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• Spins will find ways to shed energy, stop precessing and align themselves with the main magnetic field.

  
  
  
  
  
  
  
  

• Assume that each spin has same likelihood of getting back to equilibrium, therefore the rate at which spins are returning to equilibrium is proportional to the number of spins that aren't in equilibrium.

• leads time rate of change of the parallel magnetization
  

  $$\frac{dM_{z}}{dt} = \frac{1}{T_{1}} (M_{0} - M_{z}) \hspace{1.0in} (\vec{B}_{ext} \parallel \hat{z}) \\$$ (4.5)

  
  
  where $1/T_{1}$ is a rate constant that is going to be different for different tissues (CONTRAST) and depends upon how easy it is for spins to give up energy so that they can get back to equilibrium.

• (see HW) see that solution of this equation is going to involve exponentials again. Figure out specifics of solution for HW
  

  $$M_{z}(t) = M_{z}(0) e^{-t/T_{1}} + M_{0} (1 - e^{-t/T_{1}}) \hspace{0.5in} (\vec{B}_{ext} \parallel \hat{z}) \\$$ (4.6)

  
  

• What does this look like ?
  
  

  
  
  
  
  
  
  
  

• Important to understand that this is letting us know how much magnetization will be available when we take multiple measurements in an MRI experiment. (IN CLASS QUESTION)