1826
Anal. Chem. 1982, 5 4 , 1826-1828
Wavelength Accuracy of a Microprocessor-Controlled Ultraviolet/Vi sible Monochromator Malcolm W. Warren 11, James P. Avery, and Howard V. Malmstadt"' University of Illinois, School of Chemical Sciences, Department of Chemistty, 1209 W. California St., Urbana, Illinois 6 180 1
Improved wavelength accuracy Is demonstrated for a hlghresolutlon grating monochromator. The specified wavelength accuracy of fO.10 nm for a 350-nm Zerney-Turner gratlng monochromator Is Improved io f0.025 nm. Thls Improvement Is possible because the wavelength error curve Is reproducible and a smooth functlon of wavelength. The wavelength error curve Is fixed uslng a slmplex flttlng program. The flttlng program then calculates correcilon factors to be applied during the wavelength scan by the monochromator controller.
The use of high-resolution grating monochromators to isolate selected wavelengths of light is important in spectrochemical measurements. The accuracy of the monochromator drive determines the uncertainty of qualitative atomic methods. Recent improvements in monochromator controller performance have greatly improved the precision with which monochromators can select a given wavelength ( I ) . However, the precision lead screw in the 0.3-0.5 m grating monochromators limits the accuracy to about 0.1 nm. The reliability of atomic spectral line identification utilizing such compact and relatively inexpensive monochromators would be greatly enhanced if the wavelength accuracy of the readout could be improved severalfold. It was previously indicated that with the new controller it might be possible to improve accuracy by measuring the wavelength error curve and preforming appropriate corrections (1). In this paper it is demonstrated that the error in the wavelength readout can be corrected by measuring known atomic emission lines and fitting the experimental data to test error equations. The correction factors are subsequently applied during wavelength scan so that the wavelength readout is accurate to k0.02 nm for any wavelength increment within the wavelength region that correction factors have been calculated.
EXPERIMENTAL SECTION Apparatus. The monochromator test system as shown in Figure 1 is composed of a GCA/McPherson EU-700 monochromator, a monochromator controller developed by Warren et al. ( I ) , several light sources, a 1P28 photomultiplier tube, a charge-to-count converter (q to n),and a master microprocessor. The master microprocessor controls data collection and analysis by using BASIC and assembly language routines. Input and output are via a teletypewriter (TTY), video monitor (TV), and thermal plotter printer (TPP). Procedure. The emission maximum is determined five times for each atomic calibration line and averaged. The wavelength error is calculated and the error data are fit to several test equations. Spectral lines provided by each lamp (Cd-Osram, Zn-Osram, and Fe-hollow cathode lamp) were measured at different times at approximately the same temperature. The data collected are fit on the University of Illinois Cyber 175 computer using a simplex (2) fitting program (MONCOR) written by the authors to fit the test equations. The fitting program minimizes the weighted sum of the square of the residuals. The weight for each data point is proportional to the interval between the data point and its nearest neighbors. MONCOR also calculates the Present address: 75-5786 Niau Place, Kailua, Kona, HI 96740. 0003-2700/82/0354-1826$01.25/0
correction factors used in an improved version of the monochromator program.
RESULTS AND DISCUSSION The GCA/McPherson EU-700 scanning monochromator is a single-pass Czerny-Turner grating monochromator with a precision lead screw sine-bar assembly. This assembly drives the diffraction grating and provides accurate wavelength selection to kO.10 nm and resettability to fO.01 nm. It is possible to measure the apparent wavelength of a number of atomic emission lines for several elements very precisely with the monochromator controller employed (1). The apparent wavelength of each line is measured and the wavelength error calculated and plotted in Figure 2. It can be seen that the wavelength error curve is a smooth function of wavelength. The measured atomic emission maximum for each calibration l i e never varied more than kO.01 nm at constant temperature. To understand the wavelength correction factors, it is useful to examine the equations that describe the selection of a given wavelength. The relationship between the diffraction grating angle and the wavelength that passes through the exit slit is described by eq 1(3),where n = order number, X = wavelength nX = 2d sin 19 cos 4 (1) passed through exit slit, '6 = angle of grating rotation at A, d = spacing between successive grating lines in nm (847.46 nm), and 4 = exit beam angle (17.5'). If we consider Figure 3 where x = linear distance traveled by sine-bar contact pin along the lead screw, x = NP, N = number of turns of the lead screw, P = pitch of lead screw threads (32 threads/in.), r = length of sine-bar (8.420 in.), and k = cos 4 where 4 = 17.5",we can then rewrite eq 1 such that the wavelength selected is described by the physical dimensions of the monochromator and N in eq 2 or x in eq 3 A=-.-- 2dkPN
nr
or A = - 2dkx
nr
(3)
If we consider only first-order wavelengths a t constant temperature all of the independent variables in eq 2 and 3 are fixed except for Nor x , respectively. However, the pitch of the lead screw varies. All of the mechanical and optical components of the monochromator can be expected to shift different amounts with temperature due to differences in the thermal expansion coefficients. Using the physical dimensions for the monochromator and eq 3, it is possible to calculate the error in the linear distance traveled by the sine-bar contact pin along the lead screw that lead to an error of 0.10 nm. One inch in linear travel by the sine-bar contact pin along the lead screw will produce a wavelength change of 192 nm. The total error of the components used to construct the monochromator lead screw sine-bar assembly must then produce an error of less than or equal to *1/2000th of an inch in the linear distance traveled by the sine-bar contact pin if the wavelength error is to be less than fO.10 nm. 0 1982 American Chemical Society
I4NALYTICAL
CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
1827
where K , = actual value for constant in monochromator, Ki = nominal value for constant in monochromator, and {(A) = wavelength-dependent relative error for the dimensional constant. If we ignore the effects of nonperiodic effects in MASTER 8080 the diffraction grating (i.e., Roland’s ghosts), then at constant temperature f(A) should be a constant for all of the dimensions in eq 2 and 3 except for the pitch of the lead screw. For CONTROL BOAR0 example the line spacings in the diffraction grating may be slightly more or less than 847.46 nm. Changes in the length of the sine-bar can be used to compensate for the effect of constant errors. The error in the pitch of the lead screw is a result of errors in manufacturing and varies depending on Flgure 1. Experimental measurement system. the master lead screw used to cut the precision lead screw and the history of the precision lead screw itself. 0 1 5 7 ’ ’ -----I The relative error function for each monochromator dimension can be separated from the dimension in eq 3 to generate a composite relative error equation MONOCHROMATOR
FiBTl
‘ONTROLLER
I
3 ” ” ”
”
-0 151, 200
Flgure 2.
, , , , , , , ,
250
,
,
300
, ,
,
,
, , , , , , ,
350
, , ,
”
,
”
, ,,
400 450 Wavelenglh (nm I
where f ’ ( X ) = composite error equation, f’(1)N 1, fd(X) = wavelength-dependent error for d, fk(X) = wavelength-dependent error for k, and f,(h) = wavelength-dependent error for r. The measured wavelength is the equal to 2dkf ’( X)x (A) A, = (8)
,
500
550
600
nr
650
Error curve and fit to data for eq 1I. /Parabolic
Figure 3.
”
Mirrors?
Diagram precisicm lead screw and sine-bar assembly.
Form of Error Equation at Constant Temperature. The actual wavelength selected (A,) a t any temperature is equal to the wavelength indicated on counters plus the wavelength error as shown in eq 4,where X, = actual waveX, = hi - X,(X) (4) length selected, Xi = indicated wavelength (wavelength read on mechanical counters), and A,( X) = wavlelength-dependent absolute wavelength error. Note that the wavelength error is a function of the wavelength, or more precisely, a function of the sine-bar contact pin position. Rearrangement of eq 4 provides X,(X) = A, - Xi (5) The constants in eq 2 and 3 can be rewritten for a monochromator as K , = Ki(1 + f ( X ) ) (6)
x(X) is the distance traveled by the sine-bar contact pin along
the lead screw as a function of on the wavelength readout. For an ideal monochromator x(X) would be the product N and the pitch of the lead screw. Because the pitch varies along the length of the lead screw and x is not zero when N is zero, x ( h ) is composed of three components. The effective pitch of the lead screw is the instantaneous pitch at any point along the lead screw. The error in the effective pitch of the precision lead screw is dependent on several sources of error: (a) errors in the lead screw, (b) wobble in the lead screw contact pin, (c) errors in relative flatness of lead screw contact pin and sine-bar contact pin, and (d) error in correction of the constant dimensional errors with the sine-bar length adjustment. where K’ = composite nominal value for constants in eq 3, K’= 2dk/(nr),xi = linear travel due to the nominal pitch of the lead screw, xi = PiN, x,(X) = accumulated error in linear travel from N equals zero, and xo = error in linear travel when N equals zero. The accumulated error in linear travel results from the difference between the effective (or actual) pitch and the nominal pitch lead screw. Substituting eq 3 and 9 into eq 5 we obtain
where X,(zo) = error in zero-order wavelength. Using eq 3 to convert f‘(X)xo to X,(zo) and xi to Xi eq 10 becomes
+
+
X,(X) = K’f’(X)xe(h) (f’(X) - 1 ) X i X,(zo) (11) In the ideal monochromator f’(X) = 1,x , ( N ) = 0, and X,(zo) = 0. In practice X,(zo) and f ’ ( X ) are constants, while 3ce(N) is of unknown functionality. All are dependent on the mechanical components of the monochromator. Equation 11 indicates that the wavlength error curve will have three terms: (a) a constant wavelength offset (X,(zo)), (b) a linear term ((f’(X) - l)Xi), and (c) a term of unknown functionality (K’f’(X)xe(X))*
The objective when correcting errors in mechanical components is to determine a reasonable approximation to the overall error function. This requires that the zero-order error
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Anal. Chem. 1982, 54, 1828-1833
The fitting parameters KOand Kl are the only parameters that can be assigned any real significance. KOshould correspond to the error in the zero-order wavelength and K1to the error in the correction of the error in the dimensional constants by the sine-bar length adjustment. Additional terms when added to the fitting equation improves the fit, but only slightly compared to the additional computer time required to carry out the calculations. Thus it is possible to correct the wavelength of the monochromator to better than rt0.025 nm from an uncertainty of up to f0.12 nm by collecting wavelength information and fitting the wavelength error curve to a test equation. A monochromator control program has been written which automatically corrects the wavelength readout as the monochromator scans by using a look-up table to determine the correction factor. The lead screw runout usually determines the wavelength inaccuracy in quality monochromators. Therefore, this procedure herein described should provide an improvement similar to that shown for the test monochromator used in this study. For the method to provide reliable corrections the monochromator must have a wavelength setting reproducibility greater than the specified accuracy.
Table I. Results of Weighted Data Fit
KO K, K, K3
K, K5
eq 1 2
eq 1 3
1.43 E- 2 nm -5.29 E-5a -9.88 E-2' 1.39 E- 2/nm -9.16 E - l a
- 1.51 E-2 nm 5.44 E-6a 1.02 E- la 1.33 E- 2/nm - 3.75 E+ Oa 3.93 E- 3a 3.28 E-2/nm -3.53 E+Oa
K6 K, a
Unitless.
(X,(zo)), accumulated error in the linear travel from zero (x,(iV)), and the functionality the composite error (f'(X)) be determined. Results of Data Fit. Inspection of the plot of the error in the wavelength as a function of wavelength as shown in Figure 2, suggests the use of a sine curve model to fit the data. This along with eq 11 suggests eq 12 and 13 as possible test equations for the wavelength error XJX) Xe(X)
=
= KO
+ Klhi + K2 sin (&hi + K4)
KO+ KIXi + K2sin (&Xi
(12)
+ K4) + K5sin (&xi + K7)
LITERATURE CITED (1) Warren, M. W.; Avery, J. P.; Lovse, D. W.; Malmstadt, H. V. J . Autom. Chem. 1981, 3, 76. (2) Nelder, J. A.; Mead, R. Comput. J . 1965, 7 , 308. (3) GCA Corporation "Scanning Monochromator EU-700 and EUE-700 Series", Acton, MA, 1968.
(13) where KO = error in zero-order wavelength, K1 = error in constant correction, K z , K6 = sine amplitude, K3,K6 = sine angular velocity, and K4,K7 = sine phase angle. The weighted sums of the residuals according to the distance between the data point and its nearest neighbors fit eq 12 poorly in the high density data region and fit eq 13 within f0.02 nm for all data points. The fit to the experimental data for eq 13 is plotted on Figure 2.
RECEIVED for review December 24,1981. Resubmitted June 8, 1982. Accepted June 8, 1982. The authors acknowledge the support provided by Grant HEW-PHS GM 21984 and computer time on the University of Illinois Cyber 175 provided by the University of Illinois Research Board.
Markovian Statistics and Simplex Algorithm for Carbon-13 Nuclear Magnetic Resonance Spectra of Ethylene-Propylene Copolymers H.
N. Cheng
Hercules Incorporated, Research Center, Wilmington, Delaware 19899
The "C NMR spectrum of ethylene-propylene copolymers contains much lnfonatlon on polymer structure and monomer reactlvlties. I n thls work a systematlc analysis Is proposed that alms to extract the maxlmum informatlon from each spectrum. The approach used Is to model the copolymerlration of ethylene and propylene by a second-order Markovlan process and to flt the lntensltles of the NMR spectrum to the model vla a slmplex algorlthm. A computer program has been wrltten that qulckly and efflclently computes all the necessary parameters. Informatlon available Includes composltlon, comonomer sequence dlstrlbutlon, Markovlan probabllltles, and reactlvlty ratlo product ( r 2r'). The role of dlffuslon In heterogeneous copolymerlsatlon Is also dellneated. The approach is valld for ethylene-propylene copolymers that contaln no Inverted propylene sequences.
13Cnuclear magnetic resonance spectroscopy has become a powerful technique for polymer characterization ( I ) . One 0003-2700/82/0354-1828$01.25/0
fruitful area of application is in copolymer analysis where such information as polymer composition, microstructure, tacticity, sequence distribution, and propagation errors in polymerization can be readily analyzed and, with care, quantified. Ethylene-propylene copolymers are no exception; indeed, in the last 6 years several studies have been made of these copolymers (2-7). In a careful study, Carman (2) examined several ethylene-propylene rubbers and used a reaction probability model to fit all the spectral lines to the model. Since the ethylene-propylene rubbers contain inverted propylene sequences, Carman was only able to fit the data to first-order Markovian statistics. In the alternative approaches of Ray (3) and Randall ( 4 , 5 ) ,selected lines in the spectrum were taken, from which copolymer sequence distributions were calculated. The polymers studied were made with heterogeneous catalysts and were free of inverted propylene structures. For some of these polymers, it may actually be more appropriate to use a second-order Markovian model (i.e., the addition of each monomer to the polymer chain depends on both the preceding 0 1982 American Chemical Society
