Chap18 solutions

Physical Chemistry

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18 Spectroscopy 3: magnetic resonance Solutions to exercises Discussion questions E18.1(b) Before the application of a pulse the magnetization vector, M , points along the direction of the static external magnetic Feld B 0 . There are more α spins than β spins. When we apply a rotating magnetic Feld B 1 at right angles to the static Feld, the magnetization vector as seen in the rotating frame begins to precess about the B 1 Feld with angular frequency ω 1 = γ B 1 . The angle through which M rotates is θ = γ B 1 t , where t is the time for which the B 1 pulse is applied. When t = π/ 2 γ B 1 = 2 = 90 , and M has rotated into the xy plane. Now there are equal numbers of α and β spins. A 180 pulse applied for a time π/γ B 1 , rotates M antiparallel to the static Feld. Now there are more β spins than α spins. A population inversion has occurred. E18.2(b) The basic COSY experiment uses the simplest of all two-dimensional pulse sequences: a single 90 pulse to excite the spins at the end of the preparation period, and a mixing period containing just a second 90 pulse (see ±ig. 18.44 of the text). The key to the COSY technique is the effect of the second 90 pulse, which can be illustrated by consideration of the four energy levels of an AX system (as shown in ±ig. 18.12). At thermal equilibrium, the population of the α A α X level is the greatest, and that of β A β X level is the smallest; the other two levels have the same energy and an intermediate population. After the Frst 90 pulse, the spins are no longer at thermal equilibrium. If a second 90 pulse is applied at a time t 1 that is short compared to the spin-lattice relaxation time T 1 the extra input of energy causes further changes in the populations of the four states. The changes in populations will depend on how far the individual magnetizations have precessed during the evolution period. ±or simplicity, let us consider a COSY experiment in which the second 90 pulse is split into two selective pulses, one applied to X and one to A. Depending on the evolution time t 1 , the 90 pulse that excites X may leave the population differences across each of the two X transitions unchanged, inverted, or somewhere in between. Consider the extreme case in which one population difference is inverted and the other unchanged (±ig. 18.45). The 90 pulse that excites A will now generate an ±ID in which one of the two A transitions has increased in intensity, and the other has decreased. The overall effect is that precession of the X spins during the evolution period determines the amplitudes of the signals from the A spins obtained during the detection period. As the evolution time t 1 is increased, the intensities of the signals from A spins oscillate at rates determined by the frequencies of the two X transitions.
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Chap18 solutions - 18 Spectroscopy 3: magnetic resonance...

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