Relaxation field

The first approximate calculation was carried out by Debye and Hiickel and later by Onsager, who obtained the following relationship for the relative strength of the relaxation field AE/E in a very dilute solution of a single uni-univalent electrolyte... 

In the ideal case, the ionic conductivity is given by the product z,Ft/ . Because of the electrophoretic effect, the real ionic mobility differs from the ideal by A[/, and equals U° + At/,. Further, in real systems the electric field is not given by the external field E alone, but also by the relaxation field AE, and thus equals E + AE. Thus the conductivity (related to the unit external field E) is increased by the factor E + AE)/E. Consideration of both these effects leads to the following expressions for the equivalent ionic conductivity (cf. Eq. 2.4.9) ... 

The equivalent conductivity of an electrolyte solution decreases with increasing concentration due to interionic attractions described mainly by the electrophoretic and relaxation field effects 2-35>. This decrease is more pronounced if in addition the electrolyte is associated. Association of ionic salts by ion-pairing is commonly observed in solvents of low or moderate dielectric constant. The immediate goals in the analysis of conductance data are the. determination of the limiting equivalent conductance at infinite dilution, A0, and the evaluation of the association constant, KA, if ion-pairing occurs. 

Table III suggests some of the proton transfer kinetic studies one is likely to hear most about in the near future. The very first entry, colloidal suspensions, is one that Professor Langford mentioned earlier in these proceedings. In the relaxation field, one of the comparatively new developments has been the measurement of kinetics of ion transfer to and from colloidal suspensions. Yasunaga at Hiroshima University is a pioneer in this type of study (20, 21, 22). His students take materials such as iron oxides that form colloidal suspensions that do not precipitate rapidly and measure the kinetics of proton transfer to the colloidal particles using relaxation techniques such as the pressure-jump method.

Since this is essentially an engineering chapter, we shall dwell only on the last point. Prom the BPP formula, it was already qualitatively clear that, in order to become efficient and useful tools, the dispersion curves must extend over a wide interval of relaxation field values (preferably several orders of magnitude). 

B (the Zeeman field) must be made to commute rapidly between several predefined values, such as the polarization field Bp, the relaxation field and the acquisition field Ba-... 

In experiments at relaxation fields close to zero, the precision, resolution and stability of the absolute value of Br become critical and a bipolar configuration though it makes the design and implementation of the hardware more complex, it improves the precision of the whole system and offers the following advantages ... 

Improved setting precision of the relaxation field for values close to zero. 

Precise null calibration of current offsets (which can be positive as well as negative) is possible thanks to the bipolar configuration of the power supply whose range in the negative section, is much smaller than in the positive section, and can be set with a considerably higher resolution. This increases the absolute precision of the field values for very small relaxation field settings and, in practice, makes it possible to extend the NMRD profiles to fields as low as a few tens of iT. 

In a traditional magnet for NMR spectroscopy, the field Bq of the magnet is much higher than the field components originating from outside sources. Moreover, devices such as efficient NMR field stabilizers are used to suppress all interfering external fields. Consequently, the presence of such field components can be usually ignored. On the contrary, during a FFC NMR measurement the sample may be subject to very low fields (ideally down to zero) which is practically impossible when the relaxation field value becomes comparable to the environmental fields. The amplitude of such fields, if not compensated, represents the lower relaxation field limit for a reliable NMRD profile. 

When T is much larger than Ti, S B Ba, where B is the relaxation field. 

In general, an FFC relaxation rate measurement requires a series of elementary experiments in which the duration x of just one of the fixed-field intervals varies, while each of the switching intervals has always the same duration. Only in this way can one guarantee that the measured relaxation rate is correct and that it corresponds to the relaxation field present during the variable-duration interval (to be discussed later). 

The discrepancy is due to sources of errors not considered in the above discussion, such as field noise and reproducibility, thermal effects, etc. In particular, thermal effects on the magnet are important since from experience, it shows that the scatter increases when the relaxation field and/or the polarization field are close to the upper limit of the magnet. Since all such contributions are random, prolonged data accumulation reduces both the fitting errors and the scatter. 

Extremely precise field settling (well below 0.1%) is required only when switching to the acquisition field where RE pulses are to be applied and/ or the NMR signal is to be collected. In all other switching periods (for example, switching from the polarization field to the relaxation field), field-settling precision of the order of 0.1% is quite sufficient. 

PP sequence for relaxation fields smaller than half the polarization field value and the NP sequence for relaxation fields higher than that. 

Fig. 27. Thermally balanced PP and NP sequences. (PP) In the balanced PP sequence, the sample is first kept at the relaxation field By for a time — t and, then pre-polarized at the polarization field Bp for a time Tp, and finally allowed to relax for time T before the start of the detection period. The time Tp should be set to about 4Ti(Bp). As T varies during a multi-block sequence, the polarization interval position moves horizontally but the total block duration and the mean power dissipation remain constant. (NP) The balanced non-polarized sequence is conceptually similar, except for the fact that the polarization interval is replaced by a magnetization annihilation interval in which the field is zero and whose duration should be about 47 (0). In both cases, the time should be about or more than 4Ti(Br). The concept can be combined with any detection mode, not just the simple FID detection shown here.

Implementation of efficient devices for the compensation of environmental magnetic fields, both stationary and variable/alternating and a push toward more reliable measurements in the relaxation-field region of 100 Hz-10 kHz. 

In 1962 Fuoss and Onsager began a revision of their treatment of the conductance of symmetrical electrolytes. In their first paper they considered the potential of total force in the second, the relaxation field in the third, electrophoresis and in the fourth, the hydrodynamic and osmotic terms in the relaxation field (1,2,3,4). In 1965 Fuoss, Onsager, and Skinner (5) combined the results of the four papers and formulated a general conductance equation ... 

In the above treatment of the relaxation field, it has been assumed that the only motion of the central ion destroying the spherical symmetry of the ionic cloud is motion... 

Onsager considered the effect that this erratic character of the leadership would have on the time-averaged shape of the ionic cloud and therefore on the relaxation field. His final result differs from Eq. (4.309) in two respects (1) Instead of the numerical factor, there is a factor and (2) a correction factor cu/2z has to be introduced, the quantity cu being given by... 

The forces /+ and / are given by the product of the charge per mole of ions and the local field. The latter is equal to the external applied field E plus the relaxation field A.E. The relaxation field arises because of the asymmetry in the ionic atmosphere caused by the motion of the ion with respect to its atmosphere. Thus, for the cation... 

The theory of the relaxation effect makes use of the equations of hydrodynamics applied with consideration of the effects of interionic forces. It is the most difficult part of the description of non-equilibrium processes in electrolytes, and the details of the derivation are not given here. The result obtained for the relative value of the relaxation field is... 

Far from the ionisation threshold, where the escaping photoelectron has a large kinetic energy, the normal method of calculation for the continuum states is to compute the orbitals in a field determined by using the frozen orbitals of the neutral atom with one electron removed. However, if one is calculating a resonance which lies close to the threshold, this approach may fail. This happens because the escaping photoelectron moves slowly, so that the residual ion has time to relax as it escapes. It is then better to compute the orbitals in the relaxed field of the ion. This approximation is called the GRPAE or the RPAER, and is referred to as the RPAE with relaxation. 

In order to measure of equation (11b) frequency or field dependent for frequencies greater than 5 MHz, where the standard technique no longer works, Bq must be varied over the relaxation period the generation and detection of the local spin order can be achieved under any suitable high-field condition. We realized this possibility with two of our FC spectrometers by a Bq cycle with a fast field switch Bpi from the polarization to the relaxation field at the beginning of the relaxation period (t2 interval), and the inverse switch Bp from the relaxation to the detection field Bp = Bp at the end of the T2 interval. The extended Jeener sequence... 

The expression for AX/X is obtained by a series of successive approximations yielding the terms of different order which contribute to the relaxation field. The first order term arises from eqn. 5.2.13 putting all the T terms equal to zero. Fuoss and Onsager obtain, in this way, an equation equivalent to 5.2.15. The first order expression for is then replaced in the T terms and a further approximation to the perturbed distributions and ionic potentials is calculated. 

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