Problem Set #1 Rad 226a/BioE 326a
D. Spielman
[email protected]
1. Deriving the Bloch Equations.
a) Starting with
!
!
dM (t ) !
= M ( t ) × γ B ( t ),
dt
!
!
!
!
B (t ) = B1x x + B1y y + B0 z
Derive the Bloch equations in the laboratory frame without relaxation. Namely:
dM x (t )
= γ "# M y (t ) B0 − M z (t ) B1y $%
dt
dM y (t )
= γ "# M z (t ) B1x − M x (t ) B0 $%
dt
dM z (t )
= γ "# M x (t ) B1y − M y (t ) B1x $%
dt
b) Starting from the full Bloch equations in the laboratory frame:
dM x (t )
M (t )
= γ "# M y (t ) B0 − M z (t ) B1y $% − x
dt
T2
dM y (t )
M (t )
= γ "# M z (t ) B1x − M x (t ) B0 $% − y
dt
T2
dM z (t )
M (t ) − M 0
= γ "# M x (t ) B1y − M y (t ) B1x $% − z
dt
T1
Derive the Bloch equations for a frame of reference rotating at a frequency ω
clockwise about the z axis (left handed rotation).
c) Show that the following are solutions to the Bloch equations in the rotating
frame, assuming
! initial! magnetization in the transverse plane and free induction
decay (i.e. : B (t ) = B0 z ) :
−
M x (t ) = M 0 cos⎡⎣(γ B0 − ω ) t + φ ⎤⎦ e
t
M y (t ) = −M 0 sin ⎡⎣(γ B0 − ω ) t + φ ⎤⎦ e
− t T2
(
M z (t ) = M 0 1− e
− t T1
)
T2
D. Spielman
[email protected]
Problem Set #1 Rad 226a/BioE 326a
2. The Ernst Angle (de Graaf, problem 1.7)
Consider a pulse-acquire experiment consisting of an α-degree RF pulse
followed by a recovery time of TR.
a) 
Starting with the Bloch equations including T1 relaxation, derive an
expression for the recovery of the longitudinal magnetization following
a perturbation.
b)  Derive the following expression for the steady-state longitudinal
magnetization for the pulse-acquire experiment:
M z (α,T1 ) =
c) 
(
M0 1 − e
− TR T1
)
(1 − cos αe )
− TR T1
After how many Rf pulses is the longitudinal magnetization within 1%
of the steady-state magnetization when α = 40º and TR = T1?
€
d)  Find the Ernst angle (RF flip angle yielding maximum SNR per unit
time) from the steady-state expression given above.
D. Spielman
[email protected]
Problem Set #1 (cont.)
Rad 226a/BioE 326a
3. Nuclear Induction. This question concerns the paper by F. Bloch,
“Nuclear Induction”, Physical Review, 76:460-474, 1946.
a) Verify equation 16. What is the physical interpretation for this value of θ?
b)  Which of the following concepts are discussed in Bloch’s 1946 paper?
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
Continuous wave (CW) NMR
Pulsed NMR
Prepolarized NMR
Quadrature excitation
Adiabatic excitation
T1
T2
Doping samples with small amounts of paramagnetic material to reduce T1
Imaging
4. Block Equations. A sample of water is left to reach thermal equilibrium
in zero magnetic field. A magnetic field is then turned on in the x-direction.
After some time, the magnetic field is switched from the x-direction to the zdirection. Describe the behavior of the net 1H magnetic moment from the
sample over time. Can you describe the behaviour of an individual nuclear
spin? Can you explain T1 and T2?
Bz
Bx
Mx
?
My
?
Mz
?
time