Using $B_1$ to Tip the Spins

• Look at the equation of motion from Chapter 2, but look in the rotating reference frame.

  
  

  $$\left( \frac{d \vec{\mu}}{dt}\right)' = \vec{\mu} \times \left(\gamma \vec{B}_0 + \gamma \vec{B}_1 + \vec{\Omega}\right)$$ (3.15)

  
  

  choose $\vec{\Omega} = -\gamma \vec{B}_0 = -\gamma B_0 \hat{z}$ to end up with

  
  

  $$\left( \frac {d \vec{\mu}} {dt}\right)' = \vec{\mu} \times \gamma \vec{B}_1$$

  $$= \vec{\mu} \times \gamma B_1 \hat{x}'$$ (3.16)

  
  

• 

• Get same solution as in Chapter 2, but turned on side to rotate about $\vec{B}_1$
  

  $$\mu_{x'}(t) = \mu_{x'}(0)$$

  $$\mu_{y'}(t) = \mu_{y'}(0) \cos{\phi_{1}(t)} +\mu_{z'}(0)\sin{\phi_{1}(t)}$$

  $$\mu_{z'}(t) = -\mu_{y'}(0) \sin{\phi_{1}(t)} +\mu_{z'}(0)\cos{\phi_{1}(t)}$$ (3.17)

  
  
  where, if $B_1(t) = B_1$
  

  $$\phi_{1}(t) = \omega_{1}t = \gamma B_1 t$$ (3.18)

  
  
  If $B_1$ not constant in time, then need to solve an integral to determine the flip angle.

• can do rough calculations on how long it will take to flip spins as a function of $B_1$ (IN CLASS PROBLEM)

  
  
  
  
  
  
  
  

• If not on resonance that things can go bad quickly. Will make use of this later. (IN CLASS PROBLEM)