Motion of Magnetization

The measurable, macroscopic magnetization M is composed of nuclear magnetic moments. The macroscopic magnetization M can be modified or changed by magnetic fields (B0 or B1 field) and by relaxation processes (also driven by magnetic fields). The equation of motion of M(t) can be described by the Bloch equation:
If B(t) is divided into time intervals with constant amplitude Bloch`s differential equation can be integrated, and the change of M(t) can be described by 3x3 rotation matrices and a relaxation matrix:
a is the precesion angle given by α = γ|B|t, and bx, by, bz are the spatial components of the normalized (unit length) vector B. Relaxation is described by the following equation, with E1=exp(-t/T1) and E2=exp(-t/T2).
The series below depicts the motion (precession) of M for constant B = (√2,0,√2). The precession frequency of M is proportional to the magnitude of B. The magnetic field B is shown in white, the magnetization M is yellow.

Motion in a B1 field

Circularly polarized RF-fields produce a rotating magnetic field B1(t) within the transverse x-y plane. In most cases motion of magnetization is described in the rotating frame that rotates with the Larmor frequency ω=-γB0. If the frequency of the RF-field is equal to the Larmor frequency ω B1(t) is a constant vector within the x-y plane. The direction of B1(t) is given by the phase of the RF-field. If the frequency of B1(t) is not equal to the Larmor frequency B1(t) will have an additional component along z direction (in the rotating frame). This case is visualized in the part RF pulses.

The series below depicts the motion of M during an RF-pulse along x direction (B = (1,0,0), initial M = (0,0,1), rotating frame).

Motion in a B0 field

In the rotating frame the magnetization M does not precess without additional magnetic fields. Magnetic gradient fields produce an additional magnetic field oriented along z direction. The magnitude of these gradient fields varies linearly along x, y, or z direction. Deviation of the local magnetic field from the main B0 field leads to a precession of the magnetization M around the z axis.

The series below depicts the motion of M for an additional B0 field along z direction (B = (0,0,1), initial M = (1,0,0), rotating frame)

T1 and T2 relaxation

T1 relaxation leads to an exponential relaxation of the z component of the magnetization M towards its equilibrium value Mz=M0. T2 relaxation gives an exponential decay of the transverse component Mxy towards zero.

In most cases T1 is larger that T2 (except for tetrachlorocyclopropene at –89°C, see Anet Chem.Phys.Lett 1990;171:401-405) to  guaranty that |M(t)|  ≤ M0.

Motion of M during T1 and T2 relaxation (initial M = (1,0,0), T1 = 2T2, rotating frame)

Last updated: Monday, 18.04.2011