3 Multidimensional NMR Spectroscopy C Gerd Gemmecker 1999 Models used for the description of NMR experiments 1. energy level diagram: only for polarisations, not dependent phenom 2. classical treatment(BLOCH EQUATIONS: only for isolated spins(no J coupling!) 3. vektor diagram: pictorial representation of the classical approach(same limitations) 4. quantum mechanical treatment (density matrix): rather complicated; however, using appropriate simplifications and definitions-the product operators-a fairly easy and correct description of most experiments is possible 3.1. BLOCH equations The behaviour of isolated spins can be described by classical differential equations dM/dt=yM(t)x B(t)-R(M(t)-Mol [3-1 with Mo being the BOLTZMANN equilibrium magnetization and r the relaxation matrix R 01/T 2 0 0 The external magnetic field consists of the static field Bo and the oscillating r f. field Brf B(t)=Bo+ Br =Bcos(ot+φ)ex [3-3] The time-dependent behaviour of the magnetization vector corresponds to rotations in space(plus relaxation), with the Bx and B components derived from r f. pulses and B, from the static field dMz/dt =yBx My -yByMx-(Mz-MO)1 [3-4] dM/dt=yBy -Mx/2 [3-5] dMy/dt=yBZMx-y BxMz-MV/T2 [3-6] Product operators To include coupling a special quantum mechanical treatment has to be chosen for description. An operator, called the spin density matrix p(t), completely describes the state of a large ensemble of26 3 Multidimensional NMR Spectroscopy © Gerd Gemmecker, 1999 Models used for the description of NMR experiments 1. energy level diagram: only for polarisations, not for time-dependent phenomena 2. classical treatment (BLOCH EQUATIONS): only for isolated spins (no J coupling!) 3. vektor diagram: pictorial representation of the classical approach (same limitations) 4. quantum mechanical treatment (density matrix): rather complicated; however, using appropriate simplifications and definitions – the product operators – a fairly easy and correct description of most experiments is possible 3.1. BLOCH Equations The behaviour of isolated spins can be described by classical differential equations: dM/dt = gM(t) x B(t) - R{M(t) -M0 } [3-1] with M0 being the BOLTZMANN equilibrium magnetization and R the relaxation matrix: x y z R = ë ê é û ú ù 1/T2 0 0 0 1/T2 0 0 0 1/T1 The external magnetic field consists of the static field B0 and the oscillating r.f. field Brf : B(t) = B0 + Brf [3-2] Brf = B1 cos(wt + f)ex [3-3] The time-dependent behaviour of the magnetization vector corresponds to rotations in space (plus relaxation), with the Bx and By components derived from r.f. pulses and Bz from the static field: dMz /dt = gBxMy - gByMx -(Mz -M0 )/T1 [3-4] dMx /dt = gByMz - gBzMy - Mx /T2 [3-5] dMy /dt = gBzMx - gBxMz - My /T2 [3-6] Product operators To include coupling a special quantum mechanical treatment has to be chosen for description. An operator, called the spin density matrix r(t), completely describes the state of a large ensemble of