Spectral density at zero frequency

Notice the presence of a spectral density at zero frequency in i 2 arising from bz t)bz Q)dt (which evidently does not require to switch to the rotating frame). This zero frequency spectral density will be systematically encountered in transverse relaxation rates and, in the case of slow motions, explains... 

As can be seen from Eq. (1), Tic is determined by the spectral densities at the frequencies of coq, and o)h n>c- Assume here that the distributions of the spectral densities are as shown in Fig. 1 l-(b), where the frequency range of o)h> (Ocy and o)H n>c is shown by the vertical arrow. Then the Tic s of both of the non-crystalUne phases are determined by the local motion dictated by tca> independent of the other motions that have no meaningful spectral density at the frequency range indicated by the arrow. Then the two noncrystalUne phases must have the same Ti c. On the other hand, the T2C is determined by the spectral density at zero for the solid matter as pointed out at the end of Sect. 2.2. Hence, the T2c s of the amorphous phase and the crystalUne-amorphous interphase are respectively determined by the motion of B and C, because the spectral density at zero of motion A is negUgible. Since the spectral densities at zero frequency of both the B and C motions are quite different, the two phases must have quite different T2C values, [58]. 

It turns out that the secular part depends on the spectral density at zero frequency, 7(0). We can see that this makes sense as this part of transverse relaxation requires no transitions, just a field to cause a local variation in the magnetic field. Looking at the result from section 8.5.2 we see that 7(0) = 2tc, and so as the correlation time gets longer and longer, so too does the relaxation rate constant. Thus large molecules in the slow motion limit are characterised by very rapid transverse relaxation this is in contrast to longitudinal relaxation is most rapid for a particular value of the correlation time. 

Since the collective orientational correlation time depends on the structure of a liquid, it is plausible that the rate of structural evolution of the liquid is proportional to this quantity. Thus, at lower temperatures rcon is longer and therefore the structural fluctuations are slower. As a result, motional narrowing is less effective as the temperature is lowered. While less motional narrowing would normally lead to a slower decay in the time domain, in this case the spectral density goes down to zero frequency. Thus, motional narrowing can reduce the spectral density at low frequencies and thereby decrease the intermediate relaxation time. 

Chang and Woessner, using the spin-echo technique, made and TU measurements on eletal muscle (115). While the curve Is cfbse to a single exponential decay, the T2 curve can be fitted with a double exponential function. The fast and slow (f,s) components have T values of 16 and 1.6 ms, respectively, for fresh muscle. The value for the same sample Is 18 ms. Hence, T. - Tg- < Tgg and Tgg = T. The reader will recall (Section 2 that T g depends upon the spectral density at twice the Larmor frequency, T depends on J((o whereas Tgf depends upon spectral densities at zero emd at Larmor frequency and Tgg depends upon the sum J(o) + J(2a>). Thus, the experimental results obtained by Chang emd Woessner Imply that J(0)>>J(o) ) = J(2oj ). This last equality. In turn. Indicates... 

For a fixed value of Tc, the frequency dependence of either term is a Lorentzian centred at zero frequency. In the tc dependence two regimes are distinguished In the fast motion regime (coiTc spectral density is proportional to tc and does not depend on the measuring frequency a>i, whereas in the slow motion regime (a>iTc > l) it is proportional to ( Tc) i.e. the relaxation rate exhibits dispersion. 

The problem may be formulated as follows. Given random noises (t) with different correlation parameters and Tj but possessing identical spectral densities (u = 0) at zero frequency, that is,... 

The resulting transverse relaxation time also depends on the spectral density at frequency zero. 

Figuring out the frequency of a given transition is simple. On a 500MHz instrument, the frequency is 500MHz and the frequency is 125 MHz. For the single quantum Wjh transition that involves only flipping the spin of the H, the frequency of the photons that will drive this transition is 500 MHz. If the spectral density function is not zero at 500 MHz, then the Wjh transition will he efficient and the dipolar relaxation mechanism will he an efficient relaxation pathway for the H. For the transition, the frequency is 125 MHz so in this case, having spectral density at 125 MHz will make the single quantum dipolar relaxation mechanism efficient. For the double quantum W2 transition that connects the ota and pp spin state combinations, spectral density at 625 MHz (vh + Vc) is required. For the zero quantum Wg transition (also called the flip-flop transition), spectral density at 375 MHz (vh Vc) is required to make the transition efficient.

Modulation of the z component of the local field cannot infiuence the z component of the spin magnetization and thus cannot induce spin flips. However, the X and y components of the magnetization that stem from coherent superposition of spin states with lAm l = 1 are influenced, as the modulation of the z component can induce a cooperative flip-flop of two spins. The energy change during such a flip-flop is close to zero, as the two transition frequencies are almost equal. Such transverse or spin-spin relaxation with time constant 7] is thus related to spectral density /(O) at zero frequency. This spectral density 7(0) increases with increasing Tc, i.e., with a slowdown of the process. The total rate of transverse relaxation includes a contribution of spin flips, so that T2- =(T2 ) +(271)- . 

Transverse nuclear relaxation can also occur when the local fields at the nucleus fluctuate slowly, i.e. with an co frequency near zero (see also Section 3.4). Then the spectral density function will take the form... 

In this application of the BWR theory, Hudson and Lewis assume that the dominant line-broadening mechanism is provided by the modulation of a second rank tensor interaction (i.e., ZFS) higher rank tensor contributions are assumed to be negligible. R is a 7 X 7 matrix for the S = 7/2 system, with matrix elements written in terms of the spectral densities J (co, rv) (see reference [65] for details). The intensity of the i-th transition also can be calculated from the eigenvectors of R. In general, there are four transitions with non-zero intensity at any frequency, raising the prospect of a multi-exponential decay of the transverse magnetization. There is not a one-to-one correspondence between the... 

The next crucial observation is the non-identity of the longitudinal and transverse relaxation rates. It joints unambiguously to case (iii) above. The appropriate expression for the ratio of the enhancements AR of the longitudinal (i=l) and transverse (i 2) relaxation rates is a function of the spectral densities J(w) at frequencies zero, Larmor, and twice the Larmor frequency (u) 0, u)0 2(Dq) as follows ... 

Pin at frequency m. As the auto-correlation function is characterized by the correlation time Tc, where a (r) approaches zero as r Tc, the spectral density function is characterized by the frequency cuc = 1/Tc, where 7 (m) approaches zero as cu [Pg.119]

Thus, M2XC is the relaxation rate 1/ T2, which has been calculated in (3.5.6) by the BPP theory in the fast motion limit. In the slow motion limit of (3.5.6) only the spectral density (3.5.8a) at frequency zero needs to be considered, and... 

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