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Appendix ANuclear Magnetic Resonance principles and set-upA.1 Semi-classical theory of NMRIn analogy to the electron, the spinning motion of the atomic nucleus gives rise to anintrinsic angular momentum denoted by the spin quantum number I. For elements havingI > 0 the spinning motion and the charge distribution result in a nuclear magnetic momentµ, the magnitude of which is given by|µ| = gµ N√I(I + 1), (A.1)where g is the Landé factor (≈ 5.58 for the proton) and µ N is the nuclear magneton (≈5.05×10 −27 J Tesla −1 ).From a classical point of view an external magnetic field acts on the nucleus by exertinga torque in order to have it aligned with the direction of the field. In case the nucleusis not already aligned with the field, the torque on the spinning nucleus will result in aprecessional motion of the nucleus. The frequency of the precessional motion is proportionalto the externally applied field B 0 , and is called the Larmor frequency ω L , givenbyω L = γB 0 ,(A.2)where γ is the gyromagnetic ratio. The ratio is 42.58 MHz Tesla −1 for hydrogen and 6.54MHz Tesla −1 for deuterium. Suppose B 0 is directed in the z-direction, so that B 0 = B 0 e z .When an ensemble of nuclei is magnetized by B 0 , the nuclei start to precess around an axisparallel to B 0 . A net magnetization is obtained which is explained by quantum mechanics.The nuclei are forced into a certain energy state. For nuclei with spin quantum number1/2, like hydrogen nuclei, there are only two states. The energy splitting, i.e. Zeemansplitting, ∆E is proportional to B 0 :∆E = γB 0 ,(A.3)where is Planck’s constant. When the precessional axis of the nucleus is parallel tothe direction of B 0 the nucleus is in the low-energy, that is the preferred state. Whenthe precessional axis is anti-parallel to B 0 the nucleus is in the higher energy state (seeFigure A.1). Transitions between the two states are induced when the nuclei are subjectedto an EM wave with a frequency ω close or equal to the Larmor frequency ω L . A netmagnetization M is found when the number of nuclei parallel aligned to B 0 is larger thananti-parallel, so thatM = ∑ µ i = M 0 e z . (A.4)i141