The intrinsic MR signal

The MR signal generally originates from hydrogen nuclei (protons). A proton has a property of intrinsic angular momentum, often referred to as spin $\vec I$, with an associated magnetic moment $\vec\mu$:

$$\vec\mu = \gamma \vec I \quad ,$$ (1.1)

with $\gamma$ denoting the gyromagnetic ratio unique to each atom. In a single volume element corresponding to a pixel in an MR image, there are protons in abundance, each with an associated moment. The net magnetization $\vec M$ is the resultant of the individual dipole moments contained therein. In absence of a magnetic field, the orientation of these dipole moments will be random. Hence, the net magnetization will be zero. In a static magnetic induction $\vec B_0$, conventionally oriented along the spatial direction z, the energy levels of the spins are splitted. The energy separation $\Delta E$ between the levels is given by:

$$\Delta E=\hbar\gamma B_0\quad ,$$ (1.2)

with $\hbar$ Planck's constant divided by $2\pi$. The lowest energy state of the spins corresponds to an orientation along $\vec B_0$. Hence, in equilibrium and in absence of thermal energy, the net magnetization $\vec M_0$ will be orientated along $\vec B_0$. On the other hand, thermal energy drives the individual spins toward random orientations. In thermal equilibrium, these tendencies are balanced with the net magnetization being aligned parallel to $\vec B_0$ with a strength corresponding to the Boltzmann distribution of the possible energy states of the proton [5]:

$$\rho_m = {\exp(\hbar\gamma m B_0 / k_b T_s) \over \sum\limits_{m=-I}^I \exp(\hbar\gamma m B_0 / k_b T_s)} \quad ,$$ (1.3)

with m the magnetic quantum number, k_b the Boltzmann constant, and T_s the sample temperature. Usually the energy $\hbar\gamma B_0$ is much smaller than k_b T_s such that Eq. (1.3) can be approximated by:

$$\rho_m \simeq \left(1+{\hbar\gamma m B_0 \over k_b T_s}\right) / (2I+1)\quad .$$ (1.4)

The magnetization M₀ per unit volume is given by [6]:

$$N_s \sum_{m=-I}^I \rho_m \hbar\gamma m$$ M₀ = (1.5)

$$\simeq N_s {\gamma^2 \hbar^2 I(I+1) B_0 \over 3 k_b T_s}\quad ,$$ (1.6)

with N_s the number of spins at resonance per unit volume. In general, the behavior of the spins must be described by quantum mechanics. However, if the number of spins N_s is very large, classical mechanics has been shown to be suited for the description of the net magnetic moment $\vec M$ behavior [4] in a magnetic induction field $\vec B$:

$${d\vec M\over dt} = \gamma \vec M\times\vec B\quad .$$ (1.7)