Читать книгу Essential Concepts in MRI - Yang Xia - Страница 42

For spin-1/2 particles, the relaxation mechanism can be understood with a set of equations and analysis in terms of quantum transition [9, 10]. In this approach, the spin populations (the occupancies of the eigenstates of Iz with eigenvalues m=±1/2) are defined as

(3.25a)

(3.25b)

We also define the total population N₀ and the population difference n as

(3.26a)

(3.26b)

Hence, the macroscopic magnetization M is proportional to the population difference n. Using Eq. (3.21), the z component of the magnetization at time = 0 can be written as

(3.27)

Since the population is at equilibrium with the environment according to the Boltzmann distribution, the population ratio is

(3.28)

To consider the dynamics of the two populations, we define w+- as the probability of transition of a spin from |+> state to |–> state per spin per second, and w-+ as the probability of transition of a spin from |–> state to |+> state per spin per second. At equilibrium, we have

(3.29)

Combining Eq. (3.28) and Eq. (3.29), we have

(3.30)

With this equation, the changes of the spins with time can be defined as

(3.31a)

(3.31b)

Each equation in Eq. (3.31) has two terms, the increment term (the first term) and the reduction term (the second term). Given the fact that w_+- ≈ w_-+, we define

(3.32a)

(3.32b)

Note that w₀ can be considered as the probability of induced transitions, while the term ℏγB0/kBT can be considered as the probability of spontaneous transitions; their differences were distinguished first by Albert Einstein in 1916 when he published a paper on different processes occurring in the formation of an atomic spectral line in optical studies. And hence we can show that

(3.33)

where dn/dt goes to zero as n (a variable) goes to n∞ (a constant given by the population difference in the presence of B₀ but in the absence of another rf field). If we define

(3.34)

and multiplying Eq. (3.33) with (1/2)γℏ, we can derive

(3.35)

which has been defined previously as Eq. (2.15a). Therefore, we can interpret T₁ in terms of quantum transitions between the states.

The process of transverse relaxation may also be viewed as the result of quantum transitions. By defining the probability of transition between the states of operators Ix_′ and Iy_′ as w_0x′ and w_0y′, and recognizing at w0x′=w0y′=w0⊥, we can derive

(3.36)

where |M⊥|=(Mx′2+My′2)1/2. Hence,

(3.37)