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-142 Appendix A, - C D *, - D M* Fig. A.1: Zeeman splitting of energy of an ensemble of nuclei (I = 1/2) in an external magneticfield B 0 . The partitioning between the energy states is perturbed by an oscillatingfield with a frequency ω = ∆E/.The magnetization follows from Curie’s law:M 0 = N γ2 2 I(I + 1)B 0 ,3k B T(A.5)where N is the number of nuclei (or spins), k B is the Boltzmann constant, and T is thetemperature. At equilibrium the transverse components M x and M y are zero because thenuclei precess out of phase. The basis in NMR lies in the application of an oscillatingmagnetic field B 1 perpendicular to B 0 . This field is applied using an rf transmissioncoil. When the field is switched on at the resonance frequency the magnetization, i.e. thebalance between the spins in the upper- and lower-energy state is changed. Secondly thespins will start to precess in phase. In the classical view the oscillating field B 1 acts onthe magnetization vector M such that it spirals down towards the x-y plane with respectto the laboratory frame of reference. The change of the magnetization vector is describedbydM= γ(M × B 1 ). (A.6)dtThe rotating frame of reference is aligned with the z-axis, but by definition rotates withthe Larmor frequency with respect to the laboratory frame of reference. In this rotatingframe, (x′, y′, z′), the magnetization vector M rotates due to the torque in a planeperpendicular to the direction of B 1 . An rf pulse with duration τ will rotate the vectorover the so-called flip angle θ (see Figure A.2), which is given byθ = γB 1 τ.(A.7)In the NMR apparatus the magnetization from the nuclei is always measured in thetransversal plane. Suppose the magnetization is tipped to the transversal plane by applicationof a 90 ◦ pulse. The magnetization M T , which precesses with the Larmor frequency,is picked up by the rf receiver coil. The received frequencies are around the Larmor frequencyof the nuclei. In the remainder we will consider the envelope or amplitude only.The coherence in the precession of the spins is gradually lost (see Figure A.2), because due