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Principles of Magnetic Resonance: Bloch Equations and Pulsed Methods

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Principles of Magnetic Resonance: Bloch Equations and Pulsed Methods Appendix 1 Principles of magnetic resonance: Bloch equations and pulsed methods Belorizky E. and Fries P.H. In this appendix, we describe the general principles of magnetic resonance using the simple example of proton NMR, which is used in chapter 11. Further details can be found in [Abragam, 1983; Canet, 1996]. These principles are directly transposable to EPR. The transposition is made in complement 1 to chapter 10 for continuous wave EPR, and in appendix 2 to this volume for pulsed EPR. 1 - Proton NMR Interaction of a magnetic feld Bk00= B – oriented along Oz – and a proton of g p bN 81−−1 spin I = ½ which has a magnetogyric ratio gI ==' 26. 81# 0rad sT and a magnetic moment μ = γI ħI [1] is described by the Hamiltonian ĤZ = - μ . B0 [2] Two energy levels emerge EmmI=− g 'B0 , where m = ! ½ is the eigenvalue for the Îz operator (figure A1.1). Their difference is equal to DEh==gwI ''B0 0 = n0. ( Ȗ ƫ% ²², P <ò , ¨(Ȗ, ƫ% Ȗ, ƫ% Figure A1.1 - Energy levels for a proton <²² P,ò in a feld B0. © Springer Nature Switzerland AG 2020 385 P. Bertrand, Electron Paramagnetic Resonance Spectroscopy , https://doi.org/10.1007/978-3-030-39668-8 386 Electron Paramagnetic Resonance Spectroscopy - Applications For example, we obtain ν0 = 42.577 MHz for B0 = 1 T. The magnetic moment for a proton is 657-fold smaller than that for a free electron. Thus, for the same value of magnetic field, the Zeeman separations are three orders of magnitude weaker than those involved in EPR. At thermodynamic equilibrium, the ratio between the populations of the two levels can be written: N−12 DE =−exp [3] N12 cmkTB N 12 −6 For B0 = 1 T, we obtain =+16.87 # 10 at room temperature (T = 298 K). N−12 If we consider a system with N0 protons at equilibrium, the population difference is written [Volume 1, section 1.4.4]: DDE E NN12−=N−12 0 tanh . N0 [4] cm22kTB kTB In these conditions, the mean value at equilibrium Meq for the magnetisation Mz of protons in the direction of the field is given by Curie’s law: 2 2 N0 gI ' Meq = B0 [5] 4kTB gI ' DE The mean magnetic moment for a proton, which can be written , 2 2kTB is thus very small relative to its intrinsic moment μ = γI ħ /2. For resonance to be observed, transitions must be induced between the two Zeeman levels. This can be achieved by subjecting the protons to a magnetic field B1 with a direction perpendicular to B0, an amplitude B1 much smaller than B0, and which rotates with an angular velocity - ω. Resonance occurs when ω is close to ω0 = γI B0. In practice, a coil of axis Ox, in which a sinusoidal current of angu- Ox lar frequency ω circulates, creates an oscillating field BB1 = 2 1 cos wt along this axis. This field can be considered to be the sum of two fields rotating in opposing directions with angular velocities ! ω in the xy plane, of which only the component rotating at the angular frequency - ω is useful. Interaction with the variable field B1 produces transitions between the two Zeeman levels. The probability of these transitions per unit of time, which is the same in both directions, is given by time-dependent perturbation theory: p 2 2 wB=−2 gwI 1 f()w0 [6] Appendix 1 - Magnetic resonance: Bloch equations and pulsed methods 387 where f (ω - ω0) is a normalised function representing the shape of the res- onance line. As the population of the lower level is slightly greater than that of the higher level, the system of N0 protons absorbs from the radiofrequency field a power Pa given by: Pwa =−NN12 −12DE ^ h Thus, according to equation [4] 2 'w0 Pwa , N0 ^ h [7] 2kTB 2 - Motion of the macroscopic magnetisation The magnetisation M for a sample containing a large number of protons is ob- tained by adding the microscopic moments μi of the different protons: M = / mi. i When the sample is placed in a field B0, the movement of the magnetisation, which can be considered a classical vector (see for example the FIV complement in [Cohen-Tannoudji et al., 2015]), is determined by equation dM = g MB# dt I 0 By projecting this equation on the three axes of coordinates, we obtain the system: Z ] dMx ] = gI MBy 0 ] dt ] dM y ][ =−gI MBx 0 [8] ] dt ] dMz ] dt = 0 \ We will first assume that at t = 0 the magnetisation M(0) takes its equilibrium value Meqk, where k is the unit vector along Oz. In this case, the solution to this system is time-independent and is Mx = My = 0, Mz = Meq = constant. We will now assume that at t = 0 the magnetisation is separated from its equilibrium position by an angle α, for example: MMxy()00==,(00),MMeq sincaaze()= M q os [9] The solution to system [8] is then written Mtx ()= Mteq sinsawin 0 Mty ()= Mteq sincawos 0 [10] *Mtz ()= Meq cos a 388 Electron Paramagnetic Resonance Spectroscopy - Applications where ω0 = γIB0. The magnetisation M precesses around Oz with an angular velocity - ω 0, i.e., at a velocity of ω0 in the clockwise direction. Ox We now apply the radiofrequency field BB1 = 2 1 costw created by a coil of axis Ox in which a sinusoidal current of angular frequency ω, close to ω0, flows. 1 Transitions between the energy levels for the protons characterised by m =! 2 Ox are induced by the component of B1 which rotates about Oz at a frequency - ω. The projections of this component onto the (x, y, z) axes can be written: BB11xy==cosswwtB,,11−=Btin B1z 0 [11] To determine the motion of the magnetisation, it is convenient to introduce a system of coordinates R = (xʹ, yʹ, zʹ = z) which rotates about z at an angular velocity - ω, and thus follows the rotation of the field B1 defined by relations [11]. The reference frames (xʹ, yʹ, z) and (x, y, z) are respectively known as the rotating reference frame (R) and the laboratory’s reference frame (L). In the rotating reference frame (R), B1 appears to be fixed (figure A1.2). ] ] % % \ [ 5 < ȦW \ [ / Figure A1.2 - The laboratory’s reference frame (L) = (x, y, z) and the rotating reference frame (R) = (xʹ, yʹ, zʹ = z) rotating about the feld B0 // z at an angular velocity - ω. In reference frame (L), movement of M in the presence of B0 and B1 is deter- mined by the following equation: dM =+g MB# ()01B [12] ` dt jL I dM In reference frame (R), this movement is determined by , which is dt R dM ` j linked to by: ` dt jL dM dM =+W # M [13] ``dttjjLRd Appendix 1 - Magnetic resonance: Bloch equations and pulsed methods 389 where Ω = - ωk is the rotation vector of (R) relative to (L). By replacing this expression in equation [12], we obtain: dM = g MB# [14] ` dt jR Ieff where Beff is the “effective field” defined by W BBeff =+01B + [15] gI The motion of the magnetisation is much easier to determine in reference frame (R) than in reference frame (L) as Beff is constant in (R). Note that Ω and B0 have opposite directions. In the specific case where ω = ω0 = γI B0 (resonance), Beff is equal to B1 such that in the rotating reference frame, the magnetisation rotates about B1 at an angular velocity ω1 = - gI B1. If B1 is applied for a du- ration τ, M rotates by an angle α = ω1τ = - γ I B1τ from its original position. For example, if we choose a short τ (a few μs) such that ω1τ = π /2 or π, this is termed applying a radiofrequency (rf) pulse of π /2 or π, or 90 ° or 180 °. If the magnetisation initially has its equilibrium value M = Meqk, pulses of π /2 and π will align it with the axis Oyʹ and opposite to the Ozʹ = Oz axis of the rotating reference frame, respectively. After suppression of B1, M generally has a non-null component in the xy plane, Mxy = Meq sin α. In the laboratory’s reference frame, this component rotates about B0 with a frequency of -ω0. As a result, a current is induced in the detector coil placed near to the sample. The same coil is often used for both excitation and detection. 3 - Relaxation phenomena In the previous section we saw that an rf pulse applied for a short duration can cause the direction of the magnetisation M to shift from its equilibrium direc- tion Oz. We assumed that after suppression of B1, the rotation of M around B0 continues indefinitely. In fact, the magnetisation will progressively return to its equilibrium value along the z axis, and the NMR signal will disappear. This return of the magnetisation to equilibrium is known as relaxation. After an rf pulse, the Mz and Mxy components of the magnetisation M, which are parallel and perpendicular to the field B0, respectively, progressively return towards their equilibrium values: MMze" q, where Meq is given by equation [5], and Mxy " 0. This relaxation is due to the fluctuating magnetic fields acting on the protons. These magnetic fields mainly originate from the dipole-dipole 390 Electron Paramagnetic Resonance Spectroscopy - Applications interactions between the proton’s magnetic moment and the magnetic moments of neighbouring molecules, whether of nuclear or electronic origin. They can also result from hyperfine interactions or a rotational-spin effect, or even chemical displacement anisotropy interactions. The fluctuations of these magnetic fields are created by random movements of the molecules near to the protons studied.
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