Analysis of the Vibrational-Rotational Spectrum Michael G. Prais Roosevelt University, Chicago, IL 60605 Students learn about the analysis of the vihrational-rotational spectrum of a diatomic molecule as part of an experiment commonly performed in physical chemistry lahoratory. The experiment illustrates several points while leading the student to understanding connections between the quantum theory descrihed in lecture and the properties of real substances. The object of the experiment is to measure the energies of the vibrational-rotational transitions of hydrogen chloride gas and use these energies to calculate several molecular parameters, including the interatomic separation. the force constant for vibration. and the anharmonicitv constant. This experiment should alio emphasize the ralcw oropamtion. and their latiun of standard tleviuti~ms,their . . use as an indication of the quality of values of themolecular parameters. The traditional methods of analysis of the measured transition energies of a diatomic molecule found in popular textbooks ( 1 , 2 ) are an adaptation of a method descrihed by Herzberg (3).These methods can he characterized by the use of linear reeression to fit a difference in measured enereies. While the use of differences in measured energies to develop equations that can he fit with alinear least squares routine is clever, the methods have some major problems which may confuse students. First, these methods are two-step procedures that use values of some empirical constants determined bv reeression in the first steD to calculate ouantities that are fit in the second step. ~ e c a i s the e results o> the first step are values representative of a distribution of values fit by regression, theeffect of using these values in the second step is the introduction of systematic error into the calculation of the remaining empirical constants that is not removed as a random error would he. Second, a strict application of the method of Herzberg produces potentially contradictory values for some of the empirical constants by calculating those constants in each step. Finally, the methods automatically neglect some of the smaller empirical constants. In addition to describing a more appropriate way to calculate the empirical constants under the traditional method, this article presents two alternate methods to analyze the data from this experiment. The first method extends Herzherg's use of differences in measured energies to the use of successive differences of the data points for the direct calculation of all empirical constants. The values of the empirical constants from several sets of data points are then averaged to remove random errors. The second method extends Herzberg's use of the linear regression to the use of polynomial regression to fit the equation for the measured energies itself providing the full set of empirical constants in a single step. Both of these methods are straightforward, simultaneous calculations of unique values for all of the empirical constants. The calculation of the empirical constants and molecular parameters presented here also provides a measure of the accuracy and precision that is to he expected from students. Each of these methods can he understood with the standard algebra. However, some concepts from difference calculus ( 4 , s )are introduced to aid in the calculations because the rotational quantum number J is an integer independent variable. This calculus differs from the commonly taught
differential calculus in that it is based on discrete integers rather than continuous real numbers. Even so, the rules of these two calculi hear astriking resemblance. The two fitting methods resuire the evaluation of sums over a ranae of v,ilucs of rherotntional quantum numhrr that can be simplified usin. thr techniques oidiffercnce calculus. The simolified expr&ions are more compact than the straight sums, depend only on the number of data points in the range, and are evaluated in many fewer operations than the straight sums. The techniques of difference calculus also provide the basis for the direct calculation of the empirical constants using successive differences. Since the transition energies depend upon the integer quantum numher J , the use of difference calculus in the analysis of data is an additional place for the student to recognize the noncontinuous nature of quantum numbers. Vlbrallonal-Rotallonal Spectra The purpose of this spectroscopy experiment is to measure the peak locations w of infrared energy absorption and use those locations to fit a set of empirical constants wo, Bo, B1, Do, and Dl that are then used to calculate the molecular parameters.
The infrared energy absorbed during this experiment corresponds to the energy difference between vihrational-rotational states (UO, J o )and (01, JI). The transition between the two vihrational levels uo = 0 and u l = 1is of primary interest because the initial levels with uo 2 1have a relatively small population and relatively weak absorption peaks. The vihrational energy of an oscillating diatomic depends upon the freauencv of vibration v and the anharmoni. " (wavenumber) . city constant xu. The rotationalenergy of a rotatingdiatomic deoends w o n the rotational constant B., and the centrifueal stretching amitant D . T h e rotational consranti are expectvd t ~ ~ t l r o e nond the vibrational stateof rhe molecule because vibrations change the interatomic separation describing the rigid rotor. These rotational constants are typically taken to he linearly dependent on the vihrational state u. B" = B -a("
D,
=D
+ 'I*)
+ B(u + 'I2)
The parameters8 a n d D are used to calculate the interatomic separation r of the diatomic molecule, r2 = hl(8r2gcB)
the frequency (wavenumber) of vibration u, vZ = 4B3/D
and force constant k k = 4r2&2
using the reduced mass p of the diatomic. Since the interatomic distance and the force constant are determined by Volume 63
Number 9
September 1986
747
the electronic interactions and thus are independent of the nuclear masses, the expressions p B and &v2 found in the ahove expressions are constants for isotropic molecules. These expressions can he used with values of B and D from the spectra of two isotropic molecules to identify the spectra. The frequency of vibration u and the difference in the vibrational energies of the levels, that is, the band origin wo,
lower state J. This method provides precise values for B , and D,, and a statistical measure of the experimental error in these values. This third method is developed by recognizing that differences in the measured energies with either the same initial state
or the same final state .
are used to calculate the anharmonicity constant xu. The anharmonicity constant is given by a very simple expression for the transition between the vibrational levels uo = 0 and ul = 1.
Svmmetrv of the states of the rieid rotor allows two tvnes of rkationai transitions from the igitial rotational level yo = J. The first tvDe of transition carries the molecule into the rotational l e v i just ahove the illitial level in the uppervibra1. This set of transitions tional state, that is, J I = J originating from the various initial levels { J = 0, 1, ...I is called the R-branch. The second type of transition carries the molecule into the rotational level just below the initial level in the upper vibrational state, that is, J l = J - 1. This set of transitions originating from the various initial levels ( J = 1,2,. . .I is called the P-branch. This simplification allows the wavenumhers of the peaks in the R- and P-branches.
provide quantities which depend only on the rotational constants ofthe final state or the initialstate. Evaluating hoth of these differences in terms of w as a function of rn produces the same function of rn
+
to he written in terms of the rotational energy of state (u,J) in wavenumbers.
For the sake of convenience one can lahel the peaks in both the R-branch and the P-branch with the integer variable rn where rn = J 1 = 1,2,. . .for the R-branch, and rn = -J = -1, -2,. . .for the P-branch.
+
Each peak then has a unique lahel and the energies of all the peaks are described by the same expression.
The explicit dependence on the branch can be dropped since rhr peiksdthG K.hr~nchare identiiied by positiv~valursof t n . and the p w k s of the P-branch w e identified hy negative values of rn.- he equation for w(m)can he also expressed as a power series in the variable m.
Linear Fllllng Method Pooular lahoratorv texts for whvsical chemistrv. (. I ., 2). essentially from ~ e r z d e (3) r ~in aiaiyzing the measured enerries of the spectrum of a diatomic molecule. Herzberr actually describes three methods (6).The first two methods can provide precise values for B, and D, if the value of B, is used to calculate a value for D,and then the value of D, is used to calculate a new value of B , in a successive approximation scheme. These methods do not lend themselves to a calculation of a measure of the random errors in measurement. The third method fits differences in the measured energies t o a linear function of the rotational quantum number of the 748
Journal of Chemical Education
Both differences can also he expressed as a very similar function of J . FJJ
+ 1 ) - FJJ - 1 ) = AFJJ)
-
(2B, - 3DU)(2J+ 1 ) - DJ2J
+ + ...
Note that these expressions depend on the two parameters B, and D, of a single vibrational state. This allows the determinations of these two empirical constants from the differences in two measured energies. Fitting AFo(m)l(2m- 1) against (2rn - 112 gives D, as the slope and 28, - 3 0 , as the intercept. The location of the band origin wo can he determined from the sum of corresponding measured energies in the two branches.
Daniels and others (7) follow the suggestion of Herzherg in neglecting the difference in the D's so that the slope 2(B1 Bo) and intercept 2wo of the remaining linear expression can be determined. This method of calculation should not he recommended to students because it poses the question of how to proceed in the event of contradictory values of theB's and D's resulting from fitting wR(J - 1) w P ( d and from fitting AFU(m)l(2rn- 1). Once the values of Bo, B1, Do, and D I are known from the fit of AFo(rn)l(2rn- I ) , it is more accurate to use these values to calculate the terms in the above expression that depend on them and fit the sum of the known terms
-
+
against the next term in the expansion which has the form of m'(rn2 6) multivlied bv a constant. The intercept is 2wo while the slope is-not important. While this imprkement avoids contradictory values for the rotational constants, the method still calculates the empirical constants in a two-step process that allows the inaccuracies in the B's and D's to affect the value of wo. When the values of Do and Dl are small and obscured by random features of the data, the values of Do and Dl can be neglected and the values of 2 Bo and 2 B1 can be calculated as the averages of AFo(m)l(2m - 1) and AF,(rn)l(2m - I ) , respectively. The value of wo is determined as the intercept of the linear fit of
+
+
against m z ( m 2 1). Shoemaker and Garland ( 8 ) describe a linear fittine method based on the expression fdr w ( m ) in powers of m chat fits
the difference between two adjacent energies against the variable m. Aw(m) = w(m + 1) - w(m) = ( B ,
+ Bo) + ( B , - B , ) ( 2 m + 1) + . . .
The band origin wo is calculated from the equation for d m ) using these empirical coefficients. While this method avoids calculating contradictory values for the rotational constants, it is also a two-step process that introduces systematic errors into the calculation of wn. This method neglects any contribution from the D's, and thus, cannot he i s e d to Ea~cu~ate them. Difference Calculus The rerhnique.; that allow one to treat the independent variable a4 an integer are desrril~rdby difference rnlculus ( 4 , 51. Difference rnlrulus is hased on the notion of the difference in adjacent values of a function of an integer. The factorial function: ml"' = m ( m - 1 ) .
. . ( m - n + 1)
(n factors)
is the basic function in difference calculus that plays an analogous role to that of the power function x" in differential calculus. This function is so named because mcn1 = m!. The powers of m can be expressed as linear combinations of factorial functions. m=
can be evaluated by 1) replacing the power of m with factorials, 2) evaluating the sums from -N to +Nof the factorials using the summation rule from the calculus of differences, and 3) collecting powers of N. These sums for the various analyses described in this paper have been simplified using these techniques. These expressions can be programmed into least-squares routines and used to calculate the empirical constants and molecular parameters. Dlred Calculatlon Method Levine (10) describes a method for the calculation of Bo and B1 using first and second differences in the measured energies. The method calculates Bo - B, from the second differences and then uses the average of this quantity to calculate B1 Bo from the first differences. This method ignores the calculation of the D's, hut it can be extended to the calculation of all of the emoirical constants through - the use of further differences. However, i t requires the suhstitution of the averaee of the nth difference into ex~ressionsfor the (n - 1)st difrerence to get the empirical constants. This multi-step procedure again introduces systematic errors into the calculation because the average of an empirical constant is neither the true value nor a random value that can be improved by further averaging. A knowledge of difference calrulus and of the expression of the energies of transition d m ) as a power series in m, suggests that it would he possible to determine directly all of the empirical constants from successive differences in the measured energies without intermediate averaging.
+
= m12) + ,(I1 ,3 = m(31 + 3mlZl + ,(I1 = =(41 + ~ ~ 1 +3 7m(21 1 + mill m2
m4
In analogy to differential calculus there is an easily proved rule for the difference of a factorial. There is also a summation rule which is the equivalent of the rule for integration of a power.
The use of difference calculus is very appropriate for the analvsis of this exneriment because the inde~endentvariablem is an integer. The fitting methods used in this paper which require the evaluation are least-squares analyses (9), of sums over the values of m. While two of the four sums required for a linear least-squares analysis involve the dependent variable, which is a function of measured energies, the other two sums involve only functions of the independent variable m alone. The functions which are summed can be expanded into polynomials in m whose powers can then be summed from m = -N to m = +Nexcluding m = 0 where N is the number of (successive) peaks in both branches.'
The m = 0 term in the sum corresponds to a nonexistent peak and can be neglected because m("1 = 0 form = 0 except when it must be explicitly removed because ma = 1. Rather than evaluate the sums numerically as though values of f(m) were only known after measurements have been made, these sums can be performed analytically for an arbitrary N. The sums over m from -N to +Nof some power of m, such as m2,which appears after (2m is expanded,
' The same number of peaks from each branch is used for simplicity.
The successive differences are generated as columns in a difference table starting with a column of energies. Equations are presented below that allow all the empirical constants to he calculated directly from the entries in the difference table while considering the argument m to he an integer. The equation for w(m) can also he expressed in terms of the factorial function from the calculus of differences.
This notation is very convenient because the successive differences in w(m) are easily determined and expressed using the rule for the difference of a factorial which gives
The above equations can be solved for the five coefficients to give 4! C, = A4w(m)+ . . . 3! C, = A3w(m)- m A4w(m)+ . .. 2! C, = A2w(m)- m A3w((m)
+ m ( m + I) A4w(m)/2!+ . .. + + 1) A3w(m)/2! - m(m + l ) ( m+ 2) A4w(m)/3!+ .. .
l! C, = ALw(m) mA2w(m) m ( m
O! Co = w(m) - m A'w(m)
+ m(m + 1) A2w(m)/2!
- m ( m + l ) ( m+ 2) A3w(m)/3!
+ m(m + l ) ( m+ 2 ) ( m+ 3) A40(m)/4!+ .. .
A set of five coefficients can be uniquely determined from every. sequence of five adjacent energies. The fifth difference . determines the systematic error forthe five-term expansion and should be relatively small compared to the random error Volume 63 Number 9
Se~tember1986
749
in measurement of the energies. The average and standard deviation of each coefficient can be calculated from the values determined for all of the sets of five adjacent peaks. Expressions for the five empirical constants can be determined by substituting the factorial expressions for the powers of m into the expansion for w(m) in powers of m and comparing coefficients of the factorials. WQ =
C,
8, = ( C , - 2C2 B, = ( C , - C ,
+ 5C3 - 16CJ112
+ 4C,)/2
Do = CC, - 8CJI4
D, = cc, - 4CJ14
When only four coefficients are used to describe the data, take C4 to he zero so that Dl = Do = D. When only three coefficients are used, take CQ= Cp = 0 SO that D, = Do = 0. The empirical constants oo, BI, Bo, Dl, and Do can then be used to determine the molecular parameters. T h e standard dcrinriuns of thi, empirical constants nnd the molecular paramettm can be calculated t r m the standard dt.i,iatiuns of the coefficients using propagation of errors.
.I he empirical constants and mnlerular parameters for the
Polynomial Flttlng Method >
diatomic molecule(an I)? directly determined in yet another manner. T h r coeiiicients in the expansion ot'd(m) in either factorials or powers of rn can he evalunted by minimizing the square 01 thr diirerence hrtween the measurrd rnergies and the expansirm. 'l'he ci~effirientscan then be used tu calculate the rmpirical parameters. This method has the satisfyine " " feature~thatall empirical constants are calculated equivalently unlike the linear fitting method where wo is calculated after the B's and the D's are calculated. A factorial expansion is used here although a power series can be used and may prove simpler for student use. T h e factorial expansion has the benefit of working with a smaller range of numbers than the power series. This can affect the iesults of t,he ralrulatiun u,hen using a largr number of mmsured energies. The least-iuunrei method for thr sum of a set of arhitmrv functions of asingle variable reduces to the solution of a set of simultaneous equations (9).For the factorial expansion of o h ) the coefficients Co through C4 are calculated from the matrix multiplication ~~~
~
~
~~~
where ,..
Ali, =
1
mlllm"'
m=-N
is an elemental of a 5 X 5 symmetric matrix which is to be inverted, and +N
S, =
1
m"'w(m)
m=-N
is a weighted sum of the measured energies. The exact evaluation of the elements of the matrix A can he aummpiished using the summation rule from the calcnlus of diff~renres.The elemrnts are evaluated hv. I.I exoandlne . the products of factorials m(')rn'k)in powers of m, 2) replacing the resulting powers of m with one of the sums of factorials given earlier, 3) collecting terms, and 4) using the summation rule. This produces elements which are simple polynomials in the parameter N which have the same number of terms for all values of N. This work is a hit tedious for a 5 X 5 matrix hut is easily demonstrated for a 3 X 3 matrix. The matrix elements as polynomials in N for both the factorial
-
750
Journal of Chemical Education
and power expansions are available on request. The matrix can he inverted manually to give ratios of polynomials that can be simply evaluated for any number of measured energies, but it proved easier to evaluate the matrix A for a given number of measured energies and numerically invert the matrix. (There is no need for pivoting since the largest element is in the lower left corner of the inverted matrix.) The empirical constants are calculated from the coefficients Cnthrough - C . as descrihed in the ~ r e v i o u ssection. The sumof the squares of the differenceshetween the measured energies and the calculated eneraies are used to calculate the st&dard deviations of the coefficients ( 9 ) ,and these standard deviations are used to calculate the standard deviations of the empirical constants using the rules of propagation of errors. ~
~
~~
Comparison of Methods The published infrared spectra of hydrogen chloride (11, 12) can be analyzed as described in the preceding sections. To examine both the accuracy and the precision of the methods descrihed, a set of 24 energies was generated from published empirical constants (3). The data were analyzed as high-precision data good to 0.01 cm-I and as low-precision data good to 1 cm-I. Data in this precision range can be expected from t v ~ i c a instruments. l The values of the empirical constants i n d their standard deviations from each method are used to calculate the interatomic seoaration, the frequency of vibration, the force constant, th; anharmonicity constant, and their appropriate standard deviations. Popular laboratory texts (1,2) neglect the constant Do and Dl even though they are straightforward to calculate and give additional, interesting molecular parameters. For comparison the data were also analyzed with each method assuming a three-term expansion for the energies where Do = ' = n. --
r). -
The results of the calculation of the empirical constants using the linear fitting method are shown in Tahle 1. Very accurate and precise values for all the e m ~ i r i c a constants l can be determined by fitting high precision.data with a fiveterm expansion. All the empirical constants can a t least be calculated from low-precisibn data although the values of the D's are inaccurate. The inaccuracies of theD's reduce the precision in the value of wo below that found using a threeterm expansion because the inaccurately calculated values of the D's are used to determine the value of wo in the linear fitting method. These inaccuracies also severely limit the significance of the molecular parameters v and x u calculated using the D's and oo. The value of the D's and the associated molecular parameters are not given hv the three-term expansion. However, in all cases t h e caliu~ationof the interatomic separation from the B's is possible, accurate, and precise. The higher accuracy in the R's calculated using the five-term expansion will give the most accurate result for the interatomic separation. The results of the analysis of the data by direct calculation are listed in Tahle 2. Each set of five adjacent peaks in the two branches (eight sets for each branch) was used to calculate directly the five coefficients Co through C4. Examining Tahle 2 indicates that the method of direct calculation is severely inaccurate and imprecise. It is not possible to get values for the D's with even high-precision data, and i t is necessary to use high-precision data to get the B's and a . . value fir; the interatomic separation with'my accuracy. Examination of the residunls indicates that the direct ralrulation method fits the five energies exactly a t the expense of the accuracy of the empirical coefficients. The poor performance of the direct calculation method occursbeca&e the empirical constants are linearly related to the energies and their successive differences. Limited precision in the energies is propagated through to the empirical constants producing values without significance. This method requires
Means and Standard Deviations for Linear Fitting Method
Table 1.
Exact values (cm-') High-precisiondata (50.01cm-') Five-term expansion Three-term expansion Low-precisian data (fI cm-') Fiveterm expansion Three-term expansion
High-precision data (f0.01 cm-') Five-term expansion Threeterm expansion Low-precision data (+l cm-') Five-term expansion Three-term expansion
High-precision data (hO.01en-') Five-term expansion Three-term expansion Low-precision data (f1 cm-') Five-term expansion Threeterm expansion
2885.5 0.1 2885.78
Means and Standard Deviations for Direct Calculation Method '4
&
4
2885.9000
10.43995
10,13805
2885 3 2886 1
10.5 0.8 10.6 0.1
10.2 0.8 10.3 0.1
2823 340 2885
13 88 11
9 81 10
4
Residual
0.000530
0.000528
...
0.0003 0.007
0.0002 0.006
2.0949
...
...
... ...
3.8930
-0.003 0.6
0.003 0.6
...
...
4.8019
D?
Residual
0,
Law-precision data (+I cm-') Linear fining Polynomial fining
351
...
@a
6
0,
2885.9000
10.43995
10.13805
0.000530
0.000528
...
2885.8984 0.0010 2885.9 0.2
10.43985 0.00009 10.341 0.010
10.13805 0.00009 10.040 0.009
0.0005289 0.0000008
0.0005255 0.0000006
0.0021
... ...
... ...
0.6285
2888.08 0.08 2888.0 0.2
10.420 0.007 10.337 0.009
10.112 0.007 10.034 0.008
0.00045 0.00006
... ...
...
...
0.00041 0.00005
0.1658
...
...
0.5449
...
...
Means and Standard Deviations ol Molecular Parameters Five-term expansion
Polynomial fining
...
%
Threeterm expansion
r
High-precisiondata (+0.01 cm-') Linear fining
...
Means and Standard Deviations for Polynomial Flttlng Method
Table 4.
Exact values
Residual
Dq
2885.8958 0.0008 2885.890 0.002
Table 3.
Exact values (cm-')
Dn
2885.9000
Table 2.
Exan values (cm-')
BI
Bn
a"
XY
r
(11cm)
(nm)
(nm)
(11cm)
0.1274698
2988.6391
51.37
0.1274698
0127471 0.000001 0,1274707 0.0000009
2994 3 2992 4
54 2 53 2
0.1278 0.0004 0.12807 0.00010
0.1278 0.0001 0.12757 0.00007
4000 1000 3200 300
100 200
0.12809 0.00008
Volume 63
Number 9
September 1986
75 1
high precision data to produce usable values for the D's, u, and xu. The results of the calculation of the empirical parameters using the polynomial fitting method are listed in Tahle 3. The overall accuracv is similar to the linear fittine method. but the precision isslightly better except in the dkermiua: tion of wo with a three-term expansion. The overall fit of the energies is also better using the polynomial fitting method as indicated bv the residuals of fit. The resuits of the calculation of the molecular parameters using these two fitting methods are displayed in Tahle 4. The molecular parameters calculated by the polynomial fitting method are the more precise. With low-precision data, three- and five-term exnansions determine the interatomic separations to four or 'five significant figures. With highprecision data, the interatomic separation is determined to six or seven significant digits, the frequency of vibration is determined to four significant digits, and the anharmonicity constant is determined to two significant digits. I t is evident from this table that high-precision data are necessary for the calculation of significant values of the molecular parameters u and xu. While D's can be calculated with both hiph- and low-precision data, the poor quality values obtainid from the low-precision data reduce the significance of those molecular parameters. The accuracy of these results is generally one digit less than the precision. The interatomic separation can be calculated accuratelv to four dieits usine a five-term expansion t o fit low-precision data. The frequency of vihration is accurate to three digits. and the anharmonicitv constant is accurate to one dig% using a five-term expans"ion to fit high-precision data.
-
-
Conclusions The direct calculation of the expansion coefficients and empirical constants from successive differences of the measured energies produces very inaccurate and imprecise results that cannot be used to calculate significant values of the molecular parameters. The methods of analysis which use regression to fit differences in the measured energies as a
752
Journal of Chemical Education
funrtion d m or J provide much better precision and acrurathc linenr and polvnominl fittinr! methodsare both just as accurate, the p&y~omialfittingmethod produces slightly more precise results. T o get the best numerical results, fitting the measured energies with a five-term factorial expansion is necessary. The polynomial fitting method is the better method to use because it directly fits the measured energies and it calculates all of the empirical constants at the same time in the same way. While these descriptions show the precision and accuracy that an instructor can expect of hislher students, it is also important to recognize that some of the analysis can he completed before using a computer by recognizing that the quantum number J or the variable m are integers in the range -N to N - 1or -N to +N, respectivelv. One benefit of thistype of analysis is that it emphasizes the integer nature of quantum numbers in the analysis of data as well as in the derivation of formulae. The least-square fitting routines require the evaluation of sums of powers of m over the peak index m, which can be evaluated using the calculus of integer numbers-the calculus of differences. The resulting expressions become polynomials dependent on the number of peaks analyzed. Unlike the sums over m that require more terms to be evaluated when more peaks are analyzed, these polynomials have a fixed number of terms. cy. \Vh&
Literature CRed (11 Daniels. F.: Alberty; R. A.. and others. "Experimental Physical Chemistry", 7thod.; MeGraw-Hill: N e w York, 1970. (21 Shoemaker. D. P.; Garland, C. W. "Experiments in Physical Chemistry... 4th 4.; McCraw~HiIl:New York. 1981. (31 Herzberg. G. "Molecular Struefure and Maleeulsr Spectra: I. Spectra af Diatomic Molecules": Van Natrand Reinhald: N o w York, 1950. (41 Levy. H.: k m a n . F. "Finite Difference Eguations": Macmillan: Ne* York. 1961. (51 Goldberg, S. "lntraduction fo Difference Equations"; Science Editions: N e w York, 1987 ~~
~
(61 Ref3. pp 177,182, and 180 171 Ref 1.on 251 end 266. is) R& 442 and 445. (91 Bevington. P. R. ''Data Reduction and Emor Analysis for the Physical Sciences";
