The shapes and intensities of infrared absorption bands—A review

Spectroclknica Acta, 1963,

Tul.

10, pp.

1013 to 1085.

The shapes and intensities

Pergamon

Press

Ltd.

Printedin

Sorthern

Ireland

of infrared absorption bands-A

review*

K. S. SESHADRI and R. NORMAN JONES Division of Pure Chemistry, National Research Council of Canada, Ottawa, Canada (Received 23 July

1963)

CONTENTS

I. Introduction II. Factors determining the true band shape. (a) Radiation damping. (b) Doppler broadening. (c) Collision broadening. (d) General conclusions concerning the true band shape. III. Instrumental factors modifying the true band shape. The instrument functioil. Optical distortion of the band profile. Electronic distortion of the band profile. Mechanical distortion of the band profile. Errors caused by scattered radiation and related effects. Quantitative relationships among the spectrophotometer variables. IV. The det,ermination of the true band shape from the apparent band shape. (a) Analytical solution for O(Y) by Fourier analysis. (b) Curves of mixed Gauss and Cauchy character-the Voigt profile. (c) Characterization of band profiles by truncated moments. (d) An analytical solution forf(YJ by Fourier analysis. (e) Evaluationof thebandintensityparametersbynumerical integration. (f) Evaluation of the true band-intensity parameters by extrapolation. (g) Choice of units for the measurement of integrated band intensities. V. The effects of random errors on the measurement of the true and apparent absorbance. VI. Conclusions. VII. Acknowledgements. Notation.

Page 1014 1015 1016 1018 1019 1021 1025 1025 1026 1037 1041 1042 1044 1045 1045 1047 1049 1055 1057 1062 1071 1073 1079 1082

Abstract-The experimental and theoretical factors influencing the measurement of the profiles and intensities of infrared absorption bands for materials in condensed states are reviewed. The effects of t,he spectral slit function and of the time constants of the spectrophotometer recording * Published as Contribution Council of Canada.

No.

7341 from the Laboratories 1013

of the National

Research

1014

R. 8. SESHADM

and

R. NORMAN

JONES

system on the band shape are discussed, and the met,hods available for the experiment,al evaluation of these quantities are described. The theoretical and experimental problems associated with t,he determination of the true band-intensity parameters from the corresponding apparent band-intensity parameters are examined critically and the interrelationships among the various spect,rophotometric variables, including spectral slit width, amplifier time constant,, scanning rate and amplifier gain are considered both in terms of the theory of errors and with respect to general laboratory practice.

I. INTRODUCTION WHEREAS the earlier investigators of the infrared absorption spectra of polyatomic molecules were principally concerned with the nleasurement of the band positions, increasing interest is now being shown in band intensities. From vapor-phase measurements of the band intensity it is intrinsically possible to determine the dipole moment change associated with the normal vibration, and a detailed analysis of the intensities of the infrared absorption bands should reveal how the polarization of the molecule is influenced by the vibrational motions. The manner in which the gas-phase band intensities change in the liquid or solid state provides insight into the nature of the intermolecular forces in the condensed phases. The measurement of the absolute band intensities can also have important analytical applications. As yet no standards of infrared band intensity have been established, and intensity measurements on the same sample of material can vary widely when determined on different spectrophotometers. This currently restricts quantitative infrared spectrophotometry to systems in which separate calibration curves have been prepared for each instrument. Although the measurement and interpretation of infrared band intensities are beset by many difficulties, interest in this problem has been stimulated by recent progress in instrumentation, particularly by the development of simply operated high-resolution double-beam grating spectrophotometers. Several years ago WILSON and WELLS [l], and BOURGIN 123 showed that the photometric errors associated with the use of nonmonochromatic illumination could be eliminated by extrapolation techniques. Unfortunately these methods are tedious to apply and may give unsatisfactory results because the extrapolations involve measurements made under conditions where the random errors are large. With increasing interest in the subject attempts have been made to develop simple approximation Generally these methods necessitate a methods to measure band intensities. knowledge of the band shape, and involve the assumption that the band profile can be approximated by a mathematical function that can be integrated over the desired frequency range. An approximation of this kind, due to RAMSAY [3], has been widely employed, either in its original form, or as later modified by CABANA and SANDORFY [4]. There are two important problems currently engaging the attention of workers in this field. One is to obtain a more accurate mathematical expression for the [l] E. 13. WILSON JR. and A. J. WELLS, J. Ckent. Phys. 14, 578 (1946). [Z] D. G. BOURGIN, Phpx Rez?. 29, 794 (1927). [3] D.A. RAMSAY,J.A~.CIL~~.SO~. 74, 72 (1952). [4] A. CABANA and C'.SANDORFY, fi’pectrochim. Acta 16, 335 (1960).

The shapes and intenskies of infrared absorption bands

1015

experimentally observed band profile, and the ot’her is to correct this profile for the various perturbing instrumental effects. These distortions are both optical and electronic in origin, with the effect of the finite spectral slit width predominating. The environment of the absorbing molecule also modifies the band shape but it is difficult to investigate this problem until the more immediate instrumental problems have been overcome. The object of this review is to bring together the scattered literature on this It is becoming subject,, and to systematize the mathematical presentation. increasingly important that the chemical spectroscopist should have some critical appreciation of the theory underlying the measurement of band intensities, but he is often handicapped by the fact that much of the original work has been carried out by theoretical physicists and by communicafion engineers, and presented in a form that assumes detailed familiarity with quantum physics or with electronic In particular, much of the more recent work on and classical optical theory. infrared band shapes assumes prior knowledge of earlier work on the profiles of These studies in the visible region were emission lines in the visible spectrum. prompted by the problems encountered by astrophysicists in trying to interpret the emission spectra of stellar bodies as measured with energy limited spectrometers attached to telescopes. For the theoretical spectroscopist, the total area under the absorption band has primary significance, and his interest has largely been confined to the study of a relatively small number of infrared absorption bands for which this type of measurement is possible. The analytical spectroscopist is usually obliged to deal with strongly overlapping band systems where band area measurements are restricted to empirically defined area integrals. The analytical spectroscopist therefore is more concerned with peak-intensity measurements. The band width at half the maximal intensity can also be used in conjunction with the peak height as an index of the band area, and is often amenable to measurement on partly overlapping bands. The basic principles reviewed in this article apply to band areas, peak heights and half band widths. They are therefore of importance in analytical and other semi-empirical applications of infrared spectrophotometry as well as in more theoretical studies. The importance of the half band width measurement has not hitherto been fully appreciated by chemists and analytical spectroscopists, and this might be an opportune place to urge the more careful and extensive documentation of t,his important intensity parameter in future work. In this review we shall not discuss the dependence of the band intensity on the character of the associated normal vibration. This is dealt with in the recent monograph of BAUMAK (51 and in the review article of GRIBOV and SMIRNOV [6]. II. FACTORS DETERMINISG THE TRUE BAND SHAPE It is well known t’hat the absorption or emission of radiation does not occur at a discrete frequency. It is always distributed over a range of frequencies, and [5] H. I’. BAUMAN,_rlhsorf>ti0?1 ,Ypectroscopy,pp. 238-244. John Wiley, Sew I7ork, London (1962). [6] L. A. Soviet

GRIBOV

Physics

and V. 3. SXIRNOV, Uspekhi Zrspekhi, 4, 919 (1963).

Fiz. Sauk.

75, 527 (1961).

English translation,

1016

K. S. SESHATUSI and 1%. NORMAN JONXS

all infrared lines and bands have finite widths.* The three principal factors deter~lining the line width in the vapor state are (a) r&diation damping, (b) the In the liquid phase, collision broadening Doppler effect and (c) collision broadening. The basic theory is relatively so large that the two other elects can be neglected. of line broadening was developed originally with referenoe to the electronic spectra of atoms and simple molecules, snd has been reviewed by MARGEXAU and WATSOX [7] and by TOLAWSKY [8]. According to classical mechanics a vibr&ing dipole radiates energy; this energy must diminish the vibrational amplitude, resulting in a, damped vibration. The radiation emitted in such a process cannot be monochromatic and the line shape can he shown by Fourier analysis to have the form 1, -_

I-

271 (Y -

AVI,, VQ)$+ (*AY,,,)2

where I, is the intensity at the wavenumber v, AY~,~is the band width at half the maximal intensity in cm-l and ~a is the position of the band center. The mechanical frequency of the oscillator isgiven by Wo, where c is the velocity of light in the medium.? The mechanism of vibration damping is discussed here in terms of the emission process. Normally the behavior of a single oscillator can be presented with equal validity in terms of either absorption or emission. If preferred, I, in equation (I) can be regarded as a measure of the energy extracted by an absorbing system from the ra~ation field. For further consideration of this principle see Ref. [7]. It can be shown that

where m and e are the mass and charge of the oscillator. Sini5e AA Au -.- =. _-it v equation

(3) can be written in terms of wavelength:

* The term “line” is used here to describe absorption or emission of radiation between discrete xotatio~--vibration levels in both the ground and the excited states of a molecule. An infrared “band” includes all lines associatedwith a given vibrational transition. In oondensedphase spectra the line structure is commonly absent; in discussingvapor-phase spectra of small molecules at norma,lpressureswe shah usually be dealing with infrared line profiles, and for pondered-thee spectra with btrndproffles. t In this artiele frequencieswill usually be expressedby the symbol Yin units of wavenumber (cm-l). In cases where it is preferableto report frequenciesin e/s, the symbol v will be employed. Angular frequency in fad/see will be designatedby W. [7] H. MAFKJENAU and W. W. WATSON, Revs. ikl’crdernPhys. 8, 22 (1936). [Sj S. TOLANSEY, EC& .Resolz&m Spectroscopy Chap. 1. Methuen, London (1947).

The shapes and intensities of infrared absorption bands

1017

This leads to the conclusion that the half line width is invariant on a waveThe latter invariance is not in length scale and independent of the intensity. accordance with the facts, and results from the inadequacy of classical theory. In the quantum-mechanical treatment, line broadening by radiation damping is replaced by the “natural line width”, which is a consequence of the Heisenberg uncertainty principle. The intensity distribution is given by an expression similar in form to equabion (1) but with the half width term replaced by

n-here AE, and AE, are respectively the finite widths of the higher and lower energy states participating in the transition. If a molecule remains in the E, state for a finite t’ime At2, then, according to the uncertainty principle At, . AE, = & We may regard At, as the rate of spontaneous where h is Planck’s constant. transition from the state E, to all states (Ek) of lower energy and it can be shown t,hat 1 8.rr2e2 -=(7) &Cv:,k foI, k mc A& where v~,~ is the wavenumber and f2,k is the oscillator strength for the transition 2 ---f k. A similar equation (7a) can be written for the rate of spontaneous transition from the state E, to all lower energy states E,. 1 -=_

8.ir2e2

At,

From equations

(5)-(7a)

mc

XL Vf,Ji,l

(ya)

the general expression

is obtained for the half line width. For infrared fundamental bands, with which this article is mainly concerned, the lower state E, will be the ground state, and will be the only state of lower energy than E, for which the transition is allowed. Under these conditions equation (8) reduces to

or, expressed

in units of wavelength, A&,, =

(IO)

From comparison of equations (4) and (10) it will be observed that the quantummechanical expression contains additional terms equal to 3f2,1, and therefore depends on the band intensity. From equations (3) and (4) it follows that, according to classical theory, an

1018

K.

S. SESHADRI and R. NORMAN JONES

infrared line at 1000 cm-l will have a natural width of the order of 1OW cm-l. The oscillator strengths of infrared lines are considerably less than unity, so that the application of the more exact quantum-mechanical expression will further reduce the natural line width. This factor is therefore extremely small and only of academic significance to the problems dealt with in this article. The natural widths of some intense atomic emission lines in the visible region of the spectrum have been measured, where the collision broadening effects can be reduced by measurement at very low pressure, and the Doppler effect controlled by the use of atomic beams [Q, lo]. (b) Doppkr

broadening

A second factor influencing spectral line width is the thermal motion of the molecules. If all the molecules were at rest, and the natural broadening effect were neglected, the emission or absorption would be monochromatic at the waveor receding number Ye. Because of the Doppler effect, molecules approaching from the observed with velocity v will emit or absorb at the wavenumber V = vo(l f v/c, (11) where c is the velocity of light. Because of the Maxwellian velocity distribution the absorption or emission will extend over a range of wavenumber with an intensity profile conforming to a probability distribution curve. The fractional number of molecules ah/n within the velocity range C&U is given by

an -n

fl!!

[ 2~RT

1

u2

exp [-(M/2RT)v2]

dv

(12)

where &! is the molecular weight of the gas, R the gas constant and T the absolute temperature. The total intensity at the wavenumber v follows from equations (11) and (12) and can be expressed

Iv = [

2TTFJvo2] l”exp - [(G2) (v - vo12]

(13)

where

+m I, dv = 1 s --m It follows from the above that the Doppler effect produces profile and a half width given by 1,2 2RT I/” Av,,, = 2(ln 2) ~ [

NC2 1 v”

(14) a line with a Gauss

(15)

For v. = 1000 cnl-l for a vapor of molecular weight 100 at 2O”C’, a value of 1.2 x 1O-3 cm-i is obtained for the half width of the Doppler broadened line. The experimental limits of absolute wavenumber precision in the mid-infrared are of the order of 1O-2 cm-l [l l] and Doppler broadening is therefore still too small 191A. BOGROS, Compt. rend. 133,124 (1936); Ala%. phys. 17, 199 (1932). [lo] L. DOBREZOV and A. TERENIN, Naturwissenschaften 16,656 (1928). [ 111 International Union of Pure and Applied Chemistry, Tables of Wavenumbers for the Calibration of Infra-red Spectrometers. Reprinted from Pure -4ppl. Spectrosc., 1, (1961) Butterworths, p. 553.

The shapes and intensities of infrared absorption

bands

1019

influence such measurements significantly. Wavenumber differences of an order of magnitude of 1O-3 cm-l can be measured and Doppler broadening effects have been observed on infrared lines. An example is provided by the measurements of BENEDICT et al. for the 1 -+ 0 band of hydrogen chloride, measured at low pressure [la]. The Doppler broadening therefore does have significance with respect to the measurement of infrared lines of molecules in the vapor state, but does not appreciably influence measurements of the broad bands associated with condensedstate systems.

to

(c) Collision broadening Collision broadening is the principal factor influencing the half widths and profiles of bhe absorption lines in the infrared spectra of diatomic gases under normal conditions of temperature and pressure. It results from the perturbation of the energy levels of the absorbing molecules caused by the close approach of molecules of similar or different kind. This aspect of line broadening has been dealt with by LORENTZ [13] and, later, by VAN VLECK and WEISSKOPF [14]. These authors were primarily concerned with electronic processes in atoms, though their conclusions have also been shown to apply to the profiles of rotational fine structure in the vibration spectra of gases such as hydrogen chloride, hydrogen cyanide and carbon dioxide [7, 12, 151. The shape factor derived in this way has been widely assumed to apply also to the profiles of infrared absorption bands of liquids, and the validity or falsity of this assumption is basic to much of the current interest in the evaluation of band profiles for condensed phase spectra. LORENTZ assumed that the atom or molecule absorbs or emits at a discrete frequency during the time between collisions. Radiation stops abruptly when the collision occurs, and the energy becomes wholly kinetic. The time intervals between collisions are considered to be large compared with the duration of the collision. The effect of the collision is such that all pre-existing orientation is obliterated. In the original Lorentz treatment it was further assumed that after the collision the molecules were also randomly oriented with respect to the radiation field. Later VAN VLECK and WEISSKOPF suggested that after collision the molecules would tend to retain a selective low-energy orientation with respect to the field. These authors modified the original Lorentz equation to take account of this polarization, a change which also brought the Lorentz treatment into better accord with DEBYE'S theory of light scattering by dipoles. Since the conditions under which infrared spectroscopists seek to apply the Lorentz equation to condensed-phase systems are analogous rather than directly comparable with those for which it was originally derived, our main interest is to understand the nature and origin of the terms that occur in the Lorentz equation, and it is not necessary to discuss the derivation in detail. Those who may wish to do so are referred to the articles of VAN VLECK and WEISSKOPF [l4] and of MARGENAU and WATSON [7], as well as to the original Lorentz paper [l3]. [12] [13] [14] [15]

1%‘.S. BENEDICT, R. HERMAX, G. E. MOORE and S. SILVERMAN, Can. J. Phys. 34,850 (1956). H. A. LORENTZ, Koninkl. Ned. Akad. Wetenschap. Proc. 8, 591 (1906). J. H. VAN VLECK and V. F. WEISSROPF, Rem. Modern Phys. 17,227 (1945). C. HERZBERG and J. TV. T. SPINKS, Proc. Roy. 6’oc. (Lordon) A147, 434 (1934).

I(. 8. SESHADRI and R. NORMAN

1030

JONES

The problem is basically concerned with the interaction of an oscillator of natural frequency coo, mass m and charge e with an electrical field E cos cot. Classically the resultant motion is described by the differential equation

(16) for which a solution of the following

form can be assumed

x = C, exp (i03f) + C2 exp (hot)

+ C, exp (--icoot)

(17)

where

eE

c, =

m(wo2 -

(18)

~0~)

and C!, and C, are determined by the nature of the collision. At some time (t - 0) the molecule experienced a collision, as a result of which x and dx/dt will have acquired instantaneous values that will determine the subThe original postulation of LORENTZ, that the sequent behavior of the oscillator. orientation after collision is unpolarized, is equivalent to saying that x and dxldt can assume all possible values with equal probability, so that, on the average, their value is zero. Under these conditions C, and C, can be expressed in terms of Ci, (t - e), co and w,,, and a more informative equation for x is thus obtained, averaged over all directions in space. It is next necessary to modify this equation to average over the varying time This is achieved by multiplying the expression intervals from the last collision. for x by n(t), where n(t) =

1exp

(19)

V

The function n(t) is the probability that a molecule, after having one collision, will collide again after a lapsed time 8. The quantity Y is the mean time between collisions. This yields I for the time and space averaged value of x. The equation for 5 (which we do not reproduce) is a complicated algebraic function involving c1, UO, (O - wo), (o + wo) and l/v. Th’is f unction has real and imaginary parts, and from the imaginary part it can be shown that the absorption coefficient E at the frequency w is given by 27rne2 CC,---

co

l/u

mc coo [ (co - ~0~)~+ (l/~)~

l/v

1

(LO+ ~0~)~+ (l/~)~

(20)

where n is the number of molecules in the system. This, in essence, is the function that gives the Lorentz line profile, but to be useful spectroscopically it must first be modified by conversion from circular frequency to natural frequency (see-I) or wavenumber, and must also be adapted to quantum-mechanical conditions. To do this, w. is replaced by 2mii, where 87r2 Y,, is the frequency in set-l for the transition i +j: e2/m is replaced by 1p.J2 vii, 3h

in which pij is the matrix element of the electric moment for the i - j transition.

The shapes and intensities of infrared absorption bands

1021

From this we obtain x, =

8772(N, -

(1/277v)

(1/275-w)

h7,)v (V -

3hc

V,f)2 + (l/2?TV)2 -

(V +

V,j)2

+

(1/2rrw)2

1

(‘I)

where S, and hT3are the populations of the molecules in the respective states. For infrared absorption lines the half intensity line width, Av~,~ = l/m is small in comparison with the natural frequency so that the second term within the bracket of equation (21) can be neglected. Furthermore we are commonly dealing with transitions from the ground state to the first vibrationally excited state, and with pressure broadened bands where there is no individually resolved line structure: accordingly N, can be replaced by the Avogadro number, in good approximation, and N, N 0. Writing v,, for vtj we therefore obtain WI,2 (v

or, on transposing

-

vJ2

+

($Av,,,)~

1

(22)

to wavenumber, (23) ‘d” [(._vo;A~7~AvJ,2),]

’ =y Y

2

Since (1~ - ~~1) <~a, we can write v0 for v in t’he term outside bracket in equation (23), whence

,l %

=

h

4 Av112 ;

(v

-

v,J2

+

(+Av~,~)~

the square

1

in which K is a constant for the particular line or band, and the remainder of the right-hand side of equation (24) is the “shape function”. The effect of the Van Vleck and Weisskopf modification is to introduce the additional term v/v0 into equations (21) and (22), and to change the sign of the second term within the square bracket of equation (21) from minus to plus. In the mid-infrared, under the conditions normally encountered in infrared spectrophotometry, v/v0 N 1, and these changes are not significant. In the microwave region they can become important, and, in the limit where v,, + 0, the addition of the v/v0 term reduces equation (22) to the Debye expression for radiation scattering from rota,ting dipoles, whereas the original Lorentz equation reduces to zero. (d) General conclusions comerning

the true band shape

Of the three factors contributing to the line shape, it will be apparent that both radiation damping and collision broadening yield profiles of the same form. Mathemat~ically these are Cauchy functions.

in which 2b is the half band width and a/b2 the peak height, with x measured from the band center (Fig. 1). The Doppler broadening has the Gauss form ycgj = a’ exp

i

-

f

1

1062

K.

8. SESHADBI and R.

NORMAN JONES

where a’ is the peak height, and the half band width is given by 2o(ln 2)1/s (Fig.1). When expressed in terms of the constants a and b of equation (25) the Gauss expression becomes ytgj =sexp

[-X*]

(2’)

04 -

,

__--I/’ -1.6

-12

‘.

I

I

-08

I1 0

I

-04

‘\._ .__ / 1 +OB +,2

1 +04

I +16

_

Fig. 1. Comparison of the profiles of Gauss and Ceuchy curves of the same peak height and total area.

When equations

(25) and (27) are normalized

to unit peak height they become

b2 Y(c) =

b2 +

22

(28)

yfg) = exp [-WI and when normalized

to unit area (30)

with peak height (l/rb),

and Y(8) =

g&exp

( -x2/02)

(3’) (In 2)112 with peak height ___ . dJzb On normalizing

to unit peak height and unit area, the unique solutions Y(e) =

1-

1

79 (l/772) + 52

(32)

The shapes and intensities of infrared absorption bands

with half width of (l/r)

= 0.318,

1023

and

yk) = exp ( -.rrx2) with half width (In 2/n) II2 = 0.471 are obtained. These various forms of the Cauchy and Gauss distribution encountered in the analysis of band profiles.

(33)

functions

may be

Fig. 2. Comparison of the experimentally observed profile of the band at 813.7 cm-1 in the spectrum of a carbon disulf?de solution of perylene with a Cauchy curve having a = 0.931 and b2 = 1.147.

For the broad bands associated with the infrared spectra of liquid-phase systems, it may be concluded that if the band contour is determined primarily by collision processes in which the duration of the collision is small in comparison with the time between collisions, a band profile approximating to the Cauchy form will be observed.* For a collision broadened infrared line in a vapor-phase spectrum WHITE [16] has derived the exmession RT

Av,,, = 4Np2D [

l/2

__ n-iv31

(34)

where p is the collision diameter of the gas molecule, D the density, X the molecular weight and N the Avogadro number. RAMSAY [3] has noted that if values of p and D appropriate to liquid-phase systems are introduced into equation (34) a half band width of the order of 10 cm-l is obtained. This is The band at 812.7 cm-l in the comparable with values observed experimentally. spectrum of a carbon disulfide solution of perylene (Fig. 2) is a good example of a close approximation of a liquid-phase absorption band to the Lorentz profile. This spectrum is discussed further on p. 1053. * In this article the term “Lorentz curve” will be used when dealing with collision broadening effects on band shapes. The more general expression “Cauchy function” will be used in discussing the characterist,ics and behavior of equation (25). [16] H. E.

ITKITE,

Introduction

to

Atomic

Spectra,

Chap. 31.

McGraw-Hill,

New York (1934).

1024

Ii.

S. SESHADRI and R. NORMAN JONES

In the foregoing discussion it has been assumed that the absorbing molecules are all identical. In fact, liquid-phase systems often occur in which several different species of absorbing molecules are present. If the vibrational frequencies associated with the various species differ sufficiently, separate bands will be resolved for each form, but if the frequency differences are small, a single asymmetric broadened band results. The superposition of sets of distribution curves in this manner will tend to induce a Gauss component into the composite-band envelope [17, 181 (cf. page 1049). Band widening of this kind can be brought about by several causes of which we shall note only a few. The presence of mixed isotopic species will produce closely lying bands; this should be particularly in evidence for compounds containing chlorine and bromine where there are two isotopes both of which are present in comparable proportions. Certain bands in the spectra of chloroform and bromoform are notably wide and the Gauss character of their profiles has been discussed [ 191. Band broadening can also result where a complex polyatomic molecule can exist in different conformational forms. If the centers of conformational isomerism are intimately involved in the normal vibration responsible for the absorption, it is likely that two separate bands will be resolved, and many cases of this kind are known. However, if the centers of conformational isomerism are more remote from the atoms primarily involved in the normal vibration, the perturbation effect of the steric change on the band frequency may be very small, in which case it will only act to broaden the band. No examples of such an effect have yet been positively demonstrated, but it might be looked for in bands associated with the skeletal ring vibrations of cyclic molecules carrying long flexible side chains, where conformation changes in the side chain could induce small perturbations in the ring frequencies. This kind of band broadening will usually be distinguishable from that associated with isotopic effects by its much greater temperature sensitivity; the Gauss character of the profile should diminish as the temperature falls and the populations of the thermodynamically less stable conformations are reduced. Both mechanisms discussed above will lead to the presence of a relatively small number of discrete overlapping band systems. Solute-solvent interaction, however, can give rise to much more complex equilibria ranging from simple dimeric association (as in some forms of hydrogen bonding), to the loose quasioriented complexes which differ from transient collision complexes only by the fact that their lifetimes are appreciable in comparison with the vibration frequency. One might expect that the bands associated with such systems would be more pronouncedly of Gauss form because the number of participating species will be very large. “Hot bands” involving excitation from a vibrational level above the ground state may also distort the band shape, particularly for bands in the lowwavenumber range. 1171R. P. BAUMAN and S. ABRAMOWITZ, Abstracts of Symposium on Molecular Structure and Spectroscopy. The Ohio State Univer&y, Columbus, Ohio. June 1962. [l&3] H. C. VAN DE HULST and J. J. M. REESINCK, BUZZ. Astronom. Inst. Ned. 11, 121 (1947). [19] Yu. P. TSYASHCHENKO, Optica i Spekstroskopiya 9, 192 (1961). English translation, Opt. and Spec. 9, 101 (1961).

The shapes and intensities of infrared absorption bands

1025

These considerations lead us to anticipate that for many liquid-phase absorption bands both Lorentz and Gauss forms of band broadening may be operating together, so that the resultant profile will be intermediate between these two extremes. An analysis of the band profile as a function of temperature and concentration could provide information about the molecular structure in the liquid phase. Later in this review quantitative techniques for the analysis of the band profiles into their Gaussian and Lorentzian components will be considered. Before doing this, however, it is first necessary to discuss the experimental factors that affect the experimentally observed band shape, since any analysis of the band profile in terms of molecular st’ructure can only be meaningful if the instCrumental distortion has first been eliminated. III

IKSTRUMENTAL FACTORS MODIFYING THE TRUE BAND SHAPE

(a) The instrument function The general expression t’hat takes account of all the instrumentally induced distortion of the true band shape can be called the instrument function. * Although this quantity cannot be experimentally evaluated accurately, it can be described in general mathematical terms. Such a description, together with a summary of the algebra of convolutions, which is basic to the mathematical treatment of this problem, has been reported by RAUTIAN [20]. If Iv, designate the true band intensity at the wavenumber v,, and TV, the apparent intensity when the spectrophotometer is set to the scale reading v,, t)he relation between the two quantities may be formulated Tvi = K(vi)Jv,

(3.5)

where R(Y~) is some function of v, also involving the instrumental variables. Provisionally K(v,) will be accepted as a definition of the instrument function, but this definition will later be broadened to take account of the fact that the measured intensity at v, is also influenced by radiation transmitted at neighboring frequencies (p. 1028). The instrument function includes all the distorting factors involved in the measurement, but it is convenient to deal with the optical distortion separately from the electronic distortion caused by the recording system. The electronic element can be further sub-divided into one part introducing systematic errors (mainly the amplifier time constant) and a second part introducing random error associated with the signal to noise ratio. The optical factors will be considered first. * This concept of an inclusive correction function to take account of all instrumental variables is widely employed in current Russian literature under the name “Apparatnye funktsti”; it has been literally translated as “apparatus function”, and as such occurs also in the astronomical literature. WQ prefer the term “instrument function” since it is customary in English to describe a device for making spectrophotometric measurements as an “instrument” rather than an “apparatus”. [20] 8. G. RAUTIAN, Uspekhi Uspekhi 1,245 (1958). 12

B’iz. Sauk

66, 475 (1958).

English

translaCion, Soviet

Physics

1026

K.

(b) Optical distortion

s. SESHaDRI and It. NORMAN JONES

of the band projile

When a spectrophotometer scale is set to a selected wavenumber reading v,, a finite range of wavenumber is actually transmitted. This wavenumber band pass will have a characteristic intensity profile, though all the radiation will be recorded as though transmitted at a designated wavenumber. The wavenumber distribution of the energy emerging from the monochromator exit slit will depend on several factors; the most important of these are (a) the diffraction pattern

I

-U’

1/n

+ U’

Fig. 3. Diffraction pattern for non-coherent monochromatic rectangular slit.

radiation at a narrow

at the exit slit, (b) the finite widths of the entrance and exit slits and (c) the aberrations in the mirrors and lenses and the misalignment of the mirrors, lenses and slits. Before dealing with the complex effects that occur when these variables act simultaneously, the extreme cases where (a) or (b) acts alone will first be considered. (i) DiSfaction at the slits. If the mechanical slit widths are narrow in comparison with the wavelength, and the mirrors and lenses are free from aberration, the energy distribution in the exit beam will be determined by diffraction at the slit. The conventional text-book treatment of diffraction at a rectangular slit is developed in terms of illumination with non-coherent monochromatic radiation, and it is shown that if the radiation from the slit is brought to a focus, a diffraction pattern of the form shown in Fig. 3 is produced at the focal plane [al]. Since the fringe maxima are weak in comparison with the central maximum*, they can be neglected, and only the profile of the central band need be considered. The diffraction pattern shown in Fig. 3 is based on the assumption that the slit is equivalent to a self-luminous line source. The effect of using a real slit with

* For diffraction at a single slit, the ratio of the peak intensity of the nth fringe to that of the central fringe is (2/3~r)~, which corresponds to about 4.5 per cent for the first lat,eral fringe and 1 per cent for the second [21]. [21] R. A.

SAWYER, Experimental Spectroscopy, pp. 31-32.

Prent’ice-Hall,

Xew York (1944).

The shapes and intensities of infrared absorption bands

10’7

a finite width of the same order of magnitude as the wavelength has been investigated by VAN CITTERT [22]. If the illumination is non-coherent, the intensity of the central band is maximal when w = F/B, where zu is the mechanical slit width, F the focal length and B the limiting aperture of the collimator. For coherent radiation the maximum intensity is observed for w = 2F/B. In any real monochromator the optimal slit width corresponding to the most favorable combination of resolving power and light intensity will lie between these two extremes, since the illumination will be neither truly non-coherent nor truly coherent; the band profile will be similarly affected. Normally the illumination can be regarded as non-coherent, but appreciable coherency will occur if maser sources are employed. Let us first consider an idealized system using a line source. In terms of the instrumental reading, the abscissa1 scale of Fig. 3 can be replaced by the indicated wavenumber scale of the monochromator. If the instrument is scanned through the region of the central diffraction band, it will be evident from Fig. 3 that radiation of significant intensity will begin to register when the monochromator is set to a scale reading -v’; the intensity will increase to a maximum at v,, and diminish to a negligible value when the scale reads +v’. The half width of the central diffraction band (Av,) can then be expressed in units of cm-l on the wavenumber scale. In absorption spectrophotometry we are normally concerned with a related, but significantly different situation from the one described above. The slit is illuminated with continuous radiation, and the total intensity is recorded at a fixed wavenumber setting. It can readily be seen that, over a small range of wavenumber, a continuous source of uniform intensity will give rise to an infinite set of identical diffraction patterns, each of contour similar to that shown in Fig. 3, but displaced laterally with respect to one another. When the instrument scale is set at vi, some radiation of all wavenumbers from zli - v’ to v, + v’ will be transmitted, with an intensity that is approximately in inverse proportion to the displacement Iv’ - v,I. The contribution of the radiation at wavenumber v to the measured intensity at vi is a function of Iv - Y,I, which we shall designate s(v - vz) and ca,ll the spectral slit function. Where the slit is equivalent to an infinitely narrow selfluminous non-coherent source, it can be shown that sin2 [z-(v - v2)/Avi] s(v - VJ = a’ (36) (v - vJ2 in which a’ is the intensity at v, and Av, is given by v&/n for a grating spectrometer, in which n is the grating order and d the gratings spacing. For a prism instrument Avi is given by [p~~(d?z/dv)]-~ where p is the length of the prism base and n the refractive index. For finite slits of the same order of magnitude as the wavelength this relationship must be modified in accordance with the analysis of VAN CITTERT [22]. [22] P. H. VAN CITTEHT, 2. Physik. 65, 547 (1930); 1%. c'. LORD and J. R. LOOFBUROW, Practicnl

York (1948).

69, 298 (1931). r’5$ectroscopy,

See also G. R. HARRISON, p. 172, Prentice-Hall, New

102X

K. S. SESHADRI

and

R. NORMAN JONES

(ii) PI&her consideration of the instrument function. In equation (35) the instrument function was defined in terms of v, only. Equation (36) however, which expresses the contribution of slit diffraction to the instrument function, involves frequencies other than vi, and before considering further modifioations introduced by t’he use of wide slits, it is necessary to extend the definition of the instrument function. In Fig. 4 the profile of the true absorption band (Curve A) is represented by the function g(v). Considering an element of the spectrum centered about an abscissa1 value v’, it will be evident that the radiation transmitted at this wavenumber will contribute in some degree to the measured transmission over a wavenumber range determined by the instrument function, which we shall now write

Fig. 4. Convolution of the true band (Curve A) with the slit function (Curve B). The discussion concerns the conkibution of radiation of wavenumber v’ to the apparent absorption at v,. (After RAUTIAN [ZO]).

as /L(I)- vJ. The intensity contributed from v’ will be determined by Ji(v’ - v,), and it will also be proportional to the total energy transmitted at v’, and to the band width Av. We may therefore write 1;: = g(v’) . k(v’ -

vi) . Av

(37)

where I:‘, is the contribution of radiation of wavenumber v’ to the apparent absorption at v,. Since radiation of all wavenumbers for which k(v - vi) is finite will contribute in a like manner to the observed intensity at vi, the apparent absorption band profile f(vi) will be given by a function of the form f(vJ

= s__+~mY(v)uv-

vz) dv

(38)

Functions of this type are known as convolutions, and similar relationships are common in other branches of theoretical physics [ZO]. (iii) The effect of the $nite slit width. Because of the low intrinsic intensity of infrared sources, and the comparative inefficiency of infrared detectors, it is rarely

The shapes and intensities

of infrared

absorpt’ion

bands

1029

to operate an infrared spectrophotometer under conditions where the When the slits are spectral slit function is determined by diffraction alone. widened to permit the transmission of more energy, the relative importance of the diffraction term diminishes. First diffraction and later aberration effects cease to be significant, and the spectral slit function is determined purely by the geometrical properties of the slit image. With the monochromator set to the wavenumber Y,. and illuminated with an image of the rectangular entrance monochromatic radiation of wavenumber Y$, slit will be formed in the plane of the exit slit. The width of this image will be determined by the mechanical width of the entrance slit and the magnification of practicable

Fig. 5. Characteristic slit functions where the wavelength is negligible in comparison with the mechanical slit width.

the optical system. If this image is scanned by a narrow exit slit, then, in the absence of diffraction, a rectangular band pass system will be observed extending from vi - $Av, to v, + &Av, of width AvS, measured in terms of the monochromator wavenumber scale (Fig. 58). If the monochromatic image s1 (measured in cm-l) is scanned by an exit slit of width s2, the shape of the transmitted radiation band will depend on the relative magnitudes of s1 and s2. If s1 = s2 the slit function will be triangular with a half band width of Av, extending from v, - Av, to v, + Al!, as shown in Fig. 5B. If si # s2 the profile will be trapezoidal. It is illustrated in Fig. SC for the case where s2 > si. If s1 > s2 the geometry will be unchanged with a1 and s2 reversed in Fig. 5(c).

H.

1030

S. SESHADM and R. NORMAN JONES

If the entrance slit is illuminated with uniformly dispersed continuous radiation it follows, as for the diffraction case considered above, that the spectral slit function describing the frequency distribution of the energy transmitted at a fixed setting of the monochromator will have the same triangular or trapezoidal form. This may be expressed s(v -

v,) =

A-

81%

Iq -

(Iv -

A)]

(39)

for the trapezoidal case. The slit function will be zero outside the range

and if s1 > s2 it acquires a constant the trapezium where Yz If r1 = I’~, equation zero outside the range

&(8, -

s2) <: IV -

(39), reduces V, -

value of l/s, corresponding

Y,] < 11,+ $(S, -

to the triangular

8 G Iv -

to the plateau of

SJ

case, with the intensity

Y,] & v, + 5’

while within this range it takes the values s-I[1 - (IV - Y,])s-11. These expressions for the slit function assume that where the values are non-vanishing the area is normalized to unity, viz. + CC s(v - VJ dv = 1. (40) s -co In practice neither the finite spectral slit function nor the diffraction slit function is usually large enough to permit the neglect of the other, and it is necessary to deal with some appropriate combination of the two. Specific cases for grating and prism instruments will be considered later. (iv) Effects of aberration and misalignmen,t of the optics. Analytical espressions for the effects of mismatch of slit curvature, spherical aberration of the mirrors, and the presence of non-ideal prism surfaces have been discussed by BRODERSEK [23]. While it is not practical to take account of these individually, the over-all effect is usually assumed to be equivalent to a small apparent increase in the mechanical slit widths. Since the aberrational and other mechanical errors are independent of the frequency [23] they diminish the maximum resolving power by a constant amount, when measured in terms of wavelength, and therefore become proportionally more important as the wavelength becomes shorter. The slit mismatch will probably be the major misalignment error affecting the finite slit function. Where the entrance and exit slit widths are the same, the slit misalignment will round off the base and vertex of the triangular slit function, which will tend to assume more of a Gauss form. For a prism instrument of medium

[23] S. BRODERSEN, J. Opt. Sot. Am.

43, 577 (1953).

The shapes and intensities of infrared absorption bands dispersion,

PETRASH and RAUTIAN [24] have suggested a Gauss slit function

1031

of the

form* S(Y -

YJ = &

p!]1’2exp

[-f(:”

$

$1

(41)

where ZL’~is the mechanical slit width (both assumed equal) and ICis a constant term which is independent of wS, but which varies slightly with Y. In a more detailed analysis a distinction must be made between the aberration effects associated with the non-ideal opCca1 surfaces of the mirrors and the optical alignment on the one hand, and the figuring effects and mismatch of slit curvature on the other. The former are truly independent of frequency whereas the latter are not. (v) The experimental ,measurement of the spectral slit function. It follows from equation (38) that to correct the observed band intensity for slit distortion it is necessary to solve an integral equation, and this must be done separately for each point on the curve. The problem is further complicated by the fact that three variables are involved in the slit function and there is no simple method of evaluating their mutual perturbation. Theoretically the spectral slit function can only be formulated in general terms. In principle it is possible to obtain the spectral slit function experimentally by scanning a line source which is narrow in comparison with the width of the band. The observed contour of such a pseudomonochromatic source will approximate to the spectral slit function. In the ultraviolet and Raman spectrophotometry of condensed phase systems this technique can be used conveniently as atomic emission lines provide suitable narrow sources, well distributed across the spectrum. In the mid-infrared, the possible use of atomic emission lines for this purpose has not been fully investigated. There are potentially usable mercury lines at 1753.8 cm-r and 2533.9 cm-l [25], and recently HUMPHRIES [26] has reported several calculated lines in the spectrum of argon extending to 900 cm-l. Further work is needed to determine if some of these lines could be excited with sufficient intensity to make them useful for measuring spectral slit functions, and whether they have suitable profiles for this purpose. The extremely narrow line widths of maser radiation suggest that infrared maser sources could also be used in Ohis way. Both of the above proposals depend on future developments in technique. It may be possible to use the already known rotational fine structure lines of some gas-phase spectra for this purpose. There are narrow lines in the spectrum of hydrogen cyanide between 3360 and 3260 cm-l and in the carbon monoxide spectrum between 2200 and 3100 cm-l that might be used for grating spectrometers of medium resolution, though the separations between successive The band at 908.2 cm-l in the ammonia lines are only 2.5 and 3.5 cm-l respectively. * The symbolism is changed from t,hat used by the original authors. [Z-1] G. G. PETRASH and S. G. RAFTIAN, Iwhener.-Fiz. Zhw., 1,61 (1958). English translation, Natiomd Research Council of Canada Technical Translation No. 902. 1451 International Union of Pure and Applied Chemistry, Tables of Wavenumbers for the Calibration of Infra-red Spectrometers. Reprinted from Pure Appl. Spectrosc. 1, (1961) Butterworths (London). pp. 545, 574575, 584-585. Tra?as. [26] C. J. HUMPHREYS, Report to Commission 14 of the International Astronomical Union. 1.A.c’. Vol. XI (Berkeley Meeting). 1961. (In press).

1032

IL

S. SESHADRI and R. NORMAN JONES

spectrum has also been used, but t,here are theoretical objections to this, since it consists of three unresolved components. The line at 971.9 cm-1 would be theoretically more acceptable since it is a true singlet with only one allowed K-component 1271. There are also isolated bands in the spectra of some solid solutions with half widths of the order of 1 cm-l that are probably narrow enough to be used for measuring the spectral slit functions of small prism spectrometers [Z3, 291. The mercury line at 9859 cm-1 has been used by DMITRITEVSKY and NIKITIN [30] to measure the spectral slit function of a prism infrared spectrometer, but it is obviously preferable to determine the slit function closer to the wavenumber region of normal operation. The fundamental bands of HBr and DBr offer an attractive series of lines between 2750 and 1725 cm-1 with spacings of 11 cm-1 and 8 cm-l respectively [%I. Unfortunately each of these lines is an isotopic doublet. For the measurement of spectral slit functions it would be necessary to use monoisotopic species containing 79Br or *lBr only. These are technically obtainable but expensive. (vi) Calculation of the nominal spectral slit width for a grating spectrometer. Since both theoretical and experimental difficulties complicate the evaluation of the spectral slit function, approximate methods of computing the spectral half band width are commonly used, and this is accepted as a measure of the spectrometer resolution. It is reasonable to assume that if two spectrometers can be so adjusted that their spectral slit functions have the same half band width, the effect of some difference in the profile of the slit function will not significantly disturb the observed band shape, and dat,a interchange between the two spectrophotometers will be facilitated. The over-all spectral slit function half band width (measured in cm-l) will be some combination of three components contributed by the finite slit function (ss), the diffraction slit function (So), and the aberration effect (s_~).* 8 = f(Wo~_J

(42)

It has generally been conceded that a simple arit,hmetical addition of the three contributing quantities will give an unrealistically large value for s, and a number of empirical equations have been proposed for dealing with this problem. In the published literature most attention has been given to the calculation of nominal spectral slit widths for prism spectrometers, but we shall deal with the grating instrument first. The following method, which has been used in our laboratory for computing the nominal spectral slit width of a Perkin-Elmer Model 112G instrument, is due to SIEGLER [31]. * In the subsequent discussion “aberration” unless specifically indicated otherwise.

will be assumed to include “misalignment,”

[27] N. SHEPPARD. Personal communicabion (1962). [28] A. MBI and J. C. DECIUS, J. Ch.em. Phys. 28, 1003 (1958). [29] H. W. MORGAN and P. A. STAATS, Papers presented at the Symp. Molecular Structure and Spectroscopy, Colunabus, Ohio, 1961 (Abstracted in Spectrochim. Actu 1'7, 1121, 1961). [30] 0. D. DMITRIPEVSKY and V. A. NIKITIN, Optika i Spektroskopiya 8, 117 (1960). English translation, Opt. and Spec. 8, 58 (1960). [31] H. SIEOLER, Personal communicat,ion (1959).

The shapes and intensities of infrared absorption bands

The first term (sS) can be calculated ss =

from geometrical

1033

optics and is given by

v2d cos 0 ATnF ws

(43)

where LVis the number of grating passes, n the order of the spectrum, F the focal length of the spectrometer, d the grating spacing, 8 the diffraction angle and ZL’~ the mechanical slit width (both slits being assumed equal). The contribution of s,~to s is taken to be independent of sD and sg, but the latter two are not independent of each other. It has been noted above that the effect of aberration on the diffraction term will increase at the higher frequencies. Both aberration and diffraction produce a virtual widening of the mechanical slit width and a root mean square proportionality has been assumed. An effective mechanical slit width (20,~) can therefore be defined as 2l’,lr = ZL’,q+ [zou” + zuAz]l’z

(44)

in which w,, and uj, are the virtual mechanical slit widths derived from the diffraction and aberration terms. On substituting ~7,~ for uls in equation (43) an expression for the nominal spectral slit width of a grating spectrophotometer is obtained.

8 =

v2d cos

Nnl”

e * Il’eff

(45)

Before equation (45) can be applied in practice, it is necessary to express w=, w, and cos 8 in terms of more readily available parameters. The virtual mechanical slit width of the diffraction term (wD) is equal to F/BY where B is the limiting aperture of the spectrophotometer, and sin 0 = n/2vd. The paucity of narrow infrared line sources make it difficult to evaluate w,, and in our laboratory we have accepted an arbitrary value of 40 ,u for ws. PETRASH and RAUTIAN have used values between 10 ,u at 2000 cm-l and 40 p at 666 cm-l for the similar term K in calculating the spectral slit function of a prism spectrometer (equation 41) and some value in the range 10 ,u-50 p would appear reasonable [24]. Making these substitutions in equation (45) we obtain s

=

& [l - (T-J?‘:”(ws + [gj2 + wq2)

(46)

The values obtained for w,~, ws and [wn2 + wA2]l12,computed for the Perkin-Elmer 112~2instrument using a 750 line/cm double passed grating operated at a nominal spectral slit width of 1 cm-l, are given in Fig. 6. These are representative of the relative magnitudes of these terms under normal operating conditions for a medium sized grating spectrometer. (vii) Calculation of the nominal spectral slit width for a prism spectrometer. The slit-width computation for the prism instrument could be developed in a manner analogous to that for the grating instrument described above; the treatment given below differs in some respects, principally in regard to the incorporation of

K.

1034

S. SICSHAURIand R. ~oRn&N

JONES

Fig. 6. Estimated contributions of the mechanical slit x?dth (w,), the virtual diffraction slit width (wD) and the virtual aberration slit width (w_~)to the affective mechanical slit width (zo,~) for s representative medium-sized infrared grating speetrophotometer.

the aberration suggested

term, and is based on the method proposed

by ~1~~~~~~s

[3%] who

s I= s&q4 JYq&r, where F(s,) is a function of s,~. For a single traverse of a prism at minimum

(47) deviation,

we may write

(48) where A/I is the spectral slit half band width in wavelength units, &I/d0 is the dispersion of the prism, wr and w2 the mechanical widths of the entrance and exit slits and F the focal length. The angular dispersion of the prism can be shown to be dn -= $0

[l -

nz sin2 (er/2)]l12

(49)

2 sin (~$2)

where a is the apical angle of the prism [33] and n the refractive prism material. Remembering that (a) dl

a2

index

of the

dn

de =2&B (b) Av = v2 Ail and ( G) in the Littrow

mountings

[32] V. Z. WILLIAMS, Rev. Sci. Iwtr. 19, 135 (1948). 1331 R. A. SAWUER, ~~~~r~?~e~~tal s~ectr~~cop~, p. 57.

the ray traverses the prism twice,

Prentice-Hall,

New York (1944).

1035

The shapes and intensities of infrared absorpt,ion bands

n-e obtain on combination

of equations

*ss= v

[1 -

.n2

(47) and (49) w1 + w2

sin2 (a/2)]l12

8 sin (a/2) . dn/d;l

*

(50)

F

which is the expression proposed by WILLIANS for the finite slit term of equation (47). The diffraction term is given by

(51) where p is the length of the prism base. The proportionality function F(s,) was not specifically defined by WILLIAMS, but was stated to diminish from 0.9 to 0.5 as ss increased from zero to sD. Subsequently [34] WILLIAMS modified the term for ss by substituting uj for (wl + w2)/2, where 2~’is the width of the wider of the two slits; an additional aberration term was introduced by JONES and SANDORFY [35]. Taking account of these changes, writing v2(dn/dv) for (dn/dA) and introducing AV for the number of complete traversals of the optical system, we obtain for the final expression s = 11 GY

n2 sin2 (a/2)]1’2

sin (a/2)(d,n/dv)

. ‘”

F

+ F(s,J

1 EVpv(dn/dR)

+

a.4

(52)

The effect of a departure from the conditions of minimum deviat,ion has been considered by VON KEUSSLER [36].* It, has been customary in practice to choose an arbitrary constant value for F(s,), and to disregard the aberration term. The Coblentz Society, in its specifications for the publication of analytical matrices for quantitative infrared spectrophotomebry [37] has recommended various values for F(sS) ranging from 0.5 to 1.5 for different types of commercial instruments. RUSSELL and THOMPSON [38] have used F(sS) = 0.85, while JOYES and SAXDORBY [35], take F(s,) = 1, arguing that a small over-evaluation of F(s,) will compensate for the neglect of s,. Plots of ss and sg for some instruments are published in the manufacturers’ service manuals. The relative magnitudes of s,ss and sD for a Perkin-Elmer Model112 spectrometer under the normal operating conditions of our laboratory [39] are shown in Fig. 7. The difference introduced by taking F(s,) = 0.85 instead * There is a difference of a factor of 3 between the formulae of VOX KEUSSLER and of WILLIAMS; this would appear to be due to the fact that VON KEUSSLER considers a single traverse of the prism instead of the double traverse inherent in the Littrow optical system. The Vow KEUSSLER formula would apply directly to the Wadsworth mountred prism but not to the Littrow. [34] V. Z. WILLIAMS, see p. 274. of Ref. [35]. [35] R. N. JONES and C. SANDORFY, The Application of Infrared and Raman Spectrometry t,o the Elucidation of Molecular Bructure, Technique of Organic Chemistry (Edit,ed by A. WEISSBERCER) Vol. IX Chap. 4 p. 374. Interscience, New York (1956). [36] V. VON KEUSSLER, Opt% 13, 317 (1956). [37] Coblentz Society Mailing Circular No. 16, 10 November (1960). [38] R. A. RUSSELL and H. W. THOMPSON, Spectrochim. Acta 9, 133 (1957). [39] R. X. JONES, Spectrochim,. Acta 9, 235 (1957).

Ii.

1036

8. SESHADRI and R. NORMAN JONES

Fig. 7. Comparison of the contributions of the finite slit width (sS) and the diffraction term (so) to the calculated spectral slit width (.Y)for a representative infrared prism spectrophotometer.

Fig. 8. The effect of different methods of combining the spectral slit and diffraction terms on the evaluation of the nominal spectral slit width for an infrared prism spectrophotometer.

of unity for the Perkin-Elmer model-21 spectrophotometer with a sodium-chloride prism is shown in Fig. 8. The corresponding curve is given also for s = [Ss2 + SD2]l’2 which has been used by BRODERSEN [40] I. V. PEISAHHSON, 8, 57 (1960).

[23]

Optika ib’pektroskopiya

and by PEISAKHSON

(53) [40].

8, 116 (1960), English translation, Opt. alAd Spec.

The shapes and intensities of infrared absorpt,ion bands

1037

The introduction of the aberration term s, as a constant contribution to s is not very satisfactory, as it will become relatively more important as the mechanical slit width becomes narrower. It might be more realistic to introduce the aberration term int’o the El(sS) function, or to combine it with the dispersion term by a root mean-square addition. In view of the present arbitrary state of spectral slit width calculations, any reported values for s should be clearly specified as “nominal” and the equation used to compute them should be stated. (viii) The experimental measurement of the spectral slit width. As other sources of error in infrared spectrophotometry are reduced the precise evaluation of the spectral slit width becomes of increasing significance. The difficulties caused by the lack of infrared line sources have been noted in connection with the measurement of the spectral slit function. The requirements as to the line width are not quite as stringent for spectral slit width measurement as for the evaluation of the According to BRODERSEN [41] complete slit function. s:xp =

ss2 +

sD2 +

(A&J2

(54)

observed half width of a line or narrow band of where s,,~ is the experimentally true half width Aviie. If the line is scanned over a range of varying mechanical slit widths, sCXpcan be measured and sS calculated from equation (43) or (50). A plot of s&, against ss2, with s,~~as ordinate, should yield a straight line of unit slope with a positive ordinate intercept corresponding to sD2 + ( Av~,,)~, from which A su2 can be obtained if A&, is known, or if it is small enough to be neglected. separate aberration term is not included in equation (54), but if s, is added su2 + sA2 will be obtained from the extrapolation. If Av is the distance between two narrow lines that are just resolved, BRODERSEN has also suggested that 1.25(9~)~

= sR2 + sD2 + (Av~,,..)~

(55)

provided s, < Av < 2.9,.

Another method, based on the measurement of interference fringes, using an empty cell of good optical quality as a Fabry-Perot etalon has been described by COATES and HAUSDORFF [42]. (c) Electronic

distortion of the band pro$le

The finite time constant of the amplifier is the principal cause of the electronic distortion in the recording system. This has been discussed by BRODERSEN [43], ABRAMSON and MOGILEVSKI [4-j] and DMITRIYEVSKY et al. [G], among others. The treatment developed here is based on that of BRODERSEN. The output signal of a thermocouple steady,

and the desired output

or other thermo-electric

signal is superimposed

detector is never

on noise.

The limiting

[41] S. BRODERSEN, J. Opt. Sec. AWL 44, 22 (1954). [42] V. J. COATES and H. HAUSDORFF, J. Opt. Sec. Awz. 45, 425 (1955). [43] S. BRODERSEN, .I. Opt. Sot. Am. 43, 1216 (1953). [44] I. 8. ABRAMSON and A. T. MOGILEVSKI~, Iwest. Akad. NaukS.S.S. R., ser. Fiz. 19, 49 (1955). English translation, B,ulZ. Acad. Sci. U.S.S.R. 19, 48 (1955) (Columbia Technical Translations). [45] 0. D. DMITRIYEVSKY, B. S. NEPORENT and V. A. NIKITIN, Uspekhi, Piz. Nauk 64, 437 (1958). English translation No. 3.4 Infosearch Ltd., London, England.

B.

1038

S. SESHADRIand R. NORMANJONES

source of noise originates in the thermal motions of the electrons, and the magnitude of the noise signal depends on the band pass frequency range of the amplifier. Electronic filters effectively exclude noise signals that are outside of their band pass frequency range. In doing this, however, they distort the desired signal, since it also contains a.c. components of various frequencies, and the filter network attenuates some of these more than others. Granted that the inclusion of a filter network is necessary, it is desirable to know the manner and extent to which it distorts the spectrum. The function describing this distortion is the contribution of the filter system to the instrument function of t’he spectrophotometer.

Fig. 9. The RC filter net,mork.

(i) The time constant of an a.c. network. We shall consider here only the simplest case of the RC network, which qualitatively demonstrates the character of the distortion introduced by the electronic filter system. More detailed treatments will be found in text-books on communications engineering, and particular note might be taken of the articles by TOMLINSON on “Noise” [46], and by ZEPLER on “Amplifier

Design and Fault Finding”

[47].

The basic element of a low-frequency band pass filter is the resistance-capacit)ance circuit shown in Fig. 9. Since the dimensions of capacitance are p-ll-92, and of resistance pit-1, where p is the magnetic permeability in vacua, it follows that the product RC has the dimension of time: it is a measure of the time taken

for the capacitive circuit to re-establish a steady state when the input voltage is changed. If the input voltage T’, varies with time, we may write I’, =f(t), and it can be shown t’hat the output voltage (V,) after a t’ime interval (t - tl) is given by

I’, If F, is directly

=

s

kc exp

tIf(tl)

proportional

[-

J$J]

to the radiation 1

RC

dt,

incident

(56)

on the detector,

0 - tl)

expRC [

1

[46] T. B. TOMLINSON, Article on Noise in EZectronicsforSpectroscopists

(Edited by C. G. CANNON)

Chap. 7. Interscience, New York (1960). [47] E. E. ZEPLER,Article on Amplifier Design and Fault Finding in Electrorlics scopists (Edited by C. G. CANNON) Chap. 8. Interscience, New York (1960).

for Spectro-

1039

The shapes and intensities of infrared absorption bands

is a measure of the distortion introduced into the recorded spectrum by the filter network, i.e. the contribution of the network to the instrument function. It follows from equation (56) that the time/voltage profile of the output signal will approach that of the input signal as the time constant of the network diminishes with respect to the half width of the input signal pulse. An analogy with the relationship between the spectral slit width and the true band shape is apparent here. To minimize the filter distortion, RC/Gt must be made as small as possible,

IO-

-

0.8

-

0.4

-

-2

-I

0

I

2

3

4

WAVENUMBER

Fig. 10. The distortion introduced into a Gauss band profile by an RC’ filter network. J is the ratio of the input pulse width (at) to the filter time constant (T). (After BRODERSEN [43]).

where dt is the half width of the input pulse in seconds. This pulse rate will depend on both the spectral band width and the rate of scan (dv/dt) so that

A42

dt = __ dvldt

(57)

where Av;,~ is the half width of the absorption band as modified by convolution with the spectral slit function. (ii) The effect of the time constant on the band shape. BRODERSEN [43] has investigated the quantitative effect of the time constant for an emission band with a Gauss profile. His results are shown in Figs. 10 and 11 where the ordinate is proportional to the output voltage and the abscissa to the wavenumber. The particular scale units were chosen to simplify integration, as also was the selection of a Gaussian emission band rather than a Lorentzian absorption band, which would have had more direct relevance to our immediate problem. In the series of curves shown in Fig. 10, J is the ratio of the input pulse width (at) to the filter time constant. The important conclusions to be drawn from these curves are qualitative, and they would apply in varying degree to other types of filter networks.

K.

1040

s. SESHADRI

and K. NORMAN

JONES

It will be observed that whereas the spectral slit function normally induces a symmetric distortion of the true band shape *, that induced by the finite amplifier time constant is asymmetric. Increase in the scanning speed diminishes the relative height of the output voltage peak; it also broadens the band and displaces the band maximum in the direction of scan.

06

02

-3

-2

-I

0

I

2

3

WAVENUMBER

Fig. 11. The data from Fig. 10 corrected for lateral displacement of the band maximum, assuming that the displacement is proportional to RC . St. (After BRODERSEN [43].)

The displacement of the band maximum in the forward direction is approximately proportional to RC . at, provided this ratio is large. If this first-order filter distortion is allowed for, the second series of curves, shown in Fig. 11, can be computed. These illustrate more clearly the second-order asymmetric filter distortion effect for which there is no simple method of correction. Where the hrue absorption band is approximately symmetric, this type of distortion is not difficult to recognize, but if the true band is itself asymmetric, the filter distortion may A method by which the filter distortion can be evaluated easily go undetected. quantitatively and separated from the true band asymmetry will be discussed on p. 1054; this requires that the spectrum can be scanned in both directions without other change in the experimental variables. From the above discussion it is to be concluded that it is rarely possible to avoid the use of a filter network in infrared spectrophotometry, and since the second-order asymmetry cannot be completely eliminated or easily computed, care must be taken to keep it small by always operating under conditions where * In the discussion of the spectral slit function (p. 1037), it was assumed that the continuous incident radiation has uniform intensity over the finite frequency range involved. This may not be true under all circumstances, as, for example, in the edge of an atmospheric water-vapor If t,he incident radiation intensity is not absorption band in an unpurged spoctrophotometer. uniform, the slit function will also bo asymmekic.

The shapes and intensities of infrared absorption bands

Av& > -r(dv/dt) where T replaces of the filter network. *

1041

RC as a more general symbol for the time constant

(d) illechanical distortion of the band projile Inertia in the moving parts of a spectrophotometer and other imperfections in the mechanical system may also introduce distortion in the band profile, though these should be negligible in a properly adjusted instrument. They will tend to be more prevalent in double-beam than in single-beam instruments in view of the greater mechanical complexity of the mechanism. The inertial lag due to the masses of the pen carriage, the potentiometer balancing systems and the servocontrolled attenuator systems form an integral part of the servo-system. The distortion this introduces is usually considered along with the filter network distortion, but it has been discussed separately by GRIBOV [48]. Qualitatively the combined effects of these mechanical factors will resemble the filter network distortion, causing a displacement of the peak in the direction of scan, and an asymmetric distortion of the profile. The magnitude will depend on the ratio of the scanning rate to the half band width. It will not be dependent on the amplifier time constant; it should therefore be detected by scanning a band at different rates while adjusting the time constant so that T . dvldt remains constant. The band profile may also be affected by mechanical errors caused by nonlinearity in the recording system. In a single-beam instrument this could arise from non-linearity in the recording potentiometer, but this is unlikely to be significant unless bhe recorder is seriously out of adjustment. It can be checked by calibrating the recorder with an external signal supplied by a suitable function generator. The corresponding problem is more difficult to deal with in a double-beam null-type spectrophotometer where the precision of the mechanical attenuator, usually a comb, is involved. This can deteriorate deceptively as a result of minor mechanical damage or the presence of dust. GALLAWAY et al. [49] have considered this problem, and have emphasized that a systematic error in the linearity of the mechanical attenuator cannot be detected from an internal set of measurements on the spectrophotometer, and an absolute intensity calibration against an external standard must be employed. The conditions operative in an ideal linear transmittance attenuator are shown diagrammatically in Fig. 12, where the comb profile is represented as a single triangular transverse slot. The comb positions corresponding to total absorption (II,,), zero absorption (I,) and an intermediate absorption (I,) will be proportional to lateral displacements of the comb by d,,,, d, and d, from a reference position G. * Note also that similar effects would occur if the time constant of the detector were large in comparison with the pulse width. In practice however it is the electronic filter system that poses the major problems. [48] L. A. GRIBOV, Opt&a i&ektroskopiya 8, 133 (1960). English translation, Opt. andSpec. 8, 61 (1960). [49] W. S. GALLAWAY, W. KAYE and J. E. STEWART, Lecture at Fisk University Summer School, August (1961). (Unpublished). 13

1042

H. S. SESHADRI and R. NORMAN JONES

I

G Fig. 12. The effect of non-linearity in the attenuator comb on the measurement of fractional transmission in a nulltype infrared spectrophotometer.

If L is a proportionality

constant,

then

(58) and for absorbance

=logk

+1og

(do [

400)

(do - 4) 1

(59)

If the attenuator profile is non-linear, as indicated by the dotted line in Fig. 12, it may be approximated by a logarithmic function such that I, = L(d, - d,)’ and equation (59) becomes

:_o =

(do - 400) logi + T log (‘30) ( Zi [ It will be seen that both equations (59) and (60) yield linear relationships between the absorbance and the attenuator comb displacement though they differ plates or a in slope with the magnitude of r. A set of standard transmittance sector photometer [50] provides the necessary reference data from which the If the exponent is constant over the whole range of the slope can be determined. attenuator motion, the y ordinate will be expanded or contracted uniformly, but if the exponent varies significantly through different ranges of the attenuator transversal, the effect on the instrument function will be more complex, though the symmetry of the absorption band should not be changed. A similar effect can be caused by non-linearity of the detector sensitivity; this has been discussed for photo-electric detectors by CANNON and BUTTERWORTH [Gl]. log

(e) Errors

(do - 4) 1

camed by scattered radiation and related effects

In the older types of infrared spectrophotometers, scattered radiation energizing the detector was one of the most serious causes of error in the absorbance measurement; it normally increased rapidly as the wavenumber diminished. In [50] J. E. STEWART, A&. Opt. 1,75 (1962). [51] C. G. CANNON and I. S. C. BUTTERWORTH, _4naZ. Chem. 25, 168 (1953).

The shapes and intensities of infrared absorption bands

1043

the design of more modern instruments particular care has been taken to reduce the level of scattered radiation by the use of double monochromators or filters and narrow band pass chopped-beam detector systems using the Walsh principle. If these instruments are in proper adjustment, transmission errors due to scattered radiations should be less than 1 per cent under normal operating conditions. In grating instruments an effect similar to that caused by scattered radiation will be observed if the order separation is imperfect; this may be caused by improperly aligned foreprism optics or defective interference filters. The performance of a grating spectrophotometer with respect to scattered radiation and order overlap should be tested periodically. This can be done by measuring the apparent transmission through intense absorption bands at sample thicknesses that are great enough to produce virtual total absorption at the nominal wavenumber Any apparent transmission can then be attrisetting of the spectrophotometer. buted to spurious radiation. The bands listed in Table 1 have been used for this purpose in our laboratory under the conditions specified [52]. Table 1. Experiment,al determination Wavenumber (cm-l)

range

Path length Substance?

______ 3620-3600 3335-3260 3075-3045 2460-2380 1760-1675 1590-1420 1240-1190 ‘1125-1120 940-880 800-740 700-645 (820

of stray radiation*

-__

____-

(mm) 20 0.5 1 10 0.5 1 1 1 1 0.1 0.5

Phenolt Phenylacetylene Methylene dichloride Chloroform Acetone Carbon disulfide Chloroform Tetrachloroethylene Tetrachloroethylene Carbon tetrachloride Benzene

4.5

Calcium fluoride9

* Unpublished measurements of B. V. KARTHA. All measurements on liquid films unless otherwise indicated. $ 0.05 M solution in carbon tetrachloride. 3 Polished plate. t

If spurious radiation is present, the absorbance will be given by log [(T, - k)/ (T - k)] where k is a measure of the stray radiation. If T and T, are expressed as percent transmission and k is the percentage of stray radiation, the corrected absorbance will be given by log [(lo0 - k)/(T - k)] and correction tables for various values of T and E have been published by OPLER [53].* * A more precise correction is given by log [(T, the intensity of the stray radiation [54].

-

k’)/( T -

[52] B. V. KARFHA and R. N. JONES. Unpublished observations [53] A. OPLER, J. Opt. Sot. Am. 40, 401 (1950). [54] W. S. GALLAWAY. Personal communicat,ion (1962).

k”)] where k’ and k” depend on (1962).

1044

K. S. SESHADRI and R. NORMAN JONES

Errors of a somewhat similar character can occur if temperature differentials exist between various parts of the spectrophotometer. These can be particularly serious where spectra are measured at temperatures that depart appreciably from the ambient temperature of the instrument; the temperature of the rotating sector or shutter may then differ appreciably from that of the sample, and this will introduce a temperature variation at the detector pulsed to the responsive frequency of the amplifier. In this way a cold sample may induce a false zero transmission reading on the negative side of the true zero transmission reading. This error is most liable to occur where the rotating sector or mirror modulating the radiation is located between the sample and the detector. (f) Quantitative relationships

among the spectrophotometer

variables

In the measurement of an infrared spectrum the selection of the optimum operating conditions involves the choice of slit width, scanning speed, filter time constant and signal-to-noise ratio. These variables are not independent, and, in practice, the determining factor will usually be the maximum obtainable resolution consistent with an acceptable scanning speed and signal to noise ratio. The considerations entering into the choice of the optimum signal to noise ratio will be analysed in a later section on the basis of the theory of errors, but it will be useful here to summarize the inter-relationships among these quantities; the following treatment is due to DALY [55]. If V,, is the steady output voltage of the amplifier, and N the peak-to-peak noise voltage, the signal to noise ratio (R)is given by V,,/N. The output voltage varies as the amplifier gain, and as the radiant energy (E) passing through the exit slit; E in turn depends upon the square of the spectral slit width (s). According to equation (4~) s involves ss, sD and sA, but the two latter quantities are small, and for present purposes s can be taken as directly proportional to the mechanical slit width wS. We may therefore write

E ocw,~~

(61)

The peak-to-peak noise voltage (N) varies inversely as the square root of the filter time constant (T), and to maintain a given noise level, the time constant must vary inversely as the scanning speed (dv/dt), whence 1 - oc N2 7

dv

(62)

1 (63)

dtw7 so that

[55] Unicam Instruments photometer.

N2 Ltd.

Cambridge,

a;

England.

(64)

Service Manual

for S.P.

100 spectro-

The shapes and intensities of infsared absorption bands

From

operating

the following equations regulating condibions are readily derived :

(61-64)

1045

the choice of the instrumental

R = const. w,s2(dv/dt)-112

(65)

~7,~ = Gonst. R1/2(dv/dt)1/4

(66)

dV

z=

con&. ~~~~~~~~

(67)

Theoretically it would appear possible to achieve high resolution at low noise level by lengthening the time constant and scanning sufficiently slowly, but equation (66) shows that if R is to be mainbained constant, the saanning speed must vary as the fourth power of the slit, width. Therefore if the spectral slit width is halved the scanning speed must be reduced by a factor of sixteen, and the rate soon becomes impracticably slow. IV. THE DETERMINATIOK OF THE TRUE BAND SHAPE FROM THE APPARENT BAND SHAPE The ultimate objective of infrared spectrophotometry is to arrive at the true band shape and band intensity by correcting the experimentally observed band for the distortions introduced by the recording and measuring apparatus. The discussions of the preceding sect’ion have shown that our knowledge of these distortions is imperfect, and at best only semi-quantitative. Until more exact mathematical expressions can be obtained for the various elements of the instrument function, the derivation of the true curve from the experimental curve cannot be carried out analytically; it is necessary to be content with a formal statement of the relationships among the quantities involved, or to resort to numerical approximat,ions. The basic problem is to solve the integral equation f(vl)

=

[+Dg(v)l;(~

*--m

-

YJ dv

for g(v), u-here f(vZ), g(v) and Ji(v - vz) are respectively the true band profile and the instrument function. (a) Analytical

solution

for g(v) by

(‘38) the observed

band profile,

Fourier analyysis

If analytical expressions could be derived for f(vI) and k(v - vJ, equation (68) could be solved for g(s) by a Fourier transformation. This has been discussed by RAVTIAN [20] and the results only will be considered here.* The Fourier transform of a function f(x) is given by the integral 1 F(w)

=

(.27T)1/2

s

’ mf(z) _-ar

exp (ioz) dz

(69)

* Those unfamiliar with the theory of Fourier transformations will find a succinct account in Ref. [Xl. [56] S. FICH, Transient N.J. (1951).

A4naZysis in Electrical

Engir~eering

Chap. 10.

Prentice-Hall,

Englewood,

1046

and, inversely,

B.

S. SESHADRI

and It. NORMAN

the Fourier transform

JOXES

of F(w) is given by

+m 1 P(U) exp (--iox) = (2,)1/Z s _ m

&)

dco

(70)

In the current problem, the transformation involves multiplying both sides of equation (68) by exp (itox) and integrating from - 00 to + co. From this we obtain B’(W) = (27~)~‘~G(o)K(w)

(71)

whence

1 E”(o) ___-

cw

(72)

= (27T)1’2K(m)

where P(w), G(s) and K( cc)) are the Fourier transforms of ~(YJ, g(y) and k(~ - vi) respectively. Since +Oa 1 G(m) exp ( --iwv) &I (73) _ o. &)1/Z

I

dy) =

we obtain

g(y) =

$

s-

+03I+) exp ( -kov) K(w)

-m

dw

(74)

as the desired solution. From equation (71) it is to be deduced that each harmonic component of the true band, with frequency COand amplitude G(w), is changed by the instrument function into the same harmonic component of the observed band, but with a modified amplitude, determined by K(W), As an illustration the case will be considered where both the experimentally observed band and the instrument function are Cauchy distribution curves,* in which the band center is at Y,,, and the half band widths of the experimentally observed band and of the instrument function are AY, and Av, respectively, so that

and k(v where

and

are normalized

to unity.

* See footnote to p. 1023.

(76)

The shapes and intensities of infrared absorption bands The Fourier transforms

1047

of these are

F(m) :

1

(2,)l,2exp

(77)

( --oAvf)

and 1

By equations

(78)

(72) and (74) we obtain 1 G(w) = (2,)l,e exp [-(Avr

-

AQ)O

(79)

and 9(y -

y0) = & =-

+r. exp [ -(Av~ I-n

1 27-r(Y -

Avr -

-

Av&]

exp [ --~w(Y -

Av,

vJ2 + &Av, -

YJ] do

(80)

A2.‘J2

The interesting conclusion is therefore reached that if the observed band profile and the instrument function are both of Cauchy form, the true band profile will also be of Cauchy form, with a half ba,nd width such that A%(C) = AvI -

Av,

(81)

It can be shown in a similar fashion that if the observed band profile and the instrument function are both of Gauss form, the true band profile will also be Gaussian with (82) A%(O) = [( AvI)z - ( AvJ~]~/~ where Avrrco,is t,he half width of the brue Gauss band profile. (b) 0urztes of mixed Gauss and C’auchy character-The

Voigt pro$le

It will be apparent from the above discussion that the derivation of the true band shape from the experimentally observed band shape presents no serious difliculty where the convoluting functions are both of Gaussian or Cauchy form. Indeed, the relationships typified by equations (80-82) are not limited to binary convolutions, but can also be extended to more complex cases, as for example the convolution of a Gaussian true band profile with a Gaussian slit function and a Gaussian amplifier filter fun&ion. * Unfortunately, this simple situation will seldom arise in infrared spectrophotometry. In the absence of factors leading to band asymmetry, however, the true band shape may well conform to some profile of a character intermediate between * In the Russian literature the word “Reduktsiya” is widely used to describe the “deconvolution” process, and has been translated as “reduction”. The use of “reduction” in this sense is accepted practice in astronomy, but the word has been avoided in this review since this technical meaning of it may be unfamiliar to many chemical spectroscopists and liable to mis-interpretation.

K.

1048

S. SESHADRI and R. NORMAN JONES

that defined by the Cauchy and Gauss functions (cf. pp. 1023-1065). The slit function and the first-order electronic filter distortion will combine to yield an instrument function that is approximately Gaussian (cf. equations 41 and 56). Situations may therefore arise where f(vi), g( v ) and k(v - vi) could each be treated in reasonable approximation as a mixture of a Gauss and Cauchy function, with the Cauchy component probably predominating in g(v) and the Gauss in k(v - v’).

0.5 -

B

Fig. 13. Comparison of the Voigt function profile for xl/x2 = 0.6 (Curve B), with the profiles for the limiting cases corresponding to the Cauchy function (Curve A) and the Gauss function (Curve C).

These circumstances have long been recognized in connection with line shape analysis in atomic emission spectra, and have been dealt with effectively by a function developed originally by VOIGT [57], which results from the convolution of a Gauss and Cauchy curve

f(vJ = f

J+mexp [- (’,:““I.x --3c

1 Z-

s+m

2

+

(:I_

exp L-b - vJ2/x221

=x, --m

vo)2

dv

1

dv

1 + (v - VcJ2/X12

(83)

where v0 is the band center, and x1 and x2 constants. The ratio x1/x2 determines the proportionality between the Cauchy and Gauss characteristics of the profile such that if x1/x2 = 0, equation (83) reduces to a pure Gauss curve with half width Avo = 3(ln 2)lj2x2. If x1/x2 = 00 a pure Cauchy profile is obtained with half width, Avc, given by 2x1. For a given total area, rZ, such a Voigt function is completely defined by A, x1 and x2, and an example is illustrated in Fig. 13, for given values of the peak height (a’) and the half width (h). Tables from which such profiles can be computed for various values of xl/h, x1/x2, x,/h and x22/h2 have been published by VAX DE [57] W. VOIQT, Miinch. Ber. 603 (1912).

The shapes and intensities

of infrared

absorption

bands

1049

HULST and REESINCK [18], based on the computations of BORN[58], HJERTDW [59] and MINKOWSKI and BRUCK [SO].* The Voigt function has considerable versatility, and its use to approximate band shapes and instrument functions is enhanced by the fact that the convolution of two Voigt functions, individually characterized by x1’, x2’ and x1” and xZn, is also a Voigt function in which Xl = Xl' + Xl" X2 =

C(X2Y2 + (xJ2Y

(84)

Furthermore it can be shown by Fourier analysis that the convolution of several non-Voigt functions yields a curve that increasingly approximates to a Voigt profile as the number of convoluting components increases [18]. (c)

The ch.aracterization of band profiles by truncated moments

The three band profiles illustrated in Fig. 13 all have the same peak heights, and half band widths, and serve to illustrate that these intensity parameters are insufficient to define the profile of a symmetric absorption band. In practical infrared spect,rophotometry it has not been customary to describe the band profile in greater detail tha,n can be achieved with these quantities, but, with a more precise description of the profile increasing precision in measurement, becomes desirable. In practice it will also be found that only rarely are infrared bands symmetric about the maximum absorbance ordinate, and a quantitative evaluation of the skewness is also needed. These are sta,tistical problems, and can conveniently be handled by the method of truncated moment analysis. This is a well established technique, particularly for curves of Gaussian or near Gaussian form; its application to the analysis of infrared band shapes has been described recently by JONES et al. [Sl] and only the general principles will be considered here. For any distribution function that extends from - co to + co, passing through one intermediate maximum, a set of moment functions can be defined, where the rth moment of the point (x, y) about an ordinate at x’ is given by (x - x’)‘y (Fig. 14). In principle the moments may be taken about any point on the curve, but, for an absorption band with ordinates in absorbance units and abscissa in wavenumber units it is advantageous to take moments about the maximum ordinate (v,,) whence the following expression is obtained for the rth moment: PU’=jj

l I+ - Ye _-2m(Y

log (21

”

. dv

(85)

* VAN DE HULST and REESINEE use /5’in place of ,y in their tables of Voigt parameters; x is used here to avoid confusion with the statistical quant,ities ,BI and pz introduced in the following section.

1551M. BORN, Opt&. p. 482. Springer, Berlin (1933). [59] F. HJERTIHG, Astropkp. J. 88, 508 (1938). 1601 R. MINKOWSKI and H. BRUCP, 2. Pkysili. 95, 299 (1935). [61] R. N. JONES, K. S. SESHADRI, S. B. TV. JONATHAN and J. IV. HOPICINS, Can. J. Chem., 41, 750 (1963).

K. S. SESHADRI

1050

and R. NORMAN

JOKES

in which

(fW (i) Band shape analysis from the second and fowth nboments. Both the second and the fourth moments provide information about the shape of a symmetric curve, and the shape can be quantified by the quantity &,, where (87)

For a Gauss curve /I2 = 3, and symmetric be measured by yz, where yz = p2 - 3.

departures

from the Gauss profile can -

-cn

x

too

Fig. 14. Moments of a distribution curve about an ordinate at 2’. In principle is taken at the band maximum (x,,); T’ can have any value. In the case illustrated CC’ it will commonly be so chosen in the analysis of spectral band profiles.

These characteristic statistical quantities are derived from analytical expressions obtained by introducing the appropriate power functions into equation (85) and integrating from - co to + co. For infrared absorption ba,nds the ra,nge of integration will always be limited by overlap with neighboring bands, or by rapidly increasing random errors as the ordinates diminish with increasing distance from the band center. The complete moments, as defined by equation (85), are therefore unsuitable for the description of absorpt,ion band profiles, and truncated moments must be used instead. The truncated moment of the &h degree is given by J;(V P&l

= _L(Mv,,,Y

-

Yo)‘log

(21

. dv (88)

j+Jog

(gvd*v

The moments are measured successively to equal increasing distances from the band center in units of j, where j = (v - vO)/+Avl,,, and (~/+AY~,~)” is a factor

The shapes and intensities of infrared absorption bands

1051

reducing the moment to a dimensionless quantity, independent of the band width, and therefore a pure shape parameter (Fig. 15). The band shape is then characterized by a plot of the truncated second moment [p&j)] as a function ofj. In Fig. 16

j Fig. 15. Distribution

-units

curve with abscissa1 scale in j-units.

ZND MOMENT

Fig. 16. Comparison of t,he truncated second moments for the Cauchy and Gauss curves.

such plots for the Cauchy and Gauss curves are compared. It will be seen that whereas ,~&j) for the Gauss function rapidly converges to a limiting value above j = 1: the corresponding function for the C!auchy curve increases nearly linearly without limit. Plots of ,LQ(~)and &Jj), though nonlinear, show comparable differences (Figs. 15, 18). Any of these parameters might be used to differentiate

Ii.

1052

S. SESHADRI and R. NORMAN JONES

4 TH MOMENT

zo--

P,’ I

IO-

I

2

3

4

Fig. 17. Comparison of the truncated fourth moments for the Cauchy and Gauss curves.

30

P,!l

20

p2

P (I)

=4

P:(l)

IO

I

I 2

3

4

I

Fig. 18. Comparison of the band shape parameter Cauchy and Gauss curves.

&(A

for

The shapes and intensities of infrared absorption bands

1053

quantitatively between Gauss and Cauchy profiles, though in practice the &j) plot is preferred as it is more readily amenable to further analytical treatment’. and is less sensitive to the increasing random errors in the wing measurements. For the C’auchy curve, simple analytical expressions can be derived for ,~~(j), ,u4(j) and ,&, and more complex functions have also been obtained for the truncated the Voigt second moments of Voigt functions [61]. As would be anticipated, second moment plots lie within the area bounded by the Cauchy and Gauss curves of Fig. 16.

Fig. 19. Effect of the spectral slit width on the truncated second moments for the 812.7 cm-l band of perylene. Circles indicate the measured points. The values of s/Av&_ for lines A, B and C are 0.4, 0.7 and 1.0 respectively.

An illusbration of the application of truncated moment analysis to the characterization of an infrared absorption band shape is shown in Fig. 19 where the second moment plot for the perylene band at 812.7 cm-l is given for measurements at various slit widths. This band (see Fig. 1) is unusually narrow and symmetric, with wings that are free from overlapping absorption, and it is very suitable for the investigation of spectrometer performance. It will be seen from Fig. 19 that the band contour closely approaches the C’auchy form at narrow slit widths and becomes increasingly Gaussian as the slits are widened and the convoluting effect of the slit function becomes more apparent. It is tempting to treat the series of curves in Fig. 19 as Voigt, or near Voigt functions resulting from the convolution of a Lorentzian true absorption band with a Gaussian instrument function. It must be remembered however that the spectrophotometer directly measures fractional transmission; not the absorbance with which we are concerned here. If the true absorbance band envelope were Lorentzian, the

1054

K. 8. SESHADRI and R. NORMAN JONES

instrument function will be convoluted with its negative exponent, and the measured fractional transmission will be given by

(89) provided the area of the slit function is normalized to unity. The analytical treatment of the second moments of such negative exponent Cauchy and Voigt functions is being investigated, though they present more awkward mathematical problems. It is to be noted that this complication does not arise for emission bands (including Raman bands) where the emission intensity is measured directly without complication from the negative exponent dependence. In the computation of truncated band moments from experimental data it is important that the position of the maximal ordinate be determined as precisely as possible, since the moments are sensitive to small changes in this measurement. This difficulty does not occur in the determination of the complete moments where a correction for the misplacement of the maximum ordinate can be applied by means of a co-ordinate displacement based on the magnitude of the first moment. In evaluating truncated moments it has been found expedient to obtain the maximal ordinate by an objective method of computation [62]. (ii) Band nsymmetry analysis from the third moment. For bands that are symmetric about the maximum ordinate, the moments of odd order vanish, and the truncated third moment therefore provides a measure of the band skewness. Stat,ist,icians often employ t’he dimensionless quantity y1 = 3X1/(/%) = &

[

2;

1

1’2

(90)

as a measure of band skewness, in which y1 takes the sign of ,u3 to designate the direction of the skewness. The evaluation of precise truncated moments for strongly skew curves is complicated by the uncertainties that arise in the correct evaluation of j, and some arbitrary convention has to be adopted [61]. Approximate values of p3(j) however are easily obtained, and in Fig. 20 plots of pu,(j) against j are shown for three absorption bands, the third of which is appreciably more skew than the other two. In the discussion of the distortion of the band profile by the electronic filter system (p. 1040), it was noted that the second-order effect of the time constant induces a skew element into the instrument function (Fig. 11) and it is therefore of interest to find means to separate the skew elements induced experimentally from the intrinsic skewness of the band. In principle this can be done by reversing the scanning direction while maintaining all the other instrumental variables unchanged. The intrinsic skewness, together with any skewness introduced by the slit function, will remain insensitive to the direction of scan, whereas the skewness introduced by the second order time constant effect of the amplifier will reverse in sign with the reversal of the scanning direction. * If psF(j) and ,usR(j) are the observed third moments * In practice it is preferable to carry out these analyses in terms of /t&‘) reasons that are discussed in the more detailed publication [61].

x ($AY~,~)~for

[6d] R. N. JONES, K. R. SESHADRI and J. W. HOPKINS, CUTZ.J. Chem. 40, 334 (1962).

The shapes and intensities

of infrared

absorption

bands

1055

when scanning forward and backward, and ,+‘(j) and ,ugr(j) the components of the third moment associated respectively with the true band shape and the amplifier distortion, it follows that /&j)

= &SF(j)

t

and

I.4

I

3RD

p3Rm

MOMENT I

PERYLENE

ANTHRACENE

Vo=El12.7cm-’

I.2 AI/,’

=

L’,

=

p - CHLOROEENZONITRILE

726cm-’

V,

=

2234

4cm-’

2.2em-’

/2

IO

SLIT

WIDTH

5OOp

SLIT

WIDTH

5OOp

i

OS 0.6

0.4

I

o.2/,

J/

I

Fig.

2

20. Comparison

3

of the truncated third moments of differing asymmetry.

for three absorption

bands

More detailed analysis of the second and third truncated moment functions for selected standard bands should aid in the comparative evaluation of spectrophotometer performance and in the quantitative description of the effects of instrumental and environmental changes on the band contour. A computer program to obtain these data rapidly from the measured spectra, or by direct digit’al read-out, from the spect,rophot,ometer is available [63].

(d) An analytical solution for f(v,) by Fourier amlysis Referring back to the basic equation (68), a situation can be envisaged in which an approximate or exact analytical expression is available for the true distribution g( It), while the instrument function can be represented graphically and characterized by its moments. It is then of interest to evaluate how the profile of the experimentally observed band will alter with changes in the instrument function, as for example with change in the spectral slit function. It, follows from equations (69-71) that

f(vJ = [63]R.

s‘“G(o) -m

h’(w) exp (-kov)

N. JONES, K. S. SESHADRI and J. IV. HOPKINS, X.R.C.

dm Bulletin

(93) No. 9 (1962).

K. S. SESHADRI and R. NORMAN

1056

and K(w)

=

s

-2(277)1’2

JONES

+m

-

K(Y - Vi) exp [iW(Y - YL)]dv

m

1 = __ I+"[+ (%T)1’2 -m

- vi)][I + iw(v - vI) + . . . (iw)‘(v

-

J”)’ dv

T I-

(92)

1

The rth moment of the instrument function is given by

s +m

(v - vJr k(v - v,) dv

pee,=

and remembering

--OO

that PO =

equation (95) can be substituted

Ww)=

(95)

+mk(v s -cc

+CO k( v -

s -02

[

VJ dv

vr) dv

into equation (94) to give

1 (2rr)l/2

-

(iw)’ ruo,rc Ir T1

iq%p1 + . . . __

PO +

L

=-

On substituting

By differentiation

dV

1 (2n)l’Z

r=1

C

1 +a, a(<,>) 'r CAY) + (241'2 s _-oo

x y pp](96)

w

p, exp

(--iwV)

k

1 (97) dw

of equation (73)

s

+O”

-0cI

G(w) (--ito) exp (--icw) -_1

=--

dW v)l ___ dVT

{d’[g( v)]/dv’}

(2n)1/2

s +=J

G(w)

iw exp

(--iwv)

dw

(98)

+w

1 = (--lY

do

(277)1/Z -cc

and

On introducing

1+

for K(w) from equation (96) into equation (93)

= h

4dv)l -=-

(2$

s

_-m

Go

exp (--iov)

do

(99)

f rom equation (99) into equation (98) the expression for

f (vJ reduces to

f(vJ =

po(g(v) + (-1)' ;;; fi.9)

(100)

I.? by means of which the observed band shape can be obtained from the true band shape and the moment,s of the instrument function. Although, to our knowledge,

The shapes and intensities of infrared absorpt,ion bands

10.57

t,his equation has not been applied in practice, it would appear to offer a convenient method to inter-relat’e the true and apparent band profiles under appropriate circumstances. (e) The evaluation of the band intensity parameters by numerical integration Approximate evaluations of the relations between f (vJ and g(v), as obtained by numerical integration, have been applied widely to determine “true” band areas, peak heights and half band widths from experimental data. As developed originally by RAMSAY [3], a triangular instrument function and a Lorentz band profile were assumed. More recently the method has been refined by CABAWAand SANDORFY [4] wit’hout change in the basic assumptions. (i) RAMSAY’S method. If I, and I are the true incident and transmitted intensities that would be obtained with monochromatic radiation, and T, and T are the corresponding values observed for a spectrophotometer with a symmetric instrument function k(lv - vO1s) set at the scale reading v,, it’ follows that v,+s T‘ _ i-9 T,. t,

s Y.-s v,+s s Y,- 8

1,k(Iv -

v,I s) dv (101)

I,,lc((v -

v,) s) dv

where 2s is the total spectral band width. Since the instrument function is zero outside the range v, -& s, there is no objection to the formal extension of the integration range to & cc. It is also reasonable to assume that I,, is constant from v, - s to v, + s, and under these conditions T ( r, i Y, =

+m(I/IO)yk(,v ~ v,j s) dv s -cc +mk(lv -02

-

(102)

v,I s) dv

s If the true absorbance band has a Lorent’z profile and the instrument function is triangular with base 3s, and unit height,

s +m

-

L

_~ exp C-413vz

-

v#

t

b”]}k(lv -

v,l s) dv

(103)

s

RAMSAY has evaluated equation (103) by numerical integration over a range of values of a, b and s, and so obtained the profiles of the corresponding apparent absorption bands. From these data the apparent absorbance band parameters A&, ln ( To/T)max and the total areas under the apparent absorbance bands were computed. The data were then retabulated to express the true parameters, A& and ln (I,/&ax for given values of AvyiZand In ( T,! T)max over a range of s/ Av$ extending from 0.0 to 0.65. For a solute band in a dilute solution obeying the Beer-Lambert law? the area beneath the true Lorentz absorbance band will be given by

1058

K.

S. SESHADRI

and R. KORMAN JONES Since 2b = A&

where c is the concentration and 1 the path length of the solution.* and a/b2 = In (Io/l)max, equation (104) can be written A,

= l-57 . f . A$

.

(105)

max

The ratios Av&,/Av~,~ and ln MJln

(XX

are known from the numerical integrations. Equation for various values of s/A& (105) can therefore be expressed in terms of the apparent band parameters

A, = ii. K, . A$,,

(106)

max

where (107) Tables of Av;C,~IAV~,~,

and K, as functions of slAv$, and In (To/T) max are given in RAMSAY’S plots of the functions have also been published [64]. In Figs. 21-23

09

;,fl,

1,)

ZO-

/,,,‘,

I

;

08

L%#,,.

IS-

07

I.6 -

06

14-

O5 0.4 03 02 01

L”($)

mar

_ IO

,

’

I,

‘/,,

/

“/,‘, ‘/

to2 I.03 I.04 to6 co9 ’

;/ /

/

’ /

‘> 1:10: ’

KG,,

:,

’

.‘/ ‘/ ‘, ’

101

paper, and the salient

/

’ ,

08-

06m 0.4 02-

0 00

0 IO

0 20

0 30

040

0 50

0 60

Fig. 21. The effect on the intensity at the band maximum of convoluting a Cauchy curve with a triangular slit function. (Based on the computations of RAMSAY [3]). * In general the symbol b is preferable to I for the path length of the absorbing material (cf. p. 1064) but I is used here to avoid confusion with the Lorentz band width parameter. [64] R. N. JONES and C. SANDORFY, The Application of Infrared and Raman Spectrometry to the Elucidation of Molecular Structure, Technique of Organic Chemistry (Edited by A. WEISSBERGER) Vol. IX Chap. 4. pp. 282, 285. Interscience, New York (1956).

The shapes and intensities of infrared absorption bands

1059

Fig. 22. Change in the measured half band width under the conditions of Fig. 21.

0.3 0.2 0.1

0.6 04 02

SlAV,”

.2

Fig. 23. Change in the total band area under the conditions of Fig. 21.

of these relationships are illust’rated in block diagrams from which the following conclusions can be drawn: (a) The apparent peak intensity never exceeds the true peak intensity. (b) The apparent half band width is never less than the true half band width. (c) Under the experimental conditions that will normally be encountered, the t,rue band area will not exceed the apparent band area, but this may happen under conditions corresponding to t’he top right-hand section of Fig. 23, viz. for high maximum absorbance at very wide slit widths. If the true band profile is Gaussian, the area (AC) will be features

(108)

1060

in which a = A&,/2(ln

K. S.

SESHADRIand R. NORMANJONES

2)lj3 = 0*735Av~,, from which it follows that A,=UO~~.;.AV;~~.

In 3 0I

milx

(109)

For the Voigt profile the true band area cannot be expressed by a single simple formula applicable to all ratios of x1/x2, but a numerical constant falling in the range between 1.064 and 1.57 can be calculated for specific values of x1/x2 having these extreme values where x1/x2 = 0 and co respectively. These have been tabulated by VAN DE HULST and REESINCK [ 181. Numerical integrations for the convolution of Gauss and Voigt band profiles with a triangular slit function have not been made, but it is reasonable to assume that the true and apparent band parameters will show trends similar to those for the Lorentz profile. Where comparisons have been made between “true” band areas computed by RAMSAY’S equation and by the extrapolation techniques to be described in the following section, the Ramsay method almost invariably gives the higher value for the band area [38, 651. For example, MORC~LLOet al. [65] obtained a value of K = 1.27 for the C = 0 stretching band of acetophenone in chloroform solution by extrapolation to infinitely narrow slit width. Part of the discrepancy is associated with experimental uncertainty in the measurements out in the wings of the band, but MORCILLOet al. estimate that the correction for this would not raise K above 1.34. A small part of the remaining difference between 1.34 and 1.57 could result from the departure of the slit function from the triangular shape, but the greater part more probably results from the non-Lorentzian contour of the true band. In cases where the true band is not significantly asymmetric, the experimentally measured value of K obtained by extrapolation might be used to obtain the ratio of x1/x2 for the best fitting Voigt function. For the acetophenone band discussed above a K of l-27 corresponds to x1/x2 = 0.51, while a wing corrected K of 1.34 corresponds to x1/x2 = 0.79. (ii) CABANAand SANDORFY’S modijcation of RAMSAY’S method. To obtain a closer fit between the true and apparent band contours, CABANA and SANDORFY [4] measured the additional width parameters Av,.~ and AvTt, corresponding to the ordinate values r[ln (To/T)],,, and r[ln (lo/l)]max where r = 0.75, 0.25 and O-125, in addition to the r = 0.5 width of RA~CISAY.For the Lorentz profile these give the following values in terms of the Lorentz constant b. Av&

= $3 b

Au;., = $b Av;.s6 = 21/3b

(110) (111) (112)

Av;.,,~ = 21/7b (113) Tables analogous to those of Ramsay for r = 0.5 were computed for r = 0.75, 0.25 and O-125. For technical reasons the numerical integrations were carried out in horizontal instead of vertical sections, and in order to deal more effectively with [65] J. MORCILLO, J. HERRANZand M. J.

DE

LA CRUZ,Spectrochim.Acta 15, 497 (1959).

The shapes and intensities of infrared absorption bands

1061

asymmetric bands, the two sections of the band on either side of v,, were evaluated separately, using the measured values of the half fractional band widths &,la and and corresponding h’ values K,’ and K,” were &,U'l:such that Av,~ = Bv,‘~ + Bv,.~‘~ obtained. The true band area is then given by

+ 0.156,[6v$,,K,,

2j + bv;;&Kb’.,,]

It will be evident that this is a general method, capable of unlimited refinement by introducing more fractional band parameters. It treats the observed absorption band, not as a single Lorentz curve, but as a series of horizontal sections, with the profile of each section fitted by a different set of Cauchy parameters. As a practical method of quadrature for evaluating the true band area it has considerable merit, and, as used by CABANA and SANDOREY it gives areas some 10 per cent less than t,hose obtained by the original Ramsay method and therefore in better agreement with experiment. In practice it is not appreciably more difficult to apply. It is questionable however whether the Cauchy profiles of the individual sections have real physical meaning, and in this respect numerical integrations based on the Voigt profile of the complete curve would be more acceptable. Attempts have also been made to fit the absorpt’ion bands by other functions that yield expressions of the form B = K, In (T,/T),,, on integration. Several of these have been summarized by JONES and SANDORFY [66] and by ARNAUD [67]. The difficulty of evaluating the contribution of the “wings” of the band to the total area is well recognized as a major source of error in numerical integration methods, and several investigabors have proposed methods of dealing with this problem. RAMSEY assumed that the wing profiles could be treated as Lorentzian, even when the main part of the band envelope could not, and he computed tables for determining wing corrections on this basis. It is a characteristic of the Cauchy function that one half of the total area lies within the section bounded by the ordinates of the half band width (j = I of Fig. 15). There may therefore be merit to measuring this area only and doubling to obtain the total area [67, 681. The computer program developed to obtain t’he truncated band moments p. 1055 also gives the accumulat’ing areas at successive j values. These might be plotted against j to evaluate the total area by extrapolation. Such plots for C!auchy and Gauss furl&ions are shown in Fig. 24. [66] R. N. JONES and C. SANDORFY, The Application of Infrared and Raman Spectrometry to the Elucidation

of Molecular

Structure,

Techwique of Orgmic

Chemistry

WEISSBERGER) Vol. IX Chap. 4. p. 280. Interscience, New York (1956). [67] P. ARNAUD, Bull. sot. chin&. France 1037 (1961). [68] C. E. LEBERKNIGHT and J. A. ORD, Phys. Revs. 51, 430 (1937).

(Edited

by A.

1062

K.

8.

BESH~DRI

and R. NORMAN JONES

Fig. X4. Fractional areas of C’auchy and Gauss curves bollnded by ordinates at in-

creasingj values. For the C’anchy curve one half of the total area is bounded by the ordinates at j = & 1. (f)

The evaluation of the true bard intensity parameters by extrapolation

(i) Extrapolation to zero spectral slit width. In principle, the true band intensity parameters should be obtainable by measuring the apparent band at progressively diminishing spectral slit width and extrapolating to zero spectral slit width. Since neither experimental nor theoretical means exist to obtain the exact spectral slit functions such methods are empirical. Precise extrapolations will be sensitive to measurements made at the narrower slits, since the profile of the slit function may alter considerably when bhe slits are narrowed below the point where the diffraction and misalignment factors begin to dominate. At narrow slits the random errors increase due to the unavoidable fall in the signal to noise ratio, and, in compensating for this, care is necessary t’hat second-order distortion from the amplifier filter system is not introduced by the use of excessive clamping (p. 1040). Extensive analyses of Ohis problem have been made by RUSSELL and THOMPSON [38], MORCILLO et al. [65], PEISAKHSON [40] and SHCHEPKIN [69] among others. Typical plots for half band widths and peak heights measured at progressively narrowing slit widths are shown in Figs. 26 and 36. taken from the data of RUSSELL and THOMPSON. For values of s ;- Av:~, the apparent half band width is roughly with a proportionality factor close to unity. The value of A$

proportional to s, only approaches

Avile for s/Av$ c 0.2. Even using the high-resolution grating spectrophotometers currently available, it is not, practical to measure spectra routinely at nominal spectral slit widths much below 1 cm-l ; from this it follows that for bands with an apparent half width of less than 5 cn-l some form of extrapolation is necessary to [69] D. N. SHCHEPKIN, 8, 58 (1960).

Optika

i,Vpektmskopiyu

8, 118 (1960).

English translation Opt. ad

Spec.

The shapes and intensities of infrared absorption bands

CYCLOHEXANE 903

cm-’

Liquid

film

20 AVp/ cm-’ 2

IO

Fig. 25. Effect of the spectral slit width on the apparent half band width of the 903 cm-l band of cyclohexane (Taken from the data of RUSSELL and THOMPSON

r3m

CYCLOHEXANE 903 Liquid

I”

,I

,I_,

IO

cm-’ film

I,

20

I

30

S, cm-' Fig. 26. Effect of the spectral slit width on the apparent peak intensity of the 903 cm-l band of cyclohexane (Taken from the data of RUSSELL and THOIKPSON [38]).

1063

1064

li.

s.

~ESHADRI

and R. NORMAN .JOWES

obtain acceptable width measurements. Though most bands encountered in the spectra of liquids do have half widths in excess of 5 cm-l, narrower bands not uncommonly occur. The band at 812.7 cm-l in the perylene spectrum shown in Fig. 2 is the narrowest yet observed in solution-phase studies in our laboratory. though still narrower bands occur in the spectra of solids [%, 291. The variation of the peak height with the slit width follows a similar trend. The influence of the slit width on the gross appearance of the band is illustrated by the adjacent pair of bands at 903 cm-l and 861 cm-l in t,he spectrum of cpclohexane [70], illustrated in Fig. 27. The true half widths of these bands differ by a factor of

CYCLOHEXANE

Ltauld

Film

i

Fig. 27. Effect of the speckal slit width on the relative heights of the bands at 903 and 861 cm-l in the spectrum of cyclohexane. (After SLOANE [70]). The ext,rapolated half band widths are 4.9 and 8.9 cm-l respectively.

approximately two, and the relative peak intensities invert as the slit width is diminished. This example effect)ively demonstrates the importance of the ratio and the need to apply some form of extrapos/A~l(,~ in analytical spectrophot’ometry, lative correction. The integrated band area is less sensitive than either Av$ or In (To/T)max to change in the spe&ral slit width, and the application of extrapolative corrections is placed 011 a firmer theoretical basis by the use of the methods of BOURGIN [2] and of WILSON and WELLS [l]. Since the principles of these methods are fundamental to the evaluation of absolute infrared band intensities. it is important that they be examined in some detail. (ii) BOURGIN’S method for the evaluation of the true bard area. The fundamental Beer-Lambert absorption law for a solute absorption band may be expressed I,. = Iov exp ( -qbc) in which a, is the absorpt,ion coefficient,

[70] H. J. SLOANE, Appl. Spectrosc.

16,5

(115)

b the path length and c the concentration.

(1962).

The shapes and intensities of infrared absorption bands

If the ordinate intensity is expressed as fractional absorption area beneath t’he true fractional absorption curve is given by A’ =

s

[I

(116a)

[l - V/~,Ll dv

band

[l -

exp

(-ct,bc)]

dv

(1 161))

which expands to 9’ bc =

dc,2 bc s band c at’

-

k

+

!if

&2

(117)

. . . (I,,

I_?_

and, in t,he limit where bc + 0

s band

u, dv = L4

(118)

From this, the interesting conclusion is reached that by measuring the area beneath the fractional absorption curve and extrapolating to zero bc the brue integrated absorption intensity beneath the absorbance curve is obtained. Furthermore it has been shown by DENNISON [71] and by NIELSEN ebal. [72] that the area beneath the fractional absorption curve is independent of the spectral slit function, so that, provided only the slit term contributes significantly to k(v - vJ, we may substitute (T/T,), for (I/I,), in equation (I 16a) and obtain the true absorbance area from the extrapolation. This method of obtaining the true band area is attractive, since most infrared spectrophotometers record the fractional absorption directly on a linear scale. Unfortunately plots of A’ against bc are usually non-linear, and the curvature adds t,o the uncertainty of the extrapolation. * In the experimental measurements, either b or c or both may be varied. In practice it is usually more convenient t’o maintain the cell lengt(h constant and vary the concentration. Theoretically it is preferable to vary the cell length at constant concentration, since deviations from BEER’S law are far more prevalent than deviations from LAMBERT’S law, and the band intensities measured at a fixed concentration will have physical significance even for a solut’ion in which BEER’S law is not obeyed. (iii) The evaluation of LORENTZ band parameters from fractional absorption curves by the tangent construction. Although not strictly a method of obtaining true bandintensity parameters from apparent band-intensity parameters by extrapolation, it is relevant t’o consider here a novel method of measuring the half band widths and areas of Lorentz absorbance bands recent’ly developed by MECKE et al. [73]. In * For a true Lorentz absorbance band, RAMSAY [3] has shown that a linear relationship is obtained if low and medium values of log (Io/I)max are plotted against be/A’ in which case the ordinate intercept gives l/9. This method is not generally practicable since log (Io/l)max is not obtained experimentally, and it cannot be derived without making assumptions about the curve shape and slit function as described in the preceding se&ion. [71] D. M. DENNISON, Phys. Rev. 31, 503 (1928). [72] J. R. NIELSEN, V. THORNTON and E. B. DALE, Revs. Moderr/ l’hys. [73] F. LANGENBUCHER, Diplomarbeit, Freiburg i. Br. (1961).

16, 307 (1944).

1066

Ii.

S. SESHADRI and R. NORMAN JONES

common with the Bourgin method of band analysis discussed above, this permits evaluation of the absorbance band parameters A and Av$ from the fractional absorption curves obtained directly from the spectrophotometer.*

T =o To u

( )

T ( To ) u

T ( -ro ) =’ V

P’

0 -Au

P

0-

Fig. 28. Diagram illustrating the tangent construction for obtaining absorbance band areas and half widths from the fractional absorption curve. (After MECKE et al. 1731.

The Lorentz equation may be written in the form

f(v)=

1 +

where umax is the peak absorbance. simplifies to

[(Y

Writing

f(x) = The corresponding

fractional transmittance

T, =

(119)

?;),&Avl,J2 x for (v -

Y,)/+AY,,, equation

s2

(119)

(120)

band (Fig. 28) will be given by

= exp _

1

Clmsx [ 1 + x2

(121)

* It is the custom of the Freiburg school to compute band intensity parameters as wavelength functions (cf. p. 1071) and MECKE et al. describe their tangent construction with reference to a Lorentz curve with linear wavelength abscissa. As the method is based only on the general mathematical properties of the Cauchy function, it is independent of the choice of the abscissa1 units, and is presented here in terms of linear wavenumber to maintain conformity with the remainder of the text. We are very grateful to Prof. MECKE and Mr. LANGENBUCHER for the opportunity to read Mr. LANGENBUCHER’S thesis prior to publication.

The shapes and intensities of infrared absorpt’ion bands

1067

Inner tangents AP, AP’ may be drawn from the band maximum to the base where T = 1, t,ouching the curve at the points (&xi, T,) and the equation to the tangent A P will be T* = zX* where (x*, T*) is any point on the tangent. By differentiation of equation (121) dT z = exP

amas [ -il).

Substitution of the co-ordinates this into equation (122) we obtain

(122)

x( 12umax 1 + x2)--2

(123)

(x~, T,) gives the slope of AP, and on transferring (124)

T” = 2T &lax X*(1 + Xt)-2X*

At the point of contact of the curve and tangent (xt, T,) = (x*, T*), and on making this substitution in equation (124) it follows algebraically that Xt = & [5$j1’2&

[T

_ l]1’2

(125)

The selection of the valid roots depends whether ~~~~ s 2. The main practical interest is for the inner tangent in the case where amax > 2, corresponding t,o log,, (~o/nrmx = 0.87, for which the significant solution is Xt

=

[Tq+

1]1’2

[y

It is next desired to obtain the intercept PP’ = 20P. are (x,,, 1) it follows from equations (125) and (126) that

h?

=

(126) If the co-ordinates

of P

[urnax [l +(1-&)l’i]- 1) ([,+ _;2iu,ax),l,2)) (lY7) -P

Thus x0, and therefore PP,’ (127) reduces to

depends only on amax. For large values of Emal, equation (2#

a more exact approximation

I1

= (2ccmax-

l)f?

(128)

being

(x,)~ =

[a.&nr - 1 - 4(umnt _ Ide

(129)

If the absorbing material obeys the Beer-Lambert law, we may write zmax = bcdmax where b is the path length and c the concentration, and elmaxthe molecular extinction coefficient at the maximum.? With this substitution equation (128) becomes (xJ3 = (2bc.zlmax- 1)e (130) where e is the base of the natural logarithms. f For convenience E’~~~ is defined here in terms of In (TO/S!‘)y instead of loglo (!P,,/T),, to avoid carrying a numerical constant through the subsequent discussion (cf. p. 1077).

Ii.S. SESHADRI

1068

and R.

JONES

vo)/~Avllz and if the intercept

By definition .Z = (v (Fig. 28) it follows that

:.

PP’

is designated

4,

OP=x,=-

On substitution

NORMAN

Av,

(131)

A%/2 Av, = Av,,a . x,,

(132)

of x0 from equation (130) (At@

= (Av~,.J~(~~cE’~~~-

1)e

(133)

For a given absorption band, the experimental known quantities are Av,, Av,,, and bc. If (Av$ is plotted against bc for a range of measurements, the slope M will be given by M

whence

=

4Ad2 -= 4bc) Avllz =

On substituting

2(Av1,2)2~‘max e

M

1

l/2

___ e [ 2 &lrnax

this for Avl12 in the integrated Lorentz function, A,=1*57

=

1.57

(134)

1 Mdmax 1'2 1

(135) equation

(105)

M 112&‘IXlSX ~ [ 2 elmaxe [

__

2e

(136)

We therefore conclude that the area beneath the Lorentz absorbance band can be obtained from the molecular extinction coefficient at the band maximum and from the slope of the line obtained by plotting the square of the tangent intercept PP’ against bc. The half band width of the absorbance curve may also be obtained from equation (134) and in some instances this may be preferable to direct measureet al. use the Ramsay tables. ment. To obtain the true value of .z’~~~, MECKE It should be emphasized that this use of the tangent construction to obtain the band intensity is based on high values of u,,,( > 2). Although error-theory calculation (pp. 1073-1080) show that the accuracy of measurement falls under these conditions, it should be noted that the high-absorbance measurements are needed only to locate The critical measurements, which determine the position of the band maximum. the point of contact of the tangent (and hence the length of the intercept PP’), lie in a range where the absorbance errors of measurement are small. Indeed, one of the principal features of this method of curve area measurement would appear to be its convenience for dealing with just such intense bands. (iv) The method of WILSON and WELLS. If A and B are the areas beneath the true and apparent absorbance curves respectively, then (137a) (137b) where v, is the wavenumber

setting of the spectrophotometer.

The shapes

and intensities

of infrared

absorption

bands

1069

Since, by equation (68) T,, =

s band

and To,& = equation

s band

(137b) may be written

I”.

k(v-v,)

I()“k(V -

dv

(138)

v,) dv

(13%

(140) Jdit

If I, is constant over the slit width this reduces to

The denominator slit function. (I/Z,), to give

of this expression is the area (S) delineated by the spectral can be replaced by exp (-cc,,bc) and (Io/Z),z by [exp (-c~~~bc]-~

In

A-B=J

bc I band

s slit

esp ( -qbc)E(v

-

vz) dv

exp ( - cc,bc) . S

I

1dv,(142

The limit’ att,ained by A - B as bc approaches zero can be determined by the method of ~‘HOSPITAL in which both the numerator and the denominator of equation (142) are differentiated with respect to bc to give

s

exp ( - a,bc) . k(v -

lim [A -

B] = lim __

-

vJ dv

slit

exp (-tl,&bc)k(v

be- 0

cc, exp ( -qbc)

-

. k(v -

. dv,

v,)S

I

(143a)

vl) dv

s’

I

dv,

(143b)

In the limit exp ( - cr,bc) -+ 1, and therefore

lim [A -

s slit

B] =

a, . k(v -

dvl

S

be-0

and provided

vl) dv (144)

cz,,remains constant over the width of the slit

s

cr,k(v -

VJ dv = ct,

slit

-

s slit

E”S

k(v -

VJ dv

(145)

K. 8.

1070

and equation

and R. NORMAN

SESHADRI

JONES

(114) reduces to

lim [A -

B] =

be-0

I band

u, dv, -

I band

MY,dv

(146)

=o These somewhat complex arguments lead to the conclusion that provided the incident intensity is constant over the spectral band width, and the spectral slit function does not change across the absorption band, the true absorption intensit(y can be obtained by plotting the apparent absorption intensity against bc and extrapolating to zero bc. As with the Bourgin met’hod (p. 1064) either b or c may be varied and, for the same reasons, the variation of b at constant concentration is physically the more meaningful. The form of the extrapolation depends on the t#rue band profile and the profile of the slit function, but in general, it is much more nearly linear than the Bourgin extrapolation. For the convolution of the Lorentz band with the triangular slit function, RAMSAY [3] has shown that the extrapolation is very nearly linear with a small negative slope 0. which he has calculated as a function of s/A&,. The true band area can then he obtained from the expression B =A

+dOln

(147)

It is preferable in practice to obtain B by plotting B against log (To/T)max for various values of bc and determining the best straight line through the measured points with int’ercept A and slope A@ ; this is found to give more acceptable results than a statistical least-squares fit of the data. * RUSSELL and THOMPSON1381 have extended this extrapolation technique to cases where the slope is obtained experimentally without assuming the Lorentz profile. The Wilson-Wells and the Bourgin extrapolations are mathematically equivalent, and their equivalence has also been confirmed experiment,ally [74]. In summary of these extrapolation techniques, it can be stated that the corrections for the band areas are based on theory, whereas the corrections for half band widths and peak heights are still highly empirical. If proper advantage is taken of the high resolution obtainable from modern grat’ing spectrophotometers. the application of corrections for band area distortion is of doubtful significance except for the narrowest bands. This error will normally be much smaller than the errors introduced by uncertainty in the limits of integration and the experimental inaccuracy of the measurements out in the band wings. For absolute measurement of peak heights and half band widths, extrapolation to zero slit width is highly desirable, and unless this is done, peak intensity data > 0.2 can have little absolute significance. Few of t,he recorded where s/A+ * It should be observed that although both the Wilson-\Vells and the Bourgin methods are based on extrapolations of bc to zero, in the Wilson-Wells method bc may be replaced by the R h ereas the Bourgin extrapolation is experimentally observed peak intensity [log (T,/ T),,_] only valid if log (I,/l)maX is employed as the bc dependent peak intensity variable. [74] R.N.

JONES,D..LRAMSAY,

D. 9. KEIR and K. DoBRINER,.J.~~~.C~~~~.S~C. 74, 80 (19.52).

The shapes and int,ensitiesof infrared absorption bands infrared spectra in the currently anticipated that dat,e before the their permanent,

1071

of organic compounds measured in condensed phases and recorded available atlases of spectra can meet this criterion, and it is to be most of these curves will have to be remeasured at some fllture infrared band intensities and the complex band profiles can take place as acceptable physical constants of the respective substances.

(g) The choice of units for the measurement of integrated band intensities There are, as yet, no universally accepted units for the measurement of the areas beneath infrared absorption bands, and it is probably unrealistic to expect that any single unit will prove sufficiently adaptable to accommodate all the divergent demands of spect’roscopists making such measurements. In much the greater part of the published literature, band area intensities have been computed from data presented on an abscissa1 scale of wavenumber. The ordinate intensities have commonly been computed on a scale of In (T,/T) rather than log,, (T,/T), necessitating a conversion factor of 2-303 when the ordinate data are recorded as absorbance or molecular extinction coefficient (molar absorptivity). Chemical spectroscopists, concerned mainly with empirical and semi-empirical applications of band area measurements have tended to employ the unit first introduced by RAMSAY and JONES

= 2.303

‘=t,, dv I Yl

(148)

in which c is the concentration in mole/l., b the path length in cm and Ethe molecular extinction coefficient (molar absorptivity). The more theoretically oriented molecular spectroscopists, following THOMPSON[38], prefer to express the concentration in molecules per ml. and to transpose the abscissa1 scale to absolute frequency (v in set-1). This introduces both the Avogadro number (N) and the velocity of light (c) as constants in front of the integration term and, in terms of b and c as defined above, the equation becomes

MECKE et al. at Freiburg integrate over a linear scale of wavelength, expressing both the path length and the wavelength in cm and the concentration in moles/ml. whence

(150)

1072

K. S. SESHADRIand R. NORMAN JONES

OSWALD [75] has shown that the approximation

=--

103 1

bc vo2s

I In 2 0I

vd ’

(151)

AA,,2 holds t,o within a few per cent, provided il < O-01. He has comput’ed a correction factor K for bhe more exact equation

A,

I, 103 a02 In = -- I bc K IO

clv

Y

(15’7)

for the case where the band has a Lorentz profile and A&j2 > 0.01 MECKE and NOACK [76] have pointed out that when the area beneath the absorption band is expressed in this form, it is directly proportional to the atomic polarizability in accordance with dispersion theory. CRAWFORD [77] and the Minnesota school reject the “adolescent” function given in equation (148) and, following an earlier suggestion of MECKE [78], recommend the unit

(153)

This has the advantage of being more easily related to the basic molecular quantities intrinsic to the band intensity; it also facilitates the treatment of band anharmonicity and the quantitative interpretation of Fermi resonance. The dimensions of A, are cm2 molecule-l and it can therefore be treated as a cross-sectional area. These are cogent arguments for its use in dealing with the spect,ra of simple molecules, potentially susceptible to detailed theoretical analysis. but the additional complexity of the computations must be taken into consideration before judging on the suitability of this unit for more general use. Though uttered in a different context, the views expressed in Ref. [79] have relevance here. To curb the introduction of still more intensity units, the Commission on Molecular Structure and Spectroscopy of I.U.P.A.C. has provisionally recommended the use of the following units [SO]. [75] F. OSWALD, 2. Ekktrochem. 58, 345 (1954). [76] R. MECXE and K. NOACK, Chem. Ber. 93, 310 (1960). [77] B. CRAWFORD,JR., J. Chem. Phys. 29, 1043 (1958). [78] R. MECKE, 2. Physik. 101,595 (1937). [79] The Gospel according to St. Mark. Chap. 10. v. 13-15. (King James Version). [SO] International Union of Pure and Applied Chemist,ry, Compt. ren.d. vingtikme Mztrtich 1959 p. 187. Butterwortha, London.

Coxf&ence,

The shapes and intensities

(a) ATL absolute

writ.

of’ infrarecl absorption

bands

1073

This is defined as

in which b is the sample thickness in cm. n t,he concentration in molecules per cm3 This unit has the dimensions cm2 WC-l nlolecule-l, and v the frequency in set-l. and is ident8ical with that proposed in equation (149). (b) d secondary wit

with dimensions cm2 molecule-l (c) d

practical

corresponding

nit’h equat’ion (153)

z&t

in which 1 P,.

=

~

hc

log,,

IO‘ ‘-

1

i I: Y

wit#h concentration c in mole/l. of solution. b in cm and v in cm-l. This unit has dimensions IX--~ 1. mole-‘. It differs from the unit described in equation (I 4%) hy t’he omission of the numerical const,ant _“.:303. The factors for the int,erconversion of these unit,s are summarized in Table 7. X similar table of intensity unit’ interconversion fact’ors is included in the revie\\ article of GRIBOV and BMIRNOV [G] but comparison of the two Tables shows several differences. These are due partly to the fact’ that GRIBOV and SMIRXO~ have taken t,he IUPAC! Pract,ical Unit to he defined in terms of loge (1,/l) whereas it is actuallp defined in terms of log,, (10,‘1) [SO]. (‘omparison of the two Tables also indicates ambiguity concerning the definition of the Crawford unit’ (A(. in our Table and Q or lY in the table of t,he Russian workers). TYe have taken this unit, to be identical with t,he IUPAC’ Secondary Unit. ( !RAWFORII'S original paper [77] is not definite on this point : units of cm2 mole-l and cm3 mmole-l are indicat’ed hut not specified, while the unit of the IUTPAC Secondary Unit, is defined as cm2 n~olecule-1 [SO]. Our interpretation of t’he I’ra:vford unit is consistent with the st’atement given in equation (2.38) of C$RIROV nrld SNIRNOV'S paper t,hat it is equal to the TUP_-\CAbsolute Unit, Although these numerical divided by the frequency at the ljantl center in sect’. differences may not he important in theoret’iaal spect~roscopy, it, is essential that they he clarified and standardized in units that are t,o he considered for practical use. T’. THE

OF R~xuonr ERROI~S ON THE MEASUIEEMENT OF THE TRVE AND APPARENT ABS~RBSR(!E

EFFWTS

From a formal mathematical point of viejv. the argument,s developt~cl in equat,ions (K-71) suggest that, the true hand profile can he obtained with ally desired accuracy.* * Non-unlclun solrctions for q(1,) arc possible if K(w) van&w for cvrtain values of (11. Such a sit,uat,ion is ~mlikrl~ to occur in the analysis of infrared band shapes, and if it rlors, thrr~ are In&hods of tlwtling v It,h it. SW Ref. [Xl] and p. 260 of Rc+f. [ZO].

1Xl] R. P;. 16

BKACYXTLL

aml

,J. A.

ROBERTS,

Arcntrtzlircn J.

Ph!ys.

7, 60.5 (19.3-4).

1074

K. 8. SESXADRI

and

R. NORMAN

Jams

The shapes and intensities of infrared absorption bands

1075

In practice this is not so, because the reduction of the width of the instrument function leads to an increase in the random errors of measurement. This problem has been noted on several occasions in the preceding sections of this review and it is next desirable to discuss the relationships between the systematic errors and the random errors, and to derive expressions from which the optimum conditions for minimum total error can be evaluated. This problem has been considered by VAN DE HULST [SZ] and by BRODERSEN [23], among others, and more recently has been examined in considerable detail by RAUTIAN and PETRASH [20, 24, 83-861. Here we shall be concerned only with the more directly practical aspects of PETRASH and RAUTIAN’S treatment. Those interested in the underlying theory are referred to RAUTIAN’S earlier article [ 201. The measurement of an infrared band intensity from the recorder chart of a conventional double-beam spectrophotometer is illustrated diagrammatically in Fig. 29, in which L, is the baseline corresponding to 100 per cent transmission and L, mI_ I,

L3 t

_--W~-_-U

~~~ _U_._

-------7 L,

I I I

Fig. 29. Single beam located base line. L, ratio than L,. They distortion caused by

recording of an infrared absorption band. L, is an arbitrarily and L, are measured at wider slits and higher signal to noise Under these conditions the are assumed to be parallel to L,. convolution with the instrument function is assumed to be absent.

[82] H. C. VAN DE HULST, Bull. Astronom. Imt. Xed. 9, 225 (1941). [83] G. G. PETRASH, Optika i Spektroskopiya 8, 122 (1960). English translation, Opt. and Spec. 8, 60 (1960). 1. 1841G. G. PETRASH and S. G. RAUTIAN, Materialy x T’sesoiuz. Soceshcha?liia Spektroskopii, Molekuliarnaia Spektroskopiia pp. 102-106 (1957). English translation, AT’ationuZ Research CounciZ of Canada Technical Translation No. 904. [85] G. G. PETRASH and S. G. RAUTIAN, Inzhener. F&z. Zhur. 1,80 (1958). English translation, Natio?laZ Research CouwciZ of Camda TechnicaZ Translation No. 903. [86] S. G. RAUTIAN and G. G. PETRASH, Materialy IL‘Vsesoiuz. Soveshchaniia Spektroskopii, 1. Molekuliarnaia Spektroskopiia, pp. 107-111 (1957). English translation. iVatio?laZ Research CounciZ of Canada Technical Translation No. 905.

1076

K.

S.

and 1%.NORMAN JOHES

SESHADRI

is the zero transmission line. Bot3h of these are presumed to have zero slope. As we shall be concerned only with errors affecting the chart measurement, it can be further assumed that all radiation not transmitted is absorbed, so that if T is the fractional transmission, 1 - T will be the fractional absorption. To keep the problem general, the experimental measurements I,, I, and I, are made from an arbitrary baseline L,, lying below and parallel to L,. The t8ransmission in the presence of the absorbing material is L,." The true fractional absorption of the sample for radiation of wavenumber Y, can be described by the function x(Y,), where qy

) 1

=

&w + a! TO

(15-i)

in which T, is the intensity of the incident radiation, a is a measure of the constant continuous background between L, and L, and 4 is a normalizing constant to adjust the absorption at the band nla~inlurn Y* so that #g(~*) = 1. Beaause of the convolution with the instrument function, the absorption actually observed will differ from x(YJ. This difference is the principal source of the spst’ematic errors in the ~neasuremellts. For the apparent absorptio~~ we may writ,e Y(V ) J(vJ 2

+ a TO

(156)

where f(Y‘) = 9$_R(W

-

Vi) dv

(156)

or, writmingJ’(Y) for the reciprocal of the above integral,

,f(YL) =

4

(157)

F(y)

In principle, this equation holds for all points on the band profile. but it is convenient to deal with the band maximum where Y, = yO. Since. by definition $g(Y,,) = 1, equation (158) then reduces to (158)

.f(%) = f

and provides a simple expression for the systematic errors in the measureme&. In addition to this systematic error, there are random errors due to il~acc~~racies in the measurement of f(YJ, cl and 127,. These can be expressed Af(vJ

= f(vJ

- f(l*J

An = a -

Z

AT, = T, -

T,

Ay=y-5 where the bar designates the corresponding * The symbols

I$,,

(159) avemged paramet,er.

I;,, L,,L, have been changed from those u~od in the original paper.

The shapes and intensit’ies

The true absorbance*

of infrared

absorption

bands

1077

is given by D,, = -In

[1 -

X(Y)]

(160)

The symbolism can be simplified by writing s for X(Y). y for Y(Y), f forf(y) for F(V). The over-all error in the absorbance measurement can be written AD = --In (1 = ill

1 -

.T) _I- In (1 -

II,

y)

p]

x

[

=

and F

1

[

_

AY + -x_)

1

(163)

AD=

x (1

_

x)

- A(f + ~1 I g AT,,\ I \I - ; - &,, x T, I

(164)

If Al,, Al, and Al, are the errors in t’he measurements of l,, I, and I,, respectively. then AT,, = Al, - Al, and A(f + u) = Ab, - A&. Since the measurements are statistically independent. AT, and A(J’ $- a) will also be independent. Since LI and L, have zero slope, t’hey will be independent of the instrument function, and can therefore be measured at nide slits where t’he random errors are small. Consequently it, can be assumed bhat Al,, ,,1’:3j+z Al,. and that, the products Ab,Al, and Al,Al, are negligible. Writing AT,/T, for ATT,fj/Tw~:in the t,hircl term of equat’ion (164) and making the other substitut,ions noted above. AD2 may be expressed

from which the mean quadrat’ic error, P”, can be derived, where

* Hcrc. logarithms.

as on p.

1067, it, is convcment,

to express

the absorbance

in terms

of natural

K.

1078

S.

SESHADRIand R. NORMAN

JONES

In this equation the first term within parentheses describes the systematic error* and the three succeeding terms the random errors in the measurements of L,, L, and L, respectively. As so written this applies only to the error at the band maximum, but the equation can be generalized by modification of the systematic error term (cf. equations 157 and 158). To pursue the analysis further it is necessary to make assumptions about the instrumental factors affecting the various terms in equation (167). The signal to noise level in the recording of L,, L, and L,, which determines @/To2 (i = 1,2,3), will normally depend on the amplifier gain and, having regard to the interdependence of the spectrometer variables discussed on pp. 1044-1045, the noise level can be related to the square of the mechanical slit width so that AE2 iI2 --..L.

( Toa1

A

=G

(167)

where w,s’is the mechanical slit width (both assumed equal) and A is a proportionality constant. In the limit, where the slits become very wide, this relationship will no longer hold, since the recording errors will not extrapolate to zero with increasing w,, but to a constant value determined by the mechanical and electronic limitations of the spectrophotometer. This minimum error may be designated B, and PETRASH and RAUTIAN [20] suggest that [A1,2/T,2]1~2be taken as A/w,$ where ws2 & A/B and as equal to B where wsz > A/B. The q~~antities A and B can be obtained experimentally by statistical analysis of the noise level, and of AZi2 for measurements at different slit widths. It has been noted above that I, and I, can be measured at wide slits since they are independent of the instrument function. We may therefore substitute B2 for AZ,2/T,2 and AZ,2/T,,2 in equation (166). It is obviousIy more difficult to give specific values to g/x in the systematic term of equation (166), since this depends on the true shape of the band and on the instrument function. PETRASH and RAFTJAN assume that both of these are Gaussian and derive the expression

-t(q)‘]B’+;+.,,‘,,.) -x) I’([1 p2= [ (1 -x)lrl(l

(16Eo

In this equation 5 and g are constants? depending on the parameters of the Gauss functions, and x is a function of g/x such that jj/x = [l + ~~1~‘~. It is evident from equation (168) that P, as a function of both x and z will have a minimum that will correspond to particular values x, and z,~. These can be obtained by partial differentiation of equation (168) and evaluation of x, and Z, for i3P/& = 0 and 6’P/& = 0. An approximate solution gives x, = 0.5, practically independent of 5 and g. The solutions for z, are more complicated fuuotions of { and g and for further consideration of these the original papers should be consulted [24, X34-861. * In their paper [84]

PETRASHand RAUTIANreplace (1 - [Y/x])~ by the expression (1 -

f The symbol

;?(I

-

;x!‘.

< is changed from E used by the original authors.

The shapes and intensities of infrared absorption bands

1079

In the foregoing analysis no attempt was made to apply any prior analytical procedures to correct the observed band intensity for convolution with the instrument function, and the systematic error due to this effect is directly transferred into the error equation, where it contributes to g/x. In subsequent papers [85, 861. PETRASH and RAUTIAN extended the error analysis to cases where the observed curve is first “deconvoluted”, either precisely, or by an approximation due to RAYLEIGH [87], before introducing the systematic error component into the general error equation.* The results show that if precise deconvolution is carried out, there

0-

6p:

b

_) 4

a

2-

I

I

0.2

04

FRACTIONAL

I

I

0.6

0.0

ABSORPTION,

I3

X

Fig. 30. Dependence of the absorbance error on the fractional absorption. Curve a is obtained for exact deconvolution prior to the error analysis and curve b for approximate deconvolution by RAYLEIGH'S method. These are representative curves taken from a more detailed graphical analysis in Fig. 3 of Ref. [24].

is a gain in accuracy by a factor of about six, while partial deconvolution by the Rayleigh method increases the accuracy three-fold. It must be pointed out, however, that these conclusions are based on the assumption that both the true band and the instrument function have Gauss profiles ; they are therefore not directly applicable Plots of the mean quadratic error against the to practical spectrophotometry. fractional transmission for the Gauss-Gauss convoluted curves for some of the cases considered by Petrash and Rautian are shown in Fig. 30. All these curves are comparatively flat between 30 and 70 per cent transmission. Earlier calculations of BRODERSEN [23], based on a simpler analytical treatment gave similar flat curves. The choice of the optimum absorbance for minimum error is therefore not very critical. The quantity Po2,plotted in Fig. 30 is defined as P,To2 422 * See footnote to p. 1047. [87] LORD RAYLEIGH, Phil. Mag.

+s(y112’

42, 441 (1871).

1080

hI. s.

SESHADRI

ttntl Jt.

NORMAN

JONES

It is the ratio of P2 to the square of the syst,ematic error in the measurement of I, for the deconvoluted true absorption band. In a summarizing discussion of these investigat’ions [83], PETRASH concludes that to achieve a minimum total error in the true intensity measurement! the following relaGonships should be maintained : (a) The transmission through the sample should be about 50 per cent. (1,) The systematic scanning error should be about one quarter of the syst’ematic slit error. (c) The total systematic error should be about equal to the total random error. (d) The optimum time constant (T,,,) is related to the optimum slit widt’h (~7~~) and the optimum scanning speed by the formula*

.

[I

1(’,)I t IQ (1Y _ (169) 7,,, ~ _ __ 2 (0 ‘I where [ is a coefficient depending on the shape of the slit function and 11a coefficient depending on the shape of t’he t’ime con&ant funct’ion. (e) The opt’imum slit width is given by ll’,,,

=

~UhV$, -

[

Z(dv~&) l’9 1

(170)

(Av:,.Jj

in which 21is a constant and 2 the mean quadratic error when the slit’ widt’h and time const’ant are bot’h unity.l_ In applying these considerations to laboratory practice the lateral displacement of the band caused by the first-order effect of the amplifier t’ime constant must be separately allowed for (cf. 11. 1040). It is also point’ed out that if the t’ime const,ant exceeds the opt~imum, the systematic errors increase rapidly, whereas if t’he time constant is less than optimum. the increase in the random error is comparativelysmall. It is therefore preferable t’o err in the direction of too small a time constant w&h associated high noise level. The select’ion of the optimum slit widt’h depends on the band shape and on its st,ructure, as has been emphasized earlier in this review. It is not recommended that any attempt be made to apply analytical corrections for the systematic errors caused by amplifier distortion, because this offers no advantage in comparison with t,he direct reduction of the time constant,. These conclusions, based as they are, on a complex mathemat~ical analysis of the problems of infrared spect#rophot,ometry, conform closely with those arrived at more empirically by practicing chemical spectroscopist’s in t’he laboratory. Kow that instruments of high intrinsic photometric precision are becoming available, it n-ill be interesting to see whether the greater accuracy theoretically obtainable by this sophisticated treatment of the errors will justify it’s use in laboratory practice. 1’1. C’ONCLUSIONS In the course of preparing this review. it became clear that the problems of infrared spectrophotometry must be dealt with at two distinct levels of analysis. Most fundamental are the theoretical relationships between the measured absorption curve * The symbols t The symbols

in equation in equation

(169) are changed from those used by the original

author.

(170) are rhanged from those used by the original author.

The shapes and intensit,ies of infrared

absorpion

bands

1081

and the true absorption curve. These relationships can be precisely formulated mathematically in terms of the true and apparent band shape functions and the instrument function, but even if the random errors of measurement could be tot,allp eliminated. the deconvolution procedures can not’ be dealt with exactly, as the measured band envelope will still be of complex form and usually asymmetric. The key int’egral function of equat’ion (68) has not been evaluated analytically for a real absorption band, and in dealing with it we have found it necessary to resort to model soWions in which simple Gauss, Cauchy or triangular funcbions are subst’ituted for the real ones. The immediate goal of practical infrared spectrophotometry should be the suppression of the random errors of measurement t’o the point where t’hey l~ecome negligible under operating conditions where the width of the inst’rument function is small compared to the width of the absorption band. Much progress has beeu made with this problem in recent years. The fact remains however that most’ infrared spectrophotometers must still be operated routinely under conditions where the mechanical slit width distorts the spe&um. In this respect the contrast with spect’rophot’omet,rg on condensed-phase systems in the visible and medium ultraviolet’ is very striking. Once the random errors are reduced to manageable proporbions, it next, becomes of interest to invest,igate in greater detail methods for deconvoluting t*hc apparent, band to obtain the t’rue profile. The description of the band profile in terms of t,he peak height and the half width is not sufficiently exact, for this purpose ; ib bakes no account of the band asymmetry and is not definitive even for a symmetrical l)and (cf. Fig. 13). The use of t’wo separate width parameters. as proposed l)y C'ABAN.~ and SAKDORPY, provides some measure of the asymmetry, but more exact drscriptions of both the profile and the asymmetry will be required if t’he cont’onrs are to be analysctl in detail. Unless this is done we shall be rejecting a good deal of the addit~ional informat,ion made available by the newer high resolution grat’ing spectrophotometers. The analysis of the band contour and t,he asymmetry in berms of the truncated second and third moments is being developed with t)his objective in mintl. The deconvolution of an esact,ly known apparent absorption band will st’ill 1~ ineffective unless eit’her the true band profile or the inst~rument, fWction is known. so that the third component can be derived bp mathematical analysis. One n-ap of attacking this problem would be to measure the instrument function directlp b) scanning a narrow line source, and furt(her efforts should be made to develop such sources in t’he mid-infrared. An alt,ernative. and complement~ary. approach would be to measure the true profiles of a few carefxllly selected bands on spectrometers of Such bands could then very high resolubion, capable of the necessary precision. serve as absolute st’andards for t,he true band shape and could be used toget’her with the measured band to obtain the instrument funct’ion by arithmetical analpsis. Bands suitable for t’his purpose would need to be chosen with care. They should be free from sidebands, so that the wings can be measured to a considerable distance from t,he center. They should alsn be chosen from materials that can be readily made available in a high stat,e of purity. The highlv symmetric band in a carbon disulfide solution of peiylene shown in Fig. 2 meets most of these criteria. Other bands t’hat a.re currently being investigat,ed in our laboratory for this purpose

1082

K.

S. SESHADRI and

R. NORMAN

JONES

include the 726 cm-l band in a carbon disulfide solution of anthracene, which is slightly wider and less symmetric than the perylene band, and also the band at 2234*3 cm-r in a solution of p-chlorobenzonitrile in tetrachloroethylene which is pronouncedly asymmetric but exhibits no point of inflection. Such investigations as we have made can only be regarded as exploratory. A problem of this kind calls for collaborative action on the part of several suitably equipped laboratories, comparable with the cooperative efforts that have led to the establishment of acceptable standards of wavenumber calibration. Acknowledgements-We are very grateful for the helpful advice and comments we have received from N. ALPERT, W. S. GALLAWAY, J. W. HOPKINS, R. C. LORD, R. MEC~E, N. SHEPPARD and V. Z. WILLIA~W~S who have read parts of the manuscript. Our thanks are also due to our colleagues J. B. DIGIORGIO, G. A. A. NONNENMACHER and R. A. RIPLEY with whom we have had many informal discussions. and to G. BELKOV for translations from Russian. NOTATION

A A’ A B

B B

c C D

D” E

Et E E F F

4 I

0%’

J K K K L

2”’ I L

M M

Area under the true absorbance curve (log,) Area under the true fractional absorption curve Constant relating the slit width and the signal to noise ratio Area under the apparent absorbance curve (log,) Limiting aperture of the spectrometer Minimum limiting error in the recorder measurement Capacitance of the recorder filter network Constant in the derivation of the Lorentz equation Density Absorbance at v corrected for distortion by the instrument function Area under the apparent absorbance curve (log,,). Energy of an atom or molecule in the ith quantum state. Intensity of the electromagnetic field Intensity of the radiant energy incident on the detector Function defining the systematic error in the true absorbance measurement Focal length of the spectrometer Intensity of the monochromatic radiation transmitted by the absorbing material Intensity of the monochromatic radiation incident on the absorbing material The ratio of the scanning rate to the filter time constant Constant in the derivation of the Lorentz equation Constant in Ramsay’s equation for the integrated band area Correction factor in the conversion of MECKE’S intensity unit Recorder

traces on the spectrophotometer

chart

Constant in the equation for the beam attenuator displacement Molecular weight Slope of t,he extrapolation in MECKE’S tangent construction

Page 1058 1065 1078 1068 1027 1078 1038 1020 1023 1077 1050 1017 1020 1044 1076 1027 1016 1041 1039 1021 1058 1072 1075 1042 1018 1068

The shapes and intensities of infrared absorption bands

N

N

N, N P R R R

s TV

a a’ a 6 b C ii d

J* j g i _i h h

x1 1 1

1:. *

m n

n xl

P

Number of grating braversals The Avogadro number Proportional population of molecules in the ith quantum state Peak to peak noise voltage in the recording system Mean quadratic error in the true absorbance measurement Resistance of the recorder filter network The gas constant Signal to noise ratio of the recording system Area under the spectral slit function Intensity of the total radiation transmitted by the absorbing material when the spectrophotometer is set to the wavenumber v IntensiOy of the total radiation incident on the absorbing material when the spectrophotometer is set to the wavenumber v The absolute temperature Fractional transmittance Input voltage to the filter network of the recording system Output voltage from the filter network of the recording system Mean quadratic random error in the true absorbance measurement at unit slit width and unit time constant Height parameter of a Cauchy curve Height of a Gauss curve, Voigt curve or spectral slit curve Continuous absorption below the band envelope of a recorded spectrum Half-width parameter of a Cauchy or Gauss curve Path length of the absorbing material (see also 1) Concentration Velocity of light in vacua Linear displacement of the beam attenuator Line spacing of a diffraction grating Charge on an oscillator Oscillator strength for the i 4-j transition Constant in the absorbance error equation Planck’s constant Half width of the Voigt curve d(-1) Abscissa1 unit of wavenumber for the evaluation of truncated band moments Percentage transmission error caused by scattered radiation Path length of the absorbing material (see also b) Distances measured on the spectrophotometer Mass of an oscillator Refractive index of the prism material Order of a grating spectrum Number of molecules Length of the prism base

recorder chart

1083

1033 1021 1021 1044 1077 1038 1018 1044 1057 1025 1043 1018 1066 1044 1044 1081 1021 1021 1076 1021 1064 1058 1016 1041 1033 1016 1017 1078 1017 1048 1020 1050 1043 1057 1076 1016 1034 1033 1018 1027

B.

S. SESHADRI

and R.

NORMAN

JONES

Half width of the spectral slit function Contribution of aberration and misalignment to s Contribution of diffraction to s Cont8ribution of the finite slit widt’h to s Spectral width of the entrance slit Spectra,1 widt’h of the exit slit Experimentally observed spectral slit width Time Constant in the evaluation of the optimum slit width for minimum absorbance error Velocity of a radiating molecule in the direction of observat’ion Mechanical width of a single diffraction slit Mechanical widt’h of t’he ent’rance slit Mechanical width of the exit slit Virtual mechanical slit’ width contributed by aberration and misalignment Virtual mechanical slit width contribut’ed by diffraction Effective mechanical slit widt,h Opt’imum mechanical slit width for minimum error in the true absorbance measurement Mechanical slit width where it is assumed that ~(1~= 2~‘~ Baseline co-ordinate in the Me&e tangent construction Abbreviation for,f(x) in the evaluation of the absorbance error Optimum value of R’ for minimum error in the true absorbance measurement Abbreviation forf(y) in the evaluation of t’he absorbance error A fun&ion of the sysCematic error in the absorbance error equation Optimum value of 2 for minimum error in the t#rue absorbance measurement Absorption coefficient (log c) Apical angle of the prism Statistical parameter for band skewness Statistical parameter for band shape Statis6ical parameter for band skewness Parameter describing departure from Gauss profile Molecular extinction coefficient (log,,) Molecular extinction coefficienb (logy) Proport,ionality constant in bhe absorbance error equation Shape coefficient for the recorder time constant function Slope of the Wilson--Wells extrapolation Diffraction angle of a grating Time of collision in the derivation of the Lorentz equation Virtual slit constant in the Gauss slit function Wavelength Magnetic permeability in vacua Matrix element of the electric moment for the i - j transition

1032 1032 1032 1032 10’9 1029

1037 1020 1081 101x 1027

1031 1031 1033 1033 1033 1081 1031 1066

1077 1078 1077 1075 1078 1090 1034 1054 1050 1054 1050 1073 1067 107S 1081 1070 1033 1020 1031 1016 1038 1020

The shapes and intensities of infrared absorption bands

The rt,h moment of an absorption band about the maximum ordinate Wavenumber of radiation in cm-l Wavenumber at a point on an absorption or emission band Wavenumber setting on a spectrophotometer scale Wavenumber at a band maximum Frequency of radiation in set-r Shape coefficient for the slit function Collision diamet,er of a molecule in the vapor or liquid phase Width parameter in t)he Gauss equat,ion Time constant of the filter network Optimum value of 7 for minimum error in t,he true absorbance measurement Mean t’inie between molecular collisions Normalizing const’ant in the instrument function Cauchy proportionalit,y parameter in the T’oigt function Gauss proportionality parameter in the Voigt function Frequency of radiation in rad/sec Heisenberg uncertainty width of the ith quantum st’ate for an atom or molecule Half band width of t,he signal pulse applied to the filter network of a recorder amplifier VV’idth of an absorption or emission line or band at half maximal intensity A?,, as modified by convolution with the instrument function Alrl;, as modified bg convolution with the spectral slit component of the instrument function True half width of an absorption or emission line or band Half tvidt’h of a diffract’ion band Len&i of the base intercept in Me&e’s tangent construction Half width of t’he spectral slit fun&ion Funct,ion descrilung the profile of an apparent absorpCion band Funct~ion defining the proportional coruribution of the diffraction term to the spectral slit widt’h of a prism spe&rometer Fourier transform of ,f( 11~) FunctJion describing the profile of the t,rue absorption band Fourier transform of n(v) The instrurllent function Fourier transform of X(r - IV!) The sprctral slit function Funct~ioll descril)irlg the true fract’ional absorptiou Function describing the apparent fractional absorption

1085

1049 1016 1027

1027 1016 1016 1OYl 1023 1022 1041 1081 1020

1076 1048 1018 1016

1017 1039 1016 1057 1039 1037 4 lo”7 1065

102!) 101x 103-i 1015 1028 1046 102s 1046 1027 107ti 1076