Nuclear Physics

• Conservation of energy and momentum
• Conservation of charge
• Conservation of
  • Conservation of the number of nucleons. This is not universal.

Activity \(\alpha\) the average number of decays per unit time. See units of activity. As decay is spontaneous process, \[ \alpha(t) = \lambda N(t) \] where \(\lambda\) is the decay constant, and \(N(t)\) is the number of atoms.

With only a single decay path is available, the number of atoms decay exponentially: \[ \dv{N}{t} = -\lambda N(t) \implies N(t) = N_0 e^{-\lambda t}. \]

The half-life is then directly related to the decay constant \(T_{1/2} = \ln 2/\lambda\). and decay factor is the fraction of atoms that remains, namely \(e^{-\lambda t}\).

The difference in the rest energy before and after the decay is called the Q-value.

Useful to know that \[ c^2 \simeq 931 \text{ MeV/amu}. \]

Binding energy of a nucleus \(a\) is given by \[ \mathrm{BE} := Zm_p + Nm_n - m_a. \]

The binding energy per nucleon is highest at Fe.

Separation energy is the binding energy of a particular nucleon. It is measured similar to ionization energy. For the least bound neutron, sometimes called the last neutron is \[ E_S = [m_n + m(^{A-1}Z) - m(^AZ)]c^2 \]

It is a probabilistic process of tunneling out of the potential well of the strong force.

The element above the iron is technically unstable, but the energy of the alpha particle within the neucleus is so small that the probalitity of tunneling is minute compared to the highly unstable ones.

Beta decay occurs in the direction that stabilizes the nucleus. According to shell model, excess nucleon converts into the other and stabilizes the overall configuration.

• Chart of the Nuclides

The matrix with proton number \(Z\) on one axis (usually y-axis) and neutron number on the othe axis.

Separate shells for protons and neutrons, just like the electron shells in an atom.

This causes the stability of nucleus when there are roughly the same of neutrons and protons. For heavy nuclei, more nuetron is needed in order to overcome the coulombic repulsion between protons.

Spin-orbit coupling is much more significant in nucleus on the order of 1 to 10 MeV. The nucleons have strong tendency to pair up with opposite total angular momentum nucleons due to the strong nuclear interaction. For example in \(j = 5/2\) shell, 5/2 pairs with -5/2, 3/2 pairs with -3/2, 1/2 pairs with -1/2.

There only exists four stable odd-odd nuclides \(\rm ^2_1H, ^6_3Li, ^{10}_5B, ^{14}_8N\), among 165 stable nuclides, due to the strong tendency of nucleons to pair up.

The analogue to a liguid droplet.

This model gives the semi-empirical mass equation: \[ m = m_nN + m_pZ -\alpha A + \beta A^{2/3} + \gamma \frac{Z^2}{A^{1/3}} + \zeta \frac{(A - 2Z)^2}{A} + \delta \] where the third term is accounting for strong force, fourth term is for surface tension, fifth term is for coulombic force, sixth term is for the unstability due to imbalanced number of protons and nuetons, and seventh term is pairing term, it is positive if both are odd, negative if both are even, zero otherwise.

The coefficients are fitted to be:

• Magnetic field aligns nuclei.
• RF pulse tips nuclei
• Nuclei precess giving off RF signal (NMR)
  • The time for the signal to die off is \(T_2\), so this process is called \(T_2\) relaxation
• Nuclei realign with the field
  • \(T_1\) relaxation

\[ \vec{\mu}=\hbar\vec{S}\gamma \] where \(\vec{\mu}\) is the magnetic moment, \(\vec{S}\) is the spin and \(\gamma\) is the gyromagnetic ratio.

• Obtained by solving the Bloch equation.

\[ \omega = -\gamma B \] where \(\omega\) is the angular frequency at which angular momentum vector precesses, \(\gamma\) is the gyromagnetic ratio and \(B\) is the magnitude of magnetic field.

\[ P=\frac{N_{\uparrow}-N_{\downarrow}}{N_\uparrow+N_\downarrow} \] \[ N_i=N_{\rm total}\frac{e^{-E_i/kT}}{\sum_j e^{-E_j/kT}} \] \[ E_\downarrow = \hbar\omega/2\quad E_\uparrow = -\hbar\omega/2 \]

With a small angle approximation, \[ P \approx \frac{\hbar\gamma B}{2kT} \]

The signal strength in thermal equilibrium is \[ S=\frac{N\gamma^3\hbar^2B_0^2}{2kT} \]

The signal \(S(t)\) produced by precesssing nuclei is \[ S(t)=NP\sin(\theta)\gamma B\cos(\omega t)e^{-t/T_2} \] where \(N\) is the total number of nuclei, \(P\) is the polarization, \(\theta\) is the flip angle.

• Nuclear magnetization in thermal equilibrium.

\[ M_0=\sum \vec{\mu}=\mu N P\equiv \chi_0 B_0 \] where \(\chi_0\) is the magnetic nuclear susceptability.

A small inhomogeneity in the magnetic field can leads to a rapid decay.

\[ M(t)=\mu\int f(\omega) e^{-i\omega t} d\omega \] where \(f(\omega)\) is the Lorentzian distribution that represent \(\omega\) frequency distribution.

Each nucleus has \(\omega=\omega_{\text{Larmor}}+\Delta \omega_{T_2 \text{ decay}} + \Delta\omega_{B_0 \text{ inhomogeneity}}\).

Experimental decay rate \(R^*\) is given by \[ R^*=\frac{1}{T_2}+\gamma \Delta B \]

Transmit another pulse that flips the spins 180 degrees. The spin echo is observed after the same amount of time as the time from the time of aligning to the pulse.