Problem 3.4 A static field points uniformly along the positive z-axis. A classical spinning par- ticle, with positive gyromagnetic ratio y and fixed magnetic moment magnitude , has its spin initially in the direction of the static field. A circularly polarized rf field points along the axis with time-dependent amplitude Big/(t) (e.g., the rf field may be turned off at a later time) applied on-resonance starting at t 0. a) Give expressions analogous to (3.33) for all three magnetic-moment vector components in the rotating (prime) reference frame for time t > 0. Your answer will be in terms of a definite integral b) Show that the equation of motion (2.24) is satisfied by your answers in (a) for BBly c) Find the generalization of (2.35) needed for this time-dependent case. dt2 (2.35) dt2

Question

textbook: Magnetic resonance imaging, physical principles and sequence design.

Problem 3.4

eq 3.33 : μx′(t) = μx′(0)
μy′(t) = μy′(0) cos φ1(t) + μz′(0) sin φ1(t)
μz′(t) = −μy′(0) sin φ1(t) + μz′(0) cos φ1(t)

eq 2.24: d~μ/dt
= γ~μ × ~B

Problem 3.4
A static field points uniformly along the positive z-axis. A classical spinning par-
ticle, with positive gyromagnetic ratio y and fixed magnetic moment magnitude
, has its spin initially in the direction of the static field. A circularly polarized
rf field points along the axis with time-dependent amplitude Big/(t) (e.g., the
rf field may be turned off at a later time) applied on-resonance starting at t 0.
a) Give expressions analogous to (3.33) for all three magnetic-moment vector
components in the rotating (prime) reference frame for time t > 0. Your
answer will be in terms of a definite integral
b) Show that the equation of motion (2.24) is satisfied by your answers in (a)
for BBly
c) Find the generalization of (2.35) needed for this time-dependent case.
Expand
Transcribed Image Text

Problem 3.4 A static field points uniformly along the positive z-axis. A classical spinning par- ticle, with positive gyromagnetic ratio y and fixed magnetic moment magnitude , has its spin initially in the direction of the static field. A circularly polarized rf field points along the axis with time-dependent amplitude Big/(t) (e.g., the rf field may be turned off at a later time) applied on-resonance starting at t 0. a) Give expressions analogous to (3.33) for all three magnetic-moment vector components in the rotating (prime) reference frame for time t > 0. Your answer will be in terms of a definite integral b) Show that the equation of motion (2.24) is satisfied by your answers in (a) for BBly c) Find the generalization of (2.35) needed for this time-dependent case.

dt2
(2.35)
dt2
Expand
Transcribed Image Text

dt2 (2.35) dt2

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