#### Abstract

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.

#### 1. 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]. The 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. 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:

Equating the last term in equations (4) and (5), and with in (4) and in (5),using and

is obtained by equating the coefficients of the second term in equations (4) and (5):

These expressions agree with Torrey ([1], equation 59) and with Abragam ([3]. 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 [7]. 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

The steady-state magnetizations are found by setting equation (1) to zero and using Cramer’s rule [7]:where .

For , these may be simplified using equation (6):

The integrated solution equation (3) for the initial condition is

In the absence of relaxation, these are the well-known Bloch equations [8, 9]:

#### 3. Results

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

Employing the steady states of equations (17)–(19), these become

The magnetization vector relaxes to a steady state along the effective field [10, 11].

##### 3.4. Case 4: General Solution

Equations (20)–(22) may be recast using the steady states of equation (17)–(19) as

These expressions agree with equations (60)–(62) of Torrey [1] and with Abragam [3].

For a weak rf field , they reduce toin agreement with Slichter ([12], p. 35).

##### 3.5. Case 5: Equal Relaxation Rates

An exact solution of the Bloch equations is given by equations (20)–(22) using the full steady-state expressions (14)–(16). Using the steady states of equation (17)–(19), they become

#### 4. Discussion

Equation (3) describes a number of experimental situations.

##### 4.1. Case 1: Resonant Nutation

Resonant nutation (equations (28) and (29) ) was described by Torrey in his original paper [1]. The effective relaxation rate is the average of the longitudinal and transverse relaxation rates [3].

##### 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 [1] and by Morris and Chilvers [6] 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],

Accordingly, from equations (6), (7), and (17)–(19),

These values do not appear to reproduce the figures presented in [5].

##### 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 [9] as in the following example (using Hz units).

###### 4.6.1. On-Resonance Rotation

###### 4.6.2. Off-Resonance Rotation

Then, .

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.

#### 5. Conclusion

The 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

The author declares that there are no conflicts of interest.