Anal. Chem. 1982, 5 4 , 2591-2593
Table I. Quantization Noise (mV) resolution
theoretical
exptl
8-bit
11.6
11.7 2.9
10-bit
12-bit
2.9 0.72
0.73
intensity noise and absorbance noise. Intensity noise is taken as the standard deviation of the intensity of the unabsorbed primary light beam, Io, sampled at a 1120 Hz rate for 10 s. Absorbance is calculated as the log of the ratio of the unabsorbed intensity, I,, to the absorbed intensity, I , where each of these intensities is the sum of IO successive samples, taken also at a 1120 Hz rate. Thus, absorbances are calculated at a 56 Hz rate. Absorbance noise is taken as the standard deviation of 1120 such absorbances measured over a 20-s interval. [n the measurements reported here, only base line absorbance noise is measured; thus Io and I are effectively equal, except for noise. A computer program, written in Fortran, acquires 12-bit intensity data, masks the two or four least-signifkant bits to simulate 10- and 8-bit data, and computes standard deviations for each case.
RESULTS AND DISCUSSION Direct Measurement of Quantization Noise. For a 12-bit ADC with a range of 0-10 V the theoretical quantization noise is 0.722 mV. Average raw intensity data obtained experimentally at 324.7 nm showed Tl, = 63.355 mV and TS = 64.420 mV. This provided a 12-bit quantization noise value of 0.730 mV using eq 5. Table I contains a comparison of theoretical and experimiental values. The agreement is excellent, in spite of the fact that the quantization noise is in this case a very small fraction of the total noise. Reduction of Quantization Noise by Averaging. The effect of averaging on the reduction of quantization noise was determined by averaging 2 to 50 successive intensity values and computing the quantization noise from the standard deviation of the means. The experimental values are in good agreement with the theoretical prediction. Base Line Absorbance Quantization Noise. All previously reported quantization noise measurements in this work were obtained by using intensity noise measurements (in mV). It is possible to predict absorbance quantization noise values in absorbance units by using the following expression:
2591
Table 11. Double Beam Absorbance Base Line Quantization Noise no. of
quantization levels 3850 1540 530
abs std dev theoretical exptl 0.000 015 0.000 037 0.000 11
0,000 017
0.000 0 5 0.000 1 2
absorbance quantization noises agree fairly well with the theoretically predicted values. The relative contribution of quantization noise to the total system noise obviously depends on the magnitude of the other (nonquantization) noises. The base line absorbance noise of UV-visible and atomic absorption spectrometers is typically greater than 0.0001 a. Thus, if 12-bit intensity resolution is utilized, quantization noise will typically be a small fraction of the total noise. However, if the effective intensity resolution is reduced, as for example by low reference energy in UVvisible absorption or by background absorption in atomic absorption, then, of course quantization noise becomes more significant. We have found that on our continuum-source atomic absorption system which uses a 12-bit analog-to-digital converter, quantization noise accounts for less than 7% of the total base line noise when the background absorption is 1.8, corresponding to an intensity resolution of one part in 530. It should be noted that in some types of spectrometers digitization occurs after log conversion. Clearly in that case the effect of quantization would be different. The present treatment is meant to apply only to those instruments in which log conversion occurs after quantization. Finally, the opportunity usually exists to reduce quantization noise (and other noise sources as well) by averaging. If low-frequency (flicker or l/f) noise is a significant part of the total base line noise, then averaging the intensity data may reduce the relative contribution of quantization noise, because quantization noise will be reduced by averaging like any white noise, while l / f noise is not much reduced by averaging.
LITERATURE CITED (1) Horiick G.Anal. Chem. 1975, 47, 353. (2) Kelly, P. C.; Horlick, G. Anal. Chem. 1973, 45, 518. (3) Harnly, J. M.; OHaver, T. C.; Golden, B.; Wolf, W. R. Anal. Cbem. 1979, 57, 2007.
N. J. Miller-Ihli T. C. O'Haver* where Q is the quantization noise in absorbance units, S is the number of quantization levels in the intensity measurements, & refers to the dlouble beam nature of the absorbance measurement, and dz refers to the fact that ten intensities were averaged to obtain both Io and I. By use of various incident intensity (Io)values, the base line absorbance quantization noise at 324.754 mm was measured. These results appear in 'Table I1 along with the theoretical results. In general, the experimentally determined base line
University of Maryland Department of Chemistry College Park, Maryland 20742
J. M. Harnly Nutrient Composition Laboratory United States Department of Agriculture Beltsville, Maryland 20705
RECEIWDfor review June 7, 1982. Accepted August 31,1982.
Optimal Experimental Parameters for Quantitative Pulse Fourier Transform Proton Nuclear Magnetic Resonance Spectrometry Sir: In pulse Fourier transform lH NMR spectrometry, the achievement of quantitative results and/or good spectral signal to noise ratios ( S I N ) is influenced by the choice of pulse flip angle (6) and the time between consecutive pulses (T) in 0003-2700/82/0354-2591$01.25/0
relation to the lH spin-lattice relaxation time ( T l ) . It is routine practice to optimize SIN by using a small pulse flip angle and a short time T , though such a procedure will not in general (see below) yield quantitative results. Further, 0 1982 American Chemical Soclety
2582
ANALYTICAL CHEMISTRY, VOL. 54, NO. 14, DECEMBER 1982
although the conditions necessary for obtaining quantitative results are well-known (see below), these are usually defined without consideration of spectral SIN. In principle any 6 value can be used to obtain quantitative data provided it is combined with an appropriate T I T 1value. However the most commonly stated condition for achieving quantitative results contrasts with the requirements for optimizing SIN in using a large pulse flip angle (6 = 90°) and large T values (such as T / Tl = 3 or 5 for maximal errors of 5% and 1% ,respectively). Thus the question arises of how to choose experimental 6 and T / T l values such that for any experiment the desired quantitative results and the best possible spectral SIN are both achieved. An answer to this question is sought in the present work for two reasons: fmtly, the amalgamation of quantitation and maximal compatible SIN is especially appropriate in analytical applications of lH NMR; secondly, inspection of the literature indicates that, often, optimal experimental parameters have not been used. In a conventional Fourier transform 'H NMR experiment, a steady state is reached after a small number of pulses have been applied. If the time between consecutive pulses is not sufficiently long, the steady-state bulk nuclear magnetization MB, just prior to any steady-state pulse, will be less than the equilibrium magnetization M,,, Expressions for MB have been presented by a number of workers (1-6) applying a variety of assumptions. The expression in eq 1 is suitable for the present purposes
."
,
0'
" " I
05
i0
-T /
7,
83
I
0'
05 - T / i
M, = 0 M y = M ssin 6
(3)
such that relative errors in MB and My are identical. MB is plotted as a function of T / T , for various values of 6 in Figure 1,and it can be seen that the maximal deviation of Mafrom Mo occurs for the group with the longest T1value. Since a sample may contain lH groups with a wide range of Tl values, it is appropriate to define a maximal percentage error (e) as
= lOO(M0 - Ms)/Mo (4) where MB is given by eq 1for the group with longest T1value. Clearly, for a known (or more realistically, guessed) maximal T I value, there are a variety of combinations of the experimental parameters T and 6 which can be chosen to yield the desired (or acceptable) maximal error. It remains to be shown which of these combinations will produce the best signal to noise ratio in a given experimental time. Points on the curves in Figure 1 (6, T / T l )corresponding to the same E values will
!
,
90 100 ,
,
C4
The observable magnetization components are given by
I
Flgure 1. A plot of the steady-state magnetization Ma(wlth M, In eq 1 set to 100 units) and the error e (deflned by eq 4) as a functlon of TIT,, for varlous values of the pulse flip angle, 6.
where El = e-T/T1. In the derivation of this function the following have been assumed: the applied radio frequency field is collinear with the 3c axis in the rotating frame of reference with the z axis defined by the direction of the external magnetic field; transverse relaxation is sufficiently rapid that the xy component of nuclear magnetization decays to zero between pulses; and the spin-lattice relaxation for a specific proton group can be described by a single relaxation time, T l , as in
In systems where cross relaxation (7)is significant, as is usually the case in I3C NMR, eq 2 is not sufficient and the following treatment is not appropriate (8,9). For conventional lH NMR experiments, however, eq 2 is considered adequate for most systems, although deviations have been reported (10, 11).
I
50
" " I
10
, "
50
' $00
Flgure 2. A plot of (T1)"2(S/N)as a functlon of T / T 1 from eq 5, settlng k(I'),'* to unity, for varlous values of the pulse fllp angle, 6. Sets of isoaccurate polnts correspondlng to E = 1% (O),2.5% (A), 5% (U), and 10% (V)are also shown.
be termed isoaccurate points. For spectra deriving from accumulated steady-state free induction decays, the ratio of the maximal resonance peak height (S)to the root mean square noise (N) is proportional to the steady-state value of My and to the square root of the number of accumulated free induction decays, given by (I?/ T)l/*where I' is the total experimental time. A number of other factors also affect the final S I N ratio, including the choice of the data acquisition period (r (LC) has increased recently (1-4), especially with regards to optimizing separations by statistical approaches (5-9). In addition to commonly used binary solvents, ternary and quaternary systems have been described for superior isocratic separations ( I , 7, 10-12),and certain ternary solvent gradients have demonstrated better separations than corresponding binary solvent mobile phases (13, 14). Despite the expanded use of multisolvent systems, no attempt has been made to describe all the possible mobile phase classifications systematically, especially as they relate
to both solvent strength and selectivity. A comprehensive description containing these factors is currently available only for the simplest case-that of binary solvent mobile phases (15, 16). We propose here a general description of multisolvent LC mobile phases that greatly expands the opportunity for optimized solvent strength and selectivity effects. The proposed classification should enhance a more systematic investigation of these important effects on LC solute retention and facilitate current efforts in LC optimization. Although the solvent classifications discussed below are largely qualitative, more
0003-2700/82/0354-2593$01.25/0 0 1982 American Chemlcal Society
