Fundamentals - Preventive Biomechanics - page 17

Fig. 3.2 a Recovery of the longitudinal component of magnetization from the initial value
M z ð t 0 Þ to the equilibrium value M 0 at time t !1; b decay of the transverse magnetization

M ð t Þ¼ e T 2 f½ M x ð 0 Þ cosx L t M y ð 0 Þ sinx L t e x þ½ M y ð 0 Þ cosx L t þ M x ð 0 Þ sinx L t e y g
|{z}
M ? ð t Þ

þ½ M Z ð 0 Þ e T 1 þ M 0 ð 1 e T 1 Þ e z
|{z}
M jj ð t Þ

ð 3 : 21 Þ

or

M ð t Þ¼ e T 2 f½ M x ð 0 Þ e x þ M y ð 0 Þ e y cos x L t þ½ M x ð 0 Þ e x M y ð 0 Þ e y sin x L t g
| {z }
M ? ð t Þ

þf M 0 ½ M 0 M z ð 0 Þ e T 1 g e z
|{z}
M jj ð t Þ

: ð 3 : 22 Þ

The stationary solutions for t !1 derive from ( 3.20 ), ( 3.21 ) and ( 3.22 ) to (cf.
also Fig. 3.2 )

t !1 M x ð t Þ¼ 0 ;

lim

t !1 M y ð t Þ¼ 0 ;

lim

t !1 M z ð t Þ¼ M 0

lim ð 3 : 23 Þ

or in vector notation

t !1 M ð t Þ¼ M 0 e z : ð 3 : 24 Þ
Equation ( 3.22 ) describes the regrowth and decay of the net magnetization
vector in the longitudinal and transversal directions, respectively. Figure 3.1
illustrates the trajectory of the magnetization vector in the fixed reference frame.
Equations ( 3.19 ) describe the motion of the magnetic dipole moments when
initially being rotated away from the static magnetic field's longitudinal (z-)
direction and returning to the equilibrium state. Whereas ( 3.19 ) 3 describes the
recovery of the longitudinal magnetization in time, ( 3.19 a) 1 and ( 3.19 b) 2 represent

lim