Читать книгу Essential Concepts in MRI - Yang Xia - Страница 39

In the current context, the observable quantity is just the (macroscopic) magnetization M, given by

(3.17a)

(3.17b)

where N is the number of spins, and i, j, and k are the unit vectors in the Cartesian coordinates. Equation (3.17) is important because it may be shown that any state of the density matrix (defined in Appendix A2.6) for an ensemble of non-interacting spin-1/2 particles can be described using the macroscopic magnetization defined in this manner, thus permitting a classical description of simple spin systems.

In the absence of an external magnetic field, the ensemble average of the magnetization vector should be zero due to the random directions of the magnetic dipoles of the nuclei.

If a sample is immersed in an external field and in thermal equilibrium, the density operator associated with this magnetization vector is given by

(3.18)

The transverse component of M is zero due to the even distribution of the azimuthal phase angles of the precessing nuclei in the transverse plane. This corresponds to phase incoherence leading to the zero value of the off-diagonal elements of ρ,

(3.19)

The z component of the magnetization M arises from the difference in populations between the upper and lower energy states. At room temperature, the magnitude of this magnetization in the equilibrium state, M₀, can be derived as

(3.20)

For a spin-1/2 system at room temperature, the population difference between the spin-up (m=+1/2) state and the spin-down (m=−1/2) state can be calculated from the diagonal elements of ρ, as

(3.21)

For protons at B₀ = 1.4 T (60 MHz), it is equal to about 5 × 10^-6, a small value resulting from the small value of γħB₀ (Zeeman splitting) compared to k_BT (Boltzmann energy). It is this small magnetization of nuclei at room temperature that limits NMR detection sensitivity and leads to resolution limitations in MRI experiments [7, 8].