Nuclear Physics

• 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.