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Correction of finite slit width effects on Raman line widths
Spectrochimica Acta. Vol. 36A, pp. 341 f~ 344 ~ergamon press Ltd., 1980. Printed in Great Britain
Correction of finite slit width effects on Raman line widths KAZUTOSHI
National
Chemical
Laboratory
and
TANABE
for Industry (Received
l-1-5
JIRO
HIRMSHI
Honmachi,
6 March
Shibuyaku,
Tokyo
151, Japan
1979)
Abstract-An equation is derived for the correction of the finite slit width effect on line widths of Lorentzian Raman lines. It holds for up to S/S, = 0.7, more applicable than those proposed previously. A method is also urouosed for obtainine the true isotropic line width from observed parallel and perpendicular polarized spectra.
triangular shape for not so wide slits. A Gaussian slit function as well has often been used in the computation of convolution integrals in the preceding works. Therefore, we computed the convolution integrals for a Lorentzian line profile with triangular- and Gaussian slit functions.
INTRODUCTION
In general, spectral slit widths of now available Raman spectrometers are not fully small as compared with line widths of isotropic Raman spectra, and therefore, the effects of the finite slit widths can be quite large on observed line widths of narrow Raman lines. A great number of techniques [l-28] have been proposed for the correction of the slit effects on spectral line shapes, but most of them require a time-consuming numerical computation. Recently, DIJKMAN et al. [27] solved convolution integrals of a Lorentzian line profile with a triangular slit function, and derived 6, =S,Jl-2(S/&J2,
RESULTS
AND DISCUSSION
Results of the numerical integrations for the convolution of a Lorentzian line with a triangular or Gaussian slit function are summarized in Table 1. From the dependence of 6,/6, on S/6,, we found that the following equation reproduces quite well the computed values
(1)
6, = 6,[1 -(S/SJ2].
where 6,, 6, and S are the full widths (FWHH) of the true line shape, the apparent (observed) line shape, and the slit function, respectively. This does not hold for S/S, >0.5, inapplicable to measurements of narrow Raman lines. We computed convolution integrals numerically in order to examine the dependence of 6J6, on S/S,, and derived a simple and more applicable equation for the correction of the slit effect on line widths of Lorentzian Raman lines. In addition, we studied on the method to obtain the true isotropic line width from observed parallel and perpendicular polarized spectra. In this paper, we report the results of numerical integrations and the methods for the correction of the slit effects.
(2)
The ratios 6,/S,, from equations (1) and (2) are also given in Table 1. Equation (2) corresponds to the Table based
CALCULATION
Convolution integrals were computed numerically on a Lorentzian line profile and two slit functions, triangular and Gaussian. Recent theoretical studies [29-381 on vibrational or reorientational relaxation suggest that the true line profile of the Raman spectrum of a liquid is not described by the Fourier transform of a simple exponential correlation function. However, the deviation of observed Raman line shapes from a Lorentzian function seems quite small as long as ordinary liquids (except hydrogen-bonded systems) are concerned. It has been recognized that the profile of the slit function is usually described by a slightly rounded
1. Values of 8,/S, for the numerical computation on triangular and Gaussian slit functions, and for equations (1) and (2)
S/6,
Triangular
0.05
0.998
0.10
0.990
0.15
EC!.(Z)
RI.(l)
0.997
0.998
0.998
0.989
0.990
0.990
0.978
0.976
0.978
0.977
0.20
0.960
0.957
0.960
0.959
0.25
0.937
0.933
0.938
0.935
0.30
0.909
0.904
0.910
0.906
0.35
0.875
0.870
0.878
0.869
0.40
0.835
0.832
0.840
0.825
0.45
0.790
0.790
0.798
0.771
0.50
0.738
0.745
0.750
0.707
0.55
0.691
0.696
0.698
0.629
0.60
0.637
0.645
0.640
0.529
0.65
0.572
0.589
0.578
0.394
0.70
0.495
0.526
0.510
0.141
0.75
0.409
0.458
0.438
0.80
0.318
0.385
0.360
0.85
0.228
0.307
0.278
0.90
0.145
0.224
0.190
0.95
0.076
0.137
0.098
i : Imaginary 341
GZ+“SSlXl
values
obtained.
KAZUTOSHI TANABE
342
approximate of equation (1) for small values of S/S,, but is more applicable than equation (l), almost up to S/S, =0.7. The ratios S/6, =OS and 0.7 correspond to S/6, =0.68 and 1.41, respectively, which means that the working range of equation (2) is more than twice that of equation (1). Spectral slit widths of now available Raman spectrometers are of the order of 1 cm-‘, while line widths of 2 cm-’ or smaller are not peculiar in highly polarized Raman lines. Equation (2) is applicable to measurements of such narrow lines. When the depolarization ratio of a Raman line is not so small, equation (2) is not applied to the determination of the true isotropic line width from observed parallel component spectra. This is because the parallel spectrum is a sum of the isotropic and anisotropic spectra [39-401 I”“(V) =&.(v)+4(v)
(3)
and, consequently, the line shape of the parallel spectrum is not described by a Lorentzian function. We computed convolution integrals for parallel component lines consisting of various sets of I, and I,, and found the following procedures for the determination of the true isotropic line width from observed parallel and perpendicular line shapes: (1) from the line width ratio Svv,a/GvH,, and the peak intensity ratio yvH,Jyvv,a of the observed parallel (VV) and perpendicular (VH) spectra, coefficients c, and cZ are determined by the use of Table 2;
Table
2. Values
of coefficients
c1 (upper)
and c2 (lower)
as a function of 6 vv,,/S vH,a and ~vn,J~v~,~
> ,
YYHa/Yvi 0.05 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
0.931 0.044 0.932 0.043 0.934 0.042 0.936 0.040 0.939 0.038 0.942 0.036 0.946 0.033 0.949 0.030 0.953 0.027 0.958 0.024 0.962 0.020 0.966 0.017
0.10 0.857 0.103 0.859 0.101 0.863 0.098 0.867 0.094
0.15
0.777 0.185 0.780 0.182 0.785 0.176 0.791 0.169 0.800 ::o";; 0.159 0.810 0,:::; 0.148 0.886 0.821 0.076 0.134 0.894 0.834 0.068 0.119 0.846 E:E 0.105 0.912 0.860 0.051 0.091 0.921 0.874 0.043 0.077 0.930 0.888 0.036 0.063 0.939 0.903 ::',::, 0.029 0.049 0.975 0.948 0.918 0.011 0.022 0.036 0.980 0.957 0.932 0.008 0.017 0.027 0.984 0.966 0.005 0.012
a
0.20
0.25
0.30
0.686 0.309 0.690 0.303 0.697 0.293 0.706 0.278 0.718 0.259 0.732 0.239 0.747 0.218 0.764 0.193 0.783 0.169 0.801 0.146 0.821 0.122 0.842 0.098 0.862 0.078 0.882 0.060 0.903 0.042 0.923 0.027
0.581 0.504 0.586 0.491 0.596 0.471 0.608 0.445 0.624 0.414 0.642 0.379 0.663 0.341 0.685 0.301 0.709 0.262 0.735 0.223 0.760 0.186 0.787 0.150 0.814 0.119 0.842 0.087 0.868 0.064
0.452 0.851 0.460 0.824 0.472 0.784 0.490 0.732 0.511 0.671
0.895 0.042
",:Z 0.562 0.537 0.592 0.469 0.623 0.403 0.655 0.339 0.689 0.280 i:E? 0.758 0.176 0.793 0.131 0.828 0.093 0.863 0.061
and JIROHIRAISHI
(2) the apparent lated by
isotropic
6 ‘X.(1 = S,,,[c,
+
line width S,,, is calcu-
c*c%""."Yl;
(4)
(3) the true isotropic line width a,., is obtained from S,,, by the use of equation (2). Coefficients c, and c2 were determined for S/6,,, CO.7 with errors of less than 1% as shown in Table 3.
Table
3. Values of +_/6vv,, based on the numerical integration and on equation (4)
0.10
YVH,a YW,a
+
0.10
0.00
6
w,a
a,ai+‘v,a
Computed
Eq.(4i
0.20 0.40 0.60
0.858 0.864 0.875 0.896
0.859 0.863 0.875 0.895
0.10
0.30
0.00 0.20 0.40
0.458 0.495 0.589
0.460 0.493 0.592
0.50
0.40
0.00 0.20 0.40
0.432 0.470 0.571
0.433 0.468 0.574
The apparent isotropic line width S,,, can also be obtained from observed parallel and perpendicular spectra by means of graphical subtraction. However, it requires a digitized data acquisition system or a laborious manipulation procedure. On the other hand, the above procedures enable us to reach the true isotropic line width quite easily. We applied the method to Ye at 1157 cm--’ of cyclohexane, and the results are summarized in Table 4. The intensity of this line is rather low, and the line width is comparatively small. Therefore it seems quite difficult to determine the true isotropic line width by means of the extrapolation method as described below.
Table 4. Measurements
of the v4 line width of cyclohexane
S
0.5
cm-1
%I,.3
2.68
cm
%H,a
10.5 3275 147.5
-1
Crn-1
counts counts
0.255 0.045 0.945 0.034 0.187 2.54
cm-l
0.197 2.44
cm-l
Correction of finite slit width effects on Raman line widths
In the preceding studies on measurements of Raman line widths, the slit effects were often neglected by setting the slit widths so that the condition of S/S, < 0.2 might be satisfied. However, since the slit effects always give a systematic error to observed widths, it is inadequate to neglect the effects in the analysis of experimental line widths. The extrapolation technique was used for the correction of the slit effects in some of the previous studies. Here it is assumed that the true line width can be obtained by the extrapolation of observed line widths to a zero slit width 6, =&i”, 6,. This method is liable to be involved in experimental difficulties, because the value given by the extrapolation is much influenced by measurements at narrow slits of low S/N ratios, and because the extrapolation line is not linear as shown in Fig. 1. A large number of numerical methods were proposed for the correction of the slit effects on spectral line profiles. Usuallv the shape of the slit finction is assumed, and the &ue line shape may be obtained from observed spectra at wide slits of high S/N ratios. However, many of these methods require a time-consuming numerical computation, and have not so far been used in the preceding papers on measurements of Raman line widths. On the other hand, the methods based on equations (2) and (4) can give the true line width quite rapidly. Here it should be noticed that the use of equation (2) or (4) is not desirable for a large value of S/6,. This is because the magnitude of correction depends on the value of S, and the effect of the experimental uncertainty of S on the magnitude of correction becomes large for a large value of S. Therefore, equation (2) or (4) should be
applied to experimental line widths observed at as narrow slits as possible. Equations (2) and (4) have been derived empirically on the basis of the numerical computation of convolution integrals, while equation (1) was derived analytically from the solution of the convolution equation. The discrepancy between the values for equations (1) and (2) as seen in Table 1 may be attributed to some approximations used in the derivation of equation (1). DIJKMAN et al. [27] compared the values of slit correction based on equation (1) with the data reported for the slit effects in i.r. spectrophotometry. Such a comparison is inappropriate, because the slit effects in i.r. spectra depend not only on S/S, but also on the peak absorbance. Although we tried to derive an equation for the i.r. case, it seems difficult to express the correction in a simple form similar to equation (2). REFERENCES [I] H. C. BERGER and P. H. VAN CITI-ERT, 2. Phys. 79, 722 (1932); 81, 428 (1933). [2] A. C. HARDY and F. M. YOUNG, J. Opt. Sot. Am. 39, 265 (1949). [3] G. PIRLOT, Bull. Sot. Chim. Belg. 59, 352 (1950). [4] D. A. RAMSAY, J. Am. Chem. Sot. 74, 72 (1952). [5] S. BRODERSON, J. Opt. Sot. Am. 43B, 877 (1953); 44A, 22 (1954). [6] H. J. KOSTKOWSKIand A. M. BASS, J. Opt. Sot. Am.
46, 1060 (1956).
[7] S. G. RAKJTIN,Sou. Phys. Uspekhi 66, 245 (1958). [8] A. CABANA and C. SANDORFY, Spectrochim. Acta
16, 335 (1960). [9] A. L. KHIDIR and 18, 1629 (1962).
[lo]
W. F. HERGET, W. E. DEEDS, N. M. GAIL.AR, R. J. LOVELL and A. H. NELSEN, J. Opt. Sot. Am. 52,1113
(1962).
ll21 [I41 [I51 Ml [I71 [W
[I91 PO1 WI WI P31 b41 D51 [‘W
of S,/S, on SS,.
J. C. DECIUS, Spectrochim. Acta
[l l] J. S. ROLLETT and L. A. HIGGS, Proc. R. Sot. 79, 87
[13j
Fig. 1. Dependence
343
[271
(1962).
A. F.‘BONDAREV, Opt. Spectrosc. 12, 282 (1962). L. C. ALLEN, H. M. GLADNEY and S. H. GLARUM, J. Chem. Phys. 40, 3135 (1964). P. MONTIGNY, Spectrochim. Acra 20, 1373 (1964). A. RBSELER, Infrared Phys. 5, 37, 51 (1965). J. S. CHALLICE and G. M. CLARKE, Spectrochim. Acta 21, 79 (1965). R. N. JONES, R. VENKATARAGHAVAN and J. W. HOPKINS, Spectrochim. Acta 23A, 925 (1967). B. D. SAKSENA, K. C. AGARWAL, D. R. PAHWA and M. M. PRADHAN, Specrrochim. Acta 24A, 1981 (1968). J. DELTOUR, Infrared Phys. 9, 125 (1969). P. A. JANSSON, R. H. HUNT and E. K. F’LYLER, J. Opt. Sot. Am. 60, 596 (1970). J. SZOKE, Chem. Phys. Letr. 15, 404 (1972). G. HORLICK, Appl. Spectrosc. 26, 395 (1972). A. GOLDMAN and P. ALON, Appl. Spectrosc. 27, 50 (1973). G. BROUWER and J. A. J. JANSEN, Anal. Chem. 45, 2239 (1973). I. R. HILL and D. STEELE, J. Chem. Sot. Faraday II, 70, 1233 (1974). B. MOHOS, M. ZOBRIS~ and A. VON ZELEWSKY, J. Chem. Phys. 60, 4633 (1974). F. G. DIJKMAN and J. H. VAN DER MAAS, Appl. Spectrosc. 30, 545 (1976).
344
KAZ~TOSHI TANABE and JIRO HIRAISHI
[28] N. LACOME and A. LEVY, J. Mol. Spcctrosc. 71, 175 (1978). [29] A. V. RAKOV, Opt. Spectrosc. 7, 128 (1959). [30]
Y.LERoY,E.
CONSTANT~II~P.DESPLANQUES,J.Chim.
Phys. 64, 1499 (1967). [31] M. CONSTANT, J. Chem. Phys. 58, 4030 (1973). [32] G. D~GE, 2. Nuturf. 28a, 919 (1973). [33] K. A. VALIEV and E. N. IVANOV, Opr. Specvosc. 34, 87 (1973). [34] P. C. M. VAN WOERKOM, J. DE BLEYSER, M. DE ZWART and J. C. LEY~E, Chem. Phys. 4,236 (1974).
[35] M. CONSTANT, R. FAUQIJEMBERCLJE and P. DESCHEERDER,.I. Chem. Phys. 64, 667 (1976). [36] W. G. ROTHSCHILD,J. Chem. Phys. 65,455 (1976). [37] W. T. COEFFEY, Chem. Phys. Left. 52, 394 (1977). [38] F. DBGE, R. ARNDT and A. KHWEN, Chem. Phys. 21,
53 (1977). [39] F. T. BARTOLI and T. A. LITOVITZ, J. Chem. Phys. 56, 404 (1972). [40] L. A. NAFIE and W. L. PETICOLAS,J. Chem. Phys. 57, 3145 (1972).
Correction of finite slit width effects on Raman line widths KAZUTOSHI
National
Chemical
Laboratory
and
TANABE
for Industry (Received
l-1-5
JIRO
HIRMSHI
Honmachi,
6 March
Shibuyaku,
Tokyo
151, Japan
1979)
Abstract-An equation is derived for the correction of the finite slit width effect on line widths of Lorentzian Raman lines. It holds for up to S/S, = 0.7, more applicable than those proposed previously. A method is also urouosed for obtainine the true isotropic line width from observed parallel and perpendicular polarized spectra.
triangular shape for not so wide slits. A Gaussian slit function as well has often been used in the computation of convolution integrals in the preceding works. Therefore, we computed the convolution integrals for a Lorentzian line profile with triangular- and Gaussian slit functions.
INTRODUCTION
In general, spectral slit widths of now available Raman spectrometers are not fully small as compared with line widths of isotropic Raman spectra, and therefore, the effects of the finite slit widths can be quite large on observed line widths of narrow Raman lines. A great number of techniques [l-28] have been proposed for the correction of the slit effects on spectral line shapes, but most of them require a time-consuming numerical computation. Recently, DIJKMAN et al. [27] solved convolution integrals of a Lorentzian line profile with a triangular slit function, and derived 6, =S,Jl-2(S/&J2,
RESULTS
AND DISCUSSION
Results of the numerical integrations for the convolution of a Lorentzian line with a triangular or Gaussian slit function are summarized in Table 1. From the dependence of 6,/6, on S/6,, we found that the following equation reproduces quite well the computed values
(1)
6, = 6,[1 -(S/SJ2].
where 6,, 6, and S are the full widths (FWHH) of the true line shape, the apparent (observed) line shape, and the slit function, respectively. This does not hold for S/S, >0.5, inapplicable to measurements of narrow Raman lines. We computed convolution integrals numerically in order to examine the dependence of 6J6, on S/S,, and derived a simple and more applicable equation for the correction of the slit effect on line widths of Lorentzian Raman lines. In addition, we studied on the method to obtain the true isotropic line width from observed parallel and perpendicular polarized spectra. In this paper, we report the results of numerical integrations and the methods for the correction of the slit effects.
(2)
The ratios 6,/S,, from equations (1) and (2) are also given in Table 1. Equation (2) corresponds to the Table based
CALCULATION
Convolution integrals were computed numerically on a Lorentzian line profile and two slit functions, triangular and Gaussian. Recent theoretical studies [29-381 on vibrational or reorientational relaxation suggest that the true line profile of the Raman spectrum of a liquid is not described by the Fourier transform of a simple exponential correlation function. However, the deviation of observed Raman line shapes from a Lorentzian function seems quite small as long as ordinary liquids (except hydrogen-bonded systems) are concerned. It has been recognized that the profile of the slit function is usually described by a slightly rounded
1. Values of 8,/S, for the numerical computation on triangular and Gaussian slit functions, and for equations (1) and (2)
S/6,
Triangular
0.05
0.998
0.10
0.990
0.15
EC!.(Z)
RI.(l)
0.997
0.998
0.998
0.989
0.990
0.990
0.978
0.976
0.978
0.977
0.20
0.960
0.957
0.960
0.959
0.25
0.937
0.933
0.938
0.935
0.30
0.909
0.904
0.910
0.906
0.35
0.875
0.870
0.878
0.869
0.40
0.835
0.832
0.840
0.825
0.45
0.790
0.790
0.798
0.771
0.50
0.738
0.745
0.750
0.707
0.55
0.691
0.696
0.698
0.629
0.60
0.637
0.645
0.640
0.529
0.65
0.572
0.589
0.578
0.394
0.70
0.495
0.526
0.510
0.141
0.75
0.409
0.458
0.438
0.80
0.318
0.385
0.360
0.85
0.228
0.307
0.278
0.90
0.145
0.224
0.190
0.95
0.076
0.137
0.098
i : Imaginary 341
GZ+“SSlXl
values
obtained.
KAZUTOSHI TANABE
342
approximate of equation (1) for small values of S/S,, but is more applicable than equation (l), almost up to S/S, =0.7. The ratios S/6, =OS and 0.7 correspond to S/6, =0.68 and 1.41, respectively, which means that the working range of equation (2) is more than twice that of equation (1). Spectral slit widths of now available Raman spectrometers are of the order of 1 cm-‘, while line widths of 2 cm-’ or smaller are not peculiar in highly polarized Raman lines. Equation (2) is applicable to measurements of such narrow lines. When the depolarization ratio of a Raman line is not so small, equation (2) is not applied to the determination of the true isotropic line width from observed parallel component spectra. This is because the parallel spectrum is a sum of the isotropic and anisotropic spectra [39-401 I”“(V) =&.(v)+4(v)
(3)
and, consequently, the line shape of the parallel spectrum is not described by a Lorentzian function. We computed convolution integrals for parallel component lines consisting of various sets of I, and I,, and found the following procedures for the determination of the true isotropic line width from observed parallel and perpendicular line shapes: (1) from the line width ratio Svv,a/GvH,, and the peak intensity ratio yvH,Jyvv,a of the observed parallel (VV) and perpendicular (VH) spectra, coefficients c, and cZ are determined by the use of Table 2;
Table
2. Values
of coefficients
c1 (upper)
and c2 (lower)
as a function of 6 vv,,/S vH,a and ~vn,J~v~,~
> ,
YYHa/Yvi 0.05 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
0.931 0.044 0.932 0.043 0.934 0.042 0.936 0.040 0.939 0.038 0.942 0.036 0.946 0.033 0.949 0.030 0.953 0.027 0.958 0.024 0.962 0.020 0.966 0.017
0.10 0.857 0.103 0.859 0.101 0.863 0.098 0.867 0.094
0.15
0.777 0.185 0.780 0.182 0.785 0.176 0.791 0.169 0.800 ::o";; 0.159 0.810 0,:::; 0.148 0.886 0.821 0.076 0.134 0.894 0.834 0.068 0.119 0.846 E:E 0.105 0.912 0.860 0.051 0.091 0.921 0.874 0.043 0.077 0.930 0.888 0.036 0.063 0.939 0.903 ::',::, 0.029 0.049 0.975 0.948 0.918 0.011 0.022 0.036 0.980 0.957 0.932 0.008 0.017 0.027 0.984 0.966 0.005 0.012
a
0.20
0.25
0.30
0.686 0.309 0.690 0.303 0.697 0.293 0.706 0.278 0.718 0.259 0.732 0.239 0.747 0.218 0.764 0.193 0.783 0.169 0.801 0.146 0.821 0.122 0.842 0.098 0.862 0.078 0.882 0.060 0.903 0.042 0.923 0.027
0.581 0.504 0.586 0.491 0.596 0.471 0.608 0.445 0.624 0.414 0.642 0.379 0.663 0.341 0.685 0.301 0.709 0.262 0.735 0.223 0.760 0.186 0.787 0.150 0.814 0.119 0.842 0.087 0.868 0.064
0.452 0.851 0.460 0.824 0.472 0.784 0.490 0.732 0.511 0.671
0.895 0.042
",:Z 0.562 0.537 0.592 0.469 0.623 0.403 0.655 0.339 0.689 0.280 i:E? 0.758 0.176 0.793 0.131 0.828 0.093 0.863 0.061
and JIROHIRAISHI
(2) the apparent lated by
isotropic
6 ‘X.(1 = S,,,[c,
+
line width S,,, is calcu-
c*c%""."Yl;
(4)
(3) the true isotropic line width a,., is obtained from S,,, by the use of equation (2). Coefficients c, and c2 were determined for S/6,,, CO.7 with errors of less than 1% as shown in Table 3.
Table
3. Values of +_/6vv,, based on the numerical integration and on equation (4)
0.10
YVH,a YW,a
+
0.10
0.00
6
w,a
a,ai+‘v,a
Computed
Eq.(4i
0.20 0.40 0.60
0.858 0.864 0.875 0.896
0.859 0.863 0.875 0.895
0.10
0.30
0.00 0.20 0.40
0.458 0.495 0.589
0.460 0.493 0.592
0.50
0.40
0.00 0.20 0.40
0.432 0.470 0.571
0.433 0.468 0.574
The apparent isotropic line width S,,, can also be obtained from observed parallel and perpendicular spectra by means of graphical subtraction. However, it requires a digitized data acquisition system or a laborious manipulation procedure. On the other hand, the above procedures enable us to reach the true isotropic line width quite easily. We applied the method to Ye at 1157 cm--’ of cyclohexane, and the results are summarized in Table 4. The intensity of this line is rather low, and the line width is comparatively small. Therefore it seems quite difficult to determine the true isotropic line width by means of the extrapolation method as described below.
Table 4. Measurements
of the v4 line width of cyclohexane
S
0.5
cm-1
%I,.3
2.68
cm
%H,a
10.5 3275 147.5
-1
Crn-1
counts counts
0.255 0.045 0.945 0.034 0.187 2.54
cm-l
0.197 2.44
cm-l
Correction of finite slit width effects on Raman line widths
In the preceding studies on measurements of Raman line widths, the slit effects were often neglected by setting the slit widths so that the condition of S/S, < 0.2 might be satisfied. However, since the slit effects always give a systematic error to observed widths, it is inadequate to neglect the effects in the analysis of experimental line widths. The extrapolation technique was used for the correction of the slit effects in some of the previous studies. Here it is assumed that the true line width can be obtained by the extrapolation of observed line widths to a zero slit width 6, =&i”, 6,. This method is liable to be involved in experimental difficulties, because the value given by the extrapolation is much influenced by measurements at narrow slits of low S/N ratios, and because the extrapolation line is not linear as shown in Fig. 1. A large number of numerical methods were proposed for the correction of the slit effects on spectral line profiles. Usuallv the shape of the slit finction is assumed, and the &ue line shape may be obtained from observed spectra at wide slits of high S/N ratios. However, many of these methods require a time-consuming numerical computation, and have not so far been used in the preceding papers on measurements of Raman line widths. On the other hand, the methods based on equations (2) and (4) can give the true line width quite rapidly. Here it should be noticed that the use of equation (2) or (4) is not desirable for a large value of S/6,. This is because the magnitude of correction depends on the value of S, and the effect of the experimental uncertainty of S on the magnitude of correction becomes large for a large value of S. Therefore, equation (2) or (4) should be
applied to experimental line widths observed at as narrow slits as possible. Equations (2) and (4) have been derived empirically on the basis of the numerical computation of convolution integrals, while equation (1) was derived analytically from the solution of the convolution equation. The discrepancy between the values for equations (1) and (2) as seen in Table 1 may be attributed to some approximations used in the derivation of equation (1). DIJKMAN et al. [27] compared the values of slit correction based on equation (1) with the data reported for the slit effects in i.r. spectrophotometry. Such a comparison is inappropriate, because the slit effects in i.r. spectra depend not only on S/S, but also on the peak absorbance. Although we tried to derive an equation for the i.r. case, it seems difficult to express the correction in a simple form similar to equation (2). REFERENCES [I] H. C. BERGER and P. H. VAN CITI-ERT, 2. Phys. 79, 722 (1932); 81, 428 (1933). [2] A. C. HARDY and F. M. YOUNG, J. Opt. Sot. Am. 39, 265 (1949). [3] G. PIRLOT, Bull. Sot. Chim. Belg. 59, 352 (1950). [4] D. A. RAMSAY, J. Am. Chem. Sot. 74, 72 (1952). [5] S. BRODERSON, J. Opt. Sot. Am. 43B, 877 (1953); 44A, 22 (1954). [6] H. J. KOSTKOWSKIand A. M. BASS, J. Opt. Sot. Am.
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