To be able to calculate the spin-lattice relaxation rate for the nuclear spins systems of a sample it is necessary to consider the motions of the nuclear spins. As they are at the nucleus of atoms their motions are the motions of those atoms in the molecules. In the following we will consider the 1H spins with I = 1/2. These motions can be divided in two groups: internal motions and global motions. The former are the motions of hydrogen atoms inside the molecule associated with fast conformational changes involving chemical groups (e.g., CH2, CH3) in aliphatic chains, or rotations of phenyl rings around the para-axis, etc. The latter involve the motions of the molecules as a whole as can translational displacements and rotations/reorientations. In partially ordered systems like liquid- crystalline (LC) phases slow motions involving groups of molecules are identified as collective motions.
The calculus of a spin lattice relaxation can be illustrated by considering the rotations of two spins in a "spherical” molecule of radius a. To calculate Eq. 6.20 it is necessary to calculate the probability function P (^, ^0, t) that relates the angular orientation of the interproton spin vector ^0 = (60,
0) at time to with the angular orientation ^ = (в,
where DR is the rotational diffusion constant. A classical solution for this equation involves the expression of the probability function in an expansion of spherical harmonics P (^, t) = YRm l c;m(t)Y/m)(^), where the time dependent functions c;m(t) can be obtained from Eq. 6.33 using the know relation V2Y/m)(^) = —(l + 1)Y/m)(^)
c;m(t) = c;m(0)e—t/Tl (6.34)
with t—1 = DRl(l + 1)a—2.
The calculus of Eq. 6.20 can performed taking advantage of the orthogonal properties of the spherical harmonics which yields (Abragam, 1961)
Figure 6.2 Proton relaxation rates calculated with Eq. 6.37, considering r = 2 x 10-10 m and vt = «г/(2ж) = 10 kHz. (a) The relaxation rates are represented as functions of t2 for v0 = «o/(2n) = 100 MHz. (b) The relaxation rates are represented as functions of the Larmor frequency for т2 = 1/(2п 108) s (black solid line) and т2 = 1/(2п 107) s (black dashed line). Spin-spin relaxation rate and spin-lattice relaxation in the rotating frame calculated using Eqs. 6.38 and 6.39, respectively, are also presented.
with т2 = Dr/(6a2). The calculus of the spectral densities is straightforward and gives
The spin-lattice relaxation rate is
and the spin-spin relaxation rate is
The spin-lattice relaxation rate in the rotating frame is
In Fig. 6.2 are presented the relaxation rates calculates using Eqs. 6.37, 6.38, and 6.39. Kdd ~ 8 x 10-25 m3s-1.a r = 2 x 10-10 m
aFrom Eq. 6.9 using y, = yS = 2.675 221 28(81) x 108 s-1T-1, ц0 = 4n x 10-7 Hm-1, h = h/(2n)) = 1.0 5 4 5 71 628(53) x 10-34 Js, Kdd ~ 8 x 10-25 m3s-1 (IUPAC, 1997).
and v1 = ш1/(2п) = 10 kHz. As it can be observed in Fig. 6.2a the longitudinal relaxation rate is represented as a function of r2 and shows a maximum for a value of r2 & &>-1. Since ш1 т2 ^ 1, T-1 and T- are very similar within the plot's resolution and show an inflexion around ю0т2 & 1 and are proportional to t2 when ю0т2 ^ 1. In Fig. 6.2b are presented the relaxation rates calculated as functions of the Larmor frequency, for two values of t2. In both cases, it is clear that in the frequency range ю0т2 ^ 1 all relaxation rates have similar values and for values of t2 and ш1 that verify rn1x2 ^ 1 T-1 & T-1 & T- & 5t2. In the high-frequency limit &>0t2 ^ 1 T-1 & щ-2 whereas T— & T— & 3t2/2 became
independent of the Larmor frequency.
In Fig. 6.2b it is possible to observe that an increase of t2 produces an increase of the relaxation rates for frequencies ш0 ^ t-1. For frequencies ш0 ^ t-1, the spin-lattice relaxation rate has the opposite behavior as it decreases with increasing t2. In the high- frequency regime both T-1 and T- increase with t2.
This type of analysis becomes handy when considering the temperature dependence of correlation time t2. The molecular rotations are usually thermally activated processes and the rotation diffusion is usually expressed by an Arrhenius temperature dependence expression
with R = NAkB, where NA is the Avogadro's number3 and kB is the Boltzmann constant.b Ea is usually referred to as activation energy and in some cases of the order of kJ per mol. Since t2 a DR it is clear that when the temperature increases the correlation time decreases (i.e., the motions are faster).
Therefore, with respect to molecular rotations when studied by proton spin-lattice relaxation as described by Eq. 6.37, it is possible to relate an increase of temperature to a decrease of the relaxation rate at low frequencies (&>0t2 ^ 1) and to an increase of T - at high frequencies (&>0t2 ^ 1).
In the case of more than one pair of identical spins when the rotations of the molecules can be still described by a single
aAvogadro's number: NA = 6.022 141 79(30) x 1023 mol-1 (IUPAC, 1997). bkB = 1.380 650 4(24)-23 JK-1 (IUPAC, 1997).
correlation time to fast spin-spin interactions Eqs. 6.37, 6.38, and 6.39 are still valid provided that and ensemble average is performed over all interspin vectors, leading to spectral densities of the form
The relaxation mechanism expressed by Eq. 6.37 with the spectral densities given by Eq. 6.36 is often referred to as the Bloembergen-Purcell-Pound (BPP) relaxation mechanism (Bloem- bergen et al., 1948).