Relaxation Times i Relaxation = return to equilibrium (Boltzmann) after a pulse, redistribution of energy Relaxation can be described for isolated spins by the Bloch Equations, the total relaxation is determined by two characteristic time constants: Longitudinal, Spin-Lattice Relaxation Build up of longitudinal magnetisation via energy exchange between spins and their environment („lattice") enthalpy Transversal, Spin-Spin Relaxation Dephasing of transversal magnetisation without energy exchange between spins and their environment, entropy Two important relations between tTl and T2 must be remembetei ^ T2 cannot be longer than tTl: T2 < tTl y In the „extreme narrowing limit": T2 = Tt and T2 in Data Acquisition exp(-t/T1) exp(-t/T2) relaxation delay repetition time T1 governs the repetition frequency for subsequent transients Relaxation delay = 5 T1 T2 governs the decay time constant of individual FID's Optimum sensitivity of the NMR experiment is obtained ifT1 = T2 Magnetization In-Field More nuclei point in parallel to the static magnetic field. The macroscopic magnetic moment, M0 M Longitudinal Magnetization No magnetic field M = 0 I M Magnetic field Ho* 0 0 u t Spin-Lattice Relaxation Time Rx = 1/T1 [Hz] longitudinal relaxation rate constant T1 [s] longitudinal relaxation time spin-lattice relaxation time enthalpy Transverse Magnetisation -t Spin-Spin Relaxation Time R7 = 1/T7 [Hz] transverse re rate constant T2 [s] transverse relaxation time constant spin-spin relaxation time entropy Free Induction Decay FID Sü£(t)= M^- cos (ms) * exp (-t/Tj) Sim ft) = Meq, sinffflLt) t exp (-t/T2) S(t) = Steft) ■ iSUO - exp(i coit) exp (-£/T3) = £xp[-(I/T2 - i coi)t\ 11 12 Relaxation = Return to Eauilibrium thermal longitudinal > relaxaticm aon- equilibrium equilibrium state state (b) Z perturbation: itf2 pulse 2 thermal transverse (T2) and non_ equ i I ibrium lorigi ludinal (7*! > re] axation equ i I ibrium state stale Relaxation Relaxation in other types of spectroscop • spontaneous emission (not in NMR) fluorescence, phosphorescence • collisional deactivation (not in NMR, molecular tumbling does not change orientation of I, always along B0) • stimulated emission lasers magnetic interactions of I with external fluctuating mg. field (dipolar) containing many different frequencies, when it contains C0L resonance causes relaxation = emission of excess energy, transition from exited to ground state 14 Linewidth hAv1/2 Tj > h/2i T1 = lifetime of a nucleus in a certain energy state Heissenberg uncertainly principle AE At > h/27i h = 6.626 10"34 J s Avv > 1/tt High relaxation rate = short relation times = wide lines in spectra Werner Heisenberg (1901-1976) NP in physics 1932 I Correlation Time T, Correlation Time tc describes molecular tumbling 1. Look at one molecule I tc = average time during which a molecule stays in one orientation, until a collision changes its orientation small molecules, low viscosity 10"12 s polymers, high viscosity I Correlation Time x, I 2. Look at a group of molecules ( 1 mole) All molecules oriented in the same way, then Tc is time in which the orientation is dispersed to 1 rad (-60°) t < Tc molecules are close to the original orientatio t >> Tc random distribution 1/tc = tumbling rate Correlation Time xc Correlation function describes molecular tumbling Correlation function Correlation Time T, tc <> 1/cd0 poor energy transfer, T1 long, narrow lines tc = 1/g)0 effective energy transfer, T1 short, fast relaxation, wide lines I T| = viscosity, high T| = slow tumbling, long Tc, wide lines a = molecular diameter, large particles = long Tc, wide lines T = temperature, high T = fast tumbling = short Tc, narrow lines Correlation Time x, Approximate rule Tc [ps] ML in H20 at room temp Supercritical C02 is a good NMR solvent @65 °C and 65 bar has low viscosity, narrow lines short Tc = fast tumbling, small molecules, low viscosity long Tc = slow tumbling rigid molecules, high viscosity Correlation Time T The Influence of Correlation Times on Relaxation > Correlation times are not molecular constants, but depend on a number of factors, e.g. temperature, effective molecular size, solvent viscosity... > Variation of these factors may induce changes in tc of several ordes of magnitude. > These changes may lead to violation of the „extreme narrowing" conditions, and introduce the necessity for a more concise treatment of the correlation time dependence of relaxation times. 23 long tc = slow tumbling = long T1 short T2 Linewidth is given by T2 Relaxation Time T Fluctuating magnetic fields (of the right amplitude and frequency) make spins exchange energy with their environment Important mechanisms to generate these fluctuating magnetic fields are: Direct dipolar interaction of a nuclear spin with other nuclear spins Molecular motion in the presence of large chemical shieldin anisotropics Interaction of a nuclear spin with a nuclear quadrupole The individual contributions combine to make the total relaxation Dipolar Relaxation Ta DD The Direct Interaction of a Nuclear Spin with other Spins INTRAMOLECULAR The magnetic moment of a nuclear spin B influences the local field at the position of a neighbouring nucleus A: B loc(A) = Bloc,0(A) + D D denotes the dipolar coupling constant which is defined as Brownian motion of the sample containing nuclei A and B induces a fluctuation of 0 which leads in turn to a time dependent modulation of the local magnetic field Bloc(A). Dipolar Relaxation T1DD I The contribution of this modulation to the Tl relaxation of nucleus A can be expressed in terms of a characteristic time constant T1DD: Xn = molecular correlation time S = spin of nucles B yB = large magnetogyric ratio, faster relaxation, shorter 1\ DD substitution H/D nuclei with large y (e.g. H) relax nuclei with small y l/r6AB = only directly bound nuclei contribute = intramolecular (in the extreme narrowing limit) 27 Dipolar Relaxation T1DD INTERMOLECULAR I N0 = number of molecules in m3 D = difussion coefficient T = temp, high T narrows lines Protons relax both inter and intramolecularly C6H6 neat C6H6 diluted in CS2 Tj(H) = 19 s Tj(H) = 90 s 28 I: The Influence of the observed nucleus A in a A-H fragment: 31p BC 29Si 15N 103Rh Y(X) 10.84 6.73 -5.32 -2.71 -0.85 'AH [A] 1.4 1.1 1.4 1.0 1.6 (tc=10-») 8 s 5 s 33 s 17 s 48 min T1>DD (tc=10-9) 80 ms 50 ms 330 ms 170 ms 29 s II: The Influence of the neighborin g nucleus X in an A-X fragment (A=15N): X *H 31p Bc nB 51y Y(X) 26.75 10.84 6.73 8.59 7.05 'AX [A] 1.0 1.7 1.4 1.3 1.8 S(S+1) 0.75 0.75 0.75 3.75 3.75 T1>DD (^lO-11) 17 s 42 min 34 min 160 s 400 s T1;DD (tc-10-9) 170 ms 25 s 20 s 1.6 s 4s III: The Influence of the internuclear distance in a N- H fragment: 'AX [A] 1.0 2.1 2.7 (N-H) (N-C-H) (N-C-C-H) Tl,DD ftc=10-n) 8 s 24 min 110 min xc = 1011s: medium sized (in)organic molecule xc = 109s: small polymer 29 Quadrupole Induced Relaxation rT1 q The Interaction of a nuclear spin with a quadrupole momem I Nuclei I > V2 0 ^—^ © (s>.....e\ V.....6 J © © nucleus Nuclei with I> V2 possess an electric quadrupole moment eQ which is quantized according to its oriention in the electric field gradient (efg) of the electrons if the local symmetry is less than spherical. Due to strong coupling between eQ and I, the nuclear magnetic spin levels depend on both B0 and the efg. Electric quadrupole moment eQ = nonspharical distribution of the positive nuclear charge 30 Quadrupole Induced Relaxation Ta q BROWNIAN MOTION of sample molecules modulates the different mI energies which leads to a stochastic modulation of the local magnetic field Bloc(A). Tumbling = spread of energy levels 1 = 1 ........maa/w m=0 ........"maaaaa mi="^ I in solution the average transition energy does not change but the spread contributes to relaxation 31 Quadrupole Induced Relaxation Ta q The contribution to Tt(A) can be expressed in terms of a characteristic time constant Tjq (extreme narrowing limit) I Tc = correlation time / = nuclear spin Q = nuclear quadrupole moment (Q ^ 0 for I > Y2 ) qzz = electric field gradient qzz = 0 for high symmetry (spherical, CI", cubic Td, Oh, C104", S042", AsF6" yj = asymmetry parameter (yj = 0 for axial symmetry) 32 Quadrupole Induced Relaxation Ta q Nuclear Quadrupole Coupling Constant, NQCC % = £ q Q I fl Linewidth factor I. The Influence of the electric field gradient q^. 14N relaxation times: Bu4N+(Td) NaN03(D3h) NA^J (C^) MeSCNCQJ DABCOCQJ c[MHz] 0.04 0.745 1.03 3.75 4.93 Tjq 1.8 s 85 ms 29 ms 2 ms 0.6 ms 34 CSA Induced Relaxation, T1CSA I Tumbling of molecules with large chemical shielding anisotropics Important for nuclei with wide range of chemical shifts: 31P, 195Pt, 113Cd Magnetic shielding is anisotropic and may vary for different orientations of the magnetic field B0with respect to the molecular frame. Chemical Shielding Anisotropy CSA 8 * -500 8„* +14500 BROWNIAN MOTION of sample molecules induces time dependent modulation of fj and thus a stochastic fluctuation of the effective local magnetic field B01oc(A). CSA Induced Relaxation, CSA The contribution to Tt(A) can be expressed in terms of a characteristic time constant: (in the extreme narrowing limit) Tc = molecular correlation time Ad = shielding anisotropy B0 = magnetic field strength = wide lines in strong magnets !!!! 36 I: The Influence of the observed nucleus A in (Act = 100 ppm; B0 = 7 T): | A 31P 13C 15N | y(X) 10.84 6.73 -2.71 T1>CSA (tc=10-u) 130 s 340 s 35min T^csa (tc=10-9) 1.3 s 3.4 s 21s II: The Influence of the magnetic field BQ (nucleus 195Pt; Act = 1000 ppm): 1 B0[T] 4.7 7.1 11.7 17.6 | v(aH) [MHz] 200 300 500 750 TlcSA(tc=10-n) 10 s 4 s 1.6 s 0.7 s T1CSA (tc=109) 100 ms 40 ms 16 ms 7 ms III: The Influence of the shielding anisotropy (nucleus 195Pt; BQ = 7 T): Aa [ppm] 15 150 1500 15000 T1CSA (tc=10n) 5.5 h 3.3 min 2 s 20 ms T1CSA (tc=10-9) 3.3 min 2 s 20 ms 0.2 ms tc = 1011s: medium sized (in)organic molecule; tc= 109s: small polymer; Spin Rotation Induced Relaxation, Ta SR Tumbling molecule = bonding electrons move and induce magnetic field around the molecule. Important for small fast rotating molecules with high symmetry: SF6, PC13, PtL4 H^MMMH V = moment of inertia C = SR constant Tj = time in which a molecule changes its angular momentum, e.g. time between collisions Hubbard (if Tj << Tc valid for small molecules below b. p. 6kBT Contributions of CSA versus SR [Pt(P0 Inversion Pulse 90° Read Pulse r:M7(r)=M7u(l-2e-r/r') recovery of z-magnetisation Non-equilibrium z-magnetisation recorvers during delay t M2(t) is converted into observable magnetisation by the read pulse Performing a series of experiments and incrementing t allows to sample M2(t) at different times Tj is obtained from a fit of observed signal intensities as a function of t Relaxation Time T T2 relaxation occurs without energy transfer ^ „entropie process The characteristic time constant T2 is connected with the linewidth: I T T 2 2,true describes the effect of magnetic field inhomogeneities *.e. mostly bad shimming Relaxation Time T For most I = n/2-nuclei, VT2>true» 1/T2>* T2 may be determined directly from measured linewidth: nAwl/2= 1/T2«1/T2true For I = 1/2-nuclei, VT2>true < 1/T2 , T2 must be measured by a dedicated experiments (CPMG) 48