Homework-1: Spin Physics.

  1. [15 pts] Most magnetic resonance imagers operate at 1.5 Tesla, but recently clinical units have been approved that operate at 3.0 Tesla. What is the resonance frequency of the following nuclei in each of the magnetic fields?

    2.  

    1H
    23Na
    31P
     

    n = gBo
        where:
            n       =    resonance frequecy
            g       =    gymagnetic ratio for the nuclei in question
            Bo    =    magnetic field strength
     
    Resonant Frequecies
    n (MHz)
    g (MHz/T)
    Bo=1.5T
    Bo=3.0T
    1H
    42.58
    63.87
    127.7
    23Na
    11.27
    16.91
    33.8
    31P
    17.25
    25.88
    51.8
     
  2. [15 pts] What is the energy of the photon that will be absorbed by a 1H nucleus in a 1.5 Tesla magnetic field? How does this compare in energy to a 2x1019 Hz x-ray photon? What is the ionization potential for a typical organic molecule? Which of the two photons will ionize the molecule?

    3.  Given:

      Bo=1.5T
      g=42.58 MHz/T
       
    Energy absorbed by a 1H nucleus:
    EH = hn = hgB0 = 6.626 x 10-34 Js * 42.58x106 Hz/T * 1.5 T = 4.23 x 10-26 J

    Energy of a n=2 x 1019 Hz X-ray photo:
    EX = hn = 6.626 x 10-34 Js * 2 x 1019 Hz = 1.33 x 10-14 J

    How the two energies compare:
    EX / EH  = 1.33 x 10-14 J / 4.23 x 10-26 J = 3.14 x 1011
    The energy of the X-ray photon is 1011 times more than the 1H photon energy.

    Based on the figure in the notes, the ionization potential for an organic compound is approximately 6x10-19 J. Therefore only the x-ray photon can ionize the molecule.

  3. [25 pts] Based on Boltzmann statistics, which molecule in question number one will have the greater signal? At which field strength will the signal be greater? Based on natural and biological abundance, which nucleus will have the greater signal?

    4. The Bolztmann equation tells us that N-/N+ = e-DE/kT
    Where N-/N+ is the ratio of spins in the upper to those in the lower spin states. The MRI signal is proportional to

    ( N+ - N- ) = [1 - N-/N+ ] / [1 + N-/N+]
    Taking only Boltzmann statistics into account and assuming that T=310K (body temperature):

    ( N+ - N- )
    g (MHz/T)
    Bo=1.5T
    B0=3.0T
    1H
    42.58
    4.944024x10-6
    9.888048x10
    23Na
    11.27
    1.307501x10-6
    2.615002x10-6
    31P
    17.25
    2.002004x10-6
    4.004008x10-6

    In calculating the relative signals, we must take into account the natural abundances of the isotopes (NISO). In calculating the relative signals from the body we must also include the biological abundances (NBIO).

    Signal = NISO NBIO (N+ - N-)

    Relative Signals
    NISONBIO
    Bo=1.5T
    Bo=3.0T
    1H
    		99.980.63
    3.11411x10-6
    6.22822x10-6
    23Na
    		 1000.00041
    5.36075x10-10
    1.07215x10-9
    31P
    		 1000.0024
    4.80481x10-9
    9.60962x10-9

    Therefore, of the three nuclei, hydrogen will have the best signal. Of the two field strengths, 3.0 T will have the better signal. The signal will be proportional to the gyromagnetic ratio and the field strength.

  4. [15 pts] A sample has a T1 of 0.8 seconds. The net magnetization from the smaple set equal to zero and then allowed to recover towards its equilibrium value. After 1.0 seconds, what fraction of the equilibrium magnetization value will be present?

    5. Given:

      T1 = 0.8s
       
    The relationship between the equilibrium net magnetization, Mo, and the net magnetization, Mz(t), at time t is:
    Mz = Mo(1 - e-t/T1)

    Mz(t) / Mo = (1 - e-t/T1)
    Mz(t) / Mo = (1 - e-1.0/0.8)
    Mz(t) / Mo = (1 - e-1.25)
    Mz(t) / Mo = (1 - 0.2865048)
    Mz(t) / Mo = 0.71349
  5. [15 pts] A sample has a T2 of 50 ms. The net magnetization is rotated into the xy-plane and allowed to decay. Who much transverse magnetization will be present 20 ms after being placed in the plane?

    6.  Given:

      T2 = 50ms
       
    The relationship between the equilibrium net magnetization (Mxy0) and the net magnetization, Mxy(t), at time t is:
    Mxy(t) = Mxyoe-t/T2

    At 20 ms after the rotation, the amount of transverse magnetization relative to immediately after the roation:

    Mxy(t) / Mxy0 = e-20/50
    Mxy(t) / Mxy0 = e-0.4
    Mxy(t) / Mxy0 = 0.6703
  6. [15 pts] A hydrogen sample is at equilibrium in a 1.5 Tesla magnetic field. A constant B1 field of 2.34x10-4 Tesla is applied along the +x'-axis for 25 microseconds. What is the direction of the net magnetization vector after the B1 field is turned off?

    7.   Given:

      B1 = 2.34x10-4 T
      t = 25 x 10-6s
      B1 applied along the +x' axis
       
    The relationship between the rotation angle in radians (q) and the length, in seconds, that the B1 field is applied (t) is:
    q = 2p tB1g
        where:
            g       =    gymagnetic ratio for the nuclei in question
    q = 2p tB1g
    q = 2p * 25 x 10-6s * 2.34x10-4 T * 42.58MHz/T
    q = 1.565 rad.
    q = 1.565 rad. * 180o/prad = 89.673o
    The net magnetization vector will be almost along the +y' axis:  89.673o from the +z axis after the clockwise about the +x' axis.