Читать книгу Essential Concepts in MRI - Yang Xia - Страница 37

Any practical sample, no matter how small, contains an enormous number of nuclei. It is the macroscopic ensemble average of the observable quantities in which we are interested. In these ensembles, different nuclei may occupy different states |ψ>. We use the concept of sub-ensemble in which all nuclei are in identical states |ψ(t)>. The sum over all sub-ensembles, each with a classical probability p_ψ, gives the observable ensemble average in which we are interested. The averaged expectation value, by definition, is

(3.12)

where the bar refers to the statistical ensemble average, and the pair of arrow brackets, < >, represents the quantum mechanical expectation value.

Now consider spin-1/2 particles. In NMR, the dominant interaction of a spin with its environment is always via the Zeeman interaction [as in Eq. (3.5)]. This means that the natural eigenstates are those whose quantum numbers are eigenvalues of Iz, that is, |1/2> and |–1/2>.

In general, we can express any state in this basis using the Pauli’s spin matrices formalism (cf. Appendix A2.5), as

(3.13)

Since Iz=12[100−1], the ensemble averaged expectation value of Iz is determined by the difference in populations between the upper and lower energy levels, according to the Boltzmann distribution. This distribution describes the polarization of the ensemble in thermal equilibrium.

We can calculate the normalized population at thermal equilibrium as

(3.14)

where the numerators are individual populations and the denominator is the total population.

Note that k_BT is the Boltzmann energy and ħγB₀ is the Zeeman energy difference. For example, for protons in a magnetic field of strength B₀ = 7 Tesla and at room temperature (T = 300 K), we have

Since k_BT is over four orders of magnitude bigger than ħγB₀, the ratio ħγB₀/k_BT is tiny. This is a good news and bad news situation: the good news is that the exponentials in Eq. (3.14) can be simplified using the Taylor expansion since the high-order terms would be very small, while the bad news means that our signal will be very small, since the signal is proportional to the population difference.

Due to this tiny ratio between the Zeeman energy and the Boltzmann energy, we can apply the Taylor expansion (Appendix A1.1.4) to simplify the expression in Eq. (3.14) by keeping only the first two terms, as

(3.15)

This equation at the room temperature and a 7 Tesla B₀ equals approximately 0.5 ± 1.12 × 10^-5, which is almost a half and half situation between the two populations. This approximation is known as the “high-temperature approximation” in the NMR literature, except the “high temperature” in this estimation actually means the room temperature.