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Both in NMR and MRI technology always exists a static magnetic field denoted by {EQUATION()}$\mathbf{B_0}${EQUATION}, so the Larmor precession caused by this static field is somewhat trivial. Therefore, it would be desireable to eliminate this fundamental precession movement from the mathematical description. In general, this is achieved by introducing a rotating coordinate system , or with a more intuitive expression, by "rotating together with the spins".
We will use a coordinate system that rotates with angular frequency {EQUATION()}$\boldsymbol{\Omega}${EQUATION}. Mathematically the connection between the time derivative of a dynamic vector {EQUATION()}$\mathbf{b} (t)${EQUATION} in such a coordinate system and in a static system can be described by the following:
::{EQUATION()}
\label{vector_rotating_system}
\frac{ \mathrm{d} \mathbf{b}} { \mathrm{d} t } \bigg | _{\mathrm{static}} = \frac{ \mathrm{d} \mathbf{b}} { \mathrm{d} t } \bigg | _{\mathrm{rotating}} + \boldsymbol{\Omega} \times \mathbf{b}
{EQUATION}(1)::
Using (1) for the time derivative of the magnetic moment in the static field {EQUATION()}$\mathbf{B_0}${EQUATION} we get:
::{EQUATION()}
\label{moment_static_system}
\gamma \boldsymbol{\mu} \times \mathbf{B_0} = \frac{ \mathrm{d} \boldsymbol{\mu}} { \mathrm{d} t } \bigg | _{\mathrm{static}} = \frac{ \mathrm{d} \boldsymbol{\mu}} { \mathrm{d} t } \bigg | _{\mathrm{rotating}} + \boldsymbol{\Omega} \times \boldsymbol{\mu}
{EQUATION}(2)::
From this the time derivative in the rotating system:
::{EQUATION()}
\label{moment_rotating_system}
\frac{ \mathrm{d} \boldsymbol{\mu}} { \mathrm{d} t } \bigg | _{\mathrm{rotating}} = \gamma \boldsymbol{\mu} \times \mathbf{B_0} + \boldsymbol{\mu} \times \boldsymbol{\Omega} = \gamma \boldsymbol{\mu} \times \underbrace{ \bigg ( \mathbf{B_0} + \frac{ \boldsymbol{ \Omega } } { \gamma} \bigg ) }_{ \mathbf{ B_{eff}} }
{EQUATION}(3)::
As can be seen from above, the introduction of a coordinate system rotating with angular frequency {EQUATION()}$\boldsymbol{\Omega}${EQUATION} is equivalent to the alternation of the static magnetic field by {EQUATION()}$\frac{\boldsymbol{\Omega}}{\gamma}${EQUATION}. The result is called the effective magnetic field, and it has a concrete physical meaning: in the rotating system the spins seem to move like they were precessing around this effective field {EQUATION()}$\mathbf{B_{eff}}${EQUATION}.
The reader now might see one of the benefits of this description method: if we choose the angular frequency of the rotating system to be equal to the Larmor frequency, that is, if {EQUATION()}$\boldsymbol{\Omega} = \boldsymbol{\omega_0}${EQUATION}, then in this system the spins remain still, or in other words, the effective magnetic field becomes zero:
::{EQUATION()}
\label{rotating_Larmor}
\frac{ \mathrm{d} \boldsymbol{\mu}} { \mathrm{d} t } \bigg | _{\mathrm{rotating}} = \gamma \boldsymbol{\mu} \times \underbrace{ \bigg ( \mathbf{B_0} + \frac{ \boldsymbol{ \omega_0 } } { \gamma} \bigg ) }_{ \mathbf{ B_{eff}} } = \gamma \boldsymbol{\mu} \times \underbrace{ \bigg ( \mathbf{B_0} - \frac{ \gamma \mathbf{B_0} } { \gamma} \bigg ) }_{ \mathbf{ B_{eff}} } = \mathbf{0}
{EQUATION}(4)::
From now on, unless marked, we will continue the mathematical description in a rotating system.
Next, we will deal with the usual process of manipulating spins: the radiofrequency or RF excitation. First we assume that beyond the static field {EQUATION()}$\mathbf{B_0}${EQUATION} we have a linearly polarized magnetic field denoted by {EQUATION()}$\mathbf{B^{lin}_1}(t)${EQUATION} oscillating in the plane orthogonal to {EQUATION()}$\mathbf{B_0}${EQUATION} (called transverse plane) with an angular frequency {EQUATION()}$\boldsymbol{\omega}${EQUATION} and has an amplitude {EQUATION()}$b_1${EQUATION}. In the static system, this field can be written as
::{EQUATION()}
\label{circular_RF}
\mathbf{B^{lin}_1}(t) = b_1 \mathrm{cos}(\omega t) \mathbf{e_x}
{EQUATION}(5)::
Where {EQUATION()}$\mathbf{e_x}${EQUATION} refers to the unit vector at the {EQUATION()}$x${EQUATION} direction in the static system. For the rotating system, we will use the notation {EQUATION()}$\mathbf{e'_x}${EQUATION} in the same role.
Note that the directions e.g. {EQUATION()}$x'${EQUATION} or {EQUATION()}$y'${EQUATION} in the rotating frame are equivalent to a phase shift in the labor frame. For example, suppose that in a specific moment the {EQUATION()}$x${EQUATION} direction in the labor frame and the {EQUATION()}$x'${EQUATION} direction in the rotating frame are equivalent. If we now start to apply a circularly polarized RF field, with a frequency equal to the one of the rotating system and a starting direction equal to the {EQUATION()}$x${EQUATION} or {EQUATION()}$x'${EQUATION} direction then in the rotating frame the {EQUATION()}$\mathbf{B}${EQUATION} vector continiues to point to the {EQUATION()}$x'${EQUATION} direction as the frame and the field vector co-rotates. However, in the laboratory frame the field vector will of course turn. If we wait for fourth of the period of time of the rotating system, then at the moment after that the {EQUATION()}$x${EQUATION} direction of the laboratory frame will be equal to the {EQUATION()}$y'${EQUATION} direction of the rotating frame as the latter turned with 90° in this time. If we apply another circularly polarized RF pulse with the same initial direction it will also co-rotate with the rotating system but will point to its {EQUATION()}$y'$(){EQUATION} direction.
In this way we managed to apply two RF pulses with an angle of 90° between them in the rotating frame by shifting the latter with the fourth of the rotation period, i. e. with 90° phase difference. Using this time-shifting we are able to create RF pulses with arbitrary constant direction in the rotating frame by shifting them in time in the laboratory frame. The concept is shown in Figure 1.
{img fileId="3168" thumb="y" width="800" imalign="center" align="center" desc=Figure 1. Resonant pulses in the laboratory and the rotating frames. a) pulse in the x' direction of the rotating frame. b) pulse remains still in the rotating system and turns in the laboratory frame as time elapses. c) after a time fourth of the rotation period we apply another pulse with the same initial direction in the labor frame, along the y' direction in the rotating system. d) the two pulses remain still in the rotating system with a constant 90° angle between them.}
For simpler derivation, let us now define the rotating reference frame using the RF frequency {EQUATION()}$\boldsymbol{\omega}${EQUATION}, meaning {EQUATION()}$\boldsymbol{\Omega} = \boldsymbol{\omega}${EQUATION}. To write {EQUATION()}$\mathbf{B^{lin}_1}(t)${EQUATION} in the rotating system, we first express the unit vectors of the static system with the unit vectors of the rotating system. In other words:
::{EQUATION()}
\label{static_e_x}
\mathbf{e_x} = \mathbf{e'_x} \mathrm{cos}(\omega t) + \mathbf{e'_y}\mathrm{sin}(\omega t)
{EQUATION}(6)::
::{EQUATION()}
\label{static_e_y}
\mathbf{e_y} = - \mathbf{e'_x} \mathrm{sin}(\omega t) + \mathbf{e'_y}\mathrm{cos}(\omega t)
{EQUATION}(7)::
::{EQUATION()}
\label{static_e_z}
\mathbf{e_z} = \mathbf{e'_z}
{EQUATION}(8)::
Using (5) and (6) we can obtain the form of {EQUATION()}$\mathbf{B^{lin}_1}(t)${EQUATION} in the rotating system:
::{EQUATION()}
\label{b1_lin_rotate}
\mathbf{B^{lin}_1}(t) = b_1 \mathrm{cos}(\omega t) \big ( \mathbf{e'_x} \mathrm{cos}(\omega t) + \mathbf{e'_y}\mathrm{sin}(\omega t) \big )
= \frac{1}{2} b_1 \Big ( \mathbf{e'_x} \big [ 1 + \mathrm{cos} (2 \omega t) \big ] + \mathbf{e'_y} \big [ \mathrm{sin} (2 \omega t) \big ] \Big )
{EQUATION}(9)::
In (9) the effect of the terms with frequency {EQUATION()}$2 \omega${EQUATION} is averaged out for all times large compared to the RF period. Because of this, the effect of the RF field in the rotating reference can be substituted with its time averaged version:
::{EQUATION()}
\label{b1_lin_rotate_time_average}
\Big < \mathbf{B^{lin}_1}(t) \Big > = \frac{1}{2} b_1 \mathbf{e'_x}
{EQUATION}(10)::
As can be seen, the linearly polarized field will act as static in the rotating reference. Here we notice that the effect of a circularly polarized RF field would be almost the same, the only difference is that the factor {EQUATION()}$\frac{1}{2}${EQUATION} would be missing.
Now that we know the behaviour of simple RF fields in the rotating reference, let's take a look at the spin movements under an applied RF field. Suppose that we have a static field {EQUATION()}$\mathbf{B_0}${EQUATION}, a circularly polarized field {EQUATION()}$\mathbf{B_1}${EQUATION} described as above with frequency {EQUATION()}$\boldsymbol{\omega}${EQUATION}, and let's use the rotating reference also with frequency {EQUATION()}$\boldsymbol{\omega}${EQUATION}. The equation of motion for the spins is then the following:
::{EQUATION()}
\label{eq_motion_RF}
\frac{ \mathrm{d} \boldsymbol{\mu}} { \mathrm{d} t } \bigg | _{\mathrm{rotating}} = \gamma \boldsymbol{\mu} \times \mathbf{B_{eff}} = \gamma \boldsymbol{\mu} \times \Big ( \mathbf{B_0} + \frac{\boldsymbol{\omega}}{\gamma} + \mathbf{B_1} \Big )
{EQUATION}(11)::
To determine the allover angular frequency of the spins, take a look at the components of the effective magnetic field:
::{EQUATION()}
\label{B_eff_with_RF}
\mathbf{B_{eff}} = \mathbf{e'_z} \bigg ( B_0 + \frac{\omega}{\gamma} \bigg ) + \mathbf{e'_x} b_1
{EQUATION}(12)::
From this we can easily get the effective angular momentum of the spins in the rotating reference, as it equals to {EQUATION()}$- \gamma \mathbf{B_{eff}}${EQUATION}. For the consequent notation, we define another angular frequency-like quantity: {EQUATION()}$\omega_1 \equiv -\gamma b_1${EQUATION}. Note: this it NOT the same as the RF frequency {EQUATION()}$\omega${EQUATION}. From here, the derivation of the effective angular frequency of the spins is straightforward:
::{EQUATION()}
\label{omega_with_RF}
\boldsymbol{\omega}_\mathbf{eff} = \mathbf{e'_z} \Big (-\gamma B_0 - \omega \Big ) -\mathbf{e'_x} \gamma b_1 = \mathbf{e'_z} \big ( \omega_0 - \omega \big ) + \mathbf{e'_x} \omega_1
{EQUATION}(13)::
Just to be clear, we hereby repeat the meaning of all {EQUATION()}$\omega${EQUATION}-s in (13): {EQUATION()}$\omega_0${EQUATION} is the precessing frequency around {EQUATION()}$\mathbf{B_0}${EQUATION} (also called Larmor frequency), {EQUATION()}$\omega${EQUATION} is the frequency of the RF field and also of the rotating reference, and finally, {EQUATION()}$\omega_1${EQUATION} equals to {EQUATION()}$-\gamma b()_1${EQUATION}.
The case when the Larmor frequency equals to the RF frequency, that is, {EQUATION()}$\omega_0 = \omega${EQUATION} is called resonance. If this occurs, the situation becomes quite simple:
::{EQUATION()}
\label{B_eff_on_resonance}
\mathbf{B_{eff}} = \mathbf{e'_x} b_1
{EQUATION}(14)::
::{EQUATION()}
\label{omega_on_resonance}
\frac{ \mathrm{d} \boldsymbol{\mu}} { \mathrm{d} t } \bigg | _{\mathrm{rotating}} = \gamma \boldsymbol{\mu} \times \mathbf{B_{eff}} = \boldsymbol{\mu} \times \mathbf{e'_x} \gamma b_1 = - \omega_1 \boldsymbol{\mu} \times \mathbf{e'_x}
{EQUATION}(15)::
Equation (15) desrcibes a precession solely around the {EQUATION()}$x'${EQUATION} axis of the rotating frame. In other words, we can rotate the spins around the {EQUATION()}$x'${EQUATION} axis of the rotating system by applying an oscillating magnetic field on the Larmor frequency. The angle alteration after such a pulse lasts for time {EQUATION()}$\tau${EQUATION} is {EQUATION()}$\Delta \theta${EQUATION} as can be seen in (16) and in Figure 2:
::{EQUATION()}
\label{delta_theta}
\Delta \theta = \gamma b_1 \tau
{EQUATION}(16)::
{img fileId="3169" thumb="y" width="240" imalign="center" align="center" desc=Figure 2. Rotation of the magnetic moment around the x' axis in the rotating frame caused by resonant RF field.}