target="_blank" rel="nofollow" href="#fb3_img_img_09ea7d7f-3809-5417-8935-2ef6859533f6.png" alt="ModifyingAbove less-than psi Math bar pipe bar symblom upper I Subscript z Baseline Math bar pipe bar symblom psi greater-than With bar equals sigma-summation Subscript psi Baseline p Subscript psi Baseline less-than psi Math bar pipe bar symblom upper I Subscript z Baseline Math bar pipe bar symblom psi greater-than comma"/> (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
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
where the numerators are individual populations and the denominator is the total population.
Note that kBT 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 kBT is over four orders of magnitude bigger than ħγB0, the ratio ħγB0/kBT 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
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.
3.4 MEASUREMENT OF THE X COMPONENT OF ANGULAR MOMENTUM
In Appendix A2.6, we introduce the concept of density matrix operator ρ, which relates the expectation value of any operator A to the trace of the matrix product Aρ, via Eq. (A2.33). Since Ix=12[0110] in the formalism of Pauli’s spin matrices (cf. Appendix A2.5), we have, from Eq. (3.12),
where Tr represents the trace of the matrix (the sum of the diagonal elements) – see Eqs. (A2.32) and (A2.33), and the two examples at the end of Appendix A2.6. The term in brackets describes the degree of “single quantum coherence” of the ensemble, while the average (the bars) reflects the phase coherence between the +1/2 and –1/2 states. At thermal equilibrium, both a1/2*a−1/2¯ and a1/2a−1/2*¯ are zero (no transverse component), while the term (|a1/2|2¯−|a−1/2|2¯) has a stable value (the longitudinal magnetization along the z axis).
3.5 MACROSCOPIC MAGNETIZATION FOR SPIN 1/2
In the current context, the observable quantity is just the (macroscopic) magnetization M, given by
where N is the number of spins, and i, j, and k are the unit vectors in the
