On the Choice of Standards and Methods for Calibrating Prism Infrared Spectrometers

A. N. Aleksandrov and V. A. Nikitin

The rapid development of infrared spectroscopy, the elaboration of new methods of investigation, and the establishment by domestic industry of serial production of infrared spectrometers have led to a considerable expansion of the field of their application. Whereas until recently infrared spectrometers were produced only by research laboratories, in single specimens and for special purposes, serial instruments—IKS-11, IKS-6, IKS-2—are now becoming ever more widely used in factories and in scientific institutions for solving a great variety of problems in physics, chemistry, biology, medicine, production technology, etc. In connection with this, the question of the choice of standards and methods for calibrating infrared spectrometers becomes urgent.

The absence of appropriate reviews in the domestic literature has now led to a noticeable discrepancy among the results of measurements of wavelengths (or frequencies) made by different investigators, since different and not always sufficiently reliable literature data are used for calibrating instruments. Therefore, the task of selecting and establishing unified standards for calibrating infrared spectrometers acquires special importance. Our work does not claim to provide an exhaustive solution of this problem; it is only a first attempt and presupposes the further development of work in this direction with the aim of establishing all-Union standards.

Since our main topic—the calibration of infrared prism spectrometers—includes a fairly wide range of questions, we consider it advisable to discuss some of them in appendices. The appendices also include the necessary data on the refractive indices of the principal materials used in infrared spectroscopy.

A. N. Aleksandrov and V. A. Nikitin

§ 1. CHOICE OF NORMALS

1. At present the working range of prism infrared spectrometers covers the frequency interval from \(200\ \mathrm{cm}^{-1}\) to the boundary of the visible region \(14\,000\ \mathrm{cm}^{-1}\) (0.8–50.0 microns). In this interval prisms made of various materials are used, depending on their transparency and dispersion (Fig. 1). These same materials

Figure labels:
Material of prism: Glass; Quartz; LiF; CaF\(_2\); NaCl; KCl; KBr; CsBr; CsJ; KRS-5.
Most advantageous region of application (in microns): approximately 0.4; 0.2; 0.2–5; 0.3–9; 5–15; 10–20; 15–25; 15–38; 25–50; 21–40.

Fig. 1. Principal working regions of prisms used in infrared spectroscopy.

are used for the manufacture of protective windows and cuvettes for infrared analysis, but their long-wavelength transmission limit, owing to the small thickness, lies beyond the corresponding limit for prisms (Fig. 2). The designs of modern infrared spectrometers usually provide automatic recording of the spectrum and the possibility of passing from one working region to another by changing prisms. Here we shall consider mainly the questions of calibrating instruments of precisely this type.

The problem of calibrating a prism spectral instrument in frequencies (or wavelengths) can in general be solved in two ways:

The first way is to calculate the path of rays of different wavelengths according to the optical scheme of the instrument and the refractive indices of the prism material.

The second way is to calibrate the instrument by standard spectra (normals). The first way requires cumbersome calculations, does not possess high accuracy, and therefore is rarely used in practice, but in individual cases it is indispensable. The corresponding calculations can then be carried out using the values of the refractive indices of various materials given in Tables X–XVIII and in the literature data\(^{1,2}\).

The second way, which has at present become widely used, usually does not ensure sufficient accuracy of the result, mainly because of the lack of reliable standards for calibration. The difficulty lies in the fact that in the infrared

regions of the spectrum all substances in the gaseous state have absorption bands of complex structure. In practice, the number, shape, and position of the individual recorded maxima in each absorption band depend strongly on the resolving power of the instrument. In the spectra of liquids, on the contrary, broad structureless absorption bands are observed, often asymmetric because of the superposition of neighboring broad bands and having no distinct maximum. Thus, in the infrared region of the spectrum there are few objects possessing individual, narrow, symmetrical absorption lines (for example, analogous to the mercury emission lines in the visible region of the spectrum), whose maximum positions would be reproduced without distortion on instruments of different resolving power.

Fig. 2. Transmission of various materials in the infrared region of the spectrum.

The maxima of these lines would be reproduced without distortion on instruments of different resolving power.

The use of large high-resolution infrared spectrometers with diffraction gratings has increased the accuracy of measurements in the infrared region of the spectrum many times over and has made it possible to determine the true contour of absorption bands and to reveal their structure completely. It also became possible to use the results of spectral measurements on diffraction instruments for calibrating prism instruments. This was done in the best way by Ethen, Kao, and Randall,^3 who chose the resolving power of the grating instrument to be equal to the resolving power of the prism instrument, which completely eliminated the errors inevitable when comparing spectral records obtained on instruments of different resolving power, owing to shifts of maxima and changes in the shape of the absorption curve.

The most complete survey of results obtained abroad until recently is the work of Downie and co-workers,^4 which forms the basis of the first paragraph of the present article.

1. We propose to divide the “standard spectra” recommended for calibration into three classes.

For standards of class I it is advisable to take such lines, measured precisely on an instrument with a diffraction grating, or individual maxima in absorption or emission spectra, which are single, symmetrical, and sufficiently narrow. The first requirement is satisfied by those lines and maxima whose structure cannot in general be revealed by a prism instrument because of the presence of a certain practical limit of resolving power (see Appendix 1). The last requirement cannot be formulated rigorously. In practice it means that the maximum of the given line must be so sharp that locating its position does not introduce an additional error into the measurements. Examples of standards of class I may be the emission lines of mercury and the rotational structure of the absorption bands of simple molecules: HCl at \(2600\ \mathrm{cm}^{-1}\) and CO at \(2000\ \mathrm{cm}^{-1}\) (Figs. 6, 9, 11).

As standards of class II one may take such lines or maxima in absorption or emission spectra which do not satisfy one of the above requirements, but can be used for the calibration of prism instruments with the same accuracy, provided that the records obtained during calibration are compared with standard ones. As a rule, rotational structures of the absorption bands of gases and vapors (\(\mathrm{CO_2}\), \(\mathrm{H_2O}\), \(\mathrm{NH_3}\), HCl, HBr, CO, \(\mathrm{CH_4}\)) may serve as standards of both classes in the infrared region of the spectrum.

To the category of standards of class III one may assign such absorption bands which do not satisfy the requirements formulated above, but in a number of cases can be used for approximate calibration.

The position of all absorption and emission lines and bands will be given below on the frequency scale, more precisely as wavenumbers

\[ \nu\ \mathrm{cm}^{-1}=\frac{10^4}{\lambda\ (\text{in }\mu)}. \]

All frequency values have been reduced to vacuum. If no such reduction had been made in the published data used, the corresponding corrections were introduced by us on the assumption that the measurements were carried out in dry air at \(760\ \mathrm{mm}\) Hg and \(+20^\circ\mathrm{C}\). For the case of calibration of prism infrared spectrometers, the accuracy of the result under this assumption is quite sufficient (see Appendix 2).

1. Let us consider separately the working regions of various prisms and select the corresponding standards for calibration of these regions.

Glass, quartz

Glass and quartz prisms are used in the nearest infrared region, from the boundary of the visible spectrum to \(4000\ \mathrm{cm}^{-1}\) (quartz to \(3400\ \mathrm{cm}^{-1}\)), where absorption bands are observed corresponding to overtones and combination frequencies of vibrations.

groups XH. In the region above \(6500\ \mathrm{cm}^{-1}\) glass gives the best results, and below \(6500\ \mathrm{cm}^{-1}\)—quartz (cf. the dispersion values in Tables X and XI).

For calibrating this region, certain absorption bands of liquids and other objects may be used,

Fig. 3. Absorption of liquid chloroform (solid curve) and benzene (dashed curve) (IKS-11 with prism F-1)\(^5\).

Frequencies in \(\mathrm{cm}^{-1}\)

\[ \begin{array}{rrrr} (1)\ 11481 & (7)\ 8703 & (12)\ 5988 & (17)\ 4658\\ (2)\ 11334 & (8)\ 8244 & (13)\ 5921 & (18)\ 4630\\ (3)\ 10357 & (9)\ 7082 & (14)\ 5382 & (19)\ 4575\\ (4)\ 9852 & (10)\ 6671 & (15)\ 4869 & (20)\ 4380\\ (5)\ 9381 & (11)\ 6557 & (16)\ 4735 & (21)\ 4227\\ (6)\ 8787 & & & \end{array} \]

(Figs. 3, 4). For example, the second and third overtones of the CH bands of chloroform and benzene satisfy even the requirements formulated for class I standards. Their positions in the spectrum were measured rather accurately by Mecke and Oswald\(^5\); however, the paper gives no indication that the frequency values obtained were reduced to vacuum. Therefore the frequency values indicated in Fig. 3 may have a systematic error of the order of \(\pm 2\ \mathrm{cm}^{-1}\). In calibrating the IKS-11 spectrometer with prism F-1 this error may be neglected, since the intrinsic error of the instrument, as will be shown below, is here of the order of \(5\ \mathrm{cm}^{-1}\). Special designs (of the type of the five-prism glass monochromator of A. N. Terenin and N. G. Yaroslavskii\(^6\)) provide a considerably higher measurement accuracy (of the order of \(1\ \mathrm{cm}^{-1}\)), and the bands of chloroform and benzene may be used only for their approximate calibration.

Fig. 4. Absorption: a—of didymium glass \((d = 0.6\ \mathrm{mm})\) and b—of liquid 1,2,4-trichlorobenzene \((d = 0.5\ \mathrm{mm})^{7,36}\).

Frequencies in \(\mathrm{cm}^{-1}\)

(1) 14 616	(4) 11 360	(7) 6590	(10) 4644.2	(13) 4101.6
(2) 13 455	(5) 9369	(8) 6020.3	(11) 4322.9	(14) 4009.0
(3) 12 373	(6) 8194	(9) 5212	(12) 4160.3	(15) 3932.0

Fig. 5. Absorption of a polystyrene film \((d = 0.025\ \mathrm{mm})\).
\((\mathrm{LiF}\ \text{prism})^{7}\).

Frequencies in \(\mathrm{cm}^{-1}\)

(16) 3082.6	(18) 3026.5	(20) 2924.2
(17) 3060.5	(19) 3002.8	(21) 2850.0

ON THE CHOICE OF NORMALS

Table 1

Normals for calibrating the region \(4000—18\,000\ \mathrm{cm}^{-1}\)

Source	Class of normal	\(\nu_{\mathrm{vac}}\) \((\mathrm{cm}^{-1})\)	Intensity	\(\Delta\nu\) slit \((\mathrm{cm}^{-1})\)	Literature references
Hg	I	18307,8			7
Hg	I	17327,5		30	7
Hg	I	17264,4			7
K	I	13046			5
Rb	I	12820,5			5
Rb	I	12582			5
Cs	I	11736			5
Cs	I	11181			5
Hg	I	9859,4	very strong		7
Hg	I	8857,7	strong		7
Kr	I	8455,9	2000		8
Ar	II	8058,1	400	23	8
Ar	II	8034,6	500	11	8
Ar	II	8023,7	400	20	8
Ar	II	8003,4	700		8
Kr	I	7584,5	850		8
Ar	II	7476,7	800	22	8
Ar	II	7454,9	250	54	8
Ar	II	7401,1	850		8
Hg	I	7367,0	medium	56	7
Hg	I	7311,0	medium		7
Kr	II	7336,8	800	6	8
Kr	II	7330,4	1700	13	8
Kr	II	7317,4	360		8
Ar	II	7336,7	500	30	8
Ar	II	7306,6	300	21	8
Ar	II	7285,2	1000		8
Hg	I	7166,3	weak		7
Kr	I	6927,6	1100		8
Kr	II	6783,0	900	13	8
Kr	II	6770,0	250	1,5	8
Kr	II	6768,7	230		8
Kr	II	6558,0	900	40	8
Kr	II	6517,2	850	16	8
Kr	II	6501,7	350		8
Hg	I	6536,2	medium		7
Rb	I	6540,2			5
Hg	II	5908,5	medium	8	7
Hg	II	5900,9	medium	45	7
Hg	II	5855,7	medium	13	7
Hg	II	5843,0	medium		7
Kr	II	5954,5	950	24	8
Kr	II	5930,2	480	13	8
Kr	II	5917,2	1000	2,2	8
Kr	II	5915,0	700	14	8
Kr	II	5901,3	800		8

End of Table I

Source	Class of normal	$\nu_{\mathrm{vac}}\ (cm^{-1})$	Intensity	$\Delta\nu$ of slit $(cm^{-1})$	Literature references
Kr	I	5551.9	400		8
Kr	I	5501.3	1500		8
Hg	I	5074.5	very weak		7
He	I	4856.1	5000		8
Hg	I	4444.6	very weak		7
Hg	I	4299.1	very weak		7

As the most accurate and applicable for calibrating any prism spectrometers, class-I standards should include the emission lines of mercury vapor and inert gases. The corresponding data are presented in Table I (see also Fig. 6).

Fig. 6. Mercury emission lines (IKS-11 with prism F-1)*.

Frequencies in $cm^{-1}$

\[ \begin{aligned} (1)&\quad 9859.4 \\ (2)&\quad 8857.7 \\ (3)&\quad 7367.0 \\ (4)&\quad 7311.0 \\ (5)&\quad 7166.3 \end{aligned} \qquad \begin{aligned} (6)&\quad 6536.2 \\ (7)&\quad \begin{cases} 5908.5\\ 5900.9\\ 5855.7 \end{cases}\\ (8)&\quad \begin{cases} 5843.0 \end{cases} \end{aligned} \qquad \begin{aligned} (9)&\quad 5514.0 \\ (10)&\quad 5074.5 \\ (11)&\quad 4444.6 \\ (12)&\quad 4299.1 \end{aligned} \]

Some groups of lines here have been assigned to class II, in accordance with the requirements formulated above, since not every prism instrument can resolve these groups into components. A distinct separation can be achieved only with a spectral slit width not exceeding the values indicated in column 5 of the table (see Appendix 1). In compiling the table, data from works 5, 7, 8 were used; in the necessary cases the frequencies were reduced to vacuum

by the formula

\[ \nu_{\mathrm{vac}}=\nu_{\mathrm{air}}\left(1-2.86\cdot10^{-4}\right). \]

In the absence of a sufficient number of objects, in practice one may confine oneself merely to measuring the emission spectrum of mercury vapor and carry out the calibration of the instrument with high accuracy, using methods III and V of § 2. The task of calibrating the long-wavelength edge of the working region of a quartz prism is considerably facilitated if the data of Figs. 7 and 8 for the absorption bands of atmospheric moisture and methane are used as standards of class II.

LiF

With a lithium fluoride prism a broad region of the infrared spectrum, from 10,000 to 1700 cm\(^{-1}\), can be covered; there the fundamental frequencies and overtones of the vibrations of the groups XH are observed. The LiF prism gives the best results in the frequency interval 4000—1700 cm\(^{-1}\), but it is sometimes also used in the region of higher frequencies, where its dispersion is somewhat lower than that of glass and quartz. The standards

Fig. 7. Absorption of atmospheric H\(_2\)O vapor (LiF prism)\(^4\).

Frequencies in cm\(^{-1}\)

(1)	5434.6	(3)	5378.4
(2)	5414.0	(4)	5344.7
(5)	5251.7
(6)	5208.5

Fig. 8. Absorption of atmospheric H\(_2\)O vapor and methane
(\(p_4=760\) mm Hg, \(d=100\) mm, quartz prism\(^4\)).

Frequencies in cm\(^{-1}\)

(7)	4546	(12)	4260	(16)	4125	(30)	3821	(42)	3691
(8)	4422	(13)	4218	(22)	3902	(31)	3802	(43)	3675
(9)	4392	(14)	4194	(28)	3854	(38)	3736	(45)	3650
(10)	4350	(15)	4177	(29)	3838	(40)	3712	(46)	3629
(11)	4316

selected earlier for calibration of the overtone region are also applicable to the LiF prism. The comparatively high dispersion of lithium fluoride in the principal working region 4000—1700 cm\(^{-1}\) makes it possible

to achieve high resolving power and accuracy of measurements with an appropriate choice of standards. The best standards available

Fig. 9. Absorption of HCl (gas, \(p = 400\) mm Hg, \(d = 100\) mm, IKS-11, LiF prism)\(^{4,37}\).

Frequencies in cm\(^{-1}\)

(1) 3059.07	(6) 2980.90	(11) 2865.09	(16) 2752.03
(2) 3044.88	(7) 2963.24	(12) 2843.56	(17) 2727.75
(3) 3029.96	(8) 2944.81	(13) 2821.49	(18) 2703.06
(4) 3014.29	(9) 2925.78	(14) 2798.78	(19) 2677.73
(5) 2997.78	(10) 2906.25	(15) 2775.79	(20) 2651.97
			(21) 2625.74

under laboratory conditions, diatomic gases HCl, HBr, and CO (Figs. 9–11) may serve as class I standards; the rotational structure

Fig. 10. Absorption of HBr (gas, \(p = 400\) mm Hg, \(d = 100\) mm, LiF prism)\(^{4,37}\).

Frequencies in cm\(^{-1}\)

(22) 2674.66	(26) 2620.67	(30) 2541.90	(34) 2470.72
(23) 2661.91	(27) 2605.93	(31) 2524.85	(35) 2451.82
(24) 2648.69	(28) 2590.56	(32) 2507.21	(36) 2432.55
(25) 2634.83	(29) 2574.88	(33) 2488.98	(37) 2412.67

of whose absorption bands consists of a series of single, symmetrical, and narrow maxima.

The position of such maxima in the spectrum does not depend on the resolving power of the prism spectrometer \(^{22,20}\).

Under laboratory conditions hydrogen chloride is easily obtained by the action of concentrated sulfuric acid on sodium chloride with gentle heating:

\[ \mathrm{NaCl} + \mathrm{H_2SO_4} \to \mathrm{NaHSO_4} + \mathrm{HCl} \]

or by pumping concentrated hydrochloric acid through a trap cooled with liquid air. Hydrogen bromide is obtained by dripping bromine onto naphthalene:

\[ \mathrm{C_{10}H_8} + \mathrm{Br_2} \to \mathrm{C_{10}H_7Br} + \mathrm{HBr} \]

or by the action of water on phosphorus tribromide:

\[ \mathrm{PBr_3} + 3\mathrm{H_2O} \to \mathrm{H_3PO_3} + 3\mathrm{HBr}. \]

Carbon monoxide, as the only volatile product, is obtained by heating red prussiate of potash with concentrated sulfuric acid.

Fig. 11. Absorption of CO (gas, \(p = 150\) mm Hg, \(d = 100\) mm, LiF prism) \(^{4}\).

Frequencies in \(\mathrm{cm}^{-1}\)

(1) 2215,66	(12) 2179,57	(23) 2135,48	(34) 2090,56
(2) 2212,46	(13) 2176,12	(24) 2131,49	(35) 2086,27
(3) 2209,31	(14) 2172,63	(25) 2127,61	(36) 2081,95
(4) 2206,19	(15) 2169,05	(26) 2123,62	(37) 2077,57
(5) 2202,96	(16) 2165,44	(27) 2119,64	(38) 2073,19
(6) 2199,77	(17) 2161,83	(28) 2115,56	(39) 2068,69
(7) 2196,53	(18) 2158,13	(29) 2111,48	(40) 2064,31
(8) 2193,19	(19) 2154,44	(30) 2107,33	(41) 2059,79
(9) 2189,84	(20) 2150,83	(31) 2103,12	(42) 2055,31
(10) 2186,47	(21) 2147,05	(32) 2099,01	(43) 2050,72
(11) 2183,14	(22) 2139,32	(33) 2094,69	(44) 2046,14

No lesser accuracy of calibration can be obtained by measuring the spectra of ammonia, methane, and atmospheric moisture (Figs. 12–15). The absorption curves of these substances have a complex structure;

Fig. 12. Absorption of NH\(_3\) (gas, \(p = 200\) mm Hg, \(d = 100\) mm, LiF prism)\(^4\).

Frequencies in cm\(^{-1}\)

(56) 3509.1 (60) 3432.9 (69) 3276.3 (73) 3195.0
(57) 3489.9 (61) 3414.4 (70) 3256.0 (74) 3174.7
(58) 3470.7 (62) 3394.6 (71) 3236.0
(59) 3452.1 (68) 3295.4 (72) 3215.4

Fig. 13. Absorption of CH\(_4\) (gas, \(p = 170\) mm Hg, \(d = 100\) mm, LiF prism)\(^4\).

Frequencies in cm\(^{-1}\)

(1) 3165.8 (8) 3103.9 (15) 3038.2 (22) 2947.5
(2) 3157.0 (9) 3094.6 (16) 3028.3 (23) 2937.3
(3) 3148.4 (10) 3085.5 (17) 2998.9 (24) 2927.0
(4) 3139.5 (11) 3076.2 (18) 2988.4 (25) 2916.9
(5) 3130.8 (12) 3066.7 (19) 2978.5 (26) 2906.2
(6) 3121.8 (13) 3057.3 (20) 2968.0 (27) 2895.9
(7) 3112.8 (14) 3047.8 (21) 2957.8 (28) 2885.3

depending on the resolving power of the infrared spectrometer, the number and shape of the maxima obtained in the recording in these bands

Fig. 14. Absorption by atmospheric H₂O vapor (LiF prism)⁴.

Frequencies in cm⁻¹

(22) 3902	(31) 3802	(40) 3712	(45) 3650
(28) 3854	(36) 3752	(42) 3691	(46) 3629
(29) 3838	(37) 3745	(43) 3675	(52) 3586
(30) 3821	(38) 3736	(44) 3670	(53) 3566

Fig. 15. Absorption by atmospheric H₂O vapor (LiF prism)⁴.

Frequencies in cm⁻¹

(1) 2017.9	(7) 1869.5	(13) 1751.2	(19) 1675.2
(2) 1992.5	(8) 1845.5	(14) 1734.2	(20) 1662.8
(3) 1967.5	(9) 1829.5	(15) 1730.0	(21) 1645.5
(4) 1943.5	(10) 1824.8	(16) 1717.6	(22) 1635.6
(5) 1921.5	(11) 1793.5	(17) 1700.1	(23) 1617.0
(6) 1890.6	(12) 1773.5	(18) 1695.9

may be different (compare Figs. 14 and 19). Therefore they should be assigned to class II standards, and in all cases the recorded spectra obtained should be compared with the spectra shown in the figures.

On the Selection of Standards

depending on the resolving power of the infrared spectrometer, the number and shape of the maxima obtained in the recording in these bands

Fig. 14. Absorption by atmospheric vapors of H₂O (LiF prism)4.

Frequencies in cm\(^{-1}\)

(22) 3902	(31) 3802	(40) 3712	(45) 3650
(28) 3854	(36) 3752	(42) 3691	(46) 3629
(29) 3838	(37) 3745	(43) 3675	(52) 3586
(30) 3821	(38) 3736	(44) 3670	(53) 3566

Fig. 15. Absorption by atmospheric vapors of H₂O (LiF prism)4.

Frequencies in cm\(^{-1}\)

(1) 2017.9	(7) 1869.5	(13) 1751.2	(19) 1675.2
(2) 1992.5	(8) 1845.5	(14) 1734.2	(20) 1662.8
(3) 1967.5	(9) 1829.5	(15) 1730.0	(21) 1645.5
(4) 1943.5	(10) 1824.8	(16) 1717.6	(22) 1635.6
(5) 1921.5	(11) 1793.5	(17) 1700.1	(23) 1617.0
(6) 1890.6	(12) 1773.5	(18) 1695.9

may be different (compare Figs. 14 and 19). Therefore they should be assigned to class II standards, and in all cases the obtained spectral records should be compared with the spectra shown in the figures.

of fluorite should be used when comparing the spectrum records shown in Fig. 19.

NaCl

Rock-salt prisms are used most often in practice, since they make it possible to cover a broad region of the infrared spectrum from 4000 to 660 cm\(^{-1}\) with satisfactory dispersion. However, the most advantageous region for the use of NaCl prisms begins below 2000 cm\(^{-1}\). Here, for calibration, the absorption-band structures of atmospheric moisture, ammonia, and CO\(_2\) (Figs. 20, 21, and 22) are recommended as class I and II standards. Calibration of the region above 2000 cm\(^{-1}\), as well as approximate calibration of the entire working region of the NaCl prism, may be carried out using the data of Figs. 16, 23, 24, and 25 (class II and III standards).

Fig. 19. Absorption of NH\(_3\) (gas, \(p = 100\) mm Hg, \(d = 100\) mm) and atmospheric H\(_2\)O vapors (CaF\(_2\) prism).\(^4\)

Frequencies in cm\(^{-1}\)

(1) 3902	(10) 3674	(19) 3490	(29) 3296
(2) 3854	(11) 3650	(20) 3471	(30) 3276
(3) 3838	(12) 3629	(21) 3452	(31) 3256
(4) 3821	(14) 3586	(22) 3433	(32) 3236
(5) 3802	(15) 3566	(23) 3414	(33) 3216

KCl

The sylvite prism is not widely used, since in its working region of 1000–500 cm\(^{-1}\) sufficiently good results—

fluorite should be used when comparing with the spectrum records shown in Fig. 19.

NaCl

Rock-salt prisms are used most often in practice, since they make it possible to cover a broad region of the infrared spectrum from 4000 to \(660\ \mathrm{cm}^{-1}\) with satisfactory dispersion. However, the most advantageous region of application of NaCl prisms begins below \(2000\ \mathrm{cm}^{-1}\). Here, for calibration, the structures of the absorption bands of atmospheric moisture, ammonia, and \(\mathrm{CO}_2\) (Figs. 20, 21, and 22) are recommended as class I and II standards. Calibration of the region above \(2000\ \mathrm{cm}^{-1}\), as well as approximate calibration of the entire working region of the NaCl prism, may be carried out from the data of Figs. 16, 23, 24, and 25 (class II and III standards).

Fig. 19. Absorption of \(\mathrm{NH}_3\) (gas, \(p = 100\ \mathrm{mm}\) Hg, \(d = 100\ \mathrm{mm}\)) and atmospheric \(\mathrm{H}_2\mathrm{O}\) vapors (CaF\(_2\) prism).\(^4\)

Frequencies in \(\mathrm{cm}^{-1}\)

No.	Frequency	No.	Frequency	No.	Frequency	No.	Frequency
(1)	3902	(10)	3674	(19)	3490	(29)	3296
(2)	3854	(11)	3650	(20)	3471	(30)	3276
(3)	3838	(12)	3629	(21)	3452	(31)	3256
(4)	3821	(14)	3586	(22)	3433	(32)	3236
(5)	3802	(15)	3566	(23)	3414	(33)	3216

KCl

A sylvine prism is not widely used, since in its working region \(1000\text{—}500\ \mathrm{cm}^{-1}\) sufficiently good results ...

tates are achieved with NaCl and KBr prisms. The data for calibrating these two prisms can also be used for a sylvine prism.

KBr

The principal working region of potassium bromide occupies the interval from 1000 to 400 cm\(^{-1}\). When working in this region there arise

Fig. 20. Absorption by atmospheric H\(_2\)O vapors (NaCl prism)\(^4\).

Frequencies in cm\(^{-1}\)

(5) 1921.5  (15) 1735.6  (30) 1559.6  (42) 1436.6
(7) 1890.5  (16) 1717.6  (31) 1541.6  (45) 1419.6
(8) 1869.5  (17) 1699.6  (33) 1522.6  (48) 1398.6
(9) 1845.5  (18) 1684.6  (35) 1507.6  (49) 1387.6
(10) 1829.5  (19) 1670.6  (36) 1497.6  (50) 1375.7
(12) 1793.5  (23) 1636.6  (37) 1490.6  (52) 1362.7
(13) 1773.5  (25) 1617.6  (38) 1473.6  (53) 1339.7
(14) 1750.5  (28) 1577.6  (40) 1458.6

some difficulties because of the considerable amount of scattered light of high frequencies in the instrument (see Appendix 3), the comparatively low radiation energy of the sources, and the insufficient sensitivity of the receivers. Therefore, for example, on the IKS-11 spectrometer it is difficult to attain sufficiently good resolution, which complicates

Fig. 21. Absorption of NH\(_3\) (gas, \(d = 100\) mm;
a) \(p = 100\) mm Hg; b) \(p = 250\) mm Hg; NaCl prism)\(^{4}\).

Frequencies in cm\(^{-1}\)

No.	Frequency	No.	Frequency	No.	Frequency	No.	Frequency
(1)	1230.8	(15)	1046.6	(31)	892.2	(44)	788.2
(2)	1213.1	(16)	1033.5	(32)	888.1	(46)	778.3
(3)	1195.6	(17)	1027.0	(33)	872.5	(47)	776.4
(4)	1177.6	(18)	1012.3	(34)	868.0	(48)	774.1
(5)	1159.4	(19)	1005.9	(35)	853.8	(49)	770.9
(6)	1141.0	(20)	992.5	(36)	847.8	(51)	760.7
(7)	1122.7	(21)	972.2	(37)	828.0	(52)	758.8
(8)	1117.1	(23)	951.8	(38)	812.2	(54)	753.5
(9)	1103.8	(24)	948.6	(39)	810.0	(55)	748.5
(10)	1095.8	(26)	921.3	(40)	807.5	(56)	745.1
(11)	1084.8	(27)	918.6	(41)	796.1	(60)	732.3
(12)	1075.5	(28)	915.8	(42)	794.1	(61)	728.6
(13)	1066.0	(29)	912.5	(43)	791.8	(62)	723.3
(14)	1054.1	(30)	908.2

Fig. 22. Absorption of atmospheric CO₂ (NaCl prism)4.

Frequencies in cm\(^{-1}\)

(1) 720.5	(9) 688.9	(17) 675.9	(25) 658.0
(2) 700.0	(10) 687.1	(18) 674.3	(26) 656.5
(3) 698.4	(11) 685.5	(19) 672.8	(27) 655.0
(4) 696.9	(12) 683.9	(20) 668.0	(28) 653.4
(5) 695.2	(13) 682.3	(21) 664.2	(29) 651.9
(6) 693.6	(14) 680.7	(22) 662.7	(30) 650.4
(7) 691.9	(15) 679.1	(23) 661.2	(31) 648.8
(8) 690.3	(16) 677.5	(24) 659.5

Fig. 23. Absorption of atmospheric H₂O and CO₂ (NaCl prism)4.

Frequencies in cm\(^{-1}\)

(0) 3881	(2) 3619
(1) 3740	(3) 2349.3

calibration of the instrument. Potassium bromide in the region \(1000\)—\(650\ \mathrm{cm}^{-1}\) has a considerably smaller dispersion than NaCl; therefore, the rotational structure of the absorption bands of \(\mathrm{NH_3}\) and \(\mathrm{CO_2}\), which is convenient for calibrating the prism, cannot be used here, but the spectra of polystyrene, pyridine, and 1,2,4-trichlorobenzene (Fig. 24

Fig. 24. Absorption: \(a\)—polystyrene film (\(d = 0.05\ \mathrm{mm}\)),
\(b\)—1,2,4-trichlorobenzene (liquid, \(d = 0.05\ \mathrm{mm}\) and \(0.20\ \mathrm{mm}\)).

Frequencies in \(\mathrm{cm}^{-1}\)

(1) 2925.0	(5) 1603	(9) 906.2	(13) 550.7
(2) 2850.0	(6) 1494.3	(10) 699.8	(14) 486.1
(3) 1946.3	(7) 1154.5	(11) 646.0	(15) 458.9
(4) 1802	(8) 1028.4	(12) 574.7	(16) 439.4

and 25) give a quite good result over the entire working region of the KBr prism. Atmospheric moisture, \(\mathrm{CO_2}\), and methanol vapor, which give a large number of bands in the interval \(740\)—\(400\ \mathrm{cm}^{-1}\), can be used as standards of class II (Fig. 26).

KRS-5, CsBr, CsJ

Prisms made of KRS-5 (mixed crystal: 58.3% TlJ and 41.7% TlBr), cesium bromide, and cesium iodide have come into use in infrared spectrometers comparatively recently and have made it possible to extend the region of application of prism instruments to \(250\ \mathrm{cm}^{-1}\) (KRS-5 and CsBr) and to \(200\ \mathrm{cm}^{-1}\) (CsJ). Several such instruments have been described in the literature \(^{10,11,12}\), and frequency values have been obtained for absorption bands of atmospheric moisture and \(\mathrm{CO_2}\), which can

Fig. 25. Absorption of pyridine (liquid)$^{39}$

Frequencies in cm$^{-1}$

No.	Frequency	No.	Frequency	No.	Frequency	No.	Frequency
(1)	3003	(7)	1923	(13)	1441	(19)	991
(2)	3036	(8)	1872	(14)	1296	(20)	883
(3)	3004	(9)	1633	(15)	1217	(21)	749
(4)	2454	(10)	1599	(16)	1148	(22)	703
(5)	2293	(11)	1583	(17)	1068	(23)	604
(6)	1987	(12)	1482	(18)	1030	(24)	405

be used for calibration (Figs. 27 and 28). It should be noted that, when working in this region, it is necessary to take special measures to eliminate scattered radiation of high frequencies, or short wavelengths (see Appendix 3).

1. In concluding the present section, it is necessary to make several remarks on the possibility of increasing the accuracy of calibration by an appropriate choice of the method of recording spectra.

Fig. 26. Absorption by atmospheric H₂O, CO₂ and CH₃OH vapors (a cell with liquid in the illuminator unit of the spectrometer) (KBr prism)⁴.

Frequencies in cm⁻¹

(1) 736.51	(7) 548.51	(13) 472.56	(18) 439.59
(2) 720.5	(8) 526.09	(14) 457.93	(19) 436.53
(3) 611.29	(9) 502.32	(15) 453.84	(20) 425.31
(4) 599.34	(10) 491.82	(16) 444.18	(21) 423.03
(5) 582.45	(11) 492.09	(17) 442.13	(22) 419.10
(6) 574.35	(12) 474.53

Special attention should be paid to the accuracy with which reference lines or marks corresponding to definite readings on the scale of the spectrum-scanning drum are put onto the spectrum. Automatic marking devices, similar to those installed on the IKS-11 spectrometer with photographic recording, are unsuitable for purposes of accurate calibration. Our measurements on the IKS-11 showed that a significant improvement in accuracy can be achieved by a very simple

method. For this it is sufficient to disconnect the automatic contacts of the pilot-lamp circuit and to install in their place manual push-button switching outside the instrument. Then the number and position of the reference lines applied during the recording of the spectrum can be chosen in the best possible way in accordance with the characteristics of the spectrum being measured and the recording speed.

Fig. 27. Absorption by atmospheric H₂O and CO₂ (CsBr prism)^{4,10}.

Frequencies in cm⁻¹

No.	Frequency	No.	Frequency	No.	Frequency	No.	Frequency
(1)	3400	(14)	502.32	(23)	431.24	(33)	351.79
(2)	2350	(15)	492.09	(24)	425.31	(34)	349.83
(3)	1600	(16)	484.05	(25)	423.03	(35)	343.29
(4)	720.5	(17)	472.56	(26)	419.10	(37)	335.25
(5)	668	(18)	457.93	(27)	397.71	(38)	327.70
(7)	618.1	(20)	447.02	(29)	375.69	(39)	323.82
(11)	526.09	(21)	442.13	(30)	370.16	(41)	302.92
(13)	506.97	(22)	436.53	(32)	354.48	(42)	298.53
						(43)	289.57

of the spectrum. In addition, scales for readings in the intervals between the reference lines can easily be prepared (Fig. 29).

When calibrating the instrument by spectral standards, the problem always arises of finding, for each calibration band \(\nu_n\), the corresponding reading \(T_n\) on the spectrometer scale. Obviously,

that the accuracy of the graduation cannot be higher than the accuracy of the readings \(T_n\) (in comparable quantities), i.e., there exists a certain limit to the accuracy of measurements (and of graduation) for a given instrument, which we shall call the reproducibility or the intrinsic error of the instrument \(\Delta \nu_{\mathrm{pr}}\). If, with an unchanged

Fig. 28. Absorption by atmospheric vapors of \(H_2O\) and \(CO_2\) \(^{11,12}\).
\(a\)—CsJ prism in an ordinary autocollimation setup, \(b\)—CsJ prism in a double autocollimation setup according to Yushchu, \(v\)—KRS-5 prism. (Frequency values, see Fig. 27.)

mode of operation of the instrument, in repeated measurements of the spectrum several readings \(T\) have been made for one and the same band \(\nu\), and the mean value \(T_{\mathrm{mean}}\) and the mean deviation from the mean, \(\Delta T_{\mathrm{mean}}\), have been found, then \(\Delta \nu_{\mathrm{pr}}\) is determined for the region of the spectrum near \(\nu\) from the simple relation

\[ \left|\Delta \nu_{\mathrm{pr}}\right|=\frac{d\nu}{dT}\left|\Delta T\right|_{\mathrm{mean}}, \]

where \(\dfrac{d\nu}{dT}\) can be replaced by the ratio of the differences \(\dfrac{\nu_2-\nu_1}{T_2-T_1}\) of the limiting frequencies of the given spectral interval and the corresponding readings. For example, from Table II § 2 for the region \(1340\)—\(1870\ \text{cm}^{-1}\) (NaCl prism), we find \(|\Delta T|=0.2\), \(\dfrac{d\nu}{dT}\simeq \dfrac{\nu_8-\nu_{53}}{T_8-T_{53}}=3.1\), hence \(|\Delta\nu_{\text{mean}}|=\)

\(=0.6\ \text{cm}^{-1}\), and the relative error is \(\dfrac{\Delta\nu_{\text{mean}}}{\nu_{\text{mean}}}=0.04\%\). Considerable additional errors are introduced by prism changes

Fig. 29. Example of a spectrum recording with marker lines and a reading scale.
\(\mathrm{H_2O}_{\text{atmosph.}}\); IKS-11 spectrometer; NaCl prism; slit \(0.080\ \text{mm}\); speeds: spectrum scan—1, paper—3. FEOU \(1.7\ \text{V}\). Pin \(0.6\ \text{a}\).

(\(|\Delta T|\simeq 1.0\)) and fluctuations of the temperature of the working room. The corrections required in this case will be considered in § 3.

The remarks concerning the method of applying marker lines remain valid also when recording spectra with a pen (for example, with an EPP-09 self-recording instrument). But it must be said that, in comparison with photographic recording, in this case it is more difficult to attain the same degree of calibration accuracy.

§ 2. METHODS FOR PROCESSING THE RESULTS OF CALIBRATION MEASUREMENTS

If, as a result of measurements of calibration spectra, \(n\) readings \(T_n\) have been obtained on the scale of the spectrum-scanning drum, corresponding to \(n\) calibration frequencies \(\nu_n\), i.e., some function \(\nu(T)\) has been obtained, specified by a table of \(T_n\) for non-equidistant values of \(\nu_n\), then the next task is to choose a method for representing this function in a form convenient for the rapid and accurate determination of the values of \(\nu\) for any \(T\). The simplest way is to construct a smooth curve \(\nu(T)\) through the points \(\nu_n, T_n\) plotted on millimeter paper. But such a construction has substantial drawbacks. First of all, drawing a smooth curve through non-equidistant groups of points, often separated by considerable intervals, is not a simple task and inevitably introduces additional errors, especially in those cases when it is necessary to cover a wide region of the spectrum on a large scale. For example, when working with a rock-salt prism, the most commonly used frequency region is \(700\text{—}2400\ \mathrm{cm}^{-1}\). Consequently, to ensure a reading accuracy of \(1\ \mathrm{cm}^{-1}\), a frequency scale on the calibration graph \(1700\ \mathrm{mm}\) long is required. Such large sheets with calibration curves are cumbersome and inconvenient to use. Therefore, the dependence between \(\nu\) and \(T\) is more advantageously represented in the form of a table, in which the quantities \(\nu\) are given for equidistant values of \(T\) at an interval \(\Delta T\) that provides the necessary accuracy of the result under linear interpolation within the interval. Below we shall consider several methods for compiling such calibration tables.

The main sources of errors in the calibration of infrared spectral instruments are the intrinsic errors of the instrument \(\Delta \nu_{\mathrm{pr}}\) (see § 1), the inaccuracy of the frequency values of the calibration bands \(\Delta \nu_{\mathrm{pol}}\), and the errors introduced in the process of constructing calibration tables or curves \(\Delta \nu_{\mathrm{gr}}\). Approximately, it may be considered that the total error of calibration of the instrument is the sum of these quantities:

\[ \Delta \nu = \Delta \nu_{\mathrm{pr}} + \Delta \nu_{\mathrm{pol}} + \Delta \nu_{\mathrm{gr}}, \tag{1} \]

and for the relative error we have:

\[ \frac{\Delta \nu}{\nu} = \frac{\Delta \nu_{\mathrm{pr}}}{\nu} + \frac{\Delta \nu_{\mathrm{pol}}}{\nu} + \frac{\Delta \nu_{\mathrm{gr}}}{\nu}. \tag{2} \]

The relative error of the frequency values of most of the calibration bands cited in § 1 lies within the limits

\[ 0.0003\% \leq \frac{\Delta \nu_{\mathrm{pol}}}{\nu} \leq 0.01\%, \]

and it may be neglected in comparison

with a relative intrinsic error of the instrument equal, for example, for IKS-11, to 0.03–0.07%. If, in addition,

\[ \frac{\Delta \nu_{\mathrm{gr}}}{\nu} < \frac{\Delta \nu_{\mathrm{pr}}}{\nu}, \tag{3} \]

then the total error of the calibration result will assume a minimum value and will not exceed the intrinsic error of the instrument:

\[ \left(\frac{\Delta \nu}{\nu}\right)_{\min}=\frac{\Delta \nu_{\mathrm{pr}}}{\nu}. \]

Condition (3) thus makes it possible, in each particular case, to choose the most expedient calibration method.

We shall consider several methods. In method I, an increase in the accuracy of the result is achieved graphically—by constructing a smooth curve of the derivative of the function \(\nu(T)\). Method II uses a “correction curve” for the deviation of the function \(\nu(T)\) from a straight line and provides high accuracy, but is applicable only on individual portions of the spectrum. In methods III and IV, interpolation formulas are used in one form or another. The last method, V, is a combination of methods II and IV. The accuracy of the methods increases in the sequence: I, IV, V, III, and II.

Method I

Let, as a result of measurements, \(n\) pairs of values \(\nu_n, T_n\) have been obtained, from which the graph of the dependence \(T(\nu)\) has been plotted. Errors made in plotting can, to a considerable extent, be corrected in the following way[^13].

For a series of consecutive, equally spaced values of \(T\), the corresponding quantities \(\nu\) are found from the graph and a table is compiled. Finding from the table the differences \(\Delta \nu\) between pairs of neighboring values of \(\nu\), one constructs the curve of the dependence of \(\Delta \nu\) on \(T\). In those intervals where the values of \(\Delta \nu\) noticeably deviate from a smoothly increasing (or decreasing) sequence, the necessary corrections are introduced into the quantities \(\nu\), seeking the best approximation of \(\Delta \nu\) to a smooth curve. The calibration table corrected in this way provides an accuracy of the order of several units of the fourth decimal place

\[ \left(\frac{\Delta \nu_{\mathrm{gr}}}{\nu}<0.5\%\right) \]

provided that the interval \(\Delta T\) between neighboring values of \(T\) in the table is chosen sufficiently small and linear interpolation within \(\Delta T\) introduces no noticeable error.

Method II

A considerably more accurate result can be obtained by means of the method proposed by Martin[^14].

Let us consider some portion of the spectrum, comparatively uniformly filled with points \(\nu_n, T_n\), for example, an absorption region

atmospheric moisture \(1340\text{—}1870\ \mathrm{cm}^{-1}\) (see Table II). Let these points be located on the \((\nu, T)\) plane, as shown

Table II

Atmospheric moisture

Spectrometer IKS-11 No. 530032, NaCl prism

\(n\) band numbers	\(\nu_n\ (\mathrm{cm}^{-1})\)	\(T_n\) average of 6 measurements	\(\Delta T\) average
8	1869.5	1830.6	0.2
9	1845.5	1825.9	0.2
10	1829.5	1822.6	0.1
12	1793.5	1814.7	0.1
13	1773.5	1810.2	0.1
14	1750.5	1804.8	0.3
15	1735.6	1801.1	0.1
16	1717.6	1796.8	0.2
17	1699.6	1792.3	0.1
18	1684.6	1787.9	0.1
19	1670.6	1783.9	0.2
23	1636.6	1774.7	0.1
25	1617.6	1769.1	0.2
28	1577.6	1755.9	0.3
30	1559.6	1750.1	0.1
31	1541.6	1744.1	0.2
33	1522.6	1737.3	0.2
35	1507.6	1732.2	0.3
36	1497.6	1728.4	0.2
37	1490.6	1725.2	0.3
38	1473.6	1719.1	0.1
40	1458.6	1712.6	0.3
42	1436.6	1703.8	0.2
45	1419.6	1696.1	0.2
48	1395.6	1685.2	0.3
49	1387.6	1681.2	0.1
50	1375.7	1674.9	0.2
52	1362.7	1668.8	0.2
53	1339.7	1656.8	0.3
Average \(|\Delta T| = 0.2 | Average \(|\Delta T| = 0.2 | Average \(|\Delta T| = 0.2 | Average \(|\Delta T| = 0.2 | in Fig. 30. Let us choose among these points one, preferably a more reliable one, i.e. corresponding to a narrow, well-resolved peak on the absorption curve, for which the value of \(T_n\) is determined with the smallest error. Let this be the point with coordinates \(\nu_0, T_0\) (Fig. 30). Through it draw a straight line with slope \(K\), approximately equal to the average slope of the points \(\nu_n, T_n\) on the plane-

of \((\nu, T)\), i.e., in our case

\[ K \approx \frac{\nu_8-\nu_{53}}{T_8-T_{53}}, \tag{4} \]

where \(\nu_8, T_8\) and \(\nu_{53}, T_{53}\) are the coordinates of the extreme points Nos. 8 and 53 (Table II). The equation of the straight line with slope \(K\) will have the form

\[ \nu' - \nu_0 = K(T' - T_0). \tag{5} \]

Next we determine the distances \(\beta_n\) from each experimental point to the straight line (5). It is obvious that

\[ \beta_n=\nu'_n-\nu_n \tag{6} \]

or, using equation (5) and the identity \(T'_n \equiv T_n\),

\[ \beta_n=[K(T_n-T_0)+\nu_0]-\nu_n. \tag{7} \]

This dependence between \(\beta\) and \(T\) we shall call the “correction curve” for the deviation of the true calibration curve from a straight line. In Fig. 31 a “correction curve” is shown, constructed for the points of Table II.

Fig. 30. Determination of the values \(\beta_n\) in Martin’s method.

As the initial point \(\nu_0, T_0\), point No. 15 was chosen, with coordinates \(\nu_0 = 1735.6\ \text{cm}^{-1}\), \(T_0 = 1801.1\). The slope of the straight line (5) was chosen equal to 3, and the values \(\beta_n\) for each point \(\nu_n, T_n\) were calculated from the formula:

\[ \beta_n=[3(T_n-1801.1)+1735.6]-\nu_n. \]

We note that the “correction curve” can be plotted on a comparatively small scale, since inaccuracies in reading the values of \(T\) here introduce an error several times smaller in the values of \(\nu\) than occurs when readings are taken on the curve \(\nu(T)\).

Fig. 31. Form of the “correction curve” for the spectral region 1340–1870 \(\text{cm}^{-1}\) (Martin’s method).

The “correction curve” (7) makes it easy to obtain a tabular representation of the function \(\nu(T)\) of the instrument being calibrated. For this purpose, for a number of equally spaced values \(T_m\), we find the corresponding

values of \(\beta_m\) on the “correction curve,” and from (6) and (5) we obtain: \(-\beta_m+\nu'_m=\nu_m,\ \nu'_m=K(T_m-T_0)+\nu_0\), and therefore

\[ \nu_m=K(T_m-T_0)+\nu_0-\beta_m . \tag{8} \]

This is the basic equation for calculating the calibration table. As an example we give part of the calibration table (see Table III), calculated from the results of the measurements presented in Table II. At the bottom of the table the coordinates of the check point used in checking the stability of the calibration are indicated (see § 3).

Table III

Calibration table

IKS-11 spectrometer. NaCl prism

Drum divisions	Frequencies \((\text{cm}^{-1})\)	Tabulated differences
1650	1328	18
1660	1346	19
1670	1365	20
1680	1385	…
….	….	…
….	….	…

Check point (No. 30)

\[ \nu=1559.6\ \text{cm}^{-1} \]

\[ T=1750.1 \]

The principal advantage of the method set forth is the high accuracy of the result. In practice, the calibration procedure (calculations and drawing of the “correction curves”) introduces no additional error (i.e., \(\Delta\nu_{\mathrm{gr}}\) is very small), and the accuracy of the calibration result is limited only by the intrinsic error of the instrument. The method is applicable to any prismatic spectral instruments in those spectral regions where it is possible to obtain a sufficient number of points \(\nu_n, T_n\) for constructing the “correction curve” (for example, in the infrared region by measuring the structure of absorption bands of vapors of \(\mathrm{H_2O}\), \(\mathrm{NH_3}\), \(\mathrm{HCl}\), \(\mathrm{HBr}\), \(\mathrm{CO}\), etc.). If there are few such bands, or if they are separated by considerable intervals, then the method becomes inapplicable. Let us consider this case by way of an example.

Method III

Let us make use of the fact that the calibration curve \(T(\nu)\) can be approximately represented by the interpolation formula

\[ T-T_0=KZ=K\frac{1}{\nu^2-\nu_r^2}, \tag{9} \]

where \(T_0\), \(K\), and \(\nu_r\) are constants\(^{13,15}\). The best approximation is achieved by choosing the values of \(\nu_r\) according to Table IV.

Table IV

Values of the constant \(\nu_r\) in formula (9) for various prisms and spectral regions (according to\(^{4,15,16}\))

Prism material	Spectral region \((\mathrm{cm}^{-1})\)	\(\nu_r^2 \cdot 10^{-6}\)
NaCl	640—725	\(+\,0.01518\)
NaCl	725—833	\(+\,0.02666\)
NaCl	833—1014	\(+\,0.01237\)
NaCl	1014—1420	\(+\,0.00204\)
NaCl	1420—1970	\(-\,0.03323\)
LiF	2000—3200	\(-\,0.049204\)
LiF	3200—3900	\(-\,0.287009\)
LiF	3900—5000	\(-\,2.772416\)
LiF	5000—7000	\(-12.062280\)
CaF\(_2\)	1300—3500	\(+\,0.0100\)
KBr	400—1000	\(+\,0.013271\)

Let us apply formula (9) to the case of calibrating an LiF prism in the region \(2000—2600\ \mathrm{cm}^{-1}\) from the measurement results given in Table V. (Here, over a section of extent \(600\ \mathrm{cm}^{-1}\), we

Table V

Spectrometer IKS-11. LiF prism

H\(_2\)O and CO\(_2\) of the atmosphere, HCl \(p = 360\ \mathrm{mm}\) Hg, \(d = 100\ \mathrm{mm}\)

Band numbers \(n\)	Frequencies \(\nu_n\ (\mathrm{cm}^{-1})\)	Readings in drum divisions \(T_n\) mean	Object
3	1943.5	337.5	H\(_2\)O atmosph.
2	1967.5	383.0	H\(_2\)O »
1	1992.5	426.0	H\(_2\)O »
	2349.3	906.8	CO\(_2\) atmosph. (center)
22	2599.0	1136.0	HCl
21	2625.7	1156.2	HCl
20	2652.0	1176.2	HCl

we have only three points: 1992.5; 2349.3; 2599.0 cm\(^{-1}\); the preceding method is not applicable.) Let us calculate \(Z_n=\dfrac{1}{\nu_n^2-\nu_r^2}\) for all \(\nu_n\) from Table V, taking \(\nu_r^2=0.0492\cdot10^6\), and plot the points \(T_n, Z_n\) on a graph (Fig. 32). According to (9), they must lie on one straight line \(T=KZ+T_0\).

Fig. 32. Interpolation of the region 2000–2600 cm\(^{-1}\) in the calibration of an LiF prism (method III).

Indeed, it turns out that for the case analyzed the deviation of the points from the straight line does not exceed \(0.0003\cdot10^{-6}\) (in values of \(Z\)), which corresponds to \(\Delta\nu \approx 1\) cm\(^{-1}\) and does not exceed the intrinsic error of the instrument. Next, determining from the graph the values \(Z_m\) for a number of equidistant values \(T_m\) and passing to

\[ \nu_m=\left(\frac{1}{Z_m}+\nu_r^2\right)^{\frac12}, \]

one can easily construct a calibration table analogous to Table III. We note that, when using method III, the graphical construction of the straight line \(T=KZ+T_0\) is not obligatory, since the constants \(T_0\), \(K\), and \(\nu_r\) in formula (9) can be found analytically for any region of the spectrum.

Let us write (9) in the form

\[ \nu^2=\nu_r^2+\frac{K}{T-T_0}. \tag{10} \]

Comparison of this expression with Hartmann’s formula

\[ \lambda=\lambda_0+\frac{c}{d_0-d} \tag{11} \]

shows that the values of the constants \(\nu_r\), \(K\), and \(T_0\) in (10) can be found by methods analogous to Sawyer’s method\(^{17}\), p. 246, by substituting three pairs of values \(\nu_n, T_n\) (determined experimentally) into formula (10) and solving the three simultaneous equations thereby obtained\(^{16}\). For example, for the above-considered case of calibrating the region 2000–2600 cm\(^{-1}\) with an LiF prism, the system of equations may be written in the form

\[ \left. \begin{aligned} \nu_r^2&=(1992.5)^2-\frac{K}{426-T_0},\\ \nu_r^2&=(2349.3)^2-\frac{K}{906.8-T_0},\\ \nu_r^2&=(2599.0)^2-\frac{K}{1136-T_0}. \end{aligned} \right\} \tag{12} \]

Solving such a system requires rather laborious calculations, but it provides a higher accuracy of calibration than graphical construction. Solving a system of the form (12) may prove necessary in cases of calibrating spectrometers with glass and quartz prisms and with prisms made of KCl, KRS-5, KJ, CsBr, AgCl crystals, for which the value of the constant \(\nu_r\) is unknown.

We have considered several methods of calibrating prism spectral instruments that provide high accuracy over a wide spectral region. But they all have one common drawback: to carry out the calibration it is necessary to perform rather laborious measurements of the spectra of a number of compounds. In addition, the interpolation formula (9) is applicable only far from the short-wavelength absorption region of the prism material.

The method IV described below is free from these drawbacks, but gives lower accuracy (of the order of \(0.4\%\)).

Method IV

In those cases when, for one reason or another, measuring calibration spectra of different objects is difficult, the method of Mac-Kinney and Friedel\(^{15}\) may be used for approximate calibration of the instrument. The method was proposed for calibrating NaCl and KBr prisms, but it can also be extended to the case of calibrating any prism. We shall consider only the case of calibrating an NaCl prism.

To obtain the initial data here it is necessary to measure only the emission of a sodium lamp and the absorption spectra of atmospheric \(\mathrm{CO_2}\) and \(\mathrm{H_2O}\), and to make readings \(T_n\) in spectrometer scale divisions for the bands \(\nu_n\) indicated in Table VI. (The table also includes 6 frequencies of \(\mathrm{NH_3}\); their use gives some improvement of the result. Frequencies below \(668\ \mathrm{cm}^{-1}\) are used for calibrating a KBr prism.) Next one should calculate the quantities

\[ Z_n=\frac{1}{\nu_n^2-\nu_r^2}, \]

taking for \(\nu_r^2\) the value

\[ 0.015625\cdot 10^6\ \mathrm{cm}^{-2}, \]

and plot the points \(T_n, Z_n\) on a graph (Fig. 33). The points \(T_n, Z_n\) form a curve, whose equation is given by the following approximate

Fig. 33. View of the curve \(T-T_0=A\nu^2+KZ\) (method IV).

by the formula:

\[ T - T_0 = A\nu^2 + KZ, \tag{13} \]

where \(Z=\dfrac{1}{\nu^2-\nu_r^2}\). This formula is a generalization of formula (9).

Table VI

Calibration frequencies for the McKinney and Friedel method

No.	Frequencies \((\mathrm{cm}^{-1})\)	Object	Figure No.	Band No.
1	16969	\(D\)-line Na	—	—
2	3740	\(\mathrm{H_2O}\) atm.	—	—
3	2349,3	\(\mathrm{CO_2}\) »	23	3
4	1829,5	\(\mathrm{H_2O}\) »	20	10
5	1636,6	\(\mathrm{H_2O}\) »	20	23
6	1473,6	\(\mathrm{H_2O}\) »	20	38
7	1177,6	\(\mathrm{NH_3}\) »	21	4
8	1103,8	\(\mathrm{NH_3}\) »	21	9
9	1012,3	\(\mathrm{NH_3}\) »	21	18
10	948,6	\(\mathrm{NH_3}\) »	21	24
11	892,2	\(\mathrm{NH_3}\) »	21	31
12	828,0	\(\mathrm{NH_3}\) »	21	37
13	720,5	\(\mathrm{CO_2}\) »	22	1
14	668,0	\(\mathrm{CO_2}\) »	22	20
15	526,09	\(\mathrm{H_2O}\) »	27	11
16	502,32	\(\mathrm{H_2O}\) »	27	14
17	472,56	\(\mathrm{H_2O}\) »	27	17
18	457,93	\(\mathrm{H_2O}\) »	27	18
19	419,10	\(\mathrm{H_2O}\) »	27	26
20	397,71	\(\mathrm{H_2O}\) »	27	27
21	375,69	\(\mathrm{H_2O}\) »	27	29
22	370,16	\(\mathrm{H_2O}\) »	27	30

The term \(A\nu^2\) here takes into account the absorption of short-wavelength radiation by the prism material. To determine the constants \(A\) and \(K\), we shall approximately regard the branch \(BC\) of the curve \(T(Z)\) as a straight line. Continuing this straight line to its intersection with the ordinate axis, we find the difference between the ordinates of the point of intersection \(T_i\) and the reading \(T_{\mathrm{Na}}\) for the \(D\)-line of Na \((\lambda = 0{,}58932\,\mu;\ \nu = 16969\ \mathrm{cm}^{-1})\), and determine the constant \(A\) from the relation

\[ A\nu_{\mathrm{Na}}^2 = T_i - T_{\mathrm{Na}}. \]

Next, we determine from the graph the constant \(K\) as the tangent of the angle of inclination of the straight line \(BC\) to the abscissa axis, and, by substituting the found values of \(A\) and \(K\) and the quantities \(\nu_n\) and \(T_n\) for several points into formula (13), we determine the mean value of the constant \(T_0\). Thus all three constants \(A\), \(K\), and \(T_0\) are determined, and formula (13) makes it possible to find \(\nu\) from any \(T\) and to compile a calibration

table. The accuracy of the calibration in this case is no better than 0.4%. Additional computations make it possible to improve the values of the constants somewhat; in this case the accuracy of the calibration is somewhat increased (see \({}^{13,15}\)). We note that, when calibrating instruments with KBr, KRS-5, and other prisms in the region below \(1000\ \mathrm{cm}^{-1}\), the term \(A\nu^2\) in formula (13) may be discarded, since the absorption by the prism material of short-wavelength rays in this case plays no role.

Method V

Let us consider one more calibration method \({}^{18}\), which in general terms is a combination of methods II and IV and requires a smaller number of experimental points than method II.

Let us return to Fig. 33 and divide the entire working region of the prism into two parts: \(BC\)—the region where there is no absorption of short-wavelength rays by the prism material, and \(BD\)—the region where this absorption plays a noticeable role. We shall consider these regions separately.

1. Region \(BC\). Here one may disregard the term \(A\nu^2\) and apply, as a first approximation, an interpolation formula of the form

\[ T=\frac{K}{\nu^2-\nu_r^2}+T_0 . \tag{14} \]

Substituting here the measured values \(T_i\) for four calibration bands \(\nu_i\), we obtain two systems of equations with two unknowns \(K\) and \(T_0\):

\[ \left\{ \begin{aligned} T_1&=\frac{K}{\nu_1^2-\nu_r^2}+T_0,\\ T_2&=\frac{K}{\nu_2^2-\nu_r^2}+T_0, \end{aligned} \right. \qquad \left\{ \begin{aligned} T_3&=\frac{K}{\nu_3^2-\nu_r^2}+T_0,\\ T_4&=\frac{K}{\nu_4^2-\nu_r^2}+T_0. \end{aligned} \right. \]

Let \(K_1,T_{01}\) be the solutions of the first system, and \(K_2,T_{02}\) the solutions of the second system; then the mean values of the constants of formula (14) are determined from the relations

\[ \overline{K}=\frac{1}{2}(K_1+K_2),\qquad \overline{T}_0=\frac{1}{2}(T_{01}+T_{02}). \tag{15} \]

For the subsequent work we use method II, but in a somewhat modified form. Substitute \(\overline{K}\) and \(\overline{T}_0\) into formula (14) and solve it with respect to \(\nu\):

\[ \nu=\left(\frac{\overline{K}}{T-\overline{T}_0}+\nu_r^2\right)^{\frac{1}{2}} . \tag{16} \]

This equation will here play the role of straight line (5). Then, to construct the “correction curve” \(\beta'(T)\), we obtain from (6) and (16) that

\[ \beta'_n=\left(\frac{\overline{K}}{T_n-\overline{T}_0}+\nu_r^2\right)^{\frac{1}{2}}-\nu_n . \tag{17} \]

where \(\nu_n\) are calibration frequencies and \(T_n\) are the corresponding readings in divisions of the instrument scale.

The formula for calculating the calibration table will have the form

\[ \nu_m=\left(\frac{\overline{K}}{T_m-\overline{T}_0}+\nu_r^2\right)^{\frac12}-\beta'_m, \tag{18} \]

where \(\beta'_m\) are determined for each \(T_m\) from the “correction curve.”

1. Region \(BD\). As already indicated above, in order to take into account the absorption of short-wave rays by the prism material, the correction term \(A\nu^2\) must be introduced into formula (14); we note now,

Fig. 34. Dependence of \(\lg S\) on \(\lg \nu\) (method V).

Fig. 35. “Correction curve” \(\sigma(\nu)\) (method V).

that the best approximation is achieved if the exponent of \(\nu\) is regarded as an experimental quantity \([3,18]\). It is found in the following way. \(T_{n\,\mathrm{calc}}\) is calculated from (14) for several of the most reliable calibration frequencies \(\nu_n\), taking for the constants \(K\) and \(T_0\) the values from (15), and the differences between the measured values \(T_n\) and the calculated ones are formed:

\[ S_n=T_n-T_{n\,\mathrm{calc}}. \]

Next one sets

\[ S_n=A\nu_n^\alpha \]

and constructs the straight line (Fig. 34):

\[ \lg S=\alpha \lg \nu+\lg A, \]

from which the constants \(\alpha\) and \(A\) are found, and the exact values \(S_n=A\nu_n^\alpha\) are calculated for all \(\nu_n\) (or are found from the graph). After

for this they write the equation

\[ T_n=\frac{\overline{K}}{\nu_n^2-\nu_r^2}+\overline{T}_0+A\nu_n^2 \]

and from it compute \(T'_{n\,\mathrm{calc}}\) for all \(\nu_n\). The differences \(\sigma=T_n-T'_{n\,\mathrm{calc}}\) give the “correction curve” \(\sigma(\nu)\) (Fig. 35), and then, for compiling the graduation table, the equation obtained is

\[ T=\frac{\overline{K}}{\nu^2-\nu_r^2}+\overline{T}_0+A\nu^2+\sigma . \tag{19} \]

The table is compiled in values \(T_m\) for equally spaced values \(\nu_m\). In practice it is usually more convenient to use the inverse table (\(\nu\) as a function of \(T\)); it is easily obtained by recalculation. The method ensures an accuracy of \(0.1\%\) over the entire graduated interval.

§ 3. STABILITY OF THE GRADUATION

a) Influence of temperature

The appreciable dependence of the refractive indices of most crystalline materials used for making prisms on temperature (see, for example, Fig. 38) makes it necessary to take special measures to allow for, or compensate, temperature shifts of the graduation. For this purpose the prisms of modern infrared spectrometers are provided with bimetallic temperature compensators, which ensure good stability of the graduation over a certain temperature interval.

On production IKS-11 spectrometers we checked the action of the compensators. The measurements were carried out with fluctuations of the instrument temperature from 16 to \(26^\circ\mathrm{C}\). The degree of compensation could then be estimated by comparing the observed values of the temperature shifts \(\dfrac{\Delta\lambda}{\Delta t}\ \mu\cdot\mathrm{grad}^{-1}\) with those calculated for the case of absence of a compensator by the formula

\[ \Delta\lambda= \frac{ \left(\dfrac{dn}{dt}\right)_{\lambda} }{ \left(\dfrac{dn}{d\lambda}\right)_{\lambda} }\,\Delta t \tag{20} \]

(for \(\dfrac{dn}{d\lambda}\) and \(\dfrac{dn}{dt}\) the values given in Tables XII and XIV were used). Some of the results are given in Table VII. Incorrect setting of the compensators, detected by us on one prism,

Table VII

Temperature shifts of the calibration of IKS-11 spectrometers with NaCl prisms (mean values) (for $\Delta t \leqslant 6^\circ\mathrm{C}$)

Region $\lambda$ in $\mu$	Shifts $\dfrac{\Delta\lambda}{\Delta t}$, $\mu\cdot\mathrm{deg}^{-1}$: with correct installation of compensators	Shifts $\dfrac{\Delta\lambda}{\Delta t}$, $\mu\cdot\mathrm{deg}^{-1}$: calculated by (20)	Shifts $\dfrac{\Delta\lambda}{\Delta t}$, $\mu\cdot\mathrm{deg}^{-1}$: with incorrect installation of compensators
2.7	0.003	0.013	0.022
6.4	0.002	0.007	0.013
15.0	0.001	0.001	0.003

leads, as is seen from the table, to quite significant shifts. With correct installation, however, the compensation is entirely satisfactory: within $\Delta t \simeq 3^\circ\mathrm{C}$ the relative shifts $\dfrac{\Delta\lambda}{\lambda} \simeq 0.1\%$ and do not exceed, in the main working region, the intrinsic error of the instrument*).

Formula (20) can be successfully used for instruments not equipped with compensators, both for introducing corrections into measurement results at different temperatures and for establishing the permissible temperature fluctuations of the working room for any given instrument.

b) Influence of rearranging the prisms**)

It was already indicated above that on the IKS-11 spectrometer, with an unchanged operating mode, the reproducibility of readings reaches 0.2–0.3 division, which corresponds to a relative error $\dfrac{\Delta\nu}{\nu} \leqslant 0.1\%$. After rearranging the prisms, in some cases shifts of all absorption bands by 1.0–1.5 divisions are observed, which introduces an error $\dfrac{\Delta\nu}{\nu} \leqslant 0.5\%$. These shifts can be taken into account if certain absorption bands contained in the atmosphere of vapors of $\mathrm{H_2O}$ and $\mathrm{CO_2}$ are used as “control points,” the position of which is indicated in the calibration table (see

*) We note that the correct installation of the compensators corresponds to the following arrangement of the metal rods: for F-1 prisms, closer to the base of the prism—bronze, and farther away—steel-40; for LiF prisms, respectively—steel-20, zinc; for NaCl, KCl, and KBr—Invar, zinc.

**) For formulas for the exact calculation of shifts, see in [19].

Table III). When recording the spectrum after rearranging the prisms, an additional measurement of the position of the control point is made on the spectrogram. If, for example, the position of the point \(\nu = 1559.6\ \mathrm{cm}^{-1}\) (\(\mathrm{H_2O}\), No. 30, Fig. 20) during calibration was \(T = 1750.1\), and after rearranging the prism shifted to \(T' = 1751.6\), then the displacement \(\Delta T = 1.5\) may be regarded as approximately constant over a fairly wide region of the spectrum, and all readings \(T_n\) for the observed absorption bands should be corrected by the amount \(\Delta T\). A quite satisfactory correction for a NaCl prism in the principal working region \(2500\text{–}650\ \mathrm{cm}^{-1}\) is achieved, for example, by using three control points:

\[ \nu^{\mathrm{CO_2}}_1 = 2349.3\ \mathrm{cm}^{-1},\quad \nu^{\mathrm{H_2O}}_2 = 1559.6\ \mathrm{cm}^{-1},\quad \nu^{\mathrm{CO_2}}_3 = 667\ \mathrm{cm}^{-1}. \]

The standards and calibration methods considered in §§ 1, 2 make it possible to carry out an accurate calibration of any prism infrared instrument. Moreover, some of the methods of § 2 may be successfully used in the visible and ultraviolet regions of the spectrum. The advisability of applying one or another method, or one or another set of standards, is determined by the degree of accuracy that it is desired to obtain in each particular case. It is clear that, for approximate calibration by class-III standards (and for such instruments as the IKS-2 and ZRM-2 no other calibration is required), the methods of § 2 are merely an unnecessary complication. On the other hand, when accurate calibration is necessary, such a complication is fully justified (for example, on instruments of a higher class: IKS-11 and IKS-6).

In conclusion, we consider it our pleasant duty to express our deep gratitude to B. S. Neporent, N. G. Yaroslavskii, and A. V. Karyakin for discussion of the present work and a number of valuable suggestions.

APPENDICES

1. Estimation and allowance for the resolving power of an infrared spectrometer in calibration

We shall first give definitions of the quantities with which we shall have to deal.

Let a monochromator, with slit width \(S\ \mathrm{mm}\), isolate a spectral frequency interval \(\delta\nu\ \mathrm{cm}^{-1}\). Then the linear dispersion of the instrument is defined by the ratio

\[ \frac{2S}{\delta\nu}. \]

The reciprocal quantity

\[ \frac{\delta\nu}{2S} \]

is called the reciprocal linear dispersion. In what follows we shall use only the latter and denote it by \(D\) (in \(\mathrm{cm}^{-1}/\mathrm{mm}\)). By the resolution attained in spectral measurements is meant the smallest distance \(\Delta\nu\ \mathrm{cm}^{-1}\)

between two spectral lines or bands at which their maxima can still be observed separately on the given instrument. It is usually assumed that two lines or bands are at the limit of resolution if their maxima are separated by a depression equal to 0.8–0.9 of the intensity of the smaller maximum (and if this depression is at least twice the mean noise level). In the general case, the resolution depends on many factors. A discussion of these dependences lies beyond the scope of the present article (see, for example, \(^{20}\)). When calibrating an infrared spectrometer, it is sufficient to know the approximate value of the resolution for the given instrument. If the prism is used in an autocollimation arrangement at the minimum deviation for a ray \(\nu\) close to the middle of the principal working region of the prism, then the theoretical value of the resolution, called the spectral width of the slit, can be calculated for the entire working region of the prism from the approximate formula

\[ \left. \begin{aligned} \Delta \nu\ \mathrm{cm}^{-1} &= \Delta \nu_A + \Delta \nu_B = D \cdot S + \frac{\nu \cdot 10^{-4}}{m b \cdot \dfrac{dn}{d\lambda}}, \\[1.0em] D &= \frac{\nu^2\left(1-n^2\sin^2 \dfrac{A}{2}\right)^{1/2}\cdot 10^{-4}} {2m\cdot \sin \dfrac{A}{2}\cdot \dfrac{dn}{d\lambda}\cdot f}. \end{aligned} \right\} \tag{21} \]

Here \(A\) is the refracting angle of the prism, \(b\) is the length of its base in cm, \(n\) is the refractive index and \(\dfrac{dn}{d\lambda}\) is the dispersion of the prism in \(\mu^{-1}\), \(m\) is the number of passages of the ray through the prism (in the usual autocollimation arrangement with one prism \(m=2\)), \(f\) is the focal length of the collimator mirror in mm, and \(S\) is the working slit width in mm (for unequal entrance and exit slits, \(S\) must be replaced by \(\dfrac{S_1+S_2}{2}\)). The second term of the formula, \(\Delta \nu_B\), is the diffraction limit of resolution at \(S \to 0\) and does not depend on \(S\)*).

The values calculated for the IKS-11 spectrometer, \(D=\dfrac{\Delta \nu_A}{S}\ \mathrm{cm}^{-1}/\mathrm{mm}\) and \(\Delta \nu_B\ \mathrm{cm}^{-1}\), are collected in Table VIII. The expected resolution is readily determined from these values as \(\Delta \nu = D\cdot S_{\mathrm{work}}+\Delta \nu_B\). In Table IX a comparison is made of the calculated values of the resolution with those observed experimentally, which shows—

*) More precisely, we neglect this dependence. A more exact expression contains a factor \(F(S)\), varying within the limits 0.9–0.5 depending on \(S\) \(^{22,23,24}\).

Table VIII

Values of \(D\ \mathrm{cm}^{-1}/\mathrm{mm}\) and \(\Delta \nu_B\ \mathrm{cm}^{-1}\) for the IKS-11 spectrometer, calculated by (21)

\(\nu\ \mathrm{cm}^{-1}\)	Prism	\(D\)	\(\Delta\nu_B\)	Prism	\(D\)	\(\Delta\nu_B\)	Prism	\(D\)	\(\Delta\nu_B\)
13 000	F-1	617	2.6
12 000	F-1	653	2.9
11 000	F-1	675	3.3
10 000	F-1	679	3.7	LiF	994	6.6
9 000	F-1	665	4.0	LiF	830	6.2
8 000	F-1	619	4.2	LiF	680	5.6	NaCl	1040	7.4
7 000	F-1	525	4.0	LiF	510	4.8	NaCl	1140	9.5
6 000	F-1	384	3.9	LiF	344	3.8	NaCl	1190	11.3
5 000	F-1	258	2.8	LiF	212	2.8	NaCl	1110	12.2
4 500	F-1	196	2.4	LiF	157	2.3	NaCl	996	12.2
4 000	F-1	138	1.8	LiF	112	1.8	NaCl	840	11.7
3 500	F-1	—	—	LiF	75	1.3	NaCl	669	10.6
3 000	KCl			LiF	46.8	0.98	NaCl	466	8.5
2 800	KCl			LiF	38.2	0.84	NaCl	384	7.6
2 600	KCl			LiF	30.6	0.72	NaCl	314	6.6
2 400	KCl			LiF	24.2	0.61	NaCl	252	5.8
2 200	KCl			LiF	18.4	0.51	NaCl	200	5.0
2 000	KCl	219	6.7	LiF	13.8	0.42	NaCl	154	4.2
1 900	KCl	191	6.2	LiF	11.8	0.37	NaCl	132	3.8
1 800	KCl	163	5.5	LiF	10.0	0.33	NaCl	114	3.4
1 700	KCl	137	4.9	LiF	8.5	0.29	NaCl	95	3.0
1 600	KCl	116	4.3	KBr			NaCl	77.6	2.63
1 500	KCl	93	3.9	KBr			NaCl	63.0	2.32
1 400	KCl	77	3.4	KBr			NaCl	51.2	2.02
1 300	KCl	61.5	2.9	KBr			NaCl	41.5	1.74
1 200	KCl	48.5	2.45	KBr	98	4.5	NaCl	32.8	1.48
1 100	KCl	37.0	2.04	KBr	76	3.8	NaCl	25.3	1.25
1 000	KCl	27.5	1.65	KBr	55.5	3.09	NaCl	18.7	1.01
900	KCl	19.5	1.31	KBr	39.7	2.48	NaCl	13.3	0.79
800	KCl	13.5	1.01	KBr	27.6	1.93	NaCl	9.2	0.60
700	KCl	9.0	0.76	KBr	18.5	1.46	NaCl	5.9	0.44
600	KCl	5.5	0.53	KBr	11.3	1.02
500	KCl	3.1	0.34	KBr	6.3	0.68
400	KCl			KBr	3.1	0.41

shows that calculation by (21) gives a quite satisfactory qualitative estimate of the resolution. It should be noted that for the IKS-11 with F-1, LiF, and NaCl prisms, even at the minimum operating slit width, \(\Delta\nu_A > 3\Delta\nu_B\), i.e., the diffraction limit of resolution is far from being reached. This occurs mainly as a result of the low sensitivity of the infrared-radiation detector. More sensitive detectors make it possible to improve the resolution on instruments of the same type \(^{21}\) to \(\Delta\nu_A \approx 1.7\Delta\nu_B\).

Table IX

Comparison of calculated and observed resolution values
for the IKS-11 spectrometer

Prism, object	Frequencies of peaks close to the resolution limit (in cm\(^{-1}\))	Working slit width (in mm)	Calculated values \(\Delta\nu_A\)	Calculated values \(\Delta\nu_B\)	Calculated values \(\Delta\nu=\Delta\nu_A+\Delta\nu_B\)	\(\Delta\nu_{\text{obs}}\)
F-1; Hg	7367 7311	0.10	56	4	60	56
LiF; H\(_2\)O	3748 3736	0.12	11.2	1.5	12.7	12
NaCl; CO\(_2\)	2366 2340	0.09	21.5	6.3	27.8	26
LiF; H\(_2\)O	1992.5 1987.4	0.31	4.3	0.4	4.7	5.1
NaCl; NH\(_3\)	1054.1 1046.6	0.22	4.84	1.13	5.97	7.5
NaCl; NH\(_3\)	951.8 948.6	0.155	2.5	0.9	3.4	3.2

A clear illustration of the influence of the resolving power of a spectrometer on the shape and position of absorption bands is provided by recordings of the CO\(_2\) band at 2349.3 cm\(^{-1}\) with NaCl and LiF prisms. On the IKS-11 spectrometer with a LiF prism, the distance between the two absorption maxima in this band is found to be \(21.5 \pm 0.5\) cm\(^{-1}\) (Fig. 16), while with a NaCl prism it is \(26 \pm 3\) cm\(^{-1}\) (Fig. 23). The position of the center of the band, however, remains unchanged: 2349.3 cm\(^{-1}\).

2. Reduction of wave numbers to vacuum

If \(\nu'\) is the wave number measured in air, then the true value \(\nu\) in vacuum is:

\[ \nu_{\text{vac}}=\nu' - \delta . \]

The quantity \(\delta\), called the vacuum correction, is determined by the formula

\[ \delta=\nu' \cdot \frac{n_p^t-1}{n_p^t}, \]

where \(n_p^t\) is the refractive index of air at \(t^\circ\)C and \(p\) mm Hg, which is related to the refractive index of dry air, \(n_{760}^{0}\).

(for \(0^\circ\mathrm{C}\) and \(760\) mm) by the relation \(^{25}\)

\[ n_p^t - 1 = \frac{(n_{760}^0 - 1)\cdot p}{(1+0.00376t)\cdot 760}. \]

For any \(\lambda\), the refractive index \(n_{760}^0\) can be found from the formula \(^{26}\)

\[ (n_{760}^0)^2 - 1 = \left(5.7642 + \frac{0.03277}{\lambda^2 - 0.005685}\right)\cdot 10^{-4}. \]

For frequencies below \(5000\ \mathrm{cm}^{-1}\), one may use the approximate relation

\[ \delta = 2.603\cdot 10^{-4}\cdot \nu'. \]

The correction for the presence of water vapor in the air may be introduced by the formula \(^{27}\)

\[ n_{p\,(\mathrm{humid})}^t = n_{p\,(\mathrm{dry})}^t - \frac{5.5\cdot 10^{-8}\cdot \varepsilon}{1+0.00376\cdot t}, \]

where \(\varepsilon\) is the elasticity of water vapor in mm Hg.

3. Accounting for and reducing scattered light in infrared spectrometers

One of the main difficulties in calibration and measurements in the far infrared region is the presence of scattered

Fig. 36. Percentage of scattered light in the IKS-11 spectrometer with KCl and KBr prisms.

light, which is always, to a greater or lesser extent, admixed with the monochromatic radiation isolated by the spectral instrument. Since sources of infrared radiation

are incandescent bodies with an energy maximum in the region of high frequencies (short wavelengths), allowance for scattered radiation becomes especially important, since, when working on the descending branch of the source-radiation distribution curve (below \(1000\ \mathrm{cm}^{-1}\)), the scattered radiation of short wavelengths may prove quite considerable and may sometimes exceed the measured energy. In Fig. 36, by way of illustration, a graph is given showing the percentage of scattered light in the IKS-11 instrument for KBr and KCl prisms (source: a globar at \(1400^\circ\mathrm{C}\)).

Elimination of parasitic radiation can be carried out by several methods \(^{28,29,40}\), in particular by using reflecting filters—plates of LiF, CaF\(_2\), NaF, whose reflection is shown in Fig. 37. As is seen from the figure, the filters reflect up to 90% of the radiation at the maximum and are suitable over a fairly broad region of the spectrum. Very good results in eliminating scattered light when working with CsBr and CsJ prisms were obtained by using a reflecting filter installed in the illuminator block in place of a plane mirror \(^{10,11}\). The filter was made of a silver plate, which was first polished and then rubbed with fine emery. This made it possible to reduce the percentage of scattered light by a factor of 10 with a very small loss in energy (of the order of a few percent). When an additional filter of polyethylene coated with carbon black was installed, the scattered light decreased to 3% at \(40\mu\) and to 12% at \(50\mu\) (CsJ prism).

Fig. 37. Reflection of filters of LiF, CaF\(_2\) and NaF (in percent).

In those cases where complete removal of scattered light is impossible, to eliminate errors it is carefully measured and the corresponding constant corrections are introduced during subsequent work on the instrument. (When the source is replaced or the mode of its operation is changed, the corrections must be determined anew.) A method for measuring scattered light that gives quite sufficient accuracy is based on the use of filters made of various materials whose long-wavelength transmission limit lies closer than the portion of the spectrum in which the measurement is being carried out. This method does not take into account all the scattered light, since it makes it possible to measure only the light of that spectral region which is transmitted by the filter, but it provides quite sufficient accuracy, since the main share of the scattered light, as already indicated, repre-

constitutes radiation of short wavelengths. In this case, materials whose transmission is shown in Fig. 2 are used as filters, as well as glass, quartz, and mica. The long-wavelength transmission limits of the latter are respectively: 3.0 μ, 4.4 μ, and 5.3 μ (at a thickness of about 5 mm, except for mica).

4. Tables of refractive indices, dispersion, and temperature dependence of the refractive indices of crystalline materials used in infrared spectroscopy

The tables present the latest and most accurate of the data reported in the literature on the refractive indices and dispersion of various materials used in infrared spectroscopy for making prisms. These data are necessary both for carrying out the calibration of prism instruments by computational methods and for estimating resolving power, and also in the design of new optical systems. The dispersion values \(\frac{dn}{d\lambda}\) may also prove very useful in choosing a prism that best corresponds to a given spectral problem.

Fig. 38. Dependence of \(\frac{dn}{dt}\,\mathrm{grad}^{-1}\) on wavelength for NaCl.

The dependences of the temperature coefficients of the refractive indices \(\frac{dn}{dt}\) on wavelength are known only for LiF, NaCl, and KCl. For the other materials, only approximate mean values \(\left(\frac{dn}{dt}\right)_{\mathrm{avg}}\) have been determined for the infrared region, or the values \(\frac{dn_D}{dt}\) for the sodium \(D\)-line \((\lambda = 0.58932\,\mu)\).

A. N. Aleksandrov and V. A. Nikitin

Table X

Glass F-1; $n_D = 1.61290$.

$\lambda$ in $\mu$	$n$ $20^\circ\mathrm{C}$	$-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}$
0.7	1.60586
0.8	175	345
0.9	1.59881	257
1.0	652	207
1.1	463	174
1.2	303	153
1.3	156	140
1.4	022	133
1.5	1.58890	132
1.6	758	132
1.7	626	132
1.8	495	133
1.9	361	135
2.0	224	136
2.1	088	139
2.2	1.57946	144
2.3	799	150
2.4	646	157
2.5	484	165
2.6	1.57315

$$ \left(\frac{dn}{dt}\right)_{\mathrm{avg}} \approx -1\cdot 10^{-5}\ \mathrm{deg}^{-1}. $$

Table XI

Crystalline quartz; $n_D = 1.5442022$

$\lambda$ in $\mu$	$n$ $20^\circ\mathrm{C}$	$-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}$	$\lambda$ in $\mu$	$n$ $20^\circ\mathrm{C}$	$-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}$
0.7	1.540635	300	1.8	1.524145	152
0.8	1.538376	203	2.0	0972	166
0.9	6583	165	2.3	1.515610	193
1.0	5050	143	2.6	1.50986	225
1.2	2340	133	3.0	1.49953	272
1.4	1.529742	133	3.5	1.48451	335
1.6	7047	138	4.0	1.46617

$$ \frac{dn_D}{dt} = -5.4\cdot 10^{-6}\ \mathrm{deg}^{-1}. $$

(According to data$^{30}$.)

LiF; \(n_D = 1.3915^{30}\).

Table XII

\(\lambda\) in \(\mu\)	\(n\) \(23.6^\circ\mathrm{C}\)	\(-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}\)	\(-\dfrac{dn}{dt}\,10^5\,\mathrm{deg}^{-1}\)
1.0	1.38711	82	2.91
1.2	554	77
1.4	400	78
1.6	238	84
1.8	064	90
2.0	1.37875	98	2.51
2.2	669	107
2.4	446	116
2.6	203	125
2.8	1.36942	135
3.0	660	146	2.13
3.2	359	156
3.4	037	166
3.6	1.35693	178
3.8	329	189
4.0	1.34942	201	1.80
4.2	533	212
4.4	100	224
4.6	1.33645	236
4.8	165	248
5.0	1.32661	260	1.62
5.2	131	273
5.4	1.31575	285
5.6	1.30993	298
5.8	384	311
6.0	1.29745	324

(According to data\({}^{2}\).)

CaF\(_2\); \(n_D = 1.4338303\).

Table XIII

\(\lambda\) in \(\mu\)	\(n\) \(20^\circ\mathrm{C}\)	\(-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}\)	\(-\dfrac{dn}{dt}\,10^5\,\mathrm{deg}^{-1}\)
1.0	1.428923	63	1.02
1.2	7760	54
1.4	6742	48
1.6	5833	45

A. N. ALEKSANDROV AND V. A. NIKITIN

End of Table XIII

$\lambda$ in $\mu$	$n$ $23.6^\circ\mathrm{C}$	$-\dfrac{dn}{d\lambda}\,10^4\ \mu^{-1}$	$-\dfrac{dn}{dt}\,10^5\ \mathrm{deg}^{-1}$
1.8	1.424885	47
2.0	3895	50
2.3	2294	56	0.97
2.6	0525	62
3.0	1.41793	70
3.5	412	81	0.88
4.0	1.40971	93
4.5	469	105	0.84
5.0	1.39901	118
6.0	1.38562	146	0.82
7.0	1.36932	178	0.80
8.0	1.34988	212	0.79
9.0	1.32685	250	0.78

\[ \frac{dn_D}{dt}=-1\cdot 10^{-5}\ \mathrm{deg}^{-1}. \]
(According to data of $^{30}$ and $^{41}$.)

NaCl; $n_D=1.544258^{30}$.

Table XIV

$\lambda$ in $\mu$	$n$ $20^\circ\mathrm{C}$	$-\dfrac{dn}{d\lambda}\,10^4\ \mu^{-1}$	$-\dfrac{dn}{dt}\,10^5\ \mathrm{deg}^{-1}$
1.1786	1.530306	87	3.30
1.7680	1.527374	32	3.25
2.3573	1.525800	23	3.15
2.9466	1.524471	22	3.15
3.5359	1.523169	24	3.15
4.1252	1.521585	27	3.15
5.0092	1.518920	32	3.15
5.8932	1.515952	38	3.15
6.4825	1.513573	42	3.10
7.0718	1.511069	45	3.00
7.6611	1.508268	49	2.80
7.9558	1.506765	51	2.75
8.8398	1.502007	52	2.40
10.0184	1.494701	66	2.2
11.7864	1.481823	81	1.6
12.9650	1.471743	91	1.4
14.1436	1.460572	101	1.2
14.7330	1.454459	111	1.0
15.5223	1.447496	123	0.8
15.9116	1.441108	132	0.7

(According to data of $^{31,32}$.)

KCl; \(n_D = 1.490288\).

Table XV

\(\lambda\) in \(\mu\)	\(n\) \(20^\circ\mathrm{C}\)	\(-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}\)	\(-\dfrac{dn}{dt}\,10^5\,\mathrm{deg}^{-1}\)
3.5359	1.472881	18	3.28
4.7146	1.470956	19	3.20
5.3039	1.469850	20	3.15
5.8939	1.468642	22	3.10
8.2502	1.462568	30	2.92
8.8398	1.460701	33	2.87
10.018	1.45658	38	2.75
11.786	1.44908	46	2.48
12.965	1.44334	51	2.30
14.144	1.43711	57	2.06
15.912	1.42608	66	1.70
17.680	1.41392	76	1.26
20.60	1.3882	95
22.5	1.369	110

(According to data \(^{31}\).)

KBr; \(n_D = 1.5591\) \((20^\circ\mathrm{C})\).

Table XVI

\(\lambda\) in \(\mu\)	\(n\) \(22^\circ\mathrm{C}\)	\(-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}\)	\(\lambda\) in \(\mu\)	\(n\) \(22^\circ\mathrm{C}\)	\(-\dfrac{dn}{d\lambda}\,10^4\,\mu^{-1}\)
6.238	1.53523	10	17.40	1.50400	41
6.692	284	14	18.16	670	43
8.662	1.52901	18	19.01	1.49704	46
9.724	689	21	19.91	293	48
11.035	403	24	21.18	1.48664	52
11.862	199	26	21.83	307	54
14.29	1.51495	32	23.86	1.47138	61
14.98	286	34	25.14	1.46322	65

\[ \left(\frac{dn}{dt}\right)_{\mathrm{avg}} = -4.0 \cdot 10^{-5}\ \mathrm{deg}^{-1}. \]

(According to data \(^{33}\).)

KRS-5.

Table XVII

$\lambda$ in $\mu$	$n$ $27^\circ\mathrm{C}$	$\lambda$ in $\mu$	$n$ $27^\circ\mathrm{C}$
8.0	2.37653	24.0	2.32518
10.0	274	26.0	2.31493
12.0	2.36832	28.0	2.30373
14.0	317	30.0	2.29154
16.0	2.35724	32.0	2.27832
18.0	051	34.0	2.26403
20.0	2.34294	36.0	2.24862
22.0	2.33451	38.0	2.23205

(According to data$^{9}$ for the composition 58.3% TlJ, 41.7% TlBr. When the composition changes by 2.5%, the refractive index changes by up to $2\cdot 10^{-3}$.)

CsBr.

Table XVIII

$\lambda$ in $\mu$	$n$	$\lambda$ in $\mu$	$n$
9.724	1.6630	23.92	1.6324
11.035	14	25.16	1.6284
14.29	1.6561	25.97	53
14.98	49	26.60	30
15.48	39	28.33	1.6165
17.40	1.6494	29.15	31
18.16	81	29.81	06
20.57	20	30.69	1.6077
21.79	1.6386	33.11	20
22.76	60	34.48	1.5960

(According to data$^{35}$.)

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