Precession-classical description

In the classical way the first step of investigating the interaction of a spin with external magnetic field is to derive the classical connection between angular momentum and magnetic moment. The simplest model for this is the circulating point charge as shown in Figure 1.

Figure 1. Circulating point charge with charge q and velocity v.

For the magnetic moment, we will use the commonly known classical expression:

$\label{magnetic_moment_point} \boldsymbol{\mu} = \frac{1}{2} \int \mathbf{r} \times \mathbf{J} ( \mathbf{r} ) \mathrm{d^3r} = \frac{1}{2} q \mathbf{r} \times \mathbf{v} = \frac{1}{2} q v r \mathbf{n}$ (1)

Whilst for the angular momentum:

$\label{angular_moment_point} \mathbf{I} = m \mathbf{r} \times \mathbf{v} = m v r \mathbf{n}$ (2)

In (1) and (2) $\boldsymbol{\mu}$ stands for the magnetic moment, $\mathbf{J} ( \mathbf{r} )$ for the electric current density, $\mathbf{n}$ for the normal vector of the circulatory plane, and $\mathbf{I}$ for the angular momentum.

If we have not just a single point charge but an extensive rotating body with spatially varying electric charge density $\sigma ( \mathbf{r} )$ and mass density $\rho ( \mathbf{r} )$ then the expressions are as follows:

$\label{magnetic_moment_extended} \boldsymbol{\mu} = \frac{1}{2} \int \mathbf{r} \times \mathbf{J} ( \mathbf{r} ) \mathrm{d^3r} = \frac{1}{2} \int \mathbf{r} \times \sigma ( \mathbf{r} ) \mathbf{v} ( \mathbf{r} ) \mathrm{d^3r}$ (3)

$\label{angular_moment_extended} \mathbf{I} = \int \mathbf{r} \times \rho ( \mathbf{r} ) \mathbf{v} ( \mathbf{r} ) \mathrm{d^3r}$ (4)

From equations (3) and (4) follows that if we assume our extended body to have mass density and electric charge density to be proportional, which means they vary the same way in space, then the magnetic moment and the angular momentum also become proportional:

$\label{angular_and_magnetic_moment_ratio} \boldsymbol{\mu} = \frac{q}{2m} \mathbf{I} = \gamma \mathbf{I}$ (5)

Where $\gamma$ is the so-called gyromagnetic ratio which plays vital role in magnetic resonance. Its value for proton is $2\pi \times 42.58 MHz/sT = 267.54 Mrad/sT.$

To derive the equation of motion of a spin in external field we will use another two formulas: the connection between angular momentum and torque (the latter denoted by $\mathbf{N}$ ), and the torque that applies to a magnetic moment in external magnetic field.

$\label{angular_moment_torque} \frac{ \mathrm{d} \mathbf{I} } { \mathrm{d} t } = \mathbf{N}$ (6)

$\label{magnetic_moment_B} \mathbf{N} = \boldsymbol{\mu} \times \mathbf{B}$ (7)

From equations (5), (6) and (7) we can easily find that the equation of motion for the magnetic moment of a spin in external field is the following:

$\label{eq_motion_spin} \frac{ \mathrm{d} \boldsymbol{\mu} } { \mathrm{d} t } = \gamma \boldsymbol{\mu} \times \mathbf{B}$ (8)

(8) describes a clockwise precession around the magnetic field as shown in Figure 2. The angular frequency of the precession is determined by the product of the magnetic field and the gyromagnetic ratio and is called Larmor frequency:

$\label{Larmor_freq} \boldsymbol{\omega}_0 = -\gamma \mathbf{B}$ (9)

Figure 2. Magnetic moment precessing around external magnetic field

From now on without loss of generality we suppose the magnetic field to be parallel with the Z axis. With this assumption the solution of (8) by components are

$\label{eq_motion_sol_x} \mu_x(t) = \mu_x(0) \mathrm{cos} ( \omega_0 t ) + \mu_y(0) \mathrm{sin} ( \omega_0 t )$ (10)

$\label{eq_motion_sol_y} \mu_y(t) = - \mu_x(0) \mathrm{sin} ( \omega_0 t ) + \mu_y(0) \mathrm{cos} ( \omega_0 t )$ (11)

$\label{eq_motion_sol_z} \mu_z(t) = \mu_z(0)$ (12)

From (12) we can see that the $z$ component is constant in time, so we only have to deal with $x$ and $y$ components. We can do so by gathering these two components into a vector that describes the transverse projection of the magnetic moment denoted by $\boldsymbol{\mu_{\bot}}$ :

$\label{mu_transverse} \frac{ \mathrm{d} \boldsymbol{\mu_{\bot}} } { \mathrm{d} t } = \gamma \boldsymbol{\mu_{\bot}} \times \mathbf{B}$ (13)

By using complex formalism, the mathematical description of these components becomes more manageable. Let's make a time dependent complex number from these two with $\mu_x$ and $\mu_y$ stand as the real and the imaginary part respectively, and let's denote it by $\mu_+$ . In this way, the magnetic moment can be written as

$\label{moment_complex} \mu_+(t) = \mu_x(t) + \mathrm{i} \mu_y(t) = \mu_+(0) \mathrm{e}^{ \mathrm{-i} \omega_0 t}$ (14)

In this way, the two independent components are no more {EQUATION}$\mu_x${EQUATION} and {EQUATION}$\mu_y${EQUATION} but the amplitue and phase of a complex number. It is extremely important to notice that the phase in this representation is not an arbitrary, non-physical parameter like the phase of the wavefunction in basic quantum mechanics, but closely relates to the position of the precessing moment and is of utmost importance in the description of spin motion and magnetic resonance imaging.

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