Signal for an FID Experiment

Let's start to look at the MR signal from some simple experiments.

• look at the signal again, and think about how the signal will develop over time
  

  $$s(t) \propto \omega_{0} \int d^{3}r e^{-t/T_{2}(\vec{r})} {\cal B... ...ga - \omega (\vec{r}))t+\phi_{0}(\vec{r}) - \theta_{{\cal B}} (\vec{r})\right)}$$ (8.1)

  
  

• consider case where field is perfectly homogeneous
  

  $$s(t) \propto \omega_{0} e^{-t/T_{2}} e^{i\left( (\Omega - \omega_{0} )t+\phi_{... ..._{{\cal B}} \right)} \int d^{3}r {\cal B}_{\perp}(\vec{r}) M_{\perp}(\vec{r},0)$$ (8.2)

  
  

• in this case see two possiblities
  1. we are on-resonance and signal is just constant, decaying at rate set by $T_{2}$
  2. we are slightly off-resonance and signal has an oscillatory component in addition to $T_{2}$ decay
• 

• what if background field is also inhomogeneous??? expect signal to be sum over several different frequency components. spins will begin to interfere with each other, adding up to lower overall signal. As we will see later, random distribution of field variations will lead to an additional exponential decay rate characterized by $T'_{2}$

• Previously, referred to overall signal suppression envelope as having characteristic time constant $T_{2}^{*}$, which leads to
  

  $$\frac{1}{T_2^*} = \frac{1}{T_2}+\frac{1}{T_2^\prime}$$ (8.3)

  
  

  (IN-CLASS QUESTION)