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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

  • Why study nuclear physics

  • Why nuclear physics is difficult

  • Course synopsis.

  • Notation & Units

Nuclear Physics Lectures

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

  • 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

  • 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

  • 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?

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

  • 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

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

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


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


  • 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

  • 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

  • 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

nn and pp interaction same (apart from Coulomb)

“Charge symmetry”

Nuclear Physics Lectures

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


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

  • 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

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 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 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

  • 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 Term

  • Phenomenological fit to A dependence

  • Effect smaller for larger A

Nuclear Physics Lectures

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

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

Fit values in MeV


B/A (MeV)


Nuclear Physics Lectures


Valley of Stability

  • SEMF allows us to understand valley of stability.

  • Low Z, asymmetry term  Z=N

  • Higher Z, Coulomb term  N>Z.

Nuclear Physics Lectures

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