Spin-lattice and spin-spin relaxation

Transcription

1 Spin-lattice and spin-spin relaation Sequence of events in the NMR eperiment: (i) application of a 90 pulse alters the population ratios, and creates transverse magnetic field components (M () ); (ii) the magnetiation vector M recovers until it reaches its equilibrium value M 0. The components of M deca eponentiall with time constants T 1 and T 2 : dm M 0 M = dt T dm dt 1 M = T ( ) ( ) 2 The approach of the sstem to thermal equilibrium is known as relaation and T 1 and T 2 are relaation times (relaation rates R 1(2) =1/T 1(2) are also used). oth relaation times are time constants used to characterie what are assumed to be first order rate processes. Spin-lattice relaation mechanisms The spin-lattice (or longitudinal) relaation time T 1 quantifies the rate of transfer of energ from the nuclear spin sstem to the neighboring molecules (the lattice). This is relaation in the -direction and leads to restoration of oltmann equilibrium. (1) (2) 0 M 0 t 0 M 0 M 0 t NMR Spectroscop 1

2 Closel spaced energ levels the rate of spontaneous emission is negligible (~ per second) Ecited nuclear spins E "LTTICE" translations, rotations and internal motions of molecules For an given NMR transition, there will be some possible change within the lattice involving the same quantit of energ. downward flip of a nuclear spin n acceleration of some motion of the molecule in which the flip occurred T 1 values are relativel long due to the lack of means to transfer energies of NMR transitions into thermal energ. Requirements for the energ transfer: (i) (ii) (iii) the motion in the lattice (which is epected to gain or lose the energ from the nuclear spin transition) must cause a fluctuating magnetic field (effectivel acting as a local pulse ) at the site of the nuclear spin involved. as for the observe rf pulse, the local fluctuating field must have a component at the Larmor frequenc, ν 0, of the nucleus under consideration. onl and components of the local field can cause T 1 -relaation. 0 loc (t) The rf field affects all the spins similarl and in a concerted wa (i.e., the rf pulse is coherent). The local field loc is randoml different at the site of each spin at an instant (i.e, the local field is incoherent). The most common source of the local fluctuating field for spin-1/2 nuclei is direct dipolar interaction. NMR Spectroscop 2

3 + _ N S Electric dipole Magnetic dipole In single crstals: lo c 0 N S N S lo c N S N S N S r H H H θ = 0 In solution the rapid reorientation of the dipolar interaction due to molecular motions provides fluctuating fields. This relaation mechanism depends on the rate of molecular motion. Thus, T 1 is temperature dependent. r θ = 9 0 H Molecular dnamics The correlation time τ c the time taken for the (spherical) molecule to rotate b roughl 1 radian. For tpical values of the viscosit in organic solvents: τ c M w where M w is the molecular mass (in Daltons). In the solid state, where the motion is hindered, τ c is large and there are onl weak frequenc components near ω 0 (= 2πν 0 ): ω 0 τ c >> 1 T 1 is long Rapid motion, associated with short τ c, does not possess much intensit at ω 0. The etreme narrowing limit: ω 0 τ c << 1 T 1 is long The frequenc distribution is optimum for T 1 relaation in the intermediate region where 1/ τ c ~ ω 0 : ω 0 τ c 1 T 1 minimum NMR Spectroscop 3

4 T 1(2) / s 360 MH T 1 60 MH T 2 τ c / s Other mechanisms for spin-lattice relaation: (i) interaction with unpaired electrons (e.g., in paramagnetic substances). This can be ver efficient and can lead to featureless broad lines. Sometimes it can be beneficial to add a small amount of paramagnetic impurit if the relaation times are ver long. (ii) interaction with electric field gradients for quadrupolar nuclei (with spin I > 1 / 2 ). This usuall leads to short relaation times and often featureless broad lines are observed. The spin-lattice relaation time determiness what reccle dela between pulses should be used. The nuclear spin sstem must be allowed to rela back to equilibrium before the net pulse is applied and this time period is determined b T 1. NMR Spectroscop 4

5 Relationship between T 1 and chemical structure The T 1 values for 13 C nuclei directl bonded to protons var between s. Longer values ( s) are observed for quaternar carbons. This is due to the fact that the main contribution to T 1 -relaation comes from 1 H- 13 C dipole-dipole interactions. In the absence of the motional effects, directl bonded protons have the largest effect on the T 1 values of corresponding 13 C nuclei: the more hdrogen atoms are attached to a carbon the shorter T 1 is. Isooctane T 1 / s CH 3 CH 3 13 H 3 C C CH 2 C H 68 CH 3 CH 23 3 For the dipole-dipole relaation, there is a strong dependence of the relaation rate on the distance between interacting nuclei T -1 1 ~ r -6 CH. In isooctane there are man spin-pairs to consider in order to estimate the contribution of the 1 H- 13 C dipole-dipole interaction and the above relationship needs to be modified: T -1 1 ~ i r -6 CH(i). s a first approimation, we can include into consideration onl directl-bonded protons: T 1 (CH)/T 1 (CH 2 )=2. This is in agreement with the eperimental values: 23/13=1.77. Possible reasons for the small deviation from the predicted ratio are that the dipolar relaation is additionall affected b more distant protons [e.g., there are 8 and 1 geminal protons for the CH 2 and CH groups, respectivel, in isooctane], and that relaatiom mechanisms other than the dipole-dipole interaction ma also be present. In the case of the CH 3 groups, T 1 is greater than T 1 (CH)/3, showing that the methl rotation slows down considerabl the 1 H- 13 C dipole-dipole relaation b shortening the effective correlation time τ C. Dependence on the magnitude of the nuclear magnetic moment Phenanthrene 1 H 1 H 2 H 2 H 59 s 80 s µ( 1 H) / µ( 2 H) 4 NMR Spectroscop 5

6 Spin-lattice relaation time measurements The standard method for measuring T 1 is known as inversion-recover. First, a 180 inverts the magnetiation along the - ais. time period, τ, is allowed, during which spin-lattice relaation occurs causing M to go from the value of -M 0 through ero to its equilibrium value of M pulse is then applied and the FID is recorded. determination of of the T 1 value. The eperiment is repeated with different τ delas, allowing Quantitativel, the deca of M is given b the loch equation (1). Integration of Eq (1) with M = -M 0 at t = 0 gives: M = M o [1-2 ep( t/t 1 )] In practice, the following equation is used: ln (I - I τ ) = ln 2I - τ / T 1 where Iτ is the initial intensit of the signal after the 90 pulse at time τ, and I is the limiting value of I τ for a ver long interval τ. T 1 is determined from the slope of a plot of ln (I - I τ ) vs. τ. For τ = T 1 ln 2 = 0.69 T 1 Iτ = 0, i.e., T 1 can be estimated from the pulse spacing τ that shows no signal after the 90 pulse. NMR Spectroscop 6

7 180 pulse τ short τ medium τ long τ 90 pulse acquire signal and FT I = I o {1-2 ep(-τ /T 1 )} NMR Spectroscop 7

8 Spin-spin relaation mechanisms Spin-spin (or transverse) relaation time T 2 is used to qantif the rate of the deca of the magnetiation within the plane. fter a 90 pulse the nuclear spins are aligned in one direction (are said to be phase coherent), but this arrangement is graduall lost (e.g., due to field inhomogeneities and/or direct interactions between the spins without energ transfer to the lattice). 0 Resultant: M =M 0 e -t/t 2 M 0 T 2 relaation does not affect the total amount of -magnetiation, but the degree of snchroniation of the transverse magnetiation components. t T 2 is related to the linewidth at half-height ( ν 1/2 ) of the NMR signal. T 2 ~ 1/ ν 1/2 Solid-state NMR Solution NMR The spin-spin relaation is related to spin-lattice relaation, since an increase in -magnetiation without a decrease in the magnetiation in the plane is not possible: T 2 T 1 (in solutions T 2 T 1 and in solids T 2 << T 1 ) The static dipolar fields created b neighbouring dipoles: loc ~ µ / r 3 The static dipolar fields created b neighbouring dipoles are ver large in solids. Thus, the spin-spin mechanism is ver efficient T 2 ~ 1 ms in solids and polmers. NMR Spectroscop 8

9 In non-viscous solutions, the static dipolar fields average out as a result of random molecular motion, leading to T 2 T 1. When etreme narrowing condition breaks down: T 2 < T 1 Spin-spin relaation time measurements fter a 90 pulse the net magnetiation in the plane graduall decas. The deca is of the form ep( t/t * 2 ), and is due to two factors: (i) magnetic field inhomogeneit and (ii) tspin-spin relaation. n NMR signal in the time domain decaing according to ep( t/t * 2 ) gives a Lorentian peak of halfheight linewidth 1/πT * 2 after Fourier transformation. The time constant T * 2 includes both factors (i) and (ii). In order to eliminate the field inhomogeneit contribution and to measure true T 2 relaation, a spinecho pulse sequence is used. The modification of the spin-echo pulse sequence developed b Carr and Purcell: 90 τ 180 τ (1st echo) - τ 180 τ (2nd echo)... To determine T 2 we start from equation (2), which can be solved for M to give: M = M 0 ep ( -t / T 2 ) ln M = ln M 0 - t / T 2 Since intensit of the echo is proportional to M : ln I(t) = ln I 0 - t / T 2 where t = 2τ, 4τ,.... T 2 is determined from the slope of a plot of ln I(t) vs. t. NMR Spectroscop 9

10 M =M 0 90 M 180 τ 2τ 180 3τ 4τ NMR Spectroscop 10