Conceptually, this Note builds on a previous one (1), henceforth referred-to as Part I, which deals with the analysis of the effects of field noise/modulation on simple FID's and which the reader should consult also for basic assumptions and terminology.
The Hahn spin-echo (2) is extensively used in a wide range of NMR applications such as, for example, T2 measurements in LR-NMR and HR-NMR, refocusing pulses in HR-NMR, spin-diffusion measurements, sensitive plane selection and/or k-space navigation in MRI, flow imaging, receiver dead-time masking, etc. Be it protein structure determination, brain-cancer scan, NMRD profile of an elastomer, ex-situ measurement of a monument's masonry or a geo-prospecting well log, spin echoes are likely to be part of the procedure.
Understanding the statistical properties of echo fluctuations in an unstable main magnetic field is therefore of considerable importance across the whole of NMR.
This Note analyses the quantitative statistical relationship between spin echo phase instabilities and magnetic field fluctuations. Only the Hahn spin echoes induced by RF-pulses are considered, leaving for the moment out echoes induced by field-gradients which are used in many MRI techniques. Though gradient echoes sensitivity to field/gradient noise can certainly be investigated by methods similar to those adopted here, there are specifics which exceed the scope of this paper and will be investigated later. However, we will consider the generalization of the Hahn echo sequence consisting of trains of refocusing pulses, known as the Carr-Purcell-Meiboom-Gill (CPMG) sequence (3,4).
A well known, special phenomenon is the occasional exceptional sensitivity of CPMG spin echo trains to certain types of periodic magnetic field fluctuations. In his carrier as NMR service engineer, the Author has encountered practical cases when reliable CPMG measurements were all but precluded, for example, by field modulations due to i) stray fields from the electric wiring of a laboratory or ii) sample vibrations induced by a magnet cooling-water pump, floor vibrations induced by nearby heavy machinery and even by loud music. Yet, quantitative studies of the phenomenon are extremely rare so that one usually concentrates on empirical detection and suppression of such field instabilities to a point where they no longer appear to have a significant effect on the echo train. This, of course, is understandable - except that one should be able to do so on the basis of a reliable theory permitting an objective assessment of the residual disturbances. Despite this necessity, the Author has found only one serious experimental NMR study carried out long time ago by Allerhand (5). Though some of the equations he deduced are still valid, they lack the generality required to cover all present-day applications and all types of field instabilities.
In order to maintain the theoretical treatment as self-coherent as possible, we will analyze the effect of field noise on a generic n-th echo in a CPMG echo train. This then includes the special case of n=1 corresponding to a single, isolated Hahn echo and even the case with n=0 (the zero-th echo) which, as we shall see, coincides formally with a plain FID and thus represents an independent check of the validity of the results.
Particular attention will be dedicated to sequences of n CPMG echoes with equidistant timing. Such trains of echoes, with n ranging sometimes into thousands, are used in both LR (low resolution) and HR (high resolution) NMR. In the LR version one often samples just one point of each echo (synchronized with the top of its amplitude) and acquires n such samples in each CPMG scan. The HR version is more time consuming since one needs to acquire a whole FID starting at the top the last (n-th) echo. Consequently, one can acquire in each scan just one echo which, however, contains the echoes of many individual spectral lines resolvable by standard Fourier Transform techniques. In both cases, it is the phase stability of the n-th echo which is to be determined.
Types of spin-echo instabilities
Consider the CPMG sequence [(π/2)x - τ - (π)y - τ - acquisition]. The echo amplitude at time t = 2τ (the 'top' of the echo) contains a number of factors due to such diverse phenomena as transverse relaxation (T2), self-diffusion (D), homonuclear couplings (J) and 'chemical' exchange, none of which affects its RF phase. On the other hand, spin echo eliminates (refocuses) the effects of field/RF offset , field inhomogeneity, chemical shifts and heteronuclear couplings.
Any spin-echo instability therefore arises from imperfections in one or more of the re-focussing phenomena. This leads to three broad classes of spin-echo fluctuations:
1) Instabilities due to mean field/offset fluctuations which have a direct impact on the echo phase but not on its absolute magnitude. When the echo signal is acquired using a phase- & offset-insensitive detection method (diode detection, RF envelope detection or signal power detection) such instabilities don't show up - a fact which can be used to discriminate them from the other types. Instabilities of this kind (see Part I for a more detailed discussion) are the most common ones in practice and the only ones to be investigated in this Note.
2) Instabilities due to field inhomogeneity fluctuations. Such instabilities affect both echo phase and echo magnitude (in extreme cases, they may completely prevent echo formation).
3) Instabilities due to fluctuations in spin system parameters (chemical shifts and scalar couplings). This broad class of phenomena is often lumped under the very broad and generic term chemical exchange (6,7). Though, in principle, such phenomena can also be studied by theoretical methods similar to those adopted here, the topic does not match the scope of this series and shall be pursued elsewhere.
Since most modern NMR instruments employ quadrature phase detection, the echo phase and all its statistical characteristics are experimentally accessible and will be taken into consideration in our analysis.
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We have shown how the phase noise due to magnetic field instabilities propagates into isolated, as well as multiple NMR echoes. Identical effects might be due also to the phase noise of the receiver reference frequency (this, however, is normally expected to be far too small to cause any problems).
After delimiting three basic typical types of magnetic field noise (random, periodic, and mixed), we have derived specific formulae covering the individual cases. The mathematics which emerges is rather complex but, at the same time, quite fascinating.
It turns out that long trains of periodically repeated pulses like those used in the CPMG sequence interact with the corresponding frequency components of magnetic field fluctuations in a highly selective way. This has many obvious consequences in terms of understanding instrumental artifacts and proper planning of experiment involving single- and, in particular, multiple- echoes.
So far we have discussed only the artifacts due to the statistical bias of signal phase projections which can be appreciated in averaged echoes and echo trains. One should also try and establish some of the statistical characteristics of single-scan echoes and echo trains and, more generally, those characteristics which pertain to the averages obtained after a limited number of scans. The route towards such statistics has been paved but the ground which it opens is yet to be explored.
Some practical consequences of this study have been already pointed out (mostly in Section IV). Of particular practical value are the predictions (both qualitative and quantitative) regarding the field-noise contribution to the apparent value of the measured T2's and the emergence of an anomalous non-exponential behavior of CPMG-decays. Clearly, however, there is still much more to be done.