allowed values of j for a nucleon with

    l = 1, s = 1/2

  • Answer: j = 1/2 or 3/2


Total angular momentum1

Total Angular Momentum

  • Example: n = 1, l = 1, s = 1/2

  • Standard atomic notation:

    • Electron in 1p1 state

    • leading 1=principal quantum number

    • p = orbital angular quantum number

    • superscript 1 = spin quantum number

  • Standard nuclear notation:

    • Nucleon in 1p1/2 state or 1p3/2 state

    • But we know that spin orbit coupling splits the degeneracy of the 6 existing p states

    • This results in a two-fold degenerate 1p1/2 state and a four-fold degenerate 1p3/2 state

    • Which is lower in energy????


Level ordering of states

Level Ordering of States

  • Split degenerate states with higher j are always more stable than those with lower j

    • Energetically:

      1p1/2 > 1p3/2

  • Neutrons and protons fill levels independently


Example

Example

  • What is the level ordering for

    • 94Be

    • 3115P

    • 5927Co


Example1

Example

  • What is the level ordering for

    • 94Be: protons (1s21/2 2p23/2)

      neutrons (1s21/2 2p33/2)

    • 3115P: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s11/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2)

    • 5927Co: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f77/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f87/2 2p43/2)


Magnetic quantum numbers

Magnetic Quantum Numbers

  • For each of the angular momentum quantum numbers (l, s, j) there exists a magnetic analogue (ml, ms, mj) representing the resolved component of the original quantum number along the axis of the applied magnetic field

  • Each magnetic quantum number can be derived from the related quantum numbers

  • ms:

    • Magnetic spin angular momentum quantum number

    • has only 2 allowed values (s)

    • For protons, neutrons and electrons s = 1/2

    • ms = +1/2 or -1/2


Magnetic quantum numbers1

Magnetic Quantum Numbers

  • ml:

    • Magnetic orbital angular momentum quantum number

    • Has (2l+1) possible values

    • Can be positive or negative integral values

    • ml = -l, -l+1, -l+2….0…. l-2, l-1, l

    • Example: l = 3 or p orbital

      • (2*3+1) = 7 allowed values

      • ml:= -3, -2, -1, 0, 1, 2, 3

  • mj:

    • Magnetic total angular momentum quantum number

    • Has (2j+1) possible values

    • Can be positive or negative integral values

    • mj = -j, -j+1, -j+2….0…. j-2, j-1, j


Nucleon total angular momentum

Nucleon Total Angular Momentum

  • Each nucleon has an associated orbital angular momentum and an associated spin angular momentum

  • The total angular momentum quantum number of the nucleon is given by:

    j = l + s

  • The total angular momentum is:

    pj = hbar[j(j+1)]1/2

  • The observable maximum total angular momentum is

    pj = hbar x j


Summary single particle quantum numbers

Summary: Single Particle Quantum Numbers

  • Schrodinger Equation Solutions (H = E)

  • Fundamental Quantum Numbers

    n - principals - spin (1/2)

    l - orbitalj - total (l  s)

  • Magnetic Quantum Numbers

    ml - magnetic orbital (-l, -l+1….0…. l-1, l)

    ms - magnetic spin (+1/2 or -1/2)

    mj - magnetic total (-j, -j+1….0….j-1, j)


Nucleus total angular momentum

Nucleus: Total Angular Momentum

  • When two or more nucleons come together to form a nucleus the momentum components of the individual particles interact to give a resultant total angular momentum characteristic of the nucleus

  • The energy level of the nucleus as a whole is represented by the resultant (I)

  • I is historically referred to (inappropriately) as the spin of the nucleus BUT do not confuse it with the spin quantum number of a nucleon (s)

    pI = hbar [I(I+1)]1/2

  • The observable maximum value of the total nuclear angular momentum is pI = hbar x I


Nucleus total angular momentum1

Nucleus: Total Angular Momentum

  • Nucleon-nucleon coupling of the spin and orbital motions of the individual nucleons is not clearly understood

  • Two limiting coupling modes exist:

    • LS coupling

    • jj coupling (dominant)

  • In reality the coupling probably lies in between the two models


Ls coupling

Upper Case

LS Coupling

  • Also known as Russell-Saunders coupling

  • The interaction of the orbital motion of a nucleon with its own spin is considered to be weak or negligible

  • The orbital motions of different nucleons interact strongly with each other

  • The resultant total orbital angular momentum of the nucleus is represented by L and is the vector sum of the individual nucleons

    L =  li

  • Allowed values of L = 0, 1, 2, 3…

  • Common symbol notation S, P, D, F


Ls coupling1

LS Coupling

  • The spin motions of different nucleons interact strongly with each other

  • The resultant total spin angular momentum of the nucleus is represented by S and is the vector sum of the spins of the individual nucleons

    S =  si

  • The total angular momentum of the nucleus is represented by I (or sometimes J)

    I = L  S (hence the name LS coupling!)


Ls coupling application

LS Coupling - Application

  • The individual orbital and spin angular momenta of paired nucleons cancel each other and do not contribute to the total nuclear angular momentum

  • Even-even nuclei have zero nuclear spin (I=0)

  • Nuclear spin of odd-A nuclei is determined by the single unpaired nucleon

  • Odd-odd nuclei:

    I(odd-odd) = jp + jn = (lp 1/2) + (ln 1/2)


Ls coupling application1

LS Coupling - Application

Steps to determine nuclear spin state

  • Determine if nucleus is even-even, even-odd, odd-even or odd-odd

    • If even-even I=0

    • If N and/or Z are odd continue with step 2 for the odd nucleon(s)

  • Fill energy level diagram for odd nucleon

    • If odd-odd remember to fill the levels independently for protons and neutrons

  • Find the value of j for the energy level occupied by the unpaired nucleon

    • For an even-odd or odd-even system this value is the total nuclear spin

    • For an odd-odd nucleus the model can not predict the overall state; nuclear spins can range in value from

      |j1 - j2| to j1 + j2


Example2

Example

  • What is the nuclear spin for

    • 94Be: protons (1s21/2 2p23/2)

      neutrons (1s21/2 2p33/2)

    • 3115P: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s11/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2)

    • 5927Co: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f77/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f87/2 2p43/2)


Example3

Example

  • What is the nuclear spin for: (I)

    • 94Be: protons (1s21/2 2p23/2) (0)

      neutrons (1s21/2 2p33/2) (3/2)

    • 3115P: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s11/2) (1/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2) (0)

    • 5927Co: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f77/2)(7/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f87/2 2p43/2) (0)


Jj coupling

jj Coupling

  • Complementary to LS coupling

  • Considers affect of strong spin-orbit coupling for individual nucleons in a nucleus

  • Total nuclear spin: (vector sum)

    I = {j1+j2+j3…..}


Ls vs jj coupling

LS vs jj Coupling

  • These models represent two extremes of a coupling that in reality is most accurately represented as a continuum

  • In general,

    • jj coupling preferred for very heavy nuclei

    • Light nuclei are a mixture


Parity

Parity

  • A conserved quantity in nuclear reactions involving the emission of photons and nucleons

  • Nuclear property related to the symmetry properties of the wave function

  • Parity is odd(-) or even(+) based on whether or not the wave function is symmetric

  • To test for symmetry reverse the signs of all of the spatial coordinates in the function

    • If resultant solution changes sign => (-)

    • If resultant solution remains the same => (+)

  • Parity rules for combining wave functions

    • ++ = +

    • +- = -

    • -- = +


Parity1

Parity

  • Orbital angular momentum states result from solutions of the Schrodinger equation  they are wave functions with an associated parity

  • Simple rules:

    • Even-even nuclei have a ground state (+) parity

    • Even-odd and odd-even nuclei have a parity equal to that of the wave function of the unpaired nucleon

    • Odd-odd nuclei => parity is the product of the wave functions of the unpaired nucleons


Example4

Example

  • What is the parity for: (I)

    • 94Be: protons (1s21/2 2p23/2) (0)

      neutrons (1s21/2 2p33/2) (3/2)

    • 3115P: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s11/2) (1/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2) (0)

    • 5927Co: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f77/2)(7/2)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f87/2 2p43/2) (0)


Example5

Example

  • What is the parity for: (I) ()

    • 94Be: protons (1s21/2 2p23/2) (0)

      neutrons (1s21/2 2p33/2) (3/2) (-)

    • 3115P: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s11/2) (1/2) (+)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2) (0)

    • 5927Co: protons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f77/2)(7/2) (-)

      neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2 1d43/2 1f87/2 2p43/2) (0)


Nuclear spin and parity

Nuclear Spin and Parity

  • It is customary to represent the nuclear spin and parity together

  • For example in the nuclide 178O the odd nucleon is the 9th neutron which occupies the 1d5/2 state

    • Spin = 5/2

    • Parity = +

    • Standard notation to say ground state of 178O has a spin and parity of 5/2+


Example6

Example

  • What is the standard spin and parity notation for:

    • 94Be: 3/2-

    • 3115P: 1/2+

    • 5927Co: 7/2-


Nuclear charge distribution

Nuclear Charge Distribution

  • To understand the structure of the nucleus it is important to know how the protons are spatially distributed

  • The presence of a single proton displaced from the center of the nucleus is important because it can result in an effective potential experienced beyond the walls of the nucleus

  • Spherical nuclei act as magnetic monopoles

  • Deformed nuclei can have the properties of dipoles, quadrupoles, octupoles, etc

  • These electrical moments can be predicted if the nuclear spin (I) of the nucleus is known

  • Monopole (I=0), dipole (I=1/2), quadrupole (I=1)

(Most common)


Deformed nuclei

Deformed Nuclei

  • Liquid-drop model and shell model both assume a spherically symmetric nucleus

    • This is a reasonable assumption for magic number nuclei

    • But other nuclei have distorted shapes

  • Magnitude of deviation from spherical is quantized by

     = 2(a-c)/(a+c)

    where a and c are the radius with respect to the z and x axes (see diagram slide 84)

  • Two types of deformed nuclei depending on which axis is compressed

    • Prolate:  > 0

    • Oblate:  < 0

  • Maximum observed value of  =  0.6


Deformed nuclei1

z

Deformed Nuclei

Oblate: Spins on

short axis

-x

y

Prolate: Spins on

long axis


Nuclear quadrupole moment

Nuclear Quadrupole Moment

  • Nuclei with quadrupole moments (I=1) are common

  • Let a nucleus with finite quadrupole moment be represented by an ellipsoid with the semi-axis (b) parallel to the symmetry axis (Z) and the semi-axis (a) perpendicular

  • If the total charge of the nucleus (Ze) is assumed to be uniformly distributed throughout the ellipsoid, the quadrupole moment of the nucleus in the Z direction can be calculated using:

a

b

Z


Collective nuclear model

Collective Nuclear Model

  • Proposed by Bohr & Mottelsen in 1953

  • Models nucleus as a highly compressed liquid that can experience internal rotations and vibrations

  • Includes 4 discrete modes of collective motion

    • Rotate around z-axis

    • Rotate around y-axis

    • Oscillate between prolate and oblate

    • Vibrate along x-axis

  • Used to calculate allowed vibrational and rotational levels between standard nuclear levels

  • These new levels are equivalent to excited states

  • Validity depends on whether each mode of collective motion can be treated independently

  • Works best for strongly deformed nuclei (238U)


Collective nuclear model levels

Collective Nuclear Model Levels


Unified model of deformed nuclei

Unified Model of Deformed Nuclei

  • Shell model suffers from discrepancies between experimental and theoretical spin states for certain nuclei

  • Angular momentum of odd-A deformed nuclei contains components from deformed core and unpaired nucleon

  • Results in a modification of the energy levels that changes their ordering


Nilsson levels

Nilsson Levels

  • Nilsson calculated energy levels of odd-A nuclei as a function of the nuclear deformation ()

  • Each j state from the shell model is split into j+1/2 levels and may contain 2 nucleons

  • Some undeformed levels also reverse order (1f5/22p3/2)

  • Nilsson levels predict energies, angular momenta, quantum numbers, etc better for deformed nuclei than the shell model


Nilsson diagram z or n 50

Nilsson Diagram (Z or N 50)


Nilsson diagram 50 z or n 82

Nilsson Diagram (50  Z or N 82)


Example7

Example

  • Given a nuclear deformation of 0.11, find the spin and parity of 23Na using the shell model and the Nilsson diagram.

  • Which state does the chart of the nuclides confirm exists in nature?


Example8

Example

  • Given a nuclear deformation of 0.11, find the spin and parity of 23Na using the shell model and the Nilsson diagram.

  • Shell model: 5/2+

  • Nilsson: 3/2+

  • Which state does the chart of the nuclides confirm exists in nature? 3/2+


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