Quantum Mechanical Examination of Longitudinal Relaxation

• Another thing that Quantum Mechanics can give us is the probability that a particle is going to jump states
  

  $$W_{fi} = \frac{2\pi}{\hbar} \left\vert <f\vert V\vert i>\right\vert^{2}\rho(E_{f})$$ (6.9)

  
  
  Don't want do anything with this equation at this time other than to notice that we can calculate the trasition probability

• Model that we are going to work with is that there are spins and there is a lattice. The spins get back to equilibrium by trading energy with the lattice. So whenever a spin does something, the opposite transition must occur in the lattice.
  
  

  
  
  
  
  
  
  
  

• Therefore, the rate of change of excess spins depends upon how the spin-lattice system jumps from state to state
  

  $$\frac{dN_{+}}{d t} = W_{h+,l-}N_{-}n_{l} - W_{l-,h+}N_{+}n_{h}$$ (6.10)

  
  

  
  
  
  
  
  
  
  

• If we did lowest order pertubation theory, would find out that $W$s are same
  

  $$W_{h+,l-} =W_{l-,h+}\equiv W$$ (6.11)

  
  

• Finally, write the number of spins in each state in terms of the total number of spins $\pm$ the difference between the numbers of spins in each state $\Delta N$(IN CLASS PROBLEM)
  

  $$N_{\pm} = \frac{N \pm \Delta N}{2}$$ (6.12)

  
  
  
  
  
  
  
  
  
  

  Combining the previous equations (6.10), (6.11) and (6.12), gives an equation for the time rate of change of the difference between up and down spins
  

  $$\frac{d(\Delta N)}{d t} = WN(n_{l} - n_{h}) - W \Delta N (n_{l} + n_{h})$$ (6.13)

  
  

  
  
  
  
  
  
  
  

• In steady state (when all spins are aligned with main magnet), time derivative goes to zero, and can determine the equilibrium spin population difference
  

  $$(\Delta N)_0 = \frac{(n_{l} - n_{h})}{(n_{l} + n_{h})}N \hspace{0.3in}{\rm (equilibrium)}$$ (6.14)

  
  

  
  
  
  
  
  
  
  

• Can make this look very familiar by multiplying the RHS by $(n_{l} + n_{h})/(n_{l} + n_{h})$
  

  $$\frac{d(\Delta N)}{d t} = W (n_{l} + n_{h}) \left[ N \frac{(n_{l} - n_{h})}{(n_{l} + n_{h})} - \Delta N \right]$$

  $$= W (n_{l} + n_{h}) \left[ \Delta N_{0} -\Delta N \right]$$

  $$= \frac{1}{T_{1}} \left[ \Delta N_{0} -\Delta N \right]$$

  $$\; \mbox{or} \frac{d{M_z}}{dt} = \frac{1}{T_{1}} \left[ M_{0} - M_{z} \right]$$ (6.15)

  
  

  (IN CLASS PROBLEM)

  
  
  
  
  
  
  
  

• Reinforce approach taken before in determining a rate equation for longitudingal relaxation. (IN CLASS PROBLEM)

HWS 7.2 7.4 7.9 8.3 8.6 8.9