Solution of the Bloch Equations including Relaxation

The magnetization diﬀerential equations of Bloch are integrated using a matrix diagonalization method. The solution describes several limiting cases and leads to compact expressions of wide validity for a spin ensemble initially at equilibrium.


Introduction
In 1949 Torrey used Laplace transforms to provide [1] the first solution of the differential equations proposed by Bloch [2] for the magnetization components of a spin ensemble. His results are somewhat cumbersome and contain some errors. Although the problem is fundamental, a general solution including relaxation does not appear in any of the standard NMR texts, with one partial exception [3]. e problem has been revisited several times employing third-order differential equations [4,5] and Laplace transforms [6] to give unwieldy solutions using somewhat opaque derivations. e first-order differential equations are directly integrated here using a matrix diagonalization method.

Bloch Equations and Their Integration
where K � R 1 and R 2 are the longitudinal and transverse relaxation rates in s − 1 , and Δ � |ω 0 − ω rf | and ω 1 � cB 1 are the resonance frequency offset and the rf amplitude for a B 1 field along the x-axis, in radians/s. c is the (positive) gyromagnetic ratio, and M 0 is the equilibrium magnetization.
Defining a magnetization vector V � M x M y M z T , the integrated solution of equation (1) is where K � X DX − 1 . e problem is, thus, deducing the roots of K(� D), the matrix X and inverse X − 1 which diagonalize K and the steady-state magnetization vector V(ss).
For R 2 � 1s − 1 and ω 1 and/or Δ � 2π × 10 Hz, for example, the above approach is valid and avoids the explicit solution of the cubic equation (4).

2.2.
Calculation of X. X is obtained by evaluating the three cofactors of K − λ1 for λ 1 � R θ and λ 2,3 � b ± iΩ. Choosing the third row of K − λ1, the cofactors are Omitting all (small) relaxation rate difference terms (which are exactly zero for R 1 � R 2 ) and dividing all elements by ω 1 gives X: 2.3. Calculation of X − 1 . X − 1 is formed by constructing the matrix of all cofactors of X, taking the transpose, and dividing by the determinant |X| [7]. e result is ese may be rewritten in a more compact form using tan θ � (ω 1 /Δ) and Ω: Finally, we calculate the matrix A � X exp(− Dt)X − 1 : e elements of A are A 33 � cos 2 θ exp − R θ t + sin 2 θ exp − bt cos Ωt. (13)

Steady
States. e steady-state magnetizations are found by setting equation (1) to zero and using Cramer's rule [7]: where d � R 1 R 2 2 + R 1 Δ 2 + R 2 ω 2 1 . For R 2 < ω 1 and/or Δ, these may be simplified using equation (6): e on-resonance magnetization is rotated to the -y axis by the rf pulse, whereas the off-resonance magnetization undergoes an excursion that returns it to the z-axis.

Conclusion
e differential equations (1) of Bloch [2] are integrated with a matrix diagonalization method to give the solution equation (3). It correctly describes a number of experimental situations including resonant nutation, free precession and relaxation, and spin-locked relaxation. Equation (3) is exact for the case of equal longitudinal and transverse relaxation rates and leads to the general equation (34) for a spin ensemble initially at equilibrium.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.