Nuclear Phenomenology 3C24 Nuclear and Particle Physics Tricia Vahle & Simon Dean (based on Lecture Notes from Ruben Saakyan) UCL

Nuclear Notation Z – atomic number = number of protons N – neutron number = number of neutrons A – mass number = number of nucleons (Z+N) Nuclides A X ( 16 O, 40 Ca, 55 Fe etc…) –Nuclides with the same A – isobars –Nuclides with the same Z – isotopes –Nuclides with the same N – isotones

Masses and binding energies Something we know very well: –M p = MeV/c 2, M n = MeV/c 2 One might think that –M(Z,A) = Z M p + N M n - not the case !!! In real life –M(Z,A) < Z M p + N M n The mass deficit –  M(Z,A) = M(Z,A) - Z M p - N M n ––  Mc 2 – the binding energy B. –B/A – the binding energy per nucleon, the minimum energy required to remove a nucleon from the nucleus

Binding energy Binding energy per nucleon as function of A for stable nuclei

Nuclear Forces Existence of stable nuclei suggests attractive force between nucleons But they do not collapse  there must be a repulsive core at very short ranges From pp-scattering, the range of nucleon- nucleon force is short which does not correspond to the exchange of gluons

Nuclear Forces Charge symmetric pp=nn Almost charge –independent pp=nn=pn – mirror nuclei, e.g. 11 B 11 C Strongly spin-dependent –Deutron exists: pn with spin-1 –pn with spin-0 does not Nuclear forces saturate (B/A is not proportional to A) V(r) +V 0 -V 0  r = R  <<R Range~R B/A ~ V 0 r 0 Approximate description of nuclear potential

Nuclei. Shapes and sizes. Scattering experiments to find out shapes and sizes Rutherford cross-section: Taking into account spin: Mott cross-section

Nuclei. Shapes and Sizes. Nucleus is not an elementary particle Spatial extension must be taken into account If – spatial charge distribution, then we define form factor as the Fourier transform of can be extracted experimentally, then found from inverse Fourier transform

In practice d  /d  falls very rapidly with angle

Shapes and sizes Parameterised form is chosen for charge distribution, form-factor is calculated from Fourier transform A fit made to the data Resulting charge distributions can be fitted by Charge density approximately constant in the nuclear interior and falls rapidly to zero at the nuclear surface c = 1.07A 1/3 fm a = 0.54 fm

Radial charge distribution of various nuclei

Shapes and sizes Mean square radius Homogeneous charged sphere is a good approximation R charge = 1.21 A 1/3 fm If instead of electrons we will use hadrons to bombard nuclei, we can probe the nuclear density of nuclei  nucl ≈ 0.17 nucleons/fm 3 R nuclear ≈ 1.2 A 1/3 fm

Liquid drop model: semi-empirical mass formula Semi-empirical formula: theoretical basis combined with fits to experimental data Assumptions –The interior mass densities are approximately equal –Total binding energies approximately proportional their masses

Semi-empirical mass formula “0 th “term 1 st correction, volume term 2d correction, surface term 3d correction, Coulomb term

Semi-empirical mass formula 4 th correction, asymmetry term Taking into account spins and Pauli principle gives 5 th correction, pairing term Pairing term maximises the binding when both Z and N are even

Semi-empirical mass formula Constants Commonly used notation a 1 = a v, a 2 = a s, a 3 = a c, a 4 =a a, a 5 = a p The constants are obtained by fitting binding energy data Numerical values a v = 15.67, a s = 17.23, a c = 0.714, a a = 93.15, a p = 11.2 All in MeV/c 2

Nuclear stability n(p) unstable:  - (  + ) decay The maximum binding energy is around Fe and Ni Fission possible for heavy nuclei –One of decay product –  -particle ( 4 He nucleus) Spontaneous fission possible for very heavy nuclei with Z  110 –Two daughters with similar masses p-unstable n-unstable

 -decay. Phenomenology Rearranging SEMF Odd-mass and even-mass nuclei lie on different parabolas

Odd-mass nuclei Electron capture 1) 2) 3)

Even-mass nuclei  emitters lifetimes vary from ms to yrs

 -decay  -decay is energetically allowed if B(2,4) > B(Z,A) – B(Z-2,A-4) Using SEMF and assuming that along stability line Z = N B(2,4) > B(Z,A) – B(Z-2,A-4) ≈ 4 dB/dA 28.3 ≈ 4(B/A – 7.7×10 -3 A) Above A=151  -decay becomes energetically possible

 -decay Lifetimes vary from 10ns to yrs (tunneling effect) TUNELLING: T = exp(-2G) G – Gamow factor G≈2  (Z-2)/  ~ Z/  E  Small differences in E , strong effect on lifetime

Spontaneous fission Two daughter nuclei are approximately equal mass (A > 100) Example: 238 U  145 La + 90 Br + 3n (156 MeV energy release) Spontaneous fission becomes dominant only for very heavy elements A  270 SEMF: if shape is not spherical it will increase surface term and decrease Coulomb term

Deformed nuclei

Spontaneous fission The change in total energy due to deformation:  E = (1/5)  2 (2a s A 2/3 – a c Z 2 A -1/3 ) If  E < 0, the deformation is energetically favourable and fission can occur This happens if Z 2 /A  2a s /a c ≈ 48 which happens for nuclei with Z > 114 and A  270