NMR Programme

1. NMR Programme

2. NMR PG Timetable: Overview next semester

3. QM overview of an NMR experiment

4. Introduction to the Product Operator Formalism Product operators provide a convenient framework to describe a number of fundamental experiments Caveat: by no means the only or the best way Solution NMR: INEPT: HSQC, triple resonance,

5. Magnetic moment and spin • The classic magnetic moment expectation value of m

6. Bulk magnetisation M If the system consists of a large number of spin systems, the measured quantity is the ensemble average over the expectation values of the individual spins. The bulk magnetisation of an ensemble of non interacting spins is given by: in which N is the number of spins per unit volume.

7. The significance of the comutator relationships In the space of I a rotation R around an arbitrary axis n by an angle a is given by: Provided the components of I: Ii, Ij, Ik, satisfy the commutation relations:

8. Consider, for example, the rotation of the expectation value of the Cartesian components of I (i, j, k = x, y, z) under a rotation around the z-axis by an angle a. The rotated expectation value of the Ix operator is given by:

9. Baker Hausdorf lemma

11. Transverse Magnetisation • Observation Operator F+ in which K is the number of spins in the system (i.e. K = 2 for a two-spin system).

12. Time evolution of the spin ensemble: Liouville van Neuman equation • If Hamiltonian is time independent:

13. Baker-Hausdorf Lemma • where G is a hermitian operator and l a real parameter.

14. Hamiltonian of a weakly coupled 2-spin system • Interaction with the external field H0 Zeeman term • Attention here s denotes the chemical shift tensor: in liquids rotational averaging means that the tensor reduces to a number (scalar) s is a (scalar) number • w0 is defined as the Larmor frequency

15. weak coupling: n the weak coupling limit i.e. the eigenstates of the Zeeman interaction Hamiltonian are a good basis set. Hence, the indirect interaction reduces to its z-components and the so called truncated Hamiltonian reads: .

16. A pulse 1 • First transform into rotating frame • For I and S go into double rotating frame H0, with w0I= -gIH0 is removed by the propagator: with: WI,S = wI,S - w0I,S

17. A pulse 2 • In the rotating frame the r.f. field appears to be static, e.g. along the x-axis, and it couples to the x-components of the spin operators giving rise to the r.f. Hamiltonian: In the doubly rotating frame, two r.f. fields are applied such that for each nucleus there appears to be a static r.f Hamiltonian:

18. As long as the r.f. field strength is much bigger than the offset in the rotating frame |gI,sHrfI,S| >>|WI,S|the effective Hamiltonian is dominated by the r.f. field(s) and the spins evolve under the r.f. Hamiltonian i.e. are rotated around the axis of the r.f. field(s). The evolution of the density matrix proportional to Iz + Sz under a pulse of duration t I,S is according to equation the time evolution of the density matrix is given by: where aI,S = -gI,SHrf tI,Sare the flip angles The r.f. Hamiltonian acts on each spin individually, because the spin operators of the two spins commute. Using the Baker-Hausdorff lemma the expression simplifies to:

19. Evolution including J coupling • Start with an equilibrium density matrix of Suppose a 90 pulse is applied to both spins. The resulting density matrix is then given by: The transverse bulk magnetisation of spin I is proportional to and the time evolution is described by: H is the truncated Hamiltonian of the two spin system

20. The Zeeman term of spin S does not contribute to the time evolution of I+, because I+ and Sz commute. Hence, the first three terms in the brackets read: using Baker Hausdorf lemma: Expand exp in cos and sin:

21. The result leads to a number of interesting conclusions. The frequency spectrum of consists of a in-phase doublet at pJ from the frequency wI. A similar result would be obtained for the S spin. The signal structure of the I spin would be the same irrespective of S being excited. And finally in the absence of the J-coupling between the two spins, the signal of the I spin would reduce to the isolated spin case with a single line at the Larmor frequency wI of I. with I0 = -bwTr(Ix2), because Ix2 is the only operator whose trace is non zero.

22. QM overview of an NMR experiment

23. Introduction to the Product Operator Formalism Product operators provide a convenient framework to describe a number of fundamental experiments Caveat: by no means the only or the best way Solution NMR: INEPT: HSQC, triple resonance,

24. The density matrix of a spin ½ system can be expanded in Cartesian matrices that are given by: , for problems with rotational symmetry the raising and lowering operators together with Iz form more convenient basis and

25. Comutators and products

26. 2 spins

27. Evolution under Zeeman Hamiltonian Consider, for example, an isolated spin-1/2 nucleus I in an external magnetic field along the z-axis. The Cartesian spin-1/2 operators Ix, Iy, and Iz form a basis for I. Using the same notation as previously the Hamiltonian in the rotating frame is proportional to WIz with W= w - w0.The time evolution of the operators is determined by the Liouville-von Neumann equation.

28. Evolution under Zeeman Hamiltonian 2 • Hence, the general solutions are given as linear combinations of sine and cosine functions: • Alternatively this is depicted as

29. Home work The above derivation is readily extended to multispin systems. Consider, for example a weakly coupled two spin system with spins I and S. The 16 basis operators evolve under the Zeeman interaction of each spin and the indirect interaction. Since the Zeeman terms as well as the indirect interaction commute with each other, the time evolution under the entire Hamiltonian can be represented as successive evolution under the individual interactions. For the same reason the evolution of the Ix, Iy, Iz, and Sx, Sy, Sz and the IlSz and IzSl (with l = x, y, z) operators under the respective Zeeman terms is the same as in the single spin case provided the W is replaced with WI or WS, respectively.

30. Evolution of a coupled spin system under weak J coupling The indirect interaction 2pIzSz, however, mixes I and S operators, because comutators such as [2pIzSz,Ix] do not vanish. Application of the Liouville-von Neumann equation yields another set of coupled differential equations. Consider, for example, Ix and IySz.

32. Product operators rules for a pulse • Pulse around the Ix and Iy axis with flip angle a

33. Elementary example: Spin Echo First a spin echo sequence is considered. After an initial 90x° pulse the spins evolve for a period t, undergo a 180x° pulse, and evolve for another period t. For an isolated spin the chemical shift evolution proceeds as follows:

34. Spin echo in a weakly coupled system In a coupled spin system the chemical shift is refocused as well. The indirect interaction, however, evolves as follows: The generation of the IxSz term depends entirely on the presence of the pSx rotation. In the absence of the pSx rotation the sequence would result in the Iy operator. In a homonuclear spin system the pIx and pSx rotation would typically be effected by the same pulse. However, in a heteronuclear coupled spin system two pulses, one for each nucleus, would be required.

35. INEPT insensitive nuclei enhanced polarisation transfer With a delay t = 1/(4JIS) the final operator is -2IzSy. The advantage of exciting the S spin via the INEPT step rather than directly is a gain in sensitivity proportional to the gyromagnetic ratios of the two spins: gI/gS.

36. HSQC (Heteronuclear Single Quantum coherence)

37. Maths 1 • So for a Cartesian spin ½ operator

38. Maths 2

39. Rotations r= cos(a) I + sin(a) j sin(a) r a cos(a)