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.
In 1949 Torrey used Laplace transforms to provide  the first solution of the differential equations proposed by Bloch  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 . The problem has been revisited several times employing third-order differential equations [4, 5] and Laplace transforms  to give unwieldy solutions using somewhat opaque derivations. The first-order differential equations are directly integrated here using a matrix diagonalization method.
2. Bloch Equations and Their Integration
The Bloch equations for a collection of identical spins in the frame rotating at arewhere
are the longitudinal and transverse relaxation rates in , and and are the resonance frequency offset and the rf amplitude for a field along the x-axis, in radians/s. is the (positive) gyromagnetic ratio, and is the equilibrium magnetization.
Defining a magnetization vector , the integrated solution of equation (1) iswhere . The problem is, thus, deducing the roots of , the matrix and inverse which diagonalize and the steady-state magnetization vector .
2.1. Evaluation of Roots
gives the following cubic equation:
Choosing roots of the form , gives the corresponding cubic equation:
These expressions agree with Torrey (, equation 59) and with Abragam (. p. 70).
For and , for example, the above approach is valid and avoids the explicit solution of the cubic equation (4).
2.2. Calculation of
is obtained by evaluating the three cofactors of for . Choosing the third row of , the cofactors are cofactor 1 = cofactor 2 = cofactor 3 =
Omitting all (small) relaxation rate difference terms (which are exactly zero for ) and dividing all elements by gives :
2.3. Calculation of
is formed by constructing the matrix of all cofactors of , taking the transpose, and dividing by the determinant . The result is
These may be rewritten in a more compact form using and :
Finally, we calculate the matrix :
The elements of are
2.4. Steady States
For , these may be simplified using equation (6):
The integrated solution equation (3) for the initial condition is
Limiting forms of equation (3) are discussed in this section.
3.1. Case 1: Resonant Nutation
and interconvert at rate and decay to (small) steady states.
3.2. Case 2: Free Precession/ Relaxation
and interconvert at rate and decay to zero as returns to equilibrium.
3.3. Case 3: Spin-Locked Relaxation
3.4. Case 4: General Solution
For a weak rf field , they reduce toin agreement with Slichter (, p. 35).
3.5. Case 5: Equal Relaxation Rates
Equation (3) describes a number of experimental situations.
4.1. Case 1: Resonant Nutation
4.2. Case 2: Free Precession and Relaxation
In the absence of an rf field, the transverse components interconvert and relax to zero (the free induction decay) as the longitudinal component, initially zero, relaxes independently to equilibrium (equation (31)).
4.3. Case 3: Spin-Locked Relaxation
Orientation of the magnetization vector parallel to the effective field suppresses precession and results in a single-exponential approach to equilibrium, affording the longitudinal and transverse relaxation rates using equations (6) and (33) [10, 11].
4.4. Case 4: General Solution
Equation (34) presents in compact form the solutions originally given by Torrey  and by Morris and Chilvers  as Laplace expressions and the tabulations of Madhu and Kumar [4, 5] for a spin ensemble initially at equilibrium. They are valid providedwhich holds for most cases of practical interest. In the example given by Madhu and Kumar [4, 5],
These values do not appear to reproduce the figures presented in .
4.5. Case 5: Equal Relaxation Rates
The solutions (20)–(22) for equal relaxation rates are exact provided the full steady states of equation (14)–(16) are used. The inequality of Case 4 leads to the simpler expressions (36). We note also that setting results in
4.6. Neglect of Relaxation
For rf amplitude and precession terms which are large compared to relaxation rates equations (23)–(25) pertain. They are useful, for example, in describing selective (on-resonance) excitation with (off-resonance) signal suppression  as in the following example (using Hz units).
4.6.1. On-Resonance Rotation
4.6.2. Off-Resonance Rotation
For a 5 KHz offset and .
The 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.
The differential equations (1) of Bloch  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.
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
A. Abragam, The Principles of Nuclear Magnetism, Clarendon Press, Oxford, UK, 1961.
C. L. Perrin, Mathematics for Chemists, John Wiley and Sons, Hoboken, NJ, USA, 1970.
J. Cavanagh, W. J. Fairbrother, A. G. Palmer, M. Rance, and N. J. Skelton, Protein NMR Spectroscopy, Elsevier Academic Press, London, UK, 2nd edition, 2007.
R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, Oxford, UK, 1987.
C. P. Slichter, Principles of Magnetic Resonance, Springer-Verlag, New York, NY, USA, 2nd edition, 1978.