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Today’s Menu. Why study nuclear physics Why nuclear physics is difficult Course synopsis. Notation & Units. Why Study Nuclear Physics?. Understand origin of different nuclei Big bang: H, He and Li Stars: elements up to Fe Supernova: heavy elements We are all made of stardust

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Today s menu
Today’s Menu

  • Why study nuclear physics

  • Why nuclear physics is difficult

  • Course synopsis.

  • Notation & Units

Nuclear Physics Lectures

Why study nuclear physics
Why Study Nuclear Physics?

  • Understand origin of different nuclei

    • Big bang: H, He and Li

    • Stars: elements up to Fe

    • Supernova: heavy elements

  • We are all made of stardust

  • Need to know nuclear cross sections  experimental nuclear astrophysics is a hot topic.

Nuclear Physics Lectures

Practical applications
Practical Applications

  • Nuclear fission for energy generation.

    • No greenhouse gasses

    • Safety and storage of radioactive material.

  • Nuclear fusion

    • No safety issue (not a bomb)

    • Less radioactive material but still some.

  • Nuclear transmutation of radioactive waste with neutrons.

    • Turn long lived isotopes  stable or short lived.

  • Every physicist should have an informed opinion on these important issues!

Nuclear Physics Lectures

Medical applications
Medical Applications

  • Radiotherapy for cancer

    • Kill cancer cells.

    • Used for 100 years but can be improved by better delivery and dosimetery

    • Heavy ion beams can give more localised energy deposition.

  • Medical Imaging

    • MRI (Nuclear magnetic resonance)

    • X-rays (better detectors  lower doses)

    • PET

    • Many others…see Medical & Environmental short option.

Nuclear Physics Lectures

Other applications
Other Applications

  • Radioactive Dating

    • C14/C12 gives ages for dead plants/animals/people.

    • Rb/Sr gives age of earth as 4.5 Gyr.

  • Element analysis

    • Foresenic (eg date As in hair).

    • Biology (eg elements in blood cells)

    • Archaeology (eg provenance via isotope ratios).

Nuclear Physics Lectures

Why is nuclear physics hard
Why is Nuclear Physics Hard?

  • QCD theory of strong interactions  just solve the equations …

  • At short distance/large Q coupling constant small  perturbation theory ok but long distance/small Q, q  large

Nuclear Physics Lectures

Nuclear physics models
Nuclear Physics Models

  • Progress with understanding nuclear physics from QCD=0  use simple, approximate, phenomenological models.

  • Liquid Drop Model: phenomenology + QM + EM.

  • Shell Model: look at quantum states of individual nucleons  understand spin/parity magnetic moments and deviations from SEMF for binding energy.

Nuclear Physics Lectures

Course synopsis 1
Course Synopsis - 1

  • Liquid Drop Model and SEMF.

  • Applications of SEMF

    • Valley of stability.

    • abg decays.

    • Fission & fusion.

  • Shell Model

    • Magic numbers, spin/parity, magnetic moments.

Nuclear Physics Lectures

Course synopsis 2
Course Synopsis - 2

  • Cross Sections

    • Experimental definition

    • FGR theory

    • Rutherford scattering

    • Breit-Wigner resonances

  • Theory of abg decays.

  • Particle interactions in matter

    • Simple detectors for nuclear/particle physics.

Nuclear Physics Lectures

What is the use of lectures
What is the use of lectures

  • Definition of a lecture: a process whereby notes are transferred from the pages of a lecturer to the pages of the student without passing through the head of either.

  • Conclusion: to make lectures useful YOU have to participate, ask questions ! If you don’t understand something the chances are >50% of the audience doesn’t as well, so don’t be shy !

Nuclear Physics Lectures


  • To err is human … and this is a new course  lots of mistakes.

  • Please tell me about any mistakes you find in the notes (I will donate a bottle of wine to the person who finds the most mistakes!).

Nuclear Physics Lectures

The minister of science
The Minister of Science

  • This is a true story honest.

  • Once upon a time the government science minister visited the Rutherford Lab (UK national lab) and after a days visit of the lab was discussing his visit with the lab director and he said …

  • I hope that you all have a slightly better grasp of the subject by the end!

Nuclear Physics Lectures


  • Nuclei are labelled where El is the chemical symbol of the element, mass number A = number of neutrons N + number of protons Z. eg

  • Excited states labelled by * or m if they are metastable (long lived).

Nuclear Physics Lectures


  • SI units are fine for macroscopic objects like footballs but are very inconvenient for nuclei and particles  use natural units.

  • Energy: 1 eV = energy gained by electron in being accelerated by 1V.

    • 1 eV= e J.

  • Mass: MeV/c2 (or GeV/c2)

    • 1 eV/c2 = e/c2 kg.

    • Or use AMU defined by mass of 12C= 12 u

  • Momentum: MeV/c (or GeV/c)

    • 1 eV/c = e/c kg m s-1

  • Cross sections: (as big as a barn door)

    • 1 barn =10-28 m2

  • Length: fermi 1 fm = 10-15 m.

Nuclear Physics Lectures

Nuclear masses and sizes
Nuclear Masses and Sizes

  • Masses and binding energies

    • Absolute values measured with mass spectrometers.

    • Relative values from reactions and decays.

  • Nuclear Sizes

    • Measured with scattering experiments (leave discussion until after we have looked at Rutherford scattering).

    • Isotope shifts

Nuclear Physics Lectures

Nuclear mass measurements
Nuclear Mass Measurements

  • Measure relative masses by energy released in decays or reactions.

    • X  Y +Z + DE

    • Mass difference between X and Y+Z is DE/c2.

  • Absolute mass by mass spectrometers (next transparency).

  • Mass and Binding energy:

  • B = [Z MH + N Mn – M(A,Z)]/c2

Nuclear Physics Lectures

Mass spectrometer
Mass Spectrometer

  • Ion Source

  • Velocity selector  electric and magnetic forces equal.

    • qE=qvB  v=E/B

  • Momentum selector, circular orbit satisfies:

    • Mv=qBr

    • Measurement r gives M.


Velocity selector

Ion Source

Nuclear Physics Lectures

Binding energy vs a
Binding Energy vs A

  • B increases with A up to 56Fe and then slowly decreases. Why?

  • Lower values and not smooth at small A.

Nuclear Physics Lectures

Nuclear sizes isotope shift
Nuclear Sizes & Isotope Shift

  • Coulomb field modified by finite size of nucleus.

  • Assume a uniform charge distribution in the nucleus. Gauss’s law 

    integrate and apply boundary conditions

  • Difference between actual potential and Coulomb

  • Use 1st order perturbation theory

Nuclear Physics Lectures

Isotope shifts
Isotope Shifts

  • Integral gives (do it on blackboard!)

  • Isotope shift for optical spectra

  • Isotope shift for X-ray spectra (bigger effect because electrons closer to nucleus)

  • Isotope shift for X-ray spectra for muonic atoms. Effect greatly enhanced because mm~ 207 me and a0~1/m.

  • All data consistent with R=R0 A1/3 with R0=1.25fm.

Nuclear Physics Lectures

Today s menu

Data on isotope shifts of atomic spectra confirm the R = R0A1/3 . See - the isotope dependence of optical spectra (this slide)- the isotope dependence of x-ray spectra (next slide) - the isotope dependence of muonic x-ray spectra (slide after that)

Frequency shift of an optical transition in Hg at =253.7nm for different A relative to A=198.

Data obtained by laser spectroscopy.

The effect is about 1 in 107. (Note the even/odd structure.)

Bonn et al Z Phys A 276, 203 (1976)

Nuclear Physics Lectures

Today s menu

Data on the isotope shift of K X ray lines in Hg. The effect is about 1 in 106. Again the data show the R2 = A2/3dependence and the even/odd effect. Lee et al, Phys Rev C 17, 1859 (1978)

Nuclear Physics Lectures

Today s menu


Data on Isotope Shift of K Xrays from muonic atoms [in which a muon with m=207metakes the place of the atomic electron].

Because a0 ~ 1/m the effect is ~0.4%, much larger than for an electron.

The large peak is 2p3/2 to 1s1/2. The small peak is 2p1/2 to 1s1/2. The size comes from the 2j+1 statistical weight.

Shera et al Phys Rev C 14, 731 (1976)



Nuclear Physics Lectures

Today s menu

  • Aim: phenomenological understanding of nuclear binding energies as function of A & Z.

  • Nuclear density constant (see lecture 1).

  • Model effect of short range attraction due to strong interaction by liquid drop model.

  • Coulomb corrections.

  • Fermi gas model  asymmetry term.

  • QM pairing term.

  • Compare with experiment: success & failure!

Nuclear Physics Lectures

Liquid drop model nucleus
Liquid Drop Model Nucleus

  • Phenomenological model to understand binding energies.

  • Consider a liquid drop

    • Ignore gravity and assume no rotation

    • Intermolecular force repulsive at short distances, attractive at intermediate distances and negligible at large distances  constant density.

      E=-an + 4pR2T B=an-bn2/3

  • Analogy with nucleus

    • Nucleus has constant density

    • From nucleon nucleon scattering experiments: Nuclear force has short range repulsion and attractive at intermediate distances.

    • Assume charge independence of nuclear force, neutrons and protons have same strong interactions check with experiment!

Nuclear Physics Lectures

Mirror nuclei
Mirror Nuclei

  • Compare binding energies of mirror nuclei (nuclei n p). Eg 73Li and 74Be.

  • Mass difference due to n/p mass and Coulomb energy.

Nuclear Physics Lectures

Today s menu

nn and pp interaction same (apart from Coulomb)

“Charge symmetry”

Nuclear Physics Lectures

Charge symmetry and charge independence
Charge Symmetry and Charge Independence

  • Mirror nuclei showed that strong interaction is the same for nn and pp.

  • What about np ?

  • Compare energy levels in “triplets” with same A, different number of n and p. e.g.

  • Same energy levels for the same spin states  SI same for np as nn and pp.

Nuclear Physics Lectures

Charge independence
Charge Independence


2312 Mg

  • Is np force is same as nn and pp?

  • Compare energy levels in nuclei with same A.

  • Same spin/parity states have same energy.

  • np=nn=pp




Nuclear Physics Lectures

Charge independence of strong interaction
Charge Independence of Strong Interaction

  • If we correct for n/p mass difference and Coulomb interaction, then energy levels same under n p.

  • Conclusion: strong interaction same for pp, pn and nn if nucleons are in the same quantum state.

  • Beware of Pauli exclusion principle! eg why do we have bound state of pn but not pp or nn?

Nuclear Physics Lectures

Asymmetry term

IllustrationNeutron and proton states with same spacing .Crosses represent initially occupied states in ground state.If three protons were turned into neutrons the extra energy required would be 3×3 .In general if there are Z-N excess protons over neutrons the extra energy is ((Z-N)/2)2 . relative to Z=N.


  • Neutrons and protons are spin ½ fermions  obey Pauli exclusion principle.

  • If other factors were equal  ground state would have equal numbers of n & p.

Nuclear Physics Lectures

Asymmetry term1
Asymmetry Term

  • From stat. mech. density of states in 6d phase space = 1/h3

  • Integrate to get total number of protons Z, & Fermi Energy (all states filled up to this energy level).

  • Change variables p  E

Nuclear Physics Lectures

Asymmetry term2
Asymmetry Term

  • Binomial expansion keep lowest term in y/A

  • Correct functional form but too small by factor of 2. Why?

Nuclear Physics Lectures

Pairing term
Pairing Term

  • Nuclei with even number of n or even number of p more tightly bound fig.

  • Only 4 stable o-o nuclei cf 153 e-e.

  • p and n have different energy levels  small overlap of wave functions. Two p(n) in same level with opposite values of jz have AS spin state  sym spatial w.f. maximum overlap maximum binding energy because of short range attraction.

Neutron separation energy in Ba

Neutron number

Nuclear Physics Lectures

Pairing term1
Pairing Term

  • Phenomenological fit to A dependence

  • Effect smaller for larger A

Nuclear Physics Lectures

Semi empirical mass formula
Semi Empirical Mass Formula

  • Put everything together:

  • Fit to measured binding energy.

    • Fit not too bad (good to <1%).

    • Deviations are interesting  shell effects.

    • Coulomb term agrees with calculation.

    • Asymmetry term larger ?

    • Explain valley of stability.

    • Explains energetics of radioactive decays, fission and fusion.

Nuclear Physics Lectures

Today s menu

The Binding Energy per nucleon of beta-stable (odd A) nuclei.

Fit values in MeV


B/A (MeV)


Nuclear Physics Lectures


Valley of stability
Valley of Stability nuclei.

  • SEMF allows us to understand valley of stability.

  • Low Z, asymmetry term  Z=N

  • Higher Z, Coulomb term  N>Z.

Nuclear Physics Lectures