Table V. Analysis of Mixed Solutions
Present, ppm Solution 1 Albumin Glucose Oleic acid Solution 2
Average found, ppm
20
19.8 -0.1 0
0 0 10 10 0
Albumin
Glucose Oleic acid Solution 3 Albumin Glucose Oleic acid Solution 4 Albumin Glucose Oleic acid Solution 5
20 20
0.06
-0.3
2.17 2.13 0.12
We express our appreciation to Dr. H. Page Nicholson, FWQA Project Officer, for his guidance on this program.
1.20 0.12
RECEIVED for review February 8, 1971. Accepted April 28, 1971. Mention of products and manufacturers is for identification only and does not’imply endorsement by the Federal Water Quality Administration, nor of the Environmental Protection Agency. The research upon which this paper is based was performed pursuant to Contract No. 14-12-802 with the Federal Water Quality Administration, Department of the Interior.
0.25 0.56
5
0
to 20 3
8.3 20.9 3.3
1.33 1.56 0.10
The results of the completed experiment indicated that direct pyrographic analysis of multicomponent organic mixtures can be carried out in highly diluted aqueous solutions. The method permits direct qualitative and quantitative analysis of high molecular weight-low vapor pressure organic materials in mixtures, such as are found in environmental and biological systems. Combining mathematical logic with analytical procedural operations through computerized means of data handling permits fast and efficient computation of results. Application of this technique in the field of water quality control work is being explored. ACKNOWLEDGMENT
0.35
2.9 10.8 5.2
4 10
Glucose Oleic acid Solution 6 Albumin Glucose Oleic acid
0
19.0 8.7
0
Albumin
0.51 0.20
0
20 10
0.92 0.52 0.12
8.0 9.3 18.2 19.2
0
Standard deviation, PPm
analysis appears to be satisfactory considering the high order of dilution. CONCLUSIONS
Computer Controlled Fourier Transform Nuclear Magnetic Resonance System for Carbon-13 and Phosphorus-31 Spectrometry R. J. Cushley, D. R . Anderson, and S. R. Lipsky Section of Physical Sciences, Yale University School of Medicine, New Haven, Conn. 06510 A high resolution NMR spectrometer has been modified for pulse-Fourier spectrometry (FFT-NMR). Data acquisition and data handling are accomplished by means of an IBM 1800 computer with 24K of 4 psec core storage and numerous peripheral devices. The NMR free induction decay signal (up to 8192 data points) can be digitized at rates up to 20 KHz. Spectra of lac,31P, and lH nuclei have been determined and time savings of 100-fold or sensitivity enhancement of 10- to 20-fold have been realized.
THECOMPLEX Fourier relationship : f(jw) =
J m -m
f(t)e-jUt x dt
(1)
and f(t) =
J m -m
f ( i ~ ) e i “X ~dw
(2)
shows that the frequency response function and the time response function form a Fourier transform pair. Application of the Fourier transform technique to proton N M R spectrometry was demonstrated in 1966 by Ernst and Anderson ( I ) . These authors outlined the theory of a spin system (1) R. R. Ernst and W. A. Anderson, Rev. Sei. Instrum., 31, 93
(1966).
subjected to a periodic sequence of pulses. In pulse N M R spectrometry, the time response function-the free induction decay (FID) signal-can be recorded in times on the order of Tz* (transverse relaxation time plus field inhomogeneity effects), thus making possible recording of spectra with a time saving of 100-fold, or, by use of a time-averaging computer, sensitivity enhancement on the order of 20-fold compared to continuous wave (cw) measurements. We wish to describe a versatile computer arrangement for fast Fourier transform N M R spectrometry (FFT-NMR). The experimental system consists of a high resolution spectrometer modified for pulse-Fourier and directly interfaced to an IBM 1800 computer with 24K of 4 psec core storage and numerous peripheral devices for data handling and data presentation. The importance of sampling with a computer of large memory capacity is determined by the bandwidth and resolution requirements of the spectrum being measured. The sampling theorem ( 2 ) states simply that, the observed spectral bandwidth .is limited to one half the digitizing frequency. Thus, for nuclei exhibiting large chemical shifts (e.g., 13Cand 31P) digitizing rates of 10 KHz or greater are needed. Since spectral resolution varies inversely with total observation (2) H. S . Black, “Modulation Theory,” D. Van Nostrand, Princeton, N. J., 1953, Chapter 4.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
1281
R . F . GENERATOR
p+7w
-+
I
U
T
-
R E C E I V E R GATE P U L S E U
PULSE INVERTER
1 ) PHASE CORR
CRT
I
MAGNETIC TAPE
DISPLAY
I
21 I N T E R R U P T S 31 DIGITAL SWITCHES
Figure 1. Block diagram of IBM 1800-based spectrometer system for Fourier transform spectrometry FIRST
PULSE
A
'"0 fl TRANSMITTER OATlNQ PULSE I
Nth PULSE
n
I /
!
I
-eov
1
IIIY!
ECEIVER SlONAL I I M V I
1
I /
E
+IV
0
I I
I I
1
CLOCK ENABLE
r
/ /
I / I I *I
F
-12:
I
SYNC PULSES
I
1 I I
I
I
I
I / / I I /
1
I
l
l
Figure 2. Experimental timing chart time, A = l/T,b,, a memory array of 8192 points digitized at 10 KHz yields minimum line widths of 1.2 Hz. The 1800 is capable of sampling up to 8192 F I D double precision or 16,384 single precision data points at rates up to 20 KHz. Using this system, we have thus far recorded spectra of 'H, 13C, and 31P nuclei and substantial time saving or enhanced sensitivity has been realized. INSTRUMENTATION
The basic configuration of the Fourier transform spectrometer is depicted in Figure 1. It consists of a high resolution Bruker HFX-3 nuclear induction spectrometer operating at 21.5 kilogauss, equipped with a B-SV2 power amplifier for proton noise decoupling and a Bruker 20-watt power amplifier for the excitation pulse. An internal field-frequency lock is provided at 84.7 MHz ('SF) or 90.0 MHz ('H). The rf excitation pulse is created at the mixers (HP 10514) which are gated by logic levels from an IBM Electronic Contact Operate (ECO) register. The transmitter gating pulse and the resulting rf pulse are shown in parts A and B respectively, Figure 2. 1282
The FID signal is detected using the spectrometer low noise, tuned preamplifier (gain = 1000) and receiver; however, the final field demodulation stage is omitted. Another mixer is used at the output of the preamplifier to gate out the large noise burst created by the rf excitation pulse in the FID signal, thus preventing saturation of succeeding amplifier stages. The receiver gating pulse is also generated by the 1800 ECO (Figure 2C and 20). After further amplification and phase detection, the signal is conditioned for the i.5-volt input level of the 1800 ADC Multiplexer with an HP-8875A Differential Amplifier followed by a single section RC bandpass filter with selectable cutoff frequencies for use at several sampling rates. Transmission noise between spectrometer and computer (approximately 25 feet) was negligible when the R C filter was placed immediately before the ADC unit. The IBM 1800 data acquisition system was designed primarily for use in applications involving relatively slow processes running asynchronously with the computer. Because of this design concept, the ADC makes use of external sampling pulses (sync pulses) to synchronize the computer with an external process. These sync pulses were initially provided by a Tektronix Time-Mark Generator-Type 184. However, since the Time-Mark Generator is free-running and asynchronous with the computer, and the computer-generated exciting pulse with its resulting FID signal is synchronous with the computer, phase incoherence exists between the rf pulse and the onset of sampling (Le., the time between the rf pulse and the first sync pulse is random). Random phase prevents coaddition of the FID, thus resulting in decreased signal to noise ratios. In the absence of programming to circumvent this problem, the first sync pulse occurs at any time less than the digitizing period ( e . g . , 100 psec at 10 KHz) after the rf exciting pulse. This jitter has been minimized to 1 4 psec by holding the computer in a dynamic wait mode until a Time-Mark Generator initiated interrupt continues program flow. It is probably impossible to reduce this jitter further by programming means because of hardware variations in interrupt servicing response. These variations depend upon the status of the 1800 CPU instruction execution at the time the interrupt occurs. To eliminate this jitter completely, a computer controlled data clock was constructed utilizing the basic 2 MHz oscillator of the 1800. A schematic diagram of the data clock is given in Figure 3.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
*OV
SN7493 4-611
MULTIVIBRATOR
FREWEMCY SELECTION (BITS FROM 1@00)
Figure 3. IBM 1800 compatible data clock for digitization of FID's (all voltages and most grounds are deleted for simplicity) The data clock is enabled by an ECO voltage level change which is generated coincident with the trailing edge of the receiver gating pulse (see Figure 2E). The enable pulse is sensed by means of a Fairchild pA710c comparator which converts the input pulse to TTL compatible levels. The input circuit for the 2-MHz 1800 frequency is a transistor network designed as a buffer amplifier t o isolate the data clock from the 1800 logic (input impedance > 1 Megohm). The 2-MHz frequency is first counted down by a factor of 25 using two TI SN7490 decade dividers. The resultant 80-KHz frequency is counted down by a series of 8 flip-flops (two T I SN7493 4-bit binary counters) t o yield digitizing rates from 40 KHz t o 312 Hz. The desired frequency is determined by comparator sensing of 3 ECO rate selection bits, set by the controlling program. This is then translated using a TI SN7441 BCDto-Decimal decoder to NAND the particular frequency desired. The output circuit consists of a retriggerable oneshot (Fairchild TTpL-9601 Monostable Multivibrator) t o shape the sync pulse t o the proper width for the IBM 1800 ADC (-3 psec). There follows a bufTered level-translator t o give the sync pulse the necessary amplitude for the ADC (-12 t o 0 V) and an emitter-follower with a current limiting resistor at the sync pulse output jack (output impedence >1 Megohm). The results to he derived from adding the data clock are evident from Figure 4 which shows oscilloscope traces of the sync pulses. The oscilloscope was externally triggered in each case on the leading edge of the uninverted transmitter gating pulse. Figure 4A shows the relationship between the sync pulses created by the data clock and the uninverted gating pulse. Figure 4B shows the sync pulse jitter present when using the Time-Mark Generator. The exposure time for
Figure 4. Oscilloscope traces showing sync pulses
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
1283
ENTRY
A
INPUT
f l S T
DATA SWITCH 3
w/ 2
PULSE
PULSE DATA
ENTRY RELOAD RELOAD TIME DOYAIN \ F R O M DISK/
/(&\
TRANSFORM
'z
1 B IO K TMP/Acitonr 868 S i c Scan R C - IOOOHz
DOMAIN
Figure 5. Flow diagram of Fourier transform programs
Figures 4A and 4B was 0.5 sec. Finally, in Figure 4C, a 1-sec exposure of an expanded data clock generated sync pulse is shown. The pulse width is 3 microseconds and, although exposure time is twice that for Figure 4B, no jitter is apparent. Located at the spectrometer console is a 4K Fabri-Tek 1064 hardwired computer fitted with a Tektronix RM504 oscilloscope and a HP Moseley F A M X-Y recorder. The FabriTek has been digitally interfaced with the 1800 and serves as an on-line buffered CRT-plotter. SOFTWARE
A flowchart of the Fourier transform programs is depicted in Figure 5. These programs have been coded in Fortran IV and Assembler. The data acquisition routines ( N I N l O and NIN20) must be coded in Assembler t o digitize and time average data up to 20 KHz, while routines such as that used in interactive phase correction are necessarily in Assembler, so that some degree of real-time data processing may be maintained. In this case, the Assembler coded routine takes 1 sec to phase correct 4096 frequency domain data points, while the Fortran version takes over 1 min. There has also been a heavy emphasis on integer mode programming wherever loss of precision will not occur or is insignificant. All routines are stored on disc in core image format. The data acquisition and Fourier routines are divided into six core loads with each block overlaying the previous block. The computer is remotely controlled through the 1816 typewriter, interrupts, and analog inputs at the N M R console. Data acquisition is initiated by interrupting the 1800, which calls for the initial core load, INPUT. INPUT prompts aia the typewriter for all variables necessary for data acquisition 1284
400.0
450.0
Figure 6. Comparison between 31P cw spectrum of a 10% trimethylphosphite solution (A) and the correspondng Fourier transform spectrum ( B )
and transformation and stores them in Fortran Common. INPUT is then overlayed with the pulsing and digitizing core load, NINIO. Depending upon the particular pulsing program selected, data may be digitized and summed at rates u p to 10 KHz (8K two-word integers, or 16,384 one-word integers; 11 bit resolution) or at 20 KHz (4K two-word integers; 11 bit resolution). The rf pulsing is started at the NMR-CRT console by an operator initiated interrupt. The creation of the transmitter gating pulse by the 1800 ECO is a novel feature of the system since most other units use the computer output as a trigger only. The method consists of an assembly language instruction generating.the leading edge of the pulse, a precisely timed iteration creating the pulse duration, and an instruction generating the trailing edge. Pulse duration can be varied from 20 microseconds to 16 milliseconds in 500 nanosecond increments. The program can be made to generate periodic pulses, or any multiple pulse
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
70 % Dimethyl phosphite 0 . 8 Scc Scan TZ = 0.1 Sac
Figure 7. Phosphorus-31 Fourier transform spectrum of dimethylphosphite
710 Hz
I
I
combination such as D E F T (3), SEFT ( 4 ) , and ZRSE (5). The receiver gating pulse is also generated at this time; the leading edge synchronous with the leading edge of the transmitter gating pulse, and the trailing edge after a period equal to the width of the transmitter gating pulse plus some variable delay equal to the preamplifier recovery time (20 psec to 2 msec). At this point the data clock and the A D C are enabled. As each sync signal occurs, the ADC samples the FID and cycle steals the data uia data channel into one of two 128-word chained buffers in core. As each buffer is filled with incoming data, its contents are added t o the running summation in the FID array. After collecting the desired number of samples along the F I D curve, the ADC and clock are disabled. The program may be delayed at this time u p t o 60 seconds in a carefully timed loop to allow relaxation of nuclei. After the specified number of FID's are averaged, the final step in the data acquisition program is to subtract any base-line offset and normalize t o one-word integers if data have been collected in double precision mode. An error condition during data acquisition causes the computer t o print a message and take appropriate action: digitizer overload-erase memory and reinitialize for new pulse sequence; spectrometer flux stabilizer off-save F I D on disc and exit; core overflow-exit t o Fourier Transform cia the time domain output coreload, ECHO. The rationale for accepting data in which core overflow has occurred is that in all probability only a few data points near the beginning of the F I D array will overflow during the scan in which it is detected, thus causing minimal spectral distortion. After completion of data acquisition, the ECHO coreload is entered. This program first stores the F I D array on disc, and then dumps the array to cards, printer, magnetic tape, or the on-line Fabritek depending on which options have been specified. If convolution is desired ( i e . ,multiplication of the time domain by it is performed in this coreload. The data are next transformed to the frequency domain spectrum in the F F O U R coreload using the Cooley-Tukey radix two F F T algorithm (6). Because this routine is partially written in Fortran, the time required to transform 8192 real (3) E. D. Becker, J. A. Ferretti, and T. C. Farrar, J . Amer. Chem. SOC.,91,7784 (1969). (4) J. S. Waugh, J . Mol. Spectrosc., 35, 298 (1970). ( 5 ) A. Allerhand and D. W. Cochran, J. Amer. Clzeni. Soc., 92,
4482 (1970). (6) J. W. Cooley and J. W. Tukey, Math. Comput., 19, 297 (1965).
integer data points is 4 min. The resulting frequency spectrum may be presented as either the phase corrected real or amplitude (power) spectrum. This is accomplished in the PHASE coreload. In the real spectrum case, phase correction is performed with the 1800 and the operator at the NMR-CRT console in an interactive mode. A series of six potentiometers at the CRT console provide appropriate angles for phase correction. Pots 1-5 provide angles for frequency dependent phase correction, with pot 1 representing 4 at 0 frequency, pot 2, 4 at F / 4 ; pot 3, 4 at F / 2 , etc. Intermediate phase values are obtained by interpolation. Potentiometer 6 provides a frequency independent phase angle. Although all frequency correction could be done with the five frequency dependent pots, these settings need t o be changed much less often than the frequency independent phase angle; thus a separate pot is desirable for operating convenience. Finally, the frequency domain output coreload, OUTPUT, is entered. After calculating R M S base-line noise, line positions, and S/N ratios of each resonance line, the various output options may be selected. RESULTS
As stated earlier, frequency domain spectra are presented in two ways. The amplitude spectrum is given by Y(w)
=
ziY72(w)
+ Y,"w)
(3)
where Y,(w) and Y,(w) are the real and imaginary coefficients of the Fourier transform, respectively. Y(w) is the amplitude of the power spectral density of the time domain data. This method of data presentation possesses the advantage of being independent of the phase. The drawback in using this method is a broadening of resonance signals. The real spectrum is defined as Y(w)
=
Y7(w)cos 4
+ jY,(w)sin 4,
(4)
where 4 is the appropriate phase angle needed t o yield an absorption mode spectrum (1). The real spectrum gives resonance lines which more nearly approach the Lorentzian line shape. After careful evaluation of the iterative phase correction techniques sometimes employed ( I , 7), these were rejected in favor of an operator-IBM-1800 iteractive method. In this method, frequency dependent and frequency indepen(7) R. R. Ernst, J . Magn. Resoitawe, 1, 7 (1969).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
1285
I
c 60% enriched C H , C O C H , 5 m m tuba 0 . 8 sac Scan
N
c 60% onriched CH,COCH, 5 mrn tube 0 . 8 soc S c a n
Proton irradiation
v 3
Figure 8. FID and Fourier transform of I3C-enriched acetone showing effects of proton noise irradiation and convolution (lower traces) dent phase angles are supplied as a series of six potentiometer controlled voltages at the CRT console, with the phase angles being changed, a new phase correction performed, and the resulting spectrum displayed. This process is continued until a satisfactory absorption mode spectrum is obtained. Some of the advantages accrued from the use of the interactive method over iterative methods are that the interactive technique allows an absorption mode spectrum to be obtained even at low S/N ratios while the ability of iterative techniques to converge o n an absorption mode spectrum is dependent on S/N ratio and that both frequency independent and frequency dependent phase correction may be achieved much faster by the interactive method. F o r presentation of spectra we have chosen examples of 31Pand nuclei since these nuclei possess large chemical shifts and are, respectively, 6 X and 1.8 X times 1286
less sensitive than IH nuclei in natural abundance for equal numbers of nuclei at the same field strength. All spectra presented will be phase corrected real spectra. Figure 6 depicts the sensitivity enhancement realized with 31Pnuclei. Figure 6A shows a normal continuous wave (cw) spectrum of 10% trimethylphosphite in acetone. The scan time was 1000 sec and a filter time constant of 0.5 sec was used. The multiplet contains only 6 of the 10 lines associated with the phosphorus resonance. Figure 6B contains the FFT-NMR spectrum of the same sample using identical spectrometer conditions. The 10 lines of the 31Pmultiplet are clearly visible. The conditions of the FFT-NMR experiment were: total time requirement less than that of the cw experiment (868 sec cs. 1000 sec) while covering the same frequency range; use of a bandpass RC filter (fc = 1000 Ilz); and no computer manipulation (Le. convolution or apodization) on the FID signal was performed. From a comparison of spectra in Figures 6 A and 6 B a conservative 10-fold enhancement of sensitivity is claimed for 31Pnuclei. It should also be pointed out that, although trimethylphosphite contains a number of lines, it is not an ideal candidate for the FFT-NMR method. Studies using the pulse-Fourier technique indicate a Tl 3 times T2 for trimethylphosphite. The alternative advantage of using FFT-NMR over conventional methods is the large time-savings realized. In spectrum of dimethylphosphite Figure 7 is displayed a (70 % dimethylphosphite :30 % acetone). The large coupling observed is the directly bonded P-H coupling while the smaller splittings are due to the three-bond P-0-C-H coupling. The spectrum depicted in Figure 7 was recorded in 0.8 sec. Samples of 31P compounds in greater than 50% concentration (13-mm sample tubes) are routinely run in times less than 1 sec. We have studied several biological phosphates and find their salts to be particularly well suited to study by the pulseFourier method. The best conditions for enhanced sensitivity by the Fourier method arise when TI = T2. This allows extremely rapid pulse periods, hence, time-averaged data will be accumulated rapidly. We have found, for instance, that, for the disodium salt of polyadenyllic acid, Tl 2 TZ = 500 milliseconds and have determined the 31Pspectrum for a 3 x 10-5 molar solution in approximately 14 hours. We believe much lower concentrations, concentrations that are “biologically significant,” can be achieved in our laboratory. Some 13C results are presented in Figure 8. The sample consists of acetone enriched to 60 in the carbonyl position and contained in a 5-mm sample tube. The 5-mm tube is mounted co-axially in a 13-mm tube containing hexafluorobenzene for field-frequency locking. The spectrum was recorded in 0.8 sec. The upper portion of Figure 8 is comprised of the 13C F I D resulting from a 90’ pulse (pulse width = 250 psec). The resulting phase corrected real spectrum appears below left while a transformed spectrum due to the resonance decoupled from the methyl protons appears at lower right. In the lower portion of Figure 8, the effects of convolution of the time domain data are given. The time domain data are multiplied by the function e-t’T2C, where T t = 0.5 sec, and the resultant decrease of the noise present in the signal trace is apparent. The corresponding transformed spectra are included for comparison. Finally, Figure 9 shows a FFT spectrum of a proton decoupled natural abundance spectrum of diethylphthalate. The spectrum is “routine” in that no special efforts
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10,AUGUST 1971
Ar
”C Nul. Abun. DIETHYL PHTHALATE
c3.C~
I
~
4
13mm TUBE 4 0 0 sac SCAN 8192 DATA P I S . ~
~
5
CltC2
;c=
>CH2
-CH3 0
CSFB
I
H-..-..._....._..
4
1524 6
Figure 9. Natural abundance 13C Fourier transform spectrum of diethylphthalate
were taken t o optimize conditions prior t o recording the spectrum. The solution was contained in a 13-mm tube with approximately 25 C6F6added for field-frequency locking. The digitizing rate was 10 KHz with an rf pulse width of 50 psec. Pulse widths of 50 psec were generally used for 13Cnuclei since these yielded an optimum flip angle. Satellites present with some peaks are spinning sidebands. The average signal t o noise ratio for the peaks is greater than 100 t o 1. The cutoff characteristics of a single section low-pass RC filter with f c = Nyquist frequency allows down-conversion (“aliasing”) of high frequency noise. All spectra except that of natural abundance diethylphthalate (Figure 9) were determined using an RC filter. The filter used in Figure 9 was a 4-pole Tchebyscheff active filter. The active filter was constructed from two experimental components (Analog Devices N o . 9171, N o . 9172) which allowed sixfc’s to be selected under computer control in a manner identical to the rate selection in the data clock. Generally, a filter bandwith corresponding t o one-half the current digitizing rate is used. The Tchebyscheff transfer function achieves an extremely sharp roll-off per number of poles. The filter phase shift is greater than 180’ throughout the frequency range and becomes distinctly nonlinear near f c . The present method of phase correction handles such characteristics extremely well. Substitution of the active filter for the RC filter has increased S/N by a factor of two. Although we have software for forming multiple pulse sequences, a power amplifier which will yield a / 2 pulses of 10 psec or less is needed, and extensive modification of the Bruker high resolution probe is necessary to withstand the high voltages developed. Multiple pulse experiments have therefore been curtailed. However, rapid accumulation of periodic pulses is shown to give significant improvements which warrant extensive use of the pulse-Fourier system. We are currently modifying a Collins 3OL-1 linear amplifier
(500 watt) to decrease the pulse width necessary for a 90” pulse. ACKNOWLEDGMENT
We thank Mr. L. Berman for his assistance in the design of of the electronic circuitry. We should also like to express our appreciation to Dr, Dieter Ziessow for his aid in certain aspects of the computer programming. RECEIVED for review January 25, 1971. Accepted June 7, 1971. Presented at the 160th National Meeting of the American Chemical Society, Chicago, Ill., September 13-1 8, 1970. This work was supported by a National Institutes of Health Grant, RR 00356, Biotechnology Resources Branch.
CORRECTION lnterlaboratory Evaluation of a Material with Unequal Numbers of Replicates In this article by J. Mandel and R. C. Paule [ANAL.CHEM., 42, 1194 (1970)l there is a n error in the iteration formula (Eq.
ac
22). The correct formula should read -
ax
= i
The original formula gives the correct answer when convergence is obtained. In a few cases, however, divergence occurs. The above corrected formula will always converge provided that only positive values of X are permitted. This is a reasonable restriction since X is a ratio of two variances. The corrected formula also requires fewer iterations. The authors thank Mr. Lars H. Sjodahl for detection of the error. (51
- ,G)’].
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10,AUGUST 1971
1287
